3514 JOURNAL OF THE ATMOSPHERIC SCIENCES …mtmontgo/papers/nolanmontgo00.pdf3516 JOURNAL OF THE...

3514 VOLUME 57J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

q 2000 American Meteorological Society

The Algebraic Growth of Wavenumber One Disturbances in Hurricane-like Vortices

DAVID S. NOLAN AND MICHAEL T. MONTGOMERY

Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado

(Manuscript received 22 July 1999, in final form 22 December 1999)

ABSTRACT

An exact solution for the evolution of linearized perturbations of azimuthal wavenumber one on inviscidvortices was previously discovered for nondivergent vorticity dynamics on an f plane. The longtime asymptoticsfor this exact solution have been shown to allow an algebraic instability with unbounded growth even in theabsence of exponentially growing modes. The necessary requirement for this instability is that there exist a localmaximum in the basic-state angular velocity other than at the center of circulation. Hurricanes are naturallyoccurring examples of such vortices, due to the relatively calm eye and intense vorticity in the eyewall region.In this paper, the dynamics of this algebraic instability are studied in the context of the near-core dynamics ofhurricanes.

The longtime asymptotic solution can be written as a sum of three parts: a discrete mode whose amplitudegrows in time as t1/2, an excitation of the neutral pseudomode (vortex displacement) that is constant in time,and residual terms that decay in time as t21/2. A remarkable feature of the solution is that the discrete moderequires the decaying residuals to support its growth; without them it remains constant in amplitude. Theseresiduals are shown to be a collection of sheared vortex-Rossby waves that are trapped in the core of the vortex.The explicit mechanism by which these waves sustain the longtime growth of the instability is investigated. Thedecaying vortex-Rossby waves are found to continuously amplify the growing discrete mode through the pro-duction of perturbation vorticity via interaction with the basic-state vorticity gradient. This is fundamentallydifferent from the classic instability mechanism in barotropic shear flows, often interpreted in terms of discrete,counterpropagating vortex-Rossby waves.

The instability manifests itself as a growing wobble of the low-vorticity core of the vortex, resulting in a nettransport of high vorticity to the vortex center. A fully nonlinear simulation is performed to study the behaviorof the instability as its amplitude becomes large. The steady growth of the instability leads to secondaryinstabilities and vorticity mixing in the vortex core. The implications for hurricanes of the algebraic instabilityand its large-amplitude nonlinear dynamics are discussed.

1. Introduction

Unsteady, asymmetric processes near and within thecores of tropical cyclones is a topic of increasing me-teorological and geophysical interest. The growth of dis-turbances associated with barotropic and baroclinic in-stability of the symmetric hurricane vortex has beenargued as a cause for such phenomena as polygonaleyewalls, the formation of mesocyclones, and possiblyeven supercells in and near hurricane eyewalls (Schubertet al. 1999). The axisymmetrization of potential vorticity(PV) anomalies introduced by episodic convection ap-pears to be a viable mechanism for tropical cyclogenesis(Montgomery and Enagonio 1998) and intensification(Moller and Montgomery 1999). Vortex-Rossby wavesgenerated by these PV anomalies redistribute angular

Corresponding author address: Dr. David S. Nolan, Departmentof Atmospheric Science, Colorado State University, Fort Collins, CO80523.E-mail: [email protected]

momentum and vorticity near and within the core andare believed to play an important role in the dynamicsof near-core spiral bands (Guinn and Schubert 1993;Montgomery and Kallenbach 1997). Perhaps most in-triguing is the possibility that asymmetric processesmight play a role in the sudden intensity changes thatare problematic for forecasters. While the magnitude ofvorticity in inner-core asymmetries may often exceedthe modest values that limited observations have indi-cated (see, e.g., Shapiro and Montgomery 1993; Reasoret al. 2000), the linearized dynamics of such asymme-tries is without question a useful starting point in thestudy of tropical cyclone dynamics.

Working in the context of plasma physics, Smith andRosenbluth (1990) (hereafter SR90) discovered an ex-act, closed-form solution, in term of quadratures, de-scribing the evolution of linearized, azimuthal wave-number one disturbances on inviscid, two-dimensionalvortices. While it is known that all two-dimensional,inviscid vortices on an f plane are stable to exponen-tially growing disturbances of azimuthal wavenumberone (Reznik and Dewar 1994, appendix), SR90 dem-

1 NOVEMBER 2000 3515N O L A N A N D M O N T G O M E R Y

onstrated that the longtime asymptotic behavior of theirexact solution could exhibit linear growth in energy.The necessary and sufficient requirement for seculargrowth in this solution is the existence of at least onelocal maximum in the angular velocity of the basic-stateswirling flow other than at the circulation center. A re-lated asymptotic result for zero total circulation vorticesin unbounded domains was found by Llewellyn Smith(1995).

While the details of the SR90 solution will be de-scribed in the next section, we briefly describe here someof its salient features. First, the instability is inherentlylocal to the core of the vortex, in that the linear growthwill not be excited unless the initial perturbation hasnonzero vorticity inside the vortex core. Second, thelong-term solution may be written as the sum of threeparts: a growing part that is very much like a normal(discrete) mode in that its structure does not change intime and it rotates at a constant angular velocity; a re-sidual part whose amplitude decays with time; and anexcitation of a neutral mode that represents a displace-ment of the entire vortex. A remarkable feature of thesolution is that in the absence of the decaying residuals,the modal part of the solution cannot grow and ratherwill remain constant in amplitude. We will show thatthese decaying residuals are in fact vortex-Rossbywaves that are trapped in the core of the vortex by theangular velocity maximum, and that it is the continuousinteraction with these waves, via the basic-state vorticitygradient, that excites the growth of the modal part ofthe solution.

In section 2 we will review the SR90 solution andshow how some of its essential properties can be de-duced. In section 3 we introduce two basic-state vorticeswith azimuthal velocity profiles similar to those of trop-ical storms and hurricanes. Section 4 will demonstratethe appearance and dynamics of the instability by nu-merically integrating the perturbation vorticity equation.Section 5 elucidates the physical mechanism of the in-stability and the long-term growth in energy. Section 6then employs a fully nonlinear model of the two-di-mensional flow to examine the life cycle of the insta-bility, from linear growth to the formation of secondaryinstabilities and smaller-scale vortices in the eyewallregion of the vortex. Section 7 presents conclusions anddiscusses applications of the current findings to tropicalcyclone dynamics.

2. A closed-form solution for wavenumber onedisturbances

We now summarize the derivation of the SR90 so-lution and discuss its important features. The analysisthat follows applies to the two-dimensional dynamicsof incompressible, inviscid vortices on an f plane. Thestarting point is the equation for the linearized dynamicsof vertical vorticity perturbations of azimuthal wave-number one on a circular basic-state vortex,

]z ]z1 1 iVz 1 u 5 0, (2.1)1 1]t ]r

where z1 is the complex representation of the pertur-bation vorticity that varies in r and t, u1 is the associatedperturbation radial velocity (also a complex function ofr and t), V is the angular velocity of the basic-statevortex, and z is the basic-state vorticity. Equation (2.1)may be written solely in terms of the perturbationstreamfunction c1:

] 1 ] ] 1 i ]z1 iV r 2 c 2 c 5 0, (2.2)1 121 21 2]t r ]r ]r r r ]r

where we have used the invertibility relation

1 ] ] 1z 5 r 2 c , and (2.3)1 121 2r ]r ]r r

1 ]c i1u 5 2 5 2 c , (2.4)1 1r ]u r

where u is the azimuthal angle. Application to (2.2) ofa Laplace transform,

`

2ptc (r) 5 e c (r, t) dt, (2.5)p E 1

0

where the real part of p is assumed sufficiently large toensure convergence, then gives

1 ] ] 1 i ]z(p 1 iV) r 2 c 2 c 5 z (r, 0), (2.6)p p 121 2r ]r ]r r r ]r

where the z1(r, 0) denotes the initial perturbation vor-ticity. SR90 were able to explicitly integrate (2.6) andinvert the Laplace transform, yielding the streamfunc-tion in closed form1:

R

2iV(r)tc (r, t) 5 2r e [1 1 iV(r)t 2 iV(r)t]h(r) dr,1 Er

(2.7)

where R is the radius of the outer boundary of the do-main, and the impact of the initial condition is capturedin the function

r12h(r) 5 r z (r, 0) dr. (2.8)E 13r 0

Using (2.3) provides the perturbation vorticity:`]z

2iV(r)tz (r, t) 5 z (r, 0) 2 it e h(r) dr. (2.9)1 1 E]r r

At long times, the integrals in (2.7) and (2.9) are dom-

1 The result in SR90 has a slight error in that it lacks the negativesign in front of the integral.


inated by the contributions from any stationary pointswhere ]V /]r 5 0, and also by a contribution from theupper limit. For the particular case where there is asingle angular velocity maximum at r 5 rj, SR90 usedthe method of stationary phase to deduce the followinglongtime behavior of the solution:2

1/22p3pi /4 1/2c (r, t) ; e t h(r ) H(r 2 r)1 j j1 0 2|V (r )|j

2iV(r )tj3 r[V(r ) 2 V(r)]ej

h(R)2iV(R)t1 r[V(R) 2 V(r)] e9V (R)

21/21 O(t ), and (2.10)

1/22p ]z3pi /4 1/2 2iV(r )tjz (r, t) ; 2e t h(r ) H(r 2 r) e1 j j1 0 2 ]r|V (r )|j

]z h(R)2iV(R)t 21/21 e 1 O(t ),9]r V (R)

(2.11)

where H is the Heaviside step function, V0(rj) is thesecond derivative of the angular velocity at the locationof the angular velocity maximum, V9(R) is the firstderivative of the basic-state angular velocity at the outerboundary, and the remaining terms in the solution decayas t21/2.

For purposes of discussion, let us definerj1

2a 5 h(r ) 5 r z (r, 0) dr, (2.12)j E 13rj 0

1/22pb 5 , (2.13)1 0 2|V (r )|j

h(R)g 5 , (2.14)9V (R)

and rewrite (2.10) and (2.11) as3pi /4 1/2 2iV(r )tjc (r, t) ; e abt H(r 2 r)r[V(r ) 2 V(r)]e1 j j

2iV(R)t1 gr[V(R) 2 V(r)]e21/21 t f (r, t), (2.15)

]z3pi /4 1/2 2iV(r )tjz (r, t) ; 2e abt H(r 2 r) e1 j ]r

]z2iV(R)t 21/21 g e 1 t g(r, t).

]r(2.16)

2 In SR90, the expression for the longtime asymptotic streamfunc-tion [their Eq. (5)] contains an error that we have corrected. In theirsecond term, h(R) should in fact be divided by V9(R). A similarderivation for the case of a vortex with finite circulation whose an-gular velocity decreases montonically with radius was obtained inappendix A of Montgomery and Kallenbach (1997).

In (2.15) and (2.16) we have made the t21/2 decay ofthe residuals explicit and incorporated the remainingtime-dependent modulations of the streamfunction andvorticity in the functions f (r, t) and g(r, t), respectively.

Let us examine (2.15) and (2.16) to consider someof the properties of the algebraic instability. For longtimes, the streamfunction and vorticity amplitudes in-crease as t1/2, such that the perturbation kinetic energygrows linearly with time. Furthermore, the rate of theenergy growth is proportional to both a2 and b2, whichindicate the impact of the initial conditions and the shapeof the angular velocity maximum, respectively, in de-termining the rate of the long-term energy growth. Notealso that according to (2.12) there must be initial per-turbation vorticity inside the angular velocity maximumin order for growth to occur.

The perturbation vorticity and streamfunction fieldscan be divided into three parts, the first of which is agrowing perturbation similar to a normal mode in thatit has a fixed structure and rotates at a constant angularvelocity. Hereafter, we will refer to this growing part ofthe solution as the ‘‘modal’’ part. For the perturbationvorticity, this constant structure is exactly proportionalto the basic-state radial vorticity gradient up to the lo-cation of the angular velocity maximum. For the per-turbation streamfunction, the modal part is proportionalto the radius times the difference between the maximumangular velocity and the local angular velocity, againup to the location of the angular velocity maximum.

The second term in the asymptotic solution representsa neutral mode that rotates with a constant angular fre-quency equal to the angular velocity at the outer bound-ary. The perturbation vorticity for this disturbance isexactly proportional to the basic-state vorticity gradientof the flow. For the case of an unbounded domain inwhich the angular velocity vanishes at infinity, this neu-tral mode is stationary in time. A steady neutral modeof this type was first found by Michaelke and Timme(1967) (see also Gent and McWilliams 1986), and inthe case with an unbounded domain it simply representsa linear displacement of the vortex. For this reason sucha disturbance is sometimes referred to in the literatureas the ‘‘pseudomode.’’ When the domain is finite, themode interacts with the outer boundary such that it ro-tates at a constant frequency.

Since h(R) ; 1/R3, one may be tempted to think thatthe excitation of the pseudomode, as measured by g,will be zero in an unbounded domain. For vortices witha finite circulation, however, the angular velocity in thefar field has the form V ; G/2pr2, where G is the totalcirculation of the vortex. Thus as R increases the pa-rameter g instead approaches a constant:

`p2lim g 5 2 r z (r, 0) dr. (2.17)E 1GR→` 0

As it turns out, the SR90 longtime asymptotic solutionbreaks down in the case where the vortex has zero total


circulation. This case has been studied by LlewellynSmith (1995),3 where an instability results whose am-plitude grows as t/lnt. Note also that (2.17) indicatesthat initial conditions may be chosen that do not excitethe pseudomode.

At this point, little can be said about the structure ofthe decaying parts of the solution. However, due to aremarkable feature of the solution, we can draw a pre-liminary conclusion that these residual terms are essen-tial to the dynamics of the instability. If a perturbationwhose radial structure exactly matches the modal (grow-ing) part of the long-term solution is used as an initialcondition, no instability will occur! This is because, for

]zz (r, 0) 5 H(r 2 r) , (2.18)1 j ]r

we have

r rj j 21 ]z 1 ] V ]V2 3 2a 5 r dr 5 r 1 3r drE E3 3 21 2r ]r r ]r ]rj j0 0

rj1 ] ]V35 r dr 5 0,E3 1 2r ]r ]rj 0

(2.19)

where the second equality is found from writing thevorticity in terms of the angular velocity, and the lastequality comes from the definition of rj. In fact, this istrue for any perturbation that is exactly proportional tothe basic-state vorticity gradient between r 5 0 and r5 rj since the initial vorticity beyond rj is irrelevant inthe determination of a. Since the modal part of the

solution will not grow by itself, we can only concludethat the residual terms must also play a role in the al-gebraic instability. This may also be inferred from thefact that a solution growing as t1/2 cannot identicallysatisfy the vorticity equation (2.2). The pseudomode partof the solution cannot fulfill this role since it exactlysolves the vorticity equation.

In the following sections, we will use numerical sim-ulations to study the appearance and evolution of thealgebraic instability under various conditions and to de-duce the mechanism for the longtime growth of theperturbations.

3. Basic-state vortex profiles

The key element necessary for long-term algebraicgrowth is the existence of a maximum in the angularvelocity profile other than at the axis. Tropical cyclonesare naturally occuring examples of such vortices, dueto the relatively calm eye at the center of the storm andthe annulus of high vorticity around the eye caused byfrictional convergence and vortex stretching in the eye-wall. For this study we construct two azimuthal velocityprofiles, one that is meant to be representative of a mar-ginal tropical storm, and the other of a mature hurricane.In a manner similar to the method of Schubert et al.(1999), these velocity profiles are created from threeregions of constant, positive vorticity surrounded by anunbounded region of zero vorticity. The three regionsare smoothly connected to each other with cubic Her-mite polynomials, that is,

z 0 # r # r 2 d1 1 1

z S[(r 2 r 1 d )/2d ] 1 z S[(r 1 d 2 r)/2d ] r 2 d # r # r 1 d1 1 1 1 2 1 1 1 1 1 1 1

z r 1 d # r # r 2 d2 1 1 2 2z(r) 5 z S[(r 2 r 1 d )/2d ] 1 z S[(r 1 d 2 r)/2d ] r 2 d # r # r 1 d (3.1)2 2 2 2 3 2 2 2 2 2 2 2

z r 1 d # r # r 2 d3 2 2 3 3

z S[(r 2 r 1 d )/2d ] r 2 d # r # r 1 d3 3 3 3 3 3 3 30 r 1 d # r , `, 3 3

where S(x) 5 1 2 3x2 1 2x3 is the cubic Hermite poly-nomial that satisfies S(0) 5 1, S(1) 5 0, and S9(0) 5

3 The instability in this case is associated with the point at infinity,rather than with an interior extremum of V . The physical mechanismof the instability is thought to be similar to the SR90 instability, buta detailed analysis awaits future work. In the case where both a localangular velocity maximum exists and the vortex has zero circulation,both instabilities should come into play simultaneously. Section 2hconfirmes this prediction with a zero-circulation vortex that appearsto exhibit both instability mechanisms.

