# A Simple Method for Simulating Wind Profiles in the Boundary Layer of Tropical Cyclones

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## Abstract

A method to simulate characteristics of wind speed in the boundary layer of tropical cyclones in an idealized manner is developed and evaluated. The method can be used in a single-column modelling set-up with a planetary boundary-layer parametrization, or within large-eddy simulations (LES). The key step is to include terms in the horizontal velocity equations representing advection and centrifugal acceleration in tropical cyclones that occurs on scales larger than the domain size. Compared to other recently developed methods, which require two input parameters (a reference wind speed, and radius from the centre of a tropical cyclone) this new method also requires a third input parameter: the radial gradient of reference wind speed. With the new method, simulated wind profiles are similar to composite profiles from dropsonde observations; in contrast, a classic Ekman-type method tends to overpredict inflow-layer depth and magnitude, and two recently developed methods for tropical cyclone environments tend to overpredict near-surface wind speed. When used in LES, the new technique produces vertical profiles of total turbulent stress and estimated eddy viscosity that are similar to values determined from low-level aircraft flights in tropical cyclones. Temporal spectra from LES produce an inertial subrange for frequencies \(\gtrsim \)0.1 Hz, but only when the horizontal grid spacing \(\lesssim \)20 m.

## Keywords

Boundary-layer dynamics Large-eddy simulation Single-column modelling Tropical cyclone## 1 Introduction

Numerical model simulations have been used to help understand tropical cyclones for decades. Standard three-dimensional simulations can, of course, represent many dynamical processes in a tropical cyclone, including the centrifugal acceleration associated with rapidly rotating flow, and the large-scale pressure gradient acceleration that acts to counter centrifugal effects. However, substantial computational resources are often needed for three-dimensional simulations because tropical cyclones span several hundreds of kilometres in the horizontal, and require several days of model integration time. As a relatively inexpensive option, two-dimensional simulations using axisymmetric equations can be used to study many aspects of tropical cyclones. However, even axisymmetric model simulations become expensive to run, especially with continued advances in representation of physical processes such as atmospheric radiation, multi-moment microphysical schemes, and ocean feedback effects, to name but a few. Additionally, axisymmetric models are complex and difficult to modify, and interactions between the various physical parametrizations make it difficult to isolate cause and effect if tropical cyclone structure varies among different simulations.

Moreover, interest is growing in the use of large-eddy simulation (LES) to study the boundary layer of tropical cyclones (roughly, the lowest kilometre above the surface). The primary advantage of LES, of course, is that the statistical properties of turbulent flow can be predicted primarily by the model’s governing equations, with only a small role being played by a subgrid turbulence scheme. Furthermore, coherent structures within the tropical cyclone boundary layer such as quasi-two-dimensional roll vortices (e.g., Foster 2005; Morrison et al. 2005) can only be resolved using grid spacing of \(\approx \)100 m or less, i.e., typical resolution for LES. However, tropical cyclones extend hundreds of km horizontally, and so LES becomes prohibitively expensive unless the domain size is restricted, thereby making it difficult to account for the dynamical processes in rapidly rotating flow mentioned above (e.g., centrifugal acceleration).

*z*is the only coordinate. In this case, the large-scale inertial and pressure-gradient acceleration terms are included in the governing equations via mesoscale tendency terms (described in the next section) that are similar to those pioneered for boundary-layer modelling by Sommeria (1976). With only minor modifications, these mesoscale tendency terms can also be included within LES, which allows domain sizes to be only a few km in horizontal extent, making high-resolution LES (with grid spacing of

*O*(10) m or less) computationally tractable.

With these points in mind, the overall conceptual set-up for this new methodology is that the scale of the LES domain is much smaller than the scale of the entire tropical cyclone, as illustrated schematically in Fig. 1. We envision domain sizes of *O*(5 km) on a side, which is similar to the horizontal grid spacing for present-day numerical weather prediction systems (e.g., Tallapragada et al. 2014), and is comparable to typical LES domain sizes. The primary circulation of a tropical cyclone, quantified by the tangential velocity (i.e., magnitude of flow around circles centered on the tropical cyclone), is imposed as an initial condition for model simulations and, as discussed below, we consider the radial gradient of tangential velocity to be an important input parameter in our framework because it allows us to include large-scale advection tendencies as in Sommeria (1976); a key difference from Sommeria’s approach, however, is that we utilize model-predicted wind profiles in the mesoscale tendency terms, i.e., advection tendencies are not simply specified and held fixed throughout the simulation. The goal is to allow the model to predict details of the secondary circulation (via the radial velocity profile) after a user essentially specifies the primary circulation (via a few input parameters, namely, a reference wind-speed profile, and its radial gradient). This general concept is very similar to recent studies of tropical cyclones by Kepert and Wang (2001), Foster (2009), and Kepert (2012), although a key difference is that our approach is essentially one-dimensional (in height, *z*) rather than two-dimensional (radius and height). It is also not clear how several terms from these studies should be included in small-domain, three-dimensional LES (see Fig. 1). Some further comparison to previous studies is provided below (Sect. 2.3).

