1. Introduction

The tropical cyclone (TC) boundary layer is distinct from the daytime convective boundary layer due to the effect of the cyclone’s rotation on the boundary layer height and turbulence characteristics (Eliassen 1971). Turbulent mixing processes in the TC boundary layer are important contributors to the cyclone’s intensity change since it controls the depth and strength of radial inflow (Foster 2009; Zhang et al. 2015; Zhang and Pu 2017). Specifically, the inflow strength is closely tied to a boundary layer spinup paradigm of TCs (Smith and Montgomery 2015), in which the increase of the maximum tangential wind in the boundary layer occurs if the inflow is strong enough such that the fractional rate of reduction of inward displacement for an air parcel is more than the fractional rate of reduction of absolute angular momentum due to the frictional torque. Turbulence processes also affect TC size and structure by modulating the boundary layer moisture and convective activity in the outer core region (Bu et al. 2017).

However, due to safety and practical considerations it should be noted that direct in situ flux measurements in the TC boundary layer are rare. Most values are available in the outer region of TC circulations where the surface wind is relatively weak (18–30 m s⁻¹; French et al. 2007, hereafter F07; Zhang and Drennan 2012). Although recent measurements have been obtained using drones in the eyewall of major hurricanes (Cione et al. 2020, hereafter C20), it remains very rare to have observations from the surface layer (z < ~100 m) in hurricane-force winds. The scarce observations limit our understanding toward the turbulence characteristics in the TC boundary layer, which further impedes the development of planetary boundary layer (PBL) parameterizations in TC conditions. Moreover, the existing PBL parameterization schemes have generally been developed for conditions over land, or for weak winds over oceans, and directly applying these PBL schemes in high-wind conditions (like TCs) demands a closer examination.

In recognition of these issues, several studies have used large-eddy simulation (LES) to provide insight into the TC boundary layer (e.g., Zhu 2008; Rotunno et al. 2009; Green and Zhang 2015; Stern and Bryan 2018; Wu et al. 2018; Li and Pu 2021). In LES, turbulent eddies are resolved explicitly using grid spacing of O(100) m or less, and a PBL parameterization is not used. However, because TCs are so large, spanning several hundred kilometers horizontally, it can be very computationally expensive to simulate an entire TC with eddy-resolving grid spacing.

With these points in mind, a modeling framework tailored to the TC boundary layer is presented in this study that can be used for multiple purposes, including to understand the turbulence characteristics in hurricane conditions, and to assess the performance of various PBL schemes and uncertainties in the parameterizations of boundary layer processes. The framework builds upon a simple method of simulating boundary layer winds of TCs (Bryan et al. 2017, hereafter B17), which can be used for LES and also single-column modeling (SCM) that can use different PBL schemes. An important addition to this framework is the use of composite in situ thermodynamic data from dropsondes released in mature hurricanes as a reference profile, and the averaged thermodynamic profiles in the modeling framework are held nearly fixed by applying a “large-scale” nudging to these profiles. The kinematic profiles are allowed to evolve using the “mesoscale tendency” terms of B17 that account for the mesoscale pressure-gradient and centrifugal accelerations. The thermodynamic nudging allows for a simple but accurate approach to model the TC boundary layer because it circumvents the need to specify multiple complex processes such as radiation, large-scale subsidence, and microphysical processes that influence thermodynamic profiles in TCs. Output from LES can be used as a benchmark to evaluate PBL parameterizations in high-wind conditions, as demonstrated for one PBL scheme herein.

The remainder of the paper is organized as follows. Section 2 presents a composite analysis of dropsonde observations that provides the reference thermodynamic profiles used in this study. The modeling framework tailored to the TC boundary layer as well as the model setup for LES and SCM using a PBL scheme are introduced in section 3. Analyses of LES output and verification of LES results against observations are presented in section 4. The differences between a TC boundary layer and an “ordinary” shear-driven, Ekman-type boundary layer are examined in section 5. In section 6, the LES results are used to evaluate a first-order PBL scheme in TC conditions. Concluding remarks are provided in section 7.

2. Observations

a. Composite profiles for category 4–5 hurricanes

To determine average profiles in high-wind conditions of TCs to guide this study, we perform a composite analysis of dropsonde data collected during research and operational flights conducted by NOAA P3 and G-IV, NASA DC-8 and G-V, and Air Force C130 aircrafts in category 4–5 hurricanes from 1999 to 2010. The GPS dropsonde observing platform records measurements of air temperature, relative humidity, pressure, and horizontal and vertical wind speeds. Detailed instrument documentation for the dropsonde can be found in Hock and Franklin (1999). After rigorous quality control using NCAR’s Atmospheric Sounding Processing Environment (ASPEN) software along with additional manual inspection, the dataset used in this study includes observations from 570 dropsondes that were collected in 11 category 4–5 hurricanes (see Table 1).

Table 1.

A list of category 4–5 hurricanes from 1999 to 2010 in the dropsonde dataset. “No. of total sondes” refers to the number of sondes from all flights for a given storm.

