1. Introduction

The nearly circular symmetry of mature tropical cyclones, together with the observation that they can exist in a quasi-steady state for some time, provides an opportunity for a relatively simple description of their physics and structure. Further assumptions of hydrostatic and gradient wind balance and convective neutrality of the vortex lead to strong constraints on its intensity and structure, as elucidated by Kleinschmidt (1951), D. Lilly (1973, unpublished manuscript, hereafter L73), Shutts (1981), and Emanuel (1986), among others. When coupled with formulas for surface fluxes of enthalpy and momentum, the radial distribution of boundary layer gradient wind is given as a function of the local air–sea enthalpy disequilibrium, the difference in absolute temperature between the top of the boundary layer and the outflow level, and nondimensional surface exchange coefficients.

One issue that arises in constructing solutions of this kind is the specification of the so-called “outflow temperature”—the absolute temperature attained by streamlines flowing upward and outward from the storm’s core as they asymptotically level out at large radii. L73 and Emanuel (1986) both assumed that a streamline emanating from the storm’s core would asymptote to the unperturbed environmental isentropic surface whose entropy matches that of the streamline, under the assumption that the vortex is “subcritical”—that is, that internal waves are fast enough that they can propagate from the environment inward against the outflow, so that the structure of the core is, in effect, partially determined by the unperturbed environmental entropy stratification. (In supercritical flow, information about the environment cannot propagate inward against the outflow and so the interior vortex cannot “know” about the environmental stratification. In this case, some kind of shock must develop in the outflow, providing a transition from the interior flow to the environment.) In developing solutions to the interior equations, Lilly assumed that the upper tropical troposphere has sub-moist-adiabatic lapse rates, so that entropy is increasing upward, and that therefore the outflow would be in the upper troposphere, with outflow temperature increasing with decreasing entropy. By contrast, Emanuel (1986) regarded the whole ambient troposphere as being neutral to moist convection, so that any streamline reflecting elevated boundary layer entropy would have to flow out of the storm at levels above the tropopause. Since the temperature structure of the atmosphere above the tropopause is approximately isothermal, Emanuel approximated the outflow temperature as a constant. Shutts (1981) arbitrarily specified the radial profile of gradient wind in the vortex and determined the outflow temperature that was consistent with such a profile.

Whichever assumption is used, the radial structure of the solutions is sensitive to the dependence of outflow temperature on the entropy of the outflowing streamlines. In particular, the assumption of constant outflow temperature leads to a highly unrealistic radial profile of gradient wind, unless additional assumptions are made about the entropy budget of the boundary layer, as in Emanuel (1986). Moreover, attempts to find time-dependent solutions under the assumption of constant outflow temperature lead to the conclusion that all nascent vortices should decay with time, unless an empirical factor is introduced that keeps the boundary layer entropy low outside the storm’s core (Emanuel 1997). The intensity of storms also depends on the radial structure of the vortex because it influences the radial gradient of boundary layer entropy, which in turn affects the degree of air–sea thermodynamic disequilibrium at the radius of maximum winds.

The poor solutions that result when constant outflow temperature is assumed, together with the great sensitivity of the solutions to the stratification of the upper troposphere when it is assumed to be positive, motivate a reexamination of the problem. After reviewing the steady-state theory, we begin by examining the structure of the outflow in a tropical cyclone simulated using a convection-resolving axisymmetric model and show that the outflow attains an entropy stratification that appears to be independent of any small stratification that may be present in the unperturbed environment. We then postulate that this stratification arises from small-scale turbulence and tends toward a value consistent with the hypothesis that the Richardson number is near a critical value. We test this hypothesis using data generated by the numerical simulations and proceed to examine its implications for the structure and intensification of the vortex. The paper concludes with a brief summary.

2. Review of analytic axisymmetric models

Analytical models of steady, axisymmetric vortices date back to L73, Shutts (1981), and Emanuel (1986). The most sophisticated of these, developed originally by Lilly, is based on conservation of energy, entropy, and angular momentum above the boundary layer and is reviewed in Bister and Emanuel (1998) and Emanuel (2004). It makes no assumption about hydrostatic or gradient balance above the boundary layer but is difficult to solve. For this reason, and because real and numerically simulated tropical cyclones are observed to be nearly in hydrostatic and gradient wind balance above the boundary layer, we follow Emanuel (1986) in assuming that such balances apply and further that the saturation entropy above the boundary layer does not vary along surfaces of constant absolute angular momentum. Where air is flowing upward out of the boundary layer, it is likely to be saturated and the coincidence of entropy and angular momentum surfaces is then demanded by conservation of both quantities. Here and elsewhere in the vortex this state is also the condition for neutrality to slantwise moist convection (Emanuel 1983). Saturation entropy is defined as

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where T is the absolute temperature, p the pressure, q* the saturation specific humidity, c_p the heat capacity at constant pressure, R_d the gas constant of dry air, and L_υ the latent heat of vaporization. In (1) we have neglected the small effect of condensed water on heat capacity and the gas constant. Absolute angular momentum per unit mass is defined as

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where V is the azimuthal velocity, r the radius, and f the Coriolis parameter.