Compared to other recent studies of the tropical cyclone boundary layer, our method accounts for the advection of momentum by the secondary circulation of the tropical cyclone. Of note, Nakanishi and Niino (2012) and Green and Zhang (2015) considered centrifugal acceleration terms in their model equations, as does our new method, but we show herein that inclusion of radial advection is needed to produce mean wind profiles that are similar to observed profiles in tropical cyclones.

In Sect. 2, we explain the design of the mesoscale tendency terms, using a previously analyzed axisymmetric model simulation for reference. Single-column model simulations are presented and evaluated in Sect. 3, followed by large-eddy simulations in Sect. 4. We summarize this work and provide concluding remarks in Sect. 5.

## 2 Processes in the Tropical Cyclone Boundary Layer

### 2.1 Axisymmetric Model Details

To clarify the most important processes in the tropical cyclone boundary layer that must be included for an accurate simulation, we first examine an axisymmetric model simulation of an idealized tropical cyclone. This simulation produces similar flow structures compared with composite observations encompassing many tropical cyclones (e.g., Zhang et al. 2011) such as inflow-layer depth \(\approx \)1 km, surface inflow angle \(\approx \)23\(^\circ \), and a height of maximum wind speed roughly 500 m above sea level (a.s.l.), as discussed in more detail by Bryan (2012).

*r*) <250 km, and vertical grid spacing is 20 m at the surface but increases gradually to 250 m at the top of the domain (\(z = 25\) km a.s.l.).

*K*from the local vertical deformation (\(S_v\)) and moist Brunt–Vaisala frequency (\(N_m\)),

### 2.2 Mesoscale Tendency Terms

Figure 2a shows tangential velocity (\(u_\phi \), shaded) and radial velocity (\(u_r\), contours) averaged over days 8–12 of the simulation, a time period when the simulated tropical cyclone is quasi-steady (i.e., local time-tendency terms in the velocity budgets are negligible). We are here interested in the region outside the eyewall [i.e., for *r* greater than the radius of maximum wind (\(R_{{max}}\))] where the tropical cyclone boundary layer is typically characterized by radial inflow (\(u_r < 0 \)) and vertical advection is typically much smaller than radial advection.

*r*times the radial advection of absolute angular momentum, or

*f*is the Coriolis parameter (assumed constant herein). The three terms on the right side of (3) are, respectively, radial advection, centrifugal acceleration, and Coriolis acceleration.

*V*at a distance

*R*from the tropical cyclone centre, in addition to its radial gradient \(\partial V / \partial R\). Considering these as input parameters that do not change during a simulation, and understanding that the model-predicted profiles of wind speed are used to account for the secondary circulation of a tropical cyclone (via model-predicted profile of \(u_r\)), we consider the following form for the mesoscale tendency for \(u_\phi \) (\(M_\phi \)),

*V*and \(\partial V / \partial R\) may be functions of height, or could be considered constant with height for shallow domains of a few km depth. Several recent studies (Nakanishi and Niino (2012), hereafter NN12; Green and Zhang (2015), hereafter GZ15) have included centrifugal-like terms in their model equations, similar to the second term of (4). However, inclusion of an advection term (first term) makes this approach clearly different from these recent studies.

Considering now the processes that affect radial velocity \(u_r\), we return our attention to the axisymmetric model and note that tendencies from the PBL scheme (red in Fig. 2c) are again countered by a mesoscale tendency (blue in Fig. 2c). For this component of velocity, the mesoscale tendency can be considered the sum of four terms: radial advection, centrifugal acceleration, Coriolis acceleration, and radial pressure-gradient acceleration. Our formulation for the mesoscale tendency to \(u_r\) (\(M_r\)) must account for these terms, although we again consider the Coriolis term to be represented in the model equations in a standard manner, and account for the other three terms for \(M_r\), as described below.

*V*; this form is a consequence of the decision to use the model-predicted \(u_r\) to advect the user-specified gradient of angular momentum in \(M_\phi \), as discussed above.

*p*is pressure and \(\rho \) is air density. Here it becomes clear that the reference velocity

*V*is the so-called gradient wind, i.e., the tengential velocity necessary to balance the pressure gradient via Coriolis and centrifugal terms. We note that, although

*V*is assumed to be in gradient-wind balance, the model-predicted tangential velocity \(u_\phi \) does not need to obey such a relation.

*w*. We also note that the advection of \(u_r\) is a fairly small component of the radial velocity budget (Fig. 2), at least for regions outside the eyewall. Finally, as in Foster (2005), NN12, and GZ15, we neglect variations in

*r*, and simply use the constant

*R*, for the right-hand side of (8).

