## 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 *K* and the boundary layer depth scale *δ*, which they showed is

However, the tropical cyclone boundary layer^{2} 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).

^{3}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,

A limitation of (1) is that it is highly sensitive to small-amplitude short-wavelength perturbations in the pressure field, for *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

## 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.

*u*,

*w*are the radial, azimuthal, and vertical wind components, respectively;

*r*–

*z*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 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

*l*defined by

*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

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

*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

## 4. Results

### a. Time scales

We begin by examining the adjustment of an impulsively started boundary layer (i.e., one in which the initial condition is *w* field, since this represents the depth-integrated oscillation. The flow adjusts to equilibrium more quickly near the surface than aloft, but even the 1.1-km level shows only very weak oscillations after one inertial period. The corresponding evolution is shown in the linear model in Figs. 2d–f, and it is apparent that the adjustment is similar but that the oscillations are damped less rapidly. Although the oscillations have the appearance of an outward-propagating wave, most readily apparent in the *w* field, that is not a correct interpretation, for the linear model has no communication between adjacent grid columns (except for the horizontal diffusion), so radially propagating waves cannot be supported. The nature of these oscillations is made clear in Figs. 2g–l, which plot the same data except with the time axis nondimensionalized by

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 *u* are *υ* are *w* are

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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 *u* are *υ* are *w* are

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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 *u* are *υ* are *w* are

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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 m^{2} 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.

Radius–height sections of the steady-state boundary layer. For the nonlinear model, (a) *u* with contour interval *υ* with contour interval *w* with contours at

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Radius–height sections of the steady-state boundary layer. For the nonlinear model, (a) *u* with contour interval *υ* with contour interval *w* with contours at

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Radius–height sections of the steady-state boundary layer. For the nonlinear model, (a) *u* with contour interval *υ* with contour interval *w* with contours at

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

### 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 *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

Summary plots of the flow with and without sinusoidal perturbations, for the (a),(c) nonlinear and (b),(d) linearized models. Plotted in (a) are

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Summary plots of the flow with and without sinusoidal perturbations, for the (a),(c) nonlinear and (b),(d) linearized models. Plotted in (a) are

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Summary plots of the flow with and without sinusoidal perturbations, for the (a),(c) nonlinear and (b),(d) linearized models. Plotted in (a) are

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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, *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 *w* response. The filter response rises sharply once

(a) Ratio of the amplitude of the updraft maximum in the nonlinear model to the linear model, as a function of dimensionless wavelength

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

(a) Ratio of the amplitude of the updraft maximum in the nonlinear model to the linear model, as a function of dimensionless wavelength

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

(a) Ratio of the amplitude of the updraft maximum in the nonlinear model to the linear model, as a function of dimensionless wavelength

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

### 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 ^{−1} and the same three RMWs, but with decay coefficients of

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

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

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

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

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

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

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

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.

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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.

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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.

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Given the importance of the radial length scale

The radial length scale

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

The radial length scale

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

The radial length scale

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

## 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

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

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.

Hovmöller diagrams of (a) the moist heating averaged over 1–3-km heights and (b)

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Hovmöller diagrams of (a) the moist heating averaged over 1–3-km heights and (b)

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

Hovmöller diagrams of (a) the moist heating averaged over 1–3-km heights and (b)

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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 *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 ^{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

## 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),

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

The Neumann condition was criticized by Smith and Montgomery (2010), who argued that it was equivalent to the Dirichlet condition

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 ^{−1} at the inner and outer eyewalls, respectively.

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.

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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.

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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.

Citation: Journal of the Atmospheric Sciences 74, 10; 10.1175/JAS-D-17-0077.1

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

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

In the simulations presented here, the departures from gradient flow at the top of the model diminish as the domain becomes deeper. This tendency is not an artifact of the upper boundary condition, for we have seen that that condition can still allow significant agradient flow. Rather, it is the dynamics of the flow that causes it to adjust back toward gradient balance above the boundary layer. This adjustment is perhaps most easily seen using the linearized version of the model. Rosenthal (1962) and Kepert (2001) derived analytical solutions for the linearized model, under the additional assumption that the turbulent diffusivity is constant with height. Vogl and Smith (2009) also studied this system, but without finding analytic solutions. In each of these cases, the analysis begins by deriving four possible solutions to the equations. Two of these grow exponentially with height and are eliminated by the requirement that the flow remain finite as

## REFERENCES

Abarca, S. F., and M. T. Montgomery, 2013: Essential dynamics of secondary eyewall formation.

