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where ui is the velocity vector, p is the pressure, a is the specific volume definedas 1/ g where g is the density of air, and 11j is the rotation vector of the Earth.The velocities in (14.2.1) are velocities in the rotating Earth frame of reference.The independent variables are Xi = X, y, z and t where X is the east-west direction,y is the north-south direction, z is the vertical direction, and t is time.

The following three approximations have been used in the derivation of (14.2.1:

(i) Density fluctuations are small, so la' / aol and la" / aol are much less thanunity. This assumption is referred to as the Boussinesq approximation. Theconservation of fluid mass can then be written as 8 (goUj) / 8Xj = O. Thisapproximation is valid for small Mach numbers (i.e., when the wind speeds aremu~smaller than the speed of sound) in the Earth's atmosphere. Variationsof den~ for these small Mach number flows are represented by the ideal gasrelationshi~ .

(ii) The laminar molecular dissipation of velocity (i.e., v82Ui/ 8xj8xj) is ignoredbecause we will only be discussing turbulent atmospheric flows in thi.s chapter.In turbulent atmospheric flows, the turbulent stresses (represented by thesecond term on the right side of Equation (14.2.1)) are much larger than theviscous stresses. Viscous effects are, however, important close to the surfaceof leaves, cloud droplets, etc.

(ill) The averaged quantities, denoted by the overbar, change much more slowly intime and space than the perturbations. This requirement is referred to as theReynolds' averaging assumption. Pielke (1984, Figure 4-2) illustrates whenthis assumption is valid and when it is not.

Using Equation (14.2.1), a vorticity equation can be derived by applying the vectorcurl operator (Epqi8 / 8xq) such that:

(i) (ii) (iii) -(it!) -~ a -a ( -~ ) a a ( II " )at = -8XjUjUJp -£pqi8Xj ~OUiaxq -£pqi~8Xj ~OUjUi

(14.2.2)(v) (vi)

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qQO q

where cup = Epqi8(eo"Ui)/ 8xq is the density-weighted vorticity.The individual terms in (14.2.2) correspond to:

(i) the local time derivative of cup

(ii) the gradient of the grid resolvable flux of cup

(ill) the mechanism to transfer cup between the three spatial components as a resultof velocity shear (referred to as the tilting term; it should more appropriatelybe referred to as simultaneous tilting and vortex line convergence. Convergence

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620 ATMOSPHERIC VORTICES -R. PIELKE ET AL.

acts to transfer kinetic energy between the three velocity components. Term (v) isa buoyancy term that can either produce or suppress turbulence.

The ratio of term (v) to term (ui),

The solenoidal term in (14.2.2) can be utilized to describe the formation of onetype of vertically oriented circulation, which is referred to as a thermally forcedatmospheric system. As illustrated in Figure 14.4.1, dense air adjacent to less denseair at the same pressure level will result in a flow from the denser atmospheretowards the less dense region. Since density is inversely related to air temperature

is a form of the flux Richardson number. RJ < 0 corresponds to the situation wherebuoyancy produces turbulence, while RJ > 0 represents the case where buoyancysuppresses the turbulence created by shear of the mean velocity. When RJ > 0,some of the kinetic energy caused by the velocity shear generates gravity waves,rather than turbulence.

Turbulence, resulting in vortical motion, is also generated in a saturated at-mosphere if the latent heat released during condensation or deposition results ina layer in the atmosphere, or a parcel of air, which is warmer than the air in itsvicinity. The atmosphere in which this vertical instability can occur is referred toas conditionally unstable (Air Weather Service Manual Technical Report-79/0061990). Cumulus clouds are the visual manifestation when this instability is real-ized. The turbulent vortical structure in such clouds is particularly obvious in timelapse photography. Recently, the concept of slantwise instability (also referred to assymmetric instability) has been introduced to describe the generation of cumulusconvective bands along sloped ascent (Emanuel 1983).

