IL-GLOBO (1.0) - integrated Lagrangian particle model and Eulerian general circulation model GLOBO: development of the vertical diffusion module
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Geosci. Model Dev., 7, 2181–2191, 2014
www.geosci-model-dev.net/7/2181/2014/
doi:10.5194/gmd-7-2181-2014
© 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 (a.maurizi@isac.cnr.it)
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
(M)
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 www.geosci-model-dev.net/7/2181/2014/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
∂σ
3 Numerical implementation of the vertical + (2K)1/2 dW ,
∂z
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).
www.geosci-model-dev.net/7/2181/2014/ Geosci. Model Dev., 7, 2181–2191, 20142184 D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module
4 Rossi and Maurizi: IL-GLOBO: vertical diffusion module
0
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-
NLEV
dition, ∂σ =
to ensure the positivity of the interpolating .functions, (15)
285
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
where
The most common integration scheme for SDE in atmo-
290 K(z)
spheric = κumodels
transport ∗z is the Euler-Maruyama forward (16)
scheme:
for the neutral case, with proper modifications for diabatic
Σt+∆tcases.
= 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
con-
Figure Fig.
1. Schematic representation
1. Schematic representation of field
fieldvalue
valuedistributions
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
ationsamade
continuous
for ρ, thecoordinate,
derivatives ofthe quantities
σ with respect ofneeded
z are to
compute
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
convergence
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
alsofinite
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,
terpolation
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
scheme
accuracy constant
and thebetween two grid points.
computational costInoforder
thetoLagrangian
give a smoother
model (Kloeden preserves the continuity
and Platen, 1992, p. 345).of first-order
The additional derivatives.
term
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-
central
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
315
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–
time-
which implies steps are involved in
Maruyama the Eulerian
forward schemepart is
butthe
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
Equation
∂ρ (1991) cubic ∂ 2ρ
275
+ spline.
2
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)
2
for the lating cubic polynomial.
first-order derivative.This algorithm the
Following reduces
samethe considera-
number
of oscillations
tions made for ρ, thein the interpolatingoffunction
derivatives σ with compared reg-z are 1 http://www.isac.cnr.it/dinamica/projects/forecast
respecttoof where b0 is the first-order derivative of b, the Milstein scheme dpc/month
2
280 ular cubic splines and enforces the linearity when 4 points http://www.isac.cnr.it/dinamica/projects/forecasts/glob
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 www.geosci-model-dev.net/7/2181/2014/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-
∂σ
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 http://www.isac.cnr.it/dinamica/projects/forecast_dpc/month_ 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 http://www.isac.cnr.it/dinamica/projects/forecasts/glob_ of the full 3-D model. Input profiles were obtained by run-
newNH/ ning the low-resolution version of GLOBO (horizontal grid
www.geosci-model-dev.net/7/2181/2014/ Geosci. Model Dev., 7, 2181–2191, 2014diffusion module2186 5D. Rossi and A.
6
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]
50
ρ[kg/m ]
Φ/g[km]
K[m /s]
25000 50
3
∆t[s]
2
2
h expresses the re- 1 40
0.6
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
σ
merical implemen-
on and Yee, 2007). Figure Fig.2. Values
2. Valuesof ofintegration
integrationtime step ∆t
time-step 1t forforthe
thediffusivity
diffusivity pro-
profile Figure 3. Fig.Average
3. AverageGLOBO
GLOBO profiles
profilesofof ρρ (green symbols)
(green symbols) φ/gφ/g Fig. 4. Diffusivit
andand
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
their
file shownshownby the
by theredredcurve.
curve.The green
The greenline
lineshows
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
linecontribution
the contribution ofofEq.Eq.(20)
(20),and
andthethe black
black line
line lytical fits (Eqs. 23 and 24).
shown in green.
cles in the vicinity the combined
the combined conditioncondition
(Eq.(Eq.
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
1/2
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
conditions
is used
415 corresponds
to accounttofor the the
daytime conditions
specific over land.it 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.
law.(26),
The A ms−1 was
= 0.29analytical
following regular, creating
time,
(20) 385
while
(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
0
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
regular,with
creating challenging conditions for the model. For tive time-
(σIn thed case − of 1)T
IL-GLOBO,
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
455
395 0 lated strong
(ρσ = maxima
dz
ρ dσ ) andnear
usingthe ground.(24)
Equations This
andis(23),
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
−0.007theKcomputational
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 .
dz
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
g
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 www.geosci-model-dev.net/7/2181/2014/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
100
80 60
K[m /s]
10
∆t[s]
1
2
70 1 40
Table 1. RM
0.1 20
0.8 60
1.4 0
ρ[kg/m ]
K[m /s]
50
3
2
C,ρnorm
0.6 1.2 0.05
40
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
RMSE
0.03
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
σ
FigureFig.5. Dispersion
5. Dispersion experiment
experimentwith
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)
diffusivity
0.1
profile (black line)and
238
∆t profiles
and 1t profiles
n symbols) and φ/g Figure Fig.
