Effects of Surface Exchange Coefficients and Turbulence Length Scales on the Intensity and Structure of Numerically Simulated Hurricanes

George H. Bryan National Center for Atmospheric Research,* Boulder, Colorado

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Abstract

Using numerical simulations, this study examines the sensitivity of hurricane intensity and structure to changes in the surface exchange coefficients and to changes in the length scales of a turbulence parameterization. Compared to other recent articles on the topic, this study uses higher vertical resolution, more values for the turbulence length scales, a different initial environment (including higher sea surface temperature), a broader specification of surface exchange coefficients, a more realistic microphysics scheme, and a set of three-dimensional simulations. The primary conclusions from a recent study by Bryan and Rotunno are all upheld: maximum intensity is strongly affected by the horizontal turbulence length scale lh but not by the vertical turbulence length scale lυ , and the ratio of surface exchange coefficients for enthalpy and momentum, Ck /Cd , has less effect on maximum wind speed than suggested by an often-cited theoretical model. The model output is further evaluated against various metrics of hurricane intensity and structure from recent observational studies, including maximum wind speed, minimum pressure, surface wind–pressure relationships, height of maximum wind, and surface inflow angle. The model settings lh ≈ 1000 m, lυ ≈ 50 m, and Ck /Cd ≈ 0.5 produce the most reasonable match to the observational studies. This article also reconciles a recent controversy about the likely value of Ck /Cd in high wind speeds by noting that simulations in a study by Emanuel used relatively large horizontal diffusion and low sea surface temperature. The model in this study can produce category 5 hurricanes with Ck /Cd as low as 0.25.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: George H. Bryan, NCAR, 3090 Center Green Drive, Boulder, CO 80301. E-mail: gbryan@ucar.edu

Abstract

Using numerical simulations, this study examines the sensitivity of hurricane intensity and structure to changes in the surface exchange coefficients and to changes in the length scales of a turbulence parameterization. Compared to other recent articles on the topic, this study uses higher vertical resolution, more values for the turbulence length scales, a different initial environment (including higher sea surface temperature), a broader specification of surface exchange coefficients, a more realistic microphysics scheme, and a set of three-dimensional simulations. The primary conclusions from a recent study by Bryan and Rotunno are all upheld: maximum intensity is strongly affected by the horizontal turbulence length scale lh but not by the vertical turbulence length scale lυ , and the ratio of surface exchange coefficients for enthalpy and momentum, Ck /Cd , has less effect on maximum wind speed than suggested by an often-cited theoretical model. The model output is further evaluated against various metrics of hurricane intensity and structure from recent observational studies, including maximum wind speed, minimum pressure, surface wind–pressure relationships, height of maximum wind, and surface inflow angle. The model settings lh ≈ 1000 m, lυ ≈ 50 m, and Ck /Cd ≈ 0.5 produce the most reasonable match to the observational studies. This article also reconciles a recent controversy about the likely value of Ck /Cd in high wind speeds by noting that simulations in a study by Emanuel used relatively large horizontal diffusion and low sea surface temperature. The model in this study can produce category 5 hurricanes with Ck /Cd as low as 0.25.

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: George H. Bryan, NCAR, 3090 Center Green Drive, Boulder, CO 80301. E-mail: gbryan@ucar.edu

1. Introduction

Atmospheric conditions near the surface of intense (category 4–5) tropical cyclones (TCs) remain difficult to characterize. Although the magnitude of peak wind speed in TCs near the surface is fairly well documented from surface observing stations and aircraft reconnaissance (via remote sensing technology and/or dropsondes), other properties like the functional form of wind gusts with height and the variance of wind speed in time and space are less certain because of the hazards involved in data collection (e.g., Harper et al. 2010). These uncertainties create difficulties for certain disciplines—perhaps most notable being the field of civil engineering, which needs to design structures that can withstand TC winds (e.g., bridges near coastlines, offshore facilities for oil and gas companies, and proposed offshore wind-energy turbines).

The interface between the ocean and the atmosphere is also difficult to characterize. This uncertainty is often cited as holding back progress of numerical forecast model accuracy (e.g., the majority report to the National Oceanic and Atmospheric Administration 2006). One particular topic of interest for model development is the values of surface exchange coefficients for enthalpy Ck and momentum Cd (i.e., the drag coefficient) at hurricane wind speeds, which are used to determine fluxes of heat and momentum between the ocean and atmosphere in numerical models. Maximum hurricane wind speed has long been hypothesized to be proportional to (Ck/Cd)1/2 (see, e.g., the review by Emanuel 2004). However, the exact functional form between wind speed and Ck/Cd has been called into question recently based on some numerical simulations (e.g., Bryan and Rotunno 2009b; Montgomery et al. 2010) and a new analytical model (Emanuel and Rotunno 2011).

Numerical models offer promise as a tool that can decrease uncertainty about atmospheric conditions in intense hurricanes because they can provide dynamically consistent profiles of temperature, moisture, and winds that are difficult to obtain observationally. However, small-scale turbulence in hurricane boundary layers cannot be resolved in numerical models unless grid spacing is less than approximately 100 m (e.g., Rotunno et al. 2009); because such resolution is onerously expensive, the vast majority of modeling studies of hurricanes use turbulence parameterization schemes. The uncertainties in near-surface winds in TCs mentioned above make it difficult to design and evaluate these turbulence schemes, although some recent studies have been making progress by using special research datasets (e.g., Nolan et al. 2009a,b).

Numerical models have also been shown recently to produce clearly unnatural results in some cases that seem to be related to parameters in surface layer and/or turbulence parameterizations. For example, Hausman (2001) and Persing and Montgomery (2003) showed, using two different codes, that numerical models could produce unnaturally strong TCs (e.g., maximum wind speed > 140 m s−1 and minimum pressure <850 mb). The roles of horizontal turbulent exchange (i.e., mixing) between the eye and eyewall were considered by both Hausman (2001) and Persing and Montgomery (2003) as a potential explanation for the unnaturally large intensities in their model simulations. In a comprehensive examination of model settings, Bryan and Rotunno (2009b) showed that the horizontal turbulence parameterization in an axisymmetric model had a strong effect on maximum intensity, even more so than settings in the vertical turbulence (i.e., boundary layer) scheme and/or settings in the surface flux parameterizations.

This study examines a large set of numerical simulations to gain a better understanding of how TC intensity and structure are affected by changes in certain settings of the turbulence and surface-layer schemes. The primary methodology is to focus only on storms at maximum intensity, which removes the need to study the complex process of TC intensification, and allows for comparison with some steady-state theories for TC intensity. Another goal of this study is to determine which model settings yield realistic values of relatively well-observed properties of strong TCs such as maximum winds, minimum pressure, the surface wind–pressure relationship, height of maximum winds, and near-surface inflow angle. To test the generality of the results, two different sets of initial conditions are considered, one of which has a higher sea surface temperature than other recent modeling studies. The results herein also reconcile a recent controversy about a conclusion by Emanuel (1995) concerning the likely value of Ck/Cd in intense TCs.

