Meteorol Atmos Phys (2012) 115:47–56 DOI 10.1007/s00703-011-0171-7 ORIGINAL PAPER An investigation into the contraction of the hurricane radius of maximum wind Chanh Q Kieu Received: July 2011 / Accepted: 11 November 2011 / Published online: 25 November 2011 Ó 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 document some characteristic behaviors of the RMW contraction 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 intensification, and the RMW becomes nearly stationary subsequently, 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 hurricane maximum potential intensity and the hurricane eye size, an issue that has not been considered adequately in previous studies 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 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 continuous contraction during the development of hurricanes (see, e.g., Hack and Schubert 1986; Willoughby 1990; Zhu et al 2004; Knaff and Zehr 2007; Wang 2008) Early observational and modeling studies have shown that such contraction often signifies the strengthening of hurricanes; 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 similar contraction and later replaces the inner eyewall during a process called the eyewall replacement (Black and Willoughby 1992; Willoughby et al 1982; McNoldy 2004) 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 understood 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 observations of Hurricane Wilma (2005) collected by the U.S Air Force reconnaissance flight showed that Wilma’s eye contracted down to a diameter of only about 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 hurricane 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 123 48 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 observational 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 thermodynamical energy (Emanuel 1986) While Shapiro and Willoughby’s (1982) approach is seminal in explaining for the contraction of the RMW, their results 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 hurricane 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 suggestion of the different intensification rates of the tangential flows between the inner and outer region In this study, we will first document in Sect a 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 axisymmetric model developed by (Rotunno and Emanuel 1987, 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 hurricane size and the maximum potential intensity (MPI) is next discussed, and some concluding remarks are given in the final section 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 ensemble simulations are obtained using the ensemble 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 Center 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 123 C Q Kieu addition to the use of the WRF model, the axisymmetric model developed by Rotunno and Emanuel (1987) is utilized 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 representative and not model-dependent In our idealized experiments with the RE model, a simulation is initialized with a sounding profile characterizing Katrina’s environment 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 simulations 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 Figure 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 = km One notices that there is a close correlation 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 Katrina’s rapid intensification,1 i.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 axisymmetric 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 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) An investigation into the contraction of the hurricane radius 49 Fig 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 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 highresolution simulation of a hurricane-like vortex by Xu and Wang (2010, see their Figs 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 A kinematic model of the RMW contraction To examine qualitatively the picture of the RMW contraction, 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 Fig 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 R0 and Ri denote the RMW at the time t0 and ti, respectively The gray arrow denotes the radial inflow at the surface 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 contraction 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 123 50 C Q Kieu Fig 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 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 direction 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 centrifugal 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 ? 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.2 That the RMW locates consistently in the If one divides the radial domain into an outflow regime R? : {r B R0 | u(r) C 0} and an inflow regime R- : {r [ R0 | 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 123 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 quantitatively, 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: & v ¼ Xr r\R at z ẳ H; 1ị ov j ¼ or r¼R 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 basically 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 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 is not contradicting to the condition of qv/qr = because one can always construct a piecewise smooth profile that could meet both conditions in Eq 1; see Appendix for an example of such a function In the cylindrical coordinate, the evolution of the tangential wind is governed by the tangential momentum equation ffi ov ov v ov CD p ỵu ỵ ỵf ỵw ẳ 2ị v u2 ỵ v ; ot or r oz H 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 