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**Institut für Meteorologie und Klimatologie Universität Hannover**

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

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**Structure Basic equations Boussinesq-approximation and filtering**

SGS-parameterization Prandtl-layer Cloud physics Boundary conditions

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**Basic equations Navier-stokes equations**

First principle of thermodynamics and equation for any passive scalar ψ continuity equation

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**Basic equations (in “flux-form“ for incompressible flows)**

Navier-stokes equations First principle of thermodynamics and equation for any passive scalar ψ continuity equation

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**Symbols pressure density velocity components geopotential height**

Coriolis parameter alternating symbol molecular diffusivity sources or sinks velocity components spatial coordinates potential temperature passive scalar actual temperature

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**Boussinesq-approximation (I)**

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 p0 fulfills the equations shown on the right.

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**Boussinesq-approximation (II)**

Neglecting the density variations in all terms except from the buoyancy term Density variations are replaced by potential temperature variations

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**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

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**Filtering the equations (II)**

The SGS stress tensor is splitted into an isotropic and an anisotropic part: anisotropic SGS stress tensor

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**The filtered equations**

Boussinesq-approximated Reynolds equations for incompressible flows First principle of thermodynamics and equation for any passive scalar continuity equation for incompressible flows

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**The filtered equations (“flux“ form)**

Boussinesq-approximated Reynolds equations for incompressible flows First principle of thermodynamics and equation for any passive scalar continuity equation for incompressible flows

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**The parameterization model (I) (Deardorff, 1980)**

anisotropic SGS stress tensor eddy viscosity for momentum strain rate tensor turbulent kinetic energy

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**The parameterization model (II) (Deardorff, 1980)**

Eddy viscosity for heat Mixing length Characteristic grid spacing Wall adjustment factor

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**The parameterization model (III)**

Prognostic equation for the turbulent kinetic energy has to be solved: C-e-definition is no longer the original Deardorff-model but was modified by Moeng and Wyngaard

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The Poisson-equation The pressure π is computed to balance out velocity divergences (which could arise during the numerical calculation:

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The Prandtl-layer (I) velocity- and temperature gradients in the Prandl-layer Φm and Φh are the Dyer-Businger functions for momentum and heat friction velocity characteristic temperature in the Prandtl-layer

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**The Prandtl-layer (II)**

stable stratification neutral stratification unstable stratification Dyer-Businger functions for momentum and heat Richardson flux number

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Cloud Physics (I) Suppositions: liquid water content and water vapor are in thermodynamic equilibrium All thermodynamic processes are reversible Potential liquid water temperature θl and total water content q as prognostic variables For moist adiabatic processes, θl 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 potential liquid water temperature is defined as the temperature an air parcel would have if is was brought moist-adiabatically and reversibly to a reference height with pr = 1000 hPa and if all liquid water in it had been evaporated.

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**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

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**Cloud Physics (II) Equations of θl and q:**

Filtered equations of θl and q (molecular diffusion term neglected) equations of potential liquid water temperature and total water content with nu_theta and nu_q the molecular diffusivities and Q_theta and Q_q the sources and sinks. sources and sinks are divided into those caused by radiative processes and those caused by precipitaion

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Cloud Physics (III) only totally saturated or totally unsaturated grid cells are allowed in the model the specific humidity for saturated air qs is computed as follows:

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**Cloud Physics – SGS-Parameterization**

Parameterisation of Wi and Hi: The buoyancy-term in the TKE-equation is modified (the potential temperature θ is replaced by the virtual potential temperature θv): g/theta_0 * u’theta’ for the dry case is replaced by g/theta_v0 H_v,3. H_v,3 is the subgrid-scale vertical flux of virtual potential temperature (buoyancy flux)

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**Cloud Physics – SGS-Parameterization (II)**

Hv,3: subgrid-scale vertical flux of virtual potential temperature (buoyancy flux) K1, K2: Coefficients for unsaturated moist air and saturated moist air respectively K1 = - first version – for unsaturated air, second for saturated moist air

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Cloud Physics (IV) Prognosticating θl and q, the virtual potential temperature θv and the quotient of potential and actual temperature θ/T have to be computed as follows: potential liquid water temperature is defined as the temperature an air parcel would have if is was brought moist-adiabatically and reversibly to a reference height with pr = 1000 hPa and if all liquid water in it had been evaporated.

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**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 theta_l(z) lies between z+ and z-, that means that z+ is the grid level above, z- the grid level below theta_l

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**Cloud Physics – Radiation Model - Symbols**

upward and downward radiant fluxes black body radiation cloud emissivities between z1 and z2 Liquid Water Path mass exchange coefficient (empirical data) LWP=liquid water content in a column between z1 and z2

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**Cloud Physics – Precipitation Model**

Kessler scheme (Kessler, 1969) only autoconversion (production of rain by coalescence) is considered precipitation starts when a threshold value qlt is exceeded τ is a retarding time constant

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Boundary conditions

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