## 1. Introduction

Although primitive equation (PE) models will always be the most accurate, approximate balance models still serve a useful purpose in geophysical fluid dynamic (GFD) studies. Whether they are used to diagnose PE models (Möller and Jones 1998; Davis et al. 1996) or perform idealized simulations (Montgomery and Kallenbach 1997), balance models often provide fundamental insight into the dynamics of geophysical vortex flows. Previous work (Shapiro and Montgomery 1993, hereafter SM) proposed an asymmetric balance (AB) theory for rapidly rotating geophysical vortices, such as hurricanes. Unlike other balance models, the AB formulation is the only one that is formally valid when the Rossby number is large and the divergence is not small. Observations, however, have generally been inadequate to demonstrate the meteorological need for such a balance formulation and only recently has a dataset blending the vortex core and environmental flow of an intense hurricane become available for detailed analysis (Franklin et al. 1993, hereafter FLFM).

The physical basis and range of validity of the AB formulation in hurricane vortices was discussed in SM, and its dynamical consistency for linear dynamics was demonstrated by Kallenbach and Montgomery (1995) for asymmetric disturbances on a stable hurricane-like vortex in shallow water. The weakly nonlinear dynamics of the shallow water AB formulation has recently been investigated for azimuthal wavenumbers zero through four and the formulation has been shown to yield quantitatively similar results to the PE (Möller and Montgomery 1998, manuscript submitted to *J. Atmos. Sci.*). The more commonly used balance models, such as the nonlinear balance equations (BE: McWilliams 1985; Charney 1973) or the semibalance equations (SB; Raymond 1992), have received comparatively little scrutiny at the smaller mesoscales found within the near-vortex region of hurricanes. Upcoming work will compare the AB, BE, and SB models against the primitive equations in hurricane-like flows. But before embarking on this journey, it is first necessary to have a clear understanding of the regime of formal validity of these latter two balance models in hurricanes. Although the BE and SB models are popular intermediate models and are also believed useful for rapidly rotating vortices, a clear justification of the balance approximation underlying both formulations in hurricane vortices has been lacking. Beginning with a linearized PE model, section 2 presents a scale analysis that clarifies the conditions under which the balance approximation is and is not formally valid above the boundary layer in hurricanes. Section 3 augments the scale analysis with new kinematic analyses of Hurricane Gloria based on the FLFM dataset. Section 4 concludes with a discussion of the relevance of the key results to hurricane dynamics.

## 2. A priori scaling

Above the boundary layer and within 500 km from its center, a hurricane may be approximated as a symmetrically stable, circular vortex in hydrostatic and gradient balance, plus small-amplitude asymmetries (SM, Fig. 1). As a zeroth-order model of the asymmetric flow above the boundary layer we therefore consider the inviscid linearized primitive equations on a circular baroclinic vortex in gradient balance in cylindrical coordinates (radius *r,* azimuth *λ*) using pseudoheight (*z*) as the vertical coordinate. Whether the ideas developed below are generalizable to a nonlinear regime comprising episodes of extensive vorticity mixing (Schubert et al. 1998, manuscript submitted to *J. Atmos. Sci.*) is an open question.

*D*

_{V}/

*Dt*= ∂/∂

*t*+ (

*υ*

*r*)(∂/∂

*λ*) the linearized

*f*-plane momentum, continuity, and thermodynamic equations are, respectively,

*ϕ*′ denotes the perturbation geopotential incorporating the motion of the vortex (SM); (

*u*′,

*υ*′,

*w*′) denote the perturbation radial, tangential, and vertical velocity in storm-centered coordinates, respectively;

*θ*′ the perturbation potential temperature;

*υ*

*υ*

*r, z*) the basic-state tangential wind;

*N*

^{2}= (

*g*/

*θ*

_{0})(∂

*θ*

*z*) the basic-state static stability;

*Q*′ the perturbation heating rate;

*ρ*the pseudodensity;

*θ*

_{0}a reference potential temperature (300 K);

*g*the gravitational acceleration;

*ξ*

*f*+ 2

*υ*

*r*the modified Coriolis parameter; and

*η*

*f*+ ∂(

*r*

*υ*

*r*∂

*r*the basic-state absolute vertical vorticity. In this idealized model, boundary layer processes that force an interior response are incorporated into the boundary condition at

*z*= 0. Explicit formulas for the lower boundary condition have been provided elsewhere, invoking the gradient balance approximation (e.g., Ooyama 1969) or retaining full radial accelerations in the boundary layer (Shapiro 1983).

