4.7 Two phenomena
4.7.1 Crosspower spectrum and crosscorrelation
Figure 4.7 shows the crosspower (left-hand plot) and normalised
crosscorrelation function (right-hand plot) for case 5. In this case, where to
CHAPTER 4. SPECTRAL ANALYSIS - 10P 51
show_lscat_corr.pro at Thu Dec 06 12:06:32 2001 show_lscat_corr.pro at Thu Dec 06 12:08:45 2001
show_lscat_corr.pro at Thu Dec 06 12:09:19 2001 show_lscat_corr.pro at Thu Dec 06 12:39:21 2001
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.
phenomena have distinguishable phase velocities, the separation is best done using the crosspower representation.
show_lscat_crosspower.pro at Thu Dec 06 12:39:44 2001 show_lscat_corr.pro at Thu Dec 06 12:39:55 2001
Figure 4.7: Left: Crosspower, 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
CHAPTER 4. SPECTRAL ANALYSIS - 10P 52 function.
The reason for the differences in separating two phenomena is due to the fact that they add as complex numbers.
show_lscat_crosspower.pro at Thu Dec 06 12:40:23 2001 show_lscat_corr.pro at Thu Dec 06 12:40:33 2001
Figure 4.8: Left: Crosspower, right: Normalised crosscorrelation function for case 6.
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 using the time derivative of the phase as explained in subsection 2.4.5.
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.
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) where
Ssim+ =Xsim+ + iYsim+ =A+simeiΦ+sim
Ssim− =Xsim− + iYsim− =A−simeiΦ−sim (4.15) is constructed, the data is bandpass filtered to the [1 kHz, 1 MHz]
frequency range. Thereafter the separation is done using the sign of ∂tΦsim:
CHAPTER 4. SPECTRAL ANALYSIS - 10P 53
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.
∂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 cross correlation 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)
CHAPTER 4. SPECTRAL ANALYSIS - 10P 54 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 %.
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
55
Chapter 5
Transport in fusion plasmas -15p
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 V is the plasma volume [87]. The energy loss ratePL 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:
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 (see figure 5.1) is called the neoclassical transport theory [34]. 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
56
CHAPTER 5. TRANSPORT IN FUSION PLASMAS - 15P 57 energy confinement time, which is found to be very large compared to measurements.
R0 R z
j r q
poloidal toroidal
Figure 5.1: Geometry in a toroidal confinement device. The toroidal direction ϕ is the long way around the torus, the poloidal direction θ the short way.
The major radius coordinate isR(R0 being the center of the plasma column), the vertical coordinatez. The minor radius coordinateris zero in the plasma center.
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 a2.21R0.65Ptot−0.59n0.51e Bϕ0.83
Ã
ι0.42/3, (5.4) where a is the plasma minor radius, R the machine major radius, Ptot the total absorbed heating power, ne the line density, Bϕ the toroidal magnetic field and ιÃ
2/3 the rotational transform at two-thirds of the plasma minor radius [76]. The units of equation 5.4 from left to right are: s, 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 [13]. We will not describe the technique here, but just mention an application, the so-called ρ∗-scaling, ρ∗ being the normalised Larmor radius ρi/a. Different models lead to different scaling behaviours of τE:
CHAPTER 5. TRANSPORT IN FUSION PLASMAS - 15P 58
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, (5.5) where χ is the thermal diffusivity. The terms gyro-Bohm, Bohm and
stochastic 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 gyroBohm, where the turbulence correlation width is ρi, the ion Larmor radius.