Priority Project "CDC"
Conservative Dynamical Core

Last updated: 29 July 2011

Project leader: Michael Baldauf (DWD)

Introduction

The goal is to get a dynamical core with some explicit conservation properties, the most important one being mass conservation. This should be achieved with no reduction of accuracy or efficiency. A further goal (probably connected with the former) is to improve the capability of the model to work in steep terrain.

Motivation

The current COSMO-model neither with Leapfrog nor with Runge-Kutta dynamics has any explicit conservation property concerning the dynamical variables mass, momentum or energy. Conservation of these variables is one of the fundamental guiding principles in the development of dynamical cores in all branches of fluid dynamics.

Apart from this the COSMO Science Plan defines the following basic problems encountered as the spatial resolution of operational weather prediction models approaches the 1 km scale (fragment of a current draft of the COSMO Science Plan, part 5: "Dynamical core & numerics"):

The Cosmo-Model as it is applied in the COSMO Consortium will cover spatial resolutions with grid lengths from 14 to currently 2.2 km and in future (in the year about 2015) down to 1 km for operational weather forecasts. At least two problems arise:

• The explicit simulation of deep moist convection by the highly resolved model applications (grid length below 3 km) requires a closer coupling between the dynamical core and the parameterizations. This means that the dynamical core no longer has to describe solely 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 parameterizations and dynamics development.
• Second, the transition to finer resolutions has the consequence to handle with steeper slopes. This becomes more and more difficult for a terrain-following coordinate formulation. A possible alternative to the terrain-following coordinate is 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 those approaches. This means to go away from pure finite difference discretizations.

A large family of discretization schemes to achieve the goal of conservation are finite volume (FV) discretizations applied to flux form equations. These methods are well established in the Computational Fluid Dynamics community (e.g. LeVeque, 2002) and are applied more and more in global and also in regional atmospheric modelling.

Beyond the pure conservation aspect such schemes additionally describe the transport of nonlinear waves properly. The WRF model is partly formulated in flux form to conserve at least mass (Klemp, Skamarock, Dudhia (2008), Skamarock, Klemp (2008)). Also the OMEGA-model uses finite volume methods for application in operational weather forecasting and atmospheric dispersion modelling (Bacon et al. (2000)). Ideas developed there can be used for the COSMO-development.

The most elaborate FV-methods which solve the complete set of equations are based on Riemann-solvers (LeVeque, 2002). It is not clear if the ideas of time-splitting and horizontal explicit/vertical implicit discretization can be applied to these methods. But a full solution of the Riemann-problem is not always necessary. Instead a slightly simplified set of equations can be solved, where FV-discretisation is mainly used for the advection-terms as it is done in the OMEGA-model.

Alternatively a FV-discretisation of the complete compressible Euler-equation set can be combined with an efficient implicit solver as it was developed for aircraft simulations. Such a solver is available at CIRA and could be extended for atmospheric flows.

The problems discussed in the COSMO Science Plan were also successfully solved in the frame of EULAG, a computational research model for multi-scale flows, successfully applied and tested for many scales and applications. EULAG is a non-hydrostatic anelastic model employing the finite-volume non-oscillatory positive definite transport algorithm MPDATA, able to solve the flow equations both in Eulerian and Lagrangian framework.

The model equations are formulated in a generalized time-dependent curvilinear framework which allows for grid adaptivity. The model development is widely documented in the literature (for a recent review see Prusa, Smolarkiewicz and Wyszogrodzki, 2008, and references therein).

EULAG has a long record of successful applications for many flow scales an phenomena, ranging from micrometeorology and turbulence, up to global atmospheres and stellar convection. From the view point of the basic problems encountered by the development of operational numerical weather prediction models, discussed in the Science Plan and common not only for the COSMO consortium, EULAG brings a unique and well documented experience of their successful solution.

The model, practically from its beginnings, was applied to simulations of cloud microphysics and cloud dynamics, allowing –in effect- for appropriate and widely tested representation of coupling between dynamics and physics. The results are well documented in a number of publications (eg. Grabowski, 1998 and 1999, Grabowski and Smolarkiewicz 2002). The mathematically thorough and well planned structure of the model’s conservative dynamical core allows also to handle very steep slopes, which was clearly demonstrated, among others, by the experiment simulating flow around the Pentagon building (Smolarkiewicz et al. 2007).

Additionally, the practical tests show that the model scales well on the massively parallel architectures and is capable of running on many computer platforms (Prusa, Smolarkiewicz and Wyszogrodzki, 2008).

Project tasks

The main focus lies on mass conservation, because this is the most robust conserved quantity (Thuburn, 2006). Next important seems energy conservation. Instead of local energy conservation a Lagrangian conservation of potential temperature could be an intermediate second step. Conservation of momentum seems not as important for larger scale models, but becomes increasingly meaningful for local and convection resolving modelling and could possibly be achieved during this project, too.

