the wavenumber-one instability and trochoidal motion of hurricane

1 NOVEMBER 2001 3243N O L A N E T A L .

q 2001 American Meteorological Society

The Wavenumber-One Instability and Trochoidal Motion of Hurricane-like Vortices

DAVID S. NOLAN AND MICHAEL T. MONTGOMERY

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

LEWIS D. GRASSO

Cooperative Institute for Research in the Atmosphere, Fort Collins, Colorado

(Manuscript received 25 October 2000, in final form 9 May 2001)

ABSTRACT

In a previous paper, the authors discussed the dynamics of an instability that occurs in inviscid, axisymmetric,two-dimensional vortices possessing a low-vorticity core surrounded by a high-vorticity annulus. Hurricanes,with their low-vorticity cores (the eye of the storm), are naturally occurring examples of such vortices. Theinstability is for asymmetric perturbations of azimuthal wavenumber-one about the vortex, and grows in amplitudeas t1/2 for long times, despite the fact that there can be no exponentially growing wavenumber-one instabilitiesin inviscid, two-dimensional vortices. This instability is further studied in three fluid flow models: with high-resolution numerical simulations of two-dimensional flow, for linearized perturbations in an equivalent shallow-water vortex, and in a three-dimensional, baroclinic, hurricane-like vortex simulated with a high-resolutionmesoscale numerical model.

The instability is found to be robust in all of these physical models. Interestingly, the algebraic instabilitybecomes an exponential instability in the shallow-water vortex, though the structures of the algebraic andexponential modes are nearly identical. In the three-dimensional baroclinic vortex, the instability quickly leadsto substantial inner-core vorticity redistribution and mixing. The instability is associated with a displacement ofthe vortex center (as defined by either minimum pressure or streamfunction) that rotates around the vortex core,and thus offers a physical mechanism for the persistent, small-amplitude trochoidal wobble often observed inhurricane tracks. The instability also indicates that inner-core vorticity mixing will always occur in such vortices,even when the more familiar higher-wavenumber barotropic instabilities are not supported.

1. Introduction

In certain experimental apparatuses, the equations ofmotion describing the dynamics of electron plasmasguided by a magnetic field are identical to the equationsof motion for inviscid, incompressible fluid flow in twodimensions. This remarkable equivalence allows ex-perimental physicists to simulate two-dimensional flu-id flow in the laboratory, at Reynolds numbers manyorders of magnitude higher than can be simulated ac-curately with numerical models or with actual fluids.A number of such experiments have provided insightinto (nearly) inviscid two-dimensional vortex dynam-ics (Driscoll 1990; Huang et al. 1995; Driscoll et al.1996; Schecter et al. 1999). In the context of this work,Smith and Rosenbluth (1990; hereafter SR) found anexact solution, in terms of quadratures, describing theevolution of azimuthal wavenumber-one perturbationsto inviscid vortices. Furthermore, the long-time as-ymptotic limit of this solution predicts the appearance

Corresponding author address: Dr. David S. Nolan, GFDL/NOAA,P.O. Box 308, Princeton, NJ 08542.E-mail: [email protected]

of an algebraically growing instability (streamfunctionamplitudes growing like t1/2 for long times) for anyvortex that possesses an angular velocity maximumother than at the center axis.

Moderate to intense hurricanes, with their low-vor-ticity centers (the eye), are naturally occurring examplesof vortices with angular velocity maxima away fromtheir center axis. [While observationally obtained vor-ticity profiles have only recently been available—e.g.,Reasor et al. (2000); or Kossin and Eastin (2001)—anangular velocity maximum can be inferred from anyvelocity profile which is concave up, or ‘‘U-shaped,’’near the center, and many such profiles have been pre-sented, e.g., Shea and Gray (1973, Fig. 3).] Furthermore,the close relationship between wavenumber-one pertur-bations and vortex motion indicates the importance ofwavenumber-one instabilities (Willoughby 1992; Smithand Weber 1993; Reznik and Dewar 1994; Montgomeryet al. 1999). To determine whether or not the SR insta-bility could be significant in tropical cyclone dynamics,Nolan and Montgomery (2000a; hereafter NM) inves-tigated its dynamics in two-dimensional vortices withdimensional scales similar to those of a weak tropical

3244 VOLUME 58J 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

FIG. 1. Basic-state profiles for the Nolan and Montgomery (2000a) stable, hurricane-like vortex: (a) velocity, (b) vorticity, (c) angularvelocity, and (d) vorticity gradient.

storm and a medium strength hurricane. Since NMwished to observe only the algebraic wavenumber-oneinstability, both the tropical storm and hurricane-likevortices were carefully constructed to be stable to ex-ponentially growing, two-dimensional disturbances forall azimuthal wavenumbers. This was achieved in partthrough trial and error, using as a guide the stabilityanalyses of Schubert et al. (1999) for two-dimensionalvortices with stair-step vorticity profiles. The velocity,vorticity, vorticity gradient, and angular velocity pro-files of the NM hurricane-like vortex are shown in Fig.1. This vortex and modifications of it are used through-out the rest of this paper.

Nolan and Montgomery found that the instabilitydoes indeed have significant growth rates in hurricane-like vortices. Since the streamfunction and vorticity per-turbation amplitudes grow as t1/2, the perturbation ki-netic energy grows linearly in time, and NM found thata wavenumber-one perturbation in the hurricane-like

vortex could increase 20 times in energy within 24 h.(As SR showed, the growth rate of the algebraic insta-bility depends both on the shape of the angular velocityprofile and the structure of the initial perturbation, soone cannot make general statements about the growthrate of the instability.) Nolan and Montgomery furtherclarified a number of very interesting properties of theSR algebraic instability.1) The solution can be divided into three parts: (i) a

growing mode whose perturbation vorticity is ex-actly proportional to the basic-state vorticity gradientup to the radius of the maximum angular velocity(referred to hereafter as the RMV), but is zero be-yond that point, and whose contribution to the totalstreamfunction grows as t1/2; (ii) an excitation of aneutral mode, which represents a displacement of theentire vortex (a.k.a the ‘‘pseudo-mode’’); and (iii) acollection of sheared disturbances whose associatedstreamfunction decays as t21/2.


FIG. 2. The path of Hurricane Carla (1961) as reanalyzed by Jar-vinen et al. (1984), showing four trochoidal oscillations just beforelandfall. Also shown (thick line) is the ‘‘best track path,’’ displacednorthward for comparison. The range markers are at 20 n mi intervals.Original figure was adapted from Weatherwise, Oct 1961.

2) The instability can only be excited by initial con-ditions with vorticity inside the RMV. Perturbationswhose vorticity is entirely outside the RMV will notresult in instability.

3) Furthermore, algebraic growth will not result if theinitial perturbation is exactly proportional to thegrowing part of the asymptotic solution. The ‘‘grow-ing part’’ of the solution is in fact, by itself, a neutralmode. This is shown in the appendix of this paper.

4) The algebraic growth occurs due to an interaction,via the basic-state vorticity gradient, between thisneutral mode and the disturbances whose associatedstreamfunction decays as t21/2. These disturbancesare in fact a collection of sheared vortex-Rossbywaves that are trapped in the vicinity of the vortexcore by the angular velocity maximum (hereafter,

max), and their slow decay is due to their diminishedVshearing in the vicinity of max.V

5) The instability represents an oscillating displacementof the low-vorticity core of the vortex relative to thesurrounding flow, that is, a wobble of the eye of thestorm.

It is this last property of the algebraic instability thatmakes it clearly relevant to the known behaviors oftropical cyclones. The tracks of such storms often showa substantial oscillation, or wobble, with respect to sometime-averaged motion vector, referred to as the tro-choidal motion of the storm center. Such behavior hasbeen documented in a number of observational studiesdating back to the early use of radar to determine stormcenters (Jordan and Stowell 1955; Jordan 1966) and wasalso identified in satellite images (Lawrence and May-field 1977). The trochoidal motion of the center hassince been often observed in studies of the inner coresof hurricanes (Willoughby et al. 1984; Muramatsu 1986;Griffin et al. 1992; Roux and Viltard 1995), and is alsoseen in numerical simulations (Jones 1977; Abe 1987;Liu et al. 1999). The amplitude of the trochoidal oscil-lation can be quite small, so as to be lost in the temporalsmoothing process that is used to produce ‘‘best track’’records of tropical cyclones. An example of the tro-choidal oscillation is shown in Fig. 2, which shows themotion of Hurricane Carla (1961) as deduced from radarobservations and also its best track path (taken fromJarvinen et al. 1984).