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The dropsonde data are grouped as a function of the normalized radius, which is defined as the radius to the vortex center (r) normalized by the radius of maximum wind (RMW; i.e., r* = r/RMW) following Zhang et al. (2011b). Figures 1a and 1b show the radius–height plots for the composite tangential and radial winds. The RMW of the composite vortex is set to 30 km, which is approximately the mean value indicated by the stepped-frequency microwave radiometer (SFMR) and flight-level wind data (Zhang et al. 2013). The composite vortex has a maximum tangential wind of ~65 m s⁻¹ near the RMW at roughly 500–600-m height. The boundary layer inflow is strongest outside the RMW, decelerates toward the RMW, and turns into outflow above ~1 km inside the RMW. These characteristics are similar to the boundary layer structures of simulated mature hurricanes (e.g., Kepert 2001; Nolan et al. 2009) and observational composites from Zhang et al. (2011b). The vertical profiles of tangential wind, radial wind, potential temperature, and mixing ratio of water vapor are extracted where the 10-m tangential wind is roughly 25 m s⁻¹ (V25 hereafter), 35 m s⁻¹ (V35 hereafter), and 45 m s⁻¹ (V45 hereafter) (Figs. 1c–f). The vertical profiles show that the boundary layer of mature hurricanes is generally statically stable in terms of potential temperature (Fig. 1e) or virtual potential temperature (not shown) under high-wind conditions, except for a very shallow (<100 m) statically unstable layer near the surface in V25 and V35.

(top) Radius–height plot of the composite (a) tangential and (b) radial winds in the lowest 2.5 km. The vertical lines with the text V45, V35, and V25 mark the location where the 10-m tangential wind is roughly 45, 35, and 25 m s⁻¹, respectively. (middle),(bottom) The vertical profiles for (c) tangential wind (m s⁻¹), (d) radial wind (m s⁻¹), (e) potential temperature (K), and (f) specific humidity (g kg⁻¹) for V45, V35, and V25, using colors as indicated by the legend in (f).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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(top) Radius–height plot of the composite (a) tangential and (b) radial winds in the lowest 2.5 km. The vertical lines with the text V45, V35, and V25 mark the location where the 10-m tangential wind is roughly 45, 35, and 25 m s⁻¹, respectively. (middle),(bottom) The vertical profiles for (c) tangential wind (m s⁻¹), (d) radial wind (m s⁻¹), (e) potential temperature (K), and (f) specific humidity (g kg⁻¹) for V45, V35, and V25, using colors as indicated by the legend in (f).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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(top) Radius–height plot of the composite (a) tangential and (b) radial winds in the lowest 2.5 km. The vertical lines with the text V45, V35, and V25 mark the location where the 10-m tangential wind is roughly 45, 35, and 25 m s⁻¹, respectively. (middle),(bottom) The vertical profiles for (c) tangential wind (m s⁻¹), (d) radial wind (m s⁻¹), (e) potential temperature (K), and (f) specific humidity (g kg⁻¹) for V45, V35, and V25, using colors as indicated by the legend in (f).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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3. Modeling methodology

a. Framework to account for large-scale kinematic processes

The modeling framework is an extension of the “simple” method for simulating boundary layer winds in TCs that was first presented by B17 and was evaluated by Worsnop et al. (2017). The primary idea is to account for large-scale conditions via specified values for gradient wind speed V, its radial gradient ∂V/∂r, and a characteristic distance from the TC center R, as illustrated in Fig. 2. The primary merit of this approach is that it allows for controlled conditions without potentially complex feedback to the large-scale TC vortex. For example, in SCM using different PBL parameterizations, one can ensure that the simulations proceed under the same specified kinematic and thermodynamic conditions, thus ensuring a clean comparison of PBL schemes and allowing for a direct analysis of the effects of small-scale turbulence processes.

The conceptual schematic of the LES modeling framework. The LES domain, shown as a white box, is located due east of the storm center (hurricane symbol) at a distance of R. Color shading denotes the near-surface tangential wind and the maximum tangential wind (Vmax) at the RMW is marked by a black dashed circle.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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The conceptual schematic of the LES modeling framework. The LES domain, shown as a white box, is located due east of the storm center (hurricane symbol) at a distance of R. Color shading denotes the near-surface tangential wind and the maximum tangential wind (Vmax) at the RMW is marked by a black dashed circle.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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The conceptual schematic of the LES modeling framework. The LES domain, shown as a white box, is located due east of the storm center (hurricane symbol) at a distance of R. Color shading denotes the near-surface tangential wind and the maximum tangential wind (Vmax) at the RMW is marked by a black dashed circle.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Another benefit of this approach is it allows for LES with a relatively small domain of O(5) km, and thus grid spacing of O(10) m can be used with present-day computing systems. This LES setup is much more computationally tractable than simulating an entire TC that spans several hundred kilometers horizontally. Downsides of this approach include the inability to account for certain mesoscale processes, such as rainbands. It is also not possible to study rapidly evolving conditions, including TC intensification and decay. Thus, the framework clearly cannot address all types of TC boundary layers. However, the benefits of this approach include the ability to use high resolution (e.g., 10-m grid spacing) and modest supercomputing resources, and the ability to compare different modeling assumptions (e.g., different PBL schemes) in controlled and consistent conditions.

This approach of using a domain of horizontal extent of O(5) km also mimics a single grid point in a mesoscale model. PBL parameterizations in mesoscale models act only on a vertical column of information (e.g., winds and thermodynamics variables) and the SCM approach here essentially reproduces what is occurring at a single grid point in an NWP model.

