Semi-Lagrangian advection scheme with controlled damping: An alternative to nonlinear horizontal diffusion in a numerical weather prediction model

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3 SEMI-LAGRANGIAN ADVECTION SCHEME WITH CONTROLLED DAMPING 525 In this paper, for simplicity, all time-discrete aspects about diffusion will be illustrated only for two-time-level extrapolating time discretizations, but any discussion and result may be easily extended to other types of time discretization (e.g. three-time-level leap-frog schemes). Since the departure and middle points of the trajectory are generally off the model grid, an interpolation method has to be used for the evaluation of all terms in the RHS of (1). The interpolator is chosen in such a way as to be sufficiently accurate while offering acceptable computational efficiency. Typically a cubic polynomial interpolator is found to be best suited for numerical weather simulations (Staniforth and Côté, 1991). In most models, the interpolation operator I used to evaluate the RHS of (1) is constructed as a combination of cubic polynomial interpolation on 3D or 2D stencils. The SLHD interpolation operator extends this situation by the use of an additional interpolation operator with extra damping properties. The resulting operator I then becomes a combination of two interpolators: the accurate one denoted as I A and the damping one I D.Formally the SLHD operator can then be expressed as: I = (1 κ)i A + κi D = I A + κ(i D I A ). (2) Interpreting the difference I D I A as the damping contribution of the operator I D with respect to the accurate interpolation I A, the coefficient κ can be seen as a diffusion coefficient; with κ 0, no extra damping is present and the original accurate interpolation is restored. The fact that I is evaluated in grid-point space makes it possible to define κ as a general function of any atmospheric field. Ideally, the definition of κ should be inspired by fluid mechanics theories, in order to introduce a high degree of physical realism in what would then become a sophisticated diffusion scheme. However, by nature, the approach chosen here is clearly a simplified one (e.g. the notion of turbulent flux is simply ignored), and the aim of the scheme is rather to access nonlinear diffusive sources at marginal cost, in view of improving NWP applications. Consistent with this choice, the definition of κ should therefore remain relatively simple and computationally efficient. In their formulation of lateral diffusion for fluid dynamics models, Sadourny and Maynard (1997) express the dissipation of the horizontal velocity as a sum of isotropic and anisotropic parts. The isotropic part, with the physical meaning of dynamical pressure due to the sub-grid scale, plays an important role only in strong incompressibility cases. Hence, for most NWP models, the term of main relevance is the anisotropic part of the dissipation. This term is related to the tensor of the deformation rates. It can be further simplified by setting proportional to the total deformation d along the reference surface a scalar quantity having the dimension of inverse time and being defined as d = 1 2 [ (h2 h 1 u 1 h ) 1 u 2 2 x 1 h 2 h 2 x 2 h 1 + ( h1 h 2 u 1 + h ) ] 2 u 2 1/2 2. (3) x 2 h 1 h 1 x 1 h 2 Here u 1, u 2 are the x 1, x 2 components of the horizontal velocity field, with metric factors h 1 and h 2. In a way similar to Smagorinski (1963), the coefficient κ in (2) is defined as a monotonic function f(d) of the total deformation d. Regarding the value of I D as an equilibrium value, the second term in the RHS of (2) can be seen as a relaxation source term. Using the standard expression for implicit schemes with explicit relaxation coefficient (following for example the suggestion of Jarraud in Kalnay and Kanamitsu, 1988) κ can be generally expressed as a bounded function of the time step t in a following way: κ = f(d ) t 1 + f(d ) t. (4) In this equation, d represents the total deformation evaluated at time t. Note that substituting in (2) the definition (4) of κ makes the SLHD scheme equivalent to an implicit relaxation of I A toward the operator I D.This has the advantage of ensuring the stability of the scheme for any time step t while keeping a simple evaluation of the function f(d ) at time t. Consistently with the SL discretization, f(d ) should be evaluated at the origin point of the trajectory, but in order to avoid an interpolation bringing some degree of smoothing and to save computational cost, f(d ) is evaluated at the final point instead. The experimental results justify this simplification as the correlation of d(x,t) and d(x α, t) interp is usually very high, over 90%. Moreover the difference between the simplified and SL consistent discretization consists of meteorologically insignificant noise, even for active storms typically sensitive to horizontal diffusion. The damping interpolator I D now remains to be defined. Classical interpolators are always prescribed so that they become exact at the nodes, where the value of the field is known, and this is true of course even for the most damping ones among them. As a consequence, the damping character of the interpolator in a SL scheme is only statistical, because the distribution of origin points is not likely to be correlated with the computational grid. From a purely local point of view, classical interpolators are not damping when the origin point lies on a node. In order to alleviate this non-physical dependency on the actual position of the origin point, a simple solution is to define the damping interpolator as a combination of a classical interpolator, and of a local pointwise defined diffusion operator. This is the option retained in the SLHD scheme presented below, where a linear interpolator is combined with a pointwise local second-order diffusion operator. This design, using local information only, can be achieved as a very simple modification of any pre-existing SL scheme. As will be shown in the next sections, the proposed design for the nonlinear diffusion scheme, in spite of

