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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 2 Institute

of Biological, Geological and Environmental Sciences, University of Bologna, Bologna ,Italy of Climate and Atmospheric Sciences, National Research Council, Bologna, Italy

Correspondence to: A. Maurizi ([email protected]) 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 diffusion module of IL-GLOBO, a Lagrangian transport model coupled online with the Eulerian general circulation model GLOBO, is described. The module simulates the effects of turbulence on particle motion by means of a Lagrangian stochastic model (LSM) consistently with the turbulent diffusion equation used in GLOBO. The implemented LSM integrates particle trajectories, using the native σ -hybrid coordinates of the Eulerian component, and fulfils the wellmixed condition (WMC) in the general case of a variable density profile. The module is validated through a series of 1-D offline numerical experiments by assessing its accuracy in maintaining an initially well-mixed distribution in the vertical. A dynamical time-step selection algorithm with constraints related to the shape of the diffusion coefficient profile is developed and discussed. Finally, the skills of a linear interpolation and a modified Akima spline interpolation method are compared, showing that both satisfy the WMC with significant differences in computational time. A preliminary run of the fully integrated 3-D model confirms the result only for the Akima interpolation scheme while the linear interpolation does not satisfy the WMC with a reasonable choice of the minimum integration time step.

1 Introduction Global- (or hemispheric-) scale transport is recognised as an important issue in air pollution and climate change studies. Pollutants can travel across continents and have an influence even far from their source (see, for recent examples, Fiore

et al., 2011; Yu et al., 2013). Moreover, transport of volcanic emissions (e.g. the recent Eyjafjallajökull eruption) or accidental hazardous releases (like the Fukushima and Chernobyl nuclear accidents) are also important at the global scale. The natural framework for the description of tracer transport inflows is the Lagrangian approach (see, for example, the seminal works by Taylor, 1921, and Richardson, 1926). In the Lagrangian framework, the tracer transport is described by integrating the kinematic equation of motion for fluid “particles” in a given flow velocity field, provided by, e.g. a meteorological model. The turbulent motion unresolved by Eulerian equations for averaged quantities (in the Reynolds or volume-filtered sense) can be accounted for by including a stochastic component into the kinematic equation. The stochastic component can be added to the particle position to give the Lagrangian equivalent of the Eulerian advection-diffusion equation. This kind of model is usually called a random displacement model (RDM) and is suitable for dispersion over long timescales. When the stochastic component is added to the velocity, the model is usually called a random flight model (RFM), which is more suitable for shorter time dispersion. In both cases, the stochastic model formulation has to be consistent with some basic physical requirements (Thomson, 1987, 1995). Various Lagrangian transport models exist which can be used at the global scale. Some are designed specifically for the description of atmospheric chemistry (Reithmeier and Sausen, 2002; Wohltmann and Rex, 2009; Pugh et al., 2012, see, e.g.), while others focus on the transport of tracers. In the latter class, two of the most widely used models are

Published by Copernicus Publications on behalf of the European Geosciences Union.

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FLEXPART (FLEXible PARTicle dispersion model) (Stohl et al., 2005) and HYSPLIT (Hybrid Single Particle Lagrangian Integrated Trajectory Model) (Draxler and Hess, 1998), which are highly flexible and can be easily used in a variety of situations. Both are compatible with different input types (usually provided by meteorological services like the European Centre for Medium-Range Weather Forecasts (ECMWF)), relying on their own parameterisation for fields not available from the meteorological model output. Models of this kind are suited for both forward and backward dispersion studies. An alternative approach is to couple the Eulerian and Lagrangian parts online. On one hand, this makes the Eulerian fields available to the Lagrangian model at each Eulerian time step, increasing the accuracy for temporal scales shorter than the typical meteorological output interval. On the other hand, it also allows the consistent parameterisation of processes in the Eulerian and Lagrangian frameworks (e.g. the vertical dispersion in the boundary layer). Moreover, where the considered tracer may have an impact on meteorology (e.g. on radiation or cloud microphysics), online integration provides a natural way to include these effects (Baklanov et al., 2014). Online coupling also ensures the consistency of a mixed Eulerian–Lagrangian analysis of the evolution of atmospheric constituents (e.g. water or pollutants) along a trajectory (Sodemann et al., 2008; Real et al., 2010, see, e.g.). Malguzzi et al. (2011) recently developed a new global numerical weather prediction model, named GLOBO, based on a uniform latitude–longitude grid. The model is an extension to the global scale of the Bologna Limited Area Model (BOLAM) (Buzzi et al., 2004), developed and employed during the early 90s. GLOBO is used for daily forecasting at the Institute of Atmospheric Sciences and Climate of the National Research Council of Italy (ISAC-CNR) and is also used to produce monthly forecasts. Online integration with BOLAM family models has already yielded interesting results in the development of the meteorology and composition model BOLCHEM (BOLam + CHEMistry) (Mircea et al., 2008). Considering that experience, the GLOBO model constitutes the natural basis for the further development of an integrated Lagrangian model. In the following, the development of the vertical diffusion module is presented, focusing in particular on its compliance with basic theoretical requirements (Thomson, 1987, 1995, the well-mixed condition, see ) in connection with different numerical issues. In Sect. 2 the theoretical basis of the model formulation is given, while Sect. 3 describes different aspects of the numerical implementation. Finally, the model verification is presented and discussed in Sect. 4.

0, 1, . . .), are described by a set of stochastic differential equations (SDEs). The equation for the Mth order derivative M is

2

If hci ∝ hρi at some time t 0 , where hρi is the ensemble average of air density, then for all t > t 0 the two quantities must remain proportional. This condition, called well-mixed condition (WMC) after Thomson (1987), implies that hρi is also

Lagrangian stochastic model formulation

In application to dispersion in turbulent flows, Lagrangian stochastic models (LSMs), Markovian at order M(M = Geosci. Model Dev., 7, 2181–2191, 2014

(M)

dXi

= ai dt + bij dWj ,

(1) (k)

where i and j indicate the components and Xi is the kthorder time-derivative of the Lagrangian Cartesian coordinate (0) component Xi ≡ Xi . Coefficients ai and bij are called drift and Wiener coefficients, respectively. The remaining equations of the set (1 ≤ k ≤ M) are described by (k−1)

dXi

(k)

= Xi dt .

(2)

The set of equations is equivalent to the Fokker–Planck equation: M X ∂p ∂ ∂2 k =− (A (Kij p) , p) + i (k) (M) (M) ∂t ∂x ∂x k=0 ∂x i

i

(3)

j

where Aki = 1 for k < M and Aki = ai for k = M, xi is the Eulerian equivalent of Xi and Kij ≡ bik bj k /2 (Thomson, 1987). Equation (3) describes the evolution of the probability density function p(x (0) , . . ., x (M) , t), where x (k) = (k) (k) (k) (x1 , x2 , x3 ). For the evolution of (X (0) , . . ., X (M) ) to be approximated by a Markov process, the time correlation of the variable X (M+1) has to be much shorter than the characteristic evolution time of X(M) . If the model has to describe the evolution of dispersion at time t τ , where τ is the correlation time of turbulent velocity fluctuations, the process is well captured at order M = 0. When shorter times are considered, as in the case of dispersion from a single point source before the Taylor (1921) diffusive regime occurs (t ≤ τ ), order M must be increased to 1. The model of lowest order (M = 0) is referred to as random displacement model (RDM) and is sufficiently accurate to describe the transport and mixing of particles at a time and space resolution typical of a global model. The correct formulation of a RDM in a variable density flow was first obtained by Venkatram (1993) and then refined and generalised by Thomson (1995) and is briefly recalled here. Equation (3) is valid for the probability density function p of particle position with the initial condition p((x), t)|t=t0 = p((x), t0 ). Since the ensemble average concentration hci is proportional to p, Eq. (3) can be rewritten as ∂hci ∂ ∂2 =− (ai hci) + (Kij hci) . ∂t ∂xi ∂xi ∂xj

(4)

