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

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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 [FORMULA], then


where [FORMULA] is the temperature in the bubble and R is the radius of the bubble, and


where [FORMULA] is the pressure outside the bubble.


The more exact calculation modifies the value of this coefficient to 0.40. Using a value for [FORMULA], and [FORMULA], one can get for [FORMULA] and [FORMULA]. 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 [FORMULA] is the coefficient of the turbulent drag, v is the velocity of the bubble. The quantity [FORMULA] 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 [FORMULA], a consequence of the assumption 3, Eq. (16) can be arranged into a more suitable form:


The initial conditions for these equation are: at [FORMULA] [FORMULA], and [FORMULA]. 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 [FORMULA] (since for a movement with a constant speed the potential flow v and its potential [FORMULA] does not depend on time, and therefore the pressure distribution becomes [FORMULA], see Problem 2, Sect. 10, Landau & Lifshitz 1959, p. 25), and substituting [FORMULA] from the condition of hydrostatic equilibrium [FORMULA],


The acceleration "a" can be estimated with [FORMULA], [FORMULA], [FORMULA], [FORMULA], [FORMULA], so [FORMULA] and the bubble reach the constant speed during [FORMULA]. The obtained velocity of the hot bubbles [FORMULA] 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 [FORMULA] 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).

A.2. What amount of temperature surplus is necessary for the bubble to reach the solar envelope from the core?

Calculating the adiabatic temperature as


(see e.g. Lang, 1980, formula 3-296); here g is the local gravitational acceleration, and [FORMULA] is the specific heat at constant pressure. Estimating its value at [FORMULA]. At [FORMULA] the adiabatic temperature gradient is [FORMULA]. This means that the temperature difference between the adiabatic gradient and the actual one in the solar core, when going from [FORMULA] to [FORMULA] will be [FORMULA]. If one estimate the temperature difference when going downwards from the bottom of the convective zone to [FORMULA], this will be [FORMULA]. The estimation shows that it is necessary a surplus local heating [FORMULA] for the bubbles to be able to reach the bottom of the convective from [FORMULA]. This amount of heating (and more) is easily produced with electric heating induced by the interaction of tidal waves and the local magnetic field (Grandpierre, 1990). Therefore, a small amount of heating may be able to trigger the movement of the bubbles upwards and they may arrive to the bottom of the connectivezone, from where their movement is again facilitated.

A.3. What is the characteristic life-time of a hot bubble?

The bubble may get a significant energy input from electric heating (Grandpierre, 1990). Moreover, its temperature may grow exponentially when the timescale of the volume expansion is smaller than that of the local thermal instability. Now, [FORMULA] is significantly larger than [FORMULA] (Arnett & Clayton 1970) for [FORMULA], since the expanding bubble has to work against the large hydrostatic pressure of the star. The timescale of the thermal runaway is estimated to be [FORMULA] (Grandpierre, 1990) for [FORMULA] (Edwards, 1969). Even for smaller rate of energy production like [FORMULA] the runaway may be faster than volume expansion. Now the nuclear depletion lifetime of a bubble, in the absence of entrainment, can be estimated as


where Q is the heat energy produced in the dominant reaction per nucleon, and A is the number of nuclei participating in the reaction per gram, [FORMULA] is the average energy production in the bubble. With [FORMULA] amd [FORMULA] the lifetime of the bubble is [FORMULA]. During its nuclear lifetime the bubble may travel the solar radius [FORMULA] if its velocity is [FORMULA]. Therefore, even when the energy production goes largely from the triple alpha reaction, with a smaller Q, the bubble may travel from the core to the solar convective zone, especially since the triple alpha reaction prevails above [FORMULA], therefore in that case the bubble has a larger temperature surplus over its environment and so it is accelerated to a larger speed.

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© European Southern Observatory (ESO) 1999

Online publication: August 13, 199