## Abstract

The transient response of the tropical cyclone boundary layer is studied using linearized and nonlinear models, with particular focus on the frictionally forced vertical motion. The impulsively started, linearized tropical cyclone boundary layer is shown to adjust to its equilibrium solution via a series of decaying oscillations with the inertial period . In the nonlinear case, the oscillation period is slightly lengthened by inward advection of the slower-evolving flow from larger radii, but the oscillations decay more quickly. In an idealized cyclone with small sinusoidal oscillations superimposed on the gradient wind, the equilibrium nonlinear boundary layer acts as a low-pass filter with pass length scaling as , where is the 10-m frictional inflow. This filter is absent from the linearized boundary layer. The eyewall frictional updraft is similarly displaced inward of the radius of maximum winds (RMW) by a distance that scales with , owing to nonlinear overshoot of the inflowing air as it crosses the relatively sharp increase in I near the eyewall. This displacement is smaller (other things being equal) when the RMW is small, and greater when it is large, including in secondary eyewalls. The dependence of this distance on may explain, at least partially, why observed RMW are seldom less than 20 km, why storms with relatively peaked radial profiles of wind speed can intensify more rapidly, and why some secondary eyewalls initially contract rapidly with little intensification, then contract more slowly while intensifying.

## 1. Introduction

In meteorology and other branches of fluid dynamics, it is often a convenient and sufficiently accurate approximation to assume that the boundary layer is in equilibrium with the flow external to the boundary layer and with the bounding surface(s). For example, classical Ekman theory is steady state, so any use of that theory to calculate Ekman pumping, or the drift of icebergs at an angle to the wind, assumes that the time to develop an Ekman layer is less than the time scale on which the flow external to the boundary layer varies. A similar assumption is often made when studying flow around objects, namely that the mean boundary layer flow is steady. While this assumption is not always valid,1 its application where appropriate can considerably help understanding.

In the case of vortex boundary layers, Eliassen and Lystad (1977) argued that the diffusive time scale for such flows is the reciprocal of the inertial stability of the flow above the boundary layer . The relevant quantities are the turbulent diffusivity K and the boundary layer depth scale δ, which they showed is . Dimensional analysis then gives the time scale as . The same dimensional argument can be applied to the classical Ekman layer, where the adjustment time scale is . The scaling argument for the classical case has been confirmed by Lewis and Belcher (2004), who derived an analytic solution for the unsteady Ekman problem and showed that the adjustment toward the steady state proceeds via a series of decaying inertial oscillations with period . Gao and Ginis (2016) similarly noted that their diagnostic model of the tropical cyclone boundary layer reached equilibrium “within a few inertial cycles.”

However, the tropical cyclone boundary layer2 may possess other natural time scales. For example, horizontal and vertical advection are important to its dynamics (Kepert 2001; Kepert and Wang 2001), so advective time scales arise. It is therefore necessary to examine Eliassen and Lystad’s (1977) scaling arguments, for if their time-scale argument is reasonably accurate, then we can validly regard the boundary layer of the inner part of the cyclone as being slaved to the parent vortex to good approximation, leading to a significant simplification for theory. The first aim of this paper, then, is to compare the temporal evolution of the tropical cyclone boundary layer to the inertial time scale (section 4a).

Ascent through the top of the boundary layer in regions of frictional convergence is an important mechanism by which the boundary layer can affect the rest of the cyclone. For instance, frictional convergence favors the development of convection, either directly through ascent triggering convection, or through lifting high-equivalent-potential-temperature air away from the surface and reducing the stability. It is therefore important to understand what determines frictional convergence within the tropical cyclone boundary layer, especially near the eyewall(s). Kepert (2001) gives an expression for the frictional updraft derived from a linearized axisymmetric boundary layer model,

where is the updraft in the limit as , is the gradient wind and its vertical vorticity, is the drag coefficient, is the departure of the near-surface flow3 from , r is radius, and f is the Coriolis parameter. Kepert (2013) discusses some properties of this equation, including its relationship to the similar equation for the classical Ekman boundary layer. He also expands the radial derivative,

and shows that the first term dominates near an eyewall. This term consists of the radial vorticity gradient, times the (linearized) surface stress, divided by the square of the absolute vorticity. Thus, frictional updrafts in the linearized model form where there is a sufficient decrease of gradient vorticity with increasing radius. Because of the factor , even relatively weak radial vorticity gradients can produce quite strong updrafts if they are in an environment of low absolute vorticity. In the linear model, the eyewall updraft maximum typically occurs slightly outside of the radius of maximum negative radial vorticity gradient , again because of the factor . Ooyama (1969) derives a similar equation, under more restrictive assumptions, for the frictional updraft at the top of the linearized, depth-averaged boundary layer that comprises the lowest layer of his three-layer tropical cyclone model.

A limitation of (1) is that it is highly sensitive to small-amplitude short-wavelength perturbations in the pressure field, for contains . Montgomery et al. (2014) applied an equation similar to (1) to output from a NWP simulation of a tropical cyclone and found that the noise from this effect resulted in the diagnosed updraft being very different to that from their NWP simulation. They claimed to have refuted Kepert’s (2013) hypothesis on the role of the boundary layer in eyewall replacement cycles because of this sensitivity. However, Kepert and Nolan (2014) use the adjustment time scale to argue that the tropical cyclone boundary layer should be relatively insensitive to features in the gradient wind field below a certain scale, because the inflowing air would pass these features in a time less than and therefore not have time to adjust to the altered conditions associated with the feature (such as the gradient wind and its vorticity) that otherwise affect the boundary layer structure. Essentially, they argue that if is the time scale, then the appropriate length scale is U/I, where U is a scale for the strength of the frictional inflow. While a linearized TCBL model responds to small-scale features in the gradient wind field, they show that a nonlinear model, which included the full nonlinear advection terms, does not. Since the nonlinear model is insensitive to these features, and the diagnosed flow from that model satisfies Montgomery et al.’s (2014) proposed test of Kepert’s (2013) hypothesis, Kepert and Nolan (2014) conclude that Montgomery et al.’s (2014) criticism is void.

The second aim of this paper is to present a more detailed analysis of the degree to which nonlinearity in the tropical cyclone boundary layer suppresses the high sensitivity to short-scale features in pressure, or equivalently, the gradient wind (section 4b). In particular, we show that the nonlinear boundary layer model possesses an inherent low-pass filter with cutoff scale U/I.

