Proc. Indian Acad. Sci. (Math. Sci.), Vol. 103, No. 2, August 1993, pp. 197-207. �9 Printed in India.

Banerjee e t a/'s characterization theorem in thermohaline convection and its magnetorotatory extensions

JYOTI PRAKASH, AJAIB SINGH and SAVITA SHARMA Department of Mathematics, Himachal Pradesh University, Summer Hill, Shimla 171005, India

MS received 10 November 1992

Abstract. The paper mathematically establishes that magnetorotatory thermohaline convection of the Veronis 17] type cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the thermohaline Rayleigh number R . the Lewis number z, the thermal Prandtl number r the magnetic Prandtl number Ct, the Chandrasekhar number Q and the Taylor number T satisfy the inequality

2 T

when both the boundary surfaces are rigid thus achieving magnetorotatory extension of an important characterization theorem of Banerjee et al (1992) on the corresponding hydrodynamic problem. A similar characterization theorem is mathematically established in the context of the magnetorotatory thermohaline convection of the Stern (1960) type.

Keywords. Thermohaline convections; thermohaline Rayleigh number;, Veronis type, Stern type

1. Introduction

The recently established characterization theorem of C2], disproving the existence of neutral or unstable oscillatory motions of growing amplitude in a bottom heavy thermohaline convection of the Veronis type whenever the thermohaline Rayleigh number is less than a critical value and a similar theorem for a thermohaline con- vection of the Stern type [3] has brought a fresh outlook to the subject matter of double diffusive convection and paved the way for further theoretical and experimental investigations in this field of enquiry. The essence of Banerjee et ars characterization theorem is that it provides a classification of the neutral and unstable thermohaline convection of the Veronis and Stern types into two classes namely the bottom heavy class and the top heavy class and distinguishes them by means of characterization theorems which disallow the existence of oscillatory motions in the former class. The extension of these two important theorems in astrophysics and terrestrial physics, wherein the liquid concerned has the property of electrical conduction and magnetic fields and rotations are prevalent, is very much to be sought of in the present context. This paper mathematically analyses the magnetorotatory thermohaline convection of the Veronis and the Stern types wherein a uniform rotation about an axis which is parallel to gravity is superimposed on the corresponding magnetohydrodynamic

197

198 Jyoti Prakash, Ajaib Sinoh and Savita Sharma

problem [4] may be regarded as a first step of this scheme of extended investigations. Here it is mathematically shown that magnetorotatory thermohaline convection of Veronis type cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if the thermohaline Rayleigh number Rs, the Lewis number z, the thermal Prandtl number r the magnetic Prandtl number ax, the Chandrasekhar number Q and the Taylor number T satisfy the inequality

2 T , Rs~<4~4[ 1 + ~ 2 { It - ( Q t r x + ~ ) } ]

when both the boundary surfaces are rigid thus achieving magnetorotatory extension of an important theorem of Banerjee et al [2] on the corresponding hydrodynamic problem. A similar characterization theorem is mathematically established in the context of the magnetorotatory thermohaline convection of the Stern [61 type.

2. Mathemat ica l formulation and analysis

The relevant governing equations and boundary conditions of magnetorotatory thermohaline convection of the Veronis type are given in [1],

(De--a 2) D e - a Z - ~ w=-Ra20-R~a2c~+ T D ( - Q D ( D 2 - a Z ) h ~ , (1)

(D 2 - a 2 - p)O = - w, (2)

(D z - a z - P)~b = w,z (3)

(D2 - a2 - Pa ' )h~= - (4)

(D 2 -- a2 -- P)~--. -QD~-Dw, (5)

( D 2 - a 2 - - ~ ) ~ = - D ( , (6)

and w=0 0 = 0 ~=0

Dw=O a t z = 0 a n d z = l , (7) (=0

o~ =o hz = 0

where z is a real independent variable such that 0 ~< z ~< 1 and stands for vertical coordinate, d/dz denotes the derivative with respect to z. The other terms a2> O, a > O, T > O, R > O, T > O, Q > O, and R, > 0 are constants, p = p, + ip~ is a complex constant in general and stands for the complex growth rate such that p, and Pi are

