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

Apart from my laptop being stolen, there was a problem with the radii of my cells at different points (see post below). I have just sorted that out.

Updated Data

Found data which now covers the Sun's ambient conditions up to r = 0.983R. Because of this, I needed some new sexy graphs:

These are for the temperature, pressure, mass and density of the Sun.

This is a graph of the Sun's density against that of static cell's of various temperatures.

 

These two graphs plot the differences of temperature, density and pressure at the top and bottom of each cell, all over the actual value being used. One can see that at around r/R=0.95, the 1-d simulation just isn't sufficient. The rapidly decreasing pressure, temperature and density of the Sun means that the cell rapidly increases with volume, leading to greater differences between the top and bottom of the cell.

The rapidly expanding nature of the cell is illustrated above. Note the logarithmic y-axes.

The above graphs plot the motion and acceleration of two cells of different temperatures, assuming the only forces acting upon them are buoyancy and gravity.

This is the pressure scale height for the Sun.

Things to be done:

1. Work out the energy of the cell (thermal, internal, kinetic, gravitational) for each point to see which dominates. This can be done for different cell sizes/temperatures. This will give an idea of which processes govern the cell's behaviour.
2. Figure out heat loss/drag etc.

Temperature Gradient Problems

The above graph is of the absolute values of the actual temperature gradient of the sun, and of the adiabatic temperature gradient. There is a problem here in that:

For convection to occur, the absolute value of the actual temperature gradient has to be larger than the absolute value of the adiabatic temperature gradient (|dT/dr|act > |dT/dr|ad).

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Convective Instabilities in the Sun

Convection is a heat transfer process in which the heat is transferred via the motion of fluid elements. This occurs in the sun, between 0.7 and 1 solar radii, in the area known as the ‘solar convective zone’. Hotter gas rises from the radiative zone of the sun, and as it is less dense than the surrounding sun, it is forced upwards by the buoyancy force. As it is rising, this ‘cell’ of gas breaks up, and we can observe what are known as ‘granules’ on the solar surface. These make up the mottled pattern on the surface, and are comprised of hot upwelling areas, and denser down welling areas. They are of the order of ~1x10^6m in scale.

What is unknown about these granulations, is at what depths and temperatures these cells form. My project plans on shedding some light on this question via a 1.5 dimensional computer simulation. A ‘cell’ is given initial conditions (such as mass, radial position and temperature), amidst the backdrop static conditions of the Sun (mass, radial position and temperature). The differences between the cell and the surroundings will initiate a motion in the cell upwards. This cell can be followed to the surface via the simulation, and depending on whether or not it breaks up, can be compared to observed granulations.

So far, I have successfully simulated an adiabatic cell with only gravity and buoyancy acting upon it. From this, the acceleration/velocity/position of the cell can be followed as it travels to the surface. However, this is a simplification of what actually occurs, and unrealistic speeds are reached. This is because neither drag, downdraft from descending dense plasma nor heat transfer have been taken into effect. Density gradients for cells at various temperatures have also been plotted.

What remains to be achieved in this project is to add more physical factors to the motion of the cell. These can include heat transfer between the cell and its surroundings, drag, and the total amount of energy carried by the convection in the sun can be calculated. These will give a more realistic motion of the cell.

1) At what depth do the supergranules form?


2) What are the expected properties of supergranules at the surface and sub-surface?
~10% ionized hydrogen -R.F. Stein and Nordlund, Simulations of Solar Granulation
Has a strongly horizontal velocity, in order to conserve mass.
Flows up at speeds up to ~8km/s, flows down at speeds ~ -4km/s
Downward moving intergranular lane plasma has low temperature, low entropy, very low ioniztion, and high density. Upward moving granule has high temperature, high entropy, high ionization, and low density.

Upflow temperatures ~10^4K
Downflow temperatures ~6000K

Upflows occupy ~ 2/3 of area under surface
Entropy is nearly constant under surface (upflows), varies greatly (downflows)
Downflow is very turbulent, mixing a lot.

Temperature gradient for ascending uplows ~100 K/km
Near surface, upflows and downflows transport approximately equal amounts of energy. With increased depth, downflows come to dominate the energy transport.


A fluid parcel cools adiabatically until it reaches optical depths of around tau= 30 - 100, where it begins to lose heat to radiation. As it cools, it loses even more heat quicker (~200K/s)


3) What is the statistical distribution of supergranules at the surface?