In this framework, the tangential velocity outside the radius of maximum wind is assumed to be a function of radius r according to the power law, i.e., $V_T=V{\left(R/r\right)}^n$, a concept similar to the modified Rankine vortex (e.g., Hughes 1952; Mallen et al. 2005; Chen et al. 2013). In this case, $\partial V_T/\partial r=-n\left(V/R\right)$ at r = R. Furthermore, although Fig. 2 illustrates a circularly symmetric TC (for simplicity), it is possible to account for different conditions in various quadrants of a TC with this framework. For example, one could run this modeling framework twice, with the same values of R and n but different values of V to account for a TC with varying winds in different regions of a TC.

Here, we set the reference gradient wind V based on the maximum tangential wind speed in the 500–1000 m layer from the dropsonde composites at different R (Figs. 1c,d), where R is determined from the radius–height dropsonde composites (see Figs. 1a,b). Table 2 shows the detailed information for three sets of experiments at different 10-m tangential wind, i.e., V25, V35, and V45, respectively. The value of n is chosen such that the simulated wind profiles at steady state best match the dropsonde composites. In our experiments n is set to a value slightly exceeding the range of the observations for mature hurricanes, i.e., n = 0.18–0.67, in Mallen et al. (2005). However, one difference should be noted: the observed n in Mallen et al. (2005) is related to the flight-level (~700 hPa) tangential winds while n in our setting is related to the tangential winds near the top of the boundary layer (~1 km).

Table 2.

The input parameters for the LESs and single-column model simulations for V25, V35, and V45 experiments. V is gradient wind speed, R is radius from storm center, and n is a radial decay parameter, as described in the text.

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c. Details of simulations

Cloud Model 1 (CM1; Bryan and Fritsch 2002) is used in this study for all simulations, including both LES and SCM. The primary advantage of LES, of course, is that the largest turbulent eddies can be explicitly resolved, and only the smaller subgrid-scale eddies are parameterized. For this study, subgrid-scale processes for LES are parameterized using a two-part LES subgrid model following the appendix in B17, which uses a standard turbulence kinetic energy (TKE)-based scheme following Deardorff (1980) at every grid point, and a term following Sullivan et al. (1994) that acts on horizontal velocities for z < 100 m to prevent excessive shear near the surface.

The initial thermodynamic conditions (potential temperature and water vapor mixing ratio) are shown in Figs. 1e and 1f. The sea surface temperature is set to be about 2°C warmer than the near-surface air temperature. This value of air–sea temperature difference is similar to the observed climatological value in hurricane conditions (Cione et al. 2000; Cione 2015). All simulations are run for 6 h and a quasi-steady state is reached after approximately 2.5 h (see one example using V35 in Fig. 3).

Evolution of (a) the 10-m tangential wind speed and (b) boundary layer height for V35 from the LES (black, domain-averaged) and from the SCM simulation that uses BR09 with an asymptotic length scale l_∞ = 75 m (blue) and 40 m (red). The definition of boundary layer height is described in section 3d.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Evolution of (a) the 10-m tangential wind speed and (b) boundary layer height for V35 from the LES (black, domain-averaged) and from the SCM simulation that uses BR09 with an asymptotic length scale l_∞ = 75 m (blue) and 40 m (red). The definition of boundary layer height is described in section 3d.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Evolution of (a) the 10-m tangential wind speed and (b) boundary layer height for V35 from the LES (black, domain-averaged) and from the SCM simulation that uses BR09 with an asymptotic length scale l_∞ = 75 m (blue) and 40 m (red). The definition of boundary layer height is described in section 3d.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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For LES we use 512 × 512 grid points horizontally and a horizontal grid spacing (Δx) of 10 m. Of note, the domain-averaged wind profiles and turbulence properties examined here are consistent when a larger LES domain [i.e., O(10) km or O(20) km] is used (not shown). In the vertical, there are 512 grid points, with a domain depth of 3 km. The vertical grid spacing is 5 m below 2 km and increases to 12.5 m between 2 and 3 km. Rayleigh damping is applied above 2 km for horizontal winds, vertical velocity, and perturbations of potential temperature, to damp vertically propagating gravity waves. Periodic boundary conditions are used in both horizontal directions.

For SCM simulations, a very similar model setup is used except that there is only a single column and the vertical grid spacing is 50 m. The LES subgrid turbulence model is turned off and a PBL parameterization is used instead to account for all turbulent processes. The PBL scheme of Bryan and Rotunno (2009, hereafter BR09), as modified by B17, is used for all SCM simulations in this paper.

To reduce the sensitivity of the results to surface layer parameterizations, as noted in earlier studies (Braun and Tao 2000; Smith and Thomsen 2010), the GFDL surface layer scheme (Kurihara and Tuleya 1974) from HWRF is used for all of the LESs and SCM simulations. The surface drag coefficient (C_d) and surface enthalpy exchange coefficient (C_k) in the GFDL surface layer scheme have been modified in recent years based on observations in hurricanes. Figure 4 shows C_d and C_k under neutral conditions as a function of 10-m wind speed. It shows that C_d gradually increases with the 10-m wind to a maximum when 10-m wind is 30 m s⁻¹, and then gradually decreases as wind further increases, before leveling off when the 10-m wind is stronger than 50 m s⁻¹. The value of C_k gradually increases to the maximum as the 10-m wind reaches 30 m s⁻¹, and then very slowly decreases as wind further increases. The ratio C_k/C_d holds closely near ~0.5 when the 10-m wind is within 30–70 m s⁻¹, which is aligned with observations (e.g., Haus et al. 2010; Bell et al. 2012). There is no coupling to an ocean or surface wave model.

