1 Fluctuation Loss, Bohm, GyroBohm Diffusion
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13 Plasma Transport by Turbulence
The plasma is forced to move by perturbation. When the velocity is expressed by
V˜ k =
V (r, t) =
V k exp i(k · r − ωk t),
k
(13.2)
k
then V −k = V ∗k and the equation of continuity
∂n
+ ∇ · (nV ) = 0
∂t
can be written as
∂n 0
+
∂t
k
∂ n˜ k
+∇ ·
∂t
n 0 V˜ k +
n˜ k V˜ k
k
= 0.
k,k
When the first- and the second-order terms are separated, then
k
∂ n˜ k
+∇ ·
∂t
∂n 0
+∇ ·
∂t
n 0 V˜ k = 0,
(13.3)
k
n˜ k V˜ k
= 0.
(13.4)
k,k
Here we have assumed that the time derivative of n 0 is second order. The time average
of the product of (13.3) and n˜ −k becomes
γk |n k |2 + ∇n 0 · Re(n k V −k ) + n 0 k · Im(n k V −k ) = 0,
ωkr |n k |2 + ∇n 0 · Im(n k V −k ) − n 0 k · Re(n k V −k ) = 0.
(13.5)
If the time average of (13.4) is taken, we find that
∂n 0
+∇ ·
∂t
The diffusion equation is
Re(n k V −k ) exp(2γk t) = 0.
(13.6)
k
∂n 0
= ∇ · (D∇n 0 ),
∂t
and the particle flux Γ is
Γ = −D∇n 0 =
Re(n k V −k ) exp 2γk t.
k
(13.7)
13.1 Fluctuation Loss, Bohm, GyroBohm Diffusion
287
Equation (13.5) alone is not enough to determine the quantity ∇n 0 ·Re(n kV−k ) exp 2γk t.
Denote βk = n 0 k · Im(n k V −k )/∇n 0 · (Re(n k V −k )); then (13.7) is reduced to
D|∇n 0 |2 =
k
and
γk
D=
k
γk |n k |2 exp 2γk t
,
1 + βk
|n˜ k |2
1
.
2
|∇n 0 | 1 + βk
(13.8)
This is the anomalous diffusion coefficient due to fluctuation loss.
˜ k of the electric field is elecLet us consider the case in which the fluctuation E
trostatic and can be expressed by a potential φ˜ k . Then the perturbed electric field is
expressed by
˜ k = −∇ φ˜ k = −i k · φk exp i(k · r − ωk t).
E
˜ k × B drift, i.e.,
The electric field results in an E
˜ k × B)/B 2 = −i(k × b)φ˜ k /B.
V˜ k = ( E
(13.9)
where b = B/B. Equation (13.9) gives the perpendicular component of fluctuating
motion. The substitution of (13.9) into (13.3) yields
n˜ k = ∇n 0 ·
b×k
B
φk
.
ωk
(13.10)
In general ∇n 0 and b are orthogonal. Take the z axis in the direction of b and the x
axis in the direction of −∇n, i.e., let ∇n = −κn n 0 xˆ , where κn is the inverse of the
scale of the density gradient and xˆ is the unit vector in the x direction. Then (13.10)
gives
ω ∗ eφ˜ k
Te eφ˜ k
κn k y ˜
n˜ k
=
= k
,
φk = k y κ n
n0
B ωk
eBωk Te
ωk Te
where k y is y (poloidal) component of the propagation vector k. The quantity
ωk∗ ≡ k y κn
Te
eB
is the electron drift frequency. If the frequency ωk is real (i.e., if γk = 0), n˜ k and φ˜ k
have the same phase, and the fluctuation does not contribute to anomalous diffusion
as is clear from (13.8). When γk > 0, so that ω is complex, there is a phase difference
between n˜ k and φ˜ k and the fluctuation in the electric field contributes to anomalous
diffusion. (When γk < 0, the amplitude of the fluctuation is damped and does not
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13 Plasma Transport by Turbulence
contribute to diffusion.) V˜ k is expressed by
Te n˜ k ωkr + γk i
Te φ˜ k
xˆ .
