## 1. Introduction

The dynamics of atmospheric flows can be studied using primitive equation (PE) models with full physics or using filtered models, such as balanced models, for understanding the Hadley circulation (Schubert et al. 1991) or midlatitude cyclogenesis (Davis and Emanuel 1991; Montgomery and Farrell 1992). Balanced models, including those based on the standard balance equations (Charney1962) or those based on the geostrophic momentum approximation (Hoskins 1975), are filtered systems, whose accuracy lies between the primitive and the quasigeostrophic equations. For this reason they are sometimes called “intermediate models” (McWilliams and Gent 1980). Because the balanced models exclude phenomena such as gravity waves, which are often unimportant for understanding the large-scale flow, they provide for a simplified description of the evolution of weather systems. A fundamental quantity for atmospheric flows is the potential vorticity (PV), which is materially conserved in the absence of diabatic and frictional processes. In a balanced system one can use PV to diagnose the complete structure of the flow. The basis of this diagnosis is called the “invertibility principle,” which states that with a known PV distribution, a prescribed balance condition, a prescribed reference state, and with appropriate boundary conditions, it is possible to determine the complete mass and velocity fields of that balanced state (Hoskins et al. 1985).

Much of our understanding of the dynamics of tropical cyclone motion is based on results from barotropic vortex models in which the wavenumber-1 asymmetry of the relative vorticity distribution has been shown to play an important role (Fiorino and Elsberry 1989; Shapiro and Ooyama 1990; Smith et al. 1990). In the barotropic framework the vorticity asymmetries can be inverted to obtain the streamfunction asymmetries, from which the velocity of the flow across the vortex center can be determined; this velocity is to a close approximation the translation velocity of the vortex (Smith et al. 1990). A natural extension of the ideas to baroclinic vortices would point to the importance of the wavenumber-1 asymmetry in the PV distribution. With the aid of a balanced model appropriate for tropical cyclones the PV asymmetries can be inverted also to obtain the flow across the vortex axis.

Tropical cyclones are weather systems with strong upward motion, which implies that the horizontal divergence and vertical advection may be large. In addition, there is strong curvature in tropical cyclones. The presence of both strong curvature and strong divergence means that many balance conditions are not formally valid. One exception is the asymmetric balance (AB) formulation of Shapiro and Montgomery (1993, henceforth referred to as SM), which was derived recently to study the evolution of rapidly rotating vortices, including tropical cyclones. The AB formulation takes into account the full dynamics of the vortex core region and represents the outer region or far field as a quasigeostrophic flow. The theory uses the fact that in the near field all vortex asymmetries are weakcompared with the symmetric (wavenumber 0) circulation and it assumes that the wavenumber-1 component dominates the total asymmetry. Shapiro and Montgomery justify these assumptions using data from Hurricane Gloria. The major advantage of the AB theory is that, in principle, there is no restriction on the magnitude of the horizontal divergence and the vertical advection. Therefore, the theory may be at least as accurate for tropical cyclones as nonlinear balanced models, if not more so under certain circumstances. For example, in the core region of a hurricane, where divergence can be large, one could imagine that the AB theory would give more accurate results than the nonlinear balance models, as the latter formally neglect the divergent part of the wind fields.

A nested analysis of dropwindsonde and Doppler radar data for Hurricane Gloria confirms that the asymmetric divergence has a similar magnitude to the asymmetric vorticity (M. Montgomery and J. Franklin 1995, personal communication). As an example, Fig. 1 shows the azimuthal wavenumber-1 contribution of the relative vorticity and divergence of Hurricane Gloria at a height of 700 hPa. The shaded circle in the center of the hurricane shows the area where no data were available and the solid thick circle defines the radius of maximum wind. The wavenumber- 1 distribution of the vorticity (with a maximum of 500 × 10^{−6} s^{−1}) is about one order smaller in magnitude than that of the total vorticity field. The magnitude of the wavenumber-1 divergence, however, is nearly the same as that of the total divergence field and has the same size as the wavenumber-1 vorticity. The reason lies in the highly asymmetric divergence structure associated with a convective band near Gloria’s center (Franklin et al. 1993, Fig. 12).

The AB theory leads to a set of prognostic equations that, in principle, may be integrated forward in time. However, their complexity compared with the primitive equations, and that they are no more than an approximation to the latter, raises the question of the advisability of trying to solve them in time. Nevertheless, the equations may be manipulated so that the first- and second-order local time tendencies can be evaluated diagnostically at a given time. The PV provided by model output or real data can be inverted at a given time to obtain the corresponding balanced height and wind fields. It is the latter approach that is followed in the present paper, where we use a PE model to calculate the evolution of a vortex and the AB theory to diagnose and interpret the results. In this way one has the advantage of the accuracy of the primitive equations plus the simplified representation of the dynamics offered by the approximate theory. An additional advantage is that when using the AB theory diagnostically it is not necessary to predict the translation velocity and acceleration of the moving coordinate system in which the AB theory is formulated. Although the main assumption of the AB theory, that the squared local Rossby number is very small, is not strictly satisfied in these calculations, meaningful comparisons can be made between the PE results and the AB solutions. For diagnostic purposes, which we utilize here, it is quite reasonable to obtain qualitatively correct answers even if they are not quantitatively accurate. In a prognostic calculation, on the other hand, the quantitative accuracy could be very important as any errors may accumulate in time.

The AB theory assumes that the structure of the symmetric vortex is known and uses the symmetric quantities to calculate the balanced wind and mass fields of the perturbations. As discussed above, the wavenumber-1 asymmetry should be the dominant asymmetry in the vortex core. The three-dimensional AB model is applied to different situations in which a vortex is embedded in a horizontal or vertical shear flow. Calculations are carried out with or without representations of cumulus heating and surface friction. The results presented here provide insight into the influence of heating and friction on both the PV distribution and the corresponding velocity fields of a tropical cyclone. They provide also a basis for applying “PV thinking” to real data cases, complementing the recent studies of Molinari et al. (1995), Shapiro and Franklin (1995), and Wu and Emanuel (1995a,b).

In section 2 we show how the AB theory can be applied diagnostically. The fundamental assumption of the AB formulation, that the squared local Rossby number ≪1, and its limitations are discussed. The method used for PV inversion is presented, and the incorporation of a frictional boundary layer into the AB theory is described. Section 3 describes the results from calculations where an initially barotropic vortex is placed in a vertical shear flow. The results from the AB calculation are compared with those from the PE calculation. A piecewise PV inversion is performed and used to help interpret the results. In section 4 heating and friction are included in the calculations. The results are summarized and discussed in section 5. Additional discussion of the results, along with technical details of the formulation, can be found in Möller (1995). The results of our comparison between AB and PE solutions suggest that it will be necessary to make a comparison between the nonlinear balanced models and the AB theory. This is the subject of ongoing research that will be reported at a later time.

## 2. Theoretical details and method of solution

### a. Theory

The AB equations set is derived for a cylindrical coordinate system that translates with the vortex (Willoughby 1979). The pressure-based vertical coordinate, *z,* is the pseudoheight [Eq. (2.1) of Hoskins and Bretherton (1972)]. The mass and wind fields are partitioned into those associated with a translating symmetric vortex (denoted by an overbar) and weak perturbations to this vortex (denoted by a prime). The symmetric vortex, which evolves in time, is in gradient wind balance and is characterized by the potential vorticity *q**υ**η**f* + *ζ**ζ**r**υ**r*]/*r,* and the inertia parameter *ξ**f* + 2*υ**r,* where *f* is the Coriolis parameter. The symbols used are the same as those in SM.