4 526 F. VÁŇA ET AL. the above-mentioned simplifications, results in improved performances in typical NWP applications, at a relatively small increase of computational cost. 3. Implementation of the SLHD into the Aladin model 3.1. General description As an illustration of the general SLHD scheme presented in the previous section, the particular application to the limited-area spectral model Aladin will be described here. The transport scheme used for all applications reported in this paper is a two-time-level semi-implicit SL scheme (Temperton et al., 2001) with quasi-cubic interpolations (Ritchie et al., 1995), which makes a space averaging of dynamical source terms along the trajectory (Tanguay et al., 1992). The total Lagrangian source term can be split into an implicitly treated dynamical linear part LX, an explicitly-treated dynamical nonlinear residual N, and a parametrized physical part F. The time- and space-discretization of the Aladin model can therefore be summarized as X + F X O t = 1 2 ( LX O + ) 1 ( LX+ F + N 2 O + N ) F + F O. The subscripts and superscripts have the same meaning as in (1). The Helmholtz equation to be solved for X F + is then: ( X F + = 1 t ) 1 2 L (5) X O + t 2 LX O + tf O + t 2 N O }{{} I n + t 2 N F. The terms evaluated at the origin point (horizontal brace denoted by I n ) are obtained by interpolation. In the original SL scheme of Aladin, the first and third terms in the horizontal brace are computed with a quasi-cubic interpolator (Ritchie et al., 1995), while the others are obtained by simple linear interpolations. The default horizontal diffusion is then a fourth-order spectral linear diffusion scheme (Geleyn, 1998). When the SLHD scheme is activated, the quasi-cubic interpolator is replaced by the operator I defined in (2) where I A represents the original quasi-cubic interpolator. From (5) it is clear that the term ( 1 t ) 1 2 L t 2 N F is not directly affected by the SLHD as this term is not subjected to any interpolation. For simplicity, the parameter κ is taken as identical for all the prognostic variables to which the horizontal diffusion is applied. The value of κ is given by (4) with the function f(d ) defined as ( [ f(d)= ad max 1, d ]) B, (6) d 0 where a is the scaling factor for the SLHD scheme, d 0 is the threshold value of total deformation d above which an enhanced damping is activated and B is a tunable parameter for the control of this enhancement. The presence of the enhancement part allows an increased selectivity by affecting smaller areas with a stronger diffusive effect, as discussed later. In order to make the tuning of the scheme independent of the resolution and domain size, the a and d 0 parameters are expressed as functions of the spectral mesh size [ x] (i.e. the inverse of the maximum wavenumber): ( ) P [ x]ref a = 2A, (7) [ x] d 0 = D ( ) Q [ x]ref, (8) 2 [ x] where A, D are the tunable parameters, and P, Q are empirical constants for maintaining independence of A and D to the domain definition. The [ x] ref is a reference value for the mesh size. Optimal values of P and Q have been found empirically as 1.7 and 0.6 respectively within the range of useful resolutions for the Aladin model. (For grid-point resolution varying between 2.2 and 17 km with both quadratic and linear truncations, the corresponding shortest spectral wave varies approximately between 4.5 and 52 km). The value of Q is in good agreement with the conclusions of Leith (1969). There the total deformation d is expressed as proportional to the mesh size to the power 2/3 in models with truncation lying within an inertial range of a 5/3 kinetic energy power spectrum. The three tunable parameters affecting the scheme through κ are then A, B and D. In the present scheme they are by default set to 0.25, 4.0 and , respectively. The damping operator I D is defined by [ I D = I L 1 + Ɣ(h) ] 2 D 2, (9) where I L is the linear interpolator and D 2 is the nondimensional finite-difference Laplacian operator, classically computed on a three-point stencil (D 2 X) i = X i+1 2X i + X i 1. In this definition, Ɣ(h) is a function which serves to reduce the dependency of the smoothing effect of I D to the space-shift h between the origin point of the SL trajectory and the computational grid. As mentioned above, I L has no damping effect when the origin point is located at a node of the grid (h = 0), but has a maximum damping effect when the origin point is in the middle