Convective heating within the tropical cyclone eyewall intensifies the storm. Theoretical results (Schubert and Hack 1982) showing that heating inside of the RMW intensifies the storm more efficiently than that outside of the RMW, because it occurs in a region of higher inertial stability, have recently been supported by observational studies showing that tropical cyclones with heating inside of the radius of maximum winds tend to intensify more rapidly than those where it is outside (Rogers et al. 2015). Hence, if we assume that the location of the frictional eyewall updraft influences the location of such convective heating, the factors that determine the radius of the eyewall frictional updraft become important. We show in section 4c that in the nonlinear axisymmetric boundary layer, the eyewall updraft is displaced inward from the radius of maximum negative vorticity gradient in a modified Rankine vortex by the scale U/I. In particular, this inward displacement is greater for storms with large eyes and is greater for secondary eyewalls than for primary.

This displacement is also important for Kepert’s (2013) hypothesized positive feedback on secondary eyewall formation. That hypothesis proposes a positive feedback between a local vorticity perturbation to the tropical cyclone, which creates a frictionally forced updraft due to the associated increase in , which favors convection, which in turn creates a positive vorticity tendency through vortex stretching. The displacement is important because it determines the position of the updraft relative to the vorticity perturbation. Then, to the extent that the frictional convergence tends to favor convection, vortex-tube stretching in the convective updrafts will reinforce the vorticity perturbation that is responsible for the locally enhanced vorticity gradient. With the understanding of this displacement developed here, we offer explanations for further observed properties of secondary eyewall formation and the subsequent eyewall replacement cycle.

## 2. The model

The model calculates the boundary layer flow in a tropical cyclone as the response to a fixed axisymmetric pressure field representative of a tropical cyclone. Conceptually, the model used is close to that developed by Kepert and Wang (2001) and subsequently modified by Kepert (2012), but is two dimensional. Similar models have also been developed by Gao and Ginis (2014) and Williams (2015). There is no feedback from the boundary layer to the prescribed pressure field, and so such models represent one side of what is in reality a two-way interaction, as discussed by Kepert and Wang (2001). Using such a model, in which the flow above the boundary layer is held fixed, is appropriate for this study since we do not want to get adjustment time scales from the rest of the cyclone confused with those for the boundary layer. The use of a diagnostic model also enables controlled experiments, in which the pressure field imposed by the rest of the cyclone, including any small-scale features, can be precisely prescribed and held fixed. The two-dimensional model described by Kepert and Wang (2001), however, includes a small amount of implicit horizontal diffusion through its third-order upwinding horizontal advection scheme. Since the aim here is to quantify the extent to which boundary layer dynamics filter out small-scale features in the forcing gradient wind, it is necessary to use a model with more precisely known horizontal diffusion so as to not confound any damping by the boundary layer dynamics with that from the numerical diffusion. A small amount of horizontal diffusion is necessary to stabilize the numerics, but this is explicit in the new model and therefore accurately known. The new model is also formulated in such a way that it can solve the time-dependent linearized equations. While Kepert (2001) gave an analytic solution for the steady form of these, that solution required a much simpler turbulence parameterization than used in the nonlinear model. To isolate the effects of filtering by nonlinear advection, we must solve the linearized system with the same horizontal diffusion and surface flux and turbulence parameterizations as in the nonlinear, necessitating a numerical solution. The final motivation for recoding the model was to write it in two-dimensional axisymmetric form, rather than three dimensional, enabling the model to run at higher spatial resolution on very modest computer resources.

The axisymmetric equations of motion for tropical cyclone boundary layer flow in cylindrical coordinates are

where u, , and w are the radial, azimuthal, and vertical wind components, respectively; is the gradient wind, assumed constant with height; and and are the vertical and horizontal turbulent diffusivities, respectively. The radial pressure gradient force has been written in terms of the gradient wind and is assumed constant with height. We write the azimuthal flow as in (3) and (4), giving

In this equation, the terms in boxes constitute the linearized model, and (apart from the horizontal diffusion) are the time-dependent analog of those solved by Kepert (2001). The vertical velocity is obtained by vertical integrating the continuity equation

with the boundary condition at Equations (5)(7) are discretized on an unstaggered rz grid. The radial spacing is constant at 1 km in the simulations shown here, while the vertical grid is stretched using a hyperbolic sine transformation following Thompson et al. (1985), with the fluxes and mean variables staggered

The lowest tendency level is thus close to 10.0 m, allowing the use of conventional values for the 10-m drag coefficient and the highest level at 2.47 km. The domain is 400 km wide but only the innermost part is analyzed here.

The time-stepping scheme is fourth-order Runge–Kutta, and horizontal advection is calculated using a sixth-order centered scheme. Horizontal diffusion is fourth-order accurate with a constant diffusion coefficient to be discussed below. Near the domain edges, the order of accuracy is reduced in accordance with the number of available grid points for the stencil. The combination of horizontal advection and time-stepping schemes was shown to be almost nondissipative by applying it to sinusoids with wavelengths from 5 to 80 km on a domain with cyclic boundary conditions. After 24 h of advection by a 10 m s−1 flow, representative of hurricane frictional inflow, the amplitude of the sinusoid was reduced by a factor of less than for the 5-km wavelength and orders of magnitude less for the longer waves. Hence, the only horizontal diffusion of consequence is that explicitly prescribed. The coefficient was chosen to be about the smallest value that delivered numerical stability, . A higher value leads to somewhat quicker adjustment to equilibrium but does not materially affect the results. The linearized version of the model does not require horizontal diffusion for numerical stability because the radial advection is linearized but uses the same value to allow quantitative comparison of the vertical velocity response to changes in the gradient wind field in the two models.