An investigation into the contraction of the hurricane radius 51 expressed in terms of the bulk aerodynamic formula (Holton 2004) For the time being, we assume further that CD and 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 and 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 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 Vtị ẳ XtịRtị: ð3Þ Because both X(t) and R(t) are time dependent, the maximum surface wind V(t) varies with time according to: dVðtÞ dX dR ẳ RỵX : dt dt dt 4ị Next recall that since Eq has to be valid Vr, it has to be applied also at r = R(t) where qv/qr = Thus, we have dX CD pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R jUjX ỵ f ị ẳ V U2 þ V 2; dt H ð5Þ 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 = Eliminating (dX/dt)R from Eqs and leads to X dR dV CD p ẳ jUjX ỵ f ị ỵ V U2 ỵ V 2: dt dt H 6ị Because X ) f and V ) U, Eq can be approximated as: dR CD R dV % jUj ỵ : VR ỵ dt V dt H 7ị It should be emphasized that while Eq is the tangential momentum equation, Eq is simply the kinematic assumption derived from Eq Thus, the resulting Eq basically answers the question: given an initial vortex with the radial structure as described by Eq and an evolution equation as governed by Eq 2, how will the vortex evolve with time? Several conclusions can be drawn from Eq 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 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 continuously 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 contracting 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 also 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 = 103 m, R = 50 km, CD = 1.2 10-3, and dV/dt = m s-1 per hour (see Figs and 3), then the first, the second and the third term on the rhs of Eq are -5 m s-1, 3.6 m s-1 and 0.13 m s-1, respectively The contraction rate is therefore *1.4 m s-1 or roughly 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–Eliassen equation Although detailed use of the Sawyer– Eliassen equation is desired for more complete understanding 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 in more detail, the functional form for the radial wind |U| at 123 52 C Q Kieu Fig Evolutions of the RMW from the Rotunno and Emanuel’s axisymmetric hurricane model with the surface drag increasing from CD = 0.7 10-3 (far left) to 1.1 10-3 (far right) at interval of every 0.1 10-3 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 at the point p on Fig 1b, we get Vtị ẳ Vtp ị ỵ ap t ị ỵ Ot ị2 ị 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 now becomes: ap dR CD ẳ ẵbp Vp ỵ ap sị R ds H Vp ỵ ap s 9ị Equation is the governing equation for the RMW contraction around the neighborhood of the moment that we are looking for Note that ap, bp, and Vp all change at different expansion point p Therefore, Eq determines the evolution of the RMW contraction locally around An explicit integration of Eq will finally give us & ' ap s ap s CD Rsị ẳ Rp ỵ ịs Vp ỵ ị exp À½bp À Vp H with s ½0; DT ð10Þ where DT is a local neighborhood around The explicit solution Eq 10 summarizes all the conclusions obtained 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 123 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 10 is 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 CD varies from 0.7 10-3 to 1.1 10-3 are conducted.4 Figure shows 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 for 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 that 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 = ab is known (e.g., it can be computed from the time series of V up to point b) Because the breaking occurs when dR/dt = at R = Rb, we have from Eq 75 The above range of the drag coefficient CD is 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 CD is dependent upon the wind speed, the actual value of CD estimated from the model will vary with the vortex intensity (see Rotunno and Emanuel (1987) for more details) Note that the exponential decay of the RMW in Eq 10 is no longer applied near R = Rb as the expansion in Eq varies with time, i.e., all related values Vp, ap, bp are varying with the point p An investigation into the contraction of the hurricane radius Rb ¼ Ub Vb2 Vb ab ỵ jVb2 53 ! 11ị 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 Rb as given by Eq 11 at which the contraction is expected to slow down drastically afterward As long as the actual RMW of the hurricane differs from this breaking RMW, the contraction will continue as the hurricane strengthens At some point where the actual RMW and Rb are 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 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 Rotunno 2009), it is thus challenging to predict exactly the stationary value Rb in 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 * m s-1 h-1, give a value of j & 7.6 10-6 If one assumes H & 103 m as a typical depth of the PBL, CD & 7.6 10-3 m-1, consistent with the previous estimation of CD Direct estimation of the denominator in Eq 11 using the above values shows readily that ab ( jV2b, confirming our early statement of the negligible of the last term in Eq So, we can approximate Eq 11 as Rb & Ub/(jbVb) Figure shows the dependence of the critical RMW Rb on the ratio of Vb/Ub, given the value of j above, which illustrates the inverse proportionality of Rb on 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 vorticity can be balanced by the frictional dissipation 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 contains 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 for the PBL at the stationary phase is given by Vm CD Um jVm jVm ; ỵf ẳ 12ị Rb H 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 