1. Introduction

The potential vorticity conservation principle provides a basis for understanding midlatitude weather systems, both through balanced models and primitive equation models. In many tropical weather systems, such as the ITCZ and tropical cyclones, the release of latent heat plays a crucial role, so that potential vorticity is not materially conserved. However, even in these cases, the relevant dynamics is “slow manifold dynamics,” with adjusted wind and mass fields intimately related to the evolving potential vorticity field. Thus, the analysis of tropical flows in terms of potential vorticity dynamics yields insights into such processes as ITCZ breakdown, the formation of easterly waves, and into the extreme inner core structures of tropical cyclones. The primary purpose of the present paper is to study the inner core potential vorticity structure of tropical cyclones simulated with a high resolution, axisymmetric, nonhydrostatic tropical cyclone model whose thermodynamic/dynamic foundations and discretizations are based on the work of Ooyama (1990, 2001, 2002, hereafter O90, O1, O2, respectively). The model has several unique features: 1) a very accurate treatment of moist processes within the context of equilibrium thermodynamics, with a simple switch to include or exclude the effects of ice; 2) an associated potential vorticity (PV) principle and an invertibility principle, both exactly derivable from the original model equations; 3) the inclusion of the thermodynamic and dynamic effects of precipitation, in particular the vertical transport of entropy and momentum by precipitation; 4) spatial numerics based on the cubic spline transform (CST) method, which results in small computational dispersion errors and noise-free nesting.

We begin in section 2 by presenting a concise, self-contained description of the model. In section 3 and appendix B we derive the potential vorticity and invertibility principles associated with our cloudy, precipitating model atmosphere. The moist generalization of the dry Ertel potential vorticity turns out to be P = ρ⁻¹ζ · ∇θ_ρ, where ρ is the total density of moist air, ζ is the absolute vorticity vector, and θ_ρ is the virtual potential temperature. In the axisymmetric case, the vorticity vector has components in the vertical, radial, and azimuthal directions, but ∇θ_ρ has components in the vertical and radial directions only. As a consequence, the azimuthal component of ζ is lost in performing the product ζ · ∇θ_ρ, and the potential vorticity simplifies to P = ρ⁻¹ {(−∂υ/∂z) (∂θ_ρ/∂r) + [ f + ∂(rυ)/r∂r] (∂θ_ρ/∂z)}, where υ is the azimuthal component of the wind. Since the evolving hurricane remains close to a state of gradient and hydrostatic balance, υ and ρ can be expressed in terms of the pressure p. Then, since θ_ρ is a function of ρ and p only, the potential vorticity can be expressed solely in terms of the pressure. In other words, the P field contains all the required information about the balanced part of the wind and mass fields, and the invertibility principle, which determines the pressure from the potential vorticity, is an elliptic partial differential equation in the radial-vertical plane. Understanding the evolution of the P field is thus a crucial part of understanding the whole intensification problem.

In section 4 we present the results of the control experiment, paying particular attention to the potential vorticity dynamics. With 500-m horizontal and vertical resolution in the inner core, a remarkable, hollow tower potential vorticity structure emerges in the quasi–steady state, with values of potential vorticity as high as 275 PV units (PVU; where 1 PVU = 1.0 × 10⁻⁶ m² s⁻¹ K kg⁻¹) in the eyewall. This is in sharp contrast to the “zero PV picture” that emerges when equivalent potential temperature or saturation equivalent potential temperature is used as the thermodynamic variable in the definition of potential vorticity.

Section 5 contains a discussion of selected sensitivity experiments, including the important effects of ice and the effects of precipitation terms in the entropy and momentum budgets. Concluding remarks are given in section 6.

