Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue5.1999:

Advances in Water Resources, Vol. 22, No. 4, pp. 399–412, 1998
q 1998 Elsevier Science Limited
All rights reserved. Printed in Great Britain
PII: S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 0 0 1 8 - 3
0309-1708/98/$ - see front matter

Penetrative convection and multi-component
diffusion in a porous medium
John Tracey 1
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland, UK
(Received 31 May 1996; revised 12 December 1996; accepted 21 March 1997)

Linear instability and nonlinear energy stability analyses are developed for the problem
of a fluid-saturated porous layer stratified by penetrative thermal convection and two salt
concentrations. Unusual neutral curves are obtained, in particular non-perfect ‘heartshaped’ oscillatory curves that are disconnected from the stationary neutral curve. These
curves show that three critical values of the thermal Rayleigh number may be required
to fully describe the linear stability criteria. As the penetrative effect is increased, the
oscillatory curves depart more and more from a perfect heart shape. For certain values
of the parameters it is shown that the minima on the oscillatory and stationary curves
occur at the same Rayleigh number but different wavenumbers, offering the prospect of
different types of instability occurring simultaneously at different wavenumbers. A

weighted energy method is used to investigate the nonlinear stability of the problem and
yields unconditional results guaranteeing nonlinear stability for initial perturbations of
arbitrary sized amplitude. q 1998 Elsevier Science Limited. All rights reserved.

and by Murray and Chen18 to account for the effects of
temperature-dependent viscosity and volumetric expansion
coefficients and a nonlinear basic-state salinity profile.
Comparisons of theory with experimental evidence have
been difficult because of the limited amount of experimental
work in double diffusive convection in porous media. Griffiths11 considered two adjacent layers of fluid with different
temperatures and salt concentrations and obtained values of
heat and salt flux through a thin diffusive interface between
the two layers while Murray and Chen18 have considered the
effect of a time-dependent salt concentration profile in the
basic state.
The effect of a third diffusing component has received
comparatively little attention. Considering the number of
physical applications of the double diffusive problem
there must be many physical examples where a third diffusing component is present. For instance, in the viscous fluid
case, Degens et al.8 have described the waters of an East

African lake as containing many different salts and the
oceans contain many different salts with concentrations
much less than that of the predominant sodium chloride
concentration. Further applications may be found in the
area of contaminant transport. Celia et al.3 present a new
numerical procedure, an optimal test function method, for
the problem of reactive transport in porous media while
Allen and Curran1 employ a finite-element method to

1 INTRODUCTION
When two competing stratifying agencies (for instance heat
and salt concentration) are present in a viscous fluid or fluidsaturated porous layer a variety of convective phenomena
are found that do not occur in the single-component Benard
problem. Double-diffusive convection has been an active
area of research for many years in the viscous fluid case
in particular. Extensive reviews of this subject can be
found in34,13. Workers in porous media fluid mechanics
have been attracted to the problem because of its relevance
to geophysical systems such as the geothermal reservoirs
found in the Imperial Valley in California5, near Lake Kinnert in Israel28 and the Wairakei geothermal system in New

Zealand11.
The analytical treatment of the double-diffusive porous
problem is reviewed in5,20. The first study of the linear
stability of a fluid-saturated porous layer stratified by heat
and salt concentration was by Nield19. Later Taunton et al. 32
considered the onset of salt fingers. The formulation of
Nield19 has been extended by Rubin28 to introduce a nonlinear salinity profile, by Patil and Rudraiah21, who
included the effect of thermal diffusion (the Soret effect),
1
Present address: Edinburgh Petroleum Services Ltd, Research
Park, Riccarton, Edinburgh, EH14 4AP, UK.

399

400

J. Tracey

model transport of a single contaminant in porous media.
Chen et al.4 use a finite difference method to investigate

biofilm growth in tortuous porous media and solute transport has been studied by Curran and Allen7 and Allen and
Khosravani2 using finite-element collocation methods.
The linear stability of triple diffusive porous convection
was studied first by Rudraiah and Vortmeyer30 and Poulikakos25. Those authors were motivated by the work of
Griffiths10 who considered the effect of a third diffusing
component in the viscous fluid case. Subsequently the viscous fluid problem has been studied by Pearlstein et al.24
who presented a rigorous investigation of the neutral curves
which showed that the results of Griffiths10 were incomplete. In particular the conclusion of Griffiths10 that
‘‘marginal stability of oscillatory modes occurs on a hyperboloid in Rayleigh number space but the surface is very
closely approximated by its planar asymptotes for any diffusivity ratios’’ was shown to be incorrect. Pearlstein et al.24
show that for some fixed values of the diffusivity ratios,
Prandtl number and two of the three Rayleigh numbers,
three values of the third Rayleigh number may be required
in order to specify the linear stability criteria. This is due to
the existence of an isolated, symmetric heart-shaped oscillatory curve lying below the stationary curve. Pearlstein et
al.24 predict that one consequence of the perfect heart shape
of the oscillatory neutral curves is that instability could
occur at the same thermal Rayleigh number but different
wavenumbers. Tracey33 showed that all of the important
features of the triple diffusive viscous flow problem

described above are carried over to the analogous problem
in a porous medium. Of the three critical Rayleigh numbers
required to describe linear instability, two are unlikely to be
realized in practice as they lie in a region where nonlinear
effects will be of importance. In order to investigate the
nonlinear stability of the triple-diffusive problem, an application of the energy method that yields unconditional nonlinear stability is presented by Tracey33.
Pearlstein et al.24 adopted an equation of state in which
the density is linearly dependent upon temperature. The
density of a fluid will not be a linear function of temperature
in reality and so in the present work an equation of state that
is quadratic in temperature is adopted in order to investigate
the triple-diffusive problem. This quadratic temperature law
allows the introduction of the phenomenon of penetrative
convection—a term used to describe the situation when a
stable layer exists next to an unstable layer. When convection
begins in the unstable layer the motions penetrate into the
stable layer. Penetrative convection has applications in stellar
regions (see e.g. 35) and in geophysical problems, including
modelling thawing subsea permafrost (see e.g. 22) and patterned ground formation (see e.g. 9,27). Other occurrences of
penetrative convection are cited in 20,31,35.

