D 2v

Denote by Rg the distance from the TC center beyond which the geostrophic model in Eq. (5) is approximately valid; say Rg « 1000 km (Smith 1968, Kundu and Cohen 2004). Also denote by KM the vertical diffusion coefficient under geostrophic conditions and let Vg = Vgr(Rg) be the gradient tangential wind for R

Rg and

Zg = (Km/f )1/2 (Zg has the meaning of vertical length scale for the depth of the boundary layer; see below). Then one can write Eq. (2) in dimensionless form, as

22 v2 v2

ak du ak dv u + vt cos y = — — , v — vt sin y = — —, and w = 0, at z = 0 (b)

In Eqs. (6) and (7), Ro = Vg/(Rgf) is the Rossby number and r = R/Rg, z = Z/Zg, vt = Vt/Vg, u = U/Vg , v = V/Vg, w = (WRg)/(VgZg), Vgr = Vgr/Vg, k = K/Km, and a = Km/ (CD Zg Vg) are dimensionless quantities. In his model for axi-symmetric vortices, Smith (1968) allows K to be different from KM and to vary radially. In the present extension to moving vortices, we further allow K to vary azimuthally. Vertical integration of Eq. (6) under boundary conditions (7a) gives

2 dv

where wœ is the dimensionless vertical wind velocity at Z ! 1. Next we discuss how Eq. (8) is solved under the conditions (7b).

Momentum Integral Method

Similar to Smith (1968) we take the boundary layer thickness to be proportional to Zg = (KM/f )1/2, with proportionality coefficient d. In Smith's (1968) axi-symmetric formulation d exhibits only radial variation, but in the present case of moving cyclones we allow d to vary also azimuthally.

Define -q = Z/[Zg d(r,0)] = z/d(r,ff) and notice that the geostrophic model in Eq. (5) is satisfied for d = a/2; see e.g. Kundu and Cohen (2004). Following the derivations of Schlichting (1960), Mack (1962) and Smith (1968), but allowing azimuthal dependence of u and v, the solution of Eq. (8) can be approximated as u(r, y,V)=E(r, Ô)C(r, Ô,V) (a) v(r, y,V) =O(r, y,V) (b)

where E(r,8) is an unknown function, usually referred to as the amplitude coefficient, and C and O are such that (u,v) = (C,O) satisfy Eqs. (6) and (7) under geostrophic conditions (R > Rg). After some algebra one obtains

C(r, 8,j) = g(r, 8,j)vt cos 8 + f (r, 8,j)(vgr - vt sin 8) - vt cos 8 (a) Q(r, 8, j) = g(r, 8, j)(vgr - vt sin 8) -f (r, 8,j)vt cos 8 + vt sin 8 (b)

where f (r, 8,j) = -e r [a1(r, 8) sin j + a2(r, 8) cos j] (a) g(r, 8, j) = 1 - e-r [a1(r, 8) cos j + a2 (r, 8) sin j] (b)

The parameters a1(r, 8) and a2(r, 8) in Eq. (11) are calculated so that (u, v) in Eq. (9) satisfy condition (7b). This gives

L2L4 - L1L5 L1

V vgrS

vt cos 8

V vgrS J

vgr^

vt cos 8

The parameters a1 and a2 are constants independent of r and 8 (a1 = 1 and a2 = 0) only for a stationary TC and a = 0 (i.e. CD ! 1, no slip conditions). Hence, Smith's (1968) axi-symmetric formulation, where a1 and a2 are assumed constant independent of r, is theoretically correct only in the case of no slip conditions at the surface boundary.

