CAK Implementation in SPH
Last Updated: April 14, 2015Moving beyond the antigravity/spatially varying opacity approach to accelerate the winds off the stars in our SPH code, we'd like to implement CAK theory. The biggest hurdle is obtaining the velocity gradient; the velocity itself is mildly noisy (particularly in low-res sims), so the velocity gradient is even noisier. Jim MacDonald outlined a way to go about doing this in my thesis comments. It invlolves the usual methods done for determining gradients in SPH, namely using del.v = (del.(rho*v)-v.del rho)/rho where v is a vector and "." is the dot product (see bottom of page for SPH references). Here is Jim's comment:
(My equation 3.10 is rhoa(del.v)a = Sumb mb (vb-va).dela Wab, i.e. the standard divergence of the velocity equation. The only difference between the two equations is the addition of the n's, the directional vectors.)
I implemented this formalism into the SPH code and have tested it on single stars. I couldn't start the CAK theory right from the injection radius of the particles since the velocity gradient is only forward-half weighted; there are no particles closer to the stars than the newly injected particles (obviously) so the velocity gradient calcualtion has large errors. (I tested this by finding the velocity gradient of particles in a beta-law wind simulation. Away from the star Jim's method predicts the correct value, but nearer the star it goes haywire.) Therefore, I fixed the density and velocity gradient that go into the CAK caluation for R<RCAK to follow the beta law. I equate this to fixing a certain number of initializing zones in a grid based code.
The following movies show velocity vs. radius for RCAK=1.1Rstar.
(Note these are using the random injection method, not the uniform injection method.)
Low res, relaxed
Medium res, not yet relaxed
High Res: coming soon
Since the CAK force is gCAK~(dv/dr / rho)^alpha, any noise in the velocity gradient will lead to some SPH particles being accelerated more than others. This is what creates the "popcorn effect" seen in the above movies once the sim(s) relax. Individual particles are accelerated more than their neighbors, causing them to "pop up" in the vel vs. rad. plot, and then run into slower moving particles that they shock with, causing them to "fall back down".
Here are movies of the density and temperature in a random plane:
density, low res
temperature, low res
Standard SPH review papers:
Monaghan 1992
Springel 2010
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