Applied Mathematics, 2010, 1, 400-405
doi:10.4236/am.2010.15052 Published Online November 2010 (http://www.SciRP.org/journal/am)
On the Stationary Convection of Thermohaline Problems
of Veronis and Stern Types
Joginder S. Dhiman1*, Praveen K. Sharma2, Poonam Sharma1
1Department of Mathematics, Himachal Pradesh University, Summerhill, Shimla, India
2University Institute of Information Technology, Himachal Pradesh University, Summerhill, Shimla, India
Received July 28, 2010; revised September 20, 2010; accepted September 23, 2010
Abstract
The stability of thermohaline convection problems of Veronis and Stern types for stationary convection is
studied for quite general nature of boundaries. It is shown by means of an appropriately chosen linear trans-
formation, that in case of stationary convection the equations describing the eigenvalue problem for thermo-
haline convection problems are identical to equations describing the eigenvalue problem for classical Bénard
convection problem. As a consequence, the values of the critical Rayleigh numbers for the onset of station-
ary convection in thermohaline convection problems are obtained. Also, necessary conditions for the validity
of principle of exchange of stabilities for thermohaline convection problems are derived using variational
principle.
Keywords: Thermohaline Convection, Stationary Convection, Eigenvalue Problem, Principle of Exchange of
Stabilities, Rayleigh Number
1. Introduction
A problem in fluid mechanics involving the onset of
convection has been of great interest for some time.
Thermal convection occurs in nature in so many forms
and over such a wide range of scales that it could be
claimed with some justification that convection repre-
sents the most common fluid flow in the universe. The
problem of onset of thermal instability in liquid layers
heated from below originated from the experimental
works of Bénard [1]. Stimulated by Bénard experiments,
Lord Rayleigh [2] studied the Bénard problem mathe-
matically for the first time, for the idealized case of both
free boundaries and showed that the gravity dominated
thermal instability in a liquid layer heated underside,
depends upon the Rayleigh number. The theoretical
treatments of convective problems usually invoked the
so-called principle of exchange of stabilities (PES),
which is demonstrated physically as convection occur-
ring initially as a stationary convection. Alternatively, it
can be stated as “the first unstable eigen values of the
linearized system have imaginary part equal to zero”.
A broader range of dynamical behaviour is observed
in the convective instabilities that may occur when a
fluid in a gravitational field contains two components of
different diffusivities that affect the density, for example,
temperature and solute. This phenomenon is known
variously as thermohaline convection, double diffusive
convection, or thermosolutal convection. Thermohaline
convection, with its archetypal case of heat and salt, has
been extensively studied in the recent past on account of
its interesting complexities as well as its direct relevance
in many problems of practical interest. Thermohaline
convection has matured into a subject possessing funda-
mental departures from its classical counterpart, namely,
thermal convection (single diffusive convection) and is
of direct relevance to the fields of limnology, oceanog-
raphy, astrophysics, etc. The various applications of the
problem have aroused the interest of many research
workers and this led to numerous research papers in
various journals in the recent past. For a broad view of
the subject one may be referred to Turner [3] and Brandt
and Fernando [4].
Two fundamental configurations have been studied in
the context of the thermohaline instability problem, one
by Stern [5], wherein the temperature gradient is stabi-
lizing and the concentration gradient is destabilizing and
another by Veronis [6], wherein the temperature gradient
is destabilizing and the concentration gradient is stabi-
lizing. Further, one should also note the relationship of

