Generalized Stability Analyses of Asymmetric Disturbances...

1282 VOLUME 56J 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 1999 American Meteorological Society

Generalized Stability Analyses of Asymmetric Disturbances in One- and Two-CelledVortices Maintained by Radial Inflow

DAVID S. NOLAN

Department of Mathematics, Lawrence Berkeley National Laboratory, Berkeley, California

BRIAN F. FARRELL

Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachusetts

(Manuscript received 11 July 1997, in final form 27 May 1998)

ABSTRACT

The dynamics of both transient and exponentially growing disturbances in two-dimensional vortices that aremaintained by the radial inflow of a fixed cylindrical deformation field are investigated. Such deformation fieldsare chosen so that both one-celled and two-celled vortices may be studied. The linearized evolution of asymmetricperturbations is expressed in the form of a linear dynamical system dx/dt 5 Ax. The shear of the mean flowresults in a nonnormal dynamical operator A, allowing for the transient growth of perturbations even when allthe modes of the operator are decaying. It is found that one-celled vortices are stable to asymmetric perturbationsof all azimuthal wavenumbers, whereas two-celled vortices can have low-wavenumber instabilities. In all cases,generalized stability analysis of the dynamical operator identifies the perturbations that grow the fastest, bothinstantaneously and over a finite period of time. While the unstable modal perturbations necessarily convertmean-flow vorticity to perturbation vorticity, the perturbations with the fastest instantaneous growth rate usethe deformation of the mean flow to rearrange their vorticity fields into configurations with higher kinetic energy.Also found are perturbations that use a hybrid of these two mechanisms to achieve substantial energy growthover finite time periods.

Inclusion of the dynamical effects of radial inflow—vorticity advection and vorticity stretching—is found tobe extremely important in assessing the potential for transient growth and instability in these vortices. In thetwo-celled vortex, neglecting these terms destabilizes the vortex for azimuthal wavenumbers one and two. Inthe one-celled vortex, neglect of the radial inflow terms results in an overestimation of transient growth for allwavenumbers, and it is also found that for high wavenumbers the maximum transient growth decreases as thestrength of the radial inflow increases.

The effects of these perturbations through eddy flux divergences on the mean flow are also examined. In theone-celled vortex it is found that for all wavenumbers greater than one the net effect of most perturbations,regardless of their initial configuration, is to increase the kinetic energy of the mean flow. As these perturbationsare sheared over they cause upgradient eddy momentum fluxes, thereby transferring their kinetic energy to themean flow and intensifying the vortex. However, for wavenumber one in the one-celled vortex, and all wave-numbers in the two-celled vortex, it was found that nearly all perturbations have the net effect of decreasingthe kinetic energy of the mean flow. In these cases, the kinetic energy of the perturbations accumulates in nearlyneutral or unstable modal structures, so that energy acquired from the mean flow is not returned to the meanflow but instead is lost through dissipation.

1. Introduction

In this report we introduce a new approach to thestudy of the dynamics of asymmetries in primarily axi-symmetric intense atmospheric vortices—such as wa-terspouts, tornadoes, and hurricanes. For waterspoutsand tornadoes, these asymmetries include low-levelwind shear, the unsteady and asymmetric supply of their

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

angular momentum, and also the effects of turbulence.The success of recent three-dimensional numerical sim-ulations in reproducing realistic maximum tornado windspeeds (Lewellen et al. 1997), compared with the muchlower velocities predicted by theories and simulationsbased on axisymmetric models (Lilly 1969; Fiedler andRotunno 1986; Fiedler 1994), indicates that asymme-tries may play an important role in the maintenance ofan intense and robust tornadic vortex. For hurricanes,asymmetric forcing includes the beta effect, potentialvorticity anomalies in the mid- and upper levels of theatmosphere, and the shear of environmental winds; theseasymmetries are believed to play a significant role inthe track and intensity changes of these cyclones (Mol-

15 MAY 1999 1283N O L A N A N D F A R R E L L

inari 1992; Shapiro 1992; Montgomery and Farrell1993; Smith and Weber 1993).

The traditional approach to understanding the behav-ior of nearly axisymmetric vortices has relied heavilyon hydrodynamic stability analyses of their azimuthalvelocity profiles, which in practice are approximated asa function of radius, either from measurements or fromidealized functions (such as a Rankine vortex). A clas-sical stability analysis is applied to small perturbationslinearized about the mean azimuthal flow. Examples ofthis kind of analysis include those of Michaelke andTimme (1967), Rotunno (1978), Staley and Gall (1979),Gall (1983, 1985), and more recently Steffens (1988),Flierl (1988), and Peng and Williams (1991). In accor-dance with traditional notions of stability in shear flowsoriginally derived by Rayleigh (1880), most of thesereports found that a change in sign of the vorticity gra-dient of the azimuthal wind was necessary for the ex-istence of instability. Some of the analyses cited abovealso included the effects of a radially varying verticalwind that can create additional instabilities.

In recent years a new approach has been taken to thestudy of asymmetric dynamics in tropical cyclones,which relies on the analysis of such perturbations as aninitial-value problem rather than as an eigenvalue prob-lem. Carr and Williams (1989) and Smith and Mont-gomery (1995) found linearized equations for the evo-lution of asymmetric disturbances in hurricane-like vor-tex flows, while Guinn and Schubert (1993) also in-cluded the results of numerical simulations of perturbednondivergent barotropic flows in their analyses. The re-sults of these studies show that asymmetric disturbancesgenerally decay rapidly with time due to the strong shearof the vortex, and that these decay rates increase withboth the azimuthal and radial wavenumbers of the dis-turbances. Kallenbach and Montgomery (1995) alsodemonstrated how such disturbances can undergo a pe-riod of transient growth before decaying. Asymmetricdynamics have also been investigated by Willoughby(1992, 1995) for their role in determining tropical cy-clone tracks, in regard to both the northwest drift ofbarotropic vortices due to the poleward increase of theCoriolis parameter, and to how these asymmetries canbe used to determine the initial motion of the vortex inforecast models. The growth of asymmetric disturbancesnear the cores of tropical cyclones has also been ad-vocated as the cause of the polygonal eyewall phenom-enon (Schubert et al. 1999).

A simplifying assumption of great potential impor-tance to the dynamics that was made in these stabilitystudies of intense atmospheric vortices was neglectingthe convergent background flow—the radial inflow—which forms and maintains these vortices. While ini-tially this was justified by the belief that the radial ve-locities in the core of these vortices was negligible, itis now well known due to laboratory experiments (Wanand Chang 1972), theoretical analyses (Burgraff et al.1971), and axisymmetric numerical simulations (How-

ells, et al. 1988) that there is in fact a strong radialinflow near the surface in tornadic vortices, with radialvelocities typically half as large as the azimuthal ve-locities. At some point as it approaches the central axisof the vortex, the radial inflow separates from the sur-face, turns upward, and flows up and out of the vortexcore. (Note: by the ‘‘core’’ of the vortex we mean inside,and in the vicinity of, the radius of maximum winds.)Hurricanes are also known to have substantial radialinflow velocities in the surface boundary layer, fromboth observations (Franklin et al. 1993) and numericalsimulations (Shapiro 1992; Liu et al. 1997). Using avery high-resolution simulation of Hurricane Andrew(1992), Liu et al. also found small (order 1–2 m s21)radial inflow velocities outside the eyewall in the middlelevels, as did Shapiro (1992) in his three-level model.While these velocities are very small compared to themaximum azimuthal wind speeds, our results will showthat small radial inflow velocities can have a measurableimpact on asymmetric dynamics.

It is easy to imagine that this radial inflow may haveimportant effects on the stability of these vortices. Forexample, one possible effect of this radial inflow couldbe to advect growing perturbations out of the region ofmaximum shear and inside the vortex core before it cangrow substantially. In this sense, the radial inflow mayrender all growing perturbations ‘‘transient’’ by sup-pressing modal instability.

In this report we introduce a general method for de-termining the stability of two-dimensional vortex flowsthat includes the radial inflow that maintains the vortex.Furthermore, we will use an extension of classical sta-bility analysis, known as generalized stability analysis,to examine the growth of both transient and exponen-tially growing (if they exist) perturbations on the meanflow. Then we will examine specifically how these per-turbations interact with the mean flow and the effectsof radial inflow on the dynamics. In section 2 we willgive an introduction to generalized stability analysis; insection 3 we will describe our models of the idealizedvortices under consideration; in section 4 we deriveequations for the evolution of linearized asymmetricperturbations to these flows; in section 5 we will eval-uate and comment on the stability of these vortices; insection 6 we will find the transient perturbations thatgrow the most in finite times; in section 7 we will in-vestigate how these perturbations interact with the meanflow and how the presence of the radial inflow affectsthe results; and in section 8 we will discuss our con-clusions.

