understanding atypical midlevel wind speed maxima in

Understanding Atypical Midlevel Wind Speed Maxima in Hurricane Eyewalls

DANIEL P. STERN

University Corporation for Atmospheric Research, Monterey, California

JEFFREY D. KEPERT

Centre for Australian Weather and Climate Research, Melbourne, Victoria, Australia

GEORGE H. BRYAN

National Center for Atmospheric Research, Boulder, Colorado

JAMES D. DOYLE

Naval Research Laboratory, Monterey, California

(Manuscript received 18 July 2019, in final form 11 February 2020)

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

speedmaximum at 5–6-km height, alongwith a weaker boundary layermaximum. 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 . 50m s21) 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.

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 maxi-

mum wind level of horizontal winds in hurricanes,’’ and

stated that observations suggested ‘‘that a hurricane’s

maximum winds are usually found between 500 and

2000m.’’ 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 sug-

gested. More recently, Zhang et al. (2011b) used drop-

sonde composites to show that the peak tangential wind

speed occurs within the inflow layer, at a height (700m)

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:Corresponding author: Daniel P. Stern, [email protected]

MAY 2020 S TERN ET AL . 1531

DOI: 10.1175/JAS-D-19-0191.1

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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 1–2km. Extending the pioneering work

of Rosenthal (1962) and Eliassen and Lystad (1977),

Kepert (2001) developed a three-dimensional linear

analytical boundary layermodel, 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 non-

linear 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 d 5 (2K/I)1/2, in-

creasing with (vertical) turbulent diffusivity K and de-

creasing with inertial stability I2. This depth scale for the

boundary layer within a vortex, first derived byRosenthal

(1962), is a modification of the classical Ekman depth

dE5 (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 re-

duces 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 andWang (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 ra-

dial profile of tangential wind speed, the jet is strongly

supergradient. Bell and Montgomery (2008) used ob-

jective 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 ab-

solutemaximumat 550–600hPa. 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 dif-

ferent 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 cur-

rent 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 64m s21. Stern

et al. (2014) identified three obvious outlier cases within

their dataset, where the decay rate of maximum tan-

gential 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 5km, and each exhibited a local maximum at

4–4.5 km. Note that the boundary layer winds (below

roughly 1km) are poorly resolved by these Doppler an-

alyses, due to a combination of sea clutter, sidelobe ef-

fects, limited vertical resolution (500m), and the inherent

smoothing of the analysis technique (Reasor et al. 2009;

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 ten-

dency 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.

1532 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77

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.

SN11 proposed that the midlevel wind speed maxi-

mum inDennis 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 sim-

ulations 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 hypoth-

esized 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 re-

sponsible 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 cir-

culation (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 condi-

tion, the boundary layer flow beneath a Rankine vortex

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 2m s21 (with every 20m s21 thickened), and theRMWis indicated inmagenta. In (d),

supergradient winds are contoured in solid white at12,14,16, and18m s21, and subgradient winds are contoured

in dashed black at 22, 24, 26, and 28m s21.


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 os-

cillations in both the secondary and primary circula-

tions, 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 cy-

clones, 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 axisym-

metric simulations,’’ and there appears to be some evi-

dence 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 simu-

lations, Bryan and Rotunno (2009c) and Rotunno and

Bryan (2012) showed that the structure of the tangen-

tial and radial wind fields is very sensitive to the verti-

cal mixing length ly in the turbulence parameterization.

Figure 5c ofBryan andRotunno (2009c) shows alternating

inflow and outflow jets as well as a midlevel tangential

wind speed maximum, for a simulation with a constant

ly 5 50m. 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 simula-

tion 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 ly 5 50m.

