Hydrodynamic Alpha-Effect in a Convective System

Advances in Fluid Mechanics

Nonlinear Instability,Chaos and Turbulence

Volume II

Edited by:

L. DebnathUniversity of Central Florida, USA

D. N. RiahiUniversity of Illinois, USA

WITPRESS Southampton, Boston

2000

Chapter Four

Hydrodynamic alpha-effect in a convectivesystemG.V. Levina1'2, S.S. Moiseev2 & P.B. Rutkevich2

1 Institute of Continuous Media Mechanics, Ural Branchof Russian Academy of Sciences, 614013 Perm, RussiaE-Mail: [email protected]

2 Space Research Institute, Russian Academy of Sciences,117810 Moscow, RussiaE-Mail: [email protected] ; [email protected]

Abstract

A physical mechanism of large-scale structure generation in a helical turbu-lent medium is discussed. The mechanism is described in terms of mathema-tical model involving new hydrodynamic, helical-vortex, instability causedby a feedback between solenoidal components of the velocity field. The roleof a small-scale helical turbulent convection as a possible source of ener-gy for sustaining large-scale intensive vortices is analyzed in the contextof the problem on hydrodynamic alpha-effect in a convective system. Togain a better understanding of this phenomenon the results of theoreticaland numerical investigation and laboratory and field experiments as wellas observation data obtained by several Russian research teams since theearly 1980s are reviewed. Finally, recent results of numerical simulation ofhelical-vortex instability development in a nonuniformly heated medium arediscussed. Numerical analysis has shown that even in the case of laminarconvection, helical feedback can initiate some new effects. Vivid examplesof these phenomena are the formation of helical flow structure with linkageof streamlines of horizontal and vertical circulation, small convective cellmerging during formation of large-scale flow structure, and a sharp increasein the kinetic energy of formed large-scale vortex due to the action of helicalfeedback.

112 Nonlinear Instability, Chaos and Turbulence

1 Introduction

The world's advances in investigation of turbulence phenomena posea fascinating and rather intricate problem concerning formation oforganized structures with space- and time-scales essentially excee-ding those of turbulent pulsations. The keen interest of researchersinto this subject is stimulated mostly by recent findings in astro- andgeophysics which provide ample evidence for the existence of suchstructures. Among these are the large-scale motions driven by inho-mogeneous medium heating in the Sun and stars' convective zones,planets' atmospheres, and the Earth's atmosphere and oceans. Astriking example of these phenomena are tropical cyclones represen-ting large-scale long-lived helical vortices of anomalous intensity re-peatedly arising in the tropical atmosphere of the Earth.

The global system of the Earth's atmosphere circulation is knownto involve elements of even greater scale induced, for example, bya latitudinal temperature gradient. However, the intensity of theseplanetary-scale motions is relatively weak. The extremely high powerof a tropical cyclone can be explained by assuming that such a vor-tex gains energy from small-scale deep atmospheric turbulent convec-tion, which in the tropics reaches the highest intensity and extendsthroughout the troposphere layer of 14-16 km altitude. An argumentin favor of this assumption relies on the well-established fact thattropical cyclones usually axise just in the regions of strongly develo-ped convective motions within the Intertropical Convergence Zone.

A key to describing the initial stage of tropical cyclone generationis our ability to offer a reasonable explanation for the conditions andphysical mechanisms that could govern the transformation of small-scale convective atmospheric cells, each with the horizontal dimensionand lifetime typical for an ordinary cumulus cloud (several kilometersand several hours), to a rapidly rotating vortex with a specific helicalstructure, of diameter 1000 km and larger and a lifetime of up toseveral days.

According to the long-standing classical concept of turbulentflows, any large-scale formation, for instance a vortex of spontaneousor forced origin, is destroyed by turbulence. Indeed, as we know,

Nonlinear Instability, Chaos and Turbulence 113

the developed turbulence tends to restore the broken symmetry [1].Thus, for example, homogeneity or isotropy of turbulence violated onlarge scales is generally recovered on smaller scales, which is the basicpremise for the Kolmogorov-Obukhov local theory [2,3]. In this case,the interaction between large-scale disturbance and turbulence occursas a disturbance decay due to turbulent viscosity and is accompaniedby the energy transfer from the large-scale motion to small-scale tur-bulent pulsations. Under these conditions the existence of long-livedstructures with the spatial dimension L ;» Л (Л is the turbulencescale) seems to be hardly probable.

It is quite another matter when the broken symmetry is not re-stored by turbulence. Such is the case with the lack of reflectionsymmetry (mirror-invariance breakdown) which is compatible withthe theory of local structure of turbulence. Turbulence showing thisproperty is called helical. Apart from the mean energy it is also cha-racterized by nonzero pseudoscalar S = f {v'curlv)dr representing amean heticity. Helicity is one of the main characteristics of the vec-torial velocity field. First of all, the mean helicity, like energy, is aninviscid constant of motion and hence, in nonviscous flows the breakof reflection symmetry generating helicity cannot be eliminated byturbulence alone. Second, this quantity falling into the category oftopological invariants characterizes the structure of the vectorial ve-locity field and measures the degree of linkage of the vortex lines [4].Third, a nonvanishing mean helicity implying the symmetry breakwith respect to coordinate system reflections determines accordingto its sign the predominance of the left-handed or the right-handedspiral motions in the examined flow.

The sources of helical turbulence are known to be the force fieldsof a pseudovector nature, such as magnetic or Coriolis force fields.

A survey of turbulence research made in the last few decadesshows that many situations exist where a small-scale helical turbu-lence can intensify and sustain the large-scale disturbances at theexpense of the energy transmission from small to large scales.


2 Hydrodynamic alpha-effect and its manifes-tation in the planets' atmospheres

The specific properties of the small-scale helical turbulence leading tolarge-scale structure generation were first discovered and mathemati-cally substantiated in magnetohydrodynamics by Steenbeck et al. [5].This phenomenon, known as alpha-effect (a-effect), allows us to ex-plain the growth of large-scale magnetic fields in electrically conduc-ting media and forms the basis of the MHD-dynamo theory. Moti-vated by this first experience, the scientists have focused their at-tention on studying this effect more closely. The results of theirresearch efforts have been embodied in a number of published fun-damental studies on the theory of magnetic field generation in a tur-bulent medium (see, for example, monographs by Moffat [6], Parker[7], Krause & Radler [8], Vainshtein et al. [9], Zeldovich et al. [10],Ruzmaikin et al. [11]).

The role of helicity in the hydrodynamics of a non-conductingmedium has long been the subject of intense discussion dating backto the early 1970s. Thus, already in 1973 Kraichnan [12] and Brissaudet al. [13] considered the idea of helicity contribution to the energytransfer from small to large scales. Moreover, the well-known simila-rity between the equation for a magnetic field in a moving conductingmedium and the equation for vorticity of the velocity field in commonhydrodynamic flows should obviously give an additional impetus tosearching for a hydrodynamic analog to the alpha-effect. However,Krause & Rudiger [14] adduced convincing arguments for the absenceof alpha-effect in incompressible flows wi$h a homogeneous, isotropicturbulence and prohibition on the existence of generating alpha-termdue to the symmetry of the Reynolds stress tensor.

Despite this statement the scientists did not give up the attemptsto find this phenomenon. Since the early 1980s a team of researchersfrom the Space Research Institute (Moscow) has been actively in-volved in these investigations. As a result, the very first evidencesupporting the existence of alpha-effect in general hydrodynamicswas found by Moiseev et al. [15] (later abbreviated to Hot-effect inKhomenko et al. [16]). Moiseev et al. [15] treated turbulence as ho-


mogeneous, isotropic and helical and used the assumption that themedium was compressible which suggests that a nonlinear term in-corporating the Reynolds stress cannot be a symmetrical tensor.

The most noteworthy references to helicity investigations may befound in Moffat & Tsinober [17] and Prisch [18].

2.1 Equation for hydrodynamic alpha-effect in a com-pressible medium

The equation for hydrodynamic alpha-effect was derived for homoge-neous, isotropic and reflection-non-invariant turbulence [15]. Further-more, in this work, the assumption following from the very essence ofthe problem was used, i.e. the spatial, L, and temporal, T, characte-ristic scales of the examined flow are large compared with the respec-tive energy-range scales / and r of the turbulence (L^>/, T^>r). Usingthe statistical ensemble averaging and supposing that the turbulenceis Gaussian, the authors applied an effective functional technique forequation closure based on the Purutsu-Novikov formula. As a result,the mean-field hydrodynamic equations have been obtained and ithas been shown that the linearized equation for mean vorticity of thelarge-scale velocity field, и — curl(V), is analogous to that of a-effectin mean-field magnetohydrodynamics:

7 = curl(au>) + vAu>. (1)

Here, the uniform coefficients v and a are related to the random-7-

velocity field parameters: v is the turbulent viscosity, a % -{v-curlv)" 3

characterizes the mean helicity of the small-scale flow v, r is thecorrelation time, and angular brackets denote ensemble averaging.

