PUBLICATIONS. Journal of Advances in Modeling Earth Systems

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1 PUBLICATIONS Journal of Advances in Modeling Earth Systems RESEARCH ARTICLE./3MS99 Key Points: Structure of wind field and secondary circulation depends on vortex structure 3-D model is governed by vertical advection and vertical diffusion in inner core Slab model is dominated by radial and azimuthal advection in inner core Correspondence to: G. J. Williams, Citation: Williams Jr., G. J. (5), The effects of vortex structure and vortex translation on the tropical cyclone boundary layer wind field, J. Adv. Model. Earth Syst., 7, 88 4, doi:./3ms99. Received 6 DEC 3 Accepted JAN 5 Accepted article online 6 JAN 5 Published online FEB 5 Corrected 3 MAR 5 This article was corrected on 3 MAR 5. See the end of the full text for details. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial-NoDerivs License, which permits use and distribution in any medium, provided the original work is properly cited, the use is non-commercial and no modifications or adaptations are made. The effects of vortex structure and vortex translation on the tropical cyclone boundary layer wind field Gabriel J. Williams, Jr. Department of Physics and Astronomy, College of Charleston, Charleston, South Carolina, USA Abstract The effects of vortex translation and radial vortex structure in the distribution of boundary layer winds in the inner core of mature tropical cyclones are examined using a high-resolution slab model and a multilevel model. It is shown that the structure and magnitude of the wind field (and the corresponding secondary circulation) depends sensitively on the radial gradient of the gradient wind field above the boundary layer. Furthermore, it is shown that vortex translation creates low wave number asymmetries in the wind field that rotate anticyclonically with height. A budget analysis of the steady state wind field for both models was also performed in this study. Although the agradient force drives the evolution of the boundary layer wind field for both models, it is shown that the manner in which the boundary layer flow responds to this force differs between the two model representations. In particular, the inner core boundary layer flow in the slab model is dominated by the effects of horizontal advection and horizontal diffusion, leading to the development of shock structures in the model. Conversely, the inner core boundary layer flow in the multilevel model is primarily influenced by the effects of vertical advection and vertical diffusion, which eliminates shock structures in this model. These results further indicate that special care is required to ensure that qualitative applications from slab models are not unduly affected by the neglect of vertical advection.. Introduction Numerous studies have shown that the tropical cyclone boundary layer (TCBL) strongly regulates the convective and dynamical processes needed to sustain mature tropical cyclones. In particular, the TCBL dynamically controls the radial distribution of heat, moisture, and angular momentum that ascends into the eyewall clouds of a mature tropical cyclone. Therefore, an accurate understanding of the dynamics of the TCBL is a critical component in understanding the dynamics of mature tropical cyclones in general. In particular, the distribution of winds within the TCBL determines much of the overall structure of the mature tropical cyclone. In Williams et al. [3, hereinafter W3], the structure of the boundary layer wind field in Hurricane Hugo (989) was interpreted in terms of an axisymmetric slab boundary layer model. Near the inner edge of the eyewall, there were multiple updraft-downdraft couplets with the strongest updraft at 434 m height in Hugo exceeding m s. The updraft-downdraft couplet in Hugo was explained by the formation of a shock in the boundary layer radial inflow with very small radial flow on the inside edge of the shock and large radial inflow on the outside edge of the shock. In W3, a shock structure was defined as a flow discontinuity in which nonlinear horizontal advection and horizontal diffusion play dominant roles in the dynamics, similar to the shock dynamics of the viscous Burgers equation. The existence of shocks was further demonstrated through the analysis of characteristic curves where it was shown that the location of the eyewall was coincident with the intersection of characteristic curves in the numerical solution [see Williams et al., 3, Figures 9 and ]. According to the results of W3, strong inflow in the boundary layer also leads to large azimuthal wind tendency, resulting in steep gradients in the boundary layer azimuthal wind field. Since the radial derivative of azimuthal velocity is related to vertical vorticity, a thin sheet of high vorticity develops in the vicinity of the eyewall. This paper continues the work from W3 by examining the sensitivity of radial vortex structure and vortex translation speed on the structure of the TCBL wind field. According to the classic theory of shock formation, it can be shown that, for a given initial condition, the time scale of shock formation for Burgers equation is inversely proportional to the gradient of the initial radial velocity field [Whitham, 974]. If the dynamics of WILLIAMS VC 4. The Authors. 88