*ϕ*′/∂

*t*=

*ϕ*′

^{t}, as

*r*is the radius,

*λ*the azimuth,

*N*

^{2}the static stability,

*ϕ*′ the perturbation geopotential, and

*Q*′ the perturbation heating. The operator on

*ϕ*′

^{t}is a second-order linear operator in

*r*and

*z*and remains elliptic as long as the vortex is symmetrically stable. Balanced models including the AB model require the PV to be positive, that is, the vortex to be symmetrically stable, in order to be able to find a solution.

*n,*appears in the numerator. SM indicate that the assumption

^{2}

_{Ln}

*n*= 1. Thus the derivations above are valid for the wavenumber-1 asymmetry in the vortex core region, which we shall consider here to be the radii for which the Rossby number Ro [=

*υ*

*fr*)] is greater than unity.

It should be noted that the fundamental assumption of the AB theory may not be strictly satisfied in observed or model vortices. In particular, the squared local Rossby number may not be ≪1. In our model calculation it was only possible to investigate situations where ^{2}_{L}^{2}_{L}^{2}_{L}^{2}_{L}^{2}_{L}

As shown in SM, (2.1) reduces to the linearized quasigeostrophic PV equation for Ro → 0. Thus the ABtheory is valid for all the asymmetric wavenumber components in the outer region or far field, which we shall define as the region for which Ro < 1. In the so-called quasigeostrophic regime certain pairs of higher wavenumbers (≥2) can interact to generate wavenumber 1. Therefore the generalized equation system that extends into the quasigeostrophic regime must include nonlinear interactions. Following SM, the nonlinear terms of the quasigeostrophic PV equation must be added to the right-hand side of the tendency equation in the outer region. Although the terms added are nonlinear, they have no influence on the linearity of the operator on the geopotential tendency itself.

The AB theory assumes that the flow is linear in the vortex core region. This means that the nonlinear terms added to (2.1) should be small in the vortex core. In practice, it turns out that the nonlinearities become dominant near the center of the vortex because the Coriolis parameter *f* substantially underestimates the inertia frequency in the vortex core region. The largest contribution to the nonlinear terms (SM, p. 3330) scales as 1/*f*^{3}. Thus this term becomes relatively large in the inner core due to *f* being considerably smaller than the inertia frequency. In a rapidly rotating vortex the inertia stability *f*^{2} has to be replaced by *ηξ**f* in the nonlinear terms by *ηξ**ηξ**f,* so this replacement ensures that the nonlinear terms remain small in the inner core region. In the outer region, *ηξ**f,* so the replacement does not alter the nonlinear terms there. In practice we multiply the nonlinear terms by the factor *f*/*ηξ**η**ξ**ηξ*

*total*wind, it is necessary to use the PV of the linearized form of the primitive equations. The PV equation can be rewritten in terms of the geopotential

*ϕ*′ and its time derivatives as

*ϕ*′ if the operator on

*ϕ*′ is elliptic. It can be demonstrated that the neglect of the

*O*(

^{2}

_{L1}

*ϕ*′ provided that the vortex is symmetrically stable.

### b. PV inversion

The basic idea is to use the AB equations as a diagnostic tool to calculate the balanced asymmetric structure of a PE model calculation. The PE model, which is used to provide the PV for all numerical calculations, is described in Jones and Thorpe (1992). It is a hydrostatic model formulated in Cartesian coordinates, *x* and *y,* on an *f* plane. The vertical coordinate is the same pressure-based height coordinate as in the AB theory. The prognostic variables are the horizontal wind components, *u*_{zon} and *υ*_{mer}, and the potential temperature, *θ.* The vertical velocity, *w,* and the geopotential, *ϕ,* are diagnosed from the continuity and hydrostatic equations, respectively. Periodic lateral boundary conditions are used for the perturbation fields.

*ϕ*

^{′t}

_{1}

*L*

^{−1}

*O*

*ϕ*

^{′}

_{1}

*ϕ*

^{′}

_{1}

*ϕ*

^{′}

_{2}

*G*

*Q*

^{′}

_{1}

*L*

^{−1}denotes the inverse of the linear operator acting on

*ϕ*′

^{t},

*O*the operator acting on

*ϕ*′, and

*G*the operator acting on the heating function

*Q*′. The nonlinear terms of the quasigeostrophic vorticity equation are included in (2.4) through NA, which denotes the nonlinear operator acting on

*ϕ*′, where the subscripts 1 and 2 denote the wavenumbers that contribute to wavenumber 1.

*ϕ*

^{′}

_{n}

*ϕ*

^{′t}

_{n}

*ϕ*

^{′(m+1)}

_{n}

*M*

^{−1}

*q*

^{′}

_{n}

*K*

*ϕ*

^{′t(m)}

_{n}

*ϕ*

^{′tt(m)}

_{n}

*Q*

^{′}

_{n}

*Q*

^{′t}

_{n}

*ϕ*

^{′(m)}

_{n}

*m*th iteration,

*ϕ*

^{′tt}

_{n}

*Q*

^{′t}

_{n}

*M*

^{−1}the inverse of the linear operator acting on

*ϕ*

^{′(m+1)}

_{n}

*q*

^{′}

_{n}

*n*and

*K*the operator on

*ϕ*

^{′t(m)}

_{n}

*ϕ*

^{′tt(m)}

_{n}

*Q*

^{′}

_{n}

*Q*

^{′t}

_{n}

*ϕ*′

^{t}and

*ϕ*′

^{tt}for wavenumbers 1 and 2 can be calculated step-by-step as follows:

*ϕ*

^{′(m)}

_{1}

*ϕ*′

^{t},

*ϕ*′

^{tt}obtained in this way and the given PV from the PE model, it is possible to calculate the (

*m*+ 1)th guess for wavenumber 1 and 2 components of the new geopotential

*ϕ*′ from (2.5) with

*n*= 1 and

*n*= 2 respectively. The iterative method was applied with the use of underrelaxation, without which the system failed to converge.

The required translation velocities and accelerations of the coordinate system are calculated from the PE model from the positions of the geopotential minimum at the analysis time and 3 h before and after, whereas the time derivatives of the acceleration, _{x} and _{y}, turned out to be negligible. Once we have the balanced perturbation geopotential and the geopotential’s first and second time derivatives, we can calculate the perturbation wind components *u*′, *υ*′, and *w*′ from (3.7)–(3.9) of SM.

