- Stochastic polarized line formation
- 1 Introduction
- 2 The mean Stokes parameters
- 3 Numerical evaluation of the mean Stokes parameters
- 4 Second-oder moments and dispersion of the Stokes parameters
- 5 Various extensions
- 6 Summary and concluding remarks
- References
- 7 Online Material
- Appendix A: Some properties of the transport operator

A&A 453, 1095-1109 (2006)

DOI: 10.1051/0004-6361:20054532

**H. Frisch ^{1} - M. Sampoorna^{1,2,}^{} - K. N. Nagendra^{1,2}**

1 - Laboratoire Cassiopée (CNRS, UMR 6202), Observatoire de
la Côte d'Azur, BP 4229, 06304 Nice Cedex 4, France

2 - Indian Institute of Astrophysics, Koramangala Layout, Bangalore 560 034, India

Received 17 November 2005 / Accepted 27 March 2006

**Abstract**
*Context.* The Zeeman effect produced by a turbulent magnetic field or a random distribution of flux tubes is usually treated in the microturbulent or macroturbulent limits where the Zeeman propagation matrix or the Stokes parameters, respectively, are averaged over the probability distribution function of the magnetic field when computing polarized line profiles.

*Aims.* To overcome these simplifying assumptions we consider the Zeeman effect from a random magnetic field which has a finite correlation length that can be varied from zero to infinity and thus made comparable to the photon mean free-path.

*Methods.* The vector magnetic field is modeled by a Kubo-Anderson process, a piecewise constant Markov process characterized by a correlation length and a probability distribution function for the random values of the magnetic field. The micro and macro turbulent limits are recovered when the correlation goes to zero or infinity.

*Results.* An integral equation is constructed for the mean propagation operator and explicit expressions are obtained for the mean values and second-order moments of the Stokes parameters at the surface of a Milne-Eddington type atmosphere. The expression given by Landi Degl'Innocenti (1994) for the mean Stokes parameters is recovered. Mean values and rms fluctuations around the mean values are calculated numerically for a random magnetic field with isotropic Gaussian fluctuations. The effects of a finite correlation length are discussed in detail. Various extensions of the Milne-Eddington and magnetic field model are considered and the corresponding integral equations for the mean propagation operator are given.

*Conclusions.* The rms fluctuations of the Stokes parameters are shown to be very sensitive to the correlation length of the magnetic field. It is suggested to use them as a diagnostic tool to determine the scale of unresolved features in the solar atmosphere.

**Key words: **line: formation - polarization - magnetic fields - methods: analytical -
Sun: atmosphere

1 Introduction

The Zeeman effect has been used in Astrophysics for more than a century to measure magnetic fields in the Sun, stars and other
objects. The very first analyses of the Zeeman effect were carried out
with uniform magnetic fields. Together with a higher quality of data,
appeared multi-components models (Stenflo 1994), each component having
a different but uniform, or slowly varying, magnetic field. For these models,
the observable Stokes parameters are given by a conveniently weighted
average of the Stokes parameters of each component. Prompted by measurements
of asymmetrical Stokes *V* profiles, multi-components models of
another type were introduced under the name of MISMA (Sánchez Almeida
et al. 1996). In this model, each component is optically thin and the
Zeeman propagation matrix is replaced by an average over the various
components. These two types of models can be made quite
sophisticated. With the terminology used for random velocity fields
broadening, one can say that the first model is of the macroturbulent type,
since the averaging is over the radiation field, whereas the second
type of model is of the microturbulent type since the averaging is done
locally over the propagation matrix.

These two types of models may be insufficient to encompass the complexity of the solar atmosphere which shows inhomogeneities, undoubtedly related to the magnetic field structure, down to scales at the limit of the resolution power of present day telescopes. For example there is an active discussion on the fine structure of sunspot penumbrae. It seems accepted that penumbral magnetic fields have a more or less horizontal component in the form of flux tubes embedded in a more vertical background. However the diameter of these flux tubes and their spatial distribution is still a matter of controversy, the number quoted in the literature varying from 1-15 km to 100 km (Sánchez Almeida 1998; Martinez Pillet 2000; Borrero et al. 2005). In addition, because of very large kinetic and magnetic Reynolds numbers prevailing in the solar atmosphere (Childress & Gilbert 1995), turbulent magnetic and velocity fields have spectra extending over a wide range of wave-numbers. We were thus strongly motivated to consider the Zeeman effect in a medium where the magnetic field is random with a correlation length, i.e. characteristic scale of variation, comparable to radiative transfer characteristic scales. The importance of this problem has been stressed again recently (Landi Degl'Innocenti 2003; Landi Degl'Innocenti & Landolfi 2004, henceforth LL04).

The general regime, neither macro nor microturbulent, leads to polarized radiative transfer equations with random coefficients. Only a few papers have been devoted to this subject in the past (see however, Faulstich 1980; Landi Degl'Innocenti 1994, henceforth L94). Recently this field seems to be receiving some renewed interest (Caroll & Staude 2003, 2005; Silant'ev 2005). Similar problems, somewhat simpler though, have been solved in the seventies for the transfer of unpolarized radiation in the presence of a turbulent velocity with a finite correlation length (see Mihalas 1978, for a list of references). Turbulent velocity field models introduced were less or more sophisticated. The simplest one, is the Kubo-Anderson process (henceforth KAP). For radiative transfer problems, it was employed in the context of turbulent velocity fields for LTE lines (Auvergne et al. 1973) and non-LTE lines (Frisch & Frisch 1976; Froeschlé & Frisch 1980), and also in the context of random magnetic fields for the Zeeman (L94) and Hanle (Frisch 2006) effects. Actually the KAP was introduced for nuclear magnetic resonance (Anderson 1954; Kubo 1954). It was also employed to model the electric field in the stochastic Stark effect (Brissaud & Frisch 1971; Frisch & Brissaud 1971). The name Kubo-Anderson process was introduced in Auvergne et al. (1973).

