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Drift Waves
Basic theory of the drift waves in magnetized plasmas Florin Spineanu and Madalina Vlad Association EURATOM MEdC National Institute of Laser, Plasma and Radiation Physics Bucharest, Romania
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Drift Waves
Two basic elements: magnetic confinement and a gradient of density
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Generalities on drift waves
Waves sustained by the gradient of the density and propagating in the poloidal direction. The frequency is comparable to ω∗ an the poloidal wavenumber is small compared to ρ−1 s (large perpendicular wavelengths compared to ρs ). There are two branches of drift waves in the tokamak plasma: 1. Slab-like branch, which is the toroidal version of the classical sheared slab model of Pearlstein and Berk. It is characterized by rapid variation of the wave field along the magnetic line (not too small k ). The eigenmodes are not bound modes and the energy is radiated in the radial direction to the point of the ion
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Drift Waves
Landau absorbtion. 2. The toroidicity induced branch which results from the inclusion of the ion magnetic drifts ∇B and curvature. It has a bound state with slow variation along the magnetic line (slow variation on the connection length scale qR). The toroidicity induced modes do not experience shear damping. It is destabilized by any inverse electron dissipation mechanism (i.e. the electrons feed the wave).
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Drift Waves
Figure 1: schematic drift wave (Horton 1999
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Drift Waves
Figure 2: Phase shift between potential and density
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Drift Waves
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The electron distribution function
Recommend the paper Numerical solution drift kinetic eq. Santarius Hinton. ”Drift waves require the presence of two components in the plasma: 1. an adiabatic species which can come to equilibrium with the wave in a time short compared with a wave period, and 2. a hydrodynamic species which E × B drifts in the electrostatic field of the wave. ” In the dissipative trapped electron mode, • all ions are hydrodynamic • electrons are: – partly adiabatic and
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Drift Waves
– there is a component which is non-adiabatic In the absence of dissipation electrons and ions oscillate in phase under the influence of the E × B drift. When the particles are driven out-of-phase by dissipative mechanisms such as collisions or wave-particle resonances, a growing or damped mode can occur. For electrons. The drift kinetic equation is obtained from dfe dt ∂fe dx ∂fe dqα ∂fe + · + ∂t ∂x dt ∂qα dt
=
C (fe )
=
C (fe )
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Drift Waves
where qα is a generalized phase-space variable.
=
∂fe + v · ∇fe ∂t ∂fe dμ ∂fe dε ∂fe dα + + + ∂μ dt ∂ε dt ∂α dt C (fe )
where μ
≡
ε = α
≡
2 v⊥ 2B v2 |e| − Φ (for electrons) 2 me gyrophase angle
For 1. electrostatics wave instabilities, and
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2. after gyrophase averaging, one obtains |e| ∂Φ ∂f e ∂f e + v + vD + vE · ∇f e − = C (fe ) ∂t me ∂t ∂ε The bar means gyrophase averaged. (v · B) B v = B2
vDe
E×B vE = B2 2 1 v⊥ 1 × ∇B + v2 n =− Ωce 2 B Ωce =
|e| B me
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Drift Waves
The equilibrium distribution function is Maxwellian 2 n v fM e = √ exp − 3 2 vth,e ( πvth,e ) vth,e =
2Te me
The first order distribution function is a correction to the adiabatic component |e| Φ fM e + fe1 fe = Te The variables adopted from now on 2 v r, t, μ, 2
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Drift Waves
The linearized equation is
=
∂fe1 + v + vD · ∇fe1 ∂t |e| ∂Φ C (fe1 ) − fM e Te ∂t −vE · ∇fM e
The last term is the drift of the Maxwellian (equilibrium) distribution function by the electric velocity. −∇Φ × n · ∇fM e −vE · ∇fM e = − B 1 ∂Φ ∂fM e = B r∂θ ∂r where the variation of the potential corresponds to a wave propagating in the poloidal variation. The potential acts here by its variation along the poloidal θ direction.
