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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
� 2020 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS CopyrightPolicy (www.ametsoc.org/PUBSReuseLicenses).
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.
MAY 2020 S TERN ET AL . 1533
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.
1534 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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).
MAY 2020 S TERN ET AL . 1535
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.
1536 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1537
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.
1538 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1539
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.
1540 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1541
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.
1542 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1543
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.
1544 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
(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.
MAY 2020 S TERN ET AL . 1545
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.
1546 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1547
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.
1548 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1549
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.
1550 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
1552 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1553
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
1554 JOURNAL OF THE ATMOSPHER IC SC IENCES VOLUME 77
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.
MAY 2020 S TERN ET AL . 1555
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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