The condition that s* is a function of M alone, together with hydrostatic and gradient wind balance, places strong constraints on the structure of the vortex. This can be seen by integrating the thermal wind equation along angular momentum surfaces. To develop an appropriate form of the thermal wind equation, we begin by writing the equations of hydrostatic and gradient balance in pressure coordinates:

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and

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Here α is the specific volume, ϕ is the geopotential, and its gradient with respect to r in (4) is taken holding pressure constant. The thermal wind equation is obtained by differentiating (3) with respect to r and subtracting the derivative of (4) with respect to p to obtain

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Since s* is a state variable, we may express α as a function of it and of pressure,¹ p, so that by the chain rule (5) becomes

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Using one of Maxwell’s relations (Emanuel 1986),

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we may write (6) as

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But s* is a function of M alone in a slantwise neutral vortex, so (7) can be rewritten as

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Dividing each side of (8) by ∂M/∂r yields an equation for the slope of angular momentum surfaces:

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Since neither M nor ds*/dM varies on angular momentum surfaces, (9) may be directly integrated in pressure to yield

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where r_b and T_b are the radius and temperature where a given angular momentum surface intersects the top of the boundary layer. Multiplying (10) through by M and using its definition [(2)] yields

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where V_b is the azimuthal velocity where the angular momentum surface in question intersects the top of the boundary layer. This equation may be regarded as yielding the dependence of the angular velocity V/r on the effective altitude in temperature space T_b − T.

The form of (11) suggests two possible definitions of outflow temperature. The first is to define it as the temperature along an angular momentum surface where it passes through the locus of points defined by V = 0. In that case, (11) may be written as

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where T_o is the outflow temperature thus defined. A second choice is to define the outflow temperature as the temperature along an angular momentum surface in the limit as r → ∞. In that case, from (11) and using V = M/r− fr, we have

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For angular momentum surfaces originating near the storm’s core, where V/r ≫ f, the two definitions of outflow temperature leading to (12) and (13) yield almost identical results. Either way, one should think of T_o as an upper or outer boundary condition, and note that it may be a function of s*, M, and/or streamfunction.

The relation (12) or (13) yields the distribution of angular velocity at the top of the boundary layer provided that s* is a known function of M and that T_o is a known function of s* (or M). L73, Emanuel (1986), and subsequent papers have argued that the function ds*/dM is determined by the boundary layer entropy and angular momentum balance. In particular, neutrality to moist convection requires that the saturation entropy above the lifted condensation level equal the actual entropy s of the boundary layer. The budget equation for boundary layer entropy, written in angular momentum–pressure coordinates and neglecting dissipative heating, is

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where is the total time derivative of angular momentum and F is the vertical turbulent flux of entropy. (Partial derivatives with respect to time and pressure hold angular momentum constant.) The former may be derived from the azimuthal momentum equation:

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where τ_θ is the tangential stress. Substituting (15) into (14) and assuming a steady state yields

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We now make two important assumptions about the distributions of entropy and angular momentum in the turbulent boundary layer. The first is that entropy is well mixed along angular momentum surfaces, which are approximately vertical in the boundary layer, so that the second term on the left-hand side of (16) is nearly zero while at the same time ∂s/∂M ≅ ds/dM is not a function of pressure. The second assumption is that the radius of angular momentum surfaces does not vary greatly with altitude within the boundary layer. To the extent that these two approximations hold,² (16) may be integrated through the depth of the boundary layer to yield

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where F_s is the surface entropy flux,³ τ_θs is the surface azimuthal stress, and is the vertically averaged radius of angular momentum surfaces, weighted by the convergence of the turbulent flux of momentum; also, we have defined the top of the boundary layer as the level at which the turbulent fluxes of entropy and momentum vanish. But note that we would obtain a different result if the turbulent fluxes of enthalpy and angular momentum were to vanish at different altitudes. Bryan and Rotunno (2009a) showed that (17) is well satisfied in axisymmetric numerical simulations. Substitution of (17) into (12) yields

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Thus, if the surface fluxes of momentum and enthalpy are known, if the outflow temperature is a known function of entropy, and if one can relate to r_b, then the gradient wind at the top of the boundary layer V_b can be found.

To close the problem, we need to formulate surface fluxes in terms of known variables and to say something about the outflow temperature. For surface fluxes, we use the classical aerodynamic flux formulas:⁴

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where C_k and C_D are dimensionless exchange coefficients for enthalpy and momentum, ρ is the air density, |V| is the wind speed at the flux reference level, is the saturation enthalpy of the sea surface, k is the actual enthalpy of air at the reference level at which the wind speed and exchange coefficients are defined, and T_s is the sea surface temperature. Using (19) and (20) in (18) gives

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Note that |V| has dropped out, but the unsubscripted V on the left-hand side of (21) is the azimuthal wind evaluated at the flux reference height.

It has become traditional to make several more approximations in (21), and we pause here to examine their credibility. First, we might approximate V by V_b. As pointed out by Smith et al. (2008) and others, this may not be a very good approximation, especially near the radius of maximum winds, where the wind speed may appreciably exceed its gradient value. Let us suppose, for the sake of argument, that V exceeds V_b by 30%. Making the approximation that V = V_b in (21) would then yield an error of about 15%, owing to the fact that one is taking the square root of the right-hand side of (21). But in reality the error would not be that large, owing to a compensation that occurs in the first factor on the left of (21), . When the actual wind is supergradient, , and this works to compensate the fact that V > V_b in (21). (In the special case that is the radius of the angular momentum surface where it passes through the flux reference level, the compensation is exact.) Thus if we make both of the two approximations and V ≅ V_b in (21), we can write an approximate equation for the gradient wind that is likely to be in error by less than 10%:

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We emphasize that this is a relation for the gradient wind, and not the actual wind in the boundary layer, which, as noted above, can appreciably exceed its gradient value near the radius of maximum winds. We should also point out that the specification of V_b as a function of radius would allow a complete calculation of the actual boundary layer wind were it used to drive a complete boundary layer model, as in Kepert (2010). We also note that if the dissipative heating were included in the derivation of (22), then, as shown by Bister and Emanuel (1998), the T_s that appears in (22) would be replaced by T_o.

