Supercond. Sci. Technol. 10 (1997) 547–551. Printed in the UK
PII: S0953-2048(97)82236-1
Development of ‘explosion’-type
instability in superconductors with
transport current
A A Pukhov†
Scientific Centre for Applied Problems in Electrodynamics,
Russian Academy of Sciences, Izhorskaya 13/19, Moscow 127412, Russia
Received 28 February 1997, in final form 21 May 1997
Abstract. Instability of ‘explosion’ type in a superconductor carrying a transport
current is studied theoretically. Such instability arises if there is no stable normal
state of the superconductor due to its adiabatic conditions or strong dependence of
the resistivity on the temperature. It is shown that the superconductivity destruction
results from the propagation of the temperature wave of ‘explosion’ type with
rapidly increasing temperature of the normal region. The boundary between the
normal and superconducting regions moves with a constant velocity.
Group-theoretical analysis of the problem allows one to obtain analytical formulae
for the wave propagation velocity. Analytical results are confirmed by a direct
numerical simulation of the instability development. The results obtained may be of
importance for studying the cryostability of both conventional and high-temperature
superconductors.
Nomenclature
a
b
C
i
j
jc (T0 )
k
lad
L
p
q
T
Tc
Tr
T0
T∗
t
tad
u
v
vad
x
z
dimensionless parameter
dimensionless parameter
heat capacity
dimensionless current density, j/jc (T0 )
current density
critical current density
heat conductivity
characteristic length
scaling factor
scaling parameter
scaling parameter
temperature
critical temperature
resistive transition temperature,
T0 + [1 − j/jc (T0 )](Tc − T0 )
ambient temperature
characteristic temperature
time
characteristic time
dimensionless NS interface propagation velocity
NS interface propagation velocity
characteristic velocity
longitudinal coordinate
self-similar coordinate, x − ut
Greek
θ dimensionless temperature, (T − T0 )/(Tc − T0 )
ρ resistivity
† E-mail address: pukhov@theor.termo.msk.ru
c 1997 IOP Publishing Ltd
0953-2048/97/080547+05$19.50 1. Introduction
Thermal instability development in superconductors with
transport current is related to thermal bistability due to
Joule self-heating. It results from the thermal autowave
propagation in the sample.
Such autowave is the
NS interface moving with a constant velocity v and
turning the sample state from the superconducting to
normal. Phenomenon of the normal zone (NZ) nucleation
and its subsequent propagation is now well studied
[1, 2]. However, there are situations where the thermal
bistability of the superconductor ceases to exist due to
the absence (or low level) of the heat removal into the
environment (coolant, compound, etc.). In this situation the
superconductor can be in one stable state (superconducting)
only with the temperature equal to the ambient temperature
T0 whereas the stable normal state is absent. The low heat
removal leads to an unlimited Joule self-heating of the NZ
in a superconductor at temperatures higher than the resistive
transition temperature Tr .
Such situation appears in compound windings made
of conventional (low-temperature) superconductors [3, 4]
cooled by liquid helium. Due to the absence of thermal
contact with the coolant in this situation there is no steady
NZ [5], and the thermal instability development in a fully
thermally insulated superconductor is described by the nonstationary heat balance equation [6].
Similar situation appears also in high-temperature
composite superconductors [7] and superconducting films
[8] cooled by liquid nitrogen. A strong increase of the
superconductor (matrix) resistivity with the temperature
547
A A Pukhov
results in an unlimited Joule self-heating of NZ in the
superconductor at sufficiently high currents [9].
In the present paper it is shown that the unlimited NZ
temperature increase is responsible for thermal instability
development specific features of which cannot be explained
by the conventional theory of the propagation of the thermal
autowaves with a finite amplitude [1, 2]. The NZ front
propagating along the superconductor has the following
structure. The NS interface separates the superconducting
and normal regions of the sample: the temperature of
the superconducting region is constant (and equal to
the ambient temperature T0 ) whereas the normal region
temperature increases infinitely. In spite of nonstationary
normal region temperature, the NS interface propagates
with constant velocity v. Thus, the instability development
in this case occurs due to two parallel processes: expelling
of the superconducting region by the normal one and
fast heating of the normal region. The absence of the
stationary NZ and its fast (at exponential rate) heating make
it reasonable to refer to this phenomenon as the ‘explosion’type instability.
