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Dissipative instability of MHD tangential
discontinuity in magnetized plasmas with anisotropic
viscosity and thermal conductivity.
By M.S.RUDERMANy, E.VERWICHTE, R.ERD '
ELYIyy and
M.GOOSSENS
Centre for Plasma Astrophysics, K.U.Leuven, Bí3001 Heverlee, Belgium
yOn leave of Institute for Problems in Mechanics, Russian Academy of Sciences, 117526
Moscow, Russia
yyOn leave of Dept. of Astronomy, Eèotvèos University, Ludovika t'er 2, Hí1083 Budapest,
Hungary

Abstract
The stability of the MHD tangential discontinuity is studied in compressible plasmas
in the presence of anisotropic viscosity and thermal conductivity. The general disperí
sion equation is derived and solutions to this dispersion equation and stability criteria
are obtained for the limiting cases of incompressible and cold plasmas. In these two
limiting cases the effect of thermal conductivity vanishes and the solutions are only iní
fluenced by viscosity. The stability criteria for viscous plasmas are compared with those
for ideal plasmas where stability is determined by the KelviníHelmholtz velocity VKH as
a threshold for the difference in the equilibrium velocities. Viscosity turns out to have a
destabilizing influence when the viscosity coefficient takes different values at the two sides
of the discontinuity. Viscosity lowers the threshold velocity V c below the ideal Kelviní
Helmholtz velocity VKH , so that there is a range of velocities between V c and VKH where
the overstability is of a dissipative nature.
1

1. Introduction
The problem of the stability of the MHD tangential discontinuity has attracted the
attention of scientists for a few decades. This problem arises in the study of the interaction
of the solar wind with the interstellar plasma, the interaction of the solar wind with
magnetospheres of the Earth and other planets, and different magnetic configurations in
the solar atmosphere where equilibrium flows are present.
Syrovatski (1957) and Chandrasekhar (1961) studied the stability of the MHD tangení
tial discontinuity for ideal incompressible plasmas. They showed that there is a critical
value for the difference in the equilibrium velocity across the discontinuity. This critií
cal value is called the KelviníHelmholtz (KH) threshold. The tangential discontinuity is
stable for a difference in the equilibrium velocity below the KH threshold and unstable
otherwise. The instability that arises is an overstable oscillation and is called the KH
instability.
Fejer (1964) generalized the study of the KH instability to compressible plasmas. He
derived the dispersion equation and studied the particular case of a plasma that is only
slightly compressible. Subsequently many other particular cases were investigated (see,
e.g., Gerwin, 1968, McKenzie, 1970, Duhau & Gratton, 1973, Myatnizkii, 1984, Gonz'alez
& Gratton, 1994a,b, and Ruderman & Fahr, 1993, 1995). Duhau et al. (1970, 1971), Roy
Choudhury & Patel (1985), and Roy Choudhury (1986) studied the KH instability of a
tangential MHD discontinuity in plasmas with anisotropic pressure on the basis of the
Chew, Goldberger and Low equations for collisionless strongly magnetized plasmas.
So far all studies of the KH instability were carried out for ideal plasmas. However
an ideal plasma is an idealization. In real plasmas viscosity, finite resistivity and thermal
conductivity are present. In addition an exact discontinuity in the equilibrium velocity
cannot exist in a viscous plasma under general conditions. Therefore in general it does not
make sense to consider the stability of MHD tangential discontinuity in viscous plasmas.
2

However there are two exceptions to this general observation. The first exception occurs
when the coefficient of isotropic viscosity is much larger at one side of the surface of disí
continuity than at the other side. This makes it possible to take the viscosity coefficient to
be equal to zero at one side of the surface of discontinuity. A difference in the equilibrium
velocity can exist across such a surface. Such an idealized situation was considered by
Ruderman & Goossens (1995) when they studied the viscous instability of a tangential
discontinuity in an incompressible plasma. The second exception is related to a more
realistic situation. In space plasmas the magnetic field is very often sufficiently strong to
make the ion gyrofrequency ! ci much larger than the inverse mean collisonal time of ions
œ \Gamma1
i . If in addition the plasma can be considered as collisional (that is if all characteristic
scales are much larger than the mean free path of the ions), then viscosity is described
by Braginskii's tensorial expression (see Braginskii, 1965). This tensorial expression for
viscosity containes five terms. The ratio of the sum of the four last terms to the first
term is of the order of œ i ! ci . When œ i ! ci AE 1 the first term of the Braginskii's tensorial
expression very often gives a good approximation for viscosity. As a result the tensor
of viscosity is highly anisotropic. A simple derivation of the highly anisotropic tensor of
viscosity based on a qualitative physical analysis was given by Hollweg (1985). He also
gives estimations of œ i ! ci for typical conditions in the soalr corona. In accordance with
these estimations œ i ! ci ‹ 3 \Theta 10 5 in an active coronal region and œ i ! ci ‹ 7 \Theta 10 5 near
the base of a coronal hole. A property of the highly anisotropic tensor of viscosity is that
it allows a jump in the velocity across a magnetic surface since a strong magnetic field
causes charged particles to rotate around the magnetic field lines thus preventing particle
diffusion across the magnetic field lines. This implies that there is not any momentum
flux across the magnetic surfaces and different layers of plasma can slide with respect to
each other along a magnetic surface without friction.
The thermal conductivity of plasmas is due mainly to the electrons. The expression
for heat flux involves three coefficients, denoted ß k , ß? , and ßÓ by Braginskii (1965).
3

The following estimates are valid: ß?=ß k ¦ (œ e ! ce ) \Gamma2 , ßÓ=ß k ¦ (œ e ! ce ) \Gamma1 , where œ e is the
mean collisional time of electrons, and ! ce is the electron gyrofrequency. If the Coulomb
logarithm is taken to be 20, we have
œ e ‹ 10 4 T 3=2
e n \Gamma1
e s ; (1)
where T e is electron temperature and n e is electron concentration (CI units will be used
throughout). Following to Hollweg (1985) we take T e = 2 \Theta 10 6 K, n e = 3 \Theta 10 15 m \Gamma3 ,
and B = 50 G in an active region of the solar corona. Then œ e ‹ 10 \Gamma2 s, and œ e ! ce ‹ 10 7 .
Near the base of a coronal hole one might have T e = 10 6 K, n e = 10 14 m \Gamma3 , and B = 10 G.
Then œ e ‹ 10 \Gamma1 s and œ e ! ce ‹ 2 \Theta 10 7 .
Thus the parts of the heat flux vector that involves ß? and ßÓ can often been neglected,
which means that only the heat flux in the direction of magnetic field is taken into account.
As a result we obtain strongly anisotropic thermal conductivity.
The relative importance of viscosity and thermal conductivity is characterized by the
Prandtl number P r = j 0 kB=m p ß k , where j 0 is the largest coefficient of viscosity in the
Braginskii's expression for the viscosity tensor, m p is the proton mass, and kB is the
Boltzmann constant. Braginskii (1965) gives the following approximate expressions for j 0
and ß k :
j 0 ‹ kB n i T i œ i ; ß k ‹ 3k 2
B m \Gamma1
e n e T e œ e ; (2)
where T i is the temperature of ions and m e is the mass of electron. Taking n e ‹ n i and
T e ‹ T i and using (??) we obtain
P r ‹ m e œ i
3m p œ e
‹ 1
3
s
m e
m p
‹ 10 \Gamma2 : (3)
(??) shows that in plasma which consists of electrons and protons with approximately
equal temperatures P r œ 1. However this estimate leads to the conclusion that electron
thermal conductivity is more important dissipative process than ion viscosity only if fi ? ¦ 1,
where fi is the ratio of plasma pressure to magnetic pressure. In what follows we shall see
that for problem of stability of MHD tangential discontinuity the relative importance of
4