Magnetorotatory thermohaline convection 199

real constants and as a consequence the dependent variables w(z)= w,(z)+ iwi(z), O(z) = O,(z) + iOn(z), r = c~,(z) + ir ((z) = ~,(z) + i~(z), s = r 4- is and hz(z) = hz,(z)+ ih:~(z) which respectively stand for the vertical velocity, temperature, concentration, vertical vorticity, vertical current density and vertical magnetic field and are complex valued functions of the real variable z such that w,(z) wl(z), O,(z), Oi(z), ~,(z), ~i(z), ~,(z), ~l(z), ~,(z), ~i(z), h=,(z) and ha(z) are real valued functions of real variable z. Equations (1)-(7) describe the eigenvalue problem for p for given values of the other parameters and govern the magnetorotatory thermohaline convection of the Veronis type when both the boundaries are rigid. We prove the following theorems:

Qa 1 T <~ z_( a + ~) and Theorem 1. I f R~>O, R>O, Q>O, T>O, p,~>O, p~#O --~-+~-~ a

2 - ~ ) t l then anecessarycondit ion for theexis tence R,~<4n'*[l-t-~r-~-~n2 {n - ( Qa l + T / ./ _.1

of a non-trivial solution (w, O, r hz, ~, ~,p) of equations (1)-(5) and (6) together with boundary conditions (7) is that

R s < R .

Proof. Multiplying equation (1) by w* (the complex conjugate of w) throughout and integrating the resulting equation over the vertical range of z, we get

+ T w*DCdz - Q w*D(D 2 - aZ)h, dz, (8)

using equations (2)-(6) and boundary conditions on w and ~ namely w(O) = 0 = w(1) and ~(0) = 0 = ~(1), we can write

re ;o Ra 2 w*Odz = - Ra 2 O(D 2 - a 2 - p*)O* dz, (9)

- R,a' w* dz= R,a' qb(D2-a2-P- )O*dz, (10)

Io' Io ;o - w*D(D 2 - a2)hzdz = Dw*(D 2 - a2)h, dz = _ h~(D 2 - a 2)

and 1 1 1 p ,

=

d o \ a /

P* P ; ' ) , * d z ,

(12)

200 Jyoti Prakash, Ajaib Singh and Savita Sharma

so that using equations (9)-(12), it follows from equation (8) that

fo ( fo w*(D 2 -- a 2) D 2 - a 2 - wdz = - Ra 2 O(D 2 -- a 2 - p*)O*dz

;oL ( p*, /'1 + Rsa2z dp D 2 - a 2 - --)~b*dz - Q t hz( D2 - a2) z ~ Jo

(D ~_ ~ )" , ~z a s

+ T / ~ | D 2 - a 2 )8*dz+QT ~ D 2 - a 2-pal ~*dz. (13) O"

L

Integrating various terms of equation (13) by parts for an appropriate number of times and making use of boundary conditions, we have

fo fo I'lD2wl 2 + 2a21Dwl 2 + a41w[2] dz + p l - [IDwl 2 + a21wl2]dz O

;o = R a 2 lID0[ 2 + a s 1012 + p* 1012]dz

- R,a2T ~ [ID4~12 + a2lcbl2 + ~lepl21dz

Io ;o - Q i ( D 2 _ a 2 ) h z l 2 d z _ Q p , 0 1 1 - - [IDh=l 2 + a2lhzl2]dz O"

1 p , -rfo[,OC,2+a2,C:+-g,C:]dz - Q T f / [ l D ~ 1 2 + a 2 1 , 1 2 + ~ A l ~ ] 2 ] d z (14)

and equating the real and imaginary parts of both sides of equation (15) and cancelling pi (~ 0) throughout from the imaginary part. we get