The surface (a) drag coefficient and (b) enthalpy exchange coefficient under neutral conditions as a function of 10-m surface wind in the GFDL surface-layer scheme.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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The surface (a) drag coefficient and (b) enthalpy exchange coefficient under neutral conditions as a function of 10-m surface wind in the GFDL surface-layer scheme.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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The surface (a) drag coefficient and (b) enthalpy exchange coefficient under neutral conditions as a function of 10-m surface wind in the GFDL surface-layer scheme.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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4. LES results and verification

Figure 5 shows plan views of instantaneous horizontal wind speed at t = 6 h from the V45 LES output. At z = 42.5 m (Fig. 5a) an evident feature is the persistent “streaky” structures. These streaks are oriented approximately along the mean wind direction, with alternating low- and high-speed fluids. These characteristics are consistent with the findings in the shear-dominated boundary layer (e.g., Moeng and Sullivan 1994; Lorsolo et al. 2008), and show similarities to the boundary layer rolls documented in several studies of the hurricane boundary layer (e.g., Wurman and Winslow 1998; Morrison et al. 2005; Foster 2005). Farther aloft, at z = 502.5 m (Fig. 5b) the pattern appears more cellular, although elongated “steaks” are still visible, now oriented slightly more clockwise compared to low levels. It is unclear why rolls do not extend to the top of the boundary layer (i.e., up to ~1000 m) in these simulations, as has been observed in actual tropical cyclones (e.g., Lorsolo et al. 2008; Guimond et al. 2018).

Instantaneous horizontal wind speed at t = 6 h from the V45 simulation at (a) z = 42.5 m and (b) z = 502.5 m.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Instantaneous horizontal wind speed at t = 6 h from the V45 simulation at (a) z = 42.5 m and (b) z = 502.5 m.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Instantaneous horizontal wind speed at t = 6 h from the V45 simulation at (a) z = 42.5 m and (b) z = 502.5 m.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Vertical profiles of TKE (e.g., Fig. 6a) are typical for statically neutral, shear-driven boundary layers (cf. Berg et al. 2020) in that the maximum TKE is very close to the surface, and decreases asymptotically to the top of the boundary layer. In all cases, maximum TKE is at z ≈ 10 m (z/h ≈ 0.01) with a peak nondimensional value of $\text{TKE}/u_{*}^2=4.8$; this value is quite similar to other studies of shear-driven PBLs (e.g., Berg et al. 2020). Because observations from the surface layer in TCs are very rare, it is generally unclear how turbulence properties in the TC boundary layer compare to “ordinary” shear-driven boundary layers at much lower wind speeds. The close similarity of these results (Fig. 6a) to weaker-shear LES studies suggest that TC boundary layers are similar overall to other shear-driven boundary layers; this point is investigated further below.

Profiles of (a) turbulence kinetic energy (TKE) and (c) vertical turbulence momentum flux τ from LES with 10-m horizontal grid spacing. The solid line is the total, the long-dashed line is the resolved component, and the thin-dashed line is the subgrid component. (b),(d) The ratio of subgrid to total values is shown for four different grid spacings as indicated in the legend. All results are for V45.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Profiles of (a) turbulence kinetic energy (TKE) and (c) vertical turbulence momentum flux τ from LES with 10-m horizontal grid spacing. The solid line is the total, the long-dashed line is the resolved component, and the thin-dashed line is the subgrid component. (b),(d) The ratio of subgrid to total values is shown for four different grid spacings as indicated in the legend. All results are for V45.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Profiles of (a) turbulence kinetic energy (TKE) and (c) vertical turbulence momentum flux τ from LES with 10-m horizontal grid spacing. The solid line is the total, the long-dashed line is the resolved component, and the thin-dashed line is the subgrid component. (b),(d) The ratio of subgrid to total values is shown for four different grid spacings as indicated in the legend. All results are for V45.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Figure 6b shows the ratio of subgrid to total TKE for four simulations with varying horizontal grid spacing: Δx = 10, 20, 40, and 80 m, with vertical grid spacing (below 2 km) of 5, 10, 20, and 40 m, respectively. As expected, the percentage of subgrid TKE is small (less than 10%) above the near-surface layer for the highest resolution simulation. For Δx = 80 m (and larger grid spacing, not shown) the subgrid TKE is an unacceptably large percentage of the total TKE (greater than 10%), demonstrating that grid spacing of O(10) m is needed for well-resolved LES of the TC boundary layer in which the subgrid model plays a small role (except, of course, near the surface).

Profiles of vertical momentum flux τ (Fig. 6c) are also typical for shear-driven boundary layers, with an approximately linear decrease from the surface to the top of the boundary layer. The ratio of subgrid to total vertical momentum flux (Fig. 6d) shows that the subgrid model plays a very small role throughout most of the boundary layer; the exceptions are at the surface where vertical momentum flux is parameterized through Monin–Obukhov similarity theory, and near the top of the boundary layer where the subgrid ratio peaks at roughly 10% for all Δx < 40 m.

The average values of h from the final 2 h of all simulations are listed in Table 3, which show that h decreases slightly as Δx decreases for Δx ≤ 40 m. A qualitatively similar change in h with Δx was found by Berg et al. (2020) in simulations of a sheared neutral PBL with Δx as small as 2.5 m.

Table 3.

