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  • View in gallery

    Thermal forcing T ′e(r, z) = Te(r, z) − TR(z) (in K) as given in (31) for T0 = 10 K, R = 150 km, r1 = 20 km, and D = 16 km.

  • View in gallery

    Numerical solution for the forcing from Fig. 1 using αT = 10−2 min−1 and cD = 2 × 10−3. (a) Tangential wind u (solid, in m s−1) and potential temperature θ (dashed, in K, contours every 10 K), (b) potential radius Rp (solid, in km) and potential temperature θ (dashed, in K, contours every 10 K), (c) streamfunction ψ (in kg s−1), and (d) the vertical wind w (in cm s−1).

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    Eliassen secondary circulation owing to (a) diabatic heating only and (b) surface drag only. In both panels the contours depict the streamfunction ψ in kg s−1.

  • View in gallery

    Two numerical solutions with radial scale R = 150 km, (left) one in the hurricane regime (αT = 10−1 min−1, cD = 2 × 10−3) and (right) one in the monsoon regime (αT = 10−3 min−1, cD = 2 × 10−3). (a),(d) Tangential wind (solid, in m s−1) and potential temperature (dashed, in K), (b),(e) potential radius (solid, in km) and streamfunction ψ [dashed contour interval is 108 kg s−1 in (b) and 2 × 106 kg s−1 in (e)], and (c),(f) the vertical wind (in cm s−1, negative values shaded).

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    Same as Fig. 4, but with a much larger radial scale R = 1500 km. The contour interval for the dashed contours is 2 × 109 kg s−1 in (b) and 5 × 107 kg s−1 in (e).

  • View in gallery

    Semianalytical solution in the hurricane limit with the forcing from Fig. 1 and cD = 2 × 10−3. The plot conventions are like in Fig. 2.

  • View in gallery

    Radial dependence of two quantities characterizing the primary flow of the vortex: (a) the surface wind u0 and (b) the vertically averaged temperature anomaly R. In both panels the solid line represents the semianalytical solution in the hurricane limit, while the dashed lines represent five numerical solutions with different values of αT (bottom to top: αT = 1000−1, 316−1, 100−1, 32−1, 10−1 min−1).

  • View in gallery

    Schematic representation of the radial dependence of the equilibrium temperature Te (solid) and the actual temperature T (dashed) for supercritical forcing T0 > Tc0. The gray shading indicates the range available to T, and the open arrows point to the hurricane and monsoon limit, respectively. The temperature difference TeT changes sign at some outer radius ro; the radii displayed satisfy r < ro.

  • View in gallery

    (a) Surface wind and (b) departure from thermal equilibrium for 99 different vortices. The radial scale is R = 150 km for all vortices. In both panels each cross represents the numerical solution for a specific pair of values of αT and cD, and the abscissa is log10 F [with αT measured in (10 s)−1]. (a) The value plotted on the ordinate is the normalized maximum of the surface wind u0(r) over all radii rR, with the normalization factor being the corresponding maximum of the hurricane equilibrium solution. (b) The value plotted on the ordinate is log10 of the maximum value of [e(r) − (r)]/N, with the normalization factor N being the expected temperature difference for the monsoon regime according to the approximate semianalytical theory of Wirth (1998). The thin solid line in (b) depicts slope −1.

  • View in gallery

    Same as Fig. 9a, but here the value plotted on the ordinate is log10 of the maximum vertical wind w (in cm s−1). The thin solid lines depict slope +1 and −1, respectively.

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A Unified Perspective on the Dynamics of Axisymmetric Hurricanes and Monsoons

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  • 1 University of Mainz, Mainz, Germany
  • | 2 Northwest Research Associates, Bellevue, Washington
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Abstract

This paper provides a unified perspective on the dynamics of hurricane- and monsoonlike vortices by identifying them as specific limiting cases of a more general flow system. This more general system is defined as stationary axisymmetric balanced flow of a stably stratified non-Boussinesq atmosphere on the f plane. The model is based on the primitive equations assuming gradient wind balance in the radial momentum equation. The flow is forced by heating in the vortex center, which is implemented as relaxation toward a specified equilibrium temperature Te. The flow is dissipated through surface friction, and it is assumed to be almost inviscid in the interior. The heating is assumed supercritical, which means that Te does not allow a regular thermal equilibrium solution with zero surface wind, and which gives rise to a cross-vortex secondary circulation. Numerical solutions are obtained using time stepping to a steady state, where at each step the Eliassen secondary circulation is diagnosed as part of the solution strategy.

Reality and regularity of the solution is discussed, putting this work in relation to previous work. Scaling analysis suggests that for a given geometry, essential vortex properties are controlled by the ratio F = αT/cD, where αT is the rate of thermal relaxation and cD quantifies the strength of surface friction for a given surface wind. For large F, the temperature is close to Te and the vortex shows properties that can be associated with a hurricane including strong cyclonic surface winds. On the other hand, for small F, the vortex shows properties that can be associated with a monsoon; that is, the surface winds are small and the secondary circulation keeps the temperature significantly away from Te. The scaling analysis is verified by numerical solutions spanning a wide range of the parameter space. It is shown how the two limiting cases correspond with the respective approximate semianalytical theories presented previously. The results imply an important role of αT for hurricane formation.

Corresponding author address: Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz, Becherweg 21, 55099 Mainz, Germany. Email: vwirth@uni-mainz.de

Abstract

This paper provides a unified perspective on the dynamics of hurricane- and monsoonlike vortices by identifying them as specific limiting cases of a more general flow system. This more general system is defined as stationary axisymmetric balanced flow of a stably stratified non-Boussinesq atmosphere on the f plane. The model is based on the primitive equations assuming gradient wind balance in the radial momentum equation. The flow is forced by heating in the vortex center, which is implemented as relaxation toward a specified equilibrium temperature Te. The flow is dissipated through surface friction, and it is assumed to be almost inviscid in the interior. The heating is assumed supercritical, which means that Te does not allow a regular thermal equilibrium solution with zero surface wind, and which gives rise to a cross-vortex secondary circulation. Numerical solutions are obtained using time stepping to a steady state, where at each step the Eliassen secondary circulation is diagnosed as part of the solution strategy.

Reality and regularity of the solution is discussed, putting this work in relation to previous work. Scaling analysis suggests that for a given geometry, essential vortex properties are controlled by the ratio F = αT/cD, where αT is the rate of thermal relaxation and cD quantifies the strength of surface friction for a given surface wind. For large F, the temperature is close to Te and the vortex shows properties that can be associated with a hurricane including strong cyclonic surface winds. On the other hand, for small F, the vortex shows properties that can be associated with a monsoon; that is, the surface winds are small and the secondary circulation keeps the temperature significantly away from Te. The scaling analysis is verified by numerical solutions spanning a wide range of the parameter space. It is shown how the two limiting cases correspond with the respective approximate semianalytical theories presented previously. The results imply an important role of αT for hurricane formation.

Corresponding author address: Dr. Volkmar Wirth, Institute for Atmospheric Physics, University of Mainz, Becherweg 21, 55099 Mainz, Germany. Email: vwirth@uni-mainz.de

1. Introduction

Atmospheric flow is able to organize itself into distinct long-lived vortices with near circular symmetry about the vertical axis. Examples include tornados, lee vortices, hurricanes, cutoff cyclones, and monsoons of limited spatial extent, such as the North American or the Australian monsoons. These vortices have very different characteristics, owing to the different types of forcing and dissipation and to the different spatial scales involved.

