Supergradient Winds in Simulated Tropical Cyclones

Richard Rotunno aNCAR, Boulder, Colorado

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Abstract

In a previous paper a formula was derived for the maximum potential intensity of the tangential wind in a tropical cyclone called PI+. The formula, PI+2 = EPI2 + αrmwmηm, where EPI is the maximum potential intensity of the gradient wind and αrmwmηm represents the supergradient winds. The latter term is the product of the radius rm, the vertical velocity wm, the azimuthal vorticity ηm at the radius and height of the maximum tangential wind (rm, zm), and the (nearly constant) α. Examination of a series of simulations of idealized tropical cyclones indicate an increasing contribution from the supergradient-wind term to PI+ as the radius of maximum wind increases. In the present paper, the physical content of the supergradient-wind term is developed showing how it is directly related to tropical cyclone boundary layer dynamics. It is found that rmwmηmumin2zm(rm)/lυ(zm)rm, where −umin is the maximum boundary layer radial inflow velocity and lυ(z) is the vertical mixing length.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Richard Rotunno, rotunno@ucar.edu

Abstract

In a previous paper a formula was derived for the maximum potential intensity of the tangential wind in a tropical cyclone called PI+. The formula, PI+2 = EPI2 + αrmwmηm, where EPI is the maximum potential intensity of the gradient wind and αrmwmηm represents the supergradient winds. The latter term is the product of the radius rm, the vertical velocity wm, the azimuthal vorticity ηm at the radius and height of the maximum tangential wind (rm, zm), and the (nearly constant) α. Examination of a series of simulations of idealized tropical cyclones indicate an increasing contribution from the supergradient-wind term to PI+ as the radius of maximum wind increases. In the present paper, the physical content of the supergradient-wind term is developed showing how it is directly related to tropical cyclone boundary layer dynamics. It is found that rmwmηmumin2zm(rm)/lυ(zm)rm, where −umin is the maximum boundary layer radial inflow velocity and lυ(z) is the vertical mixing length.

© 2022 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Richard Rotunno, rotunno@ucar.edu

1. Introduction

Boundary layer tangential winds in excess of the tangential wind in balance with the radial pressure gradient above the boundary layer, known as “supergradient” winds, are a well-known feature of rotating-flow boundary layers (Rotunno 2014). The possible theoretical importance of supergradient winds in the tropical cyclone boundary layer was argued for in Smith et al. (2008) and the observational evidence for it reviewed by Montgomery and Smith (2017, p. 549). Emanuel (1986) developed a theory for the steady-state maximum potential gradient wind (or potential intensity; EPI). To include the supergradient wind in a formula for maximum tangential wind, Bryan and Rotunno (2009a) derived a diagnostic extension of EPI as
PI+2=EPI2+αrmwmηm,
where α is nearly a constant, rm is the radius, wm the vertical velocity, and ηm the azimuthal vorticity at the radius and height of the maximum tangential wind (rm, zm). In numerical simulations designed to conform closely to the approximations used to derive EPI, Bryan and Rotunno (2009a, their Fig. 12) found good agreement between PI+ and the numerically simulated maximum tangential wind. Motivated by this finding, the objective of the present study is to develop a better theoretical understanding of rmwmηm, with emphasis on its variation with rm.

Models for the supergradient wind are perforce boundary layer models of momentum transfer from the “free” (frictionless) atmosphere above to the frictional surface below. For rotating-flow boundary layer models, the most problematic aspect is the specification of the imposed radial pressure gradient, or equivalently, the gradient wind υg(r). In contradistinction to parallel-flow boundary layers, the secondary circulation1 of a rotating-flow boundary layer may modify the “free” atmosphere2 turning a relatively straightforward calculation of the boundary layer winds into the far-more-complex problem of solving for the “imposed” υg(r) together with the boundary layer winds (Rotunno 2014). As demonstrated by Bryan and Rotunno (2009a, their sections 4c and 5), υg(r) calculated from a numerical simulation that includes a boundary layer should not in general be the same as that from a model that does not include the boundary layer radial-momentum feedback3 on υg(r) (as in Emanuel 1986; Frisius et al. 2013; Tao et al. 2020a). With this understanding, (1) represents the sum of the gradient wind that would occur in a model absent the boundary layer radial-momentum feedback on υg(r) plus the supergradient term αrmwmηm; further elaboration on this point is in section 5.

Frisius et al. (2013) carried out the most comprehensive theoretical unification of EPI with a model for supergradient winds. Using a modified version of Emanuel’s (1986) original formulation, Frisius et al. (2013) derive υg(r) for use as an input to a boundary layer model. Frisius et al. (2013) find (their Fig. 3) that υm, the maximum supergradient wind in the boundary layer, increases with the radius of the maximum υg(r), as it varies from 5 to 50 km. Analysis of a high-resolution global model (Miyamoto et al. 2014) as well as that of idealized axisymmetric numerical simulations (Tao et al. 2020b, their Fig. 12a) finds that supergradient winds are stronger in larger simulated tropical cyclones.

Tao et al. (2020a) analyzed the quasi-steady states of a series of numerical simulations of idealized axisymmetric tropical cyclones varying physical constants such as the mixing lengths and air–sea transfer coefficients, with all cases having the same thermodynamic initial condition and sea surface temperature. For each set of physical constants, they varied the initial radius rmi of the initial tangential-wind maximum υmi in the initial profile υi(r, z) as in (1) of Xu and Wang (2018), with the same υmi in all cases. Tao et al. (2020a) found that, for each set of physical constants, the steady-state rm increases as rmi increases (consistent with Rotunno and Emanuel 1987, their Table 2) and hypothesized that the contribution from the supergradient wind to υm increases with increasing rm (p. 5). In looking at the latter result in the context of (1), the author found that αrmwmηm increases relative to EPI2 with increasing rm. It was not obvious to the author what accounts for this dependence of αrmwmηm on rm, or more generally, what were its determining factors. Montgomery and Smith (2017, p. 567) suggest that the answers lie in the boundary layer equation for the radial momentum. The present study pursues the suggested line of inquiry as it seeks to shed light on the factors determining αrmwmηm.

