A New Parametric Tropical Cyclone Tangential Wind Profile Model

Transcription

1 884 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 A New Parametric Tropical Cyclone Tangential Wind Profile Model VINCENT T. WOOD NOAA/OAR/National Severe Storms Laboratory Norman Oklahoma LUTHER W. WHITE Department of Mathematics University of Oklahoma Norman Oklahoma HUGH E. WILLOUGHBY Department of Earth and Environment Florida International University Miami Florida DAVID P. JORGENSEN NOAA/OAR/National Severe Storms Laboratory Norman Oklahoma (Manuscript received 3 April 202 in final form 2 October 202) ABSTRACT A new parametric tropical cyclone (TC) wind profile model is presented for depicting representative surface pressure profiles corresponding to multiple-maxima wind profiles that exhibit single- dual- and triplemaximum concentric-eyewall wind peaks associated with the primary (inner) secondary (first outer) and tertiary (second outer) complete rings of enhanced radar reflectivity. One profile employs five key parameters: tangential velocity maximum radius of the maximum and three different shape velocity parameters related to the shape of the profile. After tailoring the model for TC applications a gradient wind is computed from a cyclostrophic wind formulated in terms of the cyclostrophic Rossby number. A pressure via cyclostrophic balance was partitioned into separate pressure components that corresponded to multiple-maxima cyclostrophic wind profiles in order to quantitatively evaluate the significant fluctuations in central pressure deficits. The model TC intensity in terms of varying growth size and decay velocity profiles was analyzed in relation to changing each of five key parameters. Analytical results show that the first shape velocity parameter changing a sharply to broadly peaked wind profile increases the TC intensity and size by producing the corresponding central pressure fall. An increase (decrease) in the second (third) shape velocity parameter yields the pressure rise (fall) by decreasing (increasing) the inner (outer) wind profile inside (outside) the radius of the maximum. When a single-maximum tangential wind profile evolves to multiple-maxima tangential wind profiles during an eye replacement cycle the pressure falls and rises are sensitively fluctuated.. Introduction Over the years a variety of parametric wind profile models have been developed to depict representative tangential wind and/or pressure profiles within a tropical cyclone. Originally introduced to tropical meteorology by Depperman (947) a Rankine (Rankine 882) vortex model was used in numerous studies to approximate the inner core of solid-body rotation of the tropical cyclone (TC). Outside the core the tangential wind decreased Corresponding author address: Vincent T. Wood 20 David L. Boren Blvd. Room 392 Norman OK vincent.wood@noaa.gov with some values of about being inversely proportional to radius from the TC center. The velocity distributions were found to give a good approximation to the tangential wind profiles of TCs (Hughes 952; Riehl ; Malkus and Riehl 960 among others). Although the modified Rankine model had some observational support (Gray and Shea 973) the model did not allow computation of a relation between the maximum wind and minimum sea level pressure because the radial integral of the gradient wind acceleration was not bounded at a large radius (Willoughby and Rahn 2004). Schloemer (954) proposed an alternative model in which a TC surface pressure field was assumed to follow a modified rectangular hyperbolic function with DOI: 0.75/MWR-D Ó 203 American Meteorological Society

2 JUNE 203 W O O D E T A L. 885 radius and the gradient-balance wind was calculated from the radial variation of hurricane pressure. Myers (957) adopted the Schloemer model to derive a pressure wind relation in which the maximum balanced wind was proportional to the square root of the pressure difference between large radius and the TC center. Adopting the Schloemer model Holland (980) introduced an additional parameter now commonly referred to as the Holland B parameter into an inverse exponential function of radius to represent a parametric pressure field. The field in turn must be differentiated with respect to radius to obtain a gradient wind. This parameter was critical in allowing the pressure profile and thus the controlling width of the wind profile spanning the maximum to accommodate the gamut of velocity shapes typically found in mature TCs. The Holland model however suffers from two major shortcomings as noted by Willoughby and Rahn (2004): (i) the areas of strong inner winds in the eyewall and of nearly calm winds at the storm center were unrealistically wide compared to observations and (ii) the outer winds outside the eyewall decayed too rapidly with increasing radial distance from the center. Lack of extensive reconnaissance aircraft datasets prior to 980s apparently contributed to the shortcomings of the Holland model. Since then modern technologies produce better datasets that offer much potential for increased knowledge about detailed TC wind profiles including multiple-maxima concentriceyewall tangential winds. As a result the datasets led Holland et al. (200) to improve the original Holland (980) model by (i) allowing the wind equation exponent to vary in order to fit outer wind observations and (ii) adding a secondary wind profile to the preexisiting primary wind profile. Their discussion on secondary wind maxima however was superficial because it was unclear how variability of outer wind (i.e. secondary wind) profiles was related to changes in TC intensity strength and/or size during an eye replacement cycle. Two shortcomings of the Holland (980) model led Willoughby et al. (2006 hereafter WDR) to develop a different parametric model that was designed specifically to fit wind observations and has been extensively tested on reconnaissance aircraft data. The WDR model outperformed the Holland model by increasing a number of parameters to characterize and obtain an optimum fit to complex observed wind profiles. One profile was a piecewise continuous approximation defined by three different functions in different parts of the TC wind structure. Inside the eye the first function used a power of radius to control the shape of the inner profile. Within the eyewall the second function used a smooth radially varying polynomial ramp function to replace the unrealistic discontinuity (i.e. cusp) in the infinitesimal radial thickness of the Rankine tangential velocity maximum with a smooth polynomial transition joining the inner and outer profiles. Outside the eyewall the third function used a dual exponential to control two different decay rates of the outer profile: (i) the rapid wind decay close to the eyewall and (ii) the slow wind decay farther away from the eyewall. Using reconnaissance aircraft data the radial profiles of tangential wind and geopotential height were shown to be able to fit well the observed profiles of flight-level tangential wind and geopotential height. Additionally in his unpublished work with the WDR model Willoughby successfully developed multiple-maxima wind profiles that resembled to those in Hurricane Allen (980). The Holland B parameter cannot model all TCs at all times because this limited parameter ranges from 0.75 to 2.5 as noted by Willoughby and Rahn (2004) WDR Vickery and Wadhera (2008) and Vickery et al. (2009ab) in their discussion of parametric characteristics of the Holland model. The Holland B formulation is so rigid that it is difficult to have control over the shape of the different regions of the radial distribution of the gradient wind. The difficulty of the model provides motivation to apply the Wood and White (20) parametric tangential wind profile model (developed for high Rossby number vortices such as thunderstorm mesocyclones and tornadoes) to a TC and determine whether the model is capable of achieving an optimum fit to realistic TC pressure and multiple-maxima concentriceyewall tangential wind profiles with and without the eyewall replacement cycle (ERC). The objective of this study is to extend the existing parametric tangential wind profile model of Wood and White (20) to tailor the model for TC applications. In section 2 the following five key model parameters controlling the radial profile of tangential wind are described: maximum velocity radius of the maximum and three shape velocity parameters that independently control different portions of the profile. Physical interpretations of each shape velocity parameter are described. The description follows the computation of a model gradient wind from a cyclostrophic wind approximation formulated in terms of a cyclostrophic Rossby number. Section 3 discusses partitioning of the radial profile of model TC gradient and cyclostrophic winds into as many as three wind maxima. The total surface pressures are partitioned into separate components of pressures that correspond to triple wind maxima. Additionally the partitioned central surface pressure deficits corresponding to three peak tangential wind profiles are derived. Mathematical formulas for gradient and cyclostrophic vorticity for multiple wind maxima are developed and discussed in section 4.

