Particle diffusion, flows, and NLTE calculations

J.M. Fontenla

LASP-University. of Colorado

2009/3/30

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,ε)

Particles and photons interaction

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

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:

1. Binary collisions due to binary close encounters2. Vlasov terms due to long range interactions with large

number of particles

and

0)(

mm ft

fL VV

0)(

phphn fct

fL n

Distribution functions evolution

• Collisions between particles eventually drive the gas to near-Maxwell-Boltzmann distributions.

• Photon-particle interactions eventually drive radiation to Planck distribution.

and the radiative transfer equation

mmmm Cfft

fL

pV FV)(

'

''' '')(

n

n ddfRfcfccfct

fLphnnphphphphn

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).

Transport and collisional quantities

0

Ut

PFΓUUU

p

t

Qqpt H

FUU

Where the transport terms, (Γ- 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.

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

Statistical equilibrium with flows and diffusion

sillsillsilsil Rnt

n

UV

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.

Equations to solveFor 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*

kkekkkekkkkkekkjHkk CnnRnnRCnnVnt

n,11,111,1,u

w

0w1

jHH

H ann

at

aV

kkekkkekkkkkek

kHH

kH

k

CnxRnxRCnx

axnan

xant

x

,11,111,1,

kelem u

1

F

Diffusion in the case of the solar transition-region

The diffusion velocity, uk, results from analysis of the collisions and can be expressed as:

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.

kaakppTHH

kAkakpH

paTHHkH

k nnVm

kTn

nnVmezmkTn

VEg

u0

with

kaakppTHHk nnVm

kTD

0k

k

kksi n

nD uV

SRPM scheme for NLTEIterate between two calculations:

1. 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).

2. Renormalize level populations.

3. Solve simultaneously all level populations (including the continuum). Using pre-calculated diffusion terms in a logarithmic form.

4. Recalculate electron density.

Equations solved

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:

l

sil

mlmlm

k

l

m k

mmlml

l

k n

nRC

n

n

n

nRC

t

n

n

UV1

UV

sikkkkkkkkkk

kkkkkkkkkkk

nRCRCn

nRCnRCt

n

1,1,1,1,

1,1,11,1,10

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:

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.

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.)

oiijjjij

1ijiijiiijjji JnBnBnnnAnRnR ρ

With ρij 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.

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 np=nH-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.

Diffusion example using FAL 4 model C1 (Fhyd=UnH=0)

1.0

0.8

0.6

0.4

0.2

0.0

Ion

izati

on

Fracti

on

104

2 3 4 5 6 7 8 9

105

2 3 4 5 6 7 8 9

106

Temperature (K)

Carbon Ionization in Static Caselocal ionizationincluding diffusion

Upflow example using FAL 4 model COUT15 (Fhyd=UnH=1015 part. cm-2 s-1)

0.8

0.6

0.4

0.2

0.0

Ion

izati

on

Fracti

on

104

2 3 4 5 6 7 8 9

105

2 3 4 5 6 7 8 9

106

Temperature (K)

Carbon Ionization and Mass FlowStatic case (w/dif)Upflow case (w/dif)

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.