2. Model

Consider atmospheric matter to consist of dry air, airborne moisture, and precipitation, with respective densities ρ_a, ρ_m, ρ_r, so that the total density is given by ρ = ρ_a + ρ_m + ρ_r.¹ The mass density of airborne moisture is the sum of the densities of water vapor and airborne condensate, so that ρ_m = ρ_υ + ρ_c. However, the partition of ρ_m into ρ_υ and ρ_c is not considered in the prognostic stage, but only later in the diagnostic stage. In cylindrical coordinates, with the assumption of axisymmetry, the mass conservation law for dry air is ∂ρ_a/∂t + ∂(ρ_aru)/r∂r + ∂(ρ_aw)/∂z = 0, the advective form of which is (2.1). Similarly, the conservation law for ρ_r is ∂ρ_r/∂t + ∂(ρ_rru)/r∂r + ∂[ρ_r(w + W)]/∂z = Q_r, where w + W is the vertical velocity of the precipitation (so that W is the fall velocity of the precipitation relative to the dry air and airborne moisture) and Q_r is the rate of conversion from airborne moisture to precipitation. Defining the precipitation mixing ratio as μ_r = ρ_r/ρ_a, the conservation law for ρ_r can be combined with (2.1) and written in the advective form (2.3). Finally, the conservation law for the airborne moisture is ∂ρ_m/∂t + ∂(ρ_mru)/r∂r + ∂(ρ_mw + F_m)/∂z = −Q_r, where F_m is the boundary layer turbulent flux of water vapor. Adding this conservation law for ρ_m to the conservation law for ρ_r and converting the result into an advective form for the total water mixing ratio μ = (ρ_m + ρ_r)/ρ_a, we obtain (2.2). Note that, although there are four types of matter (with densities ρ_a, ρ_υ, ρ_c, ρ_r), there are only three prognostic equations, (2.1)–(2.3), for the distribution of mass. As we shall see, the separation of the airborne moisture density ρ_m into the vapor density ρ_υ and the airborne condensate density ρ_c will be accomplished diagnostically in the two alternatives of (2.14).

The total entropy density is σ = σ_a + σ_m + σ_r, consisting of the sum of the entropy densities of dry air, airborne moisture, and precipitation. Since the vertical entropy flux is given by σ_aw + σ_mw + σ_r(w + W) + ρ_aF_s = σw + σ_rW + ρ_aF_s, we can write the flux form of the entropy conservation principle as ∂σ/∂t + ∂(σru)/r∂r + ∂(σw + σ_rW + ρ_aF_s)/∂z = 0, where F_s denotes the turbulent eddy flux in the atmospheric boundary layer, and where radiative effects have been neglected. Defining s = σ/ρ_a as the “dry-air-specific” entropy of moist air, we can combine (2.1) with the above entropy conservation principle to obtain (2.4).

We next consider the momentum equations. Defining the absolute angular momentum per unit mass as m = rυ + ½fr ², we can write the absolute angular momentum budget as ∂(ρm)/∂t + ∂(ρmru)/r∂r + ∂[(ρw + ρ_rW + F_m)m]/∂z = −∂(ρ_arF_υ)/∂z, where F_υ denotes the turbulent eddy flux of υ in the boundary layer. With the aid of the continuity equation for total density, this absolute angular momentum budget can be written as (2.6). In a similar fashion we can derive the radial wind Eq. (2.5), where p = p_a + p_υ is the sum of the partial pressures of dry air and water vapor. The derivation of the vertical equation of motion is somewhat more complicated. While all the matter moves with the same horizontal velocity, the vertical velocity of dry air and airborne moisture is w, while the vertical velocity of precipitation is w + W. The prediction of both w and w + W is equivalent to predicting W, which is inconsistent with the diagnostic treatment of W through parameterized cloud microphysics. A solution to this problem, proposed in O1, is to write a single budget equation for the total vertical momentum ρw + ρ_rW, and then approximate this equation by neglecting the material derivative of W along the precipitation path, that is, by neglecting ∂W/∂t + u(∂W/∂r) + (w + W)(∂W/∂z). With this approximation our vertical momentum equation becomes (2.7). The neglect of the material derivative of W along the precipitation path is consistent with the parameterization Eq. (2.18), which gives a slow variation of W because of the small fractional power of ρ_r and the inverse square root of ρ_a. The largest errors due to this assumption are expected in a small region near the melting level, where the ice factor f_ice results in an increase of the fall velocity of the precipitation.

Consolidating our results so far, the prognostic equations for the mass density of dry air ρ_a, the total water mixing ratio μ, the precipitation mixing ratio μ_r, the specific entropy s, and the velocity components u, υ, w are

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Associated with these seven prognostic equations is a set of diagnostic equations that is required to determine ρ, σ_r, p, W, Q_r, F_m, F_s, F_u, F_υ. The diagnostic set of equations can be divided into three subsets: thermodynamic diagnosis to determine ρ, σ_r, p; precipitation microphysics diagnosis to determine W and Q_r; and air–sea interaction diagnosis to determine F_m, F_s, F_u, F_υ. The equations for the thermodynamic diagnosis are