J. S. DHIMAN ET AL. 401
the Veronis’s analysis to that of Stern’s analysis. Veronis
[6] has done the analysis of a situation that is gravita-
tionally opposite to that of Stern’s. Therefore, one can
treat the two problems by considering the same configu-
ration but with the assumption that gravity is positive
downwards in one problem and positive upward in the
other. It is important to note here that Veronis’ as well as
Stern’s work are restricted to an idealized case of dy-
namically free boundaries.
Veronis also studied that when
s
R (salinity Rayleigh
number) is sufficiently small (much less then 4
27 4 
,
the value of critical Rayleigh number c
Rfor ordinary
convection with no solute present) the effect of the solute
is to modify the results for ordinary convection by only a
small amount. As
s
Ris increased to the order ofc R, the
value of R (thermal Rayleigh number) at which the
various types of instability can first occur also increases
and as
s
Rbecomes very large the value of c R ap-
proaches asymptotic value. In other words, as s R,
1
cs
RR 

. Thus, the exact behaviour of the system as
a function of
s
R depends on

(the ratio of the diffu-
sivities). Therefore, it becomes important to study this
dependence for all combinations of boundaries. Further,
it is evident that an analogous dependence for the case of
Stern’s type configuration may also hold.
Motivated by the above discussions regarding the sta-
bility and the structures of the thermohaline convection
problems of Veronis and Stern types and their classical
counterpart, the aim of the present paper is to study the
stability of thermohaline convection problems for quite
general nature of boundaries. It is shown here that upon
using a linear transformation the equations describing the
eigenvalue problem for stationary thermohaline convec-
tion problems become identical to equations describing
the eigenvalue problem for stationary classical Bénard
convection problem. As a consequence, the values of
critical Rayleigh numbers for thermohaline convection
problems of Veronis’ and Stern’s types for different
combinations of rigid and dynamically free boundary
conditions are obtained. Furthermore, necessary condi-
tions for the validity of PES (in terms of 1
s
R 
 and
R

- Laws) in Veronis and Stern types thermohaline
convection problems are derived for sufficiently large
values of Rayleigh numbers using variational principle.
2. The Physical Configuration and the Basic
Equations
A viscous, quasi-incompressible (Boussinesq) fluid of
infinite horizontal extension and finite vertical depth is
statically confined between two horizontal boundaries
0z and zd which are respectively maintained at
uniform temperatures 0 T and 1 T and concentrations
0
C and 1 C. We mathematically analyze the onset of
convection in the system under the force field of gravity
when the temperature and concentration make opposing
contributions to the vertical density gradient. Following
the usual steps of linear stability theory, the non-dimen-
sional linearized perturbation equations governing the
physical configuration described in the foregoing para-
graph may be put in the forms [7];

22 22
22
s
p
Da Daw
RaR a







(1)


22
Da pw 


(2)
22
pw
Da








(3)
together with one of the boundary conditions;
2
0wDw   at 0zand 1z

(4a)
(both boundaries dynamically free)
or
0wDw



 at 0z and 1z

(4b)
(both boundaries rigid)
or
0wDw



 at 0z and
2
0wDw   at 1z (4c)
(lower boundary rigid and upper boundary
dynamically free)
or
0wDw



 at 1z and
2
0wDw   at 0z (4d)
(lower boundary dynamically free and upper boundary
rigid)
In the above equations, z is the real independent
variable, d Ddz  is the differentiation with respect to
z, 2 ais the square of the wave number,

is the
thermal Prandtl number,

is the Lewis number, R is
the thermal Rayleigh number,
s
R is the concentration
Rayleigh number,


ri
ppip is the complex growth
rate and ,w

and

are the perturbations in the ver-
tical velocity, temperature and concentration respectively
and are complex valued functions of the vertical coordi-
nate zonly.
3. Mathematical Analysis
The system of Equations (1)-(3) together with either of
the boundary conditions (4) constitutes an eigenvalue
problem for the complex growth rate (p) for given values
of the other parameters, namely,

, R, s Rand

and a given state of the system is stable, neutral or unsta-

J. S. DHIMAN ET AL. 402
ble according to whether r p is negative, zero or posi-
tive. Further, system of Equations (1)-(3) together with
boundary conditions (4) describes the eigenvalue prob-
lem for
1) Veronis type thermohaline convection problem, if
0Rand 0
s
R,
2) Stern type thermohaline convection problem, if
0R and 0
s
R
Furthermore, if2
00
ri
ppa , then for neutral
stability, we have0p. This is called PES.
3.1. Critical Rayleigh Numbers
When instability sets in as stationary convection, i.e.,
when PES is valid, Equations (1)-(3) for Veronis type
thermohaline configuration becomes