By combining Eq. (8)-(11) and after some algebra, one obtains the following system of differential equations in E and d:

Bi (r, 0) @E + B2(r, |E + B3(r, 0) ^ + B4(r, 0) ^ = ^(r, 0) (a) Ci (r, 0) |E + C3 (r, 0) + C4(r, 0) = Cs(r, 0) (b)

with coefficients

B5 = yA6 + dR0(rA5/R0 + A3 + Ai3 + Ai + Mn), Ci = r2R0d(Ai2 - vA^)

C3 = r2R^A2 - VgrA7), C4 = rR^A4 + Ai9 - virA8) kr2

C5 = -— A9 + rR0d [vgr(Ai8 + rAi7 + A7 - A20) - rA7/R0 - Ai5 - rAM - 2A2]

Analytical expressions for the parameters Ai-A20 in Eq. (i5) are derived in Appendix A.

The nonlinear system in Eq. (i4) can be integrated numerically to obtain E(r,0) and d(r, 0). For this purpose we use a scheme that is implicit in 0 and explicit in r. Integration starts at r = i where we set E = i and d = V2(th ese are the values under geostrophic conditions) and moves inward using a stepwise integration procedure. At each step i in r, integration with respect to 0 is performed simultaneously for all azimuthal locations using a central difference approach; see e.g. Chapra and Canale (2002).

Since the parameters in Eq. (i4) depend on E and d, a first approximation to the solution at step i is obtained by evaluating all parameters using the values of E and d from step i-i. This procedure is iterated until E and d at step i converge.

After E and d (and hence the horizontal wind components U and V) are obtained, the vertical wind velocity W is calculated using mass conservation, as

In what follows, we refer to Eqs. (6) and (7) and their solution by the momentum integral method as the Modified Smith (MS) model. In the next section we compare results for a specific storm with wind estimates from the Shapiro (i983) and Kepert (200i) formulations and with MM5 simulations, and in Sensitivity Analysis we examine how the winds generated by the MS model vary with a number of storm parameters.

Model Comparison

First we illustrate the results from the MS model and then compare the performance of that model with others for stationary and moving tropical cyclones.

Figure 1 shows the storm-relative radial and tangential wind velocity fields U and V and the vertical wind velocity W at elevations Z = 0, 0.5, 1 and 2 km obtained using the MS model. The tropical cyclone translates eastward in the northern hemisphere, with velocity Vt = 5 m/s. Asymptotically as Z ! 1, the tangential winds satisfy Holland's profile in Eq. (1) with parameters Vmax = 50 m/s, Rmax = 40 km and B = 1.6. Results are for non-slip conditions at the surface boundary (1/CD = 0) and constant vertical diffusion coefficient K = KM = 50 m2/s. This value of K is often quoted in the literature (e.g. Smith 1968, Shapiro 1983, Kepert 2001; Kepert 2006b) and is close to values extracted from MM5 simulations (M. Desflots 2006, personal communication). The Coriolis parameter f is set to 5 10—5 sec-1, which corresponds to a latitude ' of 20o North.

The model reproduces the conditions UIZ=0 = — Vt cos0, VIZ=0 = Vt sin 0, and WIZ=0 = 0 at the surface boundary; see also discussion of Fig. 2 below. Translation of the tropical cyclone causes intensification of the radial fluxes at the right and right-front of the vortex. Specifically, the maximum of U (x symbol in Fig. 1) is located at the right/right-front of motion and close to the vortex center for Z = 500 m, and moves outward while rotating clockwise as Z increases. Intensification of V is at the left-front of the vortex, with an asymmetry that decreases as one approaches gradient balance. Due to radial convergence at different altitudes, the vertical velocity W increases monotonically with Z. In addition, storm translation causes W to intensify at the right and right-front of the vortex. Similar qualitative findings on U, V and W have been reported by Kepert (2001) and Kepert and Wang (2001).

For the same storm, Fig. 2 compares the vertical profiles of the azimuthally averaged radial and tangential winds generated by the MS and Kepert (2001) models. Azimuthal averaging produces results that correspond to a stationary, and hence, axi-symmetric TC. The profiles shown in Fig. 2 are at radial distances R = Rmax = 40 km and R = 2.5Rmax =100 km from the vortex center.

