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    Figure 11 of Stern et al. (2014): Azimuthal-mean tangential wind speed for Hurricanes (a) Dennis on 10 Jul 2005, (b) Rita on 21 Sep 2005, and (c) Felix on 3 Sep 2007, and (d) for the R36A50 simulation of SN11. In all panels, the contour interval is 2 m s−1 (with every 20 m s−1 thickened), and the RMW is indicated in magenta. In (d), supergradient winds are contoured in solid white at +2, +4, +6, and +8 m s−1, and subgradient winds are contoured in dashed black at −2, −4, −6, and −8 m s−1.

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    Horizontal wind speed at 2-km height for Hurricane Patricia, composited from two WP-3D Doppler analyses respectively centered at 1733 and 2033 UTC 23 Oct 2015. Overlaid in black are horizontal trajectories of HDSS dropsondes released by the WB-57. The WB-57 flew from southeast to northwest, and the first and last sondes shown were released at 1956:43 and 2009:05 UTC, respectively. The horizontal grid spacing of the Doppler analyses is 5 km, and the analysis data are provided by NOAA/HRD. Note that this figure also appears in Doyle et al. (2017).

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    Distance–height cross sections of horizontal wind speed for Hurricane Patricia on 23 Oct 2015, from (a) WB-57 HDSS dropsondes centered at 2001 UTC and (b),(c) WP-3D Doppler analyses centered at 1733 and 2033 UTC, respectively. (d) The change in wind speed over the 3 h between the two Doppler analyses. The mean radial location of each of the 27 dropsondes used in (a) is indicated by the vertical dotted lines, and these are the same sondes shown in Fig. 2. The radial and vertical grid spacing of the Doppler analyses in (b) and (c) is 1.5 and 0.15 km, respectively. The contour interval is 5 m s−1 in all panels, with every 20 m s−1 thickened in (a)–(c), and the zero contour thickened in (d). In (d), the 60 and 80 m s−1 contours from the 1733 UTC analysis are overlaid in magenta and white, respectively.

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    Azimuthal-mean (a),(b) tangential and (c),(d) radial wind speed from the Doppler analyses centered at (left) 1733 and (right) 2033 UTC 23 Oct 2015 for Hurricane Patricia. The contour interval is 2 m s−1 in all panels, with every 20 m s−1 thickened for tangential wind speed, and the zero contour thickened for radial wind speed. In (c) and (d), contours of tangential wind speed are overlaid in blue for 60, 65, 70, and 75 m s−1.

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    Estimates of the azimuthal-mean tangential wind speed in Hurricane Patricia on 21 Oct 2015 (2 days prior to the analyses shown in previous figures) from the Doppler “swath” and “profile” analyses and for the in situ flight-level measurements from the WP-3D aircraft. Also shown is the profile for a modified Rankine vortex that is used to define the initial radial structure for the TC simulation shown in Figs. 6 and 7.

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    For the Patricia-like simulation, time series of (a) maximum wind speed, (b) minimum surface pressure, (c) RMW (defined by the azimuthal-mean tangential wind speed), and (d) height of maximum azimuthal-mean tangential wind speed. In (a), the local maximum at 10-m height (blue) is compared to the maximum of the azimuthal mean at any height (black) and the best track intensity of Patricia (magenta). In (b) and (c), the simulated values (blue) are compared to the Patricia best track and extended best track (magenta), respectively. In (d), the heights of midlevel maxima and minima (when they exist) are shown in addition to the height of the low-level absolute maximum. Note that the Patricia data are plotted from 0000 UTC 22 Oct (the first time after the analyses shown in Fig. 5) to 1800 UTC 23 Oct (the last time prior to landfall), and is shown from t = 3 to 45 h, to be consistent with the 2100 UTC 21 Oct analysis time of Fig. 5.

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    For the Patricia-like simulation at t = 36 h, azimuthal-mean (a) tangential and (b) radial wind speed. The contour interval is 2 m s−1 in both panels, with every 20 m s−1 thickened in (a), and the zero contour thickened in (b). In (b), contours of tangential wind speed are overlaid in blue for 60–85 m s−1, every 5 m s−1. In both panels, the locations of the low-level and midlevel tangential wind speed maxima are indicated by white dots, and the location of the local minimum is indicated by a white star.

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    For the (a),(c) R36A50 and (b),(d) R180A50 simulations, (top) azimuthal-mean tangential wind speed and (bottom) gradient wind speed, at times when the peak tangential wind speed is similar between the simulations. The locations of the low-level and midlevel tangential wind speed maxima (if it exists) are indicated by dots, and the location of the local minimum (if it exists) is indicated by a star. The contour interval in all panels is 2 m s−1, with every 20 m s−1 thickened. The +1 m s−1 contour of azimuthal-mean vertical velocity is overlaid on the bottom panels in magenta. Note that the radial and vertical extents of these plots is different from those in Fig. 7.

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    For the (a),(c) R36A50 and (b),(d) R180A50 simulations, (top) agradient wind speed and (bottom) azimuthal-mean radial wind speed at the same respective times as in Fig. 8. Symbols and contours are as in Fig. 8, except here the zero contour is thickened for all panels.

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    Maximum vertical shear of the tangential wind speed along the RMW vs (a) RMW and (b) Vmax. Each circle represents data from an hourly snapshot from one of the 16 simulations described in the text, and all times where Vmax exceeds 30 m s−1 are shown. Each circle in (a) is colored by its corresponding value of Vmax, and each circle in (b) is colored by its corresponding value of RMW, as indicated by the respective color bars. Note that for clarity, the color bar in (b) is capped at 50 km, and so some of the red dots correspond to RMWs of up to 100 km.

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    Azimuthal-mean tangential wind speed for l = (a) 25, (b) 50, (c) 100, and (d) 200 m. Each simulation shown here has the same initial vortex structure (R36A50), and they are compared at respective times when the peak gradient wind speeds among the simulations are approximately the same. The magenta curve in each panel is the contour of absolute angular momentum that goes through the respective location of Vmax.

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    As in Fig. 11, but for simulations with the R180A50 initial vortex. Note that the radial range shown here is different from in Fig. 11.

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    Azimuthal-mean (a),(c),(e) tangential wind speed and (b),(d),(f) agradient wind for simulations with l = 50 m and constant Cd equal to (top) 1.2 × 10−3, (middle) 2.4 × 10−3, and (bottom) 4.8 × 10−3. These simulations are compared at different times, but for the same value of maximum gradient wind speed (75 m s−1). Note that for all simulations shown in previous figures, Cd = 2.4 × 10−3 for 10-m wind speed above 25 m s−1.

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    For the R36A50 l = 25-m simulation: (a) the gradient wind speed averaged over t = 62–74 h. (b) The imposed gradient wind field used in the Kepert model, taken from the z = 0–2-km layer mean from the CM1 simulation shown in (a). Tangential wind speed in (c) the Kepert model and (d) the actual tangential wind speed in the corresponding CM1 simulation. The agradient wind speed in (e) the Kepert model and (f) the actual agradient wind speed in the corresponding CM1 simulation. The contour interval is 2 m s−1 in all panels, every 20 m s−1 is thickened in (a)–(d), and the zero contour is thickened in (e) and (f). To produce (c), the actual height-varying gradient wind from CM1 in (a) is added to the agradient wind simulated by the Kepert model in (e). Note that in (d) and (f), the CM1 fields have been interpolated onto the Kepert model grid.

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    As in Figs. 14c and 14e, but for (a),(b) a Kepert model simulation where the imposed gradient wind varies with height (taken from the CM1 simulation) and (c),(d) a Kepert model simulation where the vertical velocity is artificially scaled to agree with the low-level eyewall updraft strength in the respective CM1 simulation. Consistent with Fig. 14, the tangential winds in (b) and (d) are obtained by adding the gradient wind from the CM1 simulation (Fig. 14a) to the agradient wind from the respective Kepert model simulation in (a) and (c).

  • View in gallery

    For the R36A50 simulations, agradient wind speed in (a),(c),(e),(g) the Kepert model and (b),(d),(f),(h) the respective CM1 simulation. Each row shows simulations with a different value of l, which is given in the upper right of each panel, and increases from (top) 25 to (bottom) 200 m. Note that the time periods shown for each simulation are different, but chosen so that the maximum gradient wind speeds are similar.

  • View in gallery

    As in Fig. 16, but for tangential wind speed.

  • View in gallery

    For the R36A50 simulations, scatterplots of (a) the peak vertical diffusivity and (b) peak low-level supergradient wind speed vs the peak gradient wind speed. Each data point corresponds to an individual hourly snapshot, and only times when the peak gradient wind speed exceeds 30 m s−1 are shown. The mixing length of each simulation is indicated in the legend.

  • View in gallery

    For the set of simulations with constant Cd, the oscillation wavelength (given by the difference in heights between the low- and midlevel tangential wind speed maxima) vs the theoretical wavelength 2π2Kυ/I. Each data point corresponds to an individual hourly snapshot, and only times when the peak gradient wind speed exceeds 30 m s−1 (and a midlevel tangential wind speed maximum exists) are shown. The theoretical wavelength is evaluated at the radius of maximum gradient winds, with variables averaged over the lowest 1 km. The best-fit line is in magenta (with equation and variance explained in the box), and the 1:1 line is in black.