viscosity and thermal conductivity is characterized by the product fi \Gamma1 P r rather then P r
if fi ! ¦ 1. With the use of the same values of parameters of plasma and magnetic field that
we have taken in the previous discussion, we obtain that fi ‹ 0:016 in an active coronal
region and fi ‹ 0:007 near the base of a coronal hole. Hence typically in the solar corona
fi \Gamma1 P r ¦ 1, so that viscosity and thermal conductivity are of the same inportance.
The relative importance of viscosity and resistivity is characterized by the magnetic
Prandtl number Pm = j 0 =m p n i Ö, where Ö is the coefficient of magnetic diffusion. We take
the standard expression for Ö in fully ionized plasmas to obtain
Pm ‹ 3 \Theta 10 \Gamma10 œ i œ e n e T i : (4)
Substituting into (??) the values of parameters for an active region of the solar corona
and for the basis of a coronal hole, we get Pm ‹ 10 10 in the both cases. This estimate
shows that dissipation due to resistivity can be neglected in comparison with dissipation
due to viscosity.
Finally, the relative importance of the Hall effect to viscosity is characterized by the
dimensionless parameter œ e ! ce P \Gamma1
m . Using values of œ e ! ce and Pm calculated previously,
we obtain œ e ! ce P \Gamma1
m ‹ 10 \Gamma3 both in active region of the solar corona and near the basis
of a coronal hole. This estimate implies that we can neglect the Hall effect in comparison
with the effect of viscosity.
These observations based on dimensional analysis lead us to study the stability of an
MHD tangential discontinuity in a viscous thermal conductive plasma. We only retain the
first term in the Braginskii's expression for the tensor of viscosity and only take parallel
termal conductivity into account.
The concept of negative energy waves turned out to be very fruitful for the study of
dissipative instabilities. For instance, Ruderman & Goossens (1995) have used this coní
cept to give an interpretation of the viscous instability of an MHD tangential discontinuity
in terms of negative energy waves. The concept of negative energy waves is based on the
5

energy equation
dE
dt
= \GammaD ; (5)
where E is the soícalled linear part of wave energy and D is the dissipative function
(a detailed discussion of the concept of negative energy waves in hydrodynamics can be
found e.g. in Ostrovskii, Rybak & Tsimring, 1986). The functions E and D are not Galileií
invariant in the sense that they depend on the choice of coordinate system moving parallel
to the discontinuity. If we choose a moving coordinate system such that D ? 0, then (??)
shows that the linear part of the wave energy E decreases owing to dissipation. In case of
a monochromatic perturbation E takes the form E = Ea 2 , where a is the wave amplitude.
When E ? 0 the wave is called a positive energy wave. In accordance with (??) its
amplitude a decreases so that dissipation results in wave damping. When E ! 0 the wave
is called a negative energy wave. In accordance with (??) its amplitude a increases so
that dissipation leads to instability.
When dissipation is only present at one side of the tangential discontinuity, the choice
of the moving coordinate system in which D ? 0 is very simple. The unperturbed plasma
must be at rest at the side of discontinuity where dissipation is present in this coordinate
system. When dissipation is present at both sides of the discontinuity the choice of the
moving coordinate system where D ? 0 is much more complicated as it depends on the
ratio of the dissipative coefficients and on other plasma parameters. In fact the choice
of the coordinate system turns out to be more complicated than the study of dissipative
instability itself. In the present paper we study the stability of the MHD tangential
discontinuity with dissipation present at both sides of the discontinuity. Therefore we do
not use the concept of negative energy waves.
The paper is organized as follows. In x2 we present the set of dissipative MHD equaí
tions and boundary conditions that are used to study the stability of the MHD tangential
discontinuity. In x3 we derive the dispersion equation governing the stability of MHD taní
gential discontinuity under the assumption that perturbations are only slightly damped
6

during one wave period. In x4 and x5 we present solutions to the dispersion equation for
incompressible plasmas and cold plasmas. In x6 we present physical consideration of ideal
and dissipative instabilities in an incompressible plasma. In x7 we summarize our results.
2. Dissipative MHD equations and boundary conditions.
We consider a collisional oneífluid model of viscous thermal conductive plasmas. As
explained in the Introduction we only retain the first term in Braginskii's expression for
the tensor of viscosity and only take the parallel heat flux into account. The expressions
for the tensor of viscosity Ó
‹ and the heat flux q take the simple form:
Ó ‹ = ae 0 Ú
`
b\Omega b \Gamma 1
3
Ó
I
'
f3b:r(b:v 0 ) \Gamma r:v 0 g ; (6)
q = \Gammaß k b(b:rT 0 ) : (7)
Here Ú = j 0 =ae 0 is the kinematic coefficient of viscosity, B is the magnetic induction, ae is
the density, v is the velocity, and T is the temperature. b = B 0 =B 0 is the unit vector
along the equilibrium magnetic field, Ó
I is the unit
tensor,\Omega denotes the tensor product,
the subscript `0' refers the equilibrium quantity, and an accent denotes an Eulerian perí
turbation of any quantity. The coefficients Ú and ß k depend on the equlibrium density
and temperature. As the equlibrium density and temperature can be different at the two
sides of the discontinuity, so can Ú and ß k . For the present investigation the fact that Ú
can differ on the two sides of the discontinuity will be important. Equations (??) and
(??) are the linearized expressions for Ó ‹ and q as we only consider the linear stability of
tangential discontinuities.
?From a physical point of view the viscosity tensor (??) is characterized by the property
that at any magnetic surface the viscous stresses are normal to the surface. The expression
(??) means that the heat flux is directed along the magnetic field.
The unperturbed state is characterized by an MHD tangential discontinuity at z =
0, and all equilibrium quantities are constant at both sides of the discontinuity. The
7

equilibrium magnetic field B 0 and velocity v 0 are parallel to the plane of the discontinuity.
With the aid of (??) and (??) the linear equations of viscous thermal conductive MHD
can be written as
@ae 0
@t
+ ae 0 r:v 0 + v 0 :rae 0 = 0 ; (8)
@v 0
@t
+ (v 0 :r)v 0 = \Gamma 1
ae 0
rp 0 + 1
ïae 0
(r \Theta B 0 ) \Theta B 0
+Úfb(b:r) \Gamma 1
3 rgf3b:r(b:v 0 ) \Gamma r:v 0 g ; (9)
@B 0
@t
= r \Theta (v 0 \Theta B 0 + v 0 \Theta B 0 ) ; (10)
@p 0
@t + v 0 :rp 0 + flp 0 r:v 0 = (fl \Gamma 1)ß k (b:r) 2 T 0 ; (11)
p 0
p 0
= ae 0
ae 0
+ T 0
T 0
: (12)
Here p is the pressure, ï the magnetic permiability, and fl the adiabatic index.
The perturbed surface of the discontinuity is defined by the equation z = j(t; x; y).
The kinematic boundary conditions and the condition of continuity of stresses have to be
satisfied at this surface. In the linear approximation these boundary conditions can be
imposed at the unperturbed surface z = 0.
The linearized kinematic boundary conditions are:
w 1 = @j
@t
+ v 01 :rj ; w 2 = @j
@t
+ v 02 :rj ; (13)
where w is the z component of the velocity, and the subscript `1' and `2' refer to quantities
at z ! 0 and z ? 0 respectively.
Stresses at the discontinuity can be found with the use of (??) and take the form
F = \GammanfP 0 + ae 0 Ú(b:r(b:v 0 ) \Gamma 1
3
r:v 0 )g ; (14)
where n = (0; 0; 1) is the unit vector normal to the unperturbed surface of the discontií
nuity, and P 0 = p 0 + B 0 (b:B 0 )=ï is the Eulerian perturbation of the total pressure. The
condition of continuity of stresses can then be written as
ß
P 0 + ae 0 Úfb:r(b:v 0 ) \Gamma 1
3 r:v 0 g
Ö
= 0 ; (15)
8

