The Taylor‐Proudman Balance and the Solar Rotational Data

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Abstract
For the Sun (and because of the presence of buoyancy), the implications of the Taylor-Proudman balance (TPB, a balance between the pressure, Coriolis, and buoyancy forces in the radial and latitudinal momentum equations) differ fundamentally from those of an incompressible fluid. The TPB now only determines the latitudinal variations of the solar entropy in terms of the rotation law and known functions of r. As a consequence of the TPB, the energy equation is in fact an equation for the angular velocity, Ω(r, θ)=Ω0(ω0(r)+ω2(r)P2(cos θ)), where P2(cos θ) is the second-order Legendre polynomial. In agreement with data from the Solar Oscillations Investigations project (SOI) Michelson Doppler Imager (MDI) on board SOHO, we assume that ω0(r) is constant with r, and solve the equation for ω2(r) with a simple, heuristic expression for the convective flux [if ω0(r) is constant, then ∂Ω/∂r vanishes for θc=547, P2(cos θc)=0, in remarkable agreement with Kosovichev's results inferred from isocontours for Ω(r, θ)]. For values of the meridional motions that are not too large, solutions for ω2(r) exist that agree with these isocontours. These solutions are such that the latitudinal variation of the convective flux, arising from the latitudinal variations of the entropy, required by the TPB, are significantly reduced.