IL-GLOBO (1.0) - integrated Lagrangian particle model and Eulerian general circulation model GLOBO: development of the vertical diffusion module

Geosci. Model Dev., 7, 2181–2191, 2014
© Author(s) 2014. CC Attribution 3.0 License.

IL-GLOBO (1.0) – integrated Lagrangian particle model and
Eulerian general circulation model GLOBO: development of the
vertical diffusion module
D. Rossi1,2 and A. Maurizi2
1 Department      of Biological, Geological and Environmental Sciences, University of Bologna, Bologna ,Italy
2 Institute   of Climate and Atmospheric Sciences, National Research Council, Bologna, Italy

Correspondence to: A. Maurizi (

Received: 6 March 2014 – Published in Geosci. Model Dev. Discuss.: 30 April 2014
Revised: 26 August 2014 – Accepted: 29 August 2014 – Published: 30 September 2014

Abstract. The development and validation of the vertical dif-       et al., 2011; Yu et al., 2013). Moreover, transport of volcanic
fusion module of IL-GLOBO, a Lagrangian transport model             emissions (e.g. the recent Eyjafjallajökull eruption) or acci-
coupled online with the Eulerian general circulation model          dental hazardous releases (like the Fukushima and Chernobyl
GLOBO, is described. The module simulates the effects of            nuclear accidents) are also important at the global scale.
turbulence on particle motion by means of a Lagrangian                 The natural framework for the description of tracer trans-
stochastic model (LSM) consistently with the turbulent dif-         port inflows is the Lagrangian approach (see, for exam-
fusion equation used in GLOBO. The implemented LSM in-              ple, the seminal works by Taylor, 1921, and Richardson,
tegrates particle trajectories, using the native σ -hybrid co-      1926). In the Lagrangian framework, the tracer transport is
ordinates of the Eulerian component, and fulfils the well-          described by integrating the kinematic equation of motion
mixed condition (WMC) in the general case of a variable             for fluid “particles” in a given flow velocity field, provided
density profile. The module is validated through a series of        by, e.g. a meteorological model. The turbulent motion unre-
1-D offline numerical experiments by assessing its accuracy         solved by Eulerian equations for averaged quantities (in the
in maintaining an initially well-mixed distribution in the ver-     Reynolds or volume-filtered sense) can be accounted for by
tical. A dynamical time-step selection algorithm with con-          including a stochastic component into the kinematic equa-
straints related to the shape of the diffusion coefficient pro-     tion.
file is developed and discussed. Finally, the skills of a lin-         The stochastic component can be added to the particle
ear interpolation and a modified Akima spline interpolation         position to give the Lagrangian equivalent of the Eulerian
method are compared, showing that both satisfy the WMC              advection-diffusion equation. This kind of model is usually
with significant differences in computational time. A prelim-       called a random displacement model (RDM) and is suit-
inary run of the fully integrated 3-D model confirms the re-        able for dispersion over long timescales. When the stochas-
sult only for the Akima interpolation scheme while the linear       tic component is added to the velocity, the model is usually
interpolation does not satisfy the WMC with a reasonable            called a random flight model (RFM), which is more suit-
choice of the minimum integration time step.                        able for shorter time dispersion. In both cases, the stochas-
                                                                    tic model formulation has to be consistent with some basic
                                                                    physical requirements (Thomson, 1987, 1995).
                                                                       Various Lagrangian transport models exist which can be
1 Introduction                                                      used at the global scale. Some are designed specifically for
                                                                    the description of atmospheric chemistry (Reithmeier and
Global- (or hemispheric-) scale transport is recognised as an       Sausen, 2002; Wohltmann and Rex, 2009; Pugh et al., 2012,
important issue in air pollution and climate change studies.        see, e.g.), while others focus on the transport of tracers. In
Pollutants can travel across continents and have an influence       the latter class, two of the most widely used models are
even far from their source (see, for recent examples, Fiore

Published by Copernicus Publications on behalf of the European Geosciences Union.
2182                                                          D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module

FLEXPART (FLEXible PARTicle dispersion model) (Stohl                0, 1, . . .), are described by a set of stochastic differential
et al., 2005) and HYSPLIT (Hybrid Single Particle La-               equations (SDEs). The equation for the Mth order derivative
grangian Integrated Trajectory Model) (Draxler and Hess,            M is
1998), which are highly flexible and can be easily used in
a variety of situations. Both are compatible with different in-     dXi      = ai dt + bij dWj ,                                   (1)
put types (usually provided by meteorological services like
the European Centre for Medium-Range Weather Forecasts                                                                 (k)
                                                                    where i and j indicate the components and Xi is the kth-
(ECMWF)), relying on their own parameterisation for fields          order time-derivative of the Lagrangian Cartesian coordinate
not available from the meteorological model output. Models                               (0)
                                                                    component Xi ≡ Xi . Coefficients ai and bij are called drift
of this kind are suited for both forward and backward disper-       and Wiener coefficients, respectively. The remaining equa-
sion studies.                                                       tions of the set (1 ≤ k ≤ M) are described by
   An alternative approach is to couple the Eulerian and La-
grangian parts online. On one hand, this makes the Eulerian         dXi
                                                                       (k−1)       (k)
                                                                               = Xi dt .                                           (2)
fields available to the Lagrangian model at each Eulerian
time step, increasing the accuracy for temporal scales shorter        The set of equations is equivalent to the Fokker–Planck
than the typical meteorological output interval. On the other       equation:
hand, it also allows the consistent parameterisation of pro-
cesses in the Eulerian and Lagrangian frameworks (e.g. the                 M
                                                                    ∂p    X    ∂       k          ∂2
vertical dispersion in the boundary layer). Moreover, where            =−        (k)
                                                                                     (Ai p) +   (M)  (M)
                                                                                                         (Kij p) ,                 (3)
the considered tracer may have an impact on meteorology             ∂t    k=0 ∂x              ∂x ∂x
                                                                                     i                i      j
(e.g. on radiation or cloud microphysics), online integration
provides a natural way to include these effects (Baklanov           where Aki = 1 for k < M and Aki = ai for k = M, xi is the
et al., 2014). Online coupling also ensures the consistency         Eulerian equivalent of Xi and Kij ≡ bik bj k /2 (Thomson,
of a mixed Eulerian–Lagrangian analysis of the evolution of         1987). Equation (3) describes the evolution of the prob-
atmospheric constituents (e.g. water or pollutants) along a         ability density function p(x (0) , . . ., x (M) , t), where x (k) =
                                                                       (k) (k) (k)
trajectory (Sodemann et al., 2008; Real et al., 2010, see, e.g.).   (x1 , x2 , x3 ). For the evolution of (X (0) , . . ., X (M) ) to be
   Malguzzi et al. (2011) recently developed a new global nu-       approximated by a Markov process, the time correlation of
merical weather prediction model, named GLOBO, based on             the variable X (M+1) has to be much shorter than the charac-
a uniform latitude–longitude grid. The model is an extension        teristic evolution time of X(M) . If the model has to describe
to the global scale of the Bologna Limited Area Model (BO-          the evolution of dispersion at time t  τ , where τ is the cor-
LAM) (Buzzi et al., 2004), developed and employed during            relation time of turbulent velocity fluctuations, the process is
the early 90s. GLOBO is used for daily forecasting at the           well captured at order M = 0. When shorter times are consid-
Institute of Atmospheric Sciences and Climate of the Na-            ered, as in the case of dispersion from a single point source
tional Research Council of Italy (ISAC-CNR) and is also             before the Taylor (1921) diffusive regime occurs (t ≤ τ ), or-
used to produce monthly forecasts. Online integration with          der M must be increased to 1. The model of lowest order
BOLAM family models has already yielded interesting re-             (M = 0) is referred to as random displacement model (RDM)
sults in the development of the meteorology and composition         and is sufficiently accurate to describe the transport and mix-
model BOLCHEM (BOLam + CHEMistry) (Mircea et al.,                   ing of particles at a time and space resolution typical of a
2008). Considering that experience, the GLOBO model con-            global model.
stitutes the natural basis for the further development of an           The correct formulation of a RDM in a variable density
integrated Lagrangian model.                                        flow was first obtained by Venkatram (1993) and then re-
   In the following, the development of the vertical diffusion      fined and generalised by Thomson (1995) and is briefly re-
module is presented, focusing in particular on its compliance       called here. Equation (3) is valid for the probability den-
with basic theoretical requirements (Thomson, 1987, 1995,           sity function p of particle position with the initial condition
the well-mixed condition, see ) in connection with different        p((x), t)|t=t0 = p((x), t0 ). Since the ensemble average con-
numerical issues. In Sect. 2 the theoretical basis of the model     centration hci is proportional to p, Eq. (3) can be rewritten as
formulation is given, while Sect. 3 describes different aspects
of the numerical implementation. Finally, the model verifica-
tion is presented and discussed in Sect. 4.                         ∂hci     ∂               ∂2
                                                                         =−     (ai hci) +         (Kij hci) .                     (4)
                                                                     ∂t     ∂xi            ∂xi ∂xj

2   Lagrangian stochastic model formulation                         If hci ∝ hρi at some time t 0 , where hρi is the ensemble aver-
                                                                    age of air density, then for all t > t 0 the two quantities must
In application to dispersion in turbulent flows, Lagrangian         remain proportional. This condition, called well-mixed con-
stochastic models (LSMs), Markovian at order M(M =                  dition (WMC) after Thomson (1987), implies that hρi is also

