- Situation:

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:

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 / µ₀) 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 (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!