Atmospheric turbulence, with its large Reynolds numbers, can be considered toconsist of three regions. The turbulence production region typically occurs on thelarger spatial scale where velocity shear (term (ill) in (14.3.1)) and/or buoyancy(term (v) in (14.3.1)) provide the mechanism to transfer kinetic energy from thelarge-scale flow field to the smaller-scale turbulent fluctuations. The turbulenceis dissipated at the very small spatial scales where viscous effects become dom-inant (for the atmosphere this scale is on the order of millimetres; see Lumleyand Panofsky (1964) p.82). At the intermediate spatial scales where turbulenceis neither generated nor eliminated, the kinetic energy is transferred to smallerscales through terms (ii) and (iv) in (14.3.1). These spatial scales are referred toas the inertial subrange (Lumley and Panofsky 1964). In this subrange, the kineticenergy spectrum has a k-5/3 relationship (as predicted by Kolmogorov), where kis the wavenumber of the turbulent eddy.

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tures include extratropical and tropical cyclones, polar and subtropical anticy-clones (Pielke et al. 1987), and polar lows (Montgomery and Farrell 1992). As aresult of the influence of the Earth's rotation (i.e., the Coriolis force), these cir-culations tend to rotate counterclockwise around low pressure systems and clock-wise around high pressure systems, in the northern hemisphere, when viewed fromabove. Figure 14.6.1a presents a geostationary earth orbiting satellite (GOES) im-age of upper tropospheric water vapor and Figure 14.6.1b shows a GOES infraredimage of cloud systems, to illustrate these large-scale vortex motions.

There are several flow instabilities that generate these vortices. Inerttal insta-bility was first defined as centrifugal instability (Emanuel 1984). In a classical

two-dimensional inertial stability problem, if the angular momentum decreaseswith the radius in an axisymmetric vortex, it is inertially unstable. In the three-dimensional case using a rotating coordinate system, inertial instability occurswhen there exists a region of absolute vorticity of opposite sign to the large-scalevorticity such that an imbalance occurs between the horizontal pressure gradientforce and the Coriolis force. For zonal conditions, as shown by Holton (1979), theinertial instability condition in the northern hemisphere is f -~ < 0, where Uis the large-scale zonal wind and f = 2n sin~. The independent variable y is thenorth-south direction, and ~ is the latitude. Such an instability often occurs asair flows around high pressure systems and the pressure gradient force is not largeenough to balance the outward centripetal acceleration and the Coriolis effect.Ciesielski et al. (1989) illustrate from satellite imagery the generation of unstable,relatively small-scale horizontal vortices associated with the anticyclonic side of astrong upper tropospheric subtropical jet stream as it flows around the northsideof a subtropical high pressure system in the northern hemisphere.

Barotropic instability resulting in vortical motion develops as a result of largehorizontal gradients of velocity. Holton (1992) indicates that a necessary conditionfor this instability is that the horizontal gradient of absolute vorticity must changesign somewhere in the region. For zonal flow conditions, the barotropic instabilitycondition in the northern hemisphere is

of 02U-ay -8Y2 ~ O.

When this occurs, kinetic energy is extracted from the wind flow to generate hor-izontal waves which can break down into horizontal vortices over a time period ofa few days or less. Holton suggests that this instability mechanism is responsiblefor African wave disturbances which develop over the Sahel region, and occasion-ally intensify into tropical cyclones off the African west coast. Dobrovolskis andDiner (1990) discuss barotropic instability on Venus. In terms of potential vorticity(PV), a necessary condition for barotropic instability is that the potential vorticitymust change sign in the horizontal direction. This condition is not sufficient if theregions of positive and negative PV are too far apart, then the influence of oneregion will not affect the other and no instability will occur. The influence decays

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exponentially with the separation distance between the two PV regions (Hoskinset al. 1985).

Baroclinic instability occurs as a result of the interaction of temperature advec-tion superimposed on a velocity field. Charney (1947) introduced this concept todescribe the conversion of potential energy to kinetic energy characteristic of mid-and high-latitude extratropical cyclonic vortices. Since the horizontal temperaturegradient is largest near the polar front that demarcates air of polar origin fromthat of tropical origin, these cyclonic vortices develop along this front. The verti-cal wind shear of the horizontal wind is also large in this region as a result of thelarge horizontal pressure gradient differences in the middle and upper troposphere,which are caused by the large horizontal temperature difference between the polarand tropical air throughout the troposphere. Carlson (1991) summarizes severalof the characteristics associated with this instability, including that larger verticalwind shears yield greater growth rates and longer wavelengths, and that below acritical vertical wind shear (about 2000 km) no growth occurs. The optimal growthrate occurs for horizontal wavelengths of 3000 to 4000 kilometres which explains,for example, why mid-latitude synoptic storm systems that influence our day-to-day weather are typically this size. Examples of papers which analyze baroclinicinstability include Bannon (1989) and Reinhold (1990).