The analytical method of Tracey33 cannot be used in the
present situation as the differential equations that arise in the
eigenvalue problem here have coefficients that are functions
of the spatial variables. Instead a Chebyshev tau method is
employed. This provides quick and accurate results and in

addition yields as many eigenvalues as are required, allowing the behaviour of the growth range and the instability
mechanism to be investigated in detail.
The effect of nonlinear terms may have important consequences for the experimental realisation of the linear stability results. Work by Proctor26 and Hansen and Yuen12 on
the double-diffusive fluid problem shows that subcritical
instability can occur at values of the thermal Rayleigh
number much less than that predicted by linear theory.
Rudraiah et al.29 find a similar result in considering the
effect of rotation on the double-diffusive porous problem.
The energy method is used in this work to investigate the
nonlinear stability of the basic motionless state. The quadratic equation of state gives rise to quadratic temperature
terms in addition to convective nonlinearities. To deal with
these a weight is introduced to the temperature part of the
energy, the effect of which is to cancel out the quadratic
temperature term. The results presented here have the

important advantage that they are unconditional, i.e.,
nonlinear stability is guaranteed for initial perturbations of
arbitrary sized amplitude.

2 LINEAR STABILITY ANALYSIS
Consider a fluid-saturated porous layer lying in the infinite
three-dimensional region 0 , z , d. The lower boundary
z ¼ 0 is held at the fixed temperature T ¼ 08C while the
upper boundary z ¼ d is held at a temperature T 1 $ 48C. If
the fluid under consideration is water then penetrative convection can occur in the fluid layer. This is due to the fact
that water has a density maximum at 48C and so the lower
part of the layer is gravitationally unstable, while the upper
part is gravitationally stable. When convection occurs in the
lower part of the layer the motions will penetrate into the
upper part. Suppose further that the fluid has dissolved in it
N different chemical species. Denote the concentration of
component a by C a (a ¼ 1,…,N). The concentration of
component a at the lower and upper boundaries is held at
C l a and C u a, respectively. The density is taken to be
quadratic in the temperature field and linear in the salt concentration, i.e.

!
N
X
2
a
a
Aa (C ¹ C0 ) ,
r ¼ r0 1 ¹ A(T ¹ T0 ) þ
a¼1

a

where r 0, T 0 and C 0 (a ¼ 1,…N) are reference values of
density, temperature and salt concentration, respectively.
The constants A and A a (a ¼ 1,…,N) represent the thermal
and solute expansion coefficients, respectively.
The equations of motion which govern flow in a porous
medium are largely based on a relation which is a generalization of empirical observations (c.f.20). This relation is
known as Darcy’s Law and can be written
m

=p ¼ ¹ vrg,
k
where the variables p, m, k, v and g represent pressure,

Penetrative convection and multi-component diffusion
dynamic viscosity, permeability, seepage velocity and
gravitational acceleration, respectively. In addition to
Darcy’s Law we have the incompressibility condition and
the equations of conservation of temperature and solute.
Combining these equations with the Darcy law and the
equation of state gives the following system of 5 þ N
governing equations:
!
N
X
m
2
a
a
Aa (C ¹ C0 ) ki ,

p, i ¼ ¹ vi ¹ gr0 1 ¹ A(T ¹ T0 ) þ
k
a¼1

401

Here R and R a are the Rayleigh and salt Rayleigh numbers
and P a are salt Prandtl numbers.
The nonlinear perturbation equations are then, in nondimensional form (dropping the asterisks)
"
#
N
X
Ra fa ¹ v 2 k i ,
(7)
p, i ¼ ¹ ui ¹ 2R(y ¹ z)v þ
a¼1

ui, i ¼ 0,

(8)

(1)

v, t þ ui v, i ¼ ¹ Rw þ Dv,

(9)

vi, i ¼ 0,

(2)

Pa (f, at þ ui f, ai ) ¼ Ha Ra w þ Dfa

T, t þ vi T, i ¼ kDT,

(3)

C, at þ vi C, ai ¼ ka DCa (a ¼ 1, …, N),

(4)

where indicial notation and the Einstein summation convention have been employed. The vector k is the unit vector
in the z-direction while the variables k and k a represent
thermal and solute diffusivity, respectively. The boundary
conditions considered are
at z ¼ 0, T ¼ 08C, C a ¼ Cla (a ¼ 1, …, N), v3 ¼ 0,

(5)

at z ¼ d, T ¼ T1 $ 48C, Ca ¼ Cua (a ¼ 1, …, N), v3 ¼ 0:
The experimental realization of prescribing these boundary
conditions for the salt concentrations has been investigated
by Krishnamurti and Howard15.
a
The steady solution (vi , p, T, C ) of eqns (1)–(5) on which
we will perform a linear stability analysis is given by
T
DC a
a
z
vi ¼ 0, T ¼ 1 z, C ¼ Cla ¹
d
d
(DC a ¼ Cla ¹ Cua u, a ¼ 1, …, N),

and the steady pressure p can be obtained from eqn (1).
In order to investigate the linear stability of this basic
solution we introduce perturbations (u i, p, v, f a) to
a
(vi , p, T, C ) via
a

vi ¼ vi þ ui , p ¼ p þ p, T ¼ T þ v, C a ¼ C þ fa :
The resultant perturbation equations are non-dimensionalized using the following scalings:
2

t ¼ tp

d
k
mk
, u ¼ up , p ¼ pp , x ¼ xp d,
k
d
k

v ¼ vp T , fa ¼ (fa )p Fa , y ¼

T¼



R ¼ T1

mk
Aro gkd


1=2

Aro gkd
mk

, Fa ¼

1=2



, Ra ¼

k
Ha ¼ sgn(DCa ), Pa ¼ :
ka

a

mklDC l
Aa ro gkd


where w ¼ u 3. The boundary conditions which follow from
eqn (5) for the perturbed quantities are
w ¼ v ¼ fa (a ¼ 1, …, N) ¼ 0 at

z ¼ 0, 1:

(11)

Eqns (7)–(10) are linearized by neglecting terms containing
products of the perturbed quantities. A time dependence of
e jt is introduced by substituting
u(x, t) ¼ u(x)ejt , v(x, t) ¼ v(x)ejt , fa (x, t) ¼ fa (x)ejt
(a ¼ 1, …, N):
The pressure term is eliminated by taking curlcurl of eqn
(7) and then selecting the third component. This gives
Dw ¼ ¹ 2R(y ¹ z)Dp v ¹

N
X

Ra Dp fa ,

(12)

a¼1

jv ¼ ¹ Rw þ Dv,

(13)

*

2

2

2

(a ¼ 1, …, N),

(14)

2

where D ¼ (] /]x ) þ (] /]y ) is the horizontal Laplacian.
A normal mode representation is assumed, i.e.
w ¼ W(z) exp[i(mx þ ny)], v ¼ Q(z) exp[i(mx þ ny)],
fa ¼ Fa (z) exp[i(mx þ ny)]

(a ¼ 1, …, N),

and the following transformations are made in order to put
the equations in a form like Tracey33.
Rv → v, Ra fa → fa , R2 → R, Ha R2a → ¹ Ra

(15)

Attention is now focussed on the case of two salt fields, i.e.
N ¼ 2. The resultant equations are

4
,
T1
1=2

(10)

Pa jfa ¼ Ha Ra w þ Dfa
ð6Þ

(a ¼ 1, …, N),

,


Aa ro gkdlDC a l 1=2
,
mk

(D2 ¹ k2 )W ¹ 2(y ¹ z)k2 Q ¹ k2 F1 ¹ k2 F2 ¼ 0,

(16)

(D2 ¹ k2 )Q ¹ RW ¼ jQ,

(17)

(D2 ¹ k2 )Fa ¹ Ra W ¼ Pa jFa
2

2

(a ¼ 1, 2),

(18)

2

where k ¼ m þ n is a wavenumber and D ¼ (d/dz). The
appropriate boundary conditions are, from eqn (11),
W ¼ Q ¼ Fa (a ¼ 1, 2) ¼ 0, z ¼ 0, 1:

(19)

The presence of a z-dependent term in eqn (16) rules out
the possibility of using the analytical method of Tracey33.

402

J. Tracey

Instead, a Chebyshev tau method was used to solve the
system of eqns (16)–(19). Numerical results are presented
in Section 4.
3 NONLINEAR STABILITY ANALYSIS

If we now form eqn (26) þ eqn (29) þ l 1 eqn (27) þ l 2
eqn (28), where l 1 and l 2 are positive coupling parameters,
then,


d 1 2
l1 P1 2 l2 P2 2
hmv
ˆ iþ
kfk þ
kwk
dt 2
2
2
¼ ¹ RhMvwi þ (l1 H1 ¹ 1)R1 hwfi

An analysis of the nonlinear stability of the basic solution is
now presented by making use of the energy method. The
nonlinear perturbation equations are, from eqns (7)–(10),
p, i ¼ ¹ ui ¹ [2R(y ¹ z)v þ R1 f þ R2 w ¹ v ]ki ,

(20)

ui, i ¼ 0

(21)

v, t þ ui v, i ¼ ¹ Rw þ Dv,

(22)

P1 (f, t þ ui f, i ) ¼ H1 R1 w þ Df,

(23)

P2 (w, t þ ui w, i ) ¼ H2 R2 w þ Dw,

(24)

2

where, for later clarity of notation, the following transformations have been used
Let V denote a period cell for the solution. The boundary
conditions we consider are
w ¼ v ¼ f ¼ w ¼ 0 on z ¼ 0, 1,

(25)

and further that u i, v, f, w and p are periodic on the lateral
boundaries of V.
To commence multiply eqn (20) by u i, eqn (23) by f, eqn
(24) by w and integrate over V. Integration by parts and use
of the boundary conditions yields
0 ¼ ¹ kuk2 ¹ 2Rh(y ¹ z)vwi ¹ R1 hfwi ¹ R2 hwwi þ hv2 wi,
(26)
d P1 2
kfk ¼ H1 R1 hwfi ¹ k=fk2 ,
dt 2

ð30Þ

where M(z) ¼ m þ 2y ¹ 4z. The effect of the weight is that
the problematic hv 2wi term in eqn (26) is cancelled out by
the ¹ hwv 2i term that arises in eqn (29).
If we now define an energy
1 2
l P
lP
ˆ i þ 1 1 kfk2 þ 2 2 kwk2 ,
E(t) ¼ hmv
2
2
2

(31)

then eqn (30) shows that
dE
¼ I ¹ D,
dt
I ¼ ¹ RhMvwi þ (l1 H1 ¹ 1)R1 hwfi þ (l2 H2 ¹ 1)R2 hwwi,
2
ˆ
i þ l1 k=fk2 þ l2 k=wk2 :
D ¼ kuk2 þ hml=vl

By rearrangement,


dE
I
¼I¹D¼ ¹D 1¹
:
dt
D
If we now define
1
I
¼ max ,
H D
L

(27)

where H is the space of admissible functions, then


dE
1
# ¹D 1¹
:
dt
L

(28)

If now

R

where k·k denotes the L 2 (V) norm and hf i ¼ V f dV. In
order to deal with the hv 2wi term in eqn (26) a technique
used by Payne and Straughan23 is introduced. A weighted
energy relation is formed by multiplying eqn (22) by
(m ¹ 2z)v, where m . 2 is a coupling parameter to be
selected at our discretion, and integrating over V. Selecting
m . 2 ensures that m ¹ 2z . 0. The weighted relation is


d 1
(m ¹ 2z)v2 þ h(m ¹ 2z)vui v, i i
dt 2
¼ h(m ¹ 2z)v( ¹ Rw þ Dv)i:
Integration by parts and use of the boundary conditions
then gives


d 1 2
2
mv
ˆ
ˆ
¹ hml=vl
ˆ
i,
(29)
¼ ¹ hwv2 i ¹ Rhmwvi
dt 2
where mˆ ¼ m ¹ 2z.