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a solution of Eq. (4). Substituting c with ρ in Eq. (4) and using the continuity equation

tends, with height above the ground, to a pressure coordinate P , according to

∂hρi ∂ =− (ui hρi) , ∂t ∂xi

P = P0 σ − (P0 − PS )σ α ,

(5)

where ui is the density weighted mean velocity, defined as (Thomson, 1995): ui =

hu0 ρ 0 i hui ρi = hui i + i , hρi hρi

(6)

where P0 is a reference pressure (typically 1000 hPa), PS is the surface pressure and α is a parameter that gives the classical σ coordinate for α = 1 (Phillips, 1957). The parameter α depends on the model orography and, therefore, on resolu∂ tion. It is limited by the condition ∂p σ ≥ 0 that results in the relationship:

the following expression is obtained: α≤ ∂ ∂2 ∂ (ai hρi) + (Kij hρi) . − (ui hρi) = − ∂xi ∂xi ∂xi ∂xj

(7)

Then, integrating both sides and rearranging gives ai =

∂Kij Kij ∂hρi + + ui , ∂xj hρi ∂xj

(8)

where the non-uniqueness implied by the integration is removed considering that in the well-mixed state, the mixing ratio flux must be proportional to ui hρi. Substituting Eq. (8) into Eq. (4) gives the equivalent of Eq. (2) in Thomson (1995). At the coarse resolution typical of global models, vertical motions can be considered decoupled from the horizontal ones. Therefore, only the vertical coordinate x3 ≡ z (and X3 ≡ Z in Lagrangian terms) need to be considered. In this case, the RDM reduces to a single differential stochastic equation √ ∂K K ∂hρi dZ = w + (9) + dt + 2KdW , ∂z hρi ∂z where w ≡ u3 and K ≡ K33 . 3

3.1

Vertical coordinate

Within IL-GLOBO, the Lagrangian equations are integrated in the same coordinate system used in the Eulerian model. This choice maintains the consistency between the Lagrangian and Eulerian components and reduces the interpolation errors and computational cost. GLOBO uses a hybrid vertical coordinate system in which the terrain-following coordinate σ (0 < σ < 1) smoothly www.geosci-model-dev.net/7/2181/2014/

P0 , P0 − min(PS )

(11)

which is satisfied by the typical setting α = 2, used for a wide range of resolutions in GLOBO applications (Malguzzi et al., 2011). The vertical Lagrangian coordinate is identified by 6, corresponding to the vertical coordinate σ , and is connected to the Lagrangian vertical position Z above the ground through Eq. (10) and the hydrostatic relationship. In the meteorological component, the height above the ground z is a diagnostic quantity that can be derived from the geopotential 8 through z(σ ) = (8(σ )−8g )g −1 , where 8g is the geopotential at the height of roughness length. Since the determination of the different terms in Eq. (9) involves discrete Eulerian fields and their numerical derivatives, the choice of employing σ also has the advantage of making interpolation straightforward and consistent with the Eulerian part. Because σ (z) is not linear (σ is not a Cartesian coordinate system), the stochastic chain rule (see, e.g. Kloeden and Platen, 1992, p. 80) must be used to derive the correct form of Eq. (9) for 6, giving # " 2 ∂σ 1 ∂ ∂ 2σ (12) d6 = ω + (hρiK) + K 2 dt ∂z hρi ∂σ ∂z +

Numerical implementation of the vertical diffusion module

In its final form, IL-GLOBO is designed to be a fully online integrated model (or at least an online-access model, according to Baklanov et al., 2014), where the different components share the same “view” of the atmosphere, i.e. use the same discretisation, parameterisations, etc. The development of the vertical diffusion module is based on this principle.

(10)

∂σ (2K)1/2 dW , ∂z

where ω is the vertical velocity in the σ coordinate system and z is the Cartesian vertical coordinate. The last term in square brackets stems from the Itô–Taylor expansion of order dW 2 , which must be included for the correct description at order dt (Gardiner, 1990, p. 63). 3.2

Discretisation and interpolation

The GLOBO prognostic variables are computed on a Lorenz (1960) vertical grid: all the quantities are on “integer” levels σi , except vertical velocity, turbulent kinetic energy and mixing length and, consequently, diffusion coefficients, located at “semi-integer” levels σih (see Fig. 1). In typical applications, the GLOBO vertical grid is regularly spaced in σ (Malguzzi et al., 2011), although it is possible to use a variable grid spacing, as in its limited-area version BOLAM (Buzzi et al., 1994). Geosci. Model Dev., 7, 2181–2191, 2014

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D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module Rossi and Maurizi: IL-GLOBO: vertical diffusion module 0

285

For D (Akima, , the values the first-order are collinear 1991).ofUsing this property,derivative a linear at the lowest are computed asthe interpolating funcprofileboundary near the ground is imposed to tion by adding two fictitious points below the ground that are the two lower ∂Kwith KNLEV+1 collinear grid points−ofKthe domain. In adNLEV = .functions, (15) dition, ∂σ to ensure the positivity of the interpolating σ − σ NLEV+1 NLEV NLEV+1 the local algorithm of Fischer et al. (1991) is used, which also preserves continuitybecause of first order This isthe assumed K derivatives. is expected to be linear near the surface, according to Monin–Obukhov 3.3 Integration scheme and time-step selection

similarity theory

where

290

The most common integration scheme for SDE in atmoK(z) = κumodels spheric transport is the Euler-Maruyama forward ∗z scheme:

(16)

for the neutral case, with proper modifications for diabatic Σt+∆tcases. = Σt + a∆t + b∆W .

(17)

The second (labelled A)(12). is based The coefficients a and method b come from Equation The on the Akima (1991) cubic spline. Foris each interval it considers the previEuler-Maruyama forward scheme the simplest strong Tay295 lor approximation turns outadjacent to be of order of strong conous and theand next two intervals (for a total number of Figure Fig. 1. Schematic representation fieldvalue valuedistributions distributions 1. Schematic representation of field be- be- vergence γ = 0.5 (Kloeden and Platen, 1992, p. 305). six grid points) to compute the coefficients of the interpolattween integer (continuouslines) lines) and integer (dashed lines) levtween integer (continuous andsemi semi-integer (dashed lines) By a rather simple modification of the Euler-Maruyama ing cubic polynomial. This algorithm reduces the number of els in the GLOBO model. scheme, i.e. adding the term: levels in the GLOBO model. oscillations in the interpolating function compared to regular

for the first order derivative. Following the same consider-

1 ′ cubic and enforces the linearity when bb (∆W 2 −splines ∆t), (18) four points are 2 collinear (Akima, 1991). Using this property, a linear pro-

