On the influence of gradients in the angular velocity on the solar meridional motions

Abstract

If fluctuations in the density are neglected, the large-scale, axisymmetric azimuthal momentum equation for the solar convection zone (SCZ) contains only the velocity correlations $\left\langle {\rho u_r u_o } \right\rangle$ and $\left\langle {\rho u_\theta u_o } \right\rangle$ where u are the turbulent convective velocities and the brackets denote a large-scale average. The angular velocity, Ω, and meridional motions are expanded in Legendre polynomials and in these expansions only the two leading terms are retained (for example, $\Omega = \Omega _0 \tfrac{1}{2}\omega _0 (r) \bot \omega _2 (r)P_2 (\cos \theta )$ where θ is the polar angle). Per hemisphere, the meridional circulation is, in consequence, the superposition of two flows, characterized by one, and two cells in latitude respectively. Two equations can be derived from the azimuthal momentum equation. The first one expresses the conservation of angular momentum and essentially determines the stream function of the one-cell flow in terms of $\int_0^\pi {\langle \rho u_r u_ \odot \rangle {\text{ sin}}^{\text{2}} \theta {\text{d}}}$: the convective motions feed angular momentum to the inner regions of the SCZ and in the steady state a meridional flow must be present to remove this angular momentum. The second equation contains also the integral $\int_0^\pi {\langle \rho u_r u_ \odot \rangle {\text{ cot}}^{\text{2}} \theta {\text{d}}}$ indicative of a transport of angular momentum towards the equator.

With the help of a formalism developed earlier we evaluate, for solid body rotation, the velocity correlations $C_{r \odot } = \int_0^\pi {\langle u_r u_ \odot \rangle {\text{ sin}}^{\text{2}} \theta {\text{d}}}$ and $C_{0 \odot } = \int_0^\pi {\langle u_r u_ \odot \rangle {\text{ cot}}^{\text{2}} \theta {\text{d}}}$ for several values of an arbitrary parameter, D, left unspecified by the theory. The most striking result of these calculations is the increase of $C_{\theta \odot }$ with D. Next we calculate the turbulent viscosity coefficients defined by $C_{r \odot } = C_{r0}^O - v_{r \odot }^1 r\Omega _0 \omega _0^1 - v_{r \odot }^2 r\Omega _0 \omega _2^1 - v_{r \odot }^3 \Omega _0 \omega _2 {\text{ and }}C_{\theta \odot } = C_0^O - v_{0 \odot }^1 r\Omega _0 \omega _0^1 - v_{0 \odot }^2 r\Omega _0 \omega _2^1 - v_{\theta \odot }^3 \Omega _0 \omega _2$ whereC ⁰ _ro and C ⁰_θo are the velocity correlations for solid body rotation. In these calculations it was assumed that ω₂ was a linear function of r. The arbitrary parameter D was chosen so that the meridional flow vanishes at the surface for the rotation laws specified below. The coefficients v ⁱ _ro and v ⁱ _0o that allow for the calculation of C _ro and C _0o for any specified rotation law (with the proviso that ω₂ be linear) are the turbulent viscosity coefficients. These coefficients comply well with intuitive expectations: v ¹ _ro and −v ³ _0o are the largest in each group, and v ³ _0o is negative.

The equations for the meridional flow were first solved with ω ₀ and ω ₂ two linear functions of r (ω ¹₀ = − 2 × 10 ⁻¹² cm ⁻¹) and (ω ¹₂ = − 6 × 10 ¹² cm ⁻¹). The corresponding angular velocity increases slightly inwards at the poles and decreases at the equator in broad agreement with heliosismic observations. The computed meridional motions are far too large (≈ 150m s⁻¹). Reasonable values for the meridional motions can only be obtained if ω _o (and in consequence Ω), increase sharply with depth below the surface. The calculated meridional motion at the surface consists of a weak equatorward flow for gq < 29° and of a stronger poleward flow for θ > 29°.

In the Sun, the Taylor-Proudman balance (the Coriolis force is balanced by the pressure gradient), must be altered to include the buoyancy force. The consequences of this modification are far reaching: Ω is not required, now, to be constant along cylinders. Instead, the latitudinal dependence of the superadiabatic gradient is determined by the rotation law. For the above rotation laws, the corresponding latitudinal variations of the convective flux are of the order of 7% in the lower SCZ.