MHD Kelvin-Helmholtz instability in non-hydrostatic equilibrium

Transcription

1 Journal of Physcs: Conference Seres MHD Kelvn-Helmholt nstablty n non-hyrostatc equlbrum To cte ths artcle: Y Laghouat et al 7 J. Phys.: Conf. Ser. 64 Ve the artcle onlne for upates an enhancements. Ths content as onloae from IP aress on 9//8 at 9:4

2 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// MHD Kelvn-Helmholt nstablty n non-hyrostatc equlbrum Y Laghouat, Bouaballah, M Z an lemany 3. Laboratore LMOSI, Department of Physcs, nversty of Scences an Technology of Oran, PB 55 El Mnaouar, 3 Oran, lgera. Laboratory of Thermoynamcs an Energetcal Systems, Faculty of Physcs, STHB.PB 3 El la, Bab Eouar, 6 lgers, lgera. 3 Laboratory LEGI, Pamr team-bp 53, 38 4 Grenoble Ceex 9, France. bstract. The present or eals th the lnear stablty of a magnetohyroynamc shear flo so that a stratfe nvsc flu rotatng about a vertcal axs hen a unform magnetc fel s apple n the recton of the streamng or onal flo. In geophyscal flo, the stablty of the flo s etermne by tang nto account the nonhyrostatc conton epenng on Rcharson number R an the evaton from hyrostatc equlbrum. ccorng to Stone [], t s shon that such evaton ecreases the groth rates of three ns of nstablty hch can appear as geostrophc (G), symmetrc (S) an Kelvn- Helmholt (K-H) nstabltes. To be specfc, the evoluton of the flo s therefore consere n the lght of the nfluence of magnetc fel, partcularly, on K-H nstablty. The results of ths stuy are presente by the lnear stablty of a magnetohyroynamc, th horontal free-shear flo of stratfe flu, subject to rotaton about the vertcal axs an unform magnetc fel n the onal recton. Results are scusse an compare to prevous ors as Chanrasehar [] an Stone [].. Introucton Large-scale geophyscal flos n planetary atmosphere are often omnate by horontal basc shear flo n consequence of the nfluence of rotaton, stratfcaton an magnetc fel. In many cases the varaton of velocty plays an essental role n the ynamcs an hch may gve rse to the nstablty. Frst Chanrasehar [] has establshe the equlbrum crtera th the a of lnear theory for the stratfe non rotatng heterogeneous flu hen fferent layers are n relatve moton. It seems that the magnetc fel parallel to the streamng has a stable effect. mong the mportant ors relate to rotaton of geophyscal flu, Stone [] has gven accurate results for non geostrophc nstablty such as symmetrc nstablty, an short aves smlar as Kelvn-Helmholt nstablty n non hyrostatc contons. nother mportant ors of baroclnc nstabltes are vee by Perrehumbert et al [4] an more recently by Plougonven et al [5] have stue the moel of Eay, uner the conton of the crtcal layer. Our concern s to solve the lnear stablty problem of an unrectonal shear flo uner rotatng system. The unform magnetc fel s assume to be parallel to the streamng as n the geophyscal case so as to the unform magnetc fel combne to the rotaton of the system are expecte to have compettve effects on the stablty. We choose a hyperbolc-tangent profle as use by Dran [3], hch represents a general shear flo. s a consequence e search to unerstan the evelopment of c 7 Lt

3 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// the K-H nstablty unergong uner the conton of non hyrostatc equlbrum an th the nfluence of onal magnetc fel.. Formulaton of the problem. General equatons The MHD equatons n rotatng system for a stratfe nvsc ncompressble Boussnesq flu are expresse n tme t an rectangular coornates ( x, y, ), respectvely onal, meronal an vertcal rectons. Contnuty equatons V., () V. () V μ (3) 4π Momentum equaton, ( V. ). V ( H. ). H Ω V Π H Inucton equaton, ( V. ). H ( H. ). V λ H m (4) Contnuty of the magnetc fel,. H (5), the angular velocty, H, the magnetc fel, s the ensty of the flu, a reference ensty, μ the enotes the flu velocty, Ω ( Ω x Ω y, Ω ) ( H, H H ) Where V (, V, W ) x y magnetc permeablty of the vacuum, λ m the magnetc ffuson coeffcent an g the gravty. P μh 8π ( Ω r) Π, represents the generale pressure, here P s the statc pressure. r ( x, y, ), s the poston vector an,,, the fferental operator. y The full set of equatons () to (4) must satsfy bounary layers that t shoul be ncate latter.. Hypothess Conser a basc state of an unboune ncompressble nvsc geophyscal flu, th an ntal (,,) shear flo ( ) V n the x recton an a varable ensty (fgure ). The hole system s subject to a constant angular velocty aroun axs, Ω (,, Ω) fel (,) H. H, an a unform apple magnetc