2. Methodology

a. Model setup

This study uses Cloud Model version 1 (CM1), which is a nonhydrostatic numerical model described in Bryan and Rotunno (2009b, hereafter BR09). The axisymmetric version of CM1 is used for most simulations because of the low computational cost, which allows for greater exploration of the settings that affect simulated hurricane intensity and structure. The results from more than 400 simulations are shown in this article, all of which have been integrated for ≥12 days. Such a study using only 3D simulations would be computationally prohibitive. One set of 3D simulations is examined later in this article to test the generality of the results.

Two different model setups are used herein. For one configuration, referred to as setup A, the initial conditions and physical parameterizations are the same as the “default” configuration of BR09, which are identical to those used in the influential study by Rotunno and Emanuel (1987, hereafter RE87); see Table 1. The microphysics parameterization from RE87 is a simple scheme that does not account for ice processes and requires saturation (i.e., 100% relative humidity) in the presence of liquid water. The initial CAPE of the RE87 sounding is approximately 400 J kg−1. The nominal potential intensity (PI) for this initial environment, using the method of Bister and Emanuel (2002), is 44 m s−1 assuming Ck/Cd = 0.5, and PI is 70 m s−1 assuming Ck/Cd = 1.

Table 1.

Settings for setup A (the default for both RE87 and BR09) and setup B, where Ts is sea surface temperature, ΔT is the initial air–sea temperature difference, and “RE87” refers to the study by Rotunno and Emanuel (1987).

Table 1.

A second configuration, referred to as setup B, uses a different sounding, higher sea surface temperature Ts, and a double-moment microphysics scheme that includes ice processes (Morrison et al. 2009); see second column of Table 1. For setup B, the moist tropical (MT) sounding of Dunion (2011) is used as the initial sounding; the initial CAPE is approximately 2000 J kg−1. The larger CAPE of this sounding could be problematic in an axisymmetric model because, as discussed by RE87 (p. 548), convective updrafts could be artificially intense and might be forced to artificially cascade energy upscale; hence, 3D simulations using this setup are shown later in this article. For setup B, the sea surface temperature is 29°C, which is chosen because it gives an initial air–sea temperature difference ΔT of 2.2°C; this value of ΔT is similar to that of the RE87 setup (Table 1), and is near the climatological average value for hurricane environments (e.g., Cione et al. 2000). The nominal PI for this environment is 63 m s−1 for Ck/Cd = 0.5 and 92 m s−1 for Ck/Cd = 1.

The initial vortex for all simulations is the same as that in section 3c of BR09. The domain size and horizontal grid spacing are the same as the default setup of BR09: radial grid spacing Δr is 1 km for r < 64 km and Δr gradually increases to 15 km at the lateral boundary. To address concerns about the vertical resolution used by BR09, the simulations herein use a stretched vertical grid with much smaller values of vertical grid spacing Δz near the surface. The differences between the relatively low vertical resolution of BR09 (Res1) and the relatively high vertical resolution used herein (Res2) are provided in Table 2.

Table 2.

Settings for the relatively low vertical-resolution configuration (Res1, the BR09 default) and the relatively high vertical-resolution configuration (Res2).

Table 2.

b. Turbulence parameterization

The turbulence parameterization is the same as in BR09. Four different values of horizontal turbulence length scale lh are examined herein: lh = 3000, 1000, 300, and 0 m. Three different values of vertical turbulence length scale lυ are examined: lυ = 200 m (which was used in section 3c by BR09), 100 m, and 50 m. These values of lυ nearly cover the range from other commonly used models.1

With the high vertical resolution of Res2, it is difficult to maintain numerical stability for vertical diffusion without using a very small time step (of order 0.1 s). To overcome this problem, CM1 uses an implicit formulation for vertical diffusion that can be generalized as follows:
e1
where ϕ represents a predictive variable in the model, Δt is the time step, and Kυ is the vertical diffusion coefficient. The superscripts t and t + Δt denote values at the current time and one time step into the future, respectively. Specifically, (1) is solved at the beginning of each time step and the resulting tendencies are held fixed for the remainder of the numerical solution procedure.2 The variable γ can be set by the model user. Setting γ = 0 corresponds to the “Euler-forward” method (i.e., explicit forward-in-time integration) that is used in many models, including the RE87 axisymmetric model and the “bulk PBL” scheme in the fifth-generation Pennsylvania State University–National Center for Atmospheric Research Mesoscale Model (MM5). This method is unstable if KυΔtz2 > 0.5, and so it is only appropriate for rather coarse vertical grid spacing3 or small time steps. In fact, the RE87 model and the bulk PBL scheme in MM5 simply reduce Kυ, as necessary, to prevent instability. Alternatively, to maintain numerical stability with a reasonable time step, one can use an implicit method with 0 < γ ≤ 1. This technique is more expensive, as it requires solving a tridiagonal matrix for (1); however, it does not force model users to change Kυ or Δz. Moreover, γ ≥ 0.5 is absolutely stable (e.g., Richtmyer and Morton 1967). The CM1 default setting is γ = 1—that is, the “implicit Euler” or “Euler-backward” method, which is used for all simulations in this article. Although this method is only first-order accurate in time, it is absolutely stable and nonoscillatory (Ferziger and Perić 2002). This setting yields smooth fields near the surface with reasonable time steps, whereas lower values of γ sometimes produce obviously unphysical 2Δz oscillations. The time step for all simulations herein is 3 s, except for the lh = 0 simulations with setup B, for which the time step is 1.5 s.

c. Surface exchange coefficients

Surface fluxes of heat and momentum in the model are calculated using bulk aerodynamic formulae applied at the lowest model level, as in RE87. The surface exchange coefficients for enthalpy Ck and momentum Cd are held fixed for each individual simulation in this study to allow for comparison with past studies (e.g., Emanuel 1995; BR09), and because this approach allows for straightforward interpretation with theory. (This methodology should not imply that Ck and Cd are constant in nature; see, e.g., Andreas 2011.) The Ck is used for surface fluxes of both potential temperature and water vapor mixing ratio, and unless stated otherwise it is 1.2 × 10−3 (as in section 3c of BR09); this value is based on recent observational and laboratory studies (e.g., Drennan et al. 2007; Jeong 2008; Bell 2010), although an appropriate value for intense tropical cyclones remains uncertain. Most results are presented in terms of the ratio Ck/Cd, which is theoretically important for maximum hurricane intensity (e.g., Emanuel 2004). Dissipative heating is included in all simulations.

d. Analysis methods

All simulations are integrated for 12 days, except the lh = 0 simulations with setup A are integrated for 18 days. As in BR09, the primary focus herein is maximum hurricane intensity. The maximum azimuthal velocity at any given time υmax(t) is obtained by searching all model levels. Because intensification rate is affected by Cd (e.g., Rosenthal 1971; Montgomery et al. 2010), and because an arbitrary time period for analysis may affect results (Bryan 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.), the overall maximum intensity Vmax is determined objectively by searching for the largest 2-day running-mean value of υmax(t) using hourly output. Examples of υmax(t) and Vmax from two cases are shown in Fig. 1. All other measures of hurricane intensity and structure are 2-day averages using the same time period.

