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| 6 | <title>PALM chapter 2.0</title> |
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| 8 | <meta content="Marcus Oliver Letzel" name="AUTHOR"> |
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| 19 | <h2 style="line-height: 100%;"><font size="4">2.0 Basic techniques of |
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| 20 | the LES model and its parallelization </font> |
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| 21 | </h2> |
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| 22 | <p style="line-height: 100%;">LES models generally permit the |
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| 23 | simulation of turbulent flows, whereby those eddies, that carry the |
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| 24 | main energy are resolved by the numerical grid. Only the |
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| 25 | effect of such turbulence elements with diameter equal to or smaller |
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| 26 | than the grid spacing are parameterized in the model and |
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| 27 | by so-called subgrid-scale (SGS) transport. Larger structures are |
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| 28 | simulated directly (they are explicitly resolved) and their effects are |
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| 29 | represented by the advection terms. </p> |
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| 30 | <p style="font-style: normal; line-height: 100%;">PALM is based on the |
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| 31 | non-hydrostatic incompressible Boussinesq equations. It contains a |
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| 32 | water cycle with cloud formation and takes into account infrared |
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| 33 | radiative cooling in cloudy conditions. The model has six prognostic |
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| 34 | quantities in total u,v,w, liquid water potential temperature |
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| 35 | <font face="Thorndale, serif">Θ</font><sub>l </sub>(BETTS, |
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| 36 | 1973), total water content q and subgrid-scale turbulent kinetic energy |
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| 37 | e. The |
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| 38 | subgrid-scale turbulence is modeled according to DEARDOFF (1980) and |
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| 39 | requires the solution of an additional prognostic equation for the |
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| 40 | turbulent kinetic energy e. The long wave radiation scheme is based |
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| 41 | on the parametrization of cloud effective emissivity (e.g. Cox, 1976) |
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| 42 | and condensation is considered by a simple '0%-or-100%'-scheme, which |
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| 43 | assumes that within each grid box the air is either entirely |
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| 44 | unsaturated or entirely saturated ( see e.g., CUIJPERS and DUYNKERKE, |
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| 45 | 1993). The water cycle is closed by using a simplified version of |
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| 46 | KESSLERs scheme (KESSLER, 1965; 1969) to parameterize precipitation |
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| 47 | processes (MÜLLER and CHLOND, 1996). Incompressibility is |
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| 48 | applied by means of a Poisson equation for pressure, which is solved |
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| 49 | with a direct method (SCHUMANN and SWEET, 1988). The Poisson equation |
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| 50 | is Fourier transformed in both horizontal directions and the |
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| 51 | resulting tridiagonal matrix is solved for the transformed pressure |
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| 52 | which is then transformed back. Alternatively, a multigrid method can |
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| 53 | also be used. Lateral boundary conditions of the model are cyclic and |
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| 54 | MONIN-OBUKHOV similarity is assumed between the surface and the first |
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| 55 | computational grid level above. Alternatively, noncyclic boundary |
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| 56 | conditions |
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| 57 | (Dirichlet/Neumann) can be used along one of the |
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| 58 | horizontal directions. At the lower surface, either temperature/ |
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| 59 | humidity or their respective fluxes can be prescribed. </p> |
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| 60 | <p style="font-style: normal; line-height: 100%;">The advection terms |
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| 61 | are treated by the scheme proposed by PIACSEK and WILLIAMS (1970), |
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| 62 | which conserves the integral of linear and quadratic quantities up to |
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| 63 | very small errors. The advection of scalar quantities can optionally |
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| 64 | be performed by the monotone, locally modified version of Botts |
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| 65 | advection scheme (CHLOND, 1994). The time integration is performed |
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| 66 | with the third-order Runge-Kutta scheme. A second-order Runge-Kutta |
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| 67 | scheme, a leapfrog scheme and an Euler scheme are also implemented.</p> |
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| 68 | <p style="line-height: 100%;">By default, the time step is computed |
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| 69 | with respect to the different criteria (CFL, diffusion) and adapted |
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| 70 | automatically. In case of a non-zero geostrophic |
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| 71 | wind the coordinate system can be moved along with the mean wind in |
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| 72 | order to maximize the time step (Galilei-Transformation). </p> |
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| 73 | <p style="font-style: normal; line-height: 100%;">In principle a model |
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| 74 | run is carried out in the following way: After reading the control |
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| 75 | parameters given by the user, all prognostic variables are |
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| 76 | initialized. Initial values can be e.g. vertical profiles of the |
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| 77 | horizontal wind, calculated using a 1D subset of the 3D prognostic |
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| 78 | equation and are set in the 3D-Model as horizontally homogeneous |
