Variation of plasma parameters in the magnetic surface, a neoclassical e¤ect F. Spineanu and M. Vlad Bucharest September 11, 2017 Abstract Work Sessions of Plasma Theory. Fourth meeting: Variation of equilibrium plasma parameters in the magnetic surface in tokamak. The previous discussions have prepared the solution of the neoclassical drift-kinetic equation in various regimes. It remained to pay more attention to the collision operators and go further to viscosity and its relation with the ‡uxes. It has been formulated a request to stop for a while on the variation of the parameters in the magnetic surface, a purely neoclassical problem, which however is essential to understand spontaneous spin-up and the equilibrium ‡ows.Then some of the previous subjects will be repeated in this fourth meeting: diamagnetic ‡ow, P…rsch Schluter ‡ow, geometric inertial factor, spontaneous spin-up, magnetic pumping. We will focus on the Stringer solution of the drift-kinetic equation for the perturbation to the equilibrium function, which re‡ects the neoclassical poloidal variation. We will also mention other approaches. This text is a part of Lecture 4 from the Work Session of Plasma Theory. It is a ground for discussions, not to be taken as …nal.

1

The equilibrium poloidal ‡ows

Any poloidal ‡ow. The momentum equation for species j nmj

@uj + (uj r) uj @t

=

rpj

r

j

+ej nE + ej nuj B +Rj leads to nuj =

1 mj

j

b n

rpj

[It can be the diamagnetic current, with the signi…cance that is given by the gyration gradient of density.] 1

For reasons related to toroidal geometry this ‡ow cannot have zero divergence. Since this ‡ow is actually a current, perpendicular to the magnetic …eld Bb n, there is a non-zero divergence of this purely perpendicular electric current. Then there is another current, parallel to Bb n, whose divergence compensates for this one. This is the P…rsch Schluter current.

1.1

The P…rsch Schluter current

The drift motions of electrons and ions vdrif t;j in toroidal …eld lead to charge separation. In order to suppress this charge separation a current ‡ows along magnetic …eld lines. When there is resistivity (collisions) the neutralization of the charge separation by the parallel current is incomplete. Then there is a residual electric …eld which still remains. This is Ek and is connected with jk by 6= 0. This electric …eld induce an enhancement of the di¤usion. The enhancement comes from the radial velocity vr that exists due to the Ek coupling of the parallel electric …eld with the poloidal magnetic …eld, vr B , in the Ohm’s law rk + vr B = jk The radial velocity vr produces a radial ‡ux which multiplies the classical di¤usion.

r

= vr n. This is the factor q 2

The parallel current arising from the non-zero divergence of the diamagnetic current r j =

r? j? + rk jk

0

=

0

Now taking the perpendicular current as resulting from the diamagnetic ‡ows of electrons and ions, the parallel gradient can be written as rk jk

=

1 @ jk qR @

=

r? j? =

=

r?

e

r?

1 b n jej B

e rp

1 b n m

rp

Let us look to the last term. It is the perpendicular divergence of the diamagnetic ‡ow.

2

Note that the operator of parallel derivative is 1 @ qR @

rk

and that the perpendicular current j? is the diamagnetic current, of ions + electrons. End. This is a neoclassical e¤ect. It is the magnetic …eld that has a space variation in the perpendicular direction. First we have b n

rp

dp (b n b er ) dr dp b e dr

= =

Then, restricting the gradient to the part that contains B, we use the expression of the gradient operator expressed in the geometry of the toroidal region. This part is repeated later in this text. We have the magnitude of the magnetic …eld B0 1 + " cos and we must calculate the perpendicular divergence of the perpendicular current, which means 1 dp b r e B dr B=

and this is approximated by (B0 is constant) r

b e

B0 B

= r [b e (1 + " cos )]

Here is the essential part of the calculation: there is a divergence of the diamagnetic "‡ow" that is exclusively due to the geometry. This has consequences in the balance of ‡ows. Here it is explained how this divergence is calculated. In the orthogonal coordinates (r; ; ') we have the element of distance: 2

2

2

dl2 = (dr) + r2 (d ) + (R0 + r cos ) d'2 which gives the coe¢ cients h1

=

1

h2

=

r

h3

=

R0 + r cos

3

Then the divergence of a vector a is written r a=

1 h1 h2 h3

@ @ @ (h2 h3 a1 ) + (h1 h3 a2 ) + (h1 h2 a3 ) @r @ @'

which gives r [b e (1 + " cos )]

1 @ ((R0 + r cos ) (1 + " cos )) r (R0 + r cos ) @ i @ h 1 2 (1 + " cos ) R0 = r (R0 + r cos ) @ ( 2 sin ) = " r =

From this result we get r? j?

= = = = = =

r? (dia) = 1 b rp r? e n m 1 dp r? e ( b e ) m dr B0 1 dp e r? b B B0 dr ( 2 sin ) 1 dp " r B0 dr dp @ r (2 cos ) RB0 dr r@

Equating the two sides of the current conservation equation 1 @ jk = qR @

r e RB

dp dr

@ (2 cos ) r@

Integrating on the poloidal angle : Jk

JkP S =

"

2 dp cos B dr

This is the P…rsch Schluter current. We note 1 r B 1 " = =q B RB B B and the combination

1 dp B dr

4

is clearly coming from the diamagnetic ‡ow nudia =

1 b n m

rp

and remark that the P…rsch Schluter current is parallel with B proportional with q harmonic on proportional with the diamagnetic current. The …rst physical quantity that varies on the magnetic surface is the P…rsch Schluter current. There is a poloidal electric …eld related to this current E = It is the projection on factor of projection

1 B k B

"

2 dp cos B dr

(poloidal) of the relationship Ek = Jk =

k,

with the

Ek (B=B ) = E As mentioned above there is this electric …eld that still exists after the parallel current jk has tried to neutralize the charge separation produced by the non-zero divergence of the current of diamagnetic origin. This electric …eld is due to either …nite resistivity

=

1

, or

Landau damping

1.2

Why the equilibrium ‡ows and currents must have variations in the surface

First observation: the variations are small. They are small due to the fact that the toroidicity e¤ ect (a gemoetrical e¤ect) is small if r "= 1 R The most important part for a physical quantity is the surface-averaged part.