The idea of the KAP is to describe the atmosphere in a number of "eddies'' having lengths distributed according to a Poisson distribution with given density. It is assumed that in each eddy the magnetic field and other random parameters, such as the velocity or temperature, are constant and their values drawn at random from a probability distribution function. The mean polarized radiation field is obtained by averaging over this distribution and the distribution of the length of the eddies. A KAP is thus characterized by a correlation length and a distribution function for the values of the random variables. The correlation length and the distribution function can be selected independently. This model is fairly simple but has the correct micro and macroturbulent limits corresponding to a correlation length which is zero or infinite. As we show here, when associated to a simple atmospheric model like the Milne-Eddington model, it yields a convolution-type integral equation for the mean propagation operator from which one can deduce explicit expressions for the mean and rms fluctuations of the Stokes parameters at the surface of the atmosphere, and also for the cross-correlations between Stokes parameters. In L94, only the mean Stokes parameters at the surface are considered. It is quite clear that having explicit expressions is very useful for exploring finite correlation length effects.

In this paper the main focus is on the effects of random magnetic fields with a finite correlation length. For a full description of say, turbulent eddies or random distribution of flux tubes in a sunspot penumbrae, it is necessary to incorporate all the other relevant atmospheric parameters which typically should be described by the same type of random process as the magnetic field, in particular the same correlation length. When the magnetic field is described by a KAP, incorporating other random parameters, in particular a velocity field, also described by a KAP with the same correlation length as the magnetic field is no additional work as we explain in the Remark at the end of Sect. 2.6.

In Sect. 2 we define the random magnetic field model, establish a convolution-type integral equation for the mean propagation operator, solve it exactly for its Laplace transform and give an explicit expression for the mean value of the Stokes parameters at the surface of the atmosphere. The latter is used in Sect. 3 to study numerically the sensitivity of the mean Stokes parameters to the correlation length of a random magnetic field with isotropic Gaussian fluctuations. In Sect. 4 we establish an explicit expression for the second-order moments of the Stokes parameters and study numerically the dispersion of the Stokes parameters about their mean values. The second-order moments give also access to the mean cross-correlations between Stokes parameters. In Sect. 5 we introduce various extensions of the Milne-Eddington and magnetic field model and establish the corresponding integral equations for the mean propagation operator. A summary of the main results is presented in Sect. 6.

2 The mean Stokes parameters

2.1 The surface value of the Stokes parameters

We consider a line formed in LTE in semi-infinite one-dimensional
medium and assume that the source function is a linear function of depth.
The radiative transfer equation for the Stokes vector
for rays propagating outwards along
the normal to the surface may be written as

Here,

where

Following the usual procedure, we define the evolution operator
,
as the linear operator which transforms
into
when the source term
in Eq. (1) vanishes (Landi Degl'Innocenti 1987; see also
the appendix A).
Since photons propagate from positive *s* (inside) to *s*=0 (surface), we always take *s*'>*s*. The formal solution of the transfer equation at *s*=0 may be written as

where is the 4 4 identity matrix. We are interested in the calculation of , the mean value of over all the realizations of the random magnetic field, given by

The notation will always mean an average over all the realizations of the KAP.

2.2 The random magnetic field model

Assuming that the magnetic field
is a KAP implies that
is piecewise constant, jumping at randomly chosen points
between random values. The jumping point *s*_{i} are uniformly and
independently distributed in
with a Poisson distribution
of density
independent of *s*. In each interval
*s*_{i-1}< *s* <
*s*_{i}, the magnetic field takes a constant value
.
The
are random variables with a probability
distribution function
independent of *s*. Hence a KAP is
fully characterized by a probability distribution function and a correlation length here defined as .
We recall that for a Poisson distribution of density ,
the probability of having *r* jumps in an interval of length *L* is
.
Since
is a KAP, any element of the Zeeman propagation
matrix
is also a KAP.

The absence of memory of the Poisson process implies that a KAP is a Markov process (see the definition after Eq. (7)). The Markov property and the fact that
is piecewise constant are the two properties which allow us to obtain an integral equation
for the mean propagation operator. In addition, because
and
are chosen independent of *s*, the KAP is a stationary
process (unconditioned statistical properties are invariant under
space translations). As a consequence, the integral equation for the
mean propagation operator is of the convolution type (see
Eq. (10)). Examples of integral equations, which are not
of the convolution type because the stationarity assumption has been
relaxed, are given in Sect. 5.

2.3 The mean propagation operator

The mean value
can be calculated by
summing the contributions from realizations having *N*=0, *N*=1, *N*=2, etc.
jumping points (e.g. Brissaud & Frisch 1971). This technique yields
the mean value as sum of a series. The latter is equivalent to a Neumann series expansion of the convolution-type integral equation
(see Eq. (10)). Following Brissaud & Frisch
(1974; see also Auvergne et al. 1973) we show how to establish a integral equation for
directly. A summation method is used in Sect. 4 to calculate the second-order moments of the Stokes parameters.

When the propagation matrix is independent of space, the propagation operator
is an exponential and depends only on the difference *s*-*s*'. Henceforth referred to as
the "static'' evolution operator and denoted by
(
stands for static), it may be written as

The exponential of the constant matrix is defined in a standard way, e.g. by its power-series expansion. The operator will play an important role in the following.