where the square brackets denote the jump in a quantity across the discontinuity.
Condition (??) does not contain derivatives of the x and y components of the perturbed
velocity with respect to z. This implies that the x and y components of the perturbed
velocity can have jumps across the discontinuity in agreement with the discussion in the
Introduction.
The equations (??)--(??) and boundary conditions (??) and (??) are the basic equaí
tions for the study of the dissipative instability of the tangential discontinuity in the next
Sections.
3. Derivation of dispersion equation.
In order to derive the dispersion equation that governs the stability of the MHD
tangential discontinuity we Fourieríanalyze the perturbed quantities and take them to be
proportional to expfi(k:r \Gamma !t)g, where k = (k x ; k y ; 0), r = (x; y; z), k is real, and ! is
complex. This enables us to rewrite the equations (??)--(??) as
ae 0
dw
dz + iae 0 k:v 0
k \Gamma
i\Omega ae 0 = 0 ; (16)
\Omega v 0
k = k
ae 0
P 0 \Gamma B 0
ïae 0
(k:b)B 0
k \Gamma Úfb(b:k) \Gamma 1
3 kgQ ; (17)
\Omega w = \Gamma i
ae 0
dP 0
dz
\Gamma B 0
ïae 0
(k:b)B 0
z \Gamma iÚ
3
dQ
dz
; (18)
\Omega B 0
k = B 0 b(k:v 0
k ) \Gamma B 0 v 0
k (k:b) \Gamma iB 0 b
dw
dz
; (19)
\Omega B 0
z = \GammaB 0 w(k:b) ; (20)
\Omega p 0 + c 2
s ae 0
/
i dw
dz
\Gamma (k:v 0
k )
!
= \Gammai¼ (k:b) 2
fl \Gamma 1 (flp 0 \Gamma c 2
s ae 0 ) : (21)
In these equations we have introduced the components of the perturbed velocity and the
perturbed magnetic field that are parallel to the unperturbed plane of discontinuity
v 0
k = (u 0 ; v 0 ; 0) ; B 0
k = (B 0
x ; B 0
y ; 0) ; (22)
9

and the Doppleríshifted frequency
\Omega = ! \Gamma k:v 0 : (23)
The square of the sound speed c s is determined as c 2
s = flp 0 =ae 0 . The quantities Q and ¼
are given by
Q = 3i(k:b)(b:v 0
k ) \Gamma i(k:v 0
k ) \Gamma dw
dz
(24)
¼ = m p (fl \Gamma 1) 2 ß k
2flk B ae 0
; (25)
In these equations m p is the proton mass and kB is the Boltzmann constant. (??) was
derived from (??) by use of (??) and the fact that the equilibrium pressure, density and
temperature p 0 , ae 0 , and T 0 are related by the ideal gas law for fully ionized plasmas
p 0 = 2kB
m p
ae 0 T 0 : (26)
The Prandtl number P r measures the relative importance of viscosity and thermal
conductivity, and the Reynolds number Re = V h =Úk measures the importance of viscosity
compared to inertia. Here V h is a characteristic velocity. In what follows we assume that
Re AE 1 and P rRe AE 1. These inequalities imply that we restrict the analysis to linear
motions that are only slightly damped during a wave period. This enables us to use a
perturbation method and to treat the terms proportional to Ú and ¼ in equations (??),
(??), and (??) as small in comparison to the terms that are already present in the ideal
theory. The terms proportional to Ú and ¼ are evaluated with the use of relations obtained
in ideal MHD. All but two variables are eliminated from equations (??)--(??) and a set of
two firstíorder linear ordinary differential equations are obtained for w and P 0
w =
\Gamma\Omega ae 0 A
/
i +
Ú\Omega f\Omega 2 \Gamma 3c 2
s (k:b) 2 g
3(c 2
s + v 2
A )C
!
dP 0
dz
; (27)
dw
dz
=
\Omega ae 0 (c 2
s + v 2
A )C
(
iD
A
+
Ú\Omega 3
f\Omega 2 \Gamma 3c 2
s (k:b) 2 g
/
2c 2
s (k:b) 2
(c 2
s + v 2
A )C \Gamma k 2 \Gamma (k:b) 2
A
!
\Gamma¼\Omega
3 c 2
s (k:b) 2
(c 2
s + v 2
A )C
)
P 0 : (28)
10

The quantities A, C, and D take the form
A
=\Omega 2 \Gamma v 2
A (k:b) 2 ; C
=\Omega 2 \Gamma c 2
T (k:b) 2 ;
D
=\Omega 4 \Gamma (c 2
s + v 2
A )k
2\Omega 2 + c 2
s v 2
A k 2 (k:b) 2 ;
9
? =
? ;
(29)
and the squares of the Alfv'en and cusp speeds are given by
v 2
A = B 2
0
ïae 0
; c 2
T = c 2
s v 2
A
c 2
s + v 2
A
: (30)
In addition we obtain the approximate expression for Q:
Q = \Gamma
i\Omega f\Omega 2 \Gamma 3c 2
s (k:b) 2 g
ae 0 (c 2
s + v 2
A )C P 0 : (31)
Fourieríanalysis reduces the boundary conditions (??) and (??) to
w 1 =
\Gammai\Omega 1 j ; w 2 =
\Gammai\Omega 2 j ; (32)
ß
P 0 + Ú
3 Q
Ö
= 0 : (33)
Elimination of P 0 from (??) and (??) leads to a single secondíorder ordinary differential
equation for w
d 2 w
dz 2
\Gamma \Gamma 2 w = 0 ; (34)
where
\Gamma 2 = \Gamma 2
0 (1 + iÚK Ú + i¼K ¼ ) ; \Gamma 2
0 = \Gamma D
(c 2
s + v 2
A )C ;
K Ú
=\Omega Af\Omega 2 \Gamma 3c 2
s (k:b) 2 g 2
3(c 2
s + v 2
A )CD ; K ¼
=\Omega
3 Ac 2
s (k:b) 2
(c 2
s + v 2
A )CD :
9
? ? ? ? ? ? =
? ? ? ? ? ? ;
(35)
To ensure that the perturbations vanish far away from the discontinuity we have to
impose that \Gamma 2
0 ? 0. The assumptions Re AE 1, PeRe AE 1 lead to jÚK Ú j œ 1, j¼K ¼ j œ 1,
and subsequently to !(\Gamma 2 ) ? 0, where ! denotes the real part of a quantity. The solutions
to (??) that vanish at z ! \Gamma1 and z ! +1 respectively and satisfy the boundary
conditions (??) are
w 1 =
\Gammai\Omega 1 je \Gamma 1 z ; w 2 =
\Gammai\Omega 2 je \Gamma\Gamma 2 z ; (36)
11

where we impose that !(\Gamma 1;2 ) ? 0. The expressions for the Eulerian perturbation of total
pressure to the left (z ! \Gamma0) and the right (z ! +0) of the discontinuity immediately
follow from (??). They are
P 1 = j ae 01 A 1
\Gamma 01
`
1 \Gamma iÚ 1
2 K Ú1 \Gamma i¼ 1
2 K ¼1
'
\Gamma Ú 1
3 Q 1 ;
P 2 = \Gammaj ae 02 A 2
\Gamma 02
`
1 \Gamma iÚ 2
2 K Ú2 \Gamma i¼ 2
2 K ¼2
'
\Gamma Ú 2
3 Q 2 :
9
? ? ? ? =
? ? ? ? ;
(37)
Application of the boundary condition for the stresses (??) then gives the desired disperí
sion equation
F (!; k) j F I (!; k) + iFD (!; k) = 0 ; (38)
where
F I (!; k) = ae 01 A 1
\Gamma 01
+ ae 02 A 2
\Gamma 02
; FD (!; k) = S Ú + S ¼ ; (39)
and
S Ú = ae 01 Ú 1 A 1
2\Gamma 01
K Ú1 + ae 02 Ú 2 A 2
2\Gamma 02
K Ú2 ;
S ¼ = ae 01 ¼ 1 A 1
2\Gamma 01
K ¼1 + ae 02 ¼ 2 A 2
2\Gamma 02
K ¼2 :
9
? ? ? ? ? =
? ? ? ? ? ;
(40)
Here \Gamma 01 ? 0, \Gamma 02 ? 0. As already stated the dispersion equation (??) has been derived
by using a perturbation method so that terms proportional to Ú 2 , Ú¼, and ¼ 2 are systemí
atically neglected. The notations in (??) are obvious. The indices I and D to F I and
FD denote the ideal and dissipative parts of the leftíhand side of the dispersion equation.
When dissipation is absent so that FD = 0, the dispersion equation (??) coincides with
the dispersion equation obtained by Fejer (1964).
The objective of the paper is to show that dissipation can cause overstability of the
MHD tangential discontinuity which is stable in the absence of dissipation. We are thereí
fore interested in a situation where the discontinuity is stable in ideal MHD. The solution
to the ideal dispersion equation is a real frequency, ï
! (F I (ï!) = 0), which corresponds to
an oscillation. Dissipation produces a small imaginary correction, ifl, to the real frequency
ï
! that leads to damping or growth of the oscillation. The approximate solution to (??)
12