From the potential vorticity point of view, a necessary condition for baroclinicinstability is for the northward gradient of PV evaluated on a potential tempera-ture, 0, surface to change sign on a different 0 surface, i.e.,

andBPV

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Similar to the barotropic case, these PV regions must be close enough in thevertical to influence one another, otherwise the above criterion is not sufficient.Mathematically, barotropic and baroclinic instabilities are quite similar.

XIV.7 Cumulonimbus-Generated Mesoscale

VorticesMid-level cyclonic vortices of an average diameter of 150-300 km can be gener-ated by mesoscale cumulonimbus convective systems (Johnson et al. 1989; Bar-tels and Maddox 1991). Mesoscale convective systems in their mature stages havelarge stratiform anvils typically with latent heating at mid and upper levels inthe tropospheric and cooling at low levels. These systems are characterized by amid-tropospheric-levellow pressure anomaly with high pressure at the surface andupper troposphere. Mid-level inflow occurs with lower- and upper-level outflows.The effect of the Coriolis force on the middle-level inflow is to generate a cyclonicvortex. These vortices can generally be identified by visible satellite imagery, where

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zero. The circulation (1.1.9) around a ring centered on and perpendicular to thevortex a.xis is often nearly a constant function of radius outside the core, making ita useful global descriptor of a tornado. Circulation is the most influential parametercontrolling core diameter (Lewellen 1976), and ranges from less than 104 m2/s innarrow tornadoes to :more than 3 X 105 m2 Is in the widest tornadoes. Althougha tornado a.xis may tilt considerably from the vertical, the tilt may be considereda variant on what is essentially an upright a.xis extending from the ground to aheight of usually several kilometres.

The precise mechanisms responsible for producing the intense field of verti-cal vorticity in a tornado core remain a topic of research. However, it is nowfairly well established that the process involves first a production of relativelyweak vertical vorticity over horizontal distances at least several times the (future)tornado core diameter, followed by subsequent concentration of that vorticity bytwo-dimensional convergence in the plane orthogonal to the a.xis (see §I.3.4), asdescribed by term iii of Equation (14.2.2). That is, the most intense vorticity isneither generated in place, as by solenoidal processes, nor generated horizontally at full intensity and subsequently simply tilted into the vertical. Walko (1993)discusses the vortex line configuration in a typical mature tornado, wherein vortex

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typically occupies a region approximately 10 km wide. Production of low-levelvertical vorticity, which is required for tornado formation at the ground, appearsto require localized downward motion in order to transport middle-level horizontalmomentum to a limited region of the surface, a form of tilting horizontal vorticityinto the vertical (described by term iii of Equation 14.2.2). This was argued ontheoretical grounds by Davies-Jones (1982), and is supported by the observationthat many tornadoes develop only after a nearby precipitation-induced downdraftforms. The horizontal vorticity itself may be primarily that which existed in thepre-storm environment (Davies-Jones 1993; Walko 1993), or may be generatedsolenoidally, for example by evaporative cooling of rain (Rotunno and Klemp 1985).This second mechanism of tornado formation differs from the first in that the low-level vertical vorticity was produced by the storm itself, or by the interaction of thestorm with its environment, rather than existing before the storm. The mechanismfor concentrating that vorticity in both cases, however, is the low-level horizontal

convergence caused by the convective updraft.In an example of tornadogenesis provided by numerical simulation (Grasso

1992), a significant low pressure region with vertical continuity was found to formin the large horizontal gradients of vertical motion at low to mid levels of thestorm (see Figure 14.9.2). The interaction of the three-dimensional vorticity fieldwith the updraft allows the low pressure to form at lower levels of the storm, butstill above cloud base. In time, a low pressure tube exists from low to mid levelsto cloud base. The existence of the low at cloud base, and still in the updraftgradient, locally enhances upward vertical motion. This in turn enhances, via con-tinuity, horizontal convergence of the subcloud air. Due to the enhanced horizontalconvergence, pre-existing vertical oriented vortex lines converge beneath the low.This results in the three-dimensional vorticity vector essentially oriented verticallyupward, which grows in magnitude thus allowing further descent of the low pres-sure tube beneath cloud base. The region of low pressure now coincides with theupdraft of the developing funnel as opposed to being in the horizontal updraftgradient above cloud base. The descent of the pressure tube, and thus the funnel,interacts with and responds to surface forces allowing the descent of the funnel to