,*J. Atmos. Sci.***70**, 3216–3230, doi:10.1175/JAS-D-12-0318.1.Bell, M. M., M. T. Montgomery, and W.-C. Lee, 2012: An axisymmetric view of concentric eyewall evolution in Hurricane Rita (2005).

,*J. Atmos. Sci.***69**, 2414–2432, doi:10.1175/JAS-D-11-0167.1.Black, P. G., and Coauthors, 2007: Air–sea exchange in hurricanes: Synthesis of observations from the Coupled Boundary Layer Air–Sea Transfer experiment.

,*Bull. Amer. Meteor. Soc.***88**, 357–374, doi:10.1175/BAMS-88-3-357.Braun, S. A., and W.-K. Tao, 2000: Sensitivity of high-resolution simulations of Hurricane Bob (1991) to planetary boundary layer parameterizations.

,*Mon. Wea. Rev.***128**, 3941–3961, doi:10.1175/1520-0493(2000)129<3941:SOHRSO>2.0.CO;2.Donelan, M. A., B. K. Haus, N. Reul, W. J. Plant, M. Stiassnie, H. C. Graber, O. B. Brown, and E. S. Saltzman, 2004: On the limiting aerodynamic roughness of the ocean in very strong winds.

,*Geophys. Res. Lett.***31**, L18306, doi:10.1029/2004GL019460.Eliassen, A., and M. Lystad, 1977: The Ekman layer of a circular vortex—A numerical and theoretical study.

,*Geophys. Norv.***7**, 1–16.Frisius, T., and M. Lee, 2016: The impact of gradient wind imbalance on tropical cyclone intensification within Ooyama’s three-layer model.

,*J. Atmos. Sci.***73**, 3659–3679, doi:10.1175/JAS-D-15-0336.1.Gao, K., and I. Ginis, 2014: On the generation of roll vortices due to the inflection point instability of the hurricane boundary layer flow.

,*J. Atmos. Sci.***71**, 4292–4307, doi:10.1175/JAS-D-13-0362.1.Gao, K., and I. Ginis, 2016: On the equilibrium-state roll vortices and their effects in the hurricane boundary layer.

,*J. Atmos. Sci.***73**, 1205–1222, doi:10.1175/JAS-D-15-0089.1.Grinstein, F. F., 1994: Open boundary conditions in the simulation of subsonic turbulent shear flows.

,*J. Comput. Phys.***115**, 43–55, doi:10.1006/jcph.1994.1177.Halpern, L., 1986: Artificial boundary conditions for the linear advection-diffusion equation.

,*Math. Comput.***46**, 425–438, doi:10.1090/S0025-5718-1986-0829617-8.Huang, Y.-H., M. T. Montgomery, and C.-C. Wu, 2012: Concentric eyewall formation in Typhoon Sinlaku (2008). Part II: Axisymmetric dynamical processes.

,*J. Atmos. Sci.***69**, 662–674, doi:10.1175/JAS-D-11-0114.1.Jorgensen, D. P., 1984: Mesoscale and convective-scale characteristics of mature hurricanes. Part II: Inner core structure of Hurricane Allen (1980).

,*J. Atmos. Sci.***41**, 1287–1311, doi:10.1175/1520-0469(1984)041<1287:MACSCO>2.0.CO;2.Kepert, J. D., 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part I: Linear theory.

,*J. Atmos. Sci.***58**, 2469–2484, doi:10.1175/1520-0469(2001)058<2469:TDOBLJ>2.0.CO;2.Kepert, J. D., 2006a: Observed boundary–layer wind structure and balance in the hurricane core. Part I: Hurricane Georges.