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Understanding Atypical Midlevel Wind Speed Maxima in Hurricane Eyewalls

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  • 1 University Corporation for Atmospheric Research, Monterey, California
  • | 2 Centre for Australian Weather and Climate Research, Melbourne, Victoria, Australia
  • | 3 National Center for Atmospheric Research, Boulder, Colorado
  • | 4 Naval Research Laboratory, Monterey, California
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Abstract

In tropical cyclones (TCs), the peak wind speed is typically found near the top of the boundary layer (approximately 0.5–1 km). Recently, it was shown that in a few observed TCs, the wind speed within the eyewall can increase with height within the midtroposphere, resulting in a secondary local maximum at 4–5 km. This study presents additional evidence of such an atypical structure, using dropsonde and Doppler radar observations from Hurricane Patricia (2015). Near peak intensity, Patricia exhibited an absolute wind speed maximum at 5–6-km height, along with a weaker boundary layer maximum. Idealized simulations and a diagnostic boundary layer model are used to investigate the dynamics that result in these atypical wind profiles, which only occur in TCs that are very intense (surface wind speed > 50 m s−1) and/or very small (radius of maximum winds < 20 km). The existence of multiple maxima in wind speed is a consequence of an inertial oscillation that is driven ultimately by surface friction. The vertical oscillation in the radial velocity results in a series of unbalanced tangential wind jets, whose magnitude and structure can manifest as a midlevel wind speed maximum. The wavelength of the inertial oscillation increases with vertical mixing length l in a turbulence parameterization, and no midlevel wind speed maximum occurs when l is large. Consistent with theory, the wavelength in the simulations scales with (2K/I)1/2, where K is the (vertical) turbulent diffusivity, and I2 is the inertial stability. This scaling is used to explain why only small and/or strong TCs exhibit midlevel wind speed maxima.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel P. Stern, dstern@ucar.edu

Abstract

In tropical cyclones (TCs), the peak wind speed is typically found near the top of the boundary layer (approximately 0.5–1 km). Recently, it was shown that in a few observed TCs, the wind speed within the eyewall can increase with height within the midtroposphere, resulting in a secondary local maximum at 4–5 km. This study presents additional evidence of such an atypical structure, using dropsonde and Doppler radar observations from Hurricane Patricia (2015). Near peak intensity, Patricia exhibited an absolute wind speed maximum at 5–6-km height, along with a weaker boundary layer maximum. Idealized simulations and a diagnostic boundary layer model are used to investigate the dynamics that result in these atypical wind profiles, which only occur in TCs that are very intense (surface wind speed > 50 m s−1) and/or very small (radius of maximum winds < 20 km). The existence of multiple maxima in wind speed is a consequence of an inertial oscillation that is driven ultimately by surface friction. The vertical oscillation in the radial velocity results in a series of unbalanced tangential wind jets, whose magnitude and structure can manifest as a midlevel wind speed maximum. The wavelength of the inertial oscillation increases with vertical mixing length l in a turbulence parameterization, and no midlevel wind speed maximum occurs when l is large. Consistent with theory, the wavelength in the simulations scales with (2K/I)1/2, where K is the (vertical) turbulent diffusivity, and I2 is the inertial stability. This scaling is used to explain why only small and/or strong TCs exhibit midlevel wind speed maxima.

© 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Daniel P. Stern, dstern@ucar.edu

1. Introduction

It has long been understood that the winds in tropical cyclones (TCs) are strongest in the lower troposphere. However, it is challenging to observe this maximum, and early Doppler radar case studies suggested a peak from 1.5- to 2.5-km height (Marks and Houze 1987; Marks et al. 1992). Powell et al. (1991) wrote at the time that “very little is known about the variability of the maximum wind level of horizontal winds in hurricanes,” and stated that observations suggested “that a hurricane’s maximum winds are usually found between 500 and 2000 m.” The introduction of the GPS dropsonde in 1997 led to significant advances in our understanding of the low-level wind field within TCs. Franklin et al. (2003) used hundreds of dropsondes to demonstrate that while there is substantial variability among individual profiles (and individual storms), the winds in the eyewall are strongest on average at about 500-m height, lower than what the earlier observational studies had suggested. More recently, Zhang et al. (2011b) used dropsonde composites to show that the peak tangential wind speed occurs within the inflow layer, at a height (700 m) where the inflow is about 25% of its peak value.

Franklin et al. (2003) attributed the existence of the wind speed maximum near 500-m height to two effects: the decrease of winds with increasing height associated with thermal wind balance, and the decrease of winds with decreasing height within the boundary layer due to surface friction. However, it has subsequently become clear that this boundary layer tangential wind jet is actually a result of systematically unbalanced flow, as the gradient wind is nearly constant with height in the lowest 1–2 km. Extending the pioneering work of Rosenthal (1962) and Eliassen and Lystad (1977), Kepert (2001) developed a three-dimensional linear analytical boundary layer model, and showed that strong inward advection of angular momentum by frictionally induced inflow results in a weakly supergradient jet [consistent with Anthes (1974) and Shapiro (1983)]. In a companion study, Kepert and Wang (2001) used a nonlinear numerical model to show that the inclusion of vertical advection greatly enhances the strength of the jet,1 and they concluded that the jet is typically 10%–25% supergradient, with the jet strength increasing with TC intensity. These studies also proposed that the height of the jet is governed by a depth scale δ = (2K/I)1/2, increasing with (vertical) turbulent diffusivity K and decreasing with inertial stability I2. This depth scale for the boundary layer within a vortex, first derived by Rosenthal (1962), is a modification of the classical Ekman depth δE = (2K/f)1/2 (Ekman 1905; Holton 2004), where inertial frequency I (the square root of inertial stability) replaces the Coriolis parameter f. As I is much greater than f in the core of a tropical cyclone, storm rotation generally reduces the depth of the boundary layer.

A series of observational studies (Kepert 2006a,b; Schwendike and Kepert 2008) using dropsondes have demonstrated that the model of Kepert and Wang (2001) can generally reproduce the boundary layer profiles of tangential and radial wind of several well-observed TCs. These studies also confirmed observationally that the boundary layer jet is often supergradient, and that for strong storms characterized by a sharply peaked radial profile of tangential wind speed, the jet is strongly supergradient. Bell and Montgomery (2008) used objective analyses of dropsonde and flight-level data to evaluate gradient wind balance for Hurricane Isabel (2003), and found that the winds within the boundary layer eyewall were supergradient, by up to 15%. In simulations of Isabel, Nolan et al. (2009) found a similar magnitude of the unbalanced flow, with the boundary layer jet 15%–20% supergradient.

Franklin et al. (1993) performed an objective analysis of Hurricane Gloria (1985), primarily utilizing Doppler radar and Omega dropsondes (a predecessor to the GPS dropsonde), and investigated the structure of the wind field. They found a rather unusual vertical structure of the eyewall, with two distinct tangential wind speed maxima: a weaker maximum near 850 hPa, and an absolute maximum at 550–600 hPa. The authors speculated that this midlevel maximum resulted from subsidence and inflow from above, driven by a secondary eyewall, and that this descending inflow locally increased the winds through angular momentum advection. It was also suggested that this structure may have been related to thermal wind imbalance.

Stern and Nolan (2011, hereafter SN11) and Stern et al. (2014) used Doppler wind analyses from 39 different flights to examine the vertical structure of the eyewall, and in particular, to determine the rate at which the maximum tangential wind speed decreases with height above the boundary layer. On average, the maximum wind speed decreases by about 20% from 2- to 8-km height, with the decay rate increasing with increasing size [“size” in both SN11 and in this current study is defined by the radius of maximum winds (RMW)] and decreasing with increasing intensity. In other words, the maximum wind speed decreases more slowly with height for smaller storms, and for stronger storms. Although there is a fair amount of variability about the mean, Stern et al. (2014) found that the winds decrease with height at a similar enough rate such that the mean decay rate can be used to “predict” the actual profile for most individual cases within ±4 m s−1. Stern et al. (2014) identified three obvious outlier cases within their dataset, where the decay rate of maximum tangential wind speed was much less than average: Dennis on 10 July 2005, Rita on 21 September 2005, and Felix on 3 September 2007. Figure 1 reproduces Fig. 11 of Stern et al. (2014), which shows the radius–height structure of the azimuthal-mean tangential wind speed for these three cases. In each of these cases, the maximum winds were nearly constant or increasing with height between 2 and 5 km, and each exhibited a local maximum at 4–4.5 km. Note that the boundary layer winds (below roughly 1 km) are poorly resolved by these Doppler analyses, due to a combination of sea clutter, sidelobe effects, limited vertical resolution (500 m), and the inherent smoothing of the analysis technique (Reasor et al. 2009; Rogers et al. 2012; Lorsolo et al. 2013). Therefore, the apparent lack of any low-level wind speed maximum for Felix (Fig. 1c) is likely attributable to these deficiencies.

Fig. 1.
Fig. 1.

Figure 11 of Stern et al. (2014): Azimuthal-mean tangential wind speed for Hurricanes (a) Dennis on 10 Jul 2005, (b) Rita on 21 Sep 2005, and (c) Felix on 3 Sep 2007, and (d) for the R36A50 simulation of SN11. In all panels, the contour interval is 2 m s−1 (with every 20 m s−1 thickened), and the RMW is indicated in magenta. In (d), supergradient winds are contoured in solid white at +2, +4, +6, and +8 m s−1, and subgradient winds are contoured in dashed black at −2, −4, −6, and −8 m s−1.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

SN11 proposed that the midlevel wind speed maximum in Dennis was caused by an elevated supergradient jet, distinct from the boundary layer supergradient jet. This inference was based on the similarity in tangential wind speed structure between the idealized WRF simulations of SN11 and the Doppler analysis, as well as the clear presence of such an elevated supergradient jet in the simulations (e.g., Fig. 1d). SN11 also noted the similarity between Dennis and the analysis of Gloria by Franklin et al. (1993). With the addition of the Rita and Felix cases to their dataset, Stern et al. (2014) noted that all three storms were relatively intense (with Rita and Felix at category 5) and relatively small (with Dennis and Felix having RMWs of only 11.5 km), and hypothesized that these common characteristics among these cases may be related to their atypical midlevel wind speed maxima. It is difficult to generalize based on such a small sample of cases, and it is not possible from the observations alone to understand the dynamics responsible for the occurrence of multiple wind speed maxima. Nevertheless, the possibility that the midlevel maximum may be systematically related to storm size and intensity makes this an important issue to examine further.

Within the fluid dynamics literature, it has long been recognized that certain vortex profiles are intrinsically prone to inertial oscillations within the secondary circulation (Bödewadt 1940; Kuo 1971), in response to surface friction. These oscillations result in vertically alternating regions of inflow and outflow, which can thereby produce agradient flow, and ultimately, multiple maxima in the tangential wind speed. In a review of rotating boundary layers, Rotunno (2014) demonstrated that in the presence of a no-slip lower boundary condition, the boundary layer flow beneath a Rankine vortex erupts in an updraft near the location of the transition between potential flow and solid-body rotation, and that inertial waves may propagate within this region and into the free atmosphere above. These waves result in oscillations in both the secondary and primary circulations, as seen in Fig. 10 of Rotunno (2014). Rotunno (2014) stated that this simplified analysis is generally relevant to vortices for which the outer flow can be considered externally imposed, and not influenced by the boundary layer itself. Therefore, he considered this to be more applicable to tornadoes than to tropical cyclones, which have both a dynamic and thermodynamic feedback between the boundary layer and the outer flow. Bryan et al. (2017a) found multiple wind maxima in an axisymmetric tornado simulation and stated that this “is a common feature of strongly rotating axisymmetric simulations,” and there appears to be some evidence for this phenomenon in observed tornadoes, as can be seen in Fig. 4 and Fig. 5d of Wakimoto et al. (2012) and Fig. 4c of Wakimoto et al. (2015).

