Significant portions of this thesis examine nonlinear wave coupling in both simula- tions and experiment. One method with which to quantify this coupling is by using bispectral analysis. There is extensive literature which describes higher order spec- tral techniques including bispectral analysis. For general information see [de Witt,
2003; Kim et al., 1980; Kaup et al., 1979; Kravtchenko-Berejnoi et al., 1995] and
for plasma-specific applications see, for example,Holland et al. [2002];Moyer et al.
[2001];Yamada et al. [2008];Itoh et al.[2017]. For a KSTAR plasma specific appli-
cation, directly relevant to the work in this thesis, see [Lee et al.,2016]. An early account of the application of higher order spectral techniques to plasma physics is given in Ref. [Kim and Powers,1979]. Bispectral analysis has been previously been successfully applied to the MCI [Carbajal et al.,2014].
Any three waves interacting nonlinearly must satisfy, to good approximation, the frequency and wavenumber matching conditions
f3= f1+ f2,
k3= k1+ k2,
where waves “1” and “2” interact to produce wave “3”. To measure the amount of phase coherence between three modes that obey the above resonance conditions, one can compute the bispectrum. Defining F (f1) as the complex Fourier transform
of a quantity (for instance an electromagnetic field component) at frequency f = f1,
and F∗(f1) as its conjugate, the bispectrum is defined as
b2s(f1, f2) =| hF (f1) F (f2) F∗(f1+ f2)i |2, (2.18)
where the brackets h·i denote averaging over time. One can normalise the bispec- trum to obtain the bicoherence. This can be done in several ways [de Witt,2003;
Kravtchenko-Berejnoi et al.,1995], one of which is to use Schwartz’s inequality
b2c(f1, f2) =
| hF (f1) F (f2) F∗(f1+ f2)i |2
h| F (f1) F (f2) |2ih| F∗(f1+ f2) |2i
. (2.19) In practice, an ensemble average can replace the average over time. One thus computes several successive Fourier transforms of the same signal, sliding the Fourier transform window along the signal as we do so. These windows can overlap to some extent, which allows us to obtain improved spectral resolution, provided one is careful not to induce correlation where there is none. The number of independent Fourier transforms M , must be large enough so that the value of the bicoherence bc is statistically significant. For significant coupling the variance of bc is given by
[Kim and Powers,1979]
V ar (bc) '
1
M 1 − b
2
c . (2.20)
Some accounts differ slightly in that there is a factor 4 in the numerator of the right hand side of Eq. 2.20. Regardless of this factor, we can safely say that if bc> 1/M ,
we have a statistically significant result. All bicoherence dependant conclusions in this thesis satisfy this condition.
The bispectrum/bicoherence can either be an “auto” quantity, in which in- teractions between waves in one signal are computed, or a “cross” quantity, in which interactions between waves from two different signals are considered. The bispec- trum, Eq. 2.18, measures the extent of phase coherence due to the nonlinear coupling
between three waves that satisfy the frequency and wavenumber matching criteria above. The bicoherence, Eq. 2.19, is a normalised bispectrum bounded between 0 and 1, and quantitatively measures the fraction of the Fourier power of a signal that is due to nonlinear (specifically quadratic) interaction.
Thus, the bicoherence sheds light on nonlinear coupling; whereas the bis- pectrum yields information regarding the energy flow due to nonlinear coupling, given the wave amplitudes in the system; although it does not by itself tell use the direction of the energy flow. It is therefore useful to compute them both when diag- nosing possible nonlinear wave physics. A large value of bicoherence (close to unity) may reveal waves which have significant coupling, but do not drive additional waves in practice due to their relatively low amplitudes. This becomes apparent if one supplements the information given by the bicoherence with the bispectrum, because the latter also incorporates information about relative wave amplitudes. Conversely, plotting the bispectrum alone does not necessarily yield information about the in- trinsic strength of coupling between waves.
In a 1D3V PIC simulation, in which information on both the frequency and one dimensional wavenumber spectrum is readily available, one must decide how best to calculate and display the results of bispectral analysis. In section 4.3 of chapter 4 and section 6.3 of chapter 6, the auto-bispectrum/auto-biocherence are calculated. The signal is first Fourier transformed in space and then the bispectral analysis is performed by successive Fourier transforms in time. For purely perpen- dicular propagation and restricting the frequencies of interest to be below the lower hybrid frequency, there is approximately a one-to-one mapping between frequency and wavenumber. This allows one to select the frequency for which the spectral power of a given wavenumber is at its maximum, which, for ICE, lies along the fast Alfv´en branch. This is repeated for all wavenumbers, allowing one to calculate the bicoherence using successive Fourier transforms in time, while displaying the results of the calculation as a function of wavenumber [Irvine, 2018]. In general, PIC simulations, and certainly PIC simulations of the MCI, have much better spec- tral resolution in the wavenumber domain than they do in the frequency domain. The resulting bispectrum/bicoherence plot is significantly less coarse than if we had plotted it as a function of frequency.
A different approach is adopted in section 5.2 of chapter 5, in which the cross-bicoherence between two signals is calculated. First, a small segment of the total time series is selected and its average calculated. The result is a 1D signal containing the quantity of interest at a given time, as a function of position. The successive Fourier transforms are then performed in the spatial domain only. The
result is again a plot of the bispectrum/bicoherence as a function of wavenumber, but this is more coarse than the plot resulting from the other method of computing the bispectrum/bicoherence described in the preceding paragraph. The advantage of this approach, is that it is easier for one to diagnose at what time a nonlinear feature of interest “switches on”, which of course could be achieved by looking at the dispersion relation alone, but using this method one can ascertain what switched it on.