Amplitude Modulation And Nonlinear Self-Interactions of the Geodesic Acoustic Mode at the Edge of MAST

We studied the amplitude modulation of the radial electric field constructed from the Langmuir probe plasma potential measurements at the edge of the mega-ampere spherical tokamak (MAST). The Empirical Mode Decomposition (EMD) technique was applied, which allowed us to extract fluctuations on temporal scales of plasma turbulence, the Geodesic Acoustic Mode (GAM), and those associated with the residual poloidal flows. This decomposition preserved the nonlinear character of the signal. Hilbert transform (HT) was then used to obtain the amplitude modulation envelope of fluctuations associated with turbulence and with the GAM. We found significant spectral coherence at frequencies between 1–5 kHz, in the turbulence and the GAM envelopes and for the signal representing the low frequency zonal flows (LFZFs). We present the evidence of local and nonlocal, in frequency space, three wave interactions leading to coupling between the GAM and the low frequency (LF) part of the spectrum.


Introduction
The edge region of tokamaks, defined by the steep pressure gradient, is dominated by turbulent 13 structures of density, temperature and the electrostatic potential (see, for example, [1] and the 14 references therein) arising from resistive and/or interchange plasma instabilities. These fluctuations 15 are responsible for the intermittent turbulent radial transport, which drives core heat and particle  [7,9]. 23 Zonal flows are axisymmetric electrostatic potential modes with zero poloidal and toroidal 24 numbers, m = n = 0. In tokamak geometry, toroidal curvature couples ZF to the density perturbations 25 with poloidal mode numbers m ≥ 1 (n = 0), and with a finite frequency. This compressible component 26 of ZF is called Geodesic Acoustic Mode (GAM). The local dispersion relation for the GAM has been 27 derived from various plasma models and the leading term is ω G,l ∼ c s /R 0 , where c s is the local sound 28 speed and R 0 is the major radius [10,11]. The amplitude of the density fluctuations varies with the 29 poloidal angle θ as A ∼ sin(θ). Since its theoretical discovery, the GAM has been experimentally 30 their nonlinear character. This allows us to construct signals representing meso-scale turbulence, the 48 oscillatory GAM signal and the low frequency zonal flows, ZFZF. Hilbert transform gives nonlinear 49 envelopes for the turbulence and the GAM. The spectralcoherence of the turbulence with the GAM 50 is then examined. We find that the amplitude modulation of the turbulence and the GAM have a 51 similar behaviour at low frequencies, between 2 − 5 kHz. The auto bi-coherence reveals nonlinear 52 self-interaction of GAM and the possible coupling to these low frequency components.   54 The Mega Amp Spherical Tokamak (MAST) has a major radius R 0 ≈ 0.85 m and a minor radius 55 of a ≈ 0.65 m. The magnetic field strength is about 0.5T with the toroidal, B ζ , and the poloidal, B θ , field components giving a pitch angle of about 22 • at the edge of the device. We analyse data from an 57 Ohmic plasma discharge numbered 29150, with a line average number density, n ≈ 1.47 × 10 19 m −3 , 58 and plasma current I p = 0.43MA. No additional heating power was applied during the discharge.

59
Magnetic configuration was a double null.

60
The data was collected using a Mach type reciprocating Langmuir probe [25], on the outboard 61 mid-plane, measuring floating potential,Ṽ f as well a set of ion saturation currents (pins 2, 5 and 62 8). Figure 1 shows the schematic of the probe, with pin numbers and the relative distances between 63 them. Pins (1, 3) are positioned 0.8 cm behind pin pairs (4, 6) and (7,9). We assume that the floating 64 potentialṼ f is a good proxy of the plasma potentialṼ p . These are related byṼ p =Ṽ f + Λ, where Λ 65 is the sheath potential drop, which is a slowly varying function of the electron and ion temperatures 66 and is usually approximated by Λ ≈ 2.5T e /e [26]. It is assumed that the electrostatic potential  In practice, the iterative sifting process is performed as follows: firstly, the maxima and minima of the signal are separately connected using cubic splines to form two envelopes of the data; one that contains all of the maxima and the other, the minima. The mean of the maximum and minimum envelopes, m 1 , is calculated. For an input signal S(t), the difference, h 1 = S(t) − m 1 gives the first estimation of the envelope of S(t). However, this envelope's mean is, in general, not equal to the true local mean, especially if the data is nonlinear. The process is therefore repeated k times until the resultant, h 1k , satisfies the requirement for an IMF,  envelope and its mean after kth sifting iteration, accordingly. We then designate s 1 (t) = h 1k as the first IMF component of the data, containing the shortest period of the signal. Fluctuations at this scale are removed from the data to obtain a residual r 1 = S(t) − s 1 (t). The procedure is then repeated for the residual r 1 , treated as a new input signal. The decomposition is stopped either when the component s i , or the residue r i , become too small to be of interest, or when the residue, r i , becomes a monotonic function from which no more IMFs can be extracted. For data with a trend, the final residue should be that trend. When the process is finished, we obtain the decomposition of a signal S(t) into IMFs s i and the final residue: The IMFs may contain oscillations with different periods in one mode, and different modes can contain similar periods. This spectral leakage, or mode mixing, can be an issue, especially for short and intermittent data. We incorporate the ensemble empirical mode decomposition (EEMD) [28,29] to reduce the impact of mode mixing. This noise-assisted method adds white noise to the original data before the sifting process starts. The EMD modes are computed as normal until all of the IMFs are calculated. The original data is then reprocessed with a different noise realisation and the final IMF is averaged over all ensembles. In this work we use EEMD to decompose the radial electric field time series into a number of IMFs. We are interested in the amplitude modulation of turbulence and the and Fourier spectrum estimate (red). A significant spectral peak at ∼ 10 kHz is clearly seen. Fourier spectrum shows an internal structure of the GAM peak, with multiple modes separated by ∼ 1 kHz. A possible second harmonic is also present at ∼ 20 kHz.
Geodesic Acoustic Mode (GAM). The modes with the periods shorter than the GAM are interpreted as turbulence. An envelope of a modulated signal can be constructed using analytic signal S a : In order to quantify the nonlinear interactions between different modes we use the wavelet bi-coherence defined as whereS is a wavelet coefficient at a scale associated with a period 1/ f and at time τ. For a signal S(t), the wavelet coefficients are given bỹ where s is a temporal scale, τ is a new time label and ψ(t) is the analysing wavelet. We use Bump wavelets [30], which have better frequency resolution, but poorer time localisation compared with a standard Morlet wavelet. Given a set of wavelet coefficientsS( f , τ) the wavelet spectrum estimate is given by   largest frequency is treated as the residual noise in the data and is discarded. Similarly, we discard he smallest frequency mode, which is a nonlinear trend very close to zero. We then combine the IMFs at 126 three different frequency ranges to obtain signals of interest. The turbulence, S T (t), is a superposition 127 of IMF 2 − 4, corresponding to mean instantaneous frequencies between ∼ 25 kHz and ∼ 66 kHz. The

128
GAM is a single IMF with the mean frequency of 10 kHz, S G (t). Finally, the ZFZFs signal S Z (t) is 129 obtained by summing modes 8 − 20, with mean frequencies in the range 77 − 5000 Hz. We apply 130 an analytic signal approach to S T and S G , in order to obtain their amplitude modulation envelopes 131 E T and E G , respectively. The turbulence envelope is smoothed over 50 neighbouring points. Figure   132 4 shows signals and their upper envelopes. The GAM, its envelope and the ZFZF signal have been