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where the known functions S₁, S₂, E, C, ρ*_υ are defined in appendix A. The thermodynamic diagnosis can be interpreted conceptually as follows: Input {ρ_a, μ, μ_r, s} ⇒ Output{ρ, ρ_r, ρ_m, ρ_υ, ρ_c, T₁, T₂, T, σ_r, p_a, p_υ, p}. Specifically, starting with the values of the prognostic, thermodynamic state variables ρ_a, μ, μ_r, s, the thermodynamic diagnosis proceeds in the order given, that is, determination of the densities ρ, ρ_r, ρ_m from (2.8), the thermodynamically possible temperatures T₁, T₂, and the entropy density of precipitation σ_r from (2.9) to (2.11), the actual temperature T of the gaseous matter from (2.12), the partial pressure of dry air from (2.13), the water vapor density ρ_υ, cloud condensate density ρ_c, and water vapor partial pressure p_υ from the appropriate alternative in (2.14), and the total pressure from (2.15). It should be noted that all the other required thermodynamic functions, such as C(T), ρ*_υ(T), etc., can be determined from E(T), once it is specified. If, for all T, E(T) is specified as E_w(T), the saturation vapor pressure over liquid water, then the effects of the latent heat of fusion are not included. If a synthesized E(T) is obtained from E_w(T) and E_i(T), the saturation vapor pressure over ice, then the model includes the effects of the latent heat of fusion. As discussed in O90, the synthesized E(T) involves two specified parameters: the center of the freezing zone T_f and the width of the freezing zone ΔT_f . For our experiments with ice, we use T_f = 273.15 K and ΔT_f = 1.0 K, with E_w(T) and E_i(T) obtained from the new Goff formulas (World Meteorological Organization 1979, appendix A).

The purpose of the precipitation microphysics diagnosis is to determine the precipitation fall velocity W and the precipitation source/sink term Q_r. Our microphysics parameterization is identical to that used by Ooyama (2001), who modified the Klemp and Wilhelmson (1978) formulation to include ice. The equations for the precipitation microphysics diagnosis are

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where the constants ρ_r0, W₀, τ_auto, τ_col, τ_evap are given in appendix A. Note that the ice factor, defined as the ratio of ice to rain terminal velocity, is assumed to have the temperature dependence given by (2.16). When used in (2.18), this ice factor provides for a fivefold increase in the terminal velocity of precipitation upon descent through the melting layer. Note also that the precipitation source/sink term Q_r is the sum of three terms: the autoconversion term Q_auto, which initiates the precipitation process when the cloud condensate density ρ_c exceeds 0.001ρ_a; the collection term Q_col, which is a function of the cloud condensate and precipitation densities ρ_c and ρ_r; and the evaporation term Q_evap, which is based on the analysis of the evaporation of a stationary drop and then augmented by the dimensionless ventilation factor f_vent (with typical magnitude ∼10).

The purpose of the air–sea interaction diagnosis is to determine the boundary layer turbulent fluxes of water vapor, entropy, and radial and tangential momentum. We make the simple assumption that these fluxes vary linearly with height in the boundary layer, vanish at and above z = 1000 m, and have surface values determined by the bulk aerodynamic formulas.² For example, for the momentum fluxes F_u and F_υ appearing in (2.5) and (2.6), the surface values are

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where the subscript zero denotes a surface value and V₀ = (u²₀ + υ²₀)^1/2 is the surface wind speed. Similarly, the surface values of the water vapor and entropy fluxes are

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where μ*_υ0 and s⁽²⁾_m0 are the saturation water vapor mixing ratio and the saturation entropy at the sea surface temperature and pressure. In all experiments reported here we have assumed a sea surface temperature of 28°C. We have also assumed that the drag and exchange coefficients are equal and depend on wind speed through Deacon’s formula

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where k = 4.0 × 10⁵ m⁻¹ s. It is well known (e.g., Malkus and Riehl 1960; Ooyama 1969; Emanuel 1995; Braun and Tao 2000) that the intensity of simulated tropical cyclones is sensitive to the relative magnitude of C_D and C_H. For example, if, in place of (2.25), one chooses C_D = 0.0011 + k_DV₀ and C_H = 0.0011 + k_HV₀, very intense storms are obtained when k_D = 0 and k_H = k, while much weaker storms are obtained when k_D = k and k_H = 0. Although such variations of k_D and k_H are extreme, it is an uncomfortable fact that the behavior of C_D and C_H at high wind speeds is one of the most uncertain aspects of all tropical cyclone models. The subject is one of active research (e.g., see Lighthill 1999; Emanuel 2003). Since our emphasis in this paper is on other aspects of the tropical cyclone problem, we use (2.25) in all the experiments reported here.