Geosci. Model Dev., 7, 2181–2191, 2014                                         
D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module                                                                 2183

a solution of Eq. (4). Substituting c with ρ in Eq. (4) and       tends, with height above the ground, to a pressure coordinate
using the continuity equation                                     P , according to
∂hρi     ∂                                                        P = P0 σ − (P0 − PS )σ α ,                                  (10)
     =−     (ui hρi) ,                                     (5)
 ∂t     ∂xi
                                                                  where P0 is a reference pressure (typically 1000 hPa), PS is
where ui is the density weighted mean velocity, defined as        the surface pressure and α is a parameter that gives the clas-
(Thomson, 1995):                                                  sical σ coordinate for α = 1 (Phillips, 1957). The parameter
                                                                  α depends on the model orography and, therefore, on resolu-
       hui ρi          hu0 ρ 0 i                                                                        ∂
ui =          = hui i + i ,                                (6)    tion. It is limited by the condition ∂p σ ≥ 0 that results in the
        hρi             hρi                                       relationship:
the following expression is obtained:                                        P0
                                                                  α≤                  ,                                       (11)
   ∂                ∂               ∂2                                  P0 − min(PS )
−     (ui hρi) = −     (ai hρi) +         (Kij hρi) .      (7)
  ∂xi              ∂xi            ∂xi ∂xj                         which is satisfied by the typical setting α = 2, used for a wide
                                                                  range of resolutions in GLOBO applications (Malguzzi et al.,
Then, integrating both sides and rearranging gives                2011).
       ∂Kij Kij ∂hρi                                                 The vertical Lagrangian coordinate is identified by 6, cor-
ai =       +         + ui ,                                (8)    responding to the vertical coordinate σ , and is connected to
       ∂xj   hρi ∂xj
                                                                  the Lagrangian vertical position Z above the ground through
where the non-uniqueness implied by the integration is re-        Eq. (10) and the hydrostatic relationship. In the meteorologi-
moved considering that in the well-mixed state, the mixing        cal component, the height above the ground z is a diagnostic
ratio flux must be proportional to ui hρi. Substituting Eq. (8)   quantity that can be derived from the geopotential 8 through
into Eq. (4) gives the equivalent of Eq. (2) in Thomson           z(σ ) = (8(σ )−8g )g −1 , where 8g is the geopotential at the
(1995).                                                           height of roughness length. Since the determination of the
   At the coarse resolution typical of global models, ver-        different terms in Eq. (9) involves discrete Eulerian fields
tical motions can be considered decoupled from the hori-          and their numerical derivatives, the choice of employing σ
zontal ones. Therefore, only the vertical coordinate x3 ≡ z       also has the advantage of making interpolation straightfor-
(and X3 ≡ Z in Lagrangian terms) need to be considered. In        ward and consistent with the Eulerian part.
this case, the RDM reduces to a single differential stochastic       Because σ (z) is not linear (σ is not a Cartesian coordi-
equation                                                          nate system), the stochastic chain rule (see, e.g. Kloeden and
                                                                  Platen, 1992, p. 80) must be used to derive the correct form
              ∂K     K ∂hρi                                       of Eq. (9) for 6, giving
dZ = w +          +            dt + 2KdW ,                  (9)
               ∂z   hρi ∂z                                               "      2                                #
                                                                                  ∂σ      1 ∂                 ∂ 2σ
where w ≡ u3 and K ≡ K33 .                                        d6 = ω +                      (hρiK) + K 2 dt               (12)
                                                                                  ∂z     hρi ∂σ               ∂z
3     Numerical implementation of the vertical                          +      (2K)1/2 dW ,
      diffusion module
                                                                  where ω is the vertical velocity in the σ coordinate system
In its final form, IL-GLOBO is designed to be a fully online      and z is the Cartesian vertical coordinate. The last term in
integrated model (or at least an online-access model, accord-     square brackets stems from the Itô–Taylor expansion of order
ing to Baklanov et al., 2014), where the different compo-         dW 2 , which must be included for the correct description at
nents share the same “view” of the atmosphere, i.e. use the       order dt (Gardiner, 1990, p. 63).
same discretisation, parameterisations, etc. The development
of the vertical diffusion module is based on this principle.      3.2   Discretisation and interpolation

3.1    Vertical coordinate                                        The GLOBO prognostic variables are computed on a Lorenz
                                                                  (1960) vertical grid: all the quantities are on “integer” lev-
Within IL-GLOBO, the Lagrangian equations are integrated          els σi , except vertical velocity, turbulent kinetic energy and
in the same coordinate system used in the Eulerian model.         mixing length and, consequently, diffusion coefficients, lo-
This choice maintains the consistency between the La-             cated at “semi-integer” levels σih (see Fig. 1). In typical ap-
grangian and Eulerian components and reduces the interpo-         plications, the GLOBO vertical grid is regularly spaced in
lation errors and computational cost.                             σ (Malguzzi et al., 2011), although it is possible to use a
   GLOBO uses a hybrid vertical coordinate system in which        variable grid spacing, as in its limited-area version BOLAM
the terrain-following coordinate σ (0 < σ < 1) smoothly           (Buzzi et al., 1994).                                                  Geosci. Model Dev., 7, 2181–2191, 2014
2184                                                                            D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module
         4                                                                            Rossi and Maurizi: IL-GLOBO: vertical diffusion module
                                                                                           For D (Akima,
                                                                                                   , the values
                                                                                    are collinear           1991).ofUsingthe first-order
                                                                                                                               this property,derivative
                                                                                                                                              a linear at the lowest
                                                                                    profileboundary     are computed
                                                                                            near the ground  is imposed to  asthe interpolating func-
                                                                                    tion by adding two fictitious points below the ground that are
                                                                                           ∂Kwith the two lower
                                                                                    collinear                  KNLEV+1
                                                                                                                   grid points−ofKthe  domain. In ad-
                                                                                    dition, ∂σ             =
                                                                                            to ensure the positivity of the interpolating  .functions,          (15)
                                                                                                 NLEV+1          σNLEV+1      −   σ NLEV
                                                                                    the local algorithm of Fischer et al. (1991) is used, which
                                                                                    also preserves
                                                                                           This isthe   continuitybecause
                                                                                                     assumed        of first order
                                                                                                                                K derivatives.
                                                                                                                                    is expected to be linear near
                                                                                        the surface,
                                                                                    3.3 Integration    according
                                                                                                    scheme        to Monin–Obukhov
                                                                                                           and time-step selection                    similarity theory
                                                                                    The most common integration scheme for SDE in atmo-
                                                                              290         K(z)
                                                                                    spheric      = κumodels
                                                                                            transport ∗z    is the Euler-Maruyama forward                               (16)
                                                                                          for the neutral case, with proper modifications for diabatic
                                                                                          = Σt + a∆t + b∆W .                                       (17)
                                                                                            The second
                                                                                   The coefficients  a and method
                                                                                                            b come from(labelled   A)(12).
                                                                                                                            Equation          The on the Akima
                                                                                                                                        is based
                                                                                         (1991) cubic
                                                                                   Euler-Maruyama        spline.
                                                                                                     forward       Foris each
                                                                                                              scheme            interval
                                                                                                                         the simplest     it considers
                                                                                                                                      strong Tay-        the previ-
                                                                               295       ous and theand
                                                                                   lor approximation    next  two
                                                                                                          turns outadjacent    intervals
                                                                                                                    to be of order         (for
                                                                                                                                   of strong    a total number of
Figure Fig.
         1. Schematic    representation
              1. Schematic   representation of field
                                                              distributionsbe- be- vergence γ = 0.5 (Kloeden and Platen, 1992, p. 305).
        tween integer (continuouslines)
                                   lines) and
                                                                                         six grid points) to compute the coefficients of the interpolat-
tween integer     (continuous             andsemi  integer (dashed
                                               semi-integer        lines) lev-
                                                                 (dashed     lines) By a rather simple modification of the Euler-Maruyama
        els in the GLOBO   model.                                                        ing cubic polynomial. This algorithm reduces the number of
levels in the GLOBO model.                                                         scheme, i.e. adding the term:
                                                                                          oscillations in the interpolating function compared to regular
                                                                                    1 ′ cubic
         for the first order derivative. Following the same consider-                 bb (∆W 2 −splines
                                                                                                 ∆t),   and enforces the linearity when
                                                                                                                                    (18) four points are
                                                                                    2     collinear (Akima, 1991). Using this property, a linear pro-
    6 being
                          for ρ, thecoordinate,
                                     derivatives ofthe      quantities
                                                       σ with   respect ofneeded
                                                                            z are to
    255  computed
              the termsfromofrelationships
                                Eq. (12) similar
                                            must be  to Eq.   (13) and Eq.from
                                                          interpolated             the wherefile
                                                                            (14). 300          b′ is near   the ground
                                                                                                     the first-order        is imposed
                                                                                                                      derivative            to the scheme
                                                                                                                                 of b, the Milstein   interpolating function
             For  the  highly  varying   K  profiles,   two   different methods
Eulerian fields given at discrete levels. The computation of is obtained,                     by addingwhich istwoof order  of strong
                                                                                                                      fictitious             below γthe
                                                                                                                                   points                 = 1.ground that are
         are tested, the first with two variants. The first method in-                 It is worth   notingwith
                                                                                              collinear      that the
                                                                                                                    thestrong  order γgrid
                                                                                                                         two lower      = 1 of  the Milstein
                                                                                                                                             points     of the domain. In ad-
first- and     second-order        derivatives     of   Eulerian      model
         terpolates the function linearly at the particle position, and        quan-   scheme corresponds to the strong order γ = 1 of the Euler de-
tities isuses
                  required      in the
                       differences       implementation
                                    derivatives.                  of the(labeled
                                                  In the first variant                        dition,
                                                                           LSM. In- terministic          to ensure
                                                                                                     scheme.           the positivity
                                                                                                                 Therefore,   Milstein canofbethe   interpolating
                                                                                                                                                 regarded   as      functions,
    260  D), theandfirstderivation     algorithms
                         order two-points   derivativecan     influence
                                                          is computed   andboth               the local
                                                                                   the the correct
                                                                             kept 305                       algorithm
                                                                                                     generalization       of Fischer
                                                                                                                       of the            et al.Euler
                                                                                                                              deterministic     (1991)     is used, which also
accuracy constant
              and thebetween   two grid points.
                         computational       costInoforder
                                                                give a smoother
                                                                              model (Kloeden  preserves      the continuity
                                                                                                    and Platen,    1992, p. 345).of first-order
                                                                                                                                      The additional derivatives.
         description of the derivatives, a variant (labeled D′ ) is also               uses only already computed quantities involved in the deter-
and thus     require careful assessment.
         tested in which the three-points centered derivative is com-                  mination
                                                                                              3.3of the    drift term of scheme
                                                                                                       Integration        Equation (12).
                                                                                                                                    and Preliminary
                                                                                                                                           time-step ide- selection
    For density          and geopotential
         puted andρinterpolated       linearly at8,thelinear    interpolation
                                                         particle  position. For and alized tests do not show any appreciable accuracy improve-
    265  Ddifferences
             , the values ofderivative     are usedat the
                              first order derivative       assuming       that those
                                                               lowest boundary    310  ment with respect to the Euler-Maruyama scheme. However,
         is computed
                                                                                              The most common integration scheme for SDE in atmo-
fields are      regularas:enough. At the lower boundary, it is re- because they confirm the negligible extra computational cost
quired that                                                                            of thisspheric
                                                                                               method, the transport
                                                                                                              Milnsteinmodels       is bethe
                                                                                                                          scheme will      usedEuler–Maruyama
                                                                                                                                                  to integrate        forward
          ∂K                KNLEV+1 − KNLEV                                                   scheme:
                         =                    .                              (15)      the model.
          ∂σ   NLEV+1
                          σNLEV+1 − σNLEV                                             In the meteorology component of IL-GLOBO, the Eule-
 ∂ 2ρ                  ∂ 2ρ                                                            6t+1t =   are6                                                         (17)
                  =                    ,                                    (13) rian equations         t + a1t
                                                                                                     solved   with+   b1Wtime-step
                                                                                                                   a macro    .         ∆T , which
                            2 because K is expected to be linear near
         This  is assumed
 ∂σ 2 NLEV+1
         the surface, ∂σ       NLEV
                       according   to Monin-Obukhov similarity theory            depends basically on the horizontal resolution due to the
    270  where:                                                                        The imposed
                                                                                 limitations  coefficients         and b number.
                                                                                                          by thea Courant   come from  OtherEq.  (12). The Euler–
which implies                                                                    steps are  involved in
                                                                                       Maruyama            the Eulerian
                                                                                                        forward    schemepart is
                                                                                                                                  are simplest
                                                                                                                                       not relevant
                                                                                                                                                 strong Taylor ap-
         K(z) = κu∗ z,                                                 (16)      here. In typical implementations, ∆T ranges from 432 s for
                                                                                       proximation and turns out to be of the order                 of strong con-
  ∂ρ for the neutral case, with proper modifications for diabatic 320 362 ×vergence     242 point resolution (used for monthly forecasts1 ) to
                  =                                                                                 γ    =  0.5  (Kloeden     and  Platen,
                                                                            (14) 150 s for 1202 × 818 point resolution (used for high resolu-1992,  p. 305).
 ∂σ NLEV+1
         cases.                                                                            By forecasts
                                                                                 tion weather   a rather2 ).simple    modification
                                                                                                              The macro                  of the
                                                                                                                          time-step is taken     Euler–Maruyama
                                                                                                                                             as the
            The second     method (labeled A) is based on the Akima              upper scheme,     i.e.solution
                                                                                        limit for the     addingofthe   term:(12). The time-step
     ∂ρ (1991) cubic  ∂ 2ρ
                  + spline.
                                  For(σeach interval  it considers the pre-
                                         NLEV+1 − σNLEV )                        needed to reach the required accuracy depends on the quan-
    ∂σ vious
          NLEVand the∂σnextNLEVtwo adjacent intervals (for a total number 325 tities involved
                                                                                        1 0 in determining
                                                                                                      2            the various elements in Equa-
         of 6 grid points) to compute the coefficients of the interpo-           tion (17).bb (1W − 1t) ,                                                     (18)
for the lating  cubic polynomial.
          first-order    derivative.This   algorithm the
                                         Following     reduces
                                                            samethe considera-
         of oscillations
tions made      for ρ, thein the interpolatingoffunction
                               derivatives        σ with  compared     reg-z are 1
                                                             respecttoof               where b0 is the first-order derivative of b, the Milstein scheme    dpc/month
    280  ular cubic splines and enforces the linearity when 4 points        
computed from relationships similar to Eqs. (13) and (14).                             is obtained, which is of the order of strong convergence
   For the highly varying K profiles, two different methods                            γ = 1. It is worth noting that the strong order γ = 1 of the
are tested, the first with two variants. The first method in-                          Milstein scheme corresponds to the strong order γ = 1 of the
terpolates the function linearly at the particle position and                          Euler deterministic scheme. Therefore, Milstein can be re-
uses finite differences derivatives. In the first variant (la-                         garded as the correct generalisation of the deterministic Euler
belled D), the first-order two-point derivative is computed                            scheme (Kloeden and Platen, 1992, p. 345). The additional
and kept constant between two grid points. In order to give                            term uses only already-computed quantities involved in the
a smoother description of the derivatives, a variant (labelled                         determination of the drift term of Eq. (12). Preliminary ide-
D0 ) is also tested in which the three-point centered derivative                       alised tests do not show any appreciable accuracy improve-
is computed and interpolated linearly at the particle position.                        ment with respect to the Euler–Maruyama scheme. However,