2. Generalized stability theory

While the traditional approach to assessing the sta-bility of observed flows in fluid dynamics and meteo-rology has been has been restricted to determiningwhether the flow supports exponentially growingmodes, in recent years an alternative approach has de-


veloped. Inspired by the observation of Thompson(1887) and Orr (1907) that properly configured distur-bances in stable shear flows undergo a finite period ofenergy growth before decaying, Farrell (1982) dem-onstrated the growth of baroclinic disturbances of thistype in the atmospheric midlatitude jet. The possibilityof transient growth in the absence of exponentiallygrowing modes is expressed mathematically by the non-normality of the linearized dynamical operator, that is,the fact that the dynamical operator does not commutewith its transpose. Such transient growth may not betrivial: in Poiseuille flow at Reynolds number 5000,Butler and Farrell (1992) found disturbances that couldgrow in energy by a factor of 4897, despite the fact thatPoiseuille flow is exponentially stable at this Reynoldsnumber. This understanding has led to a general analysisof nonnormal systems that has since been used exten-sively to understand transient growth in deformation andshear flows (Farrell 1988, 1989; Butler and Farrell 1992;Farrell and Ioannou 1993a, 1993b, 1996). Considerationof both transient and exponential growth processes hascome to be called generalized stability analysis.

Let us describe in some detail how the nonnormalityof a system can lead to transient growth. First we writedown the evolution of disturbances as a linear dynamicalsystem:

dy5 Ty, (2.1)

dt

where y is a function or vector that describes the stateof the perturbation(s), and T is the time evolution op-erator. We also define a positive-definite Hermitian op-erator M such that the energy of the system can bewritten

E 5 y*My. (2.2)

We now perform the following useful change into gen-eralized velocity coordinates x:

1/2x 5 M y, (2.3)1/2 21/2A 5 M TM , (2.4)

so that in generalized velocity coordinates,

dx5 Ax, (2.5)

dt

E 5 x*x. (2.6)

Equation (2.4) is a similarity transformation, so the ei-genvalues of A are the same as the eigenvalues of T.If all the eigenvalues of T have a negative real part,then all the modes of T are decaying. So then are allthe modes of A, and we can conclude that the energyof the system goes to zero as t → `. The structure inthese generalized velocity coordinates that dominatesenergetically for large times will be the eigenvector ofA associated with the eigenvalue with the largest realpart, which we will call the least damped mode (LDM).

The structure of the LDM in the original coordinatesystem y can be found by transforming back into theoriginal coordinates, that is, inverting (2.3).

This does not address the issue of the energetics ofthe system when 0 # t , `. We can easily solve forthe instantaneous rate of change in energy:

]E ] ] ]x5 (x*x) 5 x* x 1 x*1 2 1 2]t ]t ]t ]t

† †5 x*A x 1 x*Ax 5 x*(A 1 A)x, (2.7)

where the † refers to the Hermitian matrix transpose.Because the energy operator (A† 1 A) is normal andHermitian, the eigenvector of (A† 1 A) with the largesteigenvalue will be at any instant the fastest growing (orleast decaying) perturbation. This is certain because theeigenvectors of a normal matrix are complete and or-thogonal—any other perturbation that does not projectonto the dominant eigenvector could be constructed en-tirely from the other eigenvectors, all of which growmore slowly than the first. Such fastest growing per-turbations are usually called the instantaneous optimals(IOs) of the system. By setting x 5 emax (the eigenvectorwith maximum eigenvalue) in Eq. (2.7), one can seethat this upper bound on the normalized energy growthrate is

1 ]E5 l , (2.8)max1 2E ]t

max

where lmax is the largest eigenvalue of (A† 1 A).In regards to the issue of transient growth and non-

normality, consider that when A is normal, A and A†

have the same eigenvectors. Therefore the IO will bethe same as the LDM, and its rate of change of energywill be twice the real part of the eigenvalue of the LDM.When A is nonnormal, the eigenvalues and eigenvectorsof A and the energy operator will be different, and anypositive eigenvalues of the energy operator will corre-spond to transient growth as indicated by (2.8).

One can also find the perturbations of maximumgrowth over finite time, which we call finite time op-timals (FTOs). The energy as a function of time is

†A t9 At9E(t9) 5 x*(t9)x(t9) 5 x*(0)e e x(0). (2.9)

Because the matrix product in (2.9) is normal and Her-mitian, we see again that the eigenvector correspondingto the largest eigenvalue of will have the largest†A t Ate egrowth (or smallest decay) in energy between t 5 0 andt 5 t9. We have that

E(t9)5 l , (2.10)max1 2E(0)

max

where here lmax is the maximum eigenvalue of the ma-trix product . Another approach to finding FTOs†A t Ate euses the singular value decomposition (SVD) of thepropagator matrix eAt. The propagator matrix can bedecomposed as


FIG. 1. Profiles of radial and azimuthal velocity for the idealized one-celled vortex: (a) radial velocity, (b) azimuthal velocity, (c) negativehorizontal divergence (stretching), (d) vorticity gradient.

eAt 5 UDV†, (2.11)

where U and V are unitary matrices that can be inter-preted to represent orthogonal decompositions of thedomain and the range of the propagator, and D is apositive-definite diagonal matrix, the values of whichrepresent the relative excitation of the system by theircorresponding basis vectors. We can then write

†A t At † † 2 †e e 5 VDU UDV 5 VD V . (2.12)

Clearly by previous argument the basis vector associatedwith the largest value of D will result in the most energygrowth at time t, and the magnitude of the excitationwill be the square of the largest element of D. The SVDmethod is also useful in that the maximal state of the

FTO is produced in U simultaneously with its initialcondition in V.

3. One-celled and two-celled vortices maintainedby radial inflow in a closed domain

As stated in the introduction we will investigate thedynamics of steady two-dimensional vortex flows thatare generated when a cylindrical deformation field actson a rotating fluid. In the following sections we willoutline a method for finding such steady-state solutionsgiven an arbitrary deformation field, and show how tochoose such fields so as to produce either one-celled ortwo-celled vortices.


FIG. 2. Profiles of radial and azimuthal velocity for the idealized two-celled vortex: (a) radial velocity, (b) azimuthal velocity, (c) negativehorizontal divergence (stretching), (d) vorticity gradient.

a. Formulation of the steady-state solution inarbitrary deformation fields

The classic example of one such cylindrically sym-metric flow is the well-known Burgers’ vortex solution[Burgers (1948), also known as the Burgers–Rott so-lution, see Rott (1958)]. If there exists in an unboundeddomain a cylindrical deformation field of the form

1U 5 2 ar, (3.1)

2

W 5 az, (3.2)

where U and W refer to the radial and vertical velocitiesin cylindrical coordinates, then a steady-state solutionof the Navier–Stokes equations is obtained when theazimuthal velocity is

2G ar`V(r) 5 1 2 exp , (3.3)1 2[ ]2pr 4n

where G` is the circulation at infinity and n is the ki-nematic viscosity. Note that this azimuthal velocity pro-file for small r approaches solid-body rotation, and forlarge r asymptotes to a potential vortex (V 5 G`/2pr).

Suppose instead we assume an arbitrary cylindricaldeformation field based on a radial inflow velocity fieldthat is a function of r only:

U 5 U(r). (3.4)

By continuity, we have

]W 1 ]5 2 (rU ), (3.5)

]z r ]r

so that the vertical velocity field W(r, z) may be deter-


FIG. 3. Real parts of the eigenvalues of the LDM for azimuthalwavenumbers 1–15 in the one-celled vortex: V—standard case withinflow; 1—with stretching term removed; 3—with stretching andadvection removed.

mined up to constant. Holding the U and W velocitiesfixed, we can write down a single advection–diffusionequation for the evolution of the axisymmetric azi-muthal velocity V

]V ]V ]V UV1 U 1 W 1

]t ]r ]z r

2] V 1 ]V V5 n 1 2 . (3.6)

2 21 2]r r ]r r

If the azimuthal velocity field is a function of r only,the vertical advection term may be neglected.

Now let us turn our attention to the solution of (3.6)in the finite domain 0 # r # b, with boundary conditionsV(0) 5 0 and V(b) 5 Vb. We discretize V onto equallyspaced grid points between r 5 0 and r 5 b, and writethese values as the column vector V. Using matrix rep-resentations of standard finite-difference operators, (3.6)may be represented as the inhomogeneous linear system

]V5 AV 1 B, (3.7)

]t

where B is a column vector that allows us to incorporatethe boundary conditions. Our steady-state solution isfound by solving for V such that its time derivative iszero:

V 5 2A21B. (3.8)

Examples of some solutions of (3.8) are shown in thefollowing sections.

b. One-celled vortices

While the Burgers’ vortex solution is a useful pointof reference, the fact that it resides in an unboundeddomain creates serious difficulties for the analysis of

asymmetric perturbations in this flow. The greatest dif-ficulties are those associated with the inflow of fluidthrough the edge of the domain at r 5 b. Let us insteadcreate a deformation field similar to the Burgers’ vortexdeformation field, except that its support lies entirelywithin a cylinder of radius r 5 b; we will use b 5 7for the rest of this work. Furthermore, we require thatthe radial inflow velocity transitions very smoothly tozero as we approach the outer boundary, and is nearlyzero for a substantial region near the outer boundary.An example of such a radial inflow function is given by

U(r) 5 .62mr2are (3.9)