2 the TCBL for the slab model is governed by shock structures, then this suggests that the development of shocks in the slab boundary layer model should depend upon the radial structure of the gradient wind profile. This paper addresses this hypothesis by examining the sensitivity of shock development for vortices with varying gradient wind profiles. For an axisymmetric, stationary vortex, W3 showed that the shock structures in the slab model are circular. However, if the vortex translates, asymmetries develop in the boundary layer flow and the shock may become spiral shaped. Numerical simulation of the low azimuthal wave number structure of such asymmetries was pioneered by Chow [97] and Shapiro [983]. Higher-resolution simulations of the motioninduced asymmetries for slowly moving vortices, which also resolved the vertical structure, were also presented in Kepert [], Kepert and Wang [], and Kepert [a]. The present paper will continue to address this topic by examining the evolution of the boundary layer wind field for rapidly translating vortices. As mentioned in W3, the slab boundary layer model can be regarded as a model that is at or near the bottom of a hierarchy of boundary layer models of increasing complexity. As discussed by Kepert [a, b], slab models do not capture certain important features found in multilevel models [Montgomery et al., ; Kepert, ; Kepert and Wang, ] of the TCBL, e.g., the shallow boundary layer depth found near the cyclone core and the outward radial flow just above the boundary layer. A question that should be raised is whether the essence of shock formation is a common feature of the hurricane boundary layer or simply an artifact of depth averaging in slab boundary layer models. This paper will address this topic by comparing the numerical simulations of a slab model to a multilevel model. We will see that there are important differences in the simulations produced by the slab and multilevel models. The paper is organized in the following way. Section describes the numerical models used in the present study. Section 3 discusses the evolution of a stationary vortex using both models. Section 4 examines the effects of radial vortex structure on the evolution of a stationary vortex using both models. Section 5 examines the effects of vortex translation on the evolution of a vortex using both models. The main conclusions are summarized in section 5.. The Numerical Models.. The -D Slab Boundary Layer Model In contrast to W3, the slab boundary layer model used here is a full two-dimensional model in Cartesian coordinates in order to allow for the development of boundary layer asymmetries. We consider motions in the frictional boundary layer of an incompressible fluid on an f-plane. The layer is assumed to have constant depth h, with eastward and northward velocities u(x,y,t) and v(x,y,t) that are independent of height between the surface and height h, and with vertical velocity w(x,y,t) at height h. The horizontal velocity components are discontinuous across the top of the boundary layer. In the overlying layer, frictional effects vanish, and the flow is in balance with the overlying pressure gradient force. The boundary layer flow is driven by the same pressure gradient force that occurs in the overlying fluid. The boundary layer momentum equations then take the w h ðu "uþfv x r ðf V gr r ÞV grkr uc D U u w h ðv "vþfu y r ðf V gr r ÞV grkr vc D U v h @v Þ; () () WILLIAMS VC 4. The Authors. 89

3 U:78ðu v Þ = (4) is the wind speed at m height, which is assumed to be 78% of the mean boundary layer wind speed (as supported by the dropwindsonde data of Powell et al. [3]), V gr is the specified gradient wind field, ðu; vþ " is the velocity field above the boundary layer (as prescribed by the V gr field), w 5 ðwjwjþ, and w 5 ðjwjwþ. For the calculations shown here, we have chosen the boundary layer depth as h 5, the Coriolis parameter as f55:3 5 s, and the horizontal diffusivity as K 5 m s. Concerning the dependence of C D on wind speed, we use the form given in W3 8 >< :7=U:4:764U U 5 C D 5 3 :6:546 exp U5 U 5 ; (5) >: 7:5 where the m wind speed U is expressed in m s. The initial conditions uðx; y; Þ and v(x,y,), and the forcing V gr (r) will be discussed below. The slab boundary layer model is solved using fourth-order centered differencing on the square domain 6 x; y 6 with a grid spacing of m in the region x; y, increasing to using a variable-resolution stretched grid with a constant local stretching rate. For a full derivation of the slab boundary layer model, see Williams []... The 3-D Multilevel Boundary Layer Model The 3-D boundary layer model is based on the three-dimensional primitive equations for a continuously stratified, hydrostatic, Boussinesq atmosphere in height coordinates, as given in Kepert and Wang []. The governing equations are Du Dt K K M Dv Dt K K M Dh Dt 5K K H 5 g c p h @w ; where D=Dt5@=@tu@=@xv@=@yw@=@z. The horizontal diffusion is handled following Smagorinsky et al. [965] with the diffusion coefficient given by K h 5 k ðdxþ jdj; () where k 5.4 is the von Karman constant and D the total horizontal deformation. K M and K H are the vertical turbulent exchange coefficients for momentum and heat, respectively. These latter variables are given by the Louis et al. [98] parameterization scheme recommended by Kepert []. To model the air-sea interaction, we use the standard bulk aerodynamic formulation with the surface drag coefficient given by (5). To calculate the surface drag coefficient for this model, we assume that the wind speed at m height is 78% of the mean boundary layer wind speed, defined as the mean wind speed below m (as defined by Powell et al. [3]). This is done in order to improve consistency of the parameterization with the slab model. For the upper boundary condition, it is assumed that the vertical gradients of heat, velocity, and turbulent diffusion are zero. Similar to the slab boundary layer model, the pressure field at the top of the model will be prescribed through the gradient wind field V gr. Thus, the overlying layer and V gr is specified through the upper boundary condition, whereas the surface drag is specified through the lower boundary condition. The model is horizontally unstaggered on a doubly periodic horizontal domain of 3 with a horizontal grid spacing of using fourth-order centered advection. The model consists of 6 vertically WILLIAMS VC 4. The Authors. 9

