The Structures and Dynamic of Tornado-like Vortice

8/22/2019 The Structures and Dynamic of Tornado-like Vortice

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2908 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

1999 American Meteorological Society

The Structure and Dynamics of Tornado-Like Vortices

DAVID S. NOLAN*

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

BRIAN F. FARRELL

Department of Earth and Planetary Sciences, Harvard University, Cambridge, Massachuse tts

(Manuscript received 24 November 1997, in final form 20 November 1998)

ABSTRACT

The structure and dynamics of axisymmetric tornado-like vortices are explored with a numerical model ofaxisymmetric incompressible flow based on recently developed numerical methods. The model is first shown

to compare favorably with previous results and is then used to study the effects of varying the major parameterscontrolling the vortex: the strength of the convective forcing, the strength of the rotational forcing, and themagnitude of the model eddy viscosity. Dimensional analysis of the model problem indicates that the resultsmust depend on only two dimensionless parameters. The natural choices for these two parameters are a convectiveReynolds number (based on the velocity scale associated with the convective forcing) and a parameter analogousto the swirl ratio in laboratory models. However, by examining sets of simulations with different model parametersit is found that a dimensionless parameter known as the vortex Reynolds number, which is the ratio of the far-field circulation to the eddy viscosity, is more effective than the conventional swirl ratio for predicting thestructure of the vortex.

As the value of the vortex Reynolds number is increased, it is observed that the tornado-like vortex transitionsfrom a smooth, steady flow to one with quasiperiodic oscillations. These oscillations, when present, are causedby axisymmetric disturbances propagating down toward the surface from the upper part of the domain. Attemptsto identify these oscillations with linear waves associated with the shears of the mean azimuthal and verticalwinds give mixed results.

The parameter space defined by the choices for model parameters is further explored with large sets of numericalsimulations. For much of this parameter space it is confirmed that the vortex structure and time-dependentbehavior depend strongly on the vortex Reynolds number and only weakly on the convective Reynolds number.

The authors also find that for higher convective Reynolds numbers, the maximum possible wind speed increases,and the rotational forcing necessary to achieve that wind speed decreases. Physical reasoning is used to explainthis behavior, and implications for tornado dynamics are discussed.

1. Introduction

The use of numerical models has been one of theleading contributors to advances in our understandingof the causes and dynamics of the tornado phenomenon.This method of study can generally be divided into twocategories: thunderstorm-scale simulations and tornado-scale simulations. In the former category, pioneered byKlemp and Wilhelmson (1978), three-dimensional cloud

models are used to numerically simulate the formation

* Current affiliation: Department of Atmospheric Science, Colo-rado State University, Fort Collins, Colorado.

Corresponding author address: Dr. David S. Nolan, Departmentof Atmospheric Science, Colorado State University, Fort Collins, CO80523-1371.E-mail: [email protected]

and dynamics of the thunderstorms that are responsiblefor tornado formation. While the low resolution of suchearlier models could only hope to simulate the thun-derstorm-scale environment that is believed to lead totornado formation, Wicker and Wilhelmson (1995) re-cently reported the simulation of tornado-like featureswithin a numerically modeled thunderstorm. Tornado-scale models, pioneered by Rotunno (1977), assume a

particular environment of rotation coupled with con-vection to create an intense vortex near the surface.These models are intended to provide the details of the

wind field in the tornado and an understanding of thedynamics that lead to that flow structure. Until recently,most tornado-scale numerical models assumed the tor-nado to be axisymmetric, thereby reducing the problem

from three dimensions to two dimensions in cylindricalcoordinates.

Laboratory models, pioneered by Ward (1972), were

the first to provide direct measurements of the wind field


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15 AUGUST 1999 2909N O L A N A N D F A R R E L L

FIG. 1. Illustrations of four of the stages that laboratory and nu-merical models of tornados produce as the swirl ratio is increasedfrom zero: (a) weak vortex stage; (b) surface vortex with vortexbreakdown above the surface; (c) drowned vortex jump (breakdown

just above the surface) ; a nd (d) two-celled vortex with s tagnant core.The solid lines are streamlines of meridional flow, the arrows indicatethe direction and details of the circulation, and the M marks thelocation of the maximum azimuthal (swirling) velocities.

associated with tornado-like vorticesa term thatcould apply to any vortex caused by the convergenceof rotating fluid along a boundary. (These would alsoinclude waterspouts and dust devils, for example.) Forthis reason the early numerical modelers simulated theenvironment of the laboratory apparatus rather than thatof a thunderstorm. That is, the numerically simulatedvortex was driven by forcing rotating flow into the sidesof the domain and drawing fluid out the top with somekind of outflow boundary conditions.

A significant result is that numerical models havebeen able to reproduce many of the features observedin laboratory vortices. The most universal feature of thelaboratory results is the way in which the flow throughthe vortex depends on a nondimensional parameterknown as the swirl ratio:

r0S , (1.1)

2Qh

where is the circulation of the air as it enters theconvection region in the apparatus, r0 is the radius ofthe updraft, Q is the volume flow rate per unit axiallength, and h is the depth of the inflow region (Churchet al. 1979). As the value of the swirl ratio is increased,the vortex goes through various stages, as depicted inFig. 1. For S 0.1, there is no concentrated vortex atthe surface. For larger values, a concentrated vortexdoes appear at the surface, and at some height above itthere is a vortex breakdown where the flow transi-

tions from a tight, laminar vortex to a broader, turbulentstate. The vortex breakdown (Harvey 1962; Benjamin1962; Lugt 1989) is considered to be an axisymmetricanalog to the hydraulic jump phenomenon observed inchannel flows. As S is increased, the altitude of thevortex breakdown decreases; until around S 0.45 thebreakdown is just above the surface (Church et al. 1979;Church and Snow 1993). This state has been referredto as a drowned vortex jump (DVJ; Maxworthy 1973)and is generally associated with having the highest near-surface azimuthal wind velocities. (This value of S forthe DVJ was found for the Ward vortex chamber andmay be different for other physical models.) When S isfurther increased, the vortex breakdown reaches the sur-face and the vortex changes to a two-celled structure,where there is a downward recirculation in the vortexcore and the radius of maximum winds substantiallyincreases. Still larger values ofSresult in the appearanceof multiple vortices rotating around the vortex core.

These values of S did have some dependence on theReynolds number (Re, as measured by the volume flowrate through the chamber), but this dependency decreas-es for increasing Re. Three-dimensional models havealso reproduced this multiple vortex phenomenon (Ro-tunno 1984; Lewellen 1993). A general review of lab-oratory modeling of tornado-like vortices can be foundin Church and Snow (1993).

Wilson and Rotunno (1986) used a model based onearlier work by Rotunno (1979, 1984) to address thevalidity of various approximate laminar solutions forswirling boundary layers. They found good agreementwith those solutions and with laboratory results. How-ells et al. (1988) focused instead on the dynamics over

a wide range of swirl ratios and eddy viscosities andstudied the effect of no-slip versus free-slip lowerboundary conditions. Their results demonstrated that thecombination of high swirl ratio, low eddy viscosity, andthe no-slip boundary conditions produced the highestnear-surface wind speeds. This combination (low vis-cosity but no-slip boundary conditions) may be coun-terintuitive at first, but what Howells et al. (1988) dem-onstrated was the importance of a thin but strong inwardjet at the surface that, in the no-slip case, drives thefluid above the jet much closer to r 0 than is possiblewith free-slip conditions.

