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Delta-f Gyrokinetic Equations
(via consistent derivation)
B. Scott
Max Planck Institut für Plasmaphysik
Euratom Association
D-85748 Garching, Germany
GOTiT Gyrokinetics Course, Nov 2008
Basic Ideas Covered Last Time
• required properties to satisfy phase space conservation
• particle and field energy work together in conservation
• consistency guaranteed if approximations done in Lagrangian
• geometry may be approximated if phase space rules are maintained
• traditional semi-linearised models do have consistent formulations
Outline of Delta-f Equations
• ordering on both geometry and amplitudes
— flute mode ordering as well as small fluctuation ordering
— main motivation for “fluxtube” models:
geometric separation of parallel/perp dynamics
• systematic linearisation around a background
— equilibrium, linear order, quadratic perp nonlinearities
• energy conservation
— identify conserved forms quadratic in all dependent variables
— fluctuation activity rather than formal energy
— key point: same transfer terms as total-f model
• brief discussion of energetics, turbulent cascade
— gradient drive source, dissipative sink due to collisions
foundation of delta-f model within its total-f origin
Low-frequency delta-f Gyrokinetic Formulation
• wide spectrum turbulence, usual ordering (drift scale ρs , profile scale L⊥ ):
e
ρs
dφ ∼
≡δ≪1
Te
L⊥
k⊥ ρi ∼ 1
ρ2s
c2 Te Mi
2 Te 2
ρ
= 2 2 =Z
e B
Ti i
actually allows for (e/Te )dφ ∼ 1 at k⊥ ρs ∼ δ, anticipates arbitrary k⊥ ρs
• assume df ≪ F M (background Maxwellian), linearise parallel acceleration
• keep ExB advection ({dφ, }) and “magnetic flutter” ({dAk , }) as sole nonlinearities
• keep toroidal drifts and trapping, add collisions
• parameters can be of order unity, except for δ ≪ 1
Definition of Brackets, Pieces
• gyrokinetic equation, Hamiltonian form
Bk∗
∂f
+ {H, f }
∂t
≡
∗ ∂f
Bk
cF
+ ∇H ·
· ∇f + B∗ ·
∂t
eB
∂f
∂H
∇f −
∇H
∂pz
∂pz
• total bracket {, } made up of pieces [, ]ab between pairs of coordinates Z a , Z b
• define bracket pieces,
[f, g]ab
∂f ∂g
∂g ∂f
≡
−
∂Z a ∂Z b
∂Z a ∂Z b
=0
Derivation Path
origins in total-f particles/fields Lagrangian
• particle equations of motion
Bk∗ Ṙ
cF
∂H ∗
= ∇H ·
B
+
eB
∂pz
Bk∗ ṗz = −B∗ · ∇H
• total-f equation has Hamiltonian form
Bk∗
∂f
+ {H, f }
∂t
cF
∗ ∂f
≡ Bk
+ ∇H ·
· ∇f + B∗ ·
∂t
eB
∂f
∂H
∇f −
∇H
∂pz
∂pz
• phase space grid {x z w} covers values of particle coordinates {R pz µ}
• tensor form used for background magnetic field
F=ǫ·B
cF
B =B−∇·z
eB
∗
Bk∗
1
= B · B∗
B
=0
Derivation Path
expansion of brackets, establishment of equilibrium
• total-f equation expressed as a Poisson bracket
∂f
+ {H, f } = 0
∂t
• order by order
{H, f }0 = {H0 , f0 }
{H, f }1 = {H0 , f1 } + {H1 , f0 }
• equilibrium statement: f0 = F M (H0 )
∂F M
+ {H0 , F M } = 0
∂t
{H0 , H0 }
hence
vanishes
∂F M
∂F M
{H0 , H0 } = 0
+
∂t
∂H0
=⇒
∂F M
=0
∂t
• hence F M is a proper equilibrium of the system at zeroth order
Derivation Path
linearisation of brackets, restoration of nonlinear advection comes later
• linear order
{H, f }1 = {H0 , f1 } + {H1 , f0 }
• expand it:
f =F
M
(H0 ) + dg
z2
H0 =
+ wB
2m
H = H0 + H1 (dφ, dAk )
hence
{H, f }1 = {H0 , dg} + {H1 , F M }
• finish on next page . . .
