2.0 Basic techniques of the LES model and its parallelization

LES models generally permit the simulation of turbulent flows, whereby those eddies, that carry the main energy are resolved by the numerical grid. Only the effect of such turbulence elements with diameter equal to or smaller than the grid spacing are parameterized in the model and by so-called subgrid-scale (SGS) transport. Larger structures are simulated directly (they are explicitly resolved) and their effects are represented by the advection terms.

PALM is based on the non-hydrostatic incompressible Boussinesq equations. It contains a water cycle with cloud formation and takes into account infrared radiative cooling in cloudy conditions. The model has six prognostic quantities in total – u,v,w, liquid water potential temperature Θl (BETTS, 1973), total water content q and subgrid-scale turbulent kinetic energy e. The subgrid-scale turbulence is modeled according to DEARDOFF (1980) and requires the solution of an additional prognostic equation for the turbulent kinetic energy e. The long wave radiation scheme is based on the parametrization of cloud effective emissivity (e.g. Cox, 1976) and condensation is considered by a simple '0%-or-100%'-scheme, which assumes that within each grid box the air is either entirely unsaturated or entirely saturated ( see e.g., CUIJPERS and DUYNKERKE, 1993). The water cycle is closed by using a simplified version of KESSLERs scheme (KESSLER, 1965; 1969) to parameterize precipitation processes (MÜLLER and CHLOND, 1996). Incompressibility is applied by means of a Poisson equation for pressure, which is solved with a direct method (SCHUMANN and SWEET, 1988). The Poisson equation is Fourier transformed in both horizontal directions and the resulting tridiagonal matrix is solved for the transformed pressure which is then transformed back. Alternatively, a multigrid method can also be used. Lateral boundary conditions of the model are cyclic and MONIN-OBUKHOV similarity is assumed between the surface and the first computational grid level above. Alternatively, noncyclic boundary conditions (Dirichlet/Neumann) can be used along one of the horizontal directions. At the lower surface, either temperature/ humidity or their respective fluxes can be prescribed.

The advection terms are treated by the scheme proposed by PIACSEK and WILLIAMS (1970), which conserves the integral of linear and quadratic quantities up to very small errors. The advection of scalar quantities can optionally be performed by the monotone, locally modified version of Botts advection scheme (CHLOND, 1994). The time integration is performed with the third-order Runge-Kutta scheme. A second-order Runge-Kutta scheme, a leapfrog scheme and an Euler scheme are also implemented.

By default, the time step is computed with respect to the different criteria (CFL, diffusion) and adapted automatically. In case of a non-zero geostrophic wind the coordinate system can be moved along with the mean wind in order to maximize the time step (Galilei-Transformation).

In principle a model run is carried out in the following way: After reading the control parameters given by the user, all prognostic variables are initialized. Initial values can be e.g. vertical profiles of the horizontal wind, calculated using a 1D subset of the 3D prognostic equation and are set in the 3D-Model as horizontally homogeneous initial values. Temperature profiles can only be prescribed linear (with constant gradients, which may change for different vertical height intervals) and they are assumed in the 1D-Model as stationary. After the initialization phase during which also different kinds of disturbances may be imposed to the prognostic fields, the time integration begins. Here for each individual time step the prognostic equations are successively solved for the velocity components u, v and w as well as for the potential temperature and possibly for the TKE. After the calculation of the boundary values in accordance with the given boundary conditions the provisional velocity fields are corrected with the help of the pressure solver. Following this, all diagnostic turbulence quantities including possible Prandtl-layer–quantities are computed. At the end of a time step the data output requested by the user is made (e.g. statistic of analyses for control purposes or profiles and/or graphics data). If the given end-time was reached, binary data maybe be saved for restart.

The model is based on the originally non-parallel LES model which has been operated at the institute since 1989 and which was parallelized for massively parallel computers with distributed memory using the Message-Passing-Standard MPI. It is still applicable on a single processor and also well optimized for vector machines. The parallelization takes place via a so-called domain decomposition, which divides the entire model domain into individual, vertically standing cubes, which extend from the bottom to the top of the model domain. One processor (processing element, PE) is assigned to each cube, which accomplishes the computations on all grid points of the subdomain. Users can choose between a two- and a one-dimensional domain decomposition. A 1D-decomposition is preferred on machines with a slow  network interconnection. In case of a 1D-decomposition, the grid points along x direction are distributed among the individual processors, but in y- and z-direction all respective grid points belong to the same PE.

The calculation of central differences or non-local arithmetic operations (e.g. global sums, FFT) demands communication and an appropriate data exchange between the PEs. As a substantial innovation in relation to the non-parallel model version the individual subdomains are surrounded by so-called ghost points, which contain the grid point information of the neighbor processors. The appropriate grid point values must be exchanged after each change (i.e. in particular after each time step). For this purpose MPI routines (MPI_SENDRCV) are used. For the solution of the FFT conventional (non-parallelized) procedures are used. Given that the FFTs are used in x and/or y-direction, the data which lie distributed on the individual central processing elements, have to be collected and/or relocated before. This happens by means of the routine MPI_ALLTOALLV. Certain global operations like e.g. the search for absolute maxima or minima within the 3D-arrays likewise require the employment of special MPI routines (MPI_ALLREDUCE).

Further details of the internal model structure are described in the technical/numerical documentation.

Last change: $Id: chapter_2.0.html 62 2007-03-13 02:52:40Z heinze $