3 886 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 Section 5 both uses a TC wind structural variability [similar to the Merrill (984) concept] and a TC emulator to present a discussion on the manner in which a model TC intensifies strengthens and changes size by examining the behaviors of central surface pressure deficits and radial profiles of cyclostrophic and gradient winds vorticity and surface pressure with and without an ERC. Finally conclusions and future work are presented in section 6. Our parametric TC tangential wind profile model is a part of our research to obtain an optimum fit to radial profiles of aircraft flight-level data. If the parametric model successfully replicates the general aspects of observed profiles of wind and pressure (or geopotential height) in TCs then we will eventually apply the model to azimuthal variations of high-resolution Doppler velocity signatures of vortices for various applications such as developing statistics of vortex wind profiles for use in enhancing the National Weather Service warnings. 2. Parametric modeling of TC gradient cyclostrophic winds a. Wood White parametric wind model The parametric tangential wind profile V WW formulated by Wood and White (20) for inviscid axisymmetric flow is given by V WW (r; m) 5 V X f(r; k h l) () and h l r k f(r; k h l) [ (h 2 k kr h/l ) l # k h l. 0 (2) where V WW is the tangential velocity as a function of radius r and r [ r/r X is a dimensionless radius. The Wood White profile employs a model vector of five key parameters: m 5 [V X R X k h l] T wherev X is maximum tangential velocity; R X radius of the maximum; and k h andl represent different shape velocity parameters that are related to different shapes of the velocity profile. Note that k and h are the same parameters k and n as initially developed by Wood and White (20). The parametric model () assumes a circular wind flow pattern and does not adequately depict the actual surface boundary layer winds. Surface boundary layer winds including wind reduction factor storm translation effects and the degree of overwater wind axisymmetry and asymmetry (e.g. Kepert ab) could be added to the model for use in storm surge models FIG.. Varying radial profiles of normalized tangential velocity V* as a function of (a) the growth parameter k (b) the decay parameter h and (c) the size parameter l; V X * 5 is the normalized tangential velocity peak at r 5. models of wind-driven seas and other oceanic responses to TCs. b. Physical interpretations Figure shows how each shape velocity parameter (k h l) influences the shape of a normalized velocity (V* [ V WW /V X ) profile within an axisymmetric vortex. The k parameter may be thought of the growth parameter since it controls a linearity or nonlinearity of the inner velocity profile between the vortex center (r 5 0) and r 5. When k 5 thelinearprofileisv shaped and is related to the inner core of solid-body rotation inside r 5 (Fig.a).Ask / V* is reduced within the inner core (indicated by a downward-pointing arrow in the figure) producing a transition to bowl-to- U-shaped (nonlinear) profiles with increasing width of zero tangential velocity at the vortex center. The h parameter may be thought of the decay parameter since it controls the size of the decaying outer velocity profile (Fig. b). The parameter is given as h 5 k 2 m where m 0 is the power-law exponent that

4 JUNE 203 W O O D E T A L. 887 FIG. 2. Radial profile families of V WW * for selected values of k h and l. Three different profile families in each panel are indicated by three different values of h. The thick black curve represents the profile that corresponds to h 5 k to enhance readability. The thick gray curve represents the idealized (normalized) Rankine profile for comparison. Normalized radial distance from TC center is represented by r [ r/r X. [From Wood and White (20).] governs the outer velocity profile in the Rankine vortex model (see the appendix). The higher the h value the more rapidly the tangential velocity decreases with r beyond the radius of maximum wind (RMW). The l parameter may be thought of the size parameter since it controls the radial width of the velocity profile in the annular zone of the maximum. When l 5 a broadly peaked profile results (Fig. c). As l/0 the profile transitions to a sharply peaked profile that resembles the idealized Rankine velocity profile. Both k and h alone cannot allow adjustment of the sharply profile at r 5. Although the l parameter may determine the size of both inner and outer circulations the h parameter plays an important role in determining the size of the only outer circulation in the Merrill (984) and Weatherford and Gray (988b) sense. The radial profile families of normalized tangential velocity as functions of k h and l are schematically presented in Fig. 2. In each panel of the figure three varying values of h 5 k 2 m with m