4. Diffusivity profiles
4. Diffusivity
1000 profilesused
usedininthe
the experiments.
experiments. The Thesymbols
symbols for CTfor=C0.5T =(light
0.5 (light CT C=T0.021
blue),
blue),
0.01
= 0.1
0.1 (green),CC
(green), T T==0.01
1172
0.01 (red)
(red) and
and 0.015
nate σ, and their ana- represents
represents thefrom
the data data
100from
GLOBOGLOBO andand
thethe linestheir
lines, their fitting
fittingfunction.
function. CT = C T = 0.001
0.001 (blue).
(blue). Bottom panel:
Bottom
0.001panel:
normalizedconcentration
normalised
0.021 7317
concentration pro-
pro- 0.01
60 is
K[m /s]
e colors). The ‘average’ profile is shown in red, whilethe
the“peaked”
‘peaked’ profile files for different CT (Line colors
as as
in inthe
thetop
toppanel).
panel).
is10shown files for different C (line colours 0.005
∆t[s]
The “average” profile in red, while profile
40 is
2
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
C,ρnorm
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
RMSE
ged over the whole play a linear behavior near the surface, must tend to zero 0.03
ges were performed evolves. near Equation
theFig.
boundary (12)layer was top integrated
3
and, for 4 ×
therefore, 105display
must particles a of parame-
machine. Since the initial condition was already well-mixed
0.025
and formaximum
in daytime (longi- 200 macro 5. Dispersion experiment with different choices
timeheight.
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
CTh.
determined = The actual
0.5according
(light blue), time
to average
CT = step used
(green),isCgiven
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
Eq.440(21)erties
with (the
Cthe first0.001
GLOBO
T =additional (blue).vertical
lower
Bottom level),
limit
panel: 1tand corresponds
= 0.01.
normalized
min to
Sim-
concentration pro-
e is selected, which −1
0.01
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
Then,
cores asofthe
in the other
an two Xeon
Intel
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
(C
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
sess
ing the hydrostatic thecalskill satisfied,
of this distribution
thediffusivity,
model inreal
satisfying must
thebe remain
WMC. constant
At theless as the time475 tration profile with respect to the normalized density 5profile
end
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
concentration
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
among
) 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.
limitEqua-
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
condition
reportswas concen-
thealready
GLOBO well-mixed
for each simulation are reported in Table 1 along with the putation of
tration‘averaged’
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
subsequent
In the subsequent set of setexperiments,
of experiments, thethemodel
model skill
skill in
in re-
re-
(24)
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-
producing
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
optimal
profiles setting profiles
for
of concentration wereaf-
the adap- computed D, D0 D,
niques
niquesoptimal and D Aand A described
described in Section
in Sect. 3.2. dependency
3.2.possible
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
the
first
on firstnumber
the experiment,
experiment, of the the analytical
analytical
particles, two fieldsdescribed
fields
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
finalofnormalized
the eachconcen- diffusivity
ular
profile
grid. This
resampled
provides a discrete version of
regular grid.515
the experiment
simulationsuited are
value reported
for profile in with
Table
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-
475
tion time.Simulations
The(derived
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
section
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.
al-
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-
algo-
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
www.geosci-model-dev.net/7/2181/2014/
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
8
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
Interpolation
1000algorithm Exec. time RMSE 80 Interpolation
A algorithm 1t selection
non-local Exec. time
1122 s RMSE
0.016
100 D non-local 593 s 0.022
60
K[m /s]
A 10 237 s 0.025 A local 313 s 0.042
∆t[s]
2
40
D 1 155 s 0.023 D Table 3. Execution time local and RMSE for experiments181 smade with
0.065the
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
0.016and
1.4 0
D ∆t selection algorithm. non-local 593 s 0.022
Interpolation algorithm ∆t selection exec. time RMSE
C,ρnorm
1.2
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
∆t[s]
40
2
40 100
1 Table 3. Execution 10 time and RMSE for experiments made with the
K[m /s]
∆t[s]
20
2
0.1
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.
0.1
0
C,ρnorm
1.2
(red). Top panel: Diffusivity profile as interpolated by A (black) and
1.2
∆t profiles for the different interpolation settings. Bottom panel:
C,ρnorm
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]
10
∆t[s]
2
FigureFig. D
7. Experiments
7. Experiments with
withthe
thesampled
sampled 155 s
“average”
‘average’ 0.023 distri-
diffusivity
diffusivity dis- 1 20
tribution for the
bution D ′
interpolation
for the algorithms
interpolation 162 s
algorithmsDD(blue),
(blue), D0′0.044
D (green) andAA
(green) and Figure0.18.
