NON-LINEAR FOURIER STABILITY ANALYSIS
NON-LINEAR FOURIER STABILITY ANALYSIS
Dirk K. CALLEBAUT
Physics Dept., CGB, University of Antwerp, B-2020 Antwerp, Belgium. E-mail: Dirk.Callebaut@ua.ac.be
Abstract. From the usual ‘horizontal’ Fourier perturbation analysis one term is first singled out. From this one the ‘vertical’ Fourier series is constructed. Simple combinatorial coefficients allow then immediately the nonlinear analysis of several combined series. E.g. for a cold plasma the sum of all first order density amplitudes has to be less than 37 per cent of the equilibrium density to have convergence. Taking into account the pressure decreases this limit. The method was applied to various plasma, hydrodynamical, MHD and gravitational configurations. An explanation is given for the cases where the method does not work. As an application we mention the triggering of instability of a laboratory plasma or e.g. a solar flare by several small oscillations leading in combination to a huge, even diverging, perturbation. Another application may be ball lightning.
Keywords: Plasma, stability, nonlinear, Fourier, solar eruptions PACS: 02.30.Jr, 02.30.Nw, 05.45.-a, 52.35.Sb
In the last decades non-linear theories have been flourishing, in relativity, cosmology, fluid dynamics and particularly in plasma physics, where experiments and astronomical observations request non-linear stability analyses for their explanation. We mention multiple scale analysis, soliton theory, Backlund transformations, Ablamowitz-Kaupman-Newell-Segur (AKNS) systems, Painleve analysis, etc. However, the continuation of the Fourier analysis into the non-linear domain hit on two obstacles. The first one was psychological: a Fourier analysis had to be linear. However, there is a ‘horizontal’ Fourier series (the one which is customarily used in linearized stability studies) and a ‘vertical’ Fourier series (which deals with arbitrary periodic functions; see below). For the latter the Callebaut-Fourier method gives the whole family of higher order terms corresponding to a single term of the horizontal series. Next comes the non-linear extension of the Callebaut-Fourier method which considers a sum of terms of the ‘horizontal’ Fourier series , ,.
The second obstacle is due to an inherent limitation of the Callebaut-Fourier method: there are cases where it does not work. In particular it is useless for a very simple case: sound waves. However, it works in many other situations: in plasmas, in incompressible hydrodynamics and magnetohydrodynamics, etc. Presently we understand the reason of the limitation and a further extension is in the make.
2. THE ‘HORIZONTAL’ AND THE ‘VERTICAL’ FOURIER SERIES
As stated above the horizontal Fourier series is just the usual result of the linearized perturbation equations. Here one has an infinite sum or integral over all exponentials with different frequencies and wave numbers (related by the dispersion relation) and with totally arbitrary (however ‘small’) coefficients. Now we consider a single one of those terms: together with the set of differential equations it defines a particular solution of those equations for which the first order term (with its amplitude and phase) acts like an initial condition. This particular solution may be Fourier analyzed: i.e. writing its expression using all the harmonics of the basic phase (given by the first order term). This may be obtained by iteration of the first order term, then the second order terms, etc. in the differential equations. The main and practically only condition (except some far-fetched continuity conditions which are no problem in physics) is that the function has to be periodic. It happens that in incompressible situations (hydrodynamics and magnetohydrodynamics) this periodicity is granted and agreement with certain experiments confirms the theoretical development. Moreover, the method works for plasma’s too as the electromagnetic interaction involves negligible (electron) material displacements in itself (indirectly it does of course and thus some caution is required).
Although one obtains the higher order terms associated with the given first order term (and in this respect the method may be termed non-linear) it is still a kind of Fourier analysis in which the function is given implicitly by the set of
differential equations and for which the initial or boundary conditions are replaced by the first order term.
We obtained in various cases the convergence limit i.e. the amplitude of the first order term for which the whole series is still finite: above this limit the series diverges. Even a perturbation which in its first order term looks like a harmless oscillation may inherently bear an instability.
3. THE NON-LINEAR CALLEBAUT-FOURIER ANALYSIS
Once the family of higher order terms related to a single first order term is obtained (e.g. up to a certain order) one may construct at once the non-linear analysis for several first order terms thanks to the combinatorial expressions for a power of a polynomial. This gives the full family for each first order term plus the ‘mixing’ terms with the corresponding coefficients. Clearly the analysis with these interference or mixing terms is fully non-linear. It is clear that computer algebra is indicated, certainly if one wants to take into account several higher order terms. The total amplitude of one family is obviously larger than the amplitude of its first order term. In fact this total amplitude may become infinite once the threshold is exceeded. This is even more true when considering several ‘small’ first order terms, which each still yield a finite amplitude, but together yield a very large, even a diverging, amplitude: see sections 4 and 6.