First we consider all the realizations without jumping point between 0 and *s*. For each realization
is constant in the interval [0,*s*] and the propagation operator is given by its static value. The probability that there is no jump in the
interval [0,*s*] is
.
Thus, the contribution to the mean
propagation operator from the realizations with no jump is given by:

where denotes an average involving only the probability distribution function of the magnetic field.

Next we assume that there are one or several jumping points between 0
and *s* and denote by *t* the last jumping point before *s*. For a Poisson distribution, the probability distribution of *s*-*t* is the same as the probability distribution of the intervals between successive jumps. Hence the probability that *t* falls within the
small interval
is given by the usual Poisson
formula
.

The mean of the propagation operator, when there is at least one jump, is obtained by integrating its conditional mean, *knowing*
that the last jump falls in the small interval ,
weighted
by the probability of the conditioning event. The integral is over all
possible values of *s*', that is from 0 to *s*. (Note that the
probability that the KAP has its last jump in the small interval
is proportional to ,
but the conditional
probability is, to leading order, independent of .) The
mean of the propagation operator for the case with at least one jump
may thus be written as

where denotes the conditional mean, evaluated with the conditional probability.

Two key properties are now used: (i) the Markov property of the KAP, which guarantees that, after conditioning, the "past'' (0<*t*<*s*') and the "future'' (*s*'<*t*<*s*) are independent and (ii) the
semi-group property
(see Appendix A).

Using (i) and (ii), and the fact that the propagation operator in the
interval [*s*',*s*] is just the static one, we have

We claim that

Indeed, the knowledge that a jump occurs at

Adding the contributions from Eqs. (6) and (7), we obtain a closed convolution-type integral equation for the mean propagation operator:

The stationary property implies that Eq. (10), written here for the interval [0,

Equation (10) can be solved explicitly by introducing
the Laplace transforms,

where it is assumed that to ensure convergence. The notation means that we are introducing a definition. Equation (5) implies that

Taking the Laplace transform of Eq. (10) and transforming the integral into , we obtain

which leads to

We note that the two factors in Eq. (15) commute, the product being of the form with a scalar. This can be shown by expanding the second factor in powers of or by using .

In principle, by performing an inverse Laplace transform on the r.h.s. of Eq. (15) we can obtain the mean propagation operator . Actually in our applications, only the Laplace transform is needed.

2.4 Mean values of the Stokes parameters at the surface

Returning to Eq. (4), we see that the integral in the r.h.s. is the Laplace transform of
for *p*=0
(see Eq. (11)). The mean value of the Stokes vector at the surface can thus be written as

where, according to Eq. (15),

with given by Eq. (13) with .

Equation (16), combined with Eqs. (17) and (13), yields an explicit expression for the mean value of the Stokes vector at the surface. The sole averaging which has to be performed is the averaging over in Eq. (13).

As mentioned above, this expression has first been obtained
in L94, with a stochastic magnetic field model identical to
ours, even if it is not referred to as a KAP. The proof, which is
very elegant, starts from Eq. (4). The
integral over
is first
replaced by a sum from *i*=1 to
over all the intervals
[*s*_{i-1},*s*_{i}]. Elementary algebra shows that each term in the sum is of the form

where is the constant value of Zeeman propagation matrix in the interval . The

Replacing

2.5 The macro and micro-turbulent limits

The macroturbulent limit corresponds to a correlation length
going to
infinity. In this case the magnetic field is independent of optical depth but its value is
random with a probability distribution function
.
Setting
in
Eq. (17) we obtain for the macroturbulent limit,

In the microturbulent limit, the correlation length goes to zero. Using

one obtains

The micro and macroturbulent limits can be constructed with the standard Unno-Rachkovsky solution (e.g. Rees 1987; Jefferies et al. 1989; LL04). It suffices to average over in the microturbulent limit and the Unno-Rachkovsky solution itself in the macroturbulent limit. Following L94, we can say that the result given in Eqs. (16) and (17) is a generalization of the traditional Unno-Rachkovsky solution for random magnetic fields. We can also remark that the macroturbulent limit is of the same nature as a standard multi-component model whereas the microturbulent limit is of the MISMA type. Of course, these models usually incorporate many physical processes in addition to the Zeeman effect.

2.6 Residual emergent Stokes vector

The propagation matrix will usually contain a contribution from the
background continuum opacity which we assume here to be unpolarized.
The propagation matrix is then of the form

where is the continuum opacity, assumed to be independent of frequency, the frequency integrated line opacity and the spectral line propagation matrix. We assume that the continuum and line source functions are identical and given by the Planck function. We introduce the ratio , with a constant, and the continuum optical depth which is now used as the space variable. The radiative transfer equation can then be written as

We assume that the Planck function is linear in and write with . The assumptions of a constant and a linear source function are characteristic of a Milne-Eddington model.

At the surface, the Stokes vector in the continuum is given by

With our choice for , only the first component of , i.e. the intensity component , is non zero.

Equation (16) shows that the magnetic field effects
are contained in
.
This suggests to introduce

with the Laplace transform for

where

The expression of follows from Eq. (13) where we have set and .

The mean residual Stokes vector can also be written as

where

with a scalar.

In the macroturbulent and microturbulent limits,
Eq. (29) reduces to

The microturbulent limit is readily obtained by subtracting from Eq. (22). The mean value has been investigated in some detail for random magnetic fields with isotropic and anisotropic Gaussian fluctuations in Frisch et al. (2005, henceforth Paper I) (see also Dolginov & Pavlov 1972; Domke & Pavlov 1979; Frisch et al. 2006; Sampoorna et al. 2006).

The expressions given here for the residual Stokes vector are similar
to the expressions given in Auvergne et al. (1973) for the broadening
by a turbulent velocity field. The only difference is that the line
absorption coefficient is
now a matrix instead of a simple scalar. From a numerical point of view,
it is more convenient to work with the residual Stokes vector than
with the Stokes vector itself because the averaging is done on
quantities which go to zero at large frequencies.