5 888 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 are presented for each selected values of k and l. As one progresses from the top to the bottom panels k and h remain unchanged with decreasing l. Consequently ) a broadly peaked profile undergoes a change to a sharply peaked profile and 2) the three radial profiles merge together to form the same profile inside r 5. When l is constant h changes with increasing k and the curvature of the profile progressively changes such that the V-shaped profile (left column of Fig. 2) transitions into the bowl-shaped profile (center column) and eventually into the U-shaped profile (right column). Users/analysts interested in applying this parametric model to some actual data or simulations are encouraged to experiment with each of adjustable shape velocity parameters (k h l). One s ability to fine-tune any parameters that represent physically meaningful aspects of the wind profile is much like adjusting the Holland B parameter to obtain the desired results. Once they have gained familiarity with adjustable parameters the users/analysts are able to reproduce actual or simulated TC wind profiles for a wide range of applications (e.g. hurricane risk model or model initialization for wind specification). c. Gradient versus cyclostrophic wind approximations The tangential wind in () is inappropriately scaled for TC tangential wind owing to the absence of the Coriolis parameter which is not important to the scaling argument for a tornado. To tailor the Wood White parametric wind model for TC applications we follow Willoughby (990b ) and use the fact that () can be converted to the gradient V G and cyclostrophic V C winds in terms of the cyclostrophic Rossby number [R C 5 V C /(fr)] such that V G 5 2 R C 6 2 V C s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! R C 6¼ 0 (3) 4R 2 C where f is the Coriolis parameter. Note that (3) is much simpler than gradient balance because of the following reasons. If the wind represented by () is assumed to be the gradient wind then the inward radial integral from large radius of the Coriolis term fv G (in gradient balance) becomes infinite (Willoughby 995). The difficulty of the infinite pressure difference (e.g. P c 2 P where P c is the central pressure at the storm center and P the environmental pressure at radial infinity) arises because it is the pressure not the wind that is specified at the beginning. This problem can be avoided by first computing V C (which will be described in detail in section 3) and then calculating both V G in (3) and the pressure from cyclostrophic balance which will be also discussed in that section. For the high Rossby number gradient wind approximation (3) may be rewritten equivalently as V G 5 2 R C 6 2 V C s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 4R 2 C V C ( " R C 2 4R 2 C!#) V C 2R C. (4) Here we have chosen the positive root (nonanomalous flow) and recognized that /8R 2 C. Now we substitute for R C in (4) as follows: V G V C fr V C 2 fr 5 V 2V C 2 fr C 2. (5) 2V C Scale analysis of (5) indicates that this approximation is valid only when R C. 0. Equation (5) is used to derive a parametric vorticity formula in section 4. In (3) (5) V C is expressed as with the aid of f in () V C 5 (V X g)f (6) where the product of V X and g is defined as the scaled cyclostrophic tangential wind maximum and g is a dimensionless variable computed from g [ 2 and s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 R CX 4R 2 for R CX # 0 (7a) CX g for R 2R CX. 0 (7b) CX where (7b) is obtained in a manner analogous to the development of the denominator in (4). Here R CX 5 V X /(fr X ) is the local cyclostrophic Rossby number based upon the observed RMW (or R X ) the observed maximum wind V X and the Coriolis parameter f evaluated at the TC s center position. It is used to solve Using the binomial series for the square root term and ignoring higher-order terms it can be shown that ( 6 x) a 5 6 ax 6 a(a 2 )x ax. 2!

6 JUNE 203 W O O D E T A L. 889 FIG. 3. Normalized model tangential wind profiles of V WW * (solid curve) V C * (short-dashed curve) and V G * (long-dashed curve) as functions of the normalized radial distance (r [ r/r X ) for selected values of local cyclostrophic Rossby number R CX. The model parameters that produced the V WW * profile are k 5 :5 h 5 2:0 and l 5 0:. the gradient wind relation for the cyclostrophic wind that gives the approximately correct (i.e. the observed V X ) gradient wind at the RMW when the gradient wind relation is applied to the cyclostrophic wind profile. As a practical matter it is almost as good to use V C 5 V G fr/2 [see (5)] to adjust both the cyclostrophic wind value and the radius of maximum cyclostrophic wind. In other words the radius of maximum cyclostrophic wind and the value of the maximum cyclostrophic wind are adjusted by differentiating V C with respect to r and setting the result to zero (i.e. V C / r 5 0). According to Willoughby (990b) and Willoughby and Rahn (2004) the typical local cyclostrophic Rossby numbers R CX range from 0 to 00 in TCs. As an example Fig. 3 illustrates how the different radial profiles of normalized model tangential wind vary as a function of R CX. The plots of three radial profiles of normalized model tangential wind V WW * ([V WW /V X ) V G *([V G /V X ) and V C *([V C /V X ) are compared for selected values of R CX when R X 5 V X /(fr CX ) f of s 2 at the 208N latitude are given. Additionally the assumed values of k h andl are and 0. respectively. The V WW * profile remains unchanged in all panels of Fig. 3 because () is independent of R CX.In(7a)forsmallR CX the maximum value of V C * at r 5 always is larger than the maximum V G * which in turn is always greater than the V WW * peak in normal (cyclonic) flow around a closed low pressure system. Keep in mind that this r 5 can occur no matter how very close to (or very far from)