2
2222
s
DawRa Ra

  (5)

22
Daw   (6)

22 w
Da


 (7)
together with any one of the boundary conditions (4).
Using the transformation
s
RR



 , Equations (5)-(7)
yield the following equations

2
22 2
Dawa

 (8)

22
Da w   (9)
and boundary conditions (4) become
0

 at 0z and 1z; and
either 0Dw  or 20Dw
at 0z and 1z (10)
Here,
s
R
R


 


is termed as the effective
Rayleigh number.
It is remarkable to note that Equations (8) and (9) are
identical to classical Bénard equations [8], where

plays the same role as that of R (the thermal Rayleigh
number) in Bénard convection problem. Therefore, the
results already available for the thermal convection prob-
lem [8] can easily be translated into those for the thermo-
haline convection problem of Veronis type. Consequently,
the values of critical Rayleigh number for Veronis type
thermohaline convection problem for the following three
cases of the boundary conditions can be obtained.
Following the analysis of [8], Equations (8)-(9) with
the relevant boundary conditions from (10) yield the
values of the critical Rayleigh numbers as;
Case I. When both boundaries are dynamically free.
4
27 657.51
4
c

 
this upon using the expression for c
implies that
657.51
s
c
R
R

, (11)
the same value of the critical thermal Rayleigh number
as obtained by Veronis [6] and Knobloch [9].
Case II. When both boundaries are rigid.
1707.76
s
c
R
R

. (12)
Case III. When one boundary is rigid and one is free.
1100.65
s
c
R
R

 (13)
In the results (11)-(13), the values 657.51, 1707.76,
1100.65 are respectively the values of the critical thermal
Rayleigh number for free-free, rigid-rigid, rigid-free
boundary conditions respectively for the purely thermal
problem [8].
The analogous values of critical Rayleigh number for
Stern’s type configuration can be easily obtained by re-
placing R and
s
R by R

and
s
Rrespectively, in
Equation (5) and emulating the analysis followed in Ve-
ronis type configuration. Therefore, we have
Case I. When both boundaries are dynamically free


657.57
sc
RR  . (14)
Case II. When both boundaries are rigid


1707.76
sc
RR  . (15)
Case III. When one boundary is rigid and one is free

1100.65
ss
cc
RR R    . (16)
3.2. 1
s
R 
 and R

- Laws
From results (11)-(13), one can easily see that for su-
fficiently large values of
s
R (lettings R) c R
1
s
R 
.
Since the result holds for all the cases of boundary
conditions, henceforth is referred to as 1
s
R 
-law for
Veronis type thermohaline convection problem. Also
from (14-16), we can obtain an analogous law namely;
R

-law for Stern’s type configuration for sufficiently
large values ofR.
In the following sections, we shall validates the above
stated laws using the minimum property of functional
established by variational principle.
4. A Variational Principle
Letting

2
22
F
Daw , (17)
and eliminating

from Equations (8)-(9), we get

J. S. DHIMAN ET AL. 403

22 2
DaFaw (18)
and the boundary conditions corresponding to (10) as
0wand 0F at 0z and 1z; and
either
0Dw  or 20Dw at 0z and 1z

(19)
depending on the nature of the bounding surfaces.
Multiplying Equation (18) by
F
and integrating the
resulting equation over the range of
z
, we get


11
22 2
00
12
222
0
F
DaFdzawFdz
awDawdz




(20)
Integrating by parts the above equation a suitable number
of times, using the relevant boundary conditions from
(19), we get the expression for the functional as
 

 




