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Solutions to the drift kinetic equation Florin Spineanu and Madalina Vlad National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
F. Spineanu M. Vlad – Work Sessions 3 –
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Work Sessions of Plasma Theory Abstract of the Third meeting We have done up to now a general and rather simplified excursion within the space of the neoclassical basis of transport in toroidal plasma. We have seen how the magnetic geometry imposes a set of equations of motion for the charged particles, then we have derived the drift kinetic equation; as a first departure from the kinetic description we examined basic flows (diamagnetic, Pfirsch Schluter) in the equilibrium state. We would have wanted to do more but we need to include viscosity such as to connect the friction forces to the fluxes. For this, we need to solve the drift-kinetic equation. Then we do this in two steps. First we show how the drift kinetic equation is solved for equilibrium flows (no instability yet). (Third meeting)
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Second, we will discuss more carefully the collision operators.(This will be the fourth meeting) These two parts will be of smaller extension but will prepare the discussion of the rotation of the plasma, so important for when we will turn to instabilities and turbulence.
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A short review The reason is the different way they handle the diamagnetic flow. • Geometry and particle orbits • Flows in the neoclassical equilibrium of a toroidal plasma • The neoclassical ”inertia” for plasma rotation the drift-kinetic equation and some solutions in the neoclassical equilibrium
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1
The orbit of a charged particle in toroidal geometry
The equations. For a particle in toroidal geometry, there are symmetries and then there are invariants 1 = mv2 + μB + eφ 2 2 mv⊥ μ= 2B r J = mv (1 + ε cos θ) − e Bθ hdr 0
and the fastest variable ξ : gyration angle
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NOTE that the longitudinal invariant will be found also as R2 ∇ϕ · (mv + eA) = const mvϕ R = const e 2 v dϕ = dϕ − μB − eφ m toroidal angle ψ−
J
=
ϕ
≡
From the change of variables in the Valsov equation, and after taking the gyroaverage, 1 e vD = n× μ∇B + v2 (n · ∇) n + φ Ω m
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which is written vD
v
=
×∇ −v n
=
1 v⊥ /2 + v it is almost vertical Ω R0
2
Ω 2
The equations dr dt rdθ dt
μB0 m
+ v2
=
1 − Ωc
=
1 1 dΦ0 Θv + − BT dr Ωc
R
sin θ μB0 m
+ v2 R
cos θ
where Θ = Bθ /Bϕ 1. Now we need equations for v
2 v⊥ and 2
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We must return to v
= = =
2 −2 e φ 2E − v⊥ m e 1/2 2E − 2μB − 2 φ m √ e 1/2 2 E − μB − φ m
and
2 v⊥ = μB 2 Now we take the time derivatives along the orbits
dv ∂ = + v · ∇ v dt ∂t
where for the ions (almost the motion of the plasma) v + vE + vD v=n
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vD Then
−∇φ× n vE = B 1 × μ∇B + v2 ( n = n · ∇) n Ωci
v + vE + vD · ∇v v · ∇v = n and after calculations 2
dv e −∇φ × n v⊥ =− ∇ ln B + v · ∇ ln B − ∇ φ dt 2 B m
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2 The other equation regards v⊥ /2. 2
2
d v⊥ ∂ v⊥ = + (v · ∇) dt 2 ∂t 2
v + vE + vD · ∇ (μB) = n 2
v⊥ 1 ∇ B + = v 2 B 2
v⊥ 1 −∇φ × n · ∇B + 2 B B 2
2
2
d v⊥ v⊥ 1 v⊥ 1 −∇φ × n = v ∇ B + · ∇B dt 2 2 B 2 B B
In the case of taking into account the variation in the surface of the electric potential φ = φ0 (r) + φ1 (r, θ)
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the equations for the velocities are 2 2
2
d v⊥ Bθ sin θ v⊥ v⊥ 1 dφ0 sin θ = v + dt 2 2 BT R 2 B0 dr R 2
dv e Bθ ∂φ1 1 dφ0 sin θ v⊥ Bθ sin θ =− + v − dt 2 BT R B0 dr R m BT r∂θ
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2
2.1
The basic flows at equilibrium in tokamak The diamagnetic flow
the diamagnetic flow and current in toroidal confined plasma. nvdia =
1 × ∇p n mΩc
This comes from
∂v + (v · ∇) v = −∇p + enE + env × B + R nm ∂t and take stationarity dv/dt = 0, no friction no E . Multiply what remains vectorially by B and obtain the diamagnetic flux nvdia .