The vertical diffusivity is parameterized by

with mixing length l defined by , where is the von Kármán constant, the asymptotic mixing length (Kepert and Wang 2001; Zhang and Drennan 2012) except where otherwise stated, and S is the magnitude of the vertical wind shear vector. This parameterization is therefore identical to the neutral Louis scheme discussed by Kepert (2012) and differs from the Louis scheme he recommended only in its omission of the static stability. Kepert (2012) showed that this omission has only a small effect on the simulated flow. Vertical advection uses fifth-order upwinding, with the numerical diffusion here not an issue since we are mainly concerned with the horizontal structure of the boundary layer. The outer boundary condition is zero gradient, while at , is imposed. The surface boundary conditions are that and that the surface stress is parameterized by

For simplicity, the drag coefficient is constant with value except where otherwise stated, consistent with high–wind speed measurements (Powell et al. 2003; Donelan et al. 2004; Black et al. 2007). Experiments with a wind speed–dependent did not change the conclusions. At the upper boundary, a zero-gradient boundary condition is used, similarly to Kepert and Wang (2001). This condition has been incorrectly claimed by Smith and Montgomery (2010) to be equivalent to a fixed boundary condition; their arguments are refuted in the  appendix.

Testing of the model included comparisons with other similar models: the analytic solution of Kepert (2001) for the linearized version and the Kepert and Wang (2001) model for the nonlinear version.

## 3. Experimental design

The gradient wind is prescribed by variants of the modified Rankine vortex that optionally include concentric wind maxima, as introduced by Kepert (2013). These profiles are defined in terms of vorticity and take the form

with the profiles smoothed near the transition radii , and as described by Kepert (2013), except that the blending width is reduced to . We use both the above concentric-eyewall form of the profile and the corresponding single-eyewall form, which sets Examples of each are shown in Fig. 1. Simulations of the boundary layer spinup use the latter form of this profile with , , giving a maximum wind of approximately , and a wind speed decay coefficient of . We will refer to the smoothed transitions in (13) as “vorticity steps.”

Fig. 1.

Gradient wind (red) and vorticity (blue; multiplied by 104) in two parametric wind profiles described by (13). (a) A profile with a single eyewall, , , , and blending width 10 km. (b) A profile with a double eyewall; parameters as in (a), with the addition of , , and . The red dots indicate the maximum winds, and the blue dots the local maxima of .

Fig. 1.

Gradient wind (red) and vorticity (blue; multiplied by 104) in two parametric wind profiles described by (13). (a) A profile with a single eyewall, , , , and blending width 10 km. (b) A profile with a double eyewall; parameters as in (a), with the addition of , , and . The red dots indicate the maximum winds, and the blue dots the local maxima of .

To investigate the effects of small perturbations to the vortex, we modify this profile by adding sinusoids of various amplitudes a and wavelengths λ, with the added perturbations again being defined in vorticity space. The added perturbations are smoothed to zero over the outer 20 km so that the outer boundary condition is identical in all runs. Specifically, we use

where is the profile (13) configured for a single eyewall.

## 4. Results

### a. Time scales

Fig. 2.

Hovmöller diagrams of the evolution of the boundary layer. (top) The nonlinear model plotted against time, with (a) the 10-m radial wind, (b) the 10-m azimuthal wind, and (c) the vertical velocity at 1.1-km height. (d)–(f) As in (a)–(c), respectively, but for the linearized model. (g)–(i) As in (a)–(c), respectively, except that time is nondimensionalized by the inertial period . (j)–(l) As in (g)–(i), respectively, but for the linearized model. The contour intervals for u are and for υ are . The contours for w are , with the zero contour shown in black.

Fig. 2.

Hovmöller diagrams of the evolution of the boundary layer. (top) The nonlinear model plotted against time, with (a) the 10-m radial wind, (b) the 10-m azimuthal wind, and (c) the vertical velocity at 1.1-km height. (d)–(f) As in (a)–(c), respectively, but for the linearized model. (g)–(i) As in (a)–(c), respectively, except that time is nondimensionalized by the inertial period . (j)–(l) As in (g)–(i), respectively, but for the linearized model. The contour intervals for u are and for υ are . The contours for w are , with the zero contour shown in black.

Much finer-scale oscillations are apparent in Fig. 2l near a radius of 50 km during the first two inertial periods. These appear to be an analog of the mode noted by Lewis and Belcher [2004, their (17)] and shown by them to be an unphysical consequence of the boundary conditions. Here, the weak horizontal diffusion eliminates it.

Note that these calculations are the results of adjustment from an initial condition with zero boundary layer flow. In the real atmosphere when the cyclone structure changes, the boundary layer will readjust from an initial condition much closer to the final. The time required for adjustment will be correspondingly shorter. A higher value of the horizontal diffusivity (3000 m2 s−1) also leads to more rapid adjustment, without materially affecting the final state.

Radius–height sections of the flow at the end of the period (96 h) are shown in Fig. 3, along with line plots of the 10-m horizontal wind, 1-km vertical velocity, and gradient wind and its vorticity. These are similar to previously published results from the three-dimensional model (Kepert and Wang 2001; Kepert 2010a,b) and also to radius–height sections from full-physics simulations (Braun and Tao 2000; Smith and Thomsen 2010). The difference between linear and nonlinear models is smaller here than when the analytical solution to the linear model (Kepert 2001) is used, because here both models are using the same turbulence and surface flux parameterizations, whereas the analytic solution uses K constant with height and linearizes the surface fluxes. The much larger diffusivity near the surface with vertically constant K (i.e., the analytic solution) led to smaller near-surface shear and hence a smaller departure from gradient balance.

Fig. 3.

Radius–height sections of the steady-state boundary layer. For the nonlinear model, (a) u with contour interval , (c) υ with contour interval , (e) w with contours at , and (g) (red), (blue), (green), (magenta), and (cyan). (b),(d),(f),(h) As in (a),(c),(e), and (g), respectively, but for the linearized model.

Fig. 3.

Radius–height sections of the steady-state boundary layer. For the nonlinear model, (a) u with contour interval , (c) υ with contour interval , (e) w with contours at , and (g) (red), (blue), (green), (magenta), and (cyan). (b),(d),(f),(h) As in (a),(c),(e), and (g), respectively, but for the linearized model.

### b. Space scales

We now examine the effect on the steady-state boundary layer flow of adding sinusoidal perturbations to the basic vortex. Figure 4 shows the flow summary diagrams for oscillations with a wavelength of 40 km and a vorticity amplitude of in the nonlinear and linear models. The vertical velocity responds to these oscillations in both models, but this response is substantially larger in the linear model except near the RMW. In the linear model, the amplitude decreases toward the cyclone center because of the factor in the diagnostic updraft equation [see (2)], while the nonlinear model has the opposite trend with the amplitude increasing inward (except within the eye). The nonlinear model also has the w maxima displaced inward to be mostly inward of the vorticity perturbation maxima at larger radii but becoming nearly in phase immediately outside of the RMW (Fig. 4c). In contrast, they are slightly more than a quarter wavelength outward of the vorticity maxima at all radii in the linear model (Fig. 4d) or slightly outside of the local maxima in the negative vorticity gradient , consistent with Kepert’s (2013) analysis of (2).