In the context of axisymmetric tropical cyclone simulations, Bryan and Rotunno (2009c) and Rotunno and Bryan (2012) showed that the structure of the tangential and radial wind fields is very sensitive to the vertical mixing length lυ in the turbulence parameterization. Figure 5c of Bryan and Rotunno (2009c) shows alternating inflow and outflow jets as well as a midlevel tangential wind speed maximum, for a simulation with a constant lυ = 50 m. Because the strength of inflow and outflow was much greater than is believed to occur in observed TCs, Bryan and Rotunno (2009c) concluded that this simulation was not representative of real TCs. Bryan and Rotunno (2009a) show an example of alternating jets of agradient flow in their Fig. 10, and demonstrate that parcel trajectories oscillate about a hypothetical trajectory of the balanced flow. This was shown for a simulation with an unrealistically small horizontal mixing length (and hence unrealistically strong intensity), and so it was not clear from this study whether such a structure could occur in a real tropical cyclone. More recently, Persing et al. (2013) examined and compared a pair of axisymmetric and three-dimensional idealized simulations, both using lυ = 50 m. They found multiple tangential wind speed maxima in both simulated storms when they were intense, and they identified this phenomenon to be the result of a standing centrifugal wave,2 with alternating layers of inflow and outflow that are damped as the flow approaches gradient wind balance. This explanation of multiple wind speed maxima provided by Persing et al. (2013) is similar to and consistent with that proposed by SN11. Neither of these two studies were focused on this phenomenon, however.

The purpose of our study is to comprehensively explore the dynamical mechanism that leads to a secondary maximum in tangential wind speed in the midlevel eyewall of tropical cyclones, and to understand the reasons why such a structure is rarely observed. In section 2, we present observations from Hurricane Patricia (2015) and show that it exhibited an absolute maximum in tangential wind speed at 5–6-km height, in addition to a weaker boundary layer maximum. We then use Patricia as a motivating case, and in section 3, we present an idealized simulation that qualitatively reproduces the atypical structures seen in the observed cases. In section 4, we use a series of numerical experiments to investigate the dependence of the vertical structure on size and intensity, and in section 5, we explore the sensitivity to vertical diffusion and to surface friction. In section 6, we use the boundary layer model of Kepert to demonstrate that this inertial oscillation, and the resulting midlevel wind maximum, are fundamentally due to the response of a balanced vortex to surface friction. Finally, we present a summary and our conclusions in section 7.

2. Observations from Patricia (2015)

Hurricane Patricia is the most intense recorded storm to have occurred in the Western Hemisphere (Rogers et al. 2017) and is arguably the most intense recorded TC anywhere on Earth (Velden et al. 2017). Patricia was intensively observed as part of the Office of Naval Research (ONR) Tropical Cyclone Intensity (TCI) experiment (Doyle et al. 2017), with high-altitude dropsondes released during four flights by the NASA WB-57 aircraft. The NOAA P3 aircraft also flew through Patricia at similar times as did the WB-57, on 21, 22, and 23 October. Detailed descriptions of the WB-57 and P3 flights and of the evolution of Patricia are given in Rogers et al. (2017) and Doyle et al. (2017), and here we provide a brief description of Patricia, based on the NHC report of Kimberlain et al. (2016).

Patricia originated from interactions between a tropical wave and other disturbances, and a tropical depression formed by 0600 UTC 20 October, about 330 km south-southeast of Salina Cruz, Mexico (Kimberlain et al. 2016), and the first WB-57 flight occurred on the afternoon of 20 October. Following a period of arrested development, convective activity increased over the center, and Patricia was observed to be a 50-kt tropical storm (1 kt ≈ 0.51 m s−1) by a P3 flight during the afternoon of 21 October. The environment of Patricia was extremely favorable for rapid intensification (RI), with SSTs exceeding 30°C and 850–200-hPa wind shear of 5 kt or less, and indeed Patricia intensified by 150 kt over the 54-h period beginning at 0600 UTC 21 October, with a peak intensity of 185 kt assessed at 1200 UTC 23 October. TCI missions into Patricia occurred during early RI as a 50-kt tropical storm on 21 October, and in the middle of RI as a 115-kt category 4 hurricane on 22 October. A final flight occurred on 23 October at nearly peak intensity (180 kt), but during a period of rapid weakening, which commenced in association with an eyewall replacement cycle as well as increasing vertical wind shear. Patricia made landfall near Playa Cuixmala, Mexico, as a 130-kt hurricane at 2300 UTC 23 October.

Figure 2 shows the horizontal wind speed at z = 2 km on 23 October, from the composite of two P3 Doppler analyses respectively centered at 1733 and 2033 UTC, overlaid with the horizontal trajectories of the dropsondes released by the WB-57. These 27 dropsondes were released over a 12-min period as the WB-57 overflew the center of Patricia at approximately 18.5-km height from southeast to northwest. A secondary wind maximum is evident in the Doppler composite in the southeast and northeast quadrants. Figure 3a shows horizontal wind speed measured by the dropsondes, as a function of height and distance along the SE–NW cross section. The dropsondes underwent intensive quality control, using NCAR’s Atmospheric Sounding Processing Environment (ASPEN) software along with careful manual inspection (Bell et al. 2016). As described in Doyle et al. (2017), to calculate the time-varying radial location of each sonde, we interpolate the 2-min storm center positions provided by NOAA/Hurricane Research Division (HRD) to the time of each sonde data point. For plotting, we bin the data in height for each sonde at 100-m intervals, and we use the mean radius of each sonde (negative SE, positive NW) for the entire profile. Unfortunately, the sondes sometimes failed within the eyewall, and so it is not possible to characterize the structure of the NW eyewall from the dropsondes in this case. Coverage was better in the SE eyewall, and a peak wind speed of 82 m s−1 is seen at about 6-km height, with a weaker local maximum within the boundary layer. Although there is one failed sonde just outwards of the peak winds, the next seven sondes (moving outwards) in the SE portion of the cross section all exhibit evidence of this midlevel maximum as well.

Fig. 2.
Fig. 2.

Horizontal wind speed at 2-km height for Hurricane Patricia, composited from two WP-3D Doppler analyses respectively centered at 1733 and 2033 UTC 23 Oct 2015. Overlaid in black are horizontal trajectories of HDSS dropsondes released by the WB-57. The WB-57 flew from southeast to northwest, and the first and last sondes shown were released at 1956:43 and 2009:05 UTC, respectively. The horizontal grid spacing of the Doppler analyses is 5 km, and the analysis data are provided by NOAA/HRD. Note that this figure also appears in Doyle et al. (2017).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Fig. 3.
Fig. 3.

Distance–height cross sections of horizontal wind speed for Hurricane Patricia on 23 Oct 2015, from (a) WB-57 HDSS dropsondes centered at 2001 UTC and (b),(c) WP-3D Doppler analyses centered at 1733 and 2033 UTC, respectively. (d) The change in wind speed over the 3 h between the two Doppler analyses. The mean radial location of each of the 27 dropsondes used in (a) is indicated by the vertical dotted lines, and these are the same sondes shown in Fig. 2. The radial and vertical grid spacing of the Doppler analyses in (b) and (c) is 1.5 and 0.15 km, respectively. The contour interval is 5 m s−1 in all panels, with every 20 m s−1 thickened in (a)–(c), and the zero contour thickened in (d). In (d), the 60 and 80 m s−1 contours from the 1733 UTC analysis are overlaid in magenta and white, respectively.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

The dropsondes are advected by the wind, and so they may drift substantially (10–30 km) during their 10–15-min descent from 18.5-km height to the ocean surface. At large radii, this horizontal displacement represents only a small fraction of the circumference of the TC, but near the RMW of a small storm such as Patricia, sondes may be advected more than halfway around the eyewall (as seen in Fig. 2). This raises issues of representativeness and can make it challenging to interpret the structure of the wind field from dropsondes. Fortunately, for the Patricia case, we have Doppler radar analyses (provided by NOAA/HRD) from nearby times for direct comparison. The P3 made a pass through the center from NW to SE at 1733 UTC, and a pass from SE to NW at 2033 UTC, about 30 min after the WB-57 overflew Patricia at nearly the same azimuth. Figures 3b and 3c show the distance–height Doppler analyses of wind speed from these two passes, with both cross sections oriented such that SE is to the left and NW to the right, as with the dropsonde analysis (Fig. 3a). The horizontal and vertical grid spacing of these analyses are 1.5 km and 150 m, respectively, and they effectively represent an average across a 10-km-wide path normal to the flight track (Rogers et al. 2012; Lorsolo et al. 2010). Both Doppler analyses indicate a midlevel maximum of wind speed, at 5–7-km height, in agreement with the dropsonde analysis. It can be seen that the midlevel maximum is present both to the SE and to the NW. As the midlevel maximum is present at both analysis times, we can conclude that it is likely that this type of structure can persist for at least several hours.

Patricia was undergoing rapid weakening during the period that it was sampled by the WB-57 and the P3 on 23 October, as is evident in Fig. 3d, which shows the difference in wind speed between the first and second Doppler analyses. In only 3 h, the wind speed throughout the depth of the inner eyewall decreased by 15–25 m s−1, consistent with the 50-kt decrease in best track intensity from 1800 to 2300 UTC (the time of landfall). The increase in wind speed on the NW side from 10- to 30-km radius is associated with an apparent expansion of the mid- and upper-level inner eyewall, whereas the increase in wind speed on the SE side from 30- to 50-km radius at low levels is associated with a spinup and contraction of the developing outer eyewall. Figures 4a and 4b show the azimuthal-mean tangential wind speed from the respective Doppler analyses, where we approximate the mean by averaging the two halves of each cross section. We only calculate the mean where there are valid data on both sides of the cross-section analyses, and so there are limited data within the eyewall for the 1733 UTC analysis. Nevertheless, it is apparent that there is at least a region of nearly constant azimuthal-mean tangential wind speed from 4- to 6-km height. At 2033 UTC, it is clear that the absolute maximum in azimuthal-mean tangential wind speed occurs at about 5-km height. It appears that there is a local minimum at about 2-km height, and that the RMW slopes inwards between this minimum and the maximum above. This is in contrast to the outward slope of the RMW that is typically observed in TCs (Stern and Nolan 2009; Stern et al. 2014), but is quite similar to the inward slope of the RMW in Gloria observed by Franklin et al. (1993), and seen in the simulation of SN11 (Fig. 1d).

