Parametric Instabilities

Parametric instabilities are a common wave-wave interaction. They arise when a nonlinearity such as a pressure gradient couples waves. The waves must allow frequency and wavenumber matching which are consequences of energy and momentum conservation. But parametric instabilities are not limited to waves but also arise in nonlinear resonances such as a plasma sheath. Examples of both will be shown here.

A very common parametric wave instability arises from the decay of strong electromagnetic wave, the "pump", into two electrostatic waves, an electron plasma waves and an ion acoustic waves. It is called a "decay" instability when the pump excites a lower frequency sideband, the electron plasma wave, and the difference mode, an ion acoustic wave. These three modes are the only ones in an unmagnetized plasma.

Figure 1 shows the typical spectral feature of this instability. Below a certain threshold the only wave is the electromagnetic wave which can propagate when its frequency is above the electron plasma frequency. Actually, parametric instabilities also arise when the pump does propagate, such as at the critical layer omega=omegape or in the near-zone of an antenna. The setup shown in Fig. 1a is used to observe the spectrum of a decay instability shown in Fig. 1b. The pump signal is produced by a plane grid in a uniform plasma of dimensions small compared to the free-space wavelength. The frequency is slightly above the electron plasma frequency. Langmuir waves and ion acoustic waves propagate in opposite directions to balance the wave momenta or k-vectors.

Schematic

Fig. 1. (a) Experimental setup for studying parametric instabilities. (b) High and low frequency spectra below and above the threshold of the parametric decay instability of an electromagnetic pump into electron plasma waves and ion acoustic waves.

When the pump amplitude is raised a lower frequency sideband is produced as well as a low frequency line at the difference frequency. The spatial properties of these signals reveal that they are both short-wavelength modes, an ion acoustic and an electron plasma wave. Their k-vectors are in opposite direction such that they add up to a near-zero wavenumber of the electromagnetic waves. Thus, frequency and wavenumber matching are satisfied.

Electrostatic waves become highly nonlinear when their energy density approaches the particle thermal energy density. One may therefore wonder what happens to the instability when the pump wave is further increased. This is shown in Fig. 2 where the pump is actually a propagating electromagnetic wave in the large QUIPS device. The high frequency spectrum evolves from no sideband below threshold to a single sharp lower sideband just above threshold to the generation of harmonics with line broadening which eventually form broadband turbulence of ion and electron modes. The wave amplitude saturates but the integrated sideband power continues to grow. The energy transfer causes a depletion of the pump power.

Schematic

Fig. 2. High frequency spectra of the parametric decay instability for different pump power levels. With increasing pump power first a single sideband is excited just above threshold, then harmonics grow and broaden and finally a continuous broadband decay spectrum evolves.

A second example of a parametric decay instability involves a resonance, but not a wave. It is well known that a sheath exhibits a high frequency resonance below the electron plasma frequency. It is a series resonance between the electron-depleted sheath (capacitive response) and the ambient plasma whose permittivity [1-omegape2/omega2] becomes negative (inductive response) below the electron plasma frequency. The sheath-plasma resonance can be parametrically excited by a strong pump at twice the resonance frequency. This process is not the same as the transit-time sheath-plasma instability for a positively biased electrode.

FRC

Fig. 3. Schematic of an experiment showing a parametric instability of the sheath-plasma resonance. The sheath of an rf probe is pumped at twice the sheath-plasma resonance which grows spontaneously when the pump power exceeds a certain threshold. A diagnostic electron beam (10 kV, 1 uA, 0.5 mm diam) is used to measure the sheath electric field.

Figure 3 shows an experimental setup to verify the parametric sheath-plasma instability. A coaxially fed rf probe is driven by an rf source and the reflected signal is analyzed in time or frequency domain. The rf sheath electric field can also be obtained from the deflection of a diagnostic electron beam whose diameter and density creates no perturbations.

FRC

Fig. 4. Waveforms of the pump oscillation and the parametrically excited half-frequency sheath-plasma oscillation. The growing pump amplitude exceeds the instability threshold in the middle of the trace.

The oscilloscope trace of Fig. 4 shows the pump wave whose amplitude is raised during the sweep. As the threshold amplitude is exceeded a half frequency oscillation is excited. Its frequency corresponds to the sheath-plasma resonance, established with a small amplitude scan. The physics of the nonlinearity can be explained the modulation of the sheath width by the pump and results in a Mathieu equation for the sheath-plasma oscillation.

Secondary instabilities are observed at large pump amplitudes. The field nonuniformity exerts an rf pressure on the oscillating electrons, expelling them from the rf probe. This leads to a space charge electric field to which the ions respond. The sheath widening changes the resonance frequency, detunes the sheath-plasma resonance and quenches the instability which in turn recovers the sheath. A relaxation oscillation arises on the time scale of the ion transit time through the sheath, i.e. omegapi.

FRC

Fig. 5. Ion sheath oscillations appearing in the parametric instability of the high-frequency sheath-plasma resonance. (a) Sidebands of the pump and the decay spectra with lines spaced by the ion plasma frequency. (b) Multiple sidebands created by a strong pump wave. In time the sheath-plasma oscillation consists of pulsating rf bursts.

Figure 5 displays the spectra of the observed low-frequency oscillations. They create sidebands to both the pump and the half-frequency sheath plasma oscillation. The frequency spacing of the sidebands are the same for the pump and decay modes, hence are not due to the parametric instability. The line spacing is approximately equal to the ion plasma frequency. A strong parametric instability creates multiple sidebands in the spectrum (Fig. 5b) and spiky amplitudes in time. The high frequency sheath-plasma instability also exhibits ion plasma relaxation oscillations.

References