5

1.2.1

The equation of continuity

The equations involve in this approach the neoclassical drifts of the particles =

v

b vk + vDj n (0)

(1)

+VE + VE + ::: where vDj

1

=

2 v? =2 + vk2

b n

j

R

1 2Tj b evert ej B R

it was taken

( b eR )

2Tj = v2 mj

The other velocity is electric (0)

VE

(1)

VE

=

r

=

r

(0)

B (1)

B

b n

b n

In the equation of continuity we have the divergence of the ‡ow of the particles caused by their neoclassical drift. r (nvDj ) =

1 2Tj dn0 sin ej B R dr

The divergence of the ‡ow of particles moving with the electric velocity is " # r (0) B (0) r VE = r B2 i 1 h (0) r B = r B2 1 + 2 (r B) r (0) B The last term is zero since the gradient of the potential is almost radial and ? on B. The …rst term is purely geometrical, comes from the variation of the magnitude of the magnetic …eld. It is r

1 B2

h

r

(0)

6

i B =

2 (0) v sin R E

The equation of continuity is (0)

(1)

r n1 + VE

VE +n0 =

2 R

r n0

@vkj @s 1 dpj (0) + n0 VE sin ej B dr

We NOTE that we must admit the variation in the magnetic surface of two quantities one is the density; it occurs as a perturbation n1 (r; ) to the zeroth-order, i.e. surface-averaged, density n0 (r) the other is again the parallel velocity, here vkj (r; s) where s is a variable along the line, therefore explores regions of di¤erent ; Using this form of the continuity equation Stringer obtains the P…rsch Schluter current, and this results precisely from the term where the neoclassical drift ‡ows have been expressed in terms of the gradient of the pressure (as if they would come from diamagnetic: they do not come from diamagnetic). We write the equation of continuity for electrons and for ions and substract the two equations. One obtains JkP S

=

(0)

1 dp cos B0 dr P…rsch Schluter current 2q

(0)

The two terms r VE = R2 VE sin cancel since they are identical for electrons and ions. The part that contains the neoclassical drifts is the source of the gradient of pressure dp=dr.

1.2.2

The equation of momentum conservation

The next equation is the momentum conservation where the basic ‡ow is the (0) electric velocity VE . The equation is (0)

nmi VE

@ r@

vk;i =

(Ti + Te )

" @n1 q r@

This balance of momenta involves the nonlinear static advection of the parallel velocity and the gradient of the pressure along the magnetic …eld line. This equation means that the static advection of the parallel velocity vk;i (0) by the poloidal velocity VE is balanced by the parallel gradient of the pressure projected on the poloidal direction.

7

1.2.3

The Ohm’s law

The Ohm’s law jk =

" @ q@

(1)

+

where

Te n1 jej n0

jk = JkP S The P…rsch Schluter current JkP S allows now to write the system of equations

for n1 and

n1

(1)

, perturbations on surface. 1 1 h (0) = n0 2" (0) vidia + VE cos D V E

1 dp B dr

1 dn0 n0 dr

+ where

Tj ej B

vjdia =

c2s =

q2 "2

r sin B

1 dn0 n0 dr

Ti + Te mi

and the denominator D

=

1+

vedia (0)

VE

c2s

"2 2 q2

(0)

VE We Note that the combination 1+

vidia (0)

VE and the combination 1+

vedia (0)

=1

VE

vedia (0)

VE

occur in the expression of n1 . But, in D, the posible resonance 1 only involves the electrons. It is avoided by the quantity c2s (0)

VE

"2 2 q2

c2s

=

(0)

VE 8

2

jvedia j (0)

VE

=0

which compares the poloidally projected sound velocity to the electric poloidal velocity.

1.3

A practical outcome: the radial particle ‡ux is nonuniform on the surface

The radial ‡ux of particles is calculated as an average over the magnetic surface, 2 (0; 2 ). The quantity that is averaged is the local radial ‡ux obtained as product between the density (zero-order n0 and correction for variation in the surface, n1 ( )) and the radial velocity. The radial velocity is the neoclassical drifts vdrif t;j jr plus the electric contribution produced by the variation of the potential in the surface rj

= =

nvrj Z 2 0

d (n0 + n1 ) 2

Tj 2 sin 1 @ (1) + " B0 r@ ej B0 r

!

(1 + " cos )

2

The second term in the second paranthesis is Tj 2 sin " ej B0 r

vdrif t;j jradial

consistent with the approximation adopted by Stringer for the neoclassical drift 2 expressing v? =2 + vk2 in terms of Temperature. The result rj

= =

1.4

nvrj q 2 2n0

2

1 dp B0 dr

1 D

1 6 4 B0

2 c2s q"2

(0)

2

VE

+

vidia

The equations for the currents and ‡ows

The line ds2 = h2r dr2 + h2 d

2

+ h2' d'2

The values for circular geometry are hC r

=

1

C

h

=

r

hC '

=

R0 + r cos

9

3

vjdia 7 5 (0) VE

The magnetic …eld will be assumed slightly more general than in circular surfaces, and we will return to this simple geometry by taking B (r) = B0 =const. Br

=

0 b (r) " B0 = = h q h B0 = h

B B' where

h = 1 + " cos =

R r ; "= R0 R

and q is the safety factor. The current is (HLR) Jr = 0 J =0 J'