12 to 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 = Rb such that qv/qr = (Emanuel 1986); the capital letters Um, Vm, or Rb indicate this purposely Consider next the thermodynamics equation within the PBL during the hurricane stationary phase Under this condition, we have (see Bister and Emanuel 1998) Um Cp o ln he0 ose ¼À Ts oz or ð13Þ where Ts is the surface temperature, he0 is 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 se ¼ CT jVm jCp ðln hes À ln he0 Þ Fig 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 10-6 ð14Þ where hes is the surface equivalent potential temperature, and CT is the exchange coefficient for entropy Recall that 123 54 C Q Kieu for the well-mixed PBL, qse/qz & se/H (Holton 2004), use of Eqs 13 and 14 leads to CT jVm jðln hes À ln he0 Þ 15ị Um ẳ HTs o ln he0 =or 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 Emanuel 1986) o ln hÃe ÀR2m Cp ðTPBL À Tout Þ % Vm Rb Vm ỵ fRb ị 16ị or where h*e is the saturated equivalent potential temperature along the M-surface, Tout is an average outflow temperature, and TPBL is the temperature at the top of the PBL, which could be approximated to Ts as in Emanuel (1986) Rearrange Eq 16 such that o ln he Vm % Vm ỵ fRb Þ or Rm Cp ðTPBL À Tout Þ ð17Þ Because this saturated value is originated from the top of PBL, it is reasonable to assume that h*e & he0 (see Bister and Emanuel 1998) Plugging Eq 17 into Eq 15 then gives us: CT Rb ðTs À Tout Þ jVm j Um ¼ À Cp ðln hes À ln he0 ị Ts H Vm Vm ỵ 32 fRb ị ð18Þ Substitution of Eq 18 into Eq 12 results in CT Ts Tout ị Vm ỵ fRb Cp ln hes ln he0 ị Vm2 ẳ Ts CD Vm þ 32 fRb Þ ð19Þ Note that because Vm þ fRb fRb %1 2Vm Vm ỵ 32 fRb Eq 19 can be written as fRb Vm ¼ À V2 2Vm E ð20Þ ð21Þ where VE denotes the theoretical maximum potential tangential wind according to Emanuel’s (1986) MPI theory that is defined as VE2 ¼ CT ðTs À Tout Þ Cp ðln hes À ln he0 Þ Ts CD 22ị Rearranging Eq 21 leads to Rb ẳ VE2 À Vm2 Vm f VE2 ð23Þ Several noteworthy conclusions can be obtained from Eq 23 First, the potential maximum (axisymmetric) 123 surface wind Vm must be larger than VE because Rb [ 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 Vm2 ¼ VE2 =3: For Vm2 [ VE2 =3; the smaller Rb corresponds to a stronger Vm For Vm2 \VE2 =3; Vm will increase with Rb Because the actual value of Rb is mostly smaller than 150 km, Vm is in practice sufficiently close to VE that the condition Vm2 [ VE2 =3 is dominantly satisfied As a result, the smaller RMW will correspond to the stronger intensity as documented previously (see, e.g., Willoughby and Rahn 2004) The theoretical upper limit VE corresponds to the limit at which the breaking RMW Rb is 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 23 indicates 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 causality implied in Eq 23 is 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 latitudes can be used to verify the validity of relationship Eq 23 statistically Although the role of Rb in 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, Vm must be larger than VE However, the theoretical upper-bound VE given by Eq 22 has 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 VE as 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 VE as indicated in Bryan and Rotunno (2009) The modified MPI theory proposed by Bryan and Rotunno (2009) can be incorporated in An investigation into the contraction of the hurricane radius 55 Eq 23 by simply replacing the value of VE in Eq 21 by a new value VEM given by ¼ VE2 ỵ c VEM 24ị 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 23 as it results in a negligible correction to Eq 23 The inclusion of the unbalanced part in Eq 23, i.e., replacing VE by VEM, thus allows for Vm to be substantially higher than the VE value while still maintaining the condition that Rb [ Although VEM is 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 distribution 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 gradient 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 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) axisymmetric 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 intensification 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 contraction 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 collapse 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 constraints 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 stationary 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 contraction from a full physics hurricane model A comprehensive 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, however, 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 123 56 of the secondary circulation but simply considered the maximum of the radial inflow at the location of the maximum 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 suggestions, 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 r R1 < xr vrị ẳ xR1 ỵ f rị R1 \r R2 at z ¼ H; : k=r R2 \r where R1 and R2 can 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 = Because the eyewall can be considered as a thin region, R1 * R2 *RMW and so Eq could be ensured as expected References Bister M, Emanuel K (1998) Dissipative heating and hurricane intensity Meteor Atm Phys 52:233–240 Black ML, Willoughby HE (1992) The concentric eyewall cycle of hurricane Gilbert Mon Weather Rev 120:947–957 Bryan GH, Rotunno R (2009) The maximum intensity of tropical cyclones in 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An investigation into the contraction of the hurricane radius 49 Fig a Time series of the ensemble of the RMW from the Katrina 4-day ensemble simulation (thin gray, unit: km), the ensemble mean