More graphs

Reference Graphs (Adiabatic)

Objectives - 15-10-2012

1) Correct drag force

2) Use heat transfer

3) Fill in last 5% of Sun's radius

4) Figure mechanism for cells breaking up

Simplified Motion Part Deux

I have factored in a drag force of the form:  

However, according to my current code, this generates a drag force which exceeds the net acceleration of the bubble as it floats upwards. This doesn't make sense, so I'll have to work on that.

There was a small victory on the plot front. I am now capable of saving values from a loop into an array. Huzzah!

Here is an example of the newly refined (yet still simple) "uvast.py":

In [146]: % run -i uvast.py
Mass (Mkg): 50
Radial Fraction: 0.7
Temperature (MK): 2.5
That took 1672.400 seconds

Coming soon: drag force which obeys laws of physics.

Simplified motion of Cell

After struggling valiantly with Python's 'For' and 'While' loop functions, I managed to write a code ("uvast.py") which calculates the amount of time it takes for a cell at a particular radial position and temperature to rise to 0.95 Solar Radii. There are a few things to note about this code:

1) This is a gross simplification. Neither drag, viscosity nor heat transfer has been calculated for the cell. Hence the incredible speeds achieved.

2) I have not managed to write the 'while' loop very elegantly.

3) I cannot seem to get the values for the time taken and the speeds at those times into arrays, so I am unable to plot them at this time.

Here is an example of the code's output, with a timestep of 0.01 seconds:


In [116]: % run -i uvast.py
Mass (Mkg): 50
Radial Fraction: 0.7
Temperature (MK): 2.4
That took 1989.080 seconds

Buoyancy

Now that the density of the cell can be calculated at every point (assuming constant temperature), I am now able to calculate  the following:
1) The acceleration due to gravity at every point in the Sun.
2) The force of buoyancy acting upon the cell at every point along the Radius.
3) The net acceleration of the cell at every point.

The code used for this is "buoyancy.py". It is similar to "bubble.py" in that it asks for the mass, initial position and temperature of the cell. For a cell of 50E6 kg and 3E6 K, graphs like the following are given:

The second graph is independent of the initial criteria given for the cell, as it is the gravity due to the Sun itself.

The next step is to calculate the motion of the cell. Since I now have acceleration for the cell at every point (up to 0.95 Solar Radii), I can calculate the velocity and position of the cell via integration, provided I fit a function to the acceleration. There exists a command in python which can achieve this - "curve_fit".

Graphing Densities

Data was taken from the "Standard Solar Model BS05(AGS, OP): this included solar density, solar pressure, mass and pressure for 800 points along a solar radius.

A code was then drafted which placed a 'bubble' of gas at some point on the radius, and the density was calculated via the Ideal Gas Law. This was done incorrectly for a few days, with the density of the hotter cell being calculated as much higher than the cooler surrounding sun. This occurred, as the data was not in Pascals or kg/m^3, but rather Dynes and g/cm^3. A factor of 0.6 was also added in, which represented the mean mass of the Sun.

The code used (bubble.py) asks the user for an initial mass in Mkg, inital radius and initial temperature in MK. From this, it calculates the density of the cell at every point along the radius until 0.95 solar radii. It shows the 'measured' density of the Sun above this on the same graph. It also calculates the volume of the cell at the initial point along the radius.

The output looks like this:

In [2]: % run -i bubble.py

Mass (Mkg): 500

Radial Fraction: 0.74

Temperature (MK): 2.7

For a cell of mass 500E6 kg, the volume is 4653384.02 cubic metres

http://www.sns.ias.edu/~jnb/SNdata/sndata.html#bs2005 - Data used.

First Steps

This Seán Blake's blog, which will document the progress of his Senior Sophister project entitled "Convective Instabilities in the Sun".

The objective of this project is to create a code which will simulate the upper 30% of the Sun (from tachocline to surface), and see how the behaviours of rising plasma cells are affected by their characteristics. A follow on objective of this project is to see if the depth of convective instability for cell sizes can be predicted with this model.

At the time of starting this blog, I am 2 weeks into the project. My immediate objectives are as follows:

1) Get density, gravity, temperature and pressure distribution for Sun.

2) For cell, insert initial mass, temperature and initial radial position.

3) Calculate volume/density of cell and each radial position based on ideal gas law.

4) From this, calculate buoyancy force.

5) Choose minimum timescale to calculate motion upwards of cell.

6) After each iteration, calculate anew density, buoyancy etc.

7) Discover and model what makes cells disperse/break up.

8) Make visualisation for this model.

Also to be added to this model: drag, heat transfer. So far, I am at step 4.