A number of previous studies have addressed theoscillation of a tropical cyclone’s path about its meanmotion vector. Yeh (1950) and Kuo (1950, 1969) usedclassical hydrodynamics to treat such storms as iso-lated, barotropic vortices interacting with the meanflow and planetary rotation of the surrounding envi-ronment. In addition to the well-known rightward de-flection of cyclonic vortices relative to the environ-mental flow, their solutions predicted trochoidal mo-

tion.1 Flatau and Stevens (1993) predicted that wave-number-one instabilities in a hurricane’s outflow layerwould cause trochoidal motion. Vertical shear in theenvironment has also been linked to storm motion.When the vortex is barotropic, vertical shear leads toa perturbation vorticity structure which can be inter-preted as two like-signed potential vorticity (PV)anomalies, one at the surface and one near the tro-popause, which are slightly displaced from one another.The PV anomalies tend to corotate, leading to tro-choidal motion of the low-level center (Jones 1995;Reasor and Montgomery 2001). When the vortex isbaroclinic, the upper and lower PV anomalies are ofopposite sign, and instead induce propagation ratherthan corotation (Wu and Emanuel 1993; Flatau et al.1994). For a realistic vertical structure, the impact ofvertical shear would likely be a combination of thesetwo effects.

The hurricane track motions discussed in these studies,while consistent with the general meandering motion ob-served in virtually all tropical cyclones, are fundamentallydifferent in a number of ways from the wavenumber-oneinstability and trochoidal motion presented in this paper.For the most part, the trochoidal motions discussed in theseprevious studies are of substantially longer time scales

1 The trochoidal motions predicted by Yeh (1950) and Kuo (1950,1969) are not in fact necessary consequences of the vortex-environ-ment interaction. Rather, they are a function of the initial conditions.Trochoidal motion is predicted only when the initial vortex motionis different from the environmental flow, much like the inertial mo-tions of a particle disturbed on an f plane.


(12–48 h), and substantially larger distance scales (20–200 km) than the trochoidal motion of Hurricane Carlademonstrated in Fig. 2 or predicted in NM. Second, withthe possible exception of the barotropic instability dis-cussed by Flatau and Stevens (1993), these large-scaleoscillations are a side effect of the hurricane’s interactionwith its environment, rather than fundamental to the vortexitself. Last, the wavenumber-one instability discussed byNM represents a displacement of the low-vorticity corerelative to the surrounding eyewall structure, while theprevious papers describe trochoidal motions of the entirehurricane vortex.

Preliminary nonlinear simulations of the instabilityby NM using a semispectral model showed that thecontinued growth of the instability ultimately leads toinner-core vorticity redistribution so as to remove thevorticity deficit from the center of the vortex, and per-haps ultimately to a complete redistribution of the vor-ticity into a nearly axisymmetric profile which decreasesmonotonically from the center. This is yet another wayin which the present wavenumber-one dynamics are dif-ferent from those reviewed above. When eyewall vor-ticity is mixed into the eye, the azimuthal wind profileis modified from that of a ‘‘hollow vortex’’ or ‘‘stagnantcore’’ toward a profile with solid-body rotation at thecenter. From purely kinematic effects, this results in alowering of the central pressure in the vortex. Emanuel(1997) has suggested that such inner-core vorticity mix-ing is necessary for a tropical cyclone to reach its max-imum intensity (MPI), because this lowering of the cen-tral pressure increases the equivalent potential temper-ature (ue) of the air at the surface, essentially resultingin more heat being transferred from the ocean to theatmosphere. Inner-core asymmetries and the mixing as-sociated with them have become more frequently ob-served due to the ever increasing resolution of bothobservational methods (Reasor et al. 2000; Kossin andEastin 2001; Daida and Barnes 2000) and numericalsimulations (Liu et al. 1997, 1999; Braun and Tao 2000;Braun et al. 2000; Fulton et al. 2000). As the wave-number-one instability may play a role in the redistri-bution of vorticity in the eye and eyewall regions andthus also in intensity changes, it is important to developa basic understanding of its dynamics.

In this paper, we present the results of further analysesof the wavenumber-one instability in a number of dif-ferent fluid flow models. In section 2, the nonlineardynamics of the instability in purely two-dimensionalflow is further studied with high-resolution, low-vis-cosity simulations. In section 3, the behavior and dy-namics of wavenumber-one perturbations are examinedin an equivalent vortex in a shallow-water fluid. In sec-tion 4, the wavenumber-one instability is studied usingthe dynamical core of a mesoscale numerical model(RAMS, see section 4a) to simulate the evolution of athree-dimensional, balanced vortex whose wind field ismodeled after a mature hurricane. The cumulative resultof these investigations is that the wavenumber-one in-

stability occurs in all types of vortices with low-vorticitycores, and is likely an important aspect of the inner-core dynamics of tropical cyclones.

2. High-resolution simulations in two dimensions

a. The model

Nolan and Montgomery found that the algebraic wave-number-one instability was very sensitive to the presenceof viscous dissipation. This implies that very high res-olution is necessary to capture the long-term growth ofthe instability in a fully nonlinear model. It would alsobe ideal to simulate the instability in a very large domain,so as to limit or eliminate any boundary effects. For thesereasons, we use a numerical model of incompressibletwo-dimensional fluid flow that has the special featureof adaptive mesh refinement (AMR). AMR provides forthe utilization of an arbitrary number of refined, two-waynested grids. Furthermore, these grids are not necessarilyfixed in time but rather can be moved and generatedspontaneously by the model as determined by some ar-bitrary refinement criterion; however, in all the simula-tions presented here, the refinement boxes are essentiallystationary. The model is based on a velocity-pressureformulation of the equations, is second-order accurate inspace and time, and uses flux limiters to prevent thegeneration of spurious oscillations. Further presentationsand discussions of the model are available in Almgrenet al. (1998) and Nolan et al. (2000).

b. Initial conditions

The model domain is the square region within 2400km # x, y # 400 km. The velocity profile of the originalNM vortex is modified so that its velocity field goes tozero at some finite radius, well within the confines ofthe model domain. This is achieved by multiplying theoriginal velocity profile by a suitable exponentially de-caying factor; that is,

8r*V (r) 5 V(r) exp 2 (2.1)1 2[ ]rc

where for the simulations here we use rc 5 250 km.The resulting modified velocity and vorticity profilesare shown in Fig. 3.2 As an initial perturbation to thebasic-state vortex, we use the same wavenumber-oneperturbation used throughout NM:

2 Following the suggestion of a reviewer, we examined the stabilityof this vortex using the same methodology described in NM. It turnsout that forcing the velocity profile to go to 0 near r 5 300 km causesthe appearance of unstable modes for wavenumbers n 5 2, 3, and 4.These unstable modes are due to interactions between the inner andouter vorticity regions, and appear analogous to the ‘‘type 2’’ insta-bilities discussed by Kossin et al. (2000). However, the growth ratesof these modes are relatively slow, with e-folding times of 12 h ormore.


FIG. 3. Basic-state velocity (a) and vorticity (b) profiles for thetwo-dimensional nonlinear simulations.

FIG. 4. Shaded contour plot of the initial vorticity field for thenonlinear simulations. The locations of the refinement grids are alsoshown, for (a) the entire domain, and (b) a close-up of the inner-corevorticity. Vorticity units: s21. This initial condition was used for allthe simulations.

2r 2 reyez (r, t 5 0) 5 A exp 2 , (2.2)1 1 2[ ]w /4eye

where reye 5 20 km is the center of the eyewall region,weye 5 24 km is the width of the eye-wall region, andthe maximum initial amplitude of the perturbation is10% of the local basic state flow vorticity

A 5 0.1 3 z(r ).eye (2.3)

The simulations presented here use a base grid with128 3 128 grid points, such that the base grid spacingis 6.25 km. The simulation also uses three levels ofrefinement. The first two levels have factors of 2 de-creases in the grid spacing, while the third level has afactor of 4 decrease in grid spacing, such that the gridspacing on the innermost grid is 390.625 m. The initialvorticity field, along with the locations of the refinementboxes at t 5 0, is shown in Fig. 4. The criterion usedto determine the locations of the refined grids is thatthe vorticity difference across any two grid points be

no greater than 1% of the value of the average vorticityin the eyewall region. This requirement may not alwaysbe met within the innermost grid, since the number ofrefinement boxes is limited to three in these simulations.

c. Results

We performed four simulations of the instability,varying only the viscosity for each simulation, with val-ues of n 5 10, 20, 40, and 80 m2 s21. For short times,the evolution of the vortex was nearly identical for allvalues of n, and the early evolution of the vortex withn 5 40 m2 s21 is shown every 20 min from t 5 2 h tot 5 3 h 40 min in Fig. 5. As expected, the initial per-turbation results in a steadily growing wobble of thelow-vorticity core. Associated with this wobble is a sim-


FIG. 5. Vertical vorticity fields in the inner core of the fully nonlinear simulation with y 5 40 m2 s21 at (a) 120, (b)140, (c) 160, (d) 180, (e) 200, and (f ) 220 min. Units of vorticity: s21. Contour levels and shading are as in Fig. 4.

ilar wobble in the location of the center of the vortexas defined by both the minima of the streamfunctionand pressure fields. The x coordinates of these minimaare shown as functions of time in Fig. 6. It is perhapsconfusing to see that the amplitude of the wobble onlygrows for a few hours and then begins to decay withtime. This is not due to a decline of the algebraic in-

stability mechanism, but rather is a feature of the non-linear dynamics of the instability. As time evolves thelow-vorticity core moves outward from the center notsimply as a growing wavenumber-one perturbation, butrather as a coherent ‘‘vortex hole,’’ as is evident fromFig. 5. The vorticity at the vortex center is replaced byhigher vorticity from the eyewall region, and the stream-


FIG. 6. The x coordinate of the location of the minimum stream-function (a) and pressure (b) for the high-resolution nonlinear sim-ulation with y 5 80 m2 s21.

function and pressure minima slowly migrate back to-ward the center as the low vorticity core moves away.

