DSpace at VNU: An investigation into the contraction of the hurricane radius of maximum wind

Kieu Received: 1 July 2011 / Accepted: 11 November 2011 / Published online: 25 November 2011 Springer-Verlag 2011 Abstract The radius of the maximum tangential wind RMW associated with

O R I G I N A L P A P E R

An investigation into the contraction of the hurricane radius

of maximum wind

Chanh Q Kieu

Received: 1 July 2011 / Accepted: 11 November 2011 / Published online: 25 November 2011

 Springer-Verlag 2011

Abstract The radius of the maximum tangential wind

(RMW) associated with the hurricane primary circulation

has been long known to undergo continuous contraction

during the hurricane development In this study, we

doc-ument some characteristic behaviors of the RMW

con-traction in a series of ensemble real-time simulations of

Hurricane Katrina (2005) and in idealized experiments

using the Rotunno and Emanuel (Mon Weather Rev

137:1770–1789, 1987) axisymmetric hurricane model Of

specific interest is that the contraction appears to slow

down abruptly at the middle of the hurricane

intensifica-tion, and the RMW becomes nearly stationary

subse-quently, despite the rapidly strengthening rotational flows

A kinematic model is then presented to examine such

behaviors of the RMW in which necessary conditions for

the RMW to stop contracting are examined Further use of

the Emanuel’s (J Atmos Sci 43:585–605, 1986) analytical

hurricane theory reveals a connection between the

hurri-cane maximum potential intensity and the hurrihurri-cane eye

size, an issue that has not been considered adequately in

previous studies

1 Introduction

The hurricane radius of maximum wind (RMW), which is

defined as the radius at which the tangential flow attains

its maximum value, has been long known to undergo

continuous contraction during the development of hurri-canes (see, e.g., Hack and Schubert 1986; Willoughby

1990; Zhu et al.2004; Knaff and Zehr2007; Wang2008) Early observational and modeling studies have shown that such contraction often signifies the strengthening of hur-ricanes; the smaller the RMW of a hurricane at the peak intensity, the stronger the hurricane (Willoughby and Rahn

2004) Such contraction is often accompanied by formation

of a new outer eyewall that eventually experiences a sim-ilar contraction and later replaces the inner eyewall during

a process called the eyewall replacement (Black and Wil-loughby1992; Willoughby et al 1982; McNoldy2004) Various modeling and observational studies showed that the RMW contracts very fast at first, but then slows down quickly as the RMW becomes small enough (e.g., Hack and Schubert 1986; Molinari and Vollaro 1990; Corbosiero

et al.2005; Zhu et al.2004; Kieu et al.2010; Xu and Wang

2010) However, a question that has not been well under-stood is how far the RMW of a hurricane can contract to its maximum potential capability, or put differently what is the smallest eye size a hurricane can achieve Recent observa-tions of Hurricane Wilma (2005) collected by the U.S Air Force reconnaissance flight showed that Wilma’s eye con-tracted down to a diameter of only about 5 km Such incredible small eye size was followed by a minimum central pressure at the peak intensity of 882 hPa, the lowest value for hurricanes in the Atlantic basin (Pasch et al.2009; Knaff and Zehr 2007) This shows that hurricanes tend to attain stronger intensity with smaller eye size, an intriguing observation that has not been discussed thoroughly in hur-ricane research More statistical information about such connection between hurricanes and their size can be found

in, e.g., Knaff and Zehr (2007) and Knaff et al (2007) The early theoretical study by Shapiro and Willoughby (1982) and Hack and Schubert (1986) demonstrated that

Responsible editor: M Kaplan.

C Q Kieu ( &)