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Drift Waves
Now it is assumed that all terms have time variation as exp (−iωt) and the equation becomes −iωfe1 + v + vD · ∇fe1 =
C (fe1 )
2 Te ∂Φ d ln n v 3 |e| iωΦ + 1 + ηe fM e − + 2 Te |e| B r∂θ dr vth,e 2
Two observations can be helpful. First, the toroidal variation (on the toroidal angle ϕ) must be considered also periodic and expanded as exp (−ilϕ) Second the correction function fe1 is linear in the potential Φ. Then F. Spineanu M. Vlad – Bucharest 2015 –
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Drift Waves
both can be expanded on the poloidal angle in a way that is compatible with this linear relationship Φ (r, θ, ϕ) =
∞
a m exp (imθ − ilϕ)
m=−∞
and the f1e is expanded as fe1 =
∞
fm am
m=−∞
NOTE this expansion must be retained. In the Fourier expansion of the potential Φ the coefficients am are not necessarly small, their amplitude is not ordered according to the indice m. Then the second expansion, or the distribution function fe1 in terms that each contains an amplitude am of Fourier components of the potential does not imply that it is an expansion in order of magnitude. END
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The drift kinetic equation becomes −iωfm + v + vD · ∇fm =
C (fm ) +i (ω −
ωT∗ e )
where
|e| fM e exp (imθ − ilϕ) Te
∗ ωT∗ e = ωem 1 + ηe
∗ ωem
= = =
3 v2 2 − 2 vth
Te d ln n −ky |e| B dr m Te d ln n − r |e| B dr m Te 1 − r |e| B Ln
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and ηe ≡
Ln d ln Te = d ln n LT
The collision operator is Lorentz, necessary for taking into account the pitch angle scattering, with no energy change. 1 ∂ 2 ∂f 1−ξ C (f ) = νei (v) 2 ∂ξ ∂ξ where
v ξ≡ v √ 3 3 π 1 vth,e νei (v) = 4 τe v 3 τe
= =
3 3 m2e vth,e √ 4 16 π e ni ln Λ Braginskii momentum transfer collision time
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Drift Waves
The magnetic field B =
r B0 eθ qR 1 + (r/R) cos θ B0 eϕ + 1 + (r/R) cos θ
Approximating
ε 1 q2
one has |B|
≈
B0 h (θ)
=
B0 1 + (r/R) cos θ
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the guiding center velocity is dr dt
v n
=
−
me |e| B
2 v⊥
2
+ v2
1 × ∇B n B
Adopting as coordines (r, θ, ϕ) the velocirty components 2
rdθ dt
=
Rdϕ dt
=
dr dt
=
2
1 v⊥ /2 + v r v + cos θ qR Ωce R 2 2 r 1 v⊥ /2 + v v − cos θ Rq Ωce R 2 2 1 v⊥ /2 + v sin θ Ωce R F. Spineanu M. Vlad – Bucharest 2015 –
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wher Ωce =
|e| B me
Now it is extracted by factorization from the unknown functions fm a periodic space dependence fm (r, θ) = fm exp [il (ϕ − qθ)] We express the equation in terms of fm (r, θ), fm = fm (r, θ) exp [−il (ϕ − qθ)] and in the derivatives that we must calculate in the drift-kinetic
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equation we act also on the exponential factor dθ ∂ dθ ∂fm = fm (r, θ) exp [−il (ϕ − qθ)] dt ∂θ dt ∂θ 2 2 1 r 1 v⊥ /2 + v = v + cos θ r qR Ωce R
∂ fm exp [−il (ϕ − qθ)] × ∂θ ∂ +fm exp [−il (ϕ − qθ)] ∂θ
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or dθ ∂fm dt ∂θ
=
1 r exp [−il (ϕ − qθ)] v + qR Ωce 1 ∂ fm + ilq fm × r ∂θ
2 v⊥ /2
+
R
v2
cos θ