6 being continuous quantities ationsamade for ρ, thecoordinate, derivatives ofthe σ with respect ofneeded z are to 255 computed fromof relationships to Eq. (13) and Eq.from (14). 300 the ground is imposed to the scheme interpolating function b′ is near the first-order derivative of b, the Milstein compute the terms Eq. (12) similar must be interpolated the wherefile For the highly varying K profiles, two different methods which is of order of strong convergence = 1.ground that are by adding two fictitious points below γthe Eulerian fields given at discrete levels. The computation of is obtained, are tested, the first with two variants. The first method inIt is worth notingwith that the order γgrid = 1 of the Milstein collinear thestrong two lower points of the domain. In adfirst- and second-order derivatives of Eulerian model quanterpolates the function linearly at the particle position, and scheme corresponds to the strong order γ = 1 of the Euler dedition, to ensure the positivity ofbethe interpolating functions, tities isuses alsofinite required in the implementation of the(labeled LSM. In- terministic differences derivatives. In the first variant scheme. Therefore, Milstein can regarded as the local algorithm of Fischer et al.Euler (1991) is used, which also 260 D), theand firstderivation order two-points derivativecan is computed andboth kept 305 terpolation algorithms influence the the correct generalization of the deterministic scheme constant between two grid points. order give a smoother and Platen, 1992, p. 345). The additional term preserves the continuity of first-order derivatives. accuracy and the computational costInof thetoLagrangian model (Kloeden description of the derivatives, a variant (labeled D′ ) is also uses only already computed quantities involved in the deterand thus require careful assessment. tested in which the three-points centered derivative is commination drift term of scheme Equation (12). 3.3of the Integration and Preliminary time-step ideselection For density and geopotential interpolation puted andρinterpolated linearly at8, thelinear particle position. For and alized tests do not show any appreciable accuracy improve′ central are usedat the assuming that those 265 Ddifferences , the values ofderivative first order derivative lowest boundary 310 ment with respect to the Euler-Maruyama scheme. However, The most common integration scheme for SDE in atmois computed fields are regularas:enough. At the lower boundary, it is re- because they confirm the negligible extra computational cost transport is bethe Euler–Maruyama forward of thisspheric method, the Milnsteinmodels scheme will used to integrate quired that ∂K KNLEV+1 − KNLEV scheme: the model. . (15) = ∂σ NLEV+1 2σNLEV +1 − σNLEV In the meteorology component of IL-GLOBO, the Eule∂ 2 ρ ∂ ρ 6t+1t = b1Wtime-step . (17) 315 rian equations are6 solved with+ a macro ∆T , which t + a1t = , (13) This is assumed because K is expected to be linear near 2 ∂σ 2 NLEV+1 depends basically on the horizontal resolution due to the NLEV the surface, ∂σ according to Monin-Obukhov similarity theory limitations by thea Courant OtherEq. timeThe imposed coefficients and b number. come from (12). The Euler– 270 where: steps are involved in the Eulerian part is butthe are simplest not relevant which implies Maruyama forward scheme strong Taylor aphere. In typical implementations, ∆T ranges from 432 s for (16) proximation and turns out to be of the order of strong con K(z) = κu∗ z, 242 point resolution (used for monthly forecasts1 ) to ∂ρ for the neutral case, with proper modifications for diabatic 320 362 ×vergence γ = 0.5 (Kloeden and Platen, 1992, p. 305). = (14) 150 s for 1202 × 818 point resolution (used for high resolu∂σ NLEV+1 cases. By forecasts a rather2 ).simple modification of the Euler–Maruyama tion weather The macro time-step is taken as the The second2 method (labeled A) is based on the Akima upper scheme, limit for the Equation i.e.solution addingofthe term:(12). The time-step ∂ρ (1991) cubic ∂ spline. ρ For each interval it considers the pre275 needed to reach the required accuracy depends on the quan+ (σNLEV+1 − σNLEV ) 2 two adjacent intervals (for a total number 325 tities involved ∂σ vious ∂σnextNLEV 1 0 in determining the various elements in EquaNLEVand the 2 (18) of 6 grid points) to compute the coefficients of the interpotion (17).bb (1W − 1t) , 2 cubic polynomial. algorithm the reduces the consideranumber for the lating first-order derivative.This Following same of oscillations in the interpolatingoffunction compared reg-z are 1 http://www.isac.cnr.it/dinamica/projects/forecast dpc/month tions made for ρ, the derivatives σ with respecttoof where b0 is the first-order derivative of b, the Milstein scheme 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 inMilstein 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 reuses finite differences derivatives. In the first variant (lagarded 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 ideD0 ) is also tested in which the three-point centered derivative alised tests do not show any appreciable accuracy improveis computed and interpolated linearly at the particle position. ment with respect to the Euler–Maruyama scheme. However, Geosci. Model Dev., 7, 2181–2191, 2014


D. Rossi and A. Maurizi: IL-GLOBO: vertical diffusion module because they confirm the negligible extra computational cost of this method, the Milstein scheme will be used to integrate the model. In the meteorology component of IL-GLOBO, the Eulerian equations are solved with a macro time step 1T , which depends basically on the horizontal resolution due to the limitations imposed by the Courant number. Other time steps are involved in the Eulerian part but are not relevant here. In typical implementations, 1T ranges from 432 s for 362 × 242 point resolution (used for monthly forecasts1 ) to 150 s for 1202×818 point resolution (used for high-resolution weather forecasts2 ). The macro time step is taken as the upper limit for the solution of Eq. (12). The time step needed to reach the required accuracy depends on the quantities involved in determining the various elements in Eq. (17). First, a straightforward constraint is that the time step must satisfy the relationship p ∂K −1 , 2K1t1 K (19) ∂σ (Wilson and Yee, 2007, see, e.g.), which expresses the requirement that the average root mean square step length must be much smaller than the scale of the variations of K. This gives rise to a limitation that is consistent with the surfacelayer behaviour of the diffusion coefficient, Eq. (16). The condition expressed by Eq. (19) makes 1t1 vanish for z → 0. Such behaviour ensures that the WMC is satisfied theoretically, but clearly poses problems for numerical implementation (Ermak and Nasstrom, 2000; Wilson and Yee, 2007). However, in the application of a global model, where particles can be distributed throughout the troposphere, this problem affects only a small fraction of particles in the vicinity of the surface. Therefore, it can be dealt with by selecting a 1tmin small enough for the solution to be within the accepted error and, at the same time, large enough to not impact the overall computational cost. In addition to Eq. (19), another constraint is needed to account also for the presence of maxima in the K profile, which must be present if one considers the whole atmosphere. At maxima (or minima), Eq. (19) gives an unlimited 1t1 , which is not suitable for the integration of the model as it could cause the trajectory to cross the maximum (or minimum), with a significant change in K(z) associated to a change in ∂z K sign. To avoid this problem, a further constraint is introduced, based on the normalised second-order derivative, which gives an estimation of the width of the maximum. The constraint reads 2 −1 ∂ K (20) 2K1t2 K 2 . ∂σ 1 http://www.isac.cnr.it/dinamica/projects/forecast_dpc/month_

en.htm 2 http://www.isac.cnr.it/dinamica/projects/forecasts/glob_ newNH/


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The above equation has the property of limiting 1t2 according to the sharpness of the K peak. Taking the minimum among 1T , 1t1 and 1t2 (and replacing with = CT in Eqs. 19 and 20), gives " −1 # CT ∂K −2 CT ∂ 2 K 1t = min 1T , , (21) K , 2 ∂σ 2 ∂σ 2 where the parameter CT quantifies the “much less” condition and, therefore, must be at least 0.1 or smaller. Figure 2 shows the application of Eq. (21) for a K profile representative of GLOBO (see Sect. 4) and a CT = 0.01. The 1t decreases in the presence of K gradients thanks to condition (19), and is limited around the K maximum (where ∂K/∂σ = 0) by condition (20). The maximum of 1t = 1T is attained at higher levels. 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 time step along the particle path. To overcome this problem, an additional constraint is used to make the algorithm nonlocal (or “less local”). Using the 1t0 computed at the particle position at time t, two other time steps (1t+ and 1t− ) are evaluated at the positions: 1/2

6± = 6t + a1t0 ± b1t0 .

(22)

The minimum 1t among 1t0 , 1t+ and 1t− is then used to advance the particle position 6t+1t . 3.4

Boundary conditions

The necessary boundary condition for the conservation of the probability (and therefore of the mass) is the reflective boundary (Gardiner, 1990, p. 121). Wilson and Flesch (1993) show that the elastic reflection ensures the WMC if the integration time step is small enough. However, in cases of nonhomogeneous K, numerical implementation requires that 1t vanishes as the particle approaches the boundary. For models that focus on near-surface dispersion, the time step needed to achieve the required accuracy can become very small. Ermak and Nasstrom (2000) describe a theoretically well-founded method to speed up (roughly by a factor of 10) simulations of this kind. In the case of IL-GLOBO, it will be shown that the elastic reflection condition at σ = 1, coupled with the adaptive time-step algorithm described in Sect. 3.3, can ensure a good approximation of the solution while maintaining affordable the computational cost. 4

Model verification: the well-mixed condition

In order to verify the vertical diffusion module of ILGLOBO, a series of experiments was performed with a 1-D version of the code and then tested in a preliminary version of the full 3-D model. Input profiles were obtained by running the low-resolution version of GLOBO (horizontal grid Geosci. Model Dev., 7, 2181–2191, 2014

diffusion module2186

Maurizi: IL-GLOBO: vertical diffusion module 5D. Rossi and A. 6

40000

70

35000

60

30000

80 1

the following constant value is obtained:

method is based on 400 In order to verify the vertical diffusion module of ILng variations of K ρGLOBO, 0 Rd T0 a series of experiments was performed with a 1-D ρσ = version of. the code and then tested in a preliminary version (25) path. To overcome g

Figure 3 shows the GLOBO-average profiles and their fitting functions for the density ρ and the geopotential height 8g −1 as function of σ . As far as the K profile is concerned, the function h i K(z) = Az exp −(Bz)C , (26)