As demonstrated by Moiseev et al. [15,19], eqn (1) has unstablehelical solutions growing with time for any a in the simplest caseof uniform a = const^ whereas in a flow with nonuniform helicitya — a(v) the growing solutions can be obtained only for sufficientlylarge a. This last result implies that the large-scale instability hasan excitation threshold. The generated velocity field in both cases ishelical.


Thus, in the cylindrical coordinate system {p,</>, 2}, for an axi-symmetrical distribution of helicity a = a(p) reaching the extremuma = c*o on the axis p — 0 the asymptotic solution to eqn (1) for largevalues of dimensionless parameter Ra = aoLa/u is written as [19]

« =

where La is the characteristic scale of the mean helicity variation,and 70 is the growth rate of instability

In view of expression (2) the characteristic scale of vorticity decrease,

w, is equal to

Vt, (4)

Rand the large-scale velocity field is helical, (V)u; w

It should be noted that according to expression (3) the growthof the vortex intensity could be expected only at such values of theparameter Ra that exceed its critical value Rg, namely at

(5)

Solution (2) describes the most rapidly growing velocity field modeand is applied in the vicinity of the extremum of the function ai.e. close to the center of developing vortex structure, at p<^La-

The global scale of the growing disturbance can be estimated interms of solution for а = const (see Moiseev et al. [19]):

_ „ at z < /г, р < La

at z > /1, p > La

which takes into account a considerable difference between the hori-zontal (L ~ 100 km) and vertical (h ~ 10 km) dimensions, typical

= const


for vortices of tropical cyclone type. In this case, the characteristicradial and vertical scales of the solution are, respectively,

L и 0.5La/ar0, H = h/n, (6)

where XQ is the first root of the zero order Bessel function. Theestimate for the growth rate 7 remains unchanged and the excitationthreshold is defined by relation

v > 7Г. (7)

Thus, the above analysis can be useful for geophysical applications.To evaluate the parameters at the center of vortex-forming structure,one should employ solutions (2)-(4), whereas solutions (6)-(7) arebest suited for predicting dimensions of a large-scale formation as awhole. Anticipating the main discussion that follows, we would liketo note that asymptotic solution (4) can also provide an estimate forhorizontal dimension of a vortex (see 2.2.1), but in the context oftropical cyclones arising in the Earth's atmosphere the estimation oftheir sizes in terms of formula (6) seems to be more attractive.

Later it will be demonstrated that expressions (2)-(7) are effectivetools for drawing a realistic picture of atmospheric vortices.

2.2 Intensification of vortex disturbances in the atmo-sphere

Based on the concept of hydrodynamic alpha-effect, Moiseev et al.[19] advanced the theoretical hypothesis according to which this phe-nomenon was treated as a possible physical mechanism governing theintensification of vortex disturbances in the atmosphere. By analogywith MHD-dynamo it is often called the Vortex Dynamo.

It is a well-defined fact that turbulent convection in a rota-ting medium, for example in the atmosphere, is helical and mirror-nonsymmetrical [6,7]. In nonhomogeneous atmosphere the helicityof convection can arise under the action of the Coriolis force on ahorizontal velocity component in a convective cell. As a result, theconvective cells twist about their vertical axes in such a way that


the mean helicity of convective motions is other than zero. Followingfrom Moiseev et al. [19], the mean helicity for the field of turbulentconvective velocities, v, can be estimated as

a~2Qlsin§, (8)

where Q is the angular velocity of the planet rotation, I is the charac-teristic scale of the convective cell, and Ф is the geographic latitude.In this case, eqn (1) defines the relation between the developing large-scale helical vortices and circulation in cells.

Due to an increasingly growing amount of geophysical informa-tion gained from different sources including the environmental moni-toring of the Earth, scientists are aware of various situations when theintensity of the vortex disturbances in the atmosphere increases. Inview of the above considerations, it seems quite reasonable to inter-pret this intensity increase in terms of ifa-effect. Let us discuss twostriking examples reported by Moiseev et al. [19] and Ivanov et al.[20] & Fortov et al. [21], in which the theoretical estimates obtainedby substituting the specific atmospheric parameters in solutions (2)-(7) agree fairly well with the characteristics of natural phenomena.The first example [19] concerns the problem of tropical cyclone (ty-phoon) formation in the Earth's atmosphere and the second [20,21]offers an explanation for the size and structure of large-scale distur-bances in Jovian atmosphere initiated by a fragment fall after thecollision of comet Shoemaker-Levy 9 with Jupiter in July 1994.

2.2.1 Tropical cyclones in the Earth's atmosphereA well-developed tropical cyclone is an intensive atmospheric vortex,in which the main component of velocity lies in a horizontal plane.The powerful horizontal circulation is superimposed on a weaker ver-tical circulation, the characteristic velocities of which are an order ofmagnitude lowef than those of horizontal motions. Meanwhile, thevertical circulation is of crucial importance for the existence of suchvortical system as a whole ensuring the linkage of air streamlines and,hence, the helical structure of the flow. An active zone of the formedtropical cyclone is approximately 200 km in radius. There is one more


intrinsic feature of the developed typhoon that distinguishes it fromthe midlatitude cyclones. This is the so-called "eye", which occupiesan area of 10-50 km radius in the center of the vortical formationand involves tangential motions of insignificant intensity. Tropicalcyclones usually arise in the regions of intensified convective atmo-spheric motions, so that the typhoon-forming zones are basically lo-cated between latitudes 5°-25° both to the north and to the south ofthe equator.

After many years' observation and investigation of tropical cy-clones, versatile information on this wonderful natural phenomenonhas been accumulated (see, for example, the academic monographby Riehl [22] and/or the book by French meteorologist and avia-tor Molene [23] written in the intriguing style of adventure story,which caught one of the authors' (G.L.) imagination many years ago).However, it was for a long time unclear what physical processes areinvolved in the early stages of cyclone formation: how the initialdisturbance is intensified, what causes the large-scale vertical circu-lation that feeds energy into the dynamic system of a tropical cycloneand ensures nontrivial topology of the vortex due to tangling of thestreamlines of a large-scale velocity field, fV-curlVdr ф 0, and, fi-nally, what factors are responsible for intensive global rotation of ahelical vortex. In spite of numerous attempts to investigate the prob-lem of tropical cyclone initiation, it is still far from being completelyunderstood. However, with the experience gained in this field we cansafely name, at least, two physical mechanisms essentially contribu-ting to a typhoon formation. These are the Ha-effect [19] and theMechanism of "anomalous" heat transfer acting under highly inten-sive turbulent convection and quasithermal insulation of the surfacesbounding the flow [24-28].

The Ha-e&ect was interpreted by Moiseev et al. [19] as a pos-sible mechanism for intensification of weak large-scale (hundreds ofkilometers) vortex disturbances in the tropical atmosphere at the ex-pense of energy of a thermal convection with characteristic scales ofabout 10 km. With such an approach we can not only offer a generalexplanation for the typical features of tropical cyclones but also make


reasonable quantitative estimates.These estimates were obtained by substituting in expressions (2)-

(8) the data characteristic of typhoon-birth conditions and typicalparameters of the Earth's atmosphere. The values of all pertinentquantities extracted directly from the paper of Moiseev et al. [19], aswell as calculated by the formulae from the same work, are given intable 1.

A substitution in expression (8) of the characteristic scale of con-vective velocity variation, I = 10 km, and the Earth's rotation angu-lar velocity, п и 0.75 • 10~4 s"1, yields a f m s " ^ 1ЪвтФ for themean helicity of a small-scale convection. The scale of the mean he-licity variation defined by characteristic size of the tropical cycloneformation zone can be chosen as La « 103 km.

Let us consider asymptotic solution (2)-(4) using as a coefficientof turbulent viscosity a widespread estimate, v « 103 mV"1. Bysubstituting this and the above-mentioned values into the expressionfor Ra and formula (3) and (4) determining the time of formation,T, and size, LTC, of the tropical cyclone (TC), we obtain

ч 50 km m , 0.5 hours /лХRa * 1.5 • 10* зтФ, LTC « -7=== , T = 7 « — ^ г ^ Г • 9 )

Ф sinzy>

Here, we have chosen LTC equal to the characteristic scale, pW: ofvorticity decrease defined by formula (4).

Note that the vorticity component, CJ^, in solution (2) reachesmaximum at a finite distance of po & o m the axis. Thus, for example,at Ф — 10° this distance is equal to po w Ю km. By relating, atthe nonlinear stage, the typhoon "eye" formation to this peculiarity,we can obtain an estimate for po which is comparable in the order ofmagnitude to the observation data obtained for the typhoon "eye" of

the radius,Moreover, since the growth of large-scale disturbances is possible

only at Ra > 4, then, in view of the first expression from (9), thiscondition can be met only some distance away from the equator, atФ > Ф0*. Hence, from relation (9) it follows that

at Ф^^О.г 0 . (10)

Table 1. Tropical cyclones in the Earth's atmosphere.