7 890 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 the TC center is the r. Beyond r 5 V G * increasingly deviates and rapidly decreases from V C * with the radius from the storm center. Gradient flow is a good approximation to the wind outside the vortex core but it is strictly valid only when radial accelerations of the radial wind are small. When R CX becomes large in (7b) g / indicating that two radial profiles of V C * and V G * approach the V WW * profile and nearly merge together in each panel (Fig. 3) to form one radial profile for large r. At the same time the peak values of V C * and V G * approach each other (i.e. cyclostrophic balance is obtained) at r 5. The profiles of V C * and V G * however diverge beyond r 5 for all the R CX values. The radial profiles of cyclostrophic and gradient winds (Figs. 3ab) favorably resemble those of Willoughby (995 20). 3. Partitioned pressure wind profiles a. Partitioned wind profiles for multiple-maxima eyewall tangential wind profiles Based on the seminal work of Willoughby et al. (982) intense symmetric TCs have been observed to exhibit multiple well-defined wind maxima during ERCs. Two (three) concentric eyewalls were defined by two (three) tangential wind peaks in each of the aircraft radial legs and by two (three) well-defined rings of enhanced radar reflectivity located at different radii from the centers of the TCs. An echo-free moat and a saddleshaped wind profile were situated between each pair of concentric eyewalls. Good examples of the primary secondary and tertiary tangential wind maxima within concentric eyewalls were also provided in McNoldy (2004) and Sitkowski et al. (20) which displayed flight-level tangential wind profiles observed by aircraft in Hurricanes Juliette (200) and Frances (2004) respectively. According to Sitkowski et al. (20) several TCs exhibiting multiple wind maxima show evidence of multiple ERCs. Figure 4a provides an example from Hurricane Frances (2004). The weakening primary wind maximum (identified by ) was associated with a decaying primary (inner) concentric eyewall while the intensifying secondary wind maximum (identified by 2) was associated with a developing secondary (first outer) concentric eyewall as an ERC neared its completion. The tertiary wind maximum (identified by 3) appeared to originate from spiral rainbands that ultimately arranged into a tertiary (second outer) well-defined concentric eyewall from which an additional ERC arose. There have been no documented cases of more than three well-defined tangential wind peaks in aircraft profiles of TCs. FIG. 4. Radial profiles of (a) storm-relative tangential wind (m s 2 ) and (b) surface pressure (hpa) in Hurricane Frances observed on 30 Aug The beginning and ending times (UTC) of aircraft flight-level collection data are indicated at the top-left and top-right corners respectively of (a). In (a) triple wind maxima identified by 2 and 3 are observed for each aircraft right leg [from Sitkowski et al. (20)]. In (b) arrows indicate the possible locations of kink reflected in the pressure profile. The radial distance (km) is from the TC center located at N W at 82:49 UTC. (Data obtained courtesy of NOAA/AOML/HRD.) In view of the above the triple concentric eyewall structure can be considered as a triple vortex composed of the inner vortex the first and second outer vortex configurations. We now show that the hypothesis on a three-vortex composite in TC wind profiles enables the description of complex tangential velocity distributions. By isolating the primary tangential wind profile from the secondary and tertiary tangential wind profiles the total cyclostrophic wind V C profile in (3) may be partitioned into the primary V P secondary V S and tertiary V T tangential wind profiles: V C 5 å 3 i5 V i 5 V P V S V T (8) where the subscript i represents the number of individual wind profiles [i.e. i 5 corresponding to P (primary) i 5 2toS (secondary) and i 5 3toT (tertiary)]. This subscript applies to at least one radial profile of tangential wind. The partitioned tangential wind components in (8) with the aid of () and (6) are expressed by

8 JUNE 203 W O O D E T A L. 89 V i 5 (V ix g i )f i g i [ 2 R icx 5 V ix fr ix f i 5 h l i i r k i i (h i 2 k i k i r h /l i i i ) l i s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 R icx 4R 2 icx and r i [ r R ix (9a) (9b) where the subscript X refers to the radius at which the maximum tangential wind occurs. In (9a) the primary eyewall wind component V P is the primary tangential wind in a basic-state vortex flow. The secondary V S and tertiary V T tangential wind perturbations added to the primary vortex are the perturbations that form a doubleand/or triple-peaked tangential wind profile when they are imposed on the basic state (e.g. Schubert et al. 980). In the developing stage of a secondary eyewall a broad region of nearly constant boundary layer tangential winds form underneath the new eyewall followed by increased inflow and enhanced convergence in the boundary layer (Wu et al. 202; Huang et al. 202). Using (8) the cyclostrophic wind balance for an axisymmetric TC is employed to derive a single pressure profile from the multiple-maxima wind profiles and is presented later in the subsequent subsection. b. Cyclostrophic wind balance The relationship between aircraft flight-level tangential winds and pressures in TCs is well represented using the gradient wind approximation (Willoughby 990ab 99 20). As mentioned previously in section 2c avoiding the problem of the unbounded radial integral of fv G as the outer limit of integration to compute the pressure profile approaches infinity allows one to facilitate the computation of the pressure from the cyclostrophic wind relation. Cast in terms of geopotential height Z of an isobaric surface instead of surface pressure (e.g. Willoughby and Rahn 2004) the geopotential height is obtained by integrating the cyclostrophic relationship radially inward from an environmental geopotential height Z E. The result is given by Z(r) 5 Z E g ð r V 2 C (r0 ) r 0 dr 0 (0) where Z E [5Z(r / )] is the environmental (undisturbed) geopotential height at infinitely far from the TC center g is the acceleration of gravity and r 0 is a dummy variable for the integration which is done numerically. Equation (0) can be extrapolated from the aircraft flight level to the surface under the assumption of the mean temperature T 5 (T SS T FL )/2 of a tropical atmospheric layer between the flight level (subscript FL) and the sea surface (SS) where T SS K is the typical temperature at the sea surface and T FL is the temperature at the flight level. Assuming that T is independent of r and substituting (0) into the hypsometric equation yields the surface pressure P SFC within the TC region: where " P SFC 5 P E exp R d T ð r # VC 2 (r0 ) r 0 dr 0 () gze P E 5 P FL exp. (2) R d T Here P FL is the pressure altitude at the flight level outside the TC region R d is the gas constant for dry air and P E is the asymptotic environmental surface pressure that is extrapolated from Z E in the environmental tropical atmosphere. Equation () contains the pressureadjusted flight-level tangential winds but is extrapolated to the pressure at the sea surface. In this study the Z E values of 850 and 700 hpa in the U.S. Standard Atmosphere 976 [(Committee on Extension to the Standard Atmosphere) COESA 976] are respectively 457 and 302 m. Inward radial integration of () requires a known environmental pressure P E at the sea surface and at a large radius where a zero asymptotic cyclostrophic wind occurs. Since P E is chosen to be the mean value of 04.3 hpa based on their climatological analyses of Knaff and Zehr (2007) the calculated values of T respectively are found to be 278 and 282 K for the mean temperature of the standard 700- and 850-hPa layers at Z E. This mean surface pressure is only representative of the Atlantic basin. Furthermore () involves an integral that is calculated using the trapezoidal rule (e.g. Press et al ). c. Partitioned pressure profiles for multiple-maxima eyewall tangential wind profiles In the cyclostrophic wind relationship a total surface pressure P TOT corresponds to the radial profiles of the total (summed) cyclostrophic winds and is obtained by incorporating (8) into (). As a consequence P SFC ([P TOT ) is partitioned into three different pressure components that correspond to the individual radial profiles of V P V S and V T respectively P SFC (r) [ P TOT (r) 5 P P (r)p S (r)p T (r) PE 2 (3) where the partitioned surface pressure components are expressed by

