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VOL. 56, NO. 9 1 MAY 1999J 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 1101

Dynamics of Elevated Vortices

ALAN SHAPIRO AND PAUL MARKOWSKI

School of Meteorology, University of Oklahoma, Norman, Oklahoma

(Manuscript received 26 August 1997, in final form 8 June 1998)

ABSTRACT

Theoretical hydrodynamic models for the behavior of vortices with axially varying rotation rates are presented.The flows are inviscid, axisymmetric, and incompressible. Two flow classes are considered: (i) radially unboundedsolid body–type vortices and (ii) vortex cores of finite radius embedded within radially decaying vortex profiles.

For radially unbounded solid body–type vortices with axially varying rotation rates, the von Karman–Bodewadtsimilarity principle is applicable and leads to exact nonlinear solutions of the Euler equations. A vortex overlyingnonrotating fluid, a vortex overlying a vortex of different strength, and more generally, a vortex with N horizontallayers of different rotation rate are considered. These vortices cannot exist in a steady state because continuityof pressure across the horizontal interface between the vortex layers demands that a secondary (meridional)circulation be generated. These similarity solutions are characterized by radial and azimuthal velocity fields thatincrease with radius and a vertical velocity field that is independent of radius. These solutions describe nonlinearinteractions between the vortex circulations and the vortex-induced secondary circulations, and may play a rolein the dynamics of the interior regions of broad mesoscale vortices. Decaying, amplifying, and oscillatorysolutions are found for different vertical boundary conditions and axial distributions of vorticity. The oscillatorysolutions are characterized by pulsations of vortex strength in lower and upper levels associated with periodicreversals in the sense of the secondary circulation. These solutions provide simple illustrations of the ‘‘vortexvalve effect,’’ sometimes used to explain cyclic changes in updraft and rotation strength in tornadic storms.

A linear analysis of the Euler equations is used to describe the short-time behavior of an elevated vortex offinite radius embedded within a radially decaying vortex profile (i.e., elevated Rankine-type vortices). The linearsolution describes the formation of a central updraft (as in the similarity solution) and an annular downdraftringing the periphery of the vortex core (not accounted for in the similarity solution). Downdraft strength issensitive to both the vortex core aspect ratio and outer vortex decay rate, being stronger and narrower for broadervortices and larger decay rates. It is hypothesized that this dynamically induced downdraft may facilitate thetransport of mesocyclone vorticity down to low levels in supercell thunderstorms.

1. Introduction

The behavior of axisymmetric hydrodynamic vorticeswith axially varying rotation rates is investigated. Weconsider two classes of vortex flows: (i) radially un-bounded solid body–type vortices and (ii) vortex coresof finite radius embedded within a radially decayingvortex profile. Both classes include the case of a vortexoverlying nonrotating fluid. For the first type of flows,the von Karman–Bodewadt similarity principle is ap-plicable and leads to new exact solutions of the nonlin-ear Euler equations. These similarity solutions describethe induction of secondary (meridional) circulations inradially unbounded solid body vortices, and the sub-sequent feedback of these circulations on the vortexcirculations. These solutions provide a description ofnonlinear processes that may occur in the interior re-

Corresponding author address: Dr. Alan Shapiro, University ofOklahoma, 100 E. Boyd, Room 1310, Norman, OK 73019.E-mail: [email protected]

gions of broad geophysical vortices. Decaying, ampli-fying, and oscillatory solutions are found for differentvertical boundary conditions and axial distributions ofvorticity. The oscillatory solutions are characterized bypulsations of vortex strength in lower and upper levelsassociated with periodic reversals in the sense of thesecondary circulation. The oscillatory behavior appearsto be a manifestation of the ‘‘vortex valve effect,’’ adynamical mechanism used to explain the ‘‘choking’’of updrafts in vortex chambers and the morphologicalchanges observed in some tornadic storms (Lemon etal. 1975; Davies-Jones 1986).

Our study of the second class of vortex flows is mo-tivated by a basic question concerning low-level me-socyclogenesis and associated tornadogenesis: in the ab-sence of precipitation and thermodynamic effects, howshould an isolated elevated vortex behave? To gain in-sight into this problem we perform a linear analysis ofthe Euler equations for an elevated vortex of finite coreradius (in the exact solutions described above the solidbody rotation extended to infinity—now we consider aninner core in solid body rotation embedded within a

1102 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

radially decaying outer vortex). The analysis, valid forsmall times, indicates the formation of an annular down-draft on the periphery of the vortex core. The downdraftis stronger and narrower for broader vortex cores andfor more rapid radial decay rates in the outer vortex.On the basis of these results we speculate that ‘‘internal’’storm dynamics associated with vertical gradients ofvorticity may induce or facilitate downdraft formationon the periphery of elevated mesocyclones (though pre-cipitation and asymmetric effects are obviously impor-tant as well). We speculate further that such a dynam-ically induced downdraft can transport vorticity to lowerlevels.

To see why elevated vortices cannot exist in a steadystate, consider the simplest example of a radially un-bounded vortex overlying nonrotating fluid. Continuityof pressure across the horizontal interface separating theupper vortex from the lower nonrotating fluid demandsthat an inward-directed pressure gradient force be im-pressed in the nonrotating flow as well as in the rotatingflow. As a consequence of this pressure gradient force,a secondary circulation develops such that the nonro-tating fluid moves inward and upward (assuming theflow is bounded by a lower impermeable boundary),while the vortex is displaced upward and outward (as-suming the flow is bounded either by an upper imper-meable boundary or a region of high static stability).Vortex strength subsequently decreases due to the‘‘squashing’’ of vortex lines in the thinning upper layer.If we modify the scenario so that the low-level vorticityis not zero but is smaller than in the upper layer, asecondary circulation should develop as before, but nowthe low-level vorticity can be stretched and amplified.When the strength of the low-level vortex exceeds thatof the upper-level vortex, the secondary circulationshould begin to weaken. The evolution of the secondarycirculation and its feedback on the vortex circulationthrough nonlinear processes are the subjects of this in-vestigation.

The difficulty in solving the equations of fluid motionanalytically stems from the presence of nonlinear termsassociated with fluid inertia. Exact solutions have beenobtained only in the special cases for which the non-linearity could be cast in a tractable form, typically forflows characterized by a few degrees of freedom. Theexplicit dependence of an exact solution on a few keyparameters collapses an infinite number of equivalentnumerical simulations into one solution, thereby facil-itating the analysis of flow structure and behavior. Theserare solutions are prized for their insights into funda-mental fluid flows and their utility as test solutions forthe validation of numerical flow models (Shapiro 1993).

We now briefly review some of the exact vortex so-lutions of the Navier–Stokes, Euler, and shallow waterequations. Vortex solutions and vortex dynamics in gen-eral are surveyed in Greenspan (1968), Lugt (1983),and Saffman (1992).

Exact vortex solutions of the Navier–Stokes equations

for viscous incompressible flow include the decayingline vortex (Batchelor 1967), Taylor’s decaying vortexgrid (Rosenhead 1963), the interaction of a potentialvortex with a solid boundary (Serrin 1972; Yih et al.1982; Paull and Pillow 1985), a vortex in a convergingstagnation point flow (Burgers 1948; Rott 1958, 1959),axial flow reversal in a two-celled vortex in stagnationpoint flow (Sullivan 1959), unsteady multicellular vor-tices in stagnation point flow (Bellamy-Knights 1970,1971; Hatton 1975), decaying viscous vortices satisfy-ing the Beltrami condition for alignment of the velocityand vorticity vectors (Shapiro 1993), flow due to aninfinite rotating disk (von Karman 1921), and the closelyrelated solution for the interaction of a solid body vortexwith a stationary plate (Bodewadt 1940; Zandbergenand Dijkstra 1987). We also mention Long’s well-knownviscous vortex (1958, 1961), which complemented hisinviscid theory for rotating flow drawn into a sink atthe base of a cylinder (Long 1956). However, Long’sviscous vortex is not quite a bona fide exact solution ofthe Navier–Stokes equations since an internal boundary-layer approximation was made. Similarity solutions forsteady and unsteady convective atmospheric vorticeshave been described by Gutman (1957), Kuo (1966,1967), and Bellamy-Knights and Saci (1983).

These and other exact solutions of the Navier–Stokesequations are sometimes used as proxies for tornadolikevortices (Davies-Jones 1986; Lewellen 1993). For in-stance, the paradigm of ‘‘vorticity diffusion balancingadvection’’ embodied by the viscous stagnation pointflow vortices of Burgers, Rott, Sullivan, Bellamy-Knights, and others may be relevant to the inner coreof tornadoes and other intense vortices. On the otherhand, the secondary circulations in these stagnationflow–type vortices are dynamically decoupled from theazimuthal velocity component. This decoupling readilypermits an analytic solution to be obtained, but is prob-ably not realistic for most geophysical vortices (includ-ing the tornado). We also note that the secondary cir-culations in these stagnation flow type vortices have asingularity at infinity (a similar singularity also beingpresent in the von Karman–Bodewadt-type solutionsand in the solutions described in our present study). Ifwe restrict attention to the core region, the existence ofthis singularity may not be troublesome. In contrast, theaxial singularity in Serrin’s vortex (which acts as a spin-ning wire) actually drives the flow, and is more offensivethan the stagnation flow type singularity for the studyof the core region. On the other hand, Serrin’s vortexprovides one of the few exact solutions of the Navier–Stokes equations in which both the impermeability andno-slip conditions are enforced on a rigid horizontalboundary. The near-surface flow of Serrin’s vortex be-yond the core region may be a useful analog for thefrictional boundary layer in the region of tornadoes be-yond the radius of maximum wind. Indeed, the inversedistance velocity scaling characterizing Serrin’s vortex(and other, simpler vortices such as the Rankine vortex)

1 MAY 1999 1103S H A P I R O A N D M A R K O W S K I

has been observed in real tornadoes (Wurman et al.1996).

