The Generalization of Mixing Length Theory to Rotating Convection Zones and Applications to the Sun

Abstract

The consequences of a balance between the Coriolis forces, pressure gradients and buoyancy forces in a compressible medium are investigated (the Taylor-Proudman theorem). A simple proof is given that if this balance holds, then the latitudinally dependent part of the superadiabatic gradient (∇ΔT) is determined by the angular velocity, Ω, and it is of the order of 2Ω₀ ² T/7g fo͂r rotation laws other than Ω constant along cylinders (it vanishes in this case). Here Ω₀ is the average angular velocity, T the temperature and g gravity. In the lower part of the solar convection zone (SCZ), 2Ω₀ ² T/7g is of the order of ∇ΔT _r , itself, i.e., very large.

The generalization of the mixing length theory to rotating convection zones is developed with particular emphasis on the approximations involved. The arbitrary parameters are the dimensions of the turbulent eddies. For large values of Ω₀, these eddies (with dimensions, say, ℓ _pa , ℓ _pe along and perpendicular to the axis of rotation), align themselves along the axis of rotation. Of particular importance is the “elongtion law,” i.e., the equation relating ℓ _pa /ℓ _pe to α $( = 4\Omega _0^2T/9\nabla \Delta {T_r})$, the dimensionless parameter characterizing the strength of rotation: surprisingly, there is little freedom in the choice of physically realistic elongation laws. It is shown that the Reynolds stresses transport angular momentum towards the equator and towards the deeper layers of the SCZ. This angular momentum transport tends to generate angular velocities with ω ₀(r) decreasing outwards and negative ω ₂(r) (equatorial acceleration), in conflict with a constant Ω along cylinders ($\Omega = {\omega _0}(r) + {\omega _2}(r){P_2}(\cos \phi) + {\omega _4}(r){P_4}(\cos \phi) + \cdots ;\phi$) is the polar angle and P _2n (cosθ) are Legendre polynomials). Meridional motions should change this rotation law. Large latitudinal variations (∇ _θ ΔT) in the superadiabatic gradient should, however, be present in the lower part of the SCZ. The numerical solution of the energy equation shows that large values of ∇ _θ ΔT do not necessarily imply large latitudinal variation in the energy flux.

The behaviour of the angular velocity at the boundaries of the SCZ is investigated. In order to include the overshoot region, no assumption is made concerning the radial variation of the turbulent viscosity. It is shown that for values of r slightly larger then R _c (the lower boundary), ω ₂(r) varies little with r and ω ₀(r) decreases outwards as suggested by dynamo theory.

Operated by the Assocation of Universities for Research in Astronomy. Inc. under contract with the National Science Foundation.