### c. Incorporation of friction in the AB theory

*z*=

*h,*the friction term in the momentum equations in a translating cylindrical coordinate system is given by

*C*

_{D}denotes the nondimensional drag coefficient for sea surface stress,

*u*and

*υ*are the storm-relative radial and tangential velocity components, respectively,

**u**is the corresponding velocity vector,

**c**

_{cyt}= (

*c*

_{r},

*c*

_{λ}) is the translation velocity, and

*c*

_{r}and

*c*

_{λ}the radial and azimuthal velocity components. There are several empirical determinations of

*C*

_{D}, which are all of order 1.5 × 10

^{−3}(Roll 1965; Miller 1964). In order to include the drag coefficient in the AB theory, (2.7) has to be linearized using the assumption that the asymmetries are much smaller than the symmetric components. Although the symmetric radial wind,

*u*

*υ*

*u*

*υ*

*D*

_{r}and

*D*

_{λ}, are defined as follows:

*ζ.*

## 3. Initially barotropic vortex in a vertical shear flow

### a. Primitive equation calculation

*U*

_{A}

*U*

_{0}

*U*

_{z}

*z,*

*U*

_{0}(=5 m s

^{−1}) and

*U*

_{z}(=−5 × 10

^{−4}s

^{−1}) are constants. This zonal flow is in thermal wind balance with a horizontal temperature gradient. The stably stratified temperature field has a constant static stability

*N*

^{2}= 1.5 × 10

^{−4}s

^{−2}. The latitude is 20° N, so that

*f*= 5 × 10

^{−5}s

^{−1}. Since the shear, static stability, and Coriolis parameter are all constant, this flow has uniform potential vorticity. The domain is 6480 km in the

*x*direction and 5760 km in the

*y*direction with a horizontal resolution of 15 km. The vertical resolution is 2 km and the domain height is 10 km.

Figure 2 illustrates what happens when an axisymmetric barotropic vortex (from Willoughby 1988) is superimposed on this initial state. In response to the vertical shear, the vortex tilts in the plane of the shear. Associated with the tilt there is an upward and downward projection of the PV anomalies due to the lower-and upper-level vortices respectively. The projections, illustrated by the dashed lines in Fig. 2, lead to wavenumber-1 flow asymmetries. Jones (1995) attributes the observed cyclonic rotation of the upper- and lower-level vortices about the midlevel center to advection by these flow asymmetries. The axisymmetry of the vortex is distorted among other things by the divergent circulation of the vortex. The tilt, the upward and downward projection of the PV anomalies, and the distortion of the vortex appear in a cylindrical coordinate system as a wavenumber-1 distribution. After 24 h the vortex moves about 200 km to the east.

Figure 3 shows the radial profile of the tangential wind, *υ**υ*^{2}_{L}*υ*^{2}_{L}^{2}_{L}^{2}_{L}^{2}_{L}*υ**υ*^{−1}), ^{2}_{L}

The PV of the PE model is calculated at 12 and 24 h using the vortex with maximum *υ*^{−1}. Figures 6a and 6b show the cyclonic rotation of the upper- and lower-level vortices about the midlevel center marked by the cyclone symbol. A line passes through the PV maximum at the top and bottom, showing the orientationof the tilt. After 12 h the upper-level PV anomaly is southwest of the lower PV anomaly, and after 24 h there is a larger north–south component to the tilt and the separation of the centers of the upper and lower anomalies has increased.

Since the environmental flow in the present calculation contains vertical shear, the vortex translation speed, **c**, is different at each level and we assume that the vortex moves with the speed of the vortex center at a height of 5 km. At 12 and 24 h the zonal speed is *c*_{x} = 2.5 m s^{−1}, and the meridional speed is *c*_{y} = −5.0 × 10^{−3} m s^{−1} (effectively zero). The corresponding accelerations are *;azc*_{x} = 1.7 × 10^{−6} m s^{−2} and *;azc*_{y} = 1.8 × 10^{−6} m s^{−2}. In principle, the determination of the translation speed can be problematic if **c** is different at each level, since it is then necessary to find the most appropriate level for the choice of **c**. In practice it turns out that in the baroclinic experiment described here the choice of the translation speed was not overly sensitive to the choice of the level. The reason lies in the weak tilt of the vortex so that the differences between the translation speed at each level would imply a horizontal displacement less than the horizontal grid length. With a stronger shear or a narrower vortex profile, the sensitivity to the choice of level used to define **c** would increase.

### b. Comparison with AB theory

Compared with the conventional balanced equations (McWilliams 1985), the AB theory is a new theory, and as far as we are aware, the equations have not yet been solved in three dimensions. A good test of the theoryis to compare the structure of the AB balanced fields with those obtained directly from the primitive equations. In the PE calculation in this section, the divergence is small and the wind and mass fields are broadly in balance. Thus, the PV inversion using the AB theory should be able to reproduce the height and wind fields obtained from the PE model within the restrictions imposed by the size of the squared local Rossby number.

As an example, Figs. 7 and 8 compare the wavenumber-1 distributions of the tangential and radial velocities from the AB and PE models at 12 h and at heights of 2 and 8 km. Figure 9 shows the vertical velocity at heights of 4 and 6 km, the levels at which the values are a maximum and, in the case of the PE model, equal. The radial and tangential velocity components of the AB model (*u*^{′}_{AB}*υ*^{′}_{AB}*u*^{′}_{PE}*υ*^{′}_{PE}*u*^{′}_{PE}^{−1} in the inner region (Fig. 7a) compared with *u*^{′}_{AB}^{−1} (Fig. 7b); at a height of 8 km the corresponding values are *u*^{′}_{PE}^{−1} (Fig. 7c) and *u*^{′}_{AB}^{−1} (Fig. 7d). At a height of 2 km, the maximum of *υ*^{′}_{PE}^{−1} in the inner region (Fig. 8a) compared with *υ*^{′}_{AB}^{−1} (Fig. 8b); at a height of 8 km the corresponding values are *υ*^{′}_{PE}^{−1} (Fig. 8c) and *υ*^{′}_{AB}^{−1} (Fig. 8d). The extrema of the verticalvelocity of the PE (^{′}_{PE}^{′}_{AB}*w*^{′}_{PE}^{−1} (Fig. 9a) and the corresponding *w*^{′}_{AB}^{−1} at a height of 6 km (Fig. 9b) and 0.074 m s^{−1} at a height of 4 km (Fig. 9c). The calculation at 24 h gives a very similar result (not shown).

The relative error between the amplitude maxima of the velocities in the AB and PE calculations is, as expected, comparable with the maximum ^{2}_{L}*υ*^{−1}, where the maximum of ^{2}_{L}*w*^{′}_{AB}*w*^{′}_{PE}*w*^{′}_{PE}^{−1} compared with *w*^{′}_{AB}^{−1} at a height of 4 kmand *w*^{′}_{AB}^{−1} at a height of 6 km. The maxima of the asymmetries calculated from the PE model are orientated east–west in the inner region, whereas the orientation from the AB model calculations is more north–south at each level. Near the lower part of the domain, the vertical velocity in the AB solution is twice as large as that in the PE calculation. The maximum error in *w* lies between 200- and 300- km radius, exactly where the local Rossby number has its maximum. This suggests that the AB solution is less accurate where the local Rossby number is large and that ^{2}_{L}

### c. Piecewise inversion

To understand the influence of vertical shear on the vortex and the corresponding PV distribution, it is helpful to transform the radial and tangential velocity fields into the corresponding zonal and meridional wind fields. Figure 10 shows the wavenumber-1 contributions to the horizontal wind vectors obtained from the AB calculation at heights of 2 and 8 km at 12 and 24 h. At 2 km at 12 h, the flow is from northwest to southeast in the region where the PV anomaly is positive, while at 8 km it is mainly from the southwest. After 24 h the direction of the flow across the center changes, as the vortex rotates further cyclonically. We assume that the motion of the PV anomalies occurs due to advection by the balanced part of the wind field associated with the wavenumber-1 component of the PV anomaly. Under this assumption the wind fields shown should contain components that account for both the translation and rotation. It should be possible to isolate the wind field responsible for the rotation by calculating the wind field in the translating coordinate system. Figure 11 shows the flow at 12 h in the translating system. This flow would tend to advect the surface vortex southeastwardand the upper-level vortex northwestward, resulting in an anticyclonic rotation of the upper- and lower-level vortices about the midlevel center. The apparent inconsistency with the cyclonic rotation seen in Fig. 6 is resolved by performing a piecewise PV inversion as described below.