Fig. 4.

Summary plots of the flow with and without sinusoidal perturbations, for the (a),(c) nonlinear and (b),(d) linearized models. Plotted in (a) are (red), (blue), (green), (magenta), and (cyan) for the nonlinear model, with the base vortex shown as continuous lines and the vortex with sinusoidal perturbations added shown as dashed lines. (c) The difference between the base vortex and the one with sinusoidal perturbations, with the color scale as in (a). (b),(d) As in (a) and (c), respectively, but for the linearized model. In these plots, the oscillations have a wavelength of 40 km and a vorticity amplitude of .

Fig. 4.

Summary plots of the flow with and without sinusoidal perturbations, for the (a),(c) nonlinear and (b),(d) linearized models. Plotted in (a) are (red), (blue), (green), (magenta), and (cyan) for the nonlinear model, with the base vortex shown as continuous lines and the vortex with sinusoidal perturbations added shown as dashed lines. (c) The difference between the base vortex and the one with sinusoidal perturbations, with the color scale as in (a). (b),(d) As in (a) and (c), respectively, but for the linearized model. In these plots, the oscillations have a wavelength of 40 km and a vorticity amplitude of .

In the nonlinear model, the perturbations in the gradient wind (red) and azimuthal wind (blue) very nearly cancel; that is, the oscillations in the gradient wind have almost no effect on the total azimuthal flow. This near cancellation occurs through most of the inflow layer and at all the oscillation wavelengths tested. Examination of the plots of gradient and azimuthal wind in the secondary eyewall study of Kepert and Nolan (2014, their Fig. 4) shows that the azimuthal wind is a much smoother function of radius than the gradient wind in those simulations. That smoothness is consistent with the cancellation found in the azimuthal wind response here.

Figure 5a shows the filtering of the updraft strength in the nonlinear model for a range of perturbation wavelengths. In the opening discussion, we introduced a scale for the inflow strength U, which we now take to be the 10-m inflow, , which we note varies with radius. The inertial stability I is evaluated from the vortex without oscillations and also varies with radius. The top panel shows the ratio of the amplitude of the w maxima from the nonlinear model to that from the linear model, for various wavelengths, as a function of the normalized wavelength . When the filter length scale , the nonlinear terms strongly suppress the w response. The filter response rises sharply once . The phase dispersion properties of the filtering are shown in Fig. 5b. Short waves (i.e., ) are displaced inward about , with the displacement decreasing as the filter length scale diminishes. Note that the curves do not collapse to a single curve with this scaling, as would be expected with a first-order filter, because here we have a second-order system.

Fig. 5.

(a) Ratio of the amplitude of the updraft maximum in the nonlinear model to the linear model, as a function of dimensionless wavelength , for various oscillation wavelengths. (b) Distance downstream of the vorticity gradient maximum that the updraft is in the nonlinear model, as a function of dimensionless wavelength , for various oscillation wavelengths. The oscillation wavelengths are specified by the legend in (b).

Fig. 5.

(a) Ratio of the amplitude of the updraft maximum in the nonlinear model to the linear model, as a function of dimensionless wavelength , for various oscillation wavelengths. (b) Distance downstream of the vorticity gradient maximum that the updraft is in the nonlinear model, as a function of dimensionless wavelength , for various oscillation wavelengths. The oscillation wavelengths are specified by the legend in (b).

### c. The location of the eyewall updraft in a modified Rankine vortex

The nonlinear model was run for a wide variety of vortex profiles with single eyewalls, in which the intensity, RMW, and outer profile shape were varied. Specifically, for each of three RMWs (15, 25, and 40 km), we create vortices with maximum winds of approximately 20, 30, 50, and 70 m s−1 and wind decay coefficient .4 The three 50 m s−1 vortices were also run with double and half the drag coefficient and double and half the asymptotic mixing length . Varying the drag coefficient also alters the turbulent diffusivity in the lower boundary layer, while the changes to the asymptotic mixing length also affect the diffusivity, but mainly in the middle to upper boundary layer. We also simulated six additional vortices with maximum wind speed of 50 m s−1 and the same three RMWs, but with decay coefficients of (called the flat profiles) and (called peaked).

For each run, we extract the radius of the maximum in the eyewall updraft at 500-m height. The scaling parameters of the 10-m inflow and inertial stability are evaluated at the radius of maximum negative radial vorticity gradient, which is close to the RMW.5 The height of 500 m was chosen as being reasonably close to the top of the inflow layer in these simulations. The location of the updraft was interpolated between grid points by fitting a quadratic in r through data at the peak gridpoint and its two neighbors and taking the location of the maximum in this quadratic.

The inward displacement of the updrafts is plotted against in Fig. 6a. Clearly, the results collapse to a single, nearly linear curve, with slope of order 1. These two characteristics provide strong support for Kepert and Nolan’s (2014) scaling argument, discussed in the introduction. Note that the good agreement with the scaling includes the simulations with perturbed physical parameterizations and a wide range of storm intensities.

Fig. 6.

Flow characteristics near the RMW for a variety of tropical cyclones with single eyewalls, as specified in the text. (a) Inward displacement of the eyewall updraft at 500-m height, relative to the radius of maximum . (b) Width of the eyewall updraft. (c) Inward displacement of the surface azimuthal wind maximum, relative to the radius of maximum . (d) Inward displacement of the eyewall updraft at 500-m height, relative to the radius of maximum azimuthal wind at that height. Positive values of indicate an inward displacement.

Fig. 6.

Flow characteristics near the RMW for a variety of tropical cyclones with single eyewalls, as specified in the text. (a) Inward displacement of the eyewall updraft at 500-m height, relative to the radius of maximum . (b) Width of the eyewall updraft. (c) Inward displacement of the surface azimuthal wind maximum, relative to the radius of maximum . (d) Inward displacement of the eyewall updraft at 500-m height, relative to the radius of maximum azimuthal wind at that height. Positive values of indicate an inward displacement.

