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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
b z± = v ± V~ ---, (79) 4pr
where v and b 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 b, in Equation (79View Equation), is decided by sign[- k .B0]. In other words, for an outward directed mean field B0, a negative correlation would indicate an outward directed wave vector k and vice-versa. However, it is more convenient to define the Elsässers variables in such a way that z+ always refers to waves going outward and z- to waves going inward. In order to do so, the background magnetic field B0 is artificially rotated by 180o 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 b in Alfvén units, that is we normalize it by V~ 4pr- we can use the following handy formulas relative to definitions of fields and second order moments. Fields:

± z = v ± b, (80)
1- + - v = 2 (z + z ), (81)
1 b = -(z+ - z- ). (82) 2
Second order moments:
+ - ± 1 ± 2 z and z energies ---> e = --< (z ) >, (83) 2
kinetic energy ---> ev = 1-< v2 >, (84) 2
b 1- 2 magnetic energy - --> e = 2 < b >, (85)
total energy ---> e = ev + eb, (86)
residual energy ---> er = ev- eb, (87)
cross-helicity ---> ec = 1-< v .b > . (88) 2
Normalized quantities:
e+---e-- --2ec-- normalized cross-helicity - --> sc = e+ + e- = ev + eb, (89)
ev---eb ---2er-- normalized residual-energy ---> sr = ev + eb = e+ + e- , (90)
ev 1 + sr Alfv´en ratio - --> rA = -b = ------, (91) e 1- sr
- Els¨asser ratio ---> r = e--= 1---sc. (92) E e+ 1 + sc

We expect an Alfvèn wave to satisfy the following relations:


Table 10: Expected values for Alfvèn ratio rA, normalized cross-helicity sc, and normalized residual energy s r for a pure Alfvèn wave outward or inward oriented.



Parameter Definition Expected Value



rA eV /eB 1
s c (e+ - e- )/(e+ + e-) ± 1
sr V B V B (e - e )/(e + e ) 0




13.3.2 Spectral analysis using Elsässer variables

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

e±j (fk) = 2dT-dz±j,k(dz±j,k)*, (93) n
where dz±j,k are the Fourier coefficients of the j-component among x,y, and z, n is the number of data points, dT is the sampling time, and f = k/ndT k, with k = 0, 1,2,...,n/2 is the k-th frequency. The total energy associated with the two Alfvèn modes will be the sum of the energy of the three components, i.e.,
± sum ± e (fk) = ej (fk). (94) j=x,y,z

Obviously, using Equations (93View Equation, 94View Equation), we can redefine in the frequency domain all the parameters introduced in the previous section.


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