More recent efforts to study tornadic vortices with anaxisymmetric model have been made by Fiedler (1993,

1994, 1995; hereafter F93, F94, F95, respectively)whose motivation was to determine the maximum ve-locities that a tornado-like vortex could achieve as com-pared to the velocity associated with the thermody-namic speed limit (Lilly 1969; Fiedler and Rotunno1986). This estimated speed limit is made by matchingthe minimum surface pressure, calculated from hydro-static integration of the overhead density profile, withthe central pressure deficit of a Rankine vortex or someother vortex profile. For a Rankine vortex, which hassolid-body rotation maxr/rmax out to some radius


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rmax , and a potential flow max rmax/r outside of r rmax , the pressure deficit pcyc at r 0 due to cyclo-strophic balance must be

2p drcyc

r0

r max 2 1max 2 2 r dr r dr max max2 3r rmax0 rmax2 , (1.2)max

where is the air density. Note that the size of thevortex, as indicated by rmax , does not appear in the result.The hydrostatic pressure deficit beneath a convectingupdraft may be computed from the overhead densityprofile of the updraft column:

Tp g dz g dz, (1.3)hyd

T0 0

where and T are the density and temperature devi-ations in the updraft from the ambient temperature and T, and we have used the Boussinesq approximation.The assumption that the minimum pressure achievablein the vortex is due entirely to this hydrostatic pressuredeficit gives the thermodynamic speed limit:

phyd C ,2max (1.4)

where C is some constant that depends on the vortexprofile. For a Rankine vortex, C 1.0; C 0.5 for astagnant core vortex [this can be seen by neglecting theintegral from 0 to rmax in (1.2)]; and C 0.59 for aBurgerss vortex velocity profile. F9395 used a closed

cylindrical domain, which has the advantage of pre-venting inflowoutflow boundary conditions from con-tributing to the energetics of the model. (Previous nu-merical models used fixed inflow or outflow velocities,which can act as variable energy sources.) Rather thanmodeling the temperature of the fluid, Fiedler used afixed buoyancy field at the center axis of the domain toconvectively drive the flow and force convergence atthe lower boundary. Angular momentum was suppliedby putting the entire domain into solid-body rotation.F93 and F94 explored the effects of the choice of ro-tation rate, eddy viscosity, and also the additional effectof subsidence warming in the core on the maximumwind speeds. Based on the assumption that the pressure

deficit of a drowned vortex jump can be approximatedby that of a stagnant-core vortex, F94 observed transientvortices that exceed the thermodynamic speed limit bya factor of 5. These very high transient wind speedsremind us that there is nothing about the concept orderivation of the thermodynamic speed limit that provesthat it is a real constraint on the maximum velocitiesthat can be generated in a convectively driven vortex.This is because the hydrostatic and cyclostrophic as-sumptions are clearly violated in the presence of sub-stantial vertical and horizontal accelerations. Lewellen

(1993) wrote, The tornado vortex allows a significantfraction of the potential energy of the parent storm tobe concentrated into wind kinetic energy where it cancause great damage. How exactly this occurs is still amatter of investigation.

The results of axisymmetric tornado-like vortex mod-els have been strongly confirmed by the first detailedobservations of the tornado vortex core reported byWurman et al. (1996) using the most recent advancesin Doppler radar technology. Their observations of atornado near Dimmit, Texas, during the VORTEX fieldprogram (Rasmussen et al. 1994) showed maximumwind speeds of over 70 m s1 occurring less than 200m above the ground in a tight ring around the center ofthe tornado. The structure of the azimuthal wind fieldthey observed was very consistent with those generatedby the simulations of Howells et al. (1988), F93 andF94, and also with the results we are about to show.

In this study we will examine not only the maximumwind speeds of axisymmetric tornado-like vortices, but

also how the major model parametersthe buoyancyforcing, the rotational forcing, and the eddy viscositydetermine these wind speeds as well as the structure andbehavior of the vortex. In section 2 we introduce theequations of motion and our methods of numerical in-tegration. In section 3 we show the basic features of thesimulated vortices and we introduce a way to measurethe relative values of the azimuthal and vertical windspeeds in the vortex core, which is analogous to theswirl ratio (1.1). In section 4 we discuss the dimen-sionless parameters that control the flow. In section 5we investigate how the structure and behavior of theflow in the vortex depend on these dimensionless pa-rameters. In section 6 we give some analysis of the

results, and in section 7 we discuss their implications.

2. Numerical modeling of the axisymmetricNavierStokes equations

a. The equations of motion andnondimensionalization

The incompressible NavierStokes equations for aconstant-density, constant-viscosity fluid in cylindricalcoordinates, with all azimuthally varying terms elimi-nated, are

2 2u u u 1 p 1 u u w (ru)

2 [ ] r r r r zt r z r (2.1)

2 u 1 u w (r) (2.2)

2 [ ]t r z r r r r z2w w w 1 p 1 w w

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

F (2.3)z

1 w u u w(ru) 0,

r r z r r z(2.4)


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where u is the radial velocity, is the azimuthal velocity,w is the vertical velocity, p is the pressure, Fz is a verticalforcing term (the buoyant acceleration), is the vis-cosity, and band is the density. Equation (2.4) is theincompressibility condition for axisymmetric flow.

The variables in (2.1)(2.4) are nondimensionalizedas follows:

(u, , w) U(u, , w) (2.5)

(r, z) L (r, z) (2.6)

p Pp (2.7)

F FF (2.8)z z

Lt Tt t, (2.9) U

where nondimensional variables are indicated byprimes, and we have used an advective timescale in

(2.9). For the moment we will postpone selection of themagnitudes of the dimensional parameters U, L, P, andF. Substitution of the scalings (2.5)(2.9) into the mo-mentum equations (2.1)(2.3) results in the usual non-dimensionalization of the NavierStokes equations,with /UL 1/Re (the inverse Reynolds number) mul-tiplying the diffusion terms, P/U2 multiplying the pres-sure gradient terms, and the nondimensional parameterFL/U2 multiplying the vertical forcing term .Fz

Following F93, we scale the velocities according tothe thermodynamic speed limit (1.4). We choose thelength scale to be the height of the domain, the forcingscale F to be the average of the dimensional forcingalong the r 0 axis, F Fz(0, z) , and

2U FL (2.10)

P FL (2.11)

1 . (2.12)

UL Re

Using these scalings, and dropping the primes on thenondimensional variables, we obtain the nondimension-al equations of motion:

2 2u u u p 1 u u w (ru)

2 [ ]t r z r r r r r z(2.13)

2 u 1 u w (r) (2.14)

2 [ ]t r z r r r r z2w w w p 1 w w

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

F (2.15)z

1 w u u w(ru) 0.

r r z r r z(2.16)

b. Numerical integration of the NavierStokesequations

There are two computational challenges to numericalmodeling of the equations of motion. The first is theenforcement of incompressibility (2.16); the second is

the evaluation of the nonlinear advection terms in themomentum equations (2.13)(2.15) in a manner that isaccurate and keeps the solution stable. Traditional meth-ods (Rotunno 1979; F93) have used the following tech-niques for these problems. First, the radial and verticalmomentum equations are combined into a streamfunc-tion-vorticity equation and the pressure is thereby elim-inated from the problem. This, however, creates the needto calculate the streamfunction from the vorticity, whichhas many challenges, particularly the extrapolation ofthe (unknown) vorticity to the domain boundaries. Sec-ond, the nonlinear terms in the streamfunction-vorticityequation are computed from an Arakawa-type Jacobian,which keeps the system stable as long as the usual time

step constraints are met.We have instead used a velocitypressure formula-

tion. First, we consider the equations of motion as asimplified vector equation:

u F(u) p. (2.17)

t

Here we have written the functional F(u) in place ofthe advective, diffusive, and vertical forcing terms.Since u is always nondivergent, so must be u/ t. Wealso know that an arbitrary vector field, such as onegenerated by F(u), can be written as the sum of a non-divergent part and an irrotational (and divergent) part,

the uniqueness of which is provided by the boundaryconditions. Therefore, we can conclude that for u toremain nondivergent, p must exactly cancel the ir-rotational part of F(u). Suppose we had some methodfor directly extracting the nondivergent part ofF(u). Wecall this the projection P of F(u) onto the space ofdivergence-free vector fields. The equations of motioncan then be succinctly written as u/ t P[F(u)]. Thetheory and technique behind this method were intro-duced by Chorin (1968, 1969) and have been developedextensively since then (see, e.g., the text by Peyret andTaylor 1983). Most recent projection methods are sec-ond-order accurate in space and time. In axisymmetricflows the swirling velocity is already nondivergent

and is left out of the projection.For the nonlinear advection terms, we used a Go-

dunov-type upwinding method developed by Colella(1990) and Bell et al. (1989) that is also second-orderaccurate. This method approximately solves the Rie-mann problem in the vicinity of each grid point andultimately becomes a fully explicit approximation to aCrankNicholson method. The interested reader mayfind more details in Minion (1994, 1996).