Fig. 1.
Fig. 1.

Time series of υmax(t) for Ck/Cd = 0.5 (black) and Ck/Cd = 1.5 (gray). The boxes show the time period used to calculate maximum 2-day-average azimuthal velocity Vmax.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Maximum observed intensities as a function of Ts are obtained from DeMaria and Kaplan (1994) for V10,max (maximum total wind speed at 10 m MSL) and from Holland (1997) for minimum surface pressure Pmin. Values of Pmin as a function of Ts are similar in the database used to develop the Statistical Hurricane Intensity Prediction Scheme (SHIPS; M. DeMaria 2011, personal communication). As in BR09 (p. 1775), V10,max is multiplied by a conversion factor to obtain an estimate for maximum azimuthal velocity above the surface, Vmax; based on a variety of analyses (LeeJoice 2000; Montgomery et al. 2006; Kepert 2006a,b; Zhang et al. 2011b), a value of 1.35 is used herein. Of course, maximum intensity is expected to vary with other environmental parameters, such as outflow temperature and air–sea temperature difference (e.g., Emanuel 1986). Maximum observed intensity is tied only to sea surface temperature herein for simplicity and because of the availability of thorough observational studies for the relationship between Vmax and Ts (e.g., DeMaria and Kaplan 1994; Whitney and Hobgood 1997).

e. Reevaluation of BR09’s conclusions

Before proceeding to the primary results of this study, the results of BR09’s section 3c (“Ratio of surface exchange coefficients”) are reevaluated using the model settings of this study. The most significant differences are the change in Δz near the surface (Table 2) and the level at which the surface exchange coefficients are specified (i.e., the lowest model level for winds; third row in Table 2). Results using either Res1 or Res2, and lυ = 200 m (as in BR09’s section 3c), are shown in Fig. 2a. The results are essentially the same for the two different approaches, and essentially the same as in BR09 (cf. their Fig. 6). With lυ = 50 m (which was not examined in BR09), values of Vmax are slightly lower (by as much as 12%) when using the Res2 (Fig. 2b); however, the difference occurs primarily for Ck/Cd > 1. The reason for this difference seems to be related to slightly larger vertical diffusivity and slightly smaller radius of maximum winds when using coarser vertical resolution. Results using setup B (not shown) yield the same results, although the differences for lυ = 50 m are not as pronounced (the maximum difference in Vmax is only 6%).

Fig. 2.
Fig. 2.

Sensitivity of Vmax (m s−1) to the ratio Ck/Cd using setup A with two different vertical grids. The solid lines use the same vertical resolution as BR09 (Res1) and the dashed lines use higher vertical resolution (Res2). Different colors are used for different values of lh as indicated in the legend. (a) Simulations with lυ = 200 m (as in BR09) and (b) simulations with lυ = 50 m. The gray curve shows Vmax ~ (Ck/Cd)1/2 for reference.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Overall, these tests demonstrate that all primary conclusions from BR09 are upheld when using the different model settings herein (i.e., higher vertical resolution, and surface exchange coefficients specified at 10 m MSL instead of 125 m MSL). Specifically, these results are as follows: Vmax is very sensitive to lh, but not to lυ or Δz; the theoretical response Vmax ~ (Ck/Cd)1/2 (e.g., Emanuel 1995) is only found for small lh (<100 m); and Vmax has essentially no dependence on Ck/Cd for large lh (3000 m). The higher-resolution setup (Res2) is used for the remainder of this article.

3. Comparison with E95

Figure 3 shows Vmax as a function of Ck/Cd for the various model setups used herein. The gray horizontal line denotes the estimated maximum value from observed storms for this sea surface temperature (see section 2d). For comparison, the purple lines on Fig. 3a show results from the study by Emanuel (1995, hereafter E95), which examined the effects of the ratio Ck/Cd on maximum winds in TCs. The lower line shows the results from E95 using the RE87 axisymmetric model, which did not include dissipative heating; the upper line shows these results multiplied by 1.2 to estimate the effects of dissipative heating, which theoretically increases Vmax by ~20% (Bister and Emanuel 1998). [Dissipative heating increases Vmax by 5%–15% in CM1 (not shown).] The model settings for E95’s simulations are not stated in his article, so a new set of simulations was conducted by the present author using the RE87 model; the results suggest that the model settings are the same as those in RE87 (Ts = 26°C, lh = 3000 m, and lυ = 200 m). Hence, the E95 results are plotted along with the setup A results from this study.

Fig. 3.
Fig. 3.

The quantity Vmax (m s−1) as a function of the ratio Ck/Cd from simulations that use the same value of Ck (1.2 × 10−3) but different values of Cd. The solid lines use lυ = 200 m, the dashed lines use lυ = 100 m, and the dotted lines use lυ = 50 m. Different colors are used for different values of lh as indicated in the legend. (a) Simulations with setup A, although the purple lines show the results from E95 (lower line) and results from E95 multiplied by 1.2 (upper line, which estimates the inclusion of dissipative heating), and (b) simulations with setup B. The gray line denotes the maximum observed intensity for the specified sea surface temperature.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

For Ck/Cd < 1, the E95 simulations are similar to CM1 with lh = 3000 m (red lines). The response is different for Ck/Cd > 1, and the E95 results are more like CM1 with lh = 1000 m (green lines). These differences may be attributable to different resolution, different model settings (i.e., formulation of surface exchange coefficients), or to approximations used in the RE87 model (see the appendix in BR09 for more details). Nevertheless, E95’s simulations are clearly among the set of simulations that have strong horizontal diffusion, especially for Ck/Cd < 1.

Several recent articles (e.g., Black et al. 2007; Zhang et al. 2008; Haus et al. 2010; Montgomery et al. 2010; Andreas 2011) have evaluated the conclusion from E95 that Ck/Cd is likely ~0.75 for intense TCs in nature because, according to E95, “…otherwise, the [modeled] wind speeds would be much weaker than observed.” However, if one accepts the low-lh settings examined herein, then this lower bound on Ck/Cd would be only 0.25 for setup B (Fig. 3b). Hence, the inability of E95’s simulations to produce the strongest storms in nature unless Ck/Cd exceeds roughly 0.75 is likely attributable to large horizontal diffusion (lh ≈ 3000 m) and low sea surface temperature (Ts = 26°C) in his simulations.

