# 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, 2014

2184 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, 2014

diffusion 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 the

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