Boundary layer height (m), averaged from 4 to 6 h, from resolution sensitivity simulations.

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To evaluate turbulence properties in these LES, we use available observational estimates of τ in hurricanes from NOAA P3 aircraft (F07; Z11; Zhang and Drennan 2012), which are plotted as dots in Figs. 7 and compared with nondimensional profiles of τ from LES. For Fig. 7a, profiles from all three cases (V25, V35, and V45) are nondimensionalized in the traditional way using the square of the surface friction velocity $u_{*}$. From the observational studies, estimates of $u_{*}$ are only available from the study by F07, which are all from tropical storm–force wind speeds. Actually, all values from F07 are for 10-m wind speed less than 29 m s⁻¹, and thus are most directly applicable to the V25 simulation. Results (Fig. 7a) are comparable for all simulations when normalized by $u_{*}^2$². The F07 observations in Fig. 7a tend to be larger in magnitude than the model simulations, but we suspect the observational data in this figure are biased high because of a low bias in the estimates of $u_{*}$ that was identified by Andreas et al. (2012, p. 2523).

Vertical profiles of total vertical turbulent momentum flux τ (m² s⁻²) from LES for V25 (orange), V35 (green), and V45 (black). (a) Results nondimensionalized by $u_{*}^2$ and (b) results nondimensionalized by U². Dots show observations from F07 (gray), Z11 (blue), and C20 (red).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Vertical profiles of total vertical turbulent momentum flux τ (m² s⁻²) from LES for V25 (orange), V35 (green), and V45 (black). (a) Results nondimensionalized by $u_{*}^2$ and (b) results nondimensionalized by U². Dots show observations from F07 (gray), Z11 (blue), and C20 (red).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Vertical profiles of total vertical turbulent momentum flux τ (m² s⁻²) from LES for V25 (orange), V35 (green), and V45 (black). (a) Results nondimensionalized by $u_{*}^2$ and (b) results nondimensionalized by U². Dots show observations from F07 (gray), Z11 (blue), and C20 (red).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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To bring more observational data into the analysis, and to remove the bias associated with the surface parameter $u_{*}$, we plot τ normalized by the square of mean wind speed U from each level. This normalization allows us to use the P3 measurements from Z11 as well as recent UAS measurements in hurricane eyewalls by C20; notably, these two studies include observations at hurricane-force wind speeds. Results (Fig. 7b) show that the three LESs do not compare quite as well to each other in low levels (below 400 m), but are very similar in the upper half of the boundary layer (z > 500 m). Most important for the present analysis is that the LES results compare well with the observations, especially for z < 400 m. There are some cases where observational data points are a factor of 2 larger (e.g., at z = 150 m for C20, and z = 500 m for Z11) which are difficult to explain here, but may be related to deep-convective processes in hurricanes that are not present in these simplified numerical simulations, or perhaps are related to “mesovortices” (also known as “eyewall vorticity maxima”) which are coherent vortices of O(1)-km horizontal scale that are associated with localized extreme wind gusts in hurricanes (e.g., Aberson et al. 2006; Marks et al. 2008; Stern et al. 2016).

5. Comparison to an ordinary shear-driven PBL

As noted in the introduction, PBL parameterizations have generally been developed for conditions over land, or for weak winds over oceans. It is an open question whether commonly used PBL parameterizations work well in hurricane conditions, i.e., in strong winds over the oceans. As one step toward answering this question we compare the LES from this study with a standard approach for studying shear boundary layers: the Ekman-type boundary layer. We use the term “Ekman type” in the same sense as B17, where the pressure-gradient acceleration is computing using the geostrophic wind equation, and because we are using LES, the effective eddy viscosity is not constant (as it is in the classic Ekman spiral solution). For LES, we set the pressure gradient in the x direction to +fV_geo (V_geo denotes geostrophic wind speed), and the pressure gradient in the y direction to zero, and there are no centrifugal or large-scale advective tendencies. The approach to large-scale thermodynamic tendencies is the same as before, i.e., the nudging term toward the profiles shown in Figs. 1e and 1f is retained.

Resulting mean wind profiles are presented with standard (dimensional) units as shown in Figs. 8a–c for simulations with Δx = 20 m and Δz = 10 m, where it is clear that the boundary layer depth is greater (by a factor of 2) for the Ekman-like approach as compared to the TC simulations. The much shallower boundary layer depth in TC simulations reflects the effect of TC’s strong rotation on reducing boundary layer height. In fact, it takes much longer for the Ekman-like simulations to develop a steady boundary layer, so we run them for 2 days (instead of 6 h for the TC setup) and analyze the final 2 h of every simulation. The much slower development of the Ekman-type boundary layer is attributable to the longer large-eddy turnover time, which is approximated by $h/u_{*}$. The longer run time for the simulation of the Ekman-type boundary layer explains why we use a lower resolution in this section (Δx = 20 m, instead of 10 m elsewhere in this study) to reduce computational expense. Figures 6b and 6d indicate that 20-m horizontal grid spacing maintains an acceptably small ratio of parameterized subgrid processes.

