1. Introduction
The main feature of the Mesoscale Compressible Community (MC2) model is its semi-implicit semi-Lagrangian (SISL) numerical time integration scheme developed by the late Andre Robert (Robert 1969; Kwizak and Robert 1971; Robert et al. 1972 for the SI part and Robert 1981, 1982; Robert et al. 1985; Tanguay et al. 1990 for the SL part). It includes spatial averaging of trajectory-endpoint values of the nonlinear terms evaluated at central time in addition to an off-centering of all space–time trajectory averages for the linear terms (Tanguay et al. 1992), both elements being used to reduce gravity wave noise believed connected mostly with stationary forcing, in particular but not exclusively orographic forcing, and occurring at large Courant numbers. The model behavior with respect to orographic forcing was extensively studied by Pinty et al. (1995), showing that SISL gave good results without using off-centering on all kinds of stationary nonlinear as well as linear orographic forcing provided appropriate time resolutions were used. The consequences of using large time steps, with and without off centering, were examined by Héreil and Laprise (1996).
The first comprehensive description of the model in terrain-following coordinates was given in Laprise et al. (1997). The model was later specialized to two major types of application: mesoscale forecasting (Benoit et al. 1997) and regional climate simulations (Caya and Laprise 1999). Substantial modifications were brought about in the forecast version by Thomas et al. (1997) for efficiency on various computer architectures and by Thomas et al. (1998, hereafter T98) to further improve (i) the efficiency of the SL scheme, (ii) the stability of the SI scheme over steep terrain, and (iii) the open lateral boundary conditions. In the fall of 1999, the model was used to provide daily forecasts across the European Alps at 3-km resolution during the entire special observation period (SOP; 7 September–15 November 1999) of the Mesoscale Alpine Programme (MAP). An overview of its performance during MAP is given in Benoit et al. (2002a).
In spite of the relatively good performance of the model on real as well as canonical cases, a problem handling finescale topography was becoming increasingly evident. From the beginning (Denis 1990), the model had been showing rather high sensitivity to orographic forcing. In consequence, topography used by the model was smoothed more than what is customarily done. A partial solution was to use a modified vertical coordinate as proposed by Schär et al. (2002), hereafter SLFLG). Another one was to use a modified basic state (Benoit et al. 2002b). A major step in finding a true solution to the problem was achieved when it was realized that not every model presented that much sensitivity. And recently its cause was diagnosed by Klemp et al. (2003), hereafter KSF) to be a numerical inconsistency between certain terms involving SL advection and other parts of the code.
In section 4, we unveil what we feel is the simple but true solution to this numerical inconsistency problem, and we show the obvious and immediate impacts of the correction. But first we give an up-to-date account of the model developments mentioned above, all of them mostly motivated by this search for a solution to the high spurious sensitivity of the model to finescale topography. In section 2, we review the model equations and describe the transformations of the SISL scheme to admit nonisothermal basic states and of the coordinate system to allow for more general oblique vertical coordinates. In section 3, we show how the concept of piecewise-constant finite elements is used to formalize spatial discretization and thus reduce code arbitrariness. We conclude in section 5.
2. Model equations
a. Basic features
In addition to the SISL scheme, the model has a number of other worthy characteristics. The first characteristic is a thermodynamic simplification. In effect, the equations of motion usually involved three main thermodynamic variables—temperature, pressure, and density—related diagnostically by an equation of state. Completely eliminating one of these variables simplifies the dynamic system while restraining the freedom of choice for the final set of equations. Here, density is eliminated with advantage.
The second characteristic involves a change of thermodynamic variables. With the equations written in terms of deviations, linear and nonlinear terms are easily distinguished. Previously, the basic state from which deviations could be obtained was specialized to the isothermal case and the new variables were simply taken as the perturbations of temperature and pressure. With general hydrostatic basic states, we find that the SISL scheme is more conveniently implemented using a generalization of the perturbation variables, a meaningful one. The variables namely become buoyancy and generalized pressure and the fundamental relevant parameters, the speed of sound and buoyancy frequency, are emphasized.
In the “horizontal,” that is, in the direction perpendicular to gravity, the equations are written in a so-called invariant form (Robert et al. 1972) readily admitting, with the specification of a single scaling parameter S, a choice of orthogonal coordinates: Cartesian (S = 1), spherical, and even cylindrical (rotating annulus experiments) as well as all conformal mappings (stereographic, Mercator, . . .) of the spherical earth. This feature of the model is, however, irrelevant for the problem at hand. It will not be further discussed. The equations will in fact be presented in absolute Cartesian coordinates (omitting the Coriolis force).
