## 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 *T _{e}*. Emanuel’s theory calls for strong surface winds

*u*

_{0}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

*T*. The present paper presents a generalization in the sense that we allow both nonzero surface winds and a significant deviation of

_{e}*T*from

*T*. 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.

_{e}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

*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*/

*p*

_{0}), where

*p*is pressure,

*H*= 7000 m is a constant scale height, and

*p*

_{0}= 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

*T*(

_{e}*r*,

*z*) with a constant thermal relaxation coefficient

*α*. With these assumptions the equations for steady flow readHere,

_{T}*u*,

*υ*, and

*w*denote the tangential, radial, and vertical wind, respectively, Φ is the geopotential,

*T*is the temperature,

*ρ*

_{0}(

*z*) =

*p*(

*z*)/(

*gH*),

*f*is the constant Coriolis parameter,

*g*is the acceleration due to gravity,

*T*=

_{s}*gH*/

*R*,

_{d}*R*is the gas constant for dry air,

_{d}*κ*=

*R*/

_{d}*c*= 2/7,

_{p}*c*is the specific heat at constant pressure, and

_{p}*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

*T*(

_{R}*z*) depending on altitude only. It is assumed that ∂

*T*/∂

_{e}*r*≤ 0 everywhere and that

*T*′ is small beyond a certain radius

*r*

_{0}. 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.

*is modeled through a drag lawwhere*

_{s}*u*

_{0}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

*u*

_{0}as surface wind throughout this paper. The drag coefficient

*c*is assumed to depend on

_{d}*u*

_{0}according to

^{1}with

*u*= 2.5 m s

_{s}^{−1}. In our numerical solutions we shall keep

*u*fixed and vary the strength of surface drag by varying the dimensionless parameter

_{s}*c*.

_{D}*T*toward

*T*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

_{e}*T*with ∂

_{e}*T*/∂

_{e}*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,throughout the domain.

*θ*readswith

*θ*=

_{e}*T*exp(

_{e}*κz*/

*H*). Absolute angular momentum about the axis of symmetry (henceforth referred to as angular momentum) is given byAssuming m ≥ 0 throughout the domain, one can define potential radius

*R*through

_{p}*m*=

*fR*

^{2}

_{p}/2 (Schubert and Hack 1983); choosing the positive root (as is appropriate for a radius) yieldsEquation (1) can be rewritten aswhere 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,

*DR*/

_{p}*Dt*= 0. The radial and vertical components of absolute vorticity areForming the square of (13) leads towhich can be used to rewrite (6) asPotential vorticity (PV) is given bywhere

*J*(

*a*,

*b*) = (∂

*a*/∂

*r*)(∂

*b*/∂

*z*) − (∂

*a*/∂

*z*)(∂

*b*/∂

*r*) for any

*a*(

*r*,

*z*) and

*b*(

*r*,

*z*).

*X*is given bywhere

*X*represents the effect of eddy stresses and

_{e}*X*is a response to inertial instability (if it occurs). The former is modeled as a combination of vertical and horizontal diffusion,This form of eddy viscosity can be derived from a symmetric stress tensor implying global conservation of angular momentum (cf. Becker 2001). With

_{i}*X*=

*X*, Eq. (15) can be reformulated as ∂(

_{e}*ρ*

_{0}

*m*)/∂

*t*+ div

**J**= 0, where the flux

**J**contains both advective and diffusive contributions. The coefficients

*ν*and

_{υ}*ν*are constant and small, but large enough to ensure numerical stability.

_{h}The term *X _{i}* 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

*X*turns out to be small in a sense to be discussed below, and we keep viewing our solutions as almost inviscid.

_{i}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).

*υ*and

*w*) can be described in terms of a cross-vortex mass streamfunction

*ψ*(

*r*,

*z*), defined throughthus ensuring that (5) is automatically satisfied. For

*X*= 0, Eq. (15) becomes

*J*(

*ψ*,

*m*) = 0 or, equivalently,

*J*(

*ψ*,

*R*) = 0, which means that isolines of the streamfunction

_{p}*ψ*are parallel to those of

*m*and

*R*. If

_{p}*Q*and

*X*are known, one can, by forming

*gT*

^{−1}

_{s}∂(4)/∂

*r*− ∂[(

*f*+ 2

*u*/

*r*)(1)]/∂

*z*, derive the following linear partial differential equation for

*ψ*:with the coefficientsand with the right-hand sideThe 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 aswhich 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*= 2

*R*, 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).