A simple theoretical model considered below allows
one to obtain an adequate description of the ‘explosion’type
instability development and to find analytical formulae for
the propagation velocity of the ‘explosion’-type temperature
wave versus current. Analytical results obtained are
confirmed by a direct numerical simulation of the instability
development.
2. Basic equations
The NS interface propagation along the superconductor is
described by a nonlinear heat balance equation which in
one-dimensional approximation has the following form:
∂
∂T
∂T
=
k
+ Q(T , j )
(1)
C
∂t
∂x
∂x
where t is time, x is the longitudinal coordinate, C
is the heat capacity, k is the heat conductivity of the
superconductor and Q is the specific Joule heat release
at the transport current density j . In formula (1) it is
taken into account that under adiabatic conditions the heat
removal from a superconductor is negligibly low. The
function Q(T , j ) depends on a concrete situation. For
example, the Joule heat release Q in low-temperature
composite superconductors may be written in the form:
(
0
T0 < T < Tr (j )
(2)
Q(T , j ) =
Tr (j ) < T
ρj [j − jc (T )]
where jc (T ) = jc (T0 )(Tc − T )/(Tc − T0 ) is the critical
current density, Tc is the critical temperature, Tr (j ) = T0 +
[1−j/jc (T0 )](Tc −T0 ) is the resistive transition temperature
determined by the condition jc (Tr ) = j and ρ is the matrix
resistivity which at helium temperatures (T = 4.2 K) is
usually independent of T . In the normal state T > Tc
the critical current density becomes equal to zero and the
Joule heat release Q = ρj 2 becomes independent of the
temperature. However, taking into account that unlimited
Joule self-heating starts at T > Tr (Tr < Tc ), we can assume
548
that in the ‘explosion’ mode the main heat release occurs
in the resistive state, before much Joule heat is produced.
Hence, for analysing the initial stage of the instability
development it is enough to restrict our consideration to
approximation (2) [6].
The resistivity ρ(T ) of high-temperature superconductors cooled by liquid nitrogen increases linearly with the
temperature and with a high accuracy may be estimated as

T < Tc
0
ρ(T ) =
 ρ(Tc ) + dρ (T − T∗ )
T > Tc
dT
where dρ/dT is the constant slope of ρ(T ) and T∗
is the characteristic temperature [10]. To analyse the
‘explosion’-type instability development the finite jump of
the resistivity ρ(Tc ) at T = Tc can be neglected and the
dependence ρ(T ) is assumed to be linear over the whole
interval T > T∗ . Thus, for Q we have

T < T∗
0
Q(T , j ) = dρ 2
(3)

j (T − T∗ )
T > T∗ .
dT
It will be shown below that the value T∗ introduced for
convenience does not affect the propagation velocity of the
NS interface.
For both approximations (2) and (3) the Joule heat
release increases linearly with the temperature, so the
instability development can be treated within the framework
of a unified approach. Assuming for simplicity of the
following calculations that the dependencies of C and k on
T can be neglected, it is convenient to rewrite equation (1)
in the dimensionless form:
∂ 2θ
∂θ
= 2 + f (θ)
∂t
∂ x
(
where
f (θ) =
0
θ < b/a
aθ − b
θ > b/a.
(4)
Here t and x denote the dimensionless time t/tad and
coordinate x/ lad ; tad and lad are characteristic scales
of time and length under adiabatic conditions; θ =
(T − T0 )/(Tc − T0 ) is the dimensionless temperature and
parameters a and b depend on the dimensionless current
i = j/jc (T0 ). For approximation (2) we have
tad = C(Tc − T0 )/ρjc2 (T0 )
2
= k(Tc − T0 )/ρjc2 (T0 )
lad
a=i
b = i(1 − i)
and for approximation (3) we find
tad = C
2
lad
=k
−1
dρ 2
j (T0 )
dT c
−1
dρ 2
j (T0 )
dT c
‘Explosion’-type instability in superconductors
a = i2
b = i 2 θ∗
θ∗ = (T∗ − T0 )/(Tc − T0 ).