can be written as
! = ï
! + ifl ; fl = FD (ï!)
@F I
@!
: (41)
This approximate solution and in particular the expression for the increment or decrement
fl are obtained under the condition that jflj œ jï!j. The denominator @F I
@! in (??) is
calculated at ! = ï
!. fl ? 0 corresponds to an overstable oscillation, i.e. oscillation of
which the amplitude grows exponentially in time on a time scale fl \Gamma1 .
Now we can varify the ad hoc estimate given in Introduction, that the relative imporí
tance of viscosity and thermal conductivity is characterized by the dimensionless paramí
eter fi \Gamma1 P r when fi ! ¦ 1. It follows from (??) and (??) that contributions of viscosity and
thermal conductivity to fl are proportional to S Ú and S ¼ respectively. The relative imí
portance of viscosity and thermal conductivity is characterized by the ratio S Ú =S ¼ . With
the use of (??) and (??) we obtain the estimate
S Ú
S ¼
¦
Ú\Omega 2
¼c 2
s k 2 : (42)
When fi ! ¦ 1, so that c 2
s
! ¦ v 2
A , we
have\Omega 2 ¦ v 2
A k 2 . Substituting this estimate into (??) we
finally obtain S Ú =S ¼ ¦ fi \Gamma1 P r, which means that the estimate taken ad hoc in Introducí
tion is valid.
4. Incompressible plasma
In this section the general results of the previous section are applied to the study
of the stability of the MHD tangential discontinuity in an incompressible plasma. The
approximation of incompressibility is valid in plasmas where the sound velocity is much
larger than the Alfv'en velocity (c 2
s AE v 2
A ). This approximation corresponds to taking
the limit c 2
s =v 2
A ! 1, so that acoustic signals travel with infinite speed as it were. In
13

particular we obtain for F I , S Ú , and S ¼ by means of this procedure
kF I = ae
01\Omega 1 + ae
02\Omega 2 \Gamma ae 01 v 2
A1 (k:b 1 ) 2 \Gamma ae 02 v 2
A2 (k:b 2 ) 2 ;
S Ú = \Gamma 3
4k 2
fae 01 Ú
1\Omega 1 (k:b 1 ) 4 + ae 02 Ú
2\Omega 2 (k:b 2 ) 4 g ;
S ¼ = 0 :
9
? ? ? ? ? =
? ? ? ? ? ;
(43)
It is not surprising that S ¼ vanishes for an incompressible plasma since it describes disí
sipation related to thermal conductivity. This type of dissipation comes from the energy
equation which is decoupled from the other MHD equations in an incompressible plasma.
In what follows we use the difference in velocity across the discontinuity surface V =
v 02 \Gamma v 01 and the angles ' 1 , ' 2 , and / between V and b 1 , b 2 , and k respectively. These
angles are determined by the conditions V 2 cos 2 ' 1;2 = (V:b 1;2 ) 2 , k 2 V 2 cos 2 / = (k:V) 2 ,
j' 1;2 j ß ‹
2 , and j/j ß ‹
2 . An angle is considered positive when measured counterclockwise
and negative otherwise (see figure 1). In figure 1 ' 1 ? 0, ' 2 ! 0, and / ? 0.
We first consider the tangential discontinuity in ideal MHD since we want to start from
a situation that is stable in ideal MHD and to see how dissipation can make it unstable.
The ideal dispersion equation F I (!) = 0 has two roots that can be written as
! \Sigma =
k:(ae 01 v 01 + ae 02 v 02 ) \Sigma k cos /
q
ae 01 ae 02 (V 2
KH \Gamma V 2 )
ae 01 + ae 02
; (44)
where the KelviníHelmholtz (KH) threshold velocity VKH is determined as
V 2
KH = (ae 01 + ae 02 )fae 01 v 2
A1 cos 2 (' 1 \Gamma /) + ae 02 v 2
A2 cos 2 (' 2 \Gamma /)g
ae 01 ae 02 cos 2 /
: (45)
If
V 2 ? V 2
KH ; (46)
=(!+ ) ? 0 (= denotes the imaginary part) and the discontinuity suffers the ideal KH
instability. When inequality (??) is satisfied the oscillations that propagate at the angles
\Sigma/ with respect to V have an amplitude that grows in time and are thus KH unstable,
so that (??) is a local criterion of KH instability.
In a real situation there are perturbations propagating in all directions. The tangential
discontinuity becomes unstable as soon as the local instability criterion (??) is satisfied for
14

at least one value of the angle /. This implies that the discontinuity is unstable whenever
V 2 is larger than the minimal value of the rightíhand side of (??) with respect to /. The
condition for instability can then be written as
V 2 ? V 2
KH j
(ae 01 + ae 02 )v 2
A1 v 2
A2 sin 2 (' 1 \Gamma ' 2 )
ae 01 v 2
A1 sin 2 ' 1 + ae 02 v 2
A2 sin 2 ' 2
: (47)
(??) is a global critetion of KH instability and V KH is the global KH threshold velocity.
When ' 1 = ' 2 or ' 1 = ' 2 \Sigma ‹
2 we get ï
VKH = 0 and the tangential discontinuity is
unstable for any value of V . The case ' 1 = ' 2 = 0 (vectors b 1 , b 2 , and V are collinear)
is special. The rightíhand side of (??) is now independent of / and equal to V 2
KH which
is determined by
V 2
KH = (ae 01 + ae 02 )(ae 01 v 2
A1 + ae 02 v 2
A2 )
ae 01 ae 02
: (48)
The criteria for local and global KH instabilities are the same and coincide with those
given by Syrovatskii (1957) and Chandrasekhar (1961).
The objective of the paper is to show that dissipation contrary to intuition can cause
instabilities for differences in the equilibrium velocity that are smaller than the KH threshí
old velocity VKH . For that reason we restrict the analysis to V 2 ! V 2
KH in an attempt
to study the local instability (for fixed direction of k) that is due to the presence of
dissipation. With the use of (??) and (??) we get
fl \Sigma = \Gamma 3k 2
8(ae 01 + ae 02 )
`
ae 01 Ú 1 cos 4 (' 1 \Gamma /) + ae 02 Ú 2 cos 4 (' 2 \Gamma /)
\Sigma ae 01 ae 02 V fÚ 1 cos 4 (' 1 \Gamma /) \Gamma Ú 2 cos 4 (' 2 \Gamma /)g
q
ae 01 ae 02 (V 2
KH \Gamma V 2 )
'
; (49)
where the subscripts \Sigma correspond to ! \Sigma . It is easy to see that fl + and fl \Gamma cannot be
positive simultaneously. The condition that either fl + ? 0 or fl \Gamma ? 0, so that one of the
two waves that propagate in the directions \Sigmak is unstable, can be written as
V 2 ? V 2
c j \Phi(/)V 2
KH ; (50)
where
\Phi(/) = fae 01 Ú 1 cos 4 (' 1 \Gamma /) + ae 02 Ú 2 cos 4 (' 2 \Gamma /)g 2
(ae 01 + ae 02 )fae 01 Ú 2
1 cos 8 (' 1 \Gamma /) + ae 02 Ú 2
2 cos 8 (' 2 \Gamma /)g : (51)
15

