A Consistency Condition for Wind-Field Reconstruction in a Limited Area and a Harmonic-Cosine Series Expansion

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  • 1 National Center for Atmospheric Research, Boulder, Colorado
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Abstract

This paper examines further the problem of deducing the wind field from vorticity and divergence over a limited area with prescribed winds at the boundary. An earlier work showed that the wind field in a limited area can be partitioned into internal divergent, internal rotational, and harmonic wind components. Because the harmonic wind is both nondivergent and irrotational, it is demonstrated in this paper that the two harmonic wind components at the boundary must satisfy a consistency condition. Based on this properly, a direct method is developed to solve two Laplace equations with the prescribed two harmonic wind components at the boundary. If the prescribed harmonic wind components at the boundary satisfy the consistency condition, the solution of the two Laplace equations must be nondivergent and irrotational. The direct method is shown to be highly accurate and efficient. If the prescribed wind at the boundary does not satisfy the consistency condition, this implies a mismatch between the interior vorticity and divergence and the prescribed winds at the boundary. This inconsistency must be removed before the wind field can be reconstructed. A method to remove this inconsistency is discussed.

A harmonic-cosine series expansion is also developed for a function over a limited area. The application of the harmonic-cosine series expansion to the wind-field partitioning and reconstruction problem has two distinct advantages compared with the harmonic-sine series expansion. The first is that the internal and harmonic winds can be more accurately determined at the boundary. The second is that the partitioning of the wind field into streamfunction and velocity potential can be obtained more efficiently and accurately through an iterative method.

Abstract

This paper examines further the problem of deducing the wind field from vorticity and divergence over a limited area with prescribed winds at the boundary. An earlier work showed that the wind field in a limited area can be partitioned into internal divergent, internal rotational, and harmonic wind components. Because the harmonic wind is both nondivergent and irrotational, it is demonstrated in this paper that the two harmonic wind components at the boundary must satisfy a consistency condition. Based on this properly, a direct method is developed to solve two Laplace equations with the prescribed two harmonic wind components at the boundary. If the prescribed harmonic wind components at the boundary satisfy the consistency condition, the solution of the two Laplace equations must be nondivergent and irrotational. The direct method is shown to be highly accurate and efficient. If the prescribed wind at the boundary does not satisfy the consistency condition, this implies a mismatch between the interior vorticity and divergence and the prescribed winds at the boundary. This inconsistency must be removed before the wind field can be reconstructed. A method to remove this inconsistency is discussed.

A harmonic-cosine series expansion is also developed for a function over a limited area. The application of the harmonic-cosine series expansion to the wind-field partitioning and reconstruction problem has two distinct advantages compared with the harmonic-sine series expansion. The first is that the internal and harmonic winds can be more accurately determined at the boundary. The second is that the partitioning of the wind field into streamfunction and velocity potential can be obtained more efficiently and accurately through an iterative method.

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