In the present paper we focus on the dynamics of stationary balanced vortices on the f plane with circular symmetry (also referred to as axisymmetry). Forcing is provided by heating in the vortex center. Dissipation occurs at the bottom boundary through surface drag, while the interior is assumed to be almost inviscid. These constraints still allow flows of rather different nature and include monsoon- and hurricane-like vortices. Both monsoons and hurricanes are characterized by a warm core and can be viewed as thermally forced. Although the combination of axisymmetry and vanishing interior friction can only be considered as a rough approximation for monsoons and hurricanes, these assumptions have proven fruitful as a first step toward a complete understanding of the dynamics of such systems (Emanuel 1986; Wirth 1998; Hsu and Plumb 2000). The same approach is followed here. Distinct differences between monsoons and hurricanes are their spatial scales and the location of the maximum tangential wind; the latter is close to the surface in the case of a hurricane, but close to the tropopause in the case of a monsoon.

An important issue is whether or not there exists a secondary cross-vortex circulation being directed inward–upward–outward. In case of a nonzero secondary circulation the actual temperature anomaly in the vortex interior is smaller than the assumed thermal equilibrium temperature anomaly. This is associated with diabatic heating, which can be related to the release of latent heat and concomitant precipitation. To the degree that the vortex is inviscid in the interior, parcels conserve angular momentum on the upward–outward branch of the secondary circulation. This property, called angular momentum conserving (AMC; Plumb and Hou 1992), strongly constrains the vortex; it gives rise to anticyclonic flow at the tropopause level and renders the equations nonlinear.

In our analysis we shall focus mostly on the overall features of the flow taking into consideration both the balanced vortical flow and the secondary circulation. Monsoon- and hurricane-like vortices will be identified as two specific limiting cases of the more general flow system outlined above. The controlling parameters will be identified by analyzing the time scales of the different processes involved, and the results of this analysis shall be verified through numerical solutions.

Table 1 aims to clarify the position of this work in relation to previous work. Emanuel (1986) developed a theory for the maintenance of a mature steady-state symmetric hurricane in a conditionally neutral environment. In our notation this corresponds to the actual temperature T being very close to the assumed equilibrium temperature Te. Emanuel’s theory calls for strong surface winds u0 as essential ingredient for the mechanism maintaining a hurricane against dissipation. In a separate line of development, Wirth (1998) adapted the ideas of Held and Hou (1980) and Plumb and Hou (1992) from spherical geometry to f-plane cylindrical geometry in order to arrive at a semianalytical theory for axisymmetric monsoonal systems. Surface winds are assumed to be small, which appears to be an appropriate approximation for large-scale circulations like the Hadley cell or a monsoon. In these models diabatic processes are represented by Newtonian cooling and the secondary circulation keeps the actual temperature T distinctly away from Te. The present paper presents a generalization in the sense that we allow both nonzero surface winds and a significant deviation of T from Te. This opens a new degree of freedom for the full spectrum of solutions and sheds new light on hurricane- and monsoonlike vortices as particular limits. Such a generalization was mentioned by Emanuel (1995), but that note focused on spherical geometry with zonal symmetry and was primarily concerned with the realizability of solutions expressed in terms of the horizontal gradient of boundary layer moist entropy. A somewhat different approach was taken by Fang and Tung (1996) who distinguished between dry and moist convecting regions of the atmosphere and found analytic solutions for the nonlinear Hadley circulation with an ITCZ in the limit of very fast cumulus convective adjustment.

The plan of the paper is as follows: after presenting the model equations in section 2, we discuss the numerical solution and illustrate a few general properties in section 3. Section 4 discusses threshold behavior and identifies hurricanes and monsoons as specific limits. Section 5 analyses the time scales associated with the relevant processes. This reveals the key parameters that determine whether the system approaches one of the limiting regimes or whether it is in a more general state. The scaling theory is verified by numerical solutions. Section 6 presents a summary, discussion, and our conclusions. Reality and regularity of the solution are addressed in the appendix.

2. Model equations

We consider the primitive equations on a Northern Hemisphere f plane. The flow is assumed to be rotationally symmetric about the vertical axis and balanced in the sense that the equation for radial momentum can be replaced by the gradient wind equation [see (2) below]. As coordinates we use radius r and log-p altitude z = −H ln(p/p0), where p is pressure, H = 7000 m is a constant scale height, and p0 = 1000 hPa is a constant reference pressure. The domain extends from z = 0 to z = D (with D = 16 km) in altitude and from r = 0 to r = R in radius. Diabatic heating is modeled as Newtonian relaxation toward a specified equilibrium temperature Te(r, z) with a constant thermal relaxation coefficient αT. With these assumptions the equations for steady flow read
i1520-0469-63-10-2529-e1
i1520-0469-63-10-2529-e2
i1520-0469-63-10-2529-e3
i1520-0469-63-10-2529-e4
i1520-0469-63-10-2529-e5
Here, u, υ, and w denote the tangential, radial, and vertical wind, respectively, Φ is the geopotential, T is the temperature, ρ0(z) = p(z)/(gH), f is the constant Coriolis parameter, g is the acceleration due to gravity, Ts = gH/Rd, Rd is the gas constant for dry air, κ = Rd/cp = 2/7, cp is the specific heat at constant pressure, and X represents material nonconservation of angular momentum associated with subscale motions. The term X is assumed to be small in the interior. Combining (2) and (3) yields the thermal wind equation
i1520-0469-63-10-2529-e6
We decompose the equilibrium temperature as
i1520-0469-63-10-2529-e7
with some reference temperature TR(z) depending on altitude only. It is assumed that ∂Te/∂r ≤ 0 everywhere and that T ′ is small beyond a certain radius r0. In other words, the forcing is localized and maximizes on the axis of rotation. This is in distinct contrast to related studies with spherical geometry and zonal symmetry (Held and Hou 1980; Plumb and Hou 1992; Dunkerton 1989), where the thermal forcing maximizes off the axis of rotation and T ′e = 0 right on the axis of rotation. Note also that because of strong nonlinearities our results cannot be extended to negative forcing (i.e., cooling in the vortex center) by simply changing a sign.
Surface stress Ts is modeled through a drag law
i1520-0469-63-10-2529-e8
where u0 is the tangential wind at z = 0. As explained below, z = 0 should be interpreted as the top of the boundary layer. Still, we refer to u0 as surface wind throughout this paper. The drag coefficient cd is assumed to depend on u0 according to1
i1520-0469-63-10-2529-e9
with us = 2.5 m s−1. In our numerical solutions we shall keep us fixed and vary the strength of surface drag by varying the dimensionless parameter cD.
It is helpful to view the above as a forced-dissipative system with relaxation of T toward Te playing the role of forcing and surface drag playing the role of dissipation. To be sure, the character of both the interior heating and the surface drag is relaxational. However, relaxation toward Te with ∂Te/∂r ≠ 0 is equivalent to forcing ∂u/∂z ≠ 0 and, hence, a nontrivial wind field u owing to (6). The diabatic heating Q(r, z) is part of the solution and not given a priori. In our whole study we restrict attention to a forcing that ensures that the atmosphere remains stably stratified regarding dry dynamics; that is,
i1520-0469-63-10-2529-e10
throughout the domain.
For later reference we note that potential temperature is
i1520-0469-63-10-2529-e11
and the stationary heat equation in terms of θ reads
i1520-0469-63-10-2529-e12
with θe = Te exp(κz/H). Absolute angular momentum about the axis of symmetry (henceforth referred to as angular momentum) is given by
i1520-0469-63-10-2529-e13
Assuming m ≥ 0 throughout the domain, one can define potential radius Rp through m = fR2p/2 (Schubert and Hack 1983); choosing the positive root (as is appropriate for a radius) yields
i1520-0469-63-10-2529-e14
Equation (1) can be rewritten as
i1520-0469-63-10-2529-e15
where for our axisymmetric stationary flow the material derivative reduces to D/Dt = υ∂/∂r + w∂/∂z. In the special case X = 0, this equation describes material conservation of angular momentum, Dm/Dt = 0 or, equivalently, of potential radius, DRp/Dt = 0. The radial and vertical components of absolute vorticity are
i1520-0469-63-10-2529-e16
Forming the square of (13) leads to
i1520-0469-63-10-2529-e17
which can be used to rewrite (6) as
i1520-0469-63-10-2529-e18
Potential vorticity (PV) is given by
i1520-0469-63-10-2529-e19
where J(a, b) = (∂a/∂r)(∂b/∂z) − (∂a/∂z)(∂b/∂r) for any a(r, z) and b(r, z).
The term X is given by
i1520-0469-63-10-2529-e20
where Xe represents the effect of eddy stresses and Xi is a response to inertial instability (if it occurs). The former is modeled as a combination of vertical and horizontal diffusion,
i1520-0469-63-10-2529-e21
This form of eddy viscosity can be derived from a symmetric stress tensor implying global conservation of angular momentum (cf. Becker 2001). With X = Xe, Eq. (15) can be reformulated as ∂(ρ0m)/∂t + divJ = 0, where the flux J contains both advective and diffusive contributions. The coefficients νυ and νh are constant and small, but large enough to ensure numerical stability.