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 ex-

plore the dynamical mechanism that leads to a sec-

ondary maximum in tangential wind speed in the

midlevel eyewall of tropical cyclones, and to under-

stand the reasons why such a structure is rarely ob-

served. 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 maxi-

mum. We then use Patricia as a motivating case, and in

section 3, we present an idealized simulation that qual-

itatively 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 ver-

tical 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 funda-

mentally 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 themost 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) ex-

periment (Doyle et al. 2017), with high-altitude drop-

sondes released during four flights by the NASAWB-57

aircraft. The NOAA P3 aircraft also flew through

Patricia at similar times as did theWB-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 trop-

ical wave and other disturbances, and a tropical de-

pression formed by 0600UTC 20October, about 330km

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

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 oscilla-

tion.’’ 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.


storm (1kt ’ 0.51m s21) by a P3 flight during the af-

ternoon of 21 October. The environment of Patricia was

extremely favorable for rapid intensification (RI), with

SSTs exceeding 308C 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 185kt 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 ver-

tical 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 5 2km

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 drop-

sondes 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 in-

terpolate 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 themean 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 82ms21 is seen

at about 6-kmheight, with aweaker localmaximumwithin

the boundary layer. Although there is one failed sonde just

outwards of the peak winds, the next seven sondes (mov-

ing outwards) in the SE portion of the cross section all

exhibit evidence of this midlevel maximum as well.

The dropsondes are advected by the wind, and so they

may drift substantially (10–30km) 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 representative-

ness and can make it challenging to interpret the struc-

ture of the wind field from dropsondes. Fortunately, for

the Patricia case, we have Doppler radar analyses

(provided by NOAA/HRD) from nearby times for di-

rect comparison. The P3 made a pass through the center

fromNWto SE at 1733UTC, and a pass fromSE toNWat

2033UTC, about 30min after theWB-57 overflew Patricia

at nearly the same azimuth. Figures 3b and 3c show the

distance–heightDoppler analyses ofwind 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 1.5km and 150m, 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 mid-

level maximum is present both to the SE and to the NW.

As themidlevelmaximum 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.

FIG. 2. Horizontal wind speed at 2-km height for Hurricane

Patricia, composited from two WP-3D Doppler analyses respec-

tively 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 byNOAA/HRD.

Note that this figure also appears in Doyle et al. (2017).


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 3h, the wind speed throughout

the depth of the inner eyewall decreased by 15–25ms21,

consistent with the 50-kt decrease in best track intensity

from 1800 to 2300 UTC (the time of landfall). The in-

crease 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 in-

crease 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 ob-

served by Franklin et al. (1993), and seen in the simu-

lation of SN11 (Fig. 1d).

The azimuthal-mean radial wind speed from the

Doppler analyses is shown in Figs. 4c and 4d, overlaid

FIG. 3. Distance–height cross sections of horizontal wind speed forHurricane Patricia on 23Oct 2015, from (a)WB-57

HDSS dropsondes centered at 2001 UTC and (b),(c) WP-3D Doppler analyses centered at 1733 and 2033 UTC, re-

spectively. (d)The change inwind speedover the 3 hbetween the twoDoppler analyses. Themean radial locationof 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 5m s21 in all panels, with every 20m s21 thickened in (a)–(c), and the zero contour thickened in (d). In (d), the

60 and 80m s21 contours from the 1733 UTC analysis are overlaid in magenta and white, respectively.


with select contours of the tangential wind speed. There

is a layered appearance to the radial wind within the eye-

wall; a shallow region of boundary layer inflow is toppedby

the typical low-level outflow jet, but there is another inflow

layer from 2 to 4km, and outflow once more above. Mean

inflow is not typically seen above the boundary layerwithin

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 re-

lationship between radial and tangential wind speed seen

for the boundary layer maximum. In the following sec-

tions, 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 approxi-

mately 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’’ simu-

lation, we use a domain of 2880km3 2880km3 25km,

horizontal grid spacing of 1 km in the central 320km 3320 km region (gradually stretching to 19km at the outer

portion of the domain), and 123 vertical levels, with

variable vertical grid spacing in the lowest 7km, and 250-m

grid spacing above. Formicrophysics, we use theMorrison

double-moment scheme (Morrison et al. 2009). We

use a relaxation term tomimic 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

1000m as surface pressure decreases from 1000 to

900 hPa, and a constant asymptotic vertical mixing

length l‘ 5 100m. We use a constant enthalpy ex-

change coefficient Ck 5 1.2 3 1023, and a wind speed–

dependent drag coefficient Cd that increases linearly

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 2m s21 in all panels,

with every 20m s21 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 75m s21.


from 1 3 1023 to 2.4 3 1023 for 10-m wind speed be-

tween 5 and 25m s21, and is constant above.