1 dp (1 + " cos ) b dr 1 dp b (r) =h dr 1 B' dp B' B dr

= = =

Since B' B we have

1

" J' = q

=

1

" q

h dp B0 (r) dr

which is identical with HassamKulsrud. We will …nd later below that the HK result is obtained from the Grad Shafranov equation. We will also use B=B b e + B' b e' from where we have

q B0 jBj = B 2 + B'2 = h

The perpendicular current j? comes from j

B=

10

rp

s

1+

"2 q2

where j

B0 B = j? jBj ( b er ) = j? h dp b er dr

=

from where

j? =

h 1 q B0 1 +

"2 q2

s

1+

"2 ( b er ) q2

dp dr

We notice that it is usual to work with two sets of projections of the current, (j ; j' ) and jk ; j? . The connection is ensured by the expressions

and

b e' b e? =

" B0

B = jBj

q h q B0 1+ h

b e? b e =

"2 q2

=

B' 1 =q jBj 1+

" 1 q q 1+

"2 q2

"2 q2

We use the two expressions (j ; j' ) to obtain geometrically jk as jk = j sin where cos and, as derived above

1

=q

1+

j

=

j'

=

Then jk = or jk = jk =

q "

"2 q2

+ j' cos

; sin

=

" 1 q q 1+

1 dB0 h dr q h dp (r) " B0 dr h dp (r) 1 q B0 dr 1+

"2 q2

h dp (r) q 1 q B0 dr " 1+

"2 q2

h dp (r) q 1 q B0 dr " 1+ 11

"2 q2

"2 q2

h dp (r) q 1 q B0 (r) dr " 1+

jk =

"2 q2

That we must use the Grad Shafranov equation, h" d r" B0 B0 = r q dr h q we use B =

" B0 q h

h2

dp (r) dr

and divide by B0 and h "1 d [rB (r)] = q r dr

h dp (r) B0 dr

We remind that the equilibrium equation 0=

rp + j

B

becomes the GS equation after using the Ampere’s law r

B= j

projected on the toroidal (') direction 0j

=r

Bj' =

h' @ b e' (h B ) hr h h' @r

and taking units such that

0

@ 1 @ (hr Br ) = (rB ) @ r @r

= 1, we have

1 d [rB (r)] r dr Replacing in the equilibrium equation we …nd j' =

h dp (r) " = J' B0 dr q and note that actually the Grad Shafranov equation provides us with the explicit form of the toroidal component of the current. We also note that the components of the current obey the zero-divergence condition (charge continuity) r j=0 1 @ @ @ (h h' jr ) + (hr h' j ) + (hr h j' ) = 0 hr h h' @r @ @' Here we must insert jr = 0 and assume axisymmetry @ (h j' ) = 0 @' It results

1 @ (h j ) = 0 rh @

where we use j = 0.

12

2

Poloidal nonuniformity of the pro…les

This is a neoclassical e¤ect. Stringer.

2.1

The variation of density and potential on the surface Stringer 1991

This is pfb3 1991. It is a detailed form of the PRL of 1969. Stringer calculates the correction to the distribution function that is associated with the variation of n and on magnetic surfaces. This variation is a result of toroidality. The input is therefore the drift of the particles. The treatment is drift-kinetic. (0)

fj

= fj

2 r; vk ; v? (1)

2 r; ; vk ; v? + :::

+fj (r; ) =

(0)

(1)

(r) +

(r; ) + :::

The guiding center velocity Vj

=

b vk n

+VD +V(0) + V(1) + ::: where

2 2 1 v? =2 + vk R j

VD = with

j

=

Bz b e' B

B' b ez B

ej B mj

The drift is mainly vertical, b eR

b n

Bz B

1

In any case the vertical magnetic …eld is very small

and the velocities are V(0)

=

V(1)

=

1 d (0) B dr r (1) B

13

b n

B B' " = O (") q 1

=

The diamagnetic velocities v

j

=

v Tj

=

T0j d ln n0 ej B dr 1 dT0j ej B dr

Take the parallel velocity vk =

s

2 ( mj

B

ej )

then dvk dt

= = =

1 (V r) (ej mj vk 1 mj vk

V (0)

+ B)

@ @ @ + vk + Vr r@ @lk @r

ej B @ (1) mj B' r@

2 B v? B' 2

"

(ej

V (0) vk

+ B)

sin r

This comes from dvk dt

= = =

=

@ + (V r) vk = (V r) vk @t s 2 (V r) ( B ej ) mj (V r)

1 q 2

1 2 mj

(

B

ej

1 1 (V r) ( B + ej ) m j vk

The derivation along the magnetic …eld line is @ @lk

=

rk =

=

1 @ qR @ 14

B @ B' r@

2 ( ) mj

B

ej )

The radial drift velocity takes into account the existence of the perturbation of the electric potential Vr

1 @ (1) B r@ 2 2 1 v? =2 + vk sin R j

=

The drift-kinetic equation @fj dvk @fj dv 2 @fj + (Vj r) fj + + 2 ? =0 @t @vk dt @v? dt The drift kinetic equation is linearized to order ". the result is ( (1) 2 v? 1 1 (1) " + vk2 cos fj = (0) B 2 V + vk j ej

2 v? (0) V (0) + vk fj cos 2 vth;j

ej + mj

(1)

2 v? 2

"

(0)

V

(0)

vk

@fj @vk

(0)

@fj @r

) (0)

We note the poloidal velocity, composed of the electric velocity VE and of the poloidal projection of the parallel velocity vk . This combination, representing the poloidal velocity, should be almost zero " (0) V E + vk q

0

This correction to the distribution function contains the e¤ect of the drift of the particles vD . the e¤ect of the presence of a potential constant on the magnetic surfaces (0) . the e¤ect of a variation of the electric potential in the surface, A term (1)

(1)

.