The relation given by (22) is still not a closed solution for the gradient wind, because a) k is not known a priori, and b) the outflow temperature must be specified. The first problem can be addressed by integrating (17) inward from some specified outer radius, using (19) and (20) to specify the surface fluxes and with the gradient wind specified along the way by solving (22) and using a thermodynamic relationship between enthalpy and entropy. But this procedure also requires knowledge of T_o as a function of entropy or angular momentum. It is the problem of that specification that we address here.

3. The outflow temperature

As mentioned in the introduction, previous work has treated hurricane outflow as subcritical, in the sense that internal waves can propagate inward against the outflow and thereby transmit information from the environment inward to the core. It was assumed that this subcriticality would ensure a match between the entropy stratification of the outflow and that of the unperturbed environment: air flowing out of the core would attain an altitude such that its saturation entropy matched that of the distant environment. Lilly assumed that the upper tropical troposphere has a temperature lapse rate less than moist adiabatic, so that saturation moist entropy would be increasing with altitude, and the outflow would therefore be mostly or entirely in the upper troposphere. Emanuel (1986) and subsequent work assumed that the whole tropical troposphere is nearly neutral to moist convection and thus would have nearly constant saturation entropy; boundary layer air with entropy larger than that of the unperturbed environment would therefore have to flow out of the storm above the level of the unperturbed tropopause. Since the absolute temperature can be nearly constant with height just above the tropopause, one could assume constant outflow temperature as a first approximation.

But consider the consequences of constant outflow temperature for the radial structure of the gradient wind as given by (22). Since the quantity usually increases monotonically with radius, then V would therefore also have to increase monotonically with radius, which of course does not happen. Emanuel (1986) and subsequent work addressed this obvious problem by postulating that outside the storm’s core, turbulent fluxes of entropy out of the top of the boundary layer keep the boundary layer entropy relatively low and invalidate (17) in that region. This idea is supported qualitatively by the boundary layer entropy budget of a cloud-resolving, axisymmetric model simulation by Rotunno and Emanuel (1987, hereafter RE87), which showed that outside the core, surface enthalpy fluxes are balanced mostly by convective fluxes out of the top of the boundary layer. On the other hand, the recent successful simulation of a completely dry hurricane by Mrowiec et al. (2011) calls into question whether the import of low entropy air to the boundary layer by convective downdrafts is really essential to hurricane physics.

The poor prediction of the radial structure of the gradient wind by (22) with constant T_o motivates us to reexamine the question of outflow temperature. We begin by carrying out a numerical simulation of a tropical cyclone using a nonhydrostatic, convection-permitting axisymmetric model. The model is that of RE87 modified so that the finite difference equations conserve energy; this modification generally results is slightly weaker vortices. The model is run on a uniform grid in the radius–altitude plane, with radial and vertical grid spacings of 3.75 km and 312.5 m, respectively, in a domain extending to 1500 km in radius and 25 km in altitude. The Coriolis parameter is set to 5 × 10⁻⁵ s⁻¹. To better compare with the theory developed in section 4, we here omit dissipative heating as well as the pressure dependence of the sea surface potential temperature and saturation specific humidity, so that the surface saturation entropy is constant. In RE87, the vertical mixing length was 200 m but here we set it and the horizontal mixing length to 1000 m. Experiments with smaller vertical mixing lengths show only a weak dependence of storm structure and peak wind speed, consistent with the results of Bryan and Rotunno (2009b), but the effect we wish to illustrate here is more clearly defined for the larger mixing length. As has been documented by Bryan and Rotunno (2009b), there is sensitivity to the value of the horizontal mixing length, and the value used here was chosen to give broadly reasonable results. While the formulation of turbulence is clearly an important issue in understanding tropical cyclone intensity and structure, we here focus narrowly on the issue of the outflow temperature. Also, the surface exchange coefficients, which depend on wind speed, are both capped at 3 × 10⁻³, but they approach this value at a faster rate of 6 × 10⁻⁵ (m s⁻¹)⁻¹. All other parameters are set to the values listed in Table 1 of RE87, and the same Newtonian relaxation to the initial vertical temperature distribution is used here. Note that the use of such a relaxation does not permit the environmental temperature field to adjust to the presence of the vortex as it would in a closed domain when an actual radiative transfer scheme is applied, as in Hakim (2011).

The initial atmospheric temperature is specified to lie along a pseudomoist adiabat from the lifted condensation level of air at the lowest model level to a tropopause at an altitude just below 15 km and having a temperature of −86.7°C. For simplicity, the stratosphere is initially isothermal, having the same temperature as the tropopause. The sounding is dry adiabatic from the sea surface to the lifted condensation level, and the sea surface temperature is held constant at 24.89°C, yielding a potential intensity calculated using the algorithm described by Bister and Emanuel (2002), but with the sea surface saturation enthalpy held constant, of 67.9 m s⁻¹. The initial sounding is shown in Fig. 1.