To summarize, the model consists of the seven prognostic Eqs. (2.1)–(2.7) and the eighteen diagnostic Eqs. (2.8)–(2.25). However, before discretization we make two further transformations. The first is motivated by the fact that direct use of the mixing ratio Eqs. (2.2) and (2.3) can lead to the problem of “negative water.” To prevent this, the model actually predicts ν = bhyp(μ) and ν_r = bhyp(μ_r), and then, for the purpose of thermodynamic diagnosis only, inverts these via μ = ahyp(ν) and μ_r = ahyp(ν_r), where bhyp and ahyp denote the biased hyperbolic transform and its quasi inverse, defined by

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where the bias constant μ₀ is set equal to 10⁻⁷. This procedure yields nonnegative values of μ and μ_r, which can be interpreted without difficulty by the thermodynamic diagnosis (2.8)–(2.15). In the second modification made before discretization, the prognostic equations for ρ_a, ν, ν_r, s are rewritten in terms of deviations from a resting background state that depends on z only. For example, (2.1) becomes

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where a′ = a − â, with a = ln(ρ_a/ρ_a0), â = ln(ρ̂_a/ρ_a0), and ρ̂_a(z) the specified background dry air density. This procedure of predicting deviations from a background state enhances the numerical accuracy of the model.

The domain extends from the sea surface to a height of 24 km and from the vortex center to a radius of 1536 km. The outer boundary is open, and the solutions are assumed to decay exponentially with an e-folding distance of 1400 km. The top and bottom boundaries are assumed rigid, with the boundary condition w = 0. To reduce the effect of the reflection of gravity waves off the top boundary, we have also included Rayleigh-type damping terms in (2.4)–(2.7). These terms damp the winds to zero and the specific entropy to its background value. The damping coefficient vanishes for z ≤ 18 km and increases linearly with height from 18 km to a maximum value of 0.015 s⁻¹ at 24 km. Thus, the region between 18 km and the lid at 24 km is a sponge layer that effectively damps vertically propagating gravity waves that would otherwise reenter the region below 18 km after falsely reflecting off the lid. In all the cross sections of the present paper, only the region below 18 km is displayed.

The model spatial numerics are based on the CST method described in O2. As the name implies, the CST method uses the cubic B spline as the basis function. Because the first two derivatives of the B spline are continuous, the CST method has small computational dispersion errors, similar to the Fourier spectral method. Yet, because the B spline is locally defined, the CST method allows flexibility with regard to boundary conditions. With reduced dispersion errors and flexible boundary conditions, the CST method provides for noise-free nesting. To take advantage of this, the domain is discretized into a series of nested grids. As illustrated in Fig. 1, the horizontal domain consists of six grids. The horizontal grid spacing Δr within each grid increases by a factor of 2 from the finest grid at 0.5 km to the coarsest grid at 16.0 km. The vertical domain, in contrast, consists of a single grid with 48 grid intervals and a grid spacing Δz of 0.5 km. Time integration is accomplished in a two-stage process. In the first stage all the prognostic variables are advanced explicitly using the leapfrog scheme with a small enough time step for stability of gravity waves. In the second stage these explicit predictions are implicitly adjusted via a second-order elliptic equation. This particular implementation of the semi-implicit method allows a tenfold increase of the time step. The time steps are 2.5 s on the finest grid, 5 s on the next two grids, and 10 s on the coarsest three grids.

3. Potential vorticity equation and invertibility principle

Under the assumption of axisymmetry, the radial and vertical components of the absolute vorticity vector are (ξ, ζ) = (−∂m/r∂z, ∂m/r∂r) = [−∂υ/∂z, f + ∂(rυ)/r∂r], where m = rυ + ½fr ² is the absolute angular momentum per unit mass. These are the only vorticity components that contribute to the potential vorticity. Since υ is the only velocity component appearing in the definitions of ξ and ζ, it follows that the only momentum equation involved in the PV derivation is (2.6). The derivation of the PV equation from (2.6) and the continuity equation for total density is given in appendix B. The result is

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where θ_ρ is the virtual potential temperature and P is defined below in (3.5). The effects of θ̇_ρ and ṁ appear in the first and second terms on the right-hand side of (3.1), with the former being interpreted as a measure of the variation of θ̇_ρ along the m surfaces [or equivalently the variation of θ̇_ρ along the projection of the vorticity vector onto the (r, z) plane], and with the latter being interpreted as a measure of the variation of ṁ along the θ_ρ surfaces. The last term in (3.1) is generally small and describes the effects of precipitation and boundary layer water vapor flux on the potential vorticity.