4 staggered layers (on a Lorenz grid) from the surface to m, with the midpoints of the layers at z 5.5, 75., 4.5, 5., 3.5, 435., 56.5, 75., 86.5, 35.,.5, 45., 64.5, 875.,.5, and More information on the model characteristics can be found in Williams []..3. Initial Conditions As mentioned above, the boundary layer flow for both models is driven by the same pressure gradient force that occurs in the overlying fluid. In order to compare the results from both boundary layers, we will use the same initial condition. The pressure field at the top of both models is prescribed using the analytical profile of Holland [98] p5p c ðp n p c Þexp ða=r B Þ; () where p c is the central pressure, p n is the ambient pressure, and A and B are scaling parameters. Using the gradient wind equation, it can be shown that the corresponding gradient wind profile for this pressure field is ABðpn p c Þexp ða=r B Þ V gr ðrþ5 qr B r f = rf 4 ; (3) where q is the constant air density. A and B are two parameters which determine the RMW and maximum wind speed of the vortex. It can be shown [cf. Holland, 98] that the RMW and maximum wind speed of this profile are approximately given by r max A B ; Vmax = Bðpn p c Þ : (4) qe Physically, B defines the shape or peakedness of the gradient wind profile. Thus, B controls the radial gradient of the gradient wind profile (and the inertial stability outside the radius of maximum winds), which will be an important parameter in our model experiments. As discussed in Kepert and Wang [], the Holland [98] profile has a reversed radial vorticity gradient within the RMW and thus satisfies the necessary conditions for barotropic instability. To avoid this undesirable feature, the profile is modified within the RMW as follows: VðrÞ5c rc r c 3 r 3 ; r < r max ; (5) where c, c, and c 3 are chosen to make V, dv/dr, and d V=dr continuous at the RMW. For the slab boundary layer model and the multilevel model, the velocity field is initially in gradient wind balance, where V gr (r) has the form given in (3). For the multilevel model, the initial condition for potential temperature was given by the Jordan [958] sounding and initially the wind field is in gradient wind balance with the prescribed pressure field (), except in the lowest model layer where they were reduced by 35%. This is done in order to match observations of the wind field within the surface layer [Powell et al., 3]. The numerical models were run out for h, by which time all fields had attained a steady state. In the following sections, we consider the evolution of five vortices using the slab model and the multilevel model. The parameters used in defining the vortices are defined in Table. Vortex I has a moderate radial wind profile, similar to the structure of a mature tropical cyclone. Vortex II and III are much more peaked around the radius of maximum gradient winds and they will be used to examine how radial vortex structure affects the evolution of the wind field in both models. Vortex IV and V are embedded in 5. and. m s easterly flow, respectively, and they will be used to examine how vortex translation affects the evolution of the wind field in both models. We begin by discussing our control experiment: the evolution of Vortex I. 3. The Evolution of a Stationary Vortex 3.. Results From Slab Model Figure shows the evolution of azimuthal velocity V T, radial velocity V R, vertical velocity w, and relative vorticity f for Vortex I. Within the first hour of the model simulation, there are rapid changes in the velocity and vorticity fields. The boundary layer azimuthal wind field becomes supergradient in the region 35 < r < 45 WILLIAMS VC 4. The Authors. 9

5 Table. Parameters Defining the Vortices Discussed in the Text a Vortex Maximum Wind (m s ) RMW () b Vortex Movement (m s ) I II III IV V a The maximum wind is the gradient wind that would apply for a stationary vortex in the Holland [98] parametric model. Radius of maximum winds refers to the radius of maximum gradient wind, and b is the parameter determining the amount of peakedness in the Holland [98] radial wind profile. and subgradient for r > 45. Since f5ð=rþd=drðrv T Þ, the gradients in the azimuthal velocity field lead to a vorticity maximum centered around a radius of 35. Radial inflow velocities exceeding 5 m s quickly develop and sharp gradients form in the region 35 r 45. Since w5ðh=rþd=drðrv R Þ, the gradients in the radial velocity field lead to vertical velocities of approximately 5 m s. Moreover, the gradient in radial velocity changes sign within the region of 3 r 35, leading to sharp gradients in the vertical velocity fields within this region. In W3, these features were identified as boundary layer shocks because of their resemblance to flow discontinuities associated with Burgers equation. As the vortex approaches a steady state, the maximum radial inflow reaches 3.7 m s and the maximum azimuthal wind reaches 64.8 m s, which is approximately 6% supergradient. We also note that as the radial inflow moves radially outward, we also see that the region of supergradient winds also moves radially outward. This suggests that radial inflow plays an important role in advecting the azimuthal wind maximum. In response to the evolution of the azimuthal and radial velocity fields, we see that the vertical velocity and relative vorticity fields both move radially outward as well. Finally, we note that the peak frictional convergence lies on the inside edge of the RMW, consistent with W3, and the strongest Ean pumping occurs in the vicinity of maximum vorticity gradient, consistent with the linear Ean theory of Kepert [3]. To further investigate the evolution of the boundary layer flow, we now examine the steady state budget equations for radial velocity and absolute angular momentum. It can be shown that the tendency equations for the radial velocity and absolute angular momentum in cylindrical coordinates (r,k) are given by V T [m s ] t =. h t =. h t =. h t = 3. h t =. h V R [m s ] t =. h t =. h t =. h t = 3. h t =. h t =. h t =. h t =. h t = 3. h t =. h 8 6 t =. h t =. h t =. h t = 3. h t =. h ζ [s ] w [m s ] Figure. Slab boundary layer model results for Vortex I. The four figures show the (top left) boundary layer azimuthal velocity V T, (top right) radial velocity V R, (bottom left) relative vorticity f, and (bottom right) vertical velocity w for the inner region r. The results at the five different times t:; :; :; 3:; : h are indicated by the color coding. WILLIAMS VC 4. The Authors. 9