= −ik y
V˜ k = −i(k × b)
eB Te
eB n 0
ωk∗
Then the diffusion particle flux may be obtained from (13.7) as follows:
k y γk n˜ k
ωk∗ n 0
Γ = Dκn n 0 = Re(n˜ −k V˜kx ) =
k
and
D=
k
k y γk n˜ k
κn ωk∗ n 0
2
Te
=
eB
k
n˜ k
n0
2
2
Te
n0,
eB
γk
.
κ2n
(13.11)
The anomalous diffusion coefficient due to fluctuation loss increases with time and
eventually the term with the maximum growth rate γk > 0 becomes dominant.
However, the amplitude |n˜ k | will saturate due to nonlinear effects; the saturated
amplitude will be of the order of
|n˜ k | ≈ |∇n 0 |Δx ≈
κn
n0.
kx
Δx is the correlation length of the fluctuation and the inverse of the typical wave
number k x in the x (radial) direction. Then (13.11) yields
D=
γk n˜ k
κ2n n 0
2
≈ (Δx)2 γk ≈
γk
(Δx)2
≈
,
k x2
τc
(13.12)
where τc is the autocorrelation time of the fluctuation (turbulence) and is about the
inverse of γ in the saturation phase of turbulence.
When the nondimensional coefficient inside the parentheses in (13.11) is assumed
to be at its maximum of 1/16, we have the Bohm diffusion coefficient
DB =
1 Te
.
16 eB
(13.13)
It appears that (13.13) gives the largest possible diffusion coefficient.
When the density and potential fluctuations n˜ k , φ˜ k are measured, V˜ k can be
calculated by (13.9), and the estimated outward particle flux Γ by (13.7) and diffusion
coefficient D can be compared to the values obtained by experiment. As the relation
of n˜ k and φ˜ k is given by (13.10), the phase difference will indicate whether ωk is real
(oscillatory mode) or γk > 0 (growing mode), so that this equation is very useful in
interpreting experimental results.
13.1 Fluctuation Loss, Bohm, GyroBohm Diffusion
289
Let us take an example of the fluctuation driven by ion temperature gradient drift
instability (refer to Sects. 6.8, 10.9.2). The mode is described by
φmn (r ) exp(−imθ + inz/R).
φ(r, θ, z) =
The growth rate of the fluctuation has the maximum at around kθ = (−i/r )∂/∂θ =
−m/r of [1, 2]
αθ
m
∼
|kθ | =
, αθ = 0.7 ∼ 0.8.
r
ρi
Then the correlation length Δθ in θ direction is Δθ ∼ ρi /αθ (ρi is ion Larmor radius).
The propagation constant k along the line of magnetic force near the rational
surface q(rm ) = m/n is
k = −i b · ∇ =
=
−m
r
Bθ
B
+
Bt
B
1
n
≈
R
R
n−
m
q(r )
m rq
s
kθ (r − rm )
(r − rm ) =
r R q2
Rq
where q(r ) ≡ (r/R)(Bt /Bθ ) is the safety factor (Bθ and Bt are poloidal and toroidal
fields, respectively) and s is the shear parameter (refer to Sect. 6.5.2) s ≡ rq /q. |k |
is larger than the inverse of the connection length q R of torus and is less than the
inverse of, say, the pressure gradient scale L p , that is
1
1
< |k | <
.
qR
Lp
The radial width Δr = |r − rm | of the mode near the rational surface r = rm is
roughly expected to be Δr = |r − rm | = (Rq/s)(k /kθ ) = (ρi /sαθ ) ∼ O(ρi /s).
The more accurate radial width of the eigenmode of ion temperature gradient driven
drift turbulence is given by [2, 3]
Δr = ρi
qR
s Lp
1/2
γk
ωkr
1/2
.