Exact vortex solutions of the Euler equations for in-viscid flow include the Rankine vortex (circular patchof constant vorticity), Kirchoff’s rotating elliptical vor-tex patch (Lamb 1945), elliptical vortex patch in a uni-form straining field (Moore and Saffman 1971), Hill’s(1894) propagating spherical vortex, simple configura-tions of mutually advecting line vortices (Lamb 1945;Saffman 1992; Aref et al. 1992), inviscid Beltrami flows(Lilly 1983, 1986; Davies-Jones 1985), inviscid sinkvortex (Long 1956), and vortex flows through turbo-machinery (Bragg and Hawthorne 1950). A class ofexact polynomial solutions of the shallow water equa-tions, which includes some vortex solutions, has beenstudied by Ball (1964), Thacker (1981), Cushman-Rois-in (1984, 1987), and Cushman-Roisin et al. (1985), withCushman-Roisin applying his elliptical vortex solutionto oceanic warm-core rings. The exact solutions de-scribed in our present study are very closely related tothese shallow water solutions.

The organization of this paper is as follows. In section2 we show how the Euler equations for radially un-bounded solid body–vortex-type flows reduce to a sim-pler form under the von Karman–Bodewadt similarityprinciple. In section 3 we set up the problem of an N-layer vortex flow satisfying this similarity principle andcharacterized by radial and azimuthal velocity fields thatare piecewise constant functions of height. The N-layerflow is bounded at the bottom by an impermeable plate,and is either bounded at the top by an impermeable plateor is unbounded vertically. Exact analytic solutions arederived in section 4 for two-layer (N 5 2) flows. Weconsider (i) a vortex overlying nonrotating fluid and (ii)a vortex overlying a vortex of different strength. Nu-merical results for some particular three-layer vortices(N 5 3) are presented in section 5. In section 6 we turnattention to an elevated vortex core in solid body ro-tation embedded within a radially decaying outer vortex,a specification that includes the classical (but elevated)Rankine vortex. The presence of the radial length scalegreatly complicates the analysis, and we abandon a pur-suit of nonlinear solutions in favor of a linear analysisvalid for small times. A summary and discussion followin section 7.

2. Similarity hypothesis and its consequencesThe governing equations for inviscid, axisymmetric

vortex flows are the Euler equations, expressed in cy-lindrical polar coordinates (r, f, z) as,

2]u ]u ]u y 1 ]p1 u 1 w 2 5 2 , (1)

]t ]r ]z r r ]r

]y ]y ]y uy1 u 1 w 1 5 0, (2)

]t ]r ]z r

]w ]w ]w 1 ]p1 u 1 w 5 2 . (3)

]t ]r ]z r ]zWe also consider the flow to be incompressible,

]u u ]w1 1 5 0. (4)

]r r ]z

Here u, y , and w are the radial, azimuthal (swirling),and vertical velocity components, respectively, r is the(constant) density, and p is the perturbation pressure(deviation of the total pressure from a hydrostatic ref-erence state).

The initial azimuthal velocity profile consists of solidbody rotation, y(r, z, 0) 5 V(z)r, with the angular ve-locity V(z) varying in a prescribed manner along theaxis of symmetry. The flow is unbounded in the radialdirection and there is a singularity at radial infinity. Asecondary circulation is anticipated to develop in theflow, with the azimuthal velocity profile remaining ofsolid body type, y(r, z, t) 5 V(z, t)r, with an evolvingangular velocity profile. Inspection of (1)–(4) suggeststhat if such a flow is mathematically feasible, the ve-locity field must satisfy the von Karman–Bodewadt sim-ilarity principle:

u 5 rF(z, t), y 5 rV(z, t), w 5 H(z, t). (5)

According to this scaling, the vertical velocity fieldis independent of radius. Thus, all fluid comprising ahorizontal surface is displaced vertically at the samerate, and initially horizontal material surfaces remainhorizontal for all time. For the flows considered herein,the initially horizontal interface between vortices rotat-ing with different angular velocities (or between a vor-tex overlying nonrotating air) remains horizontal. Wenote that the angular velocity V, the vertical vorticityz [[(1/r)]ry /]r 2 (1/r)]u/]f 5 2V], and the horizontaldivergence d [[(1/r)]ru/]r 5 2F] are also independentof radius.

These similarity relations (without the time depen-dence) were used to describe steady-state flows inducedby an infinite rotating disk (von Karman 1921), the flowof a rotating fluid over a stationary disk (Bodewadt1940), and flows between infinite rotating coaxial disks(Batchelor 1951; Stewartson 1953). The time-dependentrelations were applied to the development of the vonKarman–Bodewadt flows by Pearson (1965), Bodonyiand Stewartson (1977), and Bodonyi (1978). Recent in-vestigations into von Karman–Bodewadt-type flows in-dicate the nonexistence of solutions as well as the ex-istence of multiple solutions for certain parameter values(Zandbergen and Dijkstra 1987, and references therein).

In view of (4), F and the Stokes streamfunction c[defined by u 5 (1/r)]c/]z and w 5 (21/r)]c/]r] arerelated to H by

1 ]HF 5 2 , (6)

2 ]z2r

c 5 2 H. (7)2

Using (5) and (6), the equations of motion (1)–(3)reduce to


2 2r ] ]H 1 ]H ] H 1 ]p22 1 2 H 2 2V 5 2 , (8)

21 2 1 2[ ]2 ]t ]z 2 ]z ]z r ]r

]V ]H ]V5 V 2 H , (9)

]t ]z ]z

]H ]H 1 ]p1 H 5 2 . (10)

]t ]z r ]z

Since the angular velocity V is half the vertical vor-ticity, the azimuthal equation of motion (9) can also beinterpreted as the vertical vorticity equation. Equation(9) relates local changes in vertical vorticity to verticaladvection and stretching of vertical vorticity.

An equation for the azimuthal vorticity component,h [[]u/]z 2 ]w/]r 5 ]u/]z 5 (2r/2)]2H/]z2], is ob-tained from the azimuthal component of the curl of theequations of motion. Taking ]/]r of (10), yields ]2p/]r]z 5 0, that is, the radial pressure gradient force isindependent of height, a feature characteristic of bound-ary layer flows. In the context of geophysical vortices,however, the radial independence of the axial pressuregradient is an inadequacy of our similarity model (oneshared by the stagnation-type Burgers, Rott, Sullivan,and Bellamy–Knights vortices and the viscous von Kar-man–Bodewadt vortex flows). In view of this indepen-dence, the vertical derivative of (8) yields

2 2] ] ]H 1 ]H ] H22 1 2 H 2 2V 5 0. (11)

21 2 1 2[ ]]z ]t ]z 2 ]z ]z

The azimuthal vorticity equation for axisymmetric, in-compressible, inviscid flows is

2]h ]h ]h h 1 ]y1 u 1 w 2 u 2 5 0. (12)

]t ]r ]z r r ]z

For a flow satisfying the similarity hypothesis (5),u]h/]r and 2uh/r are of equal magnitude and oppositesign [(6r/4)(]H/]z)(]2H/]z2)], and (12) splits into twoequations that hold simultaneously,

2]h ]h 1 ]y1 w 2 5 0 and (13a)

]t ]z r ]z

]h hu 2 u 5 0. (13b)

]r r

According to (13a), local changes in azimuthal vorticityare forced by vertical advection of azimuthal vorticityand by differential centrifugal forcing. Equation (13b)describes a balance between the radial advection of az-imuthal vorticity and the stretching of azimuthal vor-ticity associated with radially displaced toroidal vortexlines. It can readily be shown that (11) is equivalent to(13a).

Integrating (11) with respect to z gives back (8) witha function of integration C(t) accounting for the radialpressure gradient force,

2 2] ]H 1 ]H ] H22 1 2 H 2 2V 5 2C(t). (14)

21 2 1 2]t ]z 2 ]z ]z

We can therefore think of (14) as either the verticallyintegrated azimuthal vorticity equation or as the radialequation of motion. Alternatively, we note that underthe similarity hypothesis (5), the horizontal divergenceequation (Brandes et al. 1988) reduces to

2 2]d ]d d z 121 w 1 2 5 2 ¹ p. (15)H]t ]z 2 2 r

Applying z 5 2V and d 5 2]H/]z in (15) yields (14),with C(t) identified as p/r. Thus (14) can also be2¹H

interpreted as the divergence equation.To determine the pressure, integrate (8) with respect

to r, obtaining p/r 5 Q(z, t) 1 C(t)r2/4. The functionof integration Q(z, t) is evaluated by applying this equa-tion to (10) and integrating with respect to z. In thismanner we find,

z 2 2p p ]H H r05 2 dz9 1 1 C(t) . (16)E1 2r r ]t 2 40

We will take z 5 0 to be an impermeable boundary, inwhich case p0 is the stagnation pressure. Here, C(t), theheight-independent forcing term in (14), is proportionalto the radial pressure gradient force (and is equal to

p/r).2¹H

It should be noted that our two-dimensional (z, t)partial differential equations (9) and (14), and pressureformula (16) follow from the three-dimensional (r, z, t)Euler equations without approximation. The fact thatradius does not appear in (9) or (14) confirms that exactsolutions of the Euler equations in the similarity form(5) are at least mathematically feasible. In sections 3–5, we seek exact solutions of these equations for flowswith piecewise constant vertical profiles of radial andangular velocity.