Piecewise PV inversion (Davis 1992) is a useful method for diagnosing the contributions to the total flow from individual portions of the PV distribution. Piecewise inversion has been applied to interpret baroclinic instability (Robinson 1989) as well as observations (Robinson 1988; Davis and Emanuel 1991). Davis (1992) pointed out that piecewise PV inversion is unique only for a linear inverse operator such as that in the quasigeostrophic theory. Unfortunately, quasigeostrophy is valid for large-scale motions only and becomes inaccurate for large Rossby numbers. Davis noted the ambiguity of the nonlinearity in the inversion operator for the more general Ertel PV (Ertel 1942), which is used in the standard nonlinear balance system. Thorpeand Bishop (1995) note also that linear superposition does not carry over to Ertel PV, though they hypothesize that the nonlinearity should not be significant except for anomalies that are very close together. In contrast to other balance theories (e.g., the nonlinear balance, and the semi- or quasi-balance [Raymond 1992] approximations), the AB theory has the advantage of being effectively linear (in the case presented here) and also of being able to handle large Rossby numbers Ro and large divergence. The following results will show how the AB theory can be used easily to invert the PV anomalies obtained from the PE model piecewise.

We investigate the PV-induced horizontal circulation at the top and bottom of the domain independently by dividing the PV anomaly into an upper- and a lower- level “piece.” In contrast to the inversion of the PV of the full domain, where the outer boundary is specified from the observed geopotential, the piecewise method dictates that the geopotential has to be set to zero. When considering the upper (lower)-level piece of PV, the vertical derivatives of the geopotential (i.e., the temperature) are set to zero at the bottom (top) of the domain. This is necessary because we have no way of ascertaining what portion of the observed geopotential should be associated with a given piece of PV anomaly. The lateral boundaries are assumed to be far enough away from the area of interest that the boundary conditions do not affect the result. If we set the PV anomaly in the upper region to be zero, we are able to calculate at each level the horizontal winds induced by the lower PV anomaly and vice versa. If we add the pieces of the resulting wind vectors together we should obtain the same answer as if we inverted the total PV anomaly.

The lower-level piece is defined at 0, 2, and 4 km, the upper-level piece at 6, 8, and 10 km. Figure 12 shows the winds at a height of 2 and 8 km from the piecewise PV inversion after 12 h. All results are obtained from the AB model in the translating coordinate system in order to show the influence of the PV anomalies, independently of the translation. Figures 12a and 12b show the influence of the upper-level PV anomaly at 8 and 2 km, respectively. The PV anomaly induces a southeasterly flow with a maximum of **v** = 3.8 m s^{−1} (Fig. 12a) at 8 km and a maximum of **v** = 0.9 m s^{−1} (Fig. 12b) at 2 km. Figures 12c and 12d show the influence of the lower-level PV anomaly at 8 and 2 km,respectively. The PV anomaly induces a northwesterly flow with a maximum of **v** = 0.8 m s^{−1} (Fig. 12c) at 8 km and a maximum of **v** = 9.5 m s^{−1} (Fig. 12d) at 2 km. Combining the flow at 2 km, induced by the upper-and lower-level PV anomalies, we obtain a northwesterly flow with a maximum of **v** = 8.6 m s^{−1} (Fig. 13a). At 8 km we obtain a southeasterly flow with a maximum of **v** = 3.0 m s^{−1} (Fig. 13b). A comparison between the combined flow at a height of 2 km (Fig. 13a) and the flow at 2 km, which is obtained from the total PV anomaly (Fig. 11a), shows there is very good agreement. The same is true for the height of 8 km (Figs. 11b and 13b).

To interpret these results we concentrate on the 2-km level. Analogous arguments can be used to interpret theresults at 8 km. The total flow across the PV center has the opposite sign to that one would expect from the rotation. Figure 12d shows a northwesterly flow induced by the low-level PV anomaly and which is dominant. The weaker southeasterly flow associated with the upper-level PV anomaly would explain the observed cyclonic rotation of the upper and lower vortices around the midlevel center (Fig. 12b). The flow associated with the low-level PV anomaly can be explained as follows. In our calculation we use the midlevel geopotential to define the vortex center. This center lies midway between the upper- and lower-level PV maxima (Fig. 6a). The lower-level wavenumber-1 asymmetry of PV arises due to the different location of the center and the PVmaximum. This asymmetry would essentially vanish if the vortex center coincided with the position of the lower-level PV maximum. Thus, the northwesterly flow seen at 2 km due to the lower-level PV anomaly is an artifact of the choice of the vortex center and is not physical. This flow is always in the opposite sense to the direction of the rotation and itself will rotate cyclonically around the midlevel center. In contrast, the flow at 2 km associated with the upper-level PV anomaly is due to the vortex tilt, which gives a horizontal displacement between the upper- and lower-level PV anomalies. If the vortex center coincided with the PV maximum at 2 km, the flow across the vortex center due to the upper-level anomaly would not disappear, but would be strengthened. Therefore we conclude that the large- scale rotation results from the wind field associated with the upper-level PV anomaly (Fig. 12b).

Only with the piecewise inversion is it possible to isolate the effect of the influence of the upper-level PV anomaly. If the symmetric vortex is tilted in the vertical, the decomposition of the flow depends on the choice of level one uses for defining the center. We chose the midlevel center so that the sum of each piece added up to the total flow. In doing so we neglected part of the contribution from the symmetric part of the upper-level PV anomaly, which also contributes to the rotation of the low-level vortex. Piecewise PV inversion is a useful tool for studying vortex dynamics. We have demonstrated that the AB theory can be used to perform a piecewise inversion. If attention is paid to the choice of the vortex center, piecewise inversion can be used to understand the vortex motion. The application of this tool to further datasets and further investigation of thesignificance of the choice of vortex center is a subject for future work.

### d. Comparison with quasigeostrophic theory

The quasigeostrophic theory is unable to represent the vortex core region. To confirm this we compare the velocities obtained from the PE model with those calculated diagnostically using the quasigeostrophic theory at 12 h. For both the radial and tangential velocities, *u*^{′}_{qg}*υ*^{′}_{qg}*u*^{′}_{qg}^{−1} and *υ*^{′}_{qg}^{−1}. The maxima at a height of 8 km are for both *u*^{′}_{qg}*υ*^{′}_{qg}^{−1}. Comparing the deviations of the AB theory and of the quasigeostrophic assumption with the PE calculations gives the following: at 2 km a 280% error for *u*^{′}_{qg}*υ*^{′}_{qg}*u*^{′}_{ab}*υ*^{′}_{ab}*u*^{′}_{qg}*υ*^{′}_{qg}*u*^{′}_{ab}*υ*^{′}_{ab}

## 4. Horizontal shear, diabatic heating, and friction

*U*

_{0}(=5 m s

^{−1}) is constant and

*L*

_{y}= 8640 km.This flow is illustrated in Fig. 14a. The domain is 11520 km in the

*x*direction and 8640 km in the

*y*direction with a horizontal resolution of 30 km. The vertical resolution is 2 km and the height of the domain is 14 km. The initial vorticity and tangential velocity profiles of the superposed vortex are similar to those used by Smith et al. (1990), but vary with height as shown in Fig. 14b and Fig. 14c. Note that the vortex is confined to the lower troposphere initially. Following Ooyama (1969), we assume that the latent heat released in deep convective clouds is proportional to the boundary layer convergence. Here, the flow includes a prescribed heating function that is related to the boundary layer convergence. The heating function

*Q,*which has a specified vertical profile, can be represented in the form that

*w*

_{BL}denotes the vertical velocity at the top of the boundary layer. In contrast to Ooyama (1969), who determines

*χ*from energy considerations in terms of the enthalpy of moist air, it is sufficient for our purposes to define

*χ*as 0.09 sin(

*πz*/

*H*) K m

^{−1}, where

*H*is 14 km and the maximum of heating occurs at 7 km.

