Approximate Isocontours For The Solar Angular Velocity In The Convection Zone

Abstract

The angular velocity, Ω, in the solar convection zone (SCZ) is expanded in Legendre polynomials, P n(cosθ), and the values for Ω at the equator are assumed to be given by Kosovichev's helioseismic data; here, r, θ, and φ, label the radial, latitudinal and longitudinal coordinates, respectively. The isocontours for Ω are calculated for the following two cases. (i) The angular momentum of a thin spherical shell of radius r is identical to the shell's angular momentum for solid body rotation, i.e., rotation just distributes in latitude the angular momentum of each layer. (ii) Considerations based on the Taylor–Proudman balance (a balance between the pressure, Coriolis and buoyancy forces which is expected to be amply satisfied in the SCZ), require that the radial component of the superadiabatic gradient be strongly dependent on latitude unless the coefficients in the expansion for Ω defined above satisfy a first-order differential equation, DE. The isocontours for the angular velocity determined from DE, compare remarkably well with the helioseismic data, whereas for case (i) there is a marked difference at high latitudes. The radial and latitudinal balance of angular momentum are studied. The meridional motions are determined mainly (but not entirely) by the radial balance of angular momentum, and they depend principally on the Reynolds stress, 〈ur uφ 〉. Concerning the latitudinal balance, ∂Ω/∂θ increases until the transport of angular momentum toward the poles by the meridional motions is able to balance the transport of angular momentum towards the equator by 〈 uθ uφ 〉 ( = 〈 uθ uφ 〉0 - νt sinθ∂Ω/∂θ)). Here the subscript 0 stands for solid body rotation, and νt is a turbulent viscosity coefficient. In contrast to the radial balance, the viscosity term plays a fundamental role in the latitudinal balance of angular momentum.