Fig. 4.
Fig. 4.

Azimuthal-mean (a),(b) tangential and (c),(d) radial wind speed from the Doppler analyses centered at (left) 1733 and (right) 2033 UTC 23 Oct 2015 for Hurricane Patricia. The contour interval is 2 m s−1 in all panels, with every 20 m s−1 thickened for tangential wind speed, and the zero contour thickened for radial wind speed. In (c) and (d), contours of tangential wind speed are overlaid in blue for 60, 65, 70, and 75 m s−1.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

The azimuthal-mean radial wind speed from the Doppler analyses is shown in Figs. 4c and 4d, overlaid with select contours of the tangential wind speed. There is a layered appearance to the radial wind within the eyewall; a shallow region of boundary layer inflow is topped by the typical low-level outflow jet, but there is another inflow layer from 2 to 4 km, and outflow once more above. Mean inflow is not typically seen above the boundary layer within the hurricane eyewall, and it appears that this structure of vertically oscillating radial wind is related to the unusual tangential wind speed structure. In particular, the midlevel tangential wind speed maximum is found in the region where inflow is transitioning to outflow, similar to the relationship between radial and tangential wind speed seen for the boundary layer maximum. In the following sections, we demonstrate that these atypical structures of the TC primary and secondary circulation can be reproduced in idealized simulations, and we explore the mechanisms for the occurrence of this phenomenon.

3. An idealized simulation based on Patricia

To illustrate the resemblance between simulated and observed TCs that exhibit dual eyewall wind maxima, in this section we present results from an idealized three-dimensional simulation that is designed to approximately reproduce the characteristics of Patricia. We use Cloud Model 1 (CM1; Bryan and Fritsch 2002; Bryan and Rotunno 2009c; Bryan and Morrison 2012) for all simulations in this study. For this “Patricia-like” simulation, we use a domain of 2880 km × 2880 km × 25 km, horizontal grid spacing of 1 km in the central 320 km × 320 km region (gradually stretching to 19 km at the outer portion of the domain), and 123 vertical levels, with variable vertical grid spacing in the lowest 7 km, and 250-m grid spacing above. For microphysics, we use the Morrison double-moment scheme (Morrison et al. 2009). We use a relaxation term to mimic radiative cooling (Rotunno and Emanuel 1987), and there is no parameterization of convection. For parameterizing turbulence, we use a horizontal mixing length lh that varies from 100 to 1000 m as surface pressure decreases from 1000 to 900 hPa, and a constant asymptotic vertical mixing length l = 100 m. We use a constant enthalpy exchange coefficient Ck = 1.2 × 10−3, and a wind speed–dependent drag coefficient Cd that increases linearly from 1 × 10−3 to 2.4 × 10−3 for 10-m wind speed between 5 and 25 m s−1, and is constant above.

For this simulation, we set the SST to 30.5°C based on the SHIPS analysis for Patricia, and use the observed Acapulco sounding from 1200 UTC 17 October 2015 to define a horizontally homogeneous initial thermodynamic environment. For simplicity, we model a quiescent environment, with no wind shear or mean flow. The diagnosed shear in the SHIPS analyses was generally less than 5 kt during the intensification period of Patricia, and so our no-shear simulation is still reasonably representative. We initialize an idealized vortex based on observations of the wind field when Patricia was a tropical storm. The first P3 flight into Patricia occurred around 2100 UTC 21 October, with a pass through the center at approximately 1.5-km height. Figure 5 shows three different estimates of the radial profile of azimuthal-mean tangential wind speed in Patricia: the Doppler “swath” analysis (averaged from 0.5- to 1.5-km height), the Doppler “profile” analysis (averaged from 0.45- to 1.2-km height), and the in situ flight-level winds. There is a fair amount of uncertainty among these estimates, in both the peak azimuthal-mean tangential wind speed and the RMW. This is in part due to differences in resolution and azimuthal coverage, combined with the fact that Patricia was relatively asymmetric at this time. We choose the Doppler profile analysis as most representative of the mean structure, as the resolution is better than that of the swath analysis (Rogers et al. 2012), and azimuthal coverage is greater than that of the flight-level data. Based on this Doppler analysis, we take as our initial condition a modified Rankine vortex with a maximum wind speed of 25 m s−1, an RMW of 36 km, and a decay coefficient of 0.25, and this profile is also shown in Fig. 5. The initial vertical structure of the tangential wind speed comes from the default setup of CM1, which is a maximum at the surface, linearly decaying to zero at 20-km height, as in Rotunno and Emanuel (1987). As shown in Stern (2010) and SN11, the evolution of simulated TCs is not particularly sensitive to the initial vertical structure, and simulated TCs rapidly adjust to a realistic structure following the onset of deep convection.

Fig. 5.
Fig. 5.

Estimates of the azimuthal-mean tangential wind speed in Hurricane Patricia on 21 Oct 2015 (2 days prior to the analyses shown in previous figures) from the Doppler “swath” and “profile” analyses and for the in situ flight-level measurements from the WP-3D aircraft. Also shown is the profile for a modified Rankine vortex that is used to define the initial radial structure for the TC simulation shown in Figs. 6 and 7.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Figure 6a shows the time series of the maximum surface (i.e., 10 m) wind speed, as well as the maximum azimuthal-mean tangential wind speed at any height. For comparison, we also show the best track maximum 10-m wind speed for Patricia. Figure 6b shows the minimum surface pressure for the simulation and from the best track of Patricia. As expected, this initially small storm in an extremely favorable environment rapidly intensifies, and it achieves a peak intensity that is comparable to that of Patricia. The simulated TC takes about 12 h longer than Patricia to reach peak intensity, and the peak 10-m wind speed in the simulation is about 10 m s−1 weaker than that estimated for Patricia. Though we are not attempting to simulate Patricia itself in this idealized framework, we are able to produce a simulated intensity evolution that is similar to that of the real hurricane. The RMW in the simulated storm rapidly contracts shortly after the onset of organized deep convection, which is typical of initially small storms in idealized simulations (Stern et al. 2015, 2017), and is also similar to the evolution of the RMW in Patricia (Fig. 6c).

Fig. 6.
Fig. 6.

For the Patricia-like simulation, time series of (a) maximum wind speed, (b) minimum surface pressure, (c) RMW (defined by the azimuthal-mean tangential wind speed), and (d) height of maximum azimuthal-mean tangential wind speed. In (a), the local maximum at 10-m height (blue) is compared to the maximum of the azimuthal mean at any height (black) and the best track intensity of Patricia (magenta). In (b) and (c), the simulated values (blue) are compared to the Patricia best track and extended best track (magenta), respectively. In (d), the heights of midlevel maxima and minima (when they exist) are shown in addition to the height of the low-level absolute maximum. Note that the Patricia data are plotted from 0000 UTC 22 Oct (the first time after the analyses shown in Fig. 5) to 1800 UTC 23 Oct (the last time prior to landfall), and is shown from t = 3 to 45 h, to be consistent with the 2100 UTC 21 Oct analysis time of Fig. 5.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Figure 7a shows a snapshot of the azimuthal-mean tangential wind speed at t = 36 h, which is representative of the time period when the intensity and size are most comparable to that of Patricia as it was observed on 23 October (Figs. 4a,b). There is a distinct midlevel local maximum at just over 3-km height. This type of eyewall structure is found at most times once the peak 10-m wind speed exceeds about 60 m s−1, with the height of the maximum varying from 3 to 6 km (Fig. 6d).3 The radial wind speed at t = 36 h is shown in Fig. 7b, and this can be compared to the Patricia analyses in Figs. 4c and 4d. As in the observations, there are multiple inflow and outflow layers within the eyewall, and the midlevel tangential wind speed maximum is found near the top of the midlevel inflow layer. Finally, it can also be seen that the radial structure of the tangential winds outside of the eyewall is relatively similar between the idealized simulation and the Patricia analyses (cf. Fig. 7a and Figs. 4a,b), though with given contours in the simulation found at somewhat larger radii, consistent with the slightly larger RMW. Overall, this experiment demonstrates that 1) we can reasonably reproduce the structure of Hurricane Patricia in an idealized framework, and 2) a midlevel tangential wind speed maximum exists in both the observations and the simulation. In the following two sections, we will systematically explore this phenomenon through a series of additional simulations.

Fig. 7.
Fig. 7.

For the Patricia-like simulation at t = 36 h, azimuthal-mean (a) tangential and (b) radial wind speed. The contour interval is 2 m s−1 in both panels, with every 20 m s−1 thickened in (a), and the zero contour thickened in (b). In (b), contours of tangential wind speed are overlaid in blue for 60–85 m s−1, every 5 m s−1. In both panels, the locations of the low-level and midlevel tangential wind speed maxima are indicated by white dots, and the location of the local minimum is indicated by a white star.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

4. The relationship between inner-core size and midlevel wind speed maxima

All five documented cases of midlevel eyewall wind maxima (Gloria, Dennis, Rita, Felix, and Patricia) occurred in TCs with relatively small RMWs. In contrast, in large and intense storms such as Isabel (2003) and Ivan (2005), the wind speed decreases monotonically upward above the boundary layer maximum [see Fig. 1c of Stern et al. (2014) and Fig. 6e of Nolan et al. (2009)]. This leads us to a hypothesis that size (i.e., RMW) is strongly correlated with the existence of midlevel wind speed maxima. To test this hypothesis, we examine simulations of intense TCs of varying size. As shown in SN11 and Stern et al. (2015), the quasi-steady-state size of a simulated TC can be controlled by varying either the initial RMW [as first suggested by Rotunno and Emanuel (1987)] or the initial radial profile of tangential wind speed outside of the RMW. Following those studies, we present results from four different initial RMWs (18, 36, 90, and 180 km) and four different Rankine decay coefficients (0.25, 0.5, 0.75, and 1.0), for a total of 16 simulations. For computational efficiency, these (and all subsequent) simulations use a horizontal grid spacing of 2 km and 59 vertical levels.4 Additionally, these simulations all use an SST of 28°C, and the Dunion (2011) mean sounding to define the initial thermodynamic environment.

