### 13.3 Introducing the Elsässer variables

The Alfvénic character of turbulence suggests to use the Elsässer variables to better describe the
inward and outward contributions to turbulence. Following Elsässer (1950), Dobrowolny et al. (1980b),
Goldstein et al. (1986), Grappin et al. (1989), Marsch and Tu (1989), Tu and Marsch (1990a), and Tu
et al. (1989c), Elsässer variables are defined as
where and are the proton velocity and the magnetic field measured in the s/c reference frame,
which can be looked at as an inertial reference frame. The sign in front of , in Equation (79), is decided
by . In other words, for an outward directed mean field , a negative correlation would
indicate an outward directed wave vector and vice-versa. However, it is more convenient to define the
Elsässers variables in such a way that always refers to waves going outward and to waves going
inward. In order to do so, the background magnetic field is artificially rotated by
every time it points away from the Sun, in other words, magnetic sectors are rectified (Roberts
et al., 1987a,b).

#### 13.3.1 Definitions and conservation laws

If we express in Alfvén units, that is we normalize it by we can use the following handy
formulas relative to definitions of fields and second order moments. Fields:

Second order moments:
Normalized quantities:
We expect an Alfvèn wave to satisfy the following relations:

#### 13.3.2 Spectral analysis using Elsässer variables

A spectral analysis of interplanetary data can be performed using and fields. Following Tu and
Marsch (1995a) the energy spectrum associated with these two variables can be defined in the following
way:

where are the Fourier coefficients of the j-component among , and , is the number of
data points, is the sampling time, and , with is the -th
frequency. The total energy associated with the two Alfvèn modes will be the sum of the energy of the
three components, i.e.,
Obviously, using Equations (93, 94), we can redefine in the frequency domain all the parameters
introduced in the previous section.