This function with a 5 5.0 3 1023 and m 5 2.44 31024 is shown in Fig. 1a. This particular choice for a,in conjunction with a choice of n 5 0.001 for the vis-cosity, sets the radius of maximum winds rmax 5 1 forBurgers’ vortex solution (3.3). Using this radial velocityfield and an outer boundary condition on V such thatthe circulation at the outer boundary Gb 5 2prbVb 52p (i.e., the circulation of the fluid at the edge of thedomain is equal to 2p everywhere), (3.6) results in thesolution shown in Fig. 1b. The vortex Reynolds number,as it is usually defined, is ReV 5 rbVb/n 5 1000. Thissolution, which has rmax 5 1.0 and maximum azimuthalvelocity ymax 5 0.71, is virtually identical to the Bur-gers’ solution with the same parameters despite the factthat the radial inflow velocity transitions to zero nearthe outer edge of the domain. The stretching (or negativehorizontal divergence) of the vertical velocity functionis shown in Fig. 1c, and the radial gradient of the verticalvorticity is shown in Fig. 1d.

c. Two-celled vortices

In light of observations, laboratory experiments, andboth axisymmetric and three-dimensional numericalsimulations, it is generally believed that many tornadoes(and other atmospheric vortices) have a stagnant core,in which air flows down from above along the centeraxis, diverges horizontally at the surface, and then re-circulates upward along the annulus defined by the ra-dius of maximum winds. Such a flow has come to becalled a ‘‘two-celled’’ vortex. Perhaps the most obviousexample of a two-celled vortex in nature is the hurri-cane, which has a calm eye and generally descendingmotion inside the eyewall. Much like the Burgers’ vor-tex for one-celled vortices, an analytic model for two-celled vortices was found by Sullivan (1959); unfor-tunately, it too only exists in an unbounded domain, andhas the further disadvantage that due to diffusion theazimuthal velocity in the core is substantial. We insteadpresent a simple model that produces a stagnant-corevortex in a finite domain. We define the radial velocityU(r) to have inflow outside some radius, and outflowaway from the r 5 0 axis, with a stagnation point inbetween:


FIG. 4. The LDMs for the first two azimuthal wavenumbers in the one-celled vortex: (a) contours of vorticity for k 5 1, (b) contours ofstreamfunction for k 5 1, (c) contours of vorticity for k 5 2, (d) contours of streamfunction for k 5 2. The contour intervals and decayrates (the real part of the eigenvalue) are indicated at the top of each plot.

0 r , 0.2

r 2 0.220.02 sin p 0.2 , r , 11 21.6 r 2 1

U(r) 5 0.035 cos p 2 0.15 1 , r , 31 22.0

r 2 3220.05 cos p 3 , r , 6.51 27.0

0 r . 6.5.

(3.10)

This radial velocity function is shown in Fig. 2a. Ap-plying our method, where again we have defined the

circulation at the outer boundary Gb 5 2p, and the vis-cosity n 5 0.001 so that ReV 5 1000, we find the az-imuthal velocity profile shown in Fig. 2b. The associateddeformation function is shown in Fig. 2c, while theresulting vertical vorticity gradient is shown in Fig. 2d.We can see that outside the radius of maximum windsat r 5 2, the velocity profile is nearly exactly that of apotential flow, while inside the radius of maximumwinds the velocity quickly drops to zero and the innercore is stagnant. The vorticity gradient changes sign inthe transition region 1 , r , 2, where the flow tran-sitions from the potential flow to the stagnant inner core,indicating the possibility for instability as predicted byRayleigh’s (1880) theorem for growing disturbances ininviscid rotating flows.


FIG. 5. Stability diagram for asymmetric perturbations on the two-celled vortex, showing the real parts of the LDMs for each azimuthalwavenumber from k 5 1 to k 5 15: V—standard case with inflow,1—with stretching term removed, 3—with stretching and advectionremoved.

4. The evolution of vertical vorticity perturbationsin idealized vortices with radial inflow

a. Mathematical models of wave–mean flowinteractions near and inside the vortex core

Investigations of atmospheric vortex dynamics facethe problem of using finite resources to investigate dy-namics in an unbounded domain. Most work on vortexdynamics has used ‘‘solid wall’’ boundary conditions atsome distance outside the vortex core to close the system(Howard and Gupta 1962; Staley and Gall 1979; Gall1983; Staley and Gall 1984; Gall 1985; Steffens 1988;Peng and Williams 1991). This practice is analogous tothe approximation of the midlatitude jet by boundedchannel flows, and it is based on the reasonable as-sumption that disturbances to the mean flow that mightlie outside of the ‘‘outer wall’’ will not unduly influencethe results near or inside the vortex core. The results ofsuch analyses have generally been consistent with re-sults derived from unbounded solutions based on semi-analytic methods such as contour dynamics (Rotunno1978; Flierl 1988), although Steffens (1988) did identifysome significant changes in growth rates of a particularclass of instabilities when the outer wall was movedfarther away from the vortex core. We used this tech-nique to limit the domain, and found through numericalexperimentation that the location of the outer wall didnot affect the important results (the kind of results thatare affected by the location of the outer boundary arediscussed below in section 5b).

b. The linearized evolution of vertical vorticityperturbations

The specific vortex flows we will study have beenoutlined in section 3, but generally speaking we wish

to describe the evolution of vertical vorticity pertur-bations in a swirling flow that has deformation and radialinflow that are functions of radius only. We restrict ourattention to the dynamics of the vertical vorticity com-ponent z, in cylindrical coordinates, where it is assumedto have no variation in the vertical direction:

2 2]z ]z y ]z ]w ] z 1 ]z 1 ] z1 u 1 5 z 1 n 1 1 .

2 2[ ]]t ]r r ]u ]z ]r r ]r r ]u

(4.1)

Now, we write each term in (4.1) as the sum of a radiallyvarying mean and azimuthally, radially, and temporallyvarying perturbations: u 5 U(r) 1 u9(r, u, t) , z 5 Z(r)1 z9(r, u, t), and so on for y and w. Substituting theseexpressions into (4.1), we find the first-order equationfor the perturbations:

] ] ] ]Zz9 1 U z9 1 V z9 1 u9

]t ]r ]u ]r

2 2]W ] 1 ] 1 ]5 z9 1 n 1 1 z9, (4.2)

2[ ]]z ]r r ]r r ]u

where we have written V for the mean angular velocityV/r. The last term on the left-hand side represents theconversion of mean-flow vorticity to perturbation vor-ticity, which can result in instability.

Due to the azimuthal homogeneity of the backgroundvortex, we can separate the solutions of (4.2) by writingthem as a sum of harmonically varying azimuthalwaves, that is, z9(r, u, t) 5 Sk zk(r, t)eiku, and so on forthe perturbations of u and y also. Substituting theseforms into (4.2), we create for each wavenumber k alinear equation for the evolution of the radially and tem-porally varying vorticity function zk(r, t):

] ] ]Z1 U(r) 1 ikV(r) z 1 uk k1 2]t ]r ]r

2 2]W ] z 1 ]z kk k5 z 1 n 1 2 z . (4.3)k k2 2[ ]]z ]r r ]r r

From here on we will use the convention that the termsuk, y k, zk refer to complex amplitude functions of r andt only.

c. The evaluation of velocities from the vorticity

As we can see from (4.3), when there is a nonzerobackground vorticity gradient, obtaining the evolutionof the perturbation vorticity requires knowledge of theradial velocity perturbations. Furthermore, we anticipatethe need to evaluate both uk and y k in the calculationof the perturbation kinetic energy and the eddy mo-mentum fluxes. Following Carr and Williams (1989) andSmith and Montgomery (1995), we find the velocitiesby solving for the streamfunction:


FIG. 6. Streamfunction and vorticity fields for the most unstable (least damped) perturbations at the first wavenumber with instability, k5 3, and the wavenumber with highest instability, k 5 5: (a) k 5 3 vorticity; (b) k 5 3 streamfunction; (c) k 5 5 vorticity; (d) k 5 5streamfunction.

ikuc(r, u, t) 5 c (r, t)e (4.4)O kk

21 ]c 2ikku 5 5 c (4.5)k kr ]u r

]cky 5 (4.6)k ]r

1 ] ]ukz 5 (ry ) 2k k[ ]r ]r ]u

2 2] 1 ] k5 1 2 c . (4.7)k2 2[ ]]r r ]r r

As previously discussed, we choose the boundary con-ditions of no normal flow at the outer boundary r 5 b;that is,

ck(0, t) 5 ck(b, t) 5 0. (4.8)

Given the vorticity, in the case of continuous functionsEq. (4.7) may be inverted with a Green’s function:

b

c (r, t) 5 G (r, r)z (r, t) dr. (4.9)k E k k

a

The Green’s function appropriate for this problem is[from Carr and Williams (1989), with the inner bound-ary a set to zero]


FIG. 7. Angular velocity, or phase speed, of the LDMs in thetwo-celled vortex for each azimuthal wavenumber from k 5 1 tok 5 15.