Horizontal velocity components for the LES using 20-m grid spacing with the TC setup (red) and an Ekman-like setup (black). (a)–(c) Dimensional results and (d)–(f) nondimensionalized results. Columns show (a),(d) the x component of velocity (analogous to radial velocity in the TC framework), (b),(e) the y component of velocity (analogous to tangential velocity in the TC framework), and (c),(f) the total horizontal wind speed.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Horizontal velocity components for the LES using 20-m grid spacing with the TC setup (red) and an Ekman-like setup (black). (a)–(c) Dimensional results and (d)–(f) nondimensionalized results. Columns show (a),(d) the x component of velocity (analogous to radial velocity in the TC framework), (b),(e) the y component of velocity (analogous to tangential velocity in the TC framework), and (c),(f) the total horizontal wind speed.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Horizontal velocity components for the LES using 20-m grid spacing with the TC setup (red) and an Ekman-like setup (black). (a)–(c) Dimensional results and (d)–(f) nondimensionalized results. Columns show (a),(d) the x component of velocity (analogous to radial velocity in the TC framework), (b),(e) the y component of velocity (analogous to tangential velocity in the TC framework), and (c),(f) the total horizontal wind speed.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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In addition to the deeper boundary layer, other notable differences in Figs. 8a–c include a maximum inflow at a factor-of-10-higher height (Fig. 8a) and substantially weaker near-surface wind speeds by a factor of nearly 2 (Fig. 8c). The same differences were reported in B17, where the primary contributor to higher wind speeds was identified to be the centrifugal acceleration terms in the TC framework.

Nondimensional profiles are shown in Figs. 8d–f, using the same definition of boundary layer height h as before (see section 3e) to nondimensionalize height, and using the surface friction velocity $u_{*}$ to nondimensionalize wind speeds. This analysis demonstrates that differences in the two wind profiles remain similar, i.e., for the Ekman-like simulation the height of maximum inflow remains higher above the surface, and the near-surface (for z/h < 0.1) wind speed remains weaker, although in the case the difference in the wind speed is much smaller (18% weaker in nondimensional wind speed, as opposed to a factor of ~2 weaker for dimensional wind speed).

Velocity variances are shown in Fig. 9 in the same manner as wind speed (Fig. 8), i.e., with dimensional profiles shown in the top row (Figs. 9a–c) and nondimensional profiles shown in the bottom row (Figs. 9d–f). As in Sullivan and Patton (2011), the subgrid variance is included by adding 2/3e_s to every component, essentially assuming that subgrid turbulence is isotropic. For the TC boundary layer, variances in the lower half of the boundary layer (z < 500 m) are substantially larger than those in an Ekman-type boundary layer, especially for the streamwise variance (Fig. 9b) which is a factor of 3 larger in the TC simulation. These results are consistent with the stronger low-level winds and wind shear noted earlier. Interestingly, when normalized by $u_{*}^2$ the variance profiles are fairly similar for all three components (Figs. 9d–f), although maximum nondimensional streamwise velocity variance is 20% larger in the TC simulation (Fig. 9e) near the surface. These results have implications for several applications, including for the design of wind turbines that are susceptible to TCs (e.g., Kapoor et al. 2020), and call for further studies into turbulence properties in TC boundary layers.

Turbulence velocity variances for LES using 20-m grid spacing with the TC setup (red) and an Ekman-like setup (black). (a)–(c) Dimensional results and (d)–(f) nondimensionalized results. Columns show (a),(d) the variance for the x component of velocity (analogous to radial velocity in the TC framework), (b),(e) the variance for the y component of velocity (analogous to tangential velocity in the TC framework), and (c),(f) the vertical velocity variance (solid lines) and the TKE (dashed lines).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Turbulence velocity variances for LES using 20-m grid spacing with the TC setup (red) and an Ekman-like setup (black). (a)–(c) Dimensional results and (d)–(f) nondimensionalized results. Columns show (a),(d) the variance for the x component of velocity (analogous to radial velocity in the TC framework), (b),(e) the variance for the y component of velocity (analogous to tangential velocity in the TC framework), and (c),(f) the vertical velocity variance (solid lines) and the TKE (dashed lines).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Turbulence velocity variances for LES using 20-m grid spacing with the TC setup (red) and an Ekman-like setup (black). (a)–(c) Dimensional results and (d)–(f) nondimensionalized results. Columns show (a),(d) the variance for the x component of velocity (analogous to radial velocity in the TC framework), (b),(e) the variance for the y component of velocity (analogous to tangential velocity in the TC framework), and (c),(f) the vertical velocity variance (solid lines) and the TKE (dashed lines).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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TKE profiles are shown as dashed lines in Figs. 9c and 9f. In terms of dimensional values, TKE is substantially larger (by a factor of 2–3) near the surface for the TC case. Curiously, the nondimensional TKE profiles are nearly identical (Fig. 9f).

The results in this section suggest that TC boundary layers are substantially different from ordinary shear-driven boundary layers when viewed with dimensional units, although when viewed with nondimensional scaling the differences are much smaller. Specifically, in dimensional terms, the TC boundary layer has a maximum inflow at a factor-of-10-shallower height, much stronger near-surface wind speed, and a factor-of-2-shallower boundary layer (with the caveat being that the thermodynamic conditions are held essentially fixed to TC-like conditions by the nudging term used herein). The TC boundary layer also achieves steady state in a factor-of-4-faster time in these simulations. In nondimensional terms, the differences are much smaller, though, and the nondimensional TKE profiles are nearly identical in the two frameworks. In terms of guidance for the development of PBL parameterizations, and analytic studies of TC boundary layers, these results show that nondimensional turbulence properties are quite similar to ordinary sheared boundary layers, despite the stark differences in mean wind profiles, and warrant a closer look at PBL parameterization performance in hurricane conditions.

