General Description: Model Dynamics and Numerics

see also: basic design | initial and boundary conditions | physical parameterizations | external parameters | code and parallelization | data assimilation
Last updated: September 2011

The COSMO-Model is based on the primitive hydro-thermodynamical equations describing compressible non-hydrostatic flow in a moist atmosphere without any scale approximations. A basic state is subtracted from the equations to reduce numerical errors associated with the calculation of the pressure gradient force in case of sloping coordinate surfaces. The basic state represents a time-independent dry atmosphere at rest which is prescribed to be horizontally homogeneous, vertically stratified and in hydrostatic balance.

The basic equations are written in advection form and the continuity equation is replaced by a prognostic equation for the perturbation pressure (i.e. the deviation of pressure from the reference state). The model equations are solved numerically using the traditional finite difference method. In the following we summarize the dynamical and numerical key features of the COSMO-Model.

Model Equations
  • nonhydrostatic, full compressible hydro-thermodynamical equations in advection form
  • subtraction of a hydrostatic basic state (reference atmosphere) at rest. Options for
    • a reference atmosphere defined with a constant Δt, with an increasingly negative vertical temperature gradient in the stratosphere and a limit to the vertical model extent.
    • a reference atmosphere based on a temperature profile
      T0(z) = T00 + Δt EXP(-z/h_scal),
      approaching an isothermal profile in the stratosphere and no limits of the vertical model extent.
Prognostic Variables
  • horizontal and vertical cartesian wind components (u,v,w)
  • temperature (t)
  • pressure perturbation (p', deviation from the reference state)
  • specific humidity (q_v) and specific cloud water content (q_c)
  • optionally: cloud ice content (q_i), specific water content of rain (q_r), snow (q_s) and graupel (q_g)
  • optionally: turbulent kinetic energy (tke)
Diagnostic Variables
  • Total air density
  • 2 meter temperature
  • 10 meter wind speeds
  • maximal wind gust in 10 meter
  • precipitation fluxes of rain and snow
  • and much more ....
Coordinate System
  • rotated geographical (lat/lon) coordinate system horizontally
  • generalized terrain-following height-coordinate with user defined grid stretching in the vertical. Options for
    • base-state pressure based height coordinate
    • Gal-Chen height coordinate and
    • exponential height coordinate (SLEVE) according to Schär et al. (2002)
Grid Structure
  • Arakawa C-grid, Lorenz vertical grid staggering
Spatial Discretization
  • Second-order finite differences. For the two time-level scheme also 1st and 3rd to 6th order horizontal advection (default: 5th order)
  • Option for explicit higher order vertical advection
Time Integration Default scheme:
  • 2nd and 3rd order two time-level Runge-Kutta split-explicit scheme (Wicker and Skamarock, 2002)
  • TVD (Total Variation Diminishing) variant of a 3rd order two time-level Runge-Kutta split-explicit scheme
Additional Options:
  • Second-order leapfrog HE-VI (horizontally explicit, vertically implicit) time-split integration scheme, including extensions proposed by Skamarock and Klemp (1992)
  • Option for a three time-level 3D semi-implicit scheme (Thomas et al., 2000)
Numerical Smoothing
  • 4th-order linear horizontal diffusion with option for a monotonic version including an orographic limiter
  • Rayleigh damping in upper layers
  • 2D divergence damping and off-centering in the vertical in split time steps