After a brief summary of the numerical-model setup in section 2, an analysis of the model data is presented in section 3, including the identification of relevant model-output parameters needed to estimate αrmwmηm; methods to relate these model-output parameters to rm are developed in this section through a combination of simple scaling and basic theory. A synthesis of these methods is given in section 4 and conclusions summarized in section 5.

2. Model setup

Although the present model setup is described in section 2 of Tao et al. (2020a), some basic information on the simulations is given here for convenience. The numerical model used in the latter study was version 19.7 of Cloud Model 1 (CM1) as described in Bryan and Rotunno (2009b) for axisymmetric tropical cyclone simulations. The domain is 1500 km in the radial direction with grid spacing Δr = 1 km for r < 300 km and a linear stretching to Δr = 15 km for r ≥ 300 km. The domain is 25 km deep with the lowest grid level at z = 25 m, which gradually increases to dz = 200 m up to z = 5 km above which it remains constant. The sea surface temperature Ts = 28°C and the simulations are initialized with the Jordan hurricane-season sounding. The simulations are run for 8 days until a quasi-steady state is reached for time t approximately greater than 5 days and the results are averaged over the 1-h output times for the last 24 h of the simulations.

A few further points relevant to this study are: As the horizontal and vertical mixing lengths play important roles in the present analysis, it is noted here that the simulations have the vertical mixing length specified by
lυ(z)=lκ(z+z0)l2+[κ(z+z0)]2,
where κ = 0.4 and z0 is the roughness length; Tao et al. (2020b) specify l = 100 m and z0 = 0.16 m. The control simulations for increasing rmi have the horizontal mixing length lh varying linearly from lh1(=100 m) to lh2(=1000 m) as the surface pressure goes from 1015 hPa to below 900 hPa (Stern and Bryan 2018, p. 3902); the control set is therefore designated as RX, where X stands for rmi in km units. A second set, chosen for detailed comparison with the control set, has the constant value lh1 = lh2 = 100 m and is designated as RXL. Thus, one expects horizontal diffusion to have a much bigger effect in the RX compared to the RXL simulations. Detailed analysis is restricted to the cases with the (constant) enthalpy-flux and drag coefficients Ck = Cd = 0.001. Within each set we select from Tao et al. (2020a) the cases with rmi = 60, 90, and 120 km with the “skirt” parameter (which controls the radial decay of υi(r, z) for r > rmi) B = 1.0. The initial tangential wind maximum υmi = 20 m s−1 in all cases. Figure 1 shows the initial velocity profiles at z = 0 and the associated angular-momentum profiles, M=rυ+(1/2)fr2, where the Coriolis parameter f = 0.5 × 10−4 s−1.
Fig. 1.
Fig. 1.

Initial tangential velocity profiles for rmi = 60, 90, and 120 km (solid magenta, green, and blue, respectively) and associated angular momentum (dashed) for the inner 300 km of the 1500-km domain.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

3. Analysis

To establish the framework for the present analysis, Fig. 2 shows the steady-state radial and tangential velocities (u, υ) for R120 with the position of the maximum tangential wind (rm, zm) shown by the magenta ×. This solution exhibits the familiar pattern of boundary layer radial inflow with a tangential-wind maximum nearly coincident with the u = 0 contour; the position rmin of the minimum radial velocity umin (or maximum inflow velocity) is indicated by the blue arrow. Figure 3 shows a closer-in view of the vertical velocity w and azimuthal vorticity η, which play a central role in the present analysis. Table 1 contains the model outputs that the theory developed here seeks to explain with emphasis on the increase of the supergradient wind term αrmwmηm with rm.

Fig. 2.
Fig. 2.

Steady-state radial and tangential velocities (u, υ) for R120. The contour interval (c.i.) for u is 10 m s−1 (black lines, negative values dashed) and the c.i. = 20 m s−1 for υ (magenta lines) with the maximum (113.8 m s−1) marked by the magenta × and the position rmin of umin indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Fig. 3.
Fig. 3.

Steady-state vertical velocity w (green lines; c.i. = 2 m s−1) and azimuthal vorticity η (black lines; c.i. = 0.01 s−1); zero lines not plotted. Selected streamlines (gray) bracketing (rm, zm) marked by the magenta × and the position rmin of umin indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Table 1

Model data at the location (rm, zm) of the maximum tangential wind υm; Mm is the corresponding angular momentum and Mmi is the initial angular momentum at the radius rmi of the initial maximum tangential wind.

Table 1

Filling in the formula for EPI, (1) is
PI+2=αCkCd(TbT0)(ssurfs0)+αrmwmηm,
where Tb and T0 are the temperatures at (rm, zm) and in the upper-level radial outflow at large radius, respectively, and ssurf and s0 are the saturation entropy at the sea surface temperature (Ts) and the entropy at the top of the surface layer (in practice at the first grid level, here at z = 25 m), respectively; both are evaluated at r = rm. The coefficient α = 1 without dissipation or Ts/T0 with dissipation, which is the case in this study.

Tables 1 and 2 list the parameters entering into (3). Table 2 also shows the theoretical maximum potential intensity PI+ and the EPI derived from the first term on the rhs of (3). Within cases RX and RXL, one observes a tendency for PI+ to increase and E-PI to decrease (slightly) with increasing rmi. The difference between PI+ and EPI indicates an increasing contribution from the supergradient-wind term in (3) with increasing rmi. The last column of Table 2 shows that for the more horizontally diffusive RX cases υm (Table 1) tends to be less than PI+ while the RXL cases tend to exceed slightly PI+ [consistent with Fig. 13 of Bryan (2012)]. Since Table 1 shows that rm increases with rmi, Table 2 implies that supergradient winds increase with rm.

Table 2

Inputs to and results from (3).

