Vector and scalar spherical harmonic spectral equations for rapidly rotating anisotropic alpha-effect dynamos

摘要

Spectral equations are derived for a mean field induction equation, ?B/?t-▽~2B= R▽×F with an a-effect, considered appropriate for rapid rotation, given by F = a ?B = a_1B + a_3z ?B z, where (x,?,z) are Cartesian unit vectors, a1(r,θ,φ), a3(r,θ,φ) are scalar functions of position, (r,θ,φ) are spherical polar coordinates and R is the magnetic Reynolds number. The effect of rotation on convection for different boundaries and parameters is discussed. The effect of the flow structure on a for different geostrophic and near geostrophic models is analysed. The vector spherical harmonics Y_(n,n1)~m (θ,φ) = (-1)~(n-m)[2n+ 1]~(1/2) ∑_(μ=-1,0,1)(_m~n _(μ- m)~(n1)_(-μ)~1)Y_(n1)~(m-μ)e_μ, where e_(-1) = (x - i?)/2~(1/2), e_0 =^z, e_1 = -(x + i ?)=21=2, the 2×3 matrix is a Wigner 3J coefficient and Y_n~m= Y_n~m (θ,φ) are scalar spherical harmonics, are used to derive the vector Y_(n,n1)~m forms of the induction equation for this α-effect. The solenoidal condition ▽? B = 0 is imposed by relating the Y_(n,n1)~m formalism to the toroidal–poloidal harmonic formalism, T_n~m = ▽ × (rT _n~m Y_n~m) and S_n~m = ▽ × ▽ × (rS_n~m Y_n~m). The T _n~m and S_n~m components of the induction equation are thus derived in terms of F_(n,n1)~m, the Y_(n,n1)~m components of F; F =∑_(n1=n-1)~(n+1)∑~n_(m=-n)∑_(n=0)~∞ F~m_(n,n1) Y_(n,n1)~m. These combined T_n~m /Y_(n,n1)~m, S_n~m /Y_(n,n1)~m vector spectral equations are then transformed into interaction type (a_(na)S_nS_N)_I, (a_(na)T_nT_N)_I, (a_(na)S_nT_N)_I, (a_(na)T_nS_N)_I and (a_(na)S_nS_N)_A, (a_(na)T_nT_N)_A, (a_(na)S_nT_N)_A, (a_(na)T_nS_N)_A equations for the isotropic and anisotropic components of a. As an application of the general spectral equations derived herein, the interaction equations can be specialised by restricting a_1 and a_3 to be proportional to r cos θ or cos θ, or restricting B and α to be axisymmetric. These equations are then compared to those of previous works. The differences between the equations derived herein and those of past works provide corrections and account for, at least in part, the differences in numerical solutions of the past works.