*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 Δ

*z*) the momentum balance (1) can be approximated by

_{b}*ζ*=

_{a}υ*X*. The stress

*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 forcewithin the boundary layer; expressing*

_{s}*υ*in terms of

*ψ*, the factor Δ

*z*drops out leading toThe vertical wind

_{b}*w*associated with (30) represents the same effect as Ekman pumping.

The reference temperature *T _{R}*(

*z*) is specified in terms of potential temperature

*θ*(

_{R}*z*), which is set to

*θ*= 300 K at

_{R}*z*= 0, increasing linearly with altitude with a constant lapse rate

*dθ*/

_{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

*T*= 239 K), and

_{s}*f*= 2Ω sin

*ϕ*

_{0}with Ω = 7.292 · 10

^{−5}s

^{−1}and

*ϕ*

_{0}= 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* = *T _{R}*(

*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.

*r*

_{1}= 2

*R*/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

*r*

_{1}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 Δ*z* = 470 m in the vertical and Δ*r* = 4.5 km in the horizontal. The strength of Newtonian relaxation and surface friction are given by *α _{T}* = (100 min)

^{−1}and

*c*= 2 × 10

_{D}^{−3}, respectively. Furthermore we specify

*ν*= 10 m

_{υ}^{2}s

^{−1}and

*ν*= 750 m

_{h}^{2}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

*R*(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 (

_{p}*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

*X*[see (29)] allows the inward flow to cross isosurfaces of

_{b}*m*and

*R*. The vertical wind (Fig. 2d) is upward throughout the domain; it is nonzero at

_{p}*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 *ψ _{d}* (Fig. 3a) obtained by solving (23) with

*X*= 0 and assuming

*ψ*= 0. Comparison with the total streamfunction

_{E}*ψ*(Fig. 2c) indicates that throughout most of the domain, the secondary circulation is dominated by the diabatic contribution. Surface friction gives rise to

*ψ*(Fig. 3b) obtained by solving (23) with

_{e}*F*= 0 but using the actual lower boundary condition (30). Close to the lower boundary the magnitude of

*ψ*is similar to that

_{e}*ψ*, but otherwise

_{d}*ψ*is much smaller than

_{e}*ψ*. Interior friction gives rise to

_{d}*ψ*(not shown) obtained by solving (23) with

_{ν}*Q*= 0 and assuming

*ψ*= 0;

_{E}*ψ*is more than one order of magnitude smaller than

_{ν}*ψ*, indicating that the flow is close to inviscid. Note that

_{d}*ψ*also contains the impact of our inertial relaxation if it occurs. For all vortices computed in this study the maximum amplitude of

_{ν}*ψ*is well below that of

_{ν}*ψ*or

_{d}*ψ*; in this sense the impact of inertial relaxation is considered small. The partitioning of

_{e}*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

*u*. Obviously, the result depends on the direction of integration and the choice of the boundary condition. Specifying

*u*=

*u*

_{0}(

*r*) at

*z*= 0 and integrating upward yieldsAlternatively, specifying

*u*=

*u*(

_{D}*r*) at

*z*=

*D*and integrating downward yieldsIn 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* = *T _{e}*, 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

*T*≠

*T*. Hence,

_{e}*u*

_{0}= 0 is a precondition for a true TE solution. Owing to (4) with (5) and (10),

*T*=

*T*implies (

_{e}*υ*,

*w*) = (0, 0) and provides a simple solution to the full nonlinear set of equations in the inviscid limit

*X*→ 0.

*T̂*/∂

_{c}*r*denotes a critical radial temperature gradient and

*L*=

_{R}*gD*

*f*is the external Rossby radius.