As was pointed out in Review of Boundary Layer Models, the linearized stress formulation of Kepert (2001) is accurate only for small drag coefficients CD. In the present case CD and Kepert's (2001) model correctly reproduces the non-slip condition UIZ=0 = 0 in the radial direction (see Fig. 2) but fails to reproduce the condition VIZ=0 = 0 in the tangential direction (Fig. 2). This inconsistency results in underestimation of the frictional stresses at the surface boundary and hence of the radial fluxes relative to the MS formulation; see Fig. 2 and discussion of Fig. 3 below.

Kepert's (2001) model also fails to reproduce the variation of the radial wind U with R. Observations using GPS dropsonde data (Kepert 2006a, b) show that the maximum of U increases as one approaches the center of the TC. While the MS model is consistent with this observation, the maximum values of U at 40 and 100km from Kepert's (2001) approach are about the same.

-3OO 3OO

-100

-3OO 300

////

\x\\

i/y

A

Ik

s* /jy

fox

(\v\

/ r 1

-300

V \

' /

zero vertical velocity

-300 -200 -100 0 100 200 300 -300 -200-100 0 100 200 300 -150 -100 -50 0 50 100 150

Distance from storm center (km)

Fig. 1 MS boundary layer solution for the tangential V, radial U and vertical W wind velocity fields at altitudes Z = 0, 0.5, 1, and 2 km. V and U are velocities relative to the moving vortex. The location of the maximum of U is denoted by an x symbol. The tropical cyclone translates eastwards (to the right) with velocity Vt = 5 m/s. All figures are generated under non-slip conditions at the surface boundary and using a constant vertical diffusion coefficient K = 50 m2/s. Other parameters are Vmax = 50 m/s, Rmax = 40 km and B =1.6

In both the MS and Kepert (2001) models, the depth of the boundary layer H, defined as the height Z where U ~ 0, increases with increasing R. In the MS solution, H is about 2.2 km at R = 100 km and about 1.5 km at R = 40 km, whereas

00 10 20 30 40 50 60

Fig. 2 Comparison of the axi-symmetric component of the wind field from Kepert's (2001) model and the MS model. Vertical profiles of (a) the radial wind velocity U, and (b) the tangential wind velocity V at 40 km and 100 km from the vortex center. Same storm as in Fig. 1

the corresponding values from Kepert's (2001) analysis are 1.5 and 0.8 km. Both sets of estimates are order-of-magnitude correct, with the MS values been closer to observations (e.g. Frank 1977 and Kepert 2006a).

Figures 3 and 4 compare the vertically averaged radial and tangential winds and the vertical winds obtained using the Shapiro (1983), Kepert (2001) and MS models as well as MM5 simulations. Figure 3 shows the radial variation of the axi-symmetric component, whereas Fig. 4 includes the asymmetry due to motion through contour plots. The reasons for vertically averaging U and V (over a depth of 1 km) are to ease model comparison, since Shapiro (1983) treats the BL as a slab of constant depth H =1 km, and to reduce the effect of vertical fluctuations in the MM5 solution. All storm parameters are the same as in Fig. 1, except for the drag coefficient, which is set to 0.003. This value of CD is close to values extracted from MM5 simulations for oversea conditions (M. Desflots 2006, personal communication), does not introduce significant distortions in Kepert's (2001) linear formulation, and does not cause numerical oscillations in Shapiro's (1983) approach. Also, values of CD close to 0.003 are often quoted in the literature for wind speeds in the

00 10 20 30 40 50 60

Fig. 2 Comparison of the axi-symmetric component of the wind field from Kepert's (2001) model and the MS model. Vertical profiles of (a) the radial wind velocity U, and (b) the tangential wind velocity V at 40 km and 100 km from the vortex center. Same storm as in Fig. 1