Geosci. Model Dev., 7, 2181–2191, 2014                                                                   
D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module                                                                    2185

because they confirm the negligible extra computational cost           The above equation has the property of limiting 1t2 accord-
of this method, the Milstein scheme will be used to integrate          ing to the sharpness of the K peak.
the model.                                                               Taking the minimum among 1T , 1t1 and 1t2 (and replac-
   In the meteorology component of IL-GLOBO, the Eule-                 ing  with = CT in Eqs. 19 and 20), gives
rian equations are solved with a macro time step 1T , which                       "                                 −1
                                                                                                 ∂K −2 CT ∂ 2 K
depends basically on the horizontal resolution due to the lim-                          CT
                                                                       1t = min 1T ,        K           ,                ,    (21)
itations imposed by the Courant number. Other time steps are                             2       ∂σ        2 ∂σ 2
involved in the Eulerian part but are not relevant here. In typ-
ical implementations, 1T ranges from 432 s for 362 × 242               where the parameter CT quantifies the “much less” condition
point resolution (used for monthly forecasts1 ) to 150 s for           and, therefore, must be at least 0.1 or smaller.
1202×818 point resolution (used for high-resolution weather               Figure 2 shows the application of Eq. (21) for a K profile
forecasts2 ). The macro time step is taken as the upper limit          representative of GLOBO (see Sect. 4) and a CT = 0.01. The
for the solution of Eq. (12). The time step needed to reach            1t decreases in the presence of K gradients thanks to con-
the required accuracy depends on the quantities involved in            dition (19), and is limited around the K maximum (where
determining the various elements in Eq. (17).                          ∂K/∂σ = 0) by condition (20). The maximum of 1t = 1T
   First, a straightforward constraint is that the time step must      is attained at higher levels.
satisfy the relationship                                                  It should be kept in mind that the method is based on local
                                                                       quantities and may fail if strong variations of K occur in one
                       −1                                              time step along the particle path. To overcome this problem,
p          ∂K
 2K1t1  K                      ,                               (19)   an additional constraint is used to make the algorithm non-
                                                                       local (or “less local”). Using the 1t0 computed at the particle
(Wilson and Yee, 2007, see, e.g.), which expresses the re-             position at time t, two other time steps (1t+ and 1t− ) are
quirement that the average root mean square step length must           evaluated at the positions:
be much smaller than the scale of the variations of K. This                                     1/2
gives rise to a limitation that is consistent with the surface-        6± = 6t + a1t0 ± b1t0 .                                   (22)
layer behaviour of the diffusion coefficient, Eq. (16). The            The minimum 1t among 1t0 , 1t+ and 1t− is then used to
condition expressed by Eq. (19) makes 1t1 vanish for z → 0.            advance the particle position 6t+1t .
Such behaviour ensures that the WMC is satisfied theoret-
ically, but clearly poses problems for numerical implemen-             3.4    Boundary conditions
tation (Ermak and Nasstrom, 2000; Wilson and Yee, 2007).
However, in the application of a global model, where parti-            The necessary boundary condition for the conservation of
cles can be distributed throughout the troposphere, this prob-         the probability (and therefore of the mass) is the reflective
lem affects only a small fraction of particles in the vicinity         boundary (Gardiner, 1990, p. 121). Wilson and Flesch (1993)
of the surface. Therefore, it can be dealt with by selecting a         show that the elastic reflection ensures the WMC if the inte-
1tmin small enough for the solution to be within the accepted          gration time step is small enough. However, in cases of non-
error and, at the same time, large enough to not impact the            homogeneous K, numerical implementation requires that 1t
overall computational cost.                                            vanishes as the particle approaches the boundary. For models
   In addition to Eq. (19), another constraint is needed to ac-        that focus on near-surface dispersion, the time step needed to
count also for the presence of maxima in the K profile, which          achieve the required accuracy can become very small. Ermak
must be present if one considers the whole atmosphere. At              and Nasstrom (2000) describe a theoretically well-founded
maxima (or minima), Eq. (19) gives an unlimited 1t1 , which            method to speed up (roughly by a factor of 10) simulations
is not suitable for the integration of the model as it could           of this kind.
cause the trajectory to cross the maximum (or minimum),                   In the case of IL-GLOBO, it will be shown that the elas-
with a significant change in K(z) associated to a change in            tic reflection condition at σ = 1, coupled with the adaptive
∂z K sign. To avoid this problem, a further constraint is in-          time-step algorithm described in Sect. 3.3, can ensure a good
troduced, based on the normalised second-order derivative,             approximation of the solution while maintaining affordable
which gives an estimation of the width of the maximum. The             the computational cost.
constraint reads
                       −1                                              4     Model verification: the well-mixed condition
                ∂ 2K
2K1t2  K                   .                                   (20)
                ∂σ 2
                                                                       In order to verify the vertical diffusion module of IL-
   1      GLOBO, a series of experiments was performed with a 1-D
en.htm                                                                 version of the code and then tested in a preliminary version
   2          of the full 3-D model. Input profiles were obtained by run-
newNH/                                                                 ning the low-resolution version of GLOBO (horizontal grid                                                      Geosci. Model Dev., 7, 2181–2191, 2014
diffusion module2186                                                                                          5D. Rossi and A.
                                                                                                                                Maurizi: IL-GLOBO: vertical diffusion module
                                                                                                                                                                                                                  Rossi and Ma

  that the time-step                                                                           90                                            45000                                          1.2                    90

                                                                                               80                                            40000                                                                 80
                                         100                                                                                                                                                1
                                                                                               70                                            35000                                                                 70
                     (19)                 10                                                   60                                            30000                                          0.8                    60

                                                                                                     K[m /s]

                                                                                                                                                                                                  ρ[kg/m ]