In this model, the domain is divided into rectangularcells, with velocity information stored at the cell centers


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and the scalar information stored at the cell corners. Theedges of the outermost cells are coincident with thedomain boundaries. All the simulations reported hereuse grids that are regularly spaced in each direction,with 64 64 grid points, 128 128 grid points, and,in one case, 256 128 grid points. A complete de-scription of this axisymmetric incompressible numericalmodel may be found in Nolan (1996).

3. Preliminary results

a. Comparison to previously reported results

We tested the model by comparing it to some pre-viously published results, using the same type of do-main, boundary conditions, and forcing. As discussedabove, F93 used an axisymmetric incompressible flowmodel to investigate the dynamics of vortices generatedby convection in a rotating domain. He used the stream-function-vorticity method and modified the equations sothat the domain was in solid-body rotation at a dimen-sionless rotation rate 0.2. The simulations werenondimensional, with a domain height of Z 1 andradius of R 2. This radius was chosen to be largeenough so as not to influence the behavior of the vortexnear r 0. Although our equations did not have aCoriolis term to represent solid-body rotation, we wereable to reproduce the F93 results by initializing the az-imuthal velocity with rand setting the boundariesin rotation at the same rate. We used the F93 buoyancyfield:

Fz (r, z) .2 2

20[r (z0.5) ]1.264e (3.1)

Finally, we also begin with the same value for the non-dimensional kinematic viscosity (the inverse Reynoldsnumber) 0.0005. The major and nonreproducibledifference between the F93 simulations and ours is thatF93 used a stretched coordinate system that packed thegrid points near r 0, z 0, whereas ours was designedsuch that it could only have equal gridpoint separationin each direction. F93 used 61 61 (stretched) gridpoints for most of his simulations, whereas we use 128 128 in this case.

Figure 2 shows the vertical velocities, azimuthal ve-locities, pressure, and the velocity vectors in the rzplane at t 10 for a simulation based on the F93 con-ditions. The velocity vectors in the rz plane are here-

after referred to as the meridional velocity vectors. Fig-ure 3 shows these fields at t 40. (Note that these timeswere chosen so that our results can be compared withthose of F93; however, one must also note that we areplotting , which includes the rotation of the domain,while F93 did not. Since r is small near r 0, theresults are nearly identical in the vortex core.) At t 40 we can see in the meridional vector velocity plothow the vertical flow in the vortex core separates fromthe r 0 axis, and we can also see in the verticalvelocity contour plot the appearance of a recirculation

zone above the vortex core, as indicated by the negativevelocities just above the surface. This feature is gen-erally interpreted to be an axisymmetric equivalent ofthe vortex breakdown seen in the laboratory experi-ments (cf. Fig. 1). The maximum velocities and mini-mum pressure occurring in the domain as a function oftime are shown in Fig. 4. This pressure is purely dy-namic and is defined to be zero in the upper-right-handcorner of the domain. One can see a pattern where allthree velocities episodically increase and then decreasein time. Longer simulations (not shown) revealed thatthese oscillations in the maximum velocities did notdecay completely but periodically reappeared for as longas the simulation with these parameters was integrated.The cause of these oscillations will be discussed furtherbelow in section 6. We found that the similarity of ourresults to F93 was qualitatively very high, while quan-titatively they were fairly similar, although there aresignificant differences in the maximum velocities. Our

maximum azimuthal velocity is max 1.16, which oc-curs at t 11.6; for F93, the result was max 1.26 attime t 11.

By comparing the maximum velocities at t 10 andt 40, we are inclined to draw the conclusion that thevortex is much stronger during its intensification stagethan when it has settled into a nearly steady state. Forcertain values of rotation rate and eddy viscosity, F94observed transient maximum wind speeds as much asfive times faster than the thermodynamic speed limitdefined by the stagnant-core vortex. However, it is notlikely that these transient solutions offer much guidanceto the dynamics of real tornadoes, which certainly donot form when a fixed vertical forcing field is instan-

taneously applied to a column of air in solid-body ro-tation. Furthermore, since tornadoes have been observedto produce damaging winds for as long as 45 minutes,it is a quasi-steady-state solution that should representthem best. With this in mind, our goal should be tofocus on the dynamics of quasi-steady solutions.1

b. The effect of the ceiling at z 1

It is reasonable to wonder to what extent the ceilingat z 1 effects the formation and behavior of the vortexat the lower surface. One may also speculate that therecirculation seen in the fully developed vortex owesits existence, at least in part, to the ceiling. To address

these concerns, we ran another simulation with the samegrid spacing but twice the domain size in the vertical

1 Note added in proof: Observations and numerical simulationsindicate that the highest transient wind speeds in tornadoes occur insecondary vortices, which appear episodically in the region of largeshear just inside the radius of maximum winds. In fairness, then, itis likely that Fielders (1994) study of the very high transcient windspeeds that occur in the early development of tornado-like vorticesis applicable to these secondary vortices.


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FIG. 2. Results for the F93-type simulation at t 10: (a) swirling (azimuthal) velocity; (b) vertical velocity; (c) pressure;and (d) meridional velocity vectors in the rz plane. Dashed contours indicate negative values, maxima and minima areindicated, and the contour intervals are indicated at the top of each plot. All plots show the domain for 0 r 1, 0 z 1, except for the meridional vector velocity plot, which is for 0 r 0.3, 0 z 0.3.

direction; this required using 256 grid points in the ver-tical direction. The results are shown in Fig. 5 for theazimuthal and vertical velocities at t 10 and t 40.Comparing these to the previous results at t 10 andt 40, as shown in Figs. 2 and 3, we see that the resultswith a higher ceiling are very similar. Note especiallythe appearance of a recirculation zone above the fullydeveloped drowned vortex jump vortex at t 40. Thisvortex is slightly different in that this recirculation zoneis slightly weaker and the maximum azimuthal wind

speeds are slightly higher. However, the overall resultsare essentially the same.

c. Three measured parameters to characterize thevortex

As discussed in section 1, laboratory modelers founda correlation between the structure of the tornado-likevortex and the value of the nondimensional swirl ratioS as defined by (1.1). This swirl ratio depends on the


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FIG. 3. Results for the F93-type simulation at t 40: (a) swirling (azimuthal) velocity; (b) vertical velocity; (c) pressure; and(d) meridional velocity vectors in the rz plane. Dashed contours indicate negative values, maxima and minima are indicated,and the contour intervals are indicated at the top of each plot. All plots show the domain for 0 r 1, 0 z 1, except forthe meridional vector velocity plot, which is for 0 r 0.3, 0 z 0.3.

volume flow rate of the air through the apparatus and

its required rotation rate as it enters the convection zone.The conventional definition of the swirl ratio is not ap-plicable to closed domain models such as ours and thatof F93, because the rotation and flow rates of the fluidthrough the central vortex is not under the control ofthe modeler. However, we can instead define an in-ternal swirl ratio based on the measured rotation andflow rates of the fluid through the region surroundingthe vortex. To do this, we define a control volume thatsurrounds the central vortex, such that the boundariesof the control volume represent the inflow region into

the vortex, the outflow region above the vortex, the

surface, and r 0. We define the internal swirl ratioSI as

h0

(r , z) dz 00r0

S , (3.2)I r02h0w(r, z )2r dr 0

0

where is the circulation 2rV. For our simulationshere we use r0 0.25 and h0 0.25, so that the control


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FIG. 4. Maximum velocities and minimum pressure as a function of time in the original F93-type simulation: vertical velocity (dashdot), azimuthal velocity (dashed), inward radial velocity(solid), and pressure (solid, negative).

volume is a cylinder of radius 0.25 and depth 0.25,adjacent to the corner at r 0, z 0.