The term Xi represents a so-called inertial relaxation in order to prevent significant inertial instability (∂m/∂r < 0) and, hence, the breakdown of our solution algorithm. This procedure simulates the effect of horizontal, angular momentum conserving rearrangement of parcels by subscale motions, gently relaxing the vortex toward inertially neutral. On the level of the resolved flow it appears locally as a nonconservative term. The underlying inertially neutral vortex is estimated by a minimal rearrangement of angular momentum, which renders ∂m/∂r ≥ 0 throughout the domain while conserving angular momentum globally. The resulting term Xi turns out to be small in a sense to be discussed below, and we keep viewing our solutions as almost inviscid.

We assume zero convergence of eddy stress at the lower and upper boundary of our domain; that is, X = 0 at z = 0 and z = D. This prevents the boundary layer character of the inflow and outflow, respectively. Regarding the lower boundary, it is consistent with our separate (although not explicit) treatment of the surface boundary layer. Regarding the upper boundary, our approach implies a diffusive flux of angular momentum across the domain boundary. However, this flux is small to the degree that the interior is almost inviscid, and the choice of this boundary condition does not have a significant impact on the overall character of our solutions (as was verified by explicit comparison). A more realistic treatment would have to include a stratosphere (see Dunkerton 1989, 1995; Wirth 1998).

The secondary circulation (υ and w) can be described in terms of a cross-vortex mass streamfunction ψ(r, z), defined through
i1520-0469-63-10-2529-e22
thus ensuring that (5) is automatically satisfied. For X = 0, Eq. (15) becomes J(ψ, m) = 0 or, equivalently, J(ψ, Rp) = 0, which means that isolines of the streamfunction ψ are parallel to those of m and Rp. If Q and X are known, one can, by forming gT−1s∂(4)/∂r − ∂[( f + 2u/r)(1)]/∂z, derive the following linear partial differential equation for ψ:
i1520-0469-63-10-2529-e23
with the coefficients
i1520-0469-63-10-2529-e24
i1520-0469-63-10-2529-e25
i1520-0469-63-10-2529-e26
and with the right-hand side
i1520-0469-63-10-2529-e27
The second equality in (25) is due to (6). An equation analogous to (23) was first derived by Eliassen (1952). Equation (23) is elliptic, and the vortex is symmetrically stable, as long as
i1520-0469-63-10-2529-e28
which allows one to diagnose the secondary circulation (ψ) from the knowledge of the primary vortex (u and T) and the nonconservative terms (Q and X) by inversion of an elliptic operator. At the inner (r = 0) and upper (z = D) boundary we specify ψ = 0. The outer boundary condition is less straightforward. We chose to extend the computational domain in the radial direction to r = 2R, where we require ∂ψ/∂r = 0. Our results do not sensitively depend on this boundary condition (as was verified by switching between different boundary conditions), because the extent of the computational domain is much larger than the radial scale of the forcing (see below, section 3).
The surface boundary layer cannot be treated explicitly in our model, because the assumption of gradient wind balance does not hold there. Instead, we implemented a simple parameterization of the boundary layer’s impact on the interior of the domain. Essentially, the drag law (8) is transformed into a condition for ψ at the bottom domain (i.e., at z = 0; cf. Schubert and Hack 1983; Wirth 1995). It is assumed that within the boundary layer (of thickness Δzb) the momentum balance (1) can be approximated by ζaυ = X. The stress T is equal to Ts from (8) at the bottom and zero at the top of the boundary layer. The latter is consistent with the assumption of almost inviscid flow in the interior. One obtains a mean specific force
i1520-0469-63-10-2529-e29
within the boundary layer; expressing υ in terms of ψ, the factor Δzb drops out leading to
i1520-0469-63-10-2529-e30
The vertical wind w associated with (30) represents the same effect as Ekman pumping.

The reference temperature TR(z) is specified in terms of potential temperature θR(z), which is set to θR = 300 K at z = 0, increasing linearly with altitude with a constant lapse rate R/dz = 4 K km−1. The remaining parameters are chosen as follows: g = 9.81 m s−2, R = 287 J kg−1 K−1 (implying Ts = 239 K), and f = 2Ω sinϕ0 with Ω = 7.292 · 10−5 s−1 and ϕ0 = 20° (i.e., f ≈ 0.5 × 10−4 s−1).

3. Numerical solution

The set of Eqs. (1)(5) is nonlinear and generally can be solved only by numerical means. The requirements on the numerics are substantial, because a unique implementation must be able to deal with a wide range of parameters covering distinctly different regimes (see section 4b). The method used for this investigation is the same as that described in Wirth (1998) and will only briefly be summarized here. Basically, we use the time-dependent version of our set of equations and perform a time integration until convergence is reached (cf. Dunkerton 1989). The initial state is a motionless atmosphere with T = TR(z). At each time step the knowledge of u and T allows us to diagnose the nonconservative terms X and Q, and solution of the Eliassen problem (23) yields the secondary circulation (υ, w) via (22). Although our inertial relaxation does not completely eliminate symmetric instability, it keeps its amplitude small enough such that the elliptic solver does not break down. For the actual time integration we use (1), while the associated balanced temperature is obtained from the thermal wind Eq. (6), which is integrated inward from the outer boundary. The integration is terminated when the rate of change of four different quantities (characterizing aspects of the primary vortex and the secondary circulation) are below a prespecified small fraction of their initial tendencies.

Since angular momentum conservation in the inviscid limit is an important aspect of our model, we explicitly analyzed our numerical solutions in this regard. It turned out that both explicit and implicit diffusion are small enough such that important global constraints from angular momentum conservation are observed to a satisfying degree.

The forcing used throughout this study is given by
i1520-0469-63-10-2529-e31
with r1 = 2R/15. For illustration we show this field in Fig. 1 for a particular set of parameters. It maximizes in the midtroposphere on the axis of symmetry and decays for larger radii like r−1. The radial and vertical scale of the forcing roughly correspond with the parameters r1 and D, respectively. The particular radial dependence in (31) was designed such that one obtains an inertially stable vortex in the hurricane limit.