*υ*

*z*in (2.2), (2.5) can be neglected in the first approximation. Forming ∂[

*r*(2.1)]/

*r*∂

*r*+ ∂[2.2]/

*r*∂

*λ*then gives the barotropic divergence equation:

Here *δ*′ = ∂(*ru*′)/*r*∂*r* + ∂*υ*′/*r*∂*λ* is the perturbation divergence, ∇^{2} is the horizontal Laplacian, and *υ**r* is the mean angular velocity.

The underlying assumption of BE and SB is that the horizontal divergence is small compared to the vertical vorticity. If one assumes the existence of a single horizontal length scale *L* and a single vertical length scale *H,* which characterize the flow within the near-vortex region, then the BE are justified when both the aspect ratio squared *H*^{2}/*L*^{2} and the Froude number squared *F*^{2} = *V*^{2}/*N*^{2}*H*^{2} are small compared to unity (McWilliams 1985). Here (*V, N*^{2}) denotes a characteristic horizontal velocity and a characteristic static stability, respectively. In the near-vortex region of a hurricane, *H* ≈ scale height, *L* ≈ radius of maximum tangential winds (RMW), and *V* ≈ *υ*_{max}. Although the first requirement, *H*^{2}/*L*^{2} ≪ 1, is generally met in hurricanes, we will see that the second requirement, *F*^{2} ≪ 1, is generally insufficient to justify the balance approximation in hurricanes.

*balance approximation*neglects

*D*

_{V}

*δ*′/

*Dt*and approximates (

*u*′,

*υ*′) by the rotational wind. The approximate divergence equation is then a two-dimensional Poisson equation for

*ϕ*′ given the rotational winds. This limiting equation is called the

*balance equation*. For the approximation to be valid

*D*

_{V}

*δ*′/

*Dt*must be small compared to the remaining terms in (2.6). For exactly nondivergent flow,

*δ*′ = 0, and thus

*D*

_{V}

*δ*′/

*Dt*is trivially zero. For weakly nondivergent flow, we expect

*δ*′ to be small. Specifically, we expect

*δ*

*ζ*

*ζ*′ = ∂(

*rυ*′)/

*r*∂

*r*− ∂

*u*′/

*r*∂

*λ*is the perturbation vertical vorticity. In such circumstances

*D*

_{V}

*δ*′/

*Dt*will remain small compared to the other terms in (2.6). Explicitly, for advective dynamics

*D*

_{V}

*δ*′/

*Dt*∼

*δ*

^{′}

_{n}

*n*

*υ*

*r,*where

*n*is the azimuthal wavenumber and

*δ*

^{′}

_{n}

*n*component of

*δ*′. Taking

*η*

*u*′/

*r*∂

*λ*∼

*η*

*ζ*′ as a typical term on the left-hand side of (2.6), the ratio of

*D*

_{V}

*δ*′/

*Dt*to

*η*

*ζ*′ then scales as

*ζ*

^{′}

_{n}

*n*component of

*ζ*′. Therefore, when (2.7) is satisfied and when

*n*

*υ*

*η*

*r*is not large, the neglect of

*D*

_{V}

*δ*′/

*Dt*is justified.

^{1}In these circumstances the balance approximation serves as an accurate zeroth-order approximation relating the asymmetric height and wind fields in the near-vortex region of hurricanes.

*δ*′ ∼

*ζ*′ on advective timescales, the neglect of

*D*

_{V}

*δ*′/

*Dt*is no longer justified a priori. Section 3 presents observations that suggest that

*δ*′ ∼

*ζ*′ throughout the near-vortex region of hurricanes. Anticipating that

*δ*

^{′}

_{n}

*ζ*

^{′}

_{n}

*D*

_{V}

*δ*′/

*Dt*to

*η*

*ζ*′ then scales as

*n*

*υ*

*η*

*r*is

*O*(1). An illustrative example for which these considerations are relevant corresponds to the

*stationary*asymmetric flow forced by surface friction in a steadily translating hurricane (Shapiro 1983). This asymmetric flow can be interpreted as a stationary wave response to surface friction in a steadily translating vortex.