Eventually, the low vorticity core reaches the outeredge of the eyewall, where it is sheared apart by thelarge angular velocity gradients in that part of the vortex.The low vorticity is then quickly redistributed aroundthe outer edge of the eyewall region, while the highvorticity associated with the positive part of the initialwavenumber-one perturbation finds its way to the vortexcenter. Thus, the azimuthally averaged vorticity profileis redistributed into one that is monotonically decreasingfrom the center.

The time required for this redistribution to take placeis rather sensitive to the value of the viscosity. Thevorticity fields at t 5 24 h, 36 h, and 48 h are shownfor each simulation with n 5 80, 40, 20, and 10 m2

s21, respectively, in Figs. 7 and 8. At each time, we can

see that the redistribution process is further along in thesimulations with higher viscosity. It appears that the rateat which the low-vorticity core moves outward throughthe eyewall region decreases with decreasing viscosity.The slow outward movement of the low-vorticity coreis quite similar to the behavior of vortex holes observedby Huang et al. (1995) that were produced by the growthof a wavenumber-two instability in a similar, but ex-ponentially unstable, vortex modeled by an electronplasma. In their experiments, the vortex holes remainedcoherent for hundreds of vortex circulation times, butalso slowly migrated outward from the vortex core untilthey were destroyed by the strong shear at the outeredge of the vortex. A theoretical basis for understandingthe motion of vortex holes in an inviscid fluid has beenprovided by Schecter and Dubin (1999), who showedthat vortex holes tend to move down the vortex gradientof the surrounding flow, while positive vortex anomalies(‘‘clumps’’) tend to move up the gradient. In this case,since the background vorticity is increasing outward,the hole should then move inward, or at the very least,remain trapped in the vortex core. Since the vortices domove outward, and their rate of outward motion is clear-ly dissipation-dependent, it seems that the outward mi-gration is related to dissipative effects. The hole maysurvive indefinitely in a truly inviscid fluid. Perhapsthere exists a steady, nonlinear solution of the Eulerequations that emulates the observed wobble, much asthe well-known Kirchoff’s vortex solution (Lamb 1932,section 159) for an elliptical, piecewise constant vor-ticity patch emulates the long-lived wavenumber-twoasymmetry simulated by Huang et al. (1995), often re-ferred to as the ‘‘tripole.’’

Such behavior is probably irrelevant for the innercores of hurricanes, due to asymmetries in the environ-ment, baroclinic aspects of the vortex, the effects ofconvection, and turbulent diffusion. Nonetheless wehave identified the fully nonlinear behavior of the wave-number one algebraic instability for two-dimensional,hurricane-like vortices. The instability causes the lowvorticity core of the vortex to wobble outwards fromthe center, resulting in a small-amplitude wobble of thevortex center as defined both by streamfunction andpressure minima. Eventually the instability leads to acomplete rearrangement of the inner-core vorticity, suchthat the vorticity profile is monotonically decreasingfrom the center and the core of the vortex is nearly insolid-body rotation. Similar rearrangement of inner-corevorticity has in fact been inferred from observationaldata of hurricanes (Kossin and Eastin 2001).

3. The wavenumber-one instability in a shallow-water fluid

a. The equations of motion and their numericalsolution

The equations of motion in cylindrical (r, l) coor-dinates for the evolution of a shallow-water fluid on anf plane are,


FIG. 7. Snapshots of the vorticity field at 24, 36, and 48 h for nonlinear simulations with y 5 40 and 80 m2 s21, aslabeled. Units of vorticity: s21. Contour levels and shading are as in Fig. 4.

2Du y ]h2 2 fy 1 g 5 0, (3.1)

Dt r ]r

Dy uy g ]h1 1 fu 1 5 0, and (3.2)

Dt r r ]l

Dh 1 ] 1 ]y1 h (ru) 1 5 0, (3.3)[ ]Dt r ]r r ]l

where r is the distance from the center axis, l is theazimuthal angle about the axis, u and y are the radial


FIG. 8. As in Fig. 7, but for y 5 20 and 10 m2 s21.

and azimuthal winds, h is the height of the free surface,g is the gravitational acceleration, f is the Coriolis pa-rameter, and D/Dt 5 ]/]t 1 u(]/]r) 1 (y/r)(]/]l) is thematerial derivative.

We consider the evolution of small perturbationsabout a basic-state vortex in gradient wind balance,

2]h yg 5 f y 1 , (3.4)

]r r

where 5 (r) is the basic-state azimuthal wind field,y yand 5 (r) is the basic-state height of the free surface.h hThe equations of motion (3.1)–(3.3) are linearized about


the basic state (3.4), and the perturbation u9, y9, and h9fields are then written in terms of azimuthally varyingharmonics for each azimuthal wavenumber n:

inl[u9, y9, h9] 5 [u (r, t), y (r, t), h (r, t)]e ,n n n (3.5)

where, rather than assuming exponential time depen-dencies, we have left the time dependence explicit inthe complex functions un, y n, hn. For each azimuthalwavenumber n, the equations of motion become

]u ]hn n1 inVu 2 ( f 1 2V)y 1 g 5 0, (3.6)n n]t ]r

]y ]y inn 1 inVy 1 f 1 V 1 u 1 g h 5 0, (3.7)n n n1 2]t ]r r

and

]h 1 ] in ]hn 1 inVh 1 h (ru ) 1 h y 1 u 5 0, (3.8)n n n n]t r ]r r ]r

where 5 /r is the basic-state angular velocity.V yWe solve for the stability and time evolution of the

system (3.6)–(3.8) numerically in a manner followingMontgomery and Lu (1997) and Flatau and Stevens(1989). A finite domain is prescribed from r 5 0 tosome large outer radius r 5 R. On this domain we definevalues of hn to exist at the points r 5 jDr, where j 50,1, . . . N, and Dr 5 R/N. Here un and y n values existat the ‘‘half’’ points r 5 (k 2 1/2)Dr, with k 5 1, 2,. . ., N. Derivatives computed from centered differencesbetween two adjacent grid points on the h grid fall nat-urally on the uy grid, and similar derivatives on the uygrid fall naturally on the h grid. Where data must beinterpolated from one grid to another, as in the last termsof (3.7) and (3.8), a simple average of adjacent pointsis used. Explicit boundary conditions are needed for thehn only. At the outer wall, we set hn(N) 5 0. At r 5 0the natural boundary condition for the perturbationheight must be hn(0) 5 0, so that singular behavior doesnot occur as one moves around the azimuth as r → 0.[For n 5 0 we would use the condition ]hn/]r 5 0, butno calculations with symmetric perturbations are pre-sented here.] For most of the calculations presentedhere, we use a grid spacing Dr 5 500 m and we placethe outer boundary at r 5 250 km. For all calculationswe use f 5 5.0 3 1025 s21, which corresponds to alatitude of 208N.

Gravity waves are an essential aspect of the dynamicsof the shallow water equations. As we will show below,both balanced and unbalanced initial conditions (themeaning of ‘‘balanced’’ will be discussed shortly) willproduce gravity waves that radiate outward from thevortex core. To prevent substantial gravity-wave reflec-tion from the outer boundary, we use a Rayleigh-damp-ing sponge layer at the outer limits of the domain, whichis implemented by adding the terms 2«(r)un, 2«(r)y n,and 2«(r)hn to the right-hand sides of (3.6)–(3.8), re-spectively. The damping function is defined by,

0 r , rstart r 2 rend«(r) 5 « S r # r # r (3.9)max start end1 2r 2 rend start« r . r , max end

where rstart is the beginning of the transition region intothe sponge layer, rend is the end of the transition region,S(x) 5 1 2 3x2 1 2x3 is the cubic Hermite polynomial,which has S9(0) 5 S9(1) 5 0, S(0) 5 1, S(1) 5 0, and«max 5 1/tmin is the maximum damping rate with timescale tmin. For all simulations presented here, we userstart 5 190 km, rend 5 240 km, and tmin 5 30 s.

Through standard techniques, the system of coupledlinear equations (3.6)–(3.8), represented in terms of fi-nite differences, can be expressed as a linear dynamicalsystem:

dx5 T x, (3.10)ndt

where Tn is the time evolution operator and the columnvector x contains the values of hn on the interior of theh grid and un and y n on the uy grid. This representationserves two purposes. First, the stability properties andthe eigenfunctions of the system can be determined di-rectly from numerical analysis of the matrix Tn; andsecond, numerical integration of (3.10) is straightfor-ward with either explicit (such as Runge–Kutta) or im-plicit (such as Crank–Nicholson) schemes, or even withmatrix exponentiation.

b. Potential vorticity, divergence, and balanced flow

Potential vorticity is a useful concept since it is con-served by parcels in the flow. If, in addition, the flowis in quasigeostrophic balance, the height and velocityfields can be uniquely determined from the PV. Thisresult can be generalized to the case of quasi-gradientbalance as will be discussed shortly. Indeed, whether ornot the inner-core dynamics of intense geophysical vor-tices obey similar balance principles has been a topicof some research (McWilliams 1985; Raymond 1992;Shaprio and Montgomery 1993; Montgomery and Lu1997; Moller and Montgomery 2000), and we will showthat balanced dynamics also apply to asymmetric per-turbations to our shallow-water vortex.