4 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// Fgure. Schematc representaton of the rotatng shear flo. Tanh, here an are respectvely the characterstc velocty an epth of the shear layer. The stratfe The velocty satsfy to a hyperbolc-tangent representaton of the form y, ambent buoyancy s n thermal n balance, e have, ΩD β g y, here D enotes the ervatve symbol D. The contnuty of the magnetc fel s mplctly nclue n the nucton equatons hch assume a very small magnetc ffuson coeffcent ( λ m ) so that the nfluence of the Joule effect s neglecte. When the Kelvn-Helmhol nstablty appears, the hole flo conssts n a superposton of the basc flo an the assocate perturbatons of velocty, pressure, ensty an magnetc fel..3 Basc flo From the general equatons () to (4) an by conserng the above hypothess, the ntal basc state s gven by the mean velocty fel Tanh, an the mean total pressure Π μh Ω βg Ω y x y C. 8π Where, C s a constant of ntegraton..4 Equatons of the perturbaton On the bass of prevous hypothess an neglectng the quaratc terms of perturbaton the system of equatons ()-(4) s reuce to, 3

5 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// ~ ' ~ ' Contnuty equatons v y u~ v~ ~ y (6) (7) Momentum equatons u~ u~ ~ p ~ Ωv~ (8) v~ v~ ~ μh ~ hy h x ~ p Ωu~ (9) 4π x y y ~ ~ ~ ~ μh ~ h h x p g ~ ' 4π () x ' Where ~ an p% enote respectvely, the perturbatons of both ensty an pressure. Magnetc fel equatons, ~ ~ hx hx u~ ~ H h () ~ ~ h y h y v~ H () h % h % H % t x x (3) The assocate bounary contons are % as ±. (4) 3. Stablty analyss nalyng the sturbance nto normal moes as Chanrasehar [], e see perturbaton solutons hose epenence on x, y an t s of the form, ( u~, v ~, ~ p h ~, ~,, ~ ) ( u( ), v( ), ( ), p( ), h( ), ( ) ) Exp[ ( x λy σt) ], here, λ an σ are respectvely the ave numbers n x an y rectons an the groth rate. Substtutng these expressons nto equatons (6) to (4), e obtan the follong set of equatons, ' ΩλD ( σ ) ( D) v g (5) ( u v) D λ (6) ( σ ) u ( D ) Ωv p (7) 4

6 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// μh ( σ ) v ( h λh ) Ωu λp y x 4π (8) μh ' ( σ ) ( h Dh ) Dp g x (9) 4π h x H D u σ σ H h y v σ () () H h σ () By elmnatng u, v an p from the system (5-), e are leang to a unque fferental equaton, expresse only n functon of D 4Ω ( c ) 4Ω c D 4Ωλ D D ( c ) D ( c ) ( c ) (( c ) ) ( c ) ( c ) (( c ) ) D 4Ωλ D ( D ) D K ( c ) D ( D ) (( c ) ) ( c ) 4 c c D Ω D K D K Ωλ ( D) D g D 3 ( c ) ( c ) D (3) Where K²²λ², σ μh c an. 4π For the mensonless values, e choose the follong scales, for the velocty, an for the vertcal heght, an the ave numbers. Retanng margnal stablty ( ) c an conserng λ, the equaton (3) becomes, D β only for the buoyancy n the case of the 5

7 3 D R D D D D (4) Where ² ² g R β enotes the Rcharson number, 4π μ H the lfven number an R Ω Γ s the evaton from hyrostatc equlbrum. L R o Ω an L Γ are respectvely the Rossby number an the aspect rato. By replacng the varable by the epenent varable n (4) e have, 4 4 R D D (5), here D s a smple fferental operator. The ne bounary contons become, as ±. (6) 4. Results an scusson 4. The nfluence of magnetc fel on K-H nstablty 4.. Metho of resoluton. The fferental equaton (5) s solve by analytcal metho usng the Rt-Raylegh approach, tang nto account the assocate bounary contons (6). Collat [6] has escrbe ths varatonal proceure. The soluton s approache by a fnte seres of so that N b α α, here the unnon coeffcents b are etermne by performng ther functonal J, hch can be escrbe as the energy nvolve n the phenomenon n the case of non-rotatonal shear flo ( ). By efnton e have Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// 6