(0)

@fj B @r is the radial advection due to the potential function. 15

(1)

, of the equilibrium distribution

A term

(0)

ej (1) @fj mj @vk

is the acceleration in parallel velocity produced by the electric …eld correction ej (1) after being projected along the parallel direction by B =B' . Therefore these two terms contain the e¤ect of (1) on the distribution func(0) tion (necessarily of zero-order fj since (1) is itself small). (1)

The variation of the electric potential in the surface the condition of neutrality ne = ni

is determined from

We have to obatin the densities by integrating over the velocity space Z Z 2 dvk dv? 2 can be done. The integration over v? The integration over vk is complicated by the singularities of the denominator

1 V (0) +

vk

and this integration must be treated like Landau singularity. The distribution function that is to be integrated is the Maxwell function. Then one introduces the de…nition 1 n0

Z

(0)

1

Fj

1

vk

vks

W

dvk

Ks

W vth;j

These functions are expressed through the plasma dispersion function. The relations are Ks

W vth;j

= W Ks

W vth;j

1

+Js where Js =

(

(s

2) (s

4) :::1 0

vth;j 2

s

1 2

for s odd for s even

The connection with the Plasma Dispersion Function is K0 K1

W vth;j W vth;j

1 W I W vth;j W = I vth;j =

16

1

where I (z)

=

2

1

2z exp z p +i z exp z 2

Z

z

dt exp

t2

0

NOTE that we have here the Principal value and the singularity i after integration, gives the Landau term

which,

(1)

The expression of the distribution function fj will be integrated over the velocity space to obtain the densities. Then neutrality will be invoked, obtaining an equation for the potential (1) . De…nitions V

n;j

=

V

T;j

=

T0j 1 dn0 ej B n0 dr T0j 1 dT0j ej B T0j dr (1)

Stringer …nds that the supplementary velocity Vj induced by the variation of the potential (1) in the surface is a fraction of the diamagnetic velocity (1)

Vj

"V

n;j

The density is (1)

nj n0

=

(1)

ej

Tj "

1

V (0)

V

+" exp (i )

V n;j

T;j

2

(1

Ij )

Ij

V T;j V n;j + (Ij V (0) 2V (0) V n;j +2zj2 Ij 1 + (0) V V T;j +z 2j (0) 1 + 2zj2 Ij V

1+

The new notations are zj Ij

V (0) V 2 vth;j

T;j Ij 2

1)

V (0) vth;j I (zj )

Note that zj =

(0) " q

VE

vth;j

=

V pol (projected on parallel direction) (thermal velocity) 17

#

(1)

(1)

After calculation of ne and ni it is invoked the neutrality. The equation for neutrality becomes an equation for the perturbation to the uniform electric potential on the surface: (1) (r; ).

3

Poloidal rotation Hassam Kulsrud

3.1

Basic equations

The equations nmi

@v + (v r) v @t

=

rp +j

where =

3

0

1 3

bn b n

bn b n

r B

1 3

:rv

p = n (Te + Ti )

The heat equation is expressed in terms of the entropy nT

@s + (v r) s @t

with the ‡ux of heat q= note that this is parallel qk =

=

r q

1 b rT n B k

1 rk T B

The Ohm’s law E+v

B= j

and r r

B = E =

r B =

0j

0 0

The magnetic …eld is (more general than circular) B=

0;

b (r) B0 ; h h

The following averaging operator is introduced R dS jrpj f hf i = R dS jrpj

18

:rv

The equation of continuity @ 1 @ hni + R dS @t @p jrpj

Z

dS nv = 0

The equation for the circulation

=

@ @ 1 hv Bi + R dS @t @p jrpj v

r

j b ez

Z

dS v v B

E h

3 0 B r ln B v r ln B nmi 1 X + hT B rsi mi e;i The equation for toroidal momentum Z 1 @ @ hnmi Rh v b e' i + R dS dS v (nmi Rh v b e' ) = 0 @t @p jrpj

The equation for the entropy assuming that the species is adiabatic Z @s @s 1 dS nv hni + R dS @t @p jrpj * + 2 b rT n = T +

3 0 2 (v r ln B) T

For the averaging operator we have Z 1 dp R dS dS v f = hvr f i dr jrpj

For di¤erent functions f the quantity hvr f i is derived by averaging the toroidal component of the Ohm’s law. Start from E+v B= j with

E b e' h where an external, inductive, electric …eld is considered, E, toroidal. First we multiply by B the Ohm’s law E=

r +

E B = jk jBj 19

and replace r +

E b e' h

B = E

B' h

The magnitude is B0 jBj = h

E=

D

jk Bh0

s

jk jBj

"2 q2

1+

B0 B0 E 2 = jk h h after averaging

=

s

The equation becomes

jk jBj

1+

q 1+

"2 q2

B0 h2

"2 q2 E

the previously derived expression of the parallel current h dp (r) q 1 q B0 (r) dr " 1+

jk =

"2 q2

The numerator of the expression of E is the average of s B0 "2 jk 1+ 2 h q s 1 1 dp B0 "2 q h = 1+ 2 q h q " 1 + "2 B0 dr q2

=

B0

q "

1 dp B0 dr

Then E

= =

D

jk Bh0

1 B0

q

1+

B0 h2

1 1 h2

B0

q "

"2 q2

E 1 dp B0 dr

Consider an arbitrary function f of plasma variables.