the ground, thus forming a tornado.It was once thought that the background vertical component of vorticity as-

sociated with the Earth's rotation could be concentrated into a tornado vortexby the horizontal convergence beneath a sufficiently long-lived convective updraft,but that vorticity is now generally regarded as too weak given the time available.Nevertheless, the Earth's rotation is responsible, though less directly, for estab-lishing that approximately 90% of tornadoes rotate in the same sense as the Earth(counterclockwise in the northern hemisphere, as viewed from above). The Earth'srotation establishes an ambient wind pattern, in a typical tornado-prone environ-ment, in which the vector of vertical shear of the horizontal wind rotates clockwise,in the lowest few kilometres of the atmosphere. This in turn favors developmentof counterclockwise rotation of the air in which the tornado forms (Rotunno and

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Klemp 1982; Lilly 1982). Lilly (1986) introduced the concept of helicity, which isthe vector inner product of velocity and vorticity to characterize the occurrence oflarge rotating thunderstorms.

It is possible for mechanisms other than a convective updraft to cause low-levelconvergence. For example, a precipitation-cooled downdraft can spread laterallyupon reaching the ground, and the leading edge of the advancing cold air can pro-duce a region of horizontal convergence, capable of concentrating vertical vorticity.Tornado-like vortices are sometimes observed along these convergence boundaries.Such convergence is far weaker than that fostered by deep convection, however,and is insufficient for maintaining any but weak vortices.

The maximum wind speeds commonly attained in tornadoes, and the maximumtheoretically attainable wind speeds, are both subjects of considerable interest andcontroversy. Historical estimates have placed winds as fast as the speed of sound,which would require a nearly complete vacuum at the vortex axis to provide thenecessary centripetal force. Few, if any, still believe such high estimates, and valuesin the range of 100 to 150 mls are most commonly cited for the strongest oftornadoes. Measurements of tornado winds are scant, but have yielded some usefulinformation. Analyses of motion pictures of tornadoes and of ground marks left bytornadoes (Fujita et al. 1976), studies of damage to man-made structures (Mehta1976), and direct Doppler radar measurements of tornado winds (Bluestein andUnrah 1993) generally yield estimates in or below this range (see also a reviewof tornado wind speed estimates by Golden (1976)). Nevertheless, each of thesemethods has the potential of failing to detect the actual highest wind speed withina tornado, which may be highly transient and localized, may occur a considerableheight above the ground, and may involve vertical motion not detectable in a nearlyhorizontal radar beam. While no reliable measurement has indicated tornadic;windspeeds to substantially exceed 150 mis, there still exists no proof that such windsare impossible in normal tornadic situations.

Methods for theoretically estimating tornado winds generally begin from sim-plified models which assume the vortex to be axisymmetric with a vertical axis,and consider more general vortex structures as deviations from the simple model.The simplest model is based on the assumption that the tornado vortex extendsfrom the ground up to the top of a deep convective cloud, which is warmer than itsenvironment due to latent heating from condensation. The model assumes that thepressure deficit on the vortex axis at the ground, relative to the ambient groundpressure well outside the tornado, is determined by the warmer air along the axisand the resultant reduction of the weight of the air over this location. The ro-tating vortex winds are then deduced from the pressure deficit through dynamicrelationships. This simple model, however, falls short of explaining the most intensetornadic winds.

Several modifications to the simple model have been proposed to yield estimatescloser to the maximum observed winds. Lilly (1969) hypothesized that the vortexcore may be even warmer than possible by latent heat release as a result of dry

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region near the boundary of the inner cell, a flow configuration which is unstableto axially-asymmetric perturbations. The disturbances which develop as a resultform into multiple quasi-circular vortices whose axes are located along the innercell boundary and revolve around the main tornado axis. The motion within themultiple vortices is the combined motion from the rotations of both the multi-ple vortices and the main tornado, and may exceed the rotational speeds of anaxisymmetric tornado. It has been speculated that multiple vortices add consider-ably to the destructiveness of tornadoes in part because they cause wind directionto change extremely rapidly as they move past objects on the ground. From photo-graphic evidence and damage path assessments, this higher velocity air is evidentas particularly well-defined rotating air columns within the larger tornado and arereferred to as suction vortices.