To illustrate the effect of size, we first present a comparison of two simulations, with respective initial RMWs of 36 and 180 km. We will refer to these simulations as R36A50 and R180A50, as these both have a Rankine decay coefficient of 0.5, and we name the other simulations analogously. Figures 8a and 8b show azimuthal-mean tangential wind speed for R36A50 (Fig. 8a) and R180A50 (Fig. 8b), for respective times when the maximum (Vmax) is approximately the same. As expected, the RMW (here defined as the radius of the peak azimuthal tangential wind speed at any height) is much smaller for R36A50 (12 km) than for R180A50 (44 km). Similar to the Patricia-like simulation, R36A50 exhibits a midlevel local maximum in tangential wind speed at this time (Fig. 8a), and at most times once Vmax and the peak 10-m wind speed exceed about 70 and 60 m s−1, respectively (not shown). The much larger R180A50 simulation has no such midlevel maximum at this time (Fig. 8b), nor at any other time except very briefly prior to the formation of the eyewall (not shown).

Fig. 8.
Fig. 8.

For the (a),(c) R36A50 and (b),(d) R180A50 simulations, (top) azimuthal-mean tangential wind speed and (bottom) gradient wind speed, at times when the peak tangential wind speed is similar between the simulations. The locations of the low-level and midlevel tangential wind speed maxima (if it exists) are indicated by dots, and the location of the local minimum (if it exists) is indicated by a star. The contour interval in all panels is 2 m s−1, with every 20 m s−1 thickened. The +1 m s−1 contour of azimuthal-mean vertical velocity is overlaid on the bottom panels in magenta. Note that the radial and vertical extents of these plots is different from those in Fig. 7.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

To rule out the possibility that the midlevel tangential wind speed maximum is a consequence of balanced dynamics, we examine the structure of the gradient wind speed Vg for R36A50 (Fig. 8c) and R180A50 (Fig. 8d). In both cases, the maximum gradient wind speed varies little with height within the lowest 3 km, and above 2 km, Vg decreases monotonically with increasing height. Therefore, the structure of the balanced flow does not directly result in the atypical tangential wind structure seen in R36A50. Consistent with the hypothesis of Stern et al. (2014) (and of Franklin et al. 1993), the midlevel maximum in tangential wind speed is a consequence of the systematic departure from gradient wind balance. Figures 9a and 9b show the agradient wind Vag for R36A50 and R180A50, respectively. Both simulated TCs are characterized by subgradient flow within the boundary layer outside of the eyewall, a supergradient jet within the eyewall boundary layer, and a subgradient “jet” within the eyewall just above the supergradient jet. Atop the subgradient jet is another region of weakly supergradient flow, and so there is an oscillation in the agradient flow within the eyewall.

Fig. 9.
Fig. 9.

For the (a),(c) R36A50 and (b),(d) R180A50 simulations, (top) agradient wind speed and (bottom) azimuthal-mean radial wind speed at the same respective times as in Fig. 8. Symbols and contours are as in Fig. 8, except here the zero contour is thickened for all panels.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Comparing Figs. 8a, 8c, and 9a, it is clear that the midlevel maximum in tangential wind speed for R36A50 (Fig. 8a) results from the superposition of the oscillation in agradient flow (Fig. 9a) upon the background of weakly decreasing gradient wind with increasing height (Fig. 8c). In particular, the local minimum in tangential wind speed at 2.5 km coincides with the peak subgradient jet, and the midlevel maximum in tangential wind speed at 5.5 km coincides with a midlevel supergradient jet. However, the oscillation in Vag is clearly present in both simulations, but yet the midlevel tangential wind speed maximum is only present in R36A50. In fact, the supergradient boundary layer jet is actually substantially stronger for the larger TC in R180A50 [consistent with the theory of Kepert (2001)], and the agradient jets at midlevels are comparable or stronger in R180A50 as compared to R36A50. So it is not the existence of the oscillation itself or the strength of the agradient flow that explains the differences in tangential wind structure between the two cases. Instead, it is a difference in the structure of the oscillation that results in the different tangential wind structures.

By comparing Fig. 9a to Fig. 9b, it can be seen that the wavelength of the oscillation is shorter for R36A50; that is, the minima and maxima in agradient flow are closer to each other. As a result, the vertical shear of agradient wind (along the axis of the oscillation) is greater for R36A50 as compared to R180A50. It is this vertical shear that determines the kinematic effect of the agradient flow; if the increase in Vag with height is large enough, it can overcome the decrease of Vg with height, resulting in a local maximum in tangential wind speed, as seen in R36A50 (Fig. 8a). In contrast, in R180A50, the oscillation is longer and the shear of the agradient flow is weaker, and so the weak increase in Vag with height between 5- and 9-km height (Fig. 9b) cannot overcome the decrease in Vg over this layer (Fig. 8d). An additional reason that there is no midlevel tangential wind speed maximum in R180A50 is that the layer with increasing Vag is at a greater height in R180A50 (itself also a consequence of the greater oscillation wavelength), and therefore in a region where the gradient wind is more rapidly decreasing with height, rendering it more difficult for agradient jets to become manifest in the tangential wind field.

Similar to the observations from Patricia and the Patricia-like simulation examined earlier, the TC in the R36A50 simulation exhibits an oscillation in the radial velocity within the eyewall, with vertically alternating jets of inflow and outflow (Fig. 9c). The structure of the radial velocity is closely linked to that of the agradient flow (Fig. 9a), and in turn the tangential wind (Fig. 8a). The peak subgradient flow is found where the low-level outflow is transitioning to inflow, and the peak supergradient flow is found where the midlevel inflow is transitioning to outflow. Therefore, the midlevel tangential wind speed maximum occurs at the top of the midlevel inflow jet. The TC in R180A50, in contrast, has no inflow within the eyewall above the boundary layer (Fig. 9d), even though there is an oscillation in the agradient wind. Across numerous simulations, we have found that a midlevel tangential wind speed maximum rarely occurs in the absence of midlevel eyewall inflow.

So far, we have illustrated the sensitivity of eyewall structure to inner-core size by a comparison of two simulated TCs, one with a small RMW and the other with a large RMW. Next, we can show that this behavior is a systematic function of the RMW, by evaluation of the full suite of 16 simulations described above. Figure 10a shows as a function of the RMW, the maximum vertical shear of the tangential wind speed along the RMW; in other words, the change in peak azimuthal-mean tangential wind speed with height. This is evaluated only above the boundary layer (and so excludes the primary wind speed maximum), and only when the peak tangential wind speed exceeds 30 m s−1, to ensure that the eyewall and inner core is sufficiently well developed. When the maximum shear is negative, there is (by definition) no midlevel tangential wind speed maximum. This is the case at nearly all times when the low-level RMW is larger than about 20 km. For RMWs smaller than 20 km, there is a marked change in structure, where much of the time, the peak shear is positive, indicating the existence of a midlevel local maximum in tangential wind speed. That such midlevel maxima only occur for TCs with an RMW less than about 20 km is consistent with the observed cases (Figs. 1, 3, and 4). As discussed above, this variation in the shear of the peak wind speed is due to a lengthening of the wavelength of an oscillation in agradient flow with increasing size. In section 6, we will revisit this sensitivity in structure to inner-core size, and explain why the oscillation is a function of inner-core size.

Fig. 10.
Fig. 10.

Maximum vertical shear of the tangential wind speed along the RMW vs (a) RMW and (b) Vmax. Each circle represents data from an hourly snapshot from one of the 16 simulations described in the text, and all times where Vmax exceeds 30 m s−1 are shown. Each circle in (a) is colored by its corresponding value of Vmax, and each circle in (b) is colored by its corresponding value of RMW, as indicated by the respective color bars. Note that for clarity, the color bar in (b) is capped at 50 km, and so some of the red dots correspond to RMWs of up to 100 km.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

In Fig. 10a, the data points are colored by the maximum azimuthal-mean tangential wind speed. The relationship between intensity and inner-core size is complicated, and varies among the simulations (by design) because of the different initial vortex structures. Most of the simulated storms evolve to be extremely intense, but the corresponding RMW for such intensities varies substantially (10–50 km), and the large intense simulated TCs do not exhibit midlevel wind speed maxima. Also note that there are some extremely small TCs that are relatively less intense (Vmax = 40–50 m s−1), but still have midlevel maxima. Nevertheless, intensity does influence the eyewall structure, as can be seen in Fig. 10b, which shows the peak vertical shear of the tangential wind speed as a function of Vmax, with the data points colored by the RMW. With few exceptions, a midlevel wind speed maximum does not form until Vmax exceeds 50 m s−1. This wind speed threshold is clearly a function of size; for example, when the RMW is about 20 km, midlevel maxima are only present when Vmax exceeds about 80 m s−1. Again, when the RMW is larger than 20 km, midlevel maxima rarely occur.

From these simulations, we can conclude that the existence of midlevel wind speed maxima is strongly correlated with inner-core size, and modulated by a size-dependent intensity threshold. Although we have only a few observed cases exhibiting midlevel maxima, this size-dependent relationship with intensity is consistent, as the one such TC that was not extremely intense (Dennis) had a very small RMW.

5. The influence of friction and turbulence on the existence of midlevel wind speed maxima

a. Sensitivity of eyewall structure to vertical diffusion

We demonstrated above that the structure of the eyewall tangential wind field, and in particular the existence of a midlevel local maximum, is strongly influenced by the distribution of unbalanced (agradient) flow. It is known both from theoretical studies (Kepert 2001) and numerical simulations (Rotunno and Bryan 2012) that parameterized vertical diffusion is a significant modulator of the strength and distribution of the agradient wind. Therefore, we expect that vertical diffusion may also have such an influence on the existence and character of the midlevel tangential wind speed maximum. In our simulations, we parameterize vertical turbulence processes using a Louis scheme (Louis 1979; Kepert 2012), through which we specify an asymptotic mixing length l. For all of the simulations discussed previously, we set l = 100 m, which is believed to be within the range of realistic settings (Zhang et al. 2011a; Bryan 2012; Rotunno and Bryan 2012). Here, we evaluate the sensitivity to vertical diffusion, by varying l (25, 50, 100, 200 m) for the R36A50 initial condition. Figure 11 shows the radius–height structure of the azimuthal-mean tangential wind speed for these four simulations, for respective times when the gradient wind speed is similar. We compare the simulations in this manner because the strength of the agradient wind is a strong function of the gradient wind itself (e.g., Kepert 2001; SN11), and we wish to isolate the contribution of vertical mixing length to variations in unbalanced flow. A clear systematic variation of tangential wind speed with vertical mixing length is evident, with an increase in height and a weakening in strength of the midlevel maximum for increasing l. For l = 200 m (Fig. 11d), a midlevel maximum in tangential wind speed is nonexistent. Note that the RMW is very similar among these cases [consistent with the lack of sensitivity to vertical mixing length found by Bryan and Rotunno (2009c)], and so we can easily separate the effect of size from that of vertical diffusion.5 Also of note is that as l is made small enough, a third maximum in tangential wind speed can appear (Fig. 11a). Evidently, as the wavelength of the oscillation is shortened (which can also be seen from the overlaid contours of absolute angular momentum), additional maxima in agradient wind are found at low enough heights to also overcome the background decrease in gradient wind with increasing height. Such a triple wind maximum has not been observed, perhaps because l = 25 m is an unrealistically small mixing length (Zhang et al. 2011a).

