1(:6/(77(5May 2011

President’s Address

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EURECH Fuid echaics Prie 2008 Twdiesia fuid tis d t fte crrespd t physica reaity I particuar i a spherica she twdiesia tis with a radia cpet are preveted by the Uusua Prperties f Cvecti ad Dyas i Rtatig Spherica Shes budary cditis Fuid fws are thus frced t bece tie depedet I the sipest cases the tis assue the fr f eary twdiesia prpagatig Rssby waves The physica Friedrich Busse won the EUROMECH Mechanics Prize awarded at the 7 th European echais f Rssby waves ca be uderstd the basis f the cservati f aguar Conference, in Manchester. etu Fig 1 shws a auar fuid ayer with decreasig depth fr icreasig distace fr the axis Whe a cu f fuid (aiged with the axis f rtati) ves it a shawer F H Busse 1 pace it beces cpressed ad because f the cservati f ass its et f iertia Cvecti i rtatig spherica fuid shes is a basic dyaica prcess i the descripti icreases T cserve aguar etu its rtati reative t a iertia frae f referece f heat trasprts i paets ad i stars Sice the respective fuids are fte eectricay ust decrease Reative t the rtatig syste it thus acquires aticycic vrticity The ppsite cductig the ccurrece f cvecti is frequety assciated with the geerati f agetic prcess happes whe the cu ves t a deeper pace where it gets stretched i the directi fieds (dya prcess) The iteracti betwee vecity ad agetic fieds gives rise t rich f the axis f rtati ad acquires cycic vrticity dyaica structures such as reaxati sciatis bistabe turbuet dyas ad hysteresis I the wer setch f Fig 1 we at the equatria pae f the auar ayer A pheea siusida dispaceet f the iitiay static fuid cus eads t a fw structure i the fr f cycic ad aticycic vrtices which ted t ve the cus t ew psitis as PACS ubers 4732Ef 4755pb 9260h 9530 9612Hg idicated by the dashed ie i the wer pt f the figure ie the iitia siusida dispaceet prpagates as a wave i the prgrade directi A retrgrade prpagati reative t the sese f rtati wi be btaied whe the depth f the auar ayer icreases with distace fr the Itrducti axis

Iterest i the prbe f cvecti i rtatig spheres has bee stiuated by its astrphysica appicatis ad by the search fr the rigi f geagetis i the Earth’s iquid uter cre The exprati f the sar syste ad the discvery f the agetic fieds f ther paets has apified this iterest ad i recet decades a arge uber f papers deaig with cvecti i fuid shes ad its dya acti have appeared Istead f reviewig a f this wr we ust wish t draw atteti t a uber f surprisig features that have appeared i the curse f this wr ad which are f re geera iterest i the fied f fuid dyaics I rder t ae this artice re readiy accessibe t a geera readership the use f atheatica equatis ad fruas is iiied Fr re cpete itrductis t the subect we refer t the b 1 ad the review artice 10

Thera Rssby waves A basic there f the dyaics f rtatig fuids is the PrudaTayr there which states that steady sa apitude tis f a bartrpic rtatig fuid d t vary i the directi f the axis f rtati whe viscus effects ca be egected Sa apitude eas i this cecti that the vrticity f the ti is egigibe i cparis t the rtati rate f the syste The PrudaTayr cditi is a csequece f the cpete baace betwee the Criis frce ad the gradiet This baace is as caed gestrphic baace sice it hds i gd apprxiati fr the arge scae tis i the Earth’s atsphere Fig 1 Tw setches iustratig the dyaics f Rssby waves i a auar fuid ayer with decreasig depth fr icreasig distace fr the axis

1 Institute of Physics, University of Bayreuth, D-95440 Bayreuth, Germany, [email protected]

14 15

4 5 Fig 3 Cvecti cus i a rtatig spherica fuid she fr τ 10 R 38 ×10 P 1 Dar ad ight surfaces Fig 2 Geetrica cfigurati f the rtatig auus i which cvecti is drive by cetrifuga buyacy crrespd t a cstat psitive ad egative vaue f the radia vecity

