expressions (7.20) and (7.64)). However, the general solution of these differential
equations can be represented as a combination of eigenmodes only for the case
where the operators defining these equations are Hermitian. And this is usually not
the case when the unperturbed flow velocity is included. For example, the operator
Fig. 7.18 The growth rate
of the ITG instability versus
plasma flow shear rate γ E /
V
0
0 for different values of
magnetic shear b s.
(Reproduced with
permission from [76],
© AIP Publishing 2012)
γ E = 0.1
γ E = 0.2
γ E = 0.3
γ E = 0
Fig. 7.19 Eigenfunctions of the electrostatic potential of toroidal ITG modes in a tokamak with
increasing shear of the plasma flow, γ E / V
0
0 , [77]. V. I. Dagnelie, private communication, 2020
1.0
–1.0
–0.5
0.0
0.5
1.0
1.5
2.0
2.5
dΩ /dq=0.0
3.0
1.0
–1.0
–0.5
0.0
0.5
1.0
1.5
2.0
2.5
dΩ /dq=0.1
3.0
1.0
–1.0
–0.5
0.0
0.5
1.0
1.5
2.0
2.5
dΩ /dq=0.5
3.0
Fig. 7.20 Eigenfunctions structure of the ideal ballooning mode in a tokamak with no toroidal
velocity shear (left); medium shear dΩ/dq (middle) and high shear (right), were Ω and q are the
toroidal angular velocity of the plasma and the safety factor. (Reproduced with permission from
[73], © American Physical Society 2004)
176
7 Anomalous Cross-Field Transport in Edge Plasma