Wave collapse is a most exciting phenomenon of plasma physics, first predicted in a seminal paper by the Russian scientist V.E. Zakharov in 1972. Using a fluid description of a thermal plasma and distinguishing sharply between slow and fast dynamics, Zakharov derived two coupled differential equations which describe the nonlinear evolution of so-called Langmuir waves. The Zakharov model shows how Langmuir wave packets can collapse so that the electric field reaches extreme values within a small region. In the limit where wave-particle interactions are ignored, the Zakharov equations predicts that the electric field becomes infinitely strong in a short time, called the collapse time.
In a real plasma, infinite fields are of course impossible, and it has been shown that the collapse is sooner or later halted by wave-particle interactions, and the electric field is thus prevented from growing indefinitely. Before the collapse is ultimately arrested, however, the field might reach very large values, and charged particles which pass through the field during the collapse can be accelerated to high energies. In laboratory experiments, one has observed a "burst" or "flash" of fast electrons associated with collapsing waves - producing a tail in the particle distribution.
The phenomenon of wave collapse is strongly associated with the so-called ponderomotive force. Before pursuing the concept of collapse, let us digress with a brief look at this peculiar force.
A charged particle bouncing in the field of a constant-amplitude wave will oscillate in a perfectly periodic manner. If the wave is modulated, however, so that its amplitude varies with position, the particle will experience a net force working on a time scale long compared to the wave period. This is the ponderomotive force, which can be derived by taking the time-average of the force on the particle caused by the wave field. In the 1-dimensional case, the ponderomotive force is given by
where "q", "omega", and "E0" are respectively the particle charge, the wave frequency, and the electric field amplitude, and the asterisks denote multiplication. Note that the force increases with the gradient of the electric field energy density, but decreases with the frequency. This is plausible, since a large frequency implies that the direction of the field changes before the particle has managed to move far along the modulated wave. A large spatial gradient leads to a strong ponderomotive force, since then the particle only needs to move a slight distance for the field to change. Since the force depends only on the charge of the particles, it mainly affects the electrons of a plasma.
Finally, we note that the expression for the ponderomotive force does not involve the phase of the field. This is due to the fact that we have derived the force by averaging over all phases, i.e., over one wave period.
Consider a monofrequent wave which is spatially modulated, so that its amplitude changes with position. The ponderomotive force will then push electrons away from the regions of large amplitude, and into regions of smaller amplitude. When the electrons are perturbed, an ambipolar field is set up which causes the ions to move along with the electrons, so that the plasma remains quasi-neutral. Thus, the plasma density drops in regions where the wave amplitude is large, and decreases in regions where the amplitude is small. If the wave is, e.g., a Langmuir wave, a lower hybrid or an upper hybrid wave, a drop in the background plasma density implies an increase in the refraction index. From optics it is known that wave energy tends to focus into regions of large refraction index.
The result of all this is a strong feedback mechanism; some-or-other mechanism initially modulates the wave field, so that
The Plasma-Astrophysics Group of the University of Oslo conducts research on various aspects of wave collapse. We have looked at the dynamics of individual cavitons (collapsing wave packets) for the case of Langmuir waves, upper hybrid waves and other electrostatic waves in weakly magnetized plasmas, as well as for Langmuir-type waves in multi-temperature, unmagnetized plasmas. The wave dynamics has been modelled by partial differential equations, which have been integrated by so-called spectral methods. For wave collapse applications, spectral methods are among the fastest known, and orders of magnitude faster than self-consistent particle simulations.
Many interesting results have been derived, such as recurrent wave collapse, where the caviton first tends to collapse, then expands for a certain period, once more starts to collapse, etc.. We have also performed extensive calculations on the details of transit-time interactions inside wave packets. If you are interested in further reading on our projects on wave collapse, take a look at some relevant publications, in particular the papers entitled "Nonlinear Wave Interactions in a 2-Electron Temperature Plasma", "Weakly Nonlinear High-Frequency Waves in Magnetized Plasmas", and "Electron Acceleration by Weakly Nonlinear High-Frequency Waves in Magnetized Plasmas."