Three-Dimensional Wind Field Analysis from Dual-Doppler Radar Data. Part III: The Boundary Condition: An Optimum Determination Based on a Variational Concept

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Abstract

The choice of the boundary condition when integrating the air mass continuity equation, is a major problem of the 3D wind field analysis from dual (or multiple) Doppler radar data. A zero vertical velocity at ground level seems the most natural boundary condition. Unfortunately, it is known that the integration processes is unstable with respect to this condition: it leads to errors amplifying exponentially with height. In order to overcome this difficulty various solutions have been proposed, the most recent ones using the variational analysis: (i) integrating from storm top level, (ii) integrating from storm top level while constraining the height integrated divergence to be as small as possible (Ziegier, 1978), and (iii) constraining the direct estimates of the 3D wind field to satisfy the continuity equation (Ray et al., 1980). The analysis proposed in this paper is also based upon a variational concept, but it differs in its principle from those previously cited. It consists in adjusting the boundary condition field at ground level in order to optimize the “mathematical regularity” of the vertical velocity field, followed by upward integration of the continuity equation. In such a formulation, the boundary condition at ground level is “floating” (i.e., not specified). However it is possible to require. as a subsidiary condition of the variational problem, that the vertical velocity at ground level fluctuate about zero with a specified variance σ02 (thus the condition W0=0 at ground level is statistically verified). The optimum choice of σ0 is established from considerations of statistical theory. It should be noted that the horizontal divergence (or coplane divergence) profile is unadjusted and that the equation of continuity is integrated upward from the optimum lower boundary condition to obtain W.

An application to simulated or real data helps us to appreciate the improvements brought by the present variational approach with respect to standard methods of integration: 1) the random errors are as small as in the case of an integration from storm top level, but here the boundary condition W0=0 at ground level is statistically preserved; and 2) for the integration paths where no cannot be specified (lack of data at low level), the analysis automatically generates a boundary condition which realizes the best regularity of W with respect to the neighbouring paths.

This variational analysis can be easily implemented on a computer from the program prepared for the standard integration, and it requires a short additional computation time.

Abstract

The choice of the boundary condition when integrating the air mass continuity equation, is a major problem of the 3D wind field analysis from dual (or multiple) Doppler radar data. A zero vertical velocity at ground level seems the most natural boundary condition. Unfortunately, it is known that the integration processes is unstable with respect to this condition: it leads to errors amplifying exponentially with height. In order to overcome this difficulty various solutions have been proposed, the most recent ones using the variational analysis: (i) integrating from storm top level, (ii) integrating from storm top level while constraining the height integrated divergence to be as small as possible (Ziegier, 1978), and (iii) constraining the direct estimates of the 3D wind field to satisfy the continuity equation (Ray et al., 1980). The analysis proposed in this paper is also based upon a variational concept, but it differs in its principle from those previously cited. It consists in adjusting the boundary condition field at ground level in order to optimize the “mathematical regularity” of the vertical velocity field, followed by upward integration of the continuity equation. In such a formulation, the boundary condition at ground level is “floating” (i.e., not specified). However it is possible to require. as a subsidiary condition of the variational problem, that the vertical velocity at ground level fluctuate about zero with a specified variance σ02 (thus the condition W0=0 at ground level is statistically verified). The optimum choice of σ0 is established from considerations of statistical theory. It should be noted that the horizontal divergence (or coplane divergence) profile is unadjusted and that the equation of continuity is integrated upward from the optimum lower boundary condition to obtain W.

An application to simulated or real data helps us to appreciate the improvements brought by the present variational approach with respect to standard methods of integration: 1) the random errors are as small as in the case of an integration from storm top level, but here the boundary condition W0=0 at ground level is statistically preserved; and 2) for the integration paths where no cannot be specified (lack of data at low level), the analysis automatically generates a boundary condition which realizes the best regularity of W with respect to the neighbouring paths.

This variational analysis can be easily implemented on a computer from the program prepared for the standard integration, and it requires a short additional computation time.

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