Andreas (2011) noted that neither the simulations by E95 nor BR09 produce a category 5 storm (10-m wind speed >69 m s−1) unless Ck/Cd > 0.75. However, it should be noted that a category 5 storm has never been observed for Ts = 26°C (DeMaria and Kaplan 1994; Whitney and Hobgood 1997). The Ts = 29°C simulations herein produce category 5 storms for Ck/Cd = 0.25 if horizontal diffusion is relatively weak (lh ≲ 300 m).

Clearly, the values for surface exchange coefficients are important for maximum intensity in numerical models, but the large set of simulations herein demonstrates that several other factors are also important, such as the intensity of horizontal turbulence (via lh in this model) and certain properties of the environment (e.g., sea surface temperature). Hence, the E95 simulations, which to this author’s knowledge used only one value for lh and Ts, have limited usefulness for comparison with observations. On the opposite extreme, the broader set of simulations shown in Fig. 3 is not especially useful, either, because it is unclear which model settings are realistic. The analyses in the following two sections aim to determine appropriate values for Cd, Ck, lh, and lυ by examining several metrics of TC intensity and structure.

4. Comparison with maximum observed intensity

As shown in Fig. 3, most model setups produce unnaturally strong storms. Considering first Ck/Cd ≥ 1, then only lh = 3000 m (red lines) yields reasonable intensity. However, recent observational and laboratory studies (e.g., Powell et al. 2003; Donelan et al. 2004; Drennan et al. 2007; Jeong 2008; Haus et al. 2010; Bell 2010) found that the most intense hurricanes probably have Ck ≈ 1 × 10−3 and Cd ≈ 2 × 10−3 (although the uncertainty is considerably large). Considering Ck/Cd ≈ 0.5 as a best guess for conditions in nature, then lh = 1000 m (green lines) yields the approximately correct maximum intensity with Ck/Cd ≈ 0.5 for both setups A and B. For smaller values of lh (blue and black lines), the ratio Ck/Cd needs to be ~0.25 to prevent unnaturally strong intensity. For a given lh, the parameter lυ has little influence on Vmax. A detailed analysis of the effects of horizontal and vertical diffusion on Vmax is presented by Rotunno and Bryan (2011, manuscript submitted to J. Atmos. Sci.), and readers are referred to that article for more explanation.

Figure 4 shows Pmin for the various model setups. Conclusions from above are upheld when using Pmin as a metric of storm intensity. That is, realistic maximum intensity occurs only with Ck/Cd > 1 when lh = 3000 m, but with Ck/Cd ≈ 0.5 when lh = 1000 m, or with Ck/Cd ≈ 0.25 when lh ≤ 300 m. The quantity Pmin seems to be slightly more sensitive to lυ than Vmax, although it is difficult to generalize the relationship from these results; considering only simulations with Pmin > 900 mb (i.e., the most realistic simulations), then lower lυ usually produces weaker storms.

Fig. 4.
Fig. 4.

As in Fig. 3, except for minimum surface pressure (mb).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

To better examine the interrelated effects of pressure and wind for the various model settings, Figs. 56 show scatterplots of minimum surface pressure pmin(t) and maximum 10-m wind speed υ10,max(t); each dot on these figures represents instantaneous values from a different time in the simulations. These figures use 10-m wind speed (instead of maximum tangential wind above the surface, which was used in previous figures) to facilitate comparison with observational studies. Specifically, the gray lines are wind–pressure relationships from Knaff and Zehr (2007) that are based on observed storms. Equation (8) from their study is used with two values of their normalized size parameter S; the upper curve on Figs. 56 is for S = 0.2, corresponding to relatively small storms; the lower curve is for S = 1.0, corresponding to relatively large storms.

Fig. 5.
Fig. 5.

Scatterplots of maximum 10-m wind speed υ10,max(t) vs minimum surface pressure pmin(t) using setup A for (a) (lh, lυ) = (300 m, 200 m), (b) (lh, lυ) = (1000 m, 200 m), (c) (lh, lυ) = (3000 m, 200 m), (d) (lh, lυ) = (300 m, 50 m), (e) (lh, lυ) = (1000 m, 50 m), and (f) (lh, lυ) = (3000 m, 50 m). Scatterplots are constructed using hourly data for the 2-day time period surrounding the time of maximum intensity. Different colors indicate different settings for Ck/Cd, as indicated in the legend. Gray curves show the observations-based wind–pressure relationship from Knaff and Zehr (2007) where the upper curve is for relatively small storms (S = 0.2) and the lower curve is for relatively large storms (S = 1.0). The open circles denote the maximum observed intensity for the specified sea surface temperature.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Fig. 6.
Fig. 6.

As in Fig. 5, except for setup B.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Considering setup A (Fig. 5), there is a tendency for the model to produce winds that are too strong for a given pmin. This bias is especially pronounced for relatively large lυ (Figs. 5a–c). Using lυ = 50 m (Figs. 5d–f), the model-produced wind–pressure relationship is close to the Knaff and Zehr (2007) small-storm curve, but only for Ck/Cd ≲ 0.5 (green and red dots). This analysis further supports the conclusion that Ck/Cd ≈ 0.5 is likely the best setting for axisymmetric numerical model simulations of intense TCs.

The same overall conclusions are reached when using setup B (Fig. 6), although the large-lυ cases are acceptable for this setup as long as lh ≥ 1000 m. For lυ = 50 m, the model results are closer to the large-storm relation from Knaff and Zehr (2007) as long as Ck/Cd < 0.5. These results imply that the simulated storms are larger for setup B than setup A, which is confirmed in the next section.

The maximum azimuthal velocity at any level (Vmax) is used in previous figures because it allows for comparison to previous modeling studies (e.g., E95) and to theoretical maximum wind speed (discussed in a later section). To evaluate wind speeds at the standard reporting level of 10 m MSL, the open circles on Figs. 56 denote the maximum 10-m wind speed from DeMaria and Kaplan (1994) with the Pmin values from Holland (1997) for the specified sea surface temperatures. For setup A, the settings (lh, lυ, Ck/Cd) = (1000 m, 50 m, 0.5) (green dots of Fig. 5e) produce acceptable results in the sense that the model output is close to the Knaff–Zehr wind–pressure relationship and υ10,max remains below the maximum observed value and pmin remains above the minimum observed value. For setup B, interpretation is not quite as clear. The lh = 300-m simulations with Ck/Cd = 0.25 (red dots in Figs. 6a,d) seem reasonable. However, as discussed earlier, recent observational/laboratory studies are finding Ck/Cd ≈ 0.5; assuming this value, then (lh, lυ) = (1000 m, 50 m) might be the best settings (green dots in Fig. 6e), although there is a slight overestimation of maximum intensity for this case. It seems that a value of lh somewhere between 1000 and 3000 m would produce the best results for setup B.