S9(1) 5 0. Our choices for z i, ri, and di for the twocases are listed in Table 1. These vorticity profiles andtheir associated velocity, angular velocity, and vorticitygradient profiles are shown in Figs. 1 and 2. The tropicalstorm profile has a maximum wind speed Vmax 5 20.4m s21, which occurs at a radius of maximum wind(RMW) of 52 km. The hurricane profile has Vmax 5 45.2m s21 at RMW 5 33 km. The most important distinctionbetween the two profiles is the substantially larger vor-ticity in the eyewall (maximum vorticity) region in thehurricane case, which also results in a much larger peakin angular velocity. Using the stability analyses for vor-


TABLE 1. Parameters for the model azimuthal velocity profiles.

Parameter Tropical storm Hurricane

z1 (s21)z2

z3

r1 (km)r2

r3

d1 (km)d2

d3

7.2 3 1024

9.08 3 1024

7.5 3 1025

10.050.0

140.05.0

15.040.0

7.5 3 1024

3.2 3 1023

1.5 3 1024

8.032.0

120.05.05.0

60.0

tices with stair-step vorticity profiles of Schubert et al.(1999) as a guide, both vortices have been carefullyconstructed to be stable in the inviscid limit to expo-nentially growing disturbances for all azimuthal wave-numbers.

4. Appearance and evolution of the algebraicinstability

a. The perturbation vorticity equation and itsnumerical integration

The linearized f -plane dynamics of asymmetric vor-ticity perturbations of azimuthal wavenumber n on atwo-dimensional vortex may be expressed with the sin-gle equation

2 2]z ]z ] z 1 ]z nn n n1 inVz 1 u 5 n 1 2 z , (4.1)n n n2 21 2]t ]r ]r r ]r r

where zn(r, t) and un(r, t) are complex-valued functionsof r and t whose real parts represent the perturbationvorticity and perturbation radial velocity, respectively.The right-hand side represents the effects of dissipationwhen the kinematic viscosity v is not zero. In this studywe restrict our analyses to cases with n 5 1 only, al-though our method generalizes to all azimuthal wave-numbers. The third term on the left-hand side of (4.1)requires evaluation of the perturbation radial velocityfrom the perturbation vorticity; this can be accomplishedby inverting for the streamfunction cn with the use ofa Green function, that is,

21 ]c 2innu 5 5 c , (4.2)n nr ]u r

]cny 5 , (4.3)n ]r

whereR

c (r, t) 5 G (r, r)z (r, t) dr (4.4)n E n n

0

and Gn(r, r) is the Green function for the inversion ofthe Laplacian in cylindrical coordinates for azimuthalwavenumber n, and we assume the domain is restrictedto a circle with radius R. The boundary conditions on

the perturbation velocities are such that the asymmetricflow be finite at r 5 0 and there be no flow through theouter boundary; this requires

cn(0, t) 5 cn(R, t) 5 0. (4.5)

These boundary conditions are satisfied with the fol-lowing Green function (Carr and Williams 1989):

n rn11 2n 2n11(r 2 R r ) 0 # r # r

2n2nRG (r, r) 5n n11r

n 2n 2n (r 2 R r ) r # r # R.2n2nR

(4.6)

The method for numerical integration and analysis of(4.1) is identical to that used by Nolan and Farrell(1999). The radially varying functions are convertedinto discrete vectors representing the data from r 5 Drto r 5 R 2 Dr on grid points evenly spaced Dr apart.Except where noted, all the calculations presented hereuse a domain size R 5 200 km and a grid spacing Dr5 0.25 km. Equation (4.1) is then expressed as a lineardynamical system

]zn 5 T z , (4.7)n n]t

which we then integrate with a Crank–Nicholsonscheme (see, e.g., Press et al. 1992). When the viscosityis nonzero, an additional boundary condition must beincorporated into the vorticity equation; under these cir-cumstances we use

zn(0, t) 5 zn(R, t) 5 0. (4.8)

Regularity of the solution at the axis requires the vor-ticity be zero there; the outer boundary condition ischosen for convenience.

b. Initial conditions and their evolution

We consider the evolution of wavenumber one per-turbations on both the tropical storm and hurricane vor-tices. In both cases, we use as initial conditions a Gauss-ian vorticity anomaly centered in the middle of the eye-wall region, and localized within it:

2r 2 reyez (r, 0) 5 A exp 2 , (4.9)1 5 1 2 6w /4eye

where reye 5 (r1 1 r2)/2 is the center of the eyewallregion, weye 5 r2 2 r1 is the width of the eyewall region,and the initial amplitude of the perturbation is 10% ofthe local basic-state flow vorticity

A 5 0.1z(reye). (4.10)

These perturbations are meant to be representative ofthe effect on the vorticity field of an azimuthally lo-calized burst of convection in the eyewall.

Except where noted (section 4g below), all of the


FIG. 1. Vorticity, vorticity gradient, azimuthal velocity, and angular velocity profiles for the tropical storm case.

following simulations were performed with zero vis-cosity. For long times, the absence of viscosity is prob-lematic: as the vorticity perturbations are sheared by themean flow, their spiral structures will eventually becomeunderresolved in the radial direction—this is essentiallyan aliasing problem. This occurrence can be identifiedby spurious oscillations in the perturbation energy. Fur-thermore, the time until the resolution becomes inade-quate can be estimated (Smith and Montgomery 1995,appendix):

pt ; , (4.11)

]Vmax nDr5) )6]r

which is 32 h for our hurricane simulations and 406 hfor our tropical cyclone simulations. While some of thehurricane results presented here did extend beyond thistime limit, the energy oscillations indicative of under-resolution were not observed in those cases. The timestep for all simulations was 100 s.

The initial conditions and early evolution of the per-turbation in the tropical storm case are shown in Figs.3 and 4. At t 5 2 h, we observe an important featureof the evolution. While the initial perturbation has beenadvected around the vortex by the basic-state flow, theinteraction of the perturbation velocities with the basic-state vorticity gradient has created two new vorticityanomalies, each associated with the peaks in the basic-state vorticity gradient at r 5 10 km and r 5 50 km(see Fig. 1). At later times the perturbation vorticitybecomes dominated by the anomalies at these radii. Thisis due to the rise of the modal part of the asymptoticsolution, which is exactly proportional to the vorticitygradient up to the location of the angular velocity max-imum rj 5 37.5 km.

Figure 5 shows the perturbation kinetic energy as afunction of time for 2 days of integration in the tropicalstorm case. In the early stages the perturbation kineticenergy oscillates about its initial value, but at approx-imately 15 h it begins to increase steadily, modulatedby an oscillation with a period of approximately 4.5 h.


FIG. 2. Vorticity, vorticity gradient, azimuthal velocity, and angular velocity profiles for the hurricane case.

This oscillation period is close to the vortex rotationtime (defined at the RWM) of 3.9 h, and it is causedby the periodic phase coherence of the modal (growing)part of the solution and the pseudomode. After 1 daythe mean energy growth becomes approximately linearin time.

Figures 6 and 7 show the evolution of the wave-number one perturbation in the hurricane case. The evo-lution is essentially similar, but one can see that themodal part of the solution amplifies much faster in thiscase and is already dominating the perturbation vorticityfield by t 5 2 h. At t 5 12 h and t 5 24 h the solutionis clearly dominated by the modal part of the asymptoticsolution and appears to have achieved a nearly steadystructure that rotates about the vortex center and in-creases in amplitude. The perturbation kinetic energy,shown in Fig. 8, goes through a period of rapid transientgrowth, and then settles into a linear growth pattern.The linear growth rate appears to increase slightlyaround t 5 1 day; however, this is due to a decreasedgrowth rate from 12 h # t # 24 h caused by a rebound

from the large transient growth at earlier times. Theenergy is again modulated by the interaction betweenthe growing mode and the pseudomode, whose periodof 1.2 h is nearly equal to the vortex circulation timeof 1.18 h.

In both cases one might be concerned about the im-pact of the solid-wall boundary condition at r 5 200km. Simulations with larger domains produced nearlyidentical results. As an additional check, it is possibleto integrate (4.7) as if there were no boundaries at allwhile still using a finite numerical domain. This is de-scribed in the appendix, where it is shown that the rateof longtime energy growth is nearly identical in an in-finite domain.