þ l1 k=fk2 þ l2 k=wk2 ),

where

f1 → f, f2 → w:

d P2 2
kwk ¼ H2 R2 hwwi ¹ k=wk2 ,
dt 2

2
ˆ
i
þ (l2 H2 ¹ 1)R2 hwwi ¹ (kuk2 þ hml=vl

L . 1,

(32)

(33)

then
1¹

1
¼ b . 0,
L

and so
dE
# ¹ bD:
dt

(34)

Use of the Poincare inequality (see e.g.31) results in
D $ cE:
where c is a positive constant that arises from the use of the
Poincare inequality. Therefore,
dE
# ¹ bcE,
dt

(35)

Penetrative convection and multi-component diffusion
which can be integrated to yield
E(t) # E(0)e ¹ bct :
So, if L . 1 then E(t) → 0 as t → ` at least exponentially
fast and so our steady solution is stable. This fact, together
with eqn (31), shows that kvk, kfk, kwk → 0. The lack of a
time derivative in Darcy’s law means that a similar result
cannot easily be obtained for kuk. However, if eqn (34) is
integrated with respect to time then
Zt
E(t) þ b D(s)ds # E(0),

now



R
¹ Mv þ gR1 f ¹ hR2 w k1 ¹ ui ¼ Ã, i
2

403

(40)

R
]v
ˆ ¹ 2 ¼ 0,
¹ Mw þ mDv
2
]z

(41)

gR1 w þ Df ¼ 0,

(42)

¹ hR2 w þ Dw ¼ 0,

(43)

where

0

l ¹1
l þ1
g ¼ 1p , h ¼ 2p :
2 l1
2 l2

which means that, in particular.
Z`
kuk2 ds , `:

We now consider R 1 and R 2 to be fixed and investigate the
variation of R, where now

0

So kuk 2 [ L 1 (0,`). This ensures ‘practical decay’ even
though the solution u may ‘peak’ over vanishingly small
intervals as t → `. It is difficult though to conceive of such
a situation physically.
The problem remains to find the maximum in eqn (32).
In order to clear the denominator of the maximization
problem of the coupling parameters l 1 and l 2, the following
transformations are made,
p
p
l1 f → f, l2 w → w:

Consider now the case H 1 ¼ 1, H 2 ¼ ¹ 1. This
corresponds to the situation considered in Section 2
where component 1 is stabilizing while component 2 is
destabilizing. The resultant maximisation problem to be
considered is
1
¼ max
H
L

(l1 ¹ 1)
(l2 ¹ 1)
 R1 hwfi ¹ p
 R2 hwwi
¹ RhMvwi þ p
l1
l2
2

2

2

2

ˆ
i þ k=fk þ k=wk
kuk þ hml=vl

:

(36)

!
l2 þ 1
p R2 w þ Dw ¼ 0,
¹L
2 l2

R ¼ R(m, l1 , l2 , k2 ),
where k is a wavenumber.
We will now vary each of l 1, l 2 and m in turn and find the
optimum values of these coupling parameters. Firstly, we
consider l 1, m, R 1 and R 2 to be fixed and investigate the
optimum value of l 2 by using parametric differentiation.
Let now superscripts 1 and 2 refer to a solution of eqns
(40)–(43) corresponding to parameters l 12 and l 22 respectively. We now form the inner products (eqn (40) 1, u 2),
(eqn (40) 2, u 1), (eqn (41) 1, v 2), (eqn (41) 2, v 1), (eqn (42) 1,
f 2), (Eq, (42) 2, f 1), (eqn (43) 1, w 2), and (eqn (43) 1, w 1) and
obtain the equations
¹

R1
hMv1 w2 i þ R1 ghf1 w2 i ¹ R2 hhw1 w2 i ¼ hu1 ·u2 i,
2
(45)

The Euler-Lagrange equations for this maximum are as
follows
"
!
!
#
R
l1 ¹ 1
l2 þ 1
p R1 f ¹
p R2 w ki ¹ ui ¼ Ã, i
L ¹ Mv þ
2
2 l1
2 l2
R
]v
ˆ ¹ 2 ¼ 0,
¹ L Mw þ mDv
2
]z
!
l1 ¹ 1
p R1 w þ Df ¼ 0,
L
2 l1

(44)

(37)

(38)

(39)

where à is a Lagrange multiplier introduced because u
is divergence free. At the stability limit L → 1. Setting
L ¼ 1 in the Euler-Lagrange eqns (36)–(39) will then
yield the optimum results. The equations to be solved are

¹

R2
hMv2 w1 i þ R1 ghf2 w1 i ¹ R2 hhw2 w1 i ¼ hu2 ·u1 i,
2
(46)

¹

R1 1 2
hw v i ¼ hm=v
ˆ 1 ·=v2 i,
2

(47)

¹

R2 2 1
hw v i ¼ hm=v
ˆ 2 ·=v1 i,
2

(48)

R1 ghw1 f2 i ¼ h=f1 ·=f2 i,

(49)

R1 ghw2 f1 i ¼ h=f2 ·=f1 i,

(50)

¹ R2 h1 hw1 w2 i ¼ h=w1 ·=w2 i,

(51)

¹ R2 h2 hw2 w1 i ¼ h=w2 ·=w1 i,

(52)

To proceed, form eqn (45) þ eqn (47) ¹ eqn (46) ¹ eqn
(48) þ eqn (49) ¹ eqn (50) þ eqn (51) ¹ eqn (52) to find
1
¹ hM(R1 ¹ R2 )(v1 w2 þ v2 w1 )i
2
¹ R2 (h1 ¹ h2 )hw1 w2 þ w2 w1 i:

404

J. Tracey

Divide this by l 2 ¹ l 1 and then let l 2 → l 1 to obtain,
¹

]R
]h
hMvwi ¹ 2R2
hwwi ¼ 0:
]l2
]l2

(53)