The goal of the project is to deliver a dynamical core mainly based on finite volume discretizations with the above mentioned conservation properties. The accuracy should be at least as good as the current Runge-Kutta-dynamics, whereas the efficiency should not be worse.

To this purpose two main development directions will be followed. One is based on the anelastic EULAG model (task 1), and the other one will develop a FV model based on the full compressible equations (task 2). Both branches will be connected by common developments (e.g. common advection schemes), the sharing of knowledge and experience in FV methods, and the use of common tools for evaluation (task 3).

In task 1 it is proposed to test the EULAG dynamical core as a prospective dynamical core of a future operational weather prediction model suitable for kilometer- and sub-kilometer resolutions. The aim of the task is to demonstrate that the EULAG itself, and the basic ideas defining the EULAG construction, can be used as a robust and credible base for construction of future operational weather models, and especially the COSMO model.

The successful demonstration of such a capability of the EULAG dynamical core would be of significant importance not only for the COSMO consortium, but more generally, for the whole mesoscale NWP community, as the problems discussed above are of a general, common interest for the community. The results of the whole project will, therefore, form an important input of the COSMO consortium to the NWP community knowledge.

In practice, task 1 will focus on extensive testing of the EULAG dynamical core as applied for mesoscale flows over realistic steep topography, involving also an explicit treatment of cloud processes on the model grid, with model resolutions ranging from 2.2 km to 0.5 and 0.25 km.

At this stage of the task, it is not feasible to construct a surrogate of a complete, well tuned EULAG-based NWP model involving interactive application of sophisticated parameterizations. Rather, a semi-operational model setup is planned. At first, it will use stand-alone forcing extracted from independent COSMO runs and, later, an inclusion of simplified parameterizations of basic subgrid processes working interactively with the EULAG core.

Because of the substantial difference in programmatic bases of COSMO and EULAG models (viz. weather prediction versus research in atmospheric fluid dynamics) it may be counterproductive (at the initial stage of the task) to define benchmarks emphasizing the operational NWP scores. Furthermore, the EULAG tests are planned for spatial resolutions beyond current COSMO capabilities. It is assumed, therefore, that a number of idealized and semi-idealized experiments involving flows over topography will be performed in order to compare the model results and to quantify their differences and, where possible, compare them with other field or numerical experiments.

Such tests will complement the basic experiments involving realistic flows over Alpine topography, with the semi-operational setup of EULAG. The results will be compared with observations, including radar and satellite data, in consultation with colleagues of MeteoSwiss, which will also provide necessary data for the experiments (topography, results of COSMO runs, observational data, etc).

TASK 1: The anelastic (EULAG) approach

Testing the EULAG dynamical core as a prospective dynamical core of a future operational weather prediction COSMO model for very high horizontal resolutions. These are the sub-tasks of task 1:

• Task 1.1.: Idealised tests of the EULAG dynamical core
• Task 1.2.: Tests of the EULAG dynamical core for realistic flows over the Alpine topography
• Task 1.3.: Tests of the EULAG dynamical core for realistic flows over the Alpine topography with simplified physics parameterisation
• Task 1.4.: Choice of the anelastic equation system
• Task 1.5.: Decision about continuation of task 1
• Task 1.6.: Coupling of EULAG dyn. core with COSMO via an interface
• Task 1.7.: Technical testing with COSMO by idealised cases
• Task 1.8.: Real case simulations with the preliminary coupled version
• Task 1.9.: Idealized and Real case simulations with a vertically extended domain

TASK 2: The compressible approach

These are the sub-tasks of task 2:

• Task 2.1.: Metric tensor identities
• Task 2.2.: Complete FV-solver for the EULER equations
• Task 2.3.: Fully 3D, i.e. non-direction splitted, conservative advection scheme
• Task 2.4: Decision about continuation of task 2.2
• Task 2.5: Testing the dual time stepping in COSMO
• Task 2.6: Implement the Finite Volume solver into COSMO
• Task 2.7: Perform realistic test cases

TASK 3: Assessment of dynamical cores

These are the sub-tasks of task 3:

• Task 3.1.: Maintenance of idealized test cases
• Task 3.2.: Collection and maintenance of semi-idealized test cases
• Task 3.3.: Decision tree for Steering Committee
• Task 3.4: Verification of the whole model

Links to other projects or work packages

The cooperation with the working group on physical parameterizations will be needed in 2010. Issues related to the treatment of diffusion of scalars (T, ρ, TKE, < Θ'² > ,...), and implicit diffusion.

The implication of using different prognostic variables (i.e. others than p, T) for the data assimilation (nudging scheme) should be assessed.

References