For illustration we present in Fig. 2 a numerical solution for the forcing from Fig. 1. The grid spacing is Δz = 470 m in the vertical and Δr = 4.5 km in the horizontal. The strength of Newtonian relaxation and surface friction are given by αT = (100 min)−1 and cD = 2 × 10−3, respectively. Furthermore we specify νυ = 10 m2 s−1 and νh = 750 m2 s−1. The tangential wind (Fig. 2a, solid lines) maximizes in the lower troposphere reaching values around 14 m s−1. Overall the vortex has a warm core as can be inferred from the isentropes bulging downward in the vortex center (dashed lines in Fig. 2a). The streamfunction ψ of the secondary circulation (Fig. 2c) indicates overall upward–outward flow; the streamlines are approximately parallel to the lines of constant potential radius Rp (solid lines in Fig. 2b), which is consistent with nearly inviscid flow [see our remarks right below (22)]. It follows that the secondary circulation qualifies as AMC to a good approximation. At the tropopause (z = 16 km) the tangential wind satisfies u ≈ −fr/2 for r ≤ 150 km to a very good approximation (not shown), implying that the square brackets in (1) nearly vanish. This explains how the momentum balance is achieved in the inviscid limit (X → 0) through strong nonlinearity. The major part of the inward flow occurs in the boundary layer, which in our model is located below z = 0 (not shown in the plot) and where friction Xb [see (29)] allows the inward flow to cross isosurfaces of m and Rp. The vertical wind (Fig. 2d) is upward throughout the domain; it is nonzero at z = 0, which (as pointed out before) should be interpreted as Ekman pumping at the top of the surface boundary layer. The field of diabatic heating Q (not shown) is qualitatively similar to that of w, because the isolines of potential temperature θ are close to horizontal and crossing isolines of θ requires nonzero diabatic heating.

Owing to the linearity of (23), the Eliassen secondary circulation can be partitioned into several contributions, which add up to yield the total streamfunction ψ. Diabatic heating gives rise to ψd (Fig. 3a) obtained by solving (23) with X = 0 and assuming ψE = 0. Comparison with the total streamfunction ψ (Fig. 2c) indicates that throughout most of the domain, the secondary circulation is dominated by the diabatic contribution. Surface friction gives rise to ψe (Fig. 3b) obtained by solving (23) with F = 0 but using the actual lower boundary condition (30). Close to the lower boundary the magnitude of ψe is similar to that ψd, but otherwise ψe is much smaller than ψd. Interior friction gives rise to ψν (not shown) obtained by solving (23) with Q = 0 and assuming ψE = 0; ψν is more than one order of magnitude smaller than ψd, indicating that the flow is close to inviscid. Note that ψν also contains the impact of our inertial relaxation if it occurs. For all vortices computed in this study the maximum amplitude of ψν is well below that of ψd or ψe; in this sense the impact of inertial relaxation is considered small. The partitioning of ψ into different contributions is well-defined and will be used below in our scale analysis. On the other hand, we do not attach the notion of cause and effect to the different “forcings” and the corresponding parts of the streamfunction.

4. Threshold behavior and limiting cases for supercritical forcing

The thermal wind Eq. (6) is essential for our further discussion. It can be readily integrated in the vertical and solved for u. Obviously, the result depends on the direction of integration and the choice of the boundary condition. Specifying u = u0(r) at z = 0 and integrating upward yields
i1520-0469-63-10-2529-e32
Alternatively, specifying u = uD(r) at z = D and integrating downward yields
i1520-0469-63-10-2529-e33
In both cases the positive root has been chosen corresponding to that branch of the solution, which has positive absolute angular momentum.

a. Sub- and supercritical forcing

We now ask the question whether or not the system (1)(5) allows a so-called thermal equilibrium (TE) solution, defined through T = Te, which is regular and, at the same time, has zero surface winds (see appendix). With our treatment of the lower boundary and the further assumptions made, nonzero surface winds imply nonzero vertical wind, and this results in TTe. Hence, u0 = 0 is a precondition for a true TE solution. Owing to (4) with (5) and (10), T = Te implies (υ, w) = (0, 0) and provides a simple solution to the full nonlinear set of equations in the inviscid limit X → 0.

In the appendix we derive a necessary condition for regularity of a solution with zero surface wind [see (A8)]. This turns into a necessary condition for the true TE solution to be regular:
i1520-0469-63-10-2529-e34
where the hat denotes the vertical average
i1520-0469-63-10-2529-e35
As shown in the appendix, this can essentially be replaced in the current context by requiring reality, which is guaranteed as long as
i1520-0469-63-10-2529-e36
Here, ∂c/∂r denotes a critical radial temperature gradient and LR = gD/f is the external Rossby radius.

Similar as in Plumb and Hou (1992), any Te satisfying the above constraint is called subcritical, otherwise it is called supercritical. Supercritical forcing essentially means that the radial gradient of Te is so strong that—given the zero surface wind condition—at some altitude the hydrostatic pressure gradient exceeds the threshold beyond which the anticyclonic gradient wind solution is impossible. Both (34) and (36) are inhomogeneous in Te, which means that either condition can be satisfied for any given shape of Te with finite radial derivative by choosing the amplitude small enough.

For subcritical forcing, the TE solution (32) with u0 = 0 is a universal solution valid for any value of αT and cD. Unfortunately, this solution is likely to be irrelevant, since the f-plane approximation requires synoptic or smaller scales, and on these scales any relevant thermal forcing appears to be supercritical. With our choice of parameters, the critical amplitude for the thermal forcing is Tc0 ≈ 0.5 K for R = 1000 km and Tc0 ≈ 0.005 K for R = 100 km [see (A5)]. It follows that monsoonal systems of limited spatial extent are likely to be, and hurricanes almost certainly to be, strongly supercritical (cf. Plumb and Hou 1992; Emanuel 1995; Wirth 1998). We therefore restrict our attention to strongly supercritical forcing in the remainder of this paper. In particular, the forcing for the earlier example in Fig. 2 is strongly supercritical.

b. Identification of regimes as specific limiting cases

In the case of supercritical forcing, there are basically two ways for the solution to remain real: either there must be a nonzero secondary circulation, or one has to allow nonzero surface winds. Generally, a combination of both will be realized by the actual solution.

