On the Initialization Problem: A Variational Adjustment Method

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  • 1 Illinois State Water Survey, Urbana, Ill. 61801
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Abstract

Primitive equation initialization, applicable to numerical prediction models, has been studied by using principles of variational calculus and the appropriate physical equations: two horizontal momentum, adiabatic energy, continuity, and hydrostatic. Essentially independent observations of the vector wind, geopotential height, and specific volume (though pressure and temperature), weighted according to their relative accuracies of measurement, are introduced into an adjustment functional in a least squares sense. The first variation on the functional is required to vanish and the four-dimensional Euler-Lagrange equations are solved as an elliptic boundary value problem.

The coarse 12 h observation frequency creates special problems since the variational formulation is explicit in time. A method accommodating the synoptic time scale was found whereby tendencies could be adjusted by requiring exact satisfaction of the continuity equation. This required a subsidiary variational formulation, the solution of which led to more rapid convergence to overall variational balance.

A test of the variational method using real data required satisfaction of three criteria: agreement with dynamic constraints, minimum adjustment from observed fields, and realistic patterns as could be judged from subjective pattern recognition. It was found that the criteria were satisfied and variationally optimized estimates for all dependent variables including vertical velocity were obtained.

Abstract

Primitive equation initialization, applicable to numerical prediction models, has been studied by using principles of variational calculus and the appropriate physical equations: two horizontal momentum, adiabatic energy, continuity, and hydrostatic. Essentially independent observations of the vector wind, geopotential height, and specific volume (though pressure and temperature), weighted according to their relative accuracies of measurement, are introduced into an adjustment functional in a least squares sense. The first variation on the functional is required to vanish and the four-dimensional Euler-Lagrange equations are solved as an elliptic boundary value problem.

The coarse 12 h observation frequency creates special problems since the variational formulation is explicit in time. A method accommodating the synoptic time scale was found whereby tendencies could be adjusted by requiring exact satisfaction of the continuity equation. This required a subsidiary variational formulation, the solution of which led to more rapid convergence to overall variational balance.

A test of the variational method using real data required satisfaction of three criteria: agreement with dynamic constraints, minimum adjustment from observed fields, and realistic patterns as could be judged from subjective pattern recognition. It was found that the criteria were satisfied and variationally optimized estimates for all dependent variables including vertical velocity were obtained.

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