Lab of Climate and Weather Research,

Hanoi College of Science, Vietnam National University,

334 Nguyen Trai, Hanoi 10000, Vietnam

e-mail: chanhkq@vnu.edu.vn

DOI 10.1007/s00703-011-0171-7

the rate of contraction appears to depend upon the location

of the diabatic heating force Specifically, the contraction

of the RMW is related to a much faster spinup of the

tangential flow within the inner core as compared to that in

the outer region Such rapid intensification of the tangential

flow within the inner core as compared to that in the outer

region is indeed confirmed in a comprehensive

observa-tional study by Willoughby (1990) and recently presented

in an analytical model by Kieu and Zhang (2009) As a

result, the peak of the tangential flow will continuously

shift inward until the tangential flow attains the stationary

phase as determined by the maximum available

thermo-dynamical energy (Emanuel 1986) While Shapiro and

Willoughby’s (1982) approach is seminal in explaining for

the contraction of the RMW, their results do not seem to

capture details of the RMW contraction including the

sudden slowing down of the contraction at the middle of

the hurricane intensification or connection between

hurri-cane intensity and eye size Likewise, the explicit solutions

in Kieu and Zhang (2009) provide little information about

the contraction rate of the RMW other than indirect

sug-gestion of the different intensification rates of the

tangen-tial flows between the inner and outer region

In this study, we will first document in Sect.2a number

of important properties of the RMW contraction obtained

from an ensemble of real-time cloud-resolving simulations

of Hurricane Katrina (2005) as well as from the

axisym-metric model developed by (Rotunno and Emanuel1987,

hereafter referred to as RE model) A kinematic model is

then presented to explore more quantitatively the behaviors

of the RMW contraction Connection between the

hurri-cane size and the maximum potential intensity (MPI) is

next discussed, and some concluding remarks are given in

the final section

2 Characteristics of the RMW contraction

In this study, a 100-member ensemble of cloud-resolving

simulations of Hurricane Katrina (2005) is used to examine

the behaviors of Katrina’s RMW contraction The

ensem-ble simulations are obtained using the ensemensem-ble Kalman

filter assimilating system developed originally by Snyder

and Zhang (2003), and implemented in the Weather and

Forecasting Model (WRF, version 3.1) These real-time

simulations are configured with multi-nested domains

(40.5/13.5/4.5 km), and initialized with the National

Cen-ter for Environmental Prediction (NCEP) Global Forecast

System (GFS) operational analysis valid at 0000 UTC, 25

August The ensemble is perturbed by WRF- 3DVAR and

assimilates airborne data at 1430, 1530, 1630, 1700, 1900,

and 2000 UTC Details of the simulations as well as

techniques can be found in Weng and Zhang (2011) In

addition to the use of the WRF model, the axisymmetric model developed by Rotunno and Emanuel (1987) is uti-lized to compare and isolate the fundamental axisymmetric contraction of general hurricane-like vortices This is to ensure that the key behaviors of the contraction are rep-resentative and not model-dependent In our idealized experiments with the RE model, a simulation is initialized with a sounding profile characterizing Katrina’s environ-ment at 0000 UTC, 25 Aug 2005, and a vortex with the initial maximum surface wind (VMAX) of 20 m s-1 and RMW of 90 km that are similar to Katrina’s initialization

at 0000 UTC, 25 Aug 2005

While the ensemble simulations show a wide spread of the track forecasts of Katrina (not shown), the set of sim-ulations possess a significant homogeneity in terms of temporal and spatial structure, thus offering a unique opportunity to probe in detail the general characteristics of Katrina’s intensity Figure1 shows the time evolutions of the sea level pressure and the maximum surface wind, and the corresponding evolution of the ensemble of Katrina’s RMW at z = 1 km One notices that there is a close cor-relation of the RMW and hurricane intensity during the developing period; the stronger the hurricane, the smaller the RMW

Of particular interest is that the contraction of the RMW

is not entirely proportional to Katrina’s intensity Rather, the contraction slows down rapidly at the middle of Kat-rina’s rapid intensification,1i.e., at 0900 UTC, 27 August, and the RMW is kept nearly stationary afterward despite the rapidly strengthening tangential flow The breaking of the contraction happens at the RMW of approximately

27 km among all members For the ease of later discussion,

we summarize below the main thresholds when the RMW contraction slows down abruptly at 0900 UTC, 27 August (hereafter referred to as the breaking point of the RMW contraction)

• (o1) VMAX & 65 ms-1

• (o2) PMIN & 950 hPa

• (o3) RMW & 27 km

• (o4) The breaking takes place at the middle of rapid intensification

• (o5) The RMW becomes nearly constant after the breaking point

Even though it takes much more time for the axisym-metric vortex to spin up (*10 days) as compared to the real-time simulation of Katrina (*2 days, see Fig.1), it is seen that the slowing down of the RMW contraction at

1 In this study, the ‘‘rapid intensification’’ is defined as a period during which the initial vortex amplifies rapidly from tropical storm

to hurricane stage A more careful definition of the term ‘‘rapid intensification’’ can be found, e.g., in Kaplan and DeMaria ( 2003 ).

somewhat middle of intensification is also observed in the

axisymmetric hurricane model; all show fairly consistent

behaviors of the breaking contraction except for the smaller

RMW size in the axisymmetric model (*16 km)