Shapiro’s analysis is relevant in two respects. First, unlike the naive GFD scaling, which assumes *δ*′ ∼ *F*^{2}*ζ*′ throughout the entire fluid, his calculation shows that *δ*′ near the top of the boundary layer in a translating hurricane scales with the asymmetric frictional stress. Second, from his Fig. 5 one can infer that *δ*′ ∼ *ζ*′ outside the RMW. Although asymmetric vorticity was not among the fields reported in Shapiro’s analysis, he has provided the numerical output of a representative boundary layer calculation by way of personal communication. For the case provided, the vortex’s maximum azimuthal-mean tangential velocity is 61.8 m s^{−1}, which occurs at a RMW of 35 km. The vortex’s translation speed is 10 m s^{−1}. The asymmetric convergence and relative vorticity are found to be typically dominated by the azimuthal wavenumber-one component, but the wavenumber-two component is not insignificant outside the RMW, where it contributes almost equally to the asymmetric convergence. The maximum amplitude of the wavenumber-one convergence is found to occur just inside the RMW with a value of 31.7 × 10^{−4} s^{−1}. In contrast, the maximum relative vorticity for wavenumber one occurs at the RMW with a value of 24 × 10^{−4} s^{−1}. The amplitude of asymmetric convergence is furthermore found to be of the same order or greater than the amplitude of asymmetric vorticity as far out as the vortex environment (∼500 km).

While Shapiro’s calculations only predict asymmetric convergence and vorticity *at the top* of the hurricane boundary layer, they nevertheless provide useful insight into the strength and structure of asymmetric forcing underneath a translating hurricane, which is ultimately coupled to the interior flow via cumulus convection. To the extent that convective processes in a rapidly rotating vortex project onto the slow manifold (Schubert et al. 1980; Ooyama 1982; Montgomery and Kallenbach 1997), we should not expect the naive scaling *δ*′ ∼ *F*^{2}*ζ*′ to be universally valid. Thus in the event that *δ*′ ∼ *ζ*′ in a horizontally and vertically widespread region of the vortex, an altogether different balance approximation permitting order one asymmetric divergence in its zeroth-order approximation is desirable. Asymmetric balance theory was developed with such applications in mind (SM, section 6).

## 3. Observations

FLFM described the kinematic structure of Hurricane Gloria as determined from nested analyses of Omega dropwindsonde (ODW) and airborne Doppler radar data. The data were obtained during a “synoptic-flow” experiment conducted by the Hurricane Research Division (HRD) of the Atlantic Oceanographic and Meteorological Laboratory/National Oceanic and Atmospheric Administration (AOML/NOAA) using two NOAA WP-3D research aircraft. The nested multiscale analyses simultaneously describe Gloria’s eyewall and synoptic-scale features. The combination of environmental and vortex core observations is the most comprehensive kinematic dataset obtained in a hurricane to date. Sections 2 and 3 of FLFM thoroughly discuss the data, analysis algorithm, and methodology for the Gloria analyses. Readers unfamiliar with the analyses may wish to refer to the discussion in FLFM or the appendix of Shapiro and Franklin (1995), which gives a brief overview.

Figure 1 shows the azimuthal wavenumber-one contribution to the perturbation relative vorticity (*ζ*^{′}_{1}*δ*^{′}_{1}*δ*^{′}_{1}*ζ*^{′}_{1}*δ*^{′}_{1}*ζ*^{′}_{1}*δ*^{′}_{1}*ζ*^{′}_{1}*δ*^{′}_{1}*ζ*^{′}_{1}

Section 2 showed that the neglect of the divergence tendency following the mean vortex in the divergence equation generally hinged upon the smallness of the nondimensional product (*δ*^{′}_{n}*ζ*^{′}_{n}*n**υ**η**r*). The above results show that the first term in this product, *δ*^{′}_{n}*ζ*^{′}_{n}*n**υ**η**r* for *n* = 1, in Hurricane Gloria. The *υ**η**local* Rossby number squared for azimuthal wavenumber one, *R*^{2}_{1}*R*^{2}_{1}*n**υ**η**r,* however, since it is near unity or greater over a much more extensive region (radius ∼300 km) of the hurricane.

For our final calculation, we compute each term of the left-hand side of the divergence equation (2.6) at the 850-mb and 700-mb levels. Although azimuthal wavenumber one tends to dominate the asymmetric structure, each term is evaluated with the total asymmetric fields. We evaluate *υ**δ*′/*r*∂*λ* as a proxy for *D*_{v}*δ*′/*Dt* and define this as term 1. If the asymmetric flow were stationary in the moving system this would be exact. Although a reliable estimate for ∂*δ*′/∂*t* is not available, we have no a priori reason to expect it to be significantly compensated by azimuthal advection. Term 2 is defined as −∂(*r**ξ**υ*′)/*r*∂*r,* term 3 is defined as *η**u*′/*r*∂*λ,* and term 4 is defined as (*d**dr*)×(∂*u*′/∂*λ*). Figures 4 and 5 summarize the results of these calculations at the 850-mb and 700-mb levels, respectively. From Fig. 4 we find that within the near-core region of the vortex (several RMW units) the divergence tendency is generally not small compared to either term 3 or the sum of terms 2, 3, and 4, but is of the same magnitude. Figure 5 shows similar results.