However, using eqn (41) and eqn (43) it can be shown that
2
]h k=wk2
2 ]R
hml=vl
ˆ
i
¼ ¹ 2R2
:
R
]l2
]l2 hR2

(54)

At the optimum value of l 2, (]R/]l 2) ¼ 0 and so from eqn
(54),

]h
l ¹1
¼ 2 3=2 :
]l2
4l2
]R
¼ 0 ⇒ l2 ¼ 1:
]l2

(55)

If we now fix l 2 and m and vary l 1 then a similar argument
to the above shows that
]R
]g
hMvwi þ 2R1
hwfi ¼ 0:
]l1
]l1

(61)
(62)

2
l1 þ 1
2 ]R
k=fk2 :
hml=vl
ˆ
i
¼
R
]l1 l1 (l1 ¹ 1)
So the system is singular at l 1 ¼ 1. While we cannot find a
solution for (]R/]l 1) ¼ 0, note that
l1 , 1 ⇒

]R
, 0,
]l1

which suggests that the best value is l 1 ¼ 1.
If now l 1 and l 2 are held constant and the variation in m is
considered an argument such as the above does not lead to a
result for the optimum value of m. Instead, the value
max min R(m, k2 )
k2

2

2

where D ¼ (d/dz) and k ¹ m þ n is the wavenumber.
The boundary conditions on W, Q and W follow from eqn
(25) and are
at

z ¼ 0:1:

(63)

For fixed values of R 2 eqns (60)–(63) form an eigenvalue
problem with eigenvalue R. The compound matrix method
was used to solve this. Details of this method may be found
in, e.g.31. For the maximization and minimization problems
in eqn (56) the golden section search was employed (see
e.g.6). Numerical results are presented in Section 4.

4 NUMERICAL RESULTS

Using eqn (40), eqn (42) and eqn (44), this can be rearranged to show

m$2

2
(D2 ¹ k2 )Q ¼ DQ þ RNW,
mˆ

W ¼Q¼W¼0

So,

]R
. 0,
]l1

(60)

2

and since h . 0, were have (]h/]l 2) ¼ 0. From eqn (44),

l1 . 1 ⇒

1
(D2 ¹ k2 )W ¼ Mk2 Q þ k2 W,
2

(D2 ¹ k2 )W ¼ R2 w,

]h k=wk2
¼ 0,
2
]l2 h

¹

The solenoidal term Ã;i is eliminated by taking curlcurl of
eqn (57) and then selecting the third component. Normal
modes are assumed and the transformations eqn (15) are
made, resulting in the system

(56)

is found numerically.
The optimum values l 1 ¼ l 2 ¼ 1 are substituted into eqn
(40), the effect of which is that the f equation drops out.
This means that the f equation does not provide any information and that the energy results are unlikely to be close to
the linear stability results when the terms involving f are of
any significance. We are left with three equations


R
(57)
¹ Mv ¹ R2 w ki ¹ ui ¼ Ã, i ,
2
R
]v
ˆ ¼ 0,
¹ Mw ¹ 2 þ mDv
2
]z

(58)

¹ R2 w þ Dw ¼ 0:

(59)

4.1 Linear stability
The main interest of the linear stability analysis is the nature
of the disconnected oscillatory neutral curves described in
the introduction. In the problem where the equation of state
is linear in the temperature field an analytical solution may
be obtained33 which allows the various parameter ranges to
be searched in a much less time-consuming manner than
solving the equations numerically. In the present work
attention is focussed on the situation where the salt concentration with the larger diffusion coefficient (salt field 1) is
gravitationally stable, while the salt field with the smaller
diffusion coefficient (salt field 2) is destabilising. In the
previous work the following parameters were shown to
give rise to the disconnected oscillatory neutral curves:
R1 [( ¹ 287, ¹ 285), R2 [ (259, 261), P1 ¼ 4:545454,
P2 ¼ 4:761904:
Fig. 1 shows the (R crit, R 2) stability boundary for T 1 ¼ 48C,
R 1 ¼ ¹ 286, P 1 ¼ 4.545454, P 2 ¼ 4.761904. There are
three regions of interest. To the left of the cusp (R 2 ,
260.4) there is a region of oscillatory onset. Here oscillatory instability first occurs at a smaller value of R than does
steady instability and there is a single critical value of R. To
the right of the point of infinite slope (R 2 . 261.67)
instability occurs with real growth rate. Here oscillatory
instability does not occur and again there is one critical
value of R. The intervening region is the most interesting
and is shown in the right-hand graph of Fig. 1. Here three
values of R crit are required to fully specify the linear

Penetrative convection and multi-component diffusion

405

Fig. 1. (R crit,R 2) stability boundary for T1 ¼ 48C, R 1 ¼ ¹ 286.0, P 1 ¼ 4.545454, P 2 ¼ 4.761904. The right-hand graph shows the multivalued region in more detail.

instability criteria. Oscillatory instability sets in first at the
lowest critical Rayleigh number. Then there is a region of
oscillatory instability until the middle critical Rayleigh
number is reached. At this point the system becomes

linearly stable again until the third critical Rayleigh
number is reached. Here stationary instability sets in and
the system remains linearly unstable for all higher values of
R. At higher temperatures (Fig. 2) a similar (R crit, R 2)

Fig. 2. (R crit,R 2) stability boundaries for T1 ¼ 58C, R1 ¼ ¹ 286, T1 ¼ 68C, R1 ¼ ¹ 287:5 and T 1 ¼ 78C, R 1 ¼ ¹ 291.