If one insists on the zero surface wind boundary condition (first option), there must be a secondary circulation (υ, w), which keeps T away from Te and, hence, modifies the radial slope ∂/∂r. The theory for this special case was worked out by Wirth (1998). We associate this type of flow with the monsoon regime, or monsoon limit. Numerical solutions in this regime are shown in the right column of Figs. 4 and 5, respectively. Basically, the secondary circulation must be strong enough such that the square root in (32) becomes zero at z = D out to some radius r* corresponding to zero absolute vorticity at the tropopause level. The latter, in turn, is consistent with the assumption of nearly inviscid flow in the interior, in which case the parcels conserve angular momentum while rising in the vortex center and radially spreading outward at the tropopause level (i.e., an AMC secondary circulation). Evaluating (32) at z = D using u0 = 0 yields
i1520-0469-63-10-2529-e37
where (A3) was used to express a frequently occurring combination of parameters in terms of the critical slope ∂c/∂r. Since ζa must be zero and, hence, uD = −fr/2 out to radius r*, the above equation shows that the secondary circulation must reduce the radial slope of the vertically averaged temperature to the critical slope for rr*. Having thus obtained the vertical mean temperature, a quantitative measure for the strength of the secondary circulation is obtained by invoking the conservation of energy, which translates to the so-called equal-area argument for spherical geometry (Held and Hou 1980) or its appropriate generalization to circular geometry (Wirth 1998). It was shown by Wirth (1998) that the numerical solution indeed approaches the solution from this semianalytical theory provided the surface drag is strong enough. Since this first option implies u0 = 0, it can only be used as a model for flows with small surface winds and certainly excludes hurricane-like flows.
The second option takes nonzero surface winds into account. According to (32), the presence of cyclonic surface winds u0 > 0 relaxes the constraint on the radial gradient of T. Indeed, for any supercritical forcing one could determine a surface wind u0 > 0 allowing T = Te. A straightforward way to obtain u0 in case of an AMC secondary circulation is to use (33) instead of (32). Assuming that parcels at the tropopause have risen at the origin r = 0, the AMC property renders ζa = 0 the natural boundary condition at z = D at least out to some finite radius r*. Translating this to uD = −fr/2 in (33) we obtain
i1520-0469-63-10-2529-e38
where for the second equal sign we used, again, the definition (A3) of the critical slope. For a warm core vortex (i.e., ∂/∂r ≤ 0 throughout the domain) the square root in the above expression is always real, and supercritical forcing implies cyclonic surface winds.

Hurricanes can be viewed as a limit of our system in the following sense. Thermal relaxation is very strong such that the vortex is very close to thermal equilibrium. Correspondingly, u0 can be obtained from (38) with T replaced by Te. However, the strict TE solution cannot be realized, as it would result in an inconsistency with the nonvanishing surface drag. Ekman pumping, owing to the strong surface winds from (38), implies a nonzero vertical wind in the lower troposphere which, in turn, keeps T away from Te. So there must be a nonzero AMC secondary circulation. This is consistent with the use of (38) to compute the surface wind as well as the other assumptions. Although TTe, the heating Q = αT(TeT) remains finite (thus allowing a finite w), because the coefficient αT is very large.

In the hurricane limit, r1 in (31) is equal to the radius of maximum surface wind. The secondary circulation can be obtained by computing u from (33) with uD = −fr/2 and substituting Te for T; then ψE(r) is obtained from (30) by exploiting the fact that in the interior the isolines of ψ are parallel to isolines of m or Rp, because the flow is assumed to be inviscid. This construction essentially reflects the dynamical part of the hurricane theory by Emanuel (1986) in the framework of the present model. It is illustrated in Fig. 6 in the same format as in Fig. 2. Key differences between this hurricane limit and the example from Fig. 2 are the significantly stronger surface winds and the fact that the temperature is much closer to thermal equilibrium.

Numerical solutions in the hurricane regime (large αT) are shown in the left column of Figs. 4 and 5 for relatively small and large radial scale R, respectively. These solutions are close to the corresponding semianalytical hurricane solution. To illustrate this point for R = 150 km, we compare the left column of Fig. 4 with Fig. 6. The variables quantifying the primary vortex (u and Rp) show, indeed, a close resemblance, and the same is true for the secondary circulation (streamfunction ψ), although the vertical wind w suggests that the numerical solution has not yet reached the asymptotic limit of the hurricane regime.

The transition between the hurricane and the monsoon regime is illustrated in Fig. 7, where five numerical solutions (dashed lines) are compared with the hurricane equilibrium solution (solid line). Figure 7a demonstrates that, depending on the strength of the thermal forcing, the surface wind can vary between close to zero and close to hurricane strength. Figure 7b indicates that all vortices have a warm core, but that in most cases the temperature is far from thermal equilibrium. Note also that the radial decay of the surface wind beyond the radius of maximum wind in Fig. 7a is rather modest. Evaluating (38) for T = Te with Te from (31) and R = 150 km yields u0(r) ≈ const × r−1/2 in the region r1rR. This relatively slow radial decay is consistent with observations (Mallen et al. 2005).

Figure 8 summarizes some of these results schematically. There is a whole range (gray shading) of possible solutions with the size of this range related to the strength of supercriticality. The two limits (monsoon and hurricane limit) correspond to the actual temperature T approaching either of the two boundaries of this range. In addition (not shown in the schematic), supercritical forcing implies a nonzero secondary circulation (υ, w) ≠ (0, 0). For subcritical forcing (T0Tc0, not shown), on the other hand, the size of the gray area shrinks to zero and there is a unique solution given by T = Te, u0 = 0, and (υ, w) = (0, 0).

5. Discussion of scales and characterization of regimes

What determines whether the flow is close to the monsoon limit, close to the hurricane limit, or in a more general state somewhere in between? Phrased in terms of Fig. 8, which part of the available range (gray shading) is the actual temperature (dashed line) located in? We shall address this question by a systematic analysis of the time scales of the different processes (section 5a), leading to a characterization of different regimes (section 5b). Note that in the present section we are concerned about orders of magnitude only.

a. Discussion of time scales

Instead of time scales τ we shall discuss inverse time scales or rates2 α = τ−1.

  • By design, temperature T is relaxed toward the equilibrium temperature Te with rate αT, which will be referred to as the rate of thermal forcing in the following.
  • Interior friction Xe is represented through a combination of vertical and horizontal diffusion according to (21). The relevant rate is
    i1520-0469-63-10-2529-e39
    where R and D represent the characteristic horizontal and vertical scale, respectively, of the vortex.
  • For all solutions considered in this study the diabatic part ψd of the secondary circulation extends throughout the whole troposphere (cf. Fig. 3a). The related rate of the diabatic circulation is
    i1520-0469-63-10-2529-e40
    Here, wd is the scale of the vertical wind in the upward branch3 of ψd. The second equality is obtained from the continuity Eq. (5), and υd denotes the scale of the radial in- or outflow. Approximating the heat Eq. (12) by wθ/∂z = αT(θeθ), and given that ∂θ/∂z differs only weakly from its reference value, we obtain
    i1520-0469-63-10-2529-e41
    where Δθ = Dθ/∂zθR(D) − θR(0) is approximately equal to the potential temperature difference between the top and the bottom of the domain.
  • The part ψe of the secondary circulation associated with Ekman pumping from our boundary layer parameterization modifies the vortex in the lowest part of the troposphere with a rate
    i1520-0469-63-10-2529-e42
    where we is the scale of the vertical wind due to ψe and δz ≈ 1 km is the vertical penetration of this part of the streamfunction (see Fig. 3b). The second equality is derived from the continuity equation applied to the bottom layer of depth δz, and υe denotes the scale of the radial outflow owing to ψe in that layer. Equation (30) implies wecdu0(a)−1|(z=0), and we obtain
    i1520-0469-63-10-2529-e43