The dimensionless form (4) of equation (1) simplifies
significantly real situation, since it does not take into
account the dependencies of k and C on temperature, but
allows one to analyse the details of the ‘explosion’-type
instability development analytically. It is frequently used,
since analytical results obtained using this approach have
the simplest form [2, 6, 11].
3. NS interface propagation velocity
Equation (4) does not have self-similar solution of the
moving autowave type θ(x, t) = θ(x − ut) by virtue of the
unlimited Joule self-heating of the sample normal region at
θ > b/a. However, subsequent consideration will show
that the boundary of the self-heated normal region (NS
interface) moves with a constant velocity. To illustrate
this the results of the numerical solution of equation (4)
for approximation (2) are shown in figure 1. It can
be seen that the point with a fixed temperature on the
NS interface front moves uniformly. Furthermore, the
asymptotic value of the velocity does not depend on the
initial temperature profile which means that this wave is
stable with respect to infinitely small disturbances. Similar
pattern also takes place for approximation (3). Thus, the
temperature distribution in the ‘cold’ tail of the NS interface
has the form θ (x, t) = θ(x−ut) and satisfies asymptotically
(θ 1) the following equation:
θzz + uθz + f (θ) = 0.
(5)
Here z = x − ut, u = v/vad is the dimensionless
velocity and vad = lad /tad is the characteristic velocity
under adiabatic conditions. The absence of self-similar
solution of (4) does not allow to determine the value of u
by conventional methods of the finite-amplitude autowave
propagation theory [1, 2] as the eigenvalue of equation (5).
To calculate the value of u in this case group-theoretic
arguments [12, 13] can be used. Really, equation (5) is
invariant to the family of groups of transformations
z = Lp z 0
θ = Lq θ 0
u = L−p u0
a = L−2p a 0
b = Lq−2p b0
(6)
which are the scaling groups with the scaling factor
L. Powers of scaling factor in (6) are determined from
the invariance condition of equation (5) to the group of
transformations (6), so that the new (primed symbols)
variables satisfy the same equation (5). Thus, the group
of transformations (6) contains free parameters L, p and q
which can have arbitrary values.
Figure 1. The evolution of the NS interface temperature
profile. The initial temperature distribution is chosen in the
‘stepwise’ form (curve 1). Shown profiles are separated by
time intervals 1t = 0.3tad . i = 0.5.
It is clear from physical arguments that the value of u
in (5) can depend on a and b only, i.e. u = F (a, b) where
F is some as yet undetermined function. This condition
must be invariant to the group of transformations (6), that
is u0 = F (a 0 , b0 ) [14]. Let us search the solution in the form
u ∝ a r bs , where powers r and s should be determined. So,
from (6) we find
u/a r bs = L−p+2pr+2ps−qs u0 /a 0r b0s
and from the invariance condition we obtain p(−1 + 2r +
2s) − qs = 0. Since p and q in (6) have arbitrary values,
we get a unique solution r = 1/2, s = 0. Thus, for the
propagation velocity u we have
u ∝ a 1/2 b0 ∝ (a)1/2 .
(7)
From (7) we get u ∝ (i)1/2 for approximation (2) and u ∝ i
for approximation (3). The unknown numerical factor in (7)
cannot be obtained by means of group-theoretic arguments.
To determine it some additional reasons should be invoked.
Let us consider first approximation (2). If the current is
equal to the critical one i = 1 (a = 1, b = 0) equation (4)
corresponds to the well-known Kolmogorov–Petrovsky–
Piskunov problem [15]. The propagation velocity in this
case is u = 2(df/dθ (0))1/2 = 2. Thus, for the propagation
velocity of the NS interface in the dimensional form we
find
1/2
kρ
(i)1/2 jc (T0 )
.
(8)
v=2
C
Tc − T0
Similar procedure can be applied to the approximation (3).
It follows from (7) that the velocity u does not depend
on the parameter b ∝ T∗ − T0 , so we may restrict our
consideration to the particular case b = 0 (T∗ = T0 ).
It reduces the problem to the Kolmogorov–Petrovsky–
Piskunov case again. Hence, u = 2(df/dθ (0))1/2 = 2i
and for the propagation velocity of the NS interface in
dimensional form we have
dρ 1/2
ijc (T0 )
k
.