406

J. Tracey

Fig. 3. (R,k) neutral curves for T1 ¼ 48C, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904.

stability boundary is seen, with the presence of a multivalued region (the ‘kink-region’). A similar multi-valued
curve is found when considering R 2 fixed and varying R 1.
An explanation for these stability boundaries can be
obtained by considering the R,k neutral curves.
Figs 3 and 4 show a selection of (R,k) neutral curves for
T 1 ¼ 48C, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904. Setting
T 1 ¼ 48C means that the whole of the layer is destabilising.
For R 1 ¼ ¹ 287, ¹ 286.6 the oscillatory curve is attached
to the stationary curve at two bifurcation points. As the
bifurcation points are approached along the oscillatory
curve the frequency tends to zero. At R 1 ¼ ¹286 the bifurcation points have moved closer together and finally
coalesced and a disconnected oscillatory neutral curve has
been formed. As R 1 is increased further the oscillatory curve
moves wholly beneath the stationary curve and becomes
increasingly smaller until it collapses to a point and disappears. At R 1 ¼ ¹285.1 (not shown) only the stationary curve
is found. These results are similar to those found in Tracey33.
However, in that work the disconnected oscillatory curves
were perfectly symmetric heart-shapes. It can be clearly
seen that the oscillatory curves in the present work are
not perfectly heart-shaped—the maximum of the left hand
lobe is greater than that of the right hand lobe. For values of

R 1 ¼ ¹285.4, ¹285.3 the oscillatory curve can be seen to lie
wholly below the stationary curve and so three critical
values of the thermal Raleigh number are required to fully
specify the linear stability criteria. Oscillatory instability
sets in first at the lowest critical thermal rayleigh number.
Then there is a region of oscillatory instability until the
middle critical thermal Rayleigh number is reached. At
this point the system becomes linearly stable again until
the third critical thermal Rayleigh number is reached.
Here stationary instability sets in and the system remains
linearly unstable for all higher values of R.
Fig. 5 shows two neutral curves for T 1 ¼ 58C. This results
in the lower four-fifths of the layer being destabilising. The
departure from a perfect heart shape is clearly more pronounced than for T 1 ¼ 48C. Fig. 6 shows that for T 1 ¼ 68C
the skewness has again increased. For T 1 ¼ 78C (Fig. 7), the
increased skewness results in a new effect at smaller values
of R 1. At values of R 1 ¼ ¹292, ¹291.5 the minimum on the
oscillatory curve is clearly less than the minimum on the
stationary curve. At R 1 ¼ ¹290.1 the role of minimum has
been reversed and now the minimum on the stationary curve
is smaller than the minimum on the stationary curve. At a
value of R 1 between ¹291.5 and ¹290.1 the minimum on
each curve occurs at the same value of R. This value was

Penetrative convection and multi-component diffusion

Fig. 4. (R,k) neutral curves for T1 ¼ 48C, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904.

[In fact, for

computed to be
R1 ¼ ¹ 291:067,

R1 ¼ ¹ 291:066,

with minimum thermal Rayleigh number
Rmin ¼ 228:0107,
and corresponding wavenumbers
kosc ¼ 3:77,

kstat ¼ 4:62:

Rosc
min ¼ 228:0131,

kosc ¼ 3:77,

Rstat
min ¼ 228:0090,

kstat ¼ 4:62,

Fig. 5. (R,k) neutral curves for T1 ¼ 5C, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904.

407

408

J. Tracey

Fig. 6. (R,k) neutral curves for T1 ¼ 68C, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904.

while, for
R1 ¼ ¹ 291:067,
Rosc
min ¼ 228:0107,

kosc ¼ 3:77,

Rstat
min ¼ 228:0111,

kstat ¼ 4:62:



These results predict the onset of oscillatory instability and
the onset of stationary instability at the same value of R but
different wavenumbers.
As the upper temperature is increased, the minimum on
the oscillatory curve occurs at increasingly smaller values of
the wavenumber while the minimum on the stationary curve
moves to the right as T 1 is increased.

Fig. 7. (R, k) neutral curves for T1 ¼ 78C, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904.

Penetrative convection and multi-component diffusion

Fig. 8. Graph of j r against R for T1 ¼ 48C, R1 ¼ ¹ 285:6,
R2 ¼ 261, P1 ¼ 4.545454, P 2 ¼ 4.761904, k ¼ 2.4.

One considerable advantage of the Chebyshev tau–QZ
algorithm employed here is that it yields more eigenvalues
than just the leading one. This allows us to investigate the
behaviour of the growth rate in more detail. Over the range
of values of R indicated in Figs 3–6 the leading eigenvalues
always consist of a complex conjugate pair and a real
eigenvalue. the behaviour of the real part of the growth
rate j ¼ j r þ ij i is shown in Figs 8 and 9. In both figures
the complex conjugate pair is initially the leading eigenvalue. As R is increased j r reaches a maximum for the
complex conjugate pair and then starts to decrease. The
real eigenvalue then takes over and becomes the leading
eigenvalue.
The Chebyshev tau method also yields the
eigenfunctions. Fig. 10 shows the eigenfunctions for the

Fig. 9. Graph of j r against R for T 1 ¼ 48C, R 1 ¼ ¹ 285.6, R 2 ¼
261, P 1 ¼ 4.545454, P 2 ¼ 4.761904, k ¼ 3.1.

409

Fig. 10. Plots of eigenfunctions for T 1 ¼ 78C, R 1 ¼ 228.0107,
R 1 ¼ ¹ 291.067.6, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904,
k ¼ 4.62, i.e. at the minimum of the stationary curve. The F 1 and
F 2 eigenfunctions are identical to the Q eigenfunction.

case T 1 ¼ 78C, R ¼ 228.0107, R 1 ¼ ¹ 291.067, R 2 ¼
261, k ¼ 4.62, i.e. at the minimum on the stationary curve
when the minima on the stationary and oscillatory neutral
curves coincide. The normalised eigenfunctions F 1 and F 2
are the same as the Q eigenfunction. That this should be so
can be seen from eqns (17)–(19) with j ¼ 0. These three
equations can be solved to show that
Q ¼ RF(z),

Fa ¼ Ra F(z) (a ¼ 1, 2),

where
F(z) ¼

Zz

0

ea(y ¹ z)

Zy
0

ea(y ¹ m) W(m)dmdy:

Fig. 11. Plots of eigenfunctions for T 1 ¼ 78C, R 1 ¼ 228.0107,
R 1 ¼ ¹291.067.6, R 2 ¼ 261, P 1 ¼ 4.545454, P 2 ¼ 4.761904, k ¼
3.77, i.e. at the minimum on the oscillatory curve. The F 2 eigenfunction is very similar to the F 1 curve.