2k rk11 2k 2k11(r 2 b r ) 0 # r # r

k 2k2kr bG (r, r) 5k

2k 2kr 2 bk11 r r # r # b.

k 2k2kr b

(4.10)

d. The solution and analysis of the system

We would like to express the evolution of the per-turbations as a linear dynamical system. We discretizethe domain by assigning the values of the radial func-tions to evenly spaced points from r 5 0 1 Dr to r 5b 2 Dr, each point separated by a distance Dr. Thisconverts the continuous radial functions into vectors oflength N 5 (b/Dr) 2 1. For all calculations we use Dr5 0.05 so that there are 139 grid points between theaxis and the outer boundary. We express all derivativesas matrix operators corresponding to the usual centered-difference approximations, with the exception that thefinite-difference operator used for the advection term isone-sided so that it represents a second-order upwindingadvection scheme. We must also express the Green’sfunction operation (4.9) as a matrix operation; that is,

ck 5 Gkzk, (4.11)

where k refers to the wavenumber, and the operator Gk

is defined as above according to (4.9).Finally, we take vorticity evolution equation (4.3) and

manipulate it into a form with only the time derivativeon the lhs, and install the matrix operators accordingly.The result is

dzk 5 T z , (4.12)k kdt

with

21T 5 2UD 2 ikV 1 (DZ)ikR G 1 Sk up k

2 21 2 221 n(D 1 R D 2 k R ), (4.13)

where we have written D for the matrix representingthe finite-difference calculation for the derivative withrespect to r, Dup for a similar but upwinded derivativeoperator, S for the ‘‘stretching’’ term ]W/]z, and (4.5)was used for the third term on the rhs. Operators thatrefer to functions of r only, such as the R21 operator,are simply diagonal matrices with the function valueson the diagonal. We must also incorporate into the dif-ference operators additional boundary conditions on thevorticity, which we choose to be

zk(0) 5 zk(b) 5 0. (4.14)

This condition prevents the advection and diffusion ofperturbation vorticity into the domain from the bound-aries. The solution of (4.12) in time is

zk(t) 5 .T tke z (0)k (4.15)

e. The kinetic energy of the perturbations

The kinetic energy for each perturbation in contin-uous space is

b 2 2u yk kE 5 1 2pr dr, (4.16)E 1 22 20

where the overbars refer to averages around the azimuthof the real parts of the complex velocity functions. Astreamfunction–vorticity formulation for the energy canbe found by using (4.5)–(4.9) and integrating by partsto find

b1E 5 2 c z 2pr dr. (4.17)E k k2 0

Using the fact that pq 5 (p*q 1 q*p), we can rewrite14

(4.17) asbp

E 5 2 (c*z 1 z*c )r dr. (4.18)E k k k k4 0

As discussed in section 2 it is useful to find an energymetric operator M such that the energy of the discretizedlinear dynamical system may be written E 5 z*Mz. Foreach azimuthal wavenumber k, the energy metric canbe formulated from (4.18):

2pDr †M 5 [G R 1 RG ]. (4.19)k k k4

5. Stability

a. The stability of the one-celled vortex

We first consider the one-celled vortex as describedin section 3b. The nearly identical Burgers’ vortex so-lution has been previously found to be stable to all two-dimensional disturbances by Robinson and Saffman


FIG. 8. The k 5 1 and k 5 2 IOs for the one-celled vortex: (a) k 5 1 vorticity; (b) k 5 1 streamfunction; (c) k 5 2 vorticity;(d) k 5 2 streamfunction. The contour intervals and energy growth rates (eigenvalues) are indicated at the top of each plot.

(1984); however, their analysis relied on a low-Reynoldsnumber expansion, which clearly has limited applicationto atmospheric vortices or to our one-celled vortex withReV 5 1000. Figure 3 shows the real part of the eigen-value of the LDM for each azimuthal wavenumber fromk 5 1 to k 5 15, for three cases: 1) with the radialinflow effects included, 2) with the stretching term [Sin (4.3)] eliminated, and 3) with both the stretching andradial advection terms [S and 2UDup in (4.3)] elimi-nated. We can see that for all three cases the one-celledvortex is asymptotically stable and the trend of the plotsuggests that no unstable modes exist for any azimuthalwavenumber. The decay rate of these modes is clearly

linear with respect to azimuthal wavenumber, with theexception that the k 5 1 decay rate appears exception-ally smaller than the trend of the other LDM decay rateswould indicate. This decay rate linear in k indicates thatit is the interaction with the shear of the swirling flow,rather than diffusion, that contributes most to the declineof the amplitudes of these modes.

Figure 3 also shows that the growth rates of the LDMsin all three cases are virtually identical for all k . 1.The reasons for this will be explained shortly. For k 51, however, we have two additional observations: 1)removing the stretching term substantially decreases thegrowth rate of the LDM, and 2) removing completely


FIG. 9. The IO growth rates for k 5 1 to k 5 15 in the one-celledvortex: V—standard case with inflow; 1—with stretching term re-moved; 3—with stretching and advection removed.

the effects of radial inflow greatly increases the growthrate, from 22.5 3 1023 to 22.65 3 1026 (i.e, almostto neutrality).

The two vortex flows we have constructed have cy-lindrical deformation fields that have two distinct effectson the perturbation vorticities. First, the radial inflowadvects perturbation vorticity into the vortex core (andoutward from the axis in the two-celled vortex). Second,the associated deformation field can cause either vor-ticity amplification (through stretching) or vorticity de-cay (through compression). A comparison of the am-plification rate due to vortex stretching with the actualgrowth rates suggests that the stretching/amplificationassociated with the radial inflow does play a role in thedynamics. In the case of the one-celled vortex, the max-imum stretching rate of 0.005 (see Fig. 1c) is indeedcomparable to the change that results when the stretch-ing is eliminated from the dynamics.

Figure 4 shows vorticity and streamfunction contourplots of the LDM for the first two azimuthal wave-numbers (when the effects of radial inflow are included).These two modes are considerably different. For k 51, the LDM is a dipole-like structure whose vorticitylies entirely inside the core of the vortex. For k 5 2,the LDM lies at the outer edge of the domain and spiralsinward in the same direction as the mean flow. In fact,the LDMs for all higher wavenumbers are higher-wave-number replicas of the k 5 2 LDM.

Why is the k 5 1 LDM distinct from those of allhigher wavenumbers? There are two reasons. First, k 51 perturbations experience considerably less dissipationnear r 5 0 than higher wavenumber perturbations—thisis due to the 2k2/r2 term in the diffusive part of (4.3).Second, the perturbation velocities do not go to zero atr 5 0, which is to say that only for k 5 1 can there beperturbation flow across the r 5 0 axis. In the case ofthe one-celled vortex, this allows the k 5 1 LDM to

convert more mean-flow vorticity to perturbation vor-ticity, thereby helping to sustain itself. It is for thesereasons that the decay rate of the k 5 1 LDM is sosmall, especially when the radial inflow is neglected.1

Because the higher wavenumber perturbations are there-fore generally not well suited to persist in the core ofthe vortex, their LDMs lie instead as far away from thecore as possible so that they are subjected to as littlediffusion and deformation as possible. For this reason,the structures and locations of the LDMs for k 5 2 andhigher are entirely dependent on the location of the outerboundary of the domain. The fact that the LDMs for k. 1 lie near the outer boundary also explains why elim-inating the effects of radial inflow has no noticeableeffect on the growth rates—the stretching and advectionare nearly zero in this region (see Fig. 1).

b. The stability of the two-celled vortex

We turn our analysis now to the two-celled vortexdescribed in section 3c. Figure 5 shows the real partsof the eigenvalues of the LDMs for azimuthal wave-number k 5 1 through k 5 15, where we have againalso plotted the same results with stretching removedand with all radial inflow effects removed. For the casewith inflow, there is a range of instability from k 5 3to k 5 10. This is a ‘‘classic’’ result, which has beenproduced in many previous studies of the stability ofrotating flows, particularly Staley and Gall (1979), Gall(1985), Steffens (1988), and Peng and Williams (1991).Our curve is slightly different from these earlier resultsdue to the presence of diffusion, so that the stable modesare not neutral but rather have negative growth ratesoutside the unstable range.

It is interesting to note that removing the stretchingterm increases the growth rate for k 5 1 and actuallydestabilizes the vortex for k 5 2, yet substantially de-creases the growth rates in the previously unstable range3 # k # 10. Removing both the stretching and theadvection destabilizes the vortex for k 5 1 and k 5 2,but again decreases the growth rates for 3 # k # 10.The growth rates for k . 10 are unaffected by radialinflow, and this is again due to the fact that the LDMsfor these wavenumbers lie near the outer boundarywhere there is virtually no stretching or advection.

The vorticity and streamfunction fields for the mostunstable (least damped) mode for the first azimuthalwavenumber with instability, k 5 3, and the most un-

1 Note added in proof: The LDM for k 5 1 in the one-celled vortexis in fact a slightly modified version of the stationary solution forwavenumber one perturbations found by Michaelke and Timme(1967) for unbounded domains, which is a disturbance whose stream-function is proportional to the mean flow velocity. Such a disturbancerepresents a linear displacement of the vortex center, and is sometimescalled the ‘‘pseudo-mode’’ in the literature. The modification in ourcase is caused by the introduction of an outer boundary, which causesthe mode to have a nonzero angular velocity.