Except where noted, for the remainder of the paperwe will use the results from the hurricane vortex forour analysis.

c. Evolution of special initial conditionsThe analysis in section 2 discussed how the solution

depends on the initial conditions. In particular, we made


FIG. 3. Evolution of the wavenumber one perturbation in the tropical storm case: (a) initial perturbation vorticity, (b) initialperturbation streamfunction, (c) vorticity at t 5 2 h, (d) streamfunction at t 5 2 h. Distances are in km, and contour intervalsare indicated at the top of each plot.

the following three claims: 1) if there is no initial per-turbation vorticity inside the angular velocity maximum,there will be no long-term growth; 2) if the initial per-turbation vorticity is exactly equal to the modal part ofthe solution, there will be no long-term growth; and 3)one can choose the initial conditions so as not to excitethe pseudomode.

We verify these three claims via numerical simulation.Figure 9 shows the kinetic energy as a function of timefor initial conditions corresponding to the three cases.The dashed line shows the energy for an initial conditionthat is identical to that used above, except that the initialvorticity is centered at 80 km instead of at reye. This

initial condition excites a perturbation vorticity anomalyin the vortex core, and as the initial and newly inducedanomalies are advected in and out of phase with eachother, their mutual interaction results in a periodic riseand fall of the perturbation kinetic energy. However, asthe perturbations are sheared apart, the transient growthand decay fades and we are left with a nearly constantperturbation energy. The dash–dot line shows the kineticenergy for a perturbation that is exactly proportional tothe modal part of the SR90 solution. As expected, itshows no growth and in fact is nearly constant in time(small oscillations in the energy, which are due to dis-cretization errors, are not visible in the figure). The solid


FIG. 4. As in the previous figure: (a) vorticity at t 5 12 h, (b) streamfunction at t 5 12 h, (c) vorticity at t 5 24 h, (d)streamfunction at t 5 24 h.

line shows the energy for an initial condition that hasbeen specially constructed so as not to excite the pseu-domode. This was done by using an initial conditionthat is the sum of the original initial condition (4.9),plus an identical Guassian perturbation centered at r 550 km, but opposite in sign, with its amplitude chosenso that its contribution to h(R) exactly cancels the con-tribution from the inner perturbation, such that h(R) 50. The early, small-amplitude oscillations in the energyare caused by the interactions of the inner and outervorticity anomalies, but after 12 h this interaction decaysas the outer anomaly is progressively sheared to smallerradial scales. By construction, a is the same for this

case as in the original initial condition, so that we obtainthe same linear growth rate for long times as before.

d. Comparison with the Smith and Rosenbluthsolution

In this section we will show to what extent the evo-lution of the perturbations for short and intermediatetimes matches the asymptotic solution. As previouslyobserved, the solution is dominated by the modal partafter a relatively short period of time. To demonstratethis, we show in Fig. 10 a plot for t 5 12 h of theabsolute value of the complex vorticity as a function of


FIG. 5. Wavenumber one perturbation kinetic energy vs time in thetropical storm case.

r for the numerical solution, and for the modal part ofthe asymptotic solution whose maximum amplitude hasbeen matched to that of the numerical solution. In theregion with positive vorticity gradient, the structure ofthe vorticity and the modal solution match almost per-fectly. The outer peak associated with the negative vor-ticity gradient is also present in the numerical solution,although it is not nearly as sharp; as time evolves, itdoes become higher and sharper and slowly becomesmore and more like the modal part of the asymptoticsolution.

The spatial variations beyond the outer peak suggesta possible wavelike structure to the residual part of thevorticity field. To see this more clearly, we show in Fig.11 contours of the modal part and residual parts of thevorticity in the plane. The residuals appear as a collec-tion of spiral structures, which are sheared locally bythe radial variation of the angular velocity of the basic-state flow, both inside and outside the outer peak shownin Fig. 10, which coincides with the angular velocitymaximum. Outside this radius, the vorticity perturba-tions are sheared over so as to appear as trailing spiralsrelative to the rotation of the vortex. Inside the angularvelocity maximum, the perturbations are sheared in theother direction, so as to appear as leading spirals, al-though they are in fact being symmetrized by the dif-ferential rotation of the basic-state angular velocity.

To what extent do the numerical results match thepredictions of the asymptotic solution? To answer thisquestion we compared the kinetic energy for both thetropical storm and hurricane cases to the kinetic energyassociated with the growing (modal) part of the as-ymptotic solution. This required numerical computationof a [(2.12)] and b [(2.13)] for the basic-state flow andinitial conditions. The former was computed with aniterative trapezoidal integration algorithm and the latterwas computed with finite differences. The results are

shown in Fig. 12, where we show the normalized kineticenergy as a function of time over 4 days of integrationfor both the hurricane case and the tropical storm case,and their asymptotic counterparts. While the agreementin each case is not as close as one might hope, the slopesare certainly comparable. Between t 5 72 and t 5 96h, the asymptotic solution overestimates the energygrowth rate by just 3.6% in the hurricane case and un-derestimates growth rate by 31% in the tropical stormcase. This may be due to the slower development in thetropical storm case, such that the numerical solution maynot have arrived at the purely linear growth regime pre-dicted by the asymptotics.

We can also check to what extent the modal and re-sidual parts of the solution grow and decay, respectively,like t1/2. For this analysis, we used the special initialcondition described in the previous section that has theexcitation of the pseudomode removed, since the per-sistence of the pseudomode prevented us from observ-ing the decay of the residuals. At each time step, wesubtract out from the perturbation vorticity a functionthat has the same structure as the modal part of theasymptotic solution, but with a complex amplitude fac-tor such that it matches the numerical solution exactlyat the location of the vorticity maximum near r 5 8 km(see Fig. 10).4 We treat what is left behind as the residualpart of the solution. We have then plotted in Fig. 13 themaximum amplitude of the streamfunction associatedwith these two parts of the solution as a function oftime on a log–log graph, along with reference curvesfor t1/2 and t21/2. After 72 h, the growth rate of the modalpart matches t1/2 quite well. The residuals go throughconsiderable modulations due to the existence of theexternal vorticity anomaly that was used to elimimatethe pseudomode; as these modulation decay, we do seea long-term decay that is similar to, but not exactlyparallel to, the t21/2 curve. The reasons for this discrep-ancy are not clear, as simulations with higher spatialand temporal resolution, and also with a fourth-orderRunge-Kutta integration scheme, gave nearly identicalresults.5

4 We also tested an alternative method for subtracting out the modalpart that found a complex factor that minimized the total squareddifference between the modal part and the full solution. The resultswere similar.

5 There is also another minor discrepancy between our results andthe SR90 solution. While we have shown that amplitude of the stream-function associated with the residual part of the solution is decaying,the amplitude of the residual vorticity (not shown) is not. In fact, itincreases slowly with time, due to the interaction of the residualstreamfunction field with the basic-state vorticity gradient. This isnot inconsistent with decaying streamfunction because as the residualvorticity perturbations are sheared to smaller radial scales by thebasic-state flow, their associated streamfunction field can decay evenwhile their maximum vorticity is increasing. Examination of SR90suggests their conclusion that the residual vorticity decays as t21/2

was simply an assumption based on the (correct) t21/2 decay rate ofthe streamfunction. This does not appear to be the case.


FIG. 6. Evolution of the wavenumber one perturbation in the hurricane case: (a) initial perturbation vorticity, (b) initialperturbation streamfunction, (c) vorticity at t 5 2 h, (d) streamfunction at t 5 2 h. Distances are in km, and contour intervalsare indicated at the top of each plot.

e. The residuals as vortex-Rossby waves

In the presence of a basic-state vorticity gradient,asymmetric vorticity perturbations propagate both rel-ative to the basic-state flow and also in the radial di-rection. The dynamics of such disturbances, known asvortex-Rossby waves, were elucidated by Montgomeryand Kallenbach (1997) (hereafter MK97). Using a sim-ple WKB approximation, MK97 derived the local dis-persion relation for vortex-Rossby waves:

n ]z /]r0v 5 nV 1 , (4.12)0 2 2 2R (k 1 n /R )0 0

where v is the local wave frequency, and we have usedthe following similar notations as MK97: R0 is the radiallocation of the disturbance (this was R in MK97), V 0

is the local basic-state angular velocity at R0, ]z 0/]r isthe local basic-state vorticity gradient at R0, n is theazimuthal wavenumber, and k is the local wavenumberin the radial direction.