410

J. Tracey

Fig. 12. Plot of the energy stability boundary for T 1 ¼ 48C
together with the linear instability boundaries for R 1 ¼ ¹261,
and R 1 ¼ ¹100. Also P 1 ¼ 4.545454, P 2 ¼ 4.761904.

So, when Q and F a are normalised they will be identical.
At the minimum on the oscillatory curve the eigenfunctions
for W and F a (shown in Fig. 11) are similar but not identical. This can be seen by setting j r ¼ 0 in eqns (17)–(19),
obtaining
((D2 ¹ k2 )2 þ j2i )Qr ¼ R(D2 ¹ k2 )Wr ¹ ji RWi ,
((D2 ¹ k2 )2 þ Pa j2i )Far ¼ Ra (D2 ¹ k2 )Wr ¹ ji Pa Ra Wi ,
while similar equations exist for Q i, F ai .
In Figs 10 and 11 the penetrative effect (W becoming
negative) can be clearly seen.

Fig. 13. Plot of the energy stability boundary for T 1 ¼ 58C
together with the linear instability boundaries for R 1 ¼ ¹261,
and R 1 ¼ ¹100. Also P 1 ¼ 4.545454, P 2 ¼ 4.761904.

Fig. 14. Plot of the energy stability boundary for T 1 ¼ 68C
together with the linear instability boundaries for R 1 ¼ ¹287.5,
and R 1 ¼ ¹ 100. Also P 1 ¼ 4.545454, P 2 ¼ 4.761904.

4.2 Nonlinear results
In Fig. 12 the solution of the eigenvalue problem eqns (60)–
(63) for T 1 ¼ 48C is plotted together with the (R,R 2) linear
stability boundary for R 1 ¼ ¹261 and R 1 ¼ ¹100. The
energy results are clearly closer to the linear results for
smaller (in modulus) values of R 1. This is not surprising
as the terms involving the first salt concentration drop out
of the energy analysis and so do not yield any information.
Whenever the stabilizing salt field is of any consequence the
energy results may be far from the linear results. Also at
R 1 ¼ ¹100 the disconnected oscillatory curves are not
found. The energy results do have the advantage that

Fig. 15. Plot of the energy stability boundary for T 1 ¼ 78C,
together with the linear instability boundaries for R 1 ¼ ¹291,
and R 1 ¼ ¹100. Also P 1 ¼ 4.545454, P 2 ¼ 4.761904.

Penetrative convection and multi-component diffusion
they are unconditional, i.e. for perturbations of any initial
amplitude nonlinear exponential stability is assured.
At higher temperatures (Figs 13–15) a similar result can
be seen. The energy results are closer for smaller absolute
values of R 1, however at the smaller values of R 1 the
disconnected oscillatory curves are not found.

5 DISCUSSION
In the present work a linear stability analysis is presented
that yields several interesting results. The existence of disconnected oscillatory neutral curves produces a finite interval of stable values of R, the thermal Rayleigh number, as
well as the normal semi-infinite range. The penetrative
effect has skewed these oscillatory curves away from the
perfect heart-shaped curves found in Tracey (1996). In that
work it was stated that oscillatory instability could occur at
the same thermal Rayleigh number but differing wavenumbers at the maxima on the lobes of the heart-shaped curve.
The skewed nature of the oscillatory curves produced here
by the quadratic equation of state means that this effect is
not seen. McKay and Straughan (1992) argue that the density of a fluid is never a linear function of temperature.
Consequently the perfect heart-shapes are unlikely to be
observed experimentally.
Another result which may be seen experimentally is
observed here. The effect shown in Fig. 7 whereby oscillatory and stationary instability set in at two different wavenumbers but the same thermal Rayleigh number is a
completely original phenomenon in porous convection. By
increasing R 1 one should see instability change from being
initiated by oscillatory convection with wavenumber k ¼
3.77 to a stationary instability with wavenumber k ¼ 4.62.
For values of R ¼ 228, R 1 ¼ ¹291.067 the present analysis
predicts that one should find instability occurring in two
different cell sizes, namely a smaller stationary convection
cell and a larger oscillatory convection cell.
In eqns (2.5) we regarded temperature and the normal
component of velocity as being prescribed at the boundaries. These boundary conditions are discussed by Joseph
(1976). In a porous medium the fluid will stick to a solid
wall but this effect is confined to a boundary layer whose
size is measured in pore diameters. As the wall friction does
not overtly affect the motion in the interior it is reasonable
to replace the true wall with a frictionless wall in our
analysis.
Krishnamurti and Howard (1983) discuss the experimental problems of prescribing constant concentration at the
boundaries. They suggest the use of semi-permeable membranes as boundaries through which solute can pass into the
working fluid volume. If the fluid outside the membrane is
maintained at a constant concentration then the solute
boundary condition could be realised to within a good
approximation.
There are two factors in the present problem that could
give rise to subcritical instabilities. Firstly, the nonlinear

411

term that arises in the velocity equation from the quadratic
equation of state, and secondly the competition between the
different stratifying effects. These factors show the need for
the nonlinear stability analysis of section 3. Unconditional
nonlinear stability is obtained and L 2 decay is shown for the
temperature and solute perturbations. Although the use of
Darcy’s Law means that L 2 decay for the velocity cannot be
easily proved, in section 3 it is shown that the velocity
satisfies a condition which ensures ‘‘practical decay’’.
The classical kinetic energy results presented here are
somewhat disappointing in that the energy boundary may
be far away from the linear stability boundary. As explained
in section 4 this is due to the terms involving the stabilising
salt field dropping out of the analysis. To overcome this a
generalised energy method in the vein of Mulone (1994)
may be constructed. Work to this end is in progress.

