## Review of Boundary Layer Models

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The focus of this review is on BL models for moving tropical cyclones, but studies of stationary TCs that are relevant to what follows are also mentioned.

Boundary layer models differ mainly in their treatment of altitude Z and the surface boundary conditions. In one of the earlier studies of moving TCs, Myers and Malkin (1961) used a Lagrangian parcel trajectory approach to study the horizontal winds inside the BL. The authors assume that the frictional drag force is proportional to the square of the wind speed with equal tangential and radial components. Another (implicit) assumption is that the velocity of the background flow is zero rather than equal to the translation velocity of the TC. A finding of the study is that, when a TC in the northern (southern) hemisphere moves, the radial convergence is maximum at the right-front (left-front) quadrant of the vortex and the location of this maximum rotates anticyclonically as the translation velocity Vt increases.

Based on the work of Chow (1971), Shapiro (1983) approximated the boundary layer of a moving TC by a slab of constant depth H = 1 km. The horizontal momentum equations are formulated in cylindrical coordinates that translate with the vortex and then averaged in the vertical direction. This results in a system of two partial differential equations (PDEs) that are solved numerically for the vertically averaged tangential V(R, 6) and U(R, 6) radial wind velocity as a function of radius R and the azimuth 6 relative to the direction of TC motion. Contrary to Myers and Malkin (1961), Shapiro's (1983) formulation assumes that the frictional drag force is parallel to the surface-relative flow and its magnitude is proportional to the square of the composite surface-relative wind velocity. Although the two studies use different formulations for the friction-induced convergence, they both produce maximum convergence at the right-front quadrant of vortices in the northern hemisphere. However, in Shapiro's model the location of the maximum does not depend on the translation velocity, whereas in Myers and Malkin's (1961) analysis it does.

The main limitation of Shapiro's (1983) approach is that it approximates the BL as a slab of constant depth and hence cannot resolve the variation of U and V with height. According to Anthes (1971), this leads to overestimation of the radial and vertical velocities close to the vortex core; see discussion of Fig. 3 below. Another limitation of Shapiro's analysis is that the numerical stability of the system depends on TC parameters such as the depth of the boundary layer H, the translation velocity Vt, the vertical diffusion coefficient K and the surface drag coefficient CD. This limitation becomes important when, as for example in risk studies, one needs to calculate the wind field under a wide variety of conditions.

A third BL model for moving tropical cyclones was proposed by Kepert (2001) (see also refinements in Kepert 2006b). Kepert's formulation neglects vertical advection and linearizes the horizontal advection. This produces a system of linear PDEs that is solved analytically for the radial, tangential and vertical wind velocities (U, V and W, respectively) as a function of R, 6, and Z. Kepert (2001) uses a bulk formulation of the surface stresses similar to those of Rosenthal (1962), Shapiro (1983) and Smith (1968, 2003). However, the surface boundary condition is linearized to allow analytical integration. Linearization produces inaccurate results close to the TC center (R < 2-3 RmaX) where the horizontal gradient of the wind components is high, when the vertical gradient of the horizontal wind velocity is large (this happens for large surface drag coefficient CD; see discussion on Fig. 2 below), for high translation velocities (Vt > 5 m/s), and under inertially neutral conditions (B > 1.6-1.8). Other linearizations of the horizontal momentum equations have been proposed by Haurwitz (1935), Rosenthal (1962), Miller (1965) and Elliassen and Lystad (1977), but these formulations are for stationary vortices (Vt = 0).

An order of magnitude analysis by Smith (1968) shows that in the near-core region the nonlinear terms are as important as the linear ones. To include the nonlinear terms, Smith (1968) (see also refinements in Leslie and Smith 1970 and Bode and Smith 1975) used the Karman and Pohlhausen momentum integral method to calculate the radial U(R,Z) and tangential V(R,Z) wind velocities in a stationary vortex. In the momentum integral method (Schlichting, 1960), one avoids an explicit analysis of altitude Z by assuming vertical profiles for U and V that satisfy the boundary conditions at the surface (Z = 0), and tend asymptotically to gradient balance as Z !i; see Eq. (3) below. Specifically, Smith (1968) used profiles of the Ekman type, with an amplitude coefficient E and a dimensionless BL scale thickness d as parameters. The horizontal momentum equations are vertically integrated to produce a system of ordinary differential equations that are solved numerically to obtain E and d as a function of R.

The main limitation of Smith's (1968) model is that it does not consider storm motion. Also, Smith's (1968) formulation is theoretically correct only for the case of no slip at the surface boundary (i.e. for CD ! 1); see Proposed Model. In the following section, we extend Smith's (1968) model to include storm motion and correct the formulation for the general case of stress surface boundary conditions. 