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Discretization

The model domain in PALM is discretized in space using finite differences and equidistant horizontal grid spacings (Δx, Δy). The grid can be stretched in the vertical direction well above the ABL to save computational time in the free atmosphere. 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 (see Fig. 1).

Figure 1:The Arakawa staggered C-grid. The indices i, j, k refer to grid points in x, y and z direction, respectively. Scalar quantities φ are defined at the center of the grid volume, whereas velocities are defined at the edges of the grid volumes.

It is thus possible to calculate the derivatives of the velocity components at the center of the volumes (same location as the scalars). By the same token, derivatives of scalar quantities can be calculated at the edges of the volumes. In this way it is possible to calculate derivatives over only one grid length and the effective spatial model resolution can be increased by a factor of two in comparison to non-staggered grids.

References

• Harlow FH, Welch JE. 1965. Numerical calculation of time-dependent viscous incompressible flow of fluid with free surface. Phys. Fluids 8: 2182–2189.

• Schumann U. 1977. Computational design of the basic dynamical processes of the UCLA general circulation model. in: General Circulation Models of the Atmosphere, Methods in Computational Physics. edited by: Chang J. 17. Berlin. 173–265.