5. Comparison with observed TC structure

In this section, various measures of TC structure are evaluated. This analysis addresses whether the settings determined in the previous section yield realistic structure, and some of the following analyses help narrow down likely values for these settings.

Figure 7 shows R34, which is the radius of gale force winds (34 kt; ~17.5 m s-1) at the time of Vmax. Overall, by this metric, the simulated storms are slightly larger for setup B, which supports the inference about storm size drawn in the previous section. (It is unknown why a few simulations with setup B and Ck/Cd > 1 have much larger radius of gale force winds.) Larger storms for setup B (which has a larger nominal potential intensity) is also qualitatively consistent with theory, for which length scale varies as potential intensity divided by Coriolis parameter (Emanuel 1986; Dean et al. 2009). The systematic difference in R34 between setups A and B is especially notable for Ck/Cd < 1. For both setups, lh has no systematic influence on R34. There is a slight tendency for smaller R34 as lυ decreases. The dataset analyzed by Dean et al. (2009), which is based on satellite data and in situ flight-level observations, shows that most observed storms have R34 ≲ 200 km, and almost all cases have R34 > 90 km; the model results in Fig. 7 are within this range as long as Ck/Cd ≲ 1.

Fig. 7.
Fig. 7.

As in Fig. 3, except for R34—the radius (km) of gale-force winds at 10 m MSL.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

As another measure of TC size, Fig. 8 shows Rmax, defined as the radius of Vmax. There is a clear relationship between lh and Rmax, which is explained in Rotunno and Bryan (2011, manuscript submitted to J. Atmos. Sci.). There is no clear impact on Rmax from either lυ or Ck/Cd. The analytic model of Emanuel and Rotunno (2011) also predicts no dependence on Rmax with Ck/Cd [at least, when assuming constant outer-vortex scale; see their Eq. (42)]. The same conclusions are drawn if Rmax is defined at a common level of 2 km MSL as in Stern and Nolan (2009) and Zhang et al. (2011b). Recent studies of flight-level data (usually at 700–800 mb) by Willoughby and Rahn (2004) and Mallen et al. (2005) have found that the radius of maximum winds (RMW) for strong storms (wind >50 m s−1) is usually less than ~40 km, and RMW is sometimes as small as ~10 km. The model results are reasonably consistent.

Fig. 8.
Fig. 8.

As in Fig. 3, except for Rmax—the radius (km) of Vmax.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

As it relates to the primary goal of this section, Figs. 78 demonstrate that the settings determined in the previous section—lh ≈ 1000 m, lυ ≈ 50 m, and Ck/Cd ≈ 0.5—yield reasonable values of TC size. Stated another way: there are no extraordinarily small or large storms for these model settings.

An insightful measure of TC boundary layer structure is the inflow angle, β = tan−1(u10/υ10), where u10 and υ10 are the radial and azimuthal velocities, respectively, at 10 m MSL at the location of Vmax. From a large database of dropsonde data, Powell et al. (2009) found an average value of 23°; this value is produced in the model simulations only for low values of lυ and/or low values of Ck/Cd (Fig. 9). Unreasonably small values of β are produced for Ck/Cd > 1.

Fig. 9.
Fig. 9.

As in Fig. 3, except for inflow angle β (°) at 10 m MSL at the location of maximum wind. The gray line denotes the average value from dropsondes reported by Powell et al. (2009).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Figure 10 shows zmax, which is the height of Vmax. There are three clear relationships in this figure. First, for lh and lυ held fixed, zmax increases as Ck/Cd decreases (i.e., moving from right to left). Second, for lυ held fixed, zmax tends to increase as lh increases (i.e., black lines tend to be near the bottom and red lines near the top). Third, for lh held fixed, zmax increases as lυ increases (i.e., dotted lines tend to be near the bottom and solid lines near the top). The gray line on Fig. 10 shows the value of zmax from the composite analysis of dropsonde data from Zhang et al. (2011b), using the value from their category 4–5 storms only. There are multiple model settings that yield this value (zmax = 0.9 km), but for Ck/Cd = 0.5 most model settings overestimate zmax slightly. This positive bias in zmax might be related to the use of a constant value of lυ in these simulations; it is widely acknowledged in studies of boundary layers that lυ should be smaller near the surface than near the top of the boundary layer (e.g., Wyngaard 2010, p. 234). Sensitivity simulations using a variable lυ(z) are shown in a later section. The use of constant Ck/Cd everywhere in these simulations may also be a problem; this ratio is well known to be relatively larger in weak winds (e.g., Black et al. 2007 and references therein). Nevertheless, for the settings (lh, lυ, Ck/Cd) = (1000 m, 50 m, 0.5) (green-dotted line in Fig. 10), zmax is 0.9 km for setup A and 1.0 km for setup B—both of which are close to the value from Zhang et al. (2011b).

Fig. 10.
Fig. 10.

As in Fig. 3, except for zmax—the height (km) of Vmax. The gray line denotes the average value for category 4–5 storms reported by Zhang et al. (2011b).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Overall, this analysis of TC structure supports the conclusion that the model settings lh ≈ 1000 m and Ck/Cd ≈ 0.5 produce realistic TCs with this axisymmetric model. The setting lυ ≈ 50 m produces reasonable values for inflow angle β, height of maximum winds zmax, and yields a reasonable wind–pressure relationship at the surface (previous section). The value lυ ≈ 50 m is within the range of values determined from flight-level observations in hurricane boundary layers by Zhang et al. (2011a), although their mean value for lυ is roughly 100 m.

The estimate lh ≈ 1000 m is 33% lower than the estimate of 1500 m from BR09; the difference herein is attributable mostly to the new evidence from observational and laboratory studies that find Ck/Cd ≈ 0.5 in intense hurricanes. The estimate lυ ≈ 50 m is half of the estimate from BR09; the difference herein is attributable to new observational studies (particularly Powell et al. 2009; Zhang et al. 2011b) that have helped refine this estimate.

6. Comparison with theory

Theoretical studies by Emanuel (1986, 1995) found that maximum azimuthal velocity at the top of the boundary layer should vary as follows:
e2
Using two different numerical models—one a balanced model and the other a nonhydrostatic primitive equation model—Emanuel (1995) found approximate correspondence with (2). [Technically, using values reported in Emanuel’s Tables 1–2, the E95 model gives Vmax ~ (Ck/Cd)0.59 and the RE87 model gives Vmax ~ (Ck/Cd)0.43.] BR09 found good correspondence with (2) only when using small horizontal diffusion (i.e., lh of order 100 m or less). As lh increased in BR09’s simulations, the correspondence with (2) became worse, and there was no clear relationship between Vmax and Ck/Cd for lh = 3000 m. The higher-vertical-resolution simulations herein yield the same conclusions as in BR09. Table 3 lists the exponent b for best fits of the function Vmax = a(Ck/Cd)b for all setups shown in Fig. 3. As in BR09, the exponent b increases as lh decreases. The best matches to (2) are obtained for lh = 0, especially for setup A. The same conclusions are obtained when using maximum gradient wind speed Vg from anywhere in the domain (Table 4), where Vg is calculated in the same manner as Bryan and Rotunno (2009a, p. 3047). Montgomery et al. (2010) found no support for (2), but their conclusion seems to be attributable to their shorter model integration times (Bryan 2011, manuscript submitted to Quart. J. Roy. Meteor. Soc.).
Table 3.