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Drift Waves
Next dϕ ∂fm dt ∂ϕ
= =
dϕ ∂ fm (r, θ) exp [−il (ϕ − qθ)] dt ∂ϕ 2 2 1 r 1 v⊥ /2 + v v − cos θ R Rq Ωce R
∂ fm exp [−il (ϕ − qθ)] × ∂ϕ ∂ +fm (r, θ) exp [−il (ϕ − qθ)] ∂ϕ
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or dϕ ∂fm dt ∂ϕ
=
r 1 exp [−il (ϕ − qθ)] v − Rq Ωce 1 ∂ fm − ilfm × R ∂ϕ
2 v⊥ /2
+
R
v2
cos θ
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Drift Waves
Finally the last term contains the derivation to the radial coordinate dr ∂ dr ∂fm = fm (r, θ) exp [−il (ϕ − qθ)] dt ∂r dt ∂r 2 2 1 v⊥ /2 + v sin θ = Ωce R
∂ fm (r, θ) exp [−il (ϕ − qθ)] × ∂r ∂ +fm (r, θ) exp (−ilϕ) exp [ilqθ] ∂r The last term is dq ∂ exp (ilqθ) = ilθ exp (ilqθ) ∂r dr
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Drift Waves
and the result is dr ∂fm dt ∂r
=
exp [−il (ϕ − qθ)] ×
1 Ωce
2 v⊥ /2
+
R
v2
sin θ
∂ fm (r, θ) + ilθq fm (r, θ) ∂r
We express the term produced by ∇B and curvature in the equations of motion of a particle by the variables ξ and v, v⊥ , v → (v, ξ)
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Drift Waves
as 2 v⊥ /2 + v2
= = =
1 1 2 v⊥ + 2v2 = v 2 + v2 2 2 v2 v2 1+ 2 2 v v2 2 1+ξ 2 v = vξ
Now we collect the terms.
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Drift Waves
The term with
∂ ∂θ .
dθ ∂fm exp [il (ϕ − qθ)] dt ∂θ 2 2 1 v⊥ /2 + v 1 ∂ fm r v + cos θ + ilq fm qR Ωce R r ∂θ v2 ilq rvξ ∂ fm 2 + 1 + ξ cos θ + fm qR 2Ωce R r∂θ r
=
=
The term with
∂ ∂ϕ .
exp [il (ϕ − qθ)] =
vξ −
2
rv 2R2 qΩce
dϕ ∂fm dt ∂ϕ
il ∂ fm 2 − fm 1 + ξ cos θ R∂ϕ R
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the term with
=
∂ ∂r .
dr ∂fm exp [il (ϕ − qθ)] dt ∂r 2 ∂ fm (r, θ) v 2 1 + ξ sin θ + ilθq fm (r, θ) 2Ωce R ∂r
Collecting the terms that will contribute to the drift of the first order distribution function we take separately those that multiply fm (r, θ), 2 v rvξ ilq + 1 + ξ 2 cos θ fm qR 2Ωce R r 2 rv il 2 fm + vξ − 1 + ξ cos θ − 2 2R qΩce R v2 + 1 + ξ 2 sin θ ilθq fm 2Ωce R
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Drift Waves
We note that the first terms in the first and second paranthesis can be combined rvξ ilq il ilvξ ilvξ + vξ − = − =0 qR r R R R Then the other terms lead to
=
v2 ilq 2 1 + ξ cos θ 2Ωce R r rv 2 il − 2 1 + ξ 2 cos θ − 2R qΩce R v2 2 1 + ξ sin θilθq + 2Ωce R 2 2 ε v 1 + ξ 2 q cos θ + cos θ + q rθ sin θ il 2Ωce Rr q
which multiplies fm . F. Spineanu M. Vlad – Bucharest 2015 –
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Drift Waves
The drift term becomes
= =
exp [il (ϕ − qθ)] v + vD · ∇fm dϕ ∂fm dr ∂fm dθ ∂fm + + exp [il (ϕ − qθ)] dt ∂θ dt ∂ϕ dt ∂r 2 1 v ∂ fm r 2 1 ξv + 1+ξ cos θ qR Ωce 2 R r∂θ 2 2 ε dq il 1 v 2 1 1+ξ q cos θ + cos θ + rθ sin θ fm + r Ωce 2 R q dr ∂ fm 1 v2 2 1 1+ξ sin θ + Ωce 2 R ∂r
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Drift Waves
The equation is 2 v cos θ ∂ fm rvξ 2 + 1+ξ −iω fm + qR 2Ωce R r∂θ ilv 2 2 (q cos θ + rq θ sin θ) 1 + ξ fm + 2Ωce Rr v 2 sin θ 2 ∂ fm 1+ξ + 2Ωce R ∂r + (new term) νei ∂ 2 ∂ fm 1−ξ − 2 ∂ξ ∂ξ |e| i (ω − ωT∗ e ) fM e exp [i (m − lq) θ] Te
=
The “new term” is a term of energy and comes from the variation of the perturbed distribution function in the space of the velocity. It