70 0.8

60

0.6 20000 15000

0.4

10000

2

25000

K[m /s]

h expresses the re40 1 re step length must 30 iations of K. This 20 0.1 nt with the surface 10 nt, Eq. (16). The 0.01 0 kes ∆t1 vanish for 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 is satisfied theoretσ merical implemenon and Yee, 2007). Figure Fig. 2. Values of of integration step ∆t 1t for forthe thediffusivity diffusivity pro2. Values integrationtime time-step profile model, where partifile shown by the redred curve. The green shown by the curve. The greenline lineshows shows the the contribution contribution ofof osphere, this probEq.the (19), theline bluethe linecontribution the contribution Eq.(20) (20),and andthe the black black line Eq. (19), blue ofof Eq. line cles in the vicinity the combined condition with 432s sand and C CT = 0.01). the combined condition (Eq.(Eq. 21, 21, with 1T∆T ==432 T = 0.01). t with by selecting o be within the acough to not impact the problem, an additional constraint is used to make the alof 362 × 242 cells and 50 vertical levels evenly spaced in σ ) 375 gorithm non-local (or less local). Using the ∆t0 computed at startingtheatparticle 11 March 2011 00:00 After 36 h of simulaonstraint is needed position at time t, UTC. two other time-step (∆t + and tion (12:00 UTC), averaging of σ = const surfaces were perima in the K pro∆t− ) are evaluated at the positions: formed for K, ρ and 8, obtaining vertical profiles as a funcders the whole at1/2 Σt + a∆t b∆t08 .were averaged over the whole (22) ation (19) gives tion an ofΣσ±. =Fields of 0ρ±and e integration of the domain. As far as K is concerned, averages were performed The minimum ∆t among ∆t0 , ∆t◦+ and ∆t− is then used to ross the maximum for380latitudes between +60 and −60 in daytime (lonadvance the particle position Σt+∆tNorth . n K(z) associated ◦ East) gitudes between −45 and +45 and night-time (longiproblem, a further ◦ East) conditions, over land 3.4 Boundary conditions tudes between +135 and −135 ormalized secondon of the width and of sea separately. The most intense K profile is selected necessary to boundary condition for the conservation of which The corresponds the daytime conditions over land. Prothe probability (and therefore of the mass) is the reflective files ofboundary ρ and z are rather smooth and regular over space and (Gardiner, 1990, p. 121). Wilson and Flesch (20) time, while K displays The profiles were 385 (1993) show that thelarge elasticvariability. reflection ensures the WMC if fitted with analyticaltime-step functions derived by combining hythe integration is small enough. However, inthe cases miting ∆t2 accorddrostatic equation and theK, perfect gas law. The following anof non-homogeneous numerical implementation requires ∆t vanisheswere as theused: particle approaches the boundary. For alyticalthat expressions and ∆t2 (and remodels that focus on near surface dispersion, the time-step ) and (20)), gives: achieve, the required accuracy can become very ρ(σ390) =needed ρ0 σ (Rtod 0/g+1) (23) # small. Ermak and Nasstrom (2000) describe a theoretically −1 2 K and well founded method to speed-up (roughly by a factor of 10) , (21) σ2 simulations of this kind. −R 0/g (σ 0 In thed case − of 1)T IL-GLOBO, it will be shown that the elastic z(σ ) = , (24) uch less” condition 395 reflection condition at σ = 1, coupled with the adaptive time 0 smaller. step algorithm described in Section 3.3, can ensure a good 21) for a K profile with approximation T0 = 288.0of K,the solution ρ0 = 1.2 kg maintaining m−3 and 0= while affordable ) and a CT = 0.01. −1 −0.007theKcomputational m . As acost. consequence of the hydrostatic gradients thanks to perfect gas assumption, the density ρ in sigma by expressing K maximum (where dz vertical4 units and using Eqs. (24) and (23), imum of ∆t = ∆T σ = ρ dσ the Modelρverification: well-mixed condition

Geosci. Model Dev., 7, 2181–2191, 2014

90

3

50

1.2

ρ[kg/m ]

∆t[s]

10

80

Φ/g[km]

(19)

45000

2

100

90

K[m /s]

that the time-step

Rossi and Ma

50 40 30 20

0.2 5000 0

10 0

0.2

0.4

σ

0.6

0.8

1

0

0 0.6

0.65

Fig.Average 3. Average GLOBO profilesof of ρρ (green (green symbols) andand φ/gφ/g Fig. 4. Diffusivit Figure 3. GLOBO profiles symbols) (blue symbols) as a function of verticalcoordinate coordinate σ,σand ana-anarepresents the dat (blue symbols) as a function of vertical andtheir their lytical fits (Eq. 23 and Eq. 24, lines of the same colors). The ‘average’ pro lytical fits (Eqs. 23 and 24). shown in green. by Eq. (26).

of the full 3-D model. Input profiles were obtained by running the low-resolution version of GLOBO (horizontal grid 405 of 362 × 242 cells and 50 vertical levels evenly spaced in σ) CT RMSE Time starting at 2011-03-11 00:00 UTC. After[s]36 hours of simulation (12:00 UTC), averages on σ = const surfaces were per76 profiles as a funcformed for K, ρ0.5 and Φ,0.044 obtaining vertical 435 0.1 0.037 238 tion of σ. Fields of ρ and Φ were averaged over the whole 1172 were performed 410 domain. As far0.01 as K is 0.021 concerned, averages ◦ 0.001 +600.021 7317in daytime (longifor latitude between and -60◦ North tude between -45◦ and +45◦ East) and nighttime (longitude between +135◦ and -135◦ East) conditions, over land and 440 sea separately. The most intense K profile is selected, which 415 corresponds the the daytime conditions over land.it Profiles is used to accounttofor specific K features: shouldofdisand z behaviour are rather smooth and regular over spacetend and time, play a ρlinear near the surface, must to zero while K displays a large 3variability. The profiles were fitted near thewith boundary layer top and, therefore, must display analytical functions derived combining the hydrostatic 445a maximum at height. = 0.29analytical ms−1 was equationsome and the perfectIn gasEq. law.(26), The A following expressions according were used: to average surface-layer properfirst420determined

Table 1. RMSE and execution time for different CT .

ties (the first GLOBO vertical level), and corresponds to a ρ(σ) = ρ0 σ (Rd Γ/g+1) , (23) friction velocity u∗ ' 0.7 ms−1 . Then, the other two param450 and: allowed to vary to fit the average profile giving eters were −3 −1 B = 1.3 × 10 (σ −R md Γ/gand C = 1.6. − 1)T 0 , is representative of the(24) z(σ) =the above profile Although typiΓ cal GLOBO diffusivity, real profiles can be remarkably less T0 = 288.0 K, ρ0 = 1.2 kgm− 3 and Γ = −0.007 K m−1 . regular,with creating challenging conditions for the model. For 425 As a consequence of the hydrostatic perfect gas assumpthis reason, a profile was the selected thosevertical showing 455 tion, by expressing densityamong ρ in sigma unitsiso dz ) andnear lated strong the ground.(24) This typical of strong (ρσ = maxima ρ dσ using Equations andis(23), the following constant value is obtained: convective conditions just after sunrise. Fitting Eq. (26) to this second profile gives A = 0.3 ms−1 , B = 4.0 × 10−3 m−1 ρ0 Rd T0 . (25)and ρ = and C =σ 4.5. Figure 4 reports the GLOBO “average” g 460 “peaked” K profiles as function of σ . 430

4.1

Figure 3 shows the GLOBO averaged profiles and their fitting functions for the density ρ and the geopotential height Determination the optimal setting for the Φg −1 as function of of σ.

adaptive time-step selection algorithm The first series of experiments concerns the optimisation of the adaptive scheme for 1t, i.e. the selection of the best suited value for the coefficient CT in Eq. (21). 3 In GLOBO, K also accounts for a part of the instability generated by moist convection and, therefore, it might not vanish at the boundary layer top.