In a shallow water fluid, potential vorticity can bedefined as

z 1 fq 5 , (3.11)

h

where z is the relative vertical vorticity, f is the Cor-iolis parameter, and h is the fluid depth (Pedlosky1987). If we then expand the terms on the rhs of (3.11)into basic state and perturbation parts, and neglectsecond-order terms, the perturbation PV that naturallyappears is


FIG. 9. Growth rate of the most unstable wavenumber-one modeas a function of resting depth H, for calculations with Dr 5 1 km(dash–dot, 1), Dr 5 0.5 km (dashed, x), and Dr 5 0.25 km. For Dr5 1 km, H . 64 km, and Dr 5 0.5 km, H . 256 km, growth rateswere less than 10212, that is, effectively zero.

z9 qh9q9 5 2 , (3.12)

h h

where 5 ( 1 f )/ is the basic-state PV. The vorticityq z hand divergence for asymmetric perturbations are, re-spectively,

1 ] inz 5 (ry ) 2 u , and (3.13)n n nr ]r r

1 ] ind 5 (ru ) 1 y . (3.14)n n nr ]r r

Using (3.6)–(3.8) and (3.11)–(3.14), one can derive PVand divergence forms of the equations of motion:

]q ]qn 1 inVq 1 u 5 0, and (3.15)n n]t ]r

]d in ]V 1 ]n 1 inVd 1 u h 1 r 2 (r fy )n n n1 2]t r ]r r ]r21 g¹ h 5 0, (3.16)n n

where 5 f 1 1/r ]/]r(r ) is the absolute meanh yvorticity, f 5 f 1 2 is the modified Coriolis pa-Vrameter, and the Laplacian operator for wavenumbern is,

2 2] h 1 ]h nn n2¹ h 5 1 2 h . (3.17)n n n2 2]r r ]r r

(3.15) and (3.16), along with (3.8), describe an al-ternative but complete set of equations for the asym-metric perturbations, provided one also solves ellipticequations that find the rotational and divergent partsof the velocity fields from the PV and the divergence,respectively. The system can be further simplified ifwe assume that the divergence and its material timerate of change (DV /Dt 5 ]/]t 1 in ) are small com-Vpared to the other terms in (3.16), consistent with scal-ing analysis and theory of quasi-balanced flow, as dis-cussed by McWilliams (1985), Raymond (1992), andMontgomery and Franklin (1998). Then (3.16) be-comes simply an elliptic equation for the perturbationheight field

1 ] in ]V2g¹ h 5 (r fy ) 2 u h 1 r , (3.18)n n n n1 2r ]r r ]r

where un and y n are nondivergent velocities obtainedfrom the PV relationship

2¹ c 5 z 5 hq 1 qh , and (3.19)n n n n n

in ]cnu 5 2 c , y 5 . (3.20)n n nr ]r

Note that (3.18), (3.19), and (3.20) are in fact a coupledset of linear equations, so that the exact solution for hn

and the nondivergent velocities cannot be computed im-mediately from the PV as in quasigeostrophic theory.

Rather, to compute the balanced state, we first computecn from (3.19) using hn 5 0 as a first guess. The resultingvelocities are put into (3.18) to compute the next guessfor hn, and the process is repeated. In practice we havefound convergence is always achieved in just a fewiterations.

c. The basic-state shallow-water vortex and itsstability

For our shallow-water calculations we use the samebasic-state azimuthal velocity field used throughout NM(shown in Fig. 1). We also compute the basic-stateheight field (r) in gradient wind balance with (r) [cf.h y(3.4)], given the ‘‘resting’’ height H at the outer bound-ary. Changes in H only result in an equivalent upwardor downward displacement of (r).h

We first consider the stability of the vortex by com-puting the eigenvalues and the eigenvectors of the timeevolution operator Tn. We find that, unlike the two-dimensional vortex, the shallow-water vortex does infact support unstable modes; however, the growth rates(real parts of the eigenvalues) of these modes are rel-atively weak and decrease with increasing resting depthH. This is demonstrated in Fig. 9, which shows thegrowth rates of the most unstable mode for n 5 1 forvalues of H 5 1 km, 2 km, 4 km. . . , up to 512 km.Results are presented for calculations with Dr 5 1000m, Dr 5 500 m, and Dr 5 250 m, and are very con-sistent between these three resolutions for the smallerdepths. The growth rates appear to scale with the fluiddepth as H22/3, consistent with theory that will be dis-cussed in section 3f. The e-folding times range from 9.5h for H 5 1 km to 470.7 h for H 5 512 km. For thephysically relevant depth of H 5 8 km (approximately


FIG. 10. Stability curves as a function of azimuthal wavenumbern in the NM–shallow water vortex for H 5 8 km and Dr 5 1 km(dash–dot, 1), Dr 5 0.5 km (dashed, x), and Dr 5 0.25 km (solid,V).

FIG. 11. Structure of the most unstable mode for H 5 8 km, Dr5 500 m, in terms of its (a) perturbation PV, in units of shallow-water PV, which are s21 m21; (b) perturbation height field (m); and(c) perturbation divergence (s21). The mode is normalized so thatmaximum radial velocities are 1 m s21.

one scale height of the atmosphere), the e-folding timeof the fastest growing mode is 38.75 h, which is veryslow compared to the 1.25-h circulation time of thevortex (or an equivalent hurricane).

Figure 10 shows the growth rates of the most un-stable modes as a function of n for calculations withH 5 8 km, for Dr 5 1000 m, Dr 5 500 m, and Dr5 250 m. The extremely close correlation in the growthrates of the most unstable mode for n 5 1 for the threegrid spacings indicates that, for wavenumber-one, thismode is well-resolved and robust. Also indicated areunstable modes for n 5 2 and n 5 3 whose growthrates are highly dependent on the resolution. Theseunstable modes are unphysical by-products of the dif-fusion-free, finite-difference numerical method, in thattheir vorticity fields contain delta-function-like struc-tures on the scale of the grid spacing, and their struc-tures and eigenvalues are highly sensitive to both thegrid spacing and the scheme used to interpolate be-tween the uy and h grids.3

The most unstable mode for n 5 1, H 5 8 km, andDr 5 500 m is shown in terms of qn, hn, and dn in Fig.11. The structure of the mode (which was virtually iden-tical for different grid spacings) is extremely similar tothe growing part of the SR algebraic instability—pro-portional to the basic-state vorticity gradient, up toRMV. However, there appears to be some additionalstructure to the unstable mode in the vicinity of theRMV (the thin, outer PV anomalies), which will bediscussed below. The height field is also very similarto the streamfunction field of the algebraic instability,

3 For example, cubic spline interpolation resulted in different ei-genvalues and eigenvectors for n 5 2 and n 5 3, but virtually identicalresults for n 5 1.

and there is a weak but nonzero divergence field, rough-ly 3 orders of magnitude smaller than the vorticity (whenestimating the vorticity from the PV, recall that z9 ø

q9). Perhaps most importantly, we find that the heighthand velocity fields of the unstable mode are virtuallyidentical to the same fields in balance with its PV. The


FIG. 12. Evolution of perturbation PV (q9) and height (h9) fields for the wavenumber-one, balanced perturbation localized at r 5 20 km:(a) PV, t 5 0 h; (b) height, t 5 0 h; (c) PV, t 5 2 h; and (d) height, t 5 2 h. (PV units, s21 m21 and height units, m.)

most unstable mode, computed from the linearized prim-itive equations, is a quasi-balanced flow.

d. Evolution of the wavenumber-one perturbations

We first consider the evolution of a wavenumber-oneperturbation whose wind and height fields are in balancewith its initial vorticity. The perturbation PV is definedin a manner analogous to the initial perturbation vor-ticity for the simulations in NM and in section 2 above:a Gaussian perturbation localized in the eyewall, withpeak amplitude 10% of the local basic-state flow [i.e.,(2.2)–(2.3) with q substituted for z]. For these simu-lations we used a Crank–Nicholson semi-implicit timestepping scheme with Dt 5 10 s. While the time step

necessary to accurately resolve the dynamics of gravitywaves is just ( )/Dr 5 1.79 s, we will show laterÏgHthat these waves do not play a role in the evolution ofthe asymmetric perturbations after the first 15 min. Fur-thermore, simulations with much shorter time steps gavenearly identical results.