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Drift Waves
actually originates from the first steps of derivation of the drift kinetic equation, before the averaging over the gyration motion. It comes from the term dμ d f dt dμ which will take into account the variation of the distribution function with μ or v⊥ when the magnetic momentum has a time variation. The magnetic momentum can have time variation during the motion of the particle along the magnetic line. This is described by the equations of motion of the particle dx/dt. The term is vε sin θ 1 − ξ 2 ∂ fm − qRh (θ) 2 ∂ξ
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Drift Waves
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Integral equation for the drift wave eigenmodes (Tang Rewoldt Frieman)
The distribution function obeys the Vlasov equation. Two distribution functions are mentioned: F and f . It is cited Tang NF 1978. The perturbed distribution function is f and the equation for it is
=
∂f ∂t +v · ∇f × v⊥ ∂F ∂F n 1 ∂F e + v⊥ + − ∇Φ · v 2 m ∂E B ∂μ v⊥ ∂ζ ∂f −Ω ∂ζ 0
(1)
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Drift Waves
where E
≡
μ
≡
ζ
≡
v2 (kinetic energy per unit mass) 2 2 v⊥ (magnetic moment on unit mass) 2mB gyro-phase ez + By (x) ey B =Bz k k⊥ By Bz
=
O (ε)
=
O (ε)
(1) (0) + n n By ey = ez + B
= n
(2)
(3) (4)
(5)
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Drift Waves
We NOTE that this form of the drift-kinetic equation results from the change of variable (see neoclassic) (x, v,t) → (x, ε, μ, ζ, t) and leads to
× v⊥ ∂ ∂ ∂ n ∂ =v + v⊥ + 2 ∂v ∂ε B∂μ v⊥ ∂ζ
END. The distribution function at equilibrium F contains the The distribution function F is composed of the Maxwellian part plus the correction F = FM + F (1) (6) and it has been shown that F (1) =
∂FM v⊥ sin ζ Ω ∂ (εx)
(7)
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Drift Waves
Note that we can recognize here ρ · ∇FM
(8)
the first order correction to the Maxwellian, due to the Finite Larmor Radius (FLR). The spatial derivation is made over distances comparable with ρ. This is the reason of taking the space in the derivation (εx) which is a small scale, while x is large. End. Expansion in ε. To lowest order we have f (0)
(9)
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Drift Waves
with the equation
=
v⊥ · ∇(0) f (0) e ∂FM − ∇(0) Φ(0) · v⊥ m ∂E ∂f (0) −Ω ∂ζ 0
(10)
The notations (0) ∇⊥ (1)
∇
=
∂ ∂x
(11)
∂ = ex ∂ (εx)
and ez ∇z = ∇(1) z =
∂ ∂z
(12)
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Drift Waves
We NOTE the presence of the term e (0) (0) ∂FM − ∇ Φ · v⊥ m ∂E which can be transformed writting dΦ ∂Φ − v · ∇Φ = dt ∂t and further dΦ dt ∂Φ ∂t
→
adiabatic term,
→
−iωΦ
eΦ T
END. It is time to separate in f (0) the adiabatic part which comes from the
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Drift Waves
balance between the two terms in the 0-th order equation. f
(0)
e (0) ∂FM + h(0) = Φ m ∂E (0) (0)
v⊥ · ∇
h
∂ (0) −Ω h =0 ∂ζ
(13) (14)
Now h(0) is expanded in Fourier series: in y and z it is periodic. But we also expand in x with the Fourier variable kx , ∞ 1 h(0) = dkx h (E, μ, ζ, kx ) (15) 1/2 −∞ (2π) × exp (ikx x + iky y + ikz z − iωt) Note that this is rather unusual.End. Then the equation in 0-th order for the non-adiabatic part of the