6 m s..5.5 Steady State Budget of V R TADV FR DIFF m s Steady State Budget of M a TADV FR DIFF Figure. (a) (top) Steady state budget analysis of radial velocity V R for Vortex I in the slab model. (b) (bottom) Steady state budget analysis of absolute angular momentum M a for Vortex I in the slab model. For the radial velocity and absolute angular momentum tendency, the components are radial advection (), azimuthal advection (TADV), vertical advection (), agradient force (), frictional drag force (FR), and horizontal diffusion R R fðv gr V T 5V R V R r@k V w R r V R V R a V r M r h V gr V T r C D U V R h ; r@k w h ðm am gr Þ C D U rv T h ; (6) (7) where M a 5rV T ð=þfr. Equation (6) shows that the contributions to the radial velocity tendency are, from left to right, radial advection, azimuthal advection, vertical advection, the centrifugal force, the pressure gradient force (written in terms of the gradient wind), the Coriolis force, horizontal diffusion, and surface drag. In the hurricane boundary layer, surface drag reduces the magnitude of the Coriolis force and centrifugal force, creating unbalanced flow in the boundary layer. A way to quantify this unbalanced flow in the boundary layer is through the agradient force, which is defined as the sum of the radial pressure gradient force, the Coriolis force, and centrifugal force [Huang et al., ; Smith et al., 9], i.e., 5 V T V gr fðv T V gr Þ: (8) r Figure a shows the azimuthally averaged contributions for the radial velocity tendency equation. For the radial velocity tendency, Figure a indicates that there is an approximate balance between the radial advection term () and the agradient forcing term (), as the vortex approaches a steady state. In particular, the large positive agradient force, located in the vicinity of the RMW, is associated with a large negative radial advection term. We note that the agradient force tends to be negative in the outer regions where the boundary layer flow is subgradient (V T V gr < ) and tends to be positive in inner regions where the boundary layer flow is supergradient (V T V gr > ). Thus, in the subgradient region, the effect of the agradient WILLIAMS VC 4. The Authors. 93

7 force is to decelerate an inflowing parcel toward the inner core of the vortex. In the supergradient region, the effect of the agradient force is to make an inflowing parcel slow down. Therefore, inflowing parcel from the outer regions begin to move faster than inflowing parcels within the inner region. This leads to steeper radial velocity gradients and thus stronger vertical velocities at the top of the boundary layer in the vicinity of the RMW. This is consistent with the shock interpretation given in W3, in which a Lagrangian interpretation of the model lead to intersecting characteristic curves and a nearly discontinuous behavior in radial velocity. Thus, the inward-directed agradient force leads to the development of near-surface inflow, consistent with the results of Huang et al. []. Also, note that the horizontal diffusion term is negligible everywhere except in the region of the largest gradient of radial velocity (i.e., in the vicinity of the shock). Equation (7) shows that the contributions to the absolute angular momentum tendency are, from left to right, radial advection, azimuthal advection, vertical advection, horizontal diffusion, and surface drag. Figure b shows the azimuthally averaged contributions for the absolute angular momentum tendency equation. Here we see that inward advection of angular momentum () is largely composed of horizontal diffusion (DIFF) and surface friction (FR). Since the inward advection of angular momentum is defined as V a =@r, this suggests that the inflowing air from the outer regions of the vortex advects the azimuthal wind maximum radially inward from the radius of maximum gradient wind. Moreover, as the inflowing air decelerates toward the vortex core, this leads to an increase in azimuthal wind maximum. Thus, for the slab boundary layer model, supergradient winds are generated by the inward advection of angular momentum. Since the radial advection of angular momentum outweighs surface frictional loss of angular momentum near the RMW, this indicates that horizontal diffusion must play an important role in balancing the inward advection of angular momentum in the vicinity of the RMW, as indicated in Figure b. The above analysis is consistent with the shock interpretation given in W3, in which the absolute angular momentum budget is largely balanced by radial advection and radial diffusion. When supergradient momentum develops in the hurricane boundary layer, the agradient force in the radial velocity equation is directed outward, leading to an outward-directed acceleration. According to the budget analysis in Figure, the only process that can maintain such strong radial inflow against this outward acceleration is radial advection. This implies that radial advection plays a dominant role in the dynamics of the slab boundary layer model, similar to the dynamics of Burgers equation. As inflowing air accelerates toward the core of the vortex, this inflowing air advects the azimuthal wind toward the vortex core and intensifies the azimuthal wind maximum. As supergradient winds are generated, inflowing air parcels encounter a region in which the direction of the agradient force changes sign, leading to large radial velocity gradients on the outer edge of the RMW. In this way, the shock structures observed in this model is the result of the rapid deceleration of inflowing air and the corresponding change in direction of the agradient force within the region of supergradient winds. According to the budget analysis of Figure, horizontal diffusion plays an important role in modulating the magnitude of the radial velocity gradient in the vicinity of the RMW. 3.. Results From Multilevel Model An azimuthally averaged radial cross section of the steady state flow of Vortex I is shown in Figure. We first note that the depth of the inflow layer (Figure, top) decreases with radius, from about at to below m in the eye region. This is consistent with observations [Kepert, 6a, 6b] and the linear theory of Kepert [], which shows that the boundary layer depth in the inner core of a tropical cyclone scales as I =, where I is the inertial stability. The maximum radial inflow of m s is at 75 m height and radius (which is about.5 times the RMW) and a radial outflow layer exists above the inflow layer. The frictionally forced updraft (Figure, bottom) peaks at.89 m s near the RMW at a height of approximately m with weak subsidence in the eye region. Similar to Kepert and Wang [], a boundary layer jet develops within the hurricane boundary layer (approximately z 5 6 m) at the RMW. The maximum azimuthal wind (Figure, middle) is 6.9 m s at a height of 5 m, which is about 3% supergradient. The supergradient flow maximizes within the inflow layer and extends upward into the lower portion of the outflow layer. This simulation is similar to Vortex III in Kepert and Wang []. Although many aspects of this simulation compares favorably to the slab model simulation, it is important to note that the vertical velocities produced in this model is about 5 times smaller than in the slab model. One reason for this is that the slab model does not capture the shallow boundary layer depth found near the cyclone core. Since w ð=hþð@u=@x@v=@yþ, where h is the boundary layer depth, a decreasing WILLIAMS VC 4. The Authors. 94