,*J. Atmos. Sci.***63**, 2169–2193, doi:10.1175/JAS3745.1.Kepert, J. D., 2006b: Observed boundary–layer wind structure and balance in the hurricane core. Part II: Hurricane Mitch.

,*J. Atmos. Sci.***63**, 2194–2211, doi:10.1175/JAS3746.1.Kepert, J. D., 2010a: Slab- and height-resolving models of the tropical cyclone boundary layer. Part I: Comparing the simulations.

,*Quart. J. Roy. Meteor. Soc.***136**, 1689–1699, doi:10.1002/qj.667.Kepert, J. D., 2010b: Slab- and height-resolving models of the tropical cyclone boundary layer. Part II: Why the simulations differ.

,*Quart. J. Roy. Meteor. Soc.***136**, 1700–1711, doi:10.1002/qj.685.Kepert, J. D., 2012: Choosing a boundary layer parameterization for tropical cyclone modeling.

,*Mon. Wea. Rev.***140**, 1427–1445, doi:10.1175/MWR-D-11-00217.1.Kepert, J. D., 2013: How does the boundary layer contribute to eyewall replacement cycles in axisymmetric tropical cyclones?

,*J. Atmos. Sci.***70**, 2808–2830, doi:10.1175/JAS-D-13-046.1.Kepert, J. D., and Y. Wang, 2001: The dynamics of boundary layer jets within the tropical cyclone core. Part II: Nonlinear enhancement.

,*J. Atmos. Sci.***58**, 2485–2501, doi:10.1175/1520-0469(2001)058<2485:TDOBLJ>2.0.CO;2.Kepert, J. D., and D. S. Nolan, 2014: Reply to “Comments on ‘How does the boundary layer contribute to eyewall replacement cycles in axisymmetric tropical cyclones?’”

,*J. Atmos. Sci.***71**, 4692–4704, doi:10.1175/JAS-D-14-0014.1.Kepert, J. D., J. Schwendike, and H. Ramsay, 2016: Why is the tropical cyclone boundary layer not “well mixed”?

,*J. Atmos. Sci.***73**, 957–973, doi:10.1175/JAS-D-15-0216.1.Kilroy, G., and R. K. Smith, 2013: A numerical study of rotating convection during tropical cyclogenesis.

,*Quart. J. Roy. Meteor. Soc.***139**, 1255–1269, doi:10.1002/qj.2022.Kossin, J. P., and W. H. Schubert, 2004: Mesovortices in Hurricane Isabel.

,*Bull. Amer. Meteor. Soc.***85**, 151–153, doi:10.1175/BAMS-85-2-151.Lewis, D. M., and S. E. Belcher, 2004: Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift.

,*Dyn. Atmos. Oceans***37**, 313–351, doi:10.1016/j.dynatmoce.2003.11.001.Mallen, K. J., M. T. Montgomery, and B. Wang, 2005: Reexamining the near-core radial structure of the tropical cyclone primary circulation: Implications for vortex resiliency.

,*J. Atmos. Sci.***62**, 408–425, doi:10.1175/JAS-3377.1.Montgomery, M. T., S. F. Abarca, R. K. Smith, C.-C. Wu, and Y.-H. Huang, 2014: Comments on “How does the boundary layer contribute to eyewall replacement cycles in axisymmetric tropical cyclones?”

,*J. Atmos. Sci.***71**, 4682–4691, doi:10.1175/JAS-D-13-0286.1.Nolan, D. S., Y. Moon, and D. P. Stern, 2007: Tropical cyclone intensification from asymmetric convection: Energetics and efficiency.

,*J. Atmos. Sci.***64**, 3377–3405, doi:10.1175/JAS3988.1.Nolan, D. S., D. P. Sternn, and J. A. Zhang, 2009: Evaluation of planetary boundary layer parameterizations in tropical cyclones by comparison of in situ observations and high-resolution simulations of Hurricane Isabel (2003). Part II: Inner-core boundary layer and eyewall structure.