Earth's aconditions foi

^ • - -

Angular velocity

LatitudeScale of atmospheric

convectionHelicity coefficientScale of mean heli-

city variationTurbulent viscosity

tmosrTCп

Фl

a

La

V

phereinitiation

0.75 • 10"4 s" 1

±5° ^ 25°10 km=-104 m

1.5 • втФ m s""1

103 km=106 m

103 тг s" 1

Parameters at Ф = 10°Characteristic value

of helicityDimensionless

parameterRa

0.26 m s " 1

260

Parameter

LTC

T

L«eye»

HTC

TropicalФ =

Observation

200 km

24 h

10 - 50 km

10 km

cyclone10°

Asymptoticsolution

120 km

17 h

10 km

Globalsolution

400 km

17 h

3 km

3

S

to


Substituting characteristic values in global solutions (6),(7) gives thefollowing estimates for horizontal, LTC, and vertical, Нтс-> dimen-sions of a tropical cyclone:

LTc « 400 km, HTc ~ 3 km, (11)

and the condition for excitation threshold

#"• > 1 0 ° . (12)

The characteristic time, T, of the vortex formation is still defined byexpression (9).

In table 1, the theoretical estimates are compared with the ob-servation data at the geographical latitude Ф = 10°. At this latitude,conditions (10) and (12) ensuring disturbance intensification for bothtypes of solution are satisfied due to the fact that Ф > Ф** and thevalue of dimensionless parameter Ra = 260 calculated at given Ф islaxger than R£ = 4 determined by formulae (10).

Hence, based on solutions (2)-(7) and the obtained quantitativeestimates presented in table 1, we can identify the ifa-effect as amechanism governing an exponentially rapid growth of the large-scalevortex disturbances. The generated velocity field is qualitatively andquantitatively consistent with the velocity distribution typical for atropical cyclone.

2.2.2 Large-scale disturbances in Jovian atmosphere undercollision with comet Shoemaker-Levy 9

"In July 1994 a dramatic event, the collision of comet Shoemaker-Levy 9 with Jupiter, took place. The collision was accompanied witha wide variety of diverse effects in the atmosphere, ionosphere andmagnetosphere of Jupiter. The comet impact became one of the mostgrandiose active experiments ever to be set up by Nature. We reallyhad the luck to witness the rarest cosmic phenomenon that takesplace once a thousand years." [21] The astronomical community hadprepared for this event well in advance, so that the comet collisionwith Jupiter was observed from practically all the world's observa-tories, by the Galileo, Uliss and Voyager spacecrafts and the Hubblespace telescope.


After processing all the collected data, the scientists constructeda fairly distinct picture of the comet debris influence on Jupiter'satmosphere, which, however, gave no way of explaining both thestructure and extremely large dimensions of initial atmospheric dis-turbances arising within the first few hours after the fragment fall.

According to observations summarized by Fortov et al. [21], alltrails of the comet debris impacts located in the vicinity of latitude—45°. The kinetic energy of a comet fragment measuring 1-3 kmin size and falling with the relative velocity of 60-65 kms"1 was ofthe order of 1021 — 1023J. Only 5 of 15 recorded trails producedby the largest pieces of comet debris led to essential disturbances inJupiter's atmosphere which were observed for several months. All thephotographs obtained of these trails actually refer to an initial stageof disturbance formation and have pronounced common features. Thephotos show a marked homogeneous spot in the center surroundedby a ring. This ring is the internal boundary of the ring-shaped dis-turbed area with rather dim external boundaries. Interestingly, thespatial dimensions of the characteristic formations in the atmospherepractically coincide for all large pieces of debris; thus, for example,after 1.5 hours of the fragment fall the ring radius reached approxi-mately 3000 km and its extension rate was constant, equal to « 450ms" 1 throughout the process evolution. An external boundary of theregion was 12000 km away from its center. This structure of atmo-sphere perturbation was observed for several hours after the fall of

large fragments.The analysis of dynamics of the processes developing in the re-

gions of the debris fall has shown [21] that neither the gravity-wavenor the impact-wave mechanism of disturbance generation in theplanet's cloud cover can be used adequately to explain the particu-larity of the trail structure. Moreover, the energy spent by thesemechanisms to generate disturbances similar to those observed in theJovian atmosphere should be well in excess of the energy generatedby the impact of the largest fragments.

In this situation Ivanov et al. [20] and Fortov et al. [21] suggesteduse of the previously advanced concept [19] of small-scale helical tur-


bnlence contribution to the formation of "typhoon"-type vortices inthe atmosphere for interpreting the existence of an additional energysource. Under this strategy, it can be reasonably supposed that thedebris impact acts merely as a triggering mechanism for initiation ofvortex processes, which gain energy from an extremely intensive con-vection in the Jovian atmosphere. Then, we can qualitatively explainall the specific features of impact trails, as well as derive theoreticalestimates which quantitatively agree with observation data.

The analysis of the key physical factors determining the forma-tion and evolution of a trail at the site of a large debris fall allows thescientists to differentiate three stages of its evolution [20,21]. In theinitial stage of large-scale structure formation, the major disturbingfactor is an intensive atmospheric vortex which is generated due toa trapping and twisting of the surrounding atmosphere by a risingfireball produced by a debris explosion. In the next stage this centraldisturbance is intensified at the expense of atmospheric convectionenergy. In the final stage the atmospheric vortex which has grown bythis time to dimensions comparable with the Rossby radius is sub-jected to the action of the Coriolis force and horizontal wind flowswhich disperse this disturbance.

Based on the investigations by Ivanov et al. [20] and Fortov et al.[21], we will present their interpretation of the data, observed at thestages of disturbance intensification, in terms of iJa-effect suggestedby Moiseev et al. [15,19].

Let us suppose that the main energy source for generation ofa powerful vortex is a vertical thermal convection. To estimate thecharacteristic horizontal scales of disturbances at the initial stage, weuse the asymptotic solution (2) and (3), which is valid in the vicinityof the vortex center.

In the following the characteristics of convection in Jupiter'satmosphere described by Ivanov et al. [20] and Fortov et al. [21] areused. The most intensive convective transfer occurs in the tropo-sphere zone underlying the tropopause and extending from the lowestlayers to the boundary of a cloud cover. The characteristic verticalscale of convective cells is approximately h и 100 — 150 km. In


the Jovian atmosphere we can distinguish the two latitudinal zoneswith thermcil convection of qualitatively different types: equatorial(0° < Ф < 40°) and midlatitudinal (40° < Ф < 60°). The extentof the midlatitudinal zone involving the region of debris fall is 20°in latitude or approximately 24000 km. The coefficient of turbulentviscosity, v, is chosen equal to 106 m 2 s - 1 . The imagery of Jupitercollected during its collision with the comet Shoemaker-Levy 9 showsthat within the first few hours of trail formation the entire indicatedzone was seized by developing disturbances, which allows us to use thefull size of the zone as a characteristic scale La of the mean helicityvariation. Then, choosing I = 100 km as a characteristic horizontalscale of convection, and using expression (8) to estimate the meanhelicity of small-scale convection, we obtain

a(ms~l) « 35sm# . (13)

Here, Q w 1.75 * 10~4s~x is the angular velocity of Jupiter's rotation.Then, in latitude Ф = —45°, which is the nearest to the region

of the debris fall, we have the following quantitative estimates forcoefficients involved in solutions (2),(3):

- 1a 0 «24.4 7ns- 1, Ra = 595. (14)

The value of Ra proves to be much higher than the critical valueRa

r = 4 from condition (5) leading to disturbance intensification.This suggests that the examined vortex motion in the atmospheremust grow intensively.

The scale p of the region, where the intensity of the vortex dis-turbance increases, is determined by Ivanov et al. [20] and Fortov etal. [21] from the condition

(15)

which, for example, t ~ 2 hours after the comet fragments fall yieldsthe value of radius po ~ 2000 km. On the photographs of trails formedin Jupiter's atmosphere this region is identified with the central ringwhose radius for different fragments lies in the range from 1788 to


2260 km [20]. In conditions of the Earth's atmosphere, the "eye" ofa typhoon can serve as the analog to such a structure.

The estimates for global dimensions of vortex disturbance at latertimes were obtained by Ivanov et al. [20] and Fortov et al. [21] fromsolution (6) describing a large-scale vortex whose evolution occurs atthe background of turbulent cells with the uniform helicity coefficienta.

At the values of щ and v given above, the condition for excita-tion threshold defined in this particular case by relation (7) is metfor h > 130 km, where h is the vertical scale of the atmospheric con-vection. It must be remembered that when estimating ao from (7),the characteristic horizontal scale of a turbulent cell is taken to beequal to I « 100 km. However, as indicated by Ivanov et al. [20], theestimate I ~ 400 km is also quite realistic for Jupiter's atmosphere.In this case, according to formula (8), the coefficient ao increases pro-portionally to J, leading to a corresponding decrease of the estimatefor h in condition (7).

For the selected value of La solution (6) yields L ~ 4800 km. Thisvalue agrees well with the maximum ring radius of 4649 km estimatedfrom the photos of large debris trails [20]. The characteristic verticalscale, Я , of the vortex defined by formula (6) is found to be equal to30-50 km.

