4.7 Two phenomena
4.7.1 Crosspower and crosscorrelation
Figure 4.7 shows the crosspower spectrum (left-hand plot) and normalised crosscorrelation function (right-hand plot) for case 5. In this case, where two phenomena have distinguishable phase velocities, the separation is best done using the crosspower spectrum representation.
CHAPTER 4. SPECTRAL ANALYSIS 58
Figure 4.6: Normalised crosscorrelation function for cases 1-4. Top left: Case 1, top right: Case 2, bottom left: Case 3 and bottom right: Case 4.
8
Figure 4.7: Left: Crosspower spectrum, right: Normalised crosscorrelation function for case 5.
The situation is reversed for phenomena having different time delays, see figure 4.8. Here, the separation is best using the normalised crosscorrelation function.
The reason for the differences in separating two phenomena is due to the fact that they add as complex numbers.
4.7.2 Phase separation
Having two counter propagating phenomena present in the measured signal (phase velocities of opposite signs), these phenomena can be separated
8
0
-3 3 Amplitude [a.u.]
Phase [rad]
Frequency [MHz]
-4 4
0.6
0.0 3
-3-8 8
Normalised amplitude
Phase [rad]
Time lag [ s]m
Figure 4.8: Left: Crosspower spectrum, right: Normalised crosscorrelation function for case 6.
using the time derivative of the phase as explained in chapter 2. Figure 4.9 shows the autopower spectrum of a signal constructed using counter
propagating events (case 8); the solid line is the total signal including noise, the dotted/dashed line is the signal due to events having a
positive/negative phase velocity, respectively.
Simulated autopower [a.u.]
-1 Frequency [MHz] 1 100
104
Figure 4.9: Autopower spectrum for a simulated signal composed of two counter propagating features (case 8). The solid line is the total signal (including noise), the dotted/dashed line is due to events having a posi-tive/negative phase velocity, respectively.
Once the events are created and noise is added, we are ready to use the phase separation technique on the simulated data. Before the phase derivative of
Ssim =Ssim− +Ssim+ , (4.14)
CHAPTER 4. SPECTRAL ANALYSIS 60 where
Ssim+ =Xsim+ + iYsim+ =A+simeiΦ+sim
Ssim− =Xsim− + iYsim− =A−simeiΦ−sim (4.15) is constructed, the data is band pass filtered to the [1 kHz, 1 MHz]
frequency range. Thereafter the separation is done using the sign of ∂tΦsim:
∂tΦsim <0 :
½ Ssep− =Ssim−
Ssep+ = 0 ∂tΦsim >0 :
½ Ssep− = 0
Ssep+ =Ssim+ (4.16) and the result is
Ssep=Ssep− +Ssep+ , (4.17) where
Ssep+ =Xsep+ + iYsep+ =A+sepeiΦ+sep
Ssep− =Xsep− + iYsep− =A−sepeiΦ−sep (4.18) Since we have used the phase to separate the signals, we will use the
amplitude to gauge the quality of the phase separation. The tool we have chosen is the zero time lag crosscorrelation function
C(B, C) =
PN
k=1(Bk−B)(Ck−C) rh
PN
k=1(Bk−B)2i h PN
k=1(Ck−C)2i (4.19) and the correlations we calculate are C(A+sim, A+sep),C(A+sim, A−sep),
C(A−sim, A+sep) andC(A−sim, A−sep). It is important to note that we calculate the correlations using the noise-free (i.e. ’real’) simulated signals. The correlations supply us with a quantitative measure of the phase separation;
for a successful separation, we expect the 0+ +0/0− −0 correlations to be significant and the 0+−0/0−+0 correlations to be small. This is indeed the case for the separated autopower spectrum shown in figure 4.10, where the
0+ +0/0− −0 correlations are of order 80 %, whereas the 0+−0/0−+0 correlations are about 2 %.
Separated autopower [a.u.]