The initial condition and evolution of this pertur-bation is depicted in terms of its PV and height fieldsin Figs. 12 and 13. The evolution is nearly identicalto that of the equivalent 2D vortex as shown in NM.The 12-h and 24-h PV fields are again nearly identicalto the growing part of the SR instability, proportionalto ] /]r up to RMV. The perturbation kinetic energyzversus time is shown in Fig. 14. For the first 24 h, thekinetic energy shows almost exactly the same evolution


FIG. 13. Evolution of perturbation PV (q9) and height (h9) fields for the wavenumber one, balanced perturbation localized at r 5 20 km: (a)PV, t 5 12 h; (b) height, t 5 12 h; (c) PV, t 5 24 h; and (d) height, t 5 24 h. (PV units, s21 m21 and height units, m.)

as found by NM for 2D dynamics; however, after 24h the energy growth appears to change from linear toexponential, indicating the rise of the unstable mode.The structure of the unstable mode is close to, butslightly different than the SR instability, and their dif-ferences will be shown explicitly. The high-frequencyoscillation in the perturbation kinetic energy is causedby an interaction between the rapidly rotating growinginstability and the stationary pseudo-mode (see prop-erty 2 of the SR instability described in section 1); asthese disturbances move in and out of phase with eachother the perturbation kinetic energy increases and de-creases.

Before further analyzing the differences between the

algebraic and exponential instabilities, we first considerthe evolution of perturbations from certain special initialconditions. As outlined in section 1, the SR instabilitywill not be excited if the initial perturbation vorticitylies entirely outside of the RMV; this was also con-firmed by NM with numerical calculations. NM alsoshowed that the initial condition may be adjusted so thatthere is no excitation of the pseudo-mode, thus elimi-nating the high-frequency oscillation in the perturbationkinetic energy. This is possible in 2D flow because theexcitation of the pseudo-mode is proportional to

R

2h(R) 5 r z (r, 0) dr. (3.21)E 1

0


FIG. 14. Perturbation kinetic energy as a function of time for thebalanced perturbation localized at r 5 20 km.

FIG. 15. Perturbation kinetic energies, normalized by initial values,for three special initial conditions: with the pseudo-mode removed(solid), initial PV localized at r 5 80 km (dashed), and the SR neutralmode (dash–dot).

Thus, the excitation of the pseudo-mode may be elim-inated by adding to the original initial condition (2.2)an opposite-signed vorticity anomaly, localized outsideRMV, such that h(R) 5 0 (see NM section 4c for de-tails). Here, we again replace z1 with q1 in our calcu-lations.

Results of simulations of the NM–shallow-water vor-tex with these two special initial conditions are shownin Fig. 15. When the initial perturbation PV lies entirelyoutside of the RMV, no instability is excited, and even-tually the perturbation becomes constant in amplitude.The structure of the final state, in fact, is simply thepseudo-mode. Also, with the initial PV modified so thatthe pseudo-mode is not excited, the high-frequency os-cillation in the kinetic energy is eliminated.

The SR solution shows that if the initial vorticityperturbation is exactly proportional to the basic-statevorticity gradient up to RMV, that is, z1(r, 0) } ] /]rzfor r , RMV, then the instability will not be excitedand long-time growth will not occur. Nolan and Mont-gomery confirmed this, and further demonstrated nu-merically that a wavenumber-one perturbation with

]zz (r, t) 5 C H(RMV 2 r), (3.22)1 ]r

where here H is the Heaviside step function, and C issome complex constant, is in fact a neutral mode thatrotates around the center axis with an angular velocityequal to that of max; that is, z1(r, t) 5 z1(r, 0) .2iV tmaxV eThis can be shown directly, and the proof is given inthe appendix.

Does a similar neutral mode exist in our shallow watervortex? Almost, but not quite. We simulated a pertur-bation with initial PV:

]qq (r, 0) 5 C H(RMV 2 r). (3.23)1 ]r

We then defined the initial perturbation height field withtwo distinct approaches. The first approach uses the ob-servation that if both q1 and h1 are proportional to theirrespective basic-state gradients for all r, then the timeevolution of the perturbations will be zero: this the pseu-do-mode for a shallow-water vortex. In this spirit, wedefine

]hh (r, 0) 5 C H(RMV 2 r) (3.24)1 ]r

to represent a localized displacement of the inner-core,up to RMV. [Note that if (3.23) and (3.24) hold, thenso does (3.22).] Alternatively, one can define the initialhn to be that which holds the initial qn in balance, asdescribed in section 3b. Remarkably, both of thesechoices result in virtually identical evolutions of theperturbations and their kinetic energies (KE), which isalso shown in Fig. 15 (the energies as a function of timeof these two initial conditions are so close that they areindistinguishable in the figure). This is because whenwe use (3.24), there is a discontinuity in the initial heightfield, and the solution immediately radiates a pulse ofinertia-gravity waves that quickly bring the disturbanceinto balance; the resulting change in kinetic energy,however, is negligible. For both cases, the kinetic energyremains roughly constant for the first few hours (duringwhich the perturbation rotates without a noticeablechange in structure), but then begins to increase ex-ponentially (albeit slowly, due to the slow growth rateof the unstable mode). By 48 h the perturbation KE hasincreased by a factor of only 5.44. At this time, thestructure of the disturbance is quite similar to the mostunstable mode. We note that linear growth in energydoes not seem to occur. Thus it appears that using theshallow water equivalent to the SR neutral mode inhibits


FIG. 16. Perturbation kinetic energies as a function of time fromunbalanced initial conditions: (a) evolution out to 48 h, for the per-turbations localized at r 5 20 km (solid) and at r 5 80 km (dashed),and (b) the evolution for the first hour for the perturbation localizedat r 5 20 km. Note the very different scales along the vertical axisfor the two plots, so that the long-time behavior can be analyzed in(a) and the short-time behavior can be analyzed in (b).

the algebraic instability, but nonetheless excites the mostunstable mode.

e. Evolution from unbalanced initial conditions

Just as we can generate initial conditions that arenearly balanced, we can also generate initial conditionsthat are almost completely ‘‘unbalanced.’’ This is doneby first choosing an arbitrary height perturbation, thencomputing an associated vorticity field

qz 5 h , (3.25)n nh

such that qn 5 0 everywhere [cf. (3.12)], and then com-puting the initial velocity fields from (3.19) and (3.20).We performed two simulations, one with a height per-turbation that is a Gaussian-localized at r 5 20 km anda maximum amplitude of 50 m, congruent to (2.2), anda similar perturbation localized at r 5 80 km. For thesesimulations, we used a much shorter time step, Dt 5 2s, so that the phase speeds of the resulting gravity waveswould be accurately captured. The perturbation kineticenergies as a function of time for these two simulationsare shown in Fig. 16a. The ‘‘spikes’’ in the kinetic en-ergy just after t 5 0 are associated with the rapid ap-pearance, outward propagation, and dissipation of grav-ity waves. While the freely propagating inertia-gravitywaves will not generate PV and/or balanced flow on aquiescent f plane, such is not the case when a nontrivialbasic state is present (Montgomery and Lu 1997). Byinteracting with the basic state, each of these solutionsprojects a small amount of its energy onto the balanced,‘‘slow manifold,’’ such that for the inner perturbation,the wavenumber-one instability is marginally excited,and for the outer perturbation, the final state is just thepseudo-mode as it was for the similar balanced pertur-bation above (as shown in Fig. 15). A close-up of theearly kinetic energy evolution is shown in Fig. 16b.Once the initial gravity wave packet is developed, thewaves propagate outwards with nearly constant kineticenergy, until they are dissipated in the damping regionbeyond r 5 190 km [cf. (3.9)]. The outward propagationof these waves is depicted in Fig. 17, which shows thecomplex magnitude of hn and dn at t 5 0, 30 s, 4 min,and 10 min. Observe that the outer edge of the hn wavepacket is at r 5 100 km at t 5 240 s, and at r 5 200km at t 5 600 s. From these two points we can estimatethe simulated gravity wave speed to be 277.8 m s21,less than 1% in error from the theoretical gravity wavespeed ( )1/2 5 280.1 m s21. After the waves spreadÏgHinto the damping region, only balanced flow is left be-hind.

f. Mechanisms for algebraic and exponentialinstability

Both algebraic and exponential instability mecha-nisms appear to be present in the NM–shallow-water

vortex. Exponential instability for n 5 1 perturbationsin vortices with low-vorticity cores was first observedin plasma physics experiments (Driscoll 1990), despitethe fact that linear theory predicts there can be no ex-ponentially unstable modes in a two-dimensional, in-viscid vortex for wavenumber-one (Reznik and Dewar1994).4 The SR solution showed that secular growth canoccur, but could not account for the observed exponen-tial growth. Efforts to understand this contradiction be-tween theory and experiments have since followed.

4 It should be noted that wavenumber-one instabilities were alsofound to exist in the presence of substantial viscosity by Nolan andFarrell (1999a). Most statements on the existence of exponential in-stabilities rely on analyses of the inviscid equations, and may not bevalid in the presence of viscous dissipation.