The obtained estimates show that already in the initial stageof the process the thickness of the atmosphere layer involved in thevortex motion becomes comparable with the thickness of the cloudcover. A horizontal scale of the disturbed region is more than twoorders greater than its vertical dimension. The structure of large-scale circulation formed within the atmospheric vortex should causean intensive vertical mixing, equalize the velocities and temperaturesat horizontal levels, and destroy quazistationary turbulent convectivecells. As a result, a regular energy supply to a large-scale vortexceases as soon as twenty-four hours after its initiation [20], unlike theEarth's typhoons, which keep gaining energy from the underlyingocean surface for a long time.

Theoretical predictions and observation data for Jupiter's atmo-

Table 2. Large-scale vortex disturbances in Jupiter's atmosphere.

Jupiter's atmospherezone of formation of

large-scale disturbancesAngular velocity

LatitudeScale of atmospheric

convectionHelicity coefficientScale of mean heli-

city variationTurbulent viscosity

ФI

aLa

кя

V

1.75 • 10~4 s- 1

-45°100 km =10b m

35 • sin§ m s"1

24000 km =24-106 m10c m* s~l

Parameters at Ф = -45°Characteristic value

of helicityDimensionless

parameter

«0

Ra

24.4 m s l

595

Large-scale vortexФ = -45°

Parameter

Po

L

H

Observation

1788-2260 km

4649 km

Asymptoticsolution

2000 km

—

•

Globalsolution

—

4800 km

30-50 km

o-

rif


sphere are given in table 2.Thus, it is the authors' opinion that the concept of the gene-

rating properties of small-scale helical turbulence advanced in hydro-dynamics of non-conducting medium has received bright and elegantsupport, providing a powerful tool for explanation of catastrophicprocesses in the atmosphere of the largest planet in the solar system.

This convincing example is further evidence favoring the viewthat the hydrodynamic alpha-effect is of primary importance for gene-ration of large-scale spiral vortices of "typhoon"-type in the planets'atmospheres.

3 Helical-vortex instability in a nommiformlyheated fluid

In section 2 we have discussed in detail the hydrodynamic alpha-effect for homogeneous isotropic helical turbulence in a compressiblemedium, which would seem to be the first example of this phe-nomenon discovered in hydrodynamics [15]. Moreover, further in-vestigations have demonstrated that this example is the simplest inthe sense of mathematical description. Indeed, helicity of the ve-locity field is mathematically represented by nonzero pseudoscalara ~ (v-cuWv) ф 0. For a compressible fluid due to asymmetry of theReynolds stress tensor this has found to be sufficient to obtain afteraveraging the generating alpha-term, curl{au)), for eqn (1) descri-bing the mean vorticity of the large-scale flow. It is precisely thisterm that allows us to obtain the solution ensuring an exponentialgrowth of vorticity.

For an incompressible medium the situation is quite different.If the turbulence is homogeneous and isotropic, the break of reflec-tion symmetry generating nonzero helicity of the small-scale velocityfield is not a sufficient condition for the existence of JYa-effect. Asmentioned in section 2, this is attributed to the fact that in incom-pressible flows the Reynolds stress tensor is symmetrical. Therefore,when developing an equation for mean vorticity of the velocity fieldthe pseudoscalar coefficient a must necessarily vanish. This implies


that we need some additional factors for the symmetry break enablingus to construct a generating term in the averaged equations.

The investigations of the last decade have revealed several exam-ples of ffoeffect for an incompressible fluid [29-34], in which the roleof such complicating factors is played by a prescribed large-scale flow,stable or unstable stratification, gravity force or anisotropy. Theseworks have paved the way for new trends in Hoeffect research, oneof which has given its name to the present study. A mathematical de-scription of .ffa-effect developed in these works is more sophisticatedthan for a compressible medium and involves tensor coefficientPhysically, all supplementary factors during mean-field generationare nothing but triggering mechanisms by which part of helical tur-bulence energy is pumped into large-scale motions. Now, we are wellaware of the fact [12,17,35] that such a process is expected to suppressthe energy flux into small scales. Therefore, the helical turbulenceshould obviously search for additional ways to restore the disturbedequilibrium, precisely by means of generating large-scale structures.Under these conditions some part of the turbulence energy is trans-ferred to large scales. This effect can be naturally interpreted as a

F

vortex dynamo.The first example of a vortex dynamo for homogeneous isotropic

helical turbulence in an incompressible fluid was discovered by Moi-seev et al. [29,30]. In this case, the additional factors breaking thesymmetry were the gravity force and the temperature gradient.

3.1 Mathematical model for vortex dynamo in a con-vective system

The equations for vortex dynamo [29,30] were developed for an in-compressible medium, in which a small-scale helical turbulence wasgenerated by a prescribed external force. Having chosen the gravi-ty force and the temperature gradient as supplementary symmetry-breaking factors, the authors, thereby, formulated the problem oninitiation of alpha-effect in a convective system. This example isprobably one of the most promising for various geophysical appli-cations (it will suffice to recall the examples of vortex disturbance


intensification in the atmosphere discussed in section 2). Unlike theexamples considered [19-21], in which the authors proceeded from aqualitative hypothesis that a small-scale helical atmospheric convec-tion might be a possible energy source for generation of large-scaleintensive vortices, the works by Moiseev et al. [29,30] can be viewedas the first successful attempt mathematically to substantiate thepossibility of the existence of this effect.

The classical Rayleigh-Benard convection in a plane infinite hori-zontal layer of incompressible fluid heated from below was chosen byMoiseev et al. [29,30] as an initial problem for modeling. As we know,this problem is well described by a system of Boussinesq equations[36,37]. However, in the above works [29,30] the authors introducedinto the equation of motion an additional random external force gene-rating small-scale helical turbulence. In this context, assuming thatvertical temperature gradient is small and turbulence properties aresimilar to those used for deriving the if ск-effect equation for a com-pressible medium (see section 2.1), Moiseev and his co-authors ap-plied the same averaging technique [15] to the Boussinesq system.Actually, the investigations reported by Moiseev et al. [29,30] wereconcerned with the problem of the influence of small-scale helicalturbulent noise on convection, which allowed the authors to obtaina linear mean-field equation describing generation of the large-scalevelocity field (vi).

In terms of tensor representation, this equation can be writtenin the dimensionless form as

-Aj<Vi> - < Vj >=

yfim^-kra +

<Va>, (16)

Pim = Um - Vi Vm/A , e» = {0,0,1}.X "X

Here, P r and Ra are the Prandtl and Rayleigh numbers, P j m is aprojection operator eliminating a potential part of the velocity field,


€{ the unit vector directed vertically upward, e^* the antisymmetricalLevi-Chivita tensor, A the uniform temperature gradient between thehorizontal boundaries of the layer, g the acceleration of gravity force,/? the coefficient of thermal expansion, h the layer height. Of thedimensionless parameters 5, /xi and \i2 characterizing a small-scaleturbulence, the helicity coefficient s is most interesting for furtherdiscussion. A mathematical expression of this quantity obtained byMoiseev et al. [30] is very cumbersome. Therefore, we only note herethat 5 is related in a rather complicated талпег to the coefficient ofkinematic viscosity v and thermal diffusivity x characterizing physi-cal properties of the fluid and the turbulence characteristics such asthe most energetic scale A and characteristic time r of the turbulentvelocity correlation. The parameters ^i and /Л2 represent a compli-cated combination of the same quantities i/, x-> ^ ^ d T-

A special feature of the obtained mean-field equation differentia-ting it from the equations of natural convection is the presence ofterms including tensor e ^ . These terms provide a positive feedbackbetween the solenoidal components of the velocity field. The earlierinvestigation of the ifa-effect demonstrated that these are preciselythe terms which are responsible for initiation of large-scale instability.

The best way to give a comprehensible physical interpretationof the obtained effect is to represent the vectorial field in terms oftoroidal and poloidal component [6], i.e. in the form of representationthat is frequently used in magnetohydrodynamics and is well suitedfor transformation of corresponding vector equations to the system ofequations for scalar functions. Bearing this in mind, we can expressthe vectorial velocity field in the following form:

V = V^ + Vp , VT = сиг1(еф), Vp = curl сиг1(еф).

Now the equations for the alpha-effect in a convective system canbe written as

- Д J T = -

/дI Д ) Аф = RaT + Rasfii (eV) 2 — Дх Ф , (17)\dt


Неге, Т is the temperature, ф and ф are the toroidal and poloidalpotentials of the velocity field, and A± = д?/дх2 + дР/ду2 is thetwo-dimensional Laplace operator.

Thus, the model for vortex dynamo under convection conditionsconsists of a system of dynamic equations for three modes: the tem-perature field and two solenoidal velocity fields. The system of lineareqns (17) for large-scale fields includes two different positive feed-backs. One of them acts between the poloidal component of thevelocity field and the field of temperature disturbance. It links thefirst and the second equations in system (17) ancHeads to commonconvective instability. The other, being related to specific propertiesof helical turbulence, directly links the very solenoidal componentsof the velocity field (i.e. the second and the third equations from thesystem (17)) and excites new helical-vortex instability. In this case,the toroidal and poloidal velocity fields are related only through thehelicity parameter $. If 5 ф 0, the flow shows a new topologicalproperty - the linkage of the streamlines of toroidal and poloidal cir-culation. As a result, the large-scale vectorial velocity field becomeshelical.