**Remark**

In the proof given above we have assumed for
simplicity that randomness in
,
and thus in ,
comes only from the magnetic field. If randomness comes from other
physical parameters and provided they are described with the same type
of random process as the magnetic field, in particular the same
correaltion length, all the theoretical results
given here will still hold, but the averaging over
must
be replaced by an averaging over a joint probability distribution function
,
where the
are
scalar or vector random parameters. This remark holds
also for the results in Sect. 4 on the
second-order moments.

3 Numerical evaluation of the mean Stokes parameters

In this section we use Eq. (29) to study the dependence of on the correlation length of a random magnetic field with isotropic Gaussian fluctuations. We assume that the velocity field is microturbulent. Its effects can thus be incorporated in the definition of the profile and of the Doppler width. This assumption allows us to clearly identify the effects of the random magnetic field. The function is defined in Sect. 3.1 and numerical results are presented in Sect. 3.2.

3.1 Probability distribution function of the vector magnetic field

To calculate
we must perform the averaging
over
of the r.h.s. in Eq. (30) where
.
For a random magnetic field with isotropic Gaussian
fluctuations,

where is the mean value of and , the dispersion around the mean value. The angles and are the inclination and longitude of the random magnetic field with respect to the line of sight (see Fig. 1). The direction of the mean field is defined by the angles and . The amplitudes of and are denoted by

Figure 4:
Same as Fig. 3 but for moderate turbulence (f=1).
The model parameters are: ,
,
. |

Figure 5:
Same as Fig. 4, but with the mean field
perpendicular to the direction of the LOS. |

The effects of the random magnetic field are controlled by two parameters:

where is the Zeeman shift by the mean magnetic field and the Zeeman shift by the rms fluctuations, also measured in Doppler width unit, which acts as a magnetic broadening on the -components of the Zeeman propagation matrix. The parameter , is the ratio of these two shifts. When

3.2 Numerical results. Effects of a finite correlation length

The numerical method for averaging over
is described in
Paper I, where it is applied to the calculation of
.
Although the expressions here are somewhat more
complicated, the same technique can be applied. The averaging involves
a triple integration over the variables
,
and .
The *y*-integration requires some care. It is
performed using a Gauss-Legendre quadrature formula with 10 to 30 points in a range
.
We have choosen
for *y*_{0}<1 and
for *y*_{0}>1. The
mean residual Stokes parameters are calculated in a frequency-bandwidth
with
.
All the
calculations reported here are performed with a damping parameter
*a*=0. In Paper I it is shown that the elements of
are not very sensitive to the value of *a*, unless it becomes larger
than 0.1.

Equation (29) shows that
involves the parameter
and the ratio
.
When
is small, and a fortiori
,
Eq. (28) (or (30)) shows that
can be neglected compared to the identity matrix. Hence, for small values of ,
.
Therefore for weak
lines, the Stokes parameters depend only on
(through
)
in the region of the line formation. For lines sensitive to the value of ,
the microturbulent regime is reached when
,
i.e. when the correlation length has a line
optical depth smaller than unity. These remarks are illustrated in
Fig. 2 which shows ,
the full frequency width at
half-maximum of
(the mean value of Stokes *I*),
for different choices of
and .
We have assumed
and
= 1, in order to have for Stokes *I* a single well defined peak allowing for an unambiguous definition of .
Figure 2 clearly shows that the dependence on
increases with
and that the microturbulent regime, indicated
by the fact that
reaches a constant value, sets in at roughly
.

Figure 6:
Dependence of the mean Stokes parameters on the correlation
length
of the magnetic field for a strong mean field and
weak turbulence (f=1/3). The
model parameters are: ,
,
.
The mean field
is in the direction of the LOS. The line types have the same meaning as in Fig. 3. |

Figure 7:
Same as Fig. 6, but with the mean field perpendicular to the
direction of the LOS. |

Numerical results illustrating the
dependence of
,
and
,
the
Stokes *I*, *Q* and *V* components of
,
are shown in Figs. 3 to 9 for different
values of
(10 and 100) and different magnetic field
parameters. To simplify the notation, without risk of confusion, we
have omitted the subscript "KA'' and the value
for the
components of
.
In all the figures
,
which means that the
random magnetic field broadening is of the same order as the
broadening by the combined thermal and turbulent velocities. Hence
the strength of the fluctuations is always
.
For
comparison we also show the Unno-Rachkovsky solution calculated
with the mean field ,
henceforth referred to as the mean
Unno-Rachkovsky solution and denoted UR. The relative variation
between the micro and macroturbulent limits are evaluated by
considering the ratio
where the subscript *X* stands for *I*, *Q* or *V*.

(i) Behavior of
.
All Figs. 3 to 9 clearly show that the profiles corresponding to a finite value
of
lie, as expected, between the microturbulent and
macroturbulent limits, with the microturbulent profiles being at all
frequencies broader than the macroturbulent ones, especially around
the frequencies corresponding to the -components. When
(Figs. 3 to 7), the relative variations, measured with
,
are between 10% and 20% at line center and also in
the -components, when the latter are well separated. The main
trend at line center is an increase of
with
.
The value of *f* seems to be essentially irrelevant. In Fig. 6, where the mean field is longitudinal,
shows an unpolarized -component created by the
angular averaging of the
factor in the -component
of the absorption coefficient (see Paper I). The strength of this
component is very sensitive to the angular distribution of the
magnetic field fluctuations.