FIG. 10. Vorticity and streamfunction fields for the k 5 1 FTO for t 5 8.98, and its associated RFTO: (a) k 5 1 FTOvorticity, (b) k 5 1 FTO streamfunction, (c) k 5 1 RFTO vorticity, (d) k 5 1 RFTO streamfunction.

stable wavenumber, k 5 5, are shown in Fig. 6. Thesemodes are quite similar in structure, consisting of twoconcentric rings of ellipse-shaped vorticity perturba-tions, alternating in sign, located in the transition re-gion between r 5 1 and r 5 2, with the two rings ofperturbations shifted in phase from each other such thatthe inner rings lead the outer rings by almost exactlyhalf a cycle. These modes operate with a mechanismthat is the same as the two-dimensional modes iden-tified in the earlier vortex stability studies notedabove—see, in particular, Staley and Gall (1979), Gall(1983), and Flierl (1988). That is to say, they persistand/or grow through the familiar mechanism of con-verting mean flow vorticity to perturbation vorticity

via the uk(]Z/]r) term in (4.3). In Fig. 7, the modesshow another common aspect of classical instabilitiesin that the angular velocity (phase speed) of the un-stable modes is representative of the mean flow angularvelocity in the transition zone, while the decayingmodes have a much lower angular velocity: for k 5 1and k 5 2, this is because the modes are retrogradingagainst the mean flow, while for k . 10 it is becausethe modes lie near the outer boundary where V is small.The maximum angular velocity V /r in the two-celledvortex is V 5 0.23, while halfway through the tran-sition region the angular velocity is V 5 0.15, whichcompares very favorably with the phase speeds of theunstable modes.


FIG. 11. FTO growth vs allowed growth time in the one-celledvortex for (a) azimuthal wavenumber k 5 1; (b) k 5 2.

6. Optimal growth perturbations

As Orr (1907) originally observed, the growth of aperturbation in linear, inviscid shear flow is determinedsolely by how far back against the shear the disturbanceis originally tilted. A perturbation whose phase lines aretilted all the way back becoming nearly parallel withthe flow could conceivably have unlimited growth. Inmore realistic flows this is prevented by diffusion. Withthis in mind, it is easy to imagine that the perturbationthat would grow the most in an asymptotically stableinviscid vortex without radial inflow would be a vor-ticity perturbation that spirals back against the flow atan angle such that azimuthal advection would uncoil thevorticity of the perturbation, reducing its radial wave-number until the vorticity contours were radially alignedat the moment of maximum perturbation amplitude. Vis-cosity would limit the possible maximum growth of sucha perturbation: the tighter the spiral (in either direction)made by the perturbation, the faster viscosity woulddiffuse the vorticity.

The structures of IOs are usually quite different. The

IOs seek out the part of the flow that has the most shearor deformation, and arrange the vorticity there to createthe fastest possible instantaneous growth in perturbationenergy. This is usually a structure that locally tilts backagainst the shear of the flow at an angle of 458, a ge-ometry that maximizes the eddy fluxes u9y9 , and at thesame time places the perturbation in the orientation lead-ing to the maximum rate of decrease in wavenumberfrom advection by the local difluent component of theshearing velocity field.

a. Instantaneous and finite-time optimals in theone-celled vortex

Figure 8 shows the vorticity and streamfunction fieldsfor the k 5 1 and k 5 2 IOs on our one-celled vortex.We can see these perturbations have structures verymuch like those predicted above: the vorticity contoursspiral back against the shear of the mean flow. Thestreamfunction contours show that the perturbation ve-locities flow back against the shear of the vortex at theexpected angle of 458, which maximizes the eddy fluxes,and the place where this occurs is in the vicinity of r5 1, where the maximum shear exists (note that in avortex, the ‘‘shear’’ is not the rate of change of velocity]V/]r but rather the rate of change of angular velocity]V/]r). The k 5 2 IO is essentially a higher-wavenumberreplica of the k 5 1 IO, and this was found to be truefor all wavenumbers. The normalized IO growth ratesfor k 5 1 through k 5 15 are shown in Fig. 9, againfor the three cases examined in our discussion of sta-bility: with the radial inflow, without the stretching, andwithout the radial inflow. The growth rate (with inflow)increases from 0.42 at k 5 1 to 0.55 at k 5 4 and thenquickly decreases. While such growth rates are knownin inviscid flows to increase asymptotically with wave-number toward a limit determined by the maximum de-formation of the mean flow (Howard 1972), the declineafter k 5 4 in our vortex is caused by the increasingeffects of viscosity on higher-wavenumber structures.Figure 9 also indicates that the stretching and advectionterms make very little contribution to the transientgrowth process, which is due to the fact that the de-formations of the mean radial and vertical velocities aretwo orders of magnitude smaller than the shear of themean azimuthal velocities. The fact that the eliminationof the advection terms seems to nearly cancel the effectof eliminating the stretching term is very likely due tothe fact that by incompressibility (3.5) these terms mustbe equal in magnitude.

Figures 10a,b show the vorticity and streamfunctionfor the k 5 1 FTO for a transient growth time of t 58.98. This corresponds to the time it takes for the meanflow to travel one circuit around the vortex at r 5 1. Weagain see the expected result that the perturbation spiralsback against the mean flow, and it does so almost exactlyonce. We also note, however, that the vorticity of theFTO lies not in the immediate vicinity of r 5 1 but rather


FIG. 12. Initial and fully realized states of the global optimal for k 5 1 in the one-celled vortex: (a) GO vorticity, (b) GO streamfunction,(c) RGO vorticity, (d) RGO streamfunction.

at a small distance outside of r 5 1. This displacementis indicative of the effect of radial inflow: as the vorticityis uncoiled, it is also carried inward toward r 5 1. Figures10c,d show the structure of this perturbation when it hasreached its maximum energy at t 5 8.98; we call thisstructure the realized finite time optimal (RFTO), andobtain it from the SVD of the propagator as outlined insection 2. This confirms the hypothesis that the pertur-bation reaches its maximum energy when the vorticitycontours have been completely uncoiled, and it alsoshows that the radial inflow has carried the vorticity intothe core of the vortex during this process.

Figure 11 shows the maximum transient growth in

energy as a function of time for k 5 1 and k 5 2perturbations in the one-celled vortex. For k 5 1, themaximum possible normalized energy growth is a factorof 181, which occurs at the time of t 5 131; the per-turbation that realizes this growth is called the globaloptimal (GO), and its maximal state will be called therealized global optimal (RGO). We see that the potentialfor transient growth of k 5 2 perturbations is substan-tially less than that of k 5 1 perturbations, with a max-imum potential growth factor of only 33 occurring at t5 17. The GO for k 5 1, and its realization, are shownin Fig. 12. The GO is a structure whose vorticity spiralsfar back against the mean flow, and it lies in the vicinity


FIG. 13. Vorticity and streamfunction fields for the k 5 1 and k 5 2 IOs in the two-celled vortex: (a) k 5 1 IO vorticity,(b) k 5 1 IO streamfunction, (c) k 5 2 IO vorticity, (d) k 5 2 IO streamfunction.

of r 5 5, which is the furthest extent of the field ofsubstantial inflow velocities (see Fig. 1a), while theRGO shows again that this perturbation reaches its max-imum energy when its vorticity has been advected intoa dipolelike structure in the core of the vortex.

b. IOs and FTOs in the two-celled vortex: The stableregime

As we saw above, the two-celled vortex we havegenerated is stable for azimuthal wavenumbers k 5 1,k 5 2, and k . 10. Thus we can expect that wave–meanflow interactions at these stable wavenumbers will be

dominated by the optimal transients. Figure 13 showsthe IOs for k 5 1 and k 5 2. The vorticities of theseperturbations are again arranged to create downgradienteddy fluxes at the location of the maximum shear in themean flow, which lies in the center of the transitionzone between the exterior potential flow and the stag-nant interior. Figure 14 shows the IO growth rates forazimuthal wavenumbers k 5 1 through k 5 15, includ-ing the cases with no stretching and with no radial in-flow. Comparison with Fig. 5 indicates that the transientoptimals can grow faster than their corresponding LDMsfor all wavenumbers, even when unstable modes arepresent (note that the rate of energy increase is equal


FIG. 14. Maximum instantaneous growth rates vs wavenumber for the first 15 azimuthal wavenumbers in the two-celled vortex: V—standard case with inflow, 1—with stretching term removed, 3—with stretching and advectionremoved.

to twice the modal growth rate). We also see that in thetwo-celled vortex, removal of the stretching term has asignificant effect on the growth rate; this is due to thefact that the stretching term is only one order of mag-nitude (rather than two as before) less than the shear ofthe mean azimuthal velocities. We also see again thatthe elimination of advection nearly balances the elim-ination of stretching.