Table 2

a. The radius rm

Tao et al. (2020a) find that rm is a linear function of the angular momentum at r = rm, Mm (their Figs. 3a–c) and that Mm is a linear function of Mmi (their Figs. 5d–f). It follows, therefore, that rm is a linear function Mmi for the same skirt parameter. Analysis of the present cases finds
rm=aMmi+b.
In Tao et al.’s (2020a) analyses, the Mm were based on a fixed-height measurement (zm = 1.55 km) in order that they be taken well above the boundary layer. The data shown in Table 1 are instead taken at the position of υm which occurs at different zm in the different simulations. One may verify from the data in Table 1 that a′ = 9.177 s km−1 and b′ = 1.1818 km for the RX cases and a′ = 8.161 s km−1 and b′ = −3.964 km for the RXL cases.

Xu and Wang (2018) showed in their Figs. 46 that the processes of vortex intensification take place near and radially inward of rmi due to Ekman pumping and subsequent convection (see also Peng et al. 2019). As a complement to the analysis in Xu and Wang (2018), Fig. 4 shows M(r, z, t) for the RX cases at z = zm indicated in Table 1. Although M is not conserved, its value at the quasi-steady position rm is not much different from Mmi (see also Table 1) which implies that the air arriving at rm retains a strong “memory” of the angular momentum at rmi. Theories for the gradient wind find that rmg, the radius of maximum gradient wind, should increase with the angular momentum at an outer radius (see, e.g., Fig. 7a of Tao et al. 2020b). Figure 4 suggests that rmi is the effective outer radius and that therefore, for constant υmi, Mmi increases with rmi (Table 1). It follows therefore that rmg increases with rmi (or Mmi). Boundary layer theory and simulations predict rmrmg so that one can also expect rm to increase with rmi for the same skirt parameter.

Fig. 4.
Fig. 4.

M(r, z, t) at z = zm in indicated in Table 1 for cases (a) R60, (b) R90, and (c) R120. The 100 m s−1 contour for υm is indicated in black.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

b. The product rmwmηm

The vertical velocity at (rm, zm) is related to the radially inward boundary layer flow through the continuity equation as
rmwm=0zm(rur)rmdz,
where the (small) density variations have been neglected. To make progress toward finding the minimum number of essential variables needed to explain rmwmηm, (5) is simplified by approximating the integrand by its finite-difference approximation across an annulus inside of rmin and roughly centered at rm to arrive at
rmwm0zmrminu(rmin,z)δrdz,
where rmin is the radius where the radial velocity reaches its minimum value and δr is defined as the interval over which |ru(r, z)| becomes small relative to |rminu(rmin, z)| (see, e.g., Fig. 2); although rminu(rmin, z) and δr can vary with height, they are approximated here by their surface values. Letting u(rmin, 0) ≡ umin with the foregoing approximations, (5) becomes
rmwmδrrminuminzm,
which is a simple statement that the vertical mass flux across the distance δr at z = zm equals the horizontal mass flux in the boundary layer at r = rmin through the “corner” flow illustrated by the streamlines in Fig. 3. Table 3 contains umin, rmin, and δr; given the crude approximation to the radial derivative in (5), δr is regarded as a tuning parameter to maximize agreement between ∂(ru)/∂r at (r, z) = (rm, 0) and its approximation in (6).
Table 3

Model outputs umin, inflow angle ϕ, rmin, parameters δr and λυ, vertical mixing length lυ(zm), and approximations to rmwmηm from (9) and from replacing rmin/δr by β = 3.32 and 3.76 for the RX and RXL cases, respectively. The last column is based on the approximations developed in this paper.

Table 3
The azimuthal vorticity, η = ∂u/∂z – ∂w/∂r, is to an excellent approximation η ≃ ∂u/∂z suggesting the scaling
ηmuminλυ,
where λυ is a vertical scale chosen to maximize the agreement between (8) and ηm. Further discussion of λυ is given below.
Putting (7) and (8) together gives
rmwmηmrminδrumin2zmλυ.
Table 3 shows that the approximations leading to (9) can reproduce the model results for rmwmηm shown in Table 1. The theoretical project then is to find the relation of the various parameters on the rhs of (9) to rm.

Tables 1 and 3 indicate that rmin and δr increase in more or less constant proportion to the increase of rm.4 It follows therefore that the first factor in (9), rmin/δrβ has no systematic dependence on rm. Table 3 shows the results from (9) with β ≃ 3.32 and 3.76 for the RX and RXL cases, respectively; these results indicate the major variation of rmwmηm with rm is retained. Therefore almost all the variation with rm in (9) must come from the factors umin, zm, and λυ. A boundary layer theory is required to understand the behavior of these parameters as functions of rm and is discussed next.

c. The velocity umin

The flow in the tropical cyclone boundary layer, as with all boundary layers, is determined by the imposed pressure gradient and the viscous and/or turbulent stress within it. Exact nonlinear results from rotating-flow boundary layer theory have been found and verified experimentally for two canonical cases (reviewed in Rotunno 2014), they are the boundary layer of 1) a vortex in solid-body rotation (υ = ωr) over a disc of infinite radius (Bödewadt 1940; herein referred to as the SB case) and 2) a potential vortex (υ ∝ 1/r) over a disc of finite radius (Burggraf et al. 1971; herein referred to as the PV case). Both solutions have no-slip conditions applied to the velocity at the impermeable lower boundary and constant viscosity. Approximate (but quite complex) analytical solutions for the boundary layer of a Rankine-type vortex (υr for rrm and υr−p for rrm, where 0 < p ≤ 1) with semislip (stress ∝ velocity) lower boundary conditions can be found in Kuo (1971). Kuo’s (1971) boundary layer solutions (see his Figs. 6.1–6.4) show features of the solid-body-rotation vortex for r < rm and the potential vortex for r > rm. Numerical simulations of the Navier–Stokes equations for a diffusively evolving Rankine vortex with a no-slip lower boundary are reported in Rotunno (2014). For more recent tropical cyclone–specific idealized boundary layer models see Kepert (2001), Kepert and Wang (2001), Kepert (2012, 2017), and Smith and Montgomery (2020).