The function \Phi is important since it determines whether the threshold velocity for dissií
pative instability is smaller or larger than the ideal KH threshold velocity. It is easy to
show that \Phi ß 1, i.e. V 2
c ß V 2
KH which implies that there is a range of velocities ]V c ; VKH [
in which the instability is caused by the action of viscosity. The function \Phi(/) takes its
minimal value fmin(ae 01 ; ae 02 )=(ae 01 + ae 02 )g for cos(' 1 \Gamma /) = 0 (k:b 1 = 0) if ae 01 ? ae 02 , and
for cos(' 2 \Gamma /) = 0 (k:b 2 = 0) if ae 01 ! ae 02 . The function \Phi(/) takes its maximal value 1
for cos 2 (' 1 \Gamma /)= cos 2 (' 2 \Gamma /) =
q
Ú 2 =Ú 1 .
In general we cannot find an analytical expression for the minimal value of the rightí
hand side of (??), and thus cannot obtain the global criterion for dissipative instability
in a closed form. We consider two special cases. Let us first assume that dissipation is
only present at one side of the discontinuity. Without loss of generality we can assume
that viscosity is only present below the plane of discontinuity (z ! 0), so that Ú 2 = 0. \Phi
is independent of / and takes the form
\Phi = ae 01
ae 01 + ae 02
: (52)
The global criterion of the dissipative instability is
V 2 ? V 2
c j ae 01
ae 01 + ae 02
V 2
KH : (53)
When the vectors b 1;2 and V are collinear (' 1 = ' 2 = 0) this result coincides with that
obtained by Ruderman & Goossens (1995) who considered the dissipative instability of
the MHD tangential discontinuity in an incompressible plasma with isotropic viscosity
at one side of the discontinuity. This result supports the statement by Ruderman &
Goossens (1995) that the threshold for the dissipative instability is independent of the
type of viscosity that is present at one side of the discontinuity.
Let us now look at the case where the vectors b 1;2 and V are collinear (' 1 = ' 2 = 0).
Again \Phi is independent of / and the global criterion for the dissipative instability is
V 2 ? V 2
c j (ae 01 Ú 1 + ae 02 Ú 2 ) 2
(ae 01 + ae 02 )(ae 01 Ú 2
1 + ae 02 Ú 2
2 )
V 2
KH ; (54)
16

where V 2
KH is determined by (??). The threshold velocity for global dissipative instability
V c satisfies the inequality V c ß V KH and depends on the ratio Ú 1 =Ú 2 . When Ú 1 = Ú 2 we
obtain V c = V KH . For all other combinations of Ú 1 and Ú 2 V c ! V KH , so that there is a
range of differences in the equilibrium velocity V where the MHD tangential discontinuity
is unstable when viscosity is present and stable in ideal MHD.
5. Cold plasma
In this section we consider the stability of the MHD tangential discontinuity in a cold
plasma. For the sake of simplicity we assume that ' 1 = ' 2 = 0 (so that the vectors b 1;2
and V are collinear), ae 01 = ae 02 = ae 0 , and vA1 = vA2 = vA (so that there is only a jump in
v 0 ). The expressions for F I , S Ú , and S ¼ are obtained by taking the limit c 2
s =v 2
A ! 0 and
reduce to
F I = ae 0 vA
0
@\Omega
2
1 \Gamma k 2 v 2
A cos 2 /
q
k 2 v 2
A
\Gamma\Omega 2
1
+\Omega
2
2 \Gamma k 2 v 2
A cos 2 /
q
k 2 v 2
A
\Gamma\Omega 2
2
1
A ;
S Ú = \Gamma ae 0
6vA
/
Ú
1\Omega 1(\Omega
2
1 \Gamma k 2 v 2
A cos 2 /) 2
(k 2 v 2
A
\Gamma\Omega 2
1 ) 3
2
\Gamma Ú
2\Omega 2(\Omega
2
2 \Gamma k 2 v 2
A cos 2 /) 2
(k 2 v 2
A
\Gamma\Omega 2
2 ) 3
2
!
;
S ¼ = 0:
9
? ? ? ? ? ? ? ? ? =
? ? ? ? ? ? ? ? ? ;
(55)
Here also the result S ¼ = 0 is what we expect since the temperature is equal to zero in a
cold plasma and the finite thermal conductivity does not operate.
As in Section 4 the tangential discontinuity is first considered in ideal MHD. The
equation F I (!) = 0 can be transformed to the algebraic equation
(\Omega 2
1
\Gamma\Omega 2
2
)f\Omega 2
1\Omega
2
2 \Gamma k 2 v 2
A\Omega
2
1 \Gamma k 2 v 2
A\Omega
2
2 + k 4 v 4
A cos 2 /(1 + sin 2 /)g = 0: (56)
(??) has been obtained from F I (!) = 0 with the use of squaring. Therefore (??) can
have spurious roots that do not satisfy F I (!) = 0. The root of the first factor in (??) is
spurious. The roots of the second factor are
X 2
\Sigma = k 2
`
v 2
A + 1
4 V 2 cos 2 / \Sigma vA
q
v 2
A sin 4 / + V 2 cos 2 /
'
; (57)
17

where
X = ! \Gamma 1
2 k:(v 01 + v 02 ) : (58)
When
4v 2
A ! V 2 ! 4v 2
A (1 + 2 tan 2 /) (59)
we have X 2
\Gamma ! 0 . The root of (??) is then
! = 1
2 k:(v 01 + v 02 ) + i
q
\GammaX 2
\Gamma (60)
and has a positive imaginary part. This root (??) is not spurious and satisfies the equation
F I (!) = 0. The existence of the root (??) implies that (??) is the local criterion (for fixed
k or /) for KH instability. The global criterion for KH instability is given by the left
inequality (??). The right inequality (??) shows that for V ? V KH j 2vA the waves that
propagate at angles / restricted by the inequality
j/j ? arctan
v u u t V 2 \Gamma 4v 2
A
8v 2
A
(61)
are unstable.
This result is in apparent contrediction with results obtained by Duhau & Gratton
(1973). These authors studied the KH instability of the same magnetic plasma configuí
ration but they took the finite plasma pressure into account. They in particular obtained
that in case where c 2
s ! v 2
A the MHD tangential discontinuity is unstable in two velocity
ranges: V ? 2vA , and J 1 vA ! V ! J 2 vA , where J 2 ! 2. J 1 ! 1 and J 2 !
p
2 when
c s ! 0, which means that the MHD tangential discontinuity in a cold plasma is unstaí
ble not only for V ? 2vA , but also for vA ! V ! vA
p
2. This apparent contradiction
is easily removed. The point is that the increment of the instability that is present for
J 1 vA ! V ! J 2 vA is of the order of c s k, so that this increment vanishes when c s ! 0. As
a result the tangential discontinuity is stable for vA ! V ! vA
p
2 in complete agreement
with the result of the present section.
However in reality a situation with c s = 0 does not exist, so that from a physical point
of view the approximation of a cold plasma implies that c 2
s œ v 2
A rather than c s = 0.
18

In what follows we shall study the dissipative instability of the tangential discontinuity
that takes place for V ! 2VA . There is no ideal instability of the tangential discontinuity
in the velocity range vA
p
2 ! V ! 2vA and the dissipative instability is important no
matter how small its increment is. In the velocity range vA ! V ! vA
p
2 there is an ideal
instability with a small increment of the order of c s k, so that the dissipative instability is
important only if its increment is much larger than c s k.
In order to study the instability caused by dissipation we limit the analysis to the
situation when the discontinuity is stable in ideal MHD. Hence we take V ! 2vA so that
the equation F (!) = 0 has four real roots that are determined by (??) and (??). However
only the two roots
! (\Gamma)
\Sigma = 1
2 k:(v 01 + v 02 ) \Sigma X \Gamma (62)
satisfy the equation F (!) = 0. The two remaining roots
! (+)
\Sigma = 1
2 k:(v 01 + v 02 ) \Sigma X+
are spurious. In what follows we only consider the roots ! (\Gamma)
\Sigma , so that we omit the
superscript (\Gamma). The expressions for the instability increment take the form
fl \Sigma = k 2 cos 2 /(k:V)(4v 2
A \Gamma V 2 )\Psi \Sigma
24X 2
\Gamma (2v 2
A sin 2 / + V 2 cos 2 / + 2vA
q
v 2
A sin 4 / + V 2 cos 2 /)
; (63)
where
\Psi \Sigma = Ú
1\Omega 1(\Omega
2
2 \Gamma v 2
A k 2 cos 2 /) \Gamma Ú
2\Omega 2(\Omega
2
1 \Gamma v 2
A k 2 cos 2 /) : (64)
The rightíhand side of (??) is calculated at ! = ! \Sigma , where ! \Sigma are determined by (??).
As V 2 ! 4v 2
A , the condition fl \Sigma ? 0 is equivalent to
(k:V)\Psi \Sigma ? 0 : (65)
This condition can be rewritten as
\Sigma2X \Gamma (k:V)(Ú 1 \Gamma Ú 2 )
ae
X 2
\Gamma \Gamma k 2 cos 2 /
`
v 2
A + 1
4 V 2
'oe
? k 2 V 2 cos 2 /(Ú 1 + Ú 2 )
ae
X 2
\Gamma + k 2 cos 2 /
`
v 2
A \Gamma 1
4 V 2
'oe
: (66)
19