The slowing down of the RMW contraction at the

middle of the intensification appears to be a common

characteristic of the hurricane development A recent

high-resolution simulation of a hurricane-like vortex by Xu and

Wang (2010, see their Figs.4 and 5) captures a very

similar behavior of the RMW contraction This signifies

some interesting processes behind the RMW contraction

that has not been fully understood In the next section, we

will present a kinematic model that attempts to describe the

contraction of the RMW in more specific details

3 A kinematic model of the RMW contraction

To examine qualitatively the picture of the RMW

con-traction, imagine we have an initial axisymmetric vortex

that is sufficiently defined such that the radial structure can

be approximated as seen in Fig.2, i.e., the tangential wind

increases linearly within the inner-core region and

decreases approximately inversely with radius in the outer

region Intuitively, one can anticipate from this radial

profile of the tangential flow that the radial advection tends

to accelerate of the tangential wind within the inner core at

a much faster rate than that in the outer region (provided that the advective contribution is larger than frictional dissipation) The early study by Willoughby (1990) showed that such inner-core intensification is often accompanied by the contraction of the RMW with VMAX continuously shifted inward However, it should be noted that the con-traction and the increase of VMAX are not always in phase Indeed, if this advection-based reasoning is continued, one faces a problem that the RMW would contract all the way down to the center of the vortex as long as the radial inflow

Fig 1 a Time series of the

ensemble of the RMW from the

Katrina 4-day ensemble

simulation (thin gray, unit: km),

the ensemble mean of the RMW

(bold black, unit: km), the

maximum surface wind (bold

dotted, unit: m s-1), and the

minimum sea level pressure

(bold gray) and b similar to the

a but from the deterministic

simulation of an idealized

vortex using the Rotunno and

Emanuel’s ( 1987 ) axisymmetric

hurricane model The point p in

the lower panel denotes the

location where the Taylor

expansion is carried out in

Eq 9 , and point b indicates the

breaking point where the RMW

contraction experiences the

sudden slowing down

Fig 2 Schematic evolution of the RMW at the surface The value Rb corresponds to the critical value at which the RMW stops contracting but the VMAX keeps increasing, and R0and Ridenote the RMW at the time t0and ti, respectively The gray arrow denotes the radial inflow at the surface

is maintained Of course, such collapse of the RMW is not

observed in reality, and in fact, the collapse could not occur

due to the singularity of the centrifugal force As presented

in Sect.2, the contraction always experiences a period of

slowing down and the RMW is then nearly unchanged

afterward (Fig.1)

One important characteristic of the RMW is that it does

not locate at the point where the radial inflow vanishes but

instead resides persistently in a regime where the radial

wind is inward (Fig.3) This is a very significant property

of the RMW contraction from the physical point of view if

one analyzes the projection of various forcing terms along

the tangential direction near the RMW As a parcel moves

from a very far point toward the hurricane center, the

tangential wind accelerates at a faster rate than the radial

wind over most of its trajectory in the outer region Thus,

the trajectory of the parcel has to transform gradually from

the spiral shape to a near-circular shape as it approaches the

hurricane inner region Because the RMW locates at the

inflow regime, the position of the RMW has to be located

at the radius where the frictional forcing balances the

centrifugal and Coriolis force along the tangential

direc-tion Inside the RMW, the frictional forcing tends to

dominate in the tangential direction and it thus reduces the

tangential wind down to the center Although the

centrif-ugal force also decreases with the tangential wind inside

the RMW, note that the pressure gradient has to decrease

with radius inside the RMW as well due to the existence of

the minimum pressure at the center (i.e., qp/qr ? 0 as

r? 0) As a result, the net radial forcing is outward and

the radial wind must decelerate and vanish at some point

inside the RMW.2That the RMW locates consistently in the

inflow regime implies that the frictional forcing must play some role in preventing the inward advection of the RMW

by the radial inflow

To model the behaviors of the RMW more quantita-tively, let us consider a simplified model in which an initial vortex is assumed to be symmetric and the tangential wind

v in the inner-core region satisfies the following conditions:

v¼ Xr 8 r\R

ov

orjr¼R¼ 0



where R denotes the RMW, H is the depth of the planetary boundary layer (PBL), and X represents the angular velocity of the inner core that is a function of time The vortex as given by Eq.1basically shows that the tangential wind profile increases linearly with radius and attains the maximum value at r = R This linear profile has been shown to be a very good approximation of the tangential flow within the hurricane inner-core region in numerous observational and modeling studies (see Fig.3 herein or, e.g., Rotunno and Emanuel 1987; Willoughby 1990; Liu

et al 1999) It should be mentioned that the linearity assumption of the tangential flow in Eq.1 is not con-tradicting to the condition of qv/qr = 0 because one can always construct a piecewise smooth profile that could meet both conditions in Eq.1; see Appendix 1 for an example of such a function

In the cylindrical coordinate, the evolution of the tan-gential wind is governed by the tantan-gential momentum equation

ov

otþ u ov

orþv

rþ f

þ wov

oz¼ CD

H v

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

u2þ v2

p

where u denotes the radial wind, f the Coriolis parameter,

r the radius from the vortex center, and CD is the drag coefficient Note that in the above tangential momentum equation, assumption of the well-mixed boundary layer has been explicitly used such that the frictional effects can be