The steady-state angular momentum M(r, z) in Fig. 5 suggests that the simulated flow is characterized by a Rankine-type vortex above the boundary layer. Closer examination of the velocity profile υ(r) above the boundary layer (not shown) for r > rm shows, however, that υ(r) ∝ r−n with n ≈ 0.8 (observations show n < 0.7; Mallen et al. 2005). Although qualitatively reasonable solutions for the boundary layer of this type of vortex can be found using linear theory (e.g., Kepert 2017), they are technically limited by the requirement that the secondary-circulation velocities be much smaller that the tangential velocity maximum υm. Tables 1 and 3 show that umin/υm ≈ 0.35–0.38 which is not ≪1 and important features of the (nonlinear) dynamics are not accounted for. In what follows, the case is made that, while not strictly applicable, the nonlinear theory for the boundary layer of potential vortex (n = 1.0) provides a useful interpretive tool for the present results.

Fig. 5.
Fig. 5.

M(r, z) (green; c.i. = 0.5 km2 s−1) and the dominant terms in the radial momentum equation [black, (M2Mg2)/r3; magenta, ∂τrz/∂z; c.i. =0.03 m s−2, zero line not plotted]. Position of υm indicated by the magenta × and the position rmin of umin is indicated by the blue arrow.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The most important feature of the Rankine-vortex boundary layer is the inward increase of −u(r, z) in the outer PV-like boundary layer and its inward decrease to zero in the inner SB-like boundary layer, as exemplified in Fig. 2. Figure 5 also displays the dominant terms5 in the radial momentum equation, which can be written in the form
dudt=M2Mg2r3+Du,
where Du stands for the radial diffusion terms [Eq. (13) of Rotunno and Emanuel 1987], which are dominated by the vertical diffusion of u(r, z) to the ground (or “friction”). The first term on the rhs of (10) is the difference between the outward-directed centrifugal plus Coriolis accelerations and the inward-directed pressure-gradient acceleration.

The radial momentum budget in Fig. 5 shows that the pressure-gradient term overcomes the centrifugal/Coriolis term for the outer PV-like boundary layer producing an inward radial acceleration (Fig. 2); this term changes sign for the inner SB-like boundary layer and thus produces an outward radial acceleration (cf. Fig. 6d of Smith et al. 2009). This latter outward acceleration is entirely due to M2>Mg2 at (rm, zm) illustrating that the supergradient wind is an essential feature of the boundary layer near r = rm. The frictional term opposes the inward radial velocity at and near z = 0 for both the inner-core and outer portions of the boundary layer; of special interest here is the radius at which the inward radial acceleration of the PV-like boundary layer is first cancelled by the outward frictional acceleration, indicated by the blue arrow. Not coincidentally, this radius is rmin. Figure 5 shows that rmin is also approximately the radius above the boundary layer marking the transition between the inner (SB-like) and the outer (PV-like) regions of M(r, z).

The approach here is to use results from the literature on SB-like and PV-like boundary layers to interpret the dependencies of the factors in (9) on rm and thus to the external parameters of the problem through (4). The biggest difference between the SB and PV boundary layers is the existence of a local similarity solution for the former and its nonexistence in the latter case (Rotunno 2014). In other words, for the SB case, the local radius r, viscosity K, and rotation rate ω of the flow above the boundary layer are sufficient to determine the local solution.6 In contrast, in the PV case the solution at any r depends on the upstream evolution of the boundary layer, i.e., nonlinear advection is indispensable. Figure 4 suggests one can consider the simulated tropical cyclone vortex as having three regions: an inner region of nearly solid-body rotation, a middle region of nearly constant M, and an outer region of gradually increasing M. In this latter region one can ignore advection and consider, for example, the utility of “column” models (e.g., Bryan et al. 2017). Based on the latter three-zone vortex model, the working hypothesis here is that the middle, PV-like boundary layer controls umin independently of the outer zone, while the SB-like boundary layer controls the vertical scales zm and λυ since both are defined at rm which is clearly within the SB-like region.

An analytical model for umin would consist of (10) and
dMdt=rDυ,
where Dυ stands for the diffusion terms in the azimuthal direction [Eq. (14) of Rotunno and Emanuel 1987] and the continuity equation. This model would require using a flow-dependent Kυ and horizontal viscosity. Lacking such an analytical model, one can however use the numerical model results and (10) to demonstrate the importance of nonlinear advection. Figure 6 shows the major terms in (10) at z = 0, viz.,
ududr=M2Mg2r3+τrzz;
it is clear that all three terms are of comparable magnitude. To go a step further, the radial integral of (12) at z = 0 from rmin to rmi gives
umin22rminrmi(M2Mg2r3+Du)dr+u2(rmi),
where u2(rmi) is small and can been neglected. Equation (13) yields the numerically simulated umin (Table 3) to within ±1 m s−1 (not shown) and confirms that umin is controlled by the middle region (rmin < r < rmi) of nearly constant M. This result is in broad agreement with Smith and Montgomery (2020, p. 3443) on the importance of nonlinear advection in the radial-momentum budget. Note incidentally that using the mixed-layer form of (11) to deduce u(r) in the boundary layer would fail in the limit (∂Mg/∂r) → 0 [see, e.g., Eq. (A13) of Peng et al. 2018] for a potential vortex as would any linear theory.
Fig. 6.
Fig. 6.

Terms in (12) for the (a) RX and (b) RXL cases with −udu/dr in green, (M2Mg2)/r3 in magenta, and ∂τrz/∂z in black; magnitudes in m s−2. The blue stars indicate the position of rmin in each case.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Tables 1 and 3 provide the inflow angle, ϕ ≡ tan−1(umin/υms), where υms = 0.8υm7 is the maximum tangential wind at the surface, displayed in Table 3. These values are consistent with the observationally estimated value of −22.6° ± 2.2° (Zhang and Uhlhorn 2012). Using ϕ one can recast (9) as
rmwmηm0.64β(tan ⁡ϕ)2zmλυυm2;
thus, based on (14), one expects a larger rmwmηm in cases with larger υm, other things being equal. As there is little or no systematic dependence of β, ϕ, or υm with rm within the RX and RXL cases individually, (14) implies that the dependence of rmwmηm on rm is embodied in zm/λυ.