8 6 B =.6 B =. B =.4..9 B =.6 B =. B = B =.6 B =. B = V gr [m s ] 3 I gr [s ].6.5 d/dr (M gr ) [m s ] Figure 3. Radial profiles of (left) gradient wind V gr, inertial stability I gr, and gradient of angular momentum dm a =dr for Vortex I (black line with B 5.6), II (red line with B 5.), and III (blue line with B 5.4). boundary layer depth implies reduced vertical velocities. However, a scale analysis of shows that the velocity of the flow is approximately proportional to =h so the updraft at the top of the boundary layer should not vary appreciably as the boundary layer depth decreases toward the cyclone core. Another more important reason for this is that the slab model largely ignores the effect of vertical advection and vertical diffusion within the eyewall of the vortex. To investigate the impact of vertical advection on the steady state boundary layer flow, we now examine the steady state budget equations for radial velocity and absolute angular momentum. It can be shown that the tendency equations for the radial velocity and absolute angular momentum in cylindrical coordinates (r,k,z) are given R R T c K r@k R f V T r r V R V R T R K 5V a V K h r M a r r@k @M a K () Equation (9) shows that the contributions to the radial velocity tendency are radial advection, azimuthal advection, vertical advection, the agradient force, horizontal diffusion, and vertical diffusion. Figure 4a shows the vertical profile of the radial velocity budget component for a point in the eyewall of the vortex. In the lower half of the inflow layer, we see that the agradient force produces an inward acceleration (since the flow is subgradient in this region) and is largely balanced by friction and turbulent diffusion. In the upper half of the inflow layer, the agradient force produces an outward acceleration (since the flow is supergradient in this region) and is largely balanced by vertical advection with smaller contributions from turbulent diffusion and radial advection. In particular, the vertical advection of radial velocity peaks in the upper portion of the inflow layer where it helps to maintain the inflow against the outward acceleration due to the agradient force. The radial advection of V R is largest near the surface (where the radial inflow is strongest) and tends to strengthen the inflow in the vicinity of the RMW. However, radial advection weakens (in WILLIAMS VC 4. The Authors. 95

9 V T [m s ] t =. h t =. h t =. h t = 3. h t =. h V R [m s ] t =. h t =. h t =. h t = 3. h t =. h t =. h t =. h t =. h t = 3. h t =. h 8 6 t =. h t =. h t =. h t = 3. h t =. h ζ [s ] w [m s ] Figure 4. Slab boundary layer model results for Vortex II. The four figures show the (top left) boundary layer azimuthal velocity V T, (top right) radial velocity V R, (bottom left) relative vorticity f, and (bottom right) vertical velocity w for the inner region r. The results at the five different times t:; :; :; 3:; : h are indicated by the color coding. magnitude) with height and within the outflow layer, the radial advection of V R is smaller than the remaining terms. Equation () shows that the contributions to the absolute angular momentum tendency are radial advection, azimuthal advection (which is zero for an axisymmetric vortex), vertical advection, horizontal diffusion, and vertical diffusion. Figure 3b shows the vertical profile of the azimuthal momentum budget component for a point in the eyewall of the vortex. Within the inflow layer, the azimuthal momentum budget is largely a balance between friction (which tends to spin down the flow) and inward advection of M a (which tends to spin up the flow). Vertical advection of M a also makes an important contribution to weakening the azimuthal flow within the inflow layer. Each of these three terms changes sign near the top of the inflow layer. In the outflow layer, radial advection of M a weakens the supergradient flow and is responsible for the smooth return to gradient balance that occurs within this region. This deceleration is opposed by the combined effect of vertical advection and turbulent diffusion of M a, which act to transfer supergradient momentum upward from the inflow layer into the outflow layer. It is important to note that the budget analysis for the multilevel model is substantially different than in the slab boundary layer model. In the slab boundary layer model, radial advection plays the dominant role in the dynamics of the boundary layer flow, especially near the RMW. In contrast, the budget analysis of the multilevel model suggests that vertical advection plays a dominant role in the dynamics of supergradient flow near the RMW. In the multilevel model, vertical advection acts to strengthen the inflow through nearly all of the inflow layer, and acts to weaken the outflow in the lower portion of the outflow layer. Similarly, vertical advection acts to weaken the azimuthal flow up to the height of the supergradient jet (which is located slightly below the top of the inflow layer), but above this level contributes to maintaining the supergradient flow against the outward acceleration due to gradient imbalance. In contrast, in slab models, vertical advection has a negligible impact in vicinity of the RMW. This is consistent with the conclusions of Kepert [a, b], which suggested that the vertical averaging of the nonlinear advection terms in slab model imposes errors to the slab model. The negligible role that vertical advection plays in slab models in the inner core of the vortex implies that there are important differences between slab models and multilevel models. Supergradient flow within the boundary layer is caused by the agradient force, which leads to inward advection angular momentum as WILLIAMS VC 4. The Authors. 96