The updraft width (Fig. 6b), defined as the radial distance between the two points at which the updraft strength is half its maximum, similarly tends to collapse to a single curve, but to a lesser degree, and the slope is different from 1. However, these simulations all had a blending width of 10 km in the parametric profile of the gradient wind, which places a lower bound on the updraft width. If these computations are repeated, but with the blending width reduced to 2 km, we see from Fig. 7 that the updraft width (as well as location) does indeed scale with . However, we note that observations tend to suggest that the longer blending width is more realistic (Willoughby et al. 2006; Mallen et al. 2005).

Fig. 7.

As in Figs. 6a and 6b, but with the blending width reduced to 2 km.

Fig. 7.

As in Figs. 6a and 6b, but with the blending width reduced to 2 km.

Other features of the boundary layer flow near the RMW are also displaced inward, similarly to the updraft. In particular, Powell et al. (2009) showed that friction induces an outward slope with height of the RMW. Figure 6c shows that the position of the surface RMW is mostly inside of the radius of maximum negative radial vorticity gradient and that the displacement again scales nearly linearly with . However, this displacement is less than that of the updraft, so the updraft falls inside of the surface RMW (Fig. 6d).

Finally, we investigate the radius of the frictional updraft in secondary eyewall cases. We analyze a number of runs using the concentric-eyewall parametric profile in (13) with parameters in Table 1. Figure 8 shows the inward displacement of the updraft maximum from the “step” in vorticity that defines the outer RMW, together with the updraft width. These are taken at a height of 1 km rather than 500 m because of the deeper inflow at these radii. At outer radii (large ) the updraft is located up to several tens of kilometers inward of the vorticity step. At smaller radii (small ), the updraft is located as little as 10 km inward of the step—that is, essentially on top of it. Both the width and the displacement of the outer eyewall updraft thus exhibit identical scaling to the primary eyewall.

Table 1.

Parameters for the model simulations summarized in Fig. 8. Except where otherwise specified, and .

Fig. 8.

As in Figs. 6a and 6b, but for a number of tropical cyclones with incipient or actual secondary eyewalls. Full details of the cyclone profile parameters are given in Table 1. The large triangles indicate those runs with the outer vorticity step at 75 km.

Fig. 8.

As in Figs. 6a and 6b, but for a number of tropical cyclones with incipient or actual secondary eyewalls. Full details of the cyclone profile parameters are given in Table 1. The large triangles indicate those runs with the outer vorticity step at 75 km.

Given the importance of the radial length scale , it is worth examining its value in some representative cases. Figure 9 shows this quantity for the two parametric cyclones shown in Fig. 1. The single-eyewall cyclone shows a monotonic increase, from 0 at the center to about 25 km at 200-km radius. The relationship is not monotonic with radius in the concentric-eyewall case because of the changes in inflow and inertial stability at the outer eyewall. Nevertheless, it is several times larger at the outer eyewall than at the inner eyewall.

Fig. 9.

The radial length scale for the two tropical cyclones shown in Fig. 1. Blue shows the single-eyewall case in Fig. 1a. Red shows the double eyewall case in Fig. 1b.

Fig. 9.

The radial length scale for the two tropical cyclones shown in Fig. 1. Blue shows the single-eyewall case in Fig. 1a. Red shows the double eyewall case in Fig. 1b.

## 5. Implications for tropical cyclone dynamics

### a. Implications for tropical cyclone intensification and eye diameter

Theory shows that tropical cyclones, at least when modeled as axisymmetric balanced vortices, intensify more rapidly in response to a given amount of diabatic heating when that heating falls inside of the RMW, in the region of strong inertial stability, rather than outside (Schubert and Hack 1982; Shapiro and Willoughby 1982; Nolan et al. 2007; Pendergrass and Willoughby 2009; Vigh and Schubert 2009). More recently, observational evidence has been emerging of a similar relationship. Rogers et al. (2013) compared composite analysis of the azimuthal-mean flow in intensifying and steady-state hurricanes and found that convective bursts were preferentially located inside the RMW in the intensifying storms but outside for steady-state ones. They hypothesized that this difference could be due to differences in the low-level inflow and convergence arising from differences in the inertial stability outside of the eyewall. They also noted the importance of these differences for intensification, based on the fact that heating in a region of high inertial stability leads to greater intensification in balanced axisymmetric models (Schubert and Hack 1982; Pendergrass and Willoughby 2009). Similarly, in two case studies of individual storms, Hurricane Earl had maximum near-surface convergence inside of the RMW while it was intensifying (Rogers et al. 2015, their Fig. 15), while Hurricane Edouard’s convergence maximum lay outside the RMW while it was weakening (Rogers et al. 2016, their Fig. 11).

Frictional convergence has long been assumed to be linked to the location of convection, for instance, in Ooyama’s (1969) model. This assumption requires examination, for low-level convergence beneath convection can be the cause of the convection, or the result, or a bit of both (Raymond and Herman 2012). Indeed, the use of the Sawyer–Eliassen equation to model the secondary circulation in tropical cyclones shows that either heating or friction can generate a secondary circulation. Kepert and Nolan (2014) provided evidence that frictional convergence in cyclones can strongly influence the location of convection, showing that the 3D nonlinear diagnostic boundary layer model of Kepert and Wang (2001) could accurately diagnose the radius and relative strength of the eyewall updrafts in a WRF simulation of a tropical cyclone. Frisius and Lee (2016) describe a three-layer axisymmetric model, in which the lowest layer represents the boundary layer, which relaxes some of the assumptions in Ooyama (1969) but maintains the link between frictional convergence and convection. In particular, it has the choice of three boundary layer submodels: the linearized slab used by Ooyama (1969), a nonlinear slab, and the height-parameterized model developed by Kepert (2010b). The frictional updraft peaks outside the RMW in the first of these and inside the RMW in the remaining two, with the nonlinear slab model’s updraft at smaller radius than the height-parameterized model. The location of the updraft correlates with the intensification rate and peak intensity, with the linear slab leading to slowly intensifying and weaker storms, while the nonlinear slab leads to rapidly intensifying, strong storms. Those using the height-parameterized model fall between these extremes.