Exponent b in the best fit of the function Vmax = a(Ck/Cd)b to the model results shown in Fig. 3. Values in italics indicate the best fit has a correlation coefficient R < 0.98.

Table 3.
Table 4.

As in Table 3, except for maximum gradient wind speed.

Table 4.

The theoretical relationship (2) is typically derived by assuming gradient wind balance above the boundary layer, along with several other approximations (e.g., Emanuel 1986, 1995). However, simulations with weak horizontal diffusion can have large gradient wind imbalance; for example, in the control simulation of Bryan and Rotunno (2009a) the centrifugal acceleration is twice the magnitude of the horizontal pressure gradient at the location of Vmax. To examine the magnitude of gradient wind imbalance in the present simulations, the ratio Vmax/Vg is used, where Vg for this analysis is calculated at the location of Vmax. Figure 11 shows that all simulations have at least a slight supergradient flow (Vmax/Vg > 1) at the location of Vmax, which is consistent with previous studies (e.g., Rott and Lewellen 1966; Kuo 1971; Kepert 2001; Kepert and Wang 2001; Zhang et al. 2001; Smith et al. 2008). For fixed values of lh and lυ, the imbalance increases as Ck/Cd decreases (i.e., going from right to left on Fig. 11); this result is consistent with the increasing impact of radial inflow and unbalanced flow effects as surface drag increases, as shown by Kuo (1971) and Rotunno and Bryan (2011, manuscript submitted to J. Atmos. Sci.) (their parameter K is proportional to Ck/Cd herein). For fixed values of lh and Ck/Cd, the imbalance increases as lυ decreases (i.e., solid lines tend to be near the bottom of the figure, and dotted lines tend to be near the top). For fixed values of lυ and Ck/Cd, it is difficult to draw a general conclusion about the relationship between the ratio Vmax/Vg and lh. From the conclusions of Bryan and Rotunno (2009a), one might expect Vmax/Vg to increase as lh decreases, but this is not always the case in Fig. 11.

Fig. 11.
Fig. 11.

As in Fig. 3, except for the ratio of Vmax to Vg (gradient wind speed at the location of Vmax).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

To allow a direct comparison with the results of Bryan and Rotunno (2009a), the theoretical maximum intensity (i.e., potential intensity) from Emanuel (1986), hereafter referred to as E-PI, is calculated using the relation
e3
where TB is temperature at Vmax, T0 is outflow temperature, α = Ts/T0, ssurf is moist entropy at the pressure and temperature of the ocean surface, s0 is moist entropy in the atmospheric surface layer, and all variables are calculated at the radius of Vmax. The ratio Vmax/E-PI is plotted in Fig. 12. From this figure it can be concluded that the simulations herein are consistent with those of Bryan and Rotunno (2009a); that is, for fixed lυ and Ck/Cd, Vmax in the low-lh runs exceeds E-PI by a greater amount than in the large-lh runs (i.e., black lines tend to be at the top, and red lines at the bottom). Figure 12 also shows that for Ck/Cd ≈ 2, Vmax does not appreciably exceed E-PI for any model setup, which is consistent with relatively weak supergradient flow. With regards to observed TCs, some studies have documented cases in which observed Vmax exceeds E-PI (e.g., Bell and Montgomery 2008), which, based on Fig. 12, further supports the conclusion that Ck/Cd < 1 for intense TCs in nature. Theoretically, though, it seems that E-PI should be a reasonable model for maximum intensity as long as Ck/Cd ≳ 1.5. As mentioned above, (3) is a model for maximum gradient wind speed, and so evaluation has also been done using the maximum value of Vg from anywhere in the domain; results (not shown) look much the same as Fig. 12, although the maximum value of Vg,max/E-PI is 1.75 (for Ck/Cd = 0.25).
Fig. 12.
Fig. 12.

As in Fig. 3, except for the ratio of Vmax to E-PI (the theoretical maximum azimuthal velocity assuming balanced flow).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

A more complete analytic model that includes inertial terms (i.e., that does not assume gradient wind balance) was presented by Bryan and Rotunno (2009a). They derived the relation
e4
where r is radius, η is azimuthal vorticity, w is vertical velocity, and the subscript b denotes that variables are evaluated at the location of Vmax. The ratio Vmax/PI+ is shown in Fig. 13, which shows that Vmax can exceed PI+ by a small amount in some of these simulations (up to 20%, particularly for low lh). This slight underestimation by PI+ for some cases is probably attributable to one of the approximations made in the derivation of PI+. Nevertheless, the ratio Vmax/PI+ is near 1 for all Ck/Cd and all model configurations. Hence, (4) can be utilized to understand the relationship between Vmax and Ck/Cd. Rearranging (4) using (3), one finds
e5
It is unclear how all the terms in the square brackets should vary with Ck/Cd. Using the model output, the bracketed term in (5) is found to be roughly constant with Ck/Cd for all values of lh and lυ (not shown). Hence, (5) reduces to (2) for these simulations, and therefore (2) can be valid for unbalanced as well as balanced TCs.
Fig. 13.
Fig. 13.

As in Fig. 3, except for the ratio of Vmax to PI+ (the theoretical maximum azimuthal velocity assuming unbalanced flow).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

The reason the relation (2) is not found in all simulations (e.g., Table 3) is probably attributable to the effects of horizontal diffusion in the boundary layer, which is not considered in the derivations of (3) or (4). The new analytical model by Emanuel and Rotunno (2011) incorporates the effects of turbulence, and is evaluated in Fig. 14. Specifically, their (41) is used to estimate maximum gradient wind speed Vm assuming nominal potential intensity Vp of 70 and 92 m s−1 for setups A and B, respectively (see section 2a). This analytical model reasonably predicts the maximum gradient wind in the model simulations with lh = 1000 m (green lines in Fig. 14). The underprediction for lh ≤ 300 m may be attributable to the neglect of dissipative heating and unbalanced flow effects in the derivation of Vm.

Fig. 14.
Fig. 14.