10 shown in Figures and 3. Thus, the agradient force drives the advective terms in both budgets. However, the way in which the boundary layer flow accelerates in response to the agradient force is different for both models. Since supergradient flow implies an outward acceleration due to the agradient force, the budget analysis of the multilevel model in Figure 3 indicates that the boundary layer flow within the inflow layer is maintained primarily by vertical advection and vertical diffusion (with a smaller contribution from radial advection), consistent with Kepert and Wang [] and Kepert [b]. In contrast, the budget analysis of the slab boundary layer model in Figure indicates that boundary layer flow within the inflow is maintained only by radial advection (all other terms act to weaken the inflow). This enables radial advection to play the dominant role in the dynamics of axisymmetric flow in the slab model, leading to shock development. Thus, while the shock interpretation of W3 is consistent with the slab model, it is inconsistent with the multilevel model. This is also evident by the weaker radial velocity gradients throughout the boundary layer in Figure, compared to Figure. 4. The Effects of Radial Vortex Structure on Vortex Evolution In this section, we perform a series of experiments using vortices with varying gradient wind profiles in order to examine the effects of radial vortex structure on vortex evolution. As stated in the introduction, for a given initial condition, it can be shown that the time scale of shock formation for Burgers equation is inversely proportional to the gradient of the radial velocity [Whitham, 974]. If the dynamics of the slab boundary layer model follows the dynamics of Burgers equation, then this suggests that the shock behavior discussed previously in the slab model should be much more pronounced for stronger and more compact vortices. However, because the shock interpretation of W3 seems to be inconsistent with the evolution of the wind field for the multilevel model, it is expected that the dynamics that govern the wind field in the multilevel model will be quite different than in the slab model. Another important question that can be raised is how does radial vortex structure affect the secondary circulation observed in the multilevel model? We will investigate these questions by examining the evolution of Vortex II and III. Using the form of V gr described in (3), Figure 3 shows the gradient wind profile V gr, the inertial stability I gr, and radial angular momentum gradient d=drðm a Þ for Vortex I, II, and III (which correspond to vortices with B5:6; :; :4, respectively). Since B determines the radial gradient of V gr, larger values of B are associated with larger gradients in V gr near the radius of maximum gradient winds. Furthermore, we note that as B increases, the gradient wind profile becomes inertially neutral. 4.. Results From Slab Model Figure 4 shows the evolution of velocity and vorticity fields for Vortex II. We see that there are both qualitative and quantitative differences between the evolution of Vortex I and Vortex II. As Vortex II approaches a steady state, we see that the radial inflow approaches 3 m s and contrary to the control experiment, the region of strongest radial inflow moves radially inward. As the radial inflow accelerates and moves radially inward, we see that the inflowing air advects the azimuthal wind maximum radially inward. As a result, the azimuthal wind maximum reaches 68. m s (which is % supergradient) and the azimuthal wind field becomes supergradient in the region from 7 < r < 4. In response to the larger gradients in the azimuthal and radial velocity fields, we see that the magnitude of the vertical velocity and relative vorticity field increase as well. As can be seen in Figure 4, vertical velocities exceed 8 m s within the RMW of the vortex and a stronger vorticity maximum develops in the region of the strongest Ean pumping. Figure 5 shows the azimuthally averaged contributions for the radial velocity and absolute angular momentum tendency equations. As in the control experiment, Figure 5a shows that there is an approximate balance between the agradient force term () and the radial advection term (ADV). However, the magnitude of both terms has dramatically increased. This helps to explain why the radial inflow in this experiment has increased in comparison to the control experiment. Furthermore, we also observe that the horizontal diffusion term (DIFF) has also increased in order to balance the radial velocity budget. As in the control experiment, Figure 5b shows that the inward advection of M a is balanced by frictional destruction of M a (FR) and horizontal diffusion of M a (DIFF). However, the magnitude of these terms has also dramatically increased, leading to a larger azimuthal wind maximum and a larger region of supergradient flow. Moreover, contrary to the control experiment, the dominant balance is between the horizontal diffusion of WILLIAMS VC 4. The Authors. 97

11 Steady State Budget of V R m s..5.5 TADV FR DIFF Steady State Budget of M a m s TADV FR DIFF Figure 5. (a) (top) Steady state budget analysis of radial velocity V R for Vortex II in the slab model. (b) (bottom) Steady state budget analysis of absolute angular momentum M a for Vortex II in the slab model. For the radial velocity and absolute angular momentum tendency, the components are radial advection (), azimuthal advection (TADV), vertical advection (), agradient force (), frictional drag force (FR), and horizontal diffusion (DIFF). M a and the radial advection of M a, consistent with the shock interpretation given in W3. This demonstrates that as the gradient of V gr increases, nonlinear horizontal advection and horizontal diffusion play dominant roles in the dynamics of the slab boundary layer model. These results are consistent with experiment C3in W3. V T [m s ] t =. h t =. h t =. h t = 3. h t =. h V R [m s ] t =. h t =. h t =. h t = 3. h t =. h t =. h t =. h t =. h t = 3. h t =. h 8 6 t =. h t =. h t =. h t = 3. h t =. h ζ [s ] w [m s ] Figure 6. Slab boundary layer model results for Vortex II. The four figures show the (top left) boundary layer azimuthal velocity V T, (top right) radial velocity V R, (bottom left) relative vorticity f, and (bottom right) vertical velocity w for the inner region r. The results at the five different times t:; :; :; 3:; : h are indicated by the color coding. WILLIAMS VC 4. The Authors. 98

