From left to right: slice of density, slice of temperature, projection of density along the axis, projection of density along the side,

Beta = -1.0:

Beta = +1.0:

Beta = +3.0:

Beta = -1.0, Alpha increased x10:
(projections will be done when the run completes)

Beta = +1.0, Alpha increased x10:

Beta = +3.0, Alpha increased x10:

Courtesy of Visit, two additional movies of the last run sliced in a different location than usual:

Update 8 February: Added two additional values of Beta. Original post only had Beta=+1.0
Update 16 February: Added additional run at Beta=3.0 with a larger Alpha Update 2 March: Added runs at Beta= ± 1.0 with larger Alpha

Thoughts on Energy Tracking in AstroBEAR

AstroBEAR does not track the thermal, magnetic, and kinetic energy separately (because it is trying to conserve total energy via conservative fluxing). It does however separately track density, momentum, and magnetic fields - which can be used to derive magnetic and kinetic energy - using thermal energy as a reservoir for discretization errors.

When you have flows that are dominated by non-thermal forms of energy ( \beta << 1 or M >> 1 ), discretization errors in total energy (while dynamically unimportant) can still lead to significant relative errors in thermal energy - which can be problematic if there are significant temperature-dependent microphysical processes. In those cases it might be better to solve the thermal energy equation independently and not worry about conserving total energy.

Thoughts on time-stepping

Simulations typically run for a few dynamical times - and for flows that are kinetic energy dominated ( M >> 1 and M_A >> 1 ) the computational time is independent of the flow speed (only a function of resolution N ). However, for flows that are magnetic or thermally energy dominated - the time stepping is limited by the Alfven or sound speeds respectively - and the computational time goes as \frac{1}{\min \left [M, M_A \right]} (ignoring the extra factor of N^D due to changes in the number of zones with resolution)

This can be combined as

T_c \propto \frac{1}{\min \left [1, M, M_A \right ]}

This makes simulating the RT instability M << 1 relatively computational expensive. Alfven waves can also restrict the time stepping when

M_A = \beta^{1/2} M < 1.

So simulations with modest M but very small \beta will also be relatively computational expensive.

High Alfven speeds are also somewhat easier to generate - since the magnetic fields/energy and density are not as tightly coupled as the thermal energy and density - due to material being free to leave along field lines - and the lack of flux freezing when magnetic resistivity is used.

Explicit Magnetic Resistivity

Astrobear currently implements magnetic resistivity explicitly - without subcycling - so time steps are limited by the smaller of the cell diffusion time \frac{\Delta x^2}{\eta} and the cell crossing time \frac{\Delta x}{U}. The cell diffusion time will be smaller than the cell crossing time when

\frac{\Delta x^2}{\eta} << \frac{\Delta x}{U} \rightarrow R_M < N

So the computational time to simulate a crossing time will go as

T_c = \frac{\max \left[1, \frac{N}{R_M} \right]}{\min \left [1, M, M_A \right]}

So - you pay a penalty with explicit time stepping when R_M < N that could be avoided with implicit time stepping.

Simulations involving advection around and diffusion through an obstacle

Simulations involving advection around and diffusion through an obstacle will need to run for the longer of the crossing time around - and the diffusion time within (which is longer than the crossing time by a factor of R_M) - so in the explicit case we have

T_c = \frac{ \max \left[1, \frac{N}{R_M} \right]}{\min \left [1, M, M_A \right]} \times \max \left[1, R_M \right] = \frac{\max \left[R_M, N, \frac{N}{R_M} \right]}{\min \left [1, M, M_A \right]}

Making the resistive solve implicit would reduce this to

T_c = \frac{\max \left[1, R_M \right]}{\min \left [1, M, M_A \right]}

Objectives

• Debugging
• Production runs for some setups
• Thesis writing

Main progress over break:

Writing: I've mainly been writing over the break and will probably be quite heavily focused on that for the next few weeks as well

Time-stepping issue with the Gas Drag: I'm pretty sure the time-stepping issue for the gas drag is NOT due to the equations being too stiff (not yet certain about the grain-grain collisions, still working on other stuff there). It looks like one cell eventually has a temperature that messes stuff up and leads to a runaway increase of number density (see screenshots: ​1 ​2 ​3 ​4). Need to go back to the equation next week to see if there are any caveats for certain temperature regimes that might cause this.

Next steps:

• Continue with above

Updates

​Senior Thesis has Finally Been Finished

ToDo for Paper

Preparing for submission into the astrophysical journal. Other than basic formatting and spellchecks, it seems like our paper needs three appendices. I'll make these this week, most of the information is in my senior thesis so I just need to rewrite/reformat it.

1. Gamma/Theta/Beta Derivation
   1. Derive timescales (t_C,t_G,t_T)
      • This is important for when we derive the dimensionless timescales
   2. Use these to derive gamma/theta/beta, and N_A
      • This is important for when we plot N_A in the contour plots
   3. Talk about the difference between gamma and gammaEff (gamma scales with initial climate sensitivity (dTdP|T=T_0) while gammaEff scales with climate sensitivity when temperature has increased by the temperature tolerance (dTdP|T=T0+dT)
      • This is important for plotting gammaEff on the contourplot and on the scatter plot vs the decline time. It helps reduce the scatter greatly for the latter.
2. Derive dimensionless timescale for high gamma (tau_coll=1/max(sqrt(2),theta)
   1. tauColl=1/sqrt(2) for low theta
   2. tauColl=1/theta for high theta
3. Explain scatter in gamma vs time to decline plot
   1. Show plot of gamma vs decline time compared against analytical predictions for both gamma and gammaEff, to show how the effective gamma reduces the scatter
   2. Show plot of gammaEff vs decline time colored by both T0/a/P0, to show how the points that diverge from our analytical predictions are those with low initial temperature (T0) and large orbital distance (a), yet P0 is still approximately constant for constant gamma