In the “vertical,” a terrain-following oblique coordinate of the height variety is used. The metric has now been generalized to greatly increase the flexibility in the choice of vertical coordinate definition. In particular, we may use the smooth level vertical (SLEVE) coordinate of SLFLG.
Space discretization is done using finite differences with variables distributed on a set of staggered grids particularly well suited for deriving the elliptic-type numerical equation that characterizes the SI scheme. Since the scheme is absolutely stable in principle, a balance can easily be achieved between space and time truncation errors by adjusting the time step. But even after the grid structure and local order of accuracy of finite differencing have been decided, there still remains some degree of arbitrariness for the final code. Such arbitrariness has now practically been eliminated through the application of the concept of piecewise-constant finite elements. The method can be used with advantage to construct conservative schemes on staggered grids in particular (see Laprise and Girard 1990 for further details). We cannot however claim that the resulting code is anything other than a particular set of at best second-order accurate, finite-difference equations.
Horizontal resolution is taken to be uniform in the chosen coordinate (e.g., in spherical coordinates it is uniform in latitude and longitude while in Mercator projection it is uniform in map coordinates). Vertical resolution may be varied at will using a stretching transformation (Laprise et al. 1997). It is then again uniform in the final indicial coordinate k, though this last transformation will be omitted from the following description.
b. Basic dynamic equations
c. Change of thermodynamic variables and separation of terms
It is always possible to subtract from the above equations a hydrostatic basic state. The step of identifying and separating advection and linear terms from the rest of nonlinear and source terms is a necessary one for the application of a SISL scheme. Here, we show how the scheme has been modified to allow for nonisothermal basic states leading to a set of terms linked with variable coefficients instead of constant ones. We found that redefining the thermodynamic variables greatly helped in clarifying the end result.
d. Generalized vertical coordinate
3. Space–time discretization
a. Semi-implicit semi-Lagrangian time discretization
The option of nonisothermal basic states has contributed (Benoit et al. 2002b) to the development of a satisfactory solution to the problem at hand. It has however its own drawbacks; numerical stability may be compromised (Simmons et al. 1978) for example. And, since it has little relevance to the main result of this paper, we will not discuss it further. All simulations presented here having been done using an isothermal basic state with T∗ = 273 K.
b. Staggered space discretization
The placement of grid points where dependent variables are defined is an important element of spatial discretization. In the horizontal, we use an Arakawa C type grid. In the vertical, we use a variant of the Charney–Phillips (1953), Robert (1966), vertical grid displayed in Fig. 1. Basically, N layers are considered. At their center we locate u, υ, and P (full levels). At the N − 1 interfaces (half levels) we locate B and w. This accounts for the 5N − 2 clearly internal variables and associated prognostic equations. Top and bottom boundary conditions on w account for another couple of variables and equations sufficient for closing the system when thermally isolated. In general though extra boundary conditions on B are required, and often this cannot be done directly. Here fully prognostic equations are introduced for B in the half layers touching the boundaries and possibly involving boundary-induced turbulent fluxes.
We find the dependent variables given at the following internal grid points: [u]i−1/2jk, [υ]ij−1/2k, [w]ijk−1/2, [B]ijk−1/2, [P]ijk. In the horizontal, variables having half-integer subscripts fall on frontiers of the domain in the direction concerned. Values of variables on these boundary points are given by boundary conditions. Details on how these are applied numerically are given in T98. At the top and bottom, vertical motion w is obtained diagnostically setting W = 0 in (13). At the top, this also means that wN+1/2 = 0 and therefore dw/dt = 0; PN+1/2 may then be obtained hydrostatically using BN+1/4. At the surface though, dw/dt does not generally vanish. But w1/2 is known diagnostically and therefore also implicitly dw/dt. This means that P1/2 may, indeed must, be obtained nonhydrostatically using dw/dt as diagnostically defined; P1/2 and PN+1/2 are required because extra thermodynamic equations are solved for B3/4 and BN+1/4 (and these are more accurately defined when interpreted to be located halfway in between).
c. Piecewise-constant finite elements
d. Projection operator
e. Discretization in Cartesian z coordinates
f. Discretization in oblique Z coordinates
4. Problem and solution
a. Discovery of the problem
As mentioned in the introduction, the high sensitivity of MC2 to finescale orographic forcing was a fairly well known feature, hence the tendency to rather heavily smooth the topography used by the model. This could not be better illustrated than in Benoit et al. (2002b) where the topography of the MC2 at 3-km resolution, as prepared for and used during the MAP field phase, did not seem more detailed than that of the then operational Swiss model at 14 km. Thus use of smooth topography combined with the introduction of a sponge layer at high levels, sometimes also combined with augmented horizontal diffusion coefficient on momentum, all contributed to diminish the sensitivity and hide the problem.