For this simulation, we set the SST to 30.58C 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 thermo-

dynamic 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 ide-

alized 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 azi-

muthal 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 25m s21, an RMWof 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 evo-

lution of simulated TCs is not particularly sensitive to

the initial vertical structure, and simulated TCs rap-

idly adjust to a realistic structure following the onset

of deep convection.

Figure 6a shows the time series of the maximum sur-

face (i.e., 10m) 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 min-

imum 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 10m s21 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).

Figure 7a shows a snapshot of the azimuthal-mean

tangential wind speed at t5 36h, 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 atmost times once the peak 10-mwind

speed exceeds about 60m s21, with the height of the

maximum varying from 3 to 6km (Fig. 6d).3 The radial

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 theDoppler ‘‘swath’’ and ‘‘profile’’

analyses and for the in situ flight-level measurements from theWP-

3D aircraft. Also shown is the profile for amodifiedRankine vortex

that is used to define the initial radial structure for the TC simu-

lation shown in Figs. 6 and 7.

3 Prior to the formation of a well-defined eyewall, a single mid-

level maximum or dual maxima in wind speed can occur in these

simulations (e.g., Fig. 6d prior to t 5 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.


wind speed at t5 36h 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 out-

flow layers within the eyewall, and the midlevel tan-

gential wind speedmaximum 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 simula-

tion found at somewhat larger radii, consistent with the

slightly larger RMW. Overall, this experiment dem-

onstrates that 1) we can reasonably reproduce the

structure of Hurricane Patricia in an idealized frame-

work, and 2) a midlevel tangential wind speed maxi-

mum 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.

4. The relationship between inner-core size andmidlevel 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 suchas Isabel (2003) and Ivan (2005), thewind

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 simulationsof intenseTCsof varying size.As shown

in SN11 and Stern et al. (2015), the quasi-steady-state size of

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 ex-

tended 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 5 3 to 45 h, to be consistent with the 2100 UTC 21 Oct

analysis time of Fig. 5.


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 180km) 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 2km and 59 vertical levels.4 Additionally, these

simulations all use an SST of 288C, and the Dunion

(2011) mean sounding to define the initial thermody-

namic 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 simu-

lations 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, theRMW(here defined as the radius of the

peak azimuthal tangential wind speed at any height) is

much smaller for R36A50 (12km) than for R180A50

(44km). 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

60m s21, 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).

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

speedVg for R36A50 (Fig. 8c) andR180A50 (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.

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 sub-

gradient jet, and the midlevel maximum in tangential

FIG. 7. For the Patricia-like simulation at t 5 36 h, azimuthal-

mean (a) tangential and (b) radial wind speed. The contour interval

is 2m s21 in both panels, with every 20m s21 thickened in (a), and

the zero contour thickened in (b). In (b), contours of tangential

wind speed are overlaid in blue for 60–85m s21, every 5m s21. In

both panels, the locations of the low-level and midlevel tangential

wind speedmaxima are indicated by white dots, and the location of

the local minimum is indicated by a white star.

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.


wind speed at 5.5 km coincides with a midlevel super-

gradient jet. However, the oscillation in Vag is clearly

present in both simulations, but yet the midlevel tan-

gential 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 ex-

istence 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 agra-

dient 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 inR36A50 (Fig. 8a). In contrast, inR180A50, 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 addi-

tional reason that there is no midlevel tangential wind

speed maximum in R180A50 is that the layer with in-

creasing Vag is at a greater height in R180A50 (itself

also a consequence of the greater oscillation wave-

length), 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

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 andmidlevel tangential wind speedmaxima (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 s21, with

every 20m s21 thickened. The 11m s21 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.


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 super-

gradient flow is found where the midlevel inflow is

transitioning to outflow. Therefore, the midlevel tan-

gential 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 maxi-

mum vertical shear of the tangential wind speed along

theRMW; in other words, the change in peak azimuthal-

mean tangential wind speed with height. This is evalu-

ated only above the boundary layer (and so excludes the

primary wind speed maximum), and only when the peak

tangential wind speed exceeds 30ms21, to ensure that

the eyewall and inner core is sufficiently well developed.

When the maximum shear is negative, there is (by def-

inition) 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 20km, 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 peakwind 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.

In Fig. 10a, the data points are colored by themaximum

azimuthal-mean tangential wind speed. The relationship

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.