As far as the

K(z) = Azexp

is used to accou play a linear be near the bounda maximum at so was first determ erties (the first a friction veloc rameters were B = 1.3 × 10− Although the cal GLOBO dif regular, creating this reason, a pr lated strong ma strong convectiv tion (26) on th 4.0 × 10−3 m−1 ‘averaged’ and

4.1 Determin tive time-

The first series the adaptive sc suited value for Simulations w by Equations (2 number concent

3 In GLOBO, erated by moist c boundary layer to


2187

90

∆t[s]

80 1 70

40 30 20

0.2 10 1

0

60 40 20 0

Table 1. RM

50

C,ρnorm

0.4

2

3

0.6

K[m /s]

60 ρ[kg/m ]

0.8

1000 100 10 1 0.1 1.4

2

1.2

K[m /s]

Rossi and Maurizi: IL-GLOBO: vertical diffusion module

0 0.6

1.2

0.05

1

0.045 0.04

0.8 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

Rossi and Maurizi: IL-GLOBO: vertical diffusion module 0.65

0.7

0.75

0.8 σ

0.85

0.9

0.95

1

2

K[m /s]

∆t[s]

n symbols) and φ/g 4. Diffusivity profilesused usedininthe the experiments. experiments. The Figure Fig. 4. Diffusivity profiles Thesymbols symbols 1000 nate σ, and their anarepresents thefrom data from GLOBO linestheir their fitting 100 represents the data GLOBO andand thethe lines, fittingfunction. function. 60 is e colors). The ‘average’ profile is shown in red, whilethe the“peaked” ‘peaked’ profile The “average” profile is10shown in red, while profile 40 is 1 functional form of both profiles is described shown in green. The shown in green. The functional form of both profiles is described 20 by 0.1 by Eq. (26).

1

σ

CT RMSE Time [s] FigureFig. 5. Dispersion experiment choicesofofparameparam5. Dispersion experimentwith withdifferent different choices 0.5 0.044 76 Top panel: diffusivity profile (black line)and and 1t ∆t profiles profiles eter Cter Top diffusivity profile T . panel: T. C 0.037 (black line) 238 0.1 0.5 (light blue), = 0.1 (green),CC 0.01 (red) (red) and T =(light for CTfor=C0.5 blue), CT C=T0.021 0.1 (green), and T T==0.01 1172 0.01 (blue). Bottom panel: normalizedconcentration concentration proT = 0.001 CT = C 0.001 (blue). Bottom normalised pro0.021 7317 0.001panel: files for different CT (Line colors thetop toppanel). panel). files for different C (line colours as as in inthe T

Table 1. RMSE and execution time for different CT .

Eq. (26). 1.4 0 re obtained by runsatisfied, this distribution must remain constant as the time BO (horizontal grid 1.2 evolves.0.05 Equation (12) was integrated for 4 × 105 particles As far as the Kperformed profile is concerned, the function described evenly spaced in σ) Simulations were in flow conditions and for 200 0.045 macro time-steps, each 432 s long, for a total of 1 36 hours of simulaby Eqs.K(z) (23), (24) and (26),C distributing particles with (26) numT = 86400 = Azexp −(Bz) , 0.04 s = 24 h. The actual time-step used is given by 0.8 surfaces were per465 Equation (21) with the additional lower limit ∆tmin = 0.01. ber concentration proportional ρ.0.8For0.85the0.9 WMC 0.6 0.65 0.7 to 0.75 0.95 1 to be 0.035 l profiles as a func- 435 is used to account for the specific K features: it should disSimulations were performed using 12 cores of an Intel Xeon 495 satisfied, this distribution must remain σ constant as the time 0.03 ged over the whole play a linear behavior near the surface, must tend to zero machine. Since the initial condition was already well-mixed evolves. Equation (12)layer was top integrated for 4 × 105display particles 3 ges were performed 0.025 near theFig. boundary and, therefore, must a of parame5. Dispersion experiment with different choices (C ∝ ρ), the simulation time was considered sufficient to as−1 and formaximum 200 macro timeheight. steps,Ineach 432 s(26), long, for a total of in daytime (longiat some Equation A = 0.29 ms 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 ighttime (longitude T = 86was 400 s for = 24 actual time step used by first determined to average prop0.015 CTh. = The 0.5according (light blue), CT = 0.1surface-layer (green),isCgiven = 0.01 (red) and T 470 of the simulation, final concentration profiles were computed ons, over land and erties (the first0.001 GLOBO level), and corresponds to Eq.440(21) with lower limit 1t = 0.01. SimCthe (blue).vertical Bottom panel: normalized concentration promin 0.01 T =additional in “σ volume”, i.e., c(σ) = N (σ)(∆σ)−1 , where N (σ) is the −1 e is selected, which a friction velocity u∗ ≃using 0.7 ms12 .colors Then, the other two Xeon pafor different C asof in the top panel). ulations werefiles performed cores an Intel T (Line number0.005 of particles between σ and σ + ∆σ. The skill of the 500 er land. Profiles of rameters were let to vary to fit the average profile giving 0.001 0.01 0.1 machine. Since the initial condition was already well-mixed model in reproducing the WMCCwas evaluated using the root −3 −1 ver space and time, B = 1.3 × 10 m and C = 1.6. T (C ∝ ρ), the simulation time was considered sufficient to asmean square error (RMSE) of the final normalized concenprofiles were fitted Although the above profile is representative of the typisatisfied, this distribution must remain constant as the time475 tration profile with respect to the normalized density 5profile sess thecalskill of thediffusivity, model inreal satisfying thebe WMC. At theless end ing the hydrostatic 445 GLOBO profiles can remarkably Figure 6. RMSE obtained from experiments made with 10 5 (red), 5 evolves. Equation (12) was integrated for 4 × 10 particles (derived Equation 25). experiments 5Fig. 6. using RMSE obtained from with 10 ollowing analytical of the simulation, finalchallenging concentration profiles were computed regular, creating conditions for the model. For 4 × 10 (green) and 16 × 105 (blue) particles asmade a function of C(red), T. and for 200 macro time-steps, 432 s long, for a total of 4× 105 (green) and 16 105 (blue)profiles particlesofasconcentration a function of Caf−1 ,each T . 505 Figure 5 reports the×different this reason,i.e. a profile was selected among those showing iso-the in “σ volume”, c(σ ) = N(σ )(1σ ) where N(σ ) is T = 86400 s = 24 h. The actual time-step used is given by lated strong maximum near the ground. This is skill typical ofthe ter 24 hours of simulation computed using different values number of particles between σ and σ + 1σ . The of 465 Equation conditions (21) with the lower limitEqua∆tmin = 0.01. (23) convective justadditional sunrise. Fitting of CT . The shaded region represents the interval between 3 model strong in reproducing thewere WMC wasafter evaluated using root 4.2 Evaluation of the interpolation algorithms Simulations performed using 12 cores of −1the 4.2 Evaluation the interpolation algorithms 450 tion (26) on this second profile gives A = 0.3 ms , Ban=Intel Xeon480495 standard deviationsoffrom the expected value. RMSE values mean square error of the final concenSince condition well-mixed −3 −1 4.0 × 10machine. m (RMSE) and C =the 4.5.initial Figure 4 normalised reportswas thealready GLOBO for each simulation are reported in Table 1 along with the ∝ ρ), the simulation time considered tration‘averaged’ profile(C with respect to normalised density profile to asIn thecomputation subsequent set of thethemodel skill In the subsequent setexperiments, of experiments, model skill in in rereand ‘peaked’ K the profiles as was function of σ. sufficient time. The RMSE error becomes comparable to 510 (24) skill of the model in satisfying the WMC. At the end producing the WMC was evaluated for the interpolation (derived usingsess Eq.the producing the WMC was evaluated for the interpolation tech25). the statistical′ error for CT = 0.01, which is selected astechthe of the simulation, final concentration profiles wereafcomputed 4.1 470 of the optimal setting for the adapniques D Aand A described in Section 3.2. dependency niquesoptimal D, D0 D, and described in Sect. Figure 5 Determination reports the different profiles of concentration 3.2.possible value. In order to evaluate the −1 Γ = −0.007 K m−1 . tive time-step selection algorithm in “σ volume”, i.e., c(σ) = N (σ)(∆σ) , where N (σ) is the the firstnumber experiment, the analytical fieldsdescribed described by ter 24 h of simulation computed using different values of CT . In the first experiment, the analytical fields by 485 of CIn on the of particles, two additional sets of runs T erfect gas assumpnumber of particles between σ and σ + ∆σ. The skill of the 500 Equations (23),(26) (24)with and 5 (26) with the5parameters of the ‘avwere performed with 10 and 16 × 10 particles that correThe shaded region represents the interval between 3 standard Eqs. (23), (24) and the parameters of the “average” series in ofreproducing experiments the concerns of the root gma vertical units 455 The firstmodel WMCthe wasoptimization evaluated using erage’todiffusivity profile were resampled on a 50 statistical point regspond halving doubling, deviations from the expected value. RMSE forbest eachconcendiffusivity profile wereand resampled onrespectively, a 50-point the regular grid.515 the adaptive scheme for ∆t, i.e., the the nd (23), the followmean square error (RMSE) ofselection the values finalofnormalized ular grid. This provides a discrete version of the experiment error of the base experiment. Results are reported indescribed Figure 6 suited value for profile the coefficient C into Equation 21. simulation are reported in with Table 1Talong with the computaThis provides a in discrete version of the experiment 475 tration respect the normalized density profile described the previous section, with thethe same vertical which shows that, in with the considered range, optimal CTresis were performed in flow conditions described tion time.Simulations The(derived RMSE error becomes comparable to the stain the previous section, the same vertical resolution of using Equation 25). olution of the GLOBO original fields. 490 quite independent of the number of particles. by Equations (23), (24) and (26), distributing particles with (25) tistical error forFigure CT =50.01, which is selected as the optimal the GLOBO original fields. reports the different profiles of concentration af- 505 particle number, distribution and simulation 460 number concentration proportional to ρ. For the WMC to be ItThe is worth noting that initial the time-step selection algorithm value. In order the possible dependency of CT values The time particle number, initial distribution and simulation ter to 24 evaluate hours of simulation computed using different are the same as in the experiment described in is section with the proper choice of CT ensures that the WMC also 520 3 ofiles and their fitIn GLOBO, alsoshaded accounts for a part of sets the instability genofof CTparticles, .K The represents the interval betweentime 3 are 4.1. integration is selected usinginthe local alon the number tworegion additional of runs were theThe same asreflective in thetime-step experiment described Sect. 4.1. satisfied at the boundary too, as mentioned in Secgeopotential height erated by moist convection and therefore it may not vanish at the 5 5 standard deviations the expected value. RMSE gorithm. The time-step selection algorithm requires the comperformed480with 10 and 16 × 10from particles that correspond to values The integration time step is selected using the local algotion 3.4. boundaryfor layer top. simulation are reported in Table 1 along with the each of the second orderalgorithm derivative requires of K, which is not halving and doubling, respectively, the statistical error of the rithm. putation The time-step selection the comcomputation time. The RMSE error becomes comparable to 510 possible for the D interpolation scheme. Therefore, it is esbase 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 considered range, the optimal CTthe is possible quite indepossible for the D interpolation scheme. Therefore, it is esoptimal value. In order to evaluate dependency The results of this experiment are shown in Figure 7. In the pendent of485theofnumber of number particles. timatedupper using finitethedifferences the first-order derivative. CT on the of particles, two additional sets of runs panel, integration of time-step profiles of the three 5 5 algorithm It is worthwere noting that the time-step selection The results of this experiment are shown in Fig. 7. In the performed with 10 and 16 × 10 particles that corresimulations and the Akima interpolated diffusion coefficient with the proper of Cand the WMC upper panel, are thedisplayed. integration ofnormalized the three spondchoice to halving doubling,that respectively, the is statistical 515 profile, Thetime-step lower panelprofiles shows the T ensures the base experiment. reported ininFiguresimulations 6 also satisfied error at theofreflective boundary Results too, as are mentioned and the Akima interpolated diffusion coefficient distribution of the particle after 24 hours of simulation along which shows that, in the considered range, the optimal CT profile is with the expectedThe value. Tablepanel 2 displays thethe integration time Sect. 3.4. are displayed. lower shows normalised 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/ Geosci. Model Dev., 2181–2191, 2014 with the proper choice of CT ensures that the WMC is also 520 around the region of maximum of K, 7, where it shows strong satisfied at the reflective boundary too, as mentioned in Secvariations and, on the average, is longer than the others. tion 3.4. Looking at the distribution of particles (lower panel), it reRMSE