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distribution function now reads ∂ ikx v⊥ cos ζ + iky v⊥ sin ζ − Ω h ∂ζ × exp [i (kx x + ky y)] =
(16)
0
The solution of this equation is obtained after assuming separation of the variable gyro-angle ζ from the others (E, μ, kx ), h(0) = h(0) (E, μ, kx ) g (ζ)
(17)
and the operator acts only on the the function g: iky v⊥ 1 ∂g ikx v⊥ cos ζ + sin ζ = Ω Ω g ∂ζ iv⊥ (kx cos ζ − ky sin ζ) g = exp Ω
(18) (19)
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Returning to h(0) we have (0)
h
1
=
1/2
∞
h(0) (E, μ, kx ) dkx
(20)
−∞ (2π) × exp [i (kx x + ky y + kz z − ωt + L)]
L≡
v⊥ (kx cos ζ − ky sin ζ) Ω
(21)
To determine the unknown factor h(0) (E, μ, kx ) we have to go to higher order in the expansion in ε and average the equation over the
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gyro-phase. ∂f (0) ∂t (0) · ∇(1) f (0) + v n (1) · ∇(0) f (0) +v n
=
(22)
+v⊥ · ∇(1) f (0) + v⊥ · ∇(0) f (1) (1) (1) (0) (1) × v⊥ ∂F n e (0) (0) ∂F 1 ∂F + v⊥ + − ∇ Φ · v 2 m ∂E B ∂μ v⊥ ∂ζ e ∂FM (1) (0) (1) (0) (0) (1) (0) ·∇ +n ·∇ − Φ ∇ Φ · v⊥ + v n m ∂E ∂f (1) −Ω ∂ζ 0
We NOTE the presence of the perturbation of the direction of the
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magnetic field →n (0) + n (1) n which is due to the magnetic shear By =
x B0 Ls
Alternatively, it effectively means that electromagnetic effects are taken into consideration. We NOTE that the gradient operator is perturbed too, which means that there are at least two spatial scales ∇ → ∇(0) + ∇(1)
As has been explained above, the 0-th order distribution function f (0) can be separated in
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• adiabatic part e (0) ∂FM Φ m ∂E
(23)
• and the non-adiabatic part, h(0) . This form of f (0) is introduced in the equation above, and this
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equation becomes an equation for the nonadiabaticpart −iωh(0)
(24)
e ∂Φ(0) ∂FM + m ∂t ∂E (0) · ∇(1) h(0) + v n (1) · ∇(0) h(0) +v n +v⊥ · ∇(1) h(0) + v⊥ · ∇(0) f (1) e ∂FM + Φ(0) v⊥ · ∇(1) m ∂E (1) (1) (0) (1) × v⊥ ∂F e n ∂F 1 ∂F − ∇(0) Φ(0) · v + v⊥ + 2 m ∂E B ∂μ v⊥ ∂ζ
=
∂f (1) −Ω ∂ζ 0
Now we expand the perturbation Φ(0) potential in Fourier in all
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dimensions, therefore inclusiv kx . ∞ 1 (kx ) exp (ikx x + iky y + ikz z − iωt) φ dk Φ(0) (x, y, z, t) = x 1/2 −∞ (2π) (25) The dependence of Φ(0) (x, y, z, t) on (y, z) and on time (t) is assumed periodic. This means that the Fourier amplitude φ (kx ) remains to be only dependent on the Fourier variable on x, i.e. kx . A similar procedure is applied to the function f (1) with the difference that we extract as a separate factor from the Fourier conjugate an exponential of the function L. This is connected with the structure of the Fourier expansion obtained before for the (0)-th order ∞ 1 (0) h dk (E, μ, kx ) h(0) = x 1/2 −∞ (2π) × exp [i (kx x + ky y + kz z − ωt + L)]
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Then for the first order f (1)
=
1 1/2
