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A manifest Hopf bifurcation in resistive magnetohydrodynamics
PhysicsLettersAl8o(1993) 257,258 North-Holland
PHYSICS LETTERS A
A manifest Hopf bifurcation in resistive magnetohydrodynamics H. Tasso Max-Planck-Institut für Plasmaphysik, Euratoni Association, W-8046 Garching near Munich, Germany Received 10 March 1993; accepted for publication 9 July 1993 Communicated by AR. Bishop
A special Hopf bifurcation in resistive magnetohydrodynamics can easily be identified by starting from a particularly suitable form of the linearized equations which was previously introduced by the author. As usual, the bifurcation can lead to nonlinear periodic oscillations. These oscillations can eventually be destroyed, which raises the question ofthe nature ofthe ultimate attractors.
In a previous note [l]the author proposed a general Hermitean stability condition for linearized modes in resistive magnetohydrodynamics (MHD). The condition is sufficient for stability with respect to purely growing modes. Further analysis [21 shows that the condition becomes sufficient for stability with respect to all modes if the inertial term is neglected. Furthermore, under certain restrictions, the condition can be considered as “nearly” necessary. Finally, simplified forms of the conditions were considered [3] In this Letter we consider the case for which the condition is satisfied and prove that, if the inertial term can cause some additional overstability, the modes appearing in this way fulfill the requirements ofthe center manifold theorem [4]. This means that they can beresulting stabilized a Hopfbifurcation in nonlinearly a limit cyclethrough or nonlinear periodic oscillation. The equation describing the linearized perturbations is of the type [1} N1’+P~+(Q+Q )~—0 ‘I’ a
—
‘
‘
where the operators N~P and
/
with respect to all modes if N_—O. The proof is easily given by constructing a Lyapunov function [21. If N 0, overstable modes can occur: In a special cxample [51, the overstability occurs only in the compressible case, primarily at the magnetoacoustic resonance, which implies a finite inertial term. Let us now consider the case for which (2) is satisfied but (1) is overstable for N 0. Any overstable mode of (1) is given by ~=Wexp[(iw+~’)t} where w and y are real and satisfy
(3)
,
(y2—w2)( W’~,NW) +y( W~,PW) + ( W’~,Q5W) (4)
—
9’, PW) + ( W~,Qa
—
2?w( —oW*, NW) +w( W
~/‘) (5)
—
We see from (4) and (5) and, generally, from the reality of the operators in (1) that ~*= ~ xexp[(—iw+y)t] is also an eigenmode of (1). It follows that the modes due to the inertia operator N
Q
5 are real and sym-
metric and Qa is real and antisymmetric, N and P being positive. A sufficient condition [1] with respect to purely growing modes is (~Q ~)>0 ‘2~ ‘
‘
where (a, b) is the scalar product. According to previous analysis [2], condition (2) becomes sufficient
always come in pairs with opposite sign of the real frequencies but the same growth rate, all other modes being damped because of (2). These features are precisely the principal ingredients of the center manifold theorem [4]. In summary, if (2) is satisfied, inertia-caused overstability can lead to a Hopf bifurcation resulting in a periodic nonlinear oscillation.
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
257
Volume 180, number 3
PHYSICS LETTERS A
It may be instructive to look at the following cxample, consisting of two ordinary differential equations, Tn
~
O\ 1~\ ~ (,,,.) +
(~o) (~)( +
p
.~
£1~ ~ —q~ qj ‘~Y1
(6) where n and pare positive. The eigenvalues are given by
(nQ2+pQ+q 2+q~=O. 5) The solutions of (7) are P±
means that the attractors have zero Hausdorff dimension, though the estimates [6] would deliver huge upper bounds such as lOla or much larger. In other words, nonlinear behavior and turbulence in MHD is much more configuration-dependent than in hydrodynamics. The reason is that the dimension of attractors or the number of determining modes is, to an extreme degree, configuration-dependent.
4n(t7 5+i~j~)
(8)
2n 2/4n,which fulfills condition (2), If we choose q~=p the threshold for instability is n J P2/2 = 2nq~. It is conceivable that the Mirnov or “precursor” oscillations, seen in tokamaks prior to “disruptions”, are of that kind, though they need not be related to that special instability. Pursuing this speculation, we could say that a further increase of density (or inertia) may lead to a “disruption” because the limit cycle can no longer be maintained. One of the puzzling questions is how to describe and follow such highly nonlinear systems for long times. In hydrodynamics, it is believed that such systems become turbulent or will tend to a fractal attractor with a large Hausdorff dimension increasing with the Reynolds number [6]. Estimates of the
258
Hausdorff dimension of attractors are made, however, from above [6,7] and do not necessarily reflect the real situation. Especially in MHD, this picture could be completely wrong: Nontrivial examples are known to be nonlinearly and unconditionally stable for all magnetic Reynolds numbers [3,8], which
(7)
_____
Q
6 September 1993
Let us finally note that, if condition (2) is not satisfied, other unstable eigenmodes can occur, but since the spectrum is not known, it will be very hard, in general, to check whether eigenvalues fulfill the requirements of the centerthe manifold theorem. References (1] H. Tasso, Phys. Lett.
A 147 (1990) 28. [2lH.Tasso,Phys.Lett.A 161 (1991) 289. [3] H. Tasso, Phys. Leti. A 169 (1992) 396.
14] J.E.
Marsden and M. McCracken, The Hopfbifurcation and its applications (Springer, Berlin, 1976). (51 H. Tasso and M. Cotsaftis. Plasma Phys. (part C) 7(1965) 29. [6] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics (Springer, Berlin, 1988). (7] P. Constantin, Commun. Math. Phys. 129 (1990) 241. 181 H. Tasso. Nuovo Cimento B, to be published.
