Some authors used static NLTE models to determine flare atmosphere parameters and then performed the formal solution of the radiative transfer equation with a prescribed run of the velocity to evaluate the emergent intensities. For macroscopic velocities comparable with the thermal velocity this can be a valid simplification. However, one can imagine that this approach must fail for velocities that exceed a certain limit. Thus we need a method that consistently includes the velocities. We will show that dynamic NLTE models based on ALI-approach can be satisfactorily used. Here we briefly summarize the multilevel ALI method of Rybicky & Hummer (1992) (so-called MALI) and point out the modifications needed for flare modeling.
The formal solution of radiative transfer equation (RTE) can be written down
as the lambda operator acting on the source function,
, where 
is
directional cosine and 
means frequency.
The idea of ALI method is to split the lambda operator in two parts
and to take the source function in the second term as known from previous
iteration (denoted by 
), i.e.
By choosing an appropriate approximate lambda operator 
we obtain desired accelerated lambda iteration scheme.
It was shown that the approximate lambda operator
working with an optimal convergence rate is the diagonal part of the true
lambda operator (Olson et al. 1986). To construct the
diagonal we use the method
based on Feautrier variables as proposed by Rybicky & Hummer
(1991). Static Feautrier variables can be used in the continuum
radiative transfer equation, because it is not affected by the moving
plasma.
For line radiative transfer in a moving media the Feautrier variables are
defined as (Mihalas 1978)
where 
is a distance from the line center.
The line source function has to satisfy the condition
, so these Feautrier variables
can be used only for non-overlapping lines.
Instead of the lambda operator we use the psi operator that
acts on total emissivity 
This choice appears to be the most useful for our problem formulation (for
details see Rybicky & Hummer 1992) and thus the lambda
operator will be substituted in final formulas.
The general form of equations of statistical equilibrium is (Mihalas
1978)
nl is the atomic level population of the level l and
Cll' and Rll' are, respectively, the collisional and
radiative transition
rates from level l to l'. Levels are ordered according to their
energy (i.e. El>El' for l>l'). In this exploratory work
we don't consider the non-thermal collisional rates.
If we assume the relaxation times
of collisional and radiative processes to be much shorter than
hydrodynamic ones,
the statistical equilibrium is not affected by the hydrodynamics
of the plasma.
Thus we can write the equations of statistical
equilibrium in their standard static form
Using the formulation of Rybicky & Hummer (1992) which takes
into account the angle dependence of the absorption coefficient, we express
the radiative rates as
is the specific intensity for frequency and
directional cosine . The quantities Ull' and Vll'
are defined for each transition between levels l and l'. For line
transitions there are given by
where All' and Bll' are the Einstein coefficients and
is the line profile function. Its dependence on
for a vertical macroscopic velocity v takes the form
where 
is the line center frequency.
For continuum transitions the Ull' and Vll' are given by
where all' is the photo-ionization cross section and
is the Saha-Boltzmann function (see Mihalas 1978).
The total emissivity can be expressed as
where 
is the background emissivity.
Substituting for in Eq. (4 (click here)) we obtain
the resulting formula for intensity
Note that has cancelled in the second and third term.
The momentum equation reduces to the equation of hydrostatic equilibrium
in the case of a static atmosphere. Assumption that the time changes
of the velocity are small (
)
leads to the stationary form of the momentum equation. Thus
where m is the column mass, p is the total pressure, 
is the
density and 
is the surface gravity.
The second term on the lefthand side will decrease the total particle
density N in the region with velocity as p=NkT.
In the cases when the atmosphere is not stationary the
stationary form of the momentum equation
can be less accurate than the hydrostatic equilibrium equation.
We have therefore the following possibilities to overcome this difficulty:
1) to use hydrostatic equilibrium
equation, 2) to use the stationary form of the momentum equation or 3) to use
the density as the input parameter from hydrodynamic simulations.
Substituting for in (7 (click here)) from Eq. (12 (click here)),
we obtain an iterative system of
non-linear equations for the new populations nl.
The preconditioning strategy
assures this system to be linear in nl provided we know a priori
. We use here preconditioning within the same transition only
(see Rybicky & Hummer 1992).
ESE are closed using
the charge conservation equation
where ql is the charge of the ion the level l belongs to.
The electron density is related by particle conservation equation
Using the state equation p=NkT and defining 
to be the mean
nucleon mass, we can rewrite Eq. (13 (click here))
as
The set of preconditioned ESE, constraint Eqs. (14 (click here)) and (15 (click here)), together with
momentum balance (16 (click here)) can be rewritten in a matrix form
for the vector of unknown variables
(
is the total amount of levels) as
where 
.
Although the preconditioned ESE are linear in level populations nl,
this system remains non-linear in the unknown variables due to products
of electron density with level populations (the matrix A in Eq.
(17 (click here))
depends on ). To correctly solve it we followed Heinzel
(1995) and linearized system (17 (click here))
As the matrix A depends only on the electron density, the non-zero elements
of the matrix 
appear only in its last column.
The system of linear algebraic Eqs. (18 (click here))
is solved for corrections 
. The new variables are obtained as
and iterations proceed until the changes are
small enough, i.e.
.