FIG. 17. Evolution of the perturbation height and divergence fieldsfor the unbalanced perturbation localized at r 5 20 km: (a) complexmagnitude of hn at t 5 0 (solid), 30 s (dashed), 4 min (dash–dot),and 10 min (dotted); and (b) complex magnitude of dn at t 5 30 s(dashed), 4 min (dash–dot), and 10 min (dotted; values at t 5 0 arenegligible). Height units in meters and divergence units in s21.

There are three important distinctions between thedynamics of two-dimensional and shallow-water vor-tices: the presence of freely propagating inertia-gravitywaves, the variation of the basic-state height, and thefinite value of the Rossby radius of deformation. Inertia-gravity waves clearly do not play a role in the dynamicsof either algebraic or exponential growth here, becausewe have found that within the first few hours of all thesimulations the flow evolution occurs via quasi-bal-anced flow. That the variation of the basic-state heightplays a role in the exponential instability has recentlybeen demonstrated by Finn et al. (1999a,b). They foundthat the ends of the confinement region in the electronplasma experiments are not perfectly flat, but have asmall variation with radius from the center of the ap-paratus. As a result, the vertically averaged electrondensity, which is equivalent to the two-dimensional ver-tical vorticity, is not conserved by the flow, but insteadthe potential density nL is conserved, where n is theelectron density and L 5 L(r) is the length of the con-

finement region. This obviously bears resemblance tothe conservation of potential vorticity (z 1 f )/h. Finnet al. (1999a,b) have shown that wavenumber-one isdestabilized for hollow (low-vorticity core) vorticeswhen L(r) decreases with radius, which is equivalent to

(r) increasing with radius, as is the case for a balanced,hshallow-water cyclonic vortex. This destabilization oc-curs even when the perturbations to L(r) are neglected;in most of their calculations, Finn et al. (1999a,b) con-sider the ends of the confinement region to be fixed,much like the motion of a shallow-water fluid betweentwo rigid lids of varying height.

How does the variation of (r) cause the appearancehof an exponentially unstable mode for n 5 1? Considerthe SR neutral mode (3.22), which rotates at the fre-quency max. This frequency is faster than any otherVperturbation can rotate (in the quasi-balanced regime),and as a result, the SR neutral mode cannot couple withany other perturbation. However, the radial gradient of

(r) modifies the PV gradient in such a way as to slowhdown the mode. To understand azimuthal propagationof asymmetric perturbations on a shallow-water vortex,we turn to asymmetric balance (AB) theory as devel-oped by Shapiro and Montgomery (1993) and Mont-gomery and Kallenbach (1997). It is well known thatin the core of intense vortices such as hurricanes, theRossby number is large and quasi-geostrophic theory isnot valid. However, the effect of rotation on linearized,asymmetric perturbations is not merely f but rather isdescribed by the modified Coriolis parameter f 5 f 12 and the absolute vorticity h 5 f 1 , [cf. (3.6)–V z(3.7)] both of which will be substantially larger than f(inside the RMV) due to the rapid rotation of the vortexitself. As a result, such perturbations can be describedusing a localized balance approximation, which is sim-ilar to the balance described in section 3b, but whichmay be extended through a series of approximations sothat the motions may be described in terms of a singlegeopotential f9(r, l, z, t) (in the case of a continuous,stratified fluid), much like in quasi-geostrophic theory.Using the AB formulation, Montgomery and Kallen-bach (1997) derived a local dispersion relation for thepropagation of vortex-Rossby waves in a shallow-watervortex. In particular, consider some localized PV per-turbation with azimuthal wavenumber n and radialwavenumber k. The resulting phase velocity in the az-imuthal direction is

f (]q /]r)0 0C 5 R V 1 , (3.26)pl 0 0 2 2 2 2q {k 1 n /R ) 1 g }0 0

where the zero subscripts refer to local values at somer 5 R0, and

h f0 02g 5 (3.27)0 gh0

is the square of the inverse of the local Rossby radius.Both the SR–neutral mode and the shallow-water un-


FIG. 18. Structure of the PV of the most unstable mode for H 58 km, Dr 5 0.5 km (solid, with x’s), and an amplitude-matched PVequivalent SR neutral mode (dashed, with o’s). PV units in s21 m21.

stable mode have perturbation vorticity in two localizedregions—an inner anomaly associated with the regionof positive basic-state vorticity gradient, and an outerregion associated with negative basic-state vorticity gra-dient and max (cf. Figs. 13a,c). In 2D flow, the innerVvorticity anomaly can rotate at the speed of max becauseVit is in a region of positive PV gradient; that is, it prop-agates faster than the local mean flow [cf. (3.25)]. Nolanand Montgomery found that the speed of the outer dis-turbances, which would by themselves propagate slowerthan max, are in fact increased due to local vorticityVproduction caused by the inner anomalies interactingwith the basic-state vorticity gradient. The outer anom-alies retard the speed of the inner anomalies, but theeffect is minimal due to their small scale.

Note, however, that in the shallow water case this rateof propagation is diminished because

]q 1 ]z q ]h 1 ]z5 2 , , (3.28)

]r h ]r h ]r h ]r

since is increasing monotonically from the center andhis positive everywhere. The retardation of the motionq

of the shallow-water perturbation equivalent to the SR–neutral mode [defined by (3.23)], allows it to couplewith other disturbances in the vicinity of max, and to-Vgether they form an unstable mode. In fact, the angularvelocity of the most unstable mode is 1.462 3 1023 s21,just slightly less than the maximum angular velocitywhich is 1.470 3 1023 s21. Figure 18 shows a com-parison of the complex magnitude of qn of the unstablemode and a nearly neutral perturbation (3.23) with phaseand amplitude chosen to match the most unstable modeas closely as possible. If one subtracts this nearly neutralmode (in terms of its qn and its associated balancedfields) from the unstable mode, we can see in Fig. 19the structure of the difference between them, which con-

sists of sheared disturbances in the vicinity of max. InVthe 2D case, NM showed that the velocity fields inducedby such disturbances decay in amplitude with time ast21/2, allowing for the slow but secular growth of theSR–neutral mode.5 In the shallow-water case, the nearlyneutral mode and these disturbances are phase lockedand amplifying each other, resulting in exponential in-stability.

In fact, Finn et al. (1999a,b) showed that, in the fixeddepth (no free-surface) approximation, the inner-coremode is slowed by an amount that scales as k2/3, where

L0(r)k ; (3.29)

L(0)

is a measure of the curvature of the surface normalizedby the length of the plasma chamber. The exponentialgrowth, which depends on this slowing of the mode,was also shown to scale as k2/3. As shown in Fig. 9,the wavenumber-one growth rates approximately scalewith the resting depth as H22/3, consistent with thesefindings.

The propagation speed is further diminished by theterm in the denominator, which is not present in the2g 0

purely 2D case [see Montgomery and Kallenbach (1997)or NM]. This term may modify the growth rates as theresting depth H is decreased, although we have not de-termined the explicit relationship. As the free-boundaryeffects decrease the angular velocity of the nearly neu-tral SR mode, the mode can ‘‘interact’’ more efficientlywith the basic-state flow, allowing for an even fasterexponential growth. A similar effect was also observedwhen free-boundary effects on the containment lengthL(r, l, t) were considered in the electron plasma ex-periments of Finn et al. (1999a,b).

4. The wavenumber-one instability in a three-dimensional hurricane-like vortex

We have shown that an instability similar to the SRalgebraic instability also appears in a shallow-water vor-tex with a low-vorticity core. The natural next step isto ask whether a similar instability occurs in a three-dimensional vortex with wind and temperature fieldsthat are representative of actual hurricanes. To answerthis question, we have performed fully three-dimen-sional simulations of such a vortex in a high-resolutionmesoscale model.

a. The model

The numerical model used for this simulation is ver-sion 3b of the Regional Atmospheric Modeling System(RAMS), developed at Colorado State University (Piel-ke et al. 1992). The following features of the modelwere implemented:

5 The cumulative effect of t21/2 forcing is #t21/2 dt 5 2t1/2 1 C.


FIG. 19. Spatial structure of the residual PV and balanced height fields computed from the difference between the mostunstable mode PV amplitude and phase-matched shallow-water perturbation equivalent to the SR neutral mode: (a) perturbationPV difference, in s21 m21; and (b) perturbation height difference in m.

R The fluid motions are nonhydrostatic and compress-ible (Tripoli and Cotton 1982).

R A hybrid time step scheme, where momentum is ad-vanced using a leapfrog scheme and scalars are ad-vanced using a forward scheme. Both use second-order advection.

R Vertical and horizontal turbulence are parameterizedusing a Smagorinsky (1963) deformation-based eddyviscosity with Richardson number stability modifi-cations (Lilly 1962).

R Prognostic variables are the three components of mo-mentum u, y, and w, the perturbation Exner functionp, and potential temperature u.

R The model uses an Arakawa C fully staggered grid.R Perturbation Exner function tendencies used to update

the momentum variables are computed using a timesplit scheme, similar to Klemp and Wilhelmson(1978).

R The Klemp–Wilhelmson radiation condition is usedon the lateral boundaries, in which the normal velocitycomponent specified at the lateral boundary is effec-tively advected from the interior.

