Gravity Wave–driven Flows in the Solar Tachocline. II. Stationary Flows

Abstract

The effects of gravity waves on the mean radial differential rotation profile in the solar tachocline are studied, including the effect of a uniform, toroidal magnetic field. Vertical transport of horizontal momentum arises from the radiative damping of inwardly traveling waves that are generated by low-frequency, convective fluid motions. By considering two-wave and one-wave interactions, the radiatively damped gravity waves are shown to accentuate the shear in the mean radial differential rotation. In the presence of a strong horizontal magnetic field, internal gravity waves become nearly Alfvénic and cannot propagate downward through the tachocline. For a magnetic field that is weak enough to permit wave propagation, the mean shear profile is shown to be smoother than that obtained in the case of purely hydrodynamic waves. The implications of our results for gravity-wave forcing of the internal solar rotation are discussed.

Low-frequency internal waves in magnetized rotating stellar radiation zones

“…First, once again we isolate the vertical trapping of MGI waves as soon as A < 0. Moreover, note that this result is the generalisation for of a deep stably stratified spherical shell as the one obtained by Kim & MacGregor (2003) for a reduced 2D Cartesian model of the solar tachocline. -Class IV waves are purely prograde waves (m < 0) whose characteristics change little with ν M;m , their displacement in the θ direction being very small.…”

“…Here, we follow the method that was applied for example by Kim & MacGregor (2003) to their reduced 2D Cartesian model of the magnetised solar tachocline, to derive the vertical transports of energy and angular momentum for a deep stably stratified spherical shell. First, taking the scalar product of the linearised Eulerian momentum equation with u and those of the linearised induction equation with b, we obtain the equation that rules the transport of the waves' total energy…”

Self-consistent theory of turbulent transport in the solar tachocline

“…Specifically, we express the total mass density ρ = ρ 0 (x) + ρ where ρ 0 (x) and ρ are the background and fluctuating mass densities, respectively; the total velocity u = U 0 + u, where U 0 and u are a large-scale shear flow due to differential rotation and small-scale fluctuations, and the total particle density of chemical elements n = n 0 (x) + n where n 0 (x) and n are mean and fluctuating components. Then, the main governing equations for fluctuations u, ρ , and n , involving both turbulence and gravity waves, are as follows (see, e.g., Kim & MacGregor 2003):…”

“…(1) is the small-scale forcing driving turbulence; ρ = ρ 0 (x = 0) is the constant background density [measured at the bottom of the convection zone (e.g. see Kim & MacGregor 2003)], and N = −g(∂ x ρ 0 + ρg/c 2 s )/ρ is the Brunt-Väisälä frequency, where c s is the sound speed. Note that the typical values of ν, D, µ, and N in the tachocline are 10 2 cm 2 s −1 , 10 2 cm 2 s −1 , 10 7 cm 2 s −1 , and 3 × 10 −3 s −1 , respectively.…”

Transport and instability in driven two-dimensional magnetohydrodynamic flows

“…This is investigated further below, but there is an indication that the effect of the feedback is to sharpen any pre-existing profile, increasing the flow where Ω = −U ′ = 0 and flattening the flow profile where U = 0 and |Ω| is maximal. Note that a tendency to sharpen shear profiles is found in weakly damped vortex dynamics simulations by Dritschel & McIntyre (2008), and by Kim & MacGregor (2003) in a study of gravity waves in stratified fluid, together with a bifurcation to an oscillatory state.…”