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The ZM Grid: An Alternative to the Z Grid

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  • 1 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
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Abstract

Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇2f = λf eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

Corresponding author address: Todd Ringler, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371. Email: todd@atmos.colostate.edu

Abstract

Shallow-water equations discretized on a perfect hexagonal grid are analyzed using both a momentum formulation and a vorticity-divergence formulation. The vorticity-divergence formulation uses the unstaggered Z grid that places mass, vorticity, and divergence at the centers of the hexagons. The momentum formulation uses the staggered ZM grid that places mass at the centers of the hexagons and velocity at the corners of the hexagons. It is found that the Z grid and the ZM grid are identical in their simulation of the physical modes relevant to geostrophic adjustment. Consistent with the continuous system, the simulated inertia–gravity wave phase speeds increase monotonically with increasing total wavenumber and, thus, all waves have nonzero group velocities.

Since a grid of hexagons has twice as many corners as it has centers, the ZM grid has twice as many velocity points as it has mass points. As a result, the ZM-grid velocity field is discretized at a higher resolution than the mass field and, therefore, resolves a larger region of wavenumber space than the mass field. We solve the ∇2f = λf eigenvalue problem with periodic boundary conditions on both the Z grid and ZM grid to determine the modes that can exist on each grid. The mismatch between mass and momentum leads to computational modes in the velocity field. Two techniques that can be used to control these computational modes are discussed. One technique is to use a dissipation operator that captures or “sees” the smallest-scale variations in the velocity field. The other technique is to invert elliptic equations in order to filter the high wavenumber part of the momentum field.

Results presented here lead to the conclusion that the ZM grid is an attractive alternative to the Z grid, and might be particularly useful for ocean modeling.

Corresponding author address: Todd Ringler, Department of Atmospheric Science, Colorado State University, Fort Collins, CO 80523-1371. Email: todd@atmos.colostate.edu

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