Initial sounding used in simulations with an updated version of the RE87 numerical model. The thick solid line shows temperature; dashed line shows dewpoint temperature.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Initial sounding used in simulations with an updated version of the RE87 numerical model. The thick solid line shows temperature; dashed line shows dewpoint temperature.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Initial sounding used in simulations with an updated version of the RE87 numerical model. The thick solid line shows temperature; dashed line shows dewpoint temperature.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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The simulation is initialized with a weak, warm-core vortex and is integrated forward in time until a quasi-steady state is achieved. The time evolution of the maximum wind in the model domain and the maximum wind at the lowest model level are shown in Fig. 2 and compared to the aforementioned theoretical potential intensity. As shown by Bryan and Rotunno (2009b), the intensity achieved in such simulations is usually close to the theoretical potential intensity if the horizontal mixing length is sufficiently large, as it is here. For smaller mixing lengths, the actual boundary layer wind can be appreciably larger than its gradient value near the radius of maximum winds. While the existence of supergradient winds is clearly of interest, our purpose here is to focus on those deficiencies of the existing theory that are related to outflow temperature; thus, we choose to examine a simulation that does not otherwise exhibit serious discrepancies with theory.

Evolution with time of the peak wind speed (m s⁻¹) at the lowest model level (dashed) and within the whole domain (solid) in a simulation using an updated version of the RE87 axisymmetric model. Thin horizontal line shows potential intensity.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Evolution with time of the peak wind speed (m s⁻¹) at the lowest model level (dashed) and within the whole domain (solid) in a simulation using an updated version of the RE87 axisymmetric model. Thin horizontal line shows potential intensity.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Evolution with time of the peak wind speed (m s⁻¹) at the lowest model level (dashed) and within the whole domain (solid) in a simulation using an updated version of the RE87 axisymmetric model. Thin horizontal line shows potential intensity.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Figure 3 shows the distribution in the radius–altitude plane of saturation equivalent potential temperature averaged over the last 24 h of the integration, together with the contour representing the loci of vanishing azimuthal wind, likewise averaged over 24 h. We note several features of interest. First, it is clear that while some of the contours of saturation equivalent potential temperature erupting from the boundary layer near the radius of maximum winds (about 34 km) intersect the V = 0 contour above the altitude of the ambient tropopause (also shown in Fig. 3), many such surfaces erupting outside the radius of maximum winds flow out below the ambient tropopause. Moreover, the stratification of saturation equivalent potential temperature near the V = 0 contour does not seem to be related in any obvious way to the ambient stratification, which is zero below the tropopause and large and nearly constant above it; thus we regard the outflow as self stratifying. Figure 4 shows the mass streamfunction averaged over the final 24 h of the integration, together with the V = 0 contour; clearly, much of the outflow is below the tropopause and the absolute temperature increases monotonically with the value of the streamfunction, and with decreasing saturation entropy. Remember that, according to (12), the outflow temperature is defined as the temperature at which saturation entropy (or angular momentum) surfaces pass through the V = 0 contour.

Distribution in the radius–altitude plane of saturation equivalent potential temperature (K) averaged over the last 24 h of the numerical simulation described in the text. The thick gray curve represents the V = 0 contour and the thick white line represents the altitude of the ambient tropopause. For clarity, values of saturation equivalent potential temperature have been capped at 370 K.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Distribution in the radius–altitude plane of saturation equivalent potential temperature (K) averaged over the last 24 h of the numerical simulation described in the text. The thick gray curve represents the V = 0 contour and the thick white line represents the altitude of the ambient tropopause. For clarity, values of saturation equivalent potential temperature have been capped at 370 K.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Distribution in the radius–altitude plane of saturation equivalent potential temperature (K) averaged over the last 24 h of the numerical simulation described in the text. The thick gray curve represents the V = 0 contour and the thick white line represents the altitude of the ambient tropopause. For clarity, values of saturation equivalent potential temperature have been capped at 370 K.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Mass streamfunction (black contours) and absolute temperature (K; shading) averaged over the last 24 h of the simulation described in the text. The white contour represents V = 0.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Mass streamfunction (black contours) and absolute temperature (K; shading) averaged over the last 24 h of the simulation described in the text. The white contour represents V = 0.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Mass streamfunction (black contours) and absolute temperature (K; shading) averaged over the last 24 h of the simulation described in the text. The white contour represents V = 0.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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The supposition that the outflow temperature increases with angular momentum is borne out by interpolating absolute temperature onto angular momentum surfaces. Figure 5 traces the changes in absolute temperature and azimuthal velocity along each of a family of angular momentum surfaces separated by equal increments of angular momentum. The surfaces span roughly the interval between the radius of maximum surface wind and the radius of gale-force winds. The angular momentum surfaces fall into two groups. The leftmost group, representing relatively small values of angular momentum, consists of surfaces along which the mean flow is upward, whereas the rightmost group consists of angular momentum surfaces along which the mean flow is directed downward. In between and above the boundary layer is a region of nearly constant angular momentum. The thick gray curve in Fig. 5 shows the solution to (10) with (17) for the conditions of this simulation; clearly the angular momentum surface originating from the radius of maximum winds is close to that given by thermal wind balance above the boundary layer.