12 m s..5.5 Steady State Budget of V R TADV FR DIFF m s Steady State Budget of M a TADV FR DIFF Figure 7. (a) (top) Steady state budget analysis of radial velocity V R for Vortex III in the slab model. (b) (bottom) Steady state budget analysis of absolute angular momentum M a for Vortex III in the slab model. For the radial velocity and absolute angular momentum tendency, the components are radial advection (), azimuthal advection (TADV), vertical advection (), agradient force (), frictional drag force (FR), and horizontal diffusion (DIFF). Figure 6 shows the evolution of velocity and vorticity fields for Vortex III. The results here are qualitatively similar to Vortex II, although the relative magnitudes of the terms have changed. As the vortex approaches the steady state, radial inflow approaches 35 m s and the azimuthal wind maximum approaches 7 m s. We see that the azimuthal wind maximum has been displaced further inside into the region 5 < r < 35. Furthermore, the vertical velocities approach m s. These results further indicate that the shock structure becomes more pronounced as the gradient of V gr increases. Figure 7 shows the azimuthally averaged contributions for the radial velocity and absolute angular momentum tendency equations. The results seen here are qualitatively similar to the previous vortex experiment; however, the magnitude of each term increases. We note that the radial advection terms () in radial velocity and absolute angular momentum equations are close to an order of magnitude larger than in Vortex I, leading to stronger radial inflow and stronger supergradient flow. Furthermore, for this experiment, we see that the budget of absolute angular momentum is almost completely dominated by radial advection () and horizontal diffusion (DIFF) near the RMW. These experiments demonstrate that the shock structure of these vortices become more pronounced as the gradient of V gr around the radius of maximum gradient winds increases. The most compact vortex, Vortex III, has the strongest radial inflow, strongest supergradient winds, strongest vertical velocities, and largest contraction of the azimuthal wind maximum. This is also accompanied by substantial increases (about an order of magnitude) from the nonlinear advective terms in the budget of radial velocity and absolute angular momentum. This indicates that the development of supergradient winds and radial inflow is strongly sensitive to the gradient of V gr in slab boundary layer models. This sensitivity is also consistent with the observational analyses of Hurricanes George and Mitch by Kepert [6a, 6b]. Both vortices were of similar intensity, but the steeper profile (Mitch) had almost double the near-surface inflow. The dynamics behind these experiments can be explained in terms of angular momentum considerations. At larger radii, we see that there is sufficient radial advection of M a to largely balance frictional destruction by surface drag. The rapid deceleration of air parcels toward the inner core of the TC increases the radial momentum of inflowing air parcels. As these inflowing air parcels encounter this region of strong inertial stability, the air parcels overshoot the eyewall, similar to how air parcels can overshoot a region of high WILLIAMS VC 4. The Authors. 99

13 Steady State V T Steady State V R Steady State w Steady State Asymmetric V T 5 Steady State Asymmetric V R Steady State Asymmetric w Figure 8. (a) (top left) Steady state storm-relative azimuthal velocity field V T, (top middle) radial velocity field V R, and (top right) vertical velocity field w for Vortex IV in the slab model. (b) (bottom left) Steady state storm-relative asymmetric component of the azimuthal velocity field V T, (bottom middle) radial velocity field V R, and (bottom right) vertical velocity field w for Vortex IV in the slab model. Steady State V T Steady State V R Steady State w Steady State Asymmetric V T 5 5 Steady State Asymmetric V R Steady State Asymmetric w Figure 9. (a) (top left) Steady state storm-relative azimuthal velocity field V T, (top middle) radial velocity field V R, and (top right) vertical velocity field w for Vortex V in the slab model. (b) (bottom left) Steady state storm-relative asymmetric component of the azimuthal velocity field V T, (bottom middle) radial velocity field V R, and (bottom right) vertical velocity field w for Vortex V in the slab model. WILLIAMS VC 4. The Authors.

14 Steady State for m s.5..5 Steady State for 5 m s. Steady State for m s Steady State TADV for m s x 3 Steady State TADV for 5 m s. Steady State TADV for m s Figure. (a) Steady state storm-relative radial advection term () in the radial velocity budget for (top left) Vortex I, (top middle) IV, and (top right) V in the slab model. (b) Steady state storm-relative azimuthal advection term (TADV) in the radial velocity budget for (bottom left) Vortex I, (bottom middle) IV, and (bottom right) V. Flow is given in terms of a coordinate system moving with the system. static stability in the free atmosphere. This overshoot leads to a strong contraction of the azimuthal wind maximum. As shown in this section, the radial advection of M a increases as the gradient of V gr increases, which makes this overshoot more pronounced. Moreover, as the gradient of V gr continues to increase, radial advection of M a grows substantially larger than the frictional destruction of M a. This causes advection and diffusion to dominate the dynamics of the wind field near the RMW, consistent with the shock interpretation from W Results From Multilevel Model An azimuthally averaged radial cross section of the steady state flow of Vortex II is shown in Figure 4. We see that there are both qualitative and quantitative differences between this experiment and Vortex I for this model. In this case, the radial inflow approaches 3 m s and the region of maximum radial inflow and azimuthal flow moves radially inward. Furthermore, we see that the depth of the radial inflow layer has increased and the depth of the inflow layer decreases more rapidly with radius near the RMW. The radial outflow layer also extends higher with stronger magnitude in the atmosphere. The frictionally forced updraft (Figure 4, bottom) peaks at.9 m s at a higher altitude (at approximately m). Furthermore, there is a well-defined region of subsidence within the eye region and a region of broader weak subsidence in the outer regions of the vortex. Here the maximum azimuthal wind (Figure 4, middle) increases to 68.5 m s at a height of 7 m, which is approximately % supergradient. A radial cross section of the steady state flow of Vortex III is shown in Figure 5. Here we see that as the gradient of V gr increases, the qualitative changes in the boundary layer flow are more pronounced. In this case, the radial inflow approaches 35 m s and the region of maximum radial inflow and azimuthal flow moves further inward. As in the previous case, we see that the depth of the radial inflow layer continues to increase and the depth of the inflow layer continues to decrease much more rapidly with radius near the RMW. The frictionally forced updraft (Figure 5, bottom) peaks at 3.6 m s at a height of approximately 4 m with increased subsidence within the eye region. The maximum azimuthal wind (Figure, middle) increases to 7.5 m s at a height of approximately 75 m, which is approximately 7% supergradient. The budgets of V R and M a at the RMW for Vortex I, II, and III (i.e., stationary vortices with B 5.6,.,.4) are shown in Figure 6. Although the signs and general shapes of the various terms are similar, their relative WILLIAMS VC 4. The Authors.