The idealized case with N∗ = 0.01 is generating perturbation quantities at high levels, which are rather large even for real situations. This suggested the possible benefits of introducing the option of nonisothermal basic states. This feature was indeed implemented (Benoit et al. 2002b) and essentially leads to perfect simulations when mean and basic states nearly coincide.
For real-case simulations, SLFLG showed greatly reduced numerical noise using the new SLEVE coordinate. Benoit et al. (2002b) were also able to reduce noise on real cases using nonisothermal basic states.
b. Nature of the problem
The presence of noise in models using oblique terrain-following coordinates is quite common, and causes for it may be varied. Here we are fortunate to have a canonical experiment emphasizing one specific problem. In fact, following the work of SLFLG, KSF succeeded in clearly relating the problem to a numerical inconsistency between the SL advection scheme and other parts of the code. They showed in particular that the problem was essentially resolved if less accurate quadratic interpolations, instead of the usual cubic ones, were used to obtain upstream values in just one of the equations, namely the thermodynamic equation. And we are able to verify that the problem vanishes when we replace the SL scheme by an Eulerian scheme (Fig. 2a), a result that we use here as our control. If any doubt persists, there is a classical test for numerical inconsistency that can be made, the convergence test: decrease the time step and see how the solution evolves. In fact, the time step chosen here is adequate for an Eulerian scheme but is already relatively small from the point of view of an SL scheme, and decreasing the time step by a factor of 20 (Fig. 2b) does not improve the SL solution. Hence the model is numerically inconsistent.
c. The solution
d. Impact of the solution
The expected impact of removing this numerical inconsistency on real 3D simulations is obviously a reduction in the level of noise associated with finescale orography. To document this, we show various fields resulting from three 6-h integrations made with the model over a region covering the Alps. The case is chosen from the MAP SOP and corresponds to the intensive observing period 2b (IOP-2b) event with initial time 1200 UTC 19 September 1999. The 2-km horizontal resolution used here is slightly higher than the 3-km resolution used during the MAP field phase (Benoit et al. 2002a). The horizontal grid contains 400 × 400 points and there are 44 levels. The basic state is kept isothermal with T∗ = 273 K. The physical effects package is essentially the same as that used then. This time, however, absolutely no smoothing is applied to the topography field, and the model is integrated without any horizontal diffusion and without a sponge layer aloft. For the first integration, gchen, the old method of calculating W and original Gal-Chen vertical coordinate are used. In the second integration, sleve, the SLEVE coordinate (sS1 = 10 km, sS2 = 6 km) is used. Finally the third integration, slagw, uses both the SLEVE coordinate and the new method of calculating W.
There is little need to emphasize the location and magnitude of the original noise problem (Figs. 3 and 4) as well as the impact of changing methods for calculating W and, to a smaller but significant degree, the advantage of using the SLEVE coordinate. As might be suspected the noise is greatly reduced on other fields as well. Shown in Fig. 5 are the spectra of horizontal kinetic energy and temperature variance at 5, 10, and 15 km, calculated from a reduced grid of 201 × 201 points taken from the middle of the domain using the technique described in Denis et al. (2002). We observe numerical noise reduction in every field and at all levels, in the smaller scales of course, but these include wavelengths even longer than 8–10 Δx. Not surprisingly, the precipitation field (not shown) is also affected.
e. Comparison with reality
We use the vertical velocity measured during the Merlin IV flight 26 of 19 September 1999 of the MAP field project to compare the model calculations with reality. The flight track, an Alpine transect over the upper Rhine valley, is presented in Fig. 6, with time labels, overlaid with the slagw solution for w. We focus on the portion of the flight that is level at 4880 m above sea level (flight level 160) and take the model solution at 5000 m as is, without height adjustment. Since the “raw” in situ values from the Merlin have a sampling rate of 1 Hz, which corresponds to a travel distance of ∼100 m, we apply a moving average of the N points surrounding the current data point, with N = 21; in that way, the model values, on a 2-km grid, can be better compared with the measurements. The model fields are cubically interpolated to the instantaneous 1-Hz aircraft positions, on which they form smoothly varying curves.