3. Governing equations for the N-layer vortex

We consider the special class of flows in which theazimuthal velocity is a piecewise constant function ofheight. In this case the azimuthal vorticity equation (12)can be expressed in Lagrangian form within each vortexlayer as d(h/r)/dt 5 ]V2/]z 5 0, showing that h/r(5]u/]z) is conserved. If we consider the initial radialvelocity field to be a piecewise function of height (sothat ]u/]z 5 0 initially in a vortex layer), then h/r iszero initially, and the conservation principle indicatesthat h/r (and hence ]u/]z) must be zero within eachlayer for all time (nonzero azimuthal vorticity is asso-ciated with infinite shear on the interfaces between thevortex layers). Thus, the radial velocity, angular mo-mentum, and horizontal divergence are constant withineach layer, and the vertical velocity varies linearly withheight within each layer. In general, we consider N fluidlayers in solid body rotation with different thicknesses


and rotation rates. We speculate that a vortex with acontinuous profile of angular velocity should be wellapproximated by the discrete N layer model for largeN.

With piecewise constant height dependencies for theazimuthal and radial velocity functions, and a piecewiselinear height dependence for the vertical velocity func-tion, the partial differential equations (9) and (14) [orequivalently (9) and (15)] reduce, without approxima-tion, to a system of ordinary differential equations forthe time-dependent amplitudes of the velocity functions.Low-order polynomials in the spatial coordinates (linearfor velocity, quadratic for free-surface displacement)have been used previously to obtain exact solutions ofthe nonlinear shallow water equations corresponding toelliptic paraboloidal vortices and to free oscillations inrotating elliptic paraboloidal basins (Ball 1963; Milesand Ball 1963; Ball 1964 and 1965; Thacker 1981;Cushman-Roisin 1984 and 1987; Cushman-Roisin et al.1985; Shapiro 1996). Numerical solutions of the shallowwater equations with this polynomial model were em-ployed by Tsonis et al. (1994) in a study of nonlineartime series analysis.

The vortex layers are labeled in order of increasingheight, from the lowest layer (n 5 1) to the highestlayer (n 5 N). The nth layer thickness, angular mo-mentum, and horizontal divergence functions are de-noted by Tn(t), Vn(t), and dn(t), respectively. The heightof the (n 2 1)th interface (the interface between the (n2 1)th and nth layers) is given by [ Ti. Then21z* Sn21 i51

nth vertical velocity function Hn(z, t) is related to thehorizontal divergence by dn(t) 5 2]Hn/]z. Integratingthis latter equation with respect to height within eachlayer yields Hn(z, t) 5 2zdn(t) 1 qn(t), where qn(t) isthe nth layer function of integration. Equivalently, wemay write

dz5 2zd (t) 1 q (t), z ∈ [z* , z* 1 T ). (17)n n n21 n21 ndt

Imposing the impermeability condition on the lowerboundary (z 5 0) yields q1 5 0. The requirement thatthe vertical velocity be continuous across the layer in-terfaces then yields a recursion relation for the rest ofthe functions of integration, qn 5 qn21 1 (dn 2 dn21).z*n21

Thus, the vertical velocity field can be expressed com-pletely in terms of the layer thicknesses and divergences.

With dn(t) 5 2]Hn/]z, the vertical vorticity (azi-muthal velocity) equation (9) becomes

dVn 5 2d V , n 5 1, 2, . . . , N, (18)n ndt

and the divergence equation (15) becomes2dd dn n 25 2 1 2V 2 C(t), n 5 1, 2, . . . , N. (19)ndt 2

An equation for the evolution of the nth layer thicknessTn(t) is obtained by evaluating (17) at the bottom and

top of the nth layer ( and 1 Tn, respectively)z* z*n21 n21

and subtracting the expression at the bottom from theexpression at the top,

dTn 5 2d T , n 5 1, 2, . . . , N. (20)n ndt

We suppose the flow is bounded from below by animpermeable boundary at z 5 0 and consider two pos-sible upper boundary conditions: (i) the flow is boundedat z 5 h by an impermeable boundary or (ii) the flowis unbounded vertically, though with finite vertical ve-locity at vertical infinity (necessitating zero divergencein the top layer dN(t) 5 0). In the latter case, the toplayer is displaced vertically, as a solid body, with nochange in thickness (dTN/dt 5 0), and with a verticalvelocity equal to the vertical velocity at the top of theunderlying (N 2 1)th layer. According to the azimuthalequation of motion (18), the angular velocity in the toplayer would then be unchanged [VN(t) 5 VN(0)], andthe divergence equation (19) would yield

C(t) (21)25 2V (for a vertically unbounded vortex).N

In the case of the vertically bounded vortex, the totalthickness of the vortex is constant,

N

T 5 h, (for a vertically bounded vortex), (22)O nn51

and therefore, dTn/dt 5 0, or, in view of (20),NSn51

dnTn 5 0.NSn51

Equations (18)–(20) compose 3N ordinary differen-tial equations in 3N 1 1 unknowns: Tn(t), Vn(t), dn(t),and C(t) (5 p/r). Closure is provided by boundary2¹H

data in the form of (21) [or, equivalently dN(t) 5 0] forthe vertically unbounded vortex, or (22) for the boundedvortex.

A first integral of the motion is obtained by elimi-nating dn between (18) and (20), resulting in the nth-layer potential vorticity conservation equation,d(Vn/Tn)/dt 5 0, or

V (t) V (0)n n5 , n 5 1, 2, . . . , N. (23)T (t) T (0)n n

Since Tn(t) . 0, the rotation rate Vn(t) is always of thesame sense as the initial rotation rate Vn(0).

Eliminating C(t) from Eq. (19) as applied to the nthand (n 2 1)th layers yields N 2 1 equations of the form

2 2dd dd d dn n21 n n21 2 22 1 2 2 2V 1 2V 5 0,n n21dt dt 2 2

n 5 2, 3, . . . , N. (24)

Using (20) and (23) to eliminate dn, dn21, Vn, and Vn21

in (24) in favor of the layer thicknesses, we get N 2 1coupled second-order equations in N unknowns,


2 22 21 d T 1 d T 3 1 dT 3 1 dTn21 n n21 n2 2 12 2 1 2 1 2T dt T dt 2 T dt 2 T dtn21 n n21 n

2 2V (0) V (0)n21 n2 21 2 T 2 2 T 5 0,n21 n1 2 1 2T (0) T (0)n21 n

n 5 2, 3, . . . , N. (25)

Closure is provided by the boundary data, as describedabove.

4. Two-layer vortices

a. Vortex overlying nonrotating fluid—Rigid lowerboundary

First consider the special case of a solid body vortexof infinite radial and vertical extent overlying non-rotating fluid bounded from below by a rigid hori-zontal plate. The initial radial and vertical velocitycomponents in both layers are taken to be zero. Afterthe initial time, the vortex pressure gradient (whichis impressed on the nonrotating fluid) induces a radialinflow in the nonrotating fluid. Associated with thisconverging low-level flow is a horizontally uniformvertical velocity field that increases in magnitude withheight from the lower boundary up to the vortex/non-rotating fluid interface. Since there is no upper bound-ary, the vertical motion in the vortex should not beimpeded, and the vortex can be displaced upward, asa solid body, with zero radial velocity and with avertical velocity equal to the vertical velocity at theinterface. According to the azimuthal equation of mo-tion (18), the angular velocity in the vortex V 2 wouldbe unchanged by the upward displacement, and theazimuthal velocity in the lower fluid would be zerosince it was zero initially. Thus, our special case cor-responds to N 5 2, z1 [ 0, and d 2 [ 0. From (21),C(t) is equal to , a positive constant, and the di-22V 2

vergence equation (19) for the nonrotating layer (n 51) becomes

2dd d1 1 25 2 2 2V , (26)2dt 2

which has the general solution

sinV t 2 B cosV t2 2d 5 22V . (27)1 21 2cosV t 1 B sinV t2 2

The initial condition d1(0) 5 0 implies that B 5 0, and(27) reduces to

d1 5 22V2 tanV2t. (28)

The thickness of the nonrotating layer (interface height)is obtained from (20) and (28) as

T1(t) 5 T1(0) sec2V2t. (29)

Thus, the divergence (and the radial and vertical ve-

locity components) and the interface height increaserapidly and become infinite at a finite time, T 5 p/(2V 2), equal to a quarter of the orbital period T* (52p/V 2) of upper-layer parcels about the axis of symmetry.1

This intriguing result can be compared with the sin-gular behavior of some unsteady viscous von Karman–Bodewadt-type flows. Bodonyi (1978) and Bodonyiand Stewartson (1977) report a breakdown of the nu-merical solution (verified by an asymptotic analysis)of rotating flow in which a lower disk is abruptly forcedto counterrotate. In these studies the velocity field andboundary layer depth on the lower disk grew explo-sively and became singular within half an orbital pe-riod. We speculate that the breakdown mechanism forthe viscous counterrotating flow is similar to that inour inviscid elevated vortex flow: in the absence of anupper boundary, the maintained impressing of a vortexpressure gradient force on nonrotating fluid leads toexplosive vertical accelerations. In the context of ourinviscid elevated vortex, a nonrotating fluid layer isspecified in the initial condition. In the viscous coun-terrotating flows, a level of nonrotating fluid (the exactlocation of which is influenced by diffusion) is alwayspresent.

b. Two layer vortices—Rigid lower boundary

If the lower fluid has some rotation, no matter howsmall, the singular behavior deduced above disappears.Instead, an oscillatory secondary circulation is set upin the flow in which the angular velocity and thicknessof the lower-layer vortex alternately increases and de-creases. In the absence of an upper rigid lid, the elevatedvortex oscillates vertically as a solid body (V2 5 const,d2 [ 0).