The variable P has two important properties that contribute to its fundamental importance (Schubert 2004): (i) it reduces to the classical Ertel PV in the limit of a completely dry atmosphere; (ii) it is invertible, that is, it carries all the dynamical information about the balanced part of the wind and mass fields. Other choices for the scalar field in the definition of PV are lacking in this regard. For example, if saturation equivalent potential temperature is used in place of θ_ρ, property (i) is lost, while if equivalent potential temperature is used in place of θ_ρ, property (ii) is lost.

To see that an invertibility principle exists for P, consider the situation in which the radial momentum Eq. (2.5) and the vertical momentum Eq. (2.7) reduce to the gradient wind Eq. (3.2) and the hydrostatic Eq. (3.3), respectively. Then, including the definition of virtual potential temperature (3.4) and the definition of PV (3.5), we have the system

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This is a system of four equations for the four unknowns υ, ρ, p, θ_ρ, with given P. By eliminating υ, ρ, and θ_ρ we can obtain a single partial differential equation relating the pressure p to the potential vorticity P. Although this equation is complicated in the z coordinate, simpler equivalent forms can be obtained by using either (p/p₀)^κ or θ_ρ as the vertical coordinate (Schubert et al. 2001). Note that the solution of the invertibility problem gives us the total density ρ, the total pressure p, and the virtual potential temperature θ_ρ, that is, the parts of the mass field that are of direct dynamical significance. It can be shown (Schubert 2004) that the moist invertibility principle (3.2)–(3.5) is isomorphic with the dry invertibility principle solved in previous studies (e.g., Schubert and Alworth 1987) under the interchanges ρ_a ↔ ρ, p_a ↔ p, and θ_ρ ↔ θ. Thus, previous studies of the dry invertibility principle are easily interpreted in terms of the moist invertibility principle.

In this paper we will not be concerned with actually solving the invertibility principle (3.2)–(3.5). Rather, we simply use the existence of the principle as justification for studying model output fields of P. In section 4 we will use a transformed version of (3.1) to understand the PV dynamics of a quasi-steady tropical cyclone. When attempting to obtain physical understanding of certain phenomena, it has been said (Stommel 1995) that complex models such as (2.1)–(2.27) are “not much help: like vegetable soup, they have too many ingredients to reveal which one imparts the flavor.” In the case of a tropical cyclone, it is definitely the PV that imparts the flavor.

4. Control experiment

The initial condition for all experiments is a purely azimuthal vortex in gradient and hydrostatic balance, with a horizontally uniform relative humidity field. At the surface the radial distribution of initial azimuthal velocity is 2υ_m(r/r_m)/[1 + (r/r_m)²], with the maximum wind specified as υ_m = 12 m s⁻¹ and the radius of maximum wind as r_m = 100 km. The initial azimuthal velocity is assumed to decrease linearly with height to zero at z = 18 km, and to be zero between 18 km and the model top at 24 km. Figure 1 shows this initial azimuthal wind field. The associated pressure anomaly field has a minimum of −7.1 hPa at the surface in the vortex core. The far-field temperature and humidity is taken from Jordan’s (1958) mean hurricane season sounding and is shown in Fig. 2. At the initial time this sounding is approximately valid at all radii, since the initial horizontal temperature gradient is weak, with the lower tropospheric vortex core approximately 1.1 K warmer than the far field.

To summarize the temporal and spatial evolution of this control (CNTL) experiment, Fig. 3 depicts the 3-h running mean surface tangential wind speed as a function of radius and time. Between 40 and 50 h, a very small, intense central vortex develops. As discussed in appendix C, this small, central vortex is not the result of a problem with CST numerics. In fact such small, central vortices also appear in previous high resolution axisymmetric tropical cyclone simulations (e.g., Yamasaki 1983; Willoughby et al. 1984) that use entirely different numerics. Although strong cumulus convection (referred to as a hub cloud) is sometimes observed in the center of a hurricane eye, the convection appearing near r = 0 in our CNTL experiment is unrealistically deep and intense. Beyond 80 h, however, this central vortex weakens substantially, while a secondary tangential wind maximum forms at about 30 km and propagates inward, intensifying along the way. This wind maximum is collocated with a ring of convection that eventually contracts to become the new tropical cyclone eyewall. From 100 to 180 h, the tropical cyclone gradually intensifies. As shown in Fig. 3, this overall increase of intensity is not steady but highly variable. For instance, during the 12-h period following 118 h, the maximum tangential wind oscillates over 20 m s⁻¹. This variability, which is very similar to that obtained by Willoughby et al. (1984), is caused by disturbances of the boundary layer inflow and the formation of secondary rings of convection that propagate inward. The secondary circulation of a new, inward-propagating ring and the restriction of high θ_e inflow into the core, results in the dissipation of the primary ring, and its ultimate replacement by the secondary ring.