R The top of the domain has a solid-wall boundary witha 3-km-deep friction (Rayleigh damping) layer below.

R The lower boundary condition is free-slip.

A horizontal grid spacing of 1 km is used within a domaincovering 350 3 350 km. The domain extends to a heightof 23.0 km with vertical grid spacings of 1000 m. Theinitial sounding is the Jordan (1958) mean hurricane sea-son sounding. The simulation ran for 12 h with a timestep of 10 s. The Coriolis parameter has the uniform valuef 5 5.0 3 1025, which corresponds to 208N.

Furthermore, the model was utilized without a num-ber of features typical of mesoscale simulations: no

moisture is included and there are no moist or radiativeprocesses involved. Thus the simulation is a highly ide-alized representation of the evolution of a dry, balanced,hurricane-like vortex.

b. The basic-state vortex

We wish to construct a basic state vortex that is similarto that of an actual hurricane, and shares some of thesame stability properties as the NM vortex used above.We begin by extending the NM velocity profile into thevertical, by using analytic functions to cause an appro-priate decay of the wind field with height; that is,

2.01.7zV(r, z) 5 y (r) exp 21 2[ ]z top

2.7r3 1.0 2 exp 2 , (4.1)5 1 2 6[ ]a 1 4z

where (r) is the NM velocity profile, r and z are theydistances from the center axis and the altitude, respec-tively, and ztop 5 16 km is approximately the upper limitof the wind field. The first multiplicative factor on therhs of (4.1) is a decay function, which causes the ve-locity field to decay suitably with height, as is observedin hurricanes (for examples of height–radius profiles ofobserved hurricane wind fields, see e.g., Marks 1992;Dodge et al. 1999). The second multiplicative factor hasan r dependence that causes the inner core wind field (i.e.,the eyewall) to slope outward with height, as is also ob-served. The parameter a serves only to prevent the lastbracketed term from being undefined at z 5 0; we use a5 1.0 m. The basic-state wind field is shown in Fig. 20a.


FIG. 20. Basic-state (a) azimuthal velocity in m s21, (b) potential temperature, and (c) Ertel’s PV for the balanced, hurricane-like vortex.

Next, we must find pressure temperature fields thathold this vortex in hydrostatic,

]p(r, z)5 2r(r, z)g, (4.2)

]z

and gradient wind,

2]p(r, z) V(r, z)5 r(r, z) f V(r, z) 1 , (4.3)5 6]r r

balance. This is achieved with the following iterativeprocedure. First, we initialize the axisymmetric pressureand temperature fields using the Jordan (1958) observedmean hurricane season soundings from the West Indies.Next, we compute the pressure field by integrating (4.3)inward from the outer boundary, while holding the tem-perature field at its current value. Then, we correct thetemperature (and density) fields by enforcing the hy-drostatic condition (4.2) everywhere, while holding thepressure at its current value. This process is repeated

(typically 5–7 times) until the fields converge to a so-lution. These computations are performed on a two-dimensional, evenly spaced grid with 0.5-km resolutionin both the r and z directions, with a maximum radiusof 200 km and a maximum height of 22 km. The finaldata is then interpolated onto the RAMS three-dimen-sional Cartesian grid to provide the basic-state vortex.The basic-state potential temperature and Ertel’s poten-tial vorticity fields are shown in Figs. 20b and 20c. Notethe familiar warm core and the ‘‘hollow tower’’ of PV.Although the vortex has been constructed without aboundary layer, there is a local maximum in the PVabove the surface at z 5 2.5 km; this is quite similarto the PV field produced in a recent high-resolutionsimulation of an intensifying hurricane modeled afterHurricane Andrew (1992; Chen and Yau 2001).

c. Initial conditions and resultsFor the initial conditions, we also add to this axi-

symmetric wind field a wavenumber-one perturbation


similar to those used above. We first obtain the two-dimensional wind field u9(r, l), y9(r, l) generated bythe same wavenumber-one vorticity perturbation usedin NM. We then extend the perturbation wind field intothe vertical with

[u9(r, l, z), y9(r, l, z)]

3.01.8z5 [u9(r, l), y9(r, l)] exp 2 , (4.4)1 2[ ]zmode

where we have used a similar exponential decay factorwith height as in (4.1), and zmode 5 14 km. While thepressure and temperature fields are not in balance withthis perturbation, its small amplitude prevents the re-sulting adjustment process from affecting the results sig-nificantly.

The evolution of this perturbed hurricane-like vortexwas simulated for 12 h. The initial conditions and earlyevolution of the PV fields at 30-min intervals from t 50 h to t 5 2.5 h are shown in Figs. 21 and 22, both atz 5 0.5 km and z 5 4.5 km. At z 5 0.5 km, we seethe familiar growing wobble of the low-vorticity core.At z 5 4.5 km, the amplitude of the vorticity pertur-bation and the resulting wobble are considerably less,as is the speed at which they rotate around the center,since the azimuthal winds are substantially slower atthis altitude (about 35 m s21). Unlike the two-dimen-sional simulations, the growing disturbance begins todevelop some rather complex structure within the first6–12 h, as is shown in Fig. 23. This is due to the differentrotation speeds with height of the vorticity perturba-tions, leading to stronger vertical gradients in velocityand vorticity, and ultimately to instabilities with com-plex structures in the vertical as well as horizontal di-rections. Even at this stage, however, the maximum ver-tical velocities are 25 cm s21 or less (not shown), suchthat the dynamics remain quasi-balanced. The effect ofthese secondary instabilities is to cause an even morerapid rearrangement of the inner-core vorticity. Notehow at t 5 12 h, some of the highest vorticity at z 50.5 km has arrived at the vortex center, similar to theresults for purely two-dimensional flow shown in Figs.7 and 8. At z 5 4.5 km, the instability is not nearly asrobust and the low-vorticity core remains while thewavenumber-one perturbations have been axisymmetri-zed in the near field.

The wobble of the low-level center, as indicated bythe location of the minimum pressure, can also be seenin the mesoscale simulation. This is demonstrated inFig. 24, where we show the lowest contourable pressurelevel at the lowest model level (z 5 500 m) at 30-minintervals in the simulation. These pressure contours givea clearer indication of the location of the minimum pres-sure rather than simply indicating the gridpoint locationsof the lowest pressure value, because such locations canonly fall on the 1-km-spaced grid points (consider whatthe last 12 h of data in Fig. 6 would be like if the gridspacing were three times larger in those simulations).

As we saw with the high-resolution, two-dimensionalsimulations, the amplitude of the wobble only increasesfor the first few hours, then steadily declines as theinstability transitions to nonlinear dynamics and inner-core mixing. Another imporant point is that the mini-mum pressure steadily declines as the inner-core vor-ticity rearrangement proceeds; this is exactly the processEmanuel (1997) requires for a hurricane to reach itsMPI.

5. Conclusions

We have investigated the instability of wavenumber-one perturbations to hurricane-like vortices with low-vorticity cores. We find that the instability is a robustfeature of such vortices.

In purely two-dimensional flow, the remarkable SRsolution predicts that the instability will grow in timeas t1/2, and was confirmed and studied in some detail byNM. Here, high-resolution simulations of the instabilityin purely two-dimensional flow show that the instabilityleads to a redistribution of the inner-core vorticity, withthe highest vorticity accumulating at the center of thevortex within 24 h. Given sufficient time, the low-vor-ticity core is eventually ejected from the inner core ofthe vortex and symmetrized in the surrounding envi-ronment. While we found that the timescale for thisejection process depended strongly on the viscosity,such behavior is likely only relevant for related plasmaphysics experiments and other effectively two-dimen-sional flows.

The shallow-water equations allow for the importantadditional phenomena of vortex stretching, gravitywaves, and gradient adjustment. In such an environ-ment, the wavenumber-one instability becomes an ex-ponential instability, with a structure nearly identical tothat of the algebraic instability, representing a displace-ment of the low-vorticity core relative to the surround-ing flow. The transition of the algebraic instability tothe exponential instability is caused by a slowing of thepropagation of vortex-Rossby waves, caused both bythe variation of the basic-state height, and also finiteRossby radius effects. These effects allow the SR–neu-tral mode to couple more effectively with other pertur-bations and form an unstable mode. For fluid depthsrelevant to hurricane-like vortices, the growth rates arefairly slow such that the exponential growth is com-parable to the algebraic growth on the relevant timescales of 6 to 24 h.

Finally, we have shown that a very similar instabilitywill appear when a three-dimensional, balanced vortexwith a hurricane-like wind field is initialized with awavenumber-one perturbation. For the first few hours,the evolution of the low-level PV is nearly identical towhat has been seen in 2D flow and in the shallow-waterequations. However, within 12 h the differential rotationof the vortex with height leads to three-dimensional(though still quasi-balanced) instabilities and then to a


FIG. 21. Early evolution of the Ertel’s PV in the mesoscale model simulation every 30 min, at z 5 0.5 km and z 54.5 km, as labeled. Contours at z 5 500 m are at 1.25, 2.5, 3.75, 5.0, 6.25, 7.5, 8.75, and 10.0 PV units (equal to 1026

m2 K s21 kg21). Contours at z 5 4500 m are 3, 6, 9, 12, 15, 18, and 21 PV units. The highest contour levels in eachcase are thick, so that local maxima can be identified.