In Fig. 7, we compare individually the slagw and galchen solutions with the smoothed in situ w measurements. Visually from Fig. 7, it is clear that the slagw solution is quite closer to the measurements than the galchen solution. Quantitatively, the correlation coefficients, for the flight-time period 1645–1710, are 0.26 and 0.02 respectively; as a scale of reference, the correlation coefficient between the “raw” and smoothed measurements is 0.85, and that between the two model solutions is 0.20.
Although still far from ideal, the solution presented here for the w problem in MC2 constitutes a major improvement in accuracy.
5. Summary and conclusions
SISL schemes are very efficient schemes for the numerical integration of the Euler equations of motion of the atmosphere. In principle these schemes are absolutely stable such that accuracy becomes the sole factor in determining time steps. In terrain-following coordinates though, high and steep topography has remained a challenge for both the SI and SL components. But after the work of Héreil and Laprise (1996) and T98, MC2 appeared to be well prepared for the task of high-resolution forecasting over steep mountains like the Alps.
Although sensitivity to finescale topography had been observed, it was not thought to be a serious problem. The work by SLFLG, to which we participated, was initially seen as a valuable contribution to finescale noise reduction. This remains true: the computational cost of running with the SLEVE coordinate is negligible. Furthermore the equations in the generalized coordinate, described in section 2, look simpler, and the code itself, described in section 3, is definitely simpler than the precedent (T98), especially after application of the concept of piecewise-constant finite elements.
The idealized quasi-linear case studied by SLFLG had another particularity besides the finescale structure of the relief, namely a mean state quite far from the typical isothermal basic state used in the model. A choice of basic state closer to the actual one was bound to reduce numerical errors, in a similar manner that SLEVE reduces the adverse influence of finescale terrain features on model variables aloft. This motivated us to modify the SISL scheme accordingly. But as mentioned in section 3, we did not fully implement this option of using a nonisothermal basic state for three reasons. First, as we now know, the real problem is elsewhere; second, as is well known (Simmons et al. 1978), stability of the SI scheme may then be compromised; third, after Bénard’s (2003) evaluation of our SI scheme showing that in the absence of both a time filter and off-centering the scheme was actually absolutely unstable (and in fact without off-centering the model is known to blow up sometimes), we have developed (Girard and Desgagné 2004) a more stable SI scheme using what amounts essentially to a locally defined time-dependent basic state, a more accurate scheme that does not require off-centering and will be the subject of a separate study.
The true nature—numerical inconsistency—of a noise problem has now been revealed and a solution has been found. The results presented in section 4 lead us to conclude that absolute height, although a known fixed quantity in height-based coordinates, must nevertheless be transported in a Lagrangian manner to obtain a departure height consistent with departure values of the other prognostic variables.
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A variant of the Charney–Phillips–Robert vertical discretization grid.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A variant of the Charney–Phillips–Robert vertical discretization grid.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A variant of the Charney–Phillips–Robert vertical discretization grid.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Simulations of idealized flow past topography: (a) Eulerian scheme with realistic time step (Co = 0.12), (b) inconsistent semi-Lagrangian scheme with very small time step (Co = 0.006), (c) inconsistent semi-Lagrangian scheme with Courant number equal to one (Co = 1.0), and (d) consistent semi-Lagrangian scheme with realistic time step (Co = 0.12).
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Simulations of idealized flow past topography: (a) Eulerian scheme with realistic time step (Co = 0.12), (b) inconsistent semi-Lagrangian scheme with very small time step (Co = 0.006), (c) inconsistent semi-Lagrangian scheme with Courant number equal to one (Co = 1.0), and (d) consistent semi-Lagrangian scheme with realistic time step (Co = 0.12).
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Simulations of idealized flow past topography: (a) Eulerian scheme with realistic time step (Co = 0.12), (b) inconsistent semi-Lagrangian scheme with very small time step (Co = 0.006), (c) inconsistent semi-Lagrangian scheme with Courant number equal to one (Co = 1.0), and (d) consistent semi-Lagrangian scheme with realistic time step (Co = 0.12).