The rightíhand side of (??) is positive, so that the condition (??) cannot be satisfied for
both signs in the leftíhand side. With the aid of (??) the condition that (??) is satisfied
for one choice of the sign in the leftíhand side can be written as
M 2 (1 \Gamma M 2 + ¦ 2 ) 2 ! ffi 2 ¦ 2 (M 2 + 1 \Gamma ¦ 2 ) ; (67)
where
¦ 2 cos 2 / = 1 +M 2 cos 2 / \Gamma
q
sin 4 / + 4M 2 cos 2 / : (68)
Here M is the Alfv'eníMach number and ffi is a dimensionless quantity that measures the
relative difference in viscosity at the two sides of the discontinuity. They are given by
M = V
2vA ; ffi =
fi fi fi fi
Ú 1 \Gamma Ú 2
Ú 1 + Ú 2
fi fi fi fi : (69)
The solution to (??) is
M 2 ? M 2
c (ffi; /)
j 1
2 +
2ffi 2 tan 2 /(ffi 2 \Gamma cos 2/) + [1 \Gamma ffi 2 (1 + 2 tan 2 /)]
q
(1 \Gamma ffi 2 ) 2 + 4ffi 2 sin 4 /
2(1 \Gamma ffi 2 ) 2
: (70)
In the particular case that ffi ! 1 (Ú 1 ! 0 or Ú 2 ! 0), (??) reduce to
M 2
c (1; /) = 1 + sin 2 /
4 : (71)
When ffi = 0 (Ú 1 = Ú 2 ) M c (0; /) = 1. This means that there is no instability due to the
action of viscosity for a difference in equilibrium velocity below the KH threshold when
the viscosity coefficients are equal at the two sides of the discontinuity. It can be shown
that for 0 ! ffi ! 1 the following inequalities hold
1 + sin 2 /
4 ! M 2
c (ffi; /) ! 1 : (72)
Note the two other useful identities:
M 2
c (ffi; 90 o ) = 1
1 + ffi 2
; M 2
c (ffi; 0) = 1 : (73)
20

It follows from (??) that the second identity (??) is only valid for ffi ! 1. As a consequence
the function M c (ffi; /) is discontinuous at ffi = 1, / = 0. The dependence of M c (ffi; /) on /
for different values of ffi is shown in figure 2. The right inequality of (??) shows that there
is an interval of differences in the equilibrium velocity for which a wave propagating at
the angle / with respect to the equilibrium magnetic fild is unstable in a viscous plasma
and stable in an ideal plasma.
(??) is a local criterion for dissipative instability. Figure 2 shows that M c (ffi; /) is a
nonímonotonic function of / for fixed ffi and that it attains its minimal value at / = /m .
The global criterion for the dissipative instability is
M ? M c (ffi) j M c (ffi; /m ) : (74)
where M c is the global critical Alfv'eníMach number. The angle /m is determined by the
equation
cos 2 /m (1 + sin 2 /m cos 2 /m ) = ffi 2 ; (75)
and M c (ffi) can be expressed in terms of /m as
M c = ffi
cos /m (1 + cos 2 /m ) : (76)
Figures 3 and 4 show how /m and M c depend on ffi. Both quantities are monotonically
decreasing functions of ffi.
The present analysis shows that M c ! 1 for ffi ! 1 (Ú 1 6= Ú 2 ). This means that there
is an interval of differencies in the equilibrium velocities below the KH threshold velocity,
for which the discontinuity is unstable owing to the presence of viscosity.
6. Physical discussion for an incompressible plasma
In this section we present a physical interpretation of the ideal and dissipative instaí
bilities in an incompressible plasma. For the sake of mathematical simplicity we consider
an unperturbed state with the equilibrium magneti field B 0 and flow velocity v 0 that are
21

in the x direction. In addition we restrict the analysis to twoídimensional perturbations.
Hence the perturbed magnetic field and velocity are in the xz plane, and all the perturbed
quantities are independent of y.
In what follows we use a laboratory coordinate system in which the flow velocitites
at the both sides of a tangential MHD discontinuity, v 01 and v 02 , are fixed. When the
difference between the flow velocities at the two sides of the discontinuity, V , is below the
KelviníHelmholtz threshold velocity, V ! VKH , two surface waves can propagate along the
surface of the discontinuity. We introduce a new coordinate system that moves with the
phase velocity of a surface wave in the x direction with respect to the laboratory coordinate
system. We refer to this coordinate system as a concomitant coordinate system. In the
concomitant coordinate system plasma motion perturbed by the presence of a surface
wave is steady.
Before we embark on our physical discussion of dissipative instability, it is expedient
to give a physical interpretation of the ideal KH instability and take Ú 1 = Ú 2 = 0. Let
us turn to figure 5 where the plasma motion perturbed by the presence of a surface wave
is shown. The solid line is the perturbed surface of the discontinuity. We focus on the
balance of forces acting on a fluid volume that embraces the surface of the discontinuity
(see figure 5). The centrum of curvature of the solid line that correspond to the centrum
of the fluid volume is in a point C. The radius of curvature is R. The length of the
fluid volume is 2R ffi`, and the height is 2 ffiz. The surface of the dicontinuity devides the
fluid volume in an upper and an lower part. The two parts of the fluid volume move
approximately along the arc of the circle with raduis R and centrum at C. The upper
part of the volume moves with the velocity v 02 \Gamma V ph , while the lower part moves with the
velocity v 01 \Gamma V ph , where V ph is the phase velocity of the surface wave. The centripetal
acceleration of the upper part is (v 02 \Gamma V ph ) 2 =R, that of the lower part is (v 01 \Gamma V ph ) 2 =R.
It is wellíknown that a dynamical problem can be reduced to a static problem by the
use of the inertial forces, which are equal to the product of the corresponding masses
22

and accelerations and have the directions opposite to the directions of the accelerations.
The dynamical equations are then reduced to the equations that reflet the balance of
the inertial and the active forces. The masses of the upper and lower parts of the fluid
volume equal 2ae 02 R ffi` ffiz and 2ae 01 R ffi` ffiz respectively. Therefore the projection of the
inertial force, which in our case is the centrifugal force, on the normal to the surface of
the discontinuity is given by
F cen = 2 ffi` ffizfae 01 (v 01 \Gamma V ph ) 2 + ae 02 (v 02 \Gamma V ph ) 2 g ; (77)
and is directed upwards.
Let us now calculate the projection of the active forces on the normal direction. When
doing so we only calculate forces that are of the first order with respect to the three small
quantities: ffiz, R ffi`, and R \Gamma1 . The quantity R \Gamma1 is small because we consider only small
perturbations of the surface of discontinuity. The active force consists of two parts. The
first part is due to the total pressure and the second part is due to magnetic tension.
The contribution related to the total pressure can be splitted in two parts. The first part
is due to the total equilibrium pressure P 0 , while the second part is due to the Eulerian
perturbation of the total pressure P 0 . It is straightforward to show with the use of Gauss
theorem that the resulting force of the total equilibrium pressure P 0 on any closed surface
equals zero. The projection of the force due to P 0 on the normal direction is due to
the action of P 0 on the upper and lower boundaries of the fluid volume. However the
quantity P 0 is continuous at the surface of the discontinuity and depends on z as e \Gammakjzj .
Consequantly the values of P 0 at the upper and lower boundaries are the same and P 0
does not contribute in the projection of the active force on the normal direction. In
summarizing the projection of the resulting force on the normal direction due to the total
pressure is zero.
The active force that is due to the tension of the magnetic field acts on the endíwalls
of the fluid volume (see figure 5). The force of magnetic tension that acts on each endíwall
23