4. ILLUSTRATION: ELECTRON PLASMA
As an illustration we consider the fairly simple case called the electron plasma: a uniform plasma consisting of electrons and ions, infinite in all directions. We neglect gravity, viscosity, resistivity and the magnetic contributions. The basic equations are then respectively the equation of continuity, of motion, of Poisson and the polytropic one (,
where n is the number density of the electrons, n0 their equilibrium density, v their velocity, <p is the electrical potential, p the total pressure with pi the pressure of ions which is supposed to be constant in the approximation that they are quasi immobile, e and m are the electron charge and mass, e is the permittivity, whose value in vacuum is 8.85 x 10~12 C/Vm, K and Г (polytropic exponent) are constants.
The linear perturbation is expressed as a Fourier series. We fix one term say Лехр^] as mentioned above. We thus develop only one family of higher order terms corresponding to a single Fourier term of the linearized analysis. The nonlinear terms generate then a2A2 exp[2#], a3A3 exp[3#], etc with a2, a3, • • , coefficients to be determined.
We integrated as well equation (10) to obtain a differential equation of first order and even to obtain a fully integrated equation: the same results, e.g. (11) were obtained, but the times needed for the three procedures to calculate the coefficients was variable and strongly dependent on any simplification or additional effect taken into account (e.g. when the ions are mobile as well). The series (11) is convergent provided A < e^1 (e« 2.71828 • • ) i.e. if the linearized perturbed density has an amplitude larger than 37% of the equilibrium density n0, the series is no more convergent.
A graphical method confirmed this: Indeed, summing up N terms of density, n, and plotting the result for % in the interval 0, 2% (or even 0, %) yields an oscillating graph. If there was any value in this interval for n less than zero the series has to be rejected since the electron number density may not become negative. It turns out that this graphical method confirms the radius of convergence found analytically rather well; we performed the summation in some cases up to N= 7000 to verify the result accurately. Actually when we exceeded the limit of convergence for A slightly the number of terms might be rather limited (say ten or twenty, to see whether it is going to diverge), except very close to the limit of convergence. This numerical/graphical method turned out a powerful tool in all cases where we did not have a systematic analytical expression for the coefficients.
We developed as well the theory using cosines instead of exponentials. It turned out that the radius of convergence was doubled: now the series converges for A < 2e~x, i.e. the linearized perturbed density amplitude may reach nearly 74% of the equilibrium density and still not lead to breakdown. In fact the result had to be expected: an exponetntial e* corresponds to a sine and a cosine, thus two waves instead of one, thus leading to halving the convergence limit.
Even if only the second order term is calculated it yields valuable information above the linearized theory. We suggested to verify the results experimentally e.g. by applying a (strong) external perturbation (electric field) in a Q-machine (Quiescent Plasma Machine) having a magnetized alkaline plasma or unmagnetized argon plasma of a DP-Machine (Double Plasma Machine).
5. BRIEF SURVEY OF RESULTS
The preceding results have been generalized to a multiple species plasma, by including the motion of ions as well, by including their pressure (even applying different approaches) (, ), by including the magnetic effects involved in motion too and by considering electron-positron plasma’s.
The method can be generalized to instabilities when a growth rate occurs instead of со. This was elaborated for a liquid jet involving surface tension and the result agreed very well with experiments, . Similarly it was applied to various plasma columns and extended to involve magnetic fields internally (i.e. in the plasma column) and externally (around the column).
The method was applied to gravitational problems too. E.g. an infinitely long gravitational cylinder of homogeneous density was studied to higher order [ 1 ]. The linear theory was performed in  and : they considered it as a model to investigate the stability of arms of spiral galaxies. The nonlinear theory was further extended to include a magnetic field in the gravitational cylinder and outside it. Similarly the case of a plane-parallel homogeneous gravitating medium (cf. a flat galaxy) was studied  including magnetic fields inside and outside. The results of the linear theory confirmed Jeans’ criterion, which was derived on an inappropriate basis but was made plausible on various grounds and explaining some observational data  . Again convergence and influence of various effects are calculated and discussed.
6. TRIGGERING OF INSTABILITIES FROM A SUMMATION OF SMALL
As stated above the density amplitude of the first order perturbation should reach 37 per cent of the equilibrium density in order that the resultant series should diverge and thus that the configuration should be unstable. This 37 per cent may look as a fairly large perturbation and one may doubt that it occurs often. However, if various (small) perturbations
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