The radial separation length Δrm of the adjacent rational surface rm and rm+1 is
q Δrm = q(rm+1 ) − q(rm ) =
1
1
m+1 m
1
m/n r
−
= , Δrm =
∼
=
.
n
n
n
nq
rq m
skθ
When the mode width Δr is larger than the radial separation of the rational surface
Δrm , the different modes are overlapped and the toroidal mode coupling takes place
(see Fig. 13.1). The half width Δrg of the envelope of coupled modes is estimated to
be [3–5]
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13 Plasma Transport by Turbulence
Fig. 13.1 In the upper
figure, the radial width of
eigenmode Δr is larger than
the radial separation of the
rational surfaces Δrm . A
semi-global eigenmode
structure Δrg takes place due
to the mode couplings. In the
lower figure, the radial width
of eigenmode Δr is smaller
than the radial separation of
the rational surfaces Δrm .
The modes with the radial
width Δr are independent of
each other
Δrg =
ρi L p
s
1/2
.
The radial correlation length becomes the large value of Δrg (Δrg /Δr ∼ (L p /ρi )1/2 )
and the radial propagation constant becomes kr ∼ 1/Δrg . In this case, the diffusion
coefficient D is
ρi L p ∗
T αθ
D = (Δrg )2 γk ∼
ωk ∼
.
s
eB s
where ωk∗ is the drift frequency (Sects. 6.8, 7.3). This coefficient is of the Bohm type.
When the mode width Δr is less than Δrm (weak shear case), there is no coupling
between different modes and the radial correlation length is
Δr = ρi
qR
s Lp
1/2
.
The diffusion coefficient D in this case is
D ∼ (Δr )2 ωk∗ ∼ ρ2i
qR
s Lp
kθ T
eB L p
∼
T ρi
eB L p
αθ q R
s Lp
∝
T ρi
.
eB L p
(13.14)
This is called gyro-Bohm type diffusion coefficient. It may be expected that the transport in toroidal systems becomes small in the weak shear region of negative shear
configuration near the minimum q position (refer to Sect. 15.7).
Next, let us consider stationary convective losses across the magnetic flux. Even
if fluctuations in the density and electric field are not observed at a fixed position, it is
possible that the plasma can move across the magnetic field and continuously escape.
When a stationary electric field exists and the equipotential surfaces do not coincide
13.1 Fluctuation Loss, Bohm, GyroBohm Diffusion
291
Fig. 13.2 Magnetic surface
ψ = const. and electric-field
equipotential φ = const. The
plasma moves along the
equipotential surfaces by
virtue of E × B
with the magnetic surfaces φ = const., the E × B drift is normal to the electric field
E, which itself is normal to the equipotential surface. Consequently, the plasma drifts
along the equipotential surfaces (see Fig. 13.2) which cross the magnetic surfaces.
The resultant loss is called stationary convective loss. The particle flux is given by
Γk = n 0
Ey
.
B
(13.15)
The losses due to diffusion by binary collision are proportional to B −2 , but fluctuation or convective losses are proportional to B −1 . Even if the magnetic field is
increased, the loss due to fluctuations does not decrease rapidly.
13.2 Loss by Magnetic Fluctuation
When the magnetic field in a plasma fluctuates, the lines of magnetic force will wander radially. Denote the radial shift of the field line by Δr and the radial component
of magnetic fluctuation δ B by δ Br , respectively. Then we find
L
Δr =
br dl,
0
where br = δ Br /B and l is the length along the line of magnetic force. The ensemble
average of (Δr )2 is given by
L
(Δr )2 =
L
br dl
0
0
L−l
L
=
br dl
dl
0
−l
L
=
L
dl
0
dl br (l) br (l )
0
ds br (l) br (l + s) ≈ L br2 lcorr ,
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13 Plasma Transport by Turbulence
where lcorr is
lcorr =
∞
−∞
br (l) br (l + s) ds
br2
.