Fig 3 Radial profile of the

tangential wind (bold) and

radial wind (dotted) that are

valid at z = 0.5 km from the

Rotunno and Emanuel’s ( 1987 )

axisymmetric model simulation

valid at the stationary phase

2 If one divides the radial domain into an outflow regime R?: {r B

R0| u(r) C 0} and an inflow regime R-: {r [ R 0 | u(r) \ 0}, then it

can be shown rigorously by involving the tangential momentum

equation that the stable RMW must belong to the inflow regime R-,

i.e., RMW [ R0.

expressed in terms of the bulk aerodynamic formula

(Holton2004) For the time being, we assume further that

CDand H are constant to ease our subsequent discussions

Such assumption is apparently rough as the depth of the

PBL could in fact vary with radius (Kepert and Wang

2001) Although such dependence of H on radius may have

some impacts on the exact point at which the breaking of

the contraction occurs, this is expected to affect little to the

nature of the breaking of the RMW contraction as will be

detailed later The goal here is to derive an equation that

governs the rate of change of the RMW for the simple

vortex given by Eq.1and see how far we can apply such

descriptive understanding of the idealized contraction to

the more realistic RMW contraction This allows us to

examine different factors controlling the RMW contraction

as well as criteria at which the RMW stops contracting as

observed in Sect.2

To this end, we recall that if the RMW is to contract

with time, the point R separating the inner core from the

outer region will change with time and so R depends

explicitly on time (the uppercase letters hereinafter denote

the values at the location of the RMW to distinguish with

the lowercase letters that represent field variables or

coordinates in the Eulerian frame) Let us focus on this

point R(t), which kinematically satisfies

Because both X(t) and R(t) are time dependent, the

maximum surface wind V(t) varies with time according to:

dVðtÞ

dt ¼dX

dt Rþ XdR

Next recall that since Eq.2has to be valid Vr, it has to

be applied also at r = R(t) where qv/qr = 0 Thus, we have

dX

dt R jUjðX þ f Þ ¼ CD

H V

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U2þ V2

p

where U is the radial flow at r = R, and the vertical

structure of the tangential flow is assumed to peak at the

top of PBL such that qv/qz = 0 Eliminating (dX/dt)R from

Eqs.4and5leads to

XdR

dt ¼dV

dt  jUjðX þ f Þ þCD

H V

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

U2þ V2

p

Because X f and V  U, Eq.6can be approximated

as:

dR

dt  jUj þCD

H VRþR

V

dV

It should be emphasized that while Eq.5is the tangential

momentum equation, Eq.4 is simply the kinematic

assumption derived from Eq.1 Thus, the resulting Eq.7

basically answers the question: given an initial vortex with

the radial structure as described by Eq.1 and an evolution equation as governed by Eq.2, how will the vortex evolve with time?

Several conclusions can be drawn from Eq 7 First, an important factor determining the contraction of the RMW

is indeed the inflow as expected from the earlier discussion

Of interest, however, is the second term on the rhs of Eq.7

that is associated with frictional dissipation This term turns out to be the main factor responsible for preventing the collapse of the RMW to center Although the frictional contribution is generally weak during the early hurricane development so that the RMW is allowed to shift contin-uously inward, the strong dependence of the frictional dissipation on wind speed soon results in a rapid increase

of the frictional forcing as the tangential wind becomes sufficiently large near the center This frictional dissipation

is the most effective way to keep the RMW from con-tracting Schematically, one can imagine that at any time when the RMW moves inward, the frictional dissipation will erase the newly formed peak of the tangential wind, thus keeping the RMW from contracting further

Note that although the last term on the rhs of Eq.7also prevents the contraction of the RMW, its physical roles are nonetheless less obvious as it is mainly related to the assumed linear constraint of the tangential flow within the inner core during the contraction As hurricanes intensify, i.e., V increases, the contribution of this last term will decrease and it thus becomes less important with time As

an example of the magnitude of various terms on the rhs

of Eq.7, consider typical values at the location of the RMW as U = -5 m s-1, V = 60 m s-1, H = 103m,

R = 50 km, CD= 1.2 9 10-3, and dV/dt = 1 m s-1 per hour (see Figs.1 and3), then the first, the second and the third term on the rhs of Eq.7are -5 m s-1, 3.6 m s-1and 0.13 m s-1, respectively The contraction rate is therefore