[ID2wl 2 + 2a2lDwl 2 + a'*lwl2]dz + - - [[Dwl 2 + a2lwt2]dz 0

;o = Ra 2 [ID012 + a 21012 + prlOl2]dz

fo' ;o + Q [(D2_a2)h.[2d z Qpr~l 1 _ ~ [IDh.I 2 + a 2 Ih.12]dz O

- Q T I 2 [ I D , I 2 + a 2 I ' I 2 + P T ' I , I 2 ] d z -

Maonetorotatory thermohaline convection 201

and

'fo fo s [IDwl2 + a2lwl2qdz = _ Ra 2 1012dz + R.a 2 I~blZdz

+~176 ~ ~fo Io [iDh~t 2 + aelh=12]dz +~ 1r QTal ]~]2dz ' (16) o" ff

of which equation (16) can be rearranged in the form

fo ' ' Prfo [ID2wl2+2a2jDwl2+a Iwl ] d z + a [IDwl2+a21w]2]dz

;o ~ fo' + Q I(D e - a2)hz[edz + QT [IDOl 2 + a2l~12]dz

fo~ fo = Ra 2 lID012 + a2lOI2]dz - Rsa2z [ID~bl 2 + a21Ol2]dz

-r f~ Eiocl~ + a~lCl'J+ p,-[,,e f2 IOl'd,-,,,~f: I,i~d, [IDhz[ 2 +a21h.12]dz - T l l~12dz+ 1~12dz. (17)

a a . / o a

We first note that w, 0,~b,~ and h z satisfy w(0) = 0 = w(1), 0(0) = 0 = 0(1), ~b(0) = 0 = ~b(1), ((0) = 0 = ((1) and h.(0) = 0 = h.(1) in addit ion to satisfying the governing equations and hence we have from the Rayleigh-Ritz inequality l'5]

~o fo IDwl2dz >>. n a Iwl2dz, (18)

f~ lDOledz>>, n2 f~ lOl2dz, (19)

Io ~ fo ID~bl2dz I> n 2 l~bl2dz, (20)

fo Io ID~12dz >~ n 2 1~12dz,

and

f ~ lDhzl2 dz >~ ~2 f ~ lh.12 dz, (22)

Further, since w(0) = 0 = w(1), it follows that

f:lDwl2dz=Realpartof[-;:w*D2wdz], .~1- ;3.o,~,zl,

202 Jyoti Prakash, Ajaib Sinoh and Savita Sharma

<<. Iw*llD2wldz,

~< Iwl ID2wldz,

Iwl2dzJ L Jo lD'wl2dz]'/2 (utilizing Schwartz-inequality),

"11/2 F ri ~< 1-rnLjo rl IDwl2dzJ LJo ID2wl2dz] '̀ 2

(utilizing inequality (18)),

so that we have

[.i2 ,o,,,,,d,]"' ~,__r r',,,,,,,,,,,,]"t J ~LJo which yields

L I io IDZwl2dz t> ~2 iDwl2dz, (23)

while utilizing inequality (23), we obtain

Io fo [ID2w[2 + aZlDwl2]dz >t (x2 + a2) iDwl2dz. (24)

Also, since pr >I O, we get

11 I'[Dw[ 2 4- aZlwlZ]dz i> 0. (25) pr

OrJo

Further, multiplying equetion (2) throughout by 0* (the complex conjugate 0), integrating each term of the resulting equation by parts for an appropriate number of times and making use of boundary condition on 0 namely 0(0) = 0 = 0(1), it follows that

f ~ [ lDOI 2 + a2 lOI 2 ] dz + pr f ~ [OI 2 dz = Re[ f ~ O* wdz ],

1f20.4 IO*llwldz,

Maonetorotatory thermohaline convection 203

Io 101twldz,

[ f f lOI2dz]X/2 [ f 2 Iwl2dz] 1/2

(utilizing Schwartz inequality), so that by using inequality (19) and the fact that pr i> 0, we obtain from the above equation