We shall now introduce another parameter that char-acterizes the flow. Recall that the motivation for F93and F94 was to determine the maximum wind speedsthat can be generated by a given buoyancy field. The

vertical integral along r 0 of the buoyancy function(3.1) was considered to be analogous to convectiveavailable potential energy (CAPE), which is a way ofexpressing the potential intensity of thunderstorms fromtheir environmental thermodynamic sounding (Emanuel1994). Note also that when the background vertical den-sity profile is constant, the CAPE is equal to the potentialminimum hydrostatic pressure divided by the density ,that is,

LNB pT hydCAPE g dz , (3.3)

T 0

where LNB is the level of neutral buoyancy and T(z)

is the temperature deviation from the background tem-perature T(z) of a parcel of air rising adiabatically fromthe surface to the LNB. The maximum integral of forcetime distance [integral of b(0, z)] for the forcing field(3.1) is 0.5; this is the CAPE for our model. F93 andF94 argued that the drowned vortex jump state has acyclostrophic pressure drop that is similar to that of astagnant-core vortex that has C 0.5, so the thermo-dynamic speed limit was therefore max 1. Even whenthe vortex is not in hydrostatic balance, we expect fromdimensional considerations that the square of the max-

imum azimuthal velocity would be proportional to theequivalent of the CAPE for our numerical model:

1

2b(0, z) dz C , (3.4) max0

where C is some coefficient that may also depend onthe viscosity and the rotational forcing. As discussed insection 1, this coefficient has certain well-known valuesfor various idealized velocity profiles. However, this Cis not so easily interpreted since, for a given convectiveforcing, it is proportional to the inverse of the squareof the maximum azimuthal velocity. Let us define a newvelocity coefficient, C

, as

max maxC , (3.5)

1/21 wCAPE

2 b(0, z) dz 0

so that C

is the ratio of the maximum swirling velocity

to the maximum convective velocity predicted from theCAPE equivalent.

Figure 6 shows the internal swirl ratio SI and thevelocity coefficient C

during the original F93-type sim-

ulation described above. Initially SI is not well definedsince no meridional flow exists at t 0; however, asthe vortex forms it quickly adjusts to appropriate values.Around t 10, SI wildly oscillates around a value ofapproximately 2.5 in a manner similar to the maximumvertical velocities (see Fig. 4); as the vortex settles intoa quasi-steady state these oscillations persist with much


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FIG. 5. Results of an F93-type simulation with twice the domain height: (a) contours of azimuthal velocity at t 10; (b) contours ofvertical velocity at t 10; (c) contours of azimuthal velocity at t 40; and (d) contours of vertical velocity at t 40. The domain in eachframe is from 0 r 1, 0 z 2. Dashed contours indicate negative values, maxima and minima are indicated, and the contour intervalsare indicated at the top of each plot.

smaller magnitudes and a seemingly constant frequency.

Around t 10, C shows large oscillations about a value

of 0.7 but then settles into a nearly constant value of0.8. This constancy is in contrast to the oscillations inSI, and it indicates that even while the ratio of the vol-umetric flow through the vortex core to the angular mo-mentum entering the vortex core may be varying in time,the maximum azimuthal wind speed is nearly constant.

The internal swirl ratio SI depends explicitly on thechoice for the dimensions of the control volume definedabove, and has been found to have higher values forlarger control volumes and lower values for smaller con-

trol volumes. Unfortunately, then, the internal swirl ratio

is not universal and cannot be compared to the swirlratios measured in laboratory models. An alternativeway to evaluate the structure of the flow in the vortexcore is to introduce the vortex aspect ratio, AV, whichis the ratio of the radius of the location of the maximumazimuthal wind speed (RMW) to the height of the lo-cation of the maximum azimuthal wind speed (ZMW):

RMWA . (3.6)V

ZMW

With the recent arrival of finescale Doppler radar mea-


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FIG. 6. Internal swirl ratio SI (solid line) and velocity coefficient C (dashed lined) duringthe F93-type simulation shown above.

FIG. 7. Vortex aspect ratio (RWM/ZMW) as a function of time in the F93 simulation.

surements of tornadic wind fields (Wurman et al. 1996)this parameter can be measured quite accurately in realtornadoes, since the actual values of RMW and ZMWdo not need to be measuredjust their ratio. The time

evolution of AV for the F93 simulation is shown in Fig.7. While AV is clearly meaningless until an intense vor-tex forms near the axis, it does reflect the structure ofthe vortex as it evolves: first it has low values around


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FIG. 8. Results for a simulation with 0.4 and 0.00025 at t 40: (a) swirling (azimuthal) velocity; (b) vertical velocity;(c) pressure; and (d) meridional velocity vectors in the rz plane. Dashed contours indicate negative values, maxima and minimaare indicated, and the contour intervals are indicated at the top of each plot. All plots show the domain for 0 r 1, 0 z 1, except for the meridional vector velocity plot, which is for 0 r 0.3, 0 z 0.3.

0.5 when the vortex is of the end-wall type, then

goes through some wild oscillations as the breakdownappears, and finally settles down to small oscillationsaround a value of 1.8 as the drowned vortex jump sta-bilizes. Figure 7 does show one of the limitations of AVas we have defined it: since the numerical model storesvelocities at discrete locations, the locations of the ve-locity maxima are discrete, so we see that AV goesthrough discontinous jumps between discrete values astime evolves. This should be contrasted with the verysmooth behavior of SI (see Fig. 6).

To show how these measured parameters depend on

the structure of the vortex, we consider the results of a

simulation with increased to 0.4 and decreased to0.00025. These values of and result in a vortex thatoscillates chaotically between a drowned vortex jumpand a two-celled vortex. An example of the flow fieldwhen the vortex is in a two-celled state is shown in Fig.8. There is a recirculation zone in the core of the vortex,with downward flow penetrating all the way to the sur-face. The maximum azimuthal winds are farther outfrom the axis and closer to the surface. Figure 9 showsSI, C, and AV as a function of time for this simulation.Comparison of Fig. 9 with Fig. 6 shows the strengths


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FIG. 9. (a) Internal swirl ratio SI (solid line) and velocity coefficient C (dashed lined) duringthe simulation with 0.4 and 0.00025; (b) vortex aspect ratio AV for the same

simulation.

and weaknesses of SI versus AV as indicators of thevortex flow structure. While SI is always smoothly vary-ing in time, it can become undefined and/or negative inhighly unsteady flows when the flow direction tempo-rarily reverses within the control volume, as it does inthis case around t 80. On the other hand, AV takeson only discrete values and has discontinuous behaviorin time but is well defined for all flow regimes and doesnot vary quite as wildly as SI in the high swirl regime.