Tropical cyclones are intense vortical storms which develop over the tropical oceans(Figure 14.10.1). About 80 tropical cyclones form over the tropical oceans each year(Gray 1975). They occur in regions where the sea surface temperature exceeds26°C. When the maximum wind speeds of tropical cyclones equal or exceed 65knots they are called hurricanes in the Atlantic and east Pacific, typhoons in thewestern north Pacific, and cyclones in the south Pacific. In the range between35 to 65 knots they are called tropical storms. In a hurricane, the average radialextent of hurricane force winds is about 100 km, but gale force winds (greaterthan 28 knots) may extend 500 km from the center. Extremely low surface pressureanomalies occur at the center of the vortex, which can exceed 100 mb in magnitude.The energy source for the tropical cyclone is the deep convective clouds near itscenter which develop over the rich water vapor and heat source of the tropical andsubtropical oceans. (Palmen and lliehl1957; Dunn and Miller 1960; Anthes 1982).

Most of the tropical cyclone is in gradient wind balance, where the centripetaland Coriolis forces nearly balance the horizontal pressure gradient force. However,surface friction eliminates this gradient wind balance in the lowest kilometre ortwo, causing air to spiral inward and toward low pressure. The strong cyclonictangential circulation is produced by the inward advection of air with large abso-lute angular momentum from the tropical cyclone environment. As the air spiralsinwards towards the center of the vortex, large sea surface fluxes of latent heat andto a lesser extent, sensible heat are important for providing the fuel for deep con-vection. The sensible heating from the ocean permits isothermal inflow air, ratherthan a cooling of air parcels due to adiabatic expansion as the air encounters thelower pressure of the tropical cyclone core. The deep cumulus convection occursin a ring called' the eye wall, which surrounds a region of relatively calm and clearair (referred to as the eye). The large azimuthal momentum the air has acquiredis transported upwards in the convective towers of the eye wall, resulting in a very

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favorable for genesis have been identified. A sea surface temperature of at least26.5°C is required (Palmen 1948), and the tropical disturbance needs to have deepconvective clouds (Riehl 1954). In addition to these factors, Gray (1979) identifiesthe following factors as favorable for cyclogenesis: high mid-tropospheric relativehumidity, large surface to 500 mb lapse rates of equivalent potential temperature,small tropospheric vertical wind shear, high magnitudes of low-level relative ver-tical vorticity, and sufficiently high values of the Coriolis parameter.

However, these are not sufficient conditions, and it appears that some exter-nal forcing is required: Many case studies suggest that low-level external forcingin the form of a wind speed maxima penetrating the circulation of a pre-existingdisturbance may initiate cyclogenesis (e.g., Love 1985). Such features are referredto as wind surges. There may be two distinct stages where external forcing is im-portant. A wind surge', or an externally forced increase in low level convergence,may initiate intense deep convection in a cloud cluster, which may then develop aweak mesoscale vortex (similar to those discussed in § 7) and a lowering of surfacepressure of 1-3 mb. A second surge, occurring 2-3 days later, acting on this pre-existing vortex may then lead to another convective maxima which strengthensthe vortex into a tropical storm (Zehr (1992) discusses this scenario for tropicalcyclogenesis in the western north Pacific). An element of chance is envisioned inthis process, since favorable external forcing, resulting in enhanced low-level con-vergence, is required to act on a cloud cluster which already has a pre-existingvortex. There is also observational evidence that environmental forcing at upperlevels by synoptic-scale anticyclonic vortices can playa role in cyclogenesis (Moli-nari 1988; Davidson et al. 1990). Montgomery and Farrell (1993) discuss the role ofupper level potential vorticity disturbances on the formation of tropical cyclones.