Fig. 3 Comparison of the Shapiro (i983), Kepert (200i) and MS boundary layer solutions with MM5 simulations for an axi-symmetric (stationary) TC. (a) Vertically averaged radial wind velocity U, (b) vertically averaged tangential wind velocity V and (c) vertical wind velocity W at an altitude of i km. All figures are generated using a constant vertical diffusion coefficient K = 50 m2/s and a surface drag coefficient CD = 0.003 and averaging is over a depth of i km. Other parameters are V,^ = 50 m/s, R^ = 40 km and B = i.6

Fig. 3 Comparison of the Shapiro (i983), Kepert (200i) and MS boundary layer solutions with MM5 simulations for an axi-symmetric (stationary) TC. (a) Vertically averaged radial wind velocity U, (b) vertically averaged tangential wind velocity V and (c) vertical wind velocity W at an altitude of i km. All figures are generated using a constant vertical diffusion coefficient K = 50 m2/s and a surface drag coefficient CD = 0.003 and averaging is over a depth of i km. Other parameters are V,^ = 50 m/s, R^ = 40 km and B = i.6

range encountered in hurricanes (e.g. Large and Pond i98i; Powell et al. 2003; Donelan et al. 2004).

In general, the MS model predictions are close to the MM5 simulations, except that in MM5 the vertical velocities are more peaked near R,ax (mainly due to rainband effects, which the BL models do not resolve) and the contour plots are

-300 300

200 -100 -" 0 -100-200 --300 300 200 100 I 0 -100 -200 -300 300 200 100 i 0 -100 -200 -300

^—-5—

/ ---v \

\

/ /^■'^áhíiVl i- \

/ I 1 Q} ) MY

w

I'

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300 200 100 0

-100

-200

-300 300

100 150

Fig. 4 Comparison of results form the Kepert (2001), Shapiro (1983) and MS models with MM5 simulations for a TC that translates eastwards (to the right) with velocity Vt = 5 m/s. Vertically averaged radial wind velocity U (left column), vertically averaged tangential wind velocity V (middle column) and vertical wind velocity W at an altitude of 1 km (right column). V and U are relative to the moving vortex and averaging is over a depth of 1 km. All other parameters are the same as in Fig. 3

-300 -200 -100 0 100 200 300 -300 -200 -100 0 100 200 300 -150 -100 -50

Distance from storm center (km)

100 150

Fig. 4 Comparison of results form the Kepert (2001), Shapiro (1983) and MS models with MM5 simulations for a TC that translates eastwards (to the right) with velocity Vt = 5 m/s. Vertically averaged radial wind velocity U (left column), vertically averaged tangential wind velocity V (middle column) and vertical wind velocity W at an altitude of 1 km (right column). V and U are relative to the moving vortex and averaging is over a depth of 1 km. All other parameters are the same as in Fig. 3

more erratic due to local fluctuations. Also the Kepert (2001) model reproduces well the vertically averaged tangential winds of MM5, but that model severely underestimates the radial and vertical flows, especially in the near-core region. This is due to inaccuracy of Kepert's linearization at radial distances smaller than 2-3Rmax; see Review of Boundary Layer Models. For example, flight observations show that, in the vicinity of Rmax, W is in the range 0.5-3 m/s or higher (Willoughby et al. i982; Jorgensen i984a, b; Black et al. 2002), whereas Kepert's (200i) estimates are around 0.i-0.4 m/s. In the far field (R > 2.5Rmax), Kepert's (200i) model and MM5 produce vertical velocities that are similar and in good agreement with observations (Jorgensen i984a, b; Black et al. 2002).

Shapiro's (i983) model has an opposite behavior in the near-core region, where it overpredicts the radial and vertical flows by a factor of about 2. This is consistent with the finding by Anthes (i97i) for slab-layer models; see Review of Boundary Layer Models.

Overall, we find that the MS model produces realistic estimates of the tangential, radial and vertical wind velocities and is numerically more stable and more accurate than the boundary layer models of Kepert (200i) and Shapiro (i983). Next we use the MS model to study the sensitivity of the velocity fields to various storm parameters.