Beyond 180 h, the vortex settles into a quasi steady state. Figures 4a–c depict the 180–240-h average primary and secondary circulations. The model steady-state cyclone is similar in structure to observed storms but is more intense. The tangential wind has a maximum of over 90 m s⁻¹, with the peak located beneath the eyewall at a radius of 14 km. This intense vortex is produced by the advection of high angular momentum air toward the center. This same air is later advected away from the center within the outflow branch of the secondary circulation, producing an anticyclonic vortex with a wind speed of −30 m s⁻¹ at a radius of 1250 km and a height of 14 km. The horizontal branches of the secondary circulation are concentrated into shallow layers at the surface and near the tropopause. As shown in Fig. 4b, the inflow branch is confined³ below 2 km, with a maximum inward radial flow of 31.8 m s⁻¹ at r = 21 km. The maximum inflow lies just outside the radius of maximum tangential wind. The most intense outflow is confined to a layer between 12 and 18 km. The maximum outflow of 29.5 m s⁻¹ is located at a radius of about 75 km, which is more than 50 km away from the eyewall. The two vertical branches of the secondary circulation are very different in terms of intensity and horizontal scale. Immediately above and sloping away from the region of maximum surface convergence is the ascending branch of the secondary circulation, embedded within the eyewall cloud. In the ascending branch, vertical velocities reach 5.5 m s⁻¹ just above the boundary layer and in the upper troposphere. The width of the ascending branch increases with height. In contrast, the compensating subsidence in the descending branch of the secondary circulation is very weak and extends into the far field of the domain. The transverse circulation depicted in Fig. 4 provides a deformation field in the (r, z) plane, with especially large deformation in the lower troposphere at radii between 10 and 20 km. This frontogenetic effect (Eliassen 1959; Emanuel 1997) acts to crowd together the absolute angular momentum surfaces, resulting in large vorticity.

Figure 5 shows the 180–240-h mean cross sections of T and the water vapor mixing ratio μ_υ. The maximum T ′, disregarding the effect of the very small central vortex, is approximately 17 K and is located at a height of 12 km. The axis of maximum T ′ extends both outward into the stratiform precipitation and downward along the inner edge of the eyewall. The horizontal extent of these temperature anomalies depends on the local radius of deformation, defined by (gH)^1/2{[ f + ∂(rυ)/r∂r]( f + 2υ/r)}^−1/2, where H is the equivalent depth and where the factor in the denominator is a measure of the inertial stability or “stiffness” of the vortex. When the radius of deformation is small, the vortex is stiffened such that the horizontal extent of the secondary circulation is restricted. Assuming (gH)^1/2 ≈ 60 m s⁻¹, the radius of deformation is less than 5 km along the inner edge of the eyewall; however, within the outflow, it is larger than 300 km. Therefore, we observe a narrow secondary circulation and adiabatic warming on the inner edge of the eyewall, as compared to a broad secondary circulation outside the eyewall.

The dynamics of the eye and eyewall also have a distinct influence on the distribution of water vapor, as shown in Fig. 5b. Radially, the water vapor mixing ratio μ_υ is a maximum in the eyewall, because of vertical advection within the updraft. Because of subsidence within the eye, μ_υ is a minimum along the inside of the eyewall. At the surface, the large flux of water vapor increases μ_υ outside 25 km to 23.9 g kg⁻¹, which is 5.5 g kg⁻¹ greater than the initial value. The corresponding surface relative humidity is nearly 100%. Beneath the eyewall, downdrafts produce a local surface minimum of 21.6 g kg⁻¹ at a radius of 14 km. Above the 0°C isotherm, the terminal velocity of precipitation is approximately 1.5–2 m s⁻¹, whereas below this level, it increases to as much as 8 m s⁻¹ within the eyewall. Because of the relatively small terminal velocity aloft and the intense outflow, precipitation is advected far from the eyewall, resulting in surface precipitation rates of about 10–40 mm h⁻¹. However, beneath the eyewall, W and μ_r are both large, producing a precipitation rate of over 200 mm h⁻¹.