If electrons run along the lines of magnetic force with the velocity vTe , the diffusion
coefficient De of electrons becomes [6]
De =
L 2
(Δr )2
=
b lcorr = vTe lcorr
Δt
Δt r
δ Br
B
2
.
(13.16)
We may take lcorr ∼ R in the case of tokamak.
13.3 Dimensional Analysis of Transport
The determination of scaling law between the overall energy confinement time τE
and the parameters of apparatus such as
τE = f (n, T, B, a, q, a/R)
is one of the main objectives of large experimental devices and the scaling law is
usally in the form of a power law
τE = n αn T αT B αB a αa .
Dimensional analysis of energy confinement time is discussed here according to
Connor and Taylor [7].
A: Collisionless Vlasov equation in electrostatic limit (collisionless, low beta)
We first consider the collisionless Vlasov model, in which plasma distribution function for each species is described by
ei
∂ fi
∂ fi
+ (v · ∇) f i + (E + v × B) ·
= 0,
∂t
mi
∂v
ei
f i (x, v)dv = 0.
(13.17)
(13.18)
The energy loss per unit area and unit time is given by
Q=
v
m i v2
f i dvx dv y dvz = Q(n, T, B, a).
2
We now seek all the linear transformations of the independent and dependent variables
13.3 Dimensional Analysis of Transport
293
f → α f, v → βv, x → γx,
B → δ B, t → t,
E → η E,
which leave the basic equations (13.17) and (13.18) invariant. There are three such
transformations:
A1 : f → α f,
A2 : v → βv, B → β B, t → β −1 t, E → β 2 E,
A3 : x → γx, B → γ −1 B, t → γt, E → γ −1 E.
Under these combined transformations, the heat flux transformation as Q → αβ 6 Q,
temperature as T → β 2 T , density as n → αβ 3 n. Consequently, if the heat flux is as
Q=
c pqr s n p T q B r a s ,
the requirement that it remains invariant under the transformations A1 − A3 imposes
the following restrictions on the exponents:
p = 1, 3 p + 2q + r = 6, s − r = 0,
so that the general expression for Q is restricted to
Q=
cq na 3 B 3
T
2
a B2
q
= na 3 B 3 F
T
a2 B 2
,
where F is some unknown function. The corresponding energy confinement time is
proportional to nT a/Q and so is restricted to the form
BτE = F
T
a2 B 2
.
This scaling law is the exact consequences of the model as long as the boundary
conditions do not introduce any dominant additional physical effects. Consequently,
if the scaling is assumed to follow a power law, then it must be
BτE =
T
2
a B2
q
.
Further, we made the stronger assumption that a local transport coefficient exists.
In such a case, the confinement time would be proportional to a 2 . Then exponent q
must be equal to −1; that is,
B
τE ∝ a 2 .
T
This represents the ubiquitous Bohm diffusion coefficient.
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13 Plasma Transport by Turbulence
B: Collisional Vlasov equation in the electrostatic limit (collisional, low beta)
The basic equations are
ei
∂ fi
∂ fi
+ (v · ∇) f i + (E + v × B) ·
= C( f, f ),
∂t
mi
∂v
(13.19)
where C( f, f ) is the Coulomb collisional term together with charge neutrality and
electrostatic approximation (13.18). There are two transformations which leave the
basic equations (13.19) and (13.18) invariant:
B → β B, t → β −1 t,
B1 : f → β f, v → βv,
B2 : f → γ −1 f, x → γx,
B → γ −1 B, t → γt,
E → β 2 E,
E → γ −1 E.
Then we have Q → β 7 γ −1 Q, T → β 2 T , n → β 4 γ −1 n. The constraint on exponents
is
4 p + 2q + r = 7, s − r − p = −1,
so that
Q = na 3 B 3 F
n
T
,
B 4a3 a2 B 2
n
B 4a3
BτE =
,
n
T
,
B 4a3 a2 B 2
BτE = F
p
T
a2 B 2
,
q
.
