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# A Variational Method for Analyzing Vortex Flows in Radar Scanned Tornadic Mesocyclones. Part I: Formulations and Theoretical Considerations

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• 1 NOAA/OAR/National Severe Storms Laboratory, Norman, Oklahoma
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## Abstract

A variational method is formulated with theoretical considerations for analyzing vortex flows in Doppler radar scanned tornadic mesocyclones. The method has the following features. (i) The vortex center axis (estimated as a continuous function of time and height in the four-dimensional space) is used as the vertical coordinate, so the coordinate system used for the analysis is slantwise-curvilinear and non-orthogonal in general. (ii) The vortex flow (VF), defined by the three-dimensional vector wind minus the horizontal moving velocity of vortex center axis, is expressed in terms of the covariant basis vectors (tangent to the coordinate curves), so its axisymmetric part can be properly defined in that slantwise-curvilinear coordinate system. (iii) To satisfy the mass continuity automatically, the axisymmetric part is expressed by the scalar fields of azimuthally averaged tangential velocity and cylindrical stream-function and the remaining asymmetric part is expressed by the scalar fields of stream-function and vertically integrated velocity-potential. (iv) VF-dependent covariance functions are formulated for these scalar variables and then de-convoluted to construct the square root of background error covariance matrix analytically with the latter used to transform the control vector to precondition the cost-function. (v) The de-convoluted covariance functions and their transformed control variables satisfy two required boundary conditions (that is, zero vertical velocity at the lower rigid boundary and zero cross-axis velocity along the vortex center axis), so the analyzed VF satisfies not only the mass continuity but also the two boundary conditions automatically.

Correspondence to: Dr. Qin Xu, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, Oklahoma 73072-7326. E-mail: Qin.Xu@noaa.gov

## Abstract

A variational method is formulated with theoretical considerations for analyzing vortex flows in Doppler radar scanned tornadic mesocyclones. The method has the following features. (i) The vortex center axis (estimated as a continuous function of time and height in the four-dimensional space) is used as the vertical coordinate, so the coordinate system used for the analysis is slantwise-curvilinear and non-orthogonal in general. (ii) The vortex flow (VF), defined by the three-dimensional vector wind minus the horizontal moving velocity of vortex center axis, is expressed in terms of the covariant basis vectors (tangent to the coordinate curves), so its axisymmetric part can be properly defined in that slantwise-curvilinear coordinate system. (iii) To satisfy the mass continuity automatically, the axisymmetric part is expressed by the scalar fields of azimuthally averaged tangential velocity and cylindrical stream-function and the remaining asymmetric part is expressed by the scalar fields of stream-function and vertically integrated velocity-potential. (iv) VF-dependent covariance functions are formulated for these scalar variables and then de-convoluted to construct the square root of background error covariance matrix analytically with the latter used to transform the control vector to precondition the cost-function. (v) The de-convoluted covariance functions and their transformed control variables satisfy two required boundary conditions (that is, zero vertical velocity at the lower rigid boundary and zero cross-axis velocity along the vortex center axis), so the analyzed VF satisfies not only the mass continuity but also the two boundary conditions automatically.

Correspondence to: Dr. Qin Xu, National Severe Storms Laboratory, 120 David L. Boren Blvd., Norman, Oklahoma 73072-7326. E-mail: Qin.Xu@noaa.gov
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