6. Using LES results to evaluate PBL parameterizations

Figure 10a compares K_eff from the LES (black line) in V25 and observed K_eff values determined by Zhang and Drennan (2012) (black dots) from relatively weak surface winds (18–30 m s⁻¹). As in B17, these observational data are selected when the observed wind speed at the same altitude was within 4 m s⁻¹ of the domain-averaged mean wind speed from the LES. The K_eff profile of the LES generally passes through the central part of the dot cluster at different levels, indicating a decent agreement between the LES and observations in terms of the magnitude of K_eff. However, the model values tend to be larger than observations below 150 m, and smaller than observations above 150 m. This comparison provides some confidence for the fidelity of the LES framework; however, measurements of effective eddy viscosity are scarce at higher wind speeds, which would be needed for further confidence in high-wind TC LES modeling. We also note that the effective eddy viscosity in LES is larger in magnitude for higher wind speeds (i.e., the maximum value increases from 40 m² s⁻¹ for V25 to 60 m² s⁻¹ for V45) as expected for the associated higher wind shear.

Vertical profiles of effective eddy diffusivity K (black lines) for (a) V25, (b) V35, and (c) V45. Black dots in (a) denote observational estimates from Zhang and Drennan (2012). The dashed and dotted red lines indicate analytic profiles as indicated by the legend in (a).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Vertical profiles of effective eddy diffusivity K (black lines) for (a) V25, (b) V35, and (c) V45. Black dots in (a) denote observational estimates from Zhang and Drennan (2012). The dashed and dotted red lines indicate analytic profiles as indicated by the legend in (a).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Vertical profiles of effective eddy diffusivity K (black lines) for (a) V25, (b) V35, and (c) V45. Black dots in (a) denote observational estimates from Zhang and Drennan (2012). The dashed and dotted red lines indicate analytic profiles as indicated by the legend in (a).

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Another insightful variable that can be calculated from LES output, and can be used in PBL parameterizations, is the effective turbulence length scale, calculated as

$l_{\text{eff}}={\left(K_{\text{eff}}/\sqrt{{\left(\partial \overline{u}/\partial z\right)}^2+{\left(\partial \overline{\upsilon }/\partial z\right)}^2}\right)}^{1/2},$(5)

which is plotted in Fig. 11 for all three TC LESs. The profiles (solid colored lines in Fig. 11) are very similar in these three cases, with a maximum value of approximately 40 m at z = 250 m (or roughly z = h/4). The LES-determined values of l_eff are similar to the observational estimates from Zhang and Drennan (2012) (dots in Fig. 11a), at least in the sense that there is a linear increase from the surface to about 200 m, and roughly constant values with an average of order 40 m above. However, the scatter in the observations is large, and the LES values are consistently larger than the observational estimates below 200 m.

(a) Vertical profiles of the effective mixing length l_eff from LES for V25 (orange), V35 (green), and V45 (black). Dots in (a) are observational estimates of l_eff from Zhang and Drennan (2012). (b) Dashed and dotted red lines denote analytic profiles using l_∞ = 40 m as indicated in the legend.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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(a) Vertical profiles of the effective mixing length l_eff from LES for V25 (orange), V35 (green), and V45 (black). Dots in (a) are observational estimates of l_eff from Zhang and Drennan (2012). (b) Dashed and dotted red lines denote analytic profiles using l_∞ = 40 m as indicated in the legend.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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(a) Vertical profiles of the effective mixing length l_eff from LES for V25 (orange), V35 (green), and V45 (black). Dots in (a) are observational estimates of l_eff from Zhang and Drennan (2012). (b) Dashed and dotted red lines denote analytic profiles using l_∞ = 40 m as indicated in the legend.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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The default value for l_∞ in CM1 is 75 m, which is nearly a factor of 2 larger than the LES results and observations (Fig. 11a). To see whether a smaller value matching LES and observations can produce more realistic results, we use the LES as a benchmark and run SCM simulations (see section 3d for details) using the BR09 PBL scheme. Values of eddy diffusivity K from the BR09 scheme are shown in Figs. 12a–c. Results using l_∞ = 75 m are shown in blue, which are clearly larger (by a factor of 2) than K_eff from LES. Results using l_∞ = 40 m are shown in red, which are a much closer match to the LES results. Profiles of the radial component of velocity (u) in Figs. 11d–f show a clear advantage to l_∞ = 40 m, both in terms of matching the minimum value of u near the surface and reproducing the inflow-layer depth (i.e., height where u first goes to zero). Similarly, for the tangential component of velocity (υ) in Figs. 11g–i, there is a better match to LES results for l_∞ = 40 m. Surface wind speed and boundary layer depth are also a better match to LES results with the lower value of l_∞ (Fig. 3). These results demonstrate the value of this LES framework in evaluating PBL schemes in TC conditions where observations are presently scarce or nonexistent.