5 SEMI-LAGRANGIAN ADVECTION SCHEME WITH CONTROLLED DAMPING 527 of two nodes (h = 0.5 x). Therefore Ɣ(h) is defined in such a way as to have a maximum positive value for h = 0 and to reach zero for a given value h 0 < 0.5 x. This arrangement provides I D with a more homogeneous damping with respect to h. The effect of I D on onedimensional waveswith various wavelengths is illustrated by the response for some constant values of the spaceshift h, as shown in Figure 1, where x = 1 is assumed. In Aladin we choose h 0 = 0.15 x, defining that 51% of the computational area is affected by the operator D 2. Finally, to ensure time-step independence, Ɣ is scaled to a reference time step. From experience with real tests, the performance of the SLHD is only weakly sensitive to the presence of D 2, mainly because the sum of the function Ɣ in [0, 1] is rather small. In the above, the diffusive properties of the accurate interpolation operator I A representing the inherent damping of the SL scheme have not been considered. The proposed scheme has no ambition to control this nonexcessive diffusivity of the operator I A.However,asit is known that the role of such diffusion decreases with increasing model resolution (Jakimow et al., 1992), the empirical values P and Q of the SLHD are surely also affected by this feature. (a) wave amplification factor (b) wave amplification factor Response of linear interpolator (1D wave) 0.7 h=0 0.6 h=0.01 h=0.05 h= h=0.2 h= wavelength in grid units Response of I_D interpolator (1D wave) 0.7 h=0 0.6 h=0.01 h=0.05 h= h=0.2 h= wavelength in grid units Figure 1. Response of a sinusoidal wave caused by (a) linear interpolation and (b) the I D operator used in Aladin, as a factor of the distance h from the closest grid-point on a unitary grid (mesh size equal to one) Specific implementation features This definition of the SLHD scheme has to cope with two specific problems requiring further attention. As already mentioned, the SL interpolation does not affect the whole RHS of (5), and consequently, the SLHD scheme cannot control directly all the source terms of the model. Generally, this has been found in experiments not to result in serious problems, since the non-interpolated part remains indirectly affected by the SLHD scheme during the evolution. The only exception arises when this non-interpolated part contains a forcing term which is constant in time, as e.g. the orographic forcing (which acts on horizontal and vertical velocity variables in the momentum equations). With the pure SLHD scheme, the spectrum slope for these variables near the surface is found to closely follow the one of the orography, indicating that the SLHD scheme does not offer an efficient mean to control the scale-selectivity at the end of the spectrum for these variables. This feature is illustrated in Figure 2 showing kinetic energy spectra at the lowest level of the model domain in adiabatic mode after 3 hours of forecast time; in the control experiment (cubic interpolator and fourth-order numerical diffusion), the energy spectrum slope is found to depart significantly from the orography spectrum slope near the largest resolved wavenumbers, thereby confirming the ability of the numerical diffusion to efficiently control the scaleselectivity. In contrast, experiments using either the I A and the I D interpolators (and no numerical diffusion) show that in this case, the spectrum slope is entirely controlled by the orography spectrum. Since the SLHD scheme is a combination of these two interpolators, it clearly does not allow a proper control of the selectivity near the end of the resolved spectrum. Various solutions to this problem may be proposed. A natural one would be to impose interpolations to the whole RHS of (5), but this would imply interpolations at the middle points of SL trajectories for the N term, thereby losing the advantage of spatial averaging of the source terms along the trajectory (Tanguay et al., 1992). Another solution could be to filter the model orography itself, in such a way as to impose a predefined slope for the tail of the resolved spectrum, but this would imply a deterioration of other fields (e.g. low-level temperatures). The solution finally used in Aladin was to combine the SLHD scheme with a weak and very selective sixthorder numerical (spectral) diffusion applied to momentum variables. The recourse to some numerical spectral diffusion is also justified by the second specific problem raised by the implementation of the SLHD, i.e. its limited control of the overall selectivity. As examined in the next section, the control of the SLHD scheme s selectivity is not straightforward. On one hand, the range of selectivity is almost entirely prescribed once the interpolators I A and I D have been chosen, and on the other hand, the selectivity varies when the scheme is tuned (by modifying the relative

7 SEMI-LAGRANGIAN ADVECTION SCHEME WITH CONTROLLED DAMPING 529 where the moisture field is not expected to be wellmixed, and where spurious diffusive sources mechanically result in erroneous vertical transfers of moisture. With the SLHD scheme, this problem is alleviated at least in conditions of weak flow deformation, since the diffusive source then almost vanishes. A positive impact of the SLHD scheme can therefore be expected in mountaneous areas of the forecast model, for moisture-related phenomena. Figure 3 shows an illustration of this improved physical realism in the predicted cloudiness in the Danube valley, for 15 December 2004, 06 UTC. The observation for this case was a generalized low-level cloud cover in the Danube valley (in the centre of the figure). In the forecast made with linear diffusion, most of the small-scale details, including valley clouds, are blindly eliminated by the spectral operator whereas, in agreement with observations, the