                                                                                                                                                                                                              K[m /s]
                                                                                                                                             25000                                                                 50



h expresses the re-                      1                                                      40
                                                                                                                                             20000                                                                 40
  re step length must                                                                           30
  iations of K. This                                                                                                                         15000                                          0.4                    30
                                       0.1                                                      20
 nt with the surface                                                                                                                         10000                                                                 20
                                                                                                10                                                                                          0.2
 nt, Eq. (16). The                                                                                                                           5000                                                                  10
 kes ∆t1 vanish for                   0.01                                                      0
                                                                                                                                                0                                           0                           0
                                           0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1                                                                     0    0.2   0.4       0.6   0.8     1                                0.6   0.65
 is satisfied theoret-                                                                                                                                                σ
 merical implemen-
 on and Yee, 2007). Figure Fig.2. Values
                                    2. Valuesof ofintegration
                                                     integrationtime    step ∆t
                                                                  time-step    1t forforthe
                                                                                               diffusivity     pro-
                                                                                                           profile    Figure 3. Fig.Average
                                                                                                                                      3. AverageGLOBO
                                                                                                                                                    GLOBO   profiles
                                                                                                                                                              profilesofof ρρ (green  symbols)
                                                                                                                                                                              (green symbols)        φ/gφ/g Fig. 4. Diffusivit
model, where parti-                                                                                                             (blue symbols)
                                                                                                                      (blue symbols)              as a function
                                                                                                                                           as a function     of vertical     coordinate σ,σand
                                                                                                                                                                of verticalcoordinate       andtheir ana-ana- represents the dat
                    file shownshownby the
                                       by theredredcurve.
                                                      curve.The  green
                                                               The  greenline
                                                                                   shows the
                                                                                           the contribution
                                                                                                contribution ofof
  osphere, this prob-                                                                                                           lytical fits (Eq. 23 and Eq. 24, lines of the same colors).                   The ‘average’ pro
                    Eq. (19), Eq.the
                                   (19), theline
                                      blue     bluethe
                                                          the contribution ofofEq.Eq.(20)
                                                                                             andthethe black
                                                                                                       black line
                                                                                                               line   lytical fits (Eqs. 23 and 24).
                                                                                                                                                                                                             shown in green.
 cles in the vicinity         the combined
                    the combined        conditioncondition
                                                              21, 21,
                                                                  with    1T∆T
                                                                       with     ==432 432s sand
                                                                                              and C CT =   0.01).
                                                                                                     T = 0.01).                                                                                              by Eq. (26).
 t with by selecting
                                                                                                                                of the full 3-D model. Input profiles were obtained by run-
o be within the ac-                                                                                                   Table 1. RMSE and execution time for different CT .
                                                                                                                                ning the low-resolution version of GLOBO (horizontal grid
  ough to not impact          the problem, an additional constraint is used to make the al-                                                                                                                       As far as the
                    of 362 × 242 cells and 50 vertical levels evenly spaced in σ )                                         405  of 362 × 242 cells and 50 vertical levels evenly spaced in σ)
                        375   gorithm non-local (or less local). Using the ∆t0 computed at                                                       CT
                                                                                                                                starting at 2011-03-11   RMSE
                                                                                                                                                         00:00 UTC.  Time
                                                                                                                                                                        After[s]36 hours of simula-
 onstraint is neededstartingtheatparticle
                                    11 March           2011
                                                position       00:00
                                                            at time  t, UTC.
                                                                        two other  After    36 h of
                                                                                       time-step     (∆t simula-                                                                                             K(z) = Azexp
                                                                                                           + and                tion (12:00 UTC), averages on σ = const surfaces were per-
 ima in the K pro-  tion (12:00       UTC),        averaging      of
                              ∆t− ) are evaluated at the positions:   σ  =    const    surfaces       were    per-              formed for K, ρ0.5 and Φ,0.044
                                                                                                                                                           obtaining vertical76 profiles as a func-
                                                                                                                                                                                                    435      is used to accou
 ders the whole at- formed for K, ρ and 8, obtaining vertical profiles as a func-                                                                0.1      0.037            238
                                                                                                                                tion of σ. Fields of ρ and Φ were averaged over the whole                    play a linear be
                    an ofΣσ±. =Fields
 ation (19) gives tion                 Σt + a∆t  of 0ρ±andb∆t08 .were averaged over the whole                (22)          410  domain. As far0.01as K is 0.021          1172 were performed
                                                                                                                                                          concerned, averages                                near the bounda
  e integration of the
                    domain. As far as K is concerned, averages were performed                                                   for latitude between       ◦
                                                                                                                                              0.001 +600.021  and -60◦ North
                                                                                                                                                                         7317in daytime (longi-              maximum at so
  ross the maximum            The minimum ∆t among ∆t0 , ∆t◦+ and ∆t− is then used to                                           tude between -45◦ and +45◦ East) and nighttime (longitude
                    for380latitudes     between                and −60                                                                                                                                       was first determ
  n K(z) associated           advance     the particle  +60 position   Σt+∆tNorth
                                                                                .         in daytime (lon-
                                                                                                                                between +135◦ and -135◦ East) conditions, over land and 440                  erties (the first
                    gitudes     between         −45     and   +45   ◦ East)      and    night-time        (longi-
  problem, a further                                                  ◦ East) conditions, over land                             sea separately. The most intense K profile is selected, which                a friction veloc
 ormalized second-  tudes    between
                              3.4           +135
                                      Boundary         and   −135
                                                                                                                      is used
                                                                                                                           415  corresponds
                                                                                                                                 to accounttofor the the
                                                                                                                                                      daytime   conditions
                                                                                                                                                          specific          over Profiles
                                                                                                                                                                      K features:        shouldofdis-        rameters were
 on of the width and of sea separately. The most intense K profile is selected                                        play a ρlinear
                                                                                                                                   and z behaviour
                                                                                                                                          are rather smooth
                                                                                                                                                       near theand regular
                                                                                                                                                                    surface,over   spacetend
                                                                                                                                                                                 must     and time,
                                                                                                                                                                                               to zero       B = 1.3 × 10−
                    which The        necessary to
                              corresponds            boundary     condition
                                                        the daytime               for the conservation
                                                                          conditions        over land. Pro-     of              while K displays a large 3variability. The profiles were fitted
                                                                                                                      near thewith boundary                                                                     Although the
                              the probability (and therefore of the mass) is the reflective
                    files ofboundary
                                ρ and z are         rather smooth and regular over space and                                          analytical functions derived combining the hydrostatic 445a
                                                                                                                                                layer   top    and,   therefore,     must   display
                                                                                                                                                                                                             cal GLOBO dif
                                              (Gardiner, 1990, p. 121). Wilson and Flesch                             maximum   equationsome
                                                                                                                                      at         height.
                                                                                                                                           and the  perfectIngasEq.
                                                                                                                                                                        The A              ms−1 was
                                                                                                                                                                                  = 0.29analytical
                                                                                                                                                                              following                      regular, creating
                 (20)   385
                              (1993) show that thelarge
                                        K    displays        elasticvariability.       The profiles
                                                                       reflection ensures         the WMC    wereif   first420determined
                                                                                                                                expressions according
                                                                                                                                              were used: to average surface-layer proper-                    this reason, a pr
                    fitted with      analyticaltime-step
                              the integration         functions      derived
                                                                 is small          by combining
                                                                             enough.     However, inthe        hy-
                                                                                                            cases     ties (the first GLOBO         vertical level), and corresponds to a                    lated strong ma
 miting ∆t2 accord- drostatic      equation and theK,
                              of non-homogeneous              perfect
                                                                 numericalgas law.      The following
                                                                                implementation                 an-
                                                                                                        requires              ρ(σ) = ρ0 σ (Rd Γ/g+1) ,                                     (23)
                                                                                                                      friction velocity    u∗ ' 0.7 ms−1 . Then, the other two param-                        strong convectiv
                    alyticalthat    ∆t vanisheswere
                                expressions            as theused:
                                                               particle approaches the boundary. For                                                                                            450          tion (26) on th
   and ∆t2 (and re-                                                                                                   eters were
                                                                                                                              and: allowed to vary to fit the average profile giving
                              models that focus on near surface dispersion, the time-step                                             −3      −1                                                             4.0 × 10−3 m−1
 ) and (20)), gives:            ρ0 σ (Rtod 0/g+1)
                    ρ(σ390) =needed            achieve, the required accuracy can become very                 (23)    B = 1.3 × 10 (σ −R   md Γ/gand
                                                                                                                                                  − 1)TC = 1.6.                                              ‘averaged’ and
                              small.    Ermak and Nasstrom (2000) describe a theoretically                                    z(σ) =the above profile
                                                                                                                         Although                          , is representative of the(24)    typi-
 2     −1
          #                                                                                                                                     Γ
   K                and well founded method to speed-up (roughly by a factor of 10)                                   cal GLOBO diffusivity, real profiles           can  be  remarkably       less          4.1 Determin
            ,    (21)                                                                                                              T0 = 288.0 K, ρ0 = 1.2 kgm− 3 and Γ = −0.007 K m−1 .
 σ2                           simulations of this kind.
                                    −R 0/g
                                                                                                                                creating    challenging conditions for the model. For                            tive time-
                                (σIn thed case −   of 1)T
                                                           0           it will be shown that the elastic                  425 As a consequence of the hydrostatic perfect gas assump-
                    z(σ ) =                                  ,                                                (24)    this reason,   a profile
                                                                                                                              tion, by           was the
                                                                                                                                         expressing    selected
                                                                                                                                                           densityamong   thosevertical
                                                                                                                                                                   ρ in sigma     showing unitsiso-          The first series
 uch less” condition          reflection condition         at σ = 1, coupled with the adaptive time
                        395                  0                                                                        lated strong
                                                                                                                              (ρσ = maxima
                                                                                                                                     ρ dσ   ) andnear
                                                                                                                                                  usingthe  ground.(24)
                                                                                                                                                         Equations   This
                                                                                                                                                                              typical  of strong
                                                                                                                                                                                   the follow-               the adaptive sc
   smaller.                   step algorithm described in Section 3.3, can ensure a good
 21) for a K profile                                                                                                  convective    conditions
                                                                                                                              ing constant         just
                                                                                                                                             value is    after sunrise. Fitting Eq. (26) to
                                                                                                                                                      obtained:                                              suited value for
                    with approximation
                                  T0 = 288.0of       K,the solution
                                                                ρ0 = 1.2 while     m−3
                                                                              kg maintaining     and          0=
                                                                                                     affordable                                                                                                 Simulations w
 ) and a CT = 0.01.                    −1
                                    m . As acost.            consequence of the hydrostatic                           this second profile
                                                                                                                                    ρ0 Rd T0  gives   A  =  0.3 ms−1 , B = 4.0 × 10−3 m−1
                                                                                                                              ρ  =            .                                            (25)and           by Equations (2
 gradients thanks to                                                                                                  and C =σ 4.5. Figure
                                                                                                                                        g         4 reports the GLOBO “average”
                    perfect gas assumption,                by expressing         the density ρ in sigma                                                                                         460          number concent
K maximum (where                                                                                                    “peaked” K profiles as function of σ .
 imum of ∆t = ∆T    vertical4 units Modelρverification:
                                                σ = ρ dσ the       and    using Eqs.
                                                                      well-mixed             (24) and (23),
                                                                                        condition                           430    Figure 3 shows the GLOBO averaged profiles and their fit-                     3
                                                                                                                                                                                                                   In GLOBO,
                                                                                                                                   ting functions for the density ρ and the geopotential height              erated by moist c
                       the following constant value is obtained:                                                      4.1         Determination
method is based on 400 In order to verify the vertical diffusion module of IL-                                                     Φg −1 as function of the optimal setting for the
                                                                                                                                                     of σ.                                                   boundary layer to
ng variations of K     ρGLOBO,                                                                                                    adaptive time-step selection algorithm
                         0 Rd T0 a series of experiments was performed with a 1-D
path. To overcome                                                               (25)
                 ρσ = version of. the code and then tested in a preliminary version
                                                                                                                      The first series of experiments concerns the optimisation of
                       Figure 3 shows the GLOBO-average profiles and their fitting                                    the adaptive scheme for 1t, i.e. the selection of the best
                       functions for the density ρ and the geopotential height 8g −1                                  suited value for the coefficient CT in Eq. (21).
                       as function of σ .
                          As far as the K profile is concerned, the function                                             3 In GLOBO, K also accounts for a part of the instability gener-
                                       h         i                                                                    ated by moist convection and, therefore, it might not vanish at the
                       K(z) = Az exp −(Bz)C ,                                   (26)                                  boundary layer top.