The heating increases the PV below the heating maximum and reduces the PV above it. Therefore the cyclonic anomaly is strengthened below the heating maximum and an anticyclonic PV anomaly develops above the heating maximum. Consequently heating increases the cyclonic circulation at lower levels and leads to an anticyclonic circulation at upper levels. Figures 15a–c show vertical cross sections of the PV anomalies after 6, 12, and 24 h. We noted previously that the more negative the vorticity becomes, the greater the local Rossby number becomes and the greater is the inaccuracy of the AB theory. The negative relative vorticity in the upper-level anticyclone decreases the denominator of the squared local Rossby number ^{2}_{L}^{2}_{L}^{2}_{L}

*υ*′ of the AB model became three orders of magnitude higher than that of the PE model. As the AB theory is derived for slowly evolving vortices, the acceleration term is assumed to remain small. Inclusion of the heating, however, implies the existence of fast inertial–gravity waves, which involve large accelerations. To keep Eq. (3.7) of SM consistent with the tendency equation (3.10) of SM, it is necessary to drop the first, second, and fourth terms of the right-hand side of

*D*

_{υ}

*w*′/

*Dt,*where

*w*′ is given by (3.9) of SM. Then SM’s (3.7) becomes

After 12 h the vortex has moved 200 km to the east and very slightly to the south. The translation velocity components are *c*_{x} = 4.7 m s^{−1}, *c*_{y} = 0.2 m s^{−1}, and the acceleration **;azc** is on the order of 10^{−6} m s^{−2}. The horizontal shear flow contributes to the symmetric part of the vortex. The magnitude and indeed the sign of this contribution depends on the location of the vortex center, as can be seen by referring to Fig. 14a. When the center of the vortex is south of the background flow maximum (i.e., south of the east–west line labeled 0 km in Fig. 14a), an azimuthal average of the background flow contributes a negative tangential velocity and leads in this case to a negative relative vorticity. If the negative relative vorticity exceeds a certain value, ^{2}_{L}^{2}_{L}

Results are presented at 6 and 12 h. An upper-level anticyclone is induced by the heating and increases the local Rossby number squared ^{2}_{L}^{2}_{L}*υ*′ > *υ**υ* (Ro = *υ*/*fr*). At 6 h *υ*′ = 4.9 m s^{−1} at a radius of 150 km and at 12 h *υ*′ = 5.1 m s^{−1} at a radius of 120 km, giving Ro = 0.66 after 6 h and Ro = 0.85 after 12 h. Since Ro is not small outside the vortex, we can expect the AB theory to be inaccurate at the upper levels. As in section 3, a comparison is made between the wavenumber-1 distributions of the tangential and radial wind fields of the AB and PE models. It can be assumed that even here, where heating is included, the results of the PE and AB models should be similar as long as ^{2}_{L}

The results show very clearly that the accuracy of the AB solution decreases as ^{2}_{L}^{2}_{L}^{2}_{L}*u*^{′}_{PE}^{−1} (Fig. 17a) and of *u*^{′}_{AB}^{−1} (Fig. 17b), *υ*^{′}_{PE}^{−1} (Fig. 17c), and *υ*^{′}_{AB}^{−1} (Fig. 17d) at a height of 2 km. At a height of 6 km, the PE calculations have a similar amplitude as at a height of 2 km, whereas the AB calculations show significant differences in the amplitude. At a height of 6 km *u*^{′}_{PE}^{−1} (Fig. 18a), and *u*^{′}_{AB}^{−1} (Fig. 18b), *υ*^{′}_{PE}^{−1}, and *υ*^{′}_{AB}^{−1} (Figs. 18c and 18d). After 12 h the results of the PE calculations stay the same, since there is no significant change in the background flow, and we show only the results from the AB calculations. At the heights of 2 and 6 km, the orientation of the velocity fields stays the same. At a height of 2 km there is not a large difference in the amplitudes of *u*^{′}_{AB}^{′}_{AB}*u*^{′}_{AB}^{−1} and *υ*^{′}_{AB}^{−1} (Figs. 19a and 19b). At aheight of 6 km *u*^{′}_{AB}^{−1} and *υ*^{′}_{AB}^{−1} (Figs. 19c and 19d).

Compared with the results after 6 h, the greater error after 12 h in the calculation from the AB theory is caused by the higher inaccuracy in the approximations, which is in accord with the higher local Rossby number of Fig. 16b. After 24 h the flow is not in balance, as indicated by the negative local Rossby number in Fig. 16c. In a case where we tried to initialize the vortex with a vertical shear flow, we obtained an unbalanced state after the first few hours. The solution in the AB theory diverged after the first iteration and no solution could be found. If heating is included in the calculation, the PV becomes negative after a certain period of time. Just as in the nonlinear balance equations (Davis and Emanuel 1991) the AB formulation requires the PV to be positive. In the case of the AB theory, the condition that the PV of the symmetric vortex is positive is identical to the condition that the discriminant of the operator in the geopotential tendency equation (2.1) is positive. If the PV of the symmetric vortex is negative, then the system is not elliptic and there is no balanced solution. Of course the stronger condition for the accuracy of the AB theory that the local Rossby number is small (but positive), when satisfied, guarantees that the symmetric PV is positive and that a balanced solution exists.

## 5. Summary and discussion

This paper has presented results from PV inversion using a three-dimensional diagnostic balance model for tropical cyclones. The model is based upon the asymmetric balance (AB) theory derived by Shapiro and Montgomery (1993). The main advantage of the AB theory is its ability to represent the large divergence that occurs in a hurricane. The standard balance equations neglect divergence relative to the vorticity. The AB theory takes advantage of the weakness of the asymmetries in a hurricane relative to the much stronger basic state. As the wavenumber 1 dominates the total asymmetry in the vortex core region it is sufficient to apply the AB theory for wavenumber 1 only. In the quasigeostrophic regime at large distance from the vortex, the asymmetries with wavenumbers greater than 1 must also be included.

The diagnostic application of the AB theory was explained. We showed how the two fundamental AB equations have to be manipulated and how the geopotential, its local time tendency, and its second-order time tendency can then be evaluated at a given time. We found that the nonlinear terms, which—following Shapiro and Montgomery—have to be added in the quasigeostrophic regime, became dominant near the vortex center, which contradicts the fact that the asymmetries are assumedto be weak in the vortex center. We were able to find a method to include the nonlinear terms only in the vortex far field where they are important. We described how the friction could be incorporated in the AB theory by including terms that involve bulk drag coefficients in the momentum equations, the geopotential tendency equation, and the PV equation.

The total PV was inverted using an iterative technique to solve the geopotential tendency equation and the PV equation simultaneously. A fundamental aspect of the AB theory is that it is formulated in a coordinate system that translates with the vortex. Therefore, in a baroclinic experiment where the translation speed is different at each level it is important to find the level that represents the vortex translation speed best. This is necessary to maintain accuracy of the AB theory.