C,ρnorm

0.035

7 RMSE

8


0.03 0.025 0.02 0.015 0.01

0.005 0.0

Fig. 6. RM 4 × 105 (gree

4.2 Evalu

In the subse producing t niques D, D In the fir Equations ( erage’ diffu ular grid. T described in olution of th The part time are the 4.1. The in gorithm. Th putation of possible for timated usin The results upper pane simulations profile, are distribution with the exp and RMSE The time around the variations a Looking at

2188


8 Table 2. Execution time and RMSE for experiments made with the sampled “average” diffusivity distribution and varying interpolation method.

and Maurizi: vertical diffusion module Table 3. Rossi Execution time andIL-GLOBO: RMSE for experiments made with the “peaked” diffusivity distribution, varying interpolation method and Interpolation algorithm ∆t selection exec. time RMSE 1t selection algorithm. A local 313 s 0.042 local

D

237 s 155 s 162 s

0.025 0.023 0.044

80 60 40 20 0

A Interpolation algorithm

181 s

0.065

non-local Exec. 1122 s 1t selection time non-local

D

0.016 RMSE

593 s

0.022

A local 313 s 0.042 and RMSE for experiments D Table 3. Execution time local 181 smade with 0.065the Rossi‘peaked’ and Maurizi: IL-GLOBO: vertical diffusion module diffusivity distribution, varying interpolation and A non-local 1122 s method 0.016 ∆t selection algorithm. non-local D 593 s 0.022 Interpolation algorithm ∆t selection exec. time RMSE

2

RMSE

K[m /s]

100 10 1 0.1 1.4

Exec. time

1.2

A D A D

1

local local non-local non-local

313 s 181 s 1122 s 593 s

0.042 0.065 0.016 0.022

0.8 0.9

0.95 σ

1.0 0.9

0.9540 σ

1.0 2

1000 100 10 1 Figure0.18.

K[m /s]

1

∆t[s]

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

2

∆t[s]

Normalized final concentration and expected distribution (black).

0.8

exec. time RMSE Interpolation algorithm σ A 237 s 0.025 155 s 0.023 distriD FigureFig. 7. Experiments with the sampled “average” diffusivity dis7. Experiments with the sampled ‘average’ diffusivity ′ 162 s D ′0.044 0 bution for the interpolation algorithmsDD(blue), (blue), D D (green) (green) and tribution for the interpolation algorithms andAA

K[m /s]

2

K[m /s]

1000 0.8 80 100 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 60 1000 10 40 σ 100 40 1 Table 3. Execution 10 time and RMSE for experiments made with the 20 0.1 ‘peaked’ diffusivity 20 and Fig. 7. Experiments with the sampled ‘average’ diffusivity distri1 distribution, varying interpolation method 1.4 0 0.1 bution for the interpolation algorithms D (blue), D′ (green) and∆t A selection algorithm. 0 1.2 (red). Top panel: Diffusivity profile as interpolated by A (black) and 1.2 ∆t profiles for the different interpolation settings. Bottom panel: 1 1 C,ρnorm

C,ρnorm

∆t[s]

C,ρnorm

A D 8 D0

∆t[s]

Interpolation 1000algorithm

20

C,ρnorm

Same as in Fig. 7 for experiments with the “peaked” dif8. Same as in Fig. 7 for experiments the ‘peaked’ diffusivfusivityFig. distribution. Results obtained usingwith the 0 local (left) or non(red).panel: Top panel: Diffusivity profile as interpolatedby byAA(black) (black) and (red). Top diffusivity profile as interpolated and ity 1.2 distribution. Results obtained using the local (left) or non-local Table 2. Execution time and RMSE for experiments made with local (right) 1t selection algorithm. ∆t profiles for the different interpolation settings. Bottom panel: 1t profiles forthethe different interpolation settings. Bottom panel: (right) ∆t selection algorithm. sampled ‘averaged’ diffusivity distribution, varying interpola1 Normalized final concentration expected distribution(black). (black). normalised final andand expected distribution tionconcentration method. 0.8

Interpolation algorithm exec. time RMSE 0.9 0.95 1.0 0.9 0.95 interpolation schemes, especially for81.0 D.reports Conversely, the nontime-step selection algorithms. Figure the time-step A 237 s 0.025 σ σ sults that simulations with A and D interpolation algorithms 550 local algorithm turns out to be effective in selecting theRMapand concentration profiles, while execution times and 155 sh of simulation 0.023 D distribution of thesatisfy particle 24within along with both theafter WMC the statistical limit, while the propriate time-step even in presence of strong gradients and ′ 162′ s the integration 0.044 D SEs are shown in Table 3. Although the integration timethe expected value. Table displays time andthe well 525 simulation with2 the D algorithm fails to maintain isolated This is reflected its higher computaFig. 8. Same as in maxima. Fig. 7 for experiments with the on ‘peaked’ diffusivstep profiles look very similar for the local and non-local alRMSETable obtained for the various experimental settings. state,time in particular the ground.made Additional extional cost (see Table 3). ity distribution. Results obtained using the local (left) or non-local 2.mixed Execution and RMSE near for experiments with gorithms, the small differences have a large impact on the Thethetime-step profiles are similar, except for theinterpolapro- a well periments (notdiffusivity reported) show that in order toAobtain (right) ∆t selection algorithm. sampled ‘averaged’ distribution, varying results: local algorithmonmostly in reproducing the mixed solution with D′ , resolution be doubled, tion method. 4.3theImplementation the 3-Dfails model file around the region of the maximum of K,must where it showsat least.

