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摘要
Spectral equations are derived for a mean field induction equation, partial derivative(B) over bar/partial derivative t - del(2)(B) over bar = R del x F with an alpha-effect, considered appropriate for rapid rotation, given by F = alpha.(B) over bar = a(1)(B) over bar + a(3)(z) over cap . (B) over bar(z) over cap, where ((x) over cap, (y) over cap,(z) over cap) are Cartesian unit vectors, a(1)(r, theta, phi), a(3)(r, theta, phi) are scalar functions of position, (r, theta, phi) 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 alpha for different geostrophic and near geostrophic models is analysed. The vector spherical harmonics
Y-n,n1(m)(theta, phi) = (-1)(n-m)[2n + 1](1/2) Sigma(mu=-1,0,1)(m(n) mu-m(n1) -mu(1)) Y(n1)(m-mu)e(mu),
where e(-1) = ((x) over cap - i (y) over cap)/2(1/2), e(0) = (z) over cap, e(1) = -((x) over cap + i (y) over cap)/2(1/2), the 2 x 3 matrix is a Wigner 3J coefficient and Y-n(m) = Y-n(m)(theta, phi) are scalar spherical harmonics, are used to derive the vector Y-n,n1(m) forms of the induction equation for this alpha-effect. The solenoidal condition del.(B) over bar = 0 is imposed by relating the Y-n,n1(m) formalism to the toroidal-poloidal harmonic formalism, T-n(m) = del x (rT(n)(m)Y(n)(m)) and S-n(m) = del x del x (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 = Sigma(n+1)(n1=n) Sigma(n)(m= n) Sigma(n=0Fn,n1Yn,n1m)-F-infinity-Y-m. These combined (TnYn,n1m)-Y-m, S-n(m)/Y-n,n1(m) vector spectral equations are then transformed into interaction type (a(n alpha)S(n)S(N))(I), (a(n alpha)T(n)T(N))(I), (a(n alpha)S(n)T(N))(I), (a(n alpha)T(n)S(N))(I) and (a(n alpha)S(n)S(N))(A), (a(n alpha)T(n)T(N))(A), (a(n)alpha SnTN)(A), (a(n alpha)T(n)S(N))(A) equations for the isotropic and anisotropic components of alpha. 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 theta or cos theta, or restricting (B) over bar and alpha 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.
- 出版日期2013-12-1