is given by
F t = ffiz
/
B 2
01
ï
+ B 2
02
ï
!
; (78)
and this force is perpendicular to this endíwall. We have used the equilibrium magnetic
field instead of the perturbed magnetic field when calculating F t since we want to be
consistent with linear theory. There is a small inclination of the endíwalls with respect
to the normal direction to the surface of the discontinuity at the centrum of the fluid
volume. As a result there is a nonízero projection of the magnetic tension force on the
normal direction equal F t ffi`.
Now taking into account that the projection of the force of the magnetic tension on the
normal is directed downwards, we write the balance of forces acting on the fluid volume
in the normal direction as
F cen = 2F t ffi` : (79)
With the use of (??) and (??) we obtain from (??) the equation that determines the phase
velocity of a surface wave
ae 01 (v 01 \Gamma V ph ) 2 + ae 02 (v 02 \Gamma V ph ) 2 = ae 01 v 2
A1 + ae 02 v 2
A2 : (80)
It follows from (??) that the phase velocities of the two surface waves are given by
V ph =
ae 01 v 01 + ae 02 v 02 \Sigma
q
ae 01 ae 02 (V 2
KH \Gamma V 2 )
ae 01 + ae 02
: (81)
The same result can be obtained from (??) if to take k parallel to both v 01 and v 02 and
/ = 0. We see that the phase velocities of the surface waves that can propagate on
the surface of the discontinuity are determined by the force balance in the concomitant
coordinate system. The centrifugal force related to the centripetal acceleration of a fluid
volume moving along a curved surface of the discontinuity has to be balanced by the
tension force due to the magnetic field.
The difference between the phase velocities of the two surface waves decreases when V
is increased. This difference becomes equal zero for V = VKH . If V is further increased,
24

solutions in the form of a surface wave do not anylonger exist and the KH instability of
the tangential discontinuity appears. From a physical point of view the development of
KH instability can be explained as follows. Let us consider for V ph as simply the speed of a
moving coordinate system. The value of the centrifugal force F cen is a squared function of
V ph . It takes its minimal value F min
cen at V ph = (ae 01 v 01 + ae 02 v 02 )=(ae 01 + ae 02 ). When V ! VKH
we have F min
cen ! 2F t ffi` and (??) possesses exactly two solutions. When V ? VKH we have
F min
cen ? 2F t ffi`, so that in all moving coordinate systems the centrifugal force cannot be
balanced by the force of magnetic tension in any and there is not any coordinate sytem
where the steady state can be reached.
Let us now consider the dissipative instability and take Ú 1 + Ú 2 6= 0. As we are only
interested in dissipative instability, we restrict our analysis to V ! VKH , so that there is
no KH instability in ideal MHD. We use the concomitant coordinate system in which the
plasma motion that is perturbed by a surface wave is steady when viscosity is absent. The
presence of viscosity destroys an exact balance of the centrifugal force and the magnetic
tension force, and thus causes the surface wave to damp or to grow. The growth of the
surface wave means that dissipative instability is present. Our objective here is to obtain
the criterion of the dissipative instability from a discussion of the forces that act on the
fluid volume embracing the perturbed surface of the discontinuity.
The study of the ideal KH instability was carried out in general terms. In particular,
we did not specify the shape of the perturbed surface of the discontinuity. In order
to study the dissipative instability we have to be more specific and define the shape
of the perturbed surface of the discontinuity in the concomitant coordinate system as
j = j a cos(kx). The Eulerian perturbation of total pressure modified by viscosity is
~
P 0 = P 0 + ae 0 Úfb:r(b:v 0 ) \Gamma 1
3 r:v 0 g : (82)
It is the modification of the perturbed total pressure due to viscosity that causes the
change in the shape of the perturbed surface of the discontinuity. As a result the amplitude
of oscillation of the perturbed surface of the discontinuity depends on time: j a = j a (t).
25

It is instructive to divide the total acceleration of the fluid volume a in the normal
direction in two parts: a = a cen + a vis . The centripetal acceleration a cen is caused by
the magnetic tension force and is not related to the dependence of j a on t. The viscous
acceleration a vis is related to the dependence of j a on t. This acceleration is caused by the
action of ~
P 0 on the lower and upper boundaries of the fluid volume. As we are interested
in the dependence of j a on t, we only calculate a vis .
In accordence with (??) ~
P 0
1 = ~
P 0
2 = ~
P 0
0 at z = 0. In the concomitant coordinate system
\Omega = k(V ph \Gamma v 0 ) : (83)
When c s ! 1 (an approximation of an incompressible plasma) we have the following
limiting expressions for \Gamma 0 , K Ú , and K ¼
\Gamma 0 = k ; K Ú = 3k(v 0 \Gamma V ph )
(v 0 \Gamma V ph ) 2 \Gamma v 2
A
; K ¼ = 0 : (84)
We use (??) and (??) to get from (??)
~
P 0
0 = ae 01 kj a f[(v 01 \Gamma V ph ) 2 \Gamma v 2
A1 ] cos(kx) + 3
2 Ú 1 k(v 01 \Gamma V ph ) sin(kx)g : (85)
The dependence of ~
P 0
1 and ~
P 0
2 on z is given by the functions e \Gamma 1 z and e \Gamma\Gamma 2 z respectively.
Taking into account that ffiz, Ú 1 , and Ú 2 are small, it is straighforward to obtain the
following approximate expressions that determine ~
P 0
1 at the lower boundary and ~
P 0
2 at the
upper boundary of the fluid volume
~
P 0
1 = ae 01 kj a f(1 \Gamma k ffiz)[(v 01 \Gamma V ph ) 2 \Gamma v 2
A1 ] cos(kx) + 3
2 Ú 1 k(v 01 \Gamma V ph ) sin(kx)g ;
~
P 0
2 = ae 01 kj a
ae
(1 \Gamma k ffiz)[(v 01 \Gamma V ph ) 2 \Gamma v 2
A1 ] cos(kx) + 3
2
Ú 1 k(v 01 \Gamma V ph ) sin(kx)
+ 3
2 k 2 ffiz
''
Ú 2 (v 02 \Gamma V ph )
(v 01 \Gamma V ph ) 2 \Gamma v 2
A1
(v 02 \Gamma V ph ) 2 \Gamma v 2
A2
\Gamma Ú 1 (v 01 \Gamma V ph )
#
sin(kx)
oe
:
9
? ? ? ? ? ? ? ? ? =
? ? ? ? ? ? ? ? ? ;
(86)
The force that causes the acceleration a vis consists of three contributions. The first
contribution is related to the difference in ~
P 0
1 and ~
P 0
2 at the lower and the upper boundary
of the fluid volume. The second contribution is due to a small difference in the lengths of
the upper and the lower boundaries. The third contribution is due to a small inclination
26

of the two endíwalls of the fluid volume with respect to the normal direction. It is easy
to see that the two latter contributions are of the order of j 2
a . In linear theory these two
contributions can be neglected. With the use of (??) it is straightforward to obtain from
(??) the following expression for a vis
a vis = 3k 3
4ae 01
fae 01 Ú 1 (v 01 \Gamma V ph ) + ae 02 Ú 2 (v 02 \Gamma V ph )g sin(kx) : (87)
This acceleration is directed upwards.
On the other hand the upward accelerations of the upper and lower parts of the fluid
volume are given by
a 2 = @w 2
@t
+ v 02
@w 2
@x
; a 1 = @w 1
@t
+ v 01
@w 1
@x
: (88)
With the use of (??) we then obtain for the upward acceleration of the centre of mass of
the fluid volume
a = @ 2 j
@t 2
+ 2 ae 01 (v 01 \Gamma V ph ) + ae 02 (v 02 \Gamma V ph )
ae 01 + ae 02
@ 2 j
@t@x
+ ae 01 (v 01 \Gamma V ph ) 2 + ae 02 (v 02 \Gamma V ph ) 2
ae 01 + ae 02
@ 2 j
@x 2
:
(89)
The last term in (??) represents a cen . As we only consider small viscosity, the characteristic
time of changing j a is much larger than the wave period in the laboratory coordinate
system, so that jdj a =dtj œ kV ph j a . This enables us to neglect the first term in (??) in
comparison with the second term. As a result we arrive at the following approximate
expression for a vis
a vis = \Gamma2k ae 01 (v 01 \Gamma V ph ) + ae 02 (v 02 \Gamma V ph )
ae 01 + ae 02
dj a
dt sin(kx) : (90)
We compare expressions (??) and (??) for a vis to obtain an equation for j a (t)
dj a
dt
= 3k 2 (ae 01 + ae 02 )
8ae 01
Fj a ; (91)
where
F = \Gamma ae 01 Ú 1 (v 01 \Gamma V ph ) + ae 02 Ú 2 (v 02 \Gamma V ph )
ae 01 (v 01 \Gamma V ph ) + ae 02 (v 02 \Gamma V ph ) : (92)
27