d. The height zm

Consider next the dependence of zm(rm). Looking for guidance from theory, both linear (Kepert 2001) and nonlinear (Rotunno and Bryan 2012) theory for the SB-like inner region of the simulated tropical cyclone vortex predict the depth scale δ=Kυ/ω. Kepert’s (2001) linear-theory formula, δ=2Kυ/I, where I2=r3Mg2/r yields I = 2ω for the SB vortex; Kυ is the vertical eddy viscosity. Taking into account the semislip lower boundary condition, the height of the tangential-wind maximum is given by the first of Kepert’s (2001) Eq. (27) as
zm=Kυωtan1(12/χ),
where for the SB case χ=CdV/Kυω with V a characteristic tangential velocity. The arctangent function in (15) varies from π/2 to 3π/4 as χ ranges from 0 to ∞ representing the lower boundary condition ranging from free-slip to no-slip.8 For the SB vortex ω is constant; however, the simulated tropical cyclone vortices are Rankine-like with the inner core ω varying from a constant in the inner SB vortex and diminishing as the inverse square of radius in the outer PV vortex. Figure 7 shows ω(r) at z = 3 km, where υυg. For the horizontally diffusive RX cases, ω is relatively constant for r < rm (denoted by the placement of the crosses in the radial direction), while for the RXL cases, ω is peaked just within r = rm, indicative of vorticity concentration in the eyewall. The author’s best estimate of the effective value of ω to use in (15) is given by the placement of the crosses in the ω direction in Fig. 7. To discover the relation zm(rm), one must find a relation between ω and rm. Figure 8 shows of υm/rm versus the ω estimated from Fig. 7; this analysis suggests υm/rmμω with μ varying between 1.3 and 1.4. Letting ωμ−1υm/rm and V = υm, (15) can be written as
zm=μKυrmυmtan1(12/χ),
with χ=Cdrmυm/(μKυ). As Table 1 indicates, υm is relatively constant within the RX and RXL cases individually. It follows therefore that the dependence zm(rm) by (16) depends directly on rm and indirectly on the variation of Kυ with rm.
Fig. 7.
Fig. 7.

Estimated rotation rate ω of the gradient wind at z = 3 km for the (a) RX and (b) RXL cases with X = 60 (magenta), 90 (green), and 120 (blue). The placement of the crosses in the ω direction indicates the estimated rotation rate of the inner core and their placement in the radial direction indicates rm.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

Fig. 8.
Fig. 8.

The rotation rate υm/rm (s−1) (Table 1) vs the estimates of ω (s−1) from Fig. 7. The line υm/rm = μω with μ = 1.3 and 1.4 are plotted for reference.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The theories cited above assume a constant Kυ; however, Figs. 9a and 9b show that it is highly variable, especially near the level zm (marked by the crosses) in the RXL cases. Figures 9c and 9d show that to a very good approximation
Kυ(r,z)lυ2(z)|uz|
in the boundary layer. In looking for a scaling for the vertical distance between zm and a secondary maximum aloft, Stern et al. (2020) computed 2πKυ/ω, using the maximum value of Kυ=l2Sυ where Sυ is the vertical deformation [Eq. (16) of Bryan and Rotunno 2009b] and found reasonably good agreement between the formula and the model results across simulations with different values of l and Cd (their Fig. 19). However, in the present problem, at and below zm, Kυ has significant variation, especially in the RXL cases. For Kυ and υm approximately constant, (16) implies zmrm which can be seen in Table 1 and Fig. 9 to approximately describe the horizontally diffusive RX cases. However, for the RXL cases, Table 1 and Fig. 9 suggest a stronger dependence on rm, more like zmrm.
Fig. 9.
Fig. 9.

Vertical eddy viscosity Kυ for the (a) RX (c.i. = 50 m2 s−1) and (b) RXL (c.i. = 100 m2 s−1) cases, with X = 60, 90, and 120 (magenta, green, and blue, respectively). The major contributor to Kυ, (17), for the (c) RX and (d) RXL cases with the same c.i. as in (a) and (b). The × marks the location of (rm, zm) for each case.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

In section 3b it was established that ∂u/∂z ≃ −umin/λυ, but further explanation of the origin of λυ was deferred. It may be verified by the data in Tables 1 and 3 that
λυγlυ(zm),
where γ ≃ 13 ± 1 and lυ(zm) is given by (2). A rationale for (18) is given in the last part of this section, but for now, its consequences w.r.t. to (16) are developed. Substitution of (17) and (18) into (16) gives
zm=μlυ(zm)|umin|rmγυmtan1(12/χ),
with χ=Cdγυmrm/[μlυ(zm)|umin|]. Equation (19) will be solved through iteration for zm(rm), but first consider (19) with the tan−1 factor replaced by σ. Neglecting z0 in (2) and substituting the result into (19) gives
zm=σμlκzm|umin|rml2+(κzm)2γυm,
which can be rearranged into a quartic equation for zm(rm). Equation (2) suggests the nondimensionalization (r^m,z^m)=κ(rm,zm)/l which, together with the substitution umin/υm = 0.8 tan ϕ, leads to the solution
z^m=12{[1+(2Cσ2r^m)2]1/21}1/2;C0.8μκ|tanϕ|γ,
which has the interesting property that z^mrm and z^mr^m in the limits of small and large r^m, respectively.

The data in Tables 1 and 3 have an average ϕ = −24.1°; with γ = 13.1, C = 0.0112μ. Figure 10 shows in dimensional form the exact solution of (19) and the approximate solution (21) with σ = π/2 and μ = 1.35. Evaluation of the tan−1 factor in (19) from the exact solution for μ = 1.35, or from substituting the approximate solution for zm into the formula for χ, both give σ(rm) ranging from 1.59 to 1.62 between the limits of Fig. 10. Figure 10 also plots the data points zm(rm) from Table 1. Based on the values of μ reported above, all the data points should straddle the μ = 1.35 line which is approximately the case for the RXL cases. However, the theoretical curve underestimates zm(rm) for the RX cases; in these cases there appears to be a systematic offset from the theoretical curve. In the RX cases, horizontal diffusion is as large or larger than the vertical diffusion terms in the angular momentum budget (not shown); to take the larger total diffusion into account (at least superficially), the solution with μ = 2.7 (2 × 1.35) appears to account for zm(rm) in the RX cases.