                       Geosci. Model Dev., 7, 2181–2191, 2014                                                                                  
Rossi and Maurizi: IL-GLOBO: vertical diffusion module

                                D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module                                                                                                                                                    2187
                                          Rossi and Maurizi: IL-GLOBO: vertical diffusion module

               1.2                          90                                                                                                                     1000
                                            80                                                                                                                                                                                 60

                                                                                                                                                                                                                                      K[m /s]


                                            70                                                                                                                        1                                                        40
                                                                                                                                                                                                                                                           Table 1. RM
                                                                                                                                                                    0.1                                                        20
               0.8                          60
                                                                                                                                                                    1.4                                                        0
                     ρ[kg/m ]

                                       K[m /s]



               0.6                                                                                                                                                  1.2                                                                                            0.05
                                                                                                                                                                     1                                                                                            0.045
               0.4                          30
                                                                                                                                                                    0.8                                                                                            0.04
                                            20                                                                                                                            0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95              1
               0.2                                      Rossi and Maurizi: IL-GLOBO: vertical diffusion module                                                                                                                                    7               0.035
                                            10                                                                                                                                                   σ


8          1
               0                                 0
                                                  0.6     0.65      0.7   0.75   0.8    0.85      0.9     0.95       1                                                    CT        RMSE Time [s]                                                                 0.025
                                                                                                                                            FigureFig.5. Dispersion
                                                                                                                                                         5. Dispersion experiment
                                                                                                                                                                                       withdifferent     choicesofofparame-
                                                                                                                                                                                              different choices       param-
                                                                                                                                                                          0.5       0.044               76                                                         0.02
                                                                                                                                                    T. C
                                                                                                                                            eter Cter  TopT . panel:  diffusivity
                                                                                                                                                              Top panel:           profile
                                                                                                                                                                                    0.037 (black line)
                                                                                                                                                                                      profile (black line)and
                                                                                                                                                                                                                 ∆t profiles
                                                                                                                                                                                                             and 1t   profiles
 n symbols) and φ/g Figure Fig.
                            4. Diffusivity  profiles
                                4. Diffusivity
                                           1000  profilesused
                                                                   the experiments.
                                                                        experiments. The  Thesymbols
                                                                                               symbols                                      for CTfor=C0.5T =(light
                                                                                                                                                               0.5 (light    CT C=T0.021
                                                                                                                                                                                      = 0.1
                                                                                                                                                                                     0.1     (green),CC
                                                                                                                                                                                         (green),       T T==0.01
                                                                                                                                                                                                              0.01 (red)
                                                                                                                                                                                                                    (red) and
                                                                                                                                                                                                                          and                                     0.015
 nate σ, and their ana-    represents
                    represents        thefrom
                               the data   data
                                                 GLOBOGLOBO  andand
                                                                  thethe linestheir
                                                                       lines,  their fitting
                                                                                              function.                                     CT = C    T = 0.001
                                                                                                                                                    0.001         (blue).
                                                                                                                                                             (blue).       Bottom panel:
                                                                                                                                                                                    0.021            7317
                                                                                                                                                                                                           concentration pro-
                                                                                                                                                                                                                          pro-                                     0.01
                                                                                                 60 is

                                                                                                                                 K[m /s]
 e colors).                The ‘average’  profile  is shown   in red, whilethe
                                                                                 ‘peaked’ profile                                                  files for different CT (Line  colors
                                                                                                                                                                                      as as
                                                                                                                                                                                         in inthe
                                             is10shown                                                                                      files for different   C (line   colours                                                                               0.005

                    The “average”   profile                in red,  while                     profile
                                                                                                 40 is

                                                                                                                                                                             T                                                                                        0.0
                           shown in green. The  1 functional form of both profiles is described                                                        Table 1. RMSE and execution time for different CT .
                    shown in  green. The functional
                           by Eq. (26).      0.1          form of both profiles is described     20 by
re obtained by run- Eq. (26).                           1.4                                                                 0
                                                                                                                                                       satisfied, this distribution must remain constant as the time
BO (horizontal grid                                                                                                                                                   Equation (12) was integrated for 4 × 105 particles