C: Collisionless Vlasov equation at high beta (collisionless, high beta)
The basic equations are (13.17) and Maxwell equation:
∇×E=−
∂B
, ∇ × B = μ0 j ,
∂t
j=
ei vi f i dv.
(13.20)
There are two transformations:
C1 : f → β −3 f, v → βv,
C2 : f → γ
−2
f, x → γx,
B → β B, t → β −1 t,
B→γ
−1
B, t → γt,
E → β 2 E,
E →γ
−1
The constraint on the exponents is
2 p + r − s = 2, r + 2q = 3,
and the energy confinement time is
BτE = F na 2 ,
T
a2 B 2
= F(N , β),
N ≡ na 2 .
E,
j → β j,
j → γ −2 j.
13.3 Dimensional Analysis of Transport
295
D: Collisional Vlasov equation, high beta (collisional, high beta)
The basic equations are Vlasov equation including the collisional term (13.19) and
Maxwell equation and charge neutrality (if the Debye length is negligible). The
transformation is
D1 : f → β 5 f, v → βv, x → β −4 x,
B → β 5 B, t → β −5 t,
E → β 6 E,
j → β 9 j,
and the constraint on the exponents is
2p +
5r
11
q
+
−s =
.
2
4
4
Then the energy confinement time is
BτE = F na 2 , T a 1/2 , Ba 5/4 .
E: MHD fluid models
MHD equations are
∂ρm
+ ∇ · (ρm )v = 0,
∂t
∂
+ (v · ∇) v + ∇ p − j × B = 0,
∂t
ρm
γρm
γ−1
∂
+ (v · ∇) ( pργm ) = η j 2
∂t
E + v × B = η j,
and Maxwell equation. In the case of ideal MHD fluid, there are three transformations:
E1 : n → αn,
B → α1/2 B,
E2 : v → βv, t = β −1 t,
E3 : x → γx, t → γt,
E → α1/2 E,
B → β B,
p → α p,
E → β 2 E,
j → α1/2 j,
p = β 2 p,
j → γ −1 j.
Since Q ∝ nT v → αβ 3 Q, the constraint on the exponents is
p + r/2 = 1, 2q + r = 3, s = 0,
j → β j, T → β 2 T,
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13 Plasma Transport by Turbulence
and the energy confinement time in ideal MHD case is
nT
B2
BτE = (na 2 )1/2 F
= N 1/2 F(β).
F: Resistive MHD fluid model
In the case of resistive MHD fluid, there are two transformations:
F1 : n → αn,
B → α1/2 B,
E → α1/2 E,
F2 : v → βv, x = β −4 x, t = β −5 t,
p → α p,
B → β B,
j → α1/2 j,
E → β 2 E,
p = β 2 p,
j → β 5 j, T → β 2 T,
and the constraint on the exponents is
p + r/2 = 1, 2q + r − 4s = 3.
The energy confinement time in the case of resistive MHD is
BτE
=F
n 1/2 a
na 2
, T a 1/2
B 2 a 5/2
= F1 (β, T a 1/2 ) = F2 β,
τA
τR
.
Kadomtsev’s Constraint
It is useful to discuss dimensional analysis of scaling law from Kadomtsev’s viewpoint [8]. From the variables (n, T, B, a), one can construct four independent dimensionless parameters. If we select the set of
(n, T, B, a) → β,
ρi νe i λD
,
,
a Ωe a
,
where ρi and λD are ion Larmor radius and Debye length, respectively, then the
confinement time can be written by
Ωe τE = F β,
ρi νe i λD
,
,
a Ωe a
When the Debye length is negligible or charge neutrality can be assumed, we can
drop (λD /a) and
ρi νe i
.
Ωe τE = F β, ,
a Ωe
In the case of MHD fluid models, which do not refer to particle aspects, we have
only the dimensionless parameters β and τA /τR , where τA = (an 1/2 /B)(2μ0 /m i )1/2