Figure 15 shows the maximum transient growth as afunction of time for k 5 1 and k 5 2 in the two-celledvortex. While the growth in energy by a factor of 94for k 5 1 is substantial, the growth by a factor of 2886for k 5 2 is tremendous. The initial and maximal statesof the k 5 2 GO can be seen in Fig. 16 (these structuresfor k 5 1 were similar). The initial state is the familiarreverse spiral, also displaced outward from the vortexcore, like those we saw for the one-celled vortex. Themaximal state, however, is considerably different thanthose we saw in the one-celled vortex. Rather than form-ing into a coherent quadripole structure (which we mightexpect for the maximal state of a wavenumber two op-timal), the GO evolves into a structure whose vorticityfield is a lower-wavenumber version of the unstablemodes (see Fig. 6). Why does this occur? As the GO

is deformed and advected into the vortex core, it inter-acts with the vorticity gradient of the mean flow. As itdoes so, it converts mean-flow vorticity into perturba-tion vorticity, thereby modifying the structure it wouldhave if the only effect of the mean flow were defor-mation of the perturbation vorticity field. While anyinitial perturbation in any linearized system will indeedbe dominated by the LDM in the limit as t → `, thereason that we find such large transient growth for lowwavenumbers in the two-celled vortex is that the energyacquired from the mean flow via the Orr mechanismaccumulates in these very persistent LDMs, rather thanbeing returned to the mean flow as the perturbation issheared over. This result is very similar to what wasfound by Smith and Montgomery (1995) in their anal-ysis of evolving disturbances in an unbounded Rankinevortex: as an initial disturbance is sheared over, muchof its energy can be trapped in discrete modes and persistfor long times.

c. Transient and long-term growth in the unstableregimeAre transient growth processes relevant in the unsta-

ble regime? Since the eigenvectors of Tk (for each par-


FIG. 15. FTO growth as a function of allowed growth time in thetwo-celled vortex: (a) azimuthal wavenumber k 5 1, (b) k 5 2.

ticular wavenumber k) are not orthogonal to each other,an arbitrary perturbation will almost surely project tosome extent onto the most unstable mode, thereby ex-citing an exponentially growing perturbation. The totalperturbation will then be asymptotically dominated bythe unstable LDM. For linear systems with normal dy-namical operators, the initial condition that achieves thegreatest long-term growth would not surprisingly be theLDM itself. It is interesting to note, however, in non-normal systems such as these vortex/shear flows, theperturbation that excites the LDM the most for all timesis not the LDM itself, but rather the LDM of the adjointoperator A† (hereafter referred to as the LDMA). Thiswas remarked by Farrell (1988) for neutral Rossbywaves, but the argument is general and has been furtherdiscussed by Farrell and Ioannou (1996).

To see this mathematically, first recall that the solu-tion in time to our linear dynamical system in gener-alized velocity coordinates (2.5) is

x(t) 5 eAtx(0). (6.1)

Now consider the following standard diagonalization ofthe propagator matrix

eAt 5 EeDtE21, (6.2)

where D is a diagonal matrix with the eigenvalues ofA on its diagonal and E is a matrix whose columns arethe eigenvectors of A. We may assume that the eigen-values and eigenvectors are ordered such that the LDMlies in the first column of E and its eigenvalue is thefirst element of D. For large times the resultant prop-agator matrix is dominated by the contribution of thedominant eigenmode, the LDM. Therefore,

At D t 2111lim e 5 E e E . (6.3)kl k1 1lt→`

The dominance of the LDM is indicated by the fact thatit is the only one of the eigenvectors of A that appearsin this limit. What initial condition maximizes (in anormalized sense) the response of the LDM for largetimes? It is the one that maximizes its inner productwith the rightmost term of (6.3). By Schwarz’s inequal-ity and our definition of the energy norm (2.6), this isthe complex conjugate of the rightmost term, that is,( )*. Finally, since the diagonalization of the adjoint21E1l

propagator is† †A t 21 † D t †e 5 (E ) e E , (6.4)

we observe that our desired initial condition is identicalto the LDM of the adjoint of the dynamical operator A.

Figure 17 shows the logarithm of the energy as afunction of time for three k 5 3 perturbations on thetwo-celled vortex: the LDM, the LDMA, and the FTOfor t 5 5. The LDM undergoes a steady exponentialgrowth rate from t 5 0 onward. The LDMA grows inenergy even more rapidly than the LDM for a finiteperiod of time, and then settles into the same exponentialgrowth rate, but with 4.04 times as much energy at latertimes than the LDM. The FTO has an even faster initialgrowth rate, but after longer times has less energy thanthe LDMA. Examination of the LDMA and FTO vor-ticity and streamfunction fields, as shown in Fig. 18, isrevealing. These structures are similar to the LDM inthe vortex core, but outside of r 5 2 they have thefamiliar reverse spiral. Thus we can see that their‘‘extra’’ energy growth for short times comes from thetransient growth associated with the uncoiling of thereverse spirals. We note that all the FTOs for growthtimes t . 20 (not shown) were identical to the LDMA.While for long times the structures that will dominatethe perturbations for the unstable wavenumbers are in-deed the unstable LDMs, the initial condition that leadsthe most rapid appearance of the LDM is in fact theLDMA.

7. Analysis

In this section we will examine two issues not directlyaddressed by the identification of the least damped


FIG. 16. Initial and maximal states of the GO for k 5 2 on the two-celled vortex: (a) GO vorticity, (b) GO streamfunction, (c) RGOvorticity, (d) RGO streamfunction.

modes and/or the transiently growing optimals. The firstissue will concern the exchange of energy between theperturbations and the mean flow, and the second issuewill be the effect of radial inflow on transient growthprocesses.

a. Wave–mean flow interactions

Energy is exchanged between the mean flow and theperturbations via eddy fluxes. These eddy fluxes affectthe mean flow through the divergence of the volume-averaged eddy flux terms (i.e., the Reynolds stresses)previously neglected in the mean flow equations such

as (3.6). The change to the mean azimuthal velocitycaused by the perturbation eddy flux divergences is(Carr and Williams 1989; Montgomery and Kallenbach1997)

]V 1 ]25 2 (r u9y9) 5 2u9z9 , (7.1)

2]t r ]r

where the overbars refer to azimuthal averages and theprimes refer to the perturbation velocities and vortici-ties. These terms can be easily calculated to indicate—in the linear limit where we assume that the actualchange of the mean flow is negligible—where the meanflow energy exchange is taking place and how the mean


FIG. 17. Logarithms of the energies vs time of the LDM (solid),LDMA (dashed), and the t 5 5 FTO (dash–dot) for azimuthal wave-number k 5 3 in the two-celled vortex.

flow would indeed be changed if the perturbations wereof substantial amplitude. The perturbations themselves,being asymmetric, have zero net momentum. Theychange the kinetic energy of the mean flow by rear-ranging its momentum via eddy momentum fluxes. Up-gradient (downgradient) momentum fluxes cause an in-crease (decrease) in mean flow kinetic energy. This canbe seen from examination of the equation for the rateof change of perturbation kinetic energy (for the casewe are studying: strictly two-dimensional perturbationsin cylindrically symmetric mean flows)

`dE ]U U ]V V2 25 2 u 1 y 1 uy 2 2pr dr,E 1 2[ ]dt ]r r ]r r0

(7.2)

where the overbars refer to the azimuthal averages. Forconvenience we have neglected the viscous dampingterms. Another useful form of this equation can be foundby integrating the first and third terms in the integrandof (7.2) by parts

` 2dE ] u U2 25 u 1 U 2 y 1 u9z9V 2pr dr,E 1 2[ ]dt ]r r r0

(7.3)

where we have used (7.1) to simplify the third term inthe integrand. This equation has a clear physical inter-pretation since the first and third terms now representthe products of the eddy flux divergences (which areaccelerations) multiplied by the mean flow velocities,thereby indicating the interaction of the perturbationswith the mean flow. The second term represents thework done by the mean flow against the centripetal forc-es associated with the perturbation azimuthal velocities.Note that it is always positive when U is negative. The

third term shows the direct interaction of the eddy fluxdivergence with the mean azimuthal flow. Since the rateof change of energy of the perturbations is equal andopposite to the rate of change of energy of the meanflow, this term clearly shows how upgradient (down-gradient) momentum fluxes cause an increase (decrease)in the kinetic energy of the mean flow.

The eddy flux divergence of azimuthal momentum as-sociated with the k 5 1 IO in the one-celled vortex isshown in Fig. 19a. We can see that the immediate effectof the perturbation is to decelerate the mean flow insidermax and to accelerate the flow outside rmax. The accu-mulated forcing of this perturbation over its lifetime canbe found by simultaneously integrating the evolution ofthe perturbation (4.12) and the effect on the mean flow(7.1). We performed this integration from t 5 0 to t 51000, during which the energy of this perturbation de-creased from E 5 1 to E 5 0.005. The accumulatedforcing on the mean flow is shown in Fig. 19b, wherewe can see that the net effect of the wave–mean flowinteraction for this perturbation over its lifetime is todecrease the azimuthal velocity in the vicinity of r 5 1and to increase it near r 5 0 and near r 5 2. [Note: thisresult should not be confused with the actual deviationof the mean flow velocities. The symmetric perturbationsto the vortex flow caused by the eddy flux divergenceswill themselves evolve according to an advection–dif-fusion equation similar to (3.6). The behavior of theseperturbations will be addressed in a future paper.]

The same calculation for the k 5 2 IO in the one-celled vortex gives a different result. While we can seein Fig. 20a that the instantaneous effect of this pertur-bation at t 5 0 is to decelerate the mean flow at r 51, the long-term effect shown in Fig. 20b is to acceleratethe flow near r 5 1, such that the kinetic energy of themean flow actually increases rather than decreases. Whyis this result different? Shepherd (1985) and Farrell andIoannou (1993a) have shown for inviscid linear shearflows that all perturbations, regardless of their initialconfiguration, are sheared over and give their energy tothe mean flow via eddy fluxes. Equivalent results werefound by Carr and Williams (1989) and Nolan (1996)for vortices with 1/r velocity profiles. Thus the net effectof any perturbation to these flows will ultimately be toincrease the kinetic energy of the mean flow. However,if the mean flow has a background vorticity gradient,the transient perturbations may excite neutral or nearlyneutral modes, such as the k 5 1 LDM, which thenserve as a trap for perturbation kinetic energy. In thiscase, the energy of transiently growing perturbations isnever returned to the mean flow, but rather lost throughdissipation of the perturbation instead. Our results herefor the k 5 1 and k 5 2 IOs in the one-celled vortexare examples of each of these two possible outcomes.

