Presentation on theme: "Institut für Meteorologie und Klimatologie Universität Hannover"— Presentation transcript:
1 Institut für Meteorologie und Klimatologie Universität Hannover PALM – model equationsSonja WeinbrechtInstitut für Meteorologie und Klimatologie Universität Hannover
2 Structure Basic equations Boussinesq-approximation and filtering SGS-parameterizationPrandtl-layerCloud physicsBoundary conditions
3 Basic equations Navier-stokes equations First principle of thermodynamics and equation for any passive scalar ψcontinuity equation
4 Basic equations (in “flux-form“ for incompressible flows) Navier-stokes equationsFirst principle of thermodynamics and equation for any passive scalar ψcontinuity equation
5 Symbols pressure density velocity components geopotential height Coriolis parameteralternating symbolmolecular diffusivitysources or sinksvelocity componentsspatial coordinatespotential temperaturepassive scalaractual temperature
6 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 partThe basic pressure p0 fulfills the equations shown on the right.
7 Boussinesq-approximation (II) Neglecting the density variations in all terms except from the buoyancy termDensity variations are replaced by potential temperature variations
8 Filtering the equations (I) Splitting the variables into mean part ( ¯ ) and deviation ( )’By filtering, a turbulent diffusion term comes into beingcompared with the turbulent diffusion term the molecular diffusion term can be neglectedsubgrid-scale (SGS) stress tensor
9 Filtering the equations (II) The SGS stress tensor is splitted into an isotropic and an anisotropic part:anisotropic SGS stress tensor
10 The filtered equations Boussinesq-approximated Reynolds equations for incompressible flowsFirst principle of thermodynamics and equation for any passive scalarcontinuity equation for incompressible flows
11 The filtered equations (“flux“ form) Boussinesq-approximated Reynolds equations for incompressible flowsFirst principle of thermodynamics and equation for any passive scalarcontinuity equation for incompressible flows
12 The parameterization model (I) (Deardorff, 1980) anisotropic SGS stress tensoreddy viscosity for momentumstrain rate tensorturbulent kinetic energy
13 The parameterization model (II) (Deardorff, 1980) Eddy viscosity for heatMixing lengthCharacteristic grid spacingWall adjustment factor
14 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
15 The Poisson-equationThe pressure π is computed to balance out velocity divergences (which could arise during the numerical calculation:
16 The Prandtl-layer (I)velocity- and temperature gradients in the Prandl-layerΦm and Φh are the Dyer-Businger functions for momentum and heatfriction velocitycharacteristic temperature in the Prandtl-layer
17 The Prandtl-layer (II) stable stratificationneutral stratificationunstable stratificationDyer-Businger functions for momentum and heatRichardson flux number
18 Cloud Physics (I)Suppositions: liquid water content and water vapor are in thermodynamic equilibriumAll thermodynamic processes are reversiblePotential liquid water temperature θl and total water content q as prognostic variablesFor moist adiabatic processes, θl and q are conserved.Condensation and evaporation do not have to be explicitly describedOnly totally saturated or totally unsaturated grid cells are allowed in the modelpotential 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.
19 Cloud Physics - Symbols potential liquid water temperaturetotal water contentspecific humidity, s.h. for saturated airliquid water contentlatent heat / vaporization enthalpyvirtual potential temperaturepressure (reference value)adiabatic coefficientactual liquid water temperaturesaturation vapour pressure
20 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 contentwith 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
21 Cloud Physics (III)only totally saturated or totally unsaturated grid cells are allowed in the modelthe specific humidity for saturated air qs is computed as follows:
22 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)
23 Cloud Physics – SGS-Parameterization (II) Hv,3: subgrid-scale vertical flux of virtual potential temperature (buoyancy flux)K1, K2: Coefficients forunsaturated moist airand saturated moist air respectivelyK1 = - first version – for unsaturated air, second for saturated moist air
24 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.
25 Cloud Physics – Radiation Model based on the work of Cox (1976)vertical gradients of radiant flux as sources of energythe downward radiation at the top of the model is prescribedtheta_l(z) lies between z+ and z-, that means that z+ is the grid level above, z- the grid level below theta_l
26 Cloud Physics – Radiation Model - Symbols upward and downward radiant fluxesblack body radiationcloud emissivities between z1 and z2Liquid Water Pathmass exchange coefficient (empirical data)LWP=liquid water content in a column between z1 and z2
27 Cloud Physics – Precipitation Model Kessler scheme (Kessler, 1969)only autoconversion (production of rain by coalescence) is consideredprecipitation starts when a threshold value qlt is exceededτ is a retarding time constant
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