ACKNOWLEDGEMENTS
This work was supported by the Engineering and Physical
Sciences Research Council through studentship number
94004400. I am very grateful to Professors Brian Straughan
and Ray Ogden of the University of Glasgow for their help
and encouragement.
REFERENCES
1. Allen, M.B. & Curran, M.C. Parallelizable methods for modelling flow and transport in heterogeneous porous media. Int.
Ser. Num. Math., 1993, 114 5–14.
2. Allen, M.B. & Khosravani, A. Solute transport via alternating
direction collocation using the modified method of characteristics. Adv. Water Res., 1992, 15 125–132.
3. Celia, M.A., Kindred, J.S. & Herrera, I. Contaminant transport and biodegradation. 1. A numerical model for reactive
transport in porous media. Water Resources Res., 1989, 25
1141–1148.
4. Chen, B., Cunningham, A., Ewing, R., Peralta, R. & Visser,
E. Two-dimensional modelling of microscale transport and
biotransformation in porous media. Num. Methods Partial
Differential Equations, 1994, 10 65–83.
5. Cheng, P. Heat transfer in geothermal systems. Adv. Heat
Transfer, 1989, 14 1–105.
6. W. Cheney, D. Kincaid, Numerical Mathematics and Computing, Brooks-Cole Publishing Co., Monterey, California,
1985.
7. Curran, M.C. & Allen, M.B. Parallel computing for solute
transport models via alternating direction collocation. Adv.
Water Resources, 1990, 13 70–75.
8. Degens, E.T., von Herzen, R.P., Wong, H.K., Deuser, W.G.
& Jannasch, H.W. Lake Kivu: structure, chemistry and
biology of an East African rift lake. Geol. Rundschau,
1973, 62 245–277.
9. George, J.H., Gunn, R.D. & Straughan, B. Patterned ground
formation and penetrative convection in porous media. Geophys. Astrophys. Fluid Dynamics, 1989, 46 135–158.
10. Griffiths, R.W. The influence of a third diffusing component
upon the onset of convection. J. Fluid Mech., 1979, 92 659–
670.
11. Griffiths, R.W. Layered double-diffusive convection in
porous media. J. Fluid Mech., 1981, 102 221–248.

412

J. Tracey

12. Hansen, U. & Yuen, D.A. Subcritical double-diffusive convection at infinite Prandtl number. Geophys. Astrophys. Fluid
Dynamics, 1989, 47 199–224.
13. Huppert, H.E. & Turner, J.S. Double-diffusive convection. J.
Fluid Mech., 1981, 106 299–329.
14. D.D. Joseph, Stability of Fluid Motions II, Springer-Verlag,
Berlin, Heidelberg, New York, 1976.
15. Krishnamurti, R. & Howard, L.N. Double-diffusive instability: Measurement of heat and salt fluxes. Bull. Am. Phys.
Soc., 1983, 28 1398.
16. McKay, G. & Straughan, B. Nonlinear energy stability and
convection near the density maximum. Acta Mechanica,
1992, 5 9–28.
17. Mulone, G. On the nonlinear stability of a fluid layer of a
mixture heated and salted from below. Continuum Mech.
Thermodyn., 1994, 6 161–184.
18. Murray, B.T. & Chen, C.F. Double-diffusive convection in a
porous medium. J. Fluid Mech., 1989, 201 144–166.
19. Nield, D.A. Onset of thermohaline convection in a porous
medium. Water Resources Res., 1968, 5 553–560.
20. D.A. Nield, A. Bejan, Convection in Porous Media, SpringerVerlag, New York, London, 1992.
21. Patil, P.R. & Rudraiah, N. Linear convection stability and
thermal diffusion of a horizontal quiescent layer of a two
component fluid in a porous medium. Int. J. Eng. Sci.,
1980, 18 1055–1069.
22. Payne, L.E., Song, J.C. & Straughan, B. Double diffusive
porous penetrative convection—thawing subsea permafrost.
Int. J. Eng. Sci., 1988, 26 797–809.
23. L.E. Payne, B. Straughan, Unconditional nonlinear stability
in penetrative convection. Geophys. Astrophys. Fluid
Dynamics 39 (1987) 57–63. (Also, Corrected and extended
numerical results. Geophys. Astrophys. Fluid Dynamics 43
(1988) 307–309).

24. Pearlstein, A.J., Harris, R.M. & Terrones, G. The onset of
convective instability in a triply diffusive fluid layer. J. Fluid
Mech., 1989, 202 443–465.
25. Poulikakos, D. Effect of a third diffusing component on the
onset of convection in a horizontal layer. Phys. Fluids, 1985,
28 3172–3174.
26. Proctor, M.R.E. Steady subcritical thermohaline convection.
J. Fluid Mech., 1981, 105 507–521.
27. Ray, R.J., Krantz, W.B., Caine, T.N. & Gunn, R.D. A model
for sorted patterned-ground regularity. J. Glaciol., 1983, 29
317–337.
28. Rubin, H. Effect of nonlinear stabilizing salinity profiles on
thermal convection in a porous medium layer. Water
Resources Res., 1973, 9 211–221.
29. Rudraiah, N., Shivakumara, I.S. & Friedrich, R. The effect of
rotation on linear and nonlinear double-diffusive convection
in a sparsely packed porous medium. Int. J. Heat Mass
Transfer, 1986, 29 1301–1317.
30. Rudraiah, N. & Vortmeyer, V.D. Influence of permeability
and of a third diffusing component upon the onset of convection in a porous medium. Int. J. Heat Mass Transfer,
1982, 25 457–464.
31. B. Straughan, The Energy Method, Stability and Nonlinear
Convection, Springer-Verlag Ser. in Appl. Math. Sci., 1992.
32. Taunton, J.W., Lightfoot, E.N. & Green, T. Thermohaline
instability and salt fingers in a porous medium. Phys.
Fluids, 1972, 15 748–759.
33. J. Tracey, Multi-component convection-diffusion in a porous
medium, Continuum Mech. Thermodyn. (1996) in press.
34. J.S. Turner, Buoyancy Effects in Fluids. Cambridge University Press, 1979.
35. Veronis, G. Penetrative convection. Astrophys. J., 1963, 137
641–663