Similar as in Plumb and Hou (1992), any *T _{e}* satisfying the above constraint is called subcritical, otherwise it is called supercritical. Supercritical forcing essentially means that the radial gradient of

*T*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

_{e}*T*, which means that either condition can be satisfied for any given shape of

_{e}*T*with finite radial derivative by choosing the amplitude small enough.

_{e}For subcritical forcing, the TE solution (32) with *u*_{0} = 0 is a universal solution valid for any value of *α _{T}* and

*c*. Unfortunately, this solution is likely to be irrelevant, since the

_{D}*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

*T*

_{c}_{0}≈ 0.5 K for

*R*= 1000 km and

*T*

_{c}_{0}≈ 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.

*υ*,

*w*), which keeps

*T*away from

*T*and, hence, modifies the radial slope ∂

_{e}*T̂*/∂

*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

*u*

_{0}= 0 yieldswhere (A3) was used to express a frequently occurring combination of parameters in terms of the critical slope ∂

*T̂*/∂

_{c}*r*. Since

*ζ*must be zero and, hence,

_{a}*u*= −

_{D}*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

*r*≤

*r**. 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

*u*

_{0}= 0, it can only be used as a model for flows with small surface winds and certainly excludes hurricane-like flows.

*u*

_{0}> 0 relaxes the constraint on the radial gradient of

*T*. Indeed, for any supercritical forcing one could determine a surface wind

*u*

_{0}> 0 allowing

*T*=

*T*. A straightforward way to obtain

_{e}*u*

_{0}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

*ζ*= 0 the natural boundary condition at

_{a}*z*=

*D*at least out to some finite radius

*r**. Translating this to

*u*= −

_{D}*fr*/2 in (33) we obtainwhere for the second equal sign we used, again, the definition (A3) of the critical slope. For a warm core vortex (i.e., ∂

*T̂*/∂

*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, *u*_{0} can be obtained from (38) with *T* replaced by *T _{e}*. 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

*T*. 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

_{e}*T*≈

*T*, the heating

_{e}*Q*=

*α*(

_{T}*T*−

_{e}*T*) remains finite (thus allowing a finite

*w*), because the coefficient

*α*is very large.

_{T}In the hurricane limit, *r*_{1} in (31) is equal to the radius of maximum surface wind. The secondary circulation can be obtained by computing *u* from (33) with *u _{D}* = −

*fr*/2 and substituting

*T*for

_{e}*T*; then

*ψ*(

_{E}*r*) is obtained from (30) by exploiting the fact that in the interior the isolines of

*ψ*are parallel to isolines of

*m*or

*R*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.

_{p},Numerical solutions in the hurricane regime (large *α _{T}*) 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

*R*) show, indeed, a close resemblance, and the same is true for the secondary circulation (streamfunction

_{p}*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* = *T _{e}* with

*T*from (31) and

_{e}*R*= 150 km yields

*u*

_{0}(

*r*) ≈ const ×

*r*

^{−1/2}in the region

*r*

_{1}≤

*r*≤

*R*. 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 (*T*_{0} ≤ *T _{c}*

_{0}, not shown), on the other hand, the size of the gray area shrinks to zero and there is a unique solution given by

*T*=

*T*,

_{e}*u*

_{0}= 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 rates^{2} *α* = *τ*^{−1}.

