A UNIFIED APPROACH TO THE HELIOSEISMIC FORWARD AND INVERSE PROBLEMS OF DIFFERENTIAL ROTATION

We present a general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problems for solar differential rotation. Our approach differs from previous work in two principal ways. First, in the forward problem, we represent differential rotation as the axisymmetric component of a general toroidal flow field using vector spherical harmonics. The choice of these basis functions for differential rotation over previously chosen ad hoc basis functions (e.g., trigonometric functions or Legendre functions) allows the solution to the forward problem to be written in an exceedingly simple form (eqs. [32]-[37]). More significantly, their use results in inverse problems for the set of radially dependent vector spherical harmonic expansion coefficients, which represent rotational velocity, that decouple so that each degree of differential rotation can be estimated independently from all other degrees (eqs. [56] & [61]-[63]). Second, for use in the inverse problem, we express the splitting caused by differential rotation as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem (eqs. [5] and [54]). The orthonormal polynomials are Clebsch-Gordon coefficients and the estimated expansion coefficients are called splitting coefficients. The representation of splitting with Clebsch-Gordon coefficients rather than the commonly used Legendre polynomials results in an inverse problem in which each degree of differential rotation is related to a single splitting coefficient (eq. [56]). The combined use of the vector spherical harmonics as basis functions for differential rotation and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which will greatly simplify inversion. We submit that the mathematical and computational simplicity of both the forward and inverse problems afforded by our approach argues persuasively that helioseismological investigations would be well served if the current ad hoc means of representing differential rotation and splitting would be replaced with the unified methods presented in this paper.