Tropical cyclone 'intensification' refers to strengthening of the system once ithas been designated a tropical storm. As in the case of tropical cyclogenesis, exter-nal forcing due to large-scale circulations at upper levels seems to be an importantenvironmental forcing for tropical cyclone intensity change (Holland and Merrill1984; Challa and Pfeffer 1990; Davidson et al. 1990). Another proposed explanationis convective forcing (Ooyama 1982), in which development occurs through a coop-erative feedback between latent heat release in deep cumulus convective clouds andthe developing disturbance. Emanuel (1986), and Rotunno and Emanuel (1987)emphasize the importance of ocean-atmospheric interaction, in which self-inducedlatent and sensible heat transfer from the ocean to the atmosphere act to spin upa finite amplitude disturbance. However, tropical cyclone intensity change remains

a very challenging forecasting problem.Spiral rainbands are a common feature of tropical cyclones. Inertial gravity

waves have been proposed as a mechanism for producing rainbands (Kurihara1976; Willoughby 1977). Recent observational studies have described the structureof tropical cyclone rainbands (Powell 1990; Ryan et al. 1992). The width of rain-bands is typically 10-30 km, but can be as large as 100 km. Rainbands are oftencomposed of cumulus convective cells intermingled with stratiform precipitation.

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sult of a cooperative feedback between latent heat release in convective clouds andthe developing disturbance. The convective clouds supply the heat necessary todrive the large-scale disturbance, by producing a radial pressure gradient, and thelarge-scale disturbance produces the moisture convergence necessary to drive theconvection. A positive feedback can ensue, which is eventually limited by the in-creased stability to vertical motions as the inner core warms. Models which utilizecumulus parameterization have been used to investigate mechanisms leading tothe genesis of tropical cyclones (Tuleya 1988; Challa and Pfeffer 1990) and trop-ical cyclone motion (Kurihara et al. 1990). A drawback of these models is that acumulus parameterization is used to provide the latent heating which drives themodel vortex.

The first numerical models of tropical cyclones which attempted to resolveconvective clouds were those of Yamasaki (1977), Rosenthal (1978), and Jones(1980). Although large grid increments were used (",10 km), these models stillproduced some realistic features. Willoughby et al. (1984) simulated a hurricaneusing an axisymmetric, nonhydrostatic model with horizontal grid increments of2 km, at the center of the domain. Concentric rings of convection were obtainedwhich contracted about the vortex center. Realistic features included: the outwardslope of the eyewall updraft and tangential wind maximum; the relative location ofthe updraft wind maximum and precipitation maximum; stratiform precipitationand mesoscale downdrafts outside the eye; and mid-level radial inflow.

A three-dimensional convectively explicit nonhydrostatic simulation has beenmade by Tripoli (1993). The model used nested grids (Clark and Farley 1984) witha fine grid increment of 10 km for most of the simulation, but this was reducedto 3.3 km for a six hour period. Emanating inertia gravity waves were producedby transient mesoscale convective features as the tropical cyclone developed. Itappeared that the increasingly strong inertial frequency of the core acted to in-creasingly trap the convection induced heating within the core by balancing thetangential wind against the low pressure. This process has been described in atheoretical study by Schubert and Hack (1982). A three-dimensional convectivelyexplicit nested grid simulation has also been made by Nicholls and Pielke (1993).A fine grid increment of 3 km was used throughout the simulation with a fine griddomain size of 270 X 270 km2. There were 27 vertical levels. The simulation wasinitialized with a deep vortex in gradient wind balance (maximum winds of 25m/ s). Frictional convergence led to the development of deep convection and rapidintensification. Figure 14.10.2 shows a horizontal cross section of wind vectors andtotal condensate at z = 2.5 km and at t = 20 h. At this level, a strong vortexis evident with tangential wind speeds of ",50 m/s. A well defined clear eye ofradius 15 km is apparent, which extends throughout the depth of the storm. Thestructure of the simulated tropical cyclone is in good agreement with the generalstructure of observed systems (Figure 14.10.2).

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convective clouds than models which use cumulus parameterizations. They are cer-tain to be used increasingly in the future to investigate all aspects of the life cycleof the tropical cyclone.