For the lower-layer flow (n 5 1), (19), (20), and (23)become

2dd d1 1 2 21 1 2V 2 2V 5 0, (30)2 1dt 2

1 dT1d (t) 5 2 , (31)1 T dt1

V (0)1V (t) 5 T (t) . (32)1 1T (0)1

Applying (31) and (32) in (30) results in a second-order nonlinear ordinary differential equation for thelower-layer thickness (interface height),

1 It can also be shown that the flow becomes singular within a finitetime for (i) V2 ± 0 and any choice of initial value d1(0), and for (ii)V2 5 0 and any negative initial value d1(0). In either case, the sin-gularity is associated with the nonlinear term in (26), or, equiv-2d1

alently, the (]H/]z)2 term in (8), which accounts for radial advectionof radial momentum.


FIG. 1. Evolution of nondimensional interface height T1/T1(0) (solidline) and lower layer convergence 2d1/2|V2| (line with circles) forthe vertically unconfined two-layer vortex with (a) a 5 0.1, (b) a 52.0, and (c) a 5 10.0. The lower-layer radial velocity function isgiven by F1 5 d1/2. Here a [ (0)/ is the initial ratio of (squared)2 2V V1 2

lower-layer to upper-layer angular velocity, and t [ 2|V2|t is non-dimensional time.

2 22d T 3 1 dT V (0)1 1 1 3 22 1 2 T 5 2V T . (33)1 2 12 1 2 1 2dt 2 T dt T (0)1 1

Since (33) does not explicitly involve the independentvariable, its order may be reduced by changing the de-pendent variable to P [ dT1/dt and regarding T1 as thenew independent variable,

22 2dP P V (0)1 3 22 3 1 4 T 5 4V T . (34)1 2 11 2dT T T (0)1 1 1

Equation (34) is a first-order linear equation for P2, withsolution (subject to the condition of no initial verticalmotion),

2 2 2 2dT V (0) V (0) V1 1 1 24 35 24 T 1 4 1 T1 11 2 1 2 1 2dt T (0) T (0) T (0)1 1 1

2 22 4V T ,2 1

or, after taking the root,

1/22dT |V (0)| V T (0)1 1 2 15 62 T [T 2 T (0)] 2 T .1 1 1 125 6[ ]dt T (0) V (0)1 1

(35)

The sign in (35) is determined by the requirement thatthe solution be real, that is, that [T1 2 T1(0)][ T1(0)2V2

2 (0)T1] be nonnegative. If upper-vortex rotation is2V1

initially greater in magnitude than lower-vortex rotation[ 2 (0) . 0], the solution must initially proceed2 2V V2 1

such that T1 2 T1(0) . 0, that is, the positive branchmust be chosen and the interface rises. Conversely, ifupper-vortex rotation is initially smaller in magnitudethan lower-vortex rotation [ 2 (0) , 0], the so-2 2V V2 1

lution initially proceeds on the negative branch and theinterface falls. Therefore, the 6 symbol may be replacedwith [ 2 (0)]/| 2 (0)|.2 2 2 2V V V V2 1 2 1

Separating variables in (35), integrating, and applyingthe initial condition yields

2 2 2[V 1 V (0)]T 2 2V T (0)2 1 1 2 121sin2 2[ ]T |V 2 V (0)|1 2 1

2 2V 2 V (0) p2 15 2V t 2 , (36)22 2 1 2|V 2 V (0)| 22 1

or, after rearrangement,

2T (0)1T 5 , (37)1 1 1 a 1 (1 2 a) cos2V t2

where a [ (0)/ is the ratio of the (squared) lower-2 2V V1 2

layer to upper-layer angular velocities. Applying (37)in (31) and (32), we obtain d1 and V1 as

2V (1 2 a) sin2V t2 2d 5 2 , (38)1 1 1 a 1 (1 2 a) cos2V t2

2V (0)1V 5 . (39)1 1 1 a 1 (1 2 a) cos2V t2

In contrast to the singular nature of a vertically un-bounded elevated vortex overlying nonrotating fluid,the two-layer vortex is well behaved for all time (seeFig. 1). Here the converging ‘‘in-up’’ flow induced inthe lower layer by the upper-layer vortex spins up theweak vertical vorticity in the lower layer. Eventuallythe lower-layer vortex becomes stronger than the up-per-layer vortex and the vertical pressure gradient forcereverses. In response, the secondary circulation re-verses and the lower-layer vortex spins down, even-tually becoming weaker than the upper-layer vortex.The process repeats itself and we obtain an oscillationof the interface height and lower-layer vortex strength.


The period of these oscillations, T 5 p/|V 2|, is halfthe orbital period of parcels in the upper vortex. It canbe inferred from (35) or (37) that the interface heightT1 oscillates vertically between levels T1(0) andT1(0) / (0). Thus, the amplitude of the interface2 2V V2 1

oscillation increases sharply with decreasing initial ro-tation in the lower level. For the case of strong initiallower-level rotation, the interface height rapidly dropsto a small value and maintains small values throughoutmuch of the oscillation period.

c. Two-layer vortices—Rigid upper and lowerboundaries

Now suppose that impermeable horizontal boundariesconfine the flow on the bottom (z 5 0) and at the top(z 5 h). Both layers of this two-layer vortex are initiallyin solid body rotation with different (nonzero) angularvelocities and no initial secondary circulation. The spe-cial case of a vortex overlying nonrotating fluid is ex-amined in section 4d.

Setting n 5 2 in (24) yields2 2dd dd d d2 1 2 1 2 22 1 2 2 2V 1 2V 5 0. (40)2 1dt dt 2 2

Equations (20) and (23) for n 5 1, 2 become

1 dT 1 dT1 1d 5 2 , d 5 , (41a,b)1 2T dt h 2 T dt1 1

V (0) V (0)1 2V (t) 5 T (t), V (t) 5 [h 2 T (t)],1 1 2 1T (0) h 2 T (0)1 1

(42a,b)

where we have used T2 5 h 2 T1 [obtained from (22)].Applying (41a,b) and (42a,b) in (40) yields a second-

order nonlinear ordinary differential equation for theinterface height,

22d T 3 1 1 dT1 11 22 1 21 2dt 2 h 2 T T dt1 1

2 23T V (0) T V (0)1 2 1 132 2 (h 2 T ) 1 2 (h 2 T )1 1[ ] [ ]h h 2 T (0) h T (0)1 1

5 0.

Again, changing the dependent variable to P [ dT1/dtand regarding T1 as the new independent variable yieldsa first-order linear equation for P2:

22dP 1 1 T V (0)1 22 31 3 2 P 2 4 (h 2 T )11 2 [ ]dT h 2 T T h h 2 T (0)1 1 1 1

23T V (0)1 11 4 (h 2 T ) 5 0.1[ ]h T (0)1

(43)

Solving (43) subject to the initial condition P(0) 5 0,we obtain:

2dT 41 2 25 T (h 2 T ) [T 2 T (0)]1 1 1 11 2dt h

2 2V (0) (h 2 T ) V (0) T2 1 1 13 2 .5 6[ ] [ ]h 2 T (0) T (0) T (0) h 2 T (0)1 1 1 1

(44)

The qualitative behavior of T1(t) can be deduced withanalogy to one-dimensional particle motion in a poten-tial field, that is, by regarding T1 as a particle displace-ment and (44) as an energy equation for a conservativesystem. We regard (dT1/dt)2 as the kinetic energy andthe right-hand side of (44) as (the negative of ) a non-linear potential energy function. Since (dT1/dt)2 is non-negative, the right-hand side must be nonnegative onany domain of physical interest. The behavior of thesolution depends on the nature of this nonlinear potentialenergy, especially on the points for which the potentialenergy vanishes. These points are local extrema of T1(t),and represent the turning points of the differential equa-tion. These extrema can be identified as T1 5 T1(0), h,and , whereT91

212h V (0)1T9 [ h 1 1 2 1 . (45)1 5 6[ ][ ]T (0) V (0)1 2

It is straightforward to show that if |V2(0)| , |V1(0)|,then 0 , , T1(0), whereas if |V2(0)| . |V1(0)|, thenT91h . . T1(0). If the lower layer is not rotating [V1(0)T915 0], then 5 h, whereas if the upper layer is notT91rotating [V2(0) 5 0], then 5 0. We consider the caseT91where neither V1(0) nor V2(0) are zero. Taking the rootof (44) yields

dT h 2 T1 15 62|V (0)|2 [ ]dt h 2 T (0)1

1/2(T9 2 T )[T 2 T (0)]1 1 1 13 T , (46)15 6T9T (0)1 1

where the choice of sign is determined by the require-ment that the solution be real, that is, that [ 2 T1][T1T912 T1(0)] be nonnegative. If upper-layer rotation is ini-tially stronger than lower-layer rotation [so that 2T91T1(0) . 0], the solution must proceed initially on thepositive branch of (46) and the interface rises, whereasif upper-layer rotation is initially weaker than lower-layer rotation [so that 2 T1(0) , 0], the solutionT91must proceed initially on the negative branch of (46)and the interface falls. Thereafter, in either case, thesign in (46) changes each time T1 reaches a turningpoint. The solution is such that T1 oscillates betweenT1(0) and . The solution first reaches the turningT91point T1 5 at t 5 T/2 and completes one period ofT91oscillation at t 5 T, when T1 returns to T1(0).