[fo' -I,/2 F t ' , 'l,/z (,2 +a 2) '0'2dzJ ~<LJo Iw'2dzj '

and hence f ] lID012 + a21Oi2]dz <~ (n a 1 1 1 1 + a2) f o Iwl2dz <~'~2Cni-+ ab fo IDwl 2dz,

(utilizing inequality 18) (26)

Also, multiplying equation (5) throughout by ~* (the complex conjugate of O, integrating each ,term of the resulting equation by parts for a suitable number of times and making use of boundary condition on ~ namely ~(0) = 0 = ~(1), we get

flo[lD'12+a2l'12]dz+Prf'l'12dz=Re[f~'*Dwdz+Qf~*D'~dz]~Jo =Re[-f2D[*wdz-Qf2D[*~dz ] =Re[-~2D~*wdz+Q~2~(D'-a'-P*;l)?.*dz],

(using equation (6)),

=Re[- f~ D~*wdz-Q f~ [lD~12 + a2l~12 + P*-~aa~l~12dz 1,

~< IDr

<~ f~ ID~llwldz,

IDs [Jo "l dzJ (utilizing Schwartz inequality),

204 Jyoti Prakash. Ajaib Singh and Savita Sharma

which implies, since pr >1 0, that

fo ~ fo ~ ID~12dz ~< Iwl2dz

so that using inequalities (18) and (21), we obtain from the above that

fo ';o l~12dz ~< ~'i IDwl2dz" (27)

Further, multiplying equation (4) throughout by h* (the complex conjugate of hz), integrating each term of the resulting equation by parts for a suitable number of times and making use of boundary condition on hz namely h:(0) = 0 = h~(1), we get

fo ~z+-l;o. [;o 1 [IDhzl 2 + a2lhzl 2] Ihzl2dz = Re hz* Dwdz , 17

[fo ~ = Re - Dhz.* wdz],

<<. IDhz* I lwl dz,

<~ IDhzl Iwldz,

, [ f f 71/21" 1 71/2 IDhzl2dzJ L;o Iwl2dzJ

(utilizing Schwartz inequality),

which implies, since pr >>. O, that

[ f ~ "]1/2 [- /'1 -[112 IDhzl2dzJ ~<LJo Iwl2dzJ '

so that we obtain from the preceding equation that

;o fo ~ [IDhzl 2 + a 2 Ihzl2]dz <~ Iwl2dz,

which yields with the help of inequality (18) that

f~ [IDhzl2+a2lhzl2]dz<<.~ifl t ,Dwl2dz. (28)

Maonetorotatory thermohaline convection 205

Now, utilizing inequality (20)( we have

L L I-ID~bl 2 + a 21r >/(n 2 + a 2) lr (29)

while from (16), we get

x : T t

;o]r 1 ;o[DW[2dZ_R.a2a;ol~[2d z Qa, RsaZa Rsa2r

~ [IDhzl 2 + a2[hzl2]dz. (30)

so that using inequality (30). we can write the inequality (29) as

1 (i[.2 + a 2) ~1 IDwl 2dz r(n2 + a 2) ~1 fo [lDc~12 +a21c~12] dz>>" ~ i ~ Jo - ~ . ~ Jo 1~12dz

Qol(n2 +a 2) ~1 -~. ai~a Jo [IDhzlS + a2lhziS]dz" (31)

Also, using inequalities (27) and (28), (31) yields

f / [ IDr 2 + a21r I> ( ns + a s) Rsa 2 tT

[1 Q~ n.Tlf ~ ]Dw[2dz, (32)

while equation (16) together with pr >>. 0 yields the inequality

pr Ra 2 IOISdz - Rsa 2 Ir - [IDhzl 2 + a2lhzl2]dz �9 t O (7

Vo 1 - ~ Ir a 1~12dz 40. (33)