Fig. 11.
Fig. 11.

Azimuthal-mean tangential wind speed for l = (a) 25, (b) 50, (c) 100, and (d) 200 m. Each simulation shown here has the same initial vortex structure (R36A50), and they are compared at respective times when the peak gradient wind speeds among the simulations are approximately the same. The magenta curve in each panel is the contour of absolute angular momentum that goes through the respective location of Vmax.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Figure 12 is similar to Fig. 11, but for the R180A50 initial vortex. Consistent with Fig. 8b, these large simulated TCs have monotonically decreasing tangential wind speed above the boundary layer maximum, for realistic vertical mixing lengths (Fig. 12c). However, as the mixing length is made small enough (l = 25 m, Fig. 12a), a midlevel local maximum appears. For the l = 50-m case, there is no midlevel maximum, but a region of slower decay is evident from 4 to 7 km, indicating that there is a smooth transition toward a higher-amplitude and shorter-wavelength oscillation as l decreases. It is clear that both the inner-core size and the vertical diffusion have a substantial impact on the existence and structure of the midlevel tangential wind speed maximum. However, although the size variation in these experiments spans a realistic range, the mixing-length variation extends to physically unrealistic choices (l = 25 and 200 m). Therefore, real TCs with large RMWs are not expected to exhibit multiple wind maxima, because that type of structure would require unrealistically weak turbulence.

Fig. 12.
Fig. 12.

As in Fig. 11, but for simulations with the R180A50 initial vortex. Note that the radial range shown here is different from in Fig. 11.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

b. Sensitivity of eyewall structure to surface friction

In addition to the influence of boundary layer turbulence, the distribution and (especially) the magnitude of agradient flow is also strongly modulated by the parameterization of surface friction (Bryan 2012). To examine the potential influence of surface friction on the structure of the tangential wind field, we performed 9 additional simulations, with differing constant values of Cd (note that all other simulations in this study use the standard wind speed–dependent formulation of Cd described in section 3). For these experiments, we took the R36A50 vortex (for each of l = 25, 50, and 100 m), and set Cd to be 1.2 × 10−3, 2.4 × 10−3, or 4.8 × 10−3, while adjusting Ck such that the ratio Ck/Cd = 0.5. Figure 13 shows the tangential and agradient winds for the simulations with l = 50 m and differing Cd, compared at times when the maximum gradient wind is 75 m s−1. The magnitude of agradient winds and the amplitude of the oscillation (as indicated by the contour of absolute angular momentum) increase with increasing Cd, which results in an increasingly prominent midlevel maximum of tangential wind speed (the sensitivity to Cd is qualitatively similar for the other values of l, not shown). It is also evident that both the height of the midlevel maximum as well as the wavelength of the oscillation increase with increasing Cd. Although surface friction is the ultimate driver of the inertial oscillation, a comparison of Fig. 13 to Figs. 11 and 12 indicates that variations in l have a relatively larger effect on the wavelength of the oscillation than do variations of Cd (and much of the sensitivity to Cd appears to be related to the dependence of diffusivity on Cd, not shown). Also note that Cd is better constrained observationally than is l, and in our experiments, we have varied Cd substantially further outside of the realistic range. Therefore, we believe that uncertainties in the parameterization of boundary layer turbulence are more relevant to the structure of the eyewall wind field than are uncertainties in the parameterization of surface friction.

Fig. 13.
Fig. 13.

Azimuthal-mean (a),(c),(e) tangential wind speed and (b),(d),(f) agradient wind for simulations with l = 50 m and constant Cd equal to (top) 1.2 × 10−3, (middle) 2.4 × 10−3, and (bottom) 4.8 × 10−3. These simulations are compared at different times, but for the same value of maximum gradient wind speed (75 m s−1). Note that for all simulations shown in previous figures, Cd = 2.4 × 10−3 for 10-m wind speed above 25 m s−1.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

6. Why is there an oscillation in agradient flow?

a. Insights from a boundary layer model

Based on both our numerical simulations and the observations presented here and in Stern et al. (2014), it seems apparent that midlevel tangential wind speed maxima can occur in real TCs, and that this structure ultimately results from the response of a Rankine-like vortex to surface friction. We can demonstrate this more conclusively through the use of the boundary layer model of Kepert (2017). This model, which is diagnostic, dry, and nonlinear, is similar to the model originally developed in Kepert and Wang (2001), but is axisymmetric. A radial profile of pressure that is fixed in time is imposed, and the model is integrated until a steady-state wind field is achieved. Except as noted below, the model configuration we use is identical to that described in Kepert (2017), and so we refer the reader to that study for a detailed description of the model. For our calculations with the boundary layer model, we use the same wind speed–dependent drag coefficient and parameterization of vertical turbulence processes as in the corresponding CM1 simulations.6 Note that we use a model top of 7 km in the boundary layer model, deeper than that used in Kepert (2017), so as to be able to simulate the midlevel structure. Also, to be consistent with our CM1 simulations, we use the formulation of Bryan et al. (2017b) for the length scale {lυ2=l2+[κ(z+z0)]2}, instead of the formulation of Blackadar (1962) {[lυ1=l1+(κz)1]} used in Kepert (2017). These different formulations result in minor, but noticeable differences in the heights of wind maxima and the wavelength of the oscillation (not shown).

Figure 14a shows the radius–height distribution of gradient wind speed for the R36A50 simulation with l = 25 m, averaged from t = 62–74 h. We take an average of Vg over the lowest 2 km to define a barotropic pressure profile to force the boundary layer model, and the radius–height distribution of the associated gradient wind field is shown in Fig. 14b.7 The boundary layer model is then integrated forward in time for 48 h, by which time a steady state has been achieved.8 Figure 14c shows the tangential wind speed for the Kepert model simulation, and Fig. 14d shows the corresponding tangential wind speed for the CM1 simulation. Note that for this analysis (Fig. 14c), we are adding the agradient flow calculated by the Kepert model (Fig. 14e) to the actual gradient wind structure from the CM1 simulation (Fig. 14a). The Kepert model compares reasonably well with the CM1 simulation, with respect to the vertical structure of the tangential wind. In particular, there are multiple maxima in tangential wind speed, and with similar magnitudes as compared to the CM1 simulation. Figures 14e and 14f show the agradient wind for the Kepert model and CM1 simulation, respectively, and a qualitatively similar oscillation is seen in both. Recall that the only information that the Kepert model has about the CM1 simulation is the radial profile of Vg (or equivalently, the pressure), and the parameterization of surface drag and turbulence processes. That the Kepert model can approximately reproduce the structure from the full-physics simulation demonstrates that the oscillation in agradient flow in the simulation is fundamentally a consequence of the response to surface friction of an otherwise balanced vortex.

Fig. 14.
Fig. 14.

For the R36A50 l = 25-m simulation: (a) the gradient wind speed averaged over t = 62–74 h. (b) The imposed gradient wind field used in the Kepert model, taken from the z = 0–2-km layer mean from the CM1 simulation shown in (a). Tangential wind speed in (c) the Kepert model and (d) the actual tangential wind speed in the corresponding CM1 simulation. The agradient wind speed in (e) the Kepert model and (f) the actual agradient wind speed in the corresponding CM1 simulation. The contour interval is 2 m s−1 in all panels, every 20 m s−1 is thickened in (a)–(d), and the zero contour is thickened in (e) and (f). To produce (c), the actual height-varying gradient wind from CM1 in (a) is added to the agradient wind simulated by the Kepert model in (e). Note that in (d) and (f), the CM1 fields have been interpolated onto the Kepert model grid.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

There are two notable differences between the results from the Kepert model and the CM1 simulation: the wavelength of the oscillation is somewhat shorter and the amplitude decays more rapidly with height in the Kepert model. As a result of the shortened wavelength, the secondary tangential wind speed maximum in the Kepert model is too low, and there is a third maximum in the Kepert model that is absent in the CM1 simulation. We investigated whether the use of a barotropic gradient wind field influences the oscillation, by performing a Kepert model simulation using the actual height-varying gradient wind field from the CM1 simulation instead of the barotropic field, and results from this simulation are shown in Figs. 15a and 15b. As seen by comparing Fig. 15a to Fig. 14e (agradient winds) and Fig. 15b to Fig. 14c (tangential winds), the agradient winds and resulting tangential winds change very little in the Kepert model when using a baroclinic vortex instead of a barotropic vortex. The amplitude of the oscillation weakens slightly and it decays with height slightly faster with the baroclinic vortex, but these differences are nearly imperceptible when looking at the tangential wind structure. Therefore, we can conclude that the oscillation is relatively insensitive to the vertical structure of the balanced vortex, and that the use of a barotropic vortex cannot explain the differences between the Kepert model and the CM1 simulation.

Fig. 15.
Fig. 15.

As in Figs. 14c and 14e, but for (a),(b) a Kepert model simulation where the imposed gradient wind varies with height (taken from the CM1 simulation) and (c),(d) a Kepert model simulation where the vertical velocity is artificially scaled to agree with the low-level eyewall updraft strength in the respective CM1 simulation. Consistent with Fig. 14, the tangential winds in (b) and (d) are obtained by adding the gradient wind from the CM1 simulation (Fig. 14a) to the agradient wind from the respective Kepert model simulation in (a) and (c).