When (Figs. 8 and 9), deviates strongly from the UR solution. When is longitudinal (Fig. 8), a peak appears at line center and its value is almost independent of the correlation length. As shown by the Unno-Rachkovsky solution the central component behaves essentially as , with the mean value of the absorption coefficient. At line center, when the magnetic field is random, becomes much larger than its deterministic counterpart calculated with . Hence when is fairly large, the value of the central peak may approach unity. When is in the transverse direction, one observes drastic changes between the macroturbulent and microturbulent limits which can also be explained in terms of the behavior of .

(ii) Behavior of
.
A striking feature (see
Figs. 3,
4, 6, 8) is the strong
deviation from the UR solution for strong and
moderate turbulence (see Figs. 3 and 4
with *f*=10 and *f*=1) while for weak turbulence,
stays very close to the UR solution (see Figs. 6 and 8 with *f*=1/3). The relative variations between the micro and macroturbulent limits
seems to be largely independent of the value of *f*. They are always
smaller than 10% and in general smaller than the variation of
at line center, except for the case of
Fig. 3 where they are both of the same order and
slightly less than 10%. It thus seems that
,
can
be calculated with the microturbulent limit, with reasonable
confidence, ignoring the correlation length of the magnetic field.

Figure 8:
Dependence of the mean Stokes parameters on the correlation
length
of the magnetic field for a strong line: .
The other model parameters are
and
.
They are the same as in
Fig. 6 and correspond to a weak turbulence case (f=1/3). The
mean field
is in the direction of the LOS. The line types have the same meaning as in Fig. 3. |

Figure 9:
Same as Fig. 8, but with the mean field
perpendicular to the direction of the LOS. |

(iii) Behavior of
.
Figures 5,
7, 9 show a strong deviation from the UR solution which decreases when the strength of the turbulent
fluctuations decreases. For ,
at line center
reaches 75% when *f*=1 but decreases to 20% when
*f*=1/3. For this value of ,
one can observe that the line
center is more sensitive to the correlation length than the
-components. For
and although *f*=1/3 only (see
Fig. 9),
is very sensitive to the
correlation length, at line center and also in the wings. At line
center,
is bounded by the macro and microturbulent
limits, but in the -components the behavior is not so simple
because the position of the peaks moves away from the line center when
increases. The maximum depth of the -components stays however always above the macroturbulent value. Finally we remark that for weak fluctuations (*f*=1/3),
will depart more
from the UR solution than
(compare
Figs. 6 and 7).

All the figures shown in this section confirm the remark that microturbulence is reached when .

4 Second-oder moments and dispersion of the Stokes parameters

We now examine the fluctuations of the Stokes parameters around their
mean values. For each Stokes parameter, we consider the square of
the dispersion,

where

Second-order moments are investigated in Brissaud & Frisch (1974) for systems of linear stochastic equations, but only for homogeneous systems or systems with a white noise inhomogeneous term. Here we show that explicit expressions for second-order moments can also be obtained for inhomogeneous systems with a constant inhomogeneous term. Our method is inspired by Brissaud & Frisch (1974).

When the source function
varies linearly with optical depth, one can easily obtain a vector transfer equation with a constant inhomogeneous term. It suffices to introduce
the new unknown vector

Since is non-random, and will have the same dispersion. The vector satisfies the transfer equation

where the inhomogeneous term is a constant vector. In this section, to simplify the notation, we set . The solution of Eq. (37) can be written as

where has been introduced in Sect. 2 as the propagation operator for Eq. (1).

In Sect. 4.1, we use Eq. (38) to establish a transfer equation for the tensor product and solve it for . In Sect. 4.2 we establish an explicit expression for by a summation method and use it in Sect. 4.3 to illustrate the dependence of the dispersion on the correlation length and strength of the magnetic field fluctuations.

4.1 Transfer equation for the second-order moment of the Stokes vector

To calculate the dispersions
,
we need only
,
however the latter cannot be calculated
independently of the other
.
We therefore introduce the tensor product

We associate the indices 1 to 4 to

It follows from Eq. (37) that
satisfies the transfer equation

We recall that the tensor product, also called Kronecker product (Iyanaga & Kawada 1970, p. 851), of a

A useful formula satisfied by tensor products is

provided the matrix products can be defined. It is used here several times with one of the matrix, say

It follows from Eq. (43), that Eq. (40) can be rewritten as

where

with a 16-dimension vector and a 16 16 matrix. We use calligraphic letters to denote 16 16 matrices and 16-dimension vectors (the indices run from 1 to 16).

The Green's function (or propagation operator)
associated to Eq. (44) satisfies

where is the identity matrix. The function has a static version corresponding to (i.e. ) independent of

We note that in Brissaud & Frisch (1974), is referred to as the double Green's function.

In terms of
,
the solution of Eq. (44) at the surface may be written as

Using now Eqs. (38), (43) and (45), we obtain

where

Equation (50) is the starting point for the calculation of the mean value of .

4.2 Averaging second-order moments

In this section we show that the average of
over all the realizations of the KAP can be written in the form

where is a matrix which can be written as

with

and the Laplace transform of the static double Green's function. We recall that is the Laplace transform for of the static propagation operator (see Eq. (12)). The explicit expressions of the Laplace transforms are (see Eq. (13))

We now give a proof of Eq. (54) based on the summation of a series, the

**Proof**

Taking the average of Eq. (50), we see that

with a similar definition for . To simplify the notation, we drop the superscript l on and .

We now consider an interval [0,*s*'], and examine all
the realizations of the KAP. We characterize them by the number of
jumping points *N* in the interval [0,*s*']. We stress that *s*' varies
from *s* to ,
while *s* varies from 0 to .
In
Sect. 2.3 we have already introduced the
elements needed here, namely that the probability to have no jump in
an interval of length *L* is
and that the probability
to have a jump in a small interval
around *s*_{i} is
.
The proof is based on the remarks that
and
satisfy a semi-group property and that they can be replaced by their static values if there is no jumping points between *s* and *s*'.