,*Mon. Wea. Rev.***137**, 3675–3698, doi:10.1175/2009MWR2786.1.Nolan, D. S., R. Atlas, K. T. Bhatia, and L. R. Bucci, 2013: Development and validation of a hurricane nature run using the joint OSSE nature run and the WRF model.

,*J. Adv. Model. Earth Syst.***5**, 382–405, doi:10.1002/jame.20031.Oliger, J., and A. Sundström, 1978: Theoretical and practical aspects of some initial boundary value problems in fluid dynamics.

,*SIAM J. Appl. Math.***35**, 419–446, doi:10.1137/0135035.Ooyama, K. V., 1969: Numerical simulation of the life cycle of tropical cyclones.

,*J. Atmos. Sci.***26**, 3–40, doi:10.1175/1520-0469(1969)026<0003:NSOTLC>2.0.CO;2.Ooyama, K. V., 1982: Conceptual evolution of the theory and modelling of the tropical cyclone.

,*J. Meteor. Soc. Japan***60**, 369–380, doi:10.2151/jmsj1965.60.1_369.Pendergrass, A. G., and H. E. Willoughby, 2009: Diabatically induced secondary flows in tropical cyclones. Part I: Quasi-steady forcing.

,*Mon. Wea. Rev.***137**, 805–821, doi:10.1175/2008MWR2657.1.Powell, M. D., P. J. Vickery, and T. A. Reinhold, 2003: Reduced drag coefficient for high wind speeds in tropical cyclones.

,*Nature***422**, 279–283, doi:10.1038/nature01481.Powell, M. D., E. W. Uhlhorn, and J. D. Kepert, 2009: Estimating maximum surface winds from hurricane reconnaissance measurements.

,*Wea. Forecasting***24**, 868–883, doi:10.1175/2008WAF2007087.1.Raymond, D. J., and M. J. Herman, 2012: Frictional convergence, atmospheric convection, and causality.

,*Atmósfera***25**, 253–267.Rogers, R., S. Lorsolo, P. Reasor, J. Gamache, and F. Marks, 2012: Multiscale analysis of tropical cyclone kinematic structure from airborne Doppler radar composites.

,*Mon. Wea. Rev.***140**, 77–99, doi:10.1175/MWR-D-10-05075.1.Rogers, R., P. D. Reasor, and S. Lorsolo, 2013: Airborne Doppler observations of the inner-core structural differences between intensifying and steady-state tropical cyclones.

,*Mon. Wea. Rev.***141**, 2970–2991, doi:10.1175/MWR-D-12-00357.1.Rogers, R., P. D. Reasor, and J. A. Zhang, 2015: Multiscale structure and evolution of Hurricane Earl (2010) during rapid intensification.

,*Mon. Wea. Rev.***143**, 536–562, doi:10.1175/MWR-D-14-00175.1.Rogers, R., J. A. Zhang, J. Zawislak, H. Jiang, G. Alvey, E. Zipser, and S. Stevenson, 2016: Observations of the structure and evolution of Hurricane Edouard (2014) during intensity change. Part II: Kinematic structure and the distribution of deep convection.

,*Mon. Wea. Rev.***144**, 3355–3376, doi:10.1175/MWR-D-16-0017.1.Rosenthal, S. L., 1962: A theoretical analysis of the field of motion in the hurricane boundary layer. U.S. Department of Commerce, National Hurricane Research Project Rep. 56, 12 pp. [Available online at http://www.aoml.noaa.gov/general/lib/TM/NHRP_56_1962.pdf.]

Schubert, W. H., and J. J. Hack, 1982: Inertial stability and tropical cyclone development.

,*J. Atmos. Sci.***39**, 1687–1697, doi:10.1175/1520-0469(1982)039<1687:ISATCD>2.0.CO;2.Schubert, W. H., M. T. Montgomery, R. K. Taft, T. A. Guinn, S. R. Fulton, P. Kossin, and J. P. Edwards, 1999: Polygonal eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes.