9 892 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 P P (r) 5 P E exp P S (r) 5 P E exp P T (r) 5 P E exp ( ( ( R d T R d T R d T ð r " ) V 2 P (r 0 ) V P (r 0 )V S (r 0 ) V P (r 0 )V T (r 0 ) #dr 0 (4a) r 0 ð r " ) V 2 S (r 0 ) V S (r 0 )V P (r 0 ) V S (r 0 )V T (r 0 ) #dr 0 and (4b) r 0 ð r " ) V 2 T (r 0 ) V T (r 0 )V P (r 0 ) V T (r 0 )V S (r 0 ) #dr 0. (4c) r 0 Note in (4) that P E [5P(r / )] is the extrapolated environmental surface pressure at which the winds decrease asymptotically to zero infinitely far from the TC center. On the right-hand side of (4a) VP 2 contributes most to the primary surface pressure P P. The product of V P and V S partially contributes to P P only if there is a secondary wind maximum (V SX 6¼ 0) in the radial profile of V S. Likewise the product of V P and V T contributes (though to a lesser extent) to P P only if a tertiary wind maximum (V TX 6¼ 0) occurs in the radial profile of V T. On the right-hand side of (4b) VS 2 mainly contributes to the secondary surface pressure P S while the products V S V P and V S V T each play a minor role in modulating P S provided that the primary and tertiary wind maxima are present (i.e. V PX and V TX 6¼ 0). An analogous description can be applied to (4c). In later sections the relative contributions of V P V S and V T to the total surface pressure P SFC are explored and compared to elucidate the role of the partitioned tangential wind profiles in the physical behavior of the corresponding surface pressure profiles and surface pressure deficits at the storm center. d. Partitioned central surface pressure deficits for multiple-maxima eyewall tangential wind profiles To obtain a total central surface pressure deficit DP SFC that corresponds to the sum of the three individual tangential wind profile components (V P V S andv T ) the radial gradient of P SFC can be partitioned by differentiating (3) with respect to r yielding P SFC (r) r 8 >< >: [ P TOT (r) r 5 P S (r)p T (r) P 2 E P P (r)p S (r) P 2 E P P (r) r P T (r) r P P (r)p T (r) P S (r) PE 2 r (5) where the partitioned radial surface pressure gradient components are given by P P (r) 5 P P (r) " # V 2 P (r) V P (r)v S (r) V P (r)v T (r) r R d T r (6a) P S (r) 5 P S (r) " # V 2 S (r) V S (r)v P (r) V S (r)v T (r) r R d T r and (6b) P T (r) 5 P T (r) " # V 2 T (r) V T (r)v P (r) V T (r)v S (r). r R d T r (6c) Here (6a) (6c) reveal that the radial pressure gradient fluctuations are concentrated inside the radii of multiplemaxima eyewall tangential winds. The total central surface pressure deficit DP SFC thus is obtained by integrating (5) radially inward from P E yielding DP SFC (r) [DP TOT (r) 5DP P (r) DP S (r) DP T (r) (7) where the partitioned central surface pressure deficit components are expressed by DP P (r) 5 P 2 E DP S (r) 5 P 2 E DP T (r) 5 P 2 E ð 0 ð 0 ð 0 P S (r 0 )P T (r 0 ) P P (r0 ) r 0 dr 0 (8a) P P (r 0 )P T (r 0 ) P S (r0 ) r 0 dr 0 and (8b) P P (r 0 )P S (r 0 ) P T (r0 ) r 0 dr 0. (8c)

10 JUNE 203 W O O D E T A L. 893 As discussed later in the subsequent section (7) (8) are useful in diagnosing and evaluating the significant fluctuations in partitioned central surface pressure deficits during the ERC. z i 5 V i r V " # i r 5 k i v hi 2 k i (k i 2 h i )r h /l i i i h i 2 k i k i r h /l v i i i (22) 4. Partitioned gradient and cyclostrophic vorticity for multiple-maxima eyewall tangential wind profiles For axisymmetric flow absolute vorticity z A is calculated as z A 5 z R f (9) where z R 5 V/ r V/r is the relative vorticity. The socalled gradient vorticity may be obtained by replacing the subscript R by G on the first right-hand side of (9) and is given by z G 5 V G / r V G /r. Substituting (5) into this vorticity gives z G 5 z C 2 f (20) where z C (5 V C / r V C /r) is defined as the cyclostrophic vorticity. The formulation in (20) works well within a couple hundred kilometers of the TC center. After substituting (8) into the cyclostrophic vorticity z C can easily be partitioned into three different vorticity components (z P z S z T ) that correspond to the individual radial profiles of V P V S and V T respectively and are given by z C 5 å 3 i5 z i 5 z P z S z T. (2) Following the vorticity derivation of Wood and White (20) z P z S and z T are obtained by substituting (9) into (2): where the partitioned curvature vorticity (angular velocity of rotation) v i component is given by 2 v i 5 (V ix g i ) 4 R ix h l i i r k i 2 i (h i 2 k i k i r h i /l i i ) l i 3 5. (23) The reason why k i $ in () (2) and also (23) is to avoid difficulty with a vorticity singularity at the vortex center if it is continued to r 5 0 (Wood and White 20). The exponent k i 2 inr k i2 i in (23) reveals that if 0 k i the vorticity profile produces the singularity as r/0. The profile has the undesirable property that 0 k i vorticity is infinite along the vortex axis. Difficulty with the singularity can be avoided by using k i $ only. Now (9) becomes z A 5 z G f 5 z C 5 z P z S z T. (24) Interestingly f does not appear after z C and also z P z S z T ; however the Coriolis parameter in terms of R icx is included in (9). Following the Wood and White approach a critical point of partitioned vorticity peak z i represents a location of maximum vorticity value and are given by (hi 2 k R zix 5 R i )a ki /h i i ix hi 2 k h i 2 k i a i a i. 0. (25) i Here a i is a critical solution obtained with the quadratic formula derived by Wood and White and is expressed by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a i 5 k i (2k i h i l2 i ) 2 [k i (2k i h i l 2 i )] 2 2 4k 2 i ( l2 i )(k 2 i 2 ) 2k 2. 0 i ( l2 i ) l i. 0. (26) The reason for using (25) is to locate the radius where z ix is associated with partitioned local low pressure as will be presented in this study. 5. TC simulation results This section uses a TC simulator to provide what each input parameter (V X R X k h l) may be able to deduce about TC wind structure in terms of intensity strength and size changes (e.g. Merrill 984). We performed parameter tests by varying at least one of the parameters while keeping other parameters unchanged. Finally we presented a detailed discussion on the manner in which a model TC intensifies strengthens and changes in size. Our approach was similar to that of Knaff et al. (20) who investigated the effects of fine-tuning their different parameters on the model TC intensity strength and size. Merrill (984) pioneered the use of TC wind structural variability in terms of intensity strength and size. Intensity is measured by minimum sea level pressure or maximum