**For** N = 0, we have no jump in [0,*s*'] hence no jump in [0,*s*] and [*s*,*s*'], so we can replace
and
by
and
,
respectively. We can thus write

where the exponential term is the probability that there is no jump in the intervals [0,

where the average in the r.h.s. is over .

**For** N = 1, we have one jump, say at a point *s*_{1}, within an interval
,
which can lie in either one of the intervals [0,*s*] or [*s*,*s*']. We consider
the two cases separately.

Case (a):
0<*s*_{1}<*s*<*s*'

First we use the semi-group property to write

Since there is no jump in each of the intervals [0,

where the product of exponential terms, multiplied by , is the probability of having only one jump at

The integrations over *s*_{1}, *s* and *s*' can be carried out explicitly
in terms of the Laplace transforms
and
.
The integral over *s*' is already
a Laplace transform. Changing the order of integration, the integral
can be transformed into
.
We thus obtain

where the averages are over the distribution .

Case (b):
0<*s*<*s*_{1}<*s*'

Since *s*_{1} is to the right of *s*, we now write

Proceeding exactly as above, we obtain

Transforming the integral into , integrating over

We can now construct the general formula for an arbitrary number of jumps. We denote by

Summing all the contributions from *N*=0 to infinity, we find the
result given in Eq. (54) for the matrices
and
.
The central term corresponds to the interval
[*s*_{-},*s*_{+}], the
term to its right contains the contributions of all the intervals
to the right of *s*_{+} and the term to its left the contributions of all the
intervals between 0 and *s*_{-}.

We can now write an explicit expression for
.
Since
we have assumed that the line and continuum source
functions are unpolarized,
and
.
Hence, only the
first column in the matrix
will contribute to
.
For Stokes *I* and *V* we thus have

and

where the matrix is given in Eq. (17) and the numbers refer to the matrix elements. We have similar expressions for the dispersion around the mean values of Stokes

In the microturbulent and macroturbulent limits, the expressions for
the dispersion of the Stokes parameters are simpler. In the
microturbulent limit, the dispersion is simply zero since all the
coefficients in the transfer equation are replaced by their mean
values. One is actually dealing with a deterministic problem. In the
macroturbulent limit the second order moments can be deduced from the
Unno-Rachkovsky solution which leads to

One can check that Eq. (54) with is consistent with this expression. The macroturbulent limit is interesting because it provides an upper limit for the dispersion. This point is illustrated in the next section.

We checked the result given in Eqs. (52) to (56) by applying our summation method to a scalar transfer equation where the propagation matrix
is
replaced by an absorption coefficient *K*. For this scalar problem,
the second-order moment can also be calculated with a method
introduced by Bourret et al. (1973) which relies on
the introduction of new quadratic dependent variables, chosen in such
a way that they satisfy a homogeneous system of linear stochastic
equations. This method, restricted to scalar problems, has been
applied by Auvergne et al. (1973) for the broadening of spectral lines
by a turbulent velocity field.

Once the problem of calculating the second-order moments of has been reduced to the calculation of the mean value of the r.h.s. in Eq. (50), it is very likely that methods somewhat different from the summation method presented here can be set up. In particular L94 method should work, although it could be algebraically somewhat cumbersome since it does not make direct use of the Laplace transform of the evolution operator.

4.3 Numerical evaluation of the dispersion

To calculate the dispersion of the Stokes parameters we must evaluate the elements of the matrix . The averages over (see Eq. (54)) are performed with Gauss-Legendre quadratures. The integration over the magnetic field intensity can be carried out with the same grid points as for the calculation of the mean Stokes parameters (see Sect. 3.2). The angular integrations over the polar angles and require more refined grids. Typically one needs around 30 points to calculate the dispersion while 10 or less are enough for the mean values. We note also that the width of the frequency domain must be significantly increased.

In the macroturbulent limit, the calculation of the dispersion is much
simpler since

where

We note also that all the results obtained for the second-order
moments of the Stokes parameters hold for the residual Stokes
parameters, provided we divide them by *B*_{1}^{2} (see
Eq. (52)).

Figure 10 shows
,
(
)
for the four Stokes parameters and different values of
(we use the subscript the same convention as in
Sect. 3.2). The magnetic field
parameters are
and
as in
Figs. 4 and 5 and hence correspond to a case
of moderately strong fluctuations (*f*=1). The direction of the mean
field is
and
.
Comparing with
Figs. 4 and 5 where
and
or
,
we see that
has become much smaller, as expected, and has become almost insensitive to the value of
(on the scale of
Fig. 10). Of course,
has also become somewhat smaller and remains almost independent of .
For
,
the dependence on does seem to depend on the direction of the mean field.

Figure 11:
Variation of the dispersion with the strength of the magnetic
field fluctuations. The dispersion is shown only for the
macroturbulent limit .
The line strength
and the
mean magnetic field parameters are the same as in
Fig. 10. The curves are labeled with the value of f. |

In contrast with the mean values, we see that the are very sensitive to the value of . They have their largest values in the macroturbulent limit () and go to zero in the microturbulent limit. In the macroturbulent limit, the dispersion is quite large compared to the mean value. For the mean Stokes profiles, we have seen that the microturbulent limit is essentially reached when . Figure 10 shows that the dispersion has still a significant value when . This makes the dispersion much more sensitive to the characteristic scale of the random magnetic field.