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 sim-

ulated storms evolve to be extremely intense, but the

corresponding RMW for such intensities varies sub-

stantially (10–50km), 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 5 40–50m s21), but still

have midlevel maxima. Nevertheless, intensity does in-

fluence 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

50ms21. 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 80m s21. 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, andmodulated 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 theexistence 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 ex-

istence of a midlevel local maximum, is strongly influ-

enced 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 signifi-

cant modulator of the strength and distribution of the

agradient wind. Therefore, we expect that vertical dif-

fusion 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‘ 5 100m, which is believed to be

within the range of realistic settings (Zhang et al. 2011a;

Bryan 2012; Rotunno and Bryan 2012). Here, we eval-

uate the sensitivity to vertical diffusion, by varying l‘(25, 50, 100, 200m) 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‘ 5 200m (Fig. 11d), a

midlevelmaximum in tangentialwind speed is nonexistent.

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 de-

scribed in the text, and all times where Vmax exceeds 30m s21 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.


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.5Also 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 con-

tours 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‘ 5 25m is an un-

realistically small mixing length (Zhang et al. 2011a).

Figure 12 is similar to Fig. 11, but for the R180A50

initial vortex. Consistent with Fig. 8b, these large sim-

ulated 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‘ 5 25m,

Fig. 12a), a midlevel local maximum appears. For

the l‘ 5 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

FIG. 11. Azimuthal-mean tangential wind speed for l‘ 5 (a) 25, (b) 50, (c) 100, and (d) 200m. 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.

5 Bryan and Rotunno (2009c) and some other earlier studies

used a vertical mixing length ly that was constant with height. As

described in Kepert (2012) and Bryan (2012), an asymptotic form

(with ly decreasing to zero at the surface) is more appropriate, and

this has been the default in CM1 beginning with version 16.


(l‘ 5 25 and 200m). Therefore, real TCs with large

RMWs are not expected to exhibit multiple wind max-

ima, because that type of structure would require un-

realistically weak turbulence.

b. Sensitivity of eyewall structure to surface friction

In addition to the influence of boundary layer turbu-

lence, the distribution and (especially) the magnitude of

agradient flow is also strongly modulated by the pa-

rameterization of surface friction (Bryan 2012). To ex-

amine 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 de-

scribed in section 3). For these experiments, we took the

R36A50 vortex (for each of l‘ 5 25, 50, and 100m), and

set Cd to be 1.23 1023, 2.43 1023, or 4.83 1023, while

adjusting Ck such that the ratio Ck/Cd 5 0.5. Figure 13

shows the tangential and agradient winds for the simu-

lations with l‘ 5 50m and differing Cd, compared at

times when the maximum gradient wind is 75m s21. The

magnitude of agradient winds and the amplitude of the

oscillation (as indicated by the contour of absolute an-

gular momentum) increase with increasing Cd, which

results in an increasingly prominent midlevel maximum

of tangential wind speed (the sensitivity to Cd is quali-

tatively similar for the other values of l‘, not shown). It is

also evident that both the height of the midlevel maxi-

mum aswell as the wavelength of the oscillation increase

with increasing Cd. Although surface friction is the ul-

timate 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 toCd 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

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.


eyewall wind field than are uncertainties in the param-

eterization of surface friction.

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 axisym-

metric. A radial profile of pressure that is fixed in time is

imposed, and the model is integrated until a steady-state

FIG. 13. Azimuthal-mean (a),(c),(e) tangential wind speed and (b),(d),(f) agradient wind for simulations with

l‘ 5 50m and constant Cd equal to (top) 1.2 3 1023, (middle) 2.4 3 1023, and (bottom) 4.8 3 1023. These

simulations are compared at different times, but for the same value of maximum gradient wind speed (75m s21).

Note that for all simulations shown in previous figures, Cd 5 2.4 3 1023 for 10-m wind speed above 25m s21.


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 calcu-

lations with the boundary layer model, we use the same

wind speed–dependent drag coefficient and parameter-

ization of vertical turbulence processes as in the corre-

sponding 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 fl22y 5 l22

‘ 1 [k(z1 z0)]22g,

instead of the formulation of Blackadar (1962)

f[l21y 5 l21

‘ 1 (kz)21]g used in Kepert (2017). These dif-

ferent 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‘ 5 25m, averaged from t 5 62–74 h. We take an av-

erage 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 tan-

gential 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 oscilla-

tion in agradient flow in the simulation is fundamentally a

consequence of the response to surface friction of an

otherwise balanced vortex.