The solution is obtained by separating variables in(46) and integrating,


dT1E 1/2T (h 2 T ){[T 2 T (0)][T9 2 T ]}1 1 1 1 1 1

2 sgn[T9 2 T (0)]|V (0)|1 1 25 dt, (47)E1/2[T9T (0)] [h 2 T (0)]1 1 1

where sgn is the unit sign function: sgn[ 2 T1(0)] 51T91for . T1(0), 521 for , T1(0). A partial fractionsT9 T91 1

decomposition puts the left-hand side of (47) in the formof tabulated integrals. The constant of integration is de-termined piecewise (constant for each half-period) byconsidering the initial condition and the continuity ofT1(t) at the turning points. Thus, we obtain the implicitsolution for T1(t) over a period T of this oscillation as

1/22|V (0)|h p p T9T (0)2 1 1

t 5 1 5 6[h 2 T (0)] 2 2 [h 2 T (0)](h 2 T9)1 1 1

1 F[T , T (0), T9],1 1 1 (48)

as T1 travels from T1(0) to (0 , t # T/2), andT91

1/22|V (0)|h T p p T9T (0)2 1 12 t 5 2 21 2 5 6[h 2 T (0)] 2 2 2 [h 2 T (0)](h 2 T9)1 1 1

1 F(T , T (0), T9),1 1 1 (49)

as T1 travels from back to T1(0) (T/2 , t , T), whereT91

[T9 1 T (0)]T 2 2T9T (0)1 1 1 1 121F[T , T (0), T9] 5 sin1 1 1 5 6T [T9 2 T (0)]1 1 1

1/2T9T (0) [2h 2 T (0) 2 T9](h 2 T ) 2 2[h 2 T (0)](h 2 T9)1 1 1 1 1 1 1212 sin . (50)5 6 5 6[h 2 T (0)](h 2 T9) (h 2 T )[T9 2 T (0)]1 1 1 1 1

The oscillation period is1/2

p[h 2 T (0)] T9T (0)1 1 1T 5 1 1 .7 5 6 8h|V (0)| [h 2 T (0)](h 2 T9)2 1 1

(51)

Applying (45) to (51), the oscillation period becomes

1 T (0) 1 T (0)1 1T 5 p 1 2 15 6[ ]|V (0)| h |V (0)| h2 1

hp 15 dz. (52)Eh |V(z, 0)|0

Thus T is equal to p times the mean reciprocal mag-nitude of the initial angular velocity, or half the meaninitial orbital period.

Solutions for V1 and V2 in terms of the interfaceheight follow immediately from (42a,b). Solutions ford1 and d2 are obtained from (41a,b) and (46)

1/2h 2 T (T9 2 T )[T 2 T (0)]1 1 1 1 1d 5 72|V (0)| ,1 2 5 6[ ]h 2 T (0) T9T (0)1 1 1

(53)

1/2T (T9 2 T )[T 2 T (0)]1 1 1 1 1d 5 62|V (0)| ,2 2 5 6[ ]h 2 T (0) T9T (0)1 1 1

(54)

the signs being inferred piecewise (on half-period in-tervals) from the above considerations.

Two examples are presented in Fig. 2. In both cases

the initial interface height T1(0) is set at 0.2h. In Fig.2a the lower-layer rotation is initially weaker than theupper-layer rotation and the interface quickly rises (thusstrengthening the lower-layer angular velocity and sow-ing the seeds for the eventual reversal of the secondarycirculation). The interface rapidly displaces much of theupper-layer fluid and maintains a high altitude through-out much of the period. In the case of strong low-layerrotation [Fig. 2(b)], the interface initially falls. Despitethe gentleness of the interface descent, a jet of strongradial velocity appears in the lower layer, a consequenceof mass conservation and the relative shallowness ofthe lower layer.

d. Vortex overlying nonrotating fluid—Rigid upperand lower boundaries

If there is no rotation in the lower layer initially, V1(0)5 0, and therefore V1(t) 5 0 for all time and 5 h.T91In this case, we need only consider the positive branchof (46). Separating variables and integrating, we obtain,

[h 1 T (0)]T 2 2hT (0)1 1 121sin 5 6T [h 2 T (0)]1 1

1/21/22[hT (0)] T 2 T (0) p 2h|V (0)|1 1 1 21 5 2 1 t.[ ]h 2 T (0) h 2 T 2 h 2 T (0)1 1 1

(55)

The solution for V2 follows from (42), and the solutionsfor d1 and d2 are given by the negative and positivebranches of (53) and (54), respectively, with T1(t) de-


FIG. 2. Evolution of a vertically confined two-layer vortex. Non-dimensional interface height T1/h (solid line), lower-layer conver-gence 2d1/2|V2| (line with circles), and upper-layer divergenced2/2|V2| (line with squares) are shown for an initial interface heightT1(0) of 0.2h. Initial ratio of lower-layer to upper-layer angular ve-locities V1(0)/V2(0) is (a) 0.1 and (b) 3.0. The radial velocity functionis given by F1 5 d1/2 in the lower layer and F2 5 d2/2 in the upperlayer. The angular velocity functions are proportional to the respectivelayer thicknesses. Here t [ [2h|V2(0)|]/[h 2 T1(0)]t is nondimen-sional time.

FIG. 3. Evolution of a vertically confined vortex overlying non-rotating fluid. Nondimensional interface height T1/h (solid line), low-er-layer convergence 2d1/2|V2| (line with circles), and upper-layerdivergence d2/2|V2| (line with squares) are shown for an initial in-terface height T1(0) of 0.2h. The radial velocity function is given byF1 5 d1/2 in the lower layer and F2 5 d2/2 in the upper layer. Theangular velocity function in the upper layer is proportional to theupper-layer thickness. Here t [ [2h|V2(0)|]/[h 2 T1(0)]t is nondi-mensional time.

termined implicitly from (55). The solution, depicted inFig. 3 for T1(0) 5 0.2h, shows a period of rapid initialinterface ascent followed by a long period of gentleascent. The slowness of the radial velocity decay in the

top layer is again a consequence of mass conservationand the thinness of that layer. Since there is no verticalvorticity in the lower layer, the stretching mechanismdoes not operate and there is no mechanism to reversethe sense of the secondary circulation.

e. Invariance—Three-layer and multiple-layerplanar-symmetric vortices

Equations (8)–(10) are invariant to the transforma-tion: z → 2z, H → 2H. Therefore, from symmetryconsiderations, the two-layer solutions described abovecan be reflected about the lower impermeable boundary,z 5 0, to produce analytic three-layer solutions,

u(r, 2z, t) 5 u(r, z, t), (56)

y(r, 2z, t) 5 y(r, z, t), (57)

w(r, 2z, t) 5 2w(r, z, t). (58)

In the case of the semi-infinite two-layer vortices con-fined by a lower boundary, a reflection of the solutionresults in an unconfined three-layer vortex solution de-fined piecewise on the vertical intervals, z ∈ (2`, 2T1),z ∈ (2T1, T1), and z ∈ (T1, `). In the case of two-layervortices confined between horizontal boundaries at z 50 and z 5 h, a reflection of the solution results in aconfined three-layer vortex solution defined piecewiseon the vertical intervals, z ∈ (2h, 2T1), z ∈ (2T1, T1),and z ∈ (T1, h). Further reflections of these confinedvortices about the new boundaries are possible and leadto new planar-symmetric multiple-layer vortex solu-tions. Apart from their own intrinsic interest, these an-alytic solutions can also be used to validate the nu-merical algorithms for the more general (asymmetric)multiple-layer vortices described in the next section.