Figure 11 shows the macroturbulent limit of
calculated with Eq. (70) for
(as in Fig. 10) and different
values of *f* varying between 0.5 and 4. For Stokes *I*, *Q* and *U*
the dispersion has maxima at line center and at the frequencies
corresponding to the inflexion points in the Stokes *I* profile. For
Stokes *V*, the dispersion is zero at line center for symmetry reason,
and has its maximum at the inflexion points of *I* also. The minima of
and
correspond to the zero-crossing frequencies in the mean Stokes profiles.

Starting from a case of weak fluctuations (*f*=0.5), we observe that
the dispersion increases with *f*, as expected, until say *f*=2. For
larger values of *f*, we observe a decrease of the peak value in the
wings of
,
and
,
associated to a significant broadening which reflects the fact that
the stronger the fluctuations of the magnetic field, the further out
from line center can they be felt. At line center
and
keep increasing with *f*, even beyond *f*=2. Numerical experiments, not presented, here indicate that
and
saturate to a value around 0.25 but that this phenomenon is related to the choice of
.
When the random magnetic field has a fixed direction and varies
in intensity only, the values of
and
at line center will decrease after going through a maximum. For Stokes *I*, the dispersion has a fairly complicated behavior, specially around the line center. The initial increase is also followed by some kind of saturation, but again, this is related to the choice of
.
In the wings, the behavior is essentially the same as
for the other Stokes parameters. A more detailed analysis of the
dispersion is deferred to a subsequent paper.

5 Various extensions

When some of the assumptions that were introduced to obtain explicit expressions for the mean Stokes parameters are dropped, it may still be possible to write an integral equation for the mean propagation operator. In some cases this equation can still be solved explicitly by a Laplace transform method, but in general a numerical solution is required. A few examples are given below.

5.1 Exponential source function

It follows from the solution of Eq. (1) (see
Eq. (A.11)) that the mean value
of the Stokes vector at the surface can be written as

When is linear in

Let us consider an example presented in LL04 (p. 419) in which the continuum
and line source functions are different and both have exponential
terms. The transfer equation is now of the form

The line and continuum source functions and are given by

The term can describe a chromospheric rise of temperature and the term allows for a drop of the line source function below the continuum source function at optical depths . Simple algebra (see also LL04) yields for the mean Stokes vector

where is defined in Eq. (11) and its mean value in Eq. (15).

When contains an exponential, it does not seem possible to transform the original transfer equation into a new equation with a homogeneous source term and obtain with the method described in Sect. 4 an explicit expression for the dispersion around the mean Stokes parameters.

5.2 Arbitrary depth-dependence of source function and line strength

We now assume that the line and continuum source functions and the ratio
(introduced in
Sect. 2.6) can vary with optical
depth, but not the Zeeman propagation matrix
.
This
implies that the Doppler width is taken constant. There is no hope to
obtain an exact result for the mean Stokes parameters, however
an expression given in Pecker & Schatzman (1959) for the difference
*I*_{c}(0) -*I*(0), in the case of non-polarized transfer, could be a good starting point for their numerical calculation.
For the polarized case, the expression given in the above reference becomes

Here

and

The derivation of Eq. (76) starts from the solutions of Eq. (72) for and . The main steps are the following: one combines the two terms containing and introduces with . An integration by parts then yields Eq. (76).

The mean value
still satisfies
Eq. (10) but the static
propagation operator, as shown by Eq. (78), is now a function of *s* and *s*'.

5.3 Depth-dependence of correlation length

In the preceding sections, it has been assumed that ,
the
density of the Poisson distribution, is independent of the optical
depth *s* along the line of sight. If we let
vary with depth,
the Poisson process becomes a non-homogeneous Poisson
process^{}. The probability that no jumps occur between *s* and *s*' is
.
Equation (10) becomes

with still given by Eq. (5). This integral equation can only be solved numerically.

Some other generalizations can still lead to convolution equations for the mean evolution operator. For example, if depends on the modulus of the random magnetic field or if the random magnetic field consists of several fields with different characteristic scales. Such generalizations have been considered for the statistical Stark effect (Brissaud & Frisch 1971).

5.4 Arbitrary direction of propagation

The results given in the previous sections hold for an outward directed ray normal to the surface of the atmosphere. They can easily be extended to the case of a ray making an angle with the vertical. It suffices to project on to the line of sight the quantities which describe the variations of the model along the normal to the atmosphere, such as the source function, absorption coefficients, and correlation length.

For the example treated in Sect. 5.1,
will be given by Eq. (75) with *B*_{1} changed to
and
and
changed to
and
,
where
.
For the linear source function
treated in Sect. 2.6,
the usual residual Stokes vector
will be given by Eq. (27) multiplied by
.
For the example treated in
Sect. 5.2, *w*(*s*) and
become

and

Here

For the calculation of , the correlation length should also be projected along the line of sight, which means transforming into . Thus in Eqs. (27)-(29), should be changed to . This is also the change made in LL04 (see Eq. (9.280), p. 500), where , the mean length of the eddies measured in the vertical direction, becomes along the line of sight. As a consequence, the more inclined with respect to the vertical are the rays, the closer is one to a macroturbulent type of averaging. This is consistent with a picture of random fluctuations organized in turbulent layers. Now, even in a plane parallel atmosphere, one may want to have a more or less isotropic distribution of turbulent eddies. This can be achieved by keeping the same value of (i.e. same correlation length) in all directions.

6 Summary and concluding remarks

This paper presents the first detailed investigation of the Zeeman effect created by a random magnetic field with a finite correlation length. The goal of this work is to overcome usual treatments whereby the correlation length of the magnetic field is either much smaller, or much larger, than a photon mean free-path, i.e. the microturbulent and macroturbulent limits. The random magnetic field is described by a Kubo-Anderson process which takes constant but random values on intervals of random length distributed according to a Poisson distribution of density . The random magnetic field is thus characterized by a mean correlation length defined here as and the probability distribution function of the random values taken by the magnetic field. The micro and macroturbulent limits are recovered when the correlation length goes to zero or infinity.