4. Dimensional analysis

Let us restrict ourselves to the case where there isonly one fundamental length scale L in the model prob-lem by assuming that the domain length scale and theheight and width of the forcing field Fz are all roughlythe same, as is the case for our model. (The radius ofthe domain has been chosen to be large enough so thatit does not affect the dynamics; therefore, it does not


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enter into the analysis.) We then have four dimensionalparameters: the length scale L, the forcing scale F, therotation rate , and the viscosity . We may replace theforcing scale F with a convective velocity scale U wCAPE from (3.5). The equations of motion only relatevelocities, so there are only two fundamental physicaldimensions: length and time. Application of dimen-sional analysis (the Buckingham pi theorem) requiresthat the system must depend on only two dimensionlessparameters. It is important to note that these two pa-rameters are not unique. There are in fact four possi-bilities, from which we must choose two:

ULRe , (4.1)C

LS , (4.2)d

U

2LRe , (4.3)V

2U

, (4.4)4

where we have labeled each of the dimensionless group-ings according to the following definitions: Re C is theconvective Reynolds number, a Reynolds number basedon velocity scale of the convective forcing; Sd is a swirlratio that we have labeled with the subscript dto indicatethat it comes from dimensional analysis and to differ-entiate it from the laboratory swirl ratio (1.1); Re V isthe ratio of the far-field circulation to the viscosity, oftencalled the vortex Reynolds numberand is known to con-trol the flow in laminar swirling boundary layers (see,e.g., Barcilon 1967); the physical significance of the lastparameter is not known, though it can be written asReC/Sd.

Note that any two of these parameters together shouldcontain all the information necessary to determine theoutcome of the model; that is, we expect that observ-ables such as SI or C will be functions of the twodimensionless parameters we choose (albeit differentfunctions for different choices of parameters). An ob-vious choice for one of the two dimensionless param-eters to use for our study is the convective Reynoldsnumber ReC. Since max Uthis is an appropriate Reyn-olds number for the problem. A natural choice for the

other parameter is the swirl ratio Sd since this has theobvious connection to the swirl ratio used in laboratorymodels. However, we shall see in the following sectionsthat the vortex Reynolds number Re V is in fact moreuseful for predicting the characteristics of the flow intornado-like vortices.

5. Exploration of parameter space

We now turn to the major area of investigation of thisreport: how do the structure and intensity of the vortex

depend on the dimensionless parameters? To determinethese relationships, we performed many numerical sim-ulations of the type shown above, with different valuesfor either , , or the convective velocity scale U. Dueto limits of computation time, these simulations wereperformed at a lower resolution of 64 64 grid points.Sets of simulations were performed with one of theparameters varying among the separate simulations andthe other two held fixed. The simulations in all caseswere integrated from t 0 to t 200, while the max-imum velocities, SI, RMW, and ZMW were recordedevery 10 time steps during these simulations. The meanvalues and the variances of the data in each case wereevaluated using the last 100 time units of the simula-tions.

Let us examine how the vortex changes as we varySd. Figure 10 shows the mean maximum velocities asthe domain rotation rate (equal to Sd since U L 1 in our nondimensional model) varies from 0.02 to 0.4.

The model eddy viscosity is held fixed at 0.001and the convective forcing is the F93 forcing (3.1). Forvery low rotation rates the radial and azimuthal veloc-ities are very low, while the vertical velocities are rel-atively high. For higher rotation rates all three velocitiesincrease, with the vertical velocities still exceeding theazimuthal velocities by about 50%. However, for largerrotation rates the vertical velocities decrease until theyare substantially less than the azimuthal velocities. Overthis range of higher values for the maximum azi-muthal and radial velocities decrease only slightly. Fig-ure 11a shows the mean values of SI, C, and AV versusthe rotation rate. Not surprisingly, SI increases steadilywith increasing and AV also increases, since RMW

increases while ZMW decreases with increasing . Thevelocity coefficient C

has low values for the lowestvalues of , due to the fact that an intense vortex doesnot form for very low values of the swirl ratio (see Fig.1). However, for values of where a vortex does form,C

increases to a maximum of 0.88 when 0.08 and

decreases to 0.70 when 0.4. This indicates that theefficiency with which the convective forcing is con-verted into azimuthal wind velocity decreases for largerotation rates.

How does the flow through the vortex core differbetween the low rotation-rate and high rotation-rate re-gimes? Figure 12 shows close-ups of the velocity fieldsin the vortex core for the case when 0.1 (Figs.

12a,b) and also when 0.4 (Figs. 12c,d). In the firstcase the boundary layer inflow penetrates all the wayto the r 0 axis, then turns upward into a strong axialjet along the axis. It separates from the axis around z 0.4. The maximum azimuthal velocity is V 0.83,which occurs at r 0.1 and z 0.23. In the secondcase, the boundary layer inflow also penetrates to r 0 and turns upward, but the vertical flow breaks awayfrom the vertical axis at z 0.15. The maximum azi-muthal velocity is V 0.72, occurring at r 0.12 andat the much lower altitude of z 0.08. Comparison of


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FIG. 10. Average maximum velocities as a function of the rotation rate : radial velocitiesare marked by circles, azimuthal velocities are marked by s, and vertical velocities aremarked by s.

these two results with the illustrations of the structureof laboratory vortices as a function of swirl ratio in-dicates that the results with 0.1 clearly reproducea low-swirl intense vortex as shown in Fig. 1b, whilethe results with 0.4 clearly reproduce a drowned

vortex jump as shown in Fig. 1c. (Recall from section3c that increasing to 0.4 and decreasing to 0.00025resulted in a vortex that occasionally was two celled instructure; the effect of on the vortex structure will beaddressed shortly.) The steady increases ofSI and AV asthe vortex evolves between these states show how theyare good indicators of the vortex structure.

The variance of SI is a good indicator of the unstead-iness of the structure of the flow in the vortex core.Figure 11b shows the variance of SI versus the rotationrate . We can see that for values of greater than0.25 the variance increases suddenly from negligiblevalues to values on the order of 0.5. The high varianceof SI when its average value is 2.5 or greater indicates

that the flow in the vortex core is unsteady in this re-gime. Figure 13 shows a sample of the time evolutionof the maximum velocities in the high-swirl regimeshown in Figs. 12c,d. All three velocity maxima undergosteady oscillations, with the vertical velocities varyingthe most, as much as 20%. The period of oscillation isabout six nondimensional time units, and the maximain the vertical velocities lead the maxima in the radialand azimuthal velocities by about one time unit.

So far we have shown that as we increase Sd the vortexevolves in a manner similar to laboratory vortices when

their swirl ratio S is increased, and that SI and AV areuseful indicators of the structure and behavior of theflow. In laboratory experiments it has generally beenfound that the Reynolds number of the flow had a de-creasing influence on the flow structure as it was in-

creased (Church et al. 1979). If this is the case for ourmodel, then we would expect that varying would notsignificantly affect on the flow structurethat it de-pends strongly on Sd L/U. Thus we repeat the anal-ysis varying the model eddy viscosity (equal to 1/ReCsince U L 1) while now keeping the rotation rate 0.2. Figure 14 shows the average maximum ve-locities for values of the varying from 0.0004 to 0.0022. The result is interesting in that the maxi-mum velocities change with increasing in a mannerthat is opposite to how they changed with increasingrotation rate. By comparing Figs. 15a,b to Figs. 11a,b,we can make essentially the same observation with re-gard to the internal swirl ratio SI, the velocity coefficient

C, and the vortex aspect ratio AV. We thus have theinteresting observation that increasing the eddy viscos-ity has the same effect on the structure of the vortex asdecreasing the domain rotation rate, and vice versa. Ex-amination of the flows through the vortex core in thehigh and low eddy viscosity cases (not shown) gavenearly identical results to the low and high domain ro-tation rate results (respectively) shown in Fig. 12, con-firming this observation. For the ranges of these param-eters where an intense vortex is present, C

changes

relatively little. Furthermore, we see that in both cases


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FIG. 11. (a) Average values for the internal swirl ratio SI and the velocity coefficient C as afunction of the rotation rate of the domain. Here SI is marked by the circles and CV is markedby the s. (b) Variance of SI as a function of the rotation rate of the domain.

the onset of unsteady flow in the vortex core (as mea-sured by the variance of SI) occurs when SI reaches avalue of approximately 2.3.