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Another factor that may influence the structure of the oscillation is the strength of the vertical advection. The Kepert model is forced by surface friction alone, and so the vertical velocity is underestimated compared to the CM1 simulation because the portion that is a response to diabatic heating is missing. Though the linear theory of Kepert (2001) neglects vertical advection, based on this theory, Kepert (2002) and Kepert (2006b) showed that within an updraft, the wavelength of the oscillation increases and the vertical decay rate decreases with increasing vertical velocity. Therefore, it should be expected that the (nonlinear) Kepert model will underestimate the oscillation wavelength (and overestimate the decay rate), because of the neglect of the heating-forced component of the eyewall updraft. Experiments doubling and halving the vertical advection within the Kepert model confirm this sensitivity (not shown). Figures 15c and 15d show an experiment where we scaled the vertical velocity in the Kepert model to agree with the peak azimuthal-mean low-level updraft from the CM1 simulation, and it can be seen that the structure of the oscillation in the Kepert model is brought into better agreement with CM1. In particular, the wavelength of the oscillation increases and the decay of the amplitude with height decreases, and the net effect is an increase in the heights of the second and third tangential wind speed maxima. It therefore seems that the neglect of the heating-induced component of the eyewall updraft largely explains the differences between the Kepert model and the CM1 simulation.9

To demonstrate that the Kepert model can predict the correct sensitivity of eyewall wind structure to vertical diffusivity, Figs. 16 and 17 respectively compare the agradient and tangential wind speeds from the Kepert model to CM1 for the R36A50 simulations with varying l. The sensitivity to diffusivity is similar for the corresponding set of R180A50 simulations (not shown). Note that as in Fig. 14c, for a fair comparison to the original CM1 simulations, in Fig. 17 we are taking the agradient wind speed predicted by the Kepert model and adding it to the actual height-varying gradient wind speed from the CM1 simulation. It can be seen that the overall sensitivity of tangential wind structure to diffusivity is quite similar between the Kepert model and the full-physics CM1 simulations, despite the height of the secondary wind maximum being too low in the Kepert model (a consequence of the neglect of the heating-forced updraft). Additionally, the wavelength and amplitude of the oscillations increase with storm size in the Kepert model (not shown), which is consistent with the sensitivity to size in the CM1 simulations. These results further confirm that the cause of the atypical structure of tangential wind speed seen in some CM1 simulations is the response to surface friction, and that this structure is systematically sensitive to both the size of the eyewall and the magnitude of the vertical diffusivity.

Fig. 16.
Fig. 16.

For the R36A50 simulations, agradient wind speed in (a),(c),(e),(g) the Kepert model and (b),(d),(f),(h) the respective CM1 simulation. Each row shows simulations with a different value of l, which is given in the upper right of each panel, and increases from (top) 25 to (bottom) 200 m. Note that the time periods shown for each simulation are different, but chosen so that the maximum gradient wind speeds are similar.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

Fig. 17.
Fig. 17.

As in Fig. 16, but for tangential wind speed.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

b. Insights from the linear theory of Kepert (2001)

That the wavelength of the oscillation increases systematically with increasing vertical diffusivity is consistent with the theoretical argument of Rotunno and Bryan (2012), who suggested that for a vortex in solid-body rotation, the oscillation wavelength should be proportional to the square root of the ratio of diffusivity to the angular velocity. More generally, Kepert (2001) found in his linear analytic theory that the depth of the symmetric component of the tropical cyclone boundary layer scales with 2Kυ/I, where Kυ is the vertical diffusivity and I2 is the inertial stability, defined as I2 = (f + 2V/r)(f + V/r + ∂υ/∂r).10 Kepert and Wang (2001) showed that this theoretical scaling agreed well with the height of maximum tangential wind speed in their nonlinear boundary layer model, despite the fact that the linear theory greatly underestimates the strength of the supergradient jet in the eyewall (due to the neglect of vertical advection). All else being equal, the maximum value of Kυ increases with increasing l in the Louis (1979) boundary layer parameterization, and in our simulations peak Kυ increases approximately linearly with l (Fig. 18a).11 As shown in Figs. 11, 12, 16, and 17, the wavelength of the oscillation does increase with increasing l, and so this is consistent with the theory of Kepert (2001).

Fig. 18.
Fig. 18.

For the R36A50 simulations, scatterplots of (a) the peak vertical diffusivity and (b) peak low-level supergradient wind speed vs the peak gradient wind speed. Each data point corresponds to an individual hourly snapshot, and only times when the peak gradient wind speed exceeds 30 m s−1 are shown. The mixing length of each simulation is indicated in the legend.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

We can quantitatively assess the theories of Kepert (2001) and Rotunno and Bryan (2012) by plotting the theoretical oscillation wavelength (given by 2π times the depth scale 2Kυ/I) against the actual oscillation wavelength (given by the difference in heights of the low- and midlevel tangential wind speed maxima). As shown in Fig. 19, there is a strong linear relationship between the theoretical and simulated wavelengths, although the linear theory systematically underestimates both the wavelength of the oscillation (cf. the best fit to the 1:1 line) and the height of the boundary layer tangential wind speed maximum (not shown). Using the modified depth scale of Kepert (2002, 2006b) to roughly account for vertical advection improves the predictions of the theory, though some degree of underestimate remains (not shown). Nevertheless, it appears that the oscillation wavelength in simulated TCs does indeed scale in proportion to the ratio of the square root of the diffusivity to the inertial frequency, in accordance with the predictions of the linear theory.

Fig. 19.
Fig. 19.

For the set of simulations with constant Cd, the oscillation wavelength (given by the difference in heights between the low- and midlevel tangential wind speed maxima) vs the theoretical wavelength 2π2Kυ/I. Each data point corresponds to an individual hourly snapshot, and only times when the peak gradient wind speed exceeds 30 m s−1 (and a midlevel tangential wind speed maximum exists) are shown. The theoretical wavelength is evaluated at the radius of maximum gradient winds, with variables averaged over the lowest 1 km. The best-fit line is in magenta (with equation and variance explained in the box), and the 1:1 line is in black.

Citation: Journal of the Atmospheric Sciences 77, 5; 10.1175/JAS-D-19-0191.1

These theories clearly explain why the oscillation wavelength increases with l. To understand why the oscillation wavelength also increases with TC size, note that at the RMW, I2V/r (neglecting the Coriolis term, which is a reasonable approximation in the inner core of an intense cyclone), and so the oscillation wavelength should increase in proportion to RMW, all else being equal. Finally, it might at first be expected that there should also be a strong sensitivity of the oscillation wavelength to the maximum wind speed, as inertial frequency in the eyewall will tend to increase in proportion to wind speed. However, such a sensitivity of oscillation wavelength to TC intensity is not seen in our simulations. This is because the peak diffusivity increases nearly linearly with increasing maximum wind speed in our simulations (Fig. 18a), and so the effect of intensity on inertial frequency I is approximately cancelled by its effect on diffusivity Kυ. Therefore, the depth scale from the linear theory (2Kυ/I) remains approximately constant with intensity, and so does the oscillation wavelength in our simulations. Note that although the wavelength does not change much, the amplitude of the oscillation increases strongly with intensity (Fig. 18b), and this is why there is a relationship between intensity and the vertical shear of the tangential wind speed. Ultimately, the increase of agradient flow with increasing gradient wind speed explains why the midlevel wind maximum is only seen in strong hurricanes, both in our simulations and in observations.

7. Summary and conclusions

Although in most tropical cyclones, the eyewall wind speed decreases monotonically with height above the boundary layer, some small and intense hurricanes have been observed to exhibit an additional midlevel local maximum in wind speed. Our goals in this study were to present new evidence of this atypical wind structure, to gain insight into how this phenomenon occurs, and to explain why only certain storms exhibit multiple maxima of wind speed within the eyewall.

Using dropsondes and Doppler radar analyses, we showed that Hurricane Patricia (2015) exhibited an absolute maximum in azimuthal-mean tangential wind speed at approximately 6-km height, along with a weaker maximum within the boundary layer. Patricia was both extremely strong and extremely small, which is consistent with the fact that all previous hurricanes with multiple wind speed maxima were either very intense (Rita, Gloria), very small (Dennis), or both (Felix). Based on these previous storms along with idealized simulations, the studies of SN11 and Stern et al. (2014) suggested that the midlevel wind speed maximum may be a consequence of unbalanced flow. In this study, we explored this hypothesis more thoroughly and systematically.

We were able to approximately reproduce several of the key characteristics of Patricia using an idealized three-dimensional simulation. In particular, our idealized storm rapidly intensified and contracted to become a very small category 5 hurricane, on approximately the same time scale as did Patricia. At nearly all times after the peak 10-m wind speed exceeded 60 m s−1 both a boundary layer and a midlevel tangential wind speed maximum were seen in the simulation, and this is also consistent with Patricia. Finally, the simulated midlevel tangential wind speed maximum was found near the top of an elevated inflow layer, and we showed that this characteristic was also seen in Patricia.

To explore the influence of inner-core size on the existence of the midlevel wind speed maximum, we analyzed a set of idealized simulations where we systematically varied the initial vortex structure. Focusing on two representative simulations, we showed that a storm with a small RMW exhibited a midlevel local tangential wind speed maximum, while a storm with a large RMW did not. In both simulations, the maximum gradient wind speed was nearly constant within the lowest 2 km and then decreased monotonically with height, and so differences in the balanced wind field are not directly responsible for the existence of the midlevel tangential wind speed maximum.

All tropical cyclones are characterized by some degree of unbalanced flow (e.g., Montgomery and Smith 2017), which results from accelerations forced by surface friction, turbulence, and diabatic heating. Consistently, both our small and large simulated TCs exhibited subgradient flow throughout the boundary layer outside of the eyewall, and a supergradient jet in the upper boundary layer within the eyewall. It is well understood (Kepert 2001) that this supergradient jet is the reason for the maximum of tangential wind speed within the boundary layer that is characteristic of nearly all observed hurricanes (Franklin et al. 2003; Zhang et al. 2011b). It is perhaps not as widely appreciated that this jet is part of an oscillation of the unbalanced flow, which is manifested as alternating layers of subgradient and supergradient flow within the eyewall (Kuo 1971; Bryan and Rotunno 2009b; SN11; Rotunno and Bryan 2012; Persing et al. 2013; Montgomery and Smith 2017). It is the increase of agradient wind speed with height between peaks of subgradient and supergradient flow that allows for the midlevel maximum in tangential wind speed in the small TC. Though the oscillation is also present in the large TC, its vertical wavelength is longer, and so the shear of the agradient wind speed is weaker, and is unable to overcome the decrease of the gradient wind speed with height. It is primarily for this reason that the large TC does not have a midlevel maximum in tangential wind speed.