Fig. 10.
Fig. 10.

Solution for zm(rm) from the exact (19) (solid blue line) and the approximate (21) (dashed line) for μ = 1.35 along with the RX and RXL data. Also shown is the exact solution curve for μ = 2.7.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

e. The scale λυ

In light of (17), the stress τrz can be approximated by
τrzlυ2(z)(uz)2.
Integrating (10) from z = 0 to z = zm at r = rm with Du ≃ ∂τrz/∂z and using (22) gives
lυ2(zm)(uz)zm2=u*2+U2atr=rm,
where u*2=Cdu12+υ12u1 with the subscript 1 signifying the first grid level, and U20zm[du/dtr3(M2Mg2)F]dz, where F represents horizontal diffusion and the (small) influence of filters. From (23) therefore
(uz)zm=u*2+U2lυ(zm)atr=rm.9
Comparing (23) to (8) with (18) implies
γ=uminu*2+U2.
Analysis of the numerical simulations yield u*2, U2, γ, and λυ = γlυ(zm), shown in Table 4. The values of γlυ are generally consistent with the original estimate of λυ shown in Table 3 but are on the low side, which can be traced back to the fact that (17) slightly underestimates the actual Kυ. The author has been unable to find useful estimates of the rhs of (23); however, the main virtue of (23) and (24) is that they provide a theoretical rationale for the estimate of ηm from (8) with (18), viz.,
ηmuminγlυ(zm).
Table 4

The terms on the rhs of (23), the implied value of γ from (25), and the estimated value of λυ using (8).

Table 4

4. Synthesis

Substituting (18) into (14) gives
rmwmηm0.64β(tanϕ)2zm(rm)γlυ(zm)υm2,
where zm(rm; μ) can be estimated from (21) with μ = 2.7 and 1.35 for the RX and RXL cases, respectively, and lυ(zm) from (2). Recalling that β = 3.32 and 3.76 for the RX and RXL cases, respectively, and γ = 13, the final estimate of rmwmηm based on (27) is given in the last column of Table 3. Although the numerical values differ considerably from the values of rmwmηm given in Table 1, the general increase with rm is a robust result upon which a physical interpretation can be based.

As already noted β and υm have little variation with rm within the RX and RXL cases individually. Thus the major variation with rm on the rhs of (27) comes from the ratio zm(rm)/lυ(zm) which indicates, by (7) and (8), that rmwm and ηm have inverse variations with zm and, since zm increases with rm, they have inverse variations with rm. Using the above analysis, the relation (7) can be written as rmwmβ tanϕzm(rm)υm and relation (26) as ηm ≃ tanϕυm/[γlυ(zm)]. Although zm increases with rm (Fig. 10) and lυ(zm) increases with zm by (2), the latter reaches the asymptotic limit l and thus the increase of rmwm with zm dominates the decrease of ηm with increasing lυ.

In physical terms the near constancy of ϕ within the RX and RXL cases individually, and increasing zm with rm in all cases, imply increasing horizontal mass flux through the boundary layer (−rminuminzm at r = rmin) and, by continuity, increasing vertical mass flux (rmwmδr at z = zm). As established in section 3b, rmin and δr increase with rm such that their ratio is approximately fixed and therefore they do not contribute to increasing rmwm with rm. The form of the stress (22) implies that ηm ≃ (∂u/∂z)m ∝ [lυ(zm)]−1 and that therefore the offsetting decrease of ηm with lυ(zm) in rmwmηm is limited by the approach of [lυ(zm)]−1 to a constant with increasing zm(rm).

To obtain an equation more amenable to observational analysis, (27) can be simplified using (2) (with z0 = 0) to obtain
rmwmηm0.64β(tanϕ)2κγ1+(κzml)2υm2g(β,ϕ,zm)υm2,
where the parameters κ, γ, and l are regarded as “fixed.” Substitution of (28) into (1) with υm =PI+ in (28) gives
PI+=EPI1αg.
Composite analyses (Zhang and Uhlhorn 2012; Zhang et al. 2011) as well as case studies (Kepert 2006a,b; Bell and Montgomery 2008) give a fairly consistent picture of ϕ(≃−23°) and zm (≃500 m). However, β (=rmin/δr) is harder to determine from these analyses. All the data show that rmin tracks with rm; however, δr, which is a measure of the eyewall thickness, can vary from δrrmin in the composites (Fig. 5 of Zhang et al. 2011) to something much smaller in case studies (Fig. 4 of Bell and Montgomery 2008). With α = 1.5, γ = 13.1, l = 100 m, ϕ = −24.1°, and zm = 500 m, (29) gives PI+/EPI = (1.04, 1.09, 1.15) for β = (1, 2, 3). Increasing PI+/EPI with β is consistent with the less horizontally diffusive simulations (with thinner eye walls) having larger supergradient winds (Fig. 1 of Rotunno and Bryan 2012).

A caveat to the foregoing is that the current observational estimates of Cd ≃ 0.0025 and Ck ≃ 0.001 (Bell et al. 2012) imply Ck/Cd = 0.4 instead of Ck/Cd = 1.0 used in this study. Other things being equal, a smaller Ck/Cd would decrease EPI (3) and thus increase PI+/EPI. Numerical simulations reported in (Bryan 2012, his Fig. 12) indicate a significant increase of PI+/EPI for Ck/Cd < 1.0 and small lh.

5. Summary and conclusions

Bryan and Rotunno (2009a) derive the formula (1) in which the supergradient winds are represented by the term αrmwmηm, where α is nearly a constant for a given thermodynamic environment. It is argued herein that understanding rmwmηm requires an understanding of the simulated tropical cyclone boundary layer. Based on the continuity equation and the stress model used in the numerical simulations, it is found that rmwmηmumin2zm(rm)/lυ(zm), where −umin is the maximum radial inflow velocity, lυ(z) is the vertical mixing length, (2), and (rm, zm) is the location of maximum tangential wind υm. The increase of the supergradient wind term with rm, as expressed by umin2zm(rm)/lυ(zm), is the principal focus of this paper.