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2.2
the Pfirsch Schluter flow
The parallel current arising from the non-zero divergence of the diamagnetic current ∇·j
=
0
∇⊥ · j⊥ + ∇ · j
=
0
Now taking the perpendicular current as resulting from the diamagnetic flows of electrons and ions, the parallel gradient can be
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written as ∇ · j
= = =
1 ∂ j qR ∂θ
1 × ∇p n −∇⊥ · j⊥ = −∇⊥ · e mΩ
1 × ∇p n −∇⊥ · e |e| B
Let us look to the last term. It is the perpendicular divergence of the diamagnetic flow. Note that the operator of parallel derivative is ∇ ∼
1 ∂ qR ∂θ
and that the perpendicular current j⊥ is the diamagnetic current, of ions + electrons. End.
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This is a neoclassical effect. It is the magnetic field that has a space variation in the perpendicular direction. First we have dp × ∇p = ( n× er ) n dr dp = − eθ 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. We have the magnitude of the magnetic field B0 1 + ε cos θ and we must calculate the perpendicular divergence of the B=
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perpendicular current, which means 1 ∇· − eθ B
dp dr
and this is approximated by (B0 is constant)
B0 = ∇· [ eθ (1 + ε cos θ)] ∇· eθ B Here is the essential part of the calculation: there is a divergence of the diamagnetic ”flow” that is exclusively due to the geometry. This has consequences in the balance of flows. Here it is explained how this divergence is calculated. In the orthogonal coordinates (r, θ, ϕ) we have the element of distance: dl2 = (dr)2 + r 2 (dθ)2 + (R0 + r cos θ)2 dϕ2
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which gives the coefficients h1
=
1
h2
=
r
h3
=
R0 + r cos θ
Then the divergence of a vector a is written
∂ ∂ 1 ∂ (h2 h3 a1 ) + (h1 h3 a2 ) + (h1 h2 a3 ) ∇·a= h1 h2 h3 ∂r ∂θ ∂ϕ which gives ∇· [ eθ (1 + ε cos θ)]
= = =
∂ 1 ((R0 + r cos θ) (1 + ε cos θ)) r (R0 + r cos θ) ∂θ ∂ 1 2 R0 (1 + ε cos θ) r (R0 + r cos θ) ∂θ (−2 sin θ) ε r F. Spineanu M. Vlad – Work Sessions 3 –
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because ε2 1. From this result we get −∇⊥ · j⊥
= = = = = =
−∇⊥ (dia) =
1 × ∇p −∇⊥ · e n mΩ
dp 1 (− eθ ) −∇⊥ e mΩ dr
B0 1 dp eθ ∇⊥ · B B0 dr (−2 sin θ) 1 dp ε r B0 dr r dp ∂ (2 cos θ) RB0 dr r∂θ
or, since both terms in the equations of zero-divergence of the electric
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current contain the same derivation to θ, j
3
=
1 dp cos θ −2q B0 dr Pfirsch Schluter current
The neoclassical inertia for the poloidal fluid flow
The derivation. The equation of momentum balance
∂v + (v · ∇) v = nmi ∂t
−∇p − ∇ · Π +j × B
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The Ohm’s law E + v × B = ηj and ∇×B =
μ0 j
∇×E =
0
∇·B =
0 E E = −∇φ + eϕ h (external electric field, inductive). Here φ is quasi-static, it is requested by the variation over the surface ψ, φ (r, θ). Not yet instabilities. The magnetic field is
B=
b (r) B0 , 0, h h
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B0 |B| = h
ε2 1+ 2 q
The following averaging operator is introduced dS |∇p| f f = dS |∇p|
Then
ηj Bh0 1 + B0 E=
ε2 q2
h2
The parallel current
1 q 1 dp j = h − ε 1 + ε2 B0 dr 2 q
is the parallel projection of the diamagnetic current.