22
2
2
22
2
1
22
222 2
,
DFa Fdz
awFdz
DFa FdzI
s
ay
aI
aDawdz









(21)
where, 1
I
and 2
I
are positive definite integrals, and the
limits of integration from the integral sign have been
omitted for convenience in writing.
Upon using the value of

and the positivity of
1
I
and 2
I
, Equation (21) clearly implies that
2
,
s
R
Ra

.
Hence, for large values ofs
R, we get
1
cs
RR 

. (22)
It is remarkable to note here that the result (22) is uni-
formly valid for all cases of boundary conditions.
Using the variational method of Chandrasekhar’s [8]
for thermal convection problem, we have the stationary
property of the functional  given by expression (21)
for all cases of boundary conditions given in (19) when
the quantities on right hand side are evaluated in terms of
true characteristic functions. Also the quantity on the
right hand side of (21) attains its true minimum when F
belongs to c
, i.e. the lowest characteristic value of

,
namely c
, is indeed a true minimum, i.e.
 

 

 
22
2
2
22
2
2
222
s
cc
DFa Fdz
R
RawFdz
DFa Fdz
aDawdz



 









(23)
Further, this result is also valid for all cases of the
boundary conditions (19).
5. Necessary Conditions for PES
Let us consider
cos
F
z


(24)
which obviously satisfies the boundary conditions
0F

at 12 z and 12 z (25)
We shall now consider the boundary conditions (19) in
the following forms
0wDw

 at 12 z
or 2 0wDw at 12 z
or 0wDw

 and
2
0wDw at 12 z (26a,b,c)
In (25) and (26) the origin has been shifted to the mid-
way for convenience in computation.
Equation (17) upon using (24) yields
222
()cosDawz

 (27)
Let 2
1
q and 2
2
q be the roots of the auxiliary equa-
tion of the Equation (27), hence the general solution of
Equation (27) is given by

112 22
22
cos
cosh coshz wB qzBqz
a


 
 (28)
Now, using the boundary conditions0;w0;Dw

and 20Dw

at
12
z in Equation (28), we get following three re-
spective equations as
12
12
coshcosh 0
22
qq
BB

 (29)

12
112 2
2
22
sinh sinh
22
()
qq
qBqB
Asay
a






(30)
22
12
112 2
coshcosh 0
22
qq
qBq B

 (31)
Now, solving Equations (29)-(31) for 1 Band 2 B for
different cases of boundary conditions (26), we get


2
1
2
22
0, (26)
cosh2 ,(26)
cosh2,(26)
for a
Aq
Bforb
Aq q
f
or c










(32)

J. S. DHIMAN ET AL. 404
and


1
2
2
21
0, (26)
cosh2 ,(26)
cosh2 ,(26)
for a
Aq
Bforb
Aq q
f
or c










(33)
where,
12 21
21
cosh sinhcoshsinh
2222
qqq q
qq
and
22
21 12
12 21
coshsinhcosh sinh
22 22
qq qq
qqq q


Now, substituting the values of w and F in integrals
1
Iand 2 Iand integrating, we get
 

122
222
2
1
12
2
a
IDFaFdz





 (34)

12
2
12
12
12
222 22
22
12
2cosh 2cosh1
22
2
IwFdz
qq
BB
qqa

 



 


(35)
Using the values of integrals 1
I
and 2
I
given in (34)
and (35) in inequality (23), we get

22
12
12
2
222 22
22
12
2cosh2 cosh1
22
2
2
s
c
R
R
a
qq
BB
aqqa



 









Now, utilizing sufficiently large value of s
R in the
above inequality (i.e. taking
s
R), we have
1
cs
RR 

 (36)
which is uniformly valid for all the cases of boundary
conditions.
Therefore, combining inequalities (22) and (36), we get
1
cs
RR 

 (37)
which is precisely the 1
s
R 
 - Law for Veronis type
thermohaline convection problem.
Further, replacing R and s R byR ands R
respectively, in Equation (8), emulating the proof of
s
R
1