By examining a set of simulations in which the radius of maximum tangential wind of the initial vortex rmi increases, it is found that rm increases with rmi. Based on theories for the maximum gradient wind υmg (Emanuel 1986; Frisius et al. 2013; Tao et al. 2020b), the radius rmg of the maximum gradient wind υmg should increase with increasing angular momentum at an outer radius. The present numerical simulations suggest the radius of the initial tangential wind maximum rmi is the effective outer radius. At rmi the angular momentum Mmi increases for fixed initial maximum tangential velocity; consistent with the gradient-wind theories and boundary layer theory, rmrmg so that rm increases with rmi (or Mmi).

To understand the dependence of umin2zm(rm)/lυ(zm) on rm, appeal is made to an approximate version of the simulated tropical cyclone vortex. Herein the case is made that a useful compromise between fidelity and simplicity is the Rankine vortex in which the inner core is in solid-body rotation (SB) and the outer core is characterized by a potential vortex (PV). The working hypothesis in this paper is that the mechanism(s) controlling the merger of PV outer-core and SB inner-core boundary layers is at the crux of the understanding of supergradient winds. Lacking an analytical solution for the Rankine-vortex boundary layer, the present approach is to examine the model data in light of the known properties of the SB and PV boundary layers individually.

Figure 11 is a schematic diagram of the present conceptual model10 with the key features extracted from R120L. In the PV outer core (where Mg ≃ const.), the boundary layer inflow velocity −umin is fundamentally the result of the unbalanced inward radial pressure-gradient force and nonlinear advection that accelerates the boundary layer flow inflow (cf. Burggraf et al. 1971). Analysis of the present simulations (Fig. 6) in terms of (6) and (13) support for this conclusion. A fluid parcel traversing from the PV outer-core to the SB inner-core (where dMg/dr > 0) boundary layer (i.e., from rmin to rm), by virtue of its inertia continues to flow inward while partially conserving its angular momentum, which in turn, leads to an increase of the outward-directed centrifugal acceleration w.r.t. to the inward-directed pressure-gradient acceleration in the SB vortex (cf. Bödewadt 1940). The force imbalance decelerates the inward radial flow to zero as the parcels rise up to the level of maximum tangential wind zm (Fig. 3). Although without rigorous justification, it was found that using the formulas for zm from linear and nonlinear theories of the boundary layer of a vortex in solid-body rotation (based on a constant viscosity) together with a close approximation to the model’s flow-dependent eddy viscosity (17), gives a reasonably good estimate of zm(rm) (Fig. 10). The stress model (22) introduces the factor [lυ(zm)]−1 in umin2zm(rm)/lυ(zm).

Fig. 11.
Fig. 11.

Schematic diagram of the key features of the boundary layer extracted from the R120L simulation. Shown are the angular momentum contours 2.5 and 3.0 × 106 m2 s−1, the position of maximum tangential wind (rm, zm) (magenta cross) and the location rmin of the minimum (maximum inward) radial velocity.

Citation: Journal of the Atmospheric Sciences 79, 8; 10.1175/JAS-D-21-0306.1

The physical content in rmwmηmumin2zm(rm)/lυ(zm) can be summarized as follows: Increasing zm with rm with umin ∼ ct. (within the individual sets of simulations) implies increasing horizontal mass flux through rmin and therefore, by continuity, increasing vertical mass flux at zm near r = rm (rmwm increases with rm). Although the stress model implies ηm ∝ 1/lυ(zm), the decrease of ηm with lυ(zm) is muted by the asymptotic limit lυ(z) → l for large z (≫l/κ).

Before closing, a few outstanding issues should be mentioned: As mentioned in section 1, EPI is an estimate of the gradient winds absent the boundary layer radial-momentum feedback, while αrmwmη is diagnosed from the fully interactive numerical simulations. A more complete theory yielding consistent interior and boundary layer winds at steady state would presumably come from a solution to the nonlinear partial differential equation (Tao et al. 2020b) from which (1) originates. More specifically, (1) of Tao et al. (2020b) can be written schematically as a nonlinear Helmholtz equation valid for the inviscid, adiabatic flow above the boundary layer,
Lψ=F(ψ;r,z),
where L is a Laplacian-type operator in cylindrical coordinates. The boundary conditions on ψ are on the interior domain (i.e., the lower boundary is at the top of the boundary layer). The nonlinear functional F depends on knowledge of flow within the boundary layer [e.g., M(ψ)] since the validity of (30) depends on the flow emanating upward from the boundary layer; however, the boundary layer model requires knowledge of the radial pressure gradient above the boundary layer [i.e., the solution of (30)]; hence an iteration would be required. Numerical models, such as the one analyzed here, do this iteration automatically. The author has not yet succeeded in discovering an analytical/theoretical way of unifying the interior and boundary layer solutions.

Second, a more theoretically justifiable theory for zm(rm) with a flow-dependent eddy diffusion in a Rankine-type vortex would be a welcome addition to the theory of tropical cyclone boundary layers. Third, the form of zm(rm) derived here depends on the assumed functional dependence of lυ(z) in (2); it remains to be seen what effect other equally reasonable functions lυ(z) has on the simulated zm(rm). Finally the variation of the present estimates of the supergradient wind to changes in Cd, Ck, l, and initial-vortex structure and amplitude was not addressed.

1

This is the flow in the radial-vertical plane; the primary circulation is the flow in the azimuthal direction.

2

By advecting upward an angular momentum distribution M(r) incompatible with that assumed for υg(r).

3

Figure 10 of Bryan and Rotunno (2009a) illustrates that this effect moves the maximum-tangential-wind angular-momentum surface inward with respect that implied in the EPI formulation.

4

A detailed analysis of how these parameters depend on rm would entail an analysis of how the flow-dependent diffusion and/or filters behave in cylindrical coordinates; such an analysis is not necessary to achieve the goals of the present study. For additional information on this issue, see the modeling study of Kepert (2017, his Figs. 6b–7b) which suggests eyewall width and radius increase together.

5

The horizontal diffusion terms in the radial-momentum equation are negligible in the RXL cases and are small (but nonnegligible inward of rmin) in the RX cases; horizontal diffusion affects u(r, z) in the RX cases mainly through its effect on M(r, z) in and above the boundary layer.