The Great Red Spot (GRS) is a historically fascinating vortex observed in theatmosphere of Jupiter, which has a similarity to hurricanes on the Earth. The GRShas a depth of about 200 km and is above the neutrally-stratified deep atmosphere.Horizontally, the GRS covers 20000 km in longitude so that it has a 100-to-.1 ratioin horizontal to vertical dimensions. Most current models of the GRS are based ona two-level model, where a thin upper weather layer, which contains the vortex,overlies a much deeper layer, which is meant to represent the neutrally-stratifieddeep atmosphere (Dowling and Ingersoll (1989) and references therein). Any mo-tions in the deep layer are assumed to be zonal and steady. This two-layer modelis dynamically equivalent to a one-layer shallow water model with meridionallyvarying solid bottom topography. Using the GRS cloud-top velocity data from theVoyager spacecraft, Dowling and Ingersoll (1989) derived the bottom topographyand they found that the large vortices in their shallow water model do not decayin the presence of dissipation. Instead, these vortices maintain themselves againstdissipation by absorbing the smaller vortices, and the system achieves a dynam-ical equilibrium. In the work of Dowling and Ingersoll (1989), the small vorticesarise spontaneously in the unstable shear flow, but a crucial unanswered questionfor the model and for Jupiter itself is how the cloud-top zonal velocity profile ismaintained in its present unstable state. The small vortices can also possibly beproduced by additional energy sources, which are not included in Dowling andIngersoll (1989), such as convection or other diabatic forcing. Such eddies mightbe a source of energy to both the unstable zonal flow and the large vortices.

Regarding the predictability of the emergence of the GRS-like vortex in themodel, Dowling and Ingersoll (1989) showed that vastly different initial conditionslead to a very similar final form of the vortex. In other words, the GRS-like vortexis the only equilibrium state of the system. However, it is also found in Dowling andIngersoll (1989) that the drift rate of a vortex depends sensitively on the bottomtopography.

As mentioned above, the analog to the GRS is the hurricane in the atmosphere ofthe Earth. Like the GRS, a hurricane usually has a 100-to-1 ratio in the horizontaland vertical dimensions (although this ratio changes somewhat from case to case).However, unlike the G RS which is a persistent feature on Jupiter, a hurricanehas a comparatively short existence. Unlike the GRS, hurricane genesis is highlysensitive to the initial disturbances, and, also dissimilar to the GRS, dissipateswhen this vortex moves to an environment in which its warm core structure cannotbe maintained. The continued study of the similarities and differences in these twovortical structures need to be made.

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Bienkiewicz, B. and Dudhia, P. 1993 Physical modeling of tornado-like flow andtornado effects on building loading. 7th US National Conference on WindEngineering Los Angeles, June, 1993.

Black, M.L. and Willoughby, H.E. 1992 The concentric eyewall cycle of HurricaneGilbert. Mon. Wea. Rev. 120,947-957.

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Bluestein, H.B. and Unruh, W.P. 1993 On the use of a portable FM-CW Dopplerradar for tornado research. The Tornado: Its Structure, Dynamics, Prediction,and Hazards. (eds. Church, C., Burgess, D., Doswell, C., Davies-Jones, R.)Geophysical Monograph 79, American Geophysical Union. 367-376.

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Challa, M. and Pfeffer, R.L. 1990 Formation of Atlantic hurricanes from cloudclusters and depressions. J. Atmos. Sci. 47, 909-927.

Charney, J .G. 1947 The dynamics of long waves in a baroclinic westerly current.J. Meteor. 5, 135-162.

Chopra, K.P. 1973 Atmospheric and oceanic flow problems introduced by islands.Adv. Geophys. 16, 297-421.

Church, C.R., Snow, J.T., Baker, G.L. and Agee, E.M. 1979 Characteristics oftornado-like vortices as a function of swirl ratio: A laboratory investigation.J. Atmos. Sci. 36, 1755-1776.

Ciesielski, P.E., Stevens, D.E., Johnson, R.H. and Dean, K.R. 1989 Observationalevidence for asymmetric inertial instability J. Atmos. Sci. 46, 814-831.

Clark, T .1. and Farley, R.D. 1984 Severe downslope windstorm calculations intwo and three spatial dimensions using anelastic noninteractive grid nesting:A possible mechanism for gustiness. J. Atmos. Sci. 41, 329-350.

Cotton, W.R. 1990 Storms. Geophysical Science Series, Vol. 1. ASTeR Press, FortCollins, CO, 158 pp.

Crook, N .A., Clark, T .1. and Moncrieff, M. W. 1990 The Denver cyclone. Part I:Generation in low Froude number flow. J. Atmos. Sci. 47. 2725-2742.

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Garratt, J .R. 1992 The atmospheric boundary layer. Cambridge Atmospheric andSpace Science Series, Cambridge University Press.

Golden, J .R. 1976 An assessment of wind speeds in tornadoes. Proc. Symp. onTornadoes. Lubbock, Institute for Disaster Research, Texas Tech University,107-143.

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