This apparent reciprocal relationship between the ro-tation rate and the viscosity indicates that Re V may bemuch more useful for predicting the structure of theflow in the vortex than the swirl ratio Sd. The matterremains unclear, however, since while varying changesReV and leaves Sd fixed, it also changes Re C. To see

how the flow evolves as we vary ReC, while keepingReV fixed, we var y the convective velocity scale Uwhileholding and fixed. The results are shown in Fig.16, where U increases from 0.316 to 1.414 while 0.4 and 0.001; over this range Re C increases from316 to 1414. While SI decreases by 20% as Re C goesfrom 316 to 1000, it then appears to asymptote to avalue of 2.2 for larger ReC. Furthermore, this changein SI only represents a small change in the actual struc-


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FIG. 12. Close-up of the velocity fields in the vortex core for two different domain rotation rates: (a) close-up of the azimuthalvelocity field for 0.1; (b) close-up of the meridional velocity vectors for 0.1; (c) close-up of the azimuthal velocityfield for 0.4; and (d) close-up of the meridional velocity vectors for 0.4. The contour intervals or maximum velocitiesare indicated at the top of each frame.

ture of the flow. Thus we observe only a weak depen-

dence of the vortex structure on the convective Reynoldsnumber.

6. Analysis

We have seen how, under simple convective forcingthat causes convergence, a region of fluid in solid-bodyrotation collapses into an intense vortex near the surface.The highest azimuthal velocities occur during the de-velopment of the vortex, which then settles into a quasi-steady state with maximum velocities determined pri-

marily by the strength of the convective forcing. There

are two issues that call for further investigation: 1) whatis the source of the oscillations observed in the maxi-mum velocities for certain ranges of the parameters and2) can we clarify and understand how the flow structureand behavior depend on the dimensionless parameters?We will address the former issue first.

a. Examination of the oscillations in the vortex core

In section 5 we saw that when ReV is above a certainvalue, the flow through the vortex core is unsteady.


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FIG. 13. A sample of the time evolution of the maximum velocities in an F93-type simulationwith 0.4 and 0.001: maximum inward radial velocities (solid), azimuthal velocities (dashed),and vertical velocities (dashdot).

FIG. 14. Average maximum velocities as a function of the model eddy viscosity: averagemaximum radial velocities are marked by circles, azimuthal velocities are marked by s, andvertical velocities are marked by s.


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FIG. 15. (a) Average values for the internal swirl ratio SI and the velocity coefficient C as afunction of the model eddy viscosity . Here SI is marked by the circles and C is marked by thes. (b) Variance of SI as a function of the model eddy viscosity .

Figure 13 showed that this unsteadiness is associatedwith periodic (in some cases) oscillations in the valuesof the maximum velocities. What is the cause of theseoscillations? Why are they not present in the low-swirlcase?

To address these questions we first examined closelythe velocity fields in the vortex core in the oscillatoryregime. To achieve this, we ran a 64 64 gridpointsimulation, with 0.4 and 0.001, from t 0

to t 213. We recorded the full velocity fields at 0.5time unit intervals for the last 13 time units of the sim-ulation. We then averaged the 26 fields to obtain anaverage state of the vortex during the oscillations.This allowed us to then derive deviations from this av-erage state. Upon examination of these perturbation ve-locity fields, we observe axisymmetric disturbancespropagating from the upper part of the domain downinto the vortex core. These disturbances are shown in


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FIG. 16. Average values for the internal swirl ratio SI and the velocity coefficient C as a function of thestrength of the convective forcing; SI is marked by the circles and C is marked by the s.

Figs. 17 and 18 in terms of the perturbation meridional

vector velocity fields. In Fig. 17a, we can identify aclockwise circulation centered in the vicinity ofr 0.1,z 0.25. In Fig. 17b, 0.5 time units later, this circulationcan be seen to have moved downward slightly and ismore distinct. This trend continues in the next twoframes, and in Fig. 17d at t 202 we can also see thebeginnings of a reverse (counterclockwise) circulationappearing above z 0.3. In the next series of plots(Figs. 18ad) we can see this new circulation developsand descends until it is in approximately the same lo-cation as the original disturbance. Thus the cycle repeatsitself indefinitely.

We repeated this analysis but instead for a flow witha steady vortex core by using the same parameters as

before except with 0.15, which puts the vortex inthe low-swirl regime, without oscillations in the vortexcore. Repeating the analysis above, we show in Fig. 19a series of deviations of the meridional vector velocitiesfrom the mean. In this case, we see that the perturbationsare orders of magnitude weaker than those we saw be-fore, and rather than descending toward z 0.15 wherethe maximum wind speeds occur, they are stationary.In fact, the plots show how the disturbances reversingin direction over time, like stationary waves.

Whether or not these waves propagate downward into

the location of the maximum wind speeds is determined

by the group velocities of the waves relative to the ver-tical velocity of the flow out of the vortex. When thevertical velocities are high enough to prevent thesewaves from penetrating to the surface, the vortex is saidto be supercritical, much like the flow upstream ofa hydraulic jump. When the waves do penetrate to thesurface, the vortex is considered to be subcritical.From these definitions, we would say that the vortex issupercritical in the low-swirl case and subcritical in thehigh-swirl case.

b. Eigenanalysis of axisymmetric disturbances in the

vortex core

The first place to look for a basic understanding ofsuch disturbances is in the analysis of linearized dis-turbances on the mean flow. However, the question athand is not simply one of the existence of unstablemodes. The likely source for the disturbances is theregion of highly unsteady flow where the vortex outflowsheds vortices and becomes two-dimensional (axisym-metric) turbulence (see Fig. 8b, for example). Therefore,the flow in the vortex core does not need to be unstablebut rather needs to support downward-propagating


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FIG. 17. Meridional vector velocity plots of the deviations from the average velocity field of the vortex in the oscillatory,high-swirl regime: deviation from the mean at (a) t 200.5, (b) t 201.0, (c) t 201.5, and (d) t 202.0. The plots showthe domain from 0 r 0.5, 0 z 0.5. The maximum velocity deviations are indicated at the top of each frame.

waves that reach the surface. Linear instability is, ofcourse, also a possible cause for the disturbances.

The dynamics of axisymmetric (inertial) disturbanc-es in axisymmetric swirling flows have received con-siderable attention in the field of hydrodynamic sta-bility. Early work on the stability of swirling flows toaxisymmetric disturbances were those of Rayleigh(1880), for inviscid flows, and Taylor (1923), for vis-cous flows, which are summarized in the subsequenttexts such as Lamb (1932), Chandresekhar (1961), andDrazin and Reid (1981). In our case we are particularlyinterested in the dynamics of axisymmetric distur-bances in a viscous axisymmetric flow with shear in

the azimuthal and vertical winds. We neglect the radialwind since it is clearly much smaller than the other

velocity components in the region of interest. Wewould like to show that the disturbances we observeabove are indeed associated with the shear of the ver-tical and azimuthal winds.