SLHD scheme tends to better preserve these clouds since the flow is close to stagnation there. However, beside these potential advantages, a more problematic issue is that the tuning of the SLHD scheme is not straightforward, partly due to its nonlinear character, and partly due to the fact that the damping properties are mostly determined by the choice of the interpolator I D. The selectivity of the linear interpolation is very comparable with second-order diffusion (Váňa, 2003). A higher selectivity of the scheme can be achieved by further tuning of the (6) parameters, but at the price of less active diffusion. This indicates that the diffusion activity and selectivity are in tight relation, which limits the scheme s tuning potential. Being designed in a physically inspired manner, SLHD offers the possibility of being tuned independently from the model domain definition, unlike the spectral diffusion, for which the activity at higher wavenumbers is controlled only by the model resolution. However for practical reasons it is useful to increase the diffusion activity for the smallest scales seen by model to remove the accumulated noise at the end of model spectra. As a compromise solution, the tuning of the SLHD is designed to be scale-independent for the part controlled by the parameter A from (6) and (7), while the diffusion activity is increased for the tail of the model spectra by the enhancement controlled by the parameters B and D of (6) and (8). The latter enhancement part cannot be scaleindependent as it represents a feedback via the intrinsic model resolution-dependency of d values. Still this arrangement is found significantly more consistent with the requirement that horizontal diffusion should affect mostly middle and small scales of the model when compared to the response of the spectral diffusion. Figure 4 illustrates the damping responses of the kinetic energy spectra as a function of the model scale (represented by wavenumber relative to the reference domain) for grid-point SLHD diffusion and spectral diffusion. The damping response is computed for each wave as the difference between the wave energy of the control experiment (without any horizontal diffusion) and the same from the experiment with active diffusion (a) (b) (c) Figure 3. Total cloudiness (30-hour forecast) for 15 December 2004, 06 UTC, over central Europe predicted by the Aladin model using (a) spectral diffusion, and (b) the SLHD scheme. Contour shadings: 0, 2, 4, 6 and 8 octas (from black to white). (c) is the 24-hour Cloud Microphysics picture composed from three IR channels of MSG/Seviri (converted to grey scale with enhanced contrast) valid for the same time. Here the light grey, grey and black indicate low cloud (and water surfaces (orange, blue, light and dark red in online version), cloud-free land and high-cloud areas, respectively. This figure is available in colour online at normalized by the control experiment energy. If the damping response is nearly zero, the tested diffusion scheme makes almost no impact at the considered scale. The values of damping response around one relate to the situation where the wave energy from the tested scheme is not comparable to the reference scheme, and

10 532 F. VA N A ET AL. (a) (b) (c) (d) Figure 5. MSL pressure field (24-hour forecast) by Aladin/LACE from 20 July 2001, 00 UTC, with (a) spectral diffusion and (b) SLHD. The contour interval is 2.5 hpa. (c) and (d) are corresponding east west cross-sections (cutting the cyclone centre) of potential temperature (black contours with spacing 2 K) and potential vorticity (shading and white contours). The dark shading highlights the stable areas above 1.5 PVU, and the light shading represents areas with PVU < 0 with dry symmetric instability. The spacing of the white contours follows the key on the right. This figure is available in colour online at The pressure low analyzed by Aladin/LACE is around 989 hpa, while in the linear diffusion and SLHD simulations, they are 991 hpa and 992 hpa, respectively. The global model assimilation using the same SYNOP data indicated a minimum pressure of 994 hpa (not shown). In this case, one can conclude that both diffusion schemes produced very good and reliable forecasts making forecasters confident with wind predictions near the Dalmatian coast. The good results obtained with spectral diffusion were preserved with the SLHD scheme in this case Wind storms Experience has shown that the most appropriate weather events for demonstrating the skills of the proposed Copyright 2008 Royal Meteorological Society diffusion are wind storms. Being directly influenced by the wind field, the SLHD scheme is very efficient for such situations. The simulation presented here shows a storm in the lee of the Carpathian mountains, as simulated by the Aladin/Slovakia model (operational application of Aladin in the Slovak weather service) on 17 May This model, using spectral linear horizontal diffusion, forecast a strong low-level wind jet accompanied by intense gustiness in the area of southern Slovakia and northern Hungary. According to the operational forecast from 16 May, 00 UTC, the maximum predicted winds were about m s 1 with gustiness between 30 and 45 m s 1 during the whole day. However, the observed winds were in fact much weaker, the observed 10m wind hardly exceeding 10 m s 1 with a maximum measured Q. J. R. Meteorol. Soc. 134: (2008)

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