In a cylindrical coordinate system (R, 0, Z) that follows the vortex motion, the boundary layer equations are (see Smith 1968 and Kepert 2001 for a detailed derivation),
where R is distance from the vortex center, 0 is azimuth relative to the direction of motion, f is the Coriolis parameter, U, V, and W are the storm-relative radial, tangential and vertical wind velocities, respectively, K is the vertical diffusion coefficient of the horizontal momentum, and Vgr is the tangential wind velocity under gradient wind balance; see for example Eq. (1).
As Kepert (2001) and Smith (1968), we solve Eqs. (2) for the case of a semiinfinite domain, when gradient wind balance is satisfied asymptotically as Z ! 1, dU -
The translation velocity Vt enters through the surface boundary conditions. Suppose that the vortex is translating in the positive x-direction with constant speed Vt. Using a viscous surface stress formulation similar to Smith (1968, 2003) and Kepert (2001), the conditions to be satisfied at the surface boundary (Z = 0) are:
where CD is a surface drag coefficient; see for example Rosenthal (1962), Smith (1968, 2003), Kepert (2001), and Kepert and Wang (2001). For 1/CD = 0, Eq. (4) corresponds to no slip conditions (U = — Vt cos 0, V = Vt sin 0) at the surface boundary.
For R !i, the system in Eq. (2) reduces to the classic Ekman BL equations under geostrophic conditions (Kundu and Cohen, 2004):
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