Last updated: 13 Sep 2021
To solve the adiabatic Euler equations, i.e., the balance equations for mass, momentum, and energy, a so-called dynamical core is needed. This dynamical core should have the following properties:
To achieve this goal a number of decisions have to be made. Some of them are:
Up to now the answer to these questions to achieve the above mentioned properties has led to a variety of model formulations. On the global scale the shift from spectral models to grid point models, which are considered more effective at higher resolution, has not yet been done at all the global forecasting centres. At the mesoscale, in most cases grid point models are used. Many of them use finite difference (FD) formulations, which are relatively easy to develop but are limited to structured grids. Orography can be included by a terrain-following coordinate-system and appropriate transformations. Finite element (FE) methods, originally developed for structural mechanics in engineering applications possess more freedom in using also unstructured grids but were not often used for meteorological models. One reason for that is that they inherently need implicit solvers. Currently discontinuous Galerkin (DG) methods promise a good combination of properties for accuracy and conservation and can also be applied on unstructured grids, but their application in meteorological models has just started. Finite volume (FV) methods have a long tradition in fluid mechanics especially in capturing shocks and other discontinuities by explicitly profiting from the conservative form of the equations. In principle FV-methods can be applied also to unstructured grids.
The COSMO model as it is applied in the COSMO consortium covers spatial resolutions with grid lengths from 14 km to currently 2.2 km and in the future (around the year 2015) down to about 1 km for operational weather forecasts. At these high resolutions, at least two problems arise:
First, the explicit simulation of deep moist convection by the high resolution model applications (grid length below 3 km) requires a closer coupling between the dynamical core and the parameterisations. This means that the dynamical core no longer has to describe the adiabatic equations but has to involve the diabatic terms, in particular latent heat release. Generally, this will lead to a closer collaboration between physical parameterisation and dynamics development. In the WRF model for example, the tendencies of the cloud microphysics scheme from the old time step are treated together with other physical tendencies from the current time step and in common are delivered to the dynamical core. This was up to now not possible in the COSMO model due to unstable 2 ?z patterns arising in the TKE field. Also, turbulence is a quite fast process, which therefore should closely interact with the dynamics.
Second, the transition to finer resolutions has the consequence of steeper slopes as well as more complex mountain and valley structures. This makes a terrain-following coordinate formulation more and more difficult, in particular for the COSMO model. One of the reasons for that is the current dynamical core, where the metric terms for the terrain following coordinates are treated explicitly and can therefore suffer from numerical instability. Furthermore, the metric terms must fulfil numerical constraints (metric tensor identities, Smolarkiewicz and Prusa, 2005) which are probably violated by the current model version. A possible alternative to the terrain-following coordinate would be the z-coordinate. To adequately resolve mountainous regions with large height differences between the mountain peaks and the deepest valleys, a formulation with a 3D-unstructured grid seems to be necessary if one wants to follow this approach. This is clearly beyond the scope of model development in the next few years. – In contrast, implicit (or semi-implicit) solvers are able to handle metric terms in an implicit and therefore stable manner.
In addition, direction-splitted tracer advection induces stability problems in the model. In strongly deformational flow situations a grid cell can be emptied by an advection step in one direction and therefore the specific tracer mass explodes during the compensating advection step in the other direction. This is both a problem of mass conservation and mass consistency. Such deformational flow fields occur more and more often when the complexity of the terrain increases with higher resolution. Consequently, new methods to transport additional variables will need to be developed to achieve the desired conservation properties. Traditionally, variables like moisture fields or turbulent variables (e.g., TKE) are essentially treated outside of the dynamical core. The tracer advection schemes are subdivided into two groups. Semi-Lagrange schemes (Staniforth and Côté, 1991) utilize the fact that physical quantities are transported together with the fluid parcel. They have the advantage that they can be easily formulated as full 3D schemes (which prevents from splitting errors) and have no Courant number restriction. The disadvantage of traditional semi-Lagrange schemes is the lack of conservation of the transported field. On the other hand, Eulerian methods, mainly finite volume schemes, can be easily formulated to conserve the quantity (e.g., Bott, 1989), but usually have Courant number restrictions and are harder to formulate in 3D. One exception concerning the latter point is MPDATA (Smolarkiewicz and Clark, 1986) which uses simple multidimensional upwinding and corrects the high diffusion term by the same upwinding with an artificial antidiffusive velocity. Semi-Lagrange methods formulated for volume transport can be conservative at least for 2D (horizontal) transport by the remapping procedure (Nair and Machenhauer, 2002). Finite volume schemes on the other hand can be formulated fully 3D by an appropriate flux reconstruction (Miura, 2007). Such schemes suffer less from the above mentioned splitting instability and inaccuracy.
In principle, the above mentioned limitations of the dynamical core for smaller model resolutions in complex orography can be cured by a stronger filtering of the orography. This ‘dodge’ is already used in the Alpine region for the convective-scale model applications (like COSMO-DE or COSMO-2). With an again stronger orography filtering model runs with dx=1 km over middle Europe are still possible, though this approach is unsatisfying. However, further testing is necessary to assess the true limitations in complex terrain.
Finally, the traditional distinction between tracer advection and the dynamical core has to be abandoned to get a satisfactory advection of specific masses. The advection scheme has to be consistent with the advection of total air density (Skamarock, 2006).
A transport process closely tied to advection is sedimentation of rain, snow, or graupel, with the difference that this transport does not follow the fluid parcel. Here often implicit schemes are used due to the quite high Courant numbers, particularly in the lower levels of the model.
Other transport processes like turbulent transport expressed by the divergence of diffusion fluxes falls in the area of ‘numerics and dynamics’, too (Baldauf, 2005 and 2006).
Boundary conditions (BC) for the open boundaries of a limited area model are a notoriously difficult problem. At the upper boundary at least mechanisms for damping gravity waves (whose direction of energy transport can be detrimental to the direction of the phase velocity) are necessary. Radiation conditions up to now were mainly developed for anelastic models. In compressible models the occurrence of sound waves (with a quite different dispersion relation) disturb this kind of artificial BC. In COSMO a Rayleigh damping layer is used. A new formulation (Klemp et al., 2008), which only relaxes the vertical velocity, produced reasonable results, too.
The lateral BCs of COSMO use a damping layer, too. At higher resolutions, much higher BC update frequencies than the currently used 1h (e.g. 5 min) will be needed. Though in principle not difficult, some technical adaptations to the code will be necessary.
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