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

Kepertmodel is too low, and there is a thirdmaximum in

the Kepert model that is absent in the CM1 simulation.

We investigated whether the use of a barotropic gradi-

ent 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 re-

sulting tangential winds change very little in the Kepert

model when using a baroclinic vortex instead of a bar-

otropic vortex. The amplitude of the oscillation weakens

slightly and it decays with height slightly faster with the

baroclinic vortex, but these differences are nearly im-

perceptible when looking at the tangential wind struc-

ture. 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.

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 oscilla-

tion increases and the vertical decay rate decreases with

increasing vertical velocity. Therefore, it should be ex-

pected that the (nonlinear) Kepert model will underes-

timate the oscillation wavelength (and overestimate the

decay rate), because of the neglect of the heating-forced

component of the eyewall updraft. Experiments dou-

bling and halving the vertical advection within the

Kepert model confirm this sensitivity (not shown).

6 In the boundary layer model, static stability effects are ne-

glected 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.


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 wave-

length 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

FIG. 14. For the R36A50 l‘ 5 25-m simulation: (a) the gradient wind speed averaged over t 5 62–74 h.

(b) The imposed gradient wind field used in the Kepert model, taken from the z5 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 s21 in all

panels, every 20 m s21 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.


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 vary-

ing 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 dif-

fusivity 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 consis-

tent 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 fric-

tion, and that this structure is systematically sensitive

to both the size of the eyewall and the magnitude of the

vertical diffusivity.

FIG. 15. As in Figs. 14c and 14e, but for (a),(b) aKepertmodel simulationwhere 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).

9 A reviewer (M. T. Montgomery) suggested that the Kepert

model imposes gradient wind balance at the top of the domain

(Smith andMontgomery 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 sup-

posed deficiency is not a cause of the differences between the

Kepert model and CM1.


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) 200m. Note that the time

periods shown for each simulation are different, but chosen so that the maximum gradient wind speeds

are similar.


FIG. 17. As in Fig. 16, but for tangential wind speed.


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

That the wavelength of the oscillation increases sys-

tematically with increasing vertical diffusivity is consis-

tent 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 withffiffiffiffiffiffiffiffiffiffiffiffi

2Ky/Ip

, where Ky is the vertical diffu-

sivity and I2 is the inertial stability, defined as I2 5 ( f 12V/r)( f 1 V/r 1 ›y/›r).10 Kepert and Wang (2001)

showed that this theoretical scaling agreed well with the

height of maximum tangential wind speed in their non-

linear 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 Ky increases with increasing l‘ in the Louis

(1979) boundary layer parameterization, and in our

simulations peak Ky 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 in-

creasing l‘, and so this is consistent with the theory of

Kepert (2001).

We can quantitatively assess the theories of Kepert

(2001) and Rotunno and Bryan (2012) by plotting the

theoretical oscillation wavelength (given by 2p times

the depth scaleffiffiffiffiffiffiffiffiffiffiffiffi

2Ky/Ip

) 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, al-

though 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 tan-

gential 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.

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, I’ffiffiffi

2p

V/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 toffiffiffiffiffiffiffiffiffiffiffiffiffiffi

RMWp

, 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 pro-

portion to wind speed. However, such a sensitivity of

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 30m s21 are shown. The mixing length of each

simulation is indicated in the legend.

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 andRotunno (2009c) indicates that the

peak value ofKy 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.


oscillation wavelength to TC intensity is not seen in our

simulations. This is because the peak diffusivity in-

creases nearly linearly with increasing maximum wind

speed in our simulations (Fig. 18a), and so the effect of

intensity on inertial frequency I is approximately can-

celled by its effect on diffusivity Ky. Therefore, the

depth scale from the linear theory (ffiffiffiffiffiffiffiffiffiffiffiffi

2Ky/Ip

) 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 in-

tensity (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 hurri-

canes, 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 max-

ima 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 consis-

tent 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 ide-

alized storm rapidly intensified and contracted to

become a very small category 5 hurricane, on approxi-

mately the same time scale as did Patricia. At nearly all

times after the peak 10-m wind speed exceeded 60m s21

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 sys-

tematically varied the initial vortex structure. Focusing

on two representative simulations, we showed that a

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 2pffiffiffiffiffiffiffiffiffiffiffiffi

2Ky /Ip

. Each data point corresponds to an individual hourly

snapshot, and only times when the peak gradient wind speed exceeds 30m s21 (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.