Hence, from the analysis of model (17) it follows that the formu-lated problem is essentially concerned with the flows, which might beinitiated at different combinations of instabilities currently availablein the system, and the properties of these flows.

The mathematical model (17) was theoretically investigated[30,38,39] for different types of boundary conditions in the frameworkof linear theory of stability. The results of this analysis provide uswith information typical [36,37] for the examined class of problems:the relations Racr(kx, s) (where k± is the wavenumber characterizingthe horizontal scale of disturbances), Ra%in(s) and corresponding

(s), and spectra for the growth rates of critical disturbances.The simplest of all considered cases is the problem on helical-

convective flows in a horizontal layer with free boundaries. Yet, itallows us to reveal the peculiar features caused by new instability.


Summarizing the basic results from the above-mentioned papers letus discuss in detail the effects described by model (17).

First it should be noted that in the limiting case 5 = 0 (in theabsence of helical feedback), system (17) is reduced to the Boussi-nesq equations whose solutions were carefully analyzed [36,37] bymethods of linear and nonlinear theory of hydrodynamic stability.In this case, the convective instability in the layer arises after ex-ceeding some critical Rayleigh number, Racr', and leads to formationof a flow structure consisting of numerous small cells. The cells arecreated by poloidal circulation, each having the characteristic hori-zontal dimension of the order of layer height. This particular casefully corresponds to the Rayleigh-Benard problem [36,37].

In the presence of helical feedback (s ^ 0) an increase of the he-licity parameter s leads to a decrease in the critical Rayleigh number,which means that the intensity of heating required to initiate convec-tion reduces. Quite apparently, an additional factor appears in thesystem, which causes a warm and light fluid volume to rise from theheated lower boundary. This factor could be related to a toroidal ve-locity field of each convective cell generated by helical feedback fromthe poloidal field. In this situation the minimum of neutral stabilitycurves is shifted to small wavenumbers, which implies the growth ofa typical horizontal scale of arising structures. As soon as the helicityparameter reaches some critical value, 5е7*, the wavenumber vanishes.Formally, this corresponds to an infinite horizontal dimension of a su-percritical flow and suggests qualitative changes in the flow pattern:the system of small cells should be rearranged into a single large cell,which occupies the entire available space. Moreover, according toeqns (17), the toroidal and poloidal components of the large-scalevelocity field in such vortex cell should be linked, forming thereby ahelical structure of large-scale circulation.

Hence, the action of helical-vortex instability in the convectivesystem leads to dramatic changes in the structure and dynamics offlows.


3.2 Some further theoretical investigations

Prom the outset, the scientists of the Space Research Institute haveassociated the theoretical investigations aimed at developing a mathe-matical model of the alpha-effect in hydrodynamics with possiblegeophysical applications, in particular, with the problem of tropicalcyclone formation. Even the first results of modeling the Яа-effect ina convective system [30] discussed in the previous section have convin-cingly demonstrated that the proposed approach holds much promisefor description of real atmospheric processes involving generation ofintensive large-scale vortices. This offers wide scope for further re-search and is embodied in more recent publications by Lupyan et al.[40,41], Rutkevich [42,43] and Rutkevich & Moiseev [44].

The attempts to extend approach [30] to modeling the earlystage of tropical cyclone formation naturally suggest that the ini-tial problem formulation and the basic prerequisites for derivation ofthe mean-field equations should be adapted to the specific featuresof convective heat and mass transfer in the tropical atmosphere.

A key point to derivation of the mean-field equation is to decidewhether we have enough reason to consider convection as a large-scale process on the background of prescribed small-scale turbulenceof nonconvective nature, as suggested by Moiseev et al. [30]. Thisproblem was analyzed and solved by Lupyan et al. [40]. Here, withreference made to the corresponding meteorology and geophysics li-terature, the authors based their treatment on the well-defined factthat tropical cyclones are initiated (the stage of tropical depression)in the presence of highly intensive atmospheric convection consistingof a multitude of separate cells, which are generated due to the re-lease of latent heat in the process of water condensation. Then, themotion in these cells can be reasonably treated as a small-scale con-vective turbulence. In view of the fact that convective motions inthe atmosphere are subjected to the Coriolis force, the turbulencewas considered helical [40] and modeled as that driven by a randomexternal force [30}. However, unlike Moiseev et al. [30] the para-meters of turbulent velocity field were adjusted to the parametersof the convection process by fitting the parameters of corresponding


turbulence correlation tensor to those of convection linear operator.Furthermore, taking into account the inverse influence of small-scaleconvection on the temperature gradient made a large-scale tempera-ture profile closer to a neutral one in full agreement with the theory ofconvective accommodation [45]. The efforts made in this direction ledscientists to the mean-field equation which proved to be very simpleeven compared with eqn (16) proposed by Moiseev et al. [30]. • Theanalysis of the model solutions has shown that it can be effectivelyused for qualitative description of vortex generation in the zone oftropical depression.

In the reviewed studies Moiseev et al. [29,30] and Lupyan et al.[40] formally specified helical turbulence by introducing in correla-tion tensor a term containing antisymmetrical tensor of the thirdrank €ijk~ In the sense of atmospheric phenomena this suggests thatsuch a break of reflection invariance of small-scale turbulence can beproduced by a combined action of vertical inhomogeneity (for exam-ple, by stratification sufficiently unstable for convection initiation)and the Coriolis force.

This naturally brings up the question: what is the real contribu-tion of these factors to the processes in which small-scale turbulenceexhibits generating properties; The very fact that in our interpre-tation, the initiation of large-scale vortices in the atmosphere is at-tributed to the action of the Яа-effect implicitly supports the ideathat helical turbulence involves or parametrizes all these factors. In-deed, as intuition suggests, the ascending and descending motions inthe cells of convective circulation caused by vertical inhomogeneityof the atmosphere are expected to generate small-scale helicity of thevelocity field under the action of the Coriolis force [6,8,9].

An attempt explicitly to incorporate these factors into the modelof large-scale hydrodynamic structure generation was made by Rutke-vich [42,43]. A "negative" conclusion reached by Berezin & Zhiikov[46], that initiation of convective large-scale structures in a rotatinghorizontal layer heated from below is impossible in the context of in-compressible fluid model, spurred him to reconsider the problem for-mulation. Assuming that the linear temperature gradient generated


by uniform temperature difference between the layer boundaries is un-able to ensure vertical inhomogeneity necessary for generating effects,Rutkevich [42,43] suggested the inclusion of additional inhomogeneityin the problem formulation. In practice, the best way to model therevised situation is to consider volumetric heating of the layer byinternal sources providing temperature profile in the form of squareparabola. In this formulation the problem can easily be extendedto a description of atmospheric convection gaining its energy mainlyat the expense of latent heat from vapor condensation. However, itis reasonable to suppose that an adequate model of the atmosphericconditions should consider heating from below in order to take intoaccount the effect of heating of the lower atmospheric layer due toreturn of solar radiation absorbed by the Earth's surface. The latterfactor was not considered by Rutkevich [42,43].

In view of the above reasoning, the problem examined [42,43] wasinitially defined as that of convection in a rotating horizontal layerof incompressible fluid heated by uniformly distributed internal heatsources. In this case, the vertical temperature profile in a basic state(mechanical equilibrium) has the form of a parabola whose curvatureis proportional to the power of heat generation [36]. In the equationof motion, now including the Goriolis forces the small-scale turbulenceis specified in a manner [29,30,40] discussed above, i.e. by a randomexternal force. Here, an important point is that turbulence helicityis not postulated. Then, by assuming that convection parametersare consistent with turbulence parameters, the author suggested thatunder the action of nonlinear vertical temperature gradient and theCoriolis force, the homogeneous stationary turbulence should be con-sidered as highly anisotropic.

The basic result of these theoretical investigations [42,43] was themean-field equation predicting the possibility for the onset of large-scale instability. The equation was derived in terms of procedureproposed earlier by Moiseev et al. [15,30] and Lupyan et al. [40]. Itinvolves the Reynolds stresses whose tensor structure coincides withthe structure of corresponding terms in averaged equation [40] de-veloped on the basis of helical turbulence concept. This indicates


that under examined conditions a homogeneous anisotropic small-scale turbulence becomes helical. Hence, the coefficient standing be-fore the generating term can be considered [42,43] as the coefficientof turbulence helicity. It is now proportional to the product of tem-perature gradient inhomogeneity by the turbulence intensity and theCoriolis parameter.

To summarize, the results of investigations of Rutkevich [42,43]allows us to draw a very important conclusion that the concept ofhelical turbulence actually parametrizes the joint effect of the tem-perature profile noulinearity and the Coriolis force. Moreover, thevery process of free convection under these conditions can be viewedas an effective mechanism for helical turbulence generation. Promthe viewpoint of validity of the developed theoretical models for ex-planation of natural atmospheric phenomena, this conclusion wouldprobably be the basic argument for their efficiency. However, as inany other theoretical approach the final and true assessment of theconcept on generating properties of small-scale helical turbulence andits role in constructing a physical picture of atmospheric processes canbe made only on the basis of experimental evidence.

