Abstract
Explosive phenomena such as internal disruptions and flares are difficult to explain in terms of linear instabilities. Similarly to a particle moving in a potential {alpha} x{sup 3} for which x = 0 is a linearly stable, but nonlinearly explosively unstable equilibrium point, a plasma approaching a stability limit can be expected to become explosively unstable. If an initial perturbation is not infinitesimally small, this will be the case even before the marginal point is reached. To enable such problems to be investigated, a nonlinear extension of the usual energy principle would be very helpful. This was achieved for the first time by Merkel and Schlueter within the framework of Cartesian coordinate systems. In this paper an exact coordinate system independent formulation for the Lagrangian is obtained, allowing also equilibria with flow; it has mass, entropy and flux conservation built in. For vanishing equilibrium flow velocity v(x) the Lagrangian contains, besides the potential energy, which depends on the displacements {zeta} (x, t), only quadratic but no linear terms in {zeta} representing the kinetic energy. Nonlinear stability, like linear stability, is then completely determined by the potential energy. For potential energies which consist of second- and n{sup th}-order contributions only, where
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Pfirsch, D
[1]
- Association Euratom-Max-Planck-Institut fuer Plasmaphysik, Garching (Germany)
Citation Formats
Pfirsch, D.
Nonlinear ideal MHD instabilities.
Germany: N. p.,
1992.
Web.
Pfirsch, D.
Nonlinear ideal MHD instabilities.
Germany.
Pfirsch, D.
1992.
"Nonlinear ideal MHD instabilities."
Germany.
@misc{etde_10132541,
title = {Nonlinear ideal MHD instabilities}
author = {Pfirsch, D}
abstractNote = {Explosive phenomena such as internal disruptions and flares are difficult to explain in terms of linear instabilities. Similarly to a particle moving in a potential {alpha} x{sup 3} for which x = 0 is a linearly stable, but nonlinearly explosively unstable equilibrium point, a plasma approaching a stability limit can be expected to become explosively unstable. If an initial perturbation is not infinitesimally small, this will be the case even before the marginal point is reached. To enable such problems to be investigated, a nonlinear extension of the usual energy principle would be very helpful. This was achieved for the first time by Merkel and Schlueter within the framework of Cartesian coordinate systems. In this paper an exact coordinate system independent formulation for the Lagrangian is obtained, allowing also equilibria with flow; it has mass, entropy and flux conservation built in. For vanishing equilibrium flow velocity v(x) the Lagrangian contains, besides the potential energy, which depends on the displacements {zeta} (x, t), only quadratic but no linear terms in {zeta} representing the kinetic energy. Nonlinear stability, like linear stability, is then completely determined by the potential energy. For potential energies which consist of second- and n{sup th}-order contributions only, where n is one special integer, and which can be made negative, it is shown, in full generality, that linearly marginally stable systems are nonlinearly explosively unstable even with infinitesimally small initial perturbations. Also linearly absolutely stable systems turn out to be explosively unstable, but finite initial perturbations are needed. For non-vanishing flow velocities nonlinear instabilities relating to negative-energy modes are possible. (orig.).}
place = {Germany}
year = {1992}
month = {Mar}
}
title = {Nonlinear ideal MHD instabilities}
author = {Pfirsch, D}
abstractNote = {Explosive phenomena such as internal disruptions and flares are difficult to explain in terms of linear instabilities. Similarly to a particle moving in a potential {alpha} x{sup 3} for which x = 0 is a linearly stable, but nonlinearly explosively unstable equilibrium point, a plasma approaching a stability limit can be expected to become explosively unstable. If an initial perturbation is not infinitesimally small, this will be the case even before the marginal point is reached. To enable such problems to be investigated, a nonlinear extension of the usual energy principle would be very helpful. This was achieved for the first time by Merkel and Schlueter within the framework of Cartesian coordinate systems. In this paper an exact coordinate system independent formulation for the Lagrangian is obtained, allowing also equilibria with flow; it has mass, entropy and flux conservation built in. For vanishing equilibrium flow velocity v(x) the Lagrangian contains, besides the potential energy, which depends on the displacements {zeta} (x, t), only quadratic but no linear terms in {zeta} representing the kinetic energy. Nonlinear stability, like linear stability, is then completely determined by the potential energy. For potential energies which consist of second- and n{sup th}-order contributions only, where n is one special integer, and which can be made negative, it is shown, in full generality, that linearly marginally stable systems are nonlinearly explosively unstable even with infinitesimally small initial perturbations. Also linearly absolutely stable systems turn out to be explosively unstable, but finite initial perturbations are needed. For non-vanishing flow velocities nonlinear instabilities relating to negative-energy modes are possible. (orig.).}
place = {Germany}
year = {1992}
month = {Mar}
}