PHYSICS LETTERS A
A manifest Hopf bifurcation in resistive magnetohydrodynamics H. Tasso Max-Planck-Institut für Plasmaphysik, Euratoni Association, W-8046 Garching near Munich, Germany Received 10 March 1993; accepted for publication 9 July 1993 Communicated by AR. Bishop
A special Hopf bifurcation in resistive magnetohydrodynamics can easily be identified by starting from a particularly suitable form of the linearized equations which was previously introduced by the author. As usual, the bifurcation can lead to nonlinear periodic oscillations. These oscillations can eventually be destroyed, which raises the question ofthe nature ofthe ultimate attractors.
In a previous note [l]the author proposed a general Hermitean stability condition for linearized modes in resistive magnetohydrodynamics (MHD). The condition is sufficient for stability with respect to purely growing modes. Further analysis [21 shows that the condition becomes sufficient for stability with respect to all modes if the inertial term is neglected. Furthermore, under certain restrictions, the condition can be considered as “nearly” necessary. Finally, simplified forms of the conditions were considered [3] In this Letter we consider the case for which the condition is satisfied and prove that, if the inertial term can cause some additional overstability, the modes appearing in this way fulfill the requirements ofthe center manifold theorem [4]. This means that they can beresulting stabilized a Hopfbifurcation in nonlinearly a limit cyclethrough or nonlinear periodic oscillation. The equation describing the linearized perturbations is of the type [1} N1’+P~+(Q+Q )~—0 ‘I’ a
—
‘
‘
where the operators N~P and
/
with respect to all modes if N_—O. The proof is easily given by constructing a Lyapunov function [21. If N 0, overstable modes can occur: In a special cxample [51, the overstability occurs only in the compressible case, primarily at the magnetoacoustic resonance, which implies a finite inertial term. Let us now consider the case for which (2) is satisfied but (1) is overstable for N 0. Any overstable mode of (1) is given by ~=Wexp[(iw+~’)t} where w and y are real and satisfy
(3)
,
(y2—w2)( W’~,NW) +y( W~,PW) + ( W’~,Q5W) (4)
—
9’, PW) + ( W~,Qa
—
2?w( —oW*, NW) +w( W
~/‘) (5)
—
We see from (4) and (5) and, generally, from the reality of the operators in (1) that ~*= ~ xexp[(—iw+y)t] is also an eigenmode of (1). It follows that the modes due to the inertia operator N
Q
5 are real and sym-
metric and Qa is real and antisymmetric, N and P being positive. A sufficient condition [1] with respect to purely growing modes is (~Q ~)>0 ‘2~ ‘
‘
where (a, b) is the scalar product. According to previous analysis [2], condition (2) becomes sufficient
always come in pairs with opposite sign of the real frequencies but the same growth rate, all other modes being damped because of (2). These features are precisely the principal ingredients of the center manifold theorem [4]. In summary, if (2) is satisfied, inertia-caused overstability can lead to a Hopf bifurcation resulting in a periodic nonlinear oscillation.
0375-9601/93/$ 06.00 © 1993 Elsevier Science Publishers B.V. All rights reserved.
257
Volume 180, number 3
PHYSICS LETTERS A
It may be instructive to look at the following cxample, consisting of two ordinary differential equations, Tn
~
O\ 1~\ ~ (,,,.) +
(~o) (~)( +
p
.~
£1~ ~ —q~ qj ‘~Y1
(6) where n and pare positive. The eigenvalues are given by
(nQ2+pQ+q 2+q~=O. 5) The solutions of (7) are P±
means that the attractors have zero Hausdorff dimension, though the estimates [6] would deliver huge upper bounds such as lOla or much larger. In other words, nonlinear behavior and turbulence in MHD is much more configuration-dependent than in hydrodynamics. The reason is that the dimension of attractors or the number of determining modes is, to an extreme degree, configuration-dependent.
4n(t7 5+i~j~)
(8)
2n 2/4n,which fulfills condition (2), If we choose q~=p the threshold for instability is n J P2/2 = 2nq~. It is conceivable that the Mirnov or “precursor” oscillations, seen in tokamaks prior to “disruptions”, are of that kind, though they need not be related to that special instability. Pursuing this speculation, we could say that a further increase of density (or inertia) may lead to a “disruption” because the limit cycle can no longer be maintained. One of the puzzling questions is how to describe and follow such highly nonlinear systems for long times. In hydrodynamics, it is believed that such systems become turbulent or will tend to a fractal attractor with a large Hausdorff dimension increasing with the Reynolds number [6]. Estimates of the
258
Hausdorff dimension of attractors are made, however, from above [6,7] and do not necessarily reflect the real situation. Especially in MHD, this picture could be completely wrong: Nontrivial examples are known to be nonlinearly and unconditionally stable for all magnetic Reynolds numbers [3,8], which
(7)
_____
Q
6 September 1993
Let us finally note that, if condition (2) is not satisfied, other unstable eigenmodes can occur, but since the spectrum is not known, it will be very hard, in general, to check whether eigenvalues fulfill the requirements of the centerthe manifold theorem. References (1] H. Tasso, Phys. Lett.
A 147 (1990) 28. [2lH.Tasso,Phys.Lett.A 161 (1991) 289. [3] H. Tasso, Phys. Leti. A 169 (1992) 396.
14] J.E.
Marsden and M. McCracken, The Hopfbifurcation and its applications (Springer, Berlin, 1976). (51 H. Tasso and M. Cotsaftis. Plasma Phys. (part C) 7(1965) 29. [6] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics (Springer, Berlin, 1988). (7] P. Constantin, Commun. Math. Phys. 129 (1990) 241. 181 H. Tasso. Nuovo Cimento B, to be published.