3.3 Indirect localisation
3.3.2 Single volume
Localised autopower
The current spectral density (or autopower) for a single volume can be found from equation 3.41 by assuming that kd= 0 and that we only have a single wave vector k:
I11(k, ω)∝ Z
dZS(k, Z, ω)e−w22k2(1−cos(α−θp)) (3.44) Assuming that the angles α andθp are small, we can expand the function in the exponent of equation 3.44 as
2k2[1−cos(α−θp)]≈2k2[(α−θp)2]/2 =k2(α−θp)2 (3.45) We introduce the instrumental selectivity function
χ=e−
³α−θp
∆α
´2
, (3.46)
where ∆α= ∆kk = kw2 is the transverse relative wavenumber resolution.
Using this instrumental function, the scattered power
I11(k, ω)∝
We will use this simplified equation to study how spatial resolution can be obtained indirectly. To make simulations for this purpose we need to assume a pitch angle profile and an expression for the frequency integrated local spectral density S(k, Z).
The pitch angle
For our simulations we will take the pitch angle to be described by θp(r) = arctan
an analytical profile constructed by Misguich [17], see figure 3.3. Here, ρ=r/a is the normalised minor radius coordinate, qa is the magnetic field winding number at r=a and R0 is the major radius of the plasma. The total pitch angle variation ∆θp,tot is seen to be about 15 degrees.
CHAPTER 3. SPATIAL RESOLUTION - 15P 40
Figure 3.3: Modelled pitch angle in degrees versusρ. We have usedqa = 3.3, R0 = 2.38 m and a = 0.75 m (Tore Supra parameters, see [17]).
Fluctuation profiles
The frequency integrated local spectral density is assumed to be independent of the selected wavenumber:
S(k, r) =S(r) = δn2, (3.49) where we have replaced the beam coordinate Z by the radial coordinate r.
The normalised density profile is assumed to be n(r)
n0 = 0.1 + 0.9p
(1−ρ2), (3.50)
see figure 3.4.
Figure 3.4: Modelled normalised density versus ρ.
Further, the density fluctuation profile is assumed to have the following structure:
CHAPTER 3. SPATIAL RESOLUTION - 15P 41
δn(r)
n(r) =b+c|ρ|p, (3.51)
where b, cand p are fit parameters. At present we will assume the following fit parameters: b = 0.01, c= 0.1 and p= 3, see the left-hand plot of figure 3.5.
Figure 3.5: Left: δn/n versus ρ, right: δn2 versus ρ.
Simulations
Above we have introduced spatially localised expressions for all external quantities entering equation 3.47. We set the wavenumber k to 15 cm−1 and the beam waist w to 2.7 cm. This means that the transverse relative wavenumber resolution ∆α is equal to 2.8 degrees. Figure 3.6 shows χ forα
= 0 degrees (left) and 5 degrees (right). We observe that by changing the diagnostic angle α, χ changes position in the plasma.
Figure 3.7 shows the integrand of equation 3.47 for the two cases shown in figure 3.6. We see that the 0 degree case corresponds to a signal originating in the central part of the plasma, while the 5 degree case detects edge fluctuations.
Figure 3.8 shows figures corresponding to figures 3.6 and 3.7, but now for a mini-α scan: [-5, -2.5, 0, 2.5, 5] degrees.
Finally, figure 3.9 shows the integrands in figure 3.8 integrated along ρ (=
I11).
Finally, figure 3.10 shows the effect of increasing the transverse relative wavenumber resolution ∆α from 2.8 to 28.0 degrees. The instrumental selectivity function (left) becomes extremely broad, leading to the total scattered power having no significant variation with α.
CHAPTER 3. SPATIAL RESOLUTION - 15P 42
Figure 3.6: Left: χ versus ρ for α = 0 degrees, right: χ versus ρ for α = 5 degrees (k = 15 cm−1,w = 2.7 cm).
Figure 3.7: Left: Integrand for α = 0 degrees, right: Integrand for α = 5 degrees (k = 15 cm−1,w = 2.7 cm).
Figure 3.8: Left: χ for five α values, right: Corresponding integrands (k = 15 cm−1, w = 2.7 cm).
