Random waves and dynamo action

The propagation of waves in an inviscid, electrically conducting fluid is considered. The fluid rotates with angular velocity Ω* and is permeated by a magnetic field b* which varies on the length scale L = Ql, where Q = Ω*l²/λ (l is the length scale of the waves, λ is the magnetic diffusivity) is assumed large (Q [Gt ] 1). A linearized theory is readily justified in the limit of zero Rossby number R₀ (= U₀/Ω*l, where U₀ is a typical fluid velocity) and for this case it is shown that the total wave energy of a wave train is conserved and transported at the group velocity except for that which is lost by ohmic dissipation. The analysis is extended to encompass the propagation of a sea of random waves.A hydromagnetic dynamo model is considered in which the fluid is confined between two horizontal planes perpendicular to the rotation axis a distance L₀(=O(L)) apart. Waves of given low frequency Ω*₀ (= O(R₀Q_½Ω*)) and horizontal wavenumber l⁻¹ but random orientation are excited at the lower boundary, where the kinetic energy density is 2πρU²₀. The waves are absorbed perfectly at the upper boundary, so that there is no reflexion. The linear wave energy equation remains valid in the double limit 1 [Gt ] R₀Q^½[Gt ]Q^−½, for which it is shown that dynamo action is possible provided $\Delta = L_0U^2_0/l^3\omega_0^{*2} > 1 $. When dynamo action maintains a weak magnetic field (Δ −1 [Lt ] 1) which only slightly modifies the inertial waves analytic solutions are obtained. In the case of a strong magnetic field (Δ [Gt ] 1) for which Coriolis and Lorentz forces are comparable solutions are obtained numerically. The latter class includes the more realistic case (Δ → ∞) in which the upper boundary is absent.