Like water waves, Rossby waves decay when there is frce sustaiig the agaist viscus the Rayleigh number R, the Coriolis number τ and the Prandtl number P, dissipati The pssibiity for sustainment exists i a theray ustaby stratified syste where αγβd 6 2d 2 ν thera Rssby waves ay be geerated Grwig disturbaces are btaied i a auar R = τ = P = (1) cfigurati as setched i Fig 2 whe a teperature differece T2 T 1 ad a gravity frce are νκ ν κ appied i the xdirecti I the experieta reaiati f the prbe 2 the cetrifuga frce r0 pays the re f gravity ad the teperature gradiet ust pit utward i rder t create appear as diesiess paraeters Here ν detes the ieatic viscsity f the fuid κ its the ustabe desity stratificati Fr gephysica appicatis the ppsite directis f gravity thera diffusivity ad α is its cefficiet f thera expasi Sice the Bussiesq ad teperature gradiet are re apprpriate but the atheatica prbe reais the sae apprxiati is assued the vecity fied u is seida ad the geera represetati i i bth cases ters f pida ad trida cpets ca be used The auus cfigurati f Fig 2 ay be regarded as a auar secti f a rtatig spherica she Ideed the thery deveped fr the rtatig auus ca be appied i gd u=∇×∇( vr × ) +∇ wr × (2) apprxiati t the case f cvecti i a sefgravitatig rtatig fuid sphere heated fr withi 3 As a stadard de a spherica fuid she f thicess d rtatig with a cstat 2 2 2 such that the radia vecity depeds v ae ad is give by ur = Lvr2 / ≡∂ rv / ∂−∇ r rv A aguar vecity is fte assued fr which a static state exists whe the teperature depeds typica picture f a thera Rssby wave i a rtatig spherica fuid she is shw i Fig 3 The y the distace rd fr the cetre Here we sha assue the particuar teperature ratio of the radius of the inner spherical boundary to the radius of the outer spherical surface is 2 2 distributi TS T 0 βd r 2 which crrespds t a hgeeusy heated sphere. The gravity taken to be 0.4 for all the results presented in this paper. Because f the syetry f the fied is assued t have the fr g= − dγ r Usig the egth d the tie d2ν ad the prgradey prpagatig vecity fied with respect t the equatria pae it is sufficiet t pt teperature ν γα2/ d 4 dimensionless equations can be formulated in which streaies i this pae give by r∂ v/ ∂ r = const . i rder t characterie the cvecti fw as has bee de i Figs 4 5 ad 7 Eve i the case f turbuet cvecti the part f the vecity fied that is atisyetric with respect t the equatria pae is rather sa As the Rayeigh uber R icreases beyd its critica vaue Rc the set f cvecti i the fr f thera Rssby waves is fwed by a sequece f bifurcatis siiar t thse fud i other convection problems First typica sciatis i apitude are bserved as shw i Fig 4 the ather bifurcati ay add w waveuber duatis i the aiutha directi as shw i Fig 5 Fiay a chatic state f cvecti is btaied

16 17

Chatic cvecti i rtatig spherica shes

The sequece f trasitis is as evidet i the tie depedece f average quatities such as the ctributis t the ietic eergy desity These are defied by

1 2 1 2 E= ∇×∇( v × r ) E= ∇ w × r (3a) p 2 t 2 ⌣ ⌣ 1 ⌣ 2 1 ⌣ 2 Ep = ∇×∇( v × r ) Et = ∇ w × r (3b) 2 2

5 4 where the aguar bracets idicate⌣ the average ver the fuid she v is the aiuthay Fig 5 duated shape vaciatis f cvecti fr R 29 ×10 τ 10 P 1 The pts shw streaies averaged cpet f v ad v= v − v At w supercritica Rayeigh ubers these eergy ∂v r= cst i the equatria pae ad are equidistat i tie with t 004 s that apprxiatey a fu perid is desities apprach cstat vaues crrespdig t steadiy driftig thera Rssby waves as ∂φ shw i Fig 6 cvered

