Tropical cyclone boundary layer shocks
  This paper presents numerical solutions and idealized analytical solutions of axisymmetric, $f$-plane models of the tropical cyclone boundary layer. In the numerical model, the boundary layer radial and tangential flow is forced by a specified pressure field, which can also be interpreted as a specified gradient balanced tangential wind field $v_{\rm gr}(r)$ or vorticity field $\zeta_{\rm gr}(r)$. When the specified $\zeta_{\rm gr}(r)$ field is changed from one that is radially concentrated in the inner core to one that is radially spread, the quasi-steady-state boundary layer flow transitions from a single eyewall shock-like structure to a double eyewall shock-like structure. To better understand these structures, analytical solutions are presented for two simplified versions of the model. In the simplified analytical models, which do not include horizontal diffusion, the $u(\partial u/\partial r)$ term in the radial equation of motion and the $u[f+(\partial v/\partial r)+(v/r)]$ term in the tangential equation of motion produce discontinuities in the radial and tangential wind, with associated singularities in the boundary layer pumping and the boundary layer vorticity. In the numerical model, which does include horizontal diffusion, the radial and tangential wind structures are not true discontinuities, but are shock-like, with wind changes of 20 or 30 m s$^{-1}$ over a radial distance of a few kilometers. When double shocks form, the outer shock can control the strength of the inner shock, an effect that likely plays an important role in concentric eyewall cycles. 
Christopher J. Slocum, Gabriel J. Williams, Richard K. Taft, and Wayne   H. Schubert
physics.ao-ph