Family of surfaces (thin black curves) of constant absolute angular momentum traced versus absolute temperature and azimuthal velocity. The thick gray curve shows the shape of the angular momentum intersecting the boundary layer top at the radius of maximum winds, calculated assuming thermal wind balance. The dashed vertical line represents vanishing azimuthal wind, while the dashed horizontal line shows the ambient tropopause temperature. The innermost angular momentum surface originates near the radius of maximum wind.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Family of surfaces (thin black curves) of constant absolute angular momentum traced versus absolute temperature and azimuthal velocity. The thick gray curve shows the shape of the angular momentum intersecting the boundary layer top at the radius of maximum winds, calculated assuming thermal wind balance. The dashed vertical line represents vanishing azimuthal wind, while the dashed horizontal line shows the ambient tropopause temperature. The innermost angular momentum surface originates near the radius of maximum wind.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Family of surfaces (thin black curves) of constant absolute angular momentum traced versus absolute temperature and azimuthal velocity. The thick gray curve shows the shape of the angular momentum intersecting the boundary layer top at the radius of maximum winds, calculated assuming thermal wind balance. The dashed vertical line represents vanishing azimuthal wind, while the dashed horizontal line shows the ambient tropopause temperature. The innermost angular momentum surface originates near the radius of maximum wind.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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From Fig. 5, it is clear that within the upflow, outflow temperature increases with angular momentum; slowly at first and then more rapidly as angular momentum increases. The outflow temperature of the angular momentum surface erupting at the radius of maximum surface wind is very close to the ambient tropopause temperature in this simulation, but the assumption of constant outflow temperature for angular momentum surfaces originating outside the radius of maximum winds is poor. And, since in this simulation the ambient atmosphere above the tropopause is isothermal while there is no stratification of saturation entropy below the tropopause, one cannot assume that the outflow streamlines asymptote to absolute temperatures corresponding to those of the undisturbed saturation entropy surfaces of the environment. (Doing so for this simulation would yield a constant outflow temperature.)

Figure 5 presents strong evidence that hurricane outflow is self-stratifying, at least when the ambient upper troposphere has a moist adiabatic lapse rate. But what determines the stratification of the outflow? Specification of the dependence of outflow temperature on saturation entropy (or angular momentum) is sufficient to determine the radial profile of gradient wind in the boundary layer through (22); conversely, if one had an independent means of specifying the radial profile of boundary layer gradient wind, then the outflow temperature would be determined by (22). As we can think of no independent principle to determine the radial profile of gradient wind, we focus our attention on physical constraints on the outflow stratification.

Consider what would happen if all of the outflow streamlines asymptotically approached environmental isentropic surfaces corresponding to their own values of entropy, as originally postulated. Then, referring to Fig. 3, there would be large gradients of saturation entropy, angular momentum, and streamfunction in the outflow. The Richardson number Ri would be small, suggesting that small-scale turbulence would occur, mixing velocity and entropy and thereby expanding the depth of the outflow. Such an outcome is also possible in our numerical simulation, since the subgrid-scale turbulence parameterization of RE87 is sensitive to Ri. It is also possible that horizontal mixing in the upflow contributes to the physical spacing of angular momentum and entropy surfaces.

The distribution of calculated from fields averaged over the last 24 h of the simulation is shown in Fig. 6a. Here Ri is defined as

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where Γ_m is the moist adiabatic lapse rate and U is the radial velocity. In the region where the eyewall upflow turns outward, becoming outflow, Ri is relatively small. Figure 6b shows the vertical diffusivity v as formulated in RE87, likewise averaged over the last 24 h of the simulation. This quantity is positive only where Ri is less than unity, but because it also depends on strain, its average in time may be nonzero even when the time average of Ri is greater than unity. Thus Fig. 6b demonstrates that the Richardson number is below its critical value for at least some of the time in broad sections of the outflow. Figure 7 shows a scatterplot of the numerator of (23) against the denominator, for a region bounded by the black box shown in Fig. 6a. The value of Ri seldom falls below about 1, which is shown by the black line in the figure. The critical value of Ri for the onset of turbulence in the RE87 model is 1, but several aspects of the computation performed here may distort the calculated value of Ri. First, we have neglected the contribution to Ri of vertical gradients in condensed water content, as it appears in Eq. (28) of RE87. Second, we have calculated Ri from the time-averaged wind and entropy fields rather than taking the time average of the instantaneous values of Ri. Nevertheless, Fig. 6 shows that Ri may be systematically limited by a critical value in some regions of the storm’s outflow. We can now show that this region is likely to include the eyewall. First, we neglect ∂U/∂z compared to ∂V/∂z in the denominator of (23), and substitute

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Since s* is a function of M alone,

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Using these in (23) gives

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Thus, along any given streamline, Ri will be small at small radius and where ∂M/∂z is large. This corresponds well with the location of the box in Fig. 6a. In the following sections, we will explore the implications of the hypothesis of critical Richardson number for the structure and intensity of tropical cyclones.

