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1 Fluctuation Loss, Bohm, GyroBohm Diffusion

# 1 Fluctuation Loss, Bohm, GyroBohm Diffusion

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286

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

288

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

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]

290

13 Plasma Transport by Turbulence

Fig. 13.1 In the upper

eigenmode Δr is larger than

rational surfaces Δrm . A

semi-global eigenmode

structure Δrg takes place due

to the mode couplings. In the

of eigenmode Δr is smaller

the rational surfaces Δrm .

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 ,

292

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.

294

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,

296

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

.

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

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