3.3 Laboratory modeling and campaign observations^

To support the theoretical hypothesis that hydrodynamic alphareffectis a mechanism for generation of large-scale intensive geophysical vor-tices at the expense of the energy flux from small-scale convection, aseries oflaboratory and field experiments was projected. The first ex-periments on laboratory modeling were performed in the heart of thenorthern continent in Ural. They were carried out by researchers fromone of the leading Russian scientific schools on the problems of ther-mal convection founded by Professors Gershuni and Zhukhovitsky inPerm. Field observations were accomplished directly in conditions oftropical cyclone formation in the tropical Pacific during expeditions"Typhoon-89" and "Typhoon-90".


3.3.1 Laboratory experiments in PermBy the time of the first attempts to create a laboratory model of theЯа-effect in Perm in the late 1980s, the scientists had been activelyinvolved in intensive theoretical studies into generating properties ofhelical turbulence to the detriment of experimental developments.Unfortunately, the truth is that nowadays there are only a few pub-lished experiments by Khripchenko [47], Bogatyryov [48], Bogatyryov& Smorodin [49], Levina et al. [50] worthy of note. Amongst these,particular attention should be focused on the investigation by Levinaet al, [50] for its direct bearing on the ffa-eflect. In fact, it wasinitially intended as a test for validity of the theoretical results dis-cussed above. Nevertheless, we shall briefly discuss the other results[47-49] which in spite of their nonrelevance to our immediate goalsprovide useful information for later experiments [50].

A hydrodynamic inverse energy cascade in a three-dimensionalturbulence was modeled by Khripchenko [47] for an incompressiblefluid under isothermal conditions. For experimental model the au-thor used a plane layer of conducting medium, in which a periodicalplanar vortex flow (toroidal circulation) was superimposed on theelectrovortex space periodical flow (poloidal circulation) excited byelectromagnetic forces. The total helicity of the generated small-scalevortices was nonzero, yet the general planar circulation was absent.The magnitude of the introduced small-scale helicity could be con-trolled.

The experimental investigations that followed [48,49] were madeon the same experimental setup and were aimed at creating a labo-ratory analog to the tropical cyclone using as a basis the processesobserved in a rotating cylindrical fluid layer (with a large aspect ra-tio R:h equal to 10:1) with the central zone being heated from below.Although the generating properties associated with the flow helicitywere not considered, the very formulation of the problem suggestedthat they did exist. The experiments were carried out at large valuesof the Grashof number in the range G = 104 — 106 leading to for-mation of small-scale heat carriers: thermals. The angular velocityof the layer rotation varied over a wide range. The lowest Reynolds


number used in experiments [48,49] was Re = 7 and the highest wasRe = 105.

In the absence of rotation, three types of convective structureswere observed: a radial temperature difference resulted in a large-scale advective circulation, the flow in the zone over the heater hadthe form of chaotically rising thermals, and outside the heated zonewith a size of about 1/3 of the cuvette diameter one could observethe formation of convective rolls whose axes had radial direction.

After exceeding some critical Grashof number, G0", the flow pat-tern in the rotating layer drastically changed. A large helical cyclone-type vortex developed on the background of the chaotic small-scaleflow was a new formation of high intensity standing apart from theadvective circulation cell. When interpreting the discovered effectthe author [48] advanced the idea that the small-scale helicity of therising thermals might be responsible for the onset of crisis situationfavoring the development of large-scale disturbance with a structureof helical cyclonic vortex. A thorough investigations of the veloci-ty field of this laboratory vortex [49] revealed its analogy with thevelocity field of the tropical cyclone. A qualitative analysis of theexperimental data provided an important information which allowedthe authors to interpret such peculiarities of the tropical cyclone aseye at the center, spiral bands of clouds with rainfall.

At last, the experimenters tackled the problem of Яа-effect inreal earnest. The experimental investigations undertaken by Levinafet al. [50] were directly related to the effect of small-scale helicity ongeneration and evolution of large-scale structures in turbulent con-vection.

The laboratory model used in this research consisted of a square24.5x24.5 cm2 cuvette filled with a fluid and mounted on a rotatingplatform. The angular velocity of the platform rotation about thevertical axis varied between 0.005 s~l and 0.5 s~l. A pinion geardistributing rotation over 36 identical propellers was located beneaththe cuvette bottom. This construction has the advantage that thetotal angular momentum introduced by all propellers to the fluidlayer is equal to zero but at the same time the three-dimensional


vortices generated by the propellers had helicity of identical sign.The value of helicity could be varied by changing the frequency ofpropeller rotation in the range from 1.3 s" 1 to 15 s"1. A fl^t electricheater in the form of a disk 10 cm in diameter was placed at thebottom of the cuvette. The center of the heater coincided with therotation axis of the platform. The results [50] were obtained for thelayer thickness of 10 cm.

With such a design of the experimental model it was possibleto excite in a fluid layer a cyclonic vortex [48,49] whose propertieswere carefully analyzed. In the experiments [50] such flow regimewas realized for G = 1.55-105 and Re = 70. A combination of theplatform rotation and that of propellers allowed the experimenters toinvestigate the effect of external small-scale helicity on the genera-tion and evolution of a large-scale helical vortex. Thus, the proposedmodel had a twofold purpose. It provided small-scale helicities of twodifferent origins - natural, generated according to the Яа-effect hy-pothesis by the action of the Coriolis force on a small-scale convectiveturbulence and forced, introduced by the rotation of propellers - andallowed examination of their interaction.

In these experiments the values of the Coriolis parameter and theintroduced small-scale helicity could be varied independently. Theexperiments demonstrated that introduction of external small-scalehelicity in the layer of a turbulent fluid involving a developed large-scale helical vortex leads to a change of vortex intensity. Unfortu-nately, for some reasons no attempts have been made to extend thispromising work.

3.3.2 Expeditions "Typhoon-89" and "Typhoon-90" in thetropical Pacific

In 1989-1990, under the wide program of tropical cyclone investi-gation, two scientific expeditions "ТурЬооп~89" and "Typhoon-90"on board the research ship Akademik Korolyov were organized in thetropical Pacific. Among the different problems facing the participantsin these campaigns was verification of the two theoretical hypothesesfor the role of the Яа-effect and mechanism of anomalous heat trans-fer (see section 2.2.1 and papers by Zimin et al. [24,25] & Levina et


al. [26-28]).Evidently, the task of checking the results of the theoretical and

experimental investigations against the processes occurring in natu-ral conditions was a challenge, since neither theoretical developmentsnor laboratory models could completely allow for an actual responseof heat and mass transfer and stratification in the tropical atmo-sphere. This inconsistency accounts for the absence of single valuedrelations between the quantities treated in the theoretical and la-boratory models, on the one side, and the measurements of the realprocess characteristics made in the atmosphere, on the other side.

At present the greatest amount of information about the charac-teristics of the atmosphere can be received from ground-based aero-logical and radiometric sounding. Aerological radiosondes providedata on the vertical wind velocity profiles, air temperature and humi-dity along their flight trajectories. However, a clear understandingof the mechanisms governing generation of large-scale distiurbances ispossible only with a knowledge of differential characteristics such ashelicity, energy, enstrophy, horizontal temperature profiles.

To this end, a special mathematical model has been developedto describe large-scale motions in the atmosphere during field experi-ments. It is based on the introduction of transverse spatial momentsfor physical fields into the system of equations for atmosphere hydro-dynamics. This allows us to separate the large-scale componentsfrom the general field structure [25,51,52]. The basic advantage ofthis model is that it can be easily applied to the processing of fieldexperiment data (vertical profiles of wind velocity, temperature, hu-midity, etc.) enabling us to estimate the'dynamic parameters of theatmosphere such as helicity, kinetic energy and helicity fluxes from

+

small-scale turbulence to large-scale structures.The main problem with obtaining spatial and temporal deriva-

tives of meteorological parameters is simultaneous sounding of theatmosphere at several points of space grid at different instants oftime. A successful implementation of such field experiments calls forthe coordinated participation of several research stations or ships,which is a costly operation.


This challenge was solved in campaigns "Typhoon-89" and "Typ-hoon-90" , in which a unique procedure of measurements from a singleresearch ship was proposed and tested [25,51-53]. With the help ofthis pioneering technique the scientists estimated the space and timederivatives of meteorological variables for the suggested mathematicalmodel [25,51-53]. The estimates were obtained over characteristicscales of about 50-100 km in space and 30-100 hours in time andin the most interesting situations of the "Typhoon-90" expeditionfor the time step of 6 hours. During these experiments the researchship was continuously traversing the examined closed polygon in theform of triangle or figure of eight launching aerological radiosondesat its vertices. Then, the points of measurements were uniformlydistributed over the space-time grid with the above mentioned steps.