Single-column modeling results using the BR09 PBL (red and blue lines) and LES results (black lines) for (left) V25, (center) V35, and (right) V45. Blue line shows results with l_∞ = 75 m, and red line shows results with l_∞ = 40 m. (a)–(c) Eddy viscosity K (m² s⁻¹), (d)–(f) radial wind u (m s⁻¹), and (g)–(i) tangential wind υ (m s⁻¹). The gray dashed line in (d)–(f) marks u = 0 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Single-column modeling results using the BR09 PBL (red and blue lines) and LES results (black lines) for (left) V25, (center) V35, and (right) V45. Blue line shows results with l_∞ = 75 m, and red line shows results with l_∞ = 40 m. (a)–(c) Eddy viscosity K (m² s⁻¹), (d)–(f) radial wind u (m s⁻¹), and (g)–(i) tangential wind υ (m s⁻¹). The gray dashed line in (d)–(f) marks u = 0 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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Single-column modeling results using the BR09 PBL (red and blue lines) and LES results (black lines) for (left) V25, (center) V35, and (right) V45. Blue line shows results with l_∞ = 75 m, and red line shows results with l_∞ = 40 m. (a)–(c) Eddy viscosity K (m² s⁻¹), (d)–(f) radial wind u (m s⁻¹), and (g)–(i) tangential wind υ (m s⁻¹). The gray dashed line in (d)–(f) marks u = 0 m s⁻¹.

Citation: Journal of the Atmospheric Sciences 78, 11; 10.1175/JAS-D-20-0227.1

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7. Conclusions

Understanding of turbulence characteristics in the TC boundary layer remains challenging, mostly due to insufficient observations. Additionally, planetary boundary layer (PBL) schemes in numerical models are mostly designed for nonhurricane conditions, and uncertainties of applying these PBL schemes to hurricane conditions remains poorly understood. To fill in this gap, this study presented and evaluated a modeling framework specifically for the TC boundary layer that can be used for a small-domain [O(5)-km] large-eddy simulations (LES) (which do not require PBL parameterization) and also single-column modeling (SCM) using different PBL schemes. Building upon a simple technique of simulating boundary layer winds of TCs that includes several key input parameters to represent the TC structure (B17), this framework includes a one-dimensional nudging tendency to “anchor” the moisture and temperature profiles from actual major hurricanes during the simulations, which allows us to bypass the complex settings for radiation, microphysics, and other physical processes. Simulations with this framework do not produce the excessive growth of the boundary layer height in high-wind conditions reported in B17, which did not apply thermodynamic nudging. The reference thermodynamic profiles at different high-wind conditions are derived based on a composite analysis of dropsonde observations for category 4–5 hurricanes over 1999–2010.

With this framework, steady conditions develop in a short time (~2.5 h) in both LES and SCM. Results from LESs with different horizontal grid spacing show that turbulence kinetic energy (TKE) and vertical momentum flux are mostly resolved and the subgrid-scale terms are comparably negligible only when the horizontal grid spacing is O(10) m. Profiles of vertical momentum flux, effective eddy viscosity, and turbulence length scale from LES agree well with observational values in hurricanes of various intensity. This comparison also highlights that future observations based on advanced platforms [e.g., small unmanned aircraft system (sUAS)] need to focus on the surface layer (<100 m), where observations are very scare, and the level of ~200 m, where both effective eddy viscosity and turbulence length scale are maximized according to LES.

Comparison of LES results with thermodynamic nudging but using traditional “Ekman-like” tendency terms under geostrophic conditions also show that vertical profiles of TKE and velocity variances in the TC boundary layer are similar to previous studies for “ordinary” neutral, shear-driven boundary layers. Comparison of these results to the TC framework indicates that the TC boundary layer is much shallower, develops to steady conditions much faster, and has stronger near-surface wind speed and velocity variances. These differences can be attributed to the effect of the TC’s rotation and the associated centrifugal acceleration. However, when plotted nondimensionally, the profiles of velocity variance are quite similar, and the TKE profiles are practically identical, for the Ekman-like and TC conditions.

We further demonstrated that the LES data with this framework can be used to evaluate a PBL parameterization used for hurricane simulations. In this Louis-type PBL scheme (BR09) the vertical mixing length above the surface layer is predominantly determined by a fixed asymptotic length scale l_∞. Compared to the LES results, the default setting of BR09 with l_∞ = 75 m produces an excessively deep inflow layer with weaker inflow strength due to excessive vertical mixing. By setting l_∞ = 40 m, a value more consistent with LES results, the simulated tangential and radial wind profiles from BR09 are more similar to the LES counterparts. Another important finding to note is the formulation for turbulence length scale near the surface following Mason and Thomson (1992) is a better match to both observations and LES results than the often-used Blackadar formulation (Blackadar 1962). For the K-profile parameterization (KPP), LES results suggest that the exponent in the parametric profile needs to be increased for hurricane conditions such that the maximum eddy viscosity is reduced in magnitude and located closer to the surface.

As a concluding note, it should be noted that in the spirit of a “simple” approach, this framework does not account for complex mesoscale processes and rapidly evolving conditions (e.g., rapid intensification or weakening), and cannot address all types of TC boundary layers. There are clearly advantages to studying PBL schemes in more realistic but complex conditions, such as in a long-duration “full physics” modeling system with real-data initial conditions, which this framework cannot reproduce. However, the merits of this approach include the ability to use high resolution (e.g., 10-m grid spacing) and modest supercomputing resources, and the ability to compare different underlying assumptions in PBL schemes under controlled and known hurricane conditions. In a forthcoming paper, we will document a detailed evaluation of different types of PBL schemes and the suggested revisions to these PBL schemes in hurricane conditions using this modeling framework. Examinations of the effects of the revised PBL schemes on the three-dimensional simulations of TCs are also underway.