20

We take the ' (toroidal) component of the Ohm’s law E + (v B0 )' h E + vr B h

=

j'

=

j'

and multiply by

j'

f B E f = vr f hB

f B

and average over surface j' Now we use

f B

E h

= hvr f i

h dp (r) B0 dr

" j' = q

We return to hvr f i =

j'

=

j'

E h

f B

f B E f hB

and take into account that E is already averaged. hvr f i =

1 q B " 1 E f hB f

The second term is E where

f

h dp (r) B0 dr

1 hB

1 1 1 = b(r) = hB b (r) h h

and is factored out from the averaging. After this we replace the expression of E, 1 1 1 q 1 dp 1 E hf i = B0 hf i 1 b (r) B0 h 2 " B0 dr b (r) and note that

B0 B0 =h B' = = = b b=h B 21

1

=

q "

We have 1 hf i b (r) D E 1 dp 1 q 2 B0 dr hf i 1 B0 " h2 E

=

Working the …rst term we remind that " = q

=

B B'

q 1 q h 1 = = B " B' " B0 (r) then 1 q h dp (r) B " B0 dr q h q h dp (r) = f " B0 (r) " B0 dr q 2 1 dp 2 = f h " B02 dr f

One can factorize from the averaging operator all factors that only depend on (i.e. on the radius r). 1 q h dp (r) B " B0 dr q 2 1 dp f h2 " B02 dr q 2 1 1 dp f h2 " B0 B0 dr

f = =

Finally, the expression of the average is hf vr i q = " 1 + B0

2

q "

1 1 dp f h2 B0 B0 dr D E 2

1 dp B0 dr 1 h2

hf i

or hf vr i =

q "

2

D E 8 9 1 dp = B dr 1 < 1 dp 0 f h2 + hf i 1 ; B0 : B0 dr h2 22

The order of magntitude is dB dr

dp dr

a2 R2

b2

This must be taken as a basis for the averages that will involve a function f. hf vr i =

q "

2

1 B0

"

1 dp B0 dr

f h2 +

hf i 1 h2

!#

The reason to factor out the gradient of the pressure comes from the de…nition of the average R dS v f dp R dS = hf vr i dr jrpj We introduce the notations

1 dp B0 dr

vD

It has similar parametric dependence as the diamagnetic velocity but contains the resistivity . It is the resistive classical ‡ow.Then " !# 1 q 2 hf i 2 hf vr i = vD fh 1 B0 " h2 NOTE the presence of the resistivity as FACTOR to the entire expression, i.e. again we see that vr owes its existence to the resistivity that, in the Ohm’s law, introduces the imperfect neutralization via parallel currents of the charge separation induced by the di¤erent drifts of electrons and ions. (Stringer PRL). END.

3.2

Application to adapted new form of the averaged equations

The equation of continuity @ 1 @ hni + R dS @t @p jrpj 23

Z

dS nv = 0

means R

1 dS jrpj

Z

dS nv =

dp hnvr i dr

@n 1 @ + (rn hvr i) = 0 @t r @r To calculate this averaged over the surface we use the pre…ously derived equation for f =1 and obtain 1 hvr i = vD q 2 "

1

2

2

h

1 h2

!

and introduce the notation 1

1 2"2

1

2

h

1 h2

!

then hvr i = vD 2q 2

1

The equation of continuity becomes @n 1 @ + rn vD 2q 2 @t r @r We note that

1

1

=0

is close to 1.

In a similar way, it is obtained the time evolution of the toroidal component of the ‡ow. De…ne vt hhv' i We must repeat the calculation made for vr . The velocity v' is obtained from the ' projection of the Ohm’s law, i.e. after multiplying it with b e' we take the average. We will need the component j' of the current, already derived. Finally vt

=

hhv' i q vp "

where

vE h2

1 @ B0 r@

vE =

and the poloidal rotation velocity v is expressed through the function vp that only depends on the magnetic surface ( ) v =

vp (r) h

24

The projection of the rotation velocity perpendicular on the magnetic line is v?

= vE q

h "2 q2

1+

1 1 @ q B0 =h r@ 1+

=

=

"2 q2

@ 1 r@ B

As before, together with (v ; v' ) it is possible to work with vk ; v? . The equation for a combination of vp and vt has been derived from the average of the equation for the circulation v B by Hassam and Drake. @ vt + 3 + 2q 2 @t 1 @ + r vD 2q 2 r @r 3 " 0 = vp 4 2 nmi R2 q +

1

" vp q vt +

1

3

+ 2q 2

1

" vp q

2q 2

1

vp "=q

By we note the terms related to thermal di¤usion of the adiabatic species of particles (electrons). The notations are 3

4

1 h2

=

=

2R2

*

0 1 @ 1 q 2 h 1+

"2 q2

12 + @ ln hA r@

An equation for vt results from the momentum conservation projected along the toroidal direction then averaged h @ 1 @ q i hnvt i + rn vD 2q 2 1 vt + vD 2q 2 2 vt vp =0 @t r @r "

where

2

3.3

=

1 2"2

h4

1 hh2 i

h2

Equation for poloidal rotation

The previous calculations allow to write down the equation for the evolution of the poloidaal velocity 1+

1 2q 2

@ ln vp @t

=

25

q2 1 dn vD 2 " n dr 3 0 + 4 nmi q 2 R2

0

The symbol 0 is introduced to represent the e¤ect of the thermal conductivity of the electrons 0

e

Since vt is connected with vp we have the equation for it 1 1 @ q @vt rnq 2 vD vp = @t 2n r @r "

4

Spontaneous poloidal rotation (instability)

4.1

Detailed treatment

The equation of continuity @n + r (nv) @t

=

0

nvk @n +B r + r (nv? ) @t B

=

0

As it is written it shows that we will calculate the parallel gradient of the parallel velocity. The second equation T B rn

=

nmi B v : rv

Br:

nmi

@ (B v) @t

We recognize here the momentum equation nmi

@v + nmi (v r) v = @t

r (nT )

r

considering T constant, multiply by B. This corresponds to the equation for the circulation v B. The equation of Ohm, in the absence of resistivity j=

r +v

B

= 0, multiplied by B is B r =0 the potential is constant on magnetic surfaces. B r

jk B

=

26

r? j?