,*J. Atmos. Sci.***56**, 1197–1223, doi:10.1175/1520-0469(1999)056<1197:PEAECA>2.0.CO;2.Schwendike, J., and J. D. Kepert, 2008: The boundary layer winds in Hurricanes Danielle (1998) and Isabel (2003).

,*Mon. Wea. Rev.***136**, 3168–3192, doi:10.1175/2007MWR2296.1.Shapiro, L. J., and H. E. Willoughby, 1982: The response of balanced hurricanes to local sources of heat and momentum.

,*J. Atmos. Sci.***39**, 378–394, doi:10.1175/1520-0469(1982)039<0378:TROBHT>2.0.CO;2.Smith, R. K., and S. Vogl, 2008: A simple model of the hurricane boundary layer revisited.

,*Quart. J. Roy. Meteor. Soc.***134**, 337–351, doi:10.1002/qj.216.Smith, R. K., and M. T. Montgomery, 2010: Hurricane boundary-layer theory.

,*Quart. J. Roy. Meteor. Soc.***136**, 1665–1670, doi:10.1002/qj.679.Smith, R. K., and G. L. Thomsen, 2010: Dependence of tropical-cyclone intensification on the boundary-layer representation in a numerical model.

,*Quart. J. Roy. Meteor. Soc.***136**, 1671–1685, doi:10.1002/qj.687.Stern, D. P., J. L. Vigh, D. S. Nolan, and F. Zhang, 2015: Revisiting the relationship between eyewall contraction and intensification.

,*J. Atmos. Sci.***72**, 1283–1306, doi:10.1175/JAS-D-14-0261.1.Thompson, J. F., Z. U. A. Warsi, and C. W. Mastin, 1985:

*Numerical Grid Generation: Foundations and Applications.*Elsevier, 483 pp.Vigh, J. L., and W. H. Schubert, 2009: Rapid development of the tropical cyclone warm core.

,*J. Atmos. Sci.***66**, 3335–3350, doi:10.1175/2009JAS3092.1.Vogl, S., and R. K. Smith, 2009: Limitations of a linear model for the hurricane boundary layer.

,*Quart. J. Roy. Meteor. Soc.***135**, 839–850, doi:10.1002/qj.390.Williams, G. J., Jr., 2015: The effects of vortex structure and vortex translation on the tropical cyclone boundary layer wind field.

,*J. Adv. Model. Earth Syst.***7**, 188–214, doi:10.1002/2013MS000299.Willoughby, H. E., and M. E. Rahn, 2004: Parametric presentation of the primary hurricane vortex. Part I: Observations and evaluation of the Holland (1980) model.

,*Mon. Wea. Rev.***132**, 3033–3048, doi:10.1175/MWR2831.1.Willoughby, H. E., R. W. R. Darling, and M. E. Rahn, 2006: Parametric presentation of the primary hurricane vortex. Part II: A new family of sectionally continuous profiles.

,*Mon. Wea. Rev.***134**, 1102–1120, doi:10.1175/MWR3106.1.Zhang, J. A., and W. M. Drennan, 2012: An observational study of vertical eddy diffusivity in the hurricane boundary layer.

,*J. Atmos. Sci.***69**, 3223–3236, doi:10.1175/JAS-D-11-0348.1.Zhang, J. A., R. F. Rogers, D. S. Nolan, J. Marks, and D. Frank, 2011: On the characteristic height scales of the hurricane boundary layer.

,*Mon. Wea. Rev.***139**, 2523–2535, doi:10.1175/MWR-D-10-05017.1.Zhu, Z., and P. Zhu, 2014: The role of outer rainband convection in governing the eyewall replacement cycle in numerical simulations of tropical cyclones.

,*J. Geophys. Res. Atmos.***119**, 8049–8072, doi:10.1002/2014JD021899.Zhu, Z., and P. Zhu, 2015: Sensitivities of eyewall replacement cycle to model physics, vortex structure, and background winds in numerical simulations of tropical cyclones.

,*J. Geophys. Res. Atmos.***120**, 590–622, doi:10.1002/2014JD022056.

^{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