A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part II: Tests with Analytically Formulated Vortices

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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.

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

2021

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A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part III: Sensitivities to Vortex Center Location Errors

2022

When the vortex center location is estimated from a radar-scanned tornadic mesocyclone, the estimated location is not error-free. This raises an important issue concerning the sensitivities of analyzed vortex flow (VF) fields by the VF-Var (formulated in Part I and tested in Part II) to vortex center location errors, denoted by Δxc. Numerical experiments are performed to address this issue with the following findings: The increase of |Δxc| from zero to a half of vortex core radius causes large analysis error increases in the vortex core but the increased analysis errors decrease rapidly away from the vortex core especially for dual-Doppler analyses. The increased horizontal-velocity errors in the vortex core are mainly in the Δxc-normal component, because this component varies much more rapidly than the other component along the Δxc-direction in the vortex core. The vertical variations of Δxc distort the vertical correlation structure of Δxc-dislocated VF-dependent background error covariance, which can increase the analysis errors in the vortex core. The dual-Doppler analyses have adequate accuracies outside the vortex core even when |Δxc| increases to a half of vortex core radius, while single-Doppler analyses can also have adequate accuracies outside the vortex core mainly for the single-Doppler observed velocity component. The sensitivities to Δxc are largely unaffected by the vortex slanting. The above findings are important and useful for assessing the accuracies of analyzed VFs for real radar-observed tornadic mesocyclones.

A Variational Method for Analyzing Vortex Flows in Radar-Scanned Tornadic Mesocyclones. Part IV: Applications to the 20 May 2013 Oklahoma Tornadic Mesocyclone

Nai

2022

The variational method for vortex flow (VF) analyses, called VF-Var (formulated in Part I), is applied to the 20 May 2013 Newcastle–Moore tornadic mesocyclone observed from the operational KTLX radar and an experimental phased array radar. The dual-Doppler analyzed vortex flow (VF) field reveals the following features: The axisymmetric part of the VF is a well-defined slantwise two-cell vortex in which the maximum tangential velocity is nearly 40 ms−1 at the edge of the vortex core (0.6 km from the vortex center), the central downdraft velocity reaches -35 ms−1 at 3 km height, and the surrounding-updraft velocity reaches 26 ms−1 at 5 km height. The total VF field is a loosely-defined slantwise two-cell vortex consisting of a nearly-axisymmetric vortex core (in which the ground-relative surface wind speed reaches 50 ms−1 on the southeast edge), a strong non-axisymmetric slantwise downdraft in the vortex core, and a main updraft in a banana-shape area southeast of the vortex core which extends slantwise upward and spirals cyclonically around the vortex core. The single-Doppler analysis with observations from the KTLX radar only exhibits roughly the same features as the dual-Doppler analysis but contains spurious vertical-motion structures in and around the vortex core. Analysis errors are assessed by leveraging the findings from Parts II and III, which indicate that the dual-Doppler analyzed VF is accurate enough to represent the true VF but the single-Doppler analyzed VF is not (especially for non-axisymmetric vertical motions in and around the vortex core), so the dual-Doppler analyzed VF should be useful for initializing/verifying high-resolution tornado simulations.