The set f vaciatis aifests itsef i the siusida sciatis f the ietic E eergies t describes the eergy desity f the differetia rtati which icreases st strgy with icreasig R as ca be ticed i Fig 6 This icrease is caused by the strg aiutha Reyds exerted by the cvecti eddies resutig fr their iciati with respect t radia directi as is apparet i Figs 3 thrugh 5 The shear f the differetia rtati hwever teds t ihibit cvecti i that it shears ff the cvecti eddies This is a csequece f the eary twdiesia ature f the dyaics i a rtatig syste I a rtatig syste the cvecti rs wud sipy aig theseves with the directi f the shear ad the heat trasprt wud thus reai uchaged I the rtatig sphere a precarius baace is reaied i the fr caied cvecti As shw i Fig 7 cvecti ccurs y i a restricted aiutha secti f the spherica she where the cvecti apitude is strg eugh t verce the ihibitig ifuece f the shear The axisyetric differetia rtati ctiues t be drive by the caied cvecti Fr the gestrphic a fw it des t atter whether it is drive cay r i a axisyetric fashi The advecti by the differetia rtati f the thera budary ayers which expad i the cvectig regi f Fig 4 Tie peridic vaciatis f cvecti at 5 (eft side) ad 5 (right side) fr 4 R 28 ×10 R 3 ×10 τ 10 P the she actuay stregthes the caied cvecti i that its avaiabe buyacy is r∂ v ∂ϕ = cst 1 The streaies are shw i e quarter f the equatria pae Psitive (egative) vaues are repeished idicated by sid (dashed) ies The fur quarters are equidistat i tie with t 0015 ( t 0024) i the eft (right) Istead f a caiati i space the caiati f cvecti i tie ffers ather case i the ccwise sese such that apprxiatey a fu perid is cvered by the circes pssibiity fr the precarius baace as destrated i Figs 8 ad 9 Here cvecti exists y fr a shrt perid whie the differetia rtati is sufficiety wea As the apitude f cvecti grws the differetia rtati grws eve re strgy as the Reyds stress icreases with the square f the apitude S the shearig acti beces strg eugh t cut ff cvecti w a viscus diffusi tie ust pass befre the differetia rtati has decayed sufficiety for convection to start growing again It is rearabe t see hw the chatic syste exhibits the nearly periodic relaxation oscillations shown in Fig. 8 Fig. 9 shows a sequence of plots for four istaces i tie arud a cvecti pea At first there is hardy ay cvecti the gree ies ust idicate er At the ext istace the

18 19

differetia rtati as shw by the upper rw has decayed sufficiently for the convection variatis f the teperature fied with atitude cause a differetia rtati i the fr f a columns to grow, reaching nearly their maximum amplitude in the third plot. thera wid At very w vaues f P cvecti assues the fr f iertia sciatis [5, 6, 7, 9] which d t exert sigificat Reyds stresses 8 The cvective heat trasprt i the case f caied cvecti as we as i the case f the reaxati sciatis is uch reduced reative t a case withut strg differetia rtati Here the agetic fied perates i a iprtat way By puttig braes the differetia rtati thrugh its ret frce the agetic fied perits a uch higher heat trasprt tha wud be pssibe i a eectricayisuatig fuid This is the basic reas that rapidy rtatig stars ad paets with cvectig cres exhibit agetic fieds A destrati f this effect is see

4 5 5 5 Fig 6 Tie series f eergy desities f cvecti fr P 05 τ 15 ×10 ad R ⌣3 × 10 ⌣ 32 × 10 345 × 10 5 5 6 E E E 5 4 5 × 10 7 × 10 10 (fr btt t tp) Red bue ad gree ies idicate t t ad p respectivey The critica Fig 7 caied cvecti fr R 7 × 10 τ 15 × 10 P 05 The streaies r∂ v ∂ϕ = cst (upper rw) ad Rayeigh uber fr set is Rc 215142 the isthers Θ cstat (wer rw) are shw i the equatria pae fr equidistat ties (fr eft t right) with t 003 The differential rotation has grown at the same time and begins to exert its inhibiting effect. Thus, convection is reduced at the furth istace f the sequece whie the differetia rtati reaches its axiu It shud be etied that caied cvecti ad reaxati sciatis ccur at derate Pradt ubers f the rder uity r ess At higher vaues f P the Reyds stresses are ger pwerfu eugh t geerate a strg differetia rtati Istead