5. Numerical solutions for three-layer vortices

The behavior of three-layer vortices follows from thesolution of (25) and (22) with N 5 3. It is convenientto nondimensionalize variables as

T T1 2˜ ˜T [ , T [ , t [ Vt,1 2h h

˜ ˜ ˜1 T (0) T (0) T (0)1 2 3[ 1 1 , (59)V |V (0)| |V (0)| |V (0)|1 2 3

where T3 5 1 2 T1 2 T2, and rewrite (25) as a systemof four coupled first-order equations,


˜dT1 5 P, (60)dt

˜dT2 5 Q, (61)dt

dP ˜ ˜ ˜ ˜5 T (1 2 T )A 2 T T B, (62)1 1 1 2dt

dQ ˜ ˜ ˜ ˜5 2T T A 1 T (1 2 T )B, (63)1 2 2 2dt

where

2 23 P P 1 QA [ 2˜ ˜ ˜1 2 1 2[ ]2 T 1 2 T 2 T1 1 2

2 2˜ ˜ ˜2 aT 1 g(1 2 T 2 T ) , (64)1 1 2

2 23 Q P 1 QB [ 2˜ ˜ ˜1 2 1 2[ ]2 T 1 2 T 2 T2 1 2

2 2˜ ˜ ˜2 bT 1 g(1 2 T 2 T ) , (65)2 1 2

2 2V (0)/V V (0)/V1 2a [ 2 , b [ 2 ,˜ ˜1 2 1 2T (0) T (0)1 2

2V (0)/V3g [ 2 . (66)˜ ˜1 21 2 T (0) 2 T (0)1 2

By definition, 1/V is proportional to the sum of thereciprocal potential vorticities in the three layers, whilea, b, and g are proportional to the squared potentialvorticities in the lower, middle, and upper layers, re-spectively. These definitions imply

1/2 1/2 1/22 2 21 1 5 1, (67)1 2 1 2 1 2a b g

so that only two of a, b, and g are independent. How-ever, rather than specifying a, b, g, and V directly (andthen deriving the initial angular velocity functions), wefound it convenient to calculate a, b, g, and V fromspecified values of T1(0), T2(0), V1(0), V2(0), andV3(0). It can be noted that if V1(0), V2(0), and V3(0)undergo proportional changes [so that the ratios V1(0)/V2(0) and V1(0)/V3(0) are preserved], V changes whilea, b, and g remain the same. Thus, the shape of thesolution curve is affected by the relative magnitudes ofthe rotation rates while the timescale is affected by themean rotation rate (as measured by V).

Equations (60)–(63) were integrated numerically fora variety of initial conditions with the fourth-order Run-ge–Kutta formula (Press et al. 1992). In each case theintegrations were performed from t 5 0 to t 5 15 witha nondimensional time step size Dt 5 0.01. A varietyof interesting waveforms were obtained by varying theparameter settings, but in all cases the solutions wereperiodic. We speculate that the three-layer vortex is non-

chaotic but that chaos might be possible in vortices withmore layers. Results from selected three-layer calcula-tions are depicted in Figs. 4–7. In these cases the threelayers are initially of equal thickness: T1(0) 5 T2(0) 51 2 T1(0) 2 T2(0) 5 ⅓. The parameter settings aregiven in Table 1.

In vortex A1:2:4 (Fig. 4) the initial angular velocityfunctions are specified to increase upward in a ratio of 1:2:4. There is no initial vertical motion [P(0) 5 Q(0) 50]. As in the two-layer case, the initially weakest vortexlayer (low layer in this case) rapidly thickens during thefirst half of the oscillation period and increases its angularvelocity, while the initially strongest vortex layer (upperlayer) rapidly thins and weakens its angular velocity. Thelayer of middle strength rotation (middle layer) thickensslightly at first but then thins. The motion of the interfacesduring the first half of the oscillation period is associatedwith an in-up-out secondary circulation in which the low-est layer participates in the inflow and the upper layerparticipates in the outflow. The middle layer first partic-ipates in the inflow and then participates in the outflow.The sense of the circulation reverses for all layers halfwaythrough the oscillation period. Vortex B1:2:10 (Fig. 5) issimilar to vortex A1:2:4 except the initial angular veloc-ities now increase upward in a ratio of 1:2:10. The in-creased initial discrepancy between the rotation rates inthe lowest two layers and the upper layer amplifies thesubsequent secondary circulation in all layers. Next con-sider vortex C1:2:4 (Fig. 6), which has an initially de-scending upper interface with a vertical velocity of Q(0)5 20.5 (all other initial conditions being the same as inA1:2:4). Introduction of the nonzero interface velocity cre-ates an asymmetry in the waveforms of all the variables.Compared to A1:2:4, we see that the initial descent of theupper-layer interface in C1:2:4 results in a short-term in-crease in the upper-layer thickness and a decrease in themiddle-layer thickness. However, the lower layer, respond-ing to the vortex pressure gradient, thickens as in A1:2:4. Vortex D1:2:10, the counterpart of B1:2:10 with non-zero initial vertical velocity Q(0) 5 20.5, is depicted inFig. 7.

6. Initial behavior of an elevated vortex withradial power-law decay for r . R

The similarity vortices in sections 2–5 are noteworthyin that they provide rare exact descriptions of nonlinearinteractions between vortex circulations and vortex-in-duced secondary circulations. However, the restrictivenature of the similarity scalings for the velocity andpressure fields suggest that the relevance of the simi-larity solutions to geophysical flows may be limited tothe interior portions of broad vortices. Although thegeneral validity of the similarity approach to solid bodyvortices of finite area lies beyond the scope of this in-vestigation, a specific comparison at a small time isprovided later in this section.

Mid- and lower-level mesocyclones and other me-


FIG. 4. Evolution of vortex A1:2:4, a vertically confined three-layer vortex with initially equal layerthicknesses, and initial layer angular velocities increasing upward in a ratio of 1:2:4. The two interfaceshave no initial vertical motion. (a) Nondimensional lower-layer thickness T1 (solid line), middle-layerthickness T2 (line with plus symbols), and upper-layer thickness T3 5 1 2 T1 2 T2 (line with circles).(b) Nondimensional convergence values 2d1/V, 2d2/V, and 2d3/V in the lower (solid line), middle(line with plus symbols), and upper layers (line with circles), respectively. The nth layer radial velocityfunction is given by Fn 5 dn/2 for n 5 1, 2, 3. The angular velocity functions are proportional tothe respective layer thicknesses. Here t [ Vt is nondimensional time.

TABLE 1. Parameter settings for selected three-layer vortices with T1 (0) 5 T2 (0) 5 1/3. Vortex names consist of a letter followed by theratio w1 (0)/w2 (0)/w3 (0) of initial layer angular velocities. All quantities are nondimensional.

Vortex w1 (0)/w w2 (0)/w w3 (0)/w P (0) Q (0) a b g

A1:2:4B1:2:10C1:2:4D1:2:10

0.58330.53330.58330.5333

1.16661.06661.16661.0666

2.33335.33332.33335.3333

0.00.00.00.0

0.00.0

20.520.5

6.1255.1206.1255.120

24.50020.48024.50020.480

98.000512.0098.000

512.00

soscale geophysical vortices typically consist of an iso-lated region of large vorticity embedded within a more-or-less nonrotating larger-scale environment. It is there-fore of interest to study a solid body-type vortex of finitearea embedded within a radially decaying vortex. Asbefore, we consider our vortex to be elevated, that is,both inner and outer parts of the vortex overlie a layerof nonrotating fluid. There is no initial secondary cir-culation in either the vortex or the lower fluid.

The length scale associated with a finite vortex coregreatly complicates the analysis and we abandon oursearch for exact solutions. Instead, we present a linearanalysis of the flow appropriate for small times. Thelong-term behavior will be investigated in future nu-merical simulations.

Consider a vortex with an initial azimuthal velocitydistribution y 0 [[y(r, z, 0)] given by

Vr, r # R, T # z # h,1n R

0 y (r, z) 5 VR , r . R, T # z # h,11 2r0, 0 # z , T . 1

(68)This specification includes the previous example of un-bounded solid-body rotation (n 5 21) and the classicalRankine vortex (n 5 1). For n . 1, the vertical vorticityin the outer region is negative. In the following, werestrict attention to n . 0 to ensure that the azimuthalvelocity decays with radius.


FIG. 5. As in Fig. 4 but for vortex B1:2:10, a three-layer vortex with initial layer angularvelocities increasing upward in a ratio of 1:2:10.

FIG. 6. As in Fig. 4 but for vortex C1:2:4, a three-layer vortex with an upper interfaceinitially descending with Q(0) 5 20.5.


FIG. 7. As in Fig. 4 but for vortex D1:2:10, a three-layer vortex with initial layer angularvelocities increasing upward in a ratio of 1:2:10, and with an upper interface initially descendingwith Q(0) 5 20.5.

In the absence of an initial secondary circulation, theazimuthal equation of motion (2) shows that the initialazimuthal velocity tendency is zero, while the azimuthalvorticity equation (12) yields an equation for the initialstreamfunction tendency c1 [[]c/]t(r, z, 0)],

22 1 2 1 1 0] c ] c 1 ]c ]y1 2 5 . (69)

2 2]z ]r r ]r ]z

The right-hand side of (69) vanishes everywhere exceptalong the interfacial singularity at z 5 T1. Integrating (69)an infinitesimal distance across this discontinuity, we ob-tain a jump condition for ]c1/]z (indicating a jump in theradial velocity tendency and thus a jump in the radialvelocity itself),

1 1]c ]c 21 2 0 1(r, T ) 2 (r, T ) 5 y (r, T ), (70)1 1 1]z ]z

where [ lim|«|→0(T1 1 |«|) and [ lim|«|→0(T1 2 |«|).1 2T T1 1

We take c1 itself as being continuous across the interfaceso that the normal velocity component is continuous.