These are noteworthy results because, although the near-vortex asymmetries have previously been observed to have small magnitudes in comparison to the mean vortex (SM, Fig. 1), these observations are the first to suggest that the asymmetries are not rotationally dominated in the sense that *δ*′ ≪ *ζ*′.

## 4. Discussion and conclusions

Although the balance systems of McWilliams (1985) and Raymond (1992) are popular intermediate balance models for GFD studies of stably stratified rotating flow, a formal justification of the balance equation underlying both systems when applied to the asymmetric flow in hurricanes has been lacking. This work fills this void by developing a scale analysis for linearized dynamics indicating when the balance equation is and is not valid in hurricane-like vortices. In the event that the asymmetric flow is rotationally dominated, the balance equation is shown to serve as a useful first approximation connecting the asymmetric rotational winds to the asymmetric height field. On the other hand, if the asymmetric flow evolving on the advective timescale is not rotationally dominated and the Rossby number based on the mean tangential wind and absolute vertical vorticity of the vortex is of order unity, the balance equation is shown to be formally invalid. In this case the balanced asymmetric height field cannot be obtained accurately by solving just the balance equation given the rotational winds. As with other physical approximations encountered in mathematical physics and GFD, it may turn out in practice that the balance equation gives qualitatively correct results in circumstances where it ought not be based on criteria for formal validity (e.g., Spall and McWilliams 1992). But this has yet to be demonstrated in hurricane vortices.

Boundary layer considerations of a translating hurricane forced by surface friction at the air–sea interface, as well as observations of Hurricane Gloria (1985), indicate that hurricane asymmetries are not rotationally dominated and the balance equation for the asymmetric flow is invalid in the near-vortex region. In such circumstances the alternative AB formulation appears naturally suited, at least for small but finite amplitude asymmetric disturbances on a circular vortex. These theoretical considerations are motivated in part by the desire to diagnose the influence of three-dimensional vortex Rossby waves on the hurricane vortex (Montgomery and Kallenbach 1997). For in order to assess their attendant eddy-heat and eddy-momentum flux divergences in the dynamics of the hurricane it is necessary to devise an accurate method for inferring the balanced asymmetric temperature field that contains a minimal gravity–inertia wave component. To the extent that balanced dynamics captures the essence of the hurricane this is inextricably linked to the determination of the balanced asymmetric height field.

*δ*′ = ∂(

*ru*

^{′}

_{η}

*r*∂

*r*+ ∂

*υ*

^{′}

_{ξ}

*r*∂

*λ*denotes a pseudodivergence based on the pseudomomenta

*u*

^{′}

_{η}

*η*

*r*)×(∂

*ϕ*′/∂

*λ*) and

*υ*

^{′}

_{ξ}

*ξ*

*ϕ*′/∂

*r*) of AB theory, ∂

*υ*

*z*has been neglected for simplicity, and

*u*′ and

*υ*′ denote the asymmetric radial and tangential winds in storm-centered coordinates inferred from Doppler radar. Unlike the former scheme whose convergence requires a small divergence tendency, the AB formulation permits an

*O*(1) divergence tendency but requires that the square of the local Rossby number (

*R*

^{2}

_{n}

*n*be small compared to unity. In hurricanes,

*R*

^{2}

_{n}

*R*

^{2}

_{1}

*R*

^{2}

_{1}

*δ*

^{′}

_{n}

*ζ*

^{′}

_{n}

*n*

*υ*

*η*

*r*), has been shown to generally exceed unity in a much more extensive region of the vortex. It stands to reason that the height field extracted with the balance equation may suffer more widespread distortions than the AB formulation. It remains to be determined, however, which formulation is superior in practical situations. Further examination of this problem will be reported in due course.

## Acknowledgments

This research was supported in part by the Office of Naval Research, ONR N00014-93-1-0456. MTM especially wishes to thank Dr. Dominique Möller for initially inspiring this work and encouraging its completion. JLF would like to thank Dr. Lloyd Shapiro for assistance with some of the calculations.

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^{1}

In the special event that *ζ*′ is initially zero (e.g., a frictionless symmetric vortex suddenly subject to vertical shear) the scaling (2.8) breaks down. In this case *η**u*′/*r*∂*λ* is more accurately represented as *η**nu*^{′}_{n}*r* and (2.8) is replaced by (*δ*^{′}_{n}*r*/*nu*^{′}_{n}*n**υ**η**r*), which is finite. The neglect of *D*_{V}*δ*′/*Dt* is then justified when *n**υ**η**r* is not large provided *δ*^{′}_{n}*nu*^{′}_{n}*r.*