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

2. Structure Basic equations Boussinesq-approximation and filtering
SGS-parameterization
Prandtl-layer
Cloud physics
Boundary conditions

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

4. Basic equations (in “flux-form“ for incompressible flows)
Navier-stokes equations
First principle of thermodynamics and equation for any passive scalar ψ
continuity equation

5. 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

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 part
The 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 term
Density 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 being
compared with the turbulent diffusion term the molecular diffusion term can be neglected
subgrid-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 flows
First principle of thermodynamics and equation for any passive scalar
continuity equation for incompressible flows

11. 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

12. The parameterization model (I) (Deardorff, 1980)
anisotropic SGS stress tensor
eddy viscosity for momentum
strain rate tensor
turbulent kinetic energy

13. The parameterization model (II) (Deardorff, 1980)
Eddy viscosity for heat
Mixing length
Characteristic grid spacing
Wall 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-equation
The 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-layer
Φm and Φh are the Dyer-Businger functions for momentum and heat
friction velocity
characteristic temperature in the Prandtl-layer

17. The Prandtl-layer (II)
stable stratification
neutral stratification
unstable stratification
Dyer-Businger functions for momentum and heat
Richardson flux number

18. 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.

19. 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. 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

21. 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:

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 for
unsaturated moist air
and saturated moist air respectively
K1 = - 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 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

26. 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

27. 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

28. Boundary conditions