∞
dkx f(1) (E, μ, ζ, kx )
(26)
−∞ (2π) × exp (ikx x + iky y + ikz z − iωt)
× exp (iL) The Fourier representations of the functions h(0)
(27)
Φ(0) f (1) are introduced in the equation and it is taken the average over the
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gyrophase −iω h(0)
=
(28)
By (0) +ikz v h(0) + iky v h B eφ(0) k⊥ v⊥ FM J0 +iω T Ω ∂ 2 FM e (0) exp (−iL) v⊥ cos ζ + φ m ∂E∂ (εx) (1) (1) (1) 1 1 ∂F ∂F ∂F − + sin ζ −ikx v⊥ cos ζ ∂E v⊥ ∂v⊥ v⊥ ∂ζ (1) (1) (1) 1 ∂F 1 ∂F ∂F + + cos ζ +iky v⊥ sin ζ ∂E v⊥ ∂v⊥ v⊥ ∂v⊥ 0
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with the notations k⊥ ≡
2 1/2 ky
kx2
1 ... = 2π
+ 2π
dζ (...)
(29) (30)
0
The average over the gyrophase of the expression L is 2π v ⊥ (kx cos ζ − ky sin ζ) dζ exp Ω 0 k⊥ v⊥ = 2πJ0 Ω where
k⊥ = kx2 + ky2
We are going to use the explicit form of the first correction to the Maxwellian distribution function, which is due to the Finite Larmor
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Radius F (1) =
∂FM v⊥ sin ζ Ω ∂ (εx)
The last term in the equation, containing the average over the gyrophase, can be simplified ∂ 2 FM e (0) exp (−iL) v⊥ cos ζ φ m ∂E∂ (εx)
(31)
(32)
2 ∂ 2 FM v⊥ cos ζ sin ζ −ikx Ω ∂E∂ (εx) 2 2 1 ∂FM ∂ FM v⊥ sin2 (ζ) + −iky Ω ∂E∂ (εx) Ω ∂ (εx)
This equation can be written, taking separately the last term in the last paranthesis since it is not multiplied with trigonometricfunctions
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of the gyro-phase ζ and leaving the other separately =
e (0) iky ∂FM − φ exp (−iL) (33) m Ω ∂ (εx) 2 2 v⊥ e (0) ∂ FM exp (−iL) v⊥ cos ζ − i (kx cos ζ + ky sin ζ) sin ζ + φ m ∂E∂ (εx) Ω
Further, we note that the average of the factor exp (−iL) gives a Bessel function. k⊥ v⊥ e (0) iky ∂FM J0 (34) = − φ m Ω ∂ (εx) Ω 2 e (0) ∂ FM ∂ v⊥ v⊥ (ky cos ζ − kx sin ζ) + φ sin ζ exp i m ∂E∂ (εx) ∂ζ Ω Since the average over the gyrophase of the second part is trivially
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zero, the final form of this term is =−
e (0) iky ∂FM φ J0 m Ω ∂ (εx)
k⊥ v⊥ Ω
(35)
We NOTE the nature of this term: the factors e iky 1 − φ(0) = −iky φ(0) m Ω B = velocity E × B in the radial (x) direction, in the wave field and this advects the background distribution function. The Bessel function represents the effect of the finite Larmor radius. This is the first factor Bessel function and is due to the gyro-phase average exp (−iL) Later there will be another factor Bessel function, from exp (+iL)
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END With this result we return to the equation for h(0) and obtain T (0) e φ v k ω − ω ⊥ ⊥ ∗ FM J0 h(0) = T Ω ω − k v
(36)
with the notations k
≡
ω∗T
≡
By kz + ky B 3 E − ω∗ 1 + η T /m 2 Ln d ln T = d ln n LT T 1 ω∗e ≡ −ky |e| B Ln 1 dn L−1 ≡ − n n dx η≡
(37)
(38) (39) (40)
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We NOTE that the non-adiabatic part of the perturbed distribution function exists due to a 1. drive ω − ω∗T showing that the perturbation only can exist when the wave does not follow the diamagnetic frequency 2. a propagator 1 ω − k v representing the ”integration along the particle trajectory ” operator, applied on the potential of the wave φ (x, t). END Up to here we have determined the Fourier transform of the non-adiabatic part of the potantial h(0) . Now we return to real space F. Spineanu M. Vlad – Bucharest 2015 –