11 894 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 FIG. 5. A schematic of changes in core intensity (solid curve) outer core wind strength (dotted curve) and size of the azimuthal wind s initial radial profile (dashed curve) in a TC. [From Merrill (984).] azimuthal wind in the primary vortex core; outer core wind strength is a spatially averaged tangential wind speed over an annulus between 00 and 250 km from the TC center (Weatherford and Gray 988ab); and size is the axisymmetric extent of gale-force (7 m s 2 )wind or the average radius of the outermost closed isobar (ROCI). These are illustrated in Fig. 5 as changes from an initial tangential wind profile (dashed curve). There is little confusion that arises concerning the term core intensity surrounding the RMW (Fig. 5). The term does not indicate clearly how variability in the inner wind profile and/or in the shape profile straddling the maximum is related to a change in intensity. The term could be rectified by replacing core intensity by inner core average winds. For instance tangential wind distributions have U- V- and bowl-shaped structures (e.g. Fig. a) in the inner core of intense mature hurricanes as seen in numerous examples of the flight-level profiles of tangential wind (Willoughby et al. 982 among others) as well as the radial profiles of numerically evolving tangential wind vorticity and surface pressure minimum (Schubert et al. 999; Kossin and Schubert 200; Hendricks et al. 2009). Some of the distributions in the primary eyewall exhibit the broadly profiles of tangential wind (e.g. Fig. c); others display the peaked profiles noted by WDR. We use surface pressure minimum as a proxy for intensity because it is our goal via our equation (7) to investigate how a change in the shape and distribution of tangential wind directly affects intensity strength and size. For example a transition of one shape profile into a different shape profile might produce a change in intensity strength and/or size as one goes progressively either (i) from top to bottom of Fig. 2 or from left to right or (ii) from the initial broadly peaked profile (dashed curve) to the final sharply peaked profile (solid curve) in Fig. 5. From extensive aircraft measurements Willoughby (990ab 99 20) found that the gradient wind balance was a good approximation to the azimuthally averaged tangential winds in most portions of an inner eyewall above a boundary layer where the strong radial inflows dominate. The National Oceanic and Atmospheric Administration/Atlantic Oceanographic and Meteorological Laboratory/Hurricane Research Division (NOAA/ AOML/HRD) flight-level data archive (Willoughby and Rahn 2004) provided consistent wind and geopotential height observations in hurricanes as functions of azimuth and time at fixed pressures. By averaging over azimuth and allowing for linear variation of the wind and geopotential height at fixed radii with time these data provided an improved depiction of the time-varying azimuthally symmetric structure at a single level over 4 6 h (Willoughby 990ab). The maximum of the mean wind and its radius determined in this fashion were consistent estimates of RMW and V X. The profiles also have the advantage of being close to gradient balance except (as in the boundary layer under the eyewall) where radial mean accelerations were large. It is important to recognize that the maximum azimuthally averaged wind and its radius were different from the average maximum wind and the average radius of maximum wind (i.e. mature TCs were rarely always symmetric over their lifetimes). The individual wind profiles that made up the mean varied from it in interesting ways. They were generally not in gradient balance and many (if not most) of the individual profile wind maxima were stronger than the azimuthal mean maximum but at different radii. The mean-wind profile was generally broader than the profiles that composed it because these maxima fell at different radii. As reported in the TC literature many (but not all) of the supergradient winds appeared as the results of asymmetric radial decelerations (Kepert 200; Kepert and Wang 200) when analyzed in this way. A key strength of this analysis was that the variations among the observed profiles can be treated as perturbations (nonnecessarily linear) on a well behaved nearly balanced mean vortex. Since translation of the TC and the degree of overwater wind asymmetry were not accounted for in the model the model TC was assumed to be axisymmetric and stationary in a quiescent tropical environment (i.e. no external atmospheric influences or ocean interaction). The cyclone was assumed to be positioned at 208N latitude in this study. This latitude was close to the average latitude of 23.78N based on their climatological analyses of Knaff and Zehr (2007). Note that this mean latitude is only representative of the Atlantic basin. Table lists the selected parameter values used for our 2 experiments. The main reason for presenting the