FIG. 22. Early evolution of the Ertel’s PV in the mesoscale model simulation every 30 min, at z 5 0.5 km and z 54.5 km, as labeled. Contours are as in Fig. 21.


FIG. 23. Later evolution of the Ertel’s PV in the mesoscale model simulation every 30 min, at z 5 0.5 km and z 54.5 km, as labeled. Contours are as in Fig. 21.


FIG. 24. The lowest contourable pressure levels at the lowest gridlevel (z 5 500 m) in the mesoscale model simulation of the wave-number-one instability. For each contour, the numbers indicate thetime at 30-min intervals; that is, contour 1 is at t 5 30 min, 2 is att 5 60 min, etc. Plot (a) shows the results for the first 8 times after t5 0, from t 5 30 min to t 5 4 h; the contour levels are 928.00, 927.33,926.62, 925.74, 924.98, 924.32, 923.76, and 923.34 mb, respectively.Plot (b) shows later times, from t 5 8 h 30 min to t 5 12 h; thecontour levels are 921.67, 921.54, 921.39, 921.24, 921.12, 921.04,921.00, and 921.00 mb.

rapid mixing of the inner-core vorticity. Also, the wave-number-one instability seems to be confined to the low-est few kilometers of the vortex; above, only axisym-metrization occurs. Nonetheless, the basic principleholds that a low-vorticity core, hurricane-like vortex canexperience a wavenumber-one instability, and that thisinstability leads to inner-core vorticity mixing. Initially,the mixing proceeds due to the displacement of the low-

vorticity core and a net inward flux of higher, eyewallvorticity, as was shown in the two-dimensional simu-lations of section 2. This mixing is later enhanced bysmaller-scale, three-dimensional instabilities, which arecaused by the rapidly changing horizontal and verticalstructures of the PV field.

The vortices we have studied here have been con-structed to be stable or at least only weakly unstable tohigher wavenumber perturbations. A valid question toask is whether or not, in the real atmosphere, the wave-number-one instability would be dominated by higherwavenumber instabilities. Indeed, higher wavenumberinstabilities in geophysical, hurricane-like, and tornado-like vortices have been a topic of considerable research(e.g., Rotunno 1978; Flierl 1988; Peng and Williams1991; Smyth and McWilliams 1998; Schubert et al.1999; Nolan and Farrell 1999a,b), and the authors arecurrently extending such work into the realm of three-dimensional, baroclinic, hurricane-like vortices similarto the vortex simulated in our mesoscale model (Nolanand Montgomery 2000b). However, the existence andrelevance of the wavenumber-one instability is not pre-cluded by higher-wavenumber dynamics. The higherwavenumber instabilities generally require that the eye-wall have a certain structure—that the radius of the low-vorticity core be larger than the width of the high-vor-ticity eyewall annulus, and/or that the ratio of the eye-wall to eye vorticity be above some value (see Schubertet al. 1999, section 2). The small amplitude, high fre-quency trochoidal motion of hurricanes, discussed insection 1 of this paper, is observed at least as frequentlyas higher-wavenumber instabilities and mesovorticeswithin hurricanes. The wavenumber-one instability onlyrequires the presence of an angular velocity maximumother than at the center axis, and thus can occur inhurricanes with very small eyes and also weaker storms.Furthermore, while higher-wavenumber instabilitiesmay often be present, their smaller scales makes themless effective in ameliorating the PV maximum in theeyewall region (especially in light of the steady PVgeneration caused by convection) as compared to thelarger displacements caused by the wavenumber-one in-stability (similar to the smaller total heat fluxes asso-ciated with short-wavelength baroclinic waves; see, e.g.,Welch and Tung 1998).

The wavenumber-one instability presented here, andits associated trochoidal motion, are fundamentally dif-ferent from the larger-scale, lower frequency trochoidalmotions discussed in previous studies (e.g., Yeh 1950;Kuo 1969; Abe 1987; Flatau and Stevens 1993; Jones1995). Rather than being caused by an interaction withthe hurricane’s environment, this instability is funda-mental to vortices with low-vorticity cores, and rep-resents a growing displacement of the low-vorticitycenter, which ultimately leads to inner-core vorticityand angular momentum redistribution. Such inner-coremixing also causes the minimum surface pressure to


fall, which may have an important impact on the furtherevolution of the storm.

Acknowledgments. The authors would like to thankDr. P. Reasor for his contributions to preliminary workfor this paper, and Dr. J. Kossin for his comments onthe manuscript. For continuing support with the devel-opment and utilization of the Adaptive Mesh Refine-ment Model, the authors would also like to thank Drs.A. S. Almgren and J. B. Bell, along with the Center forComputational Sciences and Engineering and the Math-ematics Department of the Lawrence Berkeley NationalLaboratory. This work was supported in part by theOffice of Naval Research Grant N00014-93-1-0456P0006 and Colorado State University.

APPENDIX

Neutral Inner-Core Modes in Two-DimensionalVortices

We show that in a two-dimensional, inviscid vortexwith an angular velocity maximum, a disturbance whoseperturbation vorticity is exactly proportional to the ba-sic-state vorticity gradient, up to the location of theangular velocity maximum, and zero elsewhere, is aneutral mode that rotates at the speed of the angularvelocity maximum. The linearized equations of motionfor inviscid perturbations are

]z ]zn 1 inVz 1 u 5 0, and (A.1)n n]t ]r

inu 5 2 c , (A.2)n nr

whereR

c (r, t) 5 z (r, t)G (r, r) dr, (A.3)n E n n

0

and in a cylindrical domain of size R, the Green functionfor wavenumber n is

n n11r r 1n 2n112 r r 0 # r # r

2n2nR 2nG (r, r) 5 (A.4)n n n11r r 1

2n n11 2 r r r # r # R2n2nR 2n

(Carr and Williams 1989).Now let us consider the case for n 5 1 and

]zz (r, 0) 5 C H(RMV 2 r), (A.5)1 ]r

where C is some complex constant, RMV is the locationof the angular velocity maximum associated with theneutral mode (if there is more than one angular velocity

maximum, there will be neutral modes associated witheach one; see the last paragraph of this appendix), andH is the Heaviside step function. Without loss of gen-erality we may consider the case with C 5 1. Then howdoes the perturbation evolve? We must solve for the c1

associated with z1:

r 2]z rr 121 2c (r) 5 2 r r dr1 E 21 2]r 2R 20

RMV 2]z rr 11 2 r drE 21 2]r 2R 2r

r RMV]z 1 ]z 121 25 2 r r dr 1 2 r drE E1 2 1 2]r 2 ]r 20 r

RMV 2]z rr1 dr 5 I 1 I 1 I .E 1 2 321 2]r 2R0

(A.6)

Let us consider the first integral,r1 ]z

2I 5 2 r dr. (A.7)1 E2r ]r0

Using the definitions of vorticity and angular velocity,one can show that

2]z ] V ]V5 r 1 3 , (A.8)

2]r ]r ]r

and thus

]z ] ]V2 3r 5 r . (A.9)1 2]r ]r ]r

Therefore,

r1 ] ]V 1 ]V3 2I 5 2 r dr 5 2 r1 E 1 22r ]r ]r 2 ]r0

1 r ]y5 y 2 . (A.10)

2 2 ]r

Using (A.9), we can write the third integral as

RMVr ] ]V3I 5 r dr 5 0, (A.11)3 E2 1 22R ]r ]r0

even for finite R, due to the definition of RMV. We nowproceed with the second integral,

RMV1 ]zI 5 2 r dr2 E2 ]rr

15 2 r[z (RMV) 2 z (r)]. (A.12)

2

Since 5 ] /]r 1 /r, we can rewrite I2 asz y y


1 ]y y ]y yI 5 2 r 1 2 1 . (A.13)2 1 2 1 2[ ]2 ]r r ]r r

r5RMV

The final step is to use the identity

]y ]V5 V 1 r 5 V , (A.14)max1 2 1 2]r ]r

r5RMV r5RMV

so that we may arrive at the solution

c (r) 5 I 1 I 5 r[V(r) 2 V ].1 1 2 max (A.15)

In fact, (A.15) only applies for r # RMV; for r . RMV,c1(r) 5 I1(r) 5 I1(RMV) 5 0.

Now, if we substitute (A.5) and (A.15) into (A.1) and(A.2), we arrive at the equation of motion

]z1 1 iV z 5 0; (A.16)max 1]t

that is, the perturbation has fixed structure and rotatesat the maximum angular velocity.

Finally, we note that these arguments do not rely onthe uniqueness of the angular velocity maximum. Thatis to say, for a vortex with an arbitrary number of an-gular velocity maxima, there will be a neutral modeassociated with each maximum. The arguments abovewill apply in each case, provided that the RMV usedthroughout is the same as that in (A.5). As SR showedin their original solution, there will also be an algebraicinstability associated with each of the angular velocitymaxima.

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