Derivation Path
linear bracket equation
• we have
∂dg
+ {H, f }1 = 0
∂t
which becomes
{H, f }1 = {H0 , dg} + {H1 , F M }
∂F M
{H1 , H0 }
{H, f }1 = {H0 , dg} +
∂H0
∂F M
{H0 , H1 } ≡ {H0 , dh}
{H, f }1 = {H0 , dg} −
∂H0
where
FM
dh ≡ dg + H1
T
• the linearised equation is
assuming
∂dg
+ {H0 , dh} = 0
∂t
FM
∂F M
=−
∂H0
T
Derivation Path
linearisation of the grid and time derivative
• velocity space grid is on z-space; let U be the grid for vk
e
z = mU + J0 Ak
c
g(z, w) = f (U, w)
• derivatives follow from . . .
which linearises to
∂f e
dg = df −
d(J0 Ak )
∂U mc
∂F M e
dg = df −
d(J0 Ak )
∂U mc
• same form as for dh manipulation:
∂F M
∂F M
= mU
∂U
∂H0
→
−
mU M
F
T
hence
dg = df + F M
eU
(J0 dAk )
T c
Derivation Path
linearisation of the time derivative, linearised gyrokinetic equation
• the time derivative is left, linearised using the Maxwellian
∂g
∂f
∂f e ∂Ak
=
−
J0
∂t
∂t
∂U c
∂t
becomes
∂dAk
∂dg
∂df
M e U
=
+F
J0
∂t
∂t
T c
∂t
• now H1 = eJ0 dφ − (U/c)eJ0 dAk , so that
FM
M e
H1 = df + F
(J0 dφ)
dh = dg +
T
T
• the linearised equation is
∂dg
+ {H0 , dh} = 0
∂t
with these definitions
dg = df + F
M
eU
(J0 dAk )
T c
dh = df + F
M
e
(J0 dφ)
T
Derivation Path
toroidal magnetic drifts: curvature terms
• curvature terms are the perpendicular drifts brackets with inhomogeneous B
Bk∗ {H0 , dh} = . . . + ∇(µB) ·
cF
cF
· ∇(dh) − ∇ · mU
eB
eB
· [H0 , dh]U x
• linearisation: pz → mU in second term
• model geometry:
F
F
= −∇ log B ·
∇·
B
B
hence since ∇B · F · ∇B vanishes,
∂H0
∂dh
F · [H0 , dh]U x =
F · ∇dh −
F · ∇H0
∂U
∂U
• so we have for the curvature terms
Bk∗ {H0 , dh}
→
mU F · ∇dh
cF
· ∇(dh)
= . . . + (µB + mU )∇(log B) ·
eB
2
Derivation Path
a simple problem with curvature terms
• anticipate the results on energetics . . . F M must commute with derivatives
• this works with the linearised parallel bracket trivially
F M [H0 , dh]U x = [H0 , F M dh]
since
[H0 , F M ]U x = 0
• with curvature terms we had to make the approximations carefully
cF
cF
F M ∇(log B) ·
· ∇(dh) = ∇(log B) ·
· ∇(F M dh)
eB
eB
since
∇(log B) ·
cF
· ∇(µB) = 0
eB
• with local fluxtube ordering, this will be satisfied trivially and generally
Fluxtube Ordering
• a fluxtube model builds flute mode ordering kk ≪ k⊥ into the coordinates
• start with a local domain in the drift plane perpendicular to B
typically, a few cm × a few cm
• extend the domain boundary points along the field line
• main complication: boundary conditions (treated on Day 4)
• here, we assume it is possible and well founded
and concentrate on the ordering on the equations
• main step: metric varies on scale a, R, constant within drift plane, varying along B
Fluxtube Rules
• field aligned coordinates {x} → {xy} × {s} with only B s ≡ B · ∇s nonvanishing
• drift plane covered by perpendicular coordinates x, y (nonlinearities, curvature terms)