(a) Square root of the Richardson number calculated from flow fields averaged over the last 24 h of the simulation described in the text. Values are bounded below by 0 and above by 3. The small black box shows the region from which the data plotted in Fig. 7 are drawn. (b) Contours of vertical diffusivity (m² s⁻¹) averaged over the last 24 h of the simulation.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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(a) Square root of the Richardson number calculated from flow fields averaged over the last 24 h of the simulation described in the text. Values are bounded below by 0 and above by 3. The small black box shows the region from which the data plotted in Fig. 7 are drawn. (b) Contours of vertical diffusivity (m² s⁻¹) averaged over the last 24 h of the simulation.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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(a) Square root of the Richardson number calculated from flow fields averaged over the last 24 h of the simulation described in the text. Values are bounded below by 0 and above by 3. The small black box shows the region from which the data plotted in Fig. 7 are drawn. (b) Contours of vertical diffusivity (m² s⁻¹) averaged over the last 24 h of the simulation.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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The buoyancy frequency squared plotted against the square of the vertical shear of the horizontal wind calculated from quantities averaged over the last 24 h of the simulation described in the text. The data are drawn from the region shown by the black box in Fig. 6. The straight line corresponds to a Richardson number of 1.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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The buoyancy frequency squared plotted against the square of the vertical shear of the horizontal wind calculated from quantities averaged over the last 24 h of the simulation described in the text. The data are drawn from the region shown by the black box in Fig. 6. The straight line corresponds to a Richardson number of 1.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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The buoyancy frequency squared plotted against the square of the vertical shear of the horizontal wind calculated from quantities averaged over the last 24 h of the simulation described in the text. The data are drawn from the region shown by the black box in Fig. 6. The straight line corresponds to a Richardson number of 1.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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4. Implications of critical Richardson number for steady-state structure

Suppose that the Richardson number is held to be at some fixed, critical value Ri_c at some radius r_t. Then (24) implies that

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Using

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we can write (25) as

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Thus the assumption of critical Richardson number at some radius r_t constrains the vertical gradient of saturation entropy there. But since s* is a state variable, its vertical gradient dictates the vertical gradient of temperature, from which one can deduce the gradient of temperature with respect to s*. Using the chain rule, we first write

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where the subscripts denote the quantity being held constant. Using the definition of saturation entropy and the Clausius–Clapeyron equation, we can write the first term as

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where R_υ is the gas constant for water vapor. Using the hydrostatic approximation for the second term in (27) and substituting (28), (27) can be written as

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For vertical gradients of s* that are consistent with (26), the second term on the right side of (29) is roughly an order of magnitude greater than the first. {Note that eliminating ds*/dM between (12) and (26) and substituting (26) into (29) shows that the last term in (29) will be much larger than the first term if . Assuming that the last term in square brackets is of order unity, this is well satisfied at hurricane wind speeds.} Using this approximation and substituting (26) into (29) gives

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This yields the dependence of T_o on saturation entropy consistent with the critical Richardson number hypothesis.

Strictly speaking, this relationship applies to the region where the Richardson number is near its critical value, and this may not correspond to the location along an angular momentum surface at which the azimuthal velocity vanishes, which according to (12) is where the outflow temperature is defined. On the other hand, the absolute temperature does not vary much from that point outward; otherwise, the alternative definition of outflow temperature based on (13) would be quite different. Thus, as long as the Richardson number is near its critical value somewhere near or outside the radius at which the azimuthal velocity vanishes, (30) should apply. On the other hand, we would not expect that the stratification is set by a critical Richardson number criterion where there is little mixing. Examination of Fig. 6b suggests that this is the case below about 7-km altitude and outside of about 70-km radius. For the present, we simply assume that (30) is valid on each angular momentum surface at r_t and return later to examine the validity of this assumption.

Using

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we may write (30) alternatively as

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This will allow us to solve for the outflow temperature in angular momentum coordinates.

A complete model for the steady-state structure of a tropical cyclone whose outflow temperature is specified by (31) may now be formulated. We begin with the definition of angular momentum, given by (2), and substitute (12) for the gradient wind at the top of the boundary layer:

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where M_b is the angular momentum at the top of the boundary layer. Substituting (19) and (20) into (17) and making use of (12) gives

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in which we have also used the approximation

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where is the saturation entropy of the sea surface and we have equated the boundary layer entropy with the saturation entropy of the free troposphere. Eliminating between (32) and (33) gives a quadratic equation for ds*/dM:

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where

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The physically relevant solution to (34) is

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Starting from some outer angular momentum surface M_o, (35) and (31) can be marched inward to yield s* and T_o as functions of M. Once these quantities have been obtained, r_b can be found using (32) and then the gradient wind can be found from the definition of angular momentum [(2)]. One complication is that depends on pressure, so that, in principle, one should also march the gradient wind equation inward to find the local surface pressure. For simplicity, we take to be a specified constant here, and we also assume that and T_b are constants. A more serious problem is that the condition of Richardson number criticality may not apply to the region of downflow, which for the aforementioned numerical simulation begins at about 120 km from the storm center. We shall return to this problem presently.

One must also satisfy boundary conditions. Given that (31) and (35) are first-order differential equations, two boundary conditions must be specified. The first might be to require the gradient wind to vanish at some outer radius r_o. Marching (35) inward will result in a monotonic increase in s*, which will eventually reduce χ in spite of decreasing M. Thus, according to (22), the gradient wind will achieve a maximum value at some particular value of M. On the other hand, the outflow temperature should achieve the ambient tropopause temperature T_t at or near the radius of maximum winds, but there is no guarantee from integrating (31) that this will be so. Thus, a “shooting” method is applied in which an outer radius is first specified, the system integrated, and the outflow temperature at the radius of maximum winds is noted. If it is not equal to T_t, the integration is restarted with a new value of r_o, and so on, until the outflow temperature at the radius of maximum winds equals T_t. Alternatively, one could simply specify r_o and adjust the value of r_t in (31) until the boundary conditions are met; this may make more sense since r_t is somewhat arbitrary in the first place and might be related, on physical grounds, to the radius of maximum wind. As discussed previously, the value of r_t will be consistent with the assumption that outflow stratification is determined by the critical Richardson number condition as long as it is near or outside the radius of vanishing azimuthal wind along angular momentum surfaces.