WMC for both interpolation schemes, especially for D. ConThe problem probably related to thethan definition of derivastrong variations and, on isthe average, is longer the othinterpolation especially D.turns Conversely, 555 A preliminary test of the for algorithms on the 3-Dnonmodel has 530 tives of K between grid In fact, although D′ computes versely, theschemes, non-local algorithm out to the be effective in ers. Looking at the distribution ofpoints. particles (lower panel), it sults thatderivatives simulations with Aorder and Dofinterpolation algorithms 550 local algorithm turns outThe to be effective selecting thepresence ap- implebeenthe performed. interpolation algorithm has been at higher approximation than D, they are selecting appropriate time step,ineven in the of can be observed that simulations with A and D interpolation both satisfy the WMCwith within the statistical limit, the the use propriate time-step in presence of strong gradients and in aeven simplified quasi-1-D form, where the diffusion not consistent a linear variation of K. while Although strongmented gradients and isolated maxima. This is reflected in its algorithms both satisfy the′ algorithm WMC within themaintain statistical limit, ′ 525 simulation failsslowly to isolatedcoefficient maxima. has Thisbeen is reflected on its higher computa-constant considered to be horizontally of Dwith canthe beDappropriate for varyingthe andwell monotone 0 higher computational cost (see Table 3). whilemixed the simulation with the D algorithm fails to maintain state, in particular near theturns ground. ex- for the tional cost (see Table 3). between grid points. IL-GLOBO uses the same parallelizafunctions like ρ and z, it out toAdditional be unsuitable the well-mixed state, in particular near the ground. periments (not reported) in order to obtainaffects aAddiwellboth the 560 tion of GLOBO, with particle exchanged between processes 535 more complex K show profilethat which, in addition, 4.3 Implementation Implementation on 3-D the 3-D model tionalmixed experiments reported) that orderFor toatobtain solution with D′ , resolution beindoubled, least.reasons, 4.3 on the model at each macro time-step. Particles are first advected horizonWiener(not stochastic term show andmust the drift term. these 0 ,scheme The problem probably related to the definition derivatally for a macro time-step using their deterministic velocity, thesolution D′isinterpolation is not used in the following exa well-mixed with D resolution must beof doubled of the algorithms on according theon3-D has (12). tives of K betweenisgrid points. In fact, although D′ computes and thentest ‘diffused’ in the vertical to Equation periments. at530least. The problem probably related to the definition of555 AApreliminary preliminary test of the algorithms themodel 3-D model has been The interpolation has been implederivatives at higher order of approximation than D, they are After 12 hThe of spinup, 5 ×algorithm 105 algorithm particles arehas released with a Thebetween second experiment concerns the although ‘peaked’ profile. In derivatives of K grid points. In fact, D0 beenperformed. performed. interpolation been implemented in a simplified quasi-1-D form, where the diffusion not540 consistent withthe variation of K. Although the use 565 vertical distribution proportional to thewhere average density prothis case, K profile is used directly, without the resamcomputes derivatives ataalinear higher order of approximation than mented in a simplified quasi-1-D form, the diffusion coefficient considered to be horizontally constant of D′ can be of appropriate for slowlySimulations varying andwith monotone file,has andbeen randomly and homogeneously distributed in the horpling the fitting function. A and D algoD, they are notlike consistent with a linear variation of K. Alcoefficient has been considered tothebesame horizontally constant betweenizontal. grid points. IL-GLOBO usescomputed parallelizafunctions z, it turns with out toboth be unsuitable for the timeParticle statistics are after 24 h from the rithms ρwere performed local and non-local 0and though the use of D can be appropriate for slowly varying between grid points. IL-GLOBO uses the same parallelisation of GLOBO, 535 more complex K profile which, inFigure addition, affectsthe both the 560 and release. with particle exchanged between processes step selection algorithm. 8 reports time-step and monotone functions like and z,term. it turns toreasons, be RMSEs un- are GLOBO, with Particles particles exchanged between processes attion eachofmacro time-step. are first were advected horizonWiener concentration stochastic termprofiles, andρthe drift For out these while execution times and A and D interpolation algorithm tested using the non′ the more complex K profile which, in addition, suitable for at each macro time step. Particles are first advected horizontally a macro time-step using their deterministic velocity, the 545 D interpolation scheme is not used in the following ex- profiles shown in Table 3. Although the integration time-step 570forlocal time-step selection. It is found that, while interpolation and then ‘diffused’ intime the vertical according todeterministic Equation (12).velocity affectsperiments. both the stochastic drift term. For the tally for a macro step their lookWiener very similar for theterm localand andthe non-local algorithms, scheme A maintains theusing WMC reasonably (RMSE=0.024), 0 interpolation After 12 time-step h of spinup, 5 ×the 105vertical particlesaccording are released a The second concerns the ‘peaked’ profile. Inthe local these reasons, the differences Dexperiment scheme is on notthe used in the and then “diffused” in toDwith Eq. (12). small have large impact results: the selection algorithm for scheme requires ex565 both vertical distribution proportional to≪5the average density pro540 this case, the K profile is used withoutthe theWMC resam-for algorithm strongly fails directly, in reproducing tremely short time-steps ∆t , see Section 4.1) in thea following experiments. After 12 h of spin-up, 5 × (10 particles are released with min file, and randomly and homogeneously distributed in thedensity hor- proof theexperiment fitting function. Simulations with A and D algoThepling second concerns the “peaked” profile. In vertical distribution proportional to the average izontal. Particle statistics computed afterdistributed 24 h from in thethe horrithmsthe were bothdirectly, local andwithout non-localthe timethis case, Kperformed profile iswith used refile, and randomly and are homogeneously release. step selection algorithm. Figure 8 reports the time-step and 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 nonalgorithms were performed with both local and non-local release. 545 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 small differences have large impact on the results: the local Geosci. Model Dev., 7, 2181–2191, 2014 algorithm strongly fails in reproducing the WMC for both

scheme A maintains the WMC reasonably (RMSE=0.024), the time-step selectionwww.geosci-model-dev.net/7/2181/2014/ algorithm for scheme D requires extremely short time-steps ( ≪ ∆tmin , see Section 4.1) in the

σ

σ 600


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

9

Fig. 9. Distribution of ∆t requirement for the condition on the first order derivative (Eq. 19) as a function of σ for interpolation algomoduleGLOBO, 2189 particle model integrated in the EulerithmsaALagrangian (left) and D (right).