                                                        1.2                                                                                            evolves.0.05
evenly spaced in σ) Simulations   As far as      the Kperformed
                                                             profile is concerned,              the function described                                                                                                                                     Fig. 6. RM
                                              were                             in    flow      conditions                                              and    for0.045 macro time-steps, each 432 s long, for a total of
                                                                                                                                                                   200                                                                                     4 × 105 (gree
36 hours of simula-                                     1
                    by Eqs.K(z)  (23),      (24) and
                                       = Azexp                (26),C distributing
                                                          −(Bz)           ,                      particles with (26)         num-                      T = 86400   0.04 s = 24 h. The actual time-step used is given by
   surfaces were per-                                   0.8
                    ber concentration proportional           0.6 0.65 0.7 to       0.75ρ.0.8For0.85the0.9 WMC    0.95 1 to be                   465    Equation  0.035
                                                                                                                                                                      (21) with the additional lower limit ∆tmin = 0.01.
 l profiles as a func- 435 is used to account for the specific K features: it should dis-
                    satisfied,      this    distribution             must       remain      σ constant as the time                                     Simulations were performed using 12 cores of an Intel Xeon 495                                      4.2 Evalu
ged over the whole            play a linear behavior near the surface, must tend                                       to zero                                     0.03
ges were performed  evolves.  near Equation
                                            boundary  (12)layer was top  integrated
                                                                                and,          for 4 ×
                                                                                         therefore,            105display
                                                                                                            must       particles  a of parame-
                                                                                                                                                       machine. Since the initial condition was already well-mixed
                    and formaximum
   in daytime (longi-            200 macro         5. Dispersion experiment with different choices
                                                                 steps,Ineach           432 s(26),  long,      for    a total  −1  of                  (C ∝ ρ), the simulation time was considered sufficient to as-                                       In the subse
                                                at some                         Equation                   A  =   0.29    ms
 ighttime (longitude
                                           ter CT . Top panel: diffusivity profile (black line) and ∆t profiles                                        sess the0.02 skill of the model in satisfying the WMC. At the end                                   producing t
                    T = 86was    400    s for
                                     first =    24
                                            determined = The        actual
                                                                  (light   blue), time
                                                                                 to   average
                                                                                      CT =   step      used
                                                                                                0.1surface-layer          prop-
                                                                                                                          =   0.01 by (red)  and                 0.015
ons, over land and
                                                                                                                       T                        470    of  the   simulation,       final concentration profiles were computed                              niques D, D
                                   with (the
                                           Cthe first0.001
                                              T =additional     (blue).vertical
                                                                            Bottom     level),
                                                                                          panel: 1tand     corresponds
                                                                                                              = 0.01.
                                                                                                       min                      to
                                                                                                                        concentration       pro-
 e is selected, which                                                             −1
                                                                                                                                                       in “σ volume”,          i.e., c(σ) = N (σ)(∆σ)−1 , where N (σ) is the                                  In the fir
                    ulations  a friction
                                   werefiles   velocity
                                              performed       u∗ ≃using
                                                  for different       C0.7  ms12 .colors
                                                                         T (Line
                                                                                         cores    asofthe
                                                                                                      in the other
                                                                                                            an        two Xeon
                                                                                                                top   panel). pa-
  er land. Profiles of        rameters were let to vary to fit the average profile giving                                                              number0.005  of particles between σ and σ + ∆σ. The skill of the 500                                Equations (
                                                                                                                                                                       0.001                         0.01                      0.1
ver space and time, machine.        Since       the     initial
                                                     −3 −1          condition            was     already         well-mixed                            model      in reproducing the WMCCwas evaluated using the root                                      erage’ diffu
                              B = 1.3 × 10 m and C = 1.6.                                                                                                                                                  T
  profiles were fitted   ∝   ρ),   the    simulation            time      was      considered
                                  Although the above profile is representative of the typi-                sufficient       to    as-                  mean square error (RMSE) of the final normalized concen-                                            ular grid. T
  ing the hydrostatic       thecalskill    satisfied,
                                          of                this distribution
                                                       model        inreal
                                                                         satisfying       must
                                                                                             thebe  remain
                                                                                                     WMC.         constant
                                                                                                                   At theless     as the time475 tration profile with respect to the normalized density 5profile
                        445         GLOBO                                    profiles can                 remarkably               5          Figure     6. RMSE obtained from experiments made with 10 5 (red),                                           described in
                                           evolves. Equation (12) was integrated for 4 × 10 particles                                                  (derived
                                                                                                                                                       5Fig.    6. using
                                                                                                                                                                     RMSE     Equation
                                                                                                                                                                                obtained       25). experiments
                                                                                                                                                                                              from                               with 10
ollowing analytical of the simulation,
                              regular, creating    finalchallenging
                                           and for 200 macro time-steps,
                                                                               conditions profiles for the were     computed
                                                                                                                model.       For
                                                                                                            432 s long, for a total of        4 ×   10    (green)     and 16      × 105 (blue)          particles asmade a function      of C(red),
                                                                                                                                                                                                                                                T.         olution of th
                              this reason,i.e.
                    in “σ volume”,               a profile
                                                       c(σ    )  was
                                                                 =      selected
                                                                      N(σ      )(1σ         −1 ,each
                                                                                           )        those showing
                                                                                                    where        N(σ      )  iso-the
                                                                                                                             is                         4×    105 (green)
                                                                                                                                                           Figure     5 reportsand 16        105 (blue)profiles
                                                                                                                                                                                     the×different                          a function of Caf-
                                                                                                                                                                                                             particlesofasconcentration          T . 505      The part
                                           T = 86400 s = 24 h. The actual time-step used is given by
                    number    lated
                                 of    strong maximum
                                      particles          between      nearσ   the ground.
                                                                              and      σ   +   1σ     This
                                                                                                        .  The is skill
                                                                                                                   typical  of  ofthe                  ter 24 hours of simulation computed using different values                                          time are the
                  (23)             465     Equation conditions
                                                            (21) with the     justadditional            lower              ∆tmin = 0.01.
                    model strong         convective
                               in reproducing               thewereWMC          wasafter      sunrise.
                                                                                          evaluated           Fitting
                                                                                                             using                                     of CT . The shaded region represents the interval between 3                                         4.1. The in
                                                                                                                     −1the      root          4.2 Evaluation                of the interpolation                  algorithms
                        450   tion (26) on this second profile gives A = 0.3 ms , Ban=Intel Xeon480495 standard
                                           Simulations                    performed           using       12   cores    of                              4.2 Evaluation
                                                                                                                                                                     deviationsoffrom    the interpolation
                                                                                                                                                                                                 the expected value. algorithms RMSE values                gorithm. Th
                    mean square            error
                              4.0 × 10machine.
                                           −3 −1
                                                 m (RMSE) andSince
                                                                 C =the  of  the
                                                                         4.5.initial  final
                                                                                  Figure      4 normalised
                                                                                                  reportswas            concen-
                                                                                                                     GLOBO        well-mixed
                                                                                                                                                       for each simulation are reported in Table 1 along with the                                          putation of
                                profile(C    with∝ ρ),
                                                and     respect       to
                                                                       K the
                                                           the simulation
                                                        ‘peaked’            profilesnormalised
                                                                                     time as was
                                                                                              function      density
                                                                                                       considered         profile to as-
                                                                                                              of σ. sufficient                In thecomputation
                                                                                                                                                        In   the subsequent   set of  setexperiments,
                                                                                                                                                                                            of   experiments,    thethemodel
                                                                                                                                                                                                                           model   skill
                                                                                                                                                                                                                                      skill in
                                                                                                                                                                                                                                             in re-
                                                                                                                                                                           time.     The     RMSE        error becomes         comparable         to 510   possible for
                    (derived usingsess        Eq.the  25).skill of the model in satisfying the WMC. At the end                                producingthe statistical′ error for CT = 0.01, which is selected astech-
                                                                                                                                                                the   WMCthe   WMC
                                                                                                                                                                                 was     was     evaluated
                                                                                                                                                                                         evaluated          for for
                                                                                                                                                                                                                 thethe   interpolation
                                                                                                                                                                                                                      interpolation          tech-
                                                                                                                                                                                                                                                the        timated usin
                        Figure4.1 470      of the simulation,
                                    5 Determination
                                       reports        the differentof the final     concentration
                                                                                profiles      setting        profiles
                                                                                                 of concentration         wereaf-
                                                                                                                  the adap-         computed             D, D0 D,
                                                                                                                                              niquesoptimal         and D Aand      A described
                                                                                                                                                                                described                in Section
                                                                                                                                                                                                   in Sect.            3.2. dependency
                                                                                                                                                                    value.     In  order      to evaluate      the                                         The results
Γ = −0.007 K m−1 .                     tivein  “σ    volume”,
                                              time-step               i.e.,
                                                                selection    c(σ)      =
                                                                                algorithm  N  (σ)(∆σ)          −1
                                                                                                                   , where       N   (σ)  is the
                    ter 24 h of simulation computed using different values of CT .                                                              485    of CIn
                                                                                                                                                  In the      T
                                                                                                                                                                 on    firstnumber
                                                                                                                                                                      the      experiment,
                                                                                                                                                                      experiment,        of  the    the   analytical
                                                                                                                                                                                              particles,     two        fieldsdescribed
                                                                                                                                                                                                                   additional     described
                                                                                                                                                                                                                                   sets of runs   by
                                                                                                                                                                                                                                                 by        upper pane
erfect gas assump-                         number of particles between σ and σ + ∆σ. The skill of the 500 Equations                                                      (23),(26)
                                                                                                                                                                                 (24)withand   5 (26) with the5parameters of the ‘av-
                    The     shaded
 gma vertical units 455 The firstmodel  region       represents
                                             series in  ofreproducing     the
                                                             experiments the      interval
                                                                                   concerns       between          3   standard               Eqs.     were
                                                                                                                                                      (23),     performed
                                                                                                                                                               (24)    and       with     10      and
                                                                                                                                                                                                 the     16  ×  10
                                                                                                                                                                                                        parameters    particles
                                                                                                                                                                                                                          of   the  that   corre-
                                                                                                                                                                                                                                     “average”             simulations
                                                                                        WMCthe       wasoptimization
                                                                                                             evaluated using    of the root             erage’todiffusivity          profile      were resampled          on a 50 statistical
                                                                                                                                                                                                                                       point reg-
                    deviations        from       the      expected          value.        RMSE                                                         spond          halving
                                                                                                                                                                            wereand        doubling,      onrespectively,
                                                                                                                                                                                                               a 50-point the                              profile, are
nd (23), the follow-          the adaptive mean     scheme
                                                      square       for
                                                                   error ∆t,     i.e., the
                                                                            (RMSE)                 the values
                                                                                             ofselection               forbest
                                                                                                                      the      eachconcen-    diffusivity
                                                                                                                                                                grid.   This
                                                                                                                                                                                provides        a  discrete    version     of
                                                                                                                                                                                                                                 regular grid.515
                                                                                                                                                                                                                               the   experiment
                    simulationsuited   are
                                        value reported
                                                  for profile    in with
                                                         the coefficient         C1Talonginto       with
                                                                                             Equation         the computa-
                                                                                                              21.                                      error of the
                                                                                                                                              This provides              base
                                                                                                                                                                      a in       experiment.
                                                                                                                                                                          discrete     version        Results
                                                                                                                                                                                                       of the    are  reported indescribed
                                                                                                                                                                                                                  experiment            Figure 6           distribution
                                           tration                           respect            the     normalized          density profile             described           the previous          section,    with   thethesame     verticalCTres-

                    tion time.Simulations
                                              RMSE     were     performed
                                                             error      becomes    in   flow     conditions
                                                                                            comparable             described
                                                                                                                    to   the     sta-         in  the  which
                                                                                                                                                         previousshows      that,
                                                                                                                                                                         section,   in with
                                                                                                                                                                                       the considered
                                                                                                                                                                                                  the    same   range,
                                                                                                                                                                                                                  vertical      optimal
                                                                                                                                                                                                                                resolution        is
                                                                                                                                                                                                                                                  of       with the exp
                                                           using Equation 25).                                                                          olution     of   the   GLOBO          original     fields.
                  (25)        by   Equations         (23),    (24)    and    (26),      distributing         particles      with                490    quite    independent         of  the    number       of particles.                                  and RMSE
                    tistical error forFigure     CT =50.01,             which         is   selected          as  the
                                                               reports the different profiles of concentration af- 505  optimal               the   GLOBO           original      fields.
                        460   number concentration proportional to ρ. For the WMC to be                                                                    ItThe    particle
                                                                                                                                                               is worth          number,
                                                                                                                                                                              noting     that initial       distribution
                                                                                                                                                                                                  the time-step       selectionand simulation
                                                                                                                                                                                                                                      algorithm               The time
                    value. In order        ter to24 evaluate            the possible
                                                       hours of simulation                        dependency
                                                                                            computed                       of CT values The time
                                                                                                                using different                           particle
                                                                                                                                                                 are  the number,
                                                                                                                                                                             same     as  initial
                                                                                                                                                                                           in   the    distribution
                                                                                                                                                                                                      experiment           and simulation
                                                                                                                                                                                                                      described        in is
  ofiles and their fit-            3                                                                                                                   with the proper choice of CT ensures that the WMC                                      also 520     around the
                    on the number          ofof
                                     In GLOBO,  CTparticles,
                                                      .K The alsoshadedtworegion
                                                                    accounts    additional represents
                                                                                  for a part      of sets      the   interval
                                                                                                               of runs
                                                                                                       the instability        were
                                                                                                                            gen-   betweentime  3 are   4.1.theThe same     asreflective
                                                                                                                                                                                 in thetime-step
                                                                                                                                                                       integration           experiment          described
                                                                                                                                                                                                          is selected     usinginthe  Sect.
                                                                                                                                                                                                                                         local 4.1.
geopotential height           erated    by   moist   5 convection       and   5therefore      it  may      not  vanish    at   the
                                                                                                                                                       satisfied     at  the                   boundary       too,  as  mentioned         in  Sec-         variations a
                    performed480with       standard
                                                10 and     deviations
                                                                16 × 10from              the expected
                                                                                  particles                      value. RMSE
                                                                                                     that correspond                to values           gorithm.
                                                                                                                                              The integration          Thetimetime-step
                                                                                                                                                                                    step      selection
                                                                                                                                                                                              is   selected  algorithm
                                                                                                                                                                                                                 using     requires
                                                                                                                                                                                                                            the    localthe   com-
                              boundaryfor    layer   top. simulation are reported in Table 1 along with the                                            tion 3.4.                                                                                           Looking at
                    halving and doubling,        each        respectively, the statistical error of the                                       rithm. putation
                                                                                                                                                         The time-step of the second
                                                                                                                                                                                   selection   orderalgorithm          of K, which
                                                                                                                                                                                                         derivative requires                 is not
                                                                                                                                                                                                                                      the com-
                                           computation time. The RMSE error becomes comparable to 510 possible for the D interpolation scheme. Therefore, it is es-
                    base experiment. Results are reported in Fig. 6 which shows                                                               putation of the second-order derivative of K, which is not
                                           the statistical error for CT = 0.01, which is selected as the                                                timated using finite differences of the first order derivative.
                    that, in the consideredoptimal value.      range,       the optimal
                                                                      In order        to evaluate     CTthe   is possible
                                                                                                                  quite inde-    dependency   possible for the D interpolation scheme. Therefore, it is es-
                                                                                                                                                        The results of this experiment are shown in Figure 7. In the
                    pendent of485theofnumber    CT on the     of number
                                                                    particles.   of particles, two additional sets of runs                    timatedupper  using     finitethedifferences
                                                                                                                                                                   panel,           integration of          the first-order
                                                                                                                                                                                                        time-step     profiles of   derivative.
                                                                                                                                                                                                                                        the three
                        It is worthwere      noting        that      the    time-step  5         selection
                                                     performed with 10 and 16 × 10 particles that corre-        5 algorithm                   The     results      of   this    experiment             are   shown
                                                                                                                                                        simulations and the Akima interpolated diffusion coefficient     in   Fig.     7. In the
                    with the proper        spondchoice to halving  of Cand  T ensures
                                                                                    doubling,that             the WMC
                                                                                                       respectively,          the is          upper
                                                                                                                                     statistical  515    panel, are
                                                                                                                                                        profile,     thedisplayed.
                                                                                                                                                                            integration     Thetime-step
                                                                                                                                                                                                    lower panelprofiles
                                                                                                                                                                                                                    shows the    ofnormalized
                                                                                                                                                                                                                                      the three
                    also satisfied error   at theofreflective
                                                          the base experiment.
                                                                           boundary Results    too, as are         reported ininFiguresimulations
                                                                                                              mentioned                         6                  and the
                                                                                                                                                        distribution            Akima
                                                                                                                                                                           of the    particle  interpolated
                                                                                                                                                                                                   after 24 hours  diffusion        coefficient
                                                                                                                                                                                                                        of simulation        along
                    Sect. 3.4.             which shows that, in the considered range, the optimal CT profile                                   is       with    the expectedThe
                                                                                                                                                          are displayed.             value.     Tablepanel
                                                                                                                                                                                            lower         2 displays
                                                                                                                                                                                                                 shows  thetheintegration
                                                                                                                                                                                                                                   normalised  time
                                   490     quite independent of the number of particles.                                                                and RMSE obtained for the various experimental settings.
                                               It is worth noting that the time-step selection algorithm                                                     The time-step profiles are similar, except for the A profile
                                           with the proper choice of CT ensures that the WMC is also 520 around the region                                                    Geosci.         Model Dev.,
                                                                                                                                                                                     of maximum            of K, 7,    2181–2191,
                                                                                                                                                                                                                    where                    2014
                                                                                                                                                                                                                               it shows strong
                                           satisfied at the reflective boundary too, as mentioned in Sec-                                               variations and, on the average, is longer than the others.
                                           tion 3.4.                                                                                                    Looking at the distribution of particles (lower panel), it re-
2188                                                                                                D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module