Figure 6 shows 180–240-h mean cross sections of the vertical component of the absolute vorticity ζ, the virtual potential temperature θ_ρ, and the potential vorticity anomaly P′ = P − P, where the far-field potential vorticity is defined by P = ( f/)(d_ρ/dz). Since the horizontal shear across the inner edge of the eyewall is very large in the middle troposphere, with tangential wind speeds increasing by 60 m s⁻¹ over only 5 km radius, there is a midtropospheric maximum of ζ and an associated maximum of P′ ≈ 275 PVU. In contrast, for the upper-tropospheric maximum of P′ ≈ 275 PVU the (∂υ/∂z) (∂θ_ρ/∂r) term in (3.5) becomes more important because of the large horizontal θ_ρ gradient resulting from the warm core of the vortex and the relatively large vertical shear of the tangential wind beneath the tropopause. In addition there is enhanced PV along the central axis, a remnant of the small central vortex. If the r axis in Fig. 6c were stretched to correspond to the z axis, the outward slope of the large PV on the inner edge of the eyewall cloud would become more apparent, and the structure would resemble a leaning tower of high PV. In three dimensions it might be viewed as a bowl of high PV.

To understand the origin of this remarkable PV structure, let us return to (3.1), neglecting the generally small last term. To rewrite this equation in a more physically revealing form, first define the potential radius R by ½fR² = m = rυ + ½fr ², so that fRṘ = ṁ where Ṙ = DR/Dt. Transforming from (r, z, t) to (R, θ_ρ, T), where T = t, the first two terms on the right-hand side of (3.1) can be written as

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and the material derivative operator as

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Note that ∂/∂R implies fixed θ_ρ, ∂/∂θ_ρ implies fixed R, and ∂/∂T implies fixed R, θ_ρ. Using (4.1)–(4.3) in (3.1), we obtain

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In the steady state and above the frictional boundary layer we have ∂P/∂T = 0 and Ṙ = 0, so that the potential vorticity Eq. (4.4) reduces to θ̇_ρ(∂P/∂θ_ρ) = P(∂θ̇_ρ/∂θ_ρ), which can also be written as

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Thus, in the steady state above the boundary layer, the ratio of θ̇_ρ to P is constant along an absolute angular momentum surface. For an angular momentum surface erupting from the boundary layer into the eyewall updraft we can assume that the integration constant required by (4.5) is set by P(R, θ_ρB) and θ̇_ρ(R, θ_ρB), the values of P and θ̇_ρ at the top of the boundary layer. Then, integration of (4.5) yields

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Thus, in the steady state there is an intimate connection between the P field and the θ̇_ρ field. In essence, the steady-state P field has become locked to the θ̇_ρ field.

Since the P field determines the primary circulation and the θ̇_ρ field, along with frictional effects, determine the secondary circulation, (4.6) can be interpreted as the fundamental relation between the primary and secondary parts of the steady-state circulation.

In the steady state, a parcel of eyewall air erupting from the boundary layer on a given angular momentum surface stays on this same outward-sloping surface as it spirals upward, crossing θ_ρ surfaces and changing its PV at the rate P(∂θ̇_ρ/∂θ_ρ), which is positive below the level of maximum θ̇_ρ and negative above this level. Note that the level of maximum θ̇_ρ is also the level of maximum P through the coupling described by (4.6). All parcels erupting from the boundary layer on the same angular momentum surface have the same Lagrangian history, but the Lagrangian histories are generally distinct on different angular momentum surfaces. Unfortunately, when applied to the CNTL experiment, the above argument is compromised by the unsteadiness illustrated in Fig. 3. However, as we shall see, the argument more accurately holds for the “no ice experiment” discussed in section 5.

It is interesting to note that the generalized moist PV, defined by (3.5), is only a modest modification of the dry Ertel PV, which is obtained by replacing ρ with ρ_a and θ_ρ with θ in (3.5). This modest modification is in sharp contrast to other proposed PV generalizations that involve use of the equivalent potential temperature or the saturation equivalent potential temperature, both of which result in drastic differences with the dry Ertel PV. To confirm that (3.5) yields PV fields that are very similar to dry Ertel PV fields, we have produced a 180–240-h mean cross section of the dry Ertel PV. This dry PV cross section (not shown) is nearly identical to Fig. 6c. Thus, the dry Ertel PV is useful for diagnostic analysis of moist models (e.g., Yau et al. 2004; Braun et al. 2006). In addition, although a hurricane is definitely a phenomenon involving moist physics, a reasonable approximation to the balanced dynamics can be constructed (e.g., Schubert and Alworth 1987; Möller and Smith 1994) using the dry PV invertibility principle and the dry PV evolution equation, as long as diabatic and frictional effects are properly included in the latter equation.

