The following two plots are for the average velocity of a cell for different initial conditions. The colour represents the average velocity in km/s (label will be added later).

The next plot has the temperature of the surrounding medium in bold white. The 3 white lines  above that are 1.25, 1.5, and 2 times the solar temperature.

The next two are for the final velocity of a cell. The white lines are the same as above.

More Gradients

1. Adiabatic1= [(gamma-1)/(gamma)] * (T/P) * dP/dr
2. Adiabatic2= [(gamma-1)/(gamma)] * (T/P) * [(-Gm/r^2) * rho]
3. Adiabatic 3 = (gamma-1) * (T/P) * (P/rho) * (drho/dr)
4. Adiabatic 4 = -g/Cp

In the above, Adiabatic1 and Adiabatic2 are exactly the same (which is why you can't see 2).

Another condition for convection is that dlnP/dlnT > (gamma-1)/(gamma). In the above graph, the green line is (gamma-1)/(gamma).

Temp Gradients

The above are actual temperature gradients calculated using the 'new' method as described below. The top plot uses every second point and the bottom plot uses every third point.

The red line on the bottom graph is the adiabatic temperature gradient.

The above plot is like the others, except it uses only every 4th point. The green line is again the adiabatic temperature gradient.

Temperature Gradients

This graph has 3 parts to it.

Black- Adiabatic.

This is the temperature gradient which equals [(gamma - 1)/(gamma)] * [(t/P)] * [(g*rho)]

Red- Actual

This was calculated using the 'old' method. This was adding and taking away 1 from each point to get x1 and x2. Their corresponding temperatures were found (y1 and y2). The slope of these were calculated: 

(y2-y1)/(x2-x1)

Blue- Actual

This was calculated from the 'new' method:

For every point along the x-axis, its preceding and following points were taken to be x1 and x2 respectively.

The y-values for each of these (y1 and y2) were found, and the slopes were calculated as above.

It can be seen that the two methods for the actual temperature gradients are pretty much equivalent.
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Adiabatic Temperature Loss of Cell
I have managed to calculate the temperature change of a cell as it travels a distance.
The plot below is of the temperature loss of a cell of initial T=3E6 Kelvin, and r_initial = 0.7R.

When factored in, this does indeed slow the cell's acceleration down, so that a cell of 2.5E6 Kelvin at r_initial = 0.7, it takes 2133.9 seconds to rise. This is still quite fast.

Temperature Gradients part 3

The above is a plot of the Temperature Gradients. The blue line is a marker at 0.714 Solar Radii. At this point, there should be a cross-over between the adiabatic and actual temperature gradients. Below this point, there should be no convection.

Temperature gradients part 2

The above is the plot of the absolute values of the Adiabatic (black) and Actual (red) temperature gradients.

From it, one can see that |dT/dr|adb > |dT/dr|act. I now believe this to be correct, whereas before, I thought this was incorrect.

http://www.ucolick.org/~krumholz/courses/fall09_ast112/notes12.pdf