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| 79 | initial values. Temperature profiles can only be prescribed linear |
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| 80 | (with constant gradients, which may change for different vertical |
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| 81 | height intervals) and they are assumed in the 1D-Model as stationary. |
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| 82 | After the initialization phase during which also different kinds of |
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| 83 | disturbances may be imposed to the prognostic fields, the time |
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| 84 | integration begins. Here for each individual time step the prognostic |
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| 85 | equations are successively solved for the velocity components u, v and |
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| 86 | w |
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| 87 | as well as for the potential temperature and possibly for the TKE. |
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| 88 | After the calculation of the boundary values in accordance with the |
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| 89 | given boundary conditions the provisional velocity fields are |
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| 90 | corrected with the help of the pressure solver. Following this, all |
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| 91 | diagnostic turbulence quantities including possible |
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| 92 | Prandtl-layerquantities are computed. At the end of a time |
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| 93 | step the data output requested by the user is made |
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| 94 | (e.g. statistic of analyses for control purposes or profiles and/or |
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| 95 | graphics data). If the given end-time was reached, binary data maybe |
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| 96 | be saved for restart. </p> |
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| 97 | <p style="font-style: normal; line-height: 100%;">The model is based |
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| 98 | on the originally non-parallel LES model which has been operated at the |
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| 99 | institute since 1989 |
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| 100 | and which was parallelized for massively parallel computers with |
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| 101 | distributed memory using the Message-Passing-Standard MPI. It is |
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| 102 | still applicable on a single processor and also well optimized for |
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| 103 | vector machines. The parallelization takes place via a so-called domain |
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| 104 | decomposition, which divides the entire model |
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| 105 | domain into individual, vertically standing cubes, which extend from |
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| 106 | the bottom to the top of the model domain. One processor (processing |
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| 107 | element, PE) is assigned to each cube, which |
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| 108 | accomplishes the computations on all grid points of the subdomain. |
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| 109 | Users can choose between a two- and a one-dimensional domain |
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| 110 | decomposition. A 1D-decomposition is preferred on machines with a |
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| 111 | slow network interconnection. In case of a 1D-decomposition, the |
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| 112 | grid points along x direction are |
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| 113 | distributed among the individual processors, but in y- and z-direction |
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| 114 | all respective grid points belong to the same PE. </p> |
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| 115 | <p style="line-height: 100%;">The calculation of central differences or |
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| 116 | non-local arithmetic operations (e.g. global |
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| 117 | sums, FFT) demands communication and an appropriate data exchange |
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| 118 | between the PEs. As a substantial innovation in relation to |
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| 119 | the non-parallel model version the individual subdomains are |
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| 120 | surrounded by so-called ghost points, which contain the grid point |
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| 121 | information of the neighbor processors. The appropriate grid point |
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| 122 | values must be exchanged after each change (i.e. in particular after |
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| 123 | each time step). For this purpose MPI routines (<tt>MPI_SENDRCV</tt>) |
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| 124 | are used. For the solution of the FFT conventional (non-parallelized) |
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| 125 | procedures are used. Given that the FFTs are used in x and/or |
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| 126 | y-direction, the data which lie distributed on the individual central |
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| 127 | processing elements, have to be collected and/or relocated before. |
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| 128 | This happens by means of the routine <tt>MPI_ALLTOALLV</tt>. Certain |
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| 129 | global operations like e.g. the search for absolute maxima or minima |
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| 130 | within the 3D-arrays likewise require the employment of special MPI |
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| 131 | routines (<tt>MPI_ALLREDUCE</tt>). </p> |
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| 132 | <p style="line-height: 100%;">Further details of the internal model |
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| 133 | structure are described in the <a href="../tec/index.html">technical/numerical |
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| 134 | documentation</a>. <br> |
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| 135 | </p> |
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| 136 | <hr><font color="#000080"><font color="#000080"><br> |
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| 137 | <a href="chapter_1.0.html"><font color="#000080"><img width="32" |
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| 143 | </font></font><br> |
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| 144 | <p style="line-height: 100%;"><span style="font-style: italic;">Last |
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| 145 | change: </span>14/04/05 (SR)<font color="#000080"><font color="#000080"><br> |
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| 146 | </font></font></p> |
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