As in Fig. 3, except for the ratio of maximum gradient wind (Vg,max) to theoretical maximum gradient wind from Emanuel and Rotunno (2011) (Vm).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

7. Sensitivity simulations

The simulations evaluated in sections 36 all use Ck = 1.2 × 10−3. To check whether conclusions are sensitive to this value, a set of simulations has been conducted using (lh, lυ) = (1000 m, 50 m) but using different values for Ck. There is negligible impact on the sensitivity of maximum 10-m wind speed (V10,max) to Ck/Cd (Fig. 15). The same conclusion holds for Vmax (not shown). (Wind speed at 10 m is used in this subsection to allow for direction comparison with the observations in DeMaria and Kaplan 1994.) Some other metrics have a small sensitivity to the choice of Ck; for example, Pmin tends to decrease as Ck increases (not shown), but the effect is only notable (~10 mb) for Ck/Cd > 1, which is likely not relevant to intense TCs in nature.

Fig. 15.
Fig. 15.

Maximum wind speed at 10 m MSL V10,max (m s−1) using lh = 1000 m and lυ = 50 m with (a) setup A and (b) setup B. Different lines are for different values of Ck, as indicated in the legend.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

As discussed in the previous section, the height of maximum winds zmax compares favorably with observations only when using lυ ≈ 50 m, which is lower (by a factor of 2) than the average value determined from flight-level observations at z ≈ 400 m by Zhang et al. (2011a). To check whether this result is attributable to the use of constant lυ, a set of simulations has been conducted using lh = 1000 m and the formulation for lυ from Blackadar (1962), lυ(z) = κz[1 + (κz/l)]−1, where κ = 0.4 is the von Kármán constant. Results using three different values of l show that maximum winds vary with Ck/Cd in essentially the same manner as shown in previous sections (Fig. 16a), although values tend to be slightly lower with the Blackadar formulation for lυ. The settings lh = 1000 m and Ck/Cd = 0.5 produce reasonable maximum intensity, as before, although with this test the value of l needs to be 100 or 200 m. Using l = 100 or 200 m also yields reasonable values for zmax (Fig. 16b). The value of lυ at 400 m MSL is listed at the bottom of Fig. 16; for l = 200 m, the vertical length scale lυ is 90 m, which is similar to the average value determined by Zhang et al. (2011a). Because of the better correspondence with observational studies, a variable lυ(z) should probably be used for future studies.

Fig. 16.
Fig. 16.

Results using setup B from simulations that use different formulations for lυ(z) as indicated in the legend: (a) maximum wind speed at 10 m MSL V10,max (m s−1); and (b) the height of maximum azimuthal velocity, zmax (km).

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

Finally, to test the sensitivity to model dimensionality, a set of three-dimensional simulations has been conducted using the same numerical model (CM1) with (lh, lυ) = (1000 m, 50 m). The 3D version of CM1 uses the same governing equations and physical parameterizations, but is integrated on a Cartesian grid. Further details are available in Bryan et al. (2010). Horizontal grid spacing is 2 km for these tests, and setup Res2 (Table 2) is used here. To allow for direct comparison with the axisymmetric model, the 3D model output is azimuthally averaged; values of wind speed at 10 m (Fig. 17) produce a similar response to Ck/Cd, although the 3D simulations are consistently weaker than the axisymmetric simulations. (Maximum instantaneous values of 10-m wind speed from any gridpoint in the 3D model fall between the two curves shown in Fig. 17.) This systematic difference between Vmax for axisymmetric and 3D simulations is consistent with results using the lower-CAPE initial condition of RE87 (Bryan et al. 2010). Further analysis is planned for a future article, although Bryan et al. (2010) speculated that momentum exchange between the eye and eyewall, which seems to be greater in the 3D runs because of their ability to resolve 3D features, probably leads to weaker intensity in 3D simulations; see also Schubert et al. (1999) and Hausman et al. (2006). Nevertheless, the 3D simulations herein confirm that lh ≈ 1000 m and Ck/Cd ≈ 0.5 yield reasonable maximum intensity, although these simulations suggest that axisymmetric models require different (i.e., larger) values of lh than 3D models.

Fig. 17.
Fig. 17.

Maximum wind speed at 10 m MSL V10,max (m s−1) using lh = 1000 m and lυ = 50 m where the solid line is from axisymmetric simulations and the dashed line is from 3D simulations. The 3D results are azimuthally averaged values.

Citation: Monthly Weather Review 140, 4; 10.1175/MWR-D-11-00231.1

8. Summary

This study examines how changes in surface exchange coefficients and turbulence length scales in a numerical model can affect maximum hurricane intensity and structure. Compared to other recent studies on the topic, this study uses higher vertical resolution, a double-moment mixed-phase microphysics scheme, an initial environment based on a recent climatological study by Dunion (2011), a higher sea surface temperature (29°C), more values for lh and lυ (the horizontal and vertical length scales, respectively), more values of Ck and Cd (the surface exchange coefficients for enthalpy and momentum, respectively), and a set of three-dimensional simulations. Most results are drawn from axisymmetric model simulations because of the lower computational cost, which allows for a thorough examination of the effects on hurricane intensity and structure from changes in the parameters Ck, Cd, lh, and/or lυ. More than 400 simulations are examined herein.

Despite all the differences in model setup compared to the recent study by Bryan and Rotunno (2009b, hereafter BR09), all primary conclusions from that study are upheld. Maximum intensity, whether examined in terms of maximum winds Vmax or minimum surface pressure Pmin, is very sensitive to lh but not to lυ. Also in agreement with BR09, it is shown that the theoretical response Vmax ~ (Ck/Cd)1/2 occurs only when horizontal turbulent diffusion is weak (lh < 100 m). Increasingly larger values of lh (up to 3000 m) yield much weaker responses of Vmax to Ck/Cd (Table 3).

From a comparison of model output to relatively well-observed metrics of hurricane structure and intensity—such as maximum winds, minimum surface pressure, height of maximum winds, surface inflow angle, and the surface wind–pressure relationship—it is concluded that the settings lh ≈ 1000 m, lυ ≈ 50 m, and Ck/Cd ≈ 0.5 produce the most reasonable results. These values are reasonably consistent with recent observational and laboratory studies (e.g., Powell et al. 2003; Donelan et al. 2004; Drennan et al. 2007; Jeong 2008; Haus et al. 2010; Bell 2010; Zhang et al. 2011b). Determination of these settings is roughly independent of the different model setups used herein (Table 1), although the setup with higher sea surface temperature (i.e., higher winds) probably requires lh somewhere between 1000 and 3000 m in the axisymmetric model to prevent excessively strong intensity. Using the variable formulation for lυ(z) from Blackadar (1962) yields some better correspondence with observations, and a value of asymptotic vertical length scale l = 200 m (lυ = 90 m at 400 m MSL) produces the best match to the observational metrics. The three-dimensional numerical simulations require lower values of lh to prevent excessively strong intensity, presumably because of their ability to produce three-dimensional motions that must be parameterized in axisymmetric models.