### 2.3 Comparison to Previous Studies

Some aspects of our mesoscale tendency terms [(4) and (10)] make our study different from other recent numerical studies of the tropical cyclone boundary layer. One major difference is that we intend to utilize these relations at a single point in the flow, in which height *z* is the only dimension (i.e., single-column modelling). Even for our large-eddy simulations, the horizontal dimensions of the domain are presumed small (e.g., Fig. 1) and so we choose to treat the mesoscale tendencies analogously for LES (details of the LES implementation of these terms are provided in Sect. 4). Consequently, tendency terms that contain radial gradients (e.g., radial advection) must be treated differently as compared to studies that have both *r* and *z* as dimensions. In fact, a major motivation for using the approximation (8) is so \(\partial u_r / \partial r\) does not need to be specified as an input parameter. For these reasons, the radial advection and centrifugal acceleration terms are subtly different compared to other seemingly similar studies of the tropical cyclone boundary layer (e.g., Kepert and Wang 2001; Foster 2009). We also note that, for radial diffusion and turbulence, we have chosen to neglect these terms because of their complex form, and the lack of a clear method to simplify them for single-column modelling, even though radial turbulence can be important in tropical cyclones (e.g., Rotunno and Bryan 2012), although typically only in the eyewall where the present approach is not valid.

We also clarify that our approximate equations are not derived via linearization of the governing equations, as in Kepert (2001) and Foster (2005). Our underlying approach is essentially a scale analysis in which relatively small terms are neglected. The budgets from a single axisymmetric simulation of an intense tropical cyclone are used here as an example (Fig. 2), but the conclusions are consistent with our previous work (e.g., Rotunno and Bryan 2012). Although we have neglected vertical advection (we reiterate: outside of the eyewall), some studies have found it to be an important contributor, especially in the \(u_\phi \) budget (e.g., Kepert and Wang 2001, their Fig. 9). This assumption is not critical to our modelling approach, as vertical advection terms could easily be added to (4) and (10) following the approach of Sommeria (1976). Vertical velocity could also be determined from the simulated flow fields [\(u_r(z)\) and \(u_\phi (z)\)] via the continuity equation, which has been used in several types of analytical and numerical models (e.g., Ooyama 1969; Emanuel 1986; Kepert 2001, 2013). We suspect that vertical advection terms are more important for weaker storms and/or broader storms than the one shown in Fig. 2, and we plan to investigate these issues in the future.

Finally, regarding the assumption that \(\partial w / \partial z\) can be neglected in (9) [in contrast to the typical assumption (e.g., Smith 1968)], we provide the following scale analysis. For a point *R* in the flow with characteristic radial velocity \(\overline{U}\), we assume its radial variation is \(\delta \overline{U} / \delta R\). Further, assume the vertical variation of *w* is \(\delta \overline{W} / H\), where *H* is the approximate depth of the inflow layer. Then the \(\partial w / \partial z\) term can be neglected in (9) when \( \delta \overline{W} / H<< \delta \overline{U} / \delta R\), and if valid, it follows from (9) that \(\delta \overline{U} / \delta R \approx - \overline{U} / R\). Using these relations, we find \(\delta \overline{W}<< - \overline{U} H / R\), and since \(H/R < 0.1\) outside of the eyewall, and \(\overline{U} \approx -10\) m s\(^{-1}\), the assumption is valid when vertical velocity at the top of the inflow layer is roughly 0.05 m s\(^{-1}\) or less. (Clearly, the assumption is not valid in tropical cyclone eyewalls, and the portions of rainbands with strong updrafts.)

### 2.4 Neglected Terms

As explained above, the mesoscale tendency terms (4) and (10) are not applicable in the eyewall of tropical cyclones, where vertical advection can be a leading-order process (e.g., Kepert and Wang 2001). They are probably also not applicable in certain parts of rainbands, especially where vertical velocity is large near the surface. Our goal has been to devise a simple set of tendency terms that can be added easily to single-column model simulations and small-domain LES for application over much (but clearly not all) of a tropical cyclone. The results reported in the following two sections demonstrate that our new approach has clear merits, especially when compared with other recently developed approaches.

*r*by

*R*.

## 3 Single Column Modelling

### 3.1 Methodology

*z*, and the horizontal velocity equations are simply

*M*terms are given by (4) and (10). We also integrate a potential temperature (\(\theta \)) equation,

*K*to small values above the PBL, and thus acts to limit growth of the boundary layer. Moisture is neglected herein for simplicity.

We integrate these equations using a third-order Runge–Kutta scheme, with vertical grid spacing of 25 m, and the domain depth of 4 km. To prevent reflection of vertically propagating waves, we apply a Rayleigh damper above 3 km. For initial conditions, we set \(u = 0\) and \(v (z) = V (z)\), with \(\theta = 300\) K at the surface and increasing linearly with height at \(5 \times 10^{-3}\) K m\(^{-1}\). We use \( f = 5 \times 10^{-5}\) s\(^{-1}\) for all simulations herein.

Surface heat flux is neglected for these simulations. The implicit assumption here is that boundary-layer turbulence in tropical cyclones is driven primarily by mean vertical wind shear near the surface. Alternatively, we might hypothesize that, in an approximately steady tropical cyclone boundary layer, the net heating via the surface heat flux is exactly canceled by a mesoscale tendency, specifically horizontal advection, that could be added to (17). For convenience, we simply neglect both effects.