- By design, temperature
*T*is relaxed toward the equilibrium temperature*T*with rate_{e}*α*, which will be referred to as the rate of thermal forcing in the following._{T} - Interior friction
*X*is represented through a combination of vertical and horizontal diffusion according to (21). The relevant rate iswhere_{e}*R*and*D*represent the characteristic horizontal and vertical scale, respectively, of the vortex. - For all solutions considered in this study the diabatic part
*ψ*of the secondary circulation extends throughout the whole troposphere (cf. Fig. 3a). The related rate of the diabatic circulation isHere,_{d}*w*is the scale of the vertical wind in the upward branch_{d}^{3}of*ψ*. The second equality is obtained from the continuity Eq. (5), and_{d}*υ*denotes the scale of the radial in- or outflow. Approximating the heat Eq. (12) by_{d}*w*∂*θ*/∂*z*=*α*(_{T}*θ*−_{e}*θ*), and given that ∂*θ*/∂*z*differs only weakly from its reference value, we obtainwhere Δ*θ*=*D*∂*θ*/∂*z*≈*θ*(_{R}*D*) −*θ*(0) is approximately equal to the potential temperature difference between the top and the bottom of the domain._{R} - The part
*ψ*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 ratewhere_{e}*w*is the scale of the vertical wind due to_{e}*ψ*and_{e}*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*υ*denotes the scale of the radial outflow owing to_{e}*ψ*in that layer. Equation (30) implies_{e}*w*∼_{e}*c*_{d}u_{0}(*Rζ*)_{a}^{−1}|_{(z=0)}, and we obtain

*α*≪

_{d}*α*is satisfied as long as the amplitude

_{T}*T*

_{0}of the forcing is much smaller than Δ

*θ*. This is, in fact, guaranteed by our choice of

*T*

_{0}not exceeding 10 K.

*m*or

*R*are almost parallel to isolines of the streamfunction

_{p}*ψ*. This corresponds to the requirement that

*ζ*≈ 0. On the other hand, at the top of the surface boundary layer we have

_{a}*ζ*≠ 0. It follows that

_{a}*υ*=

*υ*+

_{e}*υ*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

_{d}*z*= 0 in our model. In terms the scales defined above this translates to

*υ*≈

_{d}*υ*orIndeed, comparison of

_{e}*ψ*and

_{d}*ψ*from Fig. 3 with

_{e}*ψ*from Fig. 2c indicates that in this particular example the two contributions to the radial wind,

*υ*and

_{d}*υ*, are similar in magnitude and opposite to each other in the bottom part of the domain.

_{e}### 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?

*R*,

*δz,*and Δ

*θ*can be considered as constants. The remaining free parameters are

*α*and

_{T}*c*, and for each couple (

_{D}*α*,

_{T}*c*) 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 aswhere

_{D}*γ*= Δ

*θ*/(

*Rδz*) and

*ζ*

_{a}_{0}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,

*u*

_{0}, and the deviation from thermal equilibrium, (

*θ*−

_{e}*θ*). Thus, the numerator and the denominator on the right-hand side of (51) are not independent of each other. Rather, the numerator increases as

*θ*−

_{e}*θ*decreases [see (38)] such that for any given

*u*

_{0}and

*θ*−

_{e}*θ*.

To test the prediction from this heuristic argument, we considered 100 different couples (*α _{T}*,

*c*) by varying both parameters by a wide margin, thus covering over three orders of magnitude for

_{D}*α*in the range 1.67 × 10

_{T}^{−5}s

^{−1}≤

*α*≤ 1.67 × 10

_{T}^{−3}s

^{−1}combined with ten different values of

*c*in the range 0.5 × 10

_{D}^{−3}≤

*c*≤ 8.0 × 10

_{D}^{−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 *u*_{0} over all radii *r* ≤ *R* and by the corresponding maximum vertically integrated temperature difference *T̂ _{e}* −

*T̂*. These two quantities are plotted as a function of

*α*/

_{T}*c*. We did the same computations but with a larger value for

_{D}*ν*(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.

_{h}In Fig. 9a, one observes small surface wind for small values of *T̂ _{e}* −

*T̂*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.

*T̂*−

_{e}*T̂*toward zero in the hurricane limit is like 1/

*c*, the vertical wind is determined by the surface wind according to (30). The surface wind, in turn, is given by

_{D}*T*in the hurricane limit and, thus, independent of

_{e}*α*. Since ∂

_{T}*θ*/∂

*z*is approximately constant by assumption, the whole left-hand side of (52) is independent of

*α*, so

_{T}*θ*−

_{e}*θ*must scale like

*α*

^{−1}

_{T}∝ 1/

*α*is kept fixed and

_{T}*c*is varied, the vertical wind scales like

_{D}*c*in the hurricane limit. Thus, (52) indicates that

_{D}*θ*−

_{e}*θ*must scale like

*c*, which is again proportional to 1/

_{D}*θ*−

_{e}*θ*is a nonlinear function of

*α*in addition to

_{T}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 *c _{D}* and approximately