Thus, we seek solutions of2 1 2 1 1] c ] c 1 ]c

1 2 5 0, (71)2 2]z ]r r ]r

satisfying the jump condition (70) at z 5 T1. We imposethe impermeability condition on the top and bottomboundaries, assume there is no source or sink of mass

along the axis of symmetry, and let the mass flux vanishfar from the axis of symmetry,

c1(0, z) 5 c1(r, 0) 5 c1(r, h) 5 c1(`, z) 5 0. (72)

Solving (70)–(72) for c1 (see the appendix), we obtain

` 2 22V R h kpT kpz11c 5 F(r) sin sin , (73)O 2 2 1 2 1 2k p h hk51

where

` 2r kpr K (kpxR /h) r1F(r) 5 24n(n 1 1) I dx 21 E 2n12 21 2R h x R1

r kpr kpR1 2(n 1 1) I K , r , R,1 11 2 1 2R h h

r kpr kpR5 2(n 1 1) K I1 11 2 1 2R h h

r /Rr kpr I (kpxR /h)12 4n(n 1 1) K dx1 E 2n121 2R h x1

`r kpr K (kpxR /h)12 4n(n 1 1) I dx1 E 2n121 2R h xr /R


2nR2 . r . R (74)1 2r

This solution was evaluated for a range of aspectratios R/h, decay exponents n, and interface heights T1.The modified Bessel functions I1 and K1 were evaluatedwith the IMSL MATH/LIBRARY special functionsFORTRAN subroutines BSI1 and BSK1, respectively,except for large arguments (.10), where asymptoticformulas were used (Abramowitz and Stegun 1964). Theintegrals were evaluated with the trapezoidal rule. Al-though analytic forms for the initial meridional velocitytendencies u1 [[(1/r)]c1/]z] and w1 [[(21/r)]c1/]r] areavailable, it is convenient to obtain these componentsfrom c1 via finite-difference discretizations.

Results are presented for vortices overlying nonro-tating fluid of depth T1 5 0.2h. We consider a columnarvortex (R/h 5 0.5) with weak (n 5 1) and strong (n 54) outer decay (Figs. 8 and 9, respectively), and a broadvortex (R/h 5 2) with weak (n 5 1) and strong (n 54) outer decay (Figs. 10 and 11, respectively). In allcases an updraft extends across the vortex core as wellas in the nonrotating flow beneath the core. The peakupdraft speed at a fixed radius occurs at the horizonalinterface between the rotating and nonrotating fluid. Theupdraft strength and pattern (i.e., flatness of verticalvelocity isolines) is remarkably similar for vortices ofthe same aspect ratio, with greater flatness for the broad-er vortices.

The near constancy of w with respect to radius in thecore region and the relative insensitivity of the updraftto the decay exponent suggest that the similarity solu-tions presented in the previous sections may be appli-cable within the core region of the finite radius vortices,at least for a short time. This can be seen for the ex-amples considered herein by comparing Figs. 8–11 withFig. 12. This latter figure depicts the azimuthal velocityand the time tendency of the meridional velocity fieldsfor a radially unbounded (similarity) vortex overlyingnonrotating fluid (initial depth of 0.2h). Direct com-parison with the linear finite radius solutions is facili-tated by evaluating the similarity solution (55) at a smallnondimensional time Vt 5 0.247 48 (when the interfacehas risen slightly to 0.21h), expressing the meridionalvelocity fields as tendencies (u/t and w/t, at small timet) and contouring the scaled fields to match Figs. 8–11.As can be seen, the similarity solution is in good quan-titative agreement with the broad linear vortices (Figs.10 and 11) for radii extending to ;¾ of the core radius.In contrast, the similarity solution exhibits only goodqualitative agreement with the two columnar vortices,and then only out to ;½ the core radius. For radii nearand beyond the core radius, the similarity solution de-parts significantly from both the broad and columnarvortices. Since the similarity solution is independent ofa radial length scale, it is not surprising that it is inbetter agreement with the broader vortices than the co-

lumnar vortices. Indeed, it can be shown that for aninfinite aspect ratio R/h, the vertical velocity tendencyassociated with the linear solution (73) reduces to thevertical velocity tendency associated with the nonlinearsimilarity solution (55) in the limit of vanishing time.

Perhaps the most intriguing feature in Figs. 8–11 isthe annular downdraft ringing the updraft just beyondthe radius of maximum tangential wind. This downdraftis very sensitive to the aspect ratio and decay exponent,being stronger and narrower for larger R/h and largern. Indeed, for the broad vortex with strong radial decaydepicted in Fig. 11, the peak downdraft speed actuallyexceeds the peak updraft speed. Although our linearanalysis should not be used to quantify the angular mo-mentum transport in this downdraft, qualitatively we seethat the downdraft is at least ‘‘poised’’ to transport vor-tex angular momentum downward and radially inward.A complete picture of angular momentum transport inthe downdraft and its subsequent feedback on the sec-ondary circulation must await a nonlinear numericalsimulation. Of course, whether these symmetric vortex-induced downdrafts are physically realizable depends,in part, on the stability of the solutions. Of particularinterest is the stability of these vortices with anticyclonicvorticity in the outer region, that is, when Rayleigh’sstability criterion is violated. Determining the stabilitybounds of our unsteady vortices, while important, is aformidable task and must be deferred to a future study.

7. Summary and discussion

This investigation is concerned with the inviscid dy-namics of vortices with axially varying rotation rates,including the case of a vortex overlying nonrotatingfluid. We consider radially unbounded vortices in solidbody rotation and elevated Rankine-type vortices. Forthe former class of vortices, the von Karman–Bodewadtsimilarity principle is applicable and leads to exact un-steady solutions of the nonlinear Euler equations. Thesesimilarity solutions are noteworthy in that the meridi-onal circulation is not prescribed but is generated by thevortex circulation. The solutions describe decaying, am-plifying, and oscillatory behavior for both the primaryvortex and the vortex-induced secondary circulation.

The behavior of the oscillatory similarity solutions istypified by the case of a strong vortex overlying a weak-er vortex. In this case the radial pressure gradient forceinduces a converging low-level flow and an associatedupdraft that spins up the initially weak lower-layer vor-ticity to values exceeding that in the upper-layer vortex.The reversed axial distribution of vorticity (and the as-sociated reversal of the axial pressure gradient force)causes a reversal of the secondary circulation with asubsequent spindown of the low-level vorticity. For ourinviscid, unforced hydrodynamical model, the oscilla-tion proceeds ad infinitum.

The absence of a radial length scale and the unbound-ed nature of the similarity scalings (u and y increase


FIG

.8.I

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FIG. 12. Similarity solution for a radially unbounded vortex overlying nonrotating fluid withinitial depth of 0.2h. Fields are obtained from (55) at nondimensional time Vt 5 0.247 48, thatis, when the interface has risen slightly to 0.21h. (a) Azimuthal velocity y /(VR), (b) radialvelocity tendency u/t 3 10/(RV2), and (c) vertical velocity tendency w/t 3 10/(hV2). Negativecontours in (b) are dashed. Fields are scaled in the same manner and plotted with the samecontour interval as in Figs. (8)–(11).

with radius, while w and ]p/]z are independent of ra-dius) suggest that the similarity solutions may be mostrelevant to the dynamics of the interior portions of broadmesoscale vortices. The dynamics embodied by the sim-ilarity solutions might also be important as a modulatingfactor for columnar vortices embedded within a broaderparent vortex. A comparison (at a small time) betweena similarity solution and its finite radius counterpart insection 6 indicated that the similarity solution was inbetter agreement with the broader vortices than the co-lumnar vortices. However, long time numerical simu-lations of a variety of finite radius vortices will be re-quired to establish the areal and temporal bounds ofvalidity of the similarity solutions.

The oscillatory behavior of the similarity solutionsappears to be similar to the vortex valve effect some-times used to explain the cyclic appearance, demise, andreappearance of supercell characteristics in tornadicstorms and the apparent paradox of tornado formationin association with storm top collapse (Lemon et al.1975; Davies-Jones 1986). The vortex valve effect canbe demonstrated in a vortex chamber by feeding low-level rotating air into an updraft that vents through ahole at the top of the chamber. As the rotating air ap-proaches the axis of symmetry at low levels, the azi-muthal velocity increases in accordance with angularmomentum conservation. The increased low-level azi-muthal velocity is associated with a decrease in low-level pressure, a reduction of the upward axial pressuregradient force, and a ‘‘choking’’ of the updraft. Mostcolumnar geophysical vortices with axially varying ro-tation rates will not have the radial pressure gradientindependent of the axial direction. However, we believethe essential dynamical mechanism for real vortex–up-

draft oscillations is provided in its most basic form byour similarity model.

We also examine the short-term behavior of elevatedvortices with cores in solid body rotation embeddedwithin radially decaying angular momentum profiles, aclass that includes the elevated Rankine vortex. A linearanalysis of this case shows that an annular downdraftshould form on the periphery of the vortex core. Thepeak vortex-induced downdraft speed is greatest forbroad vortex cores and for large outer vortex radial-decay rates, and can exceed the peak vortex-inducedupdraft speed. Although our linear analysis should notbe used to quantify angular momentum transport, basedon the form of the downdraft we hypothesize that thevortex angular momentum can be advected downwardand radially inward. The details of this transport and itssubsequent feedback on the meridional circulation areof particular interest. For what set of parameters doesthe angular momentum remain suspended or reach theground? For what set of parameters does the radial con-vergence cause the angular momentum to spin up atmidlevels and build downward via the dynamic pipeeffect (Smith and Leslie 1978; Trapp and Davies-Jones1997) or first descend to the ground, spin up in theconverging low-level flow and then build upward? Isan oscillation set up, as in the similarity solutions?Clearly a longer-term nonlinear simulation is requiredto answer these questions.