In the two-celled vortex, the change in sign of thebackground vorticity gradient lends to the persistenceor growth of coherent structures at all wavenumbers.Through investigation of many different initial condi-


FIG. 18. Vorticity and streamfunction fields for the t 5 5 FTO and the LDMA for k 5 3 in the two-celled vortex: (a) FTO vorticity, (b)FTO streamfunction, (c) LDMA vorticity, (d) LDMA streamfunction.

tions, it appears that any perturbation initially config-ured to project favorably onto the LDM ultimately leadsto a decrease in the kinetic energy of the mean flow.However, perturbations can be chosen that will ulti-mately lead to an increase in the kinetic energy in themean flow. This is demonstrated in Fig. 21, where wehave used as an initial condition the complex conjugateof the k 5 11 IO in the two-celled vortex. Taking thecomplex conjugate results in a perturbation that spiralsin the reverse direction, so that the initial eddy mo-mentum flux is maximally upgradient rather than max-imally downgradient. Since the k 5 11 LDM is not aspersistent as those at lower wavenumbers, the structure

is sheared over and the final result is a net upgradientflux of momentum and an increase in mean flow kineticenergy. For all lower wavenumbers with this specialinitial condition the opposite result was found.

b. The effects of radial inflow on transient growth

In previous sections we determined how accountingfor the radial inflow changes the stability and optimalgrowth rates for perturbations in these vortices. How-ever, it is perhaps more important to investigate howthe stability of these vortices change as the strength ofthe deformation field changes. In the study of atmo-


FIG. 19. (a) The instantaneous eddy momentum flux divergence ofthe k 5 1 IO in the one-celled vortex. (b) The accumulated totalmomentum flux divergence from t 5 0 to t 5 1000 for the k 5 1IO.

FIG. 20. (a) The instantaneous eddy momentum flux divergence ofthe k 5 2 IO in the one-celled vortex. (b) The accumulated totalmomentum flux divergence from t 5 0 to t 5 1000 for the k 5 2IO.

spheric vortices it is practically assumed that a strongradial inflow will hold the vortex together, while a weak-er inflow may allow the vortex to break apart. In ourcase we can examine how important the radial inflowis in stabilizing the vortex, as a function of the strengthof the inflow itself. In the case of the one-celled vortex,a weaker radial inflow results in a broader, weaker vor-tex, that is, rmax increases and ymax decreases; a strongerradial inflow intensifies the vortex. In the case of thetwo-celled vortex, increasing the strength of the defor-mation field results in a sharper azimuthal velocity gra-dient in the transition zone from the potential flow tothe stagnant core.

To determine the change in the potential for wave–mean flow interaction as the radial inflow varies, wehave varied the strength of the radial inflow from halfits original value to twice its original value, while re-calculating the associated azimuthal velocity profile. Fora radial inflow of half the original strength, we obtainan azimuthal velocity profile that is of the Burger’s so-

lution type but with an increased rmax 5 1.45 and adecreased ymax 5 0.5; for twice the radial inflow wehave rmax 5 0.70 and ymax 5 1.01. For each of thesenew vortex solutions we have recalculated both the max-imum finite energy growth and this same growth whenthe radial inflow terms are neglected in the dynamics.The results of these calculations for k 5 1 and k 5 2for the one-celled vortex are shown in Fig. 22. In bothcases we see the following: first, neglecting the radialinflow terms results in a substantial overestimation ofthe maximum growth. Second, this overestimation in-creases as the strength of the radial inflow increases.Third, we find that for k 5 2 and all higher azimuthalwavenumbers (not shown), the maximum growth (in-cluding the radial terms) begins to decrease when theradial inflow becomes strong enough.

Why does the presence of radial inflow velocity sup-press transient growth? Indeed, this might seem para-doxical since increasing the radial inflow increases themaximum deformation rate of the mean flow. However,


FIG. 21. (a) The instantaneous eddy momentum flux divergence ofthe complex conjugate of the k 5 11 IO in the two-celled vortex. (b)The accumulated total momentum flux divergence from t 5 0 to t 51000.

FIG. 22. Comparisons of the maximum transient growth in the one-celled vortex when the radial inflow terms are included (V’s) or arenot included (3’s), as a function of the intensity of the radial inflowthat maintains the swirling flow: (a) for azimuthal wavenumber k 51; (b) for k 5 2.

the radial inflow also has the effect of advecting per-turbation vorticity through the region of maximumshear, thereby limiting the extent of the wave–mean flowinteraction.

Figure 23 shows similar calculations as above for k5 1 and k 5 2 in the two-celled vortex. Since the two-celled vortex is unstable for these wavenumbers whenradial inflow is neglected, we show the results only forthe case with radial inflow. In this case we see againthat the potential for transient growth begins to declinefor larger relative intensities of the radial inflow. Similarcalculations (not shown) for the high range of stablewavenumbers for the two-celled vortex, that is, k . 10,demonstrated that the maximum growth was underes-timated by neglecting the radial inflow terms. This isbecause the mean vorticity to perturbation vorticitymechanism for growth is less effective for these higherwavenumbers and the radial deformation is a significanttransient growth mechanism. However, the maximumgrowth for these wavenumbers is very small—on the

order of only a factor of 10 or less with increasing k,and the maximum growth calculated with the radial in-flow terms was only 5%–10% more than without radialinflow.

8. Summary and conclusions

We have studied the dynamics of asymmetric pertur-bations in two-dimensional vortices that are maintainedby radial inflow. The results we have found regardingstability are consistent with previous work on vorticesof various velocity profiles: the one-celled vortex is sta-ble for all azimuthal wavenumbers, while the two-celledvortex has a finite range of unstable wavenumbers dueto the change in sign of the vorticity gradient of themean flow. We also showed that the initial conditionsthat lead to the greatest perturbation energy for longtimes are not the most unstable modes but rather themost unstable modes of the adjoint operator.

For all wavenumbers in both vortices the fastest in-


FIG. 23. The maximum transient growth in the two-celled vortexas a function of the intensity of the radial inflow that maintains theswirling flow: (a) for azimuthal wavenumber k 5 1; (b) for k 5 2.

stantaneous growth rates were shown to belong not tothe LDMs but rather to the IOs. We also found that forstable wavenumbers substantial transient growth overfinite times can occur for low-wavenumber perturba-tions in both the one-celled and two-celled vortex. Theenergy source for transiently growing perturbations isthe exchange of energy with the mean flow via eddymomentum fluxes.

While the immediate effect of all growing perturba-tions is to decrease the kinetic energy of the mean flow,we found that the net effect of these perturbations onthe mean flow over their lifetimes varies from case tocase. While most initially growing perturbations resultin a long-term decrease in the mean flow kinetic energyfor all wavenumbers in the two-celled vortex, and forwavenumber k 5 1 in the one-celled vortex, we foundthat most perturbations for k . 1 in the one-celled vortexultimately increased the kinetic energy of the mean flow.

The effect of the radial inflow, generally neglectedin previous stability analyses, was also investigated. Thevortex stretching caused by the associated deformationfield was found to stabilize wavenumber 2 perturbations

in the two-celled vortex despite the fact that the effectof the deformation field is to amplify perturbation vor-ticity. It was also found that the combination of bothadvection and deformation associated with the radialinflow was necessary to stabilize the two-celled vortexto wavenumber 1 perturbations. More relevant to thecase of vortices being maintained by a radial inflow withintensity varying in time, we examined how the wave–mean flow dynamics changed as the azimuthal velocityfield was adjusted to be in balance with changing radialinflow velocities. For all wavenumbers in the one-celledvortex, and for the low, stable wavenumbers in the two-celled vortex, we found that neglecting the radial inflowterms greatly overestimated the potential for transientgrowth or even destabilized the vortex completely. Fur-thermore, these measurable effects on asymmetric dy-namics were found despite the fact that the maximumradial inflow velocities were at most only 10% of themaximum azimuthal velocities in the two-celled vortexand only 1% in the one-celled vortex.

There are two more important conclusions that canbe drawn from our analyses. First, the potential for largetransient growth of disturbances shows that a vortexneed not be formally unstable before significant asym-metric structures, such as multiple vortices or polygonaleyewalls, can appear. Such disturbances may appear if,perhaps by chance, a disturbance is introduced into thevortex that projects favorably onto the optimals tran-sients (i.e., it has an upshear tilt). Second, these favor-ably configured disturbances do not need to be intro-duced near the vortex core, but rather can be introducedin the near-vortex environment and still result in largewave–mean flow interaction (recall the global optimalsshown in Figs. 12 and 16).

Throughout this investigation we have found thatlarge transient growth in perturbation kinetic energy islimited to the lowest wavenumbers, in particular to onlyk 5 1 for the one-celled vortex and both k 5 1 and k5 2 for the two-celled vortex. Physically, k 5 1 per-turbations correspond to a linear displacement of someor all of the vorticity of the total flow. However, thistranslation is important since it corresponds to devia-tions in the path of the vortex, which has particularlyimportant applications in the forecasting of hurricanetracks. It is also generally observed that tornadoes donot travel in straight lines (see, e.g., Fujita and Smith1993) and even laboratory vortices (Wan and Chang1972; Lund and Snow 1993) ‘‘wander’’ considerably,so it is possible that the transient growth and decay ofwavenumber 1 perturbations is responsible for this phe-nomenon for both tornadoes and hurricanes. With thisin mind we can see how generalized stability analysiscould identify the characteristic features of disturbancesthat are most likely to causes significant changes in thepath and intensity of tropical cyclones and tornadoes.

