Solar glossary (Stanford solar center)
will help you.
Solar Wind Models : analytical solutions of geometrically-simplified trans-sonic, trans-Alfvénic flow
The Parker model of the solar wind plasma flow.
[Parker,E.N. (1958) ApJ, vol. 128, pp.664-676]
There are so many models and solutions proposed by Parker, E. N..
Here, the word "Parker solution" or "Parker model" is meant the steady state solution
of the spherically symmetric trans-sonic polytrope plasma flowing outward radially
from a non-rotating star.
This solution pretty well depicts basic features of the solar wind.
By taking the solar rotation and magnetic field into account, we will reach
other models [e.g. Weber and Davis, 1967 ; Sakurai, 1985].
The Parker solution family, v(r), is expressed as the contour of H(r,v),
the total specific energy
(kinetic energy, thermal enthalpy, and gravitational potential per unit mass),
The parameters with subscript 0 stand for values taken at a reference point,
generally the critical (trans-sonic) point.
The left plot is a bird-eye-view of H(r,v) drawn in x-y-z space;
x axis is for the heliocentric distance (radius),
y is for the flow speed, and
z and the colors are for the value of H
that is offset and then rescaled with logarithm for display purpose.
A movie (2.4 MB) provides 360-degree view.
There are two key parameters to determine the system;
the specific heat ratio (γ) and the typical temperature (T0).
The index γ characterizes the relationship between density and pressure (or temperature),
taking over the equation of state.
It is frequently chosen to be 1 or numbers slightly greater than 1,
so that the polytrope approximation will mimic near-isothermal high-heat conduction case
(under the absence of the heat conduction and other extra processes,
this index must be 5/3, equal to one for the monatomic or proton or electron).
Here it is set to be 1.05.
Given the solar mass and gravity constant,
the typical temperature is another key parameter
generally defined at the critical trans-sonic point.
1 MK is often chosen, for this number yields reasonably good consequences
at wide range of regions from solar surface, corona to interplanetary space.
The plot above contains all possible Parker solution types.
Among them, the solar wind of the trans-sonic outward flow
is expressed by the special one running, in the drawing above,
from left-bottom (sub-sonic) to critical point (trans-sonic)
to the upper-right (super-sonic);
the critical point is at the x-point dividing yellow and blue areas,
and the integration constant H0 at the right hand side
of the total enthalpy equation above has to be zero.
Another solution line that passes the critical point but
runs from upper-left (super-sonic near the Sun) to the x-point to bottom-right
(sub-sonic regime at region distant from the Sun)
is of few meanings for the solar wind, though,
there are lots of meanings for other systems like accretion disc.
Weber and Davis's solution
[Weber and Davis (1967) ApJ, vol. 148, pp.217-227] :
magnetized solar wind on the solar equatorial plane,
in the rotating frame.
coming soon ....
[Sakurai (1985) Astro. & Astrophys., vol. 152, pp.121-129] :
generalized W-D solution.
coming soon ....
Models for Solar Coronal Magnetic Field : vacuum limit approximations for magnetic-field dominant corona
Potential field approximation for the global solar coronal magnetic field.
[Schatten, Wilcox, and Ness (1969) Solar Physics, vol. 6, pp.442-455]
The measurement of the solar photospheric magnetic field has been long done
by many ground-based and space observatories (such as WSO of Stanford, SDO/HMI, SOHO/MDI, and SOLIS/NAO).
Because of its importance in solar physics, it is being and will be conducted.
On the contrary, even today,
it is quite difficult to measure the coronal magnetic field
and determine the three-dimensional structure
above the solar photosphere.
The PFSS (potential-field source-surface) model
is a powerful model and has really long been used
to determine the difficult-to-measure three-dimensional coronal magnetic field from
the measurement data map of the solar photospheric magnetic field.
Basically, this model takes a vacuum-limit of the MHD system,
so that the magnetic field can be expressed as the gradient
of a scalar satisfying a Laplace equation.
There are some modified versions taking into account
non-vacuum effects [e.g. Zhao et al., 2001].
In MHD simulations where the magnetic field and plasma flow
will be nonlinear-coupled,
the combination of the potential coronal magnetic field and
the Parker's plasma flow (mentioned above) is widely used
as the initial values.
To solve the Laplace equation of a scalar potential of magnetic field,
generally two methods are used;
analytical calculation in terms of
the spherical harmonics polynomial (with associated Legendre function), or
numerical iteration by differencing method.
This anaglyph image provides stereoscopic 3D view of
potential-field source-surface field.
Click links to enlarge.
You may need cyan(right)-red(left) glasses to view it properly.
In case you do not have the colored glasses,
try a monochromatic cross-your-eyes version
(you may adjust the size of this image and/or distance from your eyes to display monitor).
The former is quick and suitable for general purposes in the study of the solar corona.
shows how the global (surface) distribution is reconstructed by the spherical harmonics.
Details of computation procedure are described by
Altschuler and Newkirk [(1969) Solar Physics, vol. 9, pp.131-149].
The latter strategy, numerical iteration, is computationally very expensive.
However, this can be most adequate for calculating measurement-based
coronal magnetic field in the discretized space
such as MHD simulation grid system.
Force-Free Field approximation for local strong magnetic field.
[Nakagawa and Raadu (1972) Sol. Phys., vol. 25, pp.127-135]
coming soon ....
Nonlinear Force-Free Field (NLFFF) approximation for local strong magnetic field.
coming soon ....
last minor update: March 2011
last major update: June 2008
first version: Dec. 2002