### 3.2 Results

The first example is based on conditions in the axisymmetric model simulation from Sect 2. We choose \(R = 40\) km and, from the axisymmetric simulation at this radius, we find that constant values *V* = 38 m s\(^{-1}\) and \(\partial V/ \partial R = 8 \times 10^{-4}\) s\(^{-1}\) are reasonable matches to the gradient wind for \(z < 4\) km in the axisymmetric model [which is determined using (7) and the model-produced pressure field].

The predicted velocity profiles are shown in Fig. 5. The overall qualitative similarity between the single-column model (red curve in Fig. 5) and the axisymmetric model (grey curve) is also quite good, particular in terms of the shapes of both the \(u_r\) and \(u_\phi \) profiles. [The small-scale fluctuations in the profiles near the top of the boundary layer (at \(z = 1.4\) km in this case) are present in all of our single-column simulations, and denote where the simple PBL scheme (Eq. 1) is not used when stratification becomes large; this is a minor cosmetic feature that can be eliminated by requiring that *K* have a minimum value of *O*(1 m\(^2\) s\(^{-1}\)) (not shown).] The most noteworthy difference between the axisymmetric model and single-column model results is the magnitude of \(u_r\) near the surface, which is \(\approx \)20 % weaker for the single-column model (Fig. 5b). There are several possible reasons for these differences, such as the assumed constant values for *V* and \(\partial V / \partial R\) in this simulation, or the complete neglect of moisture, for example. It is more likely that the centrifigual acceleration is too large near the surface (\(z < 100\) m) compared to the axisymmetric model simulation because we use *V* instead of \(u_\phi \) in (4) and (10). However, we note that our results are reasonably accurate, and clearly improved on other approaches (as shown below).

We hereafter call this a classic “Ekman-type” formulation in the sense that the velocity equations include only large-scale pressure-gradient terms to oppose viscous terms, although unlike the classic analytic solution (see, e.g., Wyngaard 2010, pp. 208–209) the steady assumption is not made and viscosity is not assumed to be constant. Results using (18) and (19) [in place of \(M_r\) and \(M_\phi \) in (15) and (16)] are shown as green-dashed curves in Fig. 5. This simulation produces smaller \(u_\phi \) over all depths, and stronger radial inflow above 200 m a.s.l. that extends over a deeper layer (up to 2 km a.s.l.). The mesoscale tendency terms using the Ekman-type simulation (Fig. 6) are roughly one order of magnitude smaller than using the new method (cf. Fig. 4). A consequence of these weak tendencies is that the PBL scheme reduces \(u_\phi \) too much. In comparison, the new method has large-amplitude tendency terms that oppose the PBL tendencies. Results from our new method in comparison to results from the Ekman-type (Fig. 5) show obvious merits.

A possible shortcoming of the new method is a strong sensitivity to the input parameter \(\partial V / \partial R\). As an example, wind profiles at \(t = 12\) h from five simulations using different values of \(\partial V / \partial R\) are shown in Fig. 7; as \(\partial V / \partial R\) increases in magnitude, \(u_\phi \) decreases and \(u_r\) increases in magnitude. In fact, the largest magnitude of \(\partial V / \partial R\) produces results similar to the classic Ekman-type solution; the reason is that the \(-u_r (\partial V / \partial R)\) term becomes similar in magnitude to the \(-u_r ( V / R )\) term in (4), and thus both terms cancel each other, leaving no mechanism to oppose the PBL tendency.

*r*according to a power law

^{1}that satisfies \(u_\phi = V\) at \(r = R\), i.e.,

Thus, our input parameter \(\partial V / \partial R\) can be related to the other two input parameters (*V* and *R*) through a decay rate *n*. We note that the parameters used for the simulation in the first example give \(\partial V / \partial R = -0.8 \times (V/R)\); consistently, the simulated vortex in the axisymmetric simulation has \(u_\phi \sim r^{-0.8}\) near the surface at \(r = 40\) km. Therefore, if \(\partial V / \partial R\) is not known from observations, then a guess can be made for the decay rate *n*, and the relation \(\partial V / \partial R = -n (V/R)\) can be used once *V* and *R* are specified. Typical values for *n* for tropical cyclones of various intensity were determined by Mallen et al. (2005, their Table 2). We also note that the largest absolute value of \(\partial V / \partial R\) for Fig. 7 corresponds to \(n \approx 1\), i.e., \(u_\phi \sim r^{-1}\), a potential vortex (Burggraf et al. 1971).