With this procedure the measurements were taken under differentsynoptical conditions:

- in the regions without any visible signs of cyclonic or anticy-clonic circulation; v

- near the centers of tropical depressions, which were rapidlytransforming into tropical storms during the experiments;

- in the zones with antjcyclonic wind circulation;- on the periphery of the developed tropical cyclone.For detailed information on the collected expedition data sup-

porting the hypothesis for the mechanisms of large-scale structuregeneration, readers are referred to papers by Zimin et al. [25], Veselovet al. [51], Lazarev et al. [52], Lazarev & Moiseev [53]. The basicfindings of these expeditions were the existence of positive energy fluxfrom small to large-scale structures on the stage of tropical depression(Яа-effect) and the heat flux component aligned with the averagedtemperature gradient (mechanism of anoinalous heat transfer).

3.4 Numerical simulation of helical convective flow

Modern science commands a great variety of rather effective theo-retical and experimental techniques and facilities. Nowadays it canoffer one more powerful tool to verify theoretical hypotheses - com-puter simulation techniques, which allow extensive study of different


characteristics of phenomena under consideration.The results described in earlier parts of this section clearly demon-

strate what factors and conditions should be taken into account informulating the problem, and what aspects are of particular impor-tance in numerical modeling the Яа-effect in a convective system:

- Developed turbulent convection must be examined when small-scale heat carriers - thermals (or bubbles, plumes) exist in a fluid.

- The velocity field of the examined flow must involve all threecomponents to provide the non-zero small-scale helicity.

- The presence of the Coriolis force is a binding condition.- The vertical inhomogeneity of the medium must be essential

(which can be achieved by internal heat generation in a fluid layer orby some other means).

- The ratio of horizontal to vertical scale of the flow region isrequired to be fairly great (probably, no less than 10:1) to readilyshow the enlargement of horizontal scales of forming structures.

- The resolution on small scales must be sufficiently high to de-tect the effects produced by small-scale turbulence (first of all, thegeneration of non-zero helicity of the small-scale velocity field whichis the basic idea for the Яа-effect conception).

- The spectral flow characteristics would be obviously the mostinteresting results of numerical modeling since their analysis couldallow us to conclude whether the energy flux from small scales tolarge ones exists, how it is realized (as an inverse cascade or a "jump"#pom sxnall scales directly to the scale of optimal generation) andwhat happens to such quantities as helicity or enstrophy under theseconditions.

It is evident that the solution of the problem in this formulationis a challenging task for numerical realization, because it requiresSupercomputers and, thus, is too costly.

However, recent numerical investigations by Levina et al. [54-57]and some results first presented in this study lead to the conclusionthat a number of most interesting effects induced by helical-vortexinstability in the convective system can be examined even in the con-text of the laminar convection model using rather simple computers


of PC Pentium-type. For such modeling the calculation of one stan-dard variant of the examined problem normally took about one hour.

F

3.4.1 Numerical modeling of helical-vortex instability inlaminar convection

Consider a three-dimensional flow in a horizontal fluid layer heatedfrom below, which is described by the non-linear Boussinesq equa-tions. Let us complete the equation of motion with the term, whichphysically represents the external force generating the positive feed-back between the solenoidal components of the velocity field. Themathematical expression for this term, actually modeling the ave-raged effects caused by small-scale helical turbulence, is obtained byMoiseev et al. [29,30] (see the terms with the coefficient s in eqns(16) and (17) of this study).

Interpreting in the context of laminar convection problem theabove terms as some "model" external force, we proceed further fromthe following assumptions for the coefficients 5, fx\ and y><i, charac-terizing the small-scale helical turbulence in eqns (16),(17). Suppose,for the sake of simplicity, that /i2 = 0 (analogous reasoning was usedby Moiseev et al. [30] in the analysis of the solutions to their model),and then denote the product of Raspi by S considering it to be di-mensionless parameter, which describes the helical feedback intensity.Thus, the external force becomes the factor independent of convectionprocess.

Then, with usual vectorial notation, the system of equations forconvection can be written [54-56] as

^ + V • VV = - Vp + Д V + RaTe + S • f,eft

r + V . VT = ДГ, divV = 0, (18)at

f = e (curlV), - ^ 1 , е = {0,0,1}.

Compared with eqns (16),(17), this system, apart from <S, involvestwo new variables, pressure p and external force f responsible forhelical feedback. The square brackets denote the vector product.


The simplest mathematical formulation of the problem enablingus to study the new effects caused by the evolution of helical-vortexinstability can be obtained for the fluid flow in the cylindrical domainwith axial symmetry. In this case, all three components of the velo-city vector are maintained ensuring the non-zero helicity, whereas allphysical fields become dependent on two spatial variables only.

3.4.1.1 Numerical realization The mathematical model (18)in cylindrical coordinates {p, <p, z) transformed for numerical finitedifference calculation involves three evolutionary equations (for azi-muthal velocity v, ^-component of vorticity w^ and temperature T)and the Poisson equation for stream function Ф of vertical circulation.The full representation of the system is given by Levina et al. [55].Here, we have restricted ourselves to the relations that help us tointroduce the functions Ф and u)^

дФ 1 9 , T , du dwoz rar * oz or>

where и and w are the radial and vertical velocity components. Inthe set of three variables chosen for calculation, v characterizes thetoroidal field (horizontal circulation), and the values Ф and ш^ de-scribe the poloidal field (vertical circulation). The cylindrical layerheight h and the value /i2/# are taken as units of length and time.The Prandtl number is assumed equal to Pr = 1 throughout thecalculations.

In our calculations all the bounding surfaces of the cylinder wereassumed to be impenetrable, rigid, no-sUp. Thermal conditions in-cluded fixed temperature at the lower and upper surfaces correspon-ding to the heating from below, whereas the lateral surface was adia-batic.

The ratio of cylinder radius R to the height h for all calcula-tion variants was equal to 10:1, so that under the constraint of axialsymmetry д/д(р = 0, the computational domain represented a Rxhrectangle. In methodical calculations the discretization of compu-tational domain varied broadly, while for obtaining basic results weused the grid 100 x 20 along radius and height respectively.


valof the Rayleigh number was initiated by prescribing the point vortex,Шу Ф 0, in the center of computational domain at t — 0. The helical-vortex instability was introduced in the system by taking S Ф 0 att = 10, when the developed convective regime reached the steadystate. Thus, the helical-vortex instability was excited by coiflow.

The procedure of numerical solution is discussed in greatein the papers of Levina et al. [55,56].

3.4.1.2 Integral characteristics In order to represent the re-sults of numerical modeling, a set of characteristics have been takenwhich best illustrate the distinguishing features of the problem underconsideration.

Of principal interest in the study of the Яа-efFect is the squarevelocity value

= / V • rotVdU (19)Ju

which can be called the total flow helicity integrated over the exa-mineti computational domain. This value specifies the motion withlinked streamlines of horizontal and vertical circulation, and its signdepends on the circulation direction in the forming vortex structures.

0

Numerical modeling also allows us to trace the evolution of theother two integral characteristics: the kinetic energy Ek and the po-tential energy Ef related to temperature disturbances and therebycharacterizing the energetics of developed convective instability. The-se values are defined by the following expressions:

2dU, Ef = -Ra [ f Szdz dF, (20)

v Jo JF

where the value of temperature disturbance в is counted off from alinear temperature profile.

When calculating helicity and energetic characteristics, involvingthe definition of these values over the whole computational domain,it might also be useful to estimate their spatial distribution density.


This can be readily accomplished by decomposing the flow area intoa few sublayers. In the present study, this approach is applied forthree horizontal sublayers of equal thickness and provides us furtherinformation for analysis of the process evolution from the viewpointof its energetics. In designation the sublayers are counted off from thelower layer boundary (5ni, 25*2> Epz) and the absence of the figurein the index denotes a total value of these quantities for the wholeflow domain. In addition, the maximum and minimum of all physicalvariables are chosen from their values determined at all internal nodesof the computational grid.

3.4.1.3 Structure and energetics of large-scale helical con-vective flow In this section, the results of numerical modeling areanalyzed to determine the peculiarities of helical-convective motionsradically different from the regimes of general convection.

The numerical modeling has been carried out according to theprocedure consisting of two stages.

At the first stage, which in a sense is a test for the applied nu-merical approach, the stationary common convective flow is realizedin the absence of helical instability at S = 0. The critical Rayleighnumber numerically determined for the examined problem with theaspect ratio 10:1 is found to be equal to Rcf « 1724 and closeenough to that of the infinite horizontal layer Racr « 1708 [36]. Theevolution of convective instability at Ra = 2000 leads to the forma-tion of developed stationary convective regime (fig. la) in ten units ofdimensionless time.

In the cylindrical geometry the stationary convective motions arerealized as axially symmetrical ring-like rolls. In projection onto thecalculation domain within the plane (г,г), these motions look like thesystem of cells with vertical circulation (poloidal field) alternatingits direction from structure to structure. The number of these cellslocated along the radius appears to be nine and their horizontal scale,as expected [36,37], is comparable with the cylindrical layer height.In this case, horizontal circulation (toroidal field) is absent, becausethe azimuthal velocity component v = 0 (fig-2). Hence, under typicalconditions of natural convection no helicity generation will take place,


Ra=2000 S=0.0

a t=10

Ra*=2000 S=6.5 Ra=410 S=6.5

t=11.8 g t«10.4

h- ' •'- - J

•*:--,„- Г

t=60.0 t=40.0

Figure 1. Stream function isoHnes.