12 JUNE 203 W O O D E T A L. 895 TABLE. Model parameter values that produced the radial profiles of V i in the 2 experiments. The subscript i represents the number of individual wind profiles [i.e. i 5 corresponding to P (primary) i 5 2toS (secondary) and i 5 3toT (tertiary)]. Bold numbers for hurricanes B G indicate changes in parameter values relative to hurricane A. TC expt V ix (m s 2 ) R ix (km) k i h i l i Tropical storm V P : V PX 5 20 R PX 5 00 k P 5.0 h P 5.5 l P 5.0 Hurricane A V P : V PX 5 45 R PX 5 20 k P 5.5 h P l P Hurricane B V P : V PX 5 60 R PX 5 20 k P 5.5 h P l P Hurricane C V P : V PX 5 45 R PX 5 50 k P 5.5 h P l P Hurricane D V P : V PX 5 45 R PX 5 20 k P h P l P Hurricane E V P : V PX 5 45 R PX 5 20 k P 5.5 h P 5.8 l P Hurricane F V P : V PX 5 45 R PX 5 20 k P 5.5 h P l P 5.0 Hurricane G V P : V PX 5 45 R PX 5 20 k P 5.5 h P l P Hurricane H V P : V PX 5 45 R PX 5 20 k P 5.5 h P l P V S : V SX 5 5 R SX k S h S l S 5.0 Hurricane I V P : V PX 5 55 R PX 5 0 k P 5.5 h P l P V S : V SX 5 25 R SX 5 00 k S h S l S Hurricane J V P : V PX 5 20 R PX 5 55 k P 5.3 h P 5.7 l P V S : V SX 5 40 R SX 5 25 k S h S l S 5 0. Hurricane K < V P : V PX 5 20 R PX 5 0 k P 5.3 h P 5.7 l P V S : V : SX 5 40 R SX 5 50 k S h S l S 5 0. V T : V TX 5 2 R TX k T h T l T side-by-side panels in Figs. 6 is to compare the impact of changing one (or more) input parameter (e.g. k) on the two different TCs radial profiles of V C V G R C z A z G P SFC and DP SFC. Our parametric model is capable of extending a wind profile out to a large radius at which the tangential wind decreases asymptotically to zero. In this study the outer limit of 250 km was chosen because details in the portions of the wind and pressure structures near the TC center were displayed for ease of readability. a. Hurricanes with single-maximum eyewall tangential winds The input parameters (V PX R PX k P h P l P ) are listed in Table for initializing a single-maximum (primary) wind profile in a model mature hurricane. We set V SX and V TX equal to zero in (9) so that (8) (9) (3) (8) and (24) (25) were reduced to the following form: V C 5 V P (27) V P 5 (V PX g P )f P g P [ 2 R PCX 5 V PX fr PX f P 5 h l P P rk P P (h P 2 k P k P r h /l P P P ) l P s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! 2 R PCX 4R 2 PCX and r P [ r R PX (28a) (28b) ð r V 2 P P SFC 5 P P 5 P E exp (r0 ) R d T r 0 dr 0 (29) ð 0 P DP SFC 5DP P 5 P (r 0 ) r 0 dr 0 (30) z A 5 z C 5 z P and (3) (hp 2 k R zpx 5 R P )a kp /h P P PX. (32) h P 2 k P a P In (8) we set P S (r 0 )/P E 5 P T (r 0 )/P E 5 thus giving (30). Using the available cyclostrophic wind in (27) the gradient wind readily is calculated from (3). Figure 6 shows the radial profiles of primary cyclostrophic V C and gradient V G winds primary cyclostrophic z C and gradient z G vorticities cyclostrophic Rossby number R C surface pressure P SFC and central surface pressure deficit DP SFC in the model tropical storm and hurricane A. A tropical storm illustrates a nondeveloping TC when the shape velocity parameters of k P h P and l P (Table ) produce a nearly flat wind profile (Fig. 6a). Here V G (corresponding z G ) approaches V C (corresponding z A ) when R C is large near the TC center (Figs. 6ab). When R C decreases with large radius V G (z G ) deviates from V C (z A ) and more slowly decreases with radius than does V C (z A ). The flat wind profile of V C is supported by the fairly uniform radial pressure gradient (Fig. 6c); the central surface pressure minimum is 29 hpa. AchangeinV PX R PX k P h P andl P (Table ) evolves the tropical storm into hurricane A. As expected these TCs grow and strengthen during the transition from the

13 896 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 FIG. 6. Radial profiles of (a)(d) cyclostrophic V C and gradient V G winds (m s 2 ); (b)(e) cyclostrophic Rossby number R C absolute z A and gradient z G vorticities (s 2 ); and (c)(f) surface pressure (P SFC hpa) for a tropical storm having a nearly flat tangential wind and hurricane A having single-maximum eyewall tangential wind. In (a) arrows are indicated and described in text. In (d) and (f) gray curves represent the radial profiles from the tropical storm for comparison. In (c) and (f) DP SFC represents the surface pressure deficit between the current surface pressure and environmental pressure and not the difference between the gray and black curves. In (c) and (f) an upward-pointing arrow adjacent to number represents the radius R zpx where maximum vorticity z PX occurs. tropical storm to hurricane intensity as a comparison between the left and right panels of Fig. 6 shows. The central surface pressure minimum falls from 29 to 24 hpa; vorticity concentration initially located at the tropical storm center is progressively displaced toward R zpx 5 3 km where the strongest gradient of the inner wind profile occurs in hurricane A. At the same time the primary eyewall vorticity maximum z PX increases. Note that R zpx is calculated from (32) suggesting that z PX is associated with the central surface pressure minimum. The annular pattern of vorticity is characteristic of a twocelled structure an axial sinking motion surrounded by an annular convective-scale updraft along the inner edge of the primary eyewall that slopes radially outward with increasing altitude (Jorgensen 984). It is vital to point out that (3) is optimized to describe a relatively peaked eyewall wind maximum. Because of the way the cyclostrophic Rossby number R C changes with radius (Fig. 6b) a relatively broad cyclostrophic wind profile with its maximum at 00 km can result in dislocation of the radius of maximum gradient wind toward the TC center. This maximum occurs at 85 km where V G / r 5 0 as indicated by an upward-pointing arrow in Fig. 6a. When the V C profile is relatively flat

