Outflow Feedback in AstroBEAR

The outflow module injects a fraction of accreted mass, momentum and angular momentum back into the grid following each accretion time step and between hydro advances.

Brief Accretion Module Review

Before we discuss the details of this routine, let us briefly recall the accretion prescription here. The feedback module is currently configured to work with the Bondi accretion routine in the code (aka Krumholz accretion). This routine establishes a mass cut-off of gravitationally unstable material around the sink particle (in a control volume of radius = 4dx). Any mass greater than this cut off is removed from the ith cell in this volume (i.e. is accreted) and placed into the sink particle. The total accreted mass is then given by:

M_{acc} = \Sigma (m_{i}-m_{i, thresh})

Simply removing this mass from the cells contained within the kernel leads to an overall decrease in the momenta in the grid over the accretion step. To correct for this, the `missing' momentum is placed into the sink particle, i.e. momentum is conserved between the grid (gas) and the sink particle. For example, the components of the accreted momentum vector is:

p_{x, acc} = \Sigma p_{x,i} = M_{acc}\Sigma v_{x,i}

p_{y, acc} = \Sigma p_{y,i} = M_{acc}\Sigma v_{y,i}

p_{z, acc} = \Sigma p_{z,i} = M_{acc}\Sigma v_{z,i}

The same is so with Energy:

Now, jets are launched from rotating protostellar systems, so it makes sense to assign a 'spin' to the sink particle, induced from accreting angular momentum from the surrounding gas. While we do not explicitly conserve angular momentum in the code, angular momentum would be reduced in the grid due to the accretion step, and so to be conservative between the gas and sink particle, we assign a total accreted angular momentum to the sink as follows:

Note, this accreted angular momentum is the sink's spin angular momentum, rather than its orbital angular momentum. This is because in addition to enforcing mass, momentum, and energy conservation between the gas and the sink particle, we also conserve the COM of the system.

So to recap, mass, energy, linear and angular momentum, and the COM is conserved between the gas and the sink particle in AstroBEAR's accretion module.

Now we must transition to thinking about injecting some fraction of these conservatively accreted quantities back into the grid. How is this modeled in Astrobear (i.e. what quantities do we use at the sub-grid scale/SGS) and how are these quantities distributed in a conservative fashion throughout the kernel?

Feedback Kernel Equations

First off, injected mass will only flow back *out* into the grid (after placed into the kernel) if the injected outward directed momentum in the ith cell is greater than 2x the momentum of the ith cell (right?).. This immediately implies the need for a high injection speed of the material. So, how is this speed modeled at the SGS?

V_{out} = (\frac{G M_{sink}}{10 R_{sun}})^{1/2}

If this at first seems unphysical to you because the speed is potentially greater than the flow speed of accreted material (and should be in order to get outflow), remember this is a SGS model and we envision the gas as accelerating due to gravity as it approaches the surface of the star.

Angular momentum injection

Some preliminary thoughts here — should jet spin up or down the encompassing cloud around a star? (Likewise, it is supposed to spin down the star right, so does this imply a counter rotating jet, or a jet that spins different directions on its top or bottom?).

Current Prescription

Gets mass and momentum injection exact, but then guesses at angular momentum injection. At this step it can then mess up the linear momentum injection, which causes an iteration for the L again. This happens back and forth until some small error is achieved. Question — under what circumstance does this fail? Something about discretization error in the routine, because the kernel is only 4 zones across?

Questions / Tests

Are there any quantities that are stepped on each timestep in the kernel, or are quantities only summed/subtracted during injection/accretion steps?

Thinking how does the flow change over time such that the kernel switches from infall to outflow.

Is the amount of "push" in the outflow subject to resolution?

When we remove gas from cells, do we need to apply a pressure to those cells (if not, do we see any artificial waves generated due to the sudden pressure loss in those cells?)

How is the Bondi speed used in the accretion module?

If we can estimate the infall speed of the material around the sink as approaching the Bondi limit (is this simply pressureless collapse?), then when does V_{out} > ~ f_a V_{bondi}, where f_a is the fraction of accreted mass to be injected in outflow.

Why do we conserve the center of mass (COM) of the system — is it the location of the sink particle (if so, doesn't that coincide with the gravitational potential minimum?)

What is the spin angular momentum of the sink and how do we calculate it?

Development Progress

Algorithm Outline

AstroBEAR subroutine in scrambler.f90

1. Call AMRInit
   1. read in global data
2. Call PhysicsInit
   1. read in physics data
3. Call LevelsInit
   1. Call SinkParticleInit
      1. initializes variables for sink module
4. Call ModuleProblemInit
   1. Sets refinement criteria
   2. Call ProblemModuleInit
5. Call PhysicsFinalizeInit
   1. Uses EOS and finalizes q array
6. Begin time-stepping

Amr Subroutine in Amr_Control.f90:

1. Call ParticlePreUpdate
   1. Call ClearParticleMomementa
   2. Call CalcMoments
   3. Call SynchronizeMoments
   4. Call CheckForNewParticles
      1. If find a new particle: 1. Call CreateNewParticle, 2. Call AddNewSinkParticle
   5. Call CollectNewParticles
      1. Collects particles across processors (using mpi_allgather)
      2. Call CreateParticle: initializes variables for the new particle
      3. Call AddSinkParticle: adds to global sink list
      4. Call CreatePointGravityObject and sets these parameters
   6. Call DoAccretions
   7. Call SynchronizeAccretions
   8. Call FinalizeAccretions
2. Call ParticlePostUpdate
   1. Call SynchronizeGasForces
   2. Call AdvanceParticles
   3. Call UpdateParticles (only called when on max level)

Question, where does particle steps fit in with hydro, elliptic, radiative?

Elliptic → Hydro → Particle → Radiative?