### 3.3 Comparison to Observations and Other Methods

To further assess the fidelity of these simulations, we compare model results with composite wind-speed profiles from observed tropical cyclones based on dropsonde data from the National Oceanographic and Atmospheric Administration (NOAA). We use the quality controlled dataset of Wang et al. (2015) which contains 17 years of data from 120 tropical cyclones. (We note that dropsondes from the U.S. Air Force, and from field projects that did not use NOAA aircraft, are not included in this dataset.) For this first case, we searched for all dropsonde profiles in which the average wind speed between 500 m and 1000 m a.s.l. was between 35 and 45 m s\(^{-1}\). We used only soundings that had at least two data points in this layer and that were separated by at least 250 m, to exclude soundings with a substantial amount of missing data. We also required each sounding to have at least one data point below 100 m a.s.l., and excluded any sounding located more than 300 km from the tropical cyclone centre. From this procedure we obtained 688 soundings that were then averaged to produce composite wind profiles, using 10 m vertical grid spacing, which are shown as black curves in Fig. 8. (Our analysis does not account for observed storm motion, and does not exclude dropsondes from the eyewall of tropical cyclones.) We note that the radial velocity profile (Fig. 8b) suggests very deep inflow up to 2.7 km a.s.l., although the strongest inflow (which is most important dynamically) is confined to the lowest \(\approx \)1 km a.s.l. The level of strongest inflow is at 100 m a.s.l., and the surface (10 m a.s.l.) inflow angle is 22\(^\circ \), both of which are similar to average values from previous studies (e.g., Powell et al. 2009; Zhang et al. 2011; Zhang and Uhlhorn 2012).

The next simulation is based on this composite profile. We assume that *V* decreases linearly with height from 40 m s\(^{-1}\) at the surface to zero at 18 km a.s.l. (to roughly match the composite profile, whereas in the previous subsection we assumed *V* = constant to roughly match the gradient-wind profile from the axisymmetric model). Similarly, we assume that \(\partial V / \partial R\) decreases linearly from its maximum value at the surface to zero at 18 km a.s.l. For lack of any specific data, we retain \(\partial V/ \partial R = -8 \times 10^{-4}\) s\(^{-1}\) (at the surface), and \(R = 40\) km.

Results using the new method for mesoscale tendencies after 12 h of integration (red curves in Fig. 8) are remarkably similar to the observed profiles, especially for \(z < 800\) m. In terms of specific quantitative features, the height of maximum inflow is 90 m a.s.l., the inflow-layer depth is 1.1 km, and the surface inflow angle is 23\(^\circ \), all of which are comparable with previous observational composites (e.g. Powell et al. 2009; Zhang et al. 2011; Zhang and Uhlhorn 2012). In contrast, a simulation using the Ekman-type method (green-dashed curve in Fig. 8) produces qualitatively similar results as before, i.e., \(u_\phi \) is too low and \(u_r\) is generally too high.

*V*as in the previous paragraph) are shown as blue curves in Fig. 8. Both of these techniques produce shallower inflow layers, by approximately a factor of 2, compared to the new method and the observational composite. Further, the surface inflow angle from these techniques is smaller (by 25–50 %) than the observed average value of 23\(^\circ \) (e.g., Powell et al. 2009; Zhang and Uhlhorn 2012). The results have notably greater \(u_{\dot{\phi }}\) than the observed profile for \(z < 500\) m, and vertical wind shear (\(\partial U / \partial z\)) is 20 % larger in the surface layer (roughly, lowest 100 m a.s.l.). Very similar mean wind profiles were produced (using LES) by NN12 (their Fig. 3) and GZ15 (their Fig. 4).

Time series of some notable flow parameters are shown in Fig. 9, where the inflow-layer depth, \(z_{{infl}}\), is defined here as the lowest height at which \(u_r\) exceeds \(-3\) m s\(^{-1}\) (Fig. 9a). The height of strongest inflow, \(z_{u{{-min}}}\), is defined as the height at which \(u_r\) is a minumum (Fig. 9b), and the surface inflow angle, \(\beta _{\text {10-m}}\), is the inflow angle at 10 m a.s.l. (Fig. 9c). In all cases, the variations near the beginning of the simulations are associated with decaying inertial oscillations (e.g., Lewis and Belcher 2004). The Ekman-type method produces oscillations with the longest period, by far (\({>}\)10 h); for the NN12 and GZ15 methods, the period are very short (\(\approx \)1 h) and oscillations last for nearly 12 h. For the new method, the period is roughly 2 h, and the signal is practically zero after two oscillations. The primary conclusions from Fig. 9, though, are that the new method produces the most accurate quantitative results (\(z_{{infl}} \approx \) 1 km, \(z_{u{{-min}}} \approx \) 0.1 km, and \(\beta _{\text {10-m}} \approx \) 23\(^\circ \)) and that these values clearly converge at the analysis time used above (\(t = 12\) h).

It could be argued that the techniques of NN12 and GZ15 are preferable to simulations that neglect any inertial terms, i.e., compared to the classic Ekman-type method that only accounts for geostrophic pressure-gradient acceleration (green-dashed curve in Fig. 8). Indeed, the NN12 and GZ15 techniques produce some qualitatively accurate results, such as greatest radial velocity near the surface, and greatest wind shear in the lowest 100 m a.s.l., which is broadly consistent with the observed tropical cyclone boundary layer. Nevertheless, it seems clear that our new formulation (Eqs. 4 and 10) produces better quantitative comparison to observations. The primary reason is the inclusion of radial advection terms, as discussed above.

