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PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut.

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Presentation on theme: "PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut."— Presentation transcript:

1 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht PALM – model equations Sonja Weinbrecht Institut für Meteorologie und Klimatologie Universität Hannover

2 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Structure Basic equations Boussinesq-approximation and filtering SGS-parameterization Prandtl-layer Cloud physics Boundary conditions

3 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Basic equations 1.Navier-stokes equations 3.continuity equation 2.First principle of thermodynamics and equation for any passive scalar ψ

4 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Basic equations (in flux-form for incompressible flows) 1.Navier-stokes equations 3.continuity equation 2.First principle of thermodynamics and equation for any passive scalar ψ

5 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Symbols velocity components spatial coordinates potential temperature passive scalar actual temperature pressure density geopotential height Coriolis parameter alternating symbol molecular diffusivity sources or sinks

6 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Pressure p, density ρ, and temperature T are split into a basic part () 0, which only depends on height (except from p), and a deviation from it ()*, which is small compared with the basic part The basic pressure p 0 fulfills the equations shown on the right. Boussinesq-approximation (I)

7 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Neglecting the density variations in all terms except from the buoyancy term Density variations are replaced by potential temperature variations Boussinesq-approximation (II)

8 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Filtering the equations (I) Splitting the variables into mean part ( ¯ ) and deviation ( ) By filtering, a turbulent diffusion term comes into being compared with the turbulent diffusion term the molecular diffusion term can be neglected subgrid-scale (SGS) stress tensor

9 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht 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 Sonja Weinbrecht 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

11 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht The filtered equations (flux form) 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

12 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht The parameterization model (I) (Deardorff, 1980) strain rate tensor turbulent kinetic energy anisotropic SGS stress tensor eddy viscosity for momentum

13 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht The parameterization model (II) (Deardorff, 1980) Mixing length Characteristic grid spacing Wall adjustment factor Eddy viscosity for heat

14 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht The parameterization model (III) Prognostic equation for the turbulent kinetic energy has to be solved:

15 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht The Poisson-equation The pressure π is computed to balance out velocity divergences (which could arise during the numerical calculation:

16 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht The Prandtl-layer (I) velocity- and temperature gradients in the Prandl-layer Φ m and Φ h are the Dyer-Businger functions for momentum and heat friction velocity characteristic temperature in the Prandtl-layer

17 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht 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

18 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics (I) Suppositions: liquid water content and water vapor are in thermodynamic equilibrium All thermodynamic processes are reversible Potential liquid water temperature θ l and total water content q as prognostic variables For moist adiabatic processes, θ l and q are conserved. Condensation and evaporation do not have to be explicitly described Only totally saturated or totally unsaturated grid cells are allowed in the model

19 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics - Symbols potential liquid water temperature total water content specific humidity, s.h. for saturated air liquid water content latent heat / vaporization enthalpy virtual potential temperature pressure (reference value) adiabatic coefficient actual liquid water temperature saturation vapour pressure

20 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics (II) Filtered equations of θ l and q (molecular diffusion term neglected) Equations of θ l and q:

21 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics (III) only totally saturated or totally unsaturated grid cells are allowed in the model the specific humidity for saturated air q s is computed as follows:

22 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics – SGS-Parameterization The buoyancy-term in the TKE-equation is modified (the potential temperature θ is replaced by the virtual potential temperature θ v ): Parameterisation of W i and H i :

23 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics – SGS-Parameterization (II) H v,3 : subgrid-scale vertical flux of virtual potential temperature (buoyancy flux) K 1, K 2 : Coefficients for unsaturated moist air and saturated moist air respectively

24 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics (IV) Prognosticating θ l and q, the virtual potential temperature θ v and the quotient of potential and actual temperature θ/T have to be computed as follows:

25 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics – Radiation Model based on the work of Cox (1976) vertical gradients of radiant flux as sources of energy the downward radiation at the top of the model is prescribed

26 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics – Radiation Model - Symbols upward and downward radiant fluxes black body radiation cloud emissivities between z 1 and z 2 Liquid Water Path mass exchange coefficient (empirical data)

27 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Cloud Physics – Precipitation Model Kessler scheme (Kessler, 1969) only autoconversion (production of rain by coalescence) is considered precipitation starts when a threshold value q lt is exceeded τ is a retarding time constant

28 PALM model equations Universität Hannover Institut für Meteorologie und Klimatologie Sonja Weinbrecht Boundary conditions


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