Axisymmetric antidynamo theorems in compressible non-uniform conducting fluids

Abstract

This paper considers magnetic fields generated by dynamo action in electrically conducting fluids under axisymmetric but otherwise very general non-steady conditions, including compressible flow v , variable magnetic diffusivity, general volume shape (possibly even a union of disconnected conductors) with smooth possibly-moving boundary. We critically review previous antidynamo results and are forced to question a number of the more important ones. Using operator inequalities and maximum principles for uniformly elliptic and parabolic partial differential equations we construct a number of functions (‘comparison functions’) that bound the meridional flux function x . In particular, we derive the uniform bounds χ ≤ X ( 0 ) , 0 ≤ t ≤ τ , X ( 0 ) t τ e 1 − t / τ , t ≥ τ , Y 1 ( 0 ) F 1 ( ϖ ) e − t / τ , t ≥ 0 , where X ( t ) = max |η| and Y 1 ( t ) = max |η/ F 1 (ϖ)| at time t ; F ⩾ 0 being a prescribed function of cylindrical radius ϖ; and equality in the first bound being only possible at t = 0. These bounds prove that χ decays uniformly in space and unconditionally to zero; by advancing the time origin, that X ( t ), Y 1 ( t ) decay strictly monotonically to zero; and τ is an upper bound for the decay time. By using Schauder-type a priori estimates, spherical harmonic analysis and other techniques, it follows from these bounds that other field parameters, such as the meridional vector field B m , the external multipole moments, the toroidal current density and the net outward surface flux, all decay to zero (but not all uniformly). The decay-time bound T is directly related to the magnetic Reynolds number R (based on the speed and radius suprema, and diffusivity infimum). For free decay ( R = 0) it is seen that variations in diffusivity and volume shape cannot extend the decay time by more than a factor of about 2.5 over the poloidal free-decay time in a fixed uniform sphere (τ pol = π -2 ). However, for large R , τ may take very large values (for example τ ⩾ 10 17 diffusion time units when R « 10 2 ). The same comparison function approach applied to the azimuthal field parameter A = B ?? ϕ /ϖ leads to pointwise decay bounds analogous to those given above provided R∇ v and δη / δϖ are small (see (5.16)); but, as for χ, the decay-time bounds may sometimes be exceedingly large. These decay bounds for A also assume that given the decay of B m , the rate of generation, ( v ϕ B m A ) say, of A by differential rotation shearing the B m -lines, is reasonably modelled by a uniformly decaying function. For the special case of free decay in a uniform fluid (where υ = δη / δϖ =( υ ϕ B m A )=0) it is seen that variations in volume shape cannot extend the free-decay time by more than a factor of about 2.5 over the toroidal free-decay time in a fixed uniform sphere ( τ tor — (1 -43π) —2 ) ; and for a uniform incompressible fluid (where ∇ υ = δη / δϖ =0, but ( υ ϕ B m A ) ≢ 0) it is seen that A decays uniformly pointwise to zero. For compressible non-uniform fluids we prove more generally (i.e. regardless of ∇ υ and δϖ / δη ) that || A || 1 , the volume integral of | A | cannot grow above a finite bound determined by ( υ ϕ B m A ).When ( υ ϕ m A ) is negligible || A || 1 is shown to decay strictly monotonically. And if In | A | does not develop negative ϖ-gradients steeper than In (constant/ t ½ ) for large t , then || A || 1 decays to zero beneath a comparison function, which again may decay extremely slowly. One of our main conclusions is that axisymmetric antidynamo theorems allowing compressible flow in non-uniform fluids have not yet been shown to be generally effective, in the sense that they do not ensure decay on timescales that do not significantly exceed the relevant astroplanetary timescales, unless the compressibility and non-uniformities are specially restricted (as, for example, by Backus ( Astrophys . J . 125, 500 (1957))). Our most important results do not rely on any velocity boundary conditions and therefore apply directly to non-velocity mechanisms such as the Nernst-Ettingshausen thermomagnetic effect. The results herein include previously established antidynamo theorems as special cases, and the methods provide alternative proofs of, and strengthen, known steady antidynamo theorems.