where Δρ=ρ ( 1. Here again, we use the analogy of Eq. (7.61) to the Schrödinger
equation for an electron in a potential well / Àcosh
À2
(x/w) [6] with the energy
/ Àk
2
y , and can find the Rayleigh-Taylor (interchange) instability growth rate, γ RT m ,
versus the integer mode number m.
For the fastest-growing mode m ¼ 0, we have the following dependence of γ RT m¼0
and the corresponding eigenfunction, ψ m¼0 (x), on k y :
γ
2
RT m¼0
¼ γ
2
RT
w j k y j
w j k y j þ1
,
ψ m¼0 ðxÞ ¼ cosh
Àwjk y j
ðx=wÞ,
ð7:64Þ
where
γ
2
RT ¼
Δρ
2ρ
g
w
:
ð7:65Þ
Although this solution for the Rayleigh-Taylor (interchange) instability was
considered for somewhat idealized conditions, we will use it later to illustrate an
impact of fluid (plasma) flow velocity shear on the Rayleigh-Taylor (interchange)
instability.
Coming back to the plasma and analyzing the dissipation from Eqs. (7.56) and
(7.57) for ρ
2
s k
2
⊥ < 1 and finite, although still rather small ν k , ν k < ω Ã , we arrive at the
following modification of Eq. (7.34):
ω
2
þ γ
2
I þ
iν k
ρ 2
s k
2
⊥
ðω À ω Ã Þ ¼ 0:
ð7:66Þ
For the case of ν k > ω B,e , ρ
2
s k
2
⊥ ω Ã , this gives the so-called resistive interchange
mode
ω ¼ ω Ã þ i
ω B,e ω Ã
ν k
,
ð7:67Þ
which can be considered as a proxy for the “Resistive Ballooning Mode” (RBM)
(e.g. see Refs. [27–31] and the references therein). The eigenmode structure of the
RBM, found from numerical simulation for the DIII-D magnetic configuration, is
shown in Fig. 7.5. Once again, one can see that the mode is largely localized at the
outboard side of the torus (the “bad” curvature side).
We notice that in the context of our consideration of collisional drift waves in a
tokamak-like magnetic field, the term “ideal interchange mode” may sound strange.
However, as we have already seen (and will see later), some dispersion equations can
be rather general and, in different limits, describe different waves (e.g. recall
Eq. (7.15) which describes both the ion sound and drift waves). Similarly, considering different magnitudes of ν k (which depends not only on the plasma
collisionality but also on the parallel wavelength), our dispersion equation can
describe different modes ranging from an unstable drift wave for large ν k (recall
Eq. (7.32)) to the “ideal interchange mode” Eq. (7.59) for small ν k .
160
7 Anomalous Cross-Field Transport in Edge Plasma
equation for an electron in a potential well / Àcosh
À2
(x/w) [6] with the energy
/ Àk
2
y , and can find the Rayleigh-Taylor (interchange) instability growth rate, γ RT m ,
versus the integer mode number m.
For the fastest-growing mode m ¼ 0, we have the following dependence of γ RT m¼0
and the corresponding eigenfunction, ψ m¼0 (x), on k y :
γ
2
RT m¼0
¼ γ
2
RT
w j k y j
w j k y j þ1
,
ψ m¼0 ðxÞ ¼ cosh
Àwjk y j
ðx=wÞ,
ð7:64Þ
where
γ
2
RT ¼
Δρ
2ρ
g
w
:
ð7:65Þ
Although this solution for the Rayleigh-Taylor (interchange) instability was
considered for somewhat idealized conditions, we will use it later to illustrate an
impact of fluid (plasma) flow velocity shear on the Rayleigh-Taylor (interchange)
instability.
Coming back to the plasma and analyzing the dissipation from Eqs. (7.56) and
(7.57) for ρ
2
s k
2
⊥ < 1 and finite, although still rather small ν k , ν k < ω Ã , we arrive at the
following modification of Eq. (7.34):
ω
2
þ γ
2
I þ
iν k
ρ 2
s k
2
⊥
ðω À ω Ã Þ ¼ 0:
ð7:66Þ
For the case of ν k > ω B,e , ρ
2
s k
2
⊥ ω Ã , this gives the so-called resistive interchange
mode
ω ¼ ω Ã þ i
ω B,e ω Ã
ν k
,
ð7:67Þ
which can be considered as a proxy for the “Resistive Ballooning Mode” (RBM)
(e.g. see Refs. [27–31] and the references therein). The eigenmode structure of the
RBM, found from numerical simulation for the DIII-D magnetic configuration, is
shown in Fig. 7.5. Once again, one can see that the mode is largely localized at the
outboard side of the torus (the “bad” curvature side).
We notice that in the context of our consideration of collisional drift waves in a
tokamak-like magnetic field, the term “ideal interchange mode” may sound strange.
However, as we have already seen (and will see later), some dispersion equations can
be rather general and, in different limits, describe different waves (e.g. recall
Eq. (7.15) which describes both the ion sound and drift waves). Similarly, considering different magnitudes of ν k (which depends not only on the plasma
collisionality but also on the parallel wavelength), our dispersion equation can
describe different modes ranging from an unstable drift wave for large ν k (recall
Eq. (7.32)) to the “ideal interchange mode” Eq. (7.59) for small ν k .
160
7 Anomalous Cross-Field Transport in Edge Plasma