Equations for the structure and phase speed of axi-symmetric disturbances in a viscous flow with shear inboth the vertical and azimuthal winds are derived insection 79 of Chandresekhar (1961) for swirling flowswith azimuthal velocities of the form V(r) Ar B/r(where A and B are constants; this is the general solutionfor a steady viscous swirling flow between rotating cyl-


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FIG. 18. Meridional vector velocity plots of the deviations from the average velocity field of the vortex in the oscillatory, high-swirl regime: deviation from the mean at (a) t 202.5, (b) t 203.1, (c) t 203.6, and (d) t 204.1. The plots show thedomain from 0 r 0.5, 0 z 0.5. The maximum velocity deviations are indicated at the top of each frame.

inders). These equations describe the evolution of smalldisturbances of the form

u(r, z, t) u(r)ei(kzt), (6.1)

and so on for the perturbation azimuthal and verticalvelocities. Modifying those equations for arbitrary az-imuthal velocity profiles, we have a system of two equa-tions for the perturbation velocity functions u(r) and

, in a background flow with mean vertical velocity(r)W(r) and mean azimuthal velocity V(r):

2 2[DD* k i( kW)](DD* k )u

2W 2Vk ikru DD* [ ]r r (6.2)

dV V2[DD* k i( kW)] u, (6.3) dr r

where D / r, D* / r 1/r, and the boundary

conditions are

u DD*u 0 at r 0 (6.4)

u Du 0 at r R. (6.5)

This eigenvalue problem for can be solved by dis-cretizing the system such that the functions V, W, u, and

are replaced by vectors of finite length, and the dif-ferential operators are replaced by matrix representa-tions of their usual centered finite-difference approxi-


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FIG. 19. Meridional vector velocity plots of the deviations from the average velocity field of the vortex in the steady, low-swirl regime: deviation from the mean at (a) t 180.0, (b) t 183.0, (c) t 186.0, and (d) t 189.0. The plots show thedomain from 0 r 0.5, 0 z 0.5. The maximum velocity deviations are indicated at the top of each frame.

mations. By writing the perturbation vectors as a singlestate vector x,

ux , (6.6)[ ]v

the system of equations (6.2)(6.5) may be reduced toa single discrete eigenvalue problem,

Ax Bx, (6.7)

which can be solved with standard generalized eigen-value routines.

The mean flows V(r) and W(r) were acquired from

radial profiles of the average azimuthal and vertical ve-locity fields described in section 6a, for both the high-

swirl case with 0.4 and the low-swirl case with 0.15. We used the radial profiles of velocity at z 0.31 since the axisymmetric disturbances appeared tobe very robust at this height. These profiles are shownin Fig. 20. We first chose k 15.7, which correspondsto a vertical wavelength of 0.4, our best estimate forthe axisymmetric disturbances identified above. We setthe viscosity to be the same as in the numerical simu-lation: 0.001.

We looked for modes similar in structure to the dis-turbances we observe, with negative phase speeds and


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FIG. 20. Radial profiles of average azimuthal and vertical velocities at z 0.3125 in the two simulations considered in section 6a: (a)azimuthal velocity, high-swirl case; (b) vertical velocity, high-swirl case; (c) azimuthal velocity, low-swirl case; and (d) vertical velocity,low-swirl case.

zero or negative group velocities. The dissipation ratesfor all the modes are surprisingly high, with the imag-inary parts of their eigenvalues being on the order of0.5. The structures with the lowest dissipation rates arenot similar to the axisymmetric disturbances identified

above but rather reside far away from the axis in themost quiescent part of the flow. However, we were ableto identify modes that are similar to the disturbanceswe are looking for by selecting the one with the largestperturbation radial velocities u(r) in the vicinity of r 0.1. This mode is shown in Fig. 21 and compares wellto the disturbance shown in Fig. 17b.

While we have not identified the source of these dis-turbances as spontaneously growing unstable modes inthe vortex core, some useful information can be foundby examination of the phase speeds and group velocities

of these modes in the high- and low-swirl cases. Figure22 shows plots of the phase speed cp /k, decayrate Im{}, and the real parts of the eigenvalues , asa function ofkfor the two examples. The group velocitycg d/dk may be inferred from the plots of the

Re{}. We see that in the high-swirl case, the modeswith k l5.7 have a phase speed of 0.18 and a groupvelocity of 0.07; by comparison, the disturbances weobserved in the model had a phase speed of roughly0.04. In the low-swirl case, the phase speed is 0.05and the group velocity is 0.15. Therefore, the tendencyfor linear disturbances to travel down the vortex coreand reach the surface is much less in the low-swirl casethan in the high-swirl case. While modes with down-ward group velocities were present for all wavenumbersin both cases, such modes were not similar to the dis-


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FIG. 21. Modal structure with vertical wavelength 0.4 and maximum radial velocities at r 0.1 found from eigenanalysis of the vortex core in the high-swirl case.

turbances identified in the numerical simulations, butrather were associated with the downward recirculationoutside of the vortex. Note also that Fig. 22b does in-dicate the existence of modes near the axis with zeroor negative group velocities for wavenumbers k 12in the high-swirl case.

The large amplitudes of the downward-propagatingdisturbances shown in Figs. 17 and 18 (as much as 50%of the mean flow) suggest that their dynamics is verynonlinear. This nonlinearity may account for their per-sistence and their shorter wavelengths in comparison tothe predictions of the linear theory. It is also quite likelythat the rapid variation with height in the structure ofthe vortex is important, so that ultimately a successfulanalysis of the disturbances will require a full eigen-analysis of the flow in the entire domain.

c. The results of further explorations of parameterspace

Measurement of the internal swirl ratio SI offers away to characterize the state of the vortex as the pa-rameters are varied. We have seen that, for values of SIabove 2.5, the flow at the vortex core is highly oscil-latory, with alternating periods of strong and weak me-ridional flow through the vortex core. The internal swirlratio, and the behavior of the vortex, appears to dependvery strongly on the vortex Reynolds number and onlyweakly on the convective Reynolds number, with thisweak dependence decreasing for increasing convectiveReynolds number.

To further explore the apparently reciprocal nature ofthe rotational forcing and the eddy viscosity, we per-formed a large number of 64 64 resolution simula-tions with varying values of both and . Through theuse of the Cray T3E at the National Energy ResearchScientific Computing Center, we were able to efficiently

produce an ensemble of 120 simulations with varyingparameters. We found mean values and variances of thevelocities, the internal swirl ratio, and the velocity co-efficient for the parameter space defined by 0.06,0.09, . . . , 0.33, and 0.0006, 0.0007, . . . , 0.0017.The results are depicted in Figs. 23, 24, and 25, whichshow the mean internal swirl ratio, the variance of theinternal swirl ratio, and the mean velocity coefficient,respectively, as functions of and .

If the structure of the vortex did indeed depend solelyon the vortex Reynolds number, then we would expectthe internal swirl ratio (and vortex structure) to be purely

a function of ReV:

2LS S(Re ) S . (6.8)I I V I

If this were the case, then SI would have constant valueson rays emanating from the origin of the plane.For the most part, this is indeed what Fig. 23 indicates.

Figure 24 shows how the vortex makes the transitionfrom steady to unsteady flow for ReV 270, where thevariance in SI increases sharply. It also increases sharplyagain for ReV 400; this appears to be associated with


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FIG. 22. Plots of phase speed, decay rate, and the real parts of vs vertical wavenumber k for the modes with maximum amplitude at r 0.1: (a) phase speed and decay rate, high-swirl case; (b) , high-swirl case; (c) phase speed and decay rate, low-swirl case; and (d) ,low-swirl case.

the episodic appearances of a two-celled vortex struc-ture.

There are two caveats to the success of these results:1) examination of the contours of SI showed that theydid not extrapolate exactly back to the origin, but rather

to a point where 0 and has a small positive value;and 2) the hypothesized dependency for SI breaks downfor large and small . For the present we can onlyspeculate as to the cause of these anomalies. Note thatthe deviations from (6.8) for large and are alsoassociated with unsteady flow in the vortex core, asshown in Fig. 24. A very likely cause of this discrepancyis that 100 time units may not be long enough to com-pute the correct mean, due to the extreme variability ofSI when the vortex Reynolds number is large, as is clear-ly shown in Fig. 9a. In fact, the vortex aspect ratio was

much more consistent with a relationship such as (6.8)in this part of parameter space (not shown), probablydue to its lesser variability in the unsteady regime asshown in Fig. 9b. It is also very likely the solutions areunderresolved by the model for large Re V. We can offer

one hypothesis as to why the curves of constant SI donot extrapolate exactly back to the origin. Grid-basednumerical models of the NavierStokes equations gen-erally have a small amount of built-in numerical vis-cosity, which effectively increase the actual viscosity ofthe model. If we take this additional viscosity into ac-count, the lines of constant SI may indeed trace back tothe origin where the true viscosity is zero.