-1 Frequency [MHz] 1 100
104
Figure 4.10: Autopower spectrum for a phase separated signal composed of two counter propagating features. The solid line is the total signal, the dotted/dashed line is due to events having a positive/negative phase velocity, respectively.
Part II Experiment
62
Transport in fusion plasmas
In this chapter we summarise the transport concepts used in the thesis; the sections will necessarily be short. In section 5.1 we introduce the energy confinement time and scaling laws relating this to machine parameters.
Thereafter, in section 5.2, we describe the major transport equations and discuss the concept of anomalous transport. Sections 5.3 and 5.4 deal with two different approaches to the calculation of fluxes due to fluctuations. In section 5.5 definitions connected to turbulence theories are presented and section 5.6 contains an extremely short review on the measured properties of fluctuations in fusion plasmas.
5.1 Energy confinement
In this first section we describe a central quantity in transport analysis, the energy confinement time τE. The total energy in the plasma is
W = Z
3nTd3x= 3hnTiV, (5.1) where h·i is a spatial average and V is the plasma volume [159]. The energy loss rate PL is characterised by an energy confinement time defined by the relation
PL = W τE
(5.2) In machines currently operating, the energy loss is balanced by the
externally supplied heating power, PH (PH=PL). Using this fact along with equation 5.2, we arrive at the following expression for the energy confinement time in terms of measurable quantities:
63
CHAPTER 5. TRANSPORT IN FUSION PLASMAS 64
τE = W PH
(5.3) The theory dealing with transport of energy and particles due to binary collisions across the confining magnetic field of a toroidal device is called the neoclassical transport theory [71]. Fusion plasmas do not behave as predicted by neoclassical transport theory, in the sense that the neoclassical transport level is orders of magnitude smaller than what is experimentally observed. This also has consequences for the neoclassical energy
confinement time, which is found to be very large compared to measurements.
Since no detailed understanding of the reason for this discrepancy has emerged, we have to make empirical expressions for τE. This is usually done by scaling laws, where τE is set equal to a product of powers of various parameters involved. For the most basic approach, one uses engineering parameters, such as the toroidal magnetic field strength and the major radius of the machine in question. The scaling studies have been made using a multi-machine database consisting of hundreds or thousands of plasma discharges. An example for stellarators is the International Stellarator Scaling (ISS) from 1995:
τEISS95 = 0.079×a2.21R0.650 Ptot−0.59n0.51e Bϕ0.83ι
Ã
0.42/3, (5.4)where a is the plasma minor radius, R0 the machine major radius, Ptot the total absorbed heating power, ne the line density, Bϕ the toroidal magnetic field and ι
Ã
2/3 the rotational transform at two-thirds of the plasma minor radius (see chapter 6) [141]. The units of equation 5.4 from left to right are:s, m, m, MW, 1019 m−3 and T.
A slightly more sophisticated procedure is to make use of the invariance of the governing equations under scale transformations [38]. We will not describe the technique here, but just mention an application, the so-called ρ∗-scaling, ρ∗ being the normalised ion Larmor radius ρi/a. Different models lead to different scaling behaviours of τE, see table 5.1.
gyro-Bohm χ∝a2Bρ∗3 τE∝ρ∗−3/B ∝a5/2B Bohm χ∝a2Bρ∗2 τE∝ρ∗−2/B ∝a5/3B1/3 Stochastic χ∝a2Bρ∗ τE∝ρ∗−1/B ∝a5/6B−1/3
Table 5.1: Energy confinement time for three theories; χ is the thermal diffusivity.
The terms gyro-Bohm, Bohm and stochastic in table 5.1 refer to different
plasma transport theories. In ρ∗-experiments, other dimensionless parameters are kept constant, while ρ∗ is varied. Due to the different scaling of τE with B, such a procedure can help distinguish between different types of transport. For example, we note that the ISS95 scaling law is close to gyro-Bohm, where the turbulence correlation width is ρi, the ion Larmor radius.