Accurate computation of horizontal gradients in sigma-coordinate models is a
long-standing problem1. The horizontal gradient is numerically equivalent to the
difference between the gradient along the sigma-coordinate and the vertical gradient.
When topography is steep or when the vertical gradient is large, the horizontal gradient
becomes the difference between two large numbers. Therefore, even small errors in the
computation of either term can produce relatively large errors in the horizontal gradient.
There are at least four known numerical techniques that can help to reduce the
computational error2. MRAMS utilizes two of these. First, the atmospheric fields are
decomposed into a vertical varying base state and a perturbation from the base state.
Computation of the horizontal gradient can then be calculated from the perturbation
rather than from the total quantity. The second technique has been implemented only
recently, and involves interpolation of the field from the sigma coordinate system back to
an orthogonal cartesian system3,4. The horizontal gradient is then computed directly from
the interpolated field. Extrapolation to the cartesian grid is required when the horizontal
surface intersects topography. We have found the addition of this technique in the
turbulent diffusion code greatly reduces errors in idealized simulations. Consequently we
have adopted this technique in all of our simulations with steep topography. The new
technique is not without penalty, however. Integral properties (e.g., heat and momentum)
are not necessarily conserved. The potential conservation problem is, like the horizontal
gradient problem, well known, and various techniques have been devised to minimize the
impact6,7. We are currently exploring ways to implement some of these.
1. Haney, R. L. On the pressure gradient force over steep topography in sigma
coordinate ocean models. J. Phys. Oceanogr., 21, 610-619, (1991).
2. Song, Y. T. A general pressure gradient formulation for ocean models. Part I:
Scheme design and diagnostic analysis. Mon. Wea. Rev., 126, 3213-3229, (1998).
3. Mahrer, Y. An improved numerical approximation of the horizontal gradients in a
terrain-following coordinate system. Mon. Wea. Rev., 112, 918-922 (1984).
4. Fortunato, A. B. and Baptista, A. M. Evaluation of horizontal gradients in sigmacoordinate shallow water models. Atmos-Ocean, 34, 489-514 (1996).
5. Arakawa, A. and Suarez, M. J. Vertical differencing of the primitive equations in
sigma coordinates. Mon. Wea. Rev., 111, 34-45, (1983).
6. Arakawa, A. and Konor, C. S. Vertican differencing of the primitive equations on the
Charney-Phillips grid in hybrid -p vertical coordinates. Mon. Wea. Rev., 124, 511-528,
(1996).