Now, if permissible let R,/> R. Then in that case we derive from equation (17) and inequalities (24), (26), (32) and (33) that

fo' R'a~ fo Rsa2"c(n2 + a2) (n 2 "Jr" a 2) IDwl2dz~< n2(n 2 + a2 ) IDwlSdz- R.a2 a

1 ~2 ~ IDwlSdz' (34)

which in turn yields that

[ I ( )} L , s r "."~ l I,wl'd,~O (~: + a 2) 1 + ~ n - Q'~x - ~ ~ ( ~ + aS).j

(35)

206 Jyoti Prakash, Ajaib Singh and Savita Sharma

which implies that

R~ > a2 1 + ~ n - (Qol + , (36)

and thus we must necessarily have

R~ > 4~'[1 + ~ { ~ 2 - (Q~r, + ~ ) }] , (37)

. 7t2(7r 2 + a 2 ) 2 since the minimum value ot is 41t 4 for a 2 = It 2 > 0.

a 2

2 ~ ) } ] , then we must have

R~ < R, (38)

and this completes the proof of the theorem. This result is analogous to the corresponding result of Banerjcr et al. [2] in hydro-

dynamic thermohalinr convection of Veronis [7] type and can be stated as follows in an equivalent form namely, "Magnetorotatory hydrodynamic thermohaline convection of the Veronis [7] type cannot manifest as oscillatory motions of growing amplitude in an initially bottom heavy configuration if R~, z, a, ~1, Q T satisfy the inequality

when both the boundary surfaces are rigid". It may be noted that for the hydrodynamic case in which T = 0 and Q = 0, we recover the result due to Banerjcr et al [2] from Theorem 1.

Theorem 2. l f R <O, R s<O, T>O, Q>O, pr>lO, pi~O, ~ n4<<.(l +a) and

,R, ~< 41r, T[1 + _~{7t2 _ (Qtrl + T ) } -~ then a necessary condition for the existence

of a non-trivial solution (w,O, dp, h:, ~, ~, p) of equations (1)-(6) together with boundary conditions (7) is

tgl < jR, I.

The validity of the theorem is established by following a procedure similar to the one adopted proving Theorem 1.

The above theorem can be stated as follows in an equivalent form namely, "Magnetorotatory thermohaline convection of the Stern (1960) type cannot manifest itself as oscillatory motions of growing amplitude in an initially bottom heavy configuration if R, z, #, ct , Q and T satisfy the inequality

,,4..[1+;{..-(0o.+5)} ] when both the bounding surfaces are rigid.

Ma o n e t o r o t a t o r y thermohaline convection 207

Further, it m a y be noted that for the corresponding hydrodynamic case in which T = 0 and Q = 0, we recover the results due to Jyoti Prakash (1992) f rom Theorem 2.

Acknowledgement

One of the au thors (JP) gratefully acknowledges financial aid from the Council of Scientific and Industrial Research, New Delhi.

References

[1] Banerjee M B and Gupta J R, Studies in hydrodynamic and hydromagnetic stability, Silverline Publications, Shimla (India) (1991)

[2] Banerjee M B, Gupta J R and Jyoti Prakash, On thermohaline convection Veronis type, J. Math. Anal. Appln. (Accepted for publication, 1992)

[3] Jyoti Prakash, Ph.D. Thesis, submitted for evaluation to Himachal Pradesh University, Shimla (1992) [4] Jyoti Prakash and Ajaib Singh, Banerjee et ars characterization theorem in thermohaline convection

and its magnetohydrodynamic extensions, Indian J. Pure Appl. Math. (Communicated, 1992), [5] Schultz M H Spline Analysis (Englewood Prentice Hall Cliffs NJ) (1973) [6] Stern M E, The salt fountain and thermohaline convection, TeUus 12 (1960) 172 [7] Veronis (3, On finite amplitude instability in the thermohaline convection, J. Mar. Res. 23 (1965) I