14 JUNE 203 W O O D E T A L. 897 FIG. 7. Radial profiles of (a)(d) cyclostrophic V C and gradient V G winds (m s 2 ); (b)(e) cyclostrophic Rossby number R C absolute z A and gradient z G vorticities (s 2 ); and (c)(f) surface pressure (P SFC hpa) for hurricanes B and C having single-maximum eyewall tangential winds. In (a)(c)(d) and (f) gray curves represent the radial profiles of winds and pressures from hurricane A for comparison. In (c) and (f) DP SFC represents the surface pressure deficit between the current surface pressure and environmental pressure and not difference between the gray and black curves. In (c) and (f) an upward-pointing arrow adjacent to number represents the radius R zpx in km where maximum vorticity z PX occurs. R C / r changes the radial gradient enough to move the gradient wind maximum away from the radius where V C / r 5 0 toward the TC center. When the V C profile is sharply peaked the displacement of the radius of the gradient wind peak disappears (Fig. 6d). Thus a mathematical adjustment for the high Rossby number gradient wind approximation with its relatively broad profile is needed. Figures 7 9 provide what the model parameters (m 5 [V X R X k h l] T ) may deduce about the wind changes in a TC having only a single-maximum (primary) eyewall tangential wind. In each experiment the parameters are used to construct each idealized V C and V G profiles and the resulting R C z A z G P SFC anddp SFC profiles. As shown in Table hurricane A (Fig. 6d) is used as a reference hurricane with the constant parameters. This hurricane is compared with other experimental hurricanes (B G) when at least one of five free parameters is varied (bold numbers). Comparison of one variable parameter to the one identical reference parameter aids in determining how variations in the shape strength and/or size of a selected wind profile are related to changes in TC intensity as will be discussed in subsequent subsections.

15 898 M O N T H L Y W E A T H E R R E V I E W VOLUME 4 ) VARIABLE V X Idealized tangential wind profiles are constructed with constant R PX k P h P and l P but variable V PX from hurricane A to major hurricane B (Fig. 7a). The TC experiences increasing intensity and size as V PX increases from 45 to 60 m s 2 (indicated by bold numbers shown in Table ) and its size of the outer circulation increases (as comparison between black and gray curves in the figure illustrates). The corresponding profiles of z A z G and R C in hurricane B (Fig. 7b) are similar to those in hurricane A except that the former profiles have increased in magnitude. The corresponding central surface pressure deficit (Fig. 7c) is deepening (272 hpa) in hurricane B compared to 24 hpa in hurricane A (Fig. 6f). This is in general agreement with the analyses of Weatherford and Gray (988b) Croxford and Barnes (2002) and Knaff et al. (20) who showed that an increase in eyewall tangential wind maximum was strongly correlated with the TC intensity (decreasing surface pressure minimum). 2) VARIABLE R X A change in R PX when other parameters are held constant produces a change in the wind structure from hurricane A to hurricane C (Table ). The variable R PX plays a role in modulating the radial profiles of V C V G R C z A z G and P SFC as shown in the right panels of Fig. 7. Only the central surface pressure minimum DP SFC is unaffected (Fig. 7f). At RMW (5R PX ) 5 50 km the V G peak is about m s 2 weaker than the V C maximum or the ratio of V G to V C is /( 0:5R 2 C ) [e.g. see (4)] which is always less than.0 for R C. 0. The reason (i) why DP SFC 5243 hpa at the center of hurricane C (Fig. 7f) is slightly lower than DP SFC 524 hpa at the center of hurricane A (Fig. 6f) and (ii) the V C peak associated with hurricane C is slightly higher than that associated with hurricane A is owing to the product of V PX and g P as calculated from (28a) and also Fig. 3b. Since the difference between these pressure deficits is insignificant the deficit appears to be relatively insensitive to variations in R PX although the variable R PX appears to produce a size change of the outer wind profile. This finding was consistent with Holland (980) Weatherford and Gray (988b) and Knaff et al. (20) who showed that the variable R PX is independent of the surface pressure minimum. As a result the variable may affect a size change but does not have an effect on the intensity of a TC. 3) VARIABLE k Numerical TC simulations revealed variability in the shape of inner profiles of tangential wind vorticity and surface pressure near the TC center (Schubert et al. 999; Kossin and Schubert 200; Hendricks et al. 2009). To mimic such a shape profile we explore the role in shaping the inner wind profile by making k P variable while keeping other model variables fixed in hurricanes A and D (Table ). The initial k P and h P (5k P 0:5) values of.5 and 2.0 associated with hurricane A were increased to 3.5 and 4.0 associated with hurricane D. The reason for changing h P is to keep the outer profile unchanged. As comparison between hurricane A s inner wind profile (gray curve in Fig. 8a) and that of hurricane D illustrates a change in k P modulates the inner profiles of V C V G R C z A andz G (Figs. 8ab). In hurricane D the tangential wind distribution inside the RMW (Fig. 8a) exhibits the bowl-shaped profile; the corresponding vorticity profile (Fig. 8b) may be described as a thin annulus of strongly enhanced vorticity embedded in the primary eyewall with relatively weak vorticity in the eye and outside the eyewall. This profile results from barotropic instability near the RMW (Schubert et al. 999). The simulated profiles of tangential wind (Fig. 8a) and vorticity (Fig. 8b) resemble the radial distributions of numerically evolving tangential wind and vorticity (Kossin and Schubert 200 their Figs. 3 and ; Hendricks et al their Fig. 3) and the flight-level profiles of tangential wind and vorticity observed in Hurricanes Guillermo (997) and Gilbert (2008) of Kossin and Schubert (200 their Fig. ). Corresponding to the vorticity profile (Fig. 8b) the P SFC profile inside about r 5 20 km is flatter (black curve in Fig. 8c) than that in hurricane A (gray curve). Simultaneously DP SFC rises from 24 hpa in hurricane A to 234 hpa in hurricane D thus producing a decrease in intensity. This rise occurs when the quasi-v-shaped profile of tangential wind in hurricane A transitions into the bowl-shaped profile in hurricane D as a result of increasing k P. The shapes of the simulated pressure profile (Fig. 8c) corresponding to the changing tangential wind and corresponding vorticity profiles (Figs. 8ab) bear resemblance to those of Schubert et al. (999) Kossin and Schubert (200) and Hendricks et al. (2009). A transition from a U-shaped to a V-shaped profile of tangential wind inside the primary eyewall over a period of a few hours or less was documented by Kossin and Eastin (200) who used aircraft flight-level data to investigate the kinematic and thermodynamic distributions within the eye and eyewall of Hurricane Diana (984). The changes in the inner wind profile result in measurable and physically important differences in the eyewall vorticity (Fig. 8b) and central surface pressure (Fig. 8c). Croxford and Barnes (2002 their Fig. 3a) observed the U-shaped profiles of storm-relative tangential wind