Same as in Fig. 7 for experiments with the “peaked” dif-
(red).panel:
(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.
C,ρnorm
1t profiles forthethe different
sampled interpolation
‘averaged’ settings.
diffusivity Bottom
distribution, panel:
varying interpola- (right)
1 ∆t selection algorithm.
Normalized
normalised final final concentration
andand expected distribution(black).
(black).
tionconcentration
method. expected distribution
0.8
Interpolation algorithm exec. time RMSE 0.9 0.95
interpolation 1.0 0.9
schemes, 0.95
especially for81.0
D.reports
Conversely, the non-
A 237 s 0.025 time-step selection
σ
algorithms. Figure
σ
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-
ap-
distribution
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
′
D
the expected value. Table 162′ s the integration
0.044 SEs are shown in Table 3. Although the integration time-
525 simulation with2 the
displays
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-
diffusiv-
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
periments
sampled are similar,
(notdiffusivity
‘averaged’ reported) showexcept
that in
distribution, for theinterpola-
order
varying toAobtain
pro- a wellgorithms,
(right) the small
∆t selection differences have a large impact on the
algorithm.
file around mixed
the
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
versely,
555 theschemes,
A preliminary
non-local especially
test of the for
algorithm D.turns
Conversely,
algorithms on the
out to the
3-Dnon-
be model has
effective in
sults thatderivatives
simulations with Aorder
at higher and Dofinterpolation
approximation algorithms
than D, they 550 local
are algorithm
beenthe
selecting turns
performed. outThe
appropriate to be effective
step,ineven
interpolation
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
algorithms
not consistent
both satisfy
′
a linear
the′ algorithm
WMC within
variation of
themaintain
Although
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
horizontally
whilemixed
the simulation
state, with
in particular the D
near
0 algorithm
theturns
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
periments
535 more state,
(not complex in particular
reported) K show
profilethat near the
in order
which, ground.
to aAddi-
obtainaffects
in addition, wellboth the 560 tion of GLOBO, with particle exchanged between processes
tionalmixed
experimentsWiener(not
solution with D′ , resolution
reported)
stochastic term showandmustthethatbeindoubled,
drift orderFor
term. toatobtain
least.reasons,
these 4.3 Implementation
4.3 Implementation on the
at each macro time-step. on 3-D
the 3-D model
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-
doubled
the following ex- tally for a macro time-step using their deterministic velocity,
tives
at530least. Theof K betweenisgrid
periments.
problem points. In
probably fact, although
related D′ computes
to the definition of555 AApreliminaryand thentest
preliminary of the
‘diffused’
test of thealgorithms
in the on according
vertical
algorithms theon3-D themodel
to has (12).
Equation
3-D model has
derivatives
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
beenperformed.
In After 12
performed. The interpolation
hThe
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
K
higher variation
profile is used
order of Although
directly,
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-
diffusion
of D′ can plingbe of appropriate
the fitting for slowlySimulations
function. varying andwith monotone
A and D coefficient
algo- file,has
andbeen
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.
ParticleIL-GLOBO
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
affectsthevarying
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
andρthe
termprofiles, and drift
while it turns
z,term. For out
execution these
timestoreasons,
be RMSEs
and un- are attion
eachofmacro
GLOBO,
A and with Particles
time-step.
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
570forlocal
a macro time
time-step
time-step step. Particles
using their
selection. are first
deterministic
It is found advected
that, while velocity, horizon-
interpolation
affectsperiments.
both the lookWiener stochastic
very similar for theterm localand andthe drift term.
non-local For the
algorithms, tally
and for
then a macro
‘diffused’
scheme intime step
the vertical
A maintains theusing their
according
WMC todeterministic
reasonablyEquation (12).velocity
(RMSE=0.024),
these reasons,
The second 0 interpolation
the differences
Dexperiment concerns schemethe is on
notthe
‘peaked’ used
profile.in the
Inthe local After
and 12 time-step
then h of spinup,
“diffused” 5 ×the
in 105vertical
particlesaccording
are released toDwith
Eq. a
(12).
small have large impact results: the selection algorithm for scheme requires ex-
this case,
following
540 the K profile
algorithm
experiments. strongly is used
fails directly, withoutthe
in reproducing theWMCresam-for vertical
565 both Afterdistribution
tremely
12 h ofshort proportional
time-steps
spin-up, to≪5the
5 × (10 ∆t average
particles
min , see density
Section
are pro-
released 4.1) in thea
with
Thepling
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-
rithmsthe
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-
algorithms
545
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),
small
Geosci. differences
Model have
Dev., 7, large impact
2181–2191, on the results: the local
2014 the time-step selectionwww.geosci-model-dev.net/7/2181/2014/
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 theYou can also read
