## 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 xi = 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. 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, BX,int (x) and BX,ext (x) at z=0, using 1D field separation.

2.) Using your result for the external magnetic field part BX,ext (x) at z=0, calculate the X component of the external magnetic field part at the ionospheric level BX,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, JY,eq,Ion (x) = (2 / µ0) BX,ext, Ion (x) (in mA/m). What amount of ionospheric equivalent current IY,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 BX (x), BZ (x), BX,int (x), and BX,ext (x) at z=0.
• Plot of BX,ext, Ion (x) at z=-100 km.
• Value for IY,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