We shall now work out how the model assumptions can be formulated in terms of these inverse time scales. First, it was assumed that the static stability does not differ significantly from its reference value, which means that surfaces of potential temperature must not be significantly distorted by the action of the secondary circulation. The latter is guaranteed as long as
i1520-0469-63-10-2529-e44
Using (41), one obtains
i1520-0469-63-10-2529-e45
It follows that the condition αdαT is satisfied as long as the amplitude T0 of the forcing is much smaller than Δθ. This is, in fact, guaranteed by our choice of T0 not exceeding 10 K.
Second, our model assumes that the interior of the domain is almost inviscid such that isolines of constant m or Rp are almost parallel to isolines of the streamfunction ψ. This corresponds to the requirement that
i1520-0469-63-10-2529-e46
The smallness of interior friction has one further consequence. In the quasi-horizontal branches of the secondary circulation the rate of change of the tangential wind during spinup becomes
i1520-0469-63-10-2529-e47
Stationarity requires the right-hand side to become zero. In the outward branch at the tropopause this is satisfied because ζa ≈ 0. On the other hand, at the top of the surface boundary layer we have ζa ≠ 0. It follows that υ = υe + υd must be close to zero there; that is, the magnitude of the inflow owing to the diabatic circulation must approximately equal the magnitude of the outflow owing to the Ekman pumping. In other words, most of the inflow occurs within the surface boundary layer, which is below z = 0 in our model. In terms the scales defined above this translates to υdυe or
i1520-0469-63-10-2529-e48
Indeed, comparison of ψd and ψe from Fig. 3 with ψ from Fig. 2c indicates that in this particular example the two contributions to the radial wind, υd and υe, are similar in magnitude and opposite to each other in the bottom part of the domain.
In summary, all our steady-state solutions satisfy
i1520-0469-63-10-2529-e49

b. Characterization of regimes

What process, then, distinguishes between a monsoon- and a hurricane-like vortex? More specifically: which scales characterize the two limits (hurricane and monsoon) and how?

We are going to argue that for a given geometry the overall character of the primary vortex only depends on
i1520-0469-63-10-2529-e50
that is, on the ratio between the strength of thermal forcing and the strength of surface friction. Heuristically this can be seen as follows. With the radial scale fixed, the three parameters R, δz, and Δθ can be considered as constants. The remaining free parameters are αT and cD, and for each couple (αT, cD) one obtains a steady-state vortex. At first sight it seems that all these solutions may be different. However, this turns out not to be the case. Condition (48) can be rewritten as
i1520-0469-63-10-2529-e51
where γ = Δθ/(Rδz) and ζa0 denotes absolute vorticity at z = 0. In addition, balance in combination with the AMC condition in the free atmosphere implies a diagnostic relation between the surface wind, u0, and the deviation from thermal equilibrium, (θeθ). Thus, the numerator and the denominator on the right-hand side of (51) are not independent of each other. Rather, the numerator increases as θeθ decreases [see (38)] such that for any given F there should only be one solution for u0 and θeθ.

To test the prediction from this heuristic argument, we considered 100 different couples (αT, cD) by varying both parameters by a wide margin, thus covering over three orders of magnitude for F. More specifically, we used 10 different values of αT in the range 1.67 × 10−5 s−1αT ≤ 1.67 × 10−3 s−1 combined with ten different values of cD in the range 0.5 × 10−3cD ≤ 8.0 × 10−3, with the intervals subdivided logarithmically. The radial scale was fixed with R = 150 km. All except one set of parameters resulted in numerically stable stationary solutions.

We choose to characterize each solution by the maximum surface wind u0 over all radii rR and by the corresponding maximum vertically integrated temperature difference e. These two quantities are plotted as a function of F in Fig. 9. In both panels the different points (each characterizing a different solution) nearly collapse onto a single curve. This indicates that, indeed, these two vortex characteristics essentially depend on the ratio αT/cD. We did the same computations but with a larger value for νh (not shown). This led to an increased spread of points perpendicular to the hypothetical curve, suggesting that the remaining spread visible in Fig. 9 is associated the finite amount of viscosity that was necessary to keep the numerics stable.

In Fig. 9a, one observes small surface wind for small values of F corresponding to the monsoon regime, while the surface wind is close to the hurricane equilibrium wind for large values of F corresponding to the hurricane regime. Both are in good agreement with the prediction from the respective approximate analytical theory outlined in section 4b. For intermediate values of F there is a smooth transition between the two extremes. This intermediate regime spans somewhat more than one order of magnitude in F. The temperature difference e in Fig. 9b approaches a constant value in the monsoon limit, and this value agrees well with the prediction from the approximate semianalytical theory of Wirth (1998). On the other hand, in the hurricane limit it approaches zero, which is again in agreement with theory.

Figure 9b suggests that the approach of e toward zero in the hurricane limit is like 1/F. This, too, is consistent with theory. Consider the stationary heat equation, which can be approximated as
i1520-0469-63-10-2529-e52
For a given value of cD, the vertical wind is determined by the surface wind according to (30). The surface wind, in turn, is given by Te in the hurricane limit and, thus, independent of αT. Since ∂θ/∂z is approximately constant by assumption, the whole left-hand side of (52) is independent of αT, so θeθ must scale like α−1T ∝ 1/F. On the other hand, if αT is kept fixed and cD is varied, the vertical wind scales like cD in the hurricane limit. Thus, (52) indicates that θeθ must scale like cD, which is again proportional to 1/F.
How about the secondary circulation? We quantify its strength by the maximum (upward) vertical wind. Repeating the same procedure as before, we do not find a unique dependence on F only. This is evident in Fig. 10, where the individual points do not collapse onto one single curve. Indeed, one does not expect such simple scaling for the secondary circulation. Equation (41) yields
i1520-0469-63-10-2529-e53
and Fig. 9b indicates that θeθ is a nonlinear function of F, say G(F), such that
i1520-0469-63-10-2529-e54
This is not a function of F only, since it involves αT in addition to F.

It is interesting to further analyze the pattern of points in Fig. 10. For instance, in the monsoon limit the approximate analytical theory predicts that the maximum w is independent of the value of cD and approximately4 proportional to αT (Wirth 1998). This is reflected in the pattern of points in the lower left corner: points associated with different values of cD but the same value of αT are aligned along a horizontal line, while points associated with different values of αT but the same value of cD are along a straight line with slope +1. On the other hand, the approximate analytical theory for the hurricane limit predicts that the maximum w should be independent of αT and proportional to cD. Again, this is reflected in the pattern of points in the right half of the plot where the imagined lines corresponding to different values of αT but the same value of cD turn approximately horizontal, while the points corresponding to different values of cD but the same value of αT are approximately aligned along slope −1.

Scatterplots like those in Figs. 9 and 10, but for R = 1500 km, exhibit basic features similar to those described above (not shown). In particular, the primary vortex scales like αT/cD, but the secondary circulation does not. Overall, the transition from the monsoon to the hurricane limit appears to be more gradual and there is a slight deviation from the expected behavior in the hurricane limit.

6. Summary, discussion, and conclusions

The current paper investigates stationary axisymmetric balanced vortices on the f plane. The model is based on the primitive equations assuming gradient wind balance in the radial momentum equation. The flow is forced by heating in the vortex center, which is implemented as relaxation toward a specified equilibrium temperature Te. The flow is dissipated through surface friction, and it is assumed to be almost inviscid in the interior. The general setup is similar as in Wirth (1998) except that we do not assume that surface friction reduces the surface wind to zero. This allows much richer behavior in comparison with Wirth (1998) and includes, in particular, both monsoon- and hurricane-like vortices.

Reality and regularity of the solution was discussed putting this work in relation to previous work. Criticality of the forcing Te was defined as in Wirth (1998): For subcritical forcing it is possible to have a regular thermal equilibrium (TE) solution with zero surface wind. For supercritical forcing, on the other hand, this is no longer the case and the flow generally develops a secondary circulation as well as nonzero surface winds. Since real systems of interest appear to be strongly supercritical at small radial scales O(≲1000 km), attention was restricted to strongly supercritical conditions. Depending on the choice of parameters, the corresponding vortices may have surface winds close to zero or may have T close to Te, but not both at the same time.

