1. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka PALM – model equations Palm-Seminar Zingst July 2004 Micha Gryschka Institut für Meteorologie und Klimatologie Universität Hannover

2. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Structure Basic equations Boussinesq-approximation and filtering poisson equation for pressure Prandtl-layer how Cloud physics is imbedded

3. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Symbols velocity components spatial coordinates potential temperature passive scalar actual temperature pressure density geopotential height Coriolis parameter alternating symbol molecular diffusivity sources or sinks

4. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Basic equations 1.Navier-stokes equations 3.continuity equation 2.First principle of thermodynamics and equation for any passive scalar ψ Variety of solutions some solutions are not importent for meteorological questions some solutions cost a lot computer power (f.e. sonic waves decrease the timestep) meteorological meaningfull simplifications

5. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Boussinesq-approximation (I)

6. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka in the horizontal components: terms with * are negligible in the vertikal component: term g * / 0 is not neglibible replacing Boussinesq-approximation (II) in case of shallow convection! incrompressible (divergence free) flow (no solution for acustic waves)

7. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Are the equations directly solveable? Numerical solving of the equations implies discrete solving on a grid If the grid is small enough, the equations could be discretized directly In LES the grid is not small enough Equations have to be filtered: –large structures, resolved from the grid (and timestep) –small structures, unresolved from the grid (and timestep)

8. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Filtering the equations (I) Splitting the variables into mean part ( ¯ ) and deviation ( ) By filtering, a turbulent diffusion term comes into being subgrid-scale (SGS) stress tensor

9. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka Filtering the equations (II) The SGS stress tensor is splitted into an isotropic and an anisotropic part: anisotropic SGS stress tensor

10. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka The filtered equations 1.Boussinesq-approximated Reynolds equations for incompressible flows 3.continuity equation for incompressible flows 2.First principle of thermodynamics and equation for any passive scalar : has to be parametrized

11. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka 1.Calculating a preliminary velocity field without considering the pressure term 2.Solving the poisson equation 3.Correcting the velocitiy field with considering the pressure term The Poisson-equation (I) This strategy connects the continuity equation with the motion equations, so it's guaranteed that the flow is divergence free. Solving the poisson equation is one of the most costs of computer power! After filtering and parametrizing, only the pressure is unknown. Solution: considering the continuity equation

12. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka The Poisson-equation (II) origin of the poisson equation:

13. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka The Prandtl-layer (I) Φ m and Φ h : Dyer-Businger functions friction velocity characteristic temperature in the Prandtl-layer between ground and first grid layer

14. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka The Prandtl-layer (II) Richardson flux number Dyer-Businger functions for momentum and heat stable stratification neutral stratification unstable stratification stable stratification neutral stratification unstable stratification

15. PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie, Palm-Seminar Zingst July 2004, Micha Gryschka how Cloud Physics is imbedded prognostic equation for u i connection of dynamics and cloud physics via temperature v ( q v, q l, l ) prognostic equation for l sources and sinks: ( t l ) rad, ( t l ) prec prognostic equation for q sources and sinks: ( t q) prec q v = q - q l cloud physics model