An additional consideration of critical importance inunderstanding vortex-Rossby waves is that, much likesheared disturbances in rectilinear flows, the local radialwavenumber k changes in time according to the relation

k(t) 5 k0 2 nt(]V 0/]r), (4.13)


FIG. 7. As in the previous figure: (a) vorticity at t 5 12 h, (b) streamfunction at t 5 12 h, (c) vorticity at t 5 24 h, (d)streamfunction at t 5 24 h.

where k0 is the initial radial wavenumber and ]V 0/]ris the local radial derivative of the angular velocity.Thus, as the waves propagate, they are also sheared bythe basic-state flow, leading to an increase or decreaseof the radial wavenumber, depending on the initial in-clination of the wave. Without losing generality, we maychoose n to be always positive, and therefore the in-clinations of the waves—whether they are leading ortrailing spirals—are determined by the sign of the radialwavenumber k; positive k indicates trailing spirals andnegative k indicates leading spirals. For long enoughtimes, all waves become symmetrized, leading to a largeabsolute value of the radial wavenumber.

From the dispersion relation (4.12) and the radial

wavenumber equation (4.13), MK97 derived expres-sions for time-dependent phase and group velocities inboth the azimuthal and radial directions. Since the di-rection of energy propagation is determined by thegroup velocity, it is the radial group velocity of theresidual vortex-Rossby waves that has our greatest in-terest. Solving for Cgr 5 ]v/]k, we find

22nk(]z /]r)0C 5 . (4.14)gr 2 2 2R (k 1 n /R )0 0

Note that the radial group velocity depends on both thesign of the local vorticity gradient and the inclinationof the wave. In regions of negative vorticity gradient,the group velocity is outward for trailing spirals and


FIG. 8. Wavenumber one perturbation kinetic energy vs time in thehurricane case.

FIG. 9. Normalized perturbation kinetic energy as a function oftime in the hurricane case for three special initial conditions: novorticity in the vortex core (dashed), exactly proportional to the grow-ing part of the instability (dash–dot), and with the excitation of thepseudomode removed (solid).

FIG. 10. A comparison of the complex magnitudes of the vorticityat t 5 12 h (solid line), and a phase-and-amplitude-matched modalpart of the asymptotic solution (dashed with x’s).

inward for leading spirals. The opposite case holds truein regions of positive vorticity gradient.

The radial motions of wave packets can be identifiedwith the use of a Hovmoller diagram, as in Fig. 14,which shows contours of the complex magnitude of theresidual vorticity as a function of radius and time, wherewe have again used the initial conditions that do notexcite the pseudomode. The contour lines indicatingequal amplitude can be seen to propagate otuward be-yond r 5 28.5 km, and inward inside of r 5 28.5 km,which is the location of the angular velocity maximum.One possible interpretation of this result is that the re-sidual vorticity is simply increasing everywhere aroundr 5 28.5 km due to vorticity production via intearctionwith the basic-state vorticity gradient. However, thisseems unlikely since the vorticity gradient itself is notmaximized at r 5 28.5 km but rather is rapidly de-creasing at that radius (see Fig. 2).

On either side of the angular velocity maximum atrj, vorticity perturbations are sheared by the radial gra-dient of the angular velocity ]V /]r, which is negativebeyond rj and positive inside rj. As described above,this results in leading spirals inside rj and trailing spiralsoutside rj, which are associated with negative and pos-itive radial wavenumbers, respectively. Since the basic-state vorticity gradient is negative in the immediate vi-cinity of the angular velocity maximum, the radial groupvelocity as predicted by (4.14) is indeed outward beyondrj and inward inside rj. One can show that a local max-imum in V always coincides with a negative basic-stateradial vorticity gradient. Therefore, our analysis revealsone physical significance of the angular velocity max-imum: inside this point, the energy of the residuals istrapped; outside, the residual waves propagate away andtheir energy is deposited outside the vortex core.

f. Motion of the vortex center

In section 2, we dicusssed how a wavenumber oneperturbation that is proportional to the basic-state vor-ticity gradient represents a linear displacement of thevortex, often called the pseudomode. As we have dis-cussed in detail above, the modal (growing) part of theasymptotic solution is exactly of this form, up to thelocation of the angular velocity maximum. Thus it ap-pears that the growing part of the asymptotic solutionrepresents a linear displacement of the core of the vor-tex, relative to the surrounding flow. However, unlikethe pseudomode, this displacement is growing in time


FIG. 11. The vorticity in the hurricane case at t 5 12 h, separated into (a) the modal part, (b) the residuals.

FIG. 13. Log–log plot of the maximum amplitudes of the stream-functions of the modal part of the solution (solid) and the decayingresiduals (dashed) using the special initial conditions with no exci-tation of the pseudomode. Also shown are reference curves for t1/2

(solid) and t21/2 (dashed).

FIG. 12. A comparsion of the perturbation kinetic energy for thehurricane and tropical storm cases with the kinetic energy of thegrowing part of the asymptotic slution. Solid: hurricane, asymptoticsolution; dashed: hurricane, numerical solution; dotted: tropicalstorm, asymptotic solution; dash–dot: tropical storm, numerical so-lution.

and also rotating with a constant angular frequencyequal to the maximum angular velocity of the basic-state flow. If we define the apparent center of the vortexas the location of the minimum of the total stream-function field, we can observe its motion as the solutionevolves.

The location of the minimum streamfunction may befound rather easily due to the simple structure of theperturbation streamfunction field in the vicinity of thecenter axis (see, e.g., Figs. 7b,d). The azimuthal angleat which the minimum lies must coincide with the angle

of the minimum perturbation streamfunction; the totalstreamfunction as a function of radius is therefore com-puted along the ray emanating from the origin, whichcoincides with this angle. For increased sensitivity tothe radial location of the minimum, this radial stream-function profile is first interpolated with cubic splinesonto a local refined grid with 20 times as many gridpoints, and then the minimum is found. The motion ofthe vortex center for the first 8 h of the hurricane caseis shown in Fig. 15. The vortex center spirals cyclon-ically outward as the algebraic instability grows.


FIG. 14. Hovmoller diagram for the amplitude of the complex vor-ticity associated with the residual vortex-Rossby waves in the hur-ricane case, in the ranges h and km, showing the inward and outwardpropagation on either side of of r 5 28.5 km.

FIG. 15. Location of the vortex center in the hurricane case, markedevery 400 s, for the first 8 h. Distances are in km, and the beginningand end points are indicated.

FIG. 16. Perturbation kinetic energy as a function of time in thehurricane case, with and without disspiation: y 5 0 (solid), y 5 10m2 s21 (dashed), y 5 100 m2 s21 (dash–dot).

g. The effects of viscosity

Since the asymptotic solution is only valid for invis-cid flows, it is unknown a priori whether or not thealgebraic instability mechanism occurs in the presenceof dissipation. We performed two additional simulationsfor the hurricane case with ordinary ¹2 diffusion of theperturbation vorticity, one with n 5 10 m2 s21 and an-other with n 5 100 m2 s21. The vortex Reynolds num-bers for these two cases, defined here as RMW 3 Vmax/n ,are 1.49 3 105 and 1.49 3 104, respectively. The per-turbation energies as a function of time for these twocases, along with the energy in the inviscid case, areshown in Fig. 16. In all three cases, the short-term tran-sient growth is nearly the same for the first 12 h. Smallamounts of viscosity have only a minimal effect ontransient growth of disturbances for short times (Nolanand Farrell 1999). However, the results demonstrate thatsmall amounts of viscosity inhibit the longtime lineargrowth of the algebraic instability.

Examination of the vorticity fields at later times inthe cases with viscosity (not shown) indicates that whilethe modal part of the solution was fairly robust to thedissipation, the residual vortex-Rossby waves were sub-stantially dissipated due to their finescale structure inthe radial direction. Thus the primary effect of the dis-sipation is to damp out the decaying residuals that wehave determined to be necessary for sustained lineargrowth.

h. Results for a zero-circulation vortex

The azimuthal wind fields of hurricanes and othergeophysical vortices do not in fact extend out to infinitybut rather go to zero at some finite distance from theircenter. As discussed in section 2, the SR90 solution is

not valid for zero-circulation vortices. Nonetheless, itis difficult to see how the inner-core dynamics shouldbe changed by the structure of the mean azimuthal ve-locity in the far field. To address this issue, we per-formed an additional simulation with our hurricane vor-tex profile, modified to have zero total circulation.