Numerical solutions were obtained using time stepping to a steady state. The Eliassen secondary circulation was diagnosed explicitly as part of the solution strategy. Streamlines of the secondary circulation are approximately parallel to lines of constant angular momentum, indicating that the flow qualifies as angular momentum conserving (AMC). Both features are consistent with the assumption of negligible interior friction.

Axisymmetric hurricanes and monsoons can be identified as two regimes corresponding to two specific limits of our model. This provides a new, unified perspective on both systems. In the monsoon limit, surface friction succeeds in reducing the surface wind to zero. The actual temperature T significantly deviates from the equilibrium temperature Te owing to the secondary circulation, which is just strong enough to reduce the temperature to critical conditions. This limit was investigated in detail by Wirth (1998). In the hurricane limit, on the other hand, the relaxation toward thermal equilibrium is so strong that TTe. As a consequence, there must be strong surface winds. This limit was studied before by Emanuel (1986, 1995). In both limits there is a semianalytic theory that allows one to compute the strength of the surface wind and the secondary circulation (Wirth 1998; Emanuel 1986). Under more general conditions the system is characterized by both nonzero surface winds and a significant deviation of T from Te.

In a sense there is competition between surface friction to make the surface wind small and thermal relaxation to make T close to Te. This raises the following question: what determines whether the vortex is in the monsoon regime, in the hurricane regime, or somewhere in between? Scale analysis reveals that for a given geometry this selection is governed by the ratio F = αT/cD, where αT is the rate of thermal relaxation and cD quantifies the strength of surface friction for a given surface wind. The monsoon limit corresponds to small F, while the hurricane limit corresponds to large F. It follows that for fixed surface drag the thermal forcing time scale is the key parameter: it differs by at least one order of magnitude between the monsoon and the hurricane regime.

The scaling theory was verified through a large number of numerical solutions, for which both αT and cD were varied over a wide range. Indeed, to a good approximation both the strength of the surface wind and the deviation from the equilibrium temperature are a function of F alone. The strength of the secondary circulation, on the other hand, is not only a function of F alone, but it does show the expected behavior from the semianalytical theories in the respective limits.

One may speculate about the implications of this work for the formation of hurricanes. To be sure, the current model is too simple to describe hurricane genesis. As a possible extension one could associate the equilibrium temperature Te with the strength of the surface wind in the spirit of the Wind-Induced Surface Heat Exchange (WISHE; Emanuel et al. 1994), but this is beyond the scope of the present paper. Nevertheless, our results suggest that the establishment of a surface vortex during the transition of a tropical disturbance to a hurricane (Reasor et al. 2005) should be associated with a substantial strengthening of the thermal relaxation: given the equilibrium temperature Te and assuming fixed surface drag, the character of our steady-state vortices essentially depends on αT. Thus, our work emphasizes that, in addition to the surface wind and its coupling to the free tropospheric Te, the efficiency of convection letting T approach Te (represented by αT in our model) is an important ingredient. Its evolution must be accounted for when trying to understand the transformation of an initial perturbation into a mature hurricane. If we interpret αT as an effective relaxation rate that is roughly proportional to the fractional area of deep convection (cf. Salby and Callaghan 2004), our results are consistent with the observation that the fractional area is 100% in the eyewall of a hurricane, but much smaller in less organized convective systems. Indeed, Dunkerton (1997) found that an effective thermal relaxation rate of order 0.5 day−1 was sufficient to drive the zonally averaged Hadley circulation; this rate is consistent with a fractional coverage of 1%–2% and a convective time scale of 30 min.

Motivated by the key importance of thermal relaxation, an obvious extension of the present work would be a model that distinguishes between convecting and nonconvecting regions assigning different values of αT to either of them. For spherical geometry with zonal symmetry this allows analytic solutions in the limit of very fast convective adjustment (Fang and Tung 1996). More generally, we imagine αT to be part of the solution, in some yet-to-be-determined way (e.g., Dunkerton 1997). Other generalizations include the addition of an explicit boundary layer and a refined treatment of air–sea exchanges of momentum and moist entropy; in particular, their dependence on surface wind. Investigating such extensions is left for future research.

Overall we conclude that the dynamics of axisymmetric hurricanes and monsoons can be viewed as specific cases of a more general almost inviscid flow system with thermal forcing in the vortex center and dissipation at the bottom surface. In this framework monsoons correspond to the limit of weak thermal forcing and hurricanes to the limit of strong thermal forcing.

Acknowledgments

We sincerely thank I. M. Held and two anonymous reviewers for their constructive comments on an earlier version of this paper, which helped to improve the presentation of the material and sharpen the focus of this paper. This research was supported, in part, by the National Science Foundation and the National Oceanic and Atmospheric Administration.

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APPENDIX

Real and Regular Solutions with Zero Surface Wind

Equations (32) and (33) impose constraints on the temperature field T(r, z), because the term under the square root must not be negative for the solution to be physically viable. When concerned with a large-scale flow, it is reasonable to assume that surface drag is strong enough such as to keep the surface wind small (cf. Held and Hou 1980; Plumb and Hou 1992; Wirth 1998). This appendix explores the conditions for reality and regularity for our f-plane solutions with zero surface wind, and it puts the results into the perspective of earlier results with spherical geometry and zonal symmetry.

For zero surface wind it is convenient to use (32) with u0 = 0, yielding
i1520-0469-63-10-2529-ea1
Assuming that ∂T/∂r ≤ 0 throughout the domain, the condition for u to remain real everywhere becomes
i1520-0469-63-10-2529-ea2
where LR = gD/f (= 7943 km in our case) is the external Rossby radius based on the depth D of the domain, and where the hat denotes the vertical average [see (35)]. In other words, for the solution to be real the radial derivative of the temperature must locally be limited by the critical lower bound
i1520-0469-63-10-2529-ea3
It is instructive to consider the marginal case in which the radial temperature gradient is critical throughout a finite region (0 ≤ rr0) and zero beyond. Assuming continuity at r = r0 one obtains
i1520-0469-63-10-2529-ea4
where C = const. The corresponding balanced wind at the top of the domain (z = D) is characterized by ζa = m = 0 for 0 ≤ rr0, while at r = r0 the function ζa(r, D) is discontinuous jumping from 0 to r20f /2. The maximum amplitude of this vertically averaged temperature anomaly is given by the factor in front of the square brackets, called critical amplitude and denoted by Tc0; that is,
i1520-0469-63-10-2529-ea5

For spherical geometry with zonal symmetry there is a similar possibility for the zonal wind to become imaginary. Assuming zero surface wind and integrating the corresponding thermal wind equation [Eq. (5) in Plumb and Hou 1992] in the vertical yields a quadratic equation for the zonal wind. Requiring its solution to remain real imposes a constraint on the latitudinal temperature gradient, which is analogous to our Eq. (A2). According to this constraint, the poleward increase of the vertically averaged temperature must not exceed a certain threshold. Regarding the tropospheric general circulation, this constraint does not appear relevant, as the temperature generally decreases poleward, and this is why reality of the solution has not been an issue in related studies.A1 A physical realization of the marginal solution evokes air parcels rising on the axis of symmetry (i.e., at the pole) and traveling away from the axis of symmetry (i.e., equatorward) in the upper troposphere. The result is a planetary-scale easterly jet with zero angular momentum and zero absolute vorticity at tropopause level. While this scenario is considered unrealistic in the case of the planetary-scale general circulation with spherical geometry, it corresponds to the standard setup for our thermally forced f-plane vortices with cylindrical geometry. Thus, the spatial relation between the maximum heating and the axis of symmetry is distinctly different in the two geometries distinguished in the two columns of Table 1. This explains why different aspects are relevant in the current work and in earlier work despite a close formal analogy of the equations otherwise.