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Next we take the ϕ (toroidal) component E + (v × B0 )ϕ h E + vr Bθ h
=
ηjϕ
=
ηjϕ
As lonf as exists a toroidal current jϕ and a resistivity η = 0 there is also a radial velocity vr . Task: find the surface average of the radial velocity, multiplied by an arbitrary f , - but useful in future, function
E f ηjϕ − vr f = h Bθ f E f − = η jϕ Bθ h Bθ
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It results q 2 1 f vr = η ε B0 and for f ≡ 1,
1 dp B0 dr
2 f − fh + 1
h2
2 1 2 1 vr = vD q 2 h −1 ε h2
where vD ≡ −η
1 dp B0 dr
Again vr cannot exist but for η = 0 and comes from the diamagnetic current (which has a nonzero projection along the parallel direction). Now let see the velocities.
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Toroidal
2 q vt ≡ hvϕ = vp − vE h ε
Poloidal
vp (r) vθ = h where we have expected that vθ depends on (r, θ) and introduced another ”poloidal velocity”, vp (r) which only depends on r (the surface). the momentum equation 2 ∂ 1 1 ε 2 1 + 2q vt + vp − h 1 2 2 ∂t h 2ε q h2
2 1 ∂ 2 vp 1 1 ε 2 1 + 2q − v r vD vt + + 2q α1 − h p 1 r ∂r h2 2ε2 q ε/q h2 = F. Spineanu M. Vlad – Work Sessions 3 –
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shows that there is a nonlinear part (the second) (v · ∇) v which means momentum balance in quasi-static situation. The right hand side contains the radial velocity interacting with the toroidal magnetic field, giving a mechanical drive at a slow rate.
1 q2 1 dn ∂ ln vp 1+ 2 = − 2 vD 2q ∂t ε n dr 3 η0 − + Ξ 4 nmi q 2 R2 and we notice a factor 1 + 2q 2 which acts like a mass for the time variation of the poloidal velocity. It results purely from geometry. It shows that a poloidal flow is a pumping. F. Spineanu M. Vlad – Work Sessions 3 –
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4
4.1
The drift-kinetic equation for the neoclassical equilibrium Typical case Rosenbluth Hazeltine Hinton
The distribution function results from the equation ∂f + v · ∇f ∂t
2 d v⊥ /2 dv ∂f ∂f + + 2 /2) dt ∂v dt ∂ (v⊥ =
St (f )
The basic content of the first order drift-kinetic equation is v ∇ f (1) + vD · ∇f (0) = 0
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The short expression for this would be parallel advection of the perturbation is balanced by the radial advection of the equilibrium this is because v · ∇f
→
+ vdrif t · ∇f v n
and the difference between the two velocities is very large v
∼
106 (m/s)
vdrif t
∼
30 (m/2)
But the parallel velocity acts on functions that are almost invariant along the magnetic field (parallel) while the drift velocity exploits any radial variation of the distribution function. Then the first will act F. Spineanu M. Vlad – Work Sessions 3 –
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upon the perturbation f (1) and the second on the equilibrium fM . The distribution fnction is expanded f = f0 1 + f ρ f ∼ O l n − |e| φ exp − f0 = 3/2 T (π 2T /m) the drift-kinetic equation is ∂f0 Bθ ∂ f ∂f0 vDr + v f0 + |e| E v =0 ∂r B r∂θ ∂ since ∂ = |e| E v ∂t
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dr dt rdθ dt
=
vDr
=
vθ + vDθ
Replacing the Maxwellian fM
vDr [A1 + A2 ( − |e| φ)] + v
|e| E Bθ ∂ f − B r∂θ T
= C (f )
where A1
=
A2
=
3 T |e| φ n − + n 2T T T T2
To solve this equation, the first step consists of extracting a Spitzer function from f. This means to consider the definition of a function F. Spineanu M. Vlad – Work Sessions 3 –
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fs ,
|e| − E v fM = C E v fs T
Then we may replace the last term in the equation for f above, |e| E −fM v T by
C E v fs
On the other hand the collision operator is linearized and then the arguments can be combined. We have
C (f ) − C E v fs
= C f − E v fs
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and the equation is fM
Bθ ∂ f vDr [A1 + A2 ( − |e| φ)] + v B r∂θ
= C f − E v fs
Now it occurs the possibility to exploit the separation of time scales: collisions νef f are rare relatively to the bounce frequency ωB . This should be reflected in an expansion f = f(0) + f(1) + ... where the first order allows to neglect collisions. Bθ ∂ f(0) = −vDr [A1 + A2 ( − |e| φ)] fM v B r∂θ and we use vDr
1 ∂ hv = v Ωc r∂θ F. Spineanu M. Vlad – Work Sessions 3 –
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then 1 Bθ ∂ f(0) ∂ = − v hv [A1 + A2 ( − |e| φ)] fM v B r∂θ Ωc r∂θ with Bθ /B = b (r) /B0 independent of θ, as it is Ωc . Then f(0)
=
m hv [A1 + A2 ( − |e| φ)] fM |e| b +g (μ, , σ)
The method of solving this equation goes through the extremization of the rate of generation of entropy. For this a particular operator, dependent on four distribution function, is prepared.
