Astron. Astrophys. 348, 993-999 (1999)

## Appendix A: the travel of the hot bubbles towards the solar surface

### A.1. The velocity of the bubbles

Accepting that a hot bubble forms in the solar core, we can ask how can such a bubble survive when it starts to rise, when accelerated by the buoyant force. At first sight, it may seem that such a bubble can easily loose its surplus inner energy on its pathway by turbulent viscosity and other forms of dissipation. Moreover, heat expansion may quickly cool the bubble and so it may dissolve with its surroundings. Nevertheless, quantitative estimates show that the bubbles may easily keep their identity on travelling towards the solar surface and that if their energy surplus is significant enough, they can even reach the surface layers, where they are certainly disintegrate due to a developing shock wave ("sonic boom", Grandpierre 1981).

Let us see some equations determining the conditions of the bubbles on their way towards the surface. Following Gorbatsky (1964), if the hot bubble's surplus energy is dominated by radiation, and their initial energy surplus when they start to rise is , then

where is the temperature in the bubble and R is the radius of the bubble, and

where is the pressure outside the bubble.

The more exact calculation modifies the value of this coefficient to 0.40. Using a value for , and , one can get for and . Of course, a hotter bubble can contain the same amount of internal energy with a smaller size.

Now let us calculate the velocity of a bubble! Following Gorbatsky (1964), let us assume the following plausible assumptions:

1. the bubble keeps its spherical form

2. the region between the bubble and its environment is thin (very narrow turbulent wake)

3. the density and temperature may be regarded uniform within the bubble

4. the pressure within the bubble equals with the pressure outside it.

On its travel the bubble meets with resistance, therefore it is necessary to put a term in the equation of motion of the bubble describing it. Since the molecular viscosity is extremely low, it is enough to take into account only the turbulent drag here:

where is the coefficient of the turbulent drag, v is the velocity of the bubble. The quantity represents the so-called "induced-mass" term occuring for a body moving in a hydrodynamic medium (see Landau, Lifshitz, 1959, Sect. 11) in case of a spherical bubble. Using the equality , a consequence of the assumption 3, Eq. (16) can be arranged into a more suitable form:

The initial conditions for these equation are: at , and . As the bubble becomes accelerated to a large enough velocity, the turbulent drag and the gravitational force will balance the buoyant force and the velocity becomes steady,

Now accepting that (since for a movement with a constant speed the potential flow v and its potential does not depend on time, and therefore the pressure distribution becomes , see Problem 2, Sect. 10, Landau & Lifshitz 1959, p. 25), and substituting from the condition of hydrostatic equilibrium ,

The acceleration "a" can be estimated with , , , , , so and the bubble reach the constant speed during . The obtained velocity of the hot bubbles is much larger then the average speed of convective cells in the solar convective zone.

It has to note here that recent experimental and computational results suggest that for extremely high Rayleigh numbers the turbulent convection turns to a thready flow. "The flow is driven entirely (in the limit of infinite Ra) by these threads. The heat flux is carried by flows that maintain their identity...and can cross a convecting layer with little mixing between them. The width of the threads, in spite of entrainment, decreases with Rayleigh number instead of increasing as one might have expected on the basis of the simple `higher Ra means more turbulence means more mixing' line of argument" (Spruit, 1997).