The PALM core

The PALM model core is based on the non-hydrostatic, filtered, incompressible Navier-Stokes equations in Boussinesq-approximated form (an anelastic approximation is available as an option for simulating deep convection). By default, PALM has at least six prognostic quantities: the velocity components u, v, w on a Cartesian grid, the potential temperature θ, water vapor mixing ratio q_v and possibly a passive scalar s. Furthermore, an additional equation is solved for either the subgrid-scale turbulent kinetic energy (SGS-TKE) e (LES mode, default) or the total turbulent kinetic energy (RANS mode, see PALM-4U components).

In the LES mode, the filtering process yields four subgrid-scale (SGS) covariance terms. These SGS terms are parametrized using a 1.5-order closure after Deardorff (1980). PALM uses the modified version of Moeng and Wyngaard (1988) and Saiki et al. (2000). The closure is based on the assumption that the energy transport by SGS eddies is proportional to the local gradients of the mean quantities.

Discretization in time and space

The model domain in PALM is discretized in space using finite differences and equidistant horizontal grid spacings. The Arakawa staggered C-grid (Harlow and Welch, 1965; Arakawa and Lamb, 1977) is used, where scalar quantities are defined at the center of each grid volume, whereas velocity components are shifted by half a grid width in their respective direction so that they are defined at the edges of the grid volumes. By default, the advection terms in the prognostic equations are discretized using an upwind-biased 5th-order differencing scheme (Wicker and Skamarock, 2002) in combination with a 3rd-order Runge–Kutta time-stepping scheme after Williamson (1980).

Pressure solver

The Boussinesq approximation requires incompressibility of the flow, but the integration of the governing equations does not provide this feature. Divergence of the flow field is thus inherently produced. Hence, a predictor corrector method is used where an equation is solved for the modified perturbation pressure after every time step (e.g., Patrinos and Kistler, 1977). In case of cyclic lateral boundary conditions, the solution of the Poisson equation is achieved by using a direct fast Fourier transform (FFT). PALM provides the inefficient but less restrictive Singleton-FFT (Singleton, 1969) and the well optimized Temperton-FFT (Temperton, 1992). External FFT libraries can be used as well, with the FFTW (Frigo and Johnson, 1998) being the most efficient one. Alternatively, the iterative multigrid scheme can be used (e.g., Hackbusch, 1985). This scheme uses the Gauss–Seidel method for the inner iterations on each grid level.

Boundary conditions

PALM offers a variety of boundary conditions. Dirichlet or Neumann boundary conditions can be chosen for u, v, θ, q_v, and p^∗ at the bottom and top of the model. For the horizontal velocity components the choice of Neumann (Dirichlet) boundary conditions yields free-slip (no-slip) conditions. Neumann boundary conditions are also used for the SGS-TKE. Kinematic fluxes of heat and moisture can be prescribed at the surface instead (Neumann conditions) of temperature and humidity (Dirichlet conditions). At the top of the model, Dirichlet boundary conditions can be used with given values of the geostrophic wind. Vertical velocity is assumed to be zero at the surface and top boundaries, which implies using Neumann conditions for pressure.

Following Monin-Obukhov similarity theory (MOST) a constant flux layer can be assumed as boundary condition between the surface and the first grid level where scalars and horizontal velocities are defined. In PALM we assume that MOST can be also applied locally and we therefore calculate local fluxes, velocities, and scaling parameters. This scheme involves calculation of the Obukhov length L, which can be either done based on variables of the previous time step ("circular"), via a Newton iteration method, or via a look-up table for the stability parameter z/L.