The dual propositions that heating inside of the RMW leads to more efficient intensification, and that the frictional convergence within the boundary layer influences the location of the peak convective heating, have implications for tropical cyclone intensification. Consider two storms of the same intensity and RMW, but with different radial wind structures: one peaked and one flat. The relatively peaked storm has a compact outer circulation, and the wind speed decays rapidly outside of the RMW, while the flat storm has a broad outer circulation and slow wind speed decay. The peaked storm will have lower inertial stability outside of the inner core and, hence, stronger frictional inflow, since to first order the frictional inflow is determined by a balance between frictional destruction of absolute angular momentum and its replenishment by radial advection (Kepert 2013). Since both storms have the same inertial stability at and inside of the RMW, the inflow will overshoot farther in the peaked storm, placing the maximum frictional updraft at smaller radius and, assuming that the convection likewise is placed at smaller radius, favoring more efficient intensification. This chain of reasoning is consistent with the comparison of intensifying and nonintensifying storms by Rogers et al. (2013), for in those composites, the nonintensifiers had stronger azimuthal flow and greater inertial stability outside of the RMW, as well as weaker near-surface inflow.

However, in Rogers et al.’s (2016) analysis of Hurricane Edouard, the maximum near-surface azimuthal-mean horizontal convergence on 16 September occurs at about 1.4 times the RMW, or 45 km (see their Fig. 11). Consistent with this, the peak inflow is at over twice the RMW. These are relatively large outward displacements of these features—in particular, modeling studies that present the location of the peak inflow typically have it at less than 1.5 times the RMW in both full-physics (Braun and Tao 2000; Nolan et al. 2009; Smith and Thomsen 2010) and idealized (Kepert 2010a, 2012) models, as do other observational studies (Kepert 2006a,b; Schwendike and Kepert 2008; Zhang et al. 2011; Bell et al. 2012). Interestingly, the Doppler wind data (Rogers et al. 2016, their Fig. 2d) show some suggestion of an outer wind maximum, and the authors mention a relatively broad wind maximum and potential secondary eyewall formation. Rob Rogers and Jun Zhang (2017, personal communication) very kindly provided the azimuthal-mean Doppler radar and dropsonde analyses, with the radar data showing a distinct vorticity maximum near 60-km radius. Assuming the wind at 2-km height to be in gradient balance, we ran the boundary layer model on these data and found a secondary updraft, with associated horizontal convergence in the boundary layer. It appears, therefore, that the unusual features in Edouard’s boundary layer, noted above, are consistent with boundary layer dynamics in the presence of a weak outer wind maximum.

Stern et al. (2015) has shown recently that tropical cyclone eyes contract strongly during the first part of the intensification but much more slowly or not at all during the later part. We note also that RMWs of below about 20 km are relatively uncommon; for instance, they constitute about 10% of the flight-level profiles in Willoughby and Rahn (2004, their Fig. 5a). We have shown that the inward penetration of the frictional eyewall updraft past the RMW is smaller when the RMW is small, reducing to approximately zero for the cases considered with a 15-km RMW. Perhaps in intense storms with small RMWs, the substantial inertial stability at the RMW limits the inward penetration of the frictional updraft and hence the contraction of the eye. While it is possible that the boundary layer dynamics may contribute to both Willoughby and Rahn’s (2004) and Stern et al.’s (2015) findings, it is probably not the only cause. In particular, horizontal mixing in the vicinity of the eyewall by eyewall mesovortices (Schubert et al. 1999; Kossin and Schubert 2004) will likely also restrain eyewall contraction.

### b. Implications for eyewall replacement cycles

Recently, several papers (Huang et al. 2012; Abarca and Montgomery 2013; Kepert 2013; Kepert and Nolan 2014) have explored the role of the boundary layer in eyewall replacement cycles, including secondary eyewall formation. These papers agree on the role of frictional convergence in influencing the location and strength of convection, and that the convergence at the developing outer eyewall is located near a region of markedly supergradient flow in the upper part of the boundary layer. They further agree that an overall expansion of the wind field is an important part of the process. However, they differ on the underpinning dynamics. Huang et al. (2012) and Abarca and Montgomery (2013) discuss the wind field expansion, which they associate with increased frictional inflow, leading to the development of supergradient flow, and attribute the frictional convergence to the outward acceleration associated with the agradient force in the supergradient flow. However, they do not specify where these processes occur. Presumably they occur somewhere within the expanding wind field, but precisely where is unclear.

Kepert (2013) and Kepert and Nolan (2014) take a different view of the overall dynamics. They argue, based on the earlier three-dimensional version of the boundary layer model presented here and consideration of the updraft in the linearized model in (2), that the location of the updraft is determined by the gradient wind structure, specifically local maxima in its negative radial vorticity gradient. Montgomery et al. (2014) pointed out that numerical simulations of tropical cyclones revealed substantial finescale structure in this gradient, leading to numerous updrafts and downdrafts in a linearized boundary layer model. But Kepert and Nolan (2014) demonstrated that these vertical velocity fluctuations were absent in the nonlinear boundary layer model used by Kepert (2013) and proposed that one effect of the nonlinearity was to act as a spatial filter on the effects of short-wavelength fluctuations in the gradient wind. Section 4b of the present article more thoroughly characterizes the spatial filtering inherent in the nonlinear model.

Kepert (2013) proposed a positive feedback mechanism between boundary layer convergence and convection as a mechanism that could contribute to secondary eyewall formation, in which the dynamical connection was through the gradient vorticity . Specifically, the vorticity of the gradient wind determines the location of the frictional updraft through the mechanisms he discusses, and that frictional updraft tends to favor convection in that region. Stretching of background vorticity by convective updrafts then acts to strengthen the ambient vorticity (Kilroy and Smith 2013), feeding back to the frictional convergence. For this mechanism to work, the frictional convergence has to enhance the convection in such a way as to strengthen the preexisting vorticity “bump.” In this context, Kepert (2013) noted that the nonlinearity of the boundary layer flow was important because it displaced the frictional updraft inward, so that in his experiments it fell near the top of, rather than outside of, the vorticity step (the local maximum in ).

