15.2 The mean field reference system

The mean field reference system (see Figure 103

) reduces the problem of cross-talking between the components, due to the fact that the interplanetary magnetic field is not oriented like the axes of the reference system in which we perform the measurement. As a consequence, any component will experience a contribution from the other ones.

Let us suppose to have magnetic field data sampled in the RTN reference system. If the large-scale mean magnetic field is oriented in the $[x, y,z]$ direction, we will look for a new reference system within the RTN reference system with the $x$ axis oriented along the mean field and the other two axes lying on a plane perpendicular to this direction.

Thus, we firstly determine the direction of the unit vector parallel to the mean field, normalizing its components

ex1 = Bx/ |B |, ex2 = By/ |B |, ex3 = Bz/ |B |,

so that $^e'(e ,e ,e ) x x1 x2 x3$ is the orientation of the first axis, parallel to the ambient field. As second direction it is convenient to choose the radial direction in RTN, which is roughly the direction of the solar wind flow, $^eR(1,0, 0)$ . At this point, we compute a new direction perpendicular to the plane $^eR - ^ex$

^e'z(ez1,ez2,ez3) = ^e'x × ^eR.

Consequently, the third direction will be

' ' ' ^ey(ey1,ey2,ey3) = ^ez× ^ex.

At this point, we can rotate our data into the new reference system. Data indicated as $B(x, y,z)$ in the old reference system, will become $' ' ' ' B (x ,y,z )$ in the new reference system. The transformation is obtained applying the rotation matrix $A$

( ) ex1 ex2 ex3 A = ey1 ey2 ey3 ez1 ez2 ez3

to the vector $B$ , i.e., $' B = AB$ .

Figure 103:

Mean field reference system.