Table 2. Execution time and RMSE for experiments made with the                                                 Table 3. Rossi and Maurizi:
                                                                                                                        Execution   time andIL-GLOBO:    vertical diffusion
                                                                                                                                             RMSE for experiments           module
                                                                                                                                                                      made with  the
sampled “average” diffusivity distribution and varying interpolation                                           “peaked” diffusivity distribution, varying interpolation method and
                                                                                                                      Interpolation algorithm ∆t selection exec. time RMSE
method.                                                                                                        1t selection
                                                                                                                      A     algorithm.             local        313 s       0.042
                                                                                                                              D                                    local              181 s                        0.065
                          1000algorithm              Exec. time           RMSE                 80                 Interpolation
                                                                                                                        A       algorithm                   1t selection
                                                                                                                                                                non-local Exec. time
                                                                                                                                                                             1122  s                       RMSE
                                       100                                                                                    D                                 non-local             593 s                        0.022

                                                                                                    K[m /s]
           A                            10                   237 s         0.025                                 A                              local                    313 s     0.042

           D                             1                   155 s         0.023                                 D Table 3. Execution time local and RMSE for experiments181 smade with
         8 D0                          0.1                                                     20               Rossi‘peaked’
                                                                                                                       and Maurizi:    IL-GLOBO:       vertical diffusion  module
                                                             162 s         0.044                                 A              diffusivity distribution,
                                                                                                                                                non-local varying interpolation
                                                                                                                                                                       1122 s method
                                       1.4                                                     0
                                                                                                                 D    ∆t selection algorithm. non-local                  593  s    0.022
                                                                                                               Interpolation algorithm ∆t selection exec. time RMSE

                                                                                                               A                                         local              313 s                0.042
                                        1                                                                      D                                         local              181 s                0.065
                      1000       0.8                                  80                                       A                                       non-local           1122 s                0.016
                       100           0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1                                     D                                       non-local            593 s                0.022
                                                                      60                                    1000

                                                                                     K[m /s]
                        10                                σ


                                                                      40                                     100
                         1                                                                  Table 3. Execution 10 time and RMSE for experiments made with the

                                                                                                                                                                                                         K[m /s]

                        Fig. 7. Experiments with the sampled ‘average’                      ‘peaked’ diffusivity
                                                                             diffusivity distri-                1 distribution, varying interpolation method
                                                                                                                                                          20 and
                       1.4                                            0
                        bution for the interpolation algorithms D (blue), D′ (green) and∆t    A selection algorithm.

                        (red). Top panel: Diffusivity profile as interpolated by A (black) and
                        ∆t  profiles  for the different interpolation settings. Bottom   panel:

                         1                                                                                       1
                        Normalized final concentration and expected distribution (black).
                       0.8                                                                                                                   0.8
                             0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95           1
                 Interpolation algorithm      exec. time RMSE                                                      1000                        0.9   0.95      1.0 0.9       0.9540     1.0
                                        σ                                                                           100
                 A                              237 s         0.025                                                                                   σ                       σ

                                                                                                                                                                                       K[m /s]

FigureFig.       D
        7. Experiments
            7. Experiments with
                                        sampled 155  s
                                                 ‘average’    0.023 distri-
                                                           diffusivity   dis-                                          1                                                        20
tribution for the
       bution    D  ′
              for the             algorithms
                      interpolation             162  s
                                                   (blue), D0′0.044
                                                           D (green)   andAA
                                                               (green) and                                     Figure0.18.
                                                                                                       Same as in Fig. 7 for experiments with the “peaked” dif-
(red). Top    Top panel:    Diffusivity
                     diffusivity        profile
                                   profile  as   as interpolatedby
                                                interpolated      byAA(black)
                                                                       (black) and
                                                                               and        fusivityFig. 8. Same as in
                                                                                                     distribution.   Fig. 7 for
                                                                                                                    Results     experiments
                                                                                                                             obtained  usingwith
                                                                                                                                              thethe ‘peaked’ diffusiv-
                                                                                                                                              0 local (left) or non-
                 Table 2. Execution time and RMSE for experiments made with                       ity
                                                                                                  1.2 distribution. Results obtained using the local (left) or non-local
       ∆t  profiles  for  the different  interpolation   settings.  Bottom   panel:       local (right)   1t  selection algorithm.