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Drift Waves
and perform the inverse Fourier transform T e ω − ω ∗ exp (iky y + ikz z − iωt) h(0) = FM (41) T ω − k v ∞ 1 k⊥ v⊥ (0) φ exp (ikx x + iL) dk (k ) J × x x 0 1/2 Ω −∞ (2π) We note that at this moment we have NOT taken the average over the gyrophase in the equation for h(0) . The non-adiabatic function h(0) is introduced in the expression of
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f (0) and the gyrophase average is taken = f (0)
eΦ(0) FM (42) − T T e ω − ω∗ exp (iky y + ikz z − iωt) + FM T ω − k v ∞ v 1 k ⊥ ⊥ dkx φ(0) (kx ) J02 exp (ikx x) ×√ Ω 2π −∞
We NOTE the generation of the second factor Bessel function, coming from the average over the gyro-phase of the factor exp (iL) that is now a part of the particular form of the Fourier expansion assumed for the functions h(0) and f (1) . END. We have By x = B Ls
(43)
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It is approximated kz = 0 from which k = ky
x Ls
(44) (45)
After taking kz = 0 the Fourier phases remain to be composed of exp (iky y − iωt)
(46)
if the integration over kx is carried out. This integration is formally done and the functions are formally replaced with functions depending on x (47) Φ(0) = φ (x) exp (iky y − iωt) (0)
nj
=n j (x) exp (iky y − iωt)
(48)
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The species are j
≡
i for ions
j
≡
e for electrons
The velocity-space integrations of the result obtained above f
(49)
(0)
!
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Drift Waves
58
leads to n i
=
|e| n φ (x) (50) − Ti ∞ |e| n 1 1 √ − Z (ξ ) dkx exp (ikx x) i |x| Ti 2π −∞ vth,i ky Ls 3 × ω − ω∗i 1 − ηi Γ0 − ω∗i ηi [Γ0 + b (Γ1 − Γ0 )] 2 ∞ # 1 ω∗i ηi " 2 dkx exp (ikx x) Γ0 ξi + ξi Z (ξi ) √ − |x| 2π −∞ vth,i ky Ls ∞ 1 dkx exp (−ikx x) φ (x) ×√ 2π −∞
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and for the electrons n e
=
|e| n φ (x) Te +
+
|e| n φ (x) Te ω∗e ηe
(51)
|x| vth,e ky L s
ω − ω∗e |x| vth,e ky L s
Z (ξe )
1 Z (ξe ) − ξe − ξe2 Z (ξe ) 2
The notations are vth,i
=
vth,e
=
2Ti mi 2Te me
1/2 (52) 1/2
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Drift Waves
ξi
ξe
=
=
ω |x| vth,i ky L s
ω |x| vth,e ky L s
ω k (x)
= =
vth,i ω k (x)
vth,e
= =
parallel phase-velocity ion thermal velocity
(53)
parallel phase-velocity electron thermal velocity
Γ0
=
I0 exp (−b)
Γ1
=
I1 exp (−b)
1 2 2 k ρ b = 2 ⊥ i 2 1 2 2 = kx + ky ρi 2 vth,i ρi = Ωi
(54)
(55)
(56)
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k⊥ ρe =
3.1
k⊥ vth,e
negligible Ωe
(57)
Method of solution of the integral equation for the drift wave eigenfunction
The equation is quasi-neutrality n e − n i 0
(58)
The method of solution is the Ritz method, using a basis of functions for the perturbed potential and reducing the integral equation to an algebraic system. ∞ φn hn (x) (59) φ (x) = n=0
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Drift Waves
hn (x) ≡ $ π σ
1
1/2 n 2 n!