rian global circulation model GLOBO. Critical details of the 605 implementation have been analyzed and discussed. 1.3 The model is developed including the variable density 0.1 150 term and 1.2 the proper coordinate transformation term. The numerical1.1scheme selected to integrate the SDE is the Mil0.01 100 stein scheme, which is of order of strong convergence γ = 1. 610 595 Therefore, 1it should be regarded as the natural extension of 0.001 50 the deterministic Euler scheme, in contrast to the so-called 0.9 Euler-Maruyama scheme, which is merely the “transcrip0.0001 0 tion” of the deterministic Euler scheme, but not its equiva0.8 0.8 0.85 0.9 0.95 1 0.8 0.85 0.9 0.95 1 615 lent. σ σ 0.7 600 An adaptive time-step scheme is proposed to ensure the 0.5 0.6 0.7 0.8 0.9 1 consistency of the model implementation with the WMC reFigure 9. Distribution of 1t requirement for the conditions of the σ Fig. 9. Distribution ∆t as requirement on the first first-order derivative (Eq.of19) a functionforofthe σ condition for interpolation quirements. The time-step selection algorithm is limited not order Aderivative 19) as a function of σ for interpolation algo- Figure algorithms (left) and(Eq. D (right). only Fig. by 10. the condition imposed by the spatial of expergra- 620 10. Normalised distribution of particles the Normalized distribution of particles forfor thescale 3D 3-D experiment. rithms A (left) and D (right). dients, alsodistribution takes into account the of the of iment. The initial distribution ∝ hρi (black) andfinal thedistributions finalwidth distribuThebut initial ∝ hρi (black) andscale the obtained using thethe AA interpolation scheme (red)(red) and and the Dthe interpo605 tions maxima and minima ofinterpolation the diffusion coefficient, where obtained using scheme Dthe inlation scheme(blue). (blue).It Dashed lines show the limit of of 33ensures standard former criterium fails. is shown that thisthe algorithm terpolation scheme Dashed lines show limit standard A and 1.3 D interpolation algorithms were tested using deviation around the an initial distribution. deviations around initial distribution. that the error is the within acceptable range also at the reflectthe non-local time-step selection. It is found that while 1.2 ing boundaries. However, in case of isolated maxima, this 625 interpolation scheme A maintains the WMC reasonably scheme may fail. The implementation of aboundary non-local algoregion between σ = 0.9 and the lowest (see Fig(RMSE = 0.024), the time-step selection algorithm for 1.1 theure error isFigure within an acceptable range also atisthe reflect610 that rithm, which evaluates ∆t in 2 additional points, proposed 575 9). 10 shows the result of an experiment where scheme D requires extremely short time steps ( 1tmin , ingin boundaries. However, in schemes case of Aisolated maxima, this order to solve the problem. the WMC compliance of and D was tested with 1 in the region between σ = 0.9 and the lowest see Sect. 4.1) −5 scheme may fail. The implementation of a non-local algoTwo numerical interpolation and derivation schemes are the lower limit for ∆t changed to ∆tmin = 10 for D. It boundary (see Fig. 9). Figure 10 shows the result of an exper0.9 implemented and tested. first is based strong onpoints, the fluctuations linear in- 630 rithm, which evaluates 1tThe in the two is procan be observed that, for D additional scheme, iment where the WMC compliance of schemes A and D was areorder stillof present in the region where required timeterpolation and it issame presented in twothe versions: one posed in toKsolve the problem. tested with0.8the lower limit for 1t changed to 1tmin = 10−5 615 (D) 580 step exceeds the lower isderivation likelybetween to beschemes caused the a constant first limit. order This derivative two by grid Twokeeps numerical interpolation and are for D. It can be observed that, for the D scheme, strong fluctu′ occurrence strong(Dgradients that caninterpolated be on eventhe larger thaninin points, whileand theofother ) uses linearly derivaimplemented tested. The first is based linear 0.7 present in the same region where the required ations are still the ‘peaked’ case, near points with extremely small values of tives in theofsame interval. second in scheme (A) is based 0.5 0.6 0.7 0.8 0.9 1 terpolation K, and it is The presented two versions: one In theseAkima cases, the A scheme interpolates with a smoother time step exceeds the lower limit. σThis is likely to be caused onkeeps a K. modified (1991) interpolation algorithm withgrid a (D) a constant first order derivative between two function which reduces the problem. by the occurrence of strong gradients that can be even larger local algorithm that ensures the positivity of the interpolating 0 points, while the other (D ) uses linearly interpolated deriva- 635 than in the10.“peaked” case, near points withfor extremely small 620 function (Fischer et al., 1991). Fig. Normalized distribution of particles the 3D experiment. tives in the same interval. The second scheme (A) is based ∝ hρi and the final distributions valuesThe of initial K. Indistribution these cases, the(black) A scheme interpolates with aobIt 5is found that, although the method D′ uses derivatives 585 Conclusions on a modified Akima (1991) interpolation algorithm with a tainedfunction using thewhich A interpolation scheme (red) and the D interposmoother reduces the problem. of higher order of approximation, it creates a local inconlocal algorithm that ensures the positivity of the interpolating lation scheme (blue). Dashed lines show the limit of 3 standard sistency the of linearly interpolated function andmodel its The between development a vertical Lagrangian diffusion etThis al., 1991). function (Fischer deviation around the initial distribution. derivatives and prevents the modelthe from fulfilling WMC. is presented. constitutes first step in the building IL0 It isother found that although theA) method D uses derivatives 625 The two schemes (D and both satisfy the WMC but 5 Conclusions ofextremely higher order of approximation, it creates a local inconpeaked profiles of K may require the use of the region between σ = 0.9 and the lowest boundary (see Figsistency between the linearly interpolated function and its non-local time-step selection algorithm. The development of a vertical Lagrangian diffusion model 575 ure 9). Figure 10 shows the result of an experiment where derivatives and prevents the model from fulfilling the WMC. A test with a preliminary implementation of the fully 3D is presented. This constitutes the firstAstep building the WMC compliance of schemes and in D was testedILwith The other(IL-GLOBO) two schemesshows (D and A)while both the satisfy the WMC, but model that, A scheme display −5 GLOBO, a Lagrangian particle model integrated in the Eulethe lower limit for ∆t changed to ∆tmin = 10 for D. It peaked profiles of K may require the use of the 630 extremely a correct behavior, the D interpolation scheme requires and can becirculation observed that, forGLOBO. the D scheme, strong fluctuations rian global model Critical details of the extremely strong reduction of the integration time-step that non-local time-step selection algorithm. are still present the analysed same region the required timeimplementation haveinbeen andwhere discussed. prevents the WMC to be satisfied in reasonable A test with a preliminary implementation of time. the fully 3-D 580 step exceeds the lower limit. This is likely to be caused by the 200

590

C

C

∆t[s]

1

The model is developed with the variable density term and occurrence of strong gradients that can be even than in model (IL-GLOBO) shows that, while the A scheme displays the proper coordinate transformation term. The larger numerical behaviour, the D interpolation scheme requires an theselected ‘peaked’tocase, near points withisextremely small values of a correct scheme integrate the SDE the Milstein scheme, Code availability extremely strong reduction of the integration time step that K. In these cases, the A scheme interpolates with a smoother which is of the order of strong convergence γ = 1. Thereprevents the WMC satisfied reasonable time. function which reduces the problem. The numerical codetoofbethe verticalindiffusion module (Fortran fore, it should be regarded as the natural extension of the de635 90) is released under the GNU Public Licence and is availterministic Euler scheme, in contrast to the so-called Euler– 4 able at the BOLCHEM website . Maruyama scheme, which is merely the “transcription” of 585 5 Conclusions CodeThe availability software is packed as a library using autoconf, the deterministic Euler scheme, but not its equivalent. automake and libtools which allows for configuration AnThe adaptive time-step proposed to ensure model the development of scheme a verticalisLagrangian diffusion The numerical code of the vertical diffusion module (Fortran consistency of the model implementation re-IL- 90) is4 http://bolchem.isac.cnr.it/source is presented. This constitutes the firstwith stepthe in WMC building code Licence and is availreleased under the GNU Public 4 quirements. The time-step selection algorithm is limited not able at the BOLCHEM website . only by the condition imposed by the spatial scale of gradiThe software is packed as a library using autoconf, ents, but also by taking into account the scale of the width of automake and libtools which allows for configuration maxima and minima of the diffusion coefficient, where the 4 http://bolchem.isac.cnr.it/source_code.do former criterium fails. It is shown that this algorithm ensures www.geosci-model-dev.net/7/2181/2014/

Geosci. Model Dev., 7, 2181–2191, 2014

lent. An adap consistency quirements only by the dients, but maxima an former crite that the erro ing bounda scheme ma rithm, whic in order to Two num implemente terpolation (D) keeps a points, whi tives in the on a modifi local algori function (F It is foun of higher o sistency be derivatives The other t extremely non-local ti A test w model (ILa correct b extremely s prevents th

Code avail

The numer 90) is relea able at the The soft automake 4

http://bo

2190


and installation on a variety of systems. The code is developed in a modular way, permitting the easy improvement of physical and numerical schemes. The GLOBO model is available upon agreement with the CNR-ISAC Dynamic Meteorology Group (contact: [email protected]).

Acknowledgements. The software used for the production of this article (model development, model run, data analysis, graphics, typesetting) is free software. The authors would like to thank the whole free software community, the Free Software Foundation (http://www.fsfs.org) and the Debian Project (http://www.debian.org). This work is part of the PhD thesis of Daniele Rossi who was partially supported by the CNR and CMCC agreement: “Impatto degli ‘hot spot’ sui cambiamenti climatici a scala regionale”. Edited by: P. Jöckel

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