- Law and taking sufficiently large values of R,
we can easily prove the R  - Law for Stern type ther-
mohaline convection problem also.
6. Conclusions
We have studied the thermohaline instability of Veronis
and Stern types for the onset of stationary convection for
all possible cases of boundary conditions. It is important
to point out here that the values of critical Rayleigh
number derived by Veronis and Stern for their respective
configuration were valid for the idealized case of both
dynamically free boundaries. Since, the exact solutions
of the eigenvalue problems describing thermal/thermo-
haline convection are not obtainable in closed form for
other two cases (i.e. rigid-rigid and rigid-free) of bound-
ary combinations, therefore the values of critical
Rayleigh number for these realistic cases are not known
analytically for thermohaline convection problems.
However, to obtain the critical Rayleigh number for
thermal convection problem for these realistic cases of
boundary conditions Chandrasekhar [8] used the nume-
rical computations. In the present analysis the values of
critical Rayleigh number for all three cases of boundary
conditions have been obtained using the known results of
Chandrasekhar [8] for classical Bénard problem. Further,
the result obtained by Knobloch [9] for the idealized case
of free boundaries also follows from (37).
Veronis [6] derived an asymptotic relation for a con-
figuration which is initially gravitationally stable for the
case of both free boundaries. He remarked that for a
given very stable salt gradient (so that
s
R is sufficiently
large), c
R must have a value that is about 100 times the
value of
s
R. In other words, the destabilizing tempera-
ture gradient must exceed the stabilizing salt gradient by
a factor of 100. This result clearly violates one’s intuition,
since it means that the vertical density profile must be
highly gravitationally unstable before convection can
occur. In Section 3, this asymptotic behaviour of critical
Rayleigh number has been extended to the cases of real-
istic boundary conditions and is validated in Section 5
using the route through variational principle. This is a
necessary condition for the validity of PES for Veronis
type thermohaline convection and is named as 1
s
R 
-
Law. Analogously, a R

- Law for Stern type ther-
mohaline convection problem has also been derived here.
7. Acknowledgements
Thanks are extended to Professor J. R. Gupta for his
perspicacious comments on the subject. One of us (JSD)
gratefully acknowledges the financial support of UGC

J. S. DHIMAN ET AL. 405
under SAP.
8. References
[1] H. Bénard, “Les Tourbillons Cellulaires dans une Napple
Liquide,” Revue Generale des Sciences Pures at Appli-
qués, Vol. 11, 1900, pp. 1261-1271.
[2] L. Rayleigh, “On the Convective Current in a Horizontal
Layer of Fluid When the Higher Temperature is on the
Upper Side,” Philosophical Magazine, Vol. 32, No. 3, pp.
529-543.
[3] J. S. Turner, “Multicomponent Convection,” Annual Re-
view of Fluid Mechanics, Vol. 17, No. 1, 1985, pp. 11-44.
[4] A. Brandt and H. J. S. Fernando, “Double Diffusive Con-
vection,” American Geophysical Union, Washington, DC,
1996.
[5] M. E. Stern, “The Salt Fountain and Thermohaline Con-
vection,” Tellus, Vol. 12, No. 2, 1960, pp. 172-175.
[6] G. Veronis, “On Finite Amplitude Instability in Thermo-
haline Convection,” Journal of Marine Research, Vol. 23,
1965, pp. 1-17.
[7] J. R. Gupta, J. S. Dhiman and J. Thakur, “Thermohaline
Convection of Veronis and Stern Types Revisited,” Jour-
nal of Mathematical Analysis and Applications, Vol. 264,
No. 2, 2001, pp. 398-407.
[8] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic
Stability,” Oxford University Press, Amen House, Lon-
don, 1961.
[9] E. Knobloch, “Convection in Binary Fluids,” Physics of
Fluids, Vol. 23, No. 9, 1980, pp. 1918-1920.