The hurricane vortex was modified in the followingmanner. The total circulation G of the original hurricaneprofile was calculated. Then, we found a constant vor-ticity z* that would exactly cancel the total circulationif applied from the center out to a radius r0 5 275 km,


FIG. 17. Zero-circulation hurricane vortex and results: (a) meanazimuthal velocity profile; (b) perturbation kinetic energy vs time,for cases with (solid) and without (dashed) excitation of the pseu-domode.

that is, z* 5 G/( ). A new azimuthal velocity field2pr 0

was then computed from the modified vorticity field:

z(r) 2 z* 0 # r , r0*z (r) 5 (4.15)50 r # r , `,0

where z is the vorticity field of (3.1) for the hurricanecase. A new velocity field was computed from the mod-ified vorticity field z*, which is shown in Fig. 17a.Observe how y(r) goes to zero at r0 5 275 km.

We performed a numerical simulation of the wave-number one perturbation with the initial condition for-mulated in the same manner as in the original hurricanecase, and an additional simulation with the excitationof the pseudomode eliminated as described in section4c above. The perturbation kinetic energy for each ofthese two cases is shown as a function of time in Fig.17b. The results are very similar to the finite circulationcases, except that in the case where the pseudomode isexcited, the amplitude of the interaction with the pseu-domode is increasing in time. The perturbation vorticity

and streamfunction fields (not shown) were nearly iden-tical to those of the finite-circulation simulations.

The Llewellyn Smith (1995) analysis shows that fora monopolar vortex (no angular velocity maxima otherthan at the center axis) with zero circulation an insta-bility will occur whose amplitude grows at t/lnt andwhose structure is exactly that of the pseudomode. De-spite the fact that the Llewellyn Smith analysis does notcover the case with interior angular velocity maxima,our results suggests that an instability of this type isindeed present, because the amplitude of the pseudo-mode (as measured by the modulation of the pertur-bation energy) appears to be increasing in time. Fur-thermore, the results show that the SR90 instability isalso present in the case with zero circulation. We arriveat the interesting conclusion that each of these two in-stabilities exists in regimes where the analytic solutionsthat predicted them are not formally valid.

5. Secular-growth mechanism

In this section we explain the physical mechanism bywhich the algebraic instability grows. Before proceed-ing, we should discuss some previously offered expla-nations of the growth mechanism. SR90 already notedthat if the modal part of the solution is used as an initialcondition, longtime growth does not occur. They sug-gested that the instability must occur due to ‘‘phasecoherence,’’ rather than through the growth of an un-stable mode. Recently, Schecter (1999) has further ex-panded on this concept of phase coherence. By decom-posing a particular initial condition into a complete setof discrete and continuum modes, he showed that oneof these modes is singularly excited at the initial state.However, the energy associated with this singular ex-citation is masked through destructive interference withthe rest of the modes. As the perturbation begins toevolve, the phase dispersion among the modes allowsthe singularly excited mode to be revealed over time.

Similar explanations have been used previously toexplain the phenomenon of transient growth in linearsystems with nonnormal dynamical operators and havebeen presented for transient growth in the midlatitudejet (Farrell and Ioannou 1996) and in two-dimensionalvortices (Nolan and Farrell 1999). For transient growth,however, the constructive/destructive interference ar-gument can be complemented with more physical ar-guments, based on the rearrangement of perturbationvorticity by the shear of the basic-state flow, and theinteraction of the perturbations with the basic-state flowvia eddy momentum fluxes. These types of physicalarguments have not been made for the SR90 instability.The above-mentioned argument, while mathematicallycomplete, does not explain how the modal part of thesolution acquires energy from the basic-state flow.

Under linear dynamics the perturbations can only ac-quire energy by conversion from the basic-state flow.For asymmetric disturbances in a two-dimensional vor-


tex, the rate of perturbation energy growth can be com-puted from

`]E9 ]V25 22p u9y9r dr, (5.1)E]t ]r0

where the overbar indicates an azimuthal mean. Equa-tion (5.1) may be interpreted as the total downshearmomentum transport associated with the perturbations.As discussed in section 2, the modal part of the as-ymptotic solution cannot grow on its own; this conclu-sion can be confirmed from (5.1) because the modal partof the solution has u9y9 5 0 everywhere due to its lackof tilt relative to the basic-state flow.

Since, by itself, the modal part of the solution has

u9y9 5 0 for all r, then for long times the change inthe perturbation energy must be associated with eitherthe residual vortex-Rossby waves themselves or withtheir interaction with the modal part of the solution. Thefollowing heuristic argument demonstrates this point.Let us separate the perturbation velocity fields into theparts associated with the modal and residual parts of thesolution, that is,

1/2 21/2u9 5 t u9 1 t u9 ,mod resid

1/2 21/2y9 5 t y9 1 t y9 , (5.2)mod resid

where the time-dependent behavior is suggested by theasymptotic solution (2.15) and (2.16). If we substitute(5.2) into (5.1), we obtain

`]E9 ]V21 25 22p [tu9 y9 1 u9 y9 1 y9 u9 1 t u9 y9 ]r dr. (5.3)E mod mod mod resid mod resid resid resid]t ]r0

Using the fact that u9mody9mod 5 0 and neglecting theorder t21 contribution of the residual crossterms, we seethat the energy growth indeed comes from an interactionbetween the modal and residual parts of the solution,and that this growth rate is O(1) in time.

For a more precise understanding of the long-termgrowth mechanism, we turn next to an analysis of thevorticity production terms associated with the mode andthe residuals. Since the excitation of the pseudomodeobscures our ability to separate the solution into modaland decaying parts, we return again to the special initialcondition that has zero excitation of the pseudomode,as described in section 4c. Perturbation vorticity is cre-ated via conversion from basic-state flow vorticity bythe 2u9(]z /]r) term in the perturbation vorticity equa-tion (4.1). Again, it is useful to separate the vorticityproduction field into parts caused by the modal andresidual parts of the solution,

u9(]z /]r) 5 u9mod(]z /]r) 1 u9res(]z /]r). (5.4)

Figure 18 shows the vorticity production field associatedwith the modal part of the solution, at t 5 12 h and t5 24 h, superimposed on the vorticity field of the modalpart. The vorticity production is exactly 908 out of phasewith the vorticity itself. Thus, the vorticity productionassociated with the modal part causes azimuthal prop-agation of the modal vorticity, but neither growth nordecay. This is consistent with the fact that the modalpart of the solution has no tilt and therefore has u9y95 0.

Figure 19 shows, superimposed on the total vorticityfield, the vorticity production associated with only thedecaying residuals, at t 5 24 h and for three subsequenttimes every 800 s. The vorticity production has a strongsignal near the center axis, due to the large positive

gradient in the basic-state vorticity in that region. Ineach of the frames the vorticity production from theresiduals is strongly correlated with the vorticity of themodal part. The vorticity production from the residualsremains phase-locked with the vorticity of the growingmodal part such that a strong positive correlation per-sists, leading to steady growth in the amplitude of themode. While the amplitude of this forcing decays withthe streamfunction residuals as t21/2, their time-inte-grated effect results in t1/2 growth.

The slow decay rate of the vortex-Rossby wave re-siduals is critical in achieving longtime growth. Sincethe modal part of the solution is being forced by theresiduals, the residuals must decay in time as or moreslowly than t21 in order to realize secular growth. Pre-vious studies of the axisymmetrization of asymmetricperturbations in monopolar vortices have found consid-erably faster decay rates (Carr and Williams 1989; Smithand Montgomery 1995; Nolan 1996). In fact, as thewaves in monopolar vortices are progressively shearedto smaller and smaller scales, their dynamics becomemuch like those of disturbances in rectilinear shearflows, whose associated streamfunction amplitudes de-cay like t22. It is the presence of a local maximum inthe angular velocity that prevents such rapid decay. Atthe local maximum, the local shearing rate r(]V /]r)vanishes, allowing for substantially slower growth ofthe residual streamfunction.

This effect is remarkably similar to that found byBrunet and Warn (1990) and Brunet and Haynes (1995)in their studies of sheared vorticity perturbations in thevicinity of a local maximum in the velocity profile ofa zonal jet. In the absence of a background vorticitygradient, the streamfunction field of the perturbationswill decay only as t21/2. Since the local vorticity gra-


FIG. 18. Modal perturbation vorticity fields (contoured) and the vorticity production associated purely with the modal part (shaded), at (a)t 5 12 h, and (b) t 5 24 h. Distances are in km.

dients increase linearly with time, nonlinear effects be-come important at long times, leading to the formationof a nonlinear critical layer at the jet maximum. Thesenonlinear effects lead to secondary vortices in the vi-cinity of the critical layer. In the SR90 asymptotic so-lution the streamfunction field of the residuals and theirassociated perturbation velocities decay as t21/2, and itis the excitation of a neutral mode by these shearedperturbations that leads to the algebraic instability.Whether or not the slow decay of these disturbancesalso leads to a nonlinear critical layer in the vicinity ofthe angular velocity maximum has yet to be determined.