Acknowledgments. The authors would like to thankProf. P. Ioannou, Prof. M. Montgomery, and Dr. T.


DelSole for many helpful discussions. We would alsoespecially like to thank Dr. D. Adalsteinsson for helpingus to make tremendous improvements in the productionand appearance of our contour plots. Preliminary workfor this report was prepared as part of the Ph.D. thesisof D. Nolan while he was a student at Harvard Uni-versity, during which he was supported by NSF GrantATM-9216813 and NSF Grant 9623539. Since Novem-ber 1996 D. Nolan has been supported by the U.S. De-partment of Energy under Contract DE-AC03-76SF-00098. B. Farrell was supported by NSF Grants ATM-9216813 and 9623539.

REFERENCES

Burgers, J. M., 1948: A mathematical model illustrating the theoryof turbulence. Adv. Appl. Mech., 1, 197–199.

Burggraff, O. R., K. Stewartson, and R. Belcher, 1971: Boundarylayer induced by a potential vortex. Phys. Fluids, 14, 1821–1833.

Butler, K. M., and B. F. Farrell, 1992: Three dimensional optimalperturbations in viscous shear flows. Phys. Fluids A, 5, 1390–1393.

Carr, L. E., III, and R. T. Williams, 1989: Barotropic vortex sta-bility to perturbations from axisymmetry. J. Atmos. Sci., 46,3177–3191.

Farrell, B. F., 1982: The initial growth of disturbances in a bar-oclinic flow. J. Atmos. Sci., 39, 1663–1686., 1988: Optimal excitation of neutral Rossby waves. J. AtmosSci., 45, 163–172., 1989: Transient development in confluent and diffluent flow.J. Atmos. Sci., 46, 3279–3288., and P. J. Ioannou, 1993a: Stochastic forcing of perturbationvariance in unbouded shear and deformation flows. J. Atmos.Sci., 50, 200–211., and , 1993b: Optimal excitation of three-dimensionalperturbations in viscous constant shear flow. Phys. Fluids A,5, 1390–1400., and , 1996: Generalized stability theory. Part I: Au-tonomous operators. J. Atmos. Sci., 53, 2025–2040.

Fiedler, B. H., 1994: The thermodynamic speed limit and its vi-olation in axisymmetric numerical simulations of tornado-like vortices. Atmos.–Ocean, 32, 335–359., and R. Rotunno, 1986: A theory for the maximum wind-speeds in tornado-like vortices. J. Atmos. Sci., 43, 2328–2340.

Flierl, G. R., 1988: On the instability of geostrophic vortices. J.Fluid Mech., 197, 349–388.

Franklin, J. L., J. S. Lord, S. E. Feuer, and F. D. Marks Jr., 1993:The kinematic structure of Hurricane Gloria (1985) deter-mined from nested analyses of dropwinsonde and Dopplerradar data. Mon. Wea. Rev., 121, 2433–2451.

Fujita, T. T., and B. E. Smith, 1993: Aerial survey and photographyof tornado and microburst damage. The Tornado: Its Struc-ture, Dynamics, Prediction, and Hazards, C. Church et al.,Eds., American Geophysical Union, 1–17.

Gall, R. L., 1983: A linear analysis of the multiple vortex phe-nomenon in simulated tornadoes. J. Atmos. Sci., 40, 2010–2024., 1985: Linear dynamics of the multiple-vortex phenomenonin tornadoes. J. Atmos. Sci., 42, 761–772.

Guinn, T. A., and W. H. Schubert, 1993: Hurricane spiral bands.J. Atmos. Sci., 50, 3380–3403.

Howard, L. N., 1972: Bounds on flow quantities. Annu. Rev. FluidMech., 4, 473–493.

Howells, P. A. C., R. Rotunno, and R. K. Smith, 1988: A com-

parative study of atmosphere and laboratory-analogue nu-merical tornado-vortex models. Quart. J. Roy. Meteor. Soc.,114, 801–822.

Kallenbach, R. J., and M. T. Montgomery, 1995: Symmetrizationand hurricane motion in an asymmetric balance model. Pre-prints, 21st Conf. on Hurricanes and Tropical Meteorology,Miami, FL, Amer. Meteor. Soc., 103–105.

Leibovich, S., and K. Stewartson, 1983: A sufficient condition forthe instability of columnar vortices. J. Fluid Mech., 126, 335–356.

Lewellen, W. S., and D. C. Lewellen, 1997: Large-eddy simulationof a tornado’s interaction with the surface. J. Atmos. Sci., 54,581–605.

Liu, Y., D.-L. Zhang, and M. K. Yau, 1997: A multiscale numericalstudy of Hurricane Andrew (1992). Part I: Explicit simulationand verification. Mon. Wea. Rev., 125, 3073–3093.

Lund, D. E., and J. T. Snow, 1993: Laser doppler velocimetermeasurements in tornadolike vortices. The Tornado: ItsStructure, Dynamics, Prediction, and Hazards, C. Church, D.Burgess, C. Doswell, and R. Davies-Jones, Eds., Amer. Geo-phys. Union, 297–306.

Molinari, J., 1992: Environmental controls on eye wall cycles andintensity change in Hurricane Allen (1980). Proc. ICSU/WMO Int. Symp. on Tropical Cyclone Disasters, Beijing, Chi-na, World Meteorological Organization, 328–337.

Montgomery, M. T., and R. J. Kallenbach, 1997: A theory forvortex Rossby waves and its application to spiral bands andintensity changes in hurricanes. Quart. J. Roy. Meteor. Soc.,123, 435–465.

Nolan, D. S., 1996: Axisymmetric and asymmetric vortex dynam-ics in convergent flows. Ph.D. thesis, Harvard University, 279pp. [Available from University Microfilm, 305 N. Zeeb Rd.,Ann Arbor, MI 48106.]

Orr, W. M., 1907: Stability or instability of the steady motions ofa perfect liquid. Proc. Roy. Irish Acad., 27, 9–69.

Peng, M. S., and R. T. Williams, 1991: Stability analysis of bar-otropic vortices. Geophys. Astrophys. Fluid Dyn., 58, 263–283.

Rayleigh, Lord, 1880: On the stability and instability of certainfluid motions. Proc. London Math. Soc., 11, 57–70.

Robinson, A. C., and P. G. Saffman, 1984: Stability and structureof stretched vortices. Stud. Appl. Math., 70, 163–180.

Rott, N., 1958: On the viscous core of a line vortex. Z. AgnewMath. Mech., 9, 543–553.

Rotunno, R., 1978: A note on the stability of a cylindrical vortexsheet. J. Fluid Mech., 87, 761–771.

Schubert, W. H., M. T. Montgomery, R. K. Taft, T. A. Guinn, S.R. Fulton, J. P. Kossin, and J. P. Edwards, 1999: Polygonaleyewalls, asymmetric eye contraction, and potential vorticitymixing in hurricanes. J. Atmos. Sci., 56, 1197–1223.

Shapiro, L. J., 1992: Hurricane vortex motion in a three-layermodel. J. Atmos. Sci., 49, 401–421.

Shepherd, T. G., 1985: Time development of small disturbancesto plane Couette flow. J. Atmos. Sci., 42, 1868–1871.

Smith, G. B., II, and M. T. Montgomery, 1995: Vortex axisym-metrization: Dependence on azimuthal wavenumber or asym-metric radial structure changes. Quart. J. Roy. Meteor. Soc.,121, 1615–1650.

Smith, R. K., and H. C. Weber, 1993: An extended analytic theoryof tropical-cyclone motion in a a barotropic shear flow. Quart.J. Roy. Meteor. Soc., 119, 1149–1166.

Staley, D. O., 1985: Effect of viscosity on inertial instability ina tornado vortex. J. Atmos. Sci., 42, 293–297., and R. L. Gall, 1979: Barotropic instability in a tornadovortex. J. Atmos. Sci., 36, 973–981., and , 1984: Hydrodynamic instability of small eddiesin a tornado vortex. J. Atmos. Sci., 41, 422–429.

Steffens, J. L., 1988: The effect of vorticity-profile shape on theinstability of a two-dimensional vortex. J. Atmos. Sci., 45,254–259.


Sullivan, R. D., 1959: A two-cell vortex solution of the Navier–Stokes equations. J. Aerosp. Sci., 26, 767–768.

Thompson, W., 1887: Stability of fluid motion: Rectilinear motionof viscous fluid between two parallel planes. Philos. Mag.,24, 188–196.

Wan, C. C., and C. C. Chang, 1972: Measurement of the velocity

field in a simulated tornado-like vortex using a three-dimen-sional velocity probe. J. Atmos. Sci., 29, 116–127.

Willoughby, H. E., 1992: Linear motion of a shallow-water barotropicvortex as an initial-value problem. J. Atmos. Sci., 49, 2015–2031., 1995: Normal-mode initialization of barotropic vortex mo-tion models. J. Atmos. Sci., 52, 4501–4514.

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