6

Solutions based on linear theory are all of this type.

7

The model-derived factor 0.8 is consistent with observations (Powell 1980).

8

The nonlinear solution (Rotunno and Bryan 2012, their Table 1) shows zm(K), the nonlinear equivalent of the tan−1 function with K = 1/χ.

9

Kepert’s (2012) Eq. (12) is the analogous result for a constant-stress layer (U2 = 0) and constant lυ.

10

This schematic is similar in concept to Fig. 1b in Montgomery and Smith (2017).

Acknowledgments.

Informal reviews of the first draft of this paper by Drs. George H. Bryan, Robert G. Nystrom, and Dandan Tao are gratefully acknowledged. Formal reviews by Jeffrey D. Kepert and Michael T. Montgomery greatly improved the clarity and accuracy of the revised manuscript. Richard Rotunno is supported by the National Center for Atmospheric Research, which is a major facility sponsored by the National Science Foundation under Cooperative Agreement 1852977.

Data availability statement.

The data used in this paper are available from the author on request.

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Save
  • Bell, M. M., and M. T. Montgomery, 2008: Observed structure, evolution, and potential intensity of category 5 Hurricane Isabel (2003) from 12 to 14 September. Mon. Wea. Rev., 136, 20232046, https://doi.org/10.1175/2007MWR1858.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bell, M. M., M. T. Montgomery, and K. A. Emanuel, 2012: Air–sea enthalpy and momentum exchange at major hurricane wind speeds observed during CBLAST. J. Atmos. Sci., 69, 31973222, https://doi.org/10.1175/JAS-D-11-0276.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bödewadt, U. T., 1940: Die Drehströmung über festem Grunde. Z. Angew. Math. Mech., 20, 241253, https://doi.org/10.1002/zamm.19400200502.

  • Bryan, G. H., 2012: Effects of surface exchange coefficients and turbulence length scales on the intensity and structure of numerically simulated hurricanes. Mon. Wea. Rev., 140, 11251143, https://doi.org/10.1175/MWR-D-11-00231.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., and R. Rotunno, 2009a: Evaluation of an analytical model for the maximum intensity of tropical cyclones. J. Atmos. Sci., 66, 30423060, https://doi.org/10.1175/2009JAS3038.1.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • Bryan, G. H., and R. Rotunno, 2009b: The maximum intensity of tropical cyclones in axisymmetric numerical model simulations. Mon. Wea. Rev., 137, 17701789, https://doi.org/10.1175/2008MWR2709.1.

    • Crossref
    • Search Google Scholar
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  • Bryan, G. H., R. P. Worsnop, J. K. Lundquist, and J. Zhang, 2017: A simple method for simulating wind profiles in the boundary layer of tropical cyclones. Bound.-Layer Meteor., 162, 475502, https://doi.org/10.1007/s10546-016-0207-0.

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  • Fig. 1.

    Initial tangential velocity profiles for rmi = 60, 90, and 120 km (solid magenta, green, and blue, respectively) and associated angular momentum (dashed) for the inner 300 km of the 1500-km domain.

  • Fig. 2.

    Steady-state radial and tangential velocities (u, υ) for R120. The contour interval (c.i.) for u is 10 m s−1 (black lines, negative values dashed) and the c.i. = 20 m s−1 for υ (magenta lines) with the maximum (113.8 m s−1) marked by the magenta × and the position rmin of umin indicated by the blue arrow.

  • Fig. 3.

    Steady-state vertical velocity w (green lines; c.i. = 2 m s−1) and azimuthal vorticity η (black lines; c.i. = 0.01 s−1); zero lines not plotted. Selected streamlines (gray) bracketing (rm, zm) marked by the magenta × and the position rmin of umin indicated by the blue arrow.

  • Fig. 4.

    M(r, z, t) at z = zm in indicated in Table 1 for cases (a) R60, (b) R90, and (c) R120. The 100 m s−1 contour for υm is indicated in black.

  • Fig. 5.

    M(r, z) (green; c.i. = 0.5 km2 s−1) and the dominant terms in the radial momentum equation [black, (M2Mg2)/r3; magenta, ∂τrz/∂z; c.i. =0.03 m s−2, zero line not plotted]. Position of υm indicated by the magenta × and the position rmin of umin is indicated by the blue arrow.

  • Fig. 6.

    Terms in (12) for the (a) RX and (b) RXL cases with −udu/dr in green, (M2Mg2)/r3 in magenta, and ∂τrz/∂z in black; magnitudes in m s−2. The blue stars indicate the position of rmin in each case.

  • Fig. 7.

    Estimated rotation rate ω of the gradient wind at z = 3 km for the (a) RX and (b) RXL cases with X = 60 (magenta), 90 (green), and 120 (blue). The placement of the crosses in the ω direction indicates the estimated rotation rate of the inner core and their placement in the radial direction indicates rm.

  • Fig. 8.

    The rotation rate υm/rm (s−1) (Table 1) vs the estimates of ω (s−1) from Fig. 7. The line υm/rm = μω with μ = 1.3 and 1.4 are plotted for reference.

  • Fig. 9.

    Vertical eddy viscosity Kυ for the (a) RX (c.i. = 50 m2 s−1) and (b) RXL (c.i. = 100 m2 s−1) cases, with X = 60, 90, and 120 (magenta, green, and blue, respectively). The major contributor to Kυ, (17), for the (c) RX and (d) RXL cases with the same c.i. as in (a) and (b). The × marks the location of (rm, zm) for each case.

  • Fig. 10.

    Solution for zm(rm) from the exact (19) (solid blue line) and the approximate (21) (dashed line) for μ = 1.35 along with the RX and RXL data. Also shown is the exact solution curve for μ = 2.7.

  • Fig. 11.

    Schematic diagram of the key features of the boundary layer extracted from the R120L simulation. Shown are the angular momentum contours 2.5 and 3.0 × 106 m2 s−1, the position of maximum tangential wind (rm, zm) (magenta cross) and the location rmin of the minimum (maximum inward) radial velocity.

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