8 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// 6 4 ( ) ( ( ) R ) ² J In hch the functon 4 J s epenng on unnon constants b. J Therefore by mposng the mnmng contons on functonal J so as to, e can solve b the homogeneous system n b coeffcents. ccorng to the optmaton metho, e etermne α an α, an e reach at the convergent soluton for the orer N3. The Rcharson number R s then etermne as an egenvalue problem by settng the etermnant of the matrx coeffcents equal ero. 4.. Stablty crteron. Better results coul be obtane, of course, by usng more terms n the expanson, but the result for N3 s actually accurate for the frst egenvalue. For example, as shon n the fgure, n the absence of magnetc fel ( ), R. 5an. 7. It can be seen that the agreement beteen our results an those obtane by Dran [3], consttutes a goo approxmaton. (7) Fgure. Margnal stablty curve for the stanar case ( c ), n the absence of magnetc fel ( ). The Rcharson number R as evaluate as a functon of the mensonless onal ave number. In fgure 3, there are represente curves of margnal state ( c ) for fferent values of lfven number. For all parameter examne, t as foun that thn the nterval of,. 5 there exsts R an ncatng nstablty. n nterestng feature s the pea of the curves, that means R has a maxmum value for c c. s mentone by Dran [3] an Chanrasehar [], the maxmum value of R etermnes the ave number c of the evelopng nstablty. It s orth notng that hen R tens to ero, for c. 37 e can evaluate max. 6. The analyss reveals that for a mnmum value of c ( max ) close to c corresponng to, the moton exhbt a lmt or some threshol value. Beyon the onset of the nstablty of Kelvn- Helmholt the latter becomes non-senstve hatever the nfluence of ncreasng magnetc effect. s 7 c

9 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// the result of max. 5, the magnetc energy s close to half of the actual nertal energy an therefore the acton of magnetc fel s physcally lmte. Fgure 3. Evoluton of the margnal curves labele for fferent values of the lfven number. 4. The nfluence of rotaton on K-H nstablty 4.. Equaton of stablty (baroclnc problem). Substtutng n the equaton (3) the ne scale the onal ave number nstea of, e obtan the mensonless onal ave number. If e conser a small scale, as Stone [], e get a mensonless equaton, D [ ] ( ε R ) D D D ε D (8) Margnal stablty crtera. We notce that the term ε n the equaton (8) s smlar to n the equaton (4). The same results for the non-rotatonal system th respect to are val for the rotatonal case (K-H Baroclnc nstablty) th respect toε. The fgure 4 summares the results of the behavor of the Rcharson number R versus the parameter (non-hyrostatc contons), for fferent values of at gven. For ths flo system satsfyng a hyperbolc-tangent shear flo th nflexon pont, the results are n agreement th those precte by Stone [] for a lnear shear layer, the non-hyrostatc effect s sgnfcant for Kelvn-Helmholt nstablty, for such a small. It s mportant to notce that the curves ( ) ecrease more raply. Ω 8

10 Secon Internatonal Symposum on Instablty an Bfurcatons n Flu Dynamcs Journal of Physcs: Conference Seres 64 (7) o:.88/ /64// Fgure 4. Margnal stablty agram for fferent lfven number. In non-hyrostatc contons 5. Conclung remars Base on the lnear theory, e have extenen the Stone s moel accountng for a unform magnetc fel apple to velocty recton. ner the hypothess of non hyrostatc equlbrum of the flo an usng the hyperbolc profle as Dran, e erve the lnear stablty equaton for the vertcal perturbaton featurng K-H nstablty. By means of Rt-Raylegh metho e are leang to an egenvalue problem n orer to fn out the Rcharson number R expresse n functon of control parameters of the flo an, an the characterstc values of the nstablty as the onal ave number an the groth rate σ. Therefore for a gven ave number, e have establshe a agramm of K-H nstablty for the margnal state ( σ ), n such ay that the Rcharson number R s functon of the mofe Rossby number for fferentes values of lfven number. s a consequence, a sgnfcant property s that the stablty agram reveals that a maxmum value of the Rcharson number R max ecrease as ncrease an threfore e euce that the contrbuton of magnetc fel may able to reuce the mass transfer process, for example, n usty or pollute atmosphere. In general, e suggest applyng ths moel to specfc problems as astrophyscal stuatons, rangng from the nteracton of the solar n th the magnetospherc bounary, to the ynamcs of accreton ss an young stellar objects. References [] Stone P H 97 J. Flu. Mech [] Chanrasehar S 96 Hyroynamc an hyromagnetc stablty (Oxfor: Clarenon Press) chapter pp [3] Dran P G an Re W H 98 Hyroynamc stablty (Cambrge: nversty Press) chapter 6 pp [4] Perrehumbert R T an Sanson K L 995 Baroclnc nstablty. nnul.rev.flu Mech 7 49 [5] Plougonven R, Mura D J an Snyer C 5 J.tmos. Sc [6] Collat L 96 The numercal treatment of fferental equatons (Berln: Sprnger) 9