Armed with an understanding of the longtime growthmechanism, we can now understand the behavior of thefirst two of the three special initial conditions discussedin section 4c. First, we understand why initializing theperturbation vorticity with only the modal part does notresult in growth. By itself, the modal part is a purelyneutral structure which rotates at a frequency equal tothe angular velocity maximum. Like the pseudomode,the modal part of the asymptotic solution can be shownto exactly solve the wavenumber one perturbation vor-ticity equation. The mechanism by which this neutralmode rotates at the maximum angular velocity can bedescribed as follows. The neutral structure consists oftwo parts, an inner vorticity perturbation associated withthe region of positive vorticity gradient, and an outerperturbation associated with the region of negative vor-ticity gradient, but only up to the angular velocity max-imum (see Fig. 10). The inner perturbation rotates fasterthan the local basic-state angular velocity because itresides in a region of positive vorticity gradient [cf.MK97, Eq. (16)]. Similarly, in the absence of the innerperturbation, the outer perturbation would retrograderelative to the local angular velocity; however, it actually

propagates faster than the local basic-state flow due tothe production of perturbation vorticity induced by theinner perturbation. These disturbances modulate eachother’s phase speeds so that they rotate together at ex-actly the maximum angular velocity of the basic-statevortex. Furthermore, the far-field streamfunction con-tribution of the inner anomaly is exactly canceled bythe far-field streamfunction contribution of the outeranomaly, such that the neutral mode is localized withinthe angular velocity maximum in both streamfunctionand vorticity.

Second, we understand why initial perturbationsstrictly beyond the angular velocity maximum also donot excite the algebraic instability. Consider some ar-bitrary, wavenumber one vorticity perturbation whoseamplitude is zero inside some radius ri. As time evolves,such a disturbance will generate new perturbation vor-ticity inside ri via interaction with the basic-state flowvorticity gradient. As a simple example, let us consideran initial condition that is a d function in the wave-number one perturbation vorticity in the radial direction,localized at some radius ri, that is, z1(r, 0) 5 d(r 2 ri).The initial streamfunction field can be found from (4.4)–(4.6). For this special case, we find that for r , ri, thestreamfunction has the form c1(r, 0) 5 Cr, where C isa constant. By (4.2), the perturbation radial velocityu1(r, 0) 5 2iC, that is, is constant for r , ri. Therefore,the new vorticity 2u1(]z /]r) produced by such ananomaly will be proportional to the local basic-stateflow vorticity gradient. Inside the angular velocity max-imum, this is equivalent to the modal part of the SR90solution. Generalizing this argument to arbitrary vor-ticity perturbations that lie entirely outside the angularvelocity maximum demonstrates why such initial con-ditions do not excite the longtime algebraic growth.


FIG. 19. Total perturbation vorticity fields (contoured) and the vorticity production associated with the residual parts of the solution (shad-ed) at t 5 24 h and every 800 s thereafter: (a) t 5 24 h, (b) t 5 24.22 h, (c) t 5 24.44 h, (d) t 5 24.67 h.

Certainly, when the instability does occur, the fullperturbation vorticity field must interact with the basic-state flow. This interaction is associated with a tendencyin the basic-state azimuthal velocity V, which can becomputed from the eddy momentum flux divergence

2]V 1 ]25 2 (r u9y9) 5 2u9z9. (5.5)

2 2]r r ]r

For the inviscid, hurricane case, Fig. 20 shows the time-integrated effect of (5.5) from t 5 0 to t 5 12 h, whichwould be equal to the total change in V(r) if the line-arized perturbations were allowed to change the basic-state flow. The predicted basic-state flow change hastwo interesting features. First, it has a large positive

signal maximized at r 5 7 km, and the change in thisvicinity has a structure similar to the basic-state vorticitygradient. Second, there is a series of positive and neg-ative changes between r 5 12 and r 5 35 km; theseare due to the sheared vortex-Rossby waves in this re-gion. The positive change close to the center axis in-dicates that the primary effect of the growing wave-number one disturbances is to wipe out the angular ve-locity deficit in the core of the vortex; that is, the eddyfluxes are driving the basic-state flow toward solid-bodyrotation near the center axis. The mixing of cyclonicvorticity into the eye has been suggested to be a criticalprocess in allowing hurricanes to achieve their maxi-mum intensity (Emanuel 1997). In this particular case,


FIG. 20. Change in the mean azimuthal wind at t 5 12 h predictedfrom the eddy momentum flux divergence associated with the linearwavenumber one perturbations.

the large value of the change (15 m s21) by t 5 12 hsuggests that nonlinear dynamics would likely havecome into play by this time. While for sufficiently weak-er initial conditions this would not be the case, we showin the next section how the dynamics change as non-linear effects become important.

6. Nonlinear dynamics and secondary instability

The natural next step in our analysis is to investigatethe dynamics of the Smith and Rosenbluth instabilityin a fully nonlinear model. SR90 in fact did this, usinga particle-in-cell, point-vortex-type model for inviscid,nondivergent flow. They showed that when the ampli-tude of the wavenumber one instability becomes largeit results in the appearance of secondary instabilities inthe vortex core, due to a tightening of the vortex gradientbetween the low-vorticity core and the surrounding,higher vorticity. Here, we demonstrate similar resultsfor our hurricane-like vortex.

For fully nonlinear simulations, we used a semispec-tral model, which uses finite differences in the radialdirection and a spectral representation in the azimuthaldirection. The total flow is represented as the sum ofthe contributions from wavenumber 0 up to some trun-cation N, that is,

N

inuC(r, t, u) 5 c (r, t)e . (6.1)O nn50

Further specifics about the model can be found in ap-pendix B of Montgomery and Enagonio (1998). For ourpurposes, we again use a closed domain with a maxi-mum radius of 200 km, a radial grid spacing of 0.25km, and a spectral truncation of N 5 16. For numericalpurposes we also use a constant eddy diffusivity of 10m2 s21. Models of this type have been shown to be

excellent in reproducing the evolution of fully finitedifference and fully spectral models (with high reso-lution in two directions), for short to intermediate timesprovided that extensive mixing does not occur in thevortex core. Even when extensive mixing does occur,the semispectral model reproduces with reasonable ac-curacy the basic-state vorticity and velocity profiles ofthe end-state vortex (Montgomery et al. 2000).

The nonlinear model was initialized with the samebasic-state flow and initial conditions previously usedin the hurricane case [cf. (4.9)–(4.10)]. The evolutionof the total absolute vorticity field at early times, whilethe dynamics are still essentially linear, is shown in Fig.21. The instability manifests itself as a growing wobbleof the low-vorticity eye. In the vicinity of the low-vor-ticity region the radial vorticity gradients are indeedtightened. At later times, as shown in Fig. 22, thesetightened gradients result in the appearance of small-scale disturbances in the eyewall, likely caused by sec-ondary instabilities occurring on the locally enhancedvorticity gradients. From Fig. 22 one also observes thedevelopment of a relatively anticyclonic disturbance onthe opposite side of the eyewall region from the lowvorticity eye. Inspection of the amplitude of the wave-number one part of the nonlinear solution (not shown)indicates that the linear growth has halted by this point,and in fact the wavenumber one amplitude has begunto decline. Whether this is due to dissipation or to non-linear interactions is not certain.

Further analyses of the nonlinear dynamics of the SRinstability, while beyond the scope of this paper, areworthy of investigation. Some important questions tobe addressed follow. 1) To what extent can the second-ary instabilities account for such phenomena as locallyincreased wind speeds or the appearance of mesocy-clones in and around the hurricane eyewall? 2) What isthe final equilibrated state of a vortex that experiencesthis instability?6 3) Does the wobble reach a maximumamplitude, or does the vortex eventually mix completelyinto a monopole as was seen by Schubert et al. (1999)for exponentially unstable vortices? These issues willbe explored in forthcoming work.

7. Conclusions

The Smith and Rosenbluth exact solution and its cor-responding longtime asymptotics are useful tools for thestudy of wavenumber one asymmetries in two-dimen-sional vortices. The exact solution provides, in closed

6 Note added in proof: While the truly final equilibrated state ofvortices that undergo the algebraic instability remains for investi-gation, additional calculations performed while this article was inpress have shown that the velocity profile at the center of the vortexis always redistributed into one of solid-body rotation, as both theeddy momentum flux divergence in Fig. 20 and the vorticity fieldsin Fig. 22 clearly suggest.


