Data adjusting using variational methods
Date
1991
Authors
Zhang, Qing
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Abstract
The a-coordinate system in the numerical modeling of the atmosphere has the advantage that the earth surface is always at the coordinate surface a = 1. However, when over steeply sloping mountains, the horizontal pressure gradient P in the a-coordinate system in the equation of motion consists of two terms of almost equal magnitude but of opposite sign. Unless special care is taken, the so called catastrophic cancellation will occur which results in a sudden loss of accuracy.
Two major factors contribute to the poor accuracy of P: the inconsistency of the original data and an inadequate finite difference scheme. Through numerical experiments, we found that the data inconsistency is the main source of error among the two. The following procedures are employed to solve the problem:
• Performing data adjustment to make it consistent with the hydrostatic condition, meanwhile, the adjustment is minimized in the least squares sense using variational methods;
• Using Corby's finite difference scheme to evaluate P, which has the effect of doubling the horizontal resolution, and therefore is helpful in improving the accuracy of P.
Thorough numerical experiments are conducted to investigate the effects of the above two procedures over grids of either mountainous or flat terrains. Results show that more than 20% reduction of errors is achieved on average for both grids, but in terms of absolute value of error reduction, the procedures are more effective for mountainous terrain than for flat terrain. Significant improvement is also achieved for reducing the maximum value of errors.
Our numerical experiments also show that Corby's finite difference scheme, which is supposed to be superior to other schemes in the evaluation of Pin the a-coordinate system, is only effective when combined with the data adjustment.