Before discussing numerical solutions of (31) and (35) with the imposed boundary conditions, it is helpful to derive analytic solutions valid in the region for which V ≫ fr. In this case, it is easily shown that one can neglect the second term in the square root appearing in (35) as well as the first term on the right side of (32). Then (31) and (35) can be directly integrated and the boundary conditions

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and

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applied, where r_m is the radius of maximum winds. The resulting expression for V can be maximized with respect to r to find the radius of maximum gradient winds and the maximum gradient wind. The outer radius can also be found. Of the parameters r_t, r_m, and r_o, only one may be regarded as a free parameter; once one is specified, the other two are determined. [Note also that the quantity r_t only occurs in the combination in (31).] We choose to regard r_o as the free parameter of the problem, given that it is more or less randomly distributed in nature (Chavas and Emanuel 2010), and determine r_m and r_t as part of the solution.

The asymptotically valid solution is most easily written in terms of angular momentum:

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where M_m is the angular momentum at the radius of maximum winds, whose relationship to r_o may be found from (36) by requiring that at the outer radius:

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For a reasonably intense vortex for which V_m ≫ fr_m (which must be satisfied in any case for this asymptotic solution to be valid), and for which r_o ≫ r_m, (37) reduces approximately to

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The maximum wind speed V_m found from maximizing the radial dependence of wind speed in the solution (36) is given by

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where V_p is a velocity scale derivable strictly from environmental parameters:

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Substituting (38) into (39) gives

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Note that when C_k = C_D, the maximum intensity is reduced from the “nominal” potential intensity given by (40) by a factor of . On the other hand, the nominal potential intensity derived here does not include the effects of dissipative heating or the pressure dependence of , both of which would increase intensity; also, the environmental saturation entropy that appears in (40) is that of the actual unperturbed environment rather than the boundary layer entropy at the radius of maximum winds that appears in (22). In the limit C_D → 0, the potential intensity approaches times the nominal value given by (40) evaluated with C_k = C_D. Also note that with the approximations made in going from (37) to (38), the outer radius (or, equivalently, r_t) drops out of the expression for maximum wind speed.

Using (41) and (40) in (38) gives

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Remarkably, the relationship between the radius of maximum winds and the outer radius does not depend on the exchange coefficients, to the extent that the applied approximations are valid.

One final relationship that comes from application of the boundary conditions is

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This completes the specification of the asymptotic solution valid where V ≫ fr. There are several noteworthy aspects of this solution. First, the factor multiplying the nominal potential intensity in (41) somewhat reduces the sensitivity of the overall solution to the ratio C_k/C_D. While at first glance it appears that (41) contains a singularity, it is in fact continuous through C_k = 2C_D. Figure 8 compares the dependence of the maximum wind speed divided by on this ratio to the square root dependence that would be inferred from a naive application of (22). As noted by Bryan and Rotunno (2009b) and others, the dependence of numerically simulated peak hurricane winds on the ratio of exchange coefficients is weaker than the square root dependence in regimes for which the actual wind is approximately equal to the gradient wind (although they noted that this dependence is also a function of the turbulent mixing lengths). This can be understood by close examination of (22). As the drag coefficient increases, the multiplicative factor in front of the right side decreases, but so too does k because of increased inflow. This second effect tends to increase the surface enthalpy flux, partially offsetting the increased dissipation associated with increased drag coefficient.

Dependence (nondimensional) of wind speed on the ratio of exchange coefficients calculated from (41) (solid) compared to a square root dependence (dashed).

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Dependence (nondimensional) of wind speed on the ratio of exchange coefficients calculated from (41) (solid) compared to a square root dependence (dashed).

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Dependence (nondimensional) of wind speed on the ratio of exchange coefficients calculated from (41) (solid) compared to a square root dependence (dashed).

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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The radius of maximum winds given by (42) increases with the square of the outer radius, and linearly with the Coriolis parameter. It becomes smaller with increasing (nominal) potential intensity.

Figure 9 compares the aforementioned analytic approximate solution to a numerical solution of (31) and (35). The analytic solution is such a good approximation to the full numerical solution through the whole range of radii that one might suspect that it is an exact solution. Experiments varying the ratio of exchange coefficients indeed show that the two solutions are almost identical through a wide range of conditions, but if the potential intensity is made small enough, differences begin to appear. For hurricane-strength vortices, the analytic solution given by (36), (41), and (42) is an excellent approximation to the full solution.