*1.4 m s-1 or roughly 5 km per hour Apparently, the radial inflow and frictional forcing are the two main factors

in the contraction equation while the last term is negligible

It is clear that the above derivations are solely from the kinematic point of view and it is by no mean complete In reality, the radial inflow U at r = R is not constant but determined by various thermodynamical and dynamical constrains The inclusion of the thermodynamic processes would require a careful examination of the Sawyer–Eli-assen equation Although detailed use of the Sawyer– Eliassen equation is desired for more complete under-standing of the RMW contraction, it is sufficient to highlight again the important observation that the radial wind has to be inward at the RMW location for the contraction to occur during the hurricane development (see Fig.3)

To study the contraction according to Eq 7 in more detail, the functional form for the radial wind |U| at

r = R has to be given As mentioned above, such explicit

expression for U requires a description of the diabatic

heating distribution To limit our analysis within the

kinematic framework, let assume that U at r = R is a given

function of the RMW R as follows3: |U| = bR where b is a

positive proportional coefficient As the contraction rate

depends directly on U rather than on the variation of U with

radius, the functional form U(R) is expected to have small

change with different profiles of U(R), and therefore to the

physics of the contraction Taylor expansion of V(t) around

a given instant of time, for instance the moment tpat the

point p on Fig.1b, we get

VðtÞ ¼ VðtpÞ þ apðt  tpÞ þ Oððt  tpÞ2Þ ð8Þ

where ap is the intensification rate of the VMAX at the

point p, which can be estimated by taking a left limit up to

point p If we let s = t-tp, Eq.7now becomes:

dR

ds¼ ½bpCD

H ðVpþ apsÞ  ap

Vpþ apsR ð9Þ Equation9 is the governing equation for the RMW

contraction around the neighborhood of the moment tpthat

we are looking for Note that ap, bp, and Vpall change at

different expansion point p Therefore, Eq.9 determines

the evolution of the RMW contraction locally around tp

An explicit integration of Eq.9will finally give us

RðsÞ ¼ Rpð1 þaps

Vp

Þ exp ½bpCD

H ðVpþaps

2 Þs

where DT is a local neighborhood around tp The explicit

solution Eq.10 summarizes all the conclusions obtained

above including the role radial advection in reducing the RMW (the b parameter), the counter-effect of the frictional forcing in preventing of the RMW contraction Note that the exponential e-folding time for the RMW to decrease by a factor of e in Eq.10is Ds * H/(Hb ? CDVp), which shows that the larger the drag coefficient, the faster (longer) the contraction rate would be To verify directly the role of the frictional dissipation, a series of idealized experiments with the RE model in which the drag coefficient CDvaries from 0.7 9 10-3to 1.1 9 10-3are conducted.4Figure4shows the time series of the RMW for the above range of the drag coefficient One can easily notice that the smaller the drag coefficient is, the longer it takes for the RMW to reach the breaking point Thus, the friction does appear to play a key role in determining the contraction rate as expected Given the governing Eq.7for the contraction, it is natural

to ask a question that at what point the RMW will stop contracting Note from the observation (o4) in Sect.2that such breaking in the RMW contraction happens at the middle

of the intensification, and the RMW is virtually stationary until the hurricanes attain their peak intensity To take into account this fact, let’s assume that the rate of change of the maximum surface wind V at the breaking point b is given

a priori, i.e., (dV/dt)b= abis known (e.g., it can be computed from the time series of V up to point b) Because the breaking occurs when dR/dt = 0 at R = Rb, we have from Eq.75

Fig 4 Evolutions of the RMW

from the Rotunno and

Emanuel’s axisymmetric

hurricane model with the

surface drag increasing from

CD= 0.7 9 10-3(far left) to

1.1 9 10-3(far right) at

interval of every 0.1 9 10-3

3 Note that we only assume a functional form for U at r = R The

radial dependence of u(r) for r \ R is not necessarily of the same

linear form E.g., u(r) = br2/R would give a similar U = bR at r = R.

4 The above range of the drag coefficient CDis chosen from the requirement that the vortex has to fully develop and attain the stationary phase in the RE model Outside this range, the vortex either fails to intensify or becomes very unstable As CDis dependent upon the wind speed, the actual value of CDestimated from the model will vary with the vortex intensity (see Rotunno and Emanuel ( 1987 ) for more details).

5 Note that the exponential decay of the RMW in Eq 10 is no longer applied near R = Rbas the expansion in Eq 8 varies with time, i.e., all related values Vp, ap, b p are varying with the point p.