Not just any solution that is real can be considered as physically viable. In the past, physical viability has been related to regularity in the sense that the inviscid solution must be the frictionless limit of a solution with small, but nonzero viscosity (Hide 1969; Schneider 1977; Held and Hou 1980; Plumb and Hou 1992). In terms of angular momentum m this means that there can be no local extrema of m except on the lower boundary. To ensure this kind of regularity, at no point except on the lower boundary may both ∂m/∂r and ∂m/∂z vanish simultaneously. In particular, at a stress-free upper boundary one requires ∂m/∂r ≥ 0 or, equivalently, ζa ≥ 0 (note that we include the marginal case in our definition of regularity).

Integrating (18) from 0 to D, forming r−1∂/∂r, and using (16) yields
i1520-0469-63-10-2529-ea6
where the subscript 0 and D denotes the lower and upper boundary, respectively. Regularity implies ζaD ≥ 0. This together with the assumption of nonnegative m shows that the right-hand side of the above equation must be nonnegative:
i1520-0469-63-10-2529-ea7
For zero surface winds we have m0ζa0 = f2r2/2, and one obtains
i1520-0469-63-10-2529-ea8
as a necessary condition for regularity. This relation [which is our analog to (8) in Plumb and Hou (1992)] constrains a measure of temperature radial curvature above some threshold in the sense that positive temperature anomalies must not to be too localized in r.
We restrict our discussion to the case where (r) maximizes at r = 0, since this is satisfied for all vortices considered in this study. Integrating (A8) outward from r = 0 yields
i1520-0469-63-10-2529-ea9
where the last identity is due to (A3). It follows that regularity of a solution implies its reality. The reverse is generally not true, and it is easy to design vortices that are real but irregular.A2
Again, it is instructive to consider a solution that is marginally regular in a finite neighborhood of the origin. A physical realization can be obtained when air parcels at the origin rise to the tropopause and then spread radially outward while conserving angular momentum. The foregoing shows that this implies
i1520-0469-63-10-2529-ea10
Comparison with (A3) shows that this solution is not only marginally regular but also marginally real in the neighborhood of the origin. In this sense, marginal reality and regularity are equivalent in our present setup and the critical gradient ∂c/∂r defined in (A3) applies equally to both.

Fig. 1.
Fig. 1.

Thermal forcing T ′e(r, z) = Te(r, z) − TR(z) (in K) as given in (31) for T0 = 10 K, R = 150 km, r1 = 20 km, and D = 16 km.

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 2.
Fig. 2.

Numerical solution for the forcing from Fig. 1 using αT = 10−2 min−1 and cD = 2 × 10−3. (a) Tangential wind u (solid, in m s−1) and potential temperature θ (dashed, in K, contours every 10 K), (b) potential radius Rp (solid, in km) and potential temperature θ (dashed, in K, contours every 10 K), (c) streamfunction ψ (in kg s−1), and (d) the vertical wind w (in cm s−1).

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 3.
Fig. 3.

Eliassen secondary circulation owing to (a) diabatic heating only and (b) surface drag only. In both panels the contours depict the streamfunction ψ in kg s−1.

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 4.
Fig. 4.

Two numerical solutions with radial scale R = 150 km, (left) one in the hurricane regime (αT = 10−1 min−1, cD = 2 × 10−3) and (right) one in the monsoon regime (αT = 10−3 min−1, cD = 2 × 10−3). (a),(d) Tangential wind (solid, in m s−1) and potential temperature (dashed, in K), (b),(e) potential radius (solid, in km) and streamfunction ψ [dashed contour interval is 108 kg s−1 in (b) and 2 × 106 kg s−1 in (e)], and (c),(f) the vertical wind (in cm s−1, negative values shaded).

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 5.
Fig. 5.

Same as Fig. 4, but with a much larger radial scale R = 1500 km. The contour interval for the dashed contours is 2 × 109 kg s−1 in (b) and 5 × 107 kg s−1 in (e).

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 6.
Fig. 6.

Semianalytical solution in the hurricane limit with the forcing from Fig. 1 and cD = 2 × 10−3. The plot conventions are like in Fig. 2.

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 7.
Fig. 7.

Radial dependence of two quantities characterizing the primary flow of the vortex: (a) the surface wind u0 and (b) the vertically averaged temperature anomaly R. In both panels the solid line represents the semianalytical solution in the hurricane limit, while the dashed lines represent five numerical solutions with different values of αT (bottom to top: αT = 1000−1, 316−1, 100−1, 32−1, 10−1 min−1).

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 8.
Fig. 8.

Schematic representation of the radial dependence of the equilibrium temperature Te (solid) and the actual temperature T (dashed) for supercritical forcing T0 > Tc0. The gray shading indicates the range available to T, and the open arrows point to the hurricane and monsoon limit, respectively. The temperature difference TeT changes sign at some outer radius ro; the radii displayed satisfy r < ro.

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 9.
Fig. 9.

(a) Surface wind and (b) departure from thermal equilibrium for 99 different vortices. The radial scale is R = 150 km for all vortices. In both panels each cross represents the numerical solution for a specific pair of values of αT and cD, and the abscissa is log10 F [with αT measured in (10 s)−1]. (a) The value plotted on the ordinate is the normalized maximum of the surface wind u0(r) over all radii rR, with the normalization factor being the corresponding maximum of the hurricane equilibrium solution. (b) The value plotted on the ordinate is log10 of the maximum value of [e(r) − (r)]/N, with the normalization factor N being the expected temperature difference for the monsoon regime according to the approximate semianalytical theory of Wirth (1998). The thin solid line in (b) depicts slope −1.

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Fig. 10.
Fig. 10.

Same as Fig. 9a, but here the value plotted on the ordinate is log10 of the maximum vertical wind w (in cm s−1). The thin solid lines depict slope +1 and −1, respectively.

Citation: Journal of the Atmospheric Sciences 63, 10; 10.1175/JAS3763.1

Table 1.

Current work in relation to earlier work: The two columns refer to the different geometries and symmetries. The three rows refer to different assumptions regarding the surface wind u0 and the relation between temperature T and equilibrium temperature Te

Table 1.

1

Coupled Boundary Layer Air–Sea Transfer [CBLAST (a program of the Office of Naval Research)] observations suggest a leveling-off and possible decrease of cd at large u0 presumably due to flow separation over oceanic surface waves; we do not attempt to model this effect here.

2

The use of inverse time scales is considered to be somewhat more intuitive, since the rate α is directly related to the strength of a process, in contrast to the time scale τ, which is inversely related.

3

The vertical wind is much stronger in the upward branch than in the downward branch owing to the cylindrical geometry.

4

The relation between the maximum w and αT is linear only for strongly supercritical Te (cf. Plumb and Hou 1992), to which we restrict our attention.

A1

In Earth’s summer stratosphere, the radiative equilibrium temperature increases poleward and the constraint is relevant (cf. Shine 1987). On other planets with large obliquity (e.g., Uranus) solar radiation favors a temperature maximum at one of the poles during times at which this pole points toward the sun.

A2

Appendix B of Wirth (1998) deals with the relation between reality and regularity of the TE solution. There, a specific radial dependence of is implicitly assumed rendering the proof less general than it may appear. In contrast, here we here allow more general radial profiles.

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