A PE model was used to provide the datasets for the PV inversion. The first model calculation described an initially barotropic vortex in a vertical shear. In responseto the vertical shear, the vortex tilts and a cyclonic rotation of the upper- and lower-level vortices about the middle-level vortex center was observed. This leads to a wavenumber-1 distribution of the PV and wind fields. The AB theory was able to reproduce the height and wind fields obtained from the PE model. The results served as a test of the new AB theory. The fundamental approximation in the AB theory is the assumption that the square of the advective rate-of-change is much smaller than the inertial stability of the vortex. The ratio of the orbital frequency squared to the inertial stability is defined by the *local* Rossby number squared, ^{2}_{L}^{2}_{L}^{2}_{L}^{2}_{L}^{2}_{L}^{2}_{L}

The role played by individual portions of the PV field was investigated by performing a piecewise PV inversion. Presenting the results of the velocity fields in the moving coordinate system and using the piecewise inversion made it possible to isolate the influence of the upper-level PV anomaly on the lower-level part of the vortex and the influence of the lower-level PV anomaly on the upper-level part of the vortex. The issue regarding which level to use for defining the center of the vortex for the piecewise inversion is not fundamental to the AB theory and is, in fact, present whenever a symmetric vortex is used to define the basic state. The greater the vortex tilt, the more important this issue could be. Möller (1995, section 7.3) discusses the sensitivity of the result to the choice of level more completely. Convective coupling in a more realistic model calculation or an observed hurricane, however, would be expected to reduce the tilt and to minimize this sensitivity.

We calculated the wind fields for the same experiment using the quasigeostrophic approximation. As expected this approximation was not able to represent the vortex core region. The orientation of the velocity fields were similar, but the amplitudes were about one order of magnitude higher than in the PE calculation.

Diabatic heating and friction were included in a calculation where the vortex was embedded in a horizontal shear flow. The prescribed heating was related to the boundary layer convergence. The heating creates positive PV at low levels, thereby increasing the cyclonic circulation. At upper levels the heating creates a negative PV anomaly and an anticyclonic circulation develops. In these experiments divergence above the boundary layer was not large.

We found that the inclusion of heating resulted in values of the tangential velocity three orders of magnitude larger than the tangential velocity of the PE model. However, when the acceleration terms were omitted in the tangential velocity equation, we obtained reasonable tangential velocity fields at lower levels. Neglect of the acceleration terms is consistent with the tendencyequation. The heating produced strong vertical gradients in the tangential wind so that the PV of the symmetric vortex became negative after 24 h. A negative PV of the symmetric vortex implies a negative squared local Rossby number. As in the nonlinear balance equations, the AB formulation requires the PV to be positive in order to be able to find a solution.

It was noted previously that the local Rossby number becomes greater the more negative the relative vorticity becomes. This leads to greater inaccuracy in the AB theory. In the calculation with diabatic heating, the anticyclonic upper-level vortex became stronger with time and therefore the negative relative vorticity became greater, increasing ^{2}_{L}^{2}_{L}*υ*′ was greater than the symmetric velocity *υ**υ**υ*′. In this case the standard Rossby number Ro was the appropriate measure of the validity and accuracy as in the quasigeostrophic approximation: Ro was 0.66 after 6 h and 0.85 after 12 h. A comparison was made of the velocity components of the AB and PE model at 2 km and 6 km. The orientation of the wavenumber-1 distributions of the radial and tangential wind components were in good agreement. The amplitudes of the winds were similar for both models at a height of 2 km but differed at a height of 6 km. The relative error in the amplitudes were for the data at the height of 2 km similar to the magnitude of ^{2}_{L}

This study is the first to demonstrate the viability of the AB theory in a three-dimensional context. However, we have found problems in applying the theory when the effects of heating and friction are included. Theaccuracy of the theory is characterized by a squared local Rossby number ^{2}_{L}*local* Rossby number is small. In particular, negative PV makes it impossible to find a solution of the balance system. It should be noted, that no balance theory has a solution when the PV is negative. This negative PV is associated with the anticyclone in the upper level of the domain. Whether or not the PV is negative depends on the strength of the anticyclone. Our results show that the AB theory is not able to characterize the motion in the outflow layer in the case we studied. However, because of our crude representation of heating we cannot be sure that the strength of the anticyclone in our numerical calculation is realistic for a hurricane. In particular, our use of a fixed heating profile means that we are unable to represent the mature phase of a tropical cyclone since changes in the stability in the core of the cyclone do not alter the strength of the diabatic heating. If the anticyclone were weaker, it is possible that the PV would not have been negative, in which case we may have been able to characterize the motion in the outflow layer. Figure 1 shows that for Hurricane Gloria the asymmetric divergence had a comparable size to the asymmetric vorticity. SM showed that the squared local Rossby number (^{2}_{L}*formally* valid in regimes of large Rossby number and moderate Froude number.

## Acknowledgments

The work presented in this paper would not have been possible without the support and encouragement of Roger Smith, to whom we express our sincere thanks. We are also very grateful to Michael Montgomery and Lloyd Shapiro for their most helpful comments. We especially appreciate Lloyd Shapiro’s advice on piecewise inversion. We gratefully acknowledge James Franklin for providing Fig. 1. The comments of Dan Keyser, John Molinari, and an anonymous reviewer on an earlier version helped us to improve this manuscript. This work was supported by the German Research Council (DFG) and the U.S. Office of Naval Research through Grant N00014-95-1-0394.

