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Theory of tokamak transport. New aspects for nuclear fusion r..|INIS
Book Category:. Tokamak Physics. Magnetic Confinement Fusion. To add your own event to this list, send an email to admin fusenet. Conference "Advances and Applications in Plasma Physics". Ballooning modes are because of this more stable for small q. This concerned global magnetohydrodynamic MHD stability where the rotational transform q was the most basic stability parameter. Concerning transport, a suitable q turned out not to be enough.
This was for a while considered as a real threat to the possibility to achieve nuclear fusion. This very strong transport turned out to be due to large electrostatic vortices called Convective cells Taylor and McNamara ; Okuda and Dawson ; Sagdeev et al. Basic fundamental work on convective cells was also made by Sagdeev et al. Fortunately, it turned out that convective cells could be short circuited by magnetic shear. This opened up for new optimism about the achievement of nuclear fusion.
Another improvement of confinement could be obtained due to favourable average curvature of the magnetic field lines Hassam and Kulsrud However, this effect could be partly overcome by higher order nonlinear effects Weiland The level of transport in tokamaks is, of course, strongly dependent on the level of the turbulence. We here used the quasilinear approximation. However, as has been found later Weiland ; Weiland et al.
This means that from the transport point of view our quasilinear treatment is equivalent to that of a fully nonlinear and nonlocal turbulence code [agreement with a local turbulence code Rogers et al.
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The kernel 7 was also part of the diffusivity in the Hasegawa—Wakatani system Wakatani and Hasegawa ; Connor and Pogutse Also in this case the quasilinear approximation was found to be very good Connor and Pogutse In 8b the Doppler shift in the denominator is due to the magnetic drift of ions which is the average velocity the ions are moving with. Thus, this is a signature of the magnetic drift resonance.
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When electron trapping is added Liu ; Weiland et al. These different resonances mean that the ITG mode mainly drives ion transport and the Trapped Electron TE mode mainly drives electron transport Weiland et al. The factor in front includes off-diagonal contributions. Actually also the growth rate contributes to this. Now Eq. More general cases calculated along the same principles have turned out to usually be in good agreement with both experiments Bateman et al.
A very important aspect is here that we have not expanded in the magnetic drift Weiland ; Weiland et al. Actually, the pinch term in 8a is due to quadratic effects of the magnetic drift. As is evident this pinch will reduce the heat flux towards the axis and tends to give a transport coefficient that grows with radius.
This had previously been one of the main problems for drift wave models as also shown in Scott et al. We studied this case in Weiland with much better result than obtained in Scott et al. The result in Weiland was obtained from an interpretative code. Predictive codes Weiland and Nordman usually give a more clear radial growth of transport. The fact that the radial growth of transport coefficients is a toroidal effect was also found in kinetic simulations by LeBrun et al.
As has been emphasized in many of our papers we usually need to keep the fluid magnetic drift resonance. This is also true for the kinetic magnetic drift as pointed out in Weiland and Weiland et al.
The strong flowshear at marginal stability is due to the exact fluid resonance in 8b. We realize that the waves with the wavelength approaching system size will always hit marginal stability somewhere, thus becoming strongly damped. Thus, the form 7 of the transport kernel strongly depends on the fluid closure.
Since dissipation due to wave particle resonances strongly reduce the zonal flows, we need a reactive fluid closure to have transport kernels of the type 7. In Wakatani and Hasegawa and Connor and Pogutse the reactive closure was due to zero ion temperature but in our case it is due to detuning of the wave particle resonance by nonlinear frequency shifts Weiland et al. We also recall that the analytical solution by Chandrasekhar is only valid for constant coefficients in the Fokker—Planck equation.
We have, however, tested more general cases Holod et al. Since we here approach a stationary asymptotic state the Chandrasekhar solution will be valid asymptotically. However, the real part would still fulfil We also have interesting support for our nonlinear upshift from the fully nonlinear kinetic simulations in Dimits et al. This is from Hatzky et al. When the number of particles is insufficient the fluctuations are increased since particles can not effectively screen each others space charge.
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For larger fluctuations, we also have larger dissipation. Thus, we effectively introduce dissipation into the nonlinear particle simulations. As mentioned above our fluid model includes a full transport matrix. For drift waves, this basically means that we include ion and electron heat pinches and electron and impurity particle pinches. It follows from the second part of Eq. However, there is a built in tendency for the system to move towards a state where length scales of the dynamic gradients do not differ too much. Thus, the second term in 20 usually dominates.
Nevertheless, we should not ignore the third term. The importance of the density gradient was later verified by Austin et al. In this case, there was no heat pinch.
The fact that we here get a strong particle pinch is also important. The equations used here were from Weiland et al. This was the first model to achieve a strong particle pinch of experimental order. This problem was expressed by Wagner and Stroth Reproduced from Ma et al. As mentioned above the direction of the particle flux usually depends on which mode is driving the flux. In our case, the dominance of ITG was shown to be enhanced by strong ballooning.
There are actually two trapped electron modes which could play a role here. These are the Ubiquitous mode Coppi and Pegoraro driven by charge separation and the mode driven by compressibility and electron temperature gradient Liu ; Weiland et al. Nice overviews of these modes were shown in Garbet et al. Points correspond to experimental and lines to modelled T e profile. The modulation frequency was Shaded regions have to low signal to noise ratio. Dots are experimental and lines are from the Weiland model. Reproduced with permission from Tardini et al. As is evident from Fig. This confirms the validity of our treatment of trapped electrons.
Later tests for ion modulation have not been quite as good. However, for ion modulation we have to remember the importance of the kernel 7 which depends sensitively on the absorption of the inverse turbulent cascade by zonal flows according to Fig. This mechanism could be disturbed by a modulation in the ion heating in the experiment since it is the ions which generate the zonal flows. The zonal flow generation in our code has turned out to, in addition to the spinup of poloidal rotation in internal transport barriers and the Dimits shift, to also give us an experimental.
L — H transition Weiland In Fig.