20 21

Fig 8 Reaxati sciatis f chatic cvecti i the case 4 5 The eergy desities ⌣ ⌣ τ 10 R 65 ∙ 10 P 05 E Et (red ie) Et (bue ie) p (gree ie) ad the usset uber (bac right rdiate) are shw as fuctis f tie

Fig 10 Trasiti fr a dya state t a state f chatic reaxati sciatis fr 4 5 i Fig 10 where by chace the cvecti drive dya was ust argia such that it cud t ⌣ ⌣ τ 3 ×10 R 29 ×10 P 1 E recver after a dwward fuctuati f the agetic fied Hece the reaxati sciatis with P 04 The eergy desities Et (red ie) Et (bue ie) p (gree ie) the tta agetic eergy desity utipied their uch reduced average heat fux tae ver fr the dya state by a factr 8 (sid bac ie) ad the usset ube r u (dashed ie right rdiate) are shw as fuctis f tie

Tw distict turbuet dyas at idetica paraeter vaues Cvecti drive dyas i rtatig spherica fuid shes are fte subcritica as aready idicated i Fig 10. At slightly higher Rayleigh number, convection with a strg agetic fied persists whie the dya wi decay whe the agetic fied is artificiay reduced t say a quarter f its averaged eergy There thus exists the pssibiity f a cvecti drive dya state ad f a agetic cvecti state at idetica vaues f the extera paraeters R τ P P

Fig 9 Sequece f pts startig at t 231143 ad equidistat i tie (t 001) fr the sae case as i Fig 8 ies

f cstat uϕ ad ea teperature perturbati Θ cst i the eridia pae are shw i the eft ad right haves respectivey f the upper rw The wer rw shws crrespdig streaies r∂ v ∂ϕ = cst i the equatria pae

22 23

1 2 1 2 = ∇×∇( h × r ) = ∇ g × r (5a) p 2 t2 t ⌣ ⌣ ⌣ 1 2 1 ⌣ 2 = ∇×∇( h × r ) = ∇ g × r (5b) p2 p t2 t

The tw turbuet dya states differ strgy i their agetic eergies as we as i their ietic eergies as is apparet fr Fig 11 Whie a strg ea pida agetic fied as shw i the eft haf f Fig 11 acts as a efficiet brae the differetia rtati as easured

by Et it as ihibits cvecti The aterative dya the right side f the figure is characteried by a reativey wea ea agetic fied ad diat fuctuatig cpets Here the ietic eergy desities f cvecti are arger but the differetia rtati is sti uch weaer tha i the agetic case The bistabiity i the fr f tw differet types f dyas is t a siguar phee but exists ver a exteded regi f the paraeter space as destrated i Fig 12 A basic reas fr the cpetitiveess f bth dyas is that they exhibit eary the sae cvective heat trasprt as easured by the usset uber ui which detes the rati f the

Fig 11 Tie series f tw differet chatic attractrs are shw a D (eft cu (ab)) ad a FD dya (right cu (cd)) bth i the case R 35 × 10 6 τ 3 × 10 4 P 075 ad P 075 T he tp tw paes (ac) shw agetic eergy desities ad the btt tw paes (bd) shw ietic⌣ eergy desities⌣ i the presece f the agetic fied The cpets p are shw by thic sid ies whie Xt X p ad Xt are idicated by squares triages ad circes respectivey X stads fr either r E