Sensitivity Analysis

Figures 5, 6 and 7 show the sensitivity of MS model results to tropical cyclone characteristics: the tangential wind speed under gradient balance (parameterized here in terms of V,^, R^x and B; see Eq. (i)), the vertical diffusion coefficient K, the surface drag coefficient CD, and the translation velocity Vt.

Figures 5 shows the sensitivity of the azimuthally averaged (axi-symmetric) velocities to Vmax, R,^, B, K and CD. The base case (solid lines in Fig. 5) corresponds to the storm used in Fig. i. Sensitivity is evaluated by varying the parameters one at a time around their base-case values. V,^ affects only the amplitude of the radial profiles, whereas other parameters affect mainly the shape of the profile (e.g. rate of decay with distance) or both. The effects on the three velocity components are however not uniform. For example, R^^ has negligible influence on the peak value of the horizontal winds U and V, but affects significantly the peak vertical velocity.

Figure 6 shows the effect of CD on the wind fields for a TC that moves eastward with velocity Vt = 5 m/s. Higher values of CD correspond to increased friction at the surface boundary and stronger asymmetry of the three velocity components.

Finally, Fig. 7 shows sensitivity to the translation velocity Vt when the other parameters are fixed at Vmax = 50 m/s, Rmax= 40 km, B = i.6, K = 50 m2/s and CD = 0.003. One sees that U and W intensify in the NE quadrant relative to the direction of motion, whereas V intensifies to the left of the velocity vector. The asymmetry increases with increasing Vt.

Qualitatively similar sensitivity results are obtained when using Kepert's (200i) model (not shown here). This is expected, since Kepert's (200i) model is a linearized variant of the present MS formulation. However, Kepert's linearization has approximate validity only for large R, CD << i and small Vt; see Review of Boundary Layer Models.

Fig. 5 Sensitivity analysis of MS model results to storm parameters for a stationary TC. Vertically averaged (over a 1km depth) radial wind velocity U (left column) and tangential wind velocity V (middle column) and vertical wind velocity W at Z = 1 km (right column). Solid lines correspond to Vmax = 50 m/s, Rmax = 40 km, B = 1.6, K = 50 m2/s and non-slip conditions at the surface boundary. Each row shows results under perturbation of one parameter

Fig. 5 Sensitivity analysis of MS model results to storm parameters for a stationary TC. Vertically averaged (over a 1km depth) radial wind velocity U (left column) and tangential wind velocity V (middle column) and vertical wind velocity W at Z = 1 km (right column). Solid lines correspond to Vmax = 50 m/s, Rmax = 40 km, B = 1.6, K = 50 m2/s and non-slip conditions at the surface boundary. Each row shows results under perturbation of one parameter

Distance from storm center (km)

Fig. 6 Sensitivity of MS results to the drag coefficient CD for a tropical cyclone that translates eastwards (to the right) with velocity Vt = 5 m/s. Vertically averaged radial wind velocity U (/e/t column), vertically averaged tangential wind velocity V (midd/e co/umn), and vertical wind velocity W at an altitude of 1 km (right column). V and U are relative to the moving vortex and averaging is over a depth of 1 km. Other parameters are the same as in Fig. 4

Distance from storm center (km)

Fig. 6 Sensitivity of MS results to the drag coefficient CD for a tropical cyclone that translates eastwards (to the right) with velocity Vt = 5 m/s. Vertically averaged radial wind velocity U (/e/t column), vertically averaged tangential wind velocity V (midd/e co/umn), and vertical wind velocity W at an altitude of 1 km (right column). V and U are relative to the moving vortex and averaging is over a depth of 1 km. Other parameters are the same as in Fig. 4

Fig. 7 Sensitivity of MS model results to the translation velocity Vt. Notation and parameters are the same as in Fig. 6, except for CD = 0.003

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