The perpendicular current is extracted from the equation of momentum. For this, in contrast to previous multiplication by B we multiply vectorially by B j? =

1 B B2

T rn + r

+ nmi

dv dt

This is essentially the diamagnetic current. the equilibrium is de…ned by the functions that are ‡ux-functions n (r) Vp (r)

=

Vt

=

hv hi

hv' hi

The average over the ‡ux surface is hf i =

Z

d hf 2

the equilibrium state means @ @t R?

0 =

0 pressure is isotropic

=

0 no friction

the equations under this equilibrium assumption lead to v = v'

Vt

Vp (r) h

2qVp cos

+" Vt cos + 2qVp 1 +

1 cos 2 4

The …rst equation says that the poloidal rotation is the rotation uniform on surface Vp modulated by h = 1 + " cos Then hnRv' i = nR0 Vt vk

B B0

= Vt +

" 1 + 2q 2 Vp q

The equations @n 1 @ + (rnv r ) = 0 @t r @r 27

@ 1 @ [nVt ] + (r n [Vt v r @t r @r

=

qVp ver ]) = 0

@ " Vt + 1 + 1q 2 Vp @t q @Vt @ +v r ver [qVp ] @r @r + (magnetic pumping) 0

The velocity is associated to a ‡ux of transport of particles, across the surfaces (in radial direction). The ‡ux is generated by the collisional friction that acts perpendicular to the magnetic …eld line nvr =

R? jej B

Then the average and the variable part v r = hvr i and ver = h2 cos vr i

The origin of the poloidal spin-up : the existence of the variation of the ‡ux of transport with poloidal angle Equivalently, ver 6= 0 The equation

" @Vp 1 + 2q 2 + q @t @ +qVp (r ne vr ) @r = 0

M P Vp

the logic of the instability that consists of poloidal spin-up Assume there is a poloidal velocity. Due to the toroidality and r v=0 the poloidal rotation (with compression - distension of volume alternatively in low-…eld and high-…eld sides) necessarly is accompanied by toroidal ‡ows that ensure the preservation of the incompressibility. the toroidal ‡ows have a spatial distribution which is harmonic in the poloidal section. It is P…rsch Schluter ‡ow and current. 28

The friction R? is modulated in the surface by these ‡ows. The friction generates transport ‡uxes r which are themselves modulated in the surface but for reasons that are independent of the P…rsch-Schluter harmonic ‡ows. The radial velocity they induce is also modulated, it is r

vr

= nvr =

D (1 + cos ) r v0 a

From the combination between the two independent poloidal modulations f( )

r

Pfrirsch-Schluter ‡ow

g( )

it is induced a variation of the radial velocity ver = h2 cos vr i

This combination acts like a drive (a torque) in the equation for the poloidal velocity Vp (function of surface ). The higher the angular matching between the poloidal variation of transport rate r ( ) with the harmonic P…rsch Schluter ‡ow cos , the higher the drive of poloidal rotation. @ If the poloidal rotation is enhanced by this drive qVp @r (r ne vr ) then the amplitude of the harmonic compensatory P…rsch Schluter ‡ows increases then the poloidal drive is still higher.

5

Spontaneous spin-up (Hassam Drake)

The equations. The continuity nuk @n + r (nu? ) + B r @t B

=S

1 @ (r r @r

r)

This equation is important for the derivation of the expression for the P…rsch Schluter current. This is because it introduces the divergence of the ‡ux of particles, of the ‡ow. This is where the geometrical poloidal compression and dilation will enter the dynamics. In the term r [b e (1 + " cos )]. The momentum for all plasma (the mass is taken mi ), isothermal nmi

@u + (u r) u = @t 29

T rn + j

B

mi Su

The current conservation is essential in connecting the perpendicular current (diamagnetic) with the parallel current (P…rsch Schluter) r j=0 The Ohm’s law. Here, without the resistivity. This means that the radial velocity vr will be attributed to another reason for which the charge neutrality cannot be fully suppressed by the parallel currents. The reason may be the Landau damping which acts when the collisionality is low. This appears in Stringer where a kinetic treatment allows to calculate the variation of the density and of the potential on the magnetic surface, by integrating over the velocity space the distribution functions of electron and ions and imposing neutrality. During the integration, one has to traverse the singularity vk q" v E = 0. r +u

B=0

The magnetic …eld is B=r

r' + I ( ) r'

Relative to the work Hassam Kulsrud here it is assumed that the electrons and ions are isothermal. S (r; )

particle source

It is interesting to note how the source extracts from the momentum a part which is proportional with mi u through S. The radial ‡ux r

= =

hhe nver ii

D (r; )

@n @r

An object of study is the circulation. This is obtained taking the projection in the parallel direction of the equation of momentum conservation. It is interesting that the variation of the density in the parallel direction (for isothermal plasma) gives the pressure that opposes to the geometrical advection of the ‡ow, B (u r) u, which can be static. The @ imbalance gives @t (u B). @ (u B) + B (u r) u = @t

c2s B r ln n (parallel pressure) S u B n

(external source of momentum)

The poloidal component of the equation for the plasma momentum Bpol

nmi

du + T rn dt

=

B' j r R

30

(radial current)

We note that Bpol = r

r'

and the product j r extracts the radial current. Now comes the constraint that will provide the third equation: the total radial current traversing a magnetic surface must be zero. Integrated over a magnetic surface the zero-divergence of the current density leads to Z Z ds (j r ) = 0 ds j = jr j f lux_surf where =

ds

= then

Z

f lux_surf

ds b er

2 Rrd b er

ds 2 R Bpol jr j

nmi

du + T rn dt

=0

This equation, derived from current conservation r j=0 will be used to derive the time variation of the poloidal velocity. It is assumed …rst that the plasma velocity is smaller than the sound velocity. The equilibrium, in zero order @n0 =0 @t the equilibrium density is constant in time u?0 rn0 = 0 The perpendicular advection of the equilibrium density is zero: the density does not vary along perpendicular direction. 0 = T B0 rn0 The equilibrium density does not vary parallel with the magnetic …eld line Z 1 @n0 d =0 r @ If there is a poloidal variation of the density along the poloidal direction, the periodicity must be taken into account. 0=

r

0

+ u0 B0

31

The Ohm’s law without resistivity. This means that the lowest order density is constant on the surfaces n0 (r) and the velocity which is perpendicular on the magnetic …eld is contained in the magnetic surface. It is the electric velocity u?0

This velocity VE is poloidal.