5. Sensitivity experiments

6. Concluding remarks

We have presented results from an axisymmetric, nonhydrostatic, full-physics model of the tropical cyclone and have used the moist potential vorticity principle as a diagnostic for interpreting the quasi-steady-state structure. This analysis demonstrates how the P and θ̇_ρ fields become locked together in a thin leaning tower on the inner edge of the eyewall cloud. In addition, our sensitivity experiments indicate that accurate simulations of tropical cyclones require 1) the inclusion of ice and, in particular, the slow terminal velocities associated with frozen precipitation above the melting level; 2) the inclusion of the vertical transport of moist entropy by precipitation.

Since the numerical model used in this study is based on the equation set (2.1)–(2.25), it is interesting to comment on the usefulness of these equations as a physical model of a tropical cyclone. We certainly have the most confidence in the prognostic Eqs. (2.1)–(2.7) and the equilibrium thermodynamics (2.8)–(2.15) as being part of an accurate description of nature. Although precipitation microphysics has been parameterized as a bulk process, (2.16)–(2.18) and (2.21) have considerable observational and laboratory support (e.g., see Kessler 1969). However, (2.19) and (2.20) must be regarded as a crude parameterization of the intricate process of precipitation formation. As for the air–sea interaction parameterization (2.23)–(2.25), the most uncertain aspect is (2.25), especially the values of C_D and C_H at high wind speeds. Finally, a major limitation of the model (2.1)–(2.25) is the assumption of axisymmetry.

Consistent with the results of Persing and Montgomery (2003), our results show that the quasi-steady-state intensity for an axisymmetric storm may be much greater than predicted by energetically based, steady-state maximum potential intensity (MPI) theories. To explain this discrepancy, Persing and Montgomery have used the nonhydrostatic model developed by Rotunno and Emanuel (1987) to show that past model simulations supporting such MPI theories may have been limited by their coarse resolution and large diffusion—problems that have been overcome in both the Persing–Montgomery simulations and those presented here. At the same time it should be realized that high resolution, low diffusion, axisymmetric models probably produce overly intense vortices because of the suppression of variations in the azimuthal direction. In fact, the PV distributions produced in such models would be unstable in a combined baroclinic–barotropic sense if the models allowed variations in the azimuthal direction. Over their life cycle such instabilities would radially mix the PV, thereby reducing the maximum tangential wind (Schubert et al. 1999). In addition, although difficult to observe and thus inadequately documented, Kelvin–Helmholtz instability along the sloping inner edge of the eyewall may also play a role in limiting the maximum tangential winds.

Although the PV mixing process can be crudely parameterized in an axisymmetric model by using radial diffusion or hyperdiffusion, there are unrealistic aspects associated with this type of parameterization (Kossin and Schubert 2003). In this regard it should be recalled that our model has no explicit frictional effects above the boundary layer, although the CST method does include a third-order derivative constraint in the least squares minimization that defines the transform of spatial fields to nodal amplitudes. This third order derivative constraint is equivalent, in wavenumber space, to a sharp, sixth-order, low-pass filter that effectively eliminates small-scale errors at the resolution limit. This filter is weak compared to the usual diffusion or hyperdiffusion used in most finite difference models, so that the model flow above the boundary layer should be regarded as quite inviscid. Thus, the assumption of axisymmetry, in conjunction with the nearly inviscid nature of the flow above the boundary layer, probably leads to the overly intense vortices produced by the model.

Even with the above limitations in mind, our results help answer the question, “What is a quasi-steady-state hurricane?” It is no doubt a complicated structure involving all three flow components and many moist thermodynamic fields. But, at its core, it is an extreme structure in which the P field and the θ̇_ρ field have become intimately coupled in such a way that they vary in a similar fashion along a tightly packed group of absolute angular momentum surfaces. To capture the formation and asymmetric evolution of such extreme PV structures in full-physics 3D models is obviously a very challenging problem, but it may be necessary for accurate intensity forecasts. Perhaps this is part of a sobering realization that hurricane intensity forecasting is fundamentally much more difficult than hurricane track forecasting.