As one final test of the single-column model, we evaluate simulations at higher wind speeds. In this case, we first calculated a composite of dropsonde observations, using the same methodology as above, but for soundings in which the mean wind speed in the 500-1,000 m layer was between 55 and 65 m s\(^{-1}\) (and the same method for excluding soundings with few datapoints); 269 soundings met all criteria. Based on the average results (black curve in Fig. 10) we ran simulations using \(V = 60\) m s\(^{-1}\) at the surface decreasing to zero at 18 km a.s.l. The average radius of the dropsondes was roughly 40 km, so we set \(R = 40\) km. Simulations with the Ekman-type method, the NN12 method, and the GZ15 method exhibit the same qualitative differences from observations as before (Fig. 10).

For the new method, we ran a series of simulations with different values for \(\partial V / \partial R\), and show in Fig. 10 the case that best matches the dropsonde composite: \(\partial V / \partial R = -1.1 \times 10^{-3}\) s\(^{-1}\), corresponding to \(n = 0.7\). Results using the new technique (red in Fig. 10) again produce the best match to the observed composite. There are a few subtle differences from the dropsonde composite (\(\thickapprox 5\) m s\(^{-1}\)) that may be attributable to the inclusion of dropsondes from the eyewall of tropical cyclones. (As noted in Sect. 2, our new method is not applicable in the eye and eyewall of tropical cyclones.) Nevertheless, these analyses show that the new method accurately reproduces robust qualitative aspects of the tropical cyclone boundary layer (e.g., depth and magnitude of the inflow layer) that several previous approaches do not.

## 4 Large-Eddy Simulations

In this section we evaluate the new method when used within LES, which can provide further insight into turbulent processes within the tropical cyclone boundary layer. For example, vertical momentum fluxes (which are notoriously difficult to measure within a tropical cyclone boundary layer due to hazardous conditions) can be estimated using LES results.

Here we use the numerical model “Cloud Model version 1” (CM1) that has been used for several LES studies in recent years (e.g., Kang and Bryan 2011; Kang et al. 2012; Wang 2014; Nowotarski et al. 2014; Markowski and Bryan 2016). Details of the model used here, including a new “two part” subgrid model near the surface, are provided in the Appendix. The parametrization of surface stress uses the same scheme as in Sect. 2, although for LES we use time-averaged values of wind speed at the lowest model level to calculate an average stress, and then calculate instantaneous values following Moeng (1984).

### 4.1 Mesoscale Tendency Terms

*x*and

*y*from our Cartesian grid align with

*r*and \(\phi \), respectively, in a cylindrical grid. In other words, we locate the model domain to the east of the tropical cyclone centre, as illustrated schematically in Fig. 1. But our method for the mesoscale tendency terms for the LES model in the two horizontal directions [\(M_1^{LES}\) and \(M_2^{LES}\)] differs from NN12 and GZ15, who derived their terms from the governing equations for flow in cylindrical coordinates. We simply adopt the method developed for single-column modelling (Sect. 2) but replace the variables \(u_r\) and \(u_\phi \) with domain-average zonal \(\left\langle u\right\rangle \) and meridional \(\left\langle v\right\rangle \) components of velocity, where angled brackets denote a horizontal average over the domain at each model level. Specifically, we use,

*t*and

*z*only. (In contrast, the methods advocated by NN12 and GZ15, discussed below, are calculated independently at each gridpoint.) We intentionally chose this form based on the methodology put forth in Sect. 1 (and discussed, for example, by Sommeria 1976), i.e., that mesoscale tendency terms should account for processes on scales larger than the size of the model domain. In other words, the large-scale gradients in wind speed (which are needed for calculations of large-scale advection) are presumed to not exist on the small domain, and so we assume constant (in

*x*and

*y*) values of large-scale gradient that are specified at the beginning of a simulation; mesoscale horizontal advection terms (first terms on the right side of Eqs. 25 and 26) are then applied uniformly across the entire domain at each timestep. As demonstrated below, this methodology produces realistic wind profiles as compared to observations in the tropical cyclone boundary layer, and also avoids the potential problem of introducing an instability via the vorticity equation (NN12, GZ15). That is, because these tendencies are not functions of

*x*or

*y*, they cannot contribute to the vertical vorticity tendency.

The three terms on the right-hand side of Eq. 25 represent, respectively: mean radial advection of radial velocity in a tropical cyclone the centrifugal acceleration term associated with the mean flow in a tropical cyclone and the large-scale pressure gradient in a tropical cyclone. For (26), the two terms on the right-hand side represent, respectively: mean radial advection of tangential velocity and centrifugal acceleration. As with the single-column modelling approach, the model user must specify three parameters: *R*, *V*, and \(\partial V / \partial R\). Finally, we note that, unlike our simple framework for single-column modelling, the potential temperature field plays a direct role in the simulated flow, via the buoyancy term in the velocity equation, Eq. 31a.

### 4.2 Results

*u*,

*v*,

*w*, and \(\theta ^\prime \) (defined in the Appendix) to damp vertically propagating waves. Periodic boundary conditions are used in both horizontal directions.