= VE b e 1d 0 b e = B dr

the …rst order in " will reveal the presence of a perturbation of the density on the magnetic surface, n1 . Also we will have to work with the parallel velocity uk . @n1 @n1 + VE + n0 VE @t r@ +n0 rk uk 1 @ = S (r r ) r @r

2"

sin r

We see here that the poloidal rotation velocity VE is very active: it carries the perturbation of the density on the surface VE

@n1 r@

However we know that the poloidal velocity VE is actually the diamagnetic velocity, nvDia = 1= (m ) dp=dr. And this is equal with r , and from here we can introduce the symbol VE . To understand the third term we should remember r [b e (1 + " cos )]

1 @ ((R0 + r cos ) (1 + " cos )) r (R0 + r cos ) @ ( 2 sin ) = " r =

The factor h = 1 + " cos comes from the magnitude of the magnetic …eld B = B0 =h. The divergence is calculated for the poloidal ‡ow resulting from the electric velocity VE that carries the density n0 + n1 . Both quantities do not have variation in this order but the geometry is essential. Termenul rk uk The parallel momentum n0 mi

@uk @uk + VE @t r@ 32

=

T rk n 1

Note that it is here that the parallel viscosity the magnetic damping. Shaing, etc. End. The parallel gradient is 1 @ rk = qR @ Finally the condition

Z

d B

0

should appear to introduce

@n1 =0 r@

Since B 0 is actually constant on the surface and is taken out the integral the condition is trivially satis…ed in this order. The condition satis…ed trivially at the …rst order must be recalculated in higher order, i.e. two, "2 . The equation to be used is

or, the integral form

Z

r j

=

ds j

=0

0

f lux_surf

Z

f lux_surf

ds 2 R Bpol jr j

nmi

du + T rn dt

=0

derived from the condition of zero-divergence of the current. The part Z ds 2 R Bpol (T rn) jr j f lux_surf will be calculated as R2 ds jr j Bpol rn

1 order 1 order 1

An approximation Bpol rn

B

@n1 r@

and jr j = 2 RB In the product Bpol

nmi

du dt

Bpol

nmi

@u @t

we only retain

33

since Bpol (u r) u is of higher order. This term will provide the time variation of the poloidal velocity VE (r; t). We also have ds = 2 Rrd The integration of the …rst part is Z ds 2 du R Bpol nmi jr j dt f lux_surf Z @VE 2 Rrd B nmi 2 RB @t f lux_surf =

(2 ) rnmi

@VE @t

the integration of the second term Z ds 2 R Bpol (T rn) jr j f lux_surf Z ds 2 = T R Bpol r [n0 (r) + n1 (r; )] f lux_surf jr j Z ds 2 = T R Bpol rn1 (r; ) f lux_surf jr j we make an integration by parts and take into account the periodicity Z ds 2 2 []0 T n1 r R Bpol jr j f lux_surf Z 2 Rrd 2 = T n1 r R B b e 2 RB f lux_surf

we must …nd Z

f lux_surf

=

T

Z

ds 2 R Bpol (T rn) jr j

f lux_surf

= = Then

Tr Tr

Z

Z

Z

n1 r

ds 2 R Bpol jr j

(integration by parts)

n1 d r (hb e ) d

"

f lux_surf

2 sin r

n1

ds 2 R Bpol jr j

34

nmi

du + T rn dt

=0

(2 ) rnmi

@VE + @t

Tr

Z

d

"

2 sin r

n1

(2 ) rnmi

New notation

@VE @t @VE @t

=

0

=

2T "

Z

d sin n1 Z 1 c2s d " sin n1 r n0 2

=

n1 n0

N the equations

@N @N + VE + VE @t r@ +rk uk S 1 1 @ = (r r ) n0 n0 r @r @uk @uk + VE = @t r@ Z @VE d = c2s N @t 2

2"

sin r

c2s rk N 2"

sin r

NOTE Let us stop to make a comparison between this (Hassam Drake) system prepared for the spontaneous spin-up and the Stringer PRL system. We note that the time variation in the equation for the density, @n1 =@t, and the velocity (0) @vik

nmi VE

r@

=

(Te + Ti )

" @n1 q r@

this equation shows the balance of momentum carried by the "static advected" velocity (i.e. space variation of the velocity, (v r) v ) with the pressure. The projection is made on poloidal direction. is absent at Stringer. Since Hassam Drake work for spin-up driven by external source (poloidally asymmetric) the explicit time variation must be retained. @u The main term however in the formulation Hassam Drake is still VE r@ k @u e

(later u bE r@ k ) which is the same as in Stringer. This term will be the main part of the expansion around the equilibrium static state.