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This will serve for representation of the collisional operator. The later is the source of generation of entropy. The functions are
fa
=
fM a 1 + fa
ga
=
fM a (1 + ga )
a ≡
species, e, i
and with them we define K (f, g) = −
!
d3 v fa Ca (ga )
a
with the definition Ca (g)
=
Caa ( ga , ga ) +Cab ( ga , gb )
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K is the local rate of entropy production. ∂S = K (f, f ) ∂t hdθ ! ∂S =− d3 v fa − v E fsa Ca f − v E fs ∂t 2π a The operator of collisions is of Fokker-Planck form Cab (f ) = − d3 vb dΩ σab (Ω) × |va − vb | × [fa (va ) fb (vb ) − fa (va ) fb (vb )] wher va , vb
≡
velocities before collisions
va , vb
≡
velocities after collisions
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σab ≡ cross section To linearize the collision operator, one uses the linear expansions of f and g.
4.2
Solutions with impurities (Hirschman Sigmar)
The equation for the species a is
=
ea ∂B ∂φ ∂fa ∂fa (A) + vd,a · ∇fa + μ v E + + v n + ∂t ∂t ma ∂t ∂ ! Cab (fa , fb ) b
where
ea 1 2 φ = v + 2 ma
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with ea = Za |e| R0 (A) E eϕ = Eϕ0 R the electric field produced by induction by the tokamak transformer. The drift velocity
vd,a
=
×∇ = −v n
v Ωca
v2 1 (∇ × B)⊥ + Ωca B and, since the drift velocity will multiply the spatial gradient of the zero-order distribution function, vd,a · ∇fa0 and since fa0 = fM a has only variation in a direction perpendicular
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on the magnetic surface ∇fM a = ∇ψ we will have to use
∂fM a ∂ψ
vd,a · ∇ψ
= =
v ·∇ I v n Ωca
v v ∇ I Ωca
Ma (in this form the term vd,a · ∇f which actually is vd,a · ∇ψ ∂f∂ψ can · ∇fa1 ). be coupled to v n
The expansion fa = fa0 + fa1 + ...