Kepert’s (2013) experiments all had the outer vorticity step at 75-km radius, 3 times the radius of the primary eyewall, a value that is broadly consistent with observed secondary eyewall formations. We see in Fig. 8 that vorticity steps at this radius led to updraft maxima that are mostly 10–20 km inside of the center of the step; that is, near the top of the step if it is smoothed to a ramp. At larger radii, where is larger, the inward displacement is larger. Thus the effect on convection proposed by Kepert’s (2013), in which the frictional updraft leads to enhanced convection, will occur not near the top of the vorticity bump, but some distance inside of it. The effect of the convection on vorticity through vortex-tube stretching will similarly occur well inside of the bump. The effect will be less to amplify the bump and more to cause it to migrate inward. As it moves inward into a region of smaller , the frictional convergence, and hence the proposed positive feedback, will come more into alignment with the vorticity perturbation, and the effect will progressively shift from an inward migration to an amplification.

In the context of this scenario of an initially rapid inward translation and weak amplification tending toward a slower migration and stronger amplification, it is pertinent to reexamine model simulations of tropical cyclones undergoing eyewall replacement cycles. Figure 10 presents Hovmöller diagrams for the moist heating and vorticity of the gradient wind from the simulation of an eyewall replacement cycle by Nolan et al. (2013) that Kepert and Nolan (2014) analyzed. The linked inward propagation and amplification of both these features in the developing secondary eyewall, as well as in the initial intensification of the vortex, are clearly consistent with the proposed positive feedback. As the contraction proceeds, the radial distance between the maximum negative radial vorticity gradient (indicated by the green curve) and the moist heating (magenta curve) diminishes with time, and the eyewall shifts from relatively rapid inward translation with little amplification, to slower contraction and intensification. Zhu and Zhu (2014, 2015) similarly show that the secondary eyewall initially contracts rapidly with little intensification, then subsequently, as it moves to smaller radii, contracts more slowly and intensifies.

Fig. 10.

Hovmöller diagrams of (a) the moist heating averaged over 1–3-km heights and (b) at 2.2-km height from the WRF simulation of an eyewall replacement cycle presented by Nolan et al. (2013). The magenta curve indicates the contracting secondary eyewall, as determined from the moist heating. The green curve indicates the contracting region of significant negative radial vorticity gradient. Adapted from Kepert and Nolan (2014).

Fig. 10.

Hovmöller diagrams of (a) the moist heating averaged over 1–3-km heights and (b) at 2.2-km height from the WRF simulation of an eyewall replacement cycle presented by Nolan et al. (2013). The magenta curve indicates the contracting secondary eyewall, as determined from the moist heating. The green curve indicates the contracting region of significant negative radial vorticity gradient. Adapted from Kepert and Nolan (2014).

A further well-known effect also contributes to the rapid amplification of the gradient vorticity near the developing outer eyewall later in the process. The convection is stretching ambient vorticity, so as the ambient vorticity increases, the rate of generation of vorticity by a given convective updraft will increase.

## 6. Conclusions

The impulsively started tropical cyclone boundary layer spins up to its steady-state solution via a series of decaying inertial oscillations of period . Apart from the substitution of I for f, this evolution appears very similar to the spinup of the classical Ekman layer (Lewis and Belcher 2004). This result confirms Eliassen and Lystad’s (1977) scaling argument and supports Kepert and Nolan’s (2014) argument that the tropical cyclone boundary layer, at least in the core region, can be to good approximation regarded as being slaved to the parent vortex, in the sense that the friction-induced flow is to good approximation the steady-state boundary layer response to the pressure field from the parent vortex. Ooyama (1969) explicitly assumes the boundary layer flow to be the steady-state response to the parent vortex in his study of cyclone intensification. It seems that the role of the boundary layer is indeed as proposed by Ooyama (1969, 1982) and Frisius and Lee (2016), namely, that frictional convergence substantially influences the radial location of the convective heating, and this heating drives the secondary circulation which intensifies the vortex.

We have also confirmed the scaling argument of Kepert and Nolan (2014) that a nonlinear tropical cyclone boundary layer model should act as a low-pass filter and be relatively insensitive to oscillations in the gradient wind whose radial scale is shorter than . This points to a fundamental difference between linearized and nonlinear versions of such models; the former are sensitive to such perturbations by virtue of the term in the updraft equation, while the latter are sensitive only to perturbations of sufficiently large spatial scale. Attempts to apply linearized boundary layer models to data from models or observations should therefore first filter the data.6

It has long been known that the peak axisymmetric updraft typically falls a few kilometers within the RMW (e.g., Jorgensen 1984; Rogers et al. 2012). We have shown that the inward displacement due to friction scales as for both the primary and secondary eyewalls. Essentially, this displacement is a measure of the overshoot as the inflowing near-surface air encounters the greater inertial stability of the gradient wind at the eyewall and decelerates. It is smaller (other things being equal) when the RMW is small. The dependence of this distance on may explain, at least partially, why observed RMW are seldom less than 20 km, why storms with relatively peaked radial profiles of wind speed can intensify more rapidly, and why some secondary eyewalls initially contract rapidly with little intensification, then later contract more slowly while intensifying.

## Acknowledgments

I thank Gary Dietachmayer for advice on the numerical methods and Dave Nolan and Kevin Tory for helpful discussions. Rob Rogers and Jun Zhang kindly provided data from Hurricane Edouard and helped with its interpretation.

### APPENDIX

#### The Upper Boundary Condition

Dealing with open boundaries in computational fluid dynamics can be a difficult problem, since often only part of the flow can be contained in the model domain and we must ensure that the necessary artificial boundaries do not significantly contaminate the solution. This issue arises in the model presented here. For the upper boundary condition we have adopted the same Neumann boundary condition as used in Kepert and Wang (2001), , implemented via a single layer of virtual grid points above the domain.

We note at the outset that the presence of turbulent viscosity complicates the analysis of the boundary conditions and alters the required number of boundary conditions for a well-posed problem. For instance, for fully compressible subsonic flow the Euler equations require one and four boundary conditions on outflow and inflow, respectively, while the Navier–Stokes equations require four and five boundary conditions (Oliger and Sundström 1978; Grinstein 1994). The results of Oliger and Sundström (1978) applied to our model equations (5) and (6) indicate that two boundary conditions are needed for both inflow and outflow. They further show that the Neumann condition is appropriate on outflow boundaries and the Dirichlet condition on inflow. Similarly, Halpern (1986) has developed a family of boundary conditions for outflow boundaries with the linear advection–diffusion equation, the first member of which is the Neumann condition. We have long used the Neumann condition for the whole of the upper boundary without problems, but in response to the theory tested the use of the Dirichlet condition on inflow, retaining the Neumann condition on outflow. This change makes almost no difference to the solution, as might be expected given that the region of descent is relatively small, so for simplicity we have retained the Neumann condition.