Instantaneous fields of horizontal wind speed *U* are shown in Fig. 11. Linear “streaks” are apparent in *U* near the surface (Fig. 11a). Such structures have been observed in hurricane boundary layers (e.g., Wurman and Winslow 1998; Morrison et al. 2005; Zhang et al. 2008b; Kosiba and Wurman 2014) and are thought to be important for strong wind gusts near the surface in tropical cyclone boundary layers. Farther aloft (Fig. 11b), linear features are less obvious, although it is clear that the simulated boundary layer is turbulent.

Horizontal wind speed is plotted in Fig. 12 because it gives us an opportunity to compare with in situ low-level data collected in 2003–2004 by NOAA aircraft during the Coupled Boundary Layer Air-Sea Transfer experiment (CBLAST, Black et al. 2007). Specifically, we use data collected in Hurricanes Fabian (2003), Isabel (2003), Frances (2004), and Jeanne (2004) [see Table 1 in Zhang and Drennan (2012)]. A total of 69 “flux runs” below 800 m altitude are analyzed. Quality control and analysis procedures are explained in previous studies (e.g., French et al. 2007; Zhang et al. 2009). For Fig. 12 the black dots are in situ wind-speed measurements, averaged along the length of the flux run. We selected all cases in which this value was within 4 m s\(^{-1}\) of the average model value, \(\left\langle U \right\rangle \), at the same height in the simulation with the new mesoscale tendency terms (Fig. 12b). This procedure yields 26 observational cases. Additional measurements from these same 26 cases are included in other analyses below.

The LES results (red in Fig. 14) again have a mean positive bias compared with the observational estimates. In this case, the mean absolute difference is 22.0 m\(^2\) s\(^{-1}\). Nevertheless, we are encouraged to see comparable magnitudes in the lower half of the boundary layer (below roughly 500 m a.s.l.). An over-prediction of boundary-layer depth by our LES seems apparent when comparing the observations at \(z \approx 0.75\) km to model results. We find this over-prediction can be ameliorated by inclusion of subsidence terms to (25)–(26) (not shown).

### 4.3 Spectral Analysis

As another method to evaluate the LES, we examine temporal spectra of wind components near the surface. The observational data for this analysis were collected on 12 September 2003 in Hurricane Isabel, at an average radius of 130 km from the tropical cyclone centre, and at an average altitude of 194 m a.s.l. Mean flight-level wind speed was 33 m s\(^{-1}\). The flight leg was \(\approx \)54 km in length (\(\approx \)6 min in time), which was one of the longest flux runs during CBLAST. Power spectral density is calculated using fast Fourier transform of in situ 40-Hz data. Results are shown as black curves for the along-wind component of velocity in Fig. 15 and for vertical velocity in Fig 16. The power spectral density of both velocity components have roughly \(f^{-5/3}\) structure, indicative of a turbulent inertial subrange, for \(f > 0.1\) Hz, similar to a recent analysis (of a different dataset) by Nolan et al. (2014).

For LES, the input parameters are chosen to match data from this case. We estimate *V* using dropsonde data (Fig. 2 from Zhang and Drennan 2012); based on values near the top of the boundary layer, we choose \(V = 37\) m s\(^{-1}\). For \(\partial V / \partial R\), we use the radial gradient of wind speed from flight-level data, which is approximately \(-1.6 \times 10^{-4}\) s\(^{-1}\). Finally, the mean radius of the flight gives \(R = 130\) km.

For this simulation, the model domain extends 6 km \(\times \) 6 km horizontally, and is 4 km deep with a Rayleigh damper above 3 km. As a test of sensitivity to resolution, we use three different grid spacings: \(\Delta x =\) 31.25, 15.625, and 7.8125 m. In all cases, \(\Delta z = \Delta x / 2\). For the analysis of temporal spectra, we have wind speed every timestep from a location in the middle of the domain. To compare model results to the CBLAST observations, we use an ensemble average of power spectral density calculated using 50 %-overlapping, 6-min segments of the model time series at \(z \approx 190\) m. A 6-min segment was chosen to match the duration of the CBLAST data, resulting in 39 segments for the ensemble average.

For all three model resolutions, the power spectral densities of the along-wind component of wind speed (Fig. 15) are similar to each other in the low-frequency portion of the spectrum, from the lowest frequency (\(2.7 \times 10^{-3}\) Hz) to a critical frequency \(f_c\) above which the spectra decrease rapidly. As expected, \(f_c\) is larger for simulations with smaller grid spacing. The magnitude of \(f_c\) is apparently related to the mean advective velocity \(U_a\) divided by the smallest resolvable scale in the simulated flow, i.e., \(f_c \approx U_a / (6 \Delta )\), where \(\Delta \) is horizontal grid spacing, and \(6 \Delta \) is approximately the smallest scale that is unaffected by numerical filtering in CM1 (see, e.g., Appendix of Bryan et al. 2003).