The results in Fig. 25 for the velocity coefficient C

show a more complicated structure. The maximum az-imuthal velocities increase both for decreasing and


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FIG. 23. Contour plot of the mean value of the internal swirl ratio as a function of thedomain rotation rate and the eddy viscosity .

FIG. 24. Contour plot of the variance of the internal swirl ratio SI as a function of the domainrotation rate and the eddy viscosity . Due to the wide range in values for the variance, thecontour levels have been chosen arbitrarily to have the following values: 0.04, 0.08, . . . , 0.16;0.2, 0.4, . . . , 1.2.


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FIG. 25. Contour plot of the mean value of the velocity coefficient C

as a function of thedomain rotation rate and the eddy viscosity .

decreasing , and for each value ofthere is a particularvalue of that produces the highest azimuthal veloc-ities. By finding the ray in the plane that is coin-cident with maximum C

values, we find that the vortex

Reynolds number that maximizes the wind speeds isReV 140. The fact that C increases with increasingReC and is maximized for Re V 140 is clearly illus-trated by the results in Fig. 26, which show C

as a

function ofU and (which are proportional to Re C andReV, respectively). Here C increases with Re C becauseof the decreasing losses of angular and radial momentumin the surface boundary layer.

Why does the structure and behavior of the flow de-pend so strongly on the vortex Reynolds number (andonly weakly on Re C)? The structure of the flow dependsvery strongly on the volume rate of the flow up throughthe axis of the vortex, which is ultimately determinedby the strength of the radial inflow in the surface bound-

ary layer and whether or not this radial inflow penetratesto the axis at r 0. The surface boundary layer, in turn,can be approximated by the boundary layer beneath apotential vortex. From the point of view of theboundary layer, and are the only dimensional pa-rameters. The dynamics of swirling boundary layers andtheir strong dependence on /has been previously dis-cussed by Barcilon (1967), Burggraf et al. (1971), andSerrin (1972). Therefore, these parameters control theradial inflow, which in turn determines the structure andbehavior of the vortex.

7. Conclusions

Our results are in general agreement with previoustornado vortex research, with the important new obser-vation of the controlling effect of the vortex Reynoldsnumber on the structure and maximum wind speeds ofaxisymmetric tornado-like vortices. This result is im-portant for two reasons. First, it unifies the results ofprevious papers, such as those by Rotunno (1979), Wal-ko and Gall (1986), Howells et al. (1988), and F94 andF95, which reported a variety of results regarding vortexstructure and maximum wind speeds for a variety ofvalues for the rotational forcing and eddy viscosity. Forexample, in regards to how the model parameters af-fected the maximum wind speeds, F94 reported that,The decrease in viscosity has little effect on the max-imum windspeed of the most intense vortex, but has alarge effect on the value of 2 that produces the most

intense vortex. With 0 2.5 104

, the most intensevortex occurs with 2 0.07 . . . . With 0 1.25 104 , the most intense vortex occurs with 2 0.04. . . . Thus Fiedler observed that the maximum windspeed as a function of the rotation rate was different fordifferent viscosities; in fact, for this example the rotationrate of maximum wind speed is approximately halvedwhen the viscosity is halved. This is indeed the rela-tionship we have demonstrated and explained.

The second reason these findings are important is thatthey indicate how these vortices will behave as we in-


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FIG. 26. Contours of the velocity coefficient C

as a function of the convective velocity scale Uand the rotation rate .

crease the Reynolds number to values that apply to theatmosphere. In particular, we see two trends that aresignificant: 1) as the model eddy viscosity is decreased,the maximum possible wind speed (for the best choiceof rotation rate) increases; and 2) less and less rotational

forcing is required to achieve that maximum wind speed.This first point is significant because it means that axi-symmetric models may yet produce realistic tornadowind speeds if they can be run with more realisticReynolds numbers. The second point is particularly im-portant because one of the obvious weaknesses of cur-rent numerical tornado-vortex simulations is that theysupply the vortex with a steady, symmetric, and unlim-ited field of rotating fluid. Our results show that as theReynolds number is increased, less and less rotation willbe necessary to produce a realistic tornado vortex.

Our conclusions have some interesting implicationsfor real tornadoes and the wide variety of structures andbehaviors they exhibit. Tornado afficionados are famil-

iar with two typical tornado structures: the tight, wound-up vortex with a nearly laminar appearance (typical ofwaterspouts, for example); and the broad, diffuse vortexwith a wide base and an extremely turbulent flow (typ-ical of severe tornadoes in the central United States).In analogy with the laboratory experiments, these twocases have been recognized as vortices in the low-swirland high-swirl regimes, respectively. Our results suggestthat the structure of a tornado is determined entirely bythe angular momentum of the background rotating windfield and an as yet undetermined appropriate value for

the turbulent eddy viscosity, which may depend on var-ious factors such as surface roughness. Given these pa-rameters, the maximum wind speeds will then be theproportional to the intensity of the convection that sup-plies the vertical forcing (the overhead thunderstorm)

and to the appropriate value of the velocity coefficientC

.The structure and dynamics of actual tornadoes will

therefore depend crucially on the details of their tur-bulent swirling boundary layers. This leads to the in-evitable conclusion that a complete understanding oftornado vortices that can make more precise claims forobservables such as the maximum wind speeds will re-quire an understanding of how a fully turbulent swirlingboundary layer is different from a laminar approxima-tion.

Finally, we must reconsider our earlier claim that theupper boundary (or ceiling) of the domain does not havea significant effect on the results. The vortex produced

in the simulation with a higher ceiling did in fact havea slightly weaker recirculation zone, a smaller RMW,and a larger ZMW. In other words, the higher ceilingproduces a vortex with a smaller internal swirl ratio, SI.However, we have seen that the maximum average az-imuthal wind speed increases with decreasing SI. There-fore, we must consider the possibility that simulationswith larger closed domains will produce tornado-likevortices with higher wind speeds. Investigations on thismatter are currently under way, using numerical modelswith adaptive-mesh-refinement that will make simula-


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tions with much larger domains computationally fea-sible.

Acknowledgments. The authors would like to thankProf. H. Stone and Prof. G. Flierl for helpful discussionsregarding fluid mechanics and vortex dynamics. Wewould also like to thank Prof. A. Chorin for his exten-sive advice on numerical modeling, and Dr. M. Minionand Dr. P. Colella for their Godunov method advectionroutines and their multigrid projection solvers. We areespecially grateful to Dr. R. Kupferman for suggestingthe methods we used to generate the matrix operatorsand to enforce higher-order boundary conditions in ouranalysis of axisymmetric waves in the vortex core. Wealso gratefully acknowledge the National Energy Re-search Scientific Computing Center for providing accessto the CRAY T3E and providing support for its use.Much of the work for this report was prepared as partof the Ph.D. thesis of D. Nolan while he was a student

at Harvard University, during which he was supportedby NSF Grants ATM-9216813 and 9623539; after No-vember of 1996, D. Nolan was supported by the AppliedMathematical Sciences Subprogram of the Office of En-ergy Research, U.S. Department of Energy, under Con-tract DE-AC03-76SF-00098. B. Farrell was supportedby NSF Grant ATM-9216813 and NSF Grant 9623539.

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The Structures and Dynamic of Tornado-like Vortice

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