и

w

V

sn


Ra=2000 S=0 Ra=410 S=6.5и

4

2

О

-2

-4

Umax

уUmin \

•

0 2

Г

1

4

j

•

6 8 36 t

-0.8

Figure 2, Velocity field evolution : U&W - poloidal circulation,V- toroidal circulation.

Ra=2000 S=6.5 Ra=410 S=6.5

-2x10*

-4x10*

-6x10*

-8x10*

-1x10*

-2x10*

-3x10*

40 50 10

Figure 3. Helicity evolution.


i.e. SQ = 0.

At the second stage beginning from t = 10, a helical feedbackis introduced into the system by setting 5 ^ 0 . This means thatin modeling motions for S Ф 0 all physical fields of convective flow{S = 0) are employed as the initial distributions.

By changing Ra and 5, a few stationary helical convective regimeshave been obtained differing in the number of vortex cells formedwithin the computational domain. Let us consider two examples ofsuch flows.

In the first example, the flow structure formed by five helical vor-tex cells is observed at Ra = 2000 and S = 6.5 (fig.le). Here, thehelical-vortex instability is excited by developed convective motions.In the second example a single stationary helical vortex (fig. Ik) oc-cupying all computational domain is generated for Ra = 410 andS = 6.5. In this case, however, the helical-vortex instability is de-veloped on the background of damping convective motion, since att = 10 the introduction of non-zero helicity is accomplished simultarneously with tne reduction of the Rayleigh number to the subcritical

- . - L

value, Ra = 410.Therefore, merging of the small-scale convective cells observed

in the above examples is possibly the most significant effect disco-vered during the evolution of helicaHrortex instability. This is readi-ly demonstrated in fig.L The number of cells forming the structureof stationary helical^convective flow depends on the problem para-meters, i.e. on the intensity ratio of convective to helical-vortex in-stability.

The generation of the toroidal velocity field (horizontal circula-tion, non-typical for natural convection) due to helical feedback fromthe poloidal field of convective circulation can be considered as thefirst indication of the onset of additional instability in the system.Indeed, the azimuthal velocity component v responsible for the hori-zontal circulation (toroidal field) is absent at 5 = 0 and appears justafter introducing S Ф 0 at t > 10 (fig.2).

Furthermore, the streamlines of horizontal and vertical circula-tion are found to be linked because, as fig.3 shows, the generation


of non-zero total helicity SQ (19) begins simultaneously with the ap-pearance of the azimuthal velocity component. The motion becomeshelical and therefore, in addition to the integral characteristics ofcommon convection, we may follow the evolution of SQ.

The transformation of the flow structure is accompanied by anessential increase in the intensity of horizontal and vertical circula-tions. In the case of a stationary regime with a single large-scalehelical vortex cell, we can give a rather spectacular interpretation ofthis process. It can be traced from the evolution diagrams for threecomponents of the velocity field (fig.2). For comparison purposes, thecorresponding distributions for a purely convective flow at S = 0 aregiven in the left-hand part of this figure.

The change in the motion intensity begins with the growth ofazimuthal velocity v (toroidal field). Yet it takes some time for thepositive helical feedback to cause an essential increase in the velocitiesи and w responsible for a vertical circulation (poloidal field). In turn,an increase in the intensity of the latter leads to the growth of theazimuthal velocity field, thereby closing the feedback loop.

The generated structure (fig. Ik) is the large-scale helical vor-tex. The horizontal size of this vortex is approximately ten timesas much as the characteristic horizontal scale of each original cell inthe Rayleigh-Benard convection. A dynamic system of the vortexis generated by a powerful toroidal velocity field superimposed on apoloidal field, which is weaker but necessary for closing the feedbackloop. In this large-scale cell the maximum value of azimuthal veloci-ty, producing horizontal circulation, is several times greater than themaximal values of velocity components in the vertical circulation. Ifwe turn back to the very beginning of section 2.2.1, describing themain features of tropical cyclones, it becomes apparent that the struc-ture and dynamic characteristics of the simulated vortex reflect somepeculiarities of the field velocity structure in a tropical cyclone.

Let us now describe the energetics of the investigated processesin terms of kinetic and potential energy given in fig.4. At 0 < t < 10the system with natural convective flow has a single energy sourcewhich is the layer heating from below. The existence of the observed


Ra=2000S-0

Ra=2000

Ra=410S-6.5

10

5000

4000

3000

2000

«00

a

10 1 5 л

i20 25

Т ^ '

30 35

10

b

f

Figure 4. Kinetic and potential energy evolution.


stationary convective regime corresponds to the energy balance in thesystem: the energy supplied into the system by heating is transformedto the kinetic energy of convective motions, and the remaining partscatters due to dissipation (viscosity, boundary friction). The dis-tribution of potential energy tjrpical for natural convection is shownin fig.4b. The hottest fluid compared with a linear temperature dis-tribution is located in the upper sublayer, which is characterized bythe largest of negative potential energy values, (Eps < 0), while thecoldest (in the same sense) fluid is in the lower sublayer described bya positive value of potential energy, (Epx> 0).

The situation is changed when the helical feedback S Ф 0 is intro-duced into the system at t = 10. There appears an additional energysource, which models the averaged influence of the small-scale helicalturbulence. Its energy is spent partly to increase the potential energyEp through intensifying the poloidal velocity field related to the tem-perature disturbances. However, the greater part of the energy fromthis additional source is contributed to the kinetic flow energy. Forexample, as it is shown in fig.4a,c,e, the energy increases approxi-mately by twenty times for the first helical regime (from fig.4a tofig.4c) and approximately by a hundred times for the second (fromfig.4a to fig.4e). Particular emphasis should be placed on the factthat, at the same helicity value, S = 6.5, the kinetic energy forRa = 410 is five times as large as that for Да = 2000. Evidently,in the system with two feedbacks (convective and helical), the flowstructure in the form of a singly large-scale helical vortex (Ra = 410),is optimal from the energetics point of view. At Ra = 2000, part ofthe "helical" source energy must be expended for breaking down thestructure with developed natural convective motion (the bends of in-tegral chatacteristics curves just indicate the moments of the mergingof cells - see, respectively, the isoline patterns in figure 1).

The distribution of the potential energy demonstrates sharp quali-tative changes, which are most evident at Ra = 410. The potentialenergy of this helical regime proves to be positive in all sublayers.These changes are likely to be the result of new effects initiated byhelical feedback. One of these effects, manifesting itself at the very


beginning of the helical-vortex instability evolution, is the develop-ment of the azimuthal flow, generating the cells merging and supres-sing heat transfer through the layer by natural convection realized asa multitude of small circulation cells with up- and downward flows,and as a result decreasing energy losses due to dissipation. In thiscase, the heat can be transferred more effectively by a strongly in-tensive forced convective flow in the form of single large-scale helicalvortex. Consequently, on the one hand the generated toroidal field ofazimuthal circulation supresses natural convective motions, and onthe other hand it exerts the suction effect contributing thus to theformation of vortex cell.

Therefore, numerical modeling of helical-vortex instability in con-vective systems allows us to discover a number of new effects initiatedby helical feedback:

- generation of a toroidal velocity field in common convectivecells with poloidal circulation and, as a result, formation of a helicalflow structure with the linked streamlines of horizontal and verticalcirculation;

- merging of small-scale convective cells during formation of thelarge-scale flow structure;

- helical feedback action strongly intensifying both horizontal andvertical circulation;

- sharp increase in the kinetic energy of the vortex motion andqualitative changes in the potential energy distribution.

The results obtained can be considered as one more argumentin favor of the hydrodyn&mic alpha-effect hypothesis. However, theverification of this concept iftffl. certainly require numerical modelingof turbulent convection in the field of the Coriolis force. Such numeri-cal experiment, simulating the direct interaction between motions ofdifferent scales, is expected to give answers to key questions: whatis the scenario of small-scale helical turbulence generation, and whatis the role of small-scale helical motions in formation of large-scalevortex structures and channels for energy transfer to large scales.

Acknowledgement The authors are grateful to Professor K.-H.Radler for useful and important discussions during the 12th Winter


School on Mechanics in Perm in January of 1999. We are deeplythankful to L.V. Semoukhina and G.M. Martemyanova for their in-valuable help during preparation of the English text. We thank ouryoung colleagues - post-graduate students M.V. Starkov and A.V.Firulyov for participation in numerical modeling and shaping themanuscript. G.L. thanks Professor P.G. Frick for his interest andSupport in her investigations on large-scale structure generation un-der turbulent convection and Professor R.M. Kerr for the golden op-portunity to participate at the IUTAM/IUGG Symposium on De-velopments in Geophysical Turbulence at the National Center forAtmospheric Research in Boulder in June of 1998 when the results ofnumerical modeling were first presented.

F

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Hydrodynamic Alpha-Effect in a Convective System