Vb

V2

abþ jV2

ð11Þ

where j : CD/H, and Vb is the value of VMAX at the

breaking point Condition Eq.11 states that for each Vb

there will be a corresponding RMW Rbas given by Eq 11

at which the contraction is expected to slow down

drasti-cally afterward As long as the actual RMW of the hurricane

differs from this breaking RMW, the contraction will

con-tinue as the hurricane strengthens At some point where the

actual RMW and Rbare equal, the contraction is slowed

down and the RMW becomes unchanged despite the

increase of the tangential flow After this breaking point, the

tangential flow only amplifies in its amplitude but maintains

its stationary radial profile as seen schematically in Fig.2

It is apparent from Eq.11 that the stationary RMW

depends sensitively on the PBL parameterization, i.e., the

explicit functional form of j Given the large uncertainty in

our understanding of the PBL (see, e.g., Bryan and

Rot-unno 2009), it is thus challenging to predict exactly the

stationary value Rbin reality As the ensemble simulations

of Katrina show the consistent stationary RMW around

27 km, it is of interest to look further into what the simple

well-mixed PBL treatment can tell us about the connection

between the RMW and the hurricane intensity, using the

value of Rb= 27 km as a gauge to fix the value of j Use

of V * 65 m s-1, U * 15 m s-1, Rb* 27 km, and

ab* 1 m s-1 h-1, give a value of j & 7.6 9 10-6 If

one assumes H & 103 m as a typical depth of the PBL,

CD& 7.6 9 10-3m-1, consistent with the previous

esti-mation of CD Direct estimation of the denominator in

Eq.11 using the above values shows readily that

ab jVb, confirming our early statement of the negligible

of the last term in Eq.9 So, we can approximate Eq.11as

Rb& Ub/(jbVb) Figure 5 shows the dependence of the

critical RMW Rbon the ratio of Vb/Ub, given the value of j

above, which illustrates the inverse proportionality of Rbon

the VMAX at the breaking point That is, the larger the

maximum surface wind is, the smaller the RMW should be

in order for hurricanes to experience the abrupt slowing down It is nonetheless somewhat surprising to see that the larger the radial inflow is, the larger the breaking radius will be One could expect that the larger inflow will tend to advect more robustly the tangential wind inward, thus decreasing the RMW However, it should be recalled that the curvature vorticity is inversely proportional to radius Given a fixed value of VMAX, the RMW thus has to be larger such that the radial advection of the absolute vor-ticity can be balanced by the frictional dissipation

4 Maximum potential intensity and RMW Although the analytical dependence of the RMW on the hurricane intensity as given by Eq.10 is desirable for understanding the evolution of the RMW, this form con-tains the explicit radial inflow U that is difficult to obtain directly Under the observations (o4) and (o5) in Sect.2, it

is however possible to find a precise expression for U in terms of the thermodynamic property of the atmosphere, using the neutral slantwise assumption as in Emanuel (1986) This will allow us to link the hurricane MPI directly with the breaking RMW Rb To this end, note again that the tangential momentum Eq.6 for the PBL at the stationary phase is given by

Um

Vm

Rb

þ f

¼ CD

where the subscript m denotes the values of all variables at the stationary phase of the mature hurricanes to distinguish with the breaking moment b (see Fig.1) Note that because

of observation (o5), Rm: Rb, and so there is no need for separate variable Rm Our task now is to eliminate Um in

Eq 12to obtain a functional dependence of the maximum strength of hurricanes on the RMW Rb Note that hereafter

we will work only at the point r = Rbsuch that qv/qr = 0 (Emanuel 1986); the capital letters Um, Vm, or Rbindicate this purposely Consider next the thermodynamics equation within the PBL during the hurricane stationary phase Under this condition, we have (see Bister and Emanuel1998)

UmCpo ln he0

or ¼ 1

Ts

ose

where Ts is the surface temperature, he0is the equivalent potential temperature within the PBL, Cp is the specific heat capacity, and the surface entropy flux se per unit density is given by

se¼ CTjVmjCpðln hes ln he0Þ ð14Þ where hes is the surface equivalent potential temperature, and CTis the exchange coefficient for entropy Recall that

Fig 5 The stationary RMW Rb as a function of the ration Vb/Ub

between the tangential flow and radial flow at the breaking point as

given by Eq 11 with j = 7.6 9 10-6

for the well-mixed PBL, qse/qz & se/H (Holton2004), use

of Eqs.13and14leads to

Um¼CTjVmjðln hes ln he0Þ

Because an ascending parcel along the absolute angular

momentum surface within the eyewall is saturated,

integration of the thermal wind relationship along an

M-surface from the top of the PBL to the tropopause then

gives us (see Emanuel1986)

R2

mCp

o ln he

or ðTPBL ToutÞ  VmRbðVmþ3

2fRbÞ ð16Þ where he*is the saturated equivalent potential temperature

along the M-surface, Tout is an average outflow

temperature, and TPBLis the temperature at the top of the

PBL, which could be approximated to Ts as in Emanuel

(1986) Rearrange Eq.16such that

o ln he

R2

mCpðTPBL ToutÞðVmþ

3

2fRbÞ ð17Þ Because this saturated value is originated from the top

of PBL, it is reasonable to assume that he*&he0(see Bister

and Emanuel 1998) Plugging Eq.17 into Eq.15 then

gives us:

Um¼ CTRb

H

ðTs ToutÞ

Ts

jVmj

VmðVmþ3

2fRbÞCpðln hes ln he0Þ

ð18Þ Substitution of Eq.18into Eq.12results in

Vm2 ¼CT

CD

Cpðln hes ln he0ÞðTs ToutÞ

Ts

Vmþ fRb

ðVmþ3

2fRbÞ ð19Þ Note that because

Vmþ fRb

Vmþ3

2fRb

 1  fRb

2Vm

ð20Þ

Eq.19can be written as

Vm2 ¼ 1 fRb

2Vm

where VE denotes the theoretical maximum potential

tangential wind according to Emanuel’s (1986) MPI

theory that is defined as

VE2¼CT

CD

Cpðln hes ln he0ÞðTs ToutÞ

Ts

ð22Þ Rearranging Eq.21leads to

Rb¼ 2V

2

E V2

m

V2

E

Vm

Several noteworthy conclusions can be obtained from

Eq.23 First, the potential maximum (axisymmetric)

surface wind Vm must be larger than VE because

Rb[ 0 In particular, Eq 23 implies that the smaller the stationary RMW of a hurricane is at the breaking point, the stronger the hurricane would be at the later time when

it approaches the maximum intensity To see this point in more detail, note that if we take dRb/dVm, it can be seen that Rb has its maximum when V2

m¼ V2

E=3: For

Vm2[ V2

E=3; the smaller Rb corresponds to a stronger

Vm For V2

m\V2

E=3; Vmwill increase with Rb Because the actual value of Rbis mostly smaller than 150 km, Vmis

in practice sufficiently close to VE that the condition

Vm2[ V2

E=3 is dominantly satisfied As a result, the smaller RMW will correspond to the stronger intensity

as documented previously (see, e.g., Willoughby and Rahn2004) The theoretical upper limit VEcorresponds to the limit at which the breaking RMW Rbis equal to zero (the possibility of Vm to be greater than VE will be addressed below when the unbalanced flow is taken into account) Recall from observation (o4) that the breaking moment of the RMW contraction always happens at the middle of the intensification, well before hurricanes reach their MPI limit As a result, Eq 23 puts some causal constraints on the deterministic role of the RMW and the hurricane MPI

Second, Eq.23indicates that given the same RMW, Rb and MPI hurricanes tend to be weaker at higher latitudes due to the Coriolis factor in the denominator If the cau-sality implied in Eq.23is neglected, one can actually put this statement a bit differently that hurricanes at higher latitudes will have a smaller RMW, given the same intensity Such dependence of the hurricane size on lati-tudes can be used to verify the validity of relationship

Eq 23statistically Although the role of Rbin determining the maximum intensity is fairly small due to the weakness

of the Coriolis parameter, the above result could capture a correlation between the hurricane size and intensity that has not been derived previously

As seen from Eq.23, Vmmust be larger than VE How-ever, the theoretical upper-bound VEgiven by Eq.22has received much criticism as it is based essentially on the gradient balance assumption A recent study by Bryan and Rotunno (2009) pointed out that the gradient wind assumption is not accurate even above the PBL, particularly along the RMW up to the tropopause The unbalanced contributions are the main cause for the underestimation of the analytical MPI value VEas compared to the actual MPI value observed in a number of experiments A modified version of Emanuel’s (1986) MPI theory taking into account the imbalance of the gradient wind above the PBL results in a significantly higher bound for VEas indicated in Bryan and Rotunno (2009) The modified MPI theory pro-posed by Bryan and Rotunno (2009) can be incorporated in

Eq.23by simply replacing the value of VEin Eq.21by a

new value VEMgiven by

VEM2 ¼ V2

where the correction term c depends on the azimuthal

vorticity, vertical motion at the top of the PBL, and the

RMW; all are evaluated at r = R The slight dependence of

c on RMW will not change the implications inferred from

Eq.23as it results in a negligible correction to Eq.23 The

inclusion of the unbalanced part in Eq.23, i.e., replacing

VEby VEM, thus allows for Vm to be substantially higher

than the VEvalue while still maintaining the condition that

Rb[ 0

Although VEMis supposed to be used for estimating the

MPI, it should be noted that it is VE, not VEM,that is well

consistent with the climatology of hurricane intensity

dis-tribution despite the imperfect assumption of the gradient

wind balance As explained in Bryan and Rotunno (2009),

there is a counter-balance between the gradient balance

approximation and the assumption of PBL inviscid flow;