15 Steady State for m s. Steady State for 5 m s.4 Steady State for m s Steady State TADV for m s x 3.5 Steady State TADV for 5 m s.3 Steady State TADV for m s Figure. (a) Steady state storm-relative radial advection term () in the azimuthal velocity budget for (top left) Vortex I, (top middle) IV, and (top right) V in the slab model. (b) Steady state storm-relative azimuthal advection term (TADV) in the azimuthal velocity budget for (top left) Vortex I, (top middle) IV, and (top right) V in the slab model. Steady State Radial Velocity (m s ) Steady State Azimuthal Velocity (m s ) Steady State Vertical Velocity (m s ) Figure. Radial cross section from r < 4 and z < of the (top) steady state radial velocity V R field, (middle) azimuthal velocity V T field, and (bottom) vertical velocity w for Vortex I in the multilevel model. WILLIAMS VC 4. The Authors.

16 magnitudes and depth scale for each term has changed dramatically as B has increased. In particular, there are substantial increases in the contribution from nonlinear advective terms in the budgets. In the radial velocity budget, vertical diffusion is relatively unimportant away from the surface, and the primary balance is between the vertical advection of V R and the acceleration due to the agradient force. The increased importance of vertical advection as B is increased is primarily due to the fact that the updraft becomes much stronger and more concentrated at the RMW as the vortex becomes more compact. Furthermore, as the boundary layer depth becomes more shallow near the core of this vortex, vertical gradients increase, leading to stronger vertical advection. We also note that radial advection of V R does not increase appreciably above the surface as B increases, which is inconsistent with the shock interpretation of W3. Comparing Figure 7 with Figure 6 shows that the radial advection of V R near the surface in the multilevel model is less than half of its value in the slab model. In the absolute angular momentum budget, we see that the primary balance between inward advection of M a and vertical advection of M a is much more pronounced as B increases. This clearly shows that vertical advection is the dominant process (with turbulent diffusion playing a secondary role) that maintains radial inflow against the outward-directed acceleration due to the agradient force, consistent with Kepert and Wang []. Moreover, we note that vertical velocities are approximately 3 times smaller in this model than in the slab model. This indicates that radial velocity gradients remain much smaller than in the slab model and is much less sensitive to changes in radial vortex structure. All of these considerations indicate that shocks are not developing in the multilevel model. Based on a comparison of Figures, 4, and 5, we also see that the secondary circulation intensifies, deepens, and becomes radially constrained as the gradient of V gr increases. This can also be confirmed by examining Figure 6. As B increases, the magnitude of vertical advection of M a (corresponding to a greater vertical velocity) greatly increases, the magnitude of outward radial advection of M a (corresponding to greater outflow) greatly increases, and the depth scale of the circulation correspondingly increases. This can be explained through an analysis of gradient wind profiles. As B increases, the radial gradient of M a increases to the point where dm a =dr approaches zero outside of the RMW, as shown in Figure 3. This confines the updraft (as well as radial and vertical advection) to the vicinity of the RMW, leading to a radially constrained secondary circulation. Moreover, as shown in Figure 4, the gradient in V gr increases near the RMW as B increases, leading to strong advection of M a near the RMW. Since the radial gradient of M a is related to inertial stability (since I 5r a =@r) and since the boundary layer depth scale is proportional to I =, this implies that strong gradients of M a near the RMW will lead to a shallow boundary layer near the core. This will lead to strong vertical gradients of M a and turbulent diffusion. In this way, sharply peaked vortices with large B will produce a deeper and more intense secondary circulation than broadly peaked vortices (all other things being equal). These changes in depth scale observed as B increased can also be explained using the concept of Rossby depth in balanced vortex models. According to Schubert and McNoldy [], Rossby depth is proportional to I =. Therefore, for sharply peaked vortices (with large I near the RMW), Rossby depths near the RMW are large such that the secondary circulation is more vertically elongated and so horizontally compressed that some of the eyewall updraft can return as subsidence in the eye. Moreover, the increase of Rossby depth near the RMW allows Ean pumping to penetrate deeper into the troposphere. This explains why the depth scale of the secondary circulation (as well as the radial and vertical velocity components) continues to increase as the gradient of V gr increases. Moreover, as B increased, the vortex becomes inertially neutral outside of the RMW. 5. The Effects of Vortex Translation on Vortex Evolution As mentioned previously, the boundary layer flow may become asymmetric if the pressure field translates. We now consider the evolution of Vortex IV and V, which correspond to vortices embedded in 5. and. m s easterly flow, respectively. In order to avoid confusion, it is important to note that the analysis of this section will be in terms of storm-relative winds. 5.. Results From Slab Model The steady state flow of Vortex IV is shown in Figure 8a. Within the first hour of the model simulation, asymmetries in the wind fields begin to develop. Initially, easterly vortex translation produces significant inflow WILLIAMS VC 4. The Authors. 3