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Schematic showing the influence of vertical shear on an initially barotropic vortex. The shear flow *U**z*) is shown on the left of the figure. The solid black lines on the right of the figure show the PV of the vortex, which has been tilted in the vertical by the action of the vertical shear. The long-dashed lines illustrate the downward projection of the upper-level PV anomaly, resulting in a cyclonic circulation at the surface. The short-dashed lines illustrate the upward projection of the lower-level PV anomaly, giving a cyclonic circulation at upper levels.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Schematic showing the influence of vertical shear on an initially barotropic vortex. The shear flow *U**z*) is shown on the left of the figure. The solid black lines on the right of the figure show the PV of the vortex, which has been tilted in the vertical by the action of the vertical shear. The long-dashed lines illustrate the downward projection of the upper-level PV anomaly, resulting in a cyclonic circulation at the surface. The short-dashed lines illustrate the upward projection of the lower-level PV anomaly, giving a cyclonic circulation at upper levels.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Schematic showing the influence of vertical shear on an initially barotropic vortex. The shear flow *U**z*) is shown on the left of the figure. The solid black lines on the right of the figure show the PV of the vortex, which has been tilted in the vertical by the action of the vertical shear. The long-dashed lines illustrate the downward projection of the upper-level PV anomaly, resulting in a cyclonic circulation at the surface. The short-dashed lines illustrate the upward projection of the lower-level PV anomaly, giving a cyclonic circulation at upper levels.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial profile of (a) the tangential velocity and (b) the relative vorticity of the initial vortex in section 3.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial profile of (a) the tangential velocity and (b) the relative vorticity of the initial vortex in section 3.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial profile of (a) the tangential velocity and (b) the relative vorticity of the initial vortex in section 3.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial profile of the squared local Rossby number of the initial vortex.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial profile of the squared local Rossby number of the initial vortex.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial profile of the squared local Rossby number of the initial vortex.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the squared local Rossby number (a) after 12 h with a maximum tangential velocity of 20 m s^{−1}, (b) as (a) but after 24 h, and (c) with a maximum tangential velocity of 40 m s^{−1} after 24 h. Contour interval is 0.025 for (a) and (b) and 0.05 for (c).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the squared local Rossby number (a) after 12 h with a maximum tangential velocity of 20 m s^{−1}, (b) as (a) but after 24 h, and (c) with a maximum tangential velocity of 40 m s^{−1} after 24 h. Contour interval is 0.025 for (a) and (b) and 0.05 for (c).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the squared local Rossby number (a) after 12 h with a maximum tangential velocity of 20 m s^{−1}, (b) as (a) but after 24 h, and (c) with a maximum tangential velocity of 40 m s^{−1} after 24 h. Contour interval is 0.025 for (a) and (b) and 0.05 for (c).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the upper- and lower-level density-weighted potential vorticity fields. The density-weighted potential vorticity of the background flow has been subtracted. The solid contours are for *z* = 2 km, the dashed contours for *z* = 8 km. In both cases the zero line is dotted. Contour interval is 0.25 × 10^{−6} m^{2} K/(s kg). The position of minimum perturbation geopotential at *z* = 5 km is marked by a cyclone symbol. Only a portion of the model domain is shown: (a) 12 h and (b) 24 h. The solid line shows the orientation of the tilt.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the upper- and lower-level density-weighted potential vorticity fields. The density-weighted potential vorticity of the background flow has been subtracted. The solid contours are for *z* = 2 km, the dashed contours for *z* = 8 km. In both cases the zero line is dotted. Contour interval is 0.25 × 10^{−6} m^{2} K/(s kg). The position of minimum perturbation geopotential at *z* = 5 km is marked by a cyclone symbol. Only a portion of the model domain is shown: (a) 12 h and (b) 24 h. The solid line shows the orientation of the tilt.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the upper- and lower-level density-weighted potential vorticity fields. The density-weighted potential vorticity of the background flow has been subtracted. The solid contours are for *z* = 2 km, the dashed contours for *z* = 8 km. In both cases the zero line is dotted. Contour interval is 0.25 × 10^{−6} m^{2} K/(s kg). The position of minimum perturbation geopotential at *z* = 5 km is marked by a cyclone symbol. Only a portion of the model domain is shown: (a) 12 h and (b) 24 h. The solid line shows the orientation of the tilt.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber 1 component of the radial velocity *u* after 12 h (a) at a height of 2 km (calculated with the PE model), (b) at a height of 2 km (calculated with the AB model), (c) at a height of 8 km (calculated with the PE model), and (d) at a height of 8 km (calculated with the AB model). Contour interval is 1.0 m s^{−1} for (a) and (b) and 0.5 m s^{−1} for (c) and (d). Here (0 km, 0 km) is the vortex center; this corresponds to (−360 km, −15 km) in *x*–*y* space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber 1 component of the radial velocity *u* after 12 h (a) at a height of 2 km (calculated with the PE model), (b) at a height of 2 km (calculated with the AB model), (c) at a height of 8 km (calculated with the PE model), and (d) at a height of 8 km (calculated with the AB model). Contour interval is 1.0 m s^{−1} for (a) and (b) and 0.5 m s^{−1} for (c) and (d). Here (0 km, 0 km) is the vortex center; this corresponds to (−360 km, −15 km) in *x*–*y* space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber 1 component of the radial velocity *u* after 12 h (a) at a height of 2 km (calculated with the PE model), (b) at a height of 2 km (calculated with the AB model), (c) at a height of 8 km (calculated with the PE model), and (d) at a height of 8 km (calculated with the AB model). Contour interval is 1.0 m s^{−1} for (a) and (b) and 0.5 m s^{−1} for (c) and (d). Here (0 km, 0 km) is the vortex center; this corresponds to (−360 km, −15 km) in *x*–*y* space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 7 but for the tangential velocity *υ.*

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 7 but for the tangential velocity *υ.*

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 7 but for the tangential velocity *υ.*