Numerical solution of (31) and (35) (solid) compared to the analytic solution (dashed) described in the text. Gradient wind has been normalized by its peak value, and radius has been normalized by the radius of maximum winds.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Numerical solution of (31) and (35) (solid) compared to the analytic solution (dashed) described in the text. Gradient wind has been normalized by its peak value, and radius has been normalized by the radius of maximum winds.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Numerical solution of (31) and (35) (solid) compared to the analytic solution (dashed) described in the text. Gradient wind has been normalized by its peak value, and radius has been normalized by the radius of maximum winds.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Figure 10 compares full steady-state solutions for the radial profile of gradient wind in the steady-state model to radial profiles of azimuthal wind 12 grid points (3.75 km) above the surface, averaged over the last 24 h of the numerical integration of the RE87 model described in the previous section. The comparison is carried out for three different values of the ratio of the surface exchange coefficients of enthalpy and momentum. The theory and model solutions agree well, except in the outer region where the vertical motion is downward into the boundary layer; in such regions there is little reason to expect that the critical Richardson number argument applies. What determines the radial structure of hurricanes in the downflow region? One argument, presented by Emanuel (2004), is that the Ekman suction velocity corresponding to the radial profile of gradient wind must match the subsidence velocity that balances clear-sky radiative cooling in the regions devoid of deep convection. Also note in Fig. 10 that the match between the theoretical and model solutions is not so good when C_k/C_D = 1.5, but the predicted peak winds match the model results quite well in all three cases.

Solutions of the steady-state model described in the text (dashed) compared to the radial profiles of azimuthal wind 12 grid points above the surface, averaged over the last 24 h of three simulations using the RE87 model (solid). The three pairs of curves correspond to three different ratios of the surface exchange coefficients, as given by the values in the boxes. The outer radius of the steady-state model has been chosen in each case to yield a good match between the predicted and modeled radii of maximum winds.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Solutions of the steady-state model described in the text (dashed) compared to the radial profiles of azimuthal wind 12 grid points above the surface, averaged over the last 24 h of three simulations using the RE87 model (solid). The three pairs of curves correspond to three different ratios of the surface exchange coefficients, as given by the values in the boxes. The outer radius of the steady-state model has been chosen in each case to yield a good match between the predicted and modeled radii of maximum winds.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Solutions of the steady-state model described in the text (dashed) compared to the radial profiles of azimuthal wind 12 grid points above the surface, averaged over the last 24 h of three simulations using the RE87 model (solid). The three pairs of curves correspond to three different ratios of the surface exchange coefficients, as given by the values in the boxes. The outer radius of the steady-state model has been chosen in each case to yield a good match between the predicted and modeled radii of maximum winds.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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Some aspects of the dependence of tropical cyclones on the ratio of exchange coefficients that emerge from the steady-state model can be easily tested in the full RE87 numerical simulation. Figure 11a shows the evolution of the domain peak wind speed with time in five simulations that differ only in the ratio of the exchange coefficients. The simulations with higher values of the ratio develop more quickly and to higher intensities. Figure 11b shows the same fields but with the wind speeds normalized by

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according to (40) and (41), and time normalized by (for reasons to be described in Emanuel 2011, manuscript submitted to J. Atmos. Sci., hereafter Part II). Bear in mind that while the simulations begin with the same actual intensity, their normalized initial intensities differ. Even with this caveat, the scaled intensity evolutions collapse nearly into a common evolution. This suggests that the steady-state model successfully predicts the dependence of the steady-state gradient wind on the ratio of exchange coefficients.

(left) Evolution with time of the peak surface wind in simulations of varying surface enthalpy exchange coefficient. (right) As in the left panel, but wind speeds have been normalized using (44) and time has been normalized by the inverse square root of the enthalpy exchange coefficient.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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(left) Evolution with time of the peak surface wind in simulations of varying surface enthalpy exchange coefficient. (right) As in the left panel, but wind speeds have been normalized using (44) and time has been normalized by the inverse square root of the enthalpy exchange coefficient.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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(left) Evolution with time of the peak surface wind in simulations of varying surface enthalpy exchange coefficient. (right) As in the left panel, but wind speeds have been normalized using (44) and time has been normalized by the inverse square root of the enthalpy exchange coefficient.

Citation: Journal of the Atmospheric Sciences 68, 10; 10.1175/JAS-D-10-05024.1

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5. Summary

Analytic models of idealized, steady-state, axisymmetric tropical cyclones have assumed that the outflow streamlines asymptotically approach altitudes at which their entropy values match those of the undisturbed environment. This assumption was based on the idea that hurricane outflow is subcritical, in the sense that internal waves can communicate information about the environmental stratification inward to the vortex core. We here showed that this is not the case in numerically simulated tropical cyclones, in which the outflow entropy stratification appears to be a product of the internal storm dynamics. We postulate that the entropy stratification is determined by a requirement that the Richardson number not fall below a critical value, and analysis of numerically simulated storms suggests that this hypothesis has merit. A new steady-state model was developed based on this hypothesis and shown to produce physically realistic results; asymptotic solutions to this model are available for the case in which dissipative heating is neglected. As was true in all previous analytic models of this kind, a single radial length scale must be externally specified, but given this specification, the radial geometry of the storm is determined by the model. Given an outer radius r_o at which the storm’s gradient wind is taken to vanish, the radial profile of the gradient wind is given to a good approximation by (42), (41), and (36). The revised model exhibits a weaker dependence on the ratio of exchange coefficients than the square root dependence cited in earlier literature but is consistent with the results of numerical simulations using full-physics models. The increase in outflow temperature with radius outside the storm’s core dictates the falling off of gradient wind with radius; this radial profile of gradient wind is in good agreement with that produce by full-physics models. In Part II we shall examine the consequences of outflow self-stratification for the intensification of tropical cyclones.