The amplitude of the surface wave decreases if F ! 0 and increases if F ? 0. Hence the
criterion for the dissipative instability is given by the inequality F ? 0.
It is straightforward to show that F ? 0 for one choise of sign in (??) when the criterion
of dissipative instability (??) found in Section 4 is satisfied. If (??) is not satisfied, F ! 0
for the both choises of sign in (??). Hence we obtain the criterion of dissipative instabilí
ity from a physical discussion of the plasma motion. The present discussion shows that
the cause of the dissipative instability is the breakdown of the balance of forces acting on
the fluid volume embracing the surface of the discontinuity caused by anisotropic viscosity.
7. Conclusions
In the present paper the dissipative instability of the MHD tangential discontinuity
has been studied. The dissipative mechanisms considered here are viscosity and thermal
conductivity. Both viscosity and thermal conductivity have been assumed to be strongly
anisotropic, so that the viscous stresses are normal to the magnetic surfaces, and the
heat flux is 1ídimensional along the magnetic field lines. The general dispersion equation
determining the stability of the MHD tangential discontinuity has been derived. This
equation has been studied for the two limiting cases of an incompressible plasma and of
a cold plasma. In these two limiting cases viscosity is the only dissipative process that
affects the stability. In an incompressible plasma thermal conductivity does not affect
stability because the energy equation is decoupled from the other MHD equations. In a
cold plasma thermal conductivity does not affect stability because the temperature and,
consequently, the internal energy of the plasma is equal to zero.
In the case of an ideal plasma the stability of the MHD tangential discontinuity is
determined by the KelviníHelmholtz (KH) threshold velocity VKH . When the difference
in the equilibrium velocity V is smaller than VKH the discontinuity is stable, while it is
unstable when V is larger then VKH . The instability which is present in that case is called
28

the KelviníHelmholtz (KH) instability.
The present paper was concentrated on a situation when the MHD tangential discontií
nuity is stable in ideal MHD and intended to find out whether dissipation, contrary to the
intuitive expectation, can lead to instability. The main result is that viscosity introduces
a new threshold velocity V c for overstability which is lower than the ideal KH threshold
velocity VKH at least when the viscosity coefficient Ú takes different values at the two sides
of the discontinuity. There is a range of differencies in the equilibrium velocity between
V c and VKH when the overstability is due to dissipation, so that we would like to call it
the dissipative instability. In general we can state that dissipation destabilizes the MHD
tangential discontinuity.
Acknowledgements
This work was carried out while M.Ruderman was a guest at the Katholiek Unií
versiteit Leuven. M.Ruderman acknowledges financial support by the 'Onderzoeksfonds
K.U.Leuven', the Belgian National Fund for Scientific Research (N.F.W.O.), and the Serí
vice for Science Policy, which made his fruitful and pleasant stay at the KU Leuven
possible, and the warm hospitality of the Centrum voor PlasmaíAstrophysica of the KU
Leuven.
References
Braginskii, S.I. 1965 Transport processes in plasma. Rev. Plasma Phys. 1, 205--309.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarení
don Press.
Duhau, S., Gratton, F. & Gratton, J. 1970 Hydromagnetic oscillations of a taní
gential discontinuity in the Chew, Goldberger, and Low approximation. Phys. Fluids
29

13, 1503--1509.
Duhau, S., Gratton, F. & Gratton, J. 1971 Radiation of hydromagnetic waves
from a tangential velocity discontinuity. Phys. Fluids 14, 2067--2068.
Duhau, S. & Gratton, J. 1973 Effect of compressibility on the stability of a vortex
sheet in an ideal magnetofluid. Phys. Fluids 16, 150--152.
Fejer, J.A. 1964 Hydromagnetic stability at a fluid velocity discontinuity between comí
pressible fluids. Phys. Fluids 7, 499--503.
Gerwin 1968 Stability of interface between two fluids in relative motion. Rev. Mod.
Phys. 40, 652--658.
Gonz' alez, A.G. & Gratton, J. 1994a The KelviníHelmholtz instability in a comí
pressible plasma: the role of orientation of the magnetic field with respect to the flow.
J. Plasma Phys. 51, 43--60.
Gonz' alez, A.G. & Gratton, J. 1994b The role of density jump in the KelviníHelmí
holtz instability of a compressible plasma. J. Plasma Phys. 52, 223--244.
Hollweg, J.V. 1985 Viscosity in magnetized plasma: physical interpretation, J. G. R.
90, 7620--7622.
McKenzie, J.F. 1970 Hydromagnetic wave interaction with the magnetopause and the
bow shock. Planet. Space Sci. 18, 1í23.
MitnitskiÖ i, V.Ya. 1984 Ob ustoÖichivosti magnitogazodinamicheskih tangentsialnych
razryvov. Zh. VychislitelnoÖi Matematiki i MatematicheskoÖi Fiziki, 24, 124--131.
OstrovskiÖ i, L.A., Rybak, S.A. & Tsimring, L.Sh. 1986 Volny otritsatelnoÖi energii
v gidrodinamike. Usp. Fiz. Nauk 150, 417--437.
English translation:
Ostrovskii, L.A., Rybak, S.A. & Tsimring, L.S. 1986 Negative energy waves
in hydrodynamics. Sov. Phys. Usp. 29, 1040--1052.
Roy Choudhury, S. 1986 An analytical study of the KelviníHelmholtz instabilities of
compressible, magnetized, anisotropic, and isotropic tangential velocity discontinuií
30

ties. Phys. Fluid 29, 1509--1519.
Roy Choudhury, S. & Patel, V.L. 1985 KelviníHelmholtz instabilities of highíveloí
city magnetized, anisotropic shear layers. Phys. Fluid 28, 3292--3301.
Ruderman, M.S. & Fahr, H.J. 1993 The effect of magnetic fields on the macroscopic
instability of the heliopause. I. Parallel interstellar magnetic fields. Astron. Astrophys.
275, 635--644.
Ruderman, M.S. & Fahr, H.J. 1995 The effect of magnetic fields on the macroscopic
instability of the heliopause. II. Inclusion of solar wind magnetic fields. Astron. Astroí
phys. 299, 258--266.
Ruderman, M.S. & Goossens, M. 1995 Surface Alfv'en waves of negative energy. J.
Plasma Phys., 54, 149--155.
SyrovatskiÖ i, S.I. 1957 Magnitnaya gidrodinamika. Usp. Fiz. Nauk 62, 247--303.
31

Figure captions
Figure 1. The scetch of introduction of the angles ' 1 , ' 2 , and /. The counterclockwise
direction is considered as positive. On this figure ' 1 ? 0, ' 2 ! 0, and / ? 0.
Figure 2. The dependence of the local critical Alfv'eníMach number M c on the angle
/ between the direction of the equilibrium magnetic field and the direction of the propí
agation of the perturbation for different values of the relative difference in the viscosity
coefficient ffi in the case of a cold plasma. The values of ffi are shown under corresponding
curves.
Figure 3. The dependence of the angle /m at which the local critical Alfv'eníMach numí
ber M c attains the minimum value on the relative difference in the viscosity coefficient ffi.
Figure 4. The dependence of the global critical Alfv'eníMach numbem ï
M c on the relaí
tive difference in the viscosity coefficient ffi.
Figure 5. The scetch of the flow in the vicinity of the perturbed surface of the tangential
discontinuity (shown by solid line). The fluid volume (shaded) embracing the surface of
the discontinuity is shown together with quantities used in the physical consideration.
32