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fa0 = fM a = na (ψ)
1 a (ψ) π 2Tm a
na (ψ) = na0 exp
3/2
exp − T
a (ψ)
ma
ea φ Ta0 (ψ)
The equation for the first order
v ∂fa0 v ∇ fa1 + v ∇ I Ωca ∂ψ
=
ea E (A) fa0 −v Ta ! Cab (fa1 , fb1 ) b
Note that here we have taken ∂fa0 /∂ψ inside the paranthesis that comes from the drift velocity. Then it is convenient to separate from fa1 the part resulting from the drift motion advection of the F. Spineanu M. Vlad – Work Sessions 3 –
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equilibrium fa0 distribution function. fa1
v ∂fa0 −I Ωca ∂ψ +ga (, μ, ψ)
=
where (Hirshman Sigmar Clarke) v ∂fa0 −I Ωca ∂ψ ga
≡
diamagnetic response of the species a
≡
collisional response of the species a
Consider the surface average A (x) =
2π ∂V ∂ψ
∂V = 2π ∂ψ
A dχ ∇χ · B
dχ
1 ∇χ · B
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Returning to the drift-kinetic equation, we have
! v ∂fa0 ea E (A) fa0 + Cab (fa1 , fb1 ) = v v ∇ fa1 + I Ωca ∂ψ Ta b
We divide by v and note that the surface average is applied to 1 ∇ () dχ B · ∇χ 1 d () = dθ 1 Bθ r dl but B B dl = dlθ = rdθ Bθ Bθ Then d Bθ d = dl B rdθ
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and
1 d () = dθ 1 dl Bθ r =
1 Bθ d () dθ 1 B rdθ Bθ r 1 d dθ () B dθ
The magnitude of B is B≈
B0 1 = B0 h 1 + ε cos θ
and B is a function of θ. But it is of order ε and multiples a quantity
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which is also of order ε. Then ∇ Bθ d B rdθ d Bθ rdθ
v ∂fa0 fa1 + I Ωca ∂ψ v ∂fa0 fa1 + I Ωca ∂ψ v ∂fa0 fa1 + I Ωca ∂ψ
1 B
can be inserted in the paranthesis (A) ! 1 ea E v = fa0 + Cab (fa1 , fb1 ) v Ta b (A) ! 1 ea E v = fa0 + Cab (fa1 , fb1 ) v Ta b (A) ! B ea E v = fa0 + Cab (fa1 , fb1 ) v Ta b
Now we take the surface average #
" (A) ! v ∂fa0 d ea E B v fa1 + I = fa0 + Cab (fa1 , fb1 ) Bθ rdθ Ωca ∂ψ v Ta b
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In the left hand side we replace (...)
2π V
=
v ∂fa0 d Bθ fa1 + I = rdθ Ωca ∂ψ =
2π V 0
dθ
1 1 (...) Bθ r
d 1 dθ 1 Bθ rdθ Bθ r
v ∂fa0 fa1 + I Ωca ∂ψ
due to periodicity. This is what results " # (A) ! B ea E v fa0 + Cab (fa1 , fb1 ) =0 v Ta b
The expression of the collision operator.
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HSC write Cab (fa1 , fb1 )
=
v ∂ ∂ μv fa1 B ∂μ ∂μ def l slowing v ua1 (v) + νab − νab fa0 v2 2v slowing + 2 rba νab fa0 vth,a def l νab
The definitions The deflection frequency def l νab
Φ = νab
v
−G 3
vth,b
v
vth,b
v
vth,a
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The slowing down frequency slowing νab
Ta0 =2 Tb0
mb 1+ ma
νab
G
v vth,b v
vth,a
The frequency of collisions νab
4π e2a e2b nb0 = 3/2 √ ln Λ 3/2 ma 2 Ta
The function G (x)
where
=
Φ (x) − x dΦ(x) dx 2x2 the Chandrasekhar function
2 Φ (x) = √ π
0
x
2 dt exp −t
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In the formulas above it has been introduced 1 3 ua1 (v) = v fa1 dΩ fa0 4π dΩ ≡ solid angle in velocity space ! Bdμ 1 dΩ = π v /v σ=±1
the momentum restoring coefficient 3 slowing fb1 d v mb νba $ % rab ≡ slowing ma na0 νab The following operator is introduced
2 v fa0 Fab (v) {Fab (v)} ≡ 2 d3 v vth,a na0 F. Spineanu M. Vlad – Work Sessions 3 –
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To solve the drift-kinetic equation 2 1/2 B λ≡μ φ(ψ) − eam a and choose results fa1
=
fa1 = fa1 fa0
2 1/2 B ∂ 1 −Iv ln fa0 ea B 2 1/2 B ∂ψ ma 1 ∂fa0 +I Θ V ea B 2 1/2 ∂ψ m ⎞⎤ ⎡ a ⎛ (A)
slow slow ! ea E 2V ν 1 ν 1 a ab + 1 − def l ua1 (v) + r ab def l ⎠⎦ +Θ ⎣ 2 ⎝ def l 2 vth,a ma νa 2xa νa νab b
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The terms that contain the Heaviside function Θ only exist for circulating particles. The notations are
2 1/2 B ≡h B v xa = vth,a λc v2 V (λ) = dλ 2 v λ 2 1/2 B λc = Bc Bc (ψ) = maximum of B in a surface ψ
and νadef l
=
!
def l νab
b
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νaslow
=
!
slow νab
b
a new avareging
A A≡ h
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