It should be borne in mind that our hydrodynamicalmodel of elevated rotation of finite radius with no initialmeridional circulation is highly specialized. This modelwas chosen because it provided one of the simplest pos-sible ‘‘thought experiments’’ for studying the behaviorof elevated vortices. Evidence for a dynamically in-


duced downdraft in real mesocyclones or other geo-physical vortices will require careful analysis of highresolution four-dimensional data from observed or nu-merically modeled phenomena. We note that in Fig. 9dof Ray et al. (1981) an ‘‘unexplained’’ elongated down-draft appears in the multiple-Doppler analysis of theDel City storm at z 5 2 km ‘‘. . . in inflow air char-acterized by weak reflectivities. . . .’’ This narrownorth–south-oriented downdraft straddles a line extend-ing from approximately x 5 5, y 5 8 to x 5 5, y 515. It appears along the edge of the mesocyclone at theapproximate location of the maximum wind, and maybe a manifestation of the vortex-induced downdraft dis-cussed herein [Figs. 9e and 9f of Ray et al. (1981) alsoshow a downdraft ringing much of the mesocyclone atthe 2-km level, though much of this is likely associatedwith precipitation loading]. Brooks et al. (1993) founda downdraft in a similar position, but attributed it to thepresence of an inflow low due to high wind speeds atlow levels (Bernoulli relationship). We also note thefortuitous measurements of a dust devil that struck aninstrumented tower while data acquisition was in pro-gress (Kaimal and Businger 1970). Measurements of thehorizontal and vertical velocity components were takenat heights of 5.66 and 22.6 m. The trace of the lower-level data revealed a narrow downdraft on either sideof the dust devil updraft in the zone of radially decreas-ing tangential velocity (Fig. 2 of Kaimal and Businger1970).

Downdrafts, though not necessarily annular down-drafts, have been implicated in mesocyclonic tornado-genesis. Davies-Jones (1982a,b) argued that tilting andstretching of environmental horizontal vorticity by anupdraft alone would fail to produce appreciable rotationat low levels (i.e., only a midlevel mesocyclone wouldbe produced), since vertical vorticity is generated onlyas parcels move upward, away from the ground, in anupdraft. This was verified in the numerical simulationsof Rotunno and Klemp (1985). Davies-Jones hypothe-sized that a downdraft was necessary for the genesis ofnear-ground rotation.

Barnes (1978) and Lemon and Doswell (1979) hy-pothesized that the transition to tornadic phase in a su-percell is initiated by the rear-flank downdraft (RFD).They suggested that the RFD formed at midlevels andintensified the low-level rotation by creating strongshear (Barnes) or thermal gradients (Lemon and Dos-well) between the updraft and the RFD. Browning andDonaldson (1963) may have provided the first docu-mentation of the RFD in an early supercell study.

Three-dimensional numerical simulations (Klempand Rotunno 1983) have, however, implied the oppositecause and effect relationship; the low-level rotation in-tensifies, followed by formation of an ‘‘occlusion down-draft,’’ a smaller-scale downdraft within the RFD. Inthis scenario, Klemp and Rotunno hypothesized that theRFD is dynamically driven by a local, low-level pres-sure drop due to the intensifying rotation, which gen-

erates a downward-directed pressure gradient. Brandes(1984a,b) also presented this hypothesis based on Dopp-ler radar analysis. Our proposed mechanism of down-draft formation and downward transport of angular mo-mentum relies on the presence of strong elevated ro-tation and thus differs from Rotunno and Klemp’s(1983) occlusion downdraft, which is driven by stronglow-level rotation. The reader is referred to Klemp(1987) and Davies-Jones and Brooks (1993) for moredetailed surveys of past modeling and theoretical stud-ies.

We hypothesize that the hydrodynamic vortex-in-duced process described herein may play a role in lower-level mesocyclogenesis or tornadogenesis, eitherthrough the formation of an annular (or semiannular)downdraft or by facilitating the development of an RFD.This basic process may be important in vortex-domi-nated flows in other geophysical and engineering con-texts as well. The longer-term behavior of our idealizedvortex including the downward transport of the vortexcirculation by the annular downdraft will be examinedin future numerical simulations.

Acknowledgments. Detailed and insightful commentsby the anonymous referees led to a substantially im-proved manuscript. Discussions with Doug Lilly andKathy Kanak are gratefully acknowledged. Tom Condo,Tim Kwiatkowski, and Courtney Garrison providedcomputer assistance. This research was supported by theCenter for Analysis and Prediction of Storms (CAPS)under Grant ATM91-20009 from the National ScienceFoundation. One of us (P.M.) was supported by an AMSfellowship sponsored by GTE Federal Systems Divi-sion, Chantilly, Virginia.

APPENDIX

Derivation of the Initial StreamfunctionTendency c 1

We seek the initial streamfunction tendency c1 sat-isfying the partial differential equation (71), the jumpcondition (70), and the boundary conditions (72). To-ward that end, introduce a function C satisfying (70)and (72),

2 2C 5 2g(z)V r , r , R,

2nR2 25 2g(z)V R , r . R, (A1)1 2r

where

T1g(z) 5 1 2 z, 0 # z , T ,11 2h

T15 (h 2 z), T , z # h. (A2)1h


For later use we note that g(z) can be extended as anodd periodic function of period 2h,

` 2h kpT kpz1g(z) 5 sin sin ,O 2 2 1 2 1 2k p h hk51

0 # z # h. (A3)

Although C does not satisfy (71), c1 (5C 1 F) doessatisfy (71) provided that F satisfies

2 2] F ] F 1 ]F1 2

2 2]z ]r r ]r

2n12R25 4n(n 1 1)g(z)V , r . R,1 2r

5 0, r , R. (A4)

In terms of F, the boundary conditions (72) become

F(0, z) 5 F(r, 0) 5 F(r, h) 5 F(`, z) 5 0. (A5)

We also want to ensure that the full solution C 1 Fand its normal derivative ]C/]r 1 ]F/]r are continuousat r 5 R. Since ]C/]r obtained from (A1) is discontin-uous at r 5 R, there must be an equal and oppositediscontinuity in ]F/]r,

]F ]F1 2 2(R , z) 2 (R , z) 5 22(n 1 1)g(z)V R, (A6)

]r ]r

where R1 [ lim|«|→0(R 1 |«|), and R2 [ lim|«|→0(R 2|«|).

Applying (A3) in (A4), and expanding F in a sineseries, we obtain

` 28n(n 1 1)V h kpT kpz1F 5 f (r) sin sin , (A7)O k2 2 1 2 1 2k p h hk51

where f k(r) satisfies

2n122 2 2] f 1 ] f k p Rk k2 2 f 5 , r . R,k2 2 1 2]r r ]r h r

5 0, r , R. (A8)

Applying (A3) and (A7) in (A6), we obtain the radialjump condition,

df df Rk k1 2(R ) 2 (R ) 5 2 . (A9)dr dr 2n

The solution of (A8) in the core region is

kprf 5 c rI , r , R, (A10)k 1 11 2h

where I1 is a modified Bessel function of the first kindof order one and c1 is a constant. The second linearlydependent solution, a modified Bessel function of thethird kind of order one, K1, was rejected because of itssingular behavior at the origin. Recurrence relations,derivative formulas, and other results pertaining to mod-

ified Bessel functions are described in Abramowitz andStegun (1964) and Watson (1944).

The solution of (A8) in the outer region can be ex-pressed in terms of Lommel’s functions [section 10.7of Watson (1944)], or left in a form obtained by themethod of variation of parameters,

r /Rkpr I (kpxR /h)1f 5 2RrK dxk 1 E 2n121 2h x

1

`kpr K (kpxR /h)12 RrI dx1 E 2n121 2h xr /R

1 c rK (kpr/h), r . R. (A11)2 1

Here one of the two constants in the general solutionwas specified to make fk vanish at infinity. To verifythis behavior, write fk as a sum of indeterminate forms,apply L’Hopital’s rule, and use asymptotic and deriva-tive formulas for modified Bessel functions:

lim f kr→`

r /R ` I (kpxR /h) K (kpxR /h)1 1dx dx E E2n12 2n12x x 1 r /R5 2R lim 1

1/[rK (kpr/h)] 1/[rI (kpr/h)]r→` 1 1

2n122h R5 2 lim .

2 21 2k p rr→`

Thus fk vanishes at infinity for the cases of interest, n. 0. Moreover, since F decays by a factor of 1/r2 fasterthan C, the solution c1 5 C 1 F is dominated by Cfar from the axis of symmetry.

Continuity of fk at r 5 R and the radial jump condition(A9) yield two equations for c1 and c2,

kpR kpRc RI 5 c RK1 1 2 11 2 1 2h h

`kpR K (kpxR /h)122 R I dx, (A12)1 E 2n121 2h x1

`kpR K (kpxR /h)122R I dx0 E 2n121 2h x1

kpR kpR2 c RK 2 c RI2 0 1 01 2 1 2h h

Rh5 2 . (A13)

2kpn

Solving (A12) and (A13) and applying a formula forthe Wronskian of Bessel functions, we get


` K (kpxR /h) R kpR1c 5 2R dx 1 K , (A14)1 E 12n12 1 2x 2n h1

R kpRc 5 I . (A15)2 11 22n h

This completes the specification of the initial stream-function tendency. Collecting results, we write the so-lution as (73) and (74).

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