^{4}proportional to

*α*(Wirth 1998). This is reflected in the pattern of points in the lower left corner: points associated with different values of

_{T}*c*but the same value of

_{D}*α*are aligned along a horizontal line, while points associated with different values of

_{T}*α*but the same value of

_{T}*c*are along a straight line with slope +1. On the other hand, the approximate analytical theory for the hurricane limit predicts that the maximum

_{D}*w*should be independent of

*α*and proportional to

_{T}*c*. 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

_{D}*α*but the same value of

_{T}*c*turn approximately horizontal, while the points corresponding to different values of

_{D}*c*but the same value of

_{D}*α*are approximately aligned along slope −1.

_{T}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 *α _{T}*/

*c*, 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.

_{D}## 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 *T _{e}*. 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 *T _{e}* 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

*T*, but not both at the same time.

_{e}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 *T _{e}* 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

*T*≈

*T*. 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

_{e}*T*from

*T*.

_{e}In a sense there is competition between surface friction to make the surface wind small and thermal relaxation to make *T* close to *T _{e}*. 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

*α*/

_{T}*c*, where

_{D}*α*is the rate of thermal relaxation and

_{T}*c*quantifies the strength of surface friction for a given surface wind. The monsoon limit corresponds to small

_{D}The scaling theory was verified through a large number of numerical solutions, for which both *α _{T}* and

*c*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

_{D}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 *T _{e}* 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

*T*and assuming fixed surface drag, the character of our steady-state vortices essentially depends on

_{e}*α*. Thus, our work emphasizes that, in addition to the surface wind and its coupling to the free tropospheric

_{T}*T*, the efficiency of convection letting

_{e}*T*approach

*T*(represented by

_{e}*α*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

_{T}*α*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

_{T}^{−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 *α _{T}* 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

*α*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.

_{T}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.

*u*

_{0}= 0, yieldingAssuming that ∂

*T*/∂

*r*≤ 0 throughout the domain, the condition for

*u*to remain real everywhere becomeswhere

*L*=

_{R}*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

*r*≤

*r*

_{0}) and zero beyond. Assuming continuity at

*r*=

*r*

_{0}one obtainswhere

*C*=

*const*. The corresponding balanced wind at the top of the domain (

*z*=

*D*) is characterized by

*ζ*=

_{a}*m*= 0 for 0 ≤

*r*≤

*r*

_{0}, while at

*r*=

*r*

_{0}the function

*ζ*(

_{a}*r*,

*D*) is discontinuous jumping from 0 to

*r*

^{2}

_{0}

*f*/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

*T*

_{c}_{0}; that is,

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, *ζ _{a}* ≥ 0 (note that we include the marginal case in our definition of regularity).

*D*, forming

*r*

^{−1}∂/∂

*r*, and using (16) yieldswhere the subscript 0 and

*D*denotes the lower and upper boundary, respectively. Regularity implies

*ζ*≥ 0. This together with the assumption of nonnegative

_{aD}*m*shows that the right-hand side of the above equation must be nonnegative:For zero surface winds we have

*m*

_{0}

*ζ*

_{a}_{0}=

*f*

^{2}

*r*

^{2}/2, and one obtainsas 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*.

*T̂*(

*r*) maximizes at

*r*= 0, since this is satisfied for all vortices considered in this study. Integrating (A8) outward from

*r*= 0 yieldswhere 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}

*T̂*/∂

_{c}*r*defined in (A3) applies equally to both.

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 *u*_{0} and the relation between temperature *T* and equilibrium temperature *T _{e}*

^{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 *c _{d}* at large

*u*

_{0}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 *α _{T}* is linear only for strongly supercritical

*T*(cf. Plumb and Hou 1992), to which we restrict our attention.

_{e}^{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 *T̂* is implicitly assumed rendering the proof less general than it may appear. In contrast, here we here allow more general radial profiles.