1t profiles forthethe    different
                      sampled        interpolation
                                ‘averaged’             settings.
                                             diffusivity           Bottom
                                                           distribution,     panel:
                                                                         varying  interpola-      (right)
                                                                                                    1     ∆t selection algorithm.
normalised    final   final concentration
                                       andand    expected   distribution(black).
                       method.              expected     distribution
         Interpolation algorithm exec. time RMSE                                                    0.9      0.95
                                                                                                 interpolation          1.0 0.9
                                                                                                                  schemes,          0.95
                                                                                                                              especially  for81.0
                                                                                                                                                   Conversely,  the non-
         A                                237  s         0.025                           time-step     selection
                                                                                                                    algorithms.     Figure
                                                                                                                                                          the time-step
                 sults that simulations with A and D interpolation algorithms              550   local  algorithm    turns  out  to be  effective
                                                                                         and concentration profiles, while execution times and      in selecting  theRM-
         D      of thesatisfy
                 both   particle theafter
                                      WMC  24within
                                          155  sh of simulation
                                                         0.023       along
                                                        the statistical       with
                                                                          limit, while the       propriate   time-step    even  in presence    of  strong gradients   and
the expected      value. Table            162′ s the integration
                                                         0.044                           SEs are shown in Table 3. Although the integration time-
           525   simulation   with2 the
                                           D algorithm fails to maintaintime andthe well         isolated
                                                                                       Fig. 8. Same   as in maxima.      This is reflected
                                                                                                            Fig. 7 for experiments   with the on   its higher
                                                                                                                                              ‘peaked’         computa-
RMSETable obtained    for  the   various   experimental        settings.                 step profiles     look   very   similar   for the local and non-local al-
               2.mixed   state,time
                   Execution      in particular
                                      and RMSE near       the ground.made
                                                    for experiments        Additional
                                                                              with     ex-       tional cost
                                                                                       ity distribution.       (see
                                                                                                          Results   Table
                                                                                                                   obtained 3).
                                                                                                                             using the local (left) or non-local
   Thethetime-step     profiles
            sampled                 are  similar,
                     ‘averaged’     reported)    showexcept
                                                         that in
                                                distribution,   for  theinterpola-
                                                               varying   toAobtain
                                                                              pro- a wellgorithms,
                                                                                       (right)          the small
                                                                                               ∆t selection           differences have a large impact on the
file around      mixed
       tion method.      solution
                    region   of the  with  D′ , resolution
                                       maximum         of K,must
                                                               wherebe doubled,          results:
                                                                         it showsat least.       4.3theImplementation
                                                                                                          local algorithmonmostly the 3-Dfails
                                                                                                                                            model  in reproducing the
strong variations   The problem
                              and, on isthe   probably
                                                  average,     related    to thethan
                                                                    is longer       definition
                                                                                          the oth- of deriva-  WMC for both interpolation schemes, especially for D. Con-
ers. Looking 530    tives
                      at the of K    between grid
                                  distribution          ofpoints.
                                                              particlesIn fact,   although
                                                                             (lower     panel), D′ computes
                                                                                                     it      interpolation
                                                                                                                 555       theschemes,
                                                                                                                       A preliminary
                                                                                                                                 non-local   especially
                                                                                                                                             test   of the for
                                                                                                                                                   algorithm      D.turns
                                                                                                                                                              algorithms       on the
                                                                                                                                                                              out    to the
                                                                                                                                                                                          be      model has
                                                                                                                                                                                               effective     in
        sults thatderivatives
                       simulations         with Aorder
                                       at higher        and Dofinterpolation
                                                                     approximation    algorithms
                                                                                         than D, they  550   local
                                                                                                             are     algorithm
                                                                                                               selecting          turns
                                                                                                                              performed.   outThe
                                                                                                                                   appropriate    to be    effective
                                                                                                                                                         time               selecting
                                                                                                                                                                          algorithmin the  thepresence
                                                                                                                                                                                          has    ap- imple-
                                                                                                                                                                                                been         of
can be observed            that simulations               with A        and D interpolation
        both satisfy        the WMCwith      within       the statistical      limit,
                                                                                   K. while       the the usepropriate    time-step
                                                                                                                                  in aeven       in presence      of strong      gradients       and
                    not consistent
                  both     satisfy
                                                    a linear
                                        the′ algorithm
                                              WMC within
                                                                  variation of
                                                                              statistical      limit,          strongmented
                                                                                                                         gradients       simplified
                                                                                                                                        and     isolated  quasi-1-D
                                                                                                                                                              maxima.     form,
                                                                                                                                                                              This where      the
                                                                                                                                                                                      is reflected  diffusion
                                                                                                                                                                                                         in its
  525   simulation  of Dwith   canthe beDappropriate             failsslowly
                                                                for     to      varyingthe  andwellmonotone  isolatedcoefficient
                                                                                                                         maxima. has   Thisbeen is reflected
                                                                                                                                                       considered on its
                                                                                                                                                                       to behigher     computa-constant
         the simulation
                   state,            with
                              in particular  the    D
                                                        0  algorithm
                                                                   ground.  fails   to  maintain               higher    computational            cost    (see   Table     3).
                    functions        like ρ and         z, it            out toAdditional
                                                                                   be unsuitable  ex- for thetional cost    (see Table
                                                                                                                       between             3).
                                                                                                                                   grid points.         IL-GLOBO uses the same paralleliza-
the well-mixed
             535    more   state,
                         (not complex  in particular
                                 reported)  K show
                                                 profilethat    near     the
                                                                  in order
                                                              which,           ground.
                                                                               to            aAddi-
                                                                         in addition,           wellboth the 560 tion of GLOBO, with particle exchanged between processes
                   solution      with    D′ , resolution
                                 stochastic       term showandmustthethatbeindoubled,
                                                                        drift  orderFor
                                                                              term.     toatobtain
                                                                                             these             4.3 Implementation
                                                                                                             4.3       Implementation         on the
                                                                                                                       at each macro time-step.      on 3-D
                                                                                                                                                          the   3-D model
                                                                                                                                                            Particles    are first advected horizon-
        The problem
a well-mixed              D′isinterpolation
                    thesolution   probably
                                        with D      0 ,scheme
                                                 related       to the
                                                         resolution      definition
                                                                    is not must
                                                                             used in beof  deriva-
                                                                                       the    following ex-            tally for a macro time-step using their deterministic velocity,
at530least.  Theof K    betweenisgrid
                     problem                  points. In
                                          probably             fact, although
                                                            related                 D′ computes
                                                                         to the definition          of555 AApreliminaryand thentest
                                                                                                                   preliminary          of the
                                                                                                                                      test   of thealgorithms
                                                                                                                                                    in the         on according
                                                                                                                                                          algorithms    theon3-D  themodel
                                                                                                                                                                                        to       has (12).
                                                                                                                                                                                         3-D     model     has
derivatives of K          at   higher
                        Thebetween       order
                                second experiment  of   approximation
                                              grid points.       concerns
                                                                      In fact, than   D, they are
                                                                              the although
                                                                                    ‘peaked’       D0
                                                                                                  profile.   been
                                                                                                              In           After 12
                                                                                                                       performed.  The    interpolation
                                                                                                                                          of   spinup,
                                                                                                                                                interpolation   105 algorithm
                                                                                                                                                           5 ×algorithm       has been
                                                                                                                                                                      particles      arehas imple-
                                                                                                                                                                                          released      with a
                                                                                                                                                                                              been imple-
        not540consistent        withthe                              of K.                                   mented     in a simplified       quasi-1-D        form, where        the diffusion
computes            this case,
                derivatives          ataalinear
                                           higher   variation
                                                profile      is used
                                                         order      of       Although
                                                                        approximation      the use
                                                                                     without      the resam-
                                                                                                 than          mented
                                                                                                                 565   vertical   distribution
                                                                                                                           in a simplified            proportional
                                                                                                                                                     quasi-1-D          to thewhere
                                                                                                                                                                     form,        average      density
                                                                                                                                                                                             the          pro-
        of D′ can   plingbe of appropriate
                                  the   fitting   for    slowlySimulations
                                                   function.         varying andwith   monotone
                                                                                            A    and  D      coefficient
                                                                                                           algo-       file,has
                                                                                                                                  randomlyconsidered        to be horizontally
                                                                                                                                                   and homogeneously                     constant
                                                                                                                                                                                 distributed       in the hor-
D, they      are notlike   consistent         with a linear             variation of K.           Al-          coefficient     has been          considered        tothebesamehorizontally           constant
        functions   rithms ρwere   0and   z, it turns with
                                         performed           out toboth be unsuitable      for the time-
                                                                            local and non-local              betweenizontal.
                                                                                                                         grid points.
                                                                                                                                                statistics are usescomputed          paralleliza-
                                                                                                                                                                                    after   24 h from the
though      the   use
        more complex      of    D    can
                                K profile   be     appropriate
                                                which, inFigure          for
                                                                  addition,    slowly
                                                                                         both     the 560 and  between
                                                                                                             tion of GLOBO,  grid  points.       IL-GLOBO            uses
                                                                                                                       release. with particle exchanged between processes
                                                                                                                                                                               the   same       parallelisa-
  535               step selection         algorithm.                     8 reports           time-step
and monotone             functions
        Wiener concentration
                    stochastic             like
                                      termprofiles,     and drift
                                                          while      it turns
                                                                 z,term.    For out
                                                                    execution     these
                                                                                            be RMSEs
                                                                                          and     un- are    attion
                                                                                                                           A and        with Particles
                                                                                                                                   D interpolationparticles     exchanged
                                                                                                                                                               are first were
                                                                                                                                                           algorithm       advectedbetween
                                                                                                                                                                                  tested horizon-
                                                                                                                                                                                            using  processes
                                                                                                                                                                                                      the non-
suitable     for
        the 545  ′ the more complex K profile which, in addition,
              D interpolation
                    shown in Table       scheme        is not used
                                             3. Although                 in the following
                                                                   the integration     time-step  ex- profiles at
                                                                                                             tally each   macro
                                                                                                                         a macro     time
                                                                                                                              time-step       step.    Particles
                                                                                                                                                    using their
                                                                                                                                             selection.              are    first
                                                                                                                                                             It is found            advected
                                                                                                                                                                             that, while velocity,   horizon-
          both the  lookWiener          stochastic
                             very similar        for theterm  localand andthe    drift term.
                                                                             non-local            For the
                                                                                           algorithms,         tally
                                                                                                             and      for
                                                                                                                   then     a macro
                                                                                                                       scheme          intime      step
                                                                                                                                           the vertical
                                                                                                                                  A maintains          theusing     their
                                                                                                                                                             WMC            todeterministic
                                                                                                                                                                      reasonablyEquation       (12).velocity
these reasons,
            The second            0 interpolation
                       the differences
                               Dexperiment         concerns   schemethe    is on
                                                                         ‘peaked’    used
                                                                                                   Inthe local   After
                                                                                                               and      12 time-step
                                                                                                                     then    h of spinup,
                                                                                                                             “diffused”        5 ×the
                                                                                                                                              in      105vertical
                                                                                                                                                                         are   released toDwith
                                                                                                                                                                                             Eq.     a
                    small                        have      large    impact              results:                       the                selection       algorithm      for scheme             requires   ex-
        this case,
  540                   the K profile
                experiments.          strongly is used
                                                    fails directly,       withoutthe
                                                              in reproducing          theWMCresam-for        vertical
                                                                                                       565 both    Afterdistribution
                                                                                                                           12 h ofshort  proportional
                                                                                                                                       spin-up,              to≪5the
                                                                                                                                                       5 × (10    ∆t  average
                                                                                                                                                                      min   ,  see density
                                                                                                                                                                                   are          pro-
                                                                                                                                                                                         released 4.1)  in thea
          second of theexperiment
                           fitting function. concerns  Simulations        with A and
                                                                 the “peaked”             D algo-
                                                                                      profile.      In       file, and randomly
                                                                                                               vertical               and homogeneously
                                                                                                                           distribution        proportional to       distributed
                                                                                                                                                                         the average  in thedensity
                                                                                                                                                                                                hor- pro-
this case,         were  Kperformed
                                profile iswith    used  bothdirectly,
                                                                 local andwithout
                                                                               non-localthe    time-
                                                                                                   re-       izontal.
                                                                                                               file, and Particle
                                                                                                                           randomly statistics
                                                                                                                                           and are       computed afterdistributed
                                                                                                                                                   homogeneously                24 h from in      thethe hor-
        step selection algorithm. Figure 8 reports the time-step and                                         release.
sampling of the fitting function. Simulations with A and D                                                     izontal. Particle statistics are computed after 24 h from the
        concentration profiles, while execution times and RMSEs are                                              A and D interpolation algorithm were tested using the non-
                   were       performed           with       both      local    and    non-local               release.
        shown in Table 3. Although the integration time-step profiles 570 local time-step selection. It is found that, while interpolation
    look very similar for the local and non-local algorithms, the                                             scheme A maintains the WMC reasonably (RMSE=0.024),
Geosci.   differences
        Model         have
                Dev., 7,    large impact
                         2181–2191,      on the results: the local
                                       2014                                                                   the time-step
                                                                                                                                      algorithm for scheme D requires ex-
    algorithm strongly fails in reproducing the WMC for both                                                  tremely short time-steps ( ≪ ∆tmin , see Section 4.1) in the
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