Computational box

PISM does all simulations in a computational box which is rectangular in the PISM coordinates. The coordinate system has horizontal coordinates \(x,y\) and a vertical coordinate \(z\). The \(z\) coordinate is measured positive upward from the base of the ice.[1] The vector of gravity is in the negative \(z\) direction. The surface \(z=0\) is the base of the ice, however, and thus is usually not horizontal in the sense of being parallel to the geoid. The surface \(z=0\) is the base of the ice both when the ice is grounded and when the ice is floating.

When the ice is grounded, the true physical vertical coordinate \(z'\), namely the coordinate measure relative to a reference geoid, is given by \(z'=z+b(x,y)\) where \(b(x,y)\) is the bed topography. The surface \(z'=h(x,y)\) is the surface of the ice. In the grounded case the equation \(h(x,y)=H(x,y)+b(x,y)\) always applies if \(H(x,y)\) is the thickness of the ice.

In the floating case, the physical vertical coordinate is

\[z' = z - \frac{\rho_i}{\rho_s} H(x,y)\]

where \(\rho_i\) is the density of ice and \(\rho_s\) the density of sea water. Again \(z=0\) is the base of the ice, which is the surface

\[z' = - \frac{\rho_i}{\rho_s} H(x,y).\]

The surface of the ice is

\[h(x,y) = \left(1-\frac{\rho_i}{\rho_s}\right) H(x,y).\]

The flotation criterion \(-\frac{\rho_i}{\rho_s} H(x,y) > b(x,y)\) applies.

The computational box can extend downward into the bedrock. As \(z=0\) is the base of the ice, the bedrock corresponds to negative \(z\) values regardless of its true (i.e. \(z'\)) elevation.

The extent of the computational box, along with its bedrock extension downward, is determined by four numbers Lx, Ly, Lz, and Lbz (see Fig. 17 and Table 6). The first two of these are half-widths and have units of kilometers when set by command-line options or displayed.


Fig. 17 PISM’s computational box

Table 6 Options defining the extent of PISM’s computational box
Option Description
-Lx (km) Half-width of the computational domain (in the \(x\)-direction)
-Ly (km) Half-width of the computational domain (in the \(y\)-direction)
-Lz (meters) Height of the computational domain; must exceed maximum ice thickness
-Lbz (meters) Depth of the computational domain in the bedrock thermal layer
-x_range A,B (meters) Specify the range of \(x\) coordinates. Use this to select a subset of an input grid that isn’t in the center of a domain.
-y_range A,B (meters) Specify the range of \(y\) coordinates.

See Grid registration for details about the interpretation of \(L_x\), \(L_y\), and the way the grid spacing is computed.


[1]See On the vertical coordinate in PISM, and a critical change of variable for details.

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