Higher order perturbation analysis of plasma and gravitational waves
Loading...
Date
2004
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
Universiteit Antwerpen
Abstract
Plateau [1] initiated experimentally and theoretically the stability of a liquid
cylinder with surface tension. Lord Rayleigh [2] improved this work and
developed the linearized theory for sound waves (although the calculation
of sound velocity goes back to Newton and Laplace) and for the Plateau
experiments which was then applied to all kinds of fields: gravitation [3. 4, 5],
plasma [6], (magneto)hydrodynamics [7, 8, 9], energy principles [10]. The
linear theory flourished tremendously in the past century to a large extent
due to the goal of fusion. Soon the need for nonlinear theories was manifest
e.g. Callebaut [11, 12]. We therefore see in the previous decades a lot of
work on nonlinear theory of plasma waves and instabilities being done. These
yield either exact solutions or approximate ones. Often exact solutions are
obtained after that the equations have been approximated. We may mention
the papers with the exact solutions by Malfliet et al. [13, 14, 15], Hereman
et al. [16, 17, 18], Verheest et al. [19, 20, 21] on solitary waves and those of
Khater et al. on Backlund transformations and Painleve analysis [22, 23, 24,
25]. Amiranashvili et al. [26] gave some exact solutions for standing waves
in bounded plasmas without using the solitary wave theory but with some
boundary conditions. Callebaut and Tsintsadze [27, 28] e.g., neglecting some
higher order terms, dealt with the nonlinear bunching of Alfven waves and the
filamentation and modulation of weakly ionized magnetized plasmas. In fact,
except for the approaches leading to solitary wave solutions, the nonlinear
methods usually yield approximations and usually one has barely an idea
how long these are valid in the behavior of the plasma. The approach used
in this thesis exploits the Fourier analysis for nonlinear systems. It is rather
different from the approaches just mentioned as it allows some insight in the
convergence. Moreover, it gives useful results for the many cases where one
can not find a closed form for the solitary waves. Indeed the solitary waves
are an exceptional and rare case, comparable with a polynomial (as is clear e.g. from Malfliet’s work, see references cited) while the general solution is
an infinite series. The polynomial may use a function (e.g. tanh) instead
of the (combined) variable itself. Similarly the series may use any function
although the customary ones are exponentials and (co)sines. Infinite Fourier
series may in principle be considered as an exact solution, but in practice it
often is an approximation, which, however, allows clear insight on its validity.
The set of (partial differential) equations (e.g. equations (2.1) - (2.4)),
together with some initial and/or boundary conditions, defines a set of func
tions (which are, of course, interrelated). From the Fourier theory it is known
that if a periodic function is continuous from — oo to +oo and has a deriva
tive which is piecewise monotonous and continuous, then the function may
be developed in a Fourier series which is absolutely and uniformly convergent
in any interval. In the thesis we deal with the single variable x (= co t + k • r),
which is the combination of the four independent variables i.e. the angular
frequency ( cj = 2 tt z /, v = frequency), the time (Z), the wave vector (fc) and
space (r). Thus the conditions have to be satisfied for the function(s) of this
combined variable. In particular the function should be periodic in co t and
in k r. However, an exponential growth is easily accommodated just like
the periodic situations as was the case in a hydrodynamical problem [12].
Hence under rather general conditions it is possible to expand the functions
defined by the set of equations. When will this breakdown? E.g. when
the series diverges, i.e., physically speaking, when instability leading to dis-
rupture develops meaning that a (large) amount of energy has been made
available (either injected externally or freed by the system itself from e.g. its
potential energy or, more generally, from its free energy). The convergence
of the series puts conditions on the linear theory, mainly on its amplitude.
In fact a linearized theory can never determine its own limitations: that has
to be done by the nonlinear analysis.
In the previous works [12] it turned out that some experimental situa
tions, in particular the oscillations and instabilities of a liquid jet, could be
explained very well by this method. Moreover some cases appeared where
the nonlinear theory showed that the linear theory was good even up to de
struction of the configuration, while for wavelengths much larger than the
diameter of the jet the nonlinear terms became dominant.
Another breakdown of the method may occur e.g. when the function(s)
is (are) not periodic. However, in the linear perturbation theory one works
usually with a periodic perturbation and this generates naturally higher order
terms which are periodic too as is obvious e.g. in our present work and in
the works of Callebaut [12, 29, 30]. For a non-periodic solution one has to
take a wholly different start in the linear theory, e.g. by using a series in t
and/or re, ?/, 2, or some adequate combination of those, or if nevertheless a periodic or exponential start is used as first order term, to adapt profoundly
the nonlinear terms. Such nonlinear approaches have been elaborated in
various ways in the literature see e.g. the cited references of Malfliet et al.;
Hereman et al.; Verheest et al.; Khater et al.; Callebaut and Tsintsadze.
Description
Keywords
Perturbation Analysis, Plasma, Gravitational Waves, Hydrodynamics, Magnetic