the underestimation of the hurricane intensity by the

gra-dient balance approximation is offset by the stronger radial

inflow due to the inviscid PBL assumption Thus, it should

be more appropriate to explain the relationship Eq.23 in

the climatological framework rather than for a particular

single hurricane simulation

5 Conclusion

In this study, details of the hurricane radius of maximum

wind (RMW) contraction have been explored A number of

behaviors of the RMW contraction were documented in our

series of ensemble real-time simulations of Hurricane

Katrina (2005) using the WRF model and in the idealized

experiments with the Rotunno and Emanuel’s (1987)

axi-symmetric hurricane model Of specific interest is that the

RMW contraction shows persistently an instant of abrupt

slowing down at the middle of the hurricane rapid

inten-sification and becomes nearly stationary afterward Such

break in the RMW contraction was observed in all of our

experiments, both real-time ensemble simulations of

Katrina and idealized simulations with the axisymmetric

hurricane model

To investigate the contraction of the RMW more

quantitatively, a kinematic model has been presented The

model revealed two main processes that govern the

con-traction of the RMW: (1) the inward advection by the radial

inflow that tends to shift the RMW inward, and (2) the

frictional dissipation that tends to erase any newly formed

RMW The first appears to dominate during the early

development of hurricanes, while the second mechanism

becomes significant only at the later development near the

inner-core region after the hurricane intensity becomes sufficiently strong As soon as the frictional dissipation can balance the inward advection by the radial inflow, the contraction quickly experiences a slowing down period and the RMW can maintain a stationary value after that This explains why the RMW always contracts as hurricanes intensify, and the contraction only slows down after the hurricane intensity reaches some threshold

The role of frictional dissipation in preventing the col-lapse of the RMW was demonstrated further in our experiments with the Rotunno and Emanuel’s model By varying the drag coefficient, it was found that the RMW contraction rate is slower for the smaller drag coefficient, which is consistent with our proposed kinematic model Use of the well-mixed boundary layer approximation and the Rankine-like profile provided some additional con-straints on the dependence of the breaking point of the RMW contraction on the hurricane intensity Specifically, (1) the larger the maximal surface wind is, the smaller the RMW should be in order for hurricanes to experience the breaking contraction; and (2) the larger the radial inflow is, the larger the breaking radius would be

As the RMW contraction shows a consistent breaking at the middle of the hurricane intensification, this puts some causal constraints on the hurricane maximum potential intensity Utilizing Emanuel’s assumption of the neutral slantwise convection and the gradient wind balance, we have obtained an explicit relationship between the sta-tionary RMW and the hurricane maximum intensity; the smaller the RMW of a hurricane is at the breaking point, the stronger the hurricane intensity is at the later time The theoretical maximum potential intensity as obtained in Emanuel (1986) corresponds to a complete collapse of the RMW Furthermore, the relationship also implies that the higher the latitude at which a hurricane is located, the weaker its intensity would be, given the same inner-core size

As a final note, we should mention that the kinematic model in this study is by no mean complete Instead, the focus was solely on the direct evolution of an axisymmetric vortex from the first principle to see how far the contraction

of the RMW from this model can compare with the con-traction from a full physics hurricane model A compre-hensive treatment should include also the evolution of the radial inflow that is controlled by the diabatic heating source according to the Sawyer–Eliassen equation as well

as environmental conditions Recent studies by Wang (2009), Hill and Lackmann (2009), Xu and Wang (2010) showed the important role of moisture source to the final size of hurricanes Such outer contributions would, how-ever, require a more complete examination of the full primitive equations and the induced diabatic heating In this study, no attempt has been made to study the evolution

of the secondary circulation but simply considered the

maximum of the radial inflow at the location of the

max-imum surface wind as a given function at each instant of

time

Acknowledgments The author is grateful to Dr Fuqing Zhang and

his research group for their various in depth discussions and to two

anonymous Reviewers for their invaluable comments and

sugges-tions, which have improved the quality of this study This research

was partly supported during the author’s postdoctoral period at the

Pennsylvania State University, under Dr Fuqing Zhang’s research

grant, and partly by the Vietnam Ministry of Science and Technology

Foundation (NCCB-DHUD.2011-G/10).

Appendix

To show that the linearity assumption of the tangential flow

within the inner-core region is not contradicting to the

condition qv/qr = 0, we attempt to construct in this

appendix an example of a piecewise smooth profile that

could satisfy both the linearity and the smoothness as

follows

vðrÞ ¼

xR1þ f ðrÞ R1\r R2

8

<

where R1and R2can be regarded as the radius at the inner

and outer edge of a hurricane eyewall, and f(r) is a smooth

function that satisfies qf(r)/qr|RMW= 0 Because the

eye-wall can be considered as a thin region, R1* R2*RMW

and so Eq.1could be ensured as expected

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