Hn
√
1 2 2 σky x exp − σky x 2
(60)
where Hn is the Hermite polynomial of order n. Use of the Ritz ’s technique means simply to 1. insert this expansion of φ in series of basis functions hn (x) with coefficients φn . 2. multiply with one element of the basis hn (x)
(61)
3. integrate over x 4. multiply (just for constants) with Te ky |e| n
(62)
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It results
∞
Lnn (ω) φn = 0
(63)
n=0
The coefficients of this system of algebraic equations are
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The particular advantage of the Hermite functions is that the Fourier transform can be expressed by the same Hermite functions. The following relationship exists ∞ 1 (−i)n kx (64) dx exp (−ikx x) hn (x) = 1/2 hn 2 1/2 σk σ k y −∞ (2π) y
Important remark The three-dimensional character and the toroidicity are different things. The toroidicity usually means ion drifts and the necessity of representing quantities in the ballooning representation. This involves in the equation of the mode m the modes m + 1 and m − 1 but the coupling is linear (it is caused by the inclusion of the ion drifts, which contains the trigonometric functions cos and sin in the
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expression of ωD , where also the shear parameter appears: s). It is not the cause of the energy spectral cascade. (Only the nonlinearitry can do that.) The three-dimensional character of the dynamics of a particular mode means that the divergence of the transversal electrostatic fluctuating flow due to the E × B velocity is non-zero and then it must be completed with the consideration of the parallel component of the flow. And possibly the electron parallel dynamics is taken into account, in particular the electron response is taken adiabatic (as in Hasegawa-Mima eq.). And, possibly, the Landau damping is an active damping mechanism for the ion energy. The equation for the wave amplitude of the electron drift mode (EDM): d2 φ (x) EDM + Q (x) φ (x) = 0 2 dx F. Spineanu M. Vlad – Bucharest 2015 –
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Drift Waves
where QEDM (x)
=
−1 +
≡
a + x2 b
1 + 1/τ ω Γ0 ω − ω∗i
+ x2 −
1 + 1/τ ω Γ0 ω − ω∗i
ky vthi Ls ω
The eigenvalue is obtained from the condition √ a = −i b (2n + 1) and the corresponding eigenmodes are expressed in terms of Hermite polynomials √ 1 φ (x) = Hn iσ x exp − iσx2 2 with 1 σ= Ls ρi
ω 1 + 1/τ Γ0 ω − ω∗i
1/2
ky vthi . ω
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The following notation has been introduced ρ2i
=
ρ2i
d ln Γ0 db
This solution connects smoothly in the asymptotic region to the %x eikonal solution, φ (x) ∼ exp i kx dx . For large x the outgoing wave which was oscillatory for smaller x becomes spatially damped by ion Landau damping. The form of the solution also permits to estimate the radial extension of the eigenmode, ω xT
−1 . ky vthi Ln sometimes called ion-turning point, where the phase velocity of the electrostatic perturbation, propagating along the z direction, reaches a value which is equal to the thermal velocity of the ions. Then there is absorbtion of the electrostatic perturbation energy by the ions and
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this puts a limit to the expansion of the wave in the radial (x) direction.
4
Conclusions
Drift waves are universal. They are the basic instability in a confined system. They generate transport. We do not want transport.
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