CHAPTER 3. SPATIAL RESOLUTION - 15P 43
Figure 3.9: Total scattered power (I11) versus α(k = 15 cm−1,w= 2.7 cm).
Figure 3.10: Left: χ forα = 0 degrees versusρ, right: Total scattered power (I11) versus α (k = 15 cm−1, w= 0.27 cm).
What we have demonstrated with the above simulations is that for localisation to be possible, the following has to be true:
∆θp,tot[degrees] À∆α[degrees] = 2
kw × 180
π (3.52)
Chapter 4
Spectral analysis - 10p
In this chapter we will describe the main spectral quantities used for our analysis. The analysis procedure is very similar to the one we made in [77], and a large part of the programs have been written by S.Zoletnik.
4.1 Event creation
To assist in the interpretation of our spectral analysis, we create simulated data using finite lifetime ’events’. An event is a finite-length piece of
complex signal S. The time evolution of the event amplitude A is Gaussian;
the phase is either fixed or linearly proportional to vφt (note here that velocity is defined to be in units s−1). The total simulated signal can be written
nev
X
j=1
S =
nev
X
j=1
X+ iY =
nev
X
j=1
A×ei(vφt+φ0) (4.1) The initial time point and phase (φ0) of an event is random; the amplitude of events is fixed. Figure 4.1 shows an event versus time. We will analyse simulated signals having a length of N = 2×105 time points. Since our diagnostic samples with a rate of 20 MHz, this corresponds to 10 ms of data. We will superimpose nev = 2×103 events onto this array, at random times and with a random initial phase. These events have a lifetime of 2 µs, where the lifetime is defined to be 4 standard deviations of the Gaussian amplitude.
Since we have two measurement volumes, we would like to perform cross spectral analysis. The simulated signal in one volume is created as
described above, while the signal in the second volume can be shifted ∆t in time and ∆φ in phase.
44
CHAPTER 4. SPECTRAL ANALYSIS - 10P 45
show_lscat_rawsignal.pro at Fri Nov 16 15:01:53 2001 show_lscat_rawsignal.pro at Fri Nov 16 15:02:11 2001
Figure 4.1: Left: Real and imaginary part of a simulated event versus time, right: Amplitude and phase of the same event. In this example, the event lifetime is 4 µs, amplitude 25 and a phase velocity of 5×105 s−1. The total length of the time window is 15 µs.
Below, we will treat eight cases:
1. (∆t,∆φ, vφ) = (0,0,0) 2. (∆t,∆φ, vφ) = (1.5 µs,0,0) 3. (∆t,∆φ, vφ) = (0,1,0)
4. (∆t,∆φ, vφ) = (0,0,6×106 s−1)
5. (∆t,∆φ, vφ) = ([0.5,0.5] µs,[0,0],[6×106,−6×106] s−1) and nev = [2×103,5×102]
6. (∆t,∆φ, vφ) = ([−1.5,1.5] µs,[0,0],[6×106,6×106] s−1) and nev = [2×103,5×102]
7. (∆t,∆φ, vφ) = (0,0,0) and nev= 0
8. (∆t,∆φ, vφ) = ([0,0] µs,[0,0],[5×105,−5×105] s−1),
nev = [2×102,2×101], Aev = [25,25], lifetime 4 µs and a noise level Anoise of 2.5
Case 1 is the baseline case, events having zero phase velocity and no time or phase delay. The noise level Anoise is set to 1.5, while the event
amplitude Aev is 1.
In cases 2-4 we set ∆t, ∆φ and vφ to non-zero values in turn to observe the effects in cross spectral analysis.
Cases 5 and 6 include two phenomena, see section 4.7:
CHAPTER 4. SPECTRAL ANALYSIS - 10P 46
• In case 5 the time delays ∆t are identical, while the phase velocities vφ are equal in size but counter propagating
• In case 6 the time delays are equal in size but of opposite sign, while the phase velocities are identical
Case 7 is as the baseline case 1, except for the fact that no events are present (pure noise signal).
Finally, case 8 is used for the phase separation procedure.
In the remaining sections of this chapter, we will refer to these simulation numbers and what can be learned from their behaviour.