There is a notable difference between model spectra and the observational spectrum at low frequencies (\(f < 0.1\) Hz) which is likely related to the different measuring techniques (i.e., at a single point for the model results, and along a flight path for the observations). The difference is especially pronounced for the *w* spectra.

Most important, though, the LES spectra have \(f^{-5/3}\) structure, similar to the observational spectrum, for a certain range of frequencies (specifically, between about 0.08 Hz and the cut-off frequency \(f_c\)). This behaviour is most apparent for the two highest resolution simulations, and suggests that \(\Delta < 20\) m is needed to produce an inertial subrange in the tropical cyclone boundary layer for this model; we note that this conclusion is technically only valid for this level (190 m a.s.l.), and that higher resolution may be needed near the surface. As further evidence for the existence of an inertial subrange in our simulations, we note that the cross-stream and vertical velocity spectra have amplitude roughly four-thirds of the along-stream spectra (as expected by theory (e.g., Wyngaard 2010)) for frequencies of \(\approx \)0.08–0.5 Hz (not shown).

## 5 Summary and Conclusions

An inexpensive method to simulate wind profiles in the boundary layer of tropical cyclones is developed and evaluated. The method is intended for single-column modelling and for three-dimensional modelling with small domains (of order 5 km in extent), and can be used with a PBL parametrization or within large-eddy simulations. The key step is to account for processes that occur on scales larger than the proposed model domain.

The core of the new procedure is to add “mesoscale tendency” terms to the horizontal velocity equations to account for large-scale radial advection, centrifugal acceleration, and pressure-gradient acceleration within a tropical cyclone. The method utilizes three simple input parameters: *R*, the distance from the centre of the tropical cyclone, *V*, a reference wind profile, and \(\partial V / \partial R\), the radial gradient of *V*. Ideally, these three parameters can be determined by observations from within a tropical cyclone, or from a reference simulation (such as the axisymmetric simulation used in Sects. 2–3). The value for \(\partial V / \partial R\) is particularly difficult to estimate from observed storms, and model output can be quite sensitive to its value. But as shown in Sect. 3.2, if the tangential velocity varies as a power law, i.e., \(u_\phi \sim r^{-n}\), then the relation \(\partial V / \partial R = - n (V / R)\) can be used to estimate \(\partial V / \partial R\) once *V* and *R* are chosen, and assuming a reasonable guess can be made for *n*.

Two slightly different formulations for the mesoscale tendency terms are presented: one intended for single-column modelling, given by (4) and (10); and one formulation to be used with LES, (25)–(26). Previous methods for simulating the boundary layer of tropical cyclones were evaluated alongside the new method, including a classic Ekman-type method that adds only a large-scale (geostrophic) pressure gradient acceleration to the horizontal velocity equations, and two recently developed methods that also account for centrifugal-acceleration-like terms (Nakanishi and Niino 2012; Green and Zhang 2015). The new method, which also accounts for large-scale radial (i.e., horizontal) advection, has clear advantages: it produces realistic mean wind-speed profiles compared with composites of dropsonde observations in tropical cyclones. Also, comparison of LES results with low-level data from NOAA flights during CBLAST shows reasonable results, including an effective eddy viscosity of *O*(50 m\(^2\) s\(^{-1}\)) and temporal spectra with frequency dependence of \(f^{-5/3}\) for frequencies \(\gtrsim 0.1\) Hz.

Before concluding, we note again that simulations discussed herein neglected surface heat flux and all moist processes, for simplicity. We also do not include any mesoscale tendency term for \(\theta \), which would tend to cool the boundary layer. Consequently, \(\theta \) profiles tend to be nearly well mixed in our simulations (not shown), whereas observed \(\theta \) profiles tend to be stratified (for \(z > 100\) m) in observed tropical cyclones (e.g., Kepert et al. 2016). Despite these approximations, the modelled wind profiles compare well with observations from the tropical cyclone boundary layer. Our results suggest that effects from stratification and moisture play a relatively minor role, compared to shear-driven turbulence mechanisms, in the boundary layer of tropical cyclones. Nevertheless, stratification and moisture should be considered in future work, which would necessitate inclusion of mesoscale tendency terms (i.e., radial advection) in equations for potential temperature and water vapour, and perhaps also the evaporation of water, which Kepert et al. (2016) found to be a major contributor to the \(\theta \) profile in tropical cyclones.

## Footnotes

- 1.
The authors thank Richard Rotunno for bringing this analysis to our attention.

## Notes

### Acknowledgments

The National Center for Atmospheric Research is sponsored by the National Science Foundation. Rochelle Worsnop was supported by NSF Grant DGE-1144083. Jun Zhang was supported by NOAA HFIP Grant NA14NWS4680028. High-performance computing support was provided by NCAR’s Computational and Information Systems Laboratory (Allocation Numbers NMMM0026 and UCUB0025 on Yellowstone, ark:/85065/d7wd3xhc). The authors thank Richard Rotunno and Peter Sullivan for their informal reviews of this manuscript, as well as Kerry Emanuel, Benjamin Green, Daniel Stern, and the anonymous reviewers.

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