35

The equilibrium static state at Hassam Drake is = Fe

rk u ek e rk N

=

0

and the expansion introduces new, small, quantities b u bE ; u bk ; N

with the system

sin + rk u bk r @e uk @b uk +u bE @t r@ @b uE @t

2"b uE

=

0 b c2s rk N Z 2"c2s d b N sin r 2

= =

See the explanations below. END The functions that must be determined

N (r; ; t) ; uk (r; ; t) ; VE (r; t) The global balance is obtained by integrating over the surface @N @t @uk @t

=

S n0

=

0

1 @ n0 r@

r

R

d 2

(:::).

r

After introducing the average over surfaces, the new variables are the di¤erences that have variations in the surfaces fe = f

The source in the surface is F

S

f

1 @ r @r

(r

r)

n0

The state that is taken as reference is the absence of the poloidal rotation VEref = 0

36

and this reduces the equations to N ref e ref i.e. N

= N =

0 (no variation of the density in the surface) = Fe rk u eref k

e ref = 0 from where N e =0 rk N

The variation of the parallel velocity along the magnetic line (equivalently, in the magnetic surface) is obtained in terms of the source rk u eref k

= =

Fe

u eref k

=

qR

1 @ ref u e qR @ k

Fe

Z

d

0

Fe

Consider a perturbation of this reference state VE uk N

= VEref + VbE

= u eref +u bk k

e ref + N b = N +N

This will induce a time variation of the poloidal (electric) velocity and of the density N and of the parallel velocity. However the time variation is assumed to be slower than the sound speed @ @t

cs qR

The time variation for N is neglected and the equation for density becomes a balance sin 2" VbE + rk u bk = 0 r ref

@e uk @b uk b + VbE = c2s rk N @t r@ Z @ VbE d b sin = c2s N 2" @t 2 r

Note the preservation of the poloidal derivative of the reference parallel velocity in the second equation. This reference value of the paralel velocity is …xed by the radial ‡ux and the source of particles. It exists only because these sources and ‡uxes are NOT constant on the poloidal circumference.

37

This set of equations can be integrated. The operator that must be made explicit is rk =

1 @ qR @

Then, since VbE is constant on magnetic surfaces, the …rst equation is VbE

2"

sin r

+ rk u bk

1 @ u bk qR @ u bk

This is introduced in the second equation

=

0 or

sin = VbE 2" r cos b VE = 2qR" r ref

@ @t

2qR"

@e uk cos b VE + VbE r r@

b c2s rk N

=

c2s

=

and is integrated b c2s N

=

Z 1 @ VbE 2 (qR) " r @t qR ref u e +VbE r k 2

b 1 @N qR @

d 0 cos

0

we ignore for the moment the constant of integration which should be a function of surface. This is introduced in the equation for the time variation of VbE , the third equation Z @ VbE d b sin 2 = cs N 2" @t 2 r ( ) Z b Z sin qR d 2 1 @ VE ref 0 0 2" 2 (qR) " d cos + VbE u e = 2 r r @t r k Z b d 2 2 @ VE 1 = 4 (qR) " sin sin @t r2 2 Z qR d +2" 2 VbE sin u eref k r 2 The …rst term 2 2

4 (qR) "

@ VbE @t

!

1 r2

Z

d sin sin = 2 38

r2 4q R 2 R 2

2

@ VbE @t

!

1 1 = r2 2

2q

2

@ VbE @t

!

and the second 2"

1 + 2q 2

qR b VE r2

r R1b q VE R r r 2q b = VE r Z 2q b @ VbE d = VE sin u eref k @t r 2 =

2

In this equation we replace the reference state for the parallel velocity, which is …xed by the source and the ‡ux, both these contributions being retained with their variation along the poloidal direction 1 + 2q

2

@ VbE = VbE @t

1 2 1 2q "2 n0

Z

d S cos 2

1 1 @ n0 r @r

r

Z

d 2

r

cos

we can easily recognize that an integration by parts have been made in the right hand side. NOTE How is generated this inertia factor 1 + 2q 2 . We have seen that the …rst integration u bk =

2qR"

cos b VE r

actually obtains the P…rsch Schluter parallel current, with a poloidal ‡ow given by VbE . This PS ‡ow has a coe¢ cient q, as usual. b 1 @N The second equation has the RHS term c2s qR and this introduces the @ second q factor. They now multiply the term @b uk =@t. The enhancement of the radial di¤ usion with a factor q 2 , the known characteristics of the P…rsch Schluter "regime" has the same origin. We note however that it is not yet clear what means PS regime. END

6

Numerical study of the Stringer rotation

We have developed a numerical framework that incorporates the e¤ect of poloidal variation of the rate of particle and energy losses. The equation for poloidal rotation under the Stringer e¤ect is solved and the rate of rotation is compared with the transit time magnetic pumping damping.

39

7

Notes

The paper of Stringer 1969 calculates the radial ‡ux of particles taking into account the nonuniformity of the particle density n1 ( ) and of electric potential 1 ( ) on the magnetic surface. These result from the neoclassical drifts and the equations of continuity, the poloidal projected equation of momentum, in the presence of an equilibrium radial electric …eld represented by a potential 0 (r). In the regime of low collisions instead of collisional resistivity that permits to use the Ohm law for the parallel current projected on poloidal direction, it is invoked the kinetic process of Landau damping. The result is p 2 vth;i 1 v0 S (S + ) dn0 " i 1+ exp zi2 1 + 2 nvDi = 8 r q Uin F + L2 dr where S

1 + + 2zi2 1 + zj

This is

Uen v0

v0 q vth;j "

1 q 1 BT = vth;j " vth;j B vth;j

B = vth;j BT

the projection of the thermal (assumed parallel) velocity on the poloidal direction. And b r 0 n v0 b e = B v0 zj = vth;j Later it is found that v0 = Uni

1+

1+ 1 + 2zi2 + 2zi4

me mi

1=2

exp

zi2

Then the diamagnetic and the electric rotations are almost equal in magnitude and opposite. When vdia 1+ 0 vE the neoclassical di¤usion vanishes.

40