Gradient wind imbalance (i.e., supergradient overshoot) in the model simulations generally increases as Ck/Cd, lυ, and/or lh decreases. The inviscid, balanced model developed by Emanuel (1986, 1995) yields a reasonable upper bound on Vmax as long as Ck/Cd ≳ 1.5. The relation for maximum winds that includes unbalanced flow effects (i.e., inertial terms) presented by Bryan and Rotunno (2009a) is reasonably close to Vmax for all axisymmetric model simulations, and the analytical model for maximum gradient wind by Emanuel and Rotunno (2011) reproduces the numerical model results with lh = 1000 m. The simple relation Vmax ~ (Ck/Cd)1/2 seems to apply to simulations with strong supergradient overshoot, despite the fact that this theoretical relation is often derived by assuming balanced flow.

This study also reconciles a recent controversy in the literature about Emanuel’s (1995) conclusion that Ck/Cd is probably ~0.75 in nature, despite recent observational/laboratory data showing Ck/Cd ≈ 0.5 in intense hurricanes. Emanuel’s simulations had relatively large horizontal diffusion (probably lh ≈ 3000 m) and relatively low sea surface temperature (26°C). The simulations herein can produce category 5 storms for Ck/Cd as low as 0.25, but only when the horizontal turbulence diffusion is weak (lh ≲ 300 m).

Acknowledgments

This work was sponsored in part by the Office of Naval Research, Prime Contract N00014-10-1-0148 awarded to York University, as part of the National Oceanographic Partnership Program. Computing resources were provided by the Shared Hierarchical Academic Research Computing Network (SHARCNET; www.sharcnet.ca) and Compute/Calcul Canada; the author thanks Yongsheng Chen of York University for obtaining these resources.

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  • Schubert, W. H., M. T. Montgomery, R. K. Taft, T. A. Guinn, S. R. Fulton, J. P. Kossin, and J. P. Edwards, 1999: Polygonal eyewalls, asymmetric eye contraction, and potential vorticity mixing in hurricanes. J. Atmos. Sci., 56, 11971223.

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  • Smith, R. K., M. T. Montgomery, and S. Vogl, 2008: A critique of Emanuel’s hurricane model and potential intensity theory. Quart. J. Roy. Meteor. Soc., 134, 551561.

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  • Stern, D. P., and D. S. Nolan, 2009: Reexamining the vertical structure of tangential winds in tropical cyclones: Observations and theory. J. Atmos. Sci., 66, 35793600.

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  • Whitney, L. D., and J. S. Hobgood, 1997: The relationship between sea surface temperatures and maximum intensities of tropical cyclones in the eastern North Pacific. J. Climate, 10, 29212930.

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    • Export Citation
  • Willoughby, H. E., and M. E. Rahn, 2004: Parametric representation of the primary hurricane vortex. Part I: Observations and evaluation of the Holland (1980) model. Mon. Wea. Rev., 132, 30333048.

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  • Wyngaard, J. C., 2010: Turbulence in the Atmosphere. Cambridge University Press, 393 pp.

  • Zhang, D.-L., Y. Liu, and M. K. Yau, 2001: A multiscale numerical study of Hurricane Andrew (1992). Part IV: Unbalanced flows. Mon. Wea. Rev., 129, 92107.

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  • Zhang, J. A., P. G. Black, J. R. French, and W. M. Drennan, 2008: First direct measurements of enthalpy flux in the hurricane boundary layer: The CBLAST results. Geophys. Res. Lett., 35, L14813, doi:10.1029/2008GL034374.

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  • Zhang, J. A., F. D. Marks, M. T. Montgomery, and S. Lorsolo, 2011a: An estimation of turbulent characteristics in the low-level region of intense hurricanes Allen (1980) and Hugo (1989). Mon. Wea. Rev., 139, 14471462.

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  • Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks Jr., 2011b: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 25232535.

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1

By default, the RE87 model uses lυ = 200 m and the bulk PBL model in MM5 uses lυ = 40 m.

2

The same methodology is used for several of the planetary boundary layer schemes in the WRF model.

3

The MM5 user’s guide (chapter 8, p. 8) recommends Δz > 250 m for this scheme.

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    • Export Citation
  • Smith, R. K., M. T. Montgomery, and S. Vogl, 2008: A critique of Emanuel’s hurricane model and potential intensity theory. Quart. J. Roy. Meteor. Soc., 134, 551561.

    • Search Google Scholar
    • Export Citation
  • Stern, D. P., and D. S. Nolan, 2009: Reexamining the vertical structure of tangential winds in tropical cyclones: Observations and theory. J. Atmos. Sci., 66, 35793600.

    • Search Google Scholar
    • Export Citation
  • Whitney, L. D., and J. S. Hobgood, 1997: The relationship between sea surface temperatures and maximum intensities of tropical cyclones in the eastern North Pacific. J. Climate, 10, 29212930.

    • Search Google Scholar
    • Export Citation
  • Willoughby, H. E., and M. E. Rahn, 2004: Parametric representation of the primary hurricane vortex. Part I: Observations and evaluation of the Holland (1980) model. Mon. Wea. Rev., 132, 30333048.

    • Search Google Scholar
    • Export Citation
  • Wyngaard, J. C., 2010: Turbulence in the Atmosphere. Cambridge University Press, 393 pp.

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    • Search Google Scholar
    • Export Citation
  • Zhang, J. A., P. G. Black, J. R. French, and W. M. Drennan, 2008: First direct measurements of enthalpy flux in the hurricane boundary layer: The CBLAST results. Geophys. Res. Lett., 35, L14813, doi:10.1029/2008GL034374.

    • Search Google Scholar
    • Export Citation
  • Zhang, J. A., F. D. Marks, M. T. Montgomery, and S. Lorsolo, 2011a: An estimation of turbulent characteristics in the low-level region of intense hurricanes Allen (1980) and Hugo (1989). Mon. Wea. Rev., 139, 14471462.

    • Search Google Scholar
    • Export Citation
  • Zhang, J. A., R. F. Rogers, D. S. Nolan, and F. D. Marks Jr., 2011b: On the characteristic height scales of the hurricane boundary layer. Mon. Wea. Rev., 139, 25232535.

    • Search Google Scholar
    • Export Citation
  • Fig. 1.

    Time series of υmax(t) for Ck/Cd = 0.5 (black) and Ck/Cd = 1.5 (gray). The boxes show the time period used to calculate maximum 2-day-average azimuthal velocity Vmax.

  • Fig. 2.

    Sensitivity of Vmax (m s−1) to the ratio Ck/Cd using setup A with two different vertical grids. The solid lines use the same vertical resolution as BR09 (Res1) and the dashed lines use higher vertical resolution (Res2). Different colors are used for different values of lh as indicated in the legend. (a) Simulations with lυ = 200 m (as in BR09) and (b) simulations with lυ = 50 m. The gray curve shows Vmax ~ (Ck/Cd)1/2 for reference.