Particle diffusion, flows, and NLTE calculations

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Particle diffusion, flows, and NLTE calculations

• J.M. Fontenla
• LASP-University. of Colorado
• 2009/3/30

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In the beginning

• Vast regions of the universe contain a
  heterogeneous mix of semi-free particles (simple
  or complex) and photons.
• Particles and photons by themselves follow simple
  mass, momentum, and energy conservation.
• But in reality particles interact with both other
  particles and with photons.
• Photons do not directly interact with each other
  because the electromagnetic equations, in vacuum,
  are linear.
• In order to deal with large numbers of particles
  and photons we use distribution functions to
  describe the state of macroscopic regions of
  space-time
• f(x,t,p,E) fm(x,t,v) and fph(x,t,n,e)

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Particles and photons interaction

• If neither particles nor photons interact it
  would be (L is the Liouville operator)

and

• But when interactions need to be considered, the
  BBGKY hierarchy describes the evolution of the
  distribution function in terms of the
  two-particles distribution function, and so on.
• In dilute gases generally the following
  interactions are considered
• Binary collisions due to binary close encounters
• Vlasov terms due to long range interactions with
  large number of particles

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Distribution functions evolution
and the radiative transfer equation

• Collisions between particles eventually drive the
  gas to near-Maxwell-Boltzmann distributions.
• Photon-particle interactions eventually drive
  radiation to Planck distribution.

The gas evolution equations result from the first
3 moments of the kinetic equation, and because
the collision terms between particles cancel (but
not those of interaction with photons).
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Transport and collisional quantities
Where the transport terms, (G- I p) (viscous
tensor) and FH (heat flux) result from taking the
moments of the non-isotropic distribution
function, and the term q from the work by the
forces. The interaction terms P and Q result from
the interaction with photons (unbalanced
collisions).
Alternatively, similar equations for each species
can be derived but in these the binary-collisions
leave additional terms in the right-hand side
that describe the net effects of collisions
between particles of different species. These
additional terms for the various species must
balance.
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Thermodynamic forces and transport coefficients

• When particles departure from Maxwellian
  distributions are small they are in LTE and the
  distribution functions departures can be computed
  by linearization in terms of thermodynamic
  forces.
• Then, the transport terms can be computed as a
  linear combination of coefficients times these
  thermodynamic forces. The coefficients are the
  transport coefficients and are usually defined
  in the co-moving frame.
• Density, velocity, and temperature gradients are
  some thermodynamic forces. These result from gas
  macroscopic inhomogeneities.
• Mechanical forces (gravitational and
  electromagnetic) are also thermodynamic forces.
  These result from interaction with the external
  medium.
• Radiative interactions also give thermodynamic
  forces since they can induce departures from a
  Maxwellian distribution.
• The most usually considered transport processes
  are
• Particle diffusion of the various species
• Viscosity
• Heat conduction

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Statistical equilibrium with flows and diffusion
Taking the zero order momentum for the species s,
in the ionization stage I, and in the excitation
level l the statistical equilibrium equation is
Because transitions between levels are usually
faster than ionization/recombination we assume
the diffusion velocities for all levels of a ion
are the same. The levels equations still contain
diffusion and velocity terms, but they are minor
terms and can be evaluated directly in each
iteration of solving the equations.
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Equations to solve

• For each species and ionization stage

Or split the abundance and ionization because the
abundance equilibrium can be much slower process
since it is not affected by collisions
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Diffusion in the case of the solar
transition-region
The diffusion velocity, uk, results from analysis
of the collisions and can be expressed as
with
Where the first term (self-diffusion) is
diffusion induced by concentration gradient. The
second term is driven by all thermodynamic forces
trying to establish a concentration gradient.
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SRPM scheme for NLTE

• Iterate between two calculations
• Solve the current ion density coupling between
  points. With given radiation (J) computed from
  previous iteration. Currently using finite
  differences (could also use finite elements in
  this convection-diffusion equation).
• Renormalize level populations.
• Solve simultaneously all level populations
  (including the continuum). Using pre-calculated
  diffusion terms in a logarithmic form.
• Recalculate electron density.

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Equations solved
For the ionization, simultaneous solution at all
points of all ion densities done solving a banded
linear system with the derivative expressed by
finite differences
For the levels (including the continuum)
simultaneous solution at all points of all levels
inverting a linear system in which the formal
solution of the radiative transfer equation links
the populations at various heights
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Ionization solution issues

• The equations for all ionization stages are
  redundant. No equation is solved for the
  fully-ionized state, instead its density is made
  equal to the total element density minus all of
  the other stages densities.
• If U is zero, then a block tridiagonal matrix can
  be used, with a fast method for solution. If U is
  not zero, a five diagonal block matrix can be
  used (sometimes a four-diagonal system works
  too). There are fast solutions for this kind of
  system as well.
• The solution seek is the steady-state one.
  However, the iteration process mimics temporal
  evolution rather than a full Newton-Raphson
  procedure.
• Time-dependent simulations can use similar
  equations for computing the time-derivative.
• Eigenvalues and eigenmodes of the matrix can give
  clues about the evolution around a given state.

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Use of Net Radiative Bracket and level
populations solution
SRPM uses a modification of the Fontenla Rovira
(1985) scheme for solving the level populations.
Which is based on the formal computation of the
Net Radiative Bracket (NRB) as a function of the
level populations. (Inspired in a PhD thesis by
Domenico 1972, advisor was Skumanich.)
With ?ij the transition NRB operator and Jo the
part of the transition mean intensity that is due
to the source function of the background and to
the incident radiation.
This method converges very fast and runs
unattended, often requires a bit of damping to
avoid oscillations around the solution.
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About H and He

• A similar scheme works for H and for He.
• Diffusion coefficients derivation for these
  majority species are shown in FAL papers.
• The ionization equation is solved only for
  neutral H. assuming npnH-na closes this scalar
  equation for each height. The system does not
  require blocks but is just a banded matrix. Then,
  from na/nH the proton and electron densities are
  computed.
• The ionization equation for H, in the levels
  iteration, is solved for (npne) instead of for np
  alone. Then, a quadratic equation is solved to
  determine each, np and ne as well as na.

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Diffusion example using FAL 4 model C1
(FhydUnH0)
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Upflow example using FAL 4 model COUT15
(FhydUnH1015 part. cm-2 s-1)
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3D NLTE radiative transfer

• 3D radiative transfer (3D standard short
  characteristics) is ready and working for the
  next generation RPM3D
• A precise NLTE method for 3D is being slowly
  developed (not yet funded) for RPM3D
• Inclusion of time-dependent convection-diffusion
  for the ionization computations is needed in 3D
  simulations
• With these methods more detailed chromospheric
  models will be possible.