Anne-Marie Cumberlidge
Hi!  At present I am one of the solar physics REU students at Montana State University.  The program lasts eight weeks  from mid June to mid August 2001.  During the academic year I study Physics at Jesus College, part of Oxford University, England.
I am working under the supervision of Prof. Dana Longcope to find areas above the suns surface that have null points in the magnetic field.  It is speculated by some that these areas are connected to the presence of solar flares detected on the sun.  It is my aim to use data from magnetograms to extrapolate the observed magnetic field to find null points, and compare the presence of nulls to the subsequent observation of solar flares and other solar activity.



 Presentation

Other pages I have written (Keele University, England, 1997):



Extended version of the presentation: Electromagnetism is governed by Maxwell's Equations.

They describe how magnetic and electric fields act in both static and dynamic situations.

They provide results such as the presence of electromagnetic waves, and the lack of magnetic monopoles.

Using Maxwell's equations we can calculate the field throughout space caused by any given electric and magnetic field.


Given any magnetogram taken of the surface of the sun, it is my aim to be able to calculate the components of the magnetic field  at any point above the surface of the sun.

As a magnetogram is a 'photograph' of the sun, we consider the equations of magnetostatics.

The basic laws that govern our 'picture' from Maxwell's equations will be:

div B = 0
curl H = J + dD/dt

where B is the magnetic flux density, and J is the surface current density, and D is proportional to the electric field present.  Note in these equations I have missed out factors of pi, and the like, but they give you the idea.

If we assume that J = 0 in this situation i.e. there is no surface current density, then we obtain curl B = 0.  Using vector identities this is equivalent to being allowed to express the field in terms of a scalar potential, such that
B = -del Vm

This relation implies that Laplace equation (a separable differential equation in 3 dimensions) holds for the magnetic scalar potential (del squared fi = 0), making the problem completely analogous to that of the potential of point charges in electrostatics.  this is the equivalent of div B= 0.


An example of a typical magnetogram is as follows:




This shows different field strengths on the sun, the dark areas are negative flux through the surface, the light areas are positive flux.  It only shows the field values in cartesian co-ordinates at z=0, i.e. the surface of the sun.  (here we neglect the curvature caused by the fact that the sun is really a sphere- this can be corrected for)


To manipulate magnetograms, which provides information only on the field component in the line of sight, I used a Fourier Transform which takes the image in x-space and converts it to waves in k-space.  Any function can be thought to be comprised of a suitable linear superposition of sine and cosine functions.  The Fourier transform calculates the amount of each of the sine and cosine function that is required to make the original function.  It is simply a different representation of the same function.   In this case the transform assumes that the array we put in was periodic in its boundary conditions,  such that the array continuously repeats itself.

By using this method it is possible to extrapolate the field at the surface of the sun to different heights above the surface and know the components of the field in x, y, and z.

To get a feel for how fourier transforms work then try looking at the page developed by Kevin Cowtan working in York, England.

 For example using the image of a monotone duck, we can see a fourier transform:

The left image is of monochromatic duck
 

The right image is full Fourier Transform- the hue indicates phase of sine wave, whilst the brightness indicates amplitude of wave).  If any of the information from the fourier transform is lost, especially the phase, when we re-transform then the image will be altered- there may be noise on the picture, or blurry edges on the shape used. It is also important to ensure that both the high and low frequency components of the transform are considered.


The form of the field created at the surface of the sun when extrapolated is of  a similar form, but the further from the surface we get, the more the function is smeared, and smoothes out.  The average total field decreases as field lines join up.  This is shown below:


The scale on the left side is an indication of the total positive field summed over  the array area.  The horizontal scale gives us the distance from the plane z = 0 given in pixels.  As some indication of the scale, the arrays I work with are fixed top be have dimension 512 x 512 pixels.  The graph appears to show an exponential fall-off as we increase the distance from the solar surface.  This also implies that we will have less magnetic nulls the further from the solar surface we go.


It is also possible to plot the strength of the field in the x and y components of the field in the form of a velocity plot.  The arrows on the figure shown below indicate both the size and magnitude of the field in the plane z = 0.   When compared with the original magnetogram we see that the field is strongest between the centre of the two large areas of magnetic field in z.

The areas of magnetic null in the x and y components would be where the arrows appear essentially as dots on this plot.
 



 

By plotting the contours of the functions describing the field in x and the field in y, it is possible to see exactly where the nulls in these two components lie.  Here is an example from the magnetogram we are studying, at  the surface of the sun.  When looking carefully it can be seen that there are 12 crossing points of the two components, which can also be found using the correct program.


We can compare this to the field when it is extrapolated up to a distance of 10 pixels above the surface.  It is clear that the lines appear much smoother, though follow roughly the same shape.  We see that there are 4 crossing points of the two lines, showing it is not guaranteed that there will be the same number of nulls in every plane above the surface.  The cause in the change of number of nulls is a result of bifurcations.  They cause magnetic nulls to join and be created or annihilated.