__Computer exercise for lecture__

"Potential theory for space physics" (3 ov, 5 ECTS cr)

### (University of Helsinki, Theoretical Physics, winter 2014;
Lecturer Dos. Olaf Amm)

####
- Situation:

A meridional chain of magnetometers measures at some timestep the
follwing
magnetic disturbances:

http://space.fmi.fi/~amm/PotTheory_CEx_Hki2014_Measurements.html.

The data are given in nT.
Using a cartesian coordinate system that has its origin at the
southernmost
magnetometer station, with x pointing toward north, y toward east, and
z downward, the positions of the magnetometers are at x_{i }=
50
km* i, i=0,...,10. The measured field contains an X (northward), and Z
(downward) component, while the Y (eastward) component was zero
throughout.
We assume that all gradients in y direction, i.e., perpendicular to the
magnetometer chain, are vanishing. The Earth's main magnetic field has
been subtracted from the data.

The magnetic disturbance on the ground is partly caused by currents
in the ionosphere, and partly by currents induced into the Earth due to
the temporal variation of the former currents (see the sketch below).
The
ionosphere is assumed as an infinitely thin layer at z=-100 km, while
the
exact origin of the induced currents is not known.

#### - Tasks:

1.) Calculate the internally caused (from the induced currents) and
externally
caused (from the ionospheric currents) parts of the X component of the
ground magnetic disturbance field, B_{X,int} (x) and B_{X,ext}
(x) at z=0, using 1D field separation.
2.) Using your result for the external magnetic field part B_{X,ext}
(x) at z=0, calculate the X component of the external magnetic field
part
at the ionospheric level B_{X,ext, Ion} (x), i.e., at z=-100
km,
using 1D field continuation.

3.) Using the result of 2.), calculate the Y component of the
ionospheric
equivalent currents, J_{Y,eq,Ion }(x) = (2 / µ_{0})
B_{X,ext, Ion} (x) (in mA/m). What amount of
ionospheric equivalent current I_{Y,eq,Ion }(in A) is flowing
in
total in the direction perpendicular to the magnetometer chain?

*Optional task*: What happens if you try to continue the
external
magnetic field over the position of the ionospheric currents i.e., to a
plane z < -100 km ?

#### - Your solution should include:

- Short description of equations and methods (both numerical and
analytical)
used.
- Code listing of the program(s) used (choose whichever programming
language
you prefer).
- Plots of B
_{X} (x), B_{Z} (x), B_{X,int}
(x),
and
B_{X,ext} (x) at z=0.
- Plot of B
_{X,ext, Ion} (x) at z=-100 km.
- Value for I
_{Y,eq,Ion}

####
- Remarks:

- Think about advantages and disadvantages of the two different
field
separation
and continuation techniques presented in the lecture (Fourier method
and Green's function
method)
before you start.
- Think how to realise best the analytical solutions given in the
lecture
numerically. Standard Math packages might not do the task!
- This page can also be found on the Web at:

http://www.space.fmi.fi/~amm/PotTheory_CEx_Hki2014.html