The Neumann condition was criticized by Smith and Montgomery (2010), who argued that it was equivalent to the Dirichlet condition . In particular, they stated that “if the top of the model is taken at, or above, the level at which the frictional force vanishes, the zero-vertical-gradient condition does not allow the model to determine the flow above it, but requires the flow to return to the prescribed gradient wind with zero radial velocity” (p. 1669). We now demonstrate that the model can support substantial agradient flow at the upper boundary.

Figure A1 presents three simulations of the boundary layer for the same gradient wind, with three different configurations of the model. Similar calculations have been done on several vortex gradient wind profiles and with both the new axisymmetric and old three-dimensional versions of the model, but we choose to present this case with concentric eyewalls because the relatively large supergradient flow near the outer eyewall provides a more severe test of the upper boundary condition, especially when we place the upper boundary in the middle of that flow. The model settings are identical except for the number of vertical levels and hence the depth of the domain. The first run has 20 levels with the top at 2.47 km, as used elsewhere in this article. The second has 18 levels and the top at 1.65 km, chosen so that the model top is slightly above the top of the inflow layer, while the third has 15 levels and the top at 906 m, chosen to intersect the region of strong supergradient flow in the outer eyewall. The gradient wind profile is defined by (13) with , , , , , and the blending width , giving gradient wind maxima of 46.4 and 41.4 m s−1 at the inner and outer eyewalls, respectively.

Fig. A1.

Three simulations of the boundary layer flow with different domain depths. The domains are (a),(d),(g),(j) 20 levels, top at 2.47 km; (b),(e),(h),(k) 18 levels, top at 1.65 km; and (c),(f),(i),(l) 15 levels, top at 905 m. Only part of the domain is shown for the first of these for clarity. (a)–(i) The radial, azimuthal, and vertical velocities, respectively. (j)–(l) The gradient wind (red), 10-m −u and υ (green and blue), and w at 905 m (magenta). The inflow and azimuthal flow at the domain top level are shown as dashed green and blue curves, respectively; note that these are at different levels in the three simulations.

Fig. A1.

Three simulations of the boundary layer flow with different domain depths. The domains are (a),(d),(g),(j) 20 levels, top at 2.47 km; (b),(e),(h),(k) 18 levels, top at 1.65 km; and (c),(f),(i),(l) 15 levels, top at 905 m. Only part of the domain is shown for the first of these for clarity. (a)–(i) The radial, azimuthal, and vertical velocities, respectively. (j)–(l) The gradient wind (red), 10-m −u and υ (green and blue), and w at 905 m (magenta). The inflow and azimuthal flow at the domain top level are shown as dashed green and blue curves, respectively; note that these are at different levels in the three simulations.

It is apparent that the flow in the overlapping parts of the domains is very similar. Indeed, the maximum absolute differences between the 20- and 18-level simulations are u: 0.6 m s−1, υ: 0.7 m s−1, and w: 0.06 m s−1, while the corresponding differences between the 20- and 15-level simulations are u: 1.2 m s−1, υ: 1.2 m s−1, and w: 0.06 m s−1 The largest differences occur near the domain top where the proximity of the upper boundary condition has reduced the vertical wind shear. While we do not recommend running a boundary layer simulation with a model domain shallower than the boundary layer depth, it is nevertheless clear that the boundary conditions do precisely what such open boundary conditions are supposed to do: to not unduly influence the flow in the interior of the domain.

The bottom row of Fig. A1 also shows the flow at the model top level (dashed curves); these curves are at a different physical height in each simulation and hence differ markedly. The maximum vector departures from gradient balance at model top are 7.1, 4.7, and 2.4 m s−1 for the 15-, 18-, and 20-level simulations, respectively, all near the outer eyewall. None of these three cases satisfies Smith and Montgomery’s (2010) derivation with the second derivatives strictly equal to zero, for the frictional force (i.e., the momentum flux divergence) has not vanished even with the boundary condition . Nevertheless, these cases clearly demonstrate that the model with this boundary condition can produce marked agradient flow at the top of the domain. Such departures also occur in the original three-dimensional version of the model; see, for example, Kepert and Wang (2001, their Fig. 2).

Why does Smith and Montgomery’s (2010) analytical derivation not apply in actuality? They assume “that frictional forces can be ignored at the top of the boundary layer” (p. 1667) to remove the frictional terms, then apply the Neumann boundary condition to eliminate the vertical advection terms.A1 These eliminations, together with the assumption of a steady state, leads to the Dirichlet condition . Their key error is the elimination of the frictional terms. While these are often neglected, they are not zero, and being relatively small is not the same as being zero. Smith and Montgomery’s (2010) arguments lack rigor because they assume exact equality to zero in their mathematical derivation but relax this condition to approximate equality in some (but not all) of the ensuing discussion.

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## Footnotes

1

One counterexample can be the continental boundary layer, where the structure is strongly influenced by the diurnal variation in the land surface temperature. In this case, the flow has a memory of the past and is not determined by just the present conditions.

2

Following the arguments of Kepert et al. (2016), we consider that general definitions of the atmospheric boundary layer should also apply in tropical cyclones. Since the vortices under consideration are all axisymmetric, we use the inflow layer as a suitable proxy for the boundary layer.

3

That is, the flow at the reference height used for the drag coefficient.

4

These data are for the profile without blending; incorporation of the blending slightly modifies the RMW and maximum wind.

5

We do not use the RMW, since a second wind speed maximum does not exist in all the secondary eyewall cases considered below.

6

In contrast to multilevel nonlinear diagnostic models of the tropical cyclone boundary layer, which we have shown here tend to suppress the effects of finescale features in the gradient wind, slab models can generate finescale oscillations even when the gradient wind is smooth (Smith and Vogl 2008). Kepert (2010a,b) and Williams (2015) have argued that these oscillations are spurious. However, Abarca and Montgomery (2013) appeal to these oscillations to support their views on secondary eyewall formation.

A1

It would not be correct to use the Neumann condition to eliminate the frictional terms, for at the model top does not imply that the frictional term .