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 7 but for the vertical velocity *w* (a) at a height of 4 and 6 km (calculated with the PE model), (b) at a height of 6 km (calculated with the AB model), and (c) at a height of 4 km (calculated with the AB model). Contour interval is 0.01 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 7 but for the vertical velocity *w* (a) at a height of 4 and 6 km (calculated with the PE model), (b) at a height of 6 km (calculated with the AB model), and (c) at a height of 4 km (calculated with the AB model). Contour interval is 0.01 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 7 but for the vertical velocity *w* (a) at a height of 4 and 6 km (calculated with the PE model), (b) at a height of 6 km (calculated with the AB model), and (c) at a height of 4 km (calculated with the AB model). Contour interval is 0.01 m s^{−1}.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections calculated with the AB model. Arrows indicate the wavenumber-1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km at 12 h, maximum vector is 11.1 m s^{−1}; (b) as in (a) but at a height of 8 km, maximum vector is 3.4 m s^{−1}; (c) at a height of 2 km at 24 h, maximum vector is 16.3 m s^{−1}, and (d) as in (c) but at a height of 8 km, maximum vector is 3.7 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections calculated with the AB model. Arrows indicate the wavenumber-1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km at 12 h, maximum vector is 11.1 m s^{−1}; (b) as in (a) but at a height of 8 km, maximum vector is 3.4 m s^{−1}; (c) at a height of 2 km at 24 h, maximum vector is 16.3 m s^{−1}, and (d) as in (c) but at a height of 8 km, maximum vector is 3.7 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections calculated with the AB model. Arrows indicate the wavenumber-1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km at 12 h, maximum vector is 11.1 m s^{−1}; (b) as in (a) but at a height of 8 km, maximum vector is 3.4 m s^{−1}; (c) at a height of 2 km at 24 h, maximum vector is 16.3 m s^{−1}, and (d) as in (c) but at a height of 8 km, maximum vector is 3.7 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections in the translating coordinate system calculated with the AB model at 12 h. Arrows indicate the wavenumber- 1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km and maximum vector of 8.5 m s^{−1}, and (b) at a height of 8 km and maximum vector of 3.4 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections in the translating coordinate system calculated with the AB model at 12 h. Arrows indicate the wavenumber- 1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km and maximum vector of 8.5 m s^{−1}, and (b) at a height of 8 km and maximum vector of 3.4 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections in the translating coordinate system calculated with the AB model at 12 h. Arrows indicate the wavenumber- 1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km and maximum vector of 8.5 m s^{−1}, and (b) at a height of 8 km and maximum vector of 3.4 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections at 12 h using piecewise inversion. Arrows indicate the wavenumber 1 component of the horizontal wind. Contours of potential vorticity at the same level as the arrows with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 8 km, induced by the upper-level PV anomaly, maximum vector is 3.8 m s^{−1}; (b) at a height of 2 km, induced by the upper-level PV anomaly, maximum vector is 0.9 m s^{−1}; (c) at a height of 8 km, induced by the lower-level PV anomaly, maximum vector is 0.8 m s^{−1}; and (d) at a height of 2 km, induced by the lower-level PV anomaly, maximum vector is 9.5 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections at 12 h using piecewise inversion. Arrows indicate the wavenumber 1 component of the horizontal wind. Contours of potential vorticity at the same level as the arrows with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 8 km, induced by the upper-level PV anomaly, maximum vector is 3.8 m s^{−1}; (b) at a height of 2 km, induced by the upper-level PV anomaly, maximum vector is 0.9 m s^{−1}; (c) at a height of 8 km, induced by the lower-level PV anomaly, maximum vector is 0.8 m s^{−1}; and (d) at a height of 2 km, induced by the lower-level PV anomaly, maximum vector is 9.5 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections at 12 h using piecewise inversion. Arrows indicate the wavenumber 1 component of the horizontal wind. Contours of potential vorticity at the same level as the arrows with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 8 km, induced by the upper-level PV anomaly, maximum vector is 3.8 m s^{−1}; (b) at a height of 2 km, induced by the upper-level PV anomaly, maximum vector is 0.9 m s^{−1}; (c) at a height of 8 km, induced by the lower-level PV anomaly, maximum vector is 0.8 m s^{−1}; and (d) at a height of 2 km, induced by the lower-level PV anomaly, maximum vector is 9.5 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections at 12 h and combined flows from the piecewise inversion. Arrows indicate the wavenumber-1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km, induced by the upper- and lower-level PV anomalies, maximum vector is 8.6 m s^{−1}, and (b) at a height of 8 km, induced by the upper- and lower-level PV anomalies, maximum vector is 3.0 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections at 12 h and combined flows from the piecewise inversion. Arrows indicate the wavenumber-1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km, induced by the upper- and lower-level PV anomalies, maximum vector is 8.6 m s^{−1}, and (b) at a height of 8 km, induced by the upper- and lower-level PV anomalies, maximum vector is 3.0 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections at 12 h and combined flows from the piecewise inversion. Arrows indicate the wavenumber-1 component of the horizontal wind. Contours of potential vorticity with contour interval 0.25 × 10^{−6} m^{2} K/(s kg) (a) at a height of 2 km, induced by the upper- and lower-level PV anomalies, maximum vector is 8.6 m s^{−1}, and (b) at a height of 8 km, induced by the upper- and lower-level PV anomalies, maximum vector is 3.0 m s^{−1}. Here, (0 km, 0 km) is the midlevel vortex center.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Initial conditions for calculations with heating. (a) Horizontal shear flow in which the vortex is embedded. The vortex moves to the east and slightly to the south. Only the central part of the domain is shown. (b) Radial–height cross section of the relative vorticity (contour interval 0.5 × 10^{−5} s^{−1}). (c) Radial-height cross section of the tangential velocity in m s^{−1} of the initial vortex in section 4.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Initial conditions for calculations with heating. (a) Horizontal shear flow in which the vortex is embedded. The vortex moves to the east and slightly to the south. Only the central part of the domain is shown. (b) Radial–height cross section of the relative vorticity (contour interval 0.5 × 10^{−5} s^{−1}). (c) Radial-height cross section of the tangential velocity in m s^{−1} of the initial vortex in section 4.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Initial conditions for calculations with heating. (a) Horizontal shear flow in which the vortex is embedded. The vortex moves to the east and slightly to the south. Only the central part of the domain is shown. (b) Radial–height cross section of the relative vorticity (contour interval 0.5 × 10^{−5} s^{−1}). (c) Radial-height cross section of the tangential velocity in m s^{−1} of the initial vortex in section 4.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the PV anomalies. Contour interval is 0.2 × 10^{−6} m^{2} K/(s kg) (a) after 6 h, (b) after 12 h, and (c) after 24 h.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the PV anomalies. Contour interval is 0.2 × 10^{−6} m^{2} K/(s kg) (a) after 6 h, (b) after 12 h, and (c) after 24 h.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the PV anomalies. Contour interval is 0.2 × 10^{−6} m^{2} K/(s kg) (a) after 6 h, (b) after 12 h, and (c) after 24 h.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the squared local Rossby number (a) after 6 h, (b) after 12 h, and (c) after 24 h. Contour interval is 0.05 for (a) and (b), and 0.25 for (c).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the squared local Rossby number (a) after 6 h, (b) after 12 h, and (c) after 24 h. Contour interval is 0.05 for (a) and (b), and 0.25 for (c).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Radial–height cross sections of the squared local Rossby number (a) after 6 h, (b) after 12 h, and (c) after 24 h. Contour interval is 0.05 for (a) and (b), and 0.25 for (c).

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber-1 component after 6 h at a height of 2 km: (a) the radial velocity *u* calculated with the PE model, (b) the radial velocity *u* calculated with the AB model, (c) tangential velocity *υ* calculated with the PE model, and (d) tangential velocity *υ* calculated with the AB model. Contour interval is 0.5 m s^{−1} for (a) and (b) and 1.0 m s^{−1} for (c) and (d). Here, (0 km, 0 km) is the vortex center; this corresponds to (−920 km, −20 km) in *x*–*y* space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber-1 component after 6 h at a height of 2 km: (a) the radial velocity *u* calculated with the PE model, (b) the radial velocity *u* calculated with the AB model, (c) tangential velocity *υ* calculated with the PE model, and (d) tangential velocity *υ* calculated with the AB model. Contour interval is 0.5 m s^{−1} for (a) and (b) and 1.0 m s^{−1} for (c) and (d). Here, (0 km, 0 km) is the vortex center; this corresponds to (−920 km, −20 km) in *x*–*y* space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber-1 component after 6 h at a height of 2 km: (a) the radial velocity *u* calculated with the PE model, (b) the radial velocity *u* calculated with the AB model, (c) tangential velocity *υ* calculated with the PE model, and (d) tangential velocity *υ* calculated with the AB model. Contour interval is 0.5 m s^{−1} for (a) and (b) and 1.0 m s^{−1} for (c) and (d). Here, (0 km, 0 km) is the vortex center; this corresponds to (−920 km, −20 km) in *x*–*y* space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 17 but at a height of 6 km.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 17 but at a height of 6 km.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

As in Fig. 17 but at a height of 6 km.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber-1 component of the radial velocity *u* and tangential velocity *υ* after 12 h calculated with the AB model. At a height of 2 km (a) radial velocity and (b) tangential velocity. At a height of 6 km (c) radial velocity and (d) tangential velocity. Contour interval is 0.5 m s^{−1} for (a) and (c) and 1.0 m s^{−1} for (b) and (d). (0 km, 0 km) is the vortex center; this corresponds to (−820 km, −20 km) in x–y space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber-1 component of the radial velocity *u* and tangential velocity *υ* after 12 h calculated with the AB model. At a height of 2 km (a) radial velocity and (b) tangential velocity. At a height of 6 km (c) radial velocity and (d) tangential velocity. Contour interval is 0.5 m s^{−1} for (a) and (c) and 1.0 m s^{−1} for (b) and (d). (0 km, 0 km) is the vortex center; this corresponds to (−820 km, −20 km) in x–y space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2

Horizontal cross sections of the wavenumber-1 component of the radial velocity *u* and tangential velocity *υ* after 12 h calculated with the AB model. At a height of 2 km (a) radial velocity and (b) tangential velocity. At a height of 6 km (c) radial velocity and (d) tangential velocity. Contour interval is 0.5 m s^{−1} for (a) and (c) and 1.0 m s^{−1} for (b) and (d). (0 km, 0 km) is the vortex center; this corresponds to (−820 km, −20 km) in x–y space.

Citation: Journal of the Atmospheric Sciences 55, 2; 10.1175/1520-0469(1998)055<0259:PVIFTC>2.0.CO;2