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 de-

gree 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 sub-

gradient 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 ob-

served 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 be-

tween 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 ap-

proximately 20 km, and this is consistent with all of the

observed cases with such a maximum. In our simula-

tions, 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

50ms21. 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 maxi-

mum, 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 ra-

dial 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., in-

creased turbulent diffusivity), and this effect was re-

produced with the Kepert model. For a large mixing

length (l‘ 5 200m), the midlevel tangential wind speed

maximum is eliminated for the TCwith the small RMW,

and for a small mixing length (l‘ 5 25m), a midlevel

maximum is produced for the TC with the large RMW.

These extreme cases appear to be unrealistic, and in-

deed, this is consistent with the body of literature that

suggests that the vertical mixing length should be in the

range of 50–100m (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 2pffiffiffiffiffiffiffiffiffiffiffiffi

2Ky/Ip

, and although

this underestimates the actual oscillation wavelength for

our CM1 simulations, thewavelengths 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 approximatelyffiffiffi

2p

. These sensitivities occur

because the peak diffusivity tends to increase linearly

with the mixing length, and because the inertial fre-

quency is approximately inversely proportional to ra-

dius. Although inertial frequency at the RMW increases

approximately linearly with tangential wind speed, the

oscillation wavelength is relatively insensitive to simu-

lated 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 oscil-

lation 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 mid-

level maximum in tangential wind speed occurs.

Beyond simply providing an explanation for the ob-

served phenomenon of the midlevel wind speed maxi-

mum, 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 ap-

proximation 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 esti-

mation of surface wind speeds from flight-level winds

using standard reduction factors for such cases. In sev-

eral intense hurricanes, NHC forecast discussions and

postseason reports have noted that the peak surface

wind speed estimated by the stepped-frequency micro-

wave 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 air-

craft 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 hurri-

canes that didn’t match standard flight-level wind re-

ductions.’’ While not discounting the possibility that

such SFMR estimates are indeed biased high, there is no

physical reason that the surface wind speedmust be less

than the flight-level wind speed. The typical 90% re-

duction expected between flight level and the surface is

based on an empirically derived average from drop-

sondes (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‘ 5 50m and l‘ 5 100m)

yielded eyewall wind structures that are consistent with

observed TCs, whereas the other settings (l‘ 5 25m and

l‘ 5 200m) 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 diffu-

sivity should hold, and so it may be possible to diagnose

somemodel 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 maxi-

mum tangential wind speed in Patricia occurred at 6-km

height,13 whereas for all of our simulations with a rela-

tive 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 al-

ways contain the absolute maximum tangential wind

speed. We speculate that the reason that Patricia devi-

ates from simulations and theory in this respect is that

Patricia was observed to be rapidly weakening, in asso-

ciation 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 in-

flow that maintains the primary circulation (through

angular momentum advection) against frictional spin-

down, 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

12 https://www.nhc.noaa.gov/archive/2019/al05/al052019.discus.

031.shtml?.

13 Although it is possible that the boundary layer maximum is

underestimated due to limitations in the Doppler analysis, the

higher vertical resolution (150m) of the ‘‘profile’’ analyses exam-

ined for Patricia (compared to the 500-m-resolution ‘‘swath’’

analyses shown for Dennis, Rita, and Felix) along with the cor-

roboration from the dropsonde analysis lends confidence to the

idea that the midlevel maximum is truly stronger than the low-level

maximum in this case.


https://www.nhc.noaa.gov/archive/2019/al05/al052019.discus.031.shtml?

https://www.nhc.noaa.gov/archive/2019/al05/al052019.discus.031.shtml?

analyses. George Bryan was supported by the National

Center for Atmospheric Research, which is a major fa-

cility sponsored by the National Science Foundation

under Cooperative Agreement 1852977. We thank Pete

Finocchio and Kevin Tory for providing helpful com-

ments that improved this manuscript, and we thank

Mike Montgomery and two anonymous reviewers for

their reviews.

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