The Kubo-Anderson process has been associated to a Milne-Eddington atmospheric model with a linear source function. This combination has allowed us to construct explicit expressions that were used to study numerically the mean Stokes parameters and their dispersion at the surface of the atmosphere. The main theoretical results concern the construction of:

**(i)**- a convolution-type integral equation for the mean
propagation operator associated to the Zeeman effect which can be
solved explicitly for its Laplace transform;
**(ii)**- an explicit expression for the mean Stokes parameters at the
surface of the atmosphere which corroborates a result obtained by Landi Degl'Innocenti (1994);
**(iii)**- an explicit expression for the second-order moments of the Stokes parameters which are needed to evaluate the dispersions and cross-correlations of Stokes parameters.

Numerical investigations have been carried out for a probability
distribution function
describing a random magnetic field
with mean value
and isotropic Gaussian fluctuations with
dispersion
.
We have assumed a microturbulent
velocity with a Gaussian distribution which is equivalent to
incorporating an additional thermal broadening into the Doppler width
of the line. In agreement with the Milne-Eddington model, the ratio
of the line to continuum opacity has been
taken constant. For weak lines ( order of unity or less), the
Stokes parameters are essentially given by the profiles of the
absorption coefficients and hence depend only on
.
For
stronger lines, sensitive to the correlation length of the magnetic
field, the mean Stokes parameters lie between the micro and
macroturbulent limits. This is strictly true for Stokes *I*, because
it is a positive quantity, and at line center for Stokes *Q* and *U*. It is a bit more complicated for Stokes *V* and the -components of Stokes *Q* and *U*, because the position of the
peaks depend on the correlation length. The microturbulent limit is
reached when the correlation length is around unity in the line
optical depth unit, i.e. when
.

The numerical calculations have been performed for (a few cases with have also been considered) for different values of the mean magnetic field , dispersion and correlation length . The dispersion and mean field have been combined to construct a dimensionless parameter which measures the relative strength of the magnetic field fluctuations. The assumption that the magnetic field fluctuations are isotropic influences some of the results but not the general trends which are summarized here.

Concerning the mean values, we have found that:

**(i)**- for Stokes
*I*, the variation between the micro and macroturbulent limits is between 10% and 20%. It grows with the strength of the mean field but seems fairly insensitive to value of*f*. Departures from the UR solution (Unno-Rachkovsky solution calculated with the mean field ) can become quite large at line center when the -components are well separated, but this is partly due to the isotropy assumption; **(ii)**- Stokes
*V*shows very little dependence on the correlation length and hence, with reasonable confidence, may be calculated with the microturbulent limit. The departures from the UR solution are very large, unless*f*is significantly smaller than unity; **(iii)**- for Stokes
*Q*, the line center is quite sensitive to the correlation length of the magnetic field but only when is in the transverse direction with respect to the line of sight, or close to it. For a given random magnetic field, the departures from the UR solution are larger for Stokes*Q*than for Stokes*V*.

In addition to the magnetic field, a whole set of other atmospheric random parameters (velocities, temperatures, densities, ) are needed to properly describe a distribution of flux tubes or magnetohydrodynamic turbulence. These additional parameters should typically be described by the same type of random processes as the magnetic field, in particular the same correlation length. In this case all the theoretical results given here will hold, provided is replaced by a joint distribution function , where the stand for the other random parameters. If the random parameters have different correlation lengths, a KAP-type of modeling can still be set up. An example can be found in the case of the stochastic Stark effect (Brissaud & Frisch 1971). A composite KAP is introduced to handle simultaneously the ion and electron electric fields with their quite different characteristic lengths due to the large mass difference between the two types of particles.

M.S. is financially supported by the Council of Scientific and Industrial Research (CSIR), through a Junior Research Fellowship (JRF Grant No: 9/890(01)/2004-EMR-I), which is gratefully acknowledged. M.S. is also grateful to the Indo-FrenchSandwich Thesis Programfor making possible a visit to the Observatoire de la Côte d'Azur. Further K.N.N and M.S. are grateful to the Laboratoire Cassiopée (CNRS), the PNST (CNRS) and the French Ministère de l'Éducation Nationale for financial support during visits at the Observatoire de la Côte d'Azur where part of this work was completed. H.F. was supported by the Indo-French Center for the Promotion of Advanced Research (IFCPAR 2404-2) and by the Indian Institute of Astrophysics during her visits to Bangalore. She is grateful to J. Sánchez Almeida for stimulating discussions. The authors have also benefitted from constructive remarks from a referee.

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7 Online Material

Appendix A: Some properties of the transport operator

For the benefit of the reader we recall here some of the main
properties of the radiative transport operator for polarized transfer
(Landi Degl'Innocenti 1987). The homogeneous transfer equation
associated to Eq. (1) of the text may be written as

where

The Green's function, also called evolution or transport or
propagation operator, is here defined by

with

where is the identity operator, and the semi-group property, which can be written as

The evolution operator further satisfies two differential equations,

and

which can be derived from Eqs. (A.1) and (A.2) by taking the derivatives of Eq. (A.2) with respect to

When the propagation matrix is a constant, the evolution operator
is given by

Using Eqs. (A.3) and (A.5) one can verify that the expression

where is the prescribed value of at

Using Eq. (A.6) we can rewrite this equation as

and after integrating by parts,

When , we immediately obtain the result given in Eq. (3) of the text.

Copyright ESO 2006