FIG. 21. Evolution of the vorticity field in the early stages of the growth of the algebraic instability in a fullynonlinear model, every 20 min from t 5 120 min to t 5 220 min. Vorticity is multiplied by 103 and the contoursare labeled.

form, the wavenumber one perturbation vorticity for alltimes, in an arbitrary circular vortex, for any initial con-dition. When a local maximum in the angular velocityexists away from the center of rotation, the asymptoticsolution identifies the algebraic instability as the dom-inant structure for long times, and shows precisely howthe excitation of this instability depends on the initial

conditions and on the basic-state velocity profile. In thisreport we have elucidated the specific wave–mean flowdynamics that support the algebraic instability.

The longtime, linear growth in energy of the asymp-totic solution is caused by the continuous excitation ofa modal-type structure by decaying vortex-Rossbywaves. These waves are trapped in the vortex core by


FIG. 22. As in the previous figure, except for later times in the evolution, every 20 min from t 5 960 minto t 5 1060 min.

the local maximum in the angular velocity, which causesthe disturbances to be progressively sheared so as toappear as leading spirals relative to the basic-state flow.Such disturbances can be shown to have inward groupvelocities, so that they do not radiate to the far field ashas been demonstrated for vortex monopoles (MK97).Furthermore, the local angular velocity maximum re-sults in a diminished decay rate for the perturbation

velocity field associated with these waves. Due to thisslower decay rate, the interaction of this velocity fieldwith the basic-state flow vorticity gradient is sufficientto continually amplify the otherwise neutral modal stuc-ture in the vortex core. This mechanism is fundamen-tally different from the classic barotropic instabilitymechanism, which is often explained in terms of phase-locked, counterpropagating Rossby waves.


We found that small amounts of viscosity (comparedto the diffusion used in current numerical models) hada significant damping effect on the algebraic instability.In the hurricane case, viscosities of 10 and 100 m2 s21

halted the growth of the perturbation at t 5 40 h and t5 12 h, respectively. This suggests that numerical mod-els without very high resolution, and therefore with sub-stantial effective dissipation, will never capture the ap-pearance of the SR90 instability. Certainly, these vis-cous effects could be diminished with the use of higher-order dissipation schemes (e.g., ¹4 or ¹6 diffusion, etc.)or more sophisticated turbulence parameterizations.However, since there is actual dissipation in the atmo-sphere, turbulent and otherwise, it is also possible thatalgebraic growth for very long times may be inhibitedin reality. On the other hand, the wavenumber one neu-tral mode (the wobble) would likely be maintained, orpossibly even grow in tropical cyclones (and their sim-ulations), due to continuous excitation from episodicconvection events near the eyewall. These issues remainfor further investigation.

In recent years, minimum enstrophy (Leith 1984;Schubert et al. 1999) and maximum entropy theories(Miller et al. 1992; Whitaker and Turkington 1994;Schubert et al. 1999) have been used to predict vortexend states in two-dimensional, nearly inviscid flow. Aperceived weakness of these theories is that they predicta vortex with an initially nonmonotonic vorticity profilewill always mix into a monotonic profile, even whenthe initial state is stable to exponentially growing dis-turbances for all wavenumbers. Our results raise theinteresting possibility that any vortex with a local max-imum in its angular velocity profile may be, for practicalpurposes, algebraically unstable. If the longtime growthof the instability does ultimately result in a completemixing of the vortex core, the minimum enstrophy/max-imum entropy theories perhaps can be rehabilitated fromthis inconsistency.

Our immediate interests, however, lie in the mechan-ics of the SR90 instability and its implications for thenear-core dynamics of hurricanes. For the hurricane-likevortex profile, the algebraic instability presents itself asa steadily growing wobble of the low-vorticity regionnear the vortex center, which is associated with a netinward transport of high vorticity toward the vortex cen-ter. This is not inconsistent with the wobble of the eyeand the cycloidal track of the storm center frequentlyobserved in hurricanes. Fully nonlinear simulations per-formed with a semispectral model showed how thiswobble affects the vortex as its amplitude became large.As the low-vorticity region spirals outward, the localvorticity gradient is increased between it and the highvorticity in the eyewall region. This leads to the ap-pearance of secondary instabilities and small-scale mix-ing.7 In hurricanes, these secondary instabilities could

7 As explained in footnote 1, the vorticity in the core of the vortexis also redistributed by the instability into a monotonically decreasingvorticity profile such that the center is nearly in solid-body rotation.

be associated with locally increased wind speeds, con-vective outbreaks, or even mesocyclones.

Acknowledgments. The authors would like to thankDr. P. Reasor for his assistance in the development anduse of the nonlinear semispectral model used in section6 of this paper and Dr. L. Shapiro for encouraging usto perform the zero-circulation simulations, which pro-vided some interesting results. This work was supportedin part by the Office of Naval Research N00014-93-1-0456 P0006 and by Colorado State University.

APPENDIX

Linear Solutions in an Unbounded Domain

All the computations discussed in the body of thispaper used a finite domain with a free-slip, solid-wallboundary condition at the outer limit. Naturally, thephysical problem of interest is essentially unbounded,and in consideration of this fact we have used a nu-merical domain that is large enough so that the resultsmimic the dynamics in an unbounded domain. This hasbeen verified by performing additional calculations withincreasingly larger domains and observing that the re-sults do not change significantly as the domain sizeincreases.

Under certain conditions, however, it is possible tosimulate the dynamics of the physically unboundedproblem while using a finite numerical domain. Con-sider the time evolution equation for the perturbationvorticity (4.1), and for simplicity let us also considerthe inviscid case (n 5 0). If the initial conditions aresuch that for some radius r, such that zn(r, 0) 5 0, theonly way the vorticity at that location can ever becomenonzero is through the conversion of mean vorticity toperturbation vorticity as expressed by the term2un(]z /]r). If the mean vorticity gradient is also zeroat this location, the perturbation vorticity will remainzero.

A localized element of perturbation voriticity ‘‘gen-erates’’ a nonlocalized field of perturbation velocitiesthat sweep outward and inward from the vorticity anom-aly according to the structure of the Green function(determined by the azimuthal wavenumber and theboundary conditions). The perturbation velocity fieldgenerates new perturbation vorticity wherever it en-counters a gradient in the mean vorticity. If the meanvorticity gradient ]z /]r 5 0 for all r beyond some radiusr0, there can never be any new vorticity produced be-yond r0. Furthermore, if the perturbation vorticity is alsoinitially zero beyond that point, it will remain zero forall times. Under these circumstances the perturbationvelocity field beyond r0 has no impact on the evolutionof the vorticity inside r0. Therefore, provided these con-ditions are met, we can simulate the dynamics of theunbounded problem by using the Green function for anunbounded domain, as long as the region of relevant


FIG. A1. Comparison of the perturbation kinetic energies vs timefor the hurricane-case simulations with Green function for unbounded(solid curve) and bounded (dashed curve) domains.

FIG. A2. Streamfunction fields at t 5 7200 s (2 hours) in the simulations using (a) the bounded Green function and (b) the unboundedGreen function. The domain in each picture is restricted to 2140 km , (x, y) , 140 km.

vorticity dynamics is contained within the numericaldomain, that is, R . r0. Furthermore, this technique willalso remain valid for finite times for nonzero viscosityprovided that n is small and there is a suitable ‘‘cush-ion’’ between r0 and R.

Figure A1 shows the perturbation kinetic energy asa function of time for the hurricane case using the un-bounded Green function, compared with the same re-sults using the bounded Green function. While the mag-nitudes of the kinetic energies are remarkably consis-tent, the two solutions show slightly different modu-lation frequencies. Recall that these modulations are

caused by the interaction of the growing instability withthe pseudomode: the angular velocity of the pseudo-mode is equal to the angular velocity of the mean flowat the outer boundary, accounting for the difference be-tween the bounded and unbounded cases. The energycalculation is valid in the unbounded case because wecompute the energy from the streamfunction–vorticitycorrelation rather than from the squared velocities. Fig-ure A2 shows a comparison of the streamfunction fieldsfor both cases at t 5 2 h. While the vorticity fields inthe two cases are nearly identical (not shown), thestreamfunctions (and associated perturbation velocities)sweep out a larger area in the unbounded case, as wewould expect.

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