From our suite of simulations, we showed that the above differences in structure are a systematic effect of inner-core size; a midlevel maximum in tangential wind speed is only found when the RMW is smaller than approximately 20 km, and this is consistent with all of the observed cases with such a maximum. In our simulations, there is also a dependence on wind speed, and the midlevel maximum is usually absent when the peak azimuthal-mean tangential wind speed is less than 50 m s−1. It is also apparent that the stronger the peak wind speed, the larger the RMW can be while still exhibiting a midlevel local tangential wind speed maximum, and this too is consistent with the observed cases. Note that although midlevel wind speed maxima are apparently rare in real TCs, the oscillation in agradient flow within the hurricane eyewall is likely ubiquitous, but not easily observed.

The fundamental cause of the oscillation in both radial wind speed and agradient flow within the eyewall is surface friction, which we demonstrated using the diagnostic boundary layer model of Kepert (2017). Knowing only the pressure field, the Kepert model was able to qualitatively reproduce the structure of the eyewall wind field from the respective CM1 simulations. In the full-physics CM1 simulations, the wavelength of the oscillation increased and the amplitude decreased as the chosen vertical mixing length increased (i.e., increased turbulent diffusivity), and this effect was reproduced with the Kepert model. For a large mixing length (l = 200 m), the midlevel tangential wind speed maximum is eliminated for the TC with the small RMW, and for a small mixing length (l = 25 m), a midlevel maximum is produced for the TC with the large RMW. These extreme cases appear to be unrealistic, and indeed, this is consistent with the body of literature that suggests that the vertical mixing length should be in the range of 50–100 m (Zhang et al. 2011a; Bryan 2012).

The linear analytical theory of Kepert predicts an oscillatory solution for the radial and agradient winds of an otherwise balanced tropical cyclone in response to surface friction. The theory predicts that the vertical wavelength of the oscillation is 2π2Kυ/I, and although this underestimates the actual oscillation wavelength for our CM1 simulations, the wavelengths do indeed scale in approximate accordance with the theory. Specifically, doubling either the vertical mixing length or the RMW results in an increase of the oscillation wavelength by a factor of approximately 2. These sensitivities occur because the peak diffusivity tends to increase linearly with the mixing length, and because the inertial frequency is approximately inversely proportional to radius. Although inertial frequency at the RMW increases approximately linearly with tangential wind speed, the oscillation wavelength is relatively insensitive to simulated TC intensity, because the peak diffusivity also tends to increase approximately linearly with tangential wind speed, and so these two effects largely cancel each other. Therefore, although the amplitude of the oscillation increases with TC intensity, the wavelength is determined by the size of the RMW and the vertical mixing length. When the oscillation amplitude is large enough and/or the wavelength is small enough, a midlevel maximum in tangential wind speed occurs.

Beyond simply providing an explanation for the observed phenomenon of the midlevel wind speed maximum, the systematic modulation of the eyewall inertial oscillation by storm size and intensity may have broader implications for our overall understanding of tropical cyclone structure. For example, it is sometimes assumed that aircraft reconnaissance occurs above the region of significant agradient flow and so the flight-level wind speed can be equated to the gradient wind speed (e.g., Knaff et al. 2011). While this may be a reasonable approximation most of the time, our results indicate that the inertial oscillation can sometimes have substantial amplitude well above the boundary layer, and so it is possible for the flight-level winds to be subgradient (e.g., Fig. 16d) or supergradient (regardless of whether or not there is a midlevel wind speed maximum).

In turn, the implicit assumption of gradient wind balance at flight level could potentially affect the estimation of surface wind speeds from flight-level winds using standard reduction factors for such cases. In several intense hurricanes, NHC forecast discussions and postseason reports have noted that the peak surface wind speed estimated by the stepped-frequency microwave radiometer (SFMR) exceeded the peak measured flight-level wind speed, and they concluded that the SFMR estimates in intense storms may therefore be biased high. For example, in Hurricane Dorian (2019), the 2300 EDT 31 August discussion12 stated “Both aircraft measured peak flight-level winds that support an initial intensity of 130 kt. There have been some higher surface wind estimates from the SFMR, but these data are questionable based on our experience of very high SFMR-measured wind speeds in recent strong hurricanes that didn’t match standard flight-level wind reductions.” While not discounting the possibility that such SFMR estimates are indeed biased high, there is no physical reason that the surface wind speed must be less than the flight-level wind speed. The typical 90% reduction expected between flight level and the surface is based on an empirically derived average from dropsondes (Franklin et al. 2003), and the actual ratio of peak wind speeds can vary both randomly and systematically (Powell et al. 2009). Given the possibility that an aircraft may be sampling at a level of subgradient flow, it is at least plausible that the peak surface wind speed can exceed the peak flight-level wind speed, and so we therefore recommend caution in concluding that the deviations from expected relationships seen in some intense hurricanes are indicative of biases in the SFMR estimates.

Our findings also have implications for boundary layer parameterizations used in numerical weather prediction (NWP) models. The simulations with intermediate vertical mixing lengths (l = 50 m and l = 100 m) yielded eyewall wind structures that are consistent with observed TCs, whereas the other settings (l = 25 m and l = 200 m) appeared to result in unrealistic structures. Though operational NWP models generally use more sophisticated boundary layer parameterizations than the simple mixing-length scheme in CM1, the same sensitivity of tangential wind structure to vertical diffusivity should hold, and so it may be possible to diagnose some model biases by examining the vertical structure of the eyewall tangential winds.

In closing, we note that there is one key aspect of the structure of Hurricane Patricia that is not reproduced in any of our idealized simulations. The absolute maximum tangential wind speed in Patricia occurred at 6-km height,13 whereas for all of our simulations with a relative midlevel maximum, the strongest speeds were still found within the boundary layer. The theory of Kepert also predicts that the amplitude of the oscillation of unbalanced flow should decay with increasing height, which would mean that the boundary layer should always contain the absolute maximum tangential wind speed. We speculate that the reason that Patricia deviates from simulations and theory in this respect is that Patricia was observed to be rapidly weakening, in association with an eyewall replacement cycle. As can be seen from the radial velocity fields, the boundary layer inflow to Patricia’s inner eyewall substantially weakened during the period in which it was observed. It is this inflow that maintains the primary circulation (through angular momentum advection) against frictional spindown, and so when the inflow is suppressed, rapid spindown can occur. We hypothesize that because the spindown occurs first within the boundary layer, the boundary layer tangential wind jet in Patricia weakened more rapidly than did the midlevel jet, which allowed the absolute maximum to occur at midlevels.

Acknowledgments

This research was supported by the Chief of Naval Research through the NRL Base Program (PE 0601153N), as well as the Office of Naval Research TCI Departmental Research Initiative (PE 0601153N). We also acknowledge and thank the entire TCI team including the YES, Inc., instrument team, the NASA WB-57 research flight team, and the dropsonde quality control effort led by Michael Bell. We thank NOAA/HRD for making available the Doppler radar analyses. George Bryan was supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977. We thank Pete Finocchio and Kevin Tory for providing helpful comments that improved this manuscript, and we thank Mike Montgomery and two anonymous reviewers for their reviews.

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1

As pointed out by a reviewer (M. T. Montgomery), nonlinear depth-averaged (or “slab”) boundary layer models are able to produce strongly agradient flow in the absence (by definition) of vertical advection (e.g., Shapiro 1983; Smith and Vogl 2008). One might therefore be inclined to conclude that radial advection of radial momentum alone is the essential process resulting in such tangential wind speed jets. However, as demonstrated by Kepert (2010a,b) and Williams (2015), slab models have an intrinsic tendency to substantially overestimate the strength of the boundary layer inflow and agradient flow, and this necessarily means that the influence of radial advection on the strength of the agradient winds is overstated in these models.

2

Although the centrifugal force is dominant within the hurricane eyewall, the Coriolis force still plays a role, and so we choose to use the more general (and more widely used) term “inertial oscillation.” We view this as interchangeable with “centrifugal wave” in the context of this study, and so the explanations of SN11 and Persing et al. (2013) are essentially equivalent.

3

Prior to the formation of a well-defined eyewall, a single midlevel maximum or dual maxima in wind speed can occur in these simulations (e.g., Fig. 6d prior to t = 18 h), but this is distinct from the phenomenon we are examining in this study (and may be an artifact of the initial convective adjustment of the imposed vortex structure), and so we exclude these times from most of our analyses.

4

For all of these simulations described in sections 4 and 5, we used v19.1 of CM1. For the Patricia-like simulation described in section 3, we used v19.4.

5

Bryan and Rotunno (2009c) and some other earlier studies used a vertical mixing length lυ that was constant with height. As described in Kepert (2012) and Bryan (2012), an asymptotic form (with lυ decreasing to zero at the surface) is more appropriate, and this has been the default in CM1 beginning with version 16.

6

In the boundary layer model, static stability effects are neglected in the parameterization of turbulence, whereas these are included in the CM1 simulations. As shown in Kepert (2012), this choice has only a small effect.

7

To avoid instability that can occur in the boundary layer model when it is forced by an intense vortex profile, we apply a low-pass filter as well as impose a floor on the minimum value of vorticity. The resulting profile of gradient wind is very similar to the original CM1 output, although the maximum speed is slightly reduced.

8

A steady state is actually reached well prior to 48 h, and the wind field changes very little after about 12 h.

9

A reviewer (M. T. Montgomery) suggested that the Kepert model imposes gradient wind balance at the top of the domain (Smith and Montgomery 2010), and that this could be a cause of the differences between the Kepert model solution and the CM1 simulation. As demonstrated conclusively in Kepert (2017), the boundary layer model in no way imposes gradient wind balance as an upper boundary condition, and the model can and does produce substantial agradient flow at the upper boundary when this boundary is located where the flow is unbalanced (their Fig. A1). Therefore, the model is not limited in this respect, and this supposed deficiency is not a cause of the differences between the Kepert model and CM1.

10

The length scale of Kepert (2001) is actually equivalent to that of Rotunno and Bryan (2012) when evaluated at the RMW and neglecting f.

11

Equation (17) of Bryan and Rotunno (2009c) indicates that the peak value of Kυ in CM1 increases with l2, all else being equal. The approximately linear dependence that we see instead implies that the vertical shear decreases approximately linearly with l (for a given wind speed), as a compensating response to the increased diffusivity.

13

Although it is possible that the boundary layer maximum is underestimated due to limitations in the Doppler analysis, the higher vertical resolution (150 m) of the “profile” analyses examined for Patricia (compared to the 500-m-resolution “swath” analyses shown for Dennis, Rita, and Felix) along with the corroboration from the dropsonde analysis lends confidence to the idea that the midlevel maximum is truly stronger than the low-level maximum in this case.

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