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
(a) Topography field, contours every 250 m; (b) gchen, inconsistent SL scheme in original terrain-following coordinate; (c) sleve, inconsistent SL scheme in SLEVE coordinate; and (d) slagw, consistent SL scheme in SLEVE coordinate. (b)–(d) show a 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km, vertical motion field at 10 km, and contours every 50 cm s−1.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
(a) Topography field, contours every 250 m; (b) gchen, inconsistent SL scheme in original terrain-following coordinate; (c) sleve, inconsistent SL scheme in SLEVE coordinate; and (d) slagw, consistent SL scheme in SLEVE coordinate. (b)–(d) show a 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km, vertical motion field at 10 km, and contours every 50 cm s−1.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
(a) Topography field, contours every 250 m; (b) gchen, inconsistent SL scheme in original terrain-following coordinate; (c) sleve, inconsistent SL scheme in SLEVE coordinate; and (d) slagw, consistent SL scheme in SLEVE coordinate. (b)–(d) show a 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km, vertical motion field at 10 km, and contours every 50 cm s−1.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km: (a), (c), (e) vertical motion fields at 5 km over a subdomain, and (b), (d), (f) 15-km-high cross sections along arrow, with contours every m s−1. (a), (b) gchen: inconsistent semi-Lagrangian scheme in original coordinate; (c), (d) sleve: inconsistent semi-Lagrangian scheme in SLEVE coordinate; and (e), (f) slagw: consistent semi-Lagrangian scheme in SLEVE coordinate.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km: (a), (c), (e) vertical motion fields at 5 km over a subdomain, and (b), (d), (f) 15-km-high cross sections along arrow, with contours every m s−1. (a), (b) gchen: inconsistent semi-Lagrangian scheme in original coordinate; (c), (d) sleve: inconsistent semi-Lagrangian scheme in SLEVE coordinate; and (e), (f) slagw: consistent semi-Lagrangian scheme in SLEVE coordinate.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km: (a), (c), (e) vertical motion fields at 5 km over a subdomain, and (b), (d), (f) 15-km-high cross sections along arrow, with contours every m s−1. (a), (b) gchen: inconsistent semi-Lagrangian scheme in original coordinate; (c), (d) sleve: inconsistent semi-Lagrangian scheme in SLEVE coordinate; and (e), (f) slagw: consistent semi-Lagrangian scheme in SLEVE coordinate.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km: Spectra of (left) horizontal kinetic energy and (right) temperature variance at three levels in the atmosphere (5, 10, and 15 km). The spectra were calculated from reduced grids of 201 × 201 points taken in the middle of the domain.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km: Spectra of (left) horizontal kinetic energy and (right) temperature variance at three levels in the atmosphere (5, 10, and 15 km). The spectra were calculated from reduced grids of 201 × 201 points taken in the middle of the domain.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
A 6-h forecast valid 1800 UTC 19 Sep 1999, made with MC2 at a resolution of 2 km: Spectra of (left) horizontal kinetic energy and (right) temperature variance at three levels in the atmosphere (5, 10, and 15 km). The spectra were calculated from reduced grids of 201 × 201 points taken in the middle of the domain.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Merlin IV flight 26 track segment (flight level: 160–4880 m above sea level), with time labels (hh:mm multiplied by 10, e.g., 171.00 = 17:10 = 17h 10min) every other minute overlaid on the MC2 vertical velocity at 5000 m above sea level (slagw run). Gray shade interval is 1 m s−1.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Merlin IV flight 26 track segment (flight level: 160–4880 m above sea level), with time labels (hh:mm multiplied by 10, e.g., 171.00 = 17:10 = 17h 10min) every other minute overlaid on the MC2 vertical velocity at 5000 m above sea level (slagw run). Gray shade interval is 1 m s−1.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Merlin IV flight 26 track segment (flight level: 160–4880 m above sea level), with time labels (hh:mm multiplied by 10, e.g., 171.00 = 17:10 = 17h 10min) every other minute overlaid on the MC2 vertical velocity at 5000 m above sea level (slagw run). Gray shade interval is 1 m s−1.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Comparison between Merlin IV averaged in situ vertical velocity measurements (gray line) and MC2 results interpolated to the flight track (heavy lines): (a) slagw run and (b) galchen run.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Comparison between Merlin IV averaged in situ vertical velocity measurements (gray line) and MC2 results interpolated to the flight track (heavy lines): (a) slagw run and (b) galchen run.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
Comparison between Merlin IV averaged in situ vertical velocity measurements (gray line) and MC2 results interpolated to the flight track (heavy lines): (a) slagw run and (b) galchen run.
Citation: Monthly Weather Review 133, 6; 10.1175/MWR2931.1