(9)
v=2
C
dT
549
A A Pukhov
4. Discussion
Figure 2. The propagation velocity of the NS interface v
versus i for approximations (2) and (3). : numerical
calculations; full curve (1): formula (8); full curve 2:
formula (9).
Shown in figure 2 is the comparison of the numerical
calculation of v(i) with analytical formulae (8) and (9).
In conclusion of this section an additional specific
feature of the ‘explosion’-type instability development
should be noted. In the case of bistability the steady
propagating front of the NS interface forms from initial
temperature profile at exponential rate (several ‘heat
relaxation’ times [1, 2]). On the other hand the steady
propagation velocity of the ‘explosion’-type NS interface
sets in more slowly. To illustrate this let us consider the
simplest case of a = 1 and b = 0:
∂ 2θ
∂θ
= 2 + θ.
∂t
∂ x
(10)
It means that i = 1 for approximation (2) or i = 1 and
θ∗ = 0 for approximation (3). Asymptotic solution of
equation (10) at x 1 and at t 1 when the details
of the initial temperature distribution θ(x, 0) become not
significant has the form:
θ (x, t) = A(4πt)−1/2 exp(−x 2 /4t + t)
where
Z
A=
(11)
+∞
θ(x, 0) dx.
−∞
From (11) we find that the coordinate x0 of the point on
the NS interface front with fixed temperature θ(x0 , t) =
constant varies as x02 = 4t 2 − 2t log(t) + · · · and its velocity
varies as dx0 /dt = 2 − 1/2t + · · · (here the higher order
terms are omitted). Naturally, the asymptotic value of the
velocity dx0 /dt (+∞) = 2 coincides with the value of the
NS interface propagation velocity u = 2 determined by
formula (8) at i = 1. The power-law rate of the velocity
relaxation is related to a positive definition of the function
f (θ ) in (4), i.e. in the end, to the unlimited NZ self-heating.
550
Thus, the ‘explosion’-type instability development in a
superconductor carrying a transport current has a number of
specific features related to the absence of the steady normal
state. In this situation the superconductor is monostable and
has only one steady state, namely superconducting. The
monostability can occur if the heat removal is negligibly
low and the instability development is determined largely
by the Joule self-heating. If under the action of external
disturbances the NZ nucleates in the sample, then the
superconductor monostability leads to the unlimited Joule
self-heating. Exponential rate of the temperature increase in
the NZ results in peculiarities of the instability development
which can be naturally called ‘explosion’-type instability.
The boundaries of the self-heated NZ (NS interfaces)
move with a constant velocity despite of the non-similarity
character of the instability development.
Considered
approximations (2), (3) simplify significantly real situation,
but allow one to describe adequately specific features of the
instability development and to obtain analytical formulae
for v(i). The fact that the velocity of the ‘explosion’type NS interface propagation is constant seems to be
independent of the specific choice of mathematical model
for a monostable superconductor description. In particular,
similar situation has been described in [9] where the
instability development is analysed near the transition from
bistability mode to monostability mode with a finite heat
removal from a superconductor comparable to Joule heat
release. Such bifurcation occurs at a sufficiently high
transport current exceeding the ‘maximum equilibrium’
current [2, 9], above which it was found numerically that
the NS interface moves at a constant velocity [9].
In general, the instability development is initiated by
external thermal disturbances. This process has a threshold
character, i.e. there is a critical value of the disturbance
energy Ec beyond which the instability sets in. The value
of the critical energy Ec for approximation (2) has been
found in [6] by using group-theoretic arguments similar to
ones used here to determine the propagation velocity v.
The group-theoretic analysis of autowave processes
in superconductors considered here is of some interest
for a number of nonlinear active media of different
nature. Specific features of the ‘explosion’-type instability
development in superconductors pointed out are close
to the so-called ‘sharpening’ regimes in active media
with the strong dependence of the heat conductivity
and heat capacity on the temperature [16]. However,
as distinguished from the ‘explosion’-type instability
development, ‘sharpening’ regimes are characterized by a
faster than the exponential rate temperature increase, when
the medium temperature tends to infinity within a finite time
interval.
Acknowledgments
This work was supported in part by the Russian State
Program on Superconductivity under Project No 96083 and
by the Russian Foundation for Basic Research under Project
No 96-02-18949.
‘Explosion’-type instability in superconductors
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