Large amplitude Alfvenic disturbances of arbitrary spatial shape and polarization are described by MHD equations, without resort to the usual assumption of planarity. However, because of their nonplanar nature, the direction of propagation of these disturbances cannot, in general, be determined by looking for minima in a variance matrix constructed from observed field fluctuations. When such minima exist, one is observing that subset of interplanetary Alfven waves that is essentially planar

Barish, F. D.

Goldstein, M. L.

Klimas, A. J.

English

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On the Theory of Large Amplitude Alfven Waves
by
M. L. Goldstein, A. J. Klimas, and F. D Barish
NASA/Goddard Space Flight Center
Greenbelt, Maryland 20771
"I
On the Theory of Large Amplitude AUVn Waves**
M. L. Goldstein*, A. J. Klimas and F. D Barish*
NASA/Goddard Space Flight Center
Greenbelt, Maryland 20771
*NAS/NRC Postdoctoral Resident Research Associate
+NAS/NRC Senior Postdoctoral Resident Research Associate
Large amplitude Alfvn waves have been shown to constitute an
important component of the MvHD turbulence that has been detected by
spacecraft magnetometer and plasma probes (v.,e.g., Belcher and Davis,
1971 and Daily, 1973).
The theory of this wave mode has also been studied (v. Barnes and
Hollweg, 1974 and references contained there). Two features of the
Alfv6n mode have been particularly useful in analyzing experimental
data. The first is the relationship between the fluctuation magnetic
field AB and the fluctuation fluid velocity v. This relationship,which
defines the Alfv6n mode, is
v = + AB ()
- 4W ) 1/2()
The second feature was used extensively by Daily (1973) to determine
the propagation direction of interplanetary Alfvn waves. The
technique, briefly described, is to construct a variance matrix of the
fluctuating components of the interplanetary field B. One then examines
this variance matrix to determine whether or not it contains a direction
of minimum variance. This direction, when it can be found, defines the
propagation direction (to within a sign) of a planar Alfvn wave. (The
reader is referred to Daily (973 for a more comprehensive discussion of
SPresented at the Solar Wind Conference, Asilomar, California; March 1974.
-1-
the method). This method only defines the propagation direction of
plane waves. It is not clear that one should expect Alfvn waves
observed at 1 AU to be predominately planar.
The full MHD equations that describe the propagation of nonplanar,
large amplitude, Alfvn waves in an expanding solar wind which contains
a spiral mean magnetic field are nonlinear. One might expect the
solutions of these.equations to include significant mode coupling
which is not included in linear treatments (v., e.g., V81k, Morfill, Alpers
and Lee, 1974 ) in which Alfvn waves formed near the sun refract into
plane waves as they propagate toward and beyond 1 AU.
Another feature of observed interplanetary A.lfv6n waves is that
they are often not even approximately periodic (v. Fig. 1 in Belcher
and Davis 1971). Although such disturbances can be described in terms
of a Fourier.decompositionit would be useful to be able to discuss
the propagation properties of these nonlinear wave packets as a whole,
without resort to a Fourier analysis with its attendant problems of
determining the importance of mode coupling. It is also not necessary
to assume that the solutions are plane waves (cf. Barnes and Hollweg,
1974).
Wave packet solutions can be directly obtained from the MID
equations. In the discussion below we derive solutions of the MHD
equations which describe the propagation of nonplanar, large amplitude
Alfvn waves in an infinite, homogeneous plasma, with no background
flow. Extension of these ideas to the more complicated geometry of a
radially expanding solar wind containing a background spiral field is
-2-
deferred to a later paper (Barish, Goldstein, and Klimas, in preparation).
We begin with the usual MHD equations:
+ V (p v) = 0 (2)a t
P( _+v* = *P+ 1 (VxB)xB (3)
- = V x (vx B) (4)
at
where p and P are the particle mass density and fluid pressure tensor,
respectively. We restrict ourselves to the MHD mode in which p and P
have no fluctuating components.
Because we are interested in Alfvn waves, we substitute equation
(1) into (2) - (4) and find that B2 is constant for the Alfvn mode.
It is also straightforward to show that if one operates on equation (3)
with /t, that the set of equations (1) - (4) reduces to
2
a - ( - 0 VV) AB (5)
at2 - 47rp
where Bo and AB are the mean and fluctuating magnetic fields,
respectively. The interpretation of this wave equation is clear. Non-
planar, arbitrary amplitude Alfvn waves propagate along B with a group
velocity VA = Bo/(47p)1 / 2 . The mode is incompressible (p and P are
constant) and the magnitude of B is conserved.
One can construct explicit solutions to these equations. If we
use the spherical coordinate system defined in the figure 1, then the
magnitude ofAB is given by
AB -2B o cos0; /2 5 5T (6)
The equation for 0 and b found from the divergence condition is
-3-
-2 sin 20-- + 2 cos 2 sin + 2 cos 2 cos
si2O@ sin 4' - -- +2 o 20CO} - .
=sin 20 [sin - - cos (7)IE x 5 y
where 0 and 4 are functions of (x,y,z). One can see that the
polarization and spatial extent of these waves can be quite general.
In particular, one can look for solutions which are plane polarized
( 4 = 0 throughout the disturbance). In this case the characteristic
equations which follow from equation (7) can be solved for a variety
of boundary conditions on 0 . A typical solution is, qualitatively, a
single wave crest of arbitrary steepness and amplitude, which decays
to the background field, Bo, in the direction parallel to B o, but does
not decay in a plane which intersects B.
One can in principle find solutions to equation (7) which are
spatially confined in three dimensions. The technique is to leave 4 as
an arbitrary function of space and prescribe the spatial dependence of
0. For example, a spatially confined wave packet can be described by
0-(xyz) - exp - (x2 2 +z2(8)2 11 2
where 0 < e <1 parametrises the amplitude of the perturbation. Substituting
(8) into (7) yields a complicated partial differential equatLon for
(x,y,z). While it does not appear possible to solve analytically for
S(x,y,z), a solution can, in principle, be obtained numerically.
Conclusions
Large amplitude Alfv6nic disturbances of arbitrary spatial shape
and polarization are described by the MHD equations, without resort to
the usual assumption of planarity. Such disturbances propagate along
the mean field with a group velocity, VA = Bo (4p)-1/2. Their Alfv6nic
-4-
nature is characterized by constancy of IBI and the relation (I)
between v and AB. However, because of their nonplanar nature, the
direction of propagation of these disturbances cannot, in general, be
determined by looking for minima in a variance matrix constructed from
observed field fluctuations. When such minima exist (Daily, 1973), one
is observing that subset of interplanetary Alfv6n waves that is essen-
tially planar. An extension of this work that describes the propagation
of these MHD waves in a model solar wind is in preparation.
References
1) Barnes and J. Hollweg, 1974 preprint.
2) J. W. Belcher and L. Davis, Jr., 1971, J. of Geophys. Res., 76, 3534,
(see figure 1, p. 3537).
3) W. D. Daily, 1973, J. of Geophys. Res., 78, 2043.
4) H. J. V81k, F. Morfill, W. Alpers and M. A. Lee, 1974,
Astrophys. & Space Science, in press.
-5-
pRECEDING PAGE BLANK NOT i±LLIDJ
-z
-z A B (x,y,z)
(xBy,z)
I
y ' I
-,j XY
BI
z
A spherical coordinate SJstem in which 0 (x, y, z) and d (x, y, z)
define the direction of A B witb respect to Bo. The magnitude of
AB follows from the cocstanv,/ of' , and is given by equation
(6in the text.
NASA-GSFC
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On the theory of large amplitude Alfven waves

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