D. Lortz

On the stability of steady finite amplitude convection

The static state of a horizontal layer of fluid heated from below may become unstable. If the layer is infinitely large in horizontal extent, the Boussinesq equations admit many different steady solutions. A systematic method is presented here which yields the finite-amplitude steady solutions by means of successive approximations. It turns out that not every solution of the linear problem is an approximation to the non-linear problem, yet there are still an infinite number of finite amplitude solutions. A similar procedure has been applied to the stability problem for these steady finite amplitude solutions with the result that three-dimensional solutions are unstable but there is a class of two-dimensional flows which are stable. The problem has been treated for both rigid and free boundaries.

EXACT SOLUTIONS OF THE HYDROMAGNETIC DYNAMO PROBLEM.

Exact solutions of the steady-state hydromagnetic equations with a solenoidal velocity field are derived for appropriate boundary conditions in a helical geometry.

Über die Existenz toroidaler magnetohydrostatischer Gleichgewichte ohne Rotationstransformation

ZusammenfassungEs wurde die existenz von Lösungen der Gleichungen des magneto-hydrostatischen Gleichgewichts unter folgenden Bedingungen untersucht. Das MagnetfeldB ist auf einer dreifach zusammenhängenden analytischen toroidalen FlächeT tangential, die zu einer Meridionalebene spiegelsymmetrisch liegt. Der Druck als hinreichend glatte Funktion vonq=ϕ|B|−1dl und der azimutale magnetische FlussF sind vorgegeben. Es zeigte sich, dass ein Gleichgewicht existiert, falls der Druckgradient nicht zu gross ist. Diese Gleichgewichtskonfiguration liegt symmetrisch zur Ebene der Spiegelsymmetrie und besitzt nur geschlossene Magnetfeldlinien. Die in den Anwendungen ausserdem erwünschte Forderung, dass die Flächenq=const dreifachen toroidalen Zusammenhang haben, ist erfüllbar, indem die FlächeT so gewählt wird, dass das durchT undF bestimmte Vakuummagnetfeld diese Eigenschaft hat.AbstractThe magnetohydrostatic boundary value problem is investigated for a toroidal boundary surfaceT which is symmetric with respect to a plane, but which need not be continuously symmetric. The existence of an equilibrium solution is proved for the boundary condition that the magnetic fieldB be tangential atT. The pressure as a sufficiently smooth function ofq=ϕ|B|−1dl and the azimuthal magnetic flux can be chosen arbitrarily.

On the Decay of Symmetric Toroidal Dynamo Fields

The “anti-dynamo” theorems for toroidal magnetic fields with axisymmetry and plane symmetry are generalized to the case of a compressible, time-dependent flow in a fluid with arbitrary conductivity

A stability criterion for steady finite amplitude convection with an external magnetic field

In a horizontal layer of fluid, thermal expansion or the presence of dissolved salt may cause a density gradient opposite to the direction of gravity. In such cases, when the buoyancy forces are sufficient to overcome the dissipative effects, the static state becomes unstable and convective motions arise. If the layer is infinitely large in horizontal extent, the non-linear convection problem is highly degenerate, admitting many different steady-state solutions. A general necessary criterion for stability of such non-linear steady solutions is developed here for the case in which a homogeneous vertical magnetic field acts on the fluid. The criterion is demonstrated for two rigid bounding surfaces which are perfect thermal and electrical conductors, but it is applicable to more general kinds of boundary conditions.

A Simple Stationary Dynamo Model

Solutions of the stationary dynamo equations are derived such that outside a torus the magnetic field is the axisymmetric vacuum field of a circular loop, while inside the torus in the limit of large aspect ratio both the velocity and the magnetic fields have helical symmetry.

On the stability of two-dimensional convection

ZusammenfassungDie Stabilität einer zweidimensionalen stationären Konvektionsströmung wird für den Grenzfall grosser Prandtlzahl untersucht. Es ergibt sich ein notwendiges Stabilitätskriterium, das für beliebige Rayleighzahlen gültig ist.

ASYMPTOTIC MAGNETIC SURFACES.

Toroidal magnetic fields are considered. On rather weak assumptions it is shown that single-valued formal solutions of the equation B. grad F=0 exist, and that the asymptotic magnetic surfaces F=const. are uniquely determined up to any order. Recurrence formulae are derived which allow calculation of any order of F from its lower orders, and explicit expressions are given for F0 which depend on whether the rotational transform is of the same or of higher order than the perturbing field. For the special cases of stellarator-like vacuum fields a necessary condition is derived for the asymptotic magnetic surfaces to be toroidally closed. As another application the lowest order of an adiabatic invariant is constructed for the longitudinal guiding centre motion.

Stabilization of axisymmetric modes in plasma--vacuum equilibria by current reversal

Two‐dimensional plasma–vacuum equilibria are constructed by solving a free‐boundary problem for small deviations from a circular cross section, and their two‐dimensional stability is studied by examining that eigenmode that reduces to the marginal rigid shift in the circular limit (all other two‐dimensional modes are stable). The vacuum is assumed to extend to infinity (no wall), N axial external currents are assumed to be symmetrically placed far away from the plasma, and the plasma current density is assumed to be a linear function of the poloidal magnetic flux. It is found that instability always results for N=2 (elliptical corrugation), but that stability may result for N≥3 if the axial plasma current density changes sign across some pressure surface. It is concluded that axially symmetric modes in toroidal equilibria with large aspect ratio can be stabilized by current reversal, even if no conducting wall surrounds the vacuum.

On the Stability of Axisymmetric MHD Modes

The stability of axisymmetric ideal MHD equilibria which are symmetric with respect to the equatorial plane is considered. It is found that for external axisymmetric modes which are antisymmetric with respect to the equatorial plane and for profiles such that the current density vanishes at the free plasma boundary the stability problem reduces to a classical interior-exterior scalar eigenvalue problem. Because of the separation property the resulting stability condition is necessary and sufficient and is thus more stringent than criteria derived by choosing special test functions, e.g. the vertical shift condition.