Speed of sound at the Sun surface

Question

The gas velocity at the Sun's surface is subsonic and At 1 AU from the sun, the solar wind is supersonic. Like Parker wind solution.

Why the velocity at the Sun's surface is subsonic?

(In the isothermal condition but I think also in general)

Answer 1

It is not obvious that convection in the Sun must be subsonic, because the ratio of speeds depends on temperature, which drops to 6000 K at the surface, before the plasma becomes transparent and radiative heat transport takes over from convection.

The speed of sound is simply $\sqrt{5P/3\rho }\sim \sqrt{T}$, and the polytropic rule says that $\rho \sim {{T}^{3/2}}$.

The speed of convection currents cannot be calculated with any precision, but it may be estimated via glorified dimensional analysis. Naive dimensional analysis is ambiguous because too many parameters seem relevant: the luminosity (Q), radius (R), density $(\rho )$, coefficient of thermal expansion (a), specific heat capacity $({{C}_{P}})$, and acceleration of gravity (g). However, a detailed look at convection narrows down the relevant combination. When a blob of plasma absorbs heat, it becomes buoyant, and the heat is converted to kinetic energy as the blob rises through a distance $Z={{C}_{P}}/ag=T{{C}_{P}}/g$. Kolmogorov’s theory of turbulence identifies power per unit mass $(Q/Z{{R}^{2}}\rho )$ as the key combination. The meaningful combination with the units of speed is $\sqrt[3]{Q/ZR\rho }\sim {{T}^{-5/6}}$. According to this argument, convection gets faster and overtakes the speed of sound as you approach the surface outbound.

Caveat: Rotation and MHD both tend to restrain convection. Coriolis forces deflect radial up- & down-drafts, and conductive plasma cannot flow across magnetic field lines without distorting them, with resulting pushback. The qualitative effect is to slow convective motions while increasing temperature differences between up- & down-drafts. There is no consensus scaling law for such conditions. The naive argument may only be valid for convection along the polar axis, i.e., the axis of rotation.