Here the atter paraeter detes the agetic Pradt uber which is defied as the rati betwee ieatic viscsity ad agetic diffusivity P ν/λ The agetic diffusivity itsef is defied as the iverse f the eectrica cductivity ties the agetic pereabiity λ σ = −1 − 1 The bistabe cexistece f a agetic cvecti state ad a dya state is typica fr a Fig 12 The upper rw shws the hysteresis effect i the rati f agetic t ietic eergy E at τ 3 × 10 4 ( a) as a subcritica bifurcati as it is as fud i the case f shear fw istabiity fucti f the Pradt uber i the case f R 35 × 10 6 PP 05 (b) as a fucti f the rati PP i the case re surprisig is the fact that tw differet turbuet dya states ca exist at idetica f R 35 × 10 6 P 075 ad (c) as a fucti f the Rayeigh uber i the case P 075 P 15 Fu ad vaues f the extera paraeters as has bee shw i 11 ad is destrated by the tw epty circes idicate FD ad D dyas respectivey The critica vaue f R fr the set f thera cvecti fr exapes shw i Fig 11 Here the represetati f the agetic fux desity B i ters f the cases shw i (c) is Rc 659145 A trasiti fr FD t D dyas as PP decreases i (b) is expected but pida ad trida cpets B=∇×∇( hr × ) +∇ gr × (4) has bee used ad agetic is t idicated wig t ac f data The wer rw shws the vaue ui f the usset uber at r ri fr the sae dya cases Vaues fr agetic cvecti are idicated by a fied square fr cparis eergy desities have bee defied i aagy t expressis (3)

ea teperature gradiets at the ier budary f the she i the presece ad i the absece f cvecti The wer haf f Fig 12 t y destrates the surprisig cicidece f the heat trasprts f the tw dya types but as shws that these heat trasprts exceed by far that fud i the absece f a agetic fied

24 25

Ccudig rears 10 F H Busse Cvective fws i rapidy rtatig spheres ad their dya acti Phys Fuids 14 13011314 (2002) The existece f tw distict turbuet states is a rare phee athugh exapes exist i agetic hydrdyaics see fr istace 12 13 I agethydrdyaics the 11 R D Siitev ad F H Busse Bistabiity ad hysteresis f dipar dyas geerated agetic fied ffers ew degrees f freed which aw the cexistece f re tha e by turbuet cvecti i rtatig spherica shes Eur Phys ett 85 (2009) 19001 turbuet state Iitia cditis thus deterie which f the cpetig states is actuay reaied 12 F Raveet are A Chiffaude ad F Daviaud utistabiity ad ery Effect The pssibiity f bistabiity cud be f iterest fr the iterpretati f paetary ad i a Highy Turbuet Fw Experieta Evidece fr a Gba bifurcati Phys Rev stear agetis agetic hysteresis effects assciated with stear sciatis cud evetuay ett 93 164501 (2004) be expaied i this way Ceagues ivved i uerica siuatis f cvecti drive dyas shud be reided that their sutis ay deped quite strgy the iitia [13] uica ad D P athrp Bistabiity ad hystreresis i a highy turbuet swirig cditis used fw Physica A49 162166 (2005)

Acwedgets The hep f Dr R Siitev i creatig the figures ad usefu cets f Prf W Pesch are gratefuy acwedged

Refereces 1 E Dry A Sward eds atheatica Aspects f atura Dyas CRC Press Tayr & Fracis Grup 2007

2 F H Busse C R Carriga abratry siuati f thera cvecti i rtatig paets ad stars SCIECE 191 8183 (1976)

3 F H Busse Thera istabiities i rapidy rtatig systes Fuid ech 44 441460 (1970)

4 F H Busse Asypttic thery f cvecti i a rtatig cyidrica auus Fuid ech 173 545556 (1986)

5 hag cupig betwee the Picar equati ad the heat equati Fuid ech 268 211229 (1994)

6 hag ad F H Busse the set f cvecti i a rtatig spherica shes Gephys Astrphys Fuid Dy 39 119147 (1987)

7 Ardes F H Busse ad Wicht Thera cvecti i rtatig spherica shes Phys Earth Paet Iter 99 5567 (1997)

[8] H. P. Greenspan, On the nonlinear interaction of inertial modes. J. Fluid Mech., 36, 257- 264 (1969)

9 R D Siitev ad F H Busse Patters f cvecti i rtatig spherica shes ew Phys 5 9719720 (2003)

26 27

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“8th EUROMECH Fluid Mechanics Conference (EFMC8)” EUROMECH Colloquia in 2011-2012

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