• parallel coordinate is s (parallel bracket)
• geometry depends only on s, drifts on fluctuations F · ∇ involve only x, y
• in this context F M is part of the geometry, varying through B = B(s)
F M now commutes with F · ∇ trivially
• derivatives involving the curvature ∇B · F has scale R, still involve s
• effect on Jacobian: Bk∗ → B since ∇B · F · B vanishes
• curvature terms are recast, rest remains
Derivation Path
curvature operator
• curvature terms
Bk∗ {H0 , dh}
cF
= . . . + (µB + mU )∇(log B) ·
· ∇(dh)
eB
2
• curvature operator used in fluid models
cF
K() = −∇(log B ) · 2 · ∇()
B
2
• definition as a vector/gradient contraction or divergence
x ∂f
∂
∂
x
K(f ) = K
+K
=
(K f ) +
(Ky f )
∂x
∂y
∂x
∂y
y ∂f
Kx,y = K(x, y)
• divergence requirement trivially satisfied due to fluxtube ordering
Derivation Path
recasting of curvature terms
• curvature terms
Bk∗ {H0 , dh}
• curvature operator
cF
= . . . + (µB + mU )∇(log B) ·
· ∇(dh)
eB
2
cF
K() = −∇(log B ) · 2 · ∇()
B
2
• curvature terms in fluxtube ordering
µB + mU 2
K(dh)
{H0 , dh} = . . . −
2e
Derivation Path
nonlinear advection
• nonlinear drift advection term
c F
∇H1 ·
· ∇dg
e B2
• under fluxtube ordering only x, y derivatives appear
• express in terms of dh since [H1 , H1 ]xy vanishes
c F xy
c F xy
[H1 , dg]xy =
[H1 , dh]xy
2
2
e B
e B
• geometric quantity is cF xy /B 2 , stream function is ψ ≡ H1 /e
• parallel bracket remains linearised
Derivation Path
linear drive
• to conserve free energy F M commutes with [, ]xy hence is independent of x
• linear gradient drive inserted separately (as fixed part of df )
[ψ, dh]xy
→
[ψ, dh]xy + F M [ψ, log f0 ]xy
• express in terms of background density, temperature scale lengths
∇ log f0 = ∇ log n0 +
M
T ∂F
3
−
F M ∂T
2
∇ log T0
• for a Maxwellian ∂F M /∂T = (mz 2 /2T )(F M /T )
we have all the terms now, next comes the field equations
Derivation Path
field equations: polarisation
• linearised polarisation equation from total-f model with M = e2 F M /T
(required as gyrokinetic Hamiltonian is only to 1st order in fields)
XZ
dW [eJ0 f + (J0 MJ0 − J0 M) φ] = 0
sp
• merely assume F M does not contribute to the eJ0 f term
XZ
sp
2
e
dW e(J0 df ) + F M
J02 − 1 dφ = 0
T
• due to antisymmetry of the z(J0 dAk ) term, this is equivalently
XZ
sp
2
e
J02 − 1 dφ = 0
dW e(J0 dg) + F M
T
Derivation Path
field equations: induction
• gyrokinetic Ampere’s law from total-f model
ωp2
c2
−
∇2⊥
!
Z
e
4π X
dW pz (J0 f )
Ak =
c sp
m
X 4πe2
ωp2
Ak =
(J0 f J0 )Ak
2
c2
mc
sp
• linearisations: F M does not contribute to current, only F M contributes to ωp
(required as gyrokinetic Hamiltonian is only to 1st order in fields)
ωp2
c2
−
∇2⊥
!
Z
4π X
dW ez(J0 dg)
dAk =
c sp
X 4πe2
ωp2
2
M
J0 dAk
dAk =
F
2
c2
mc
sp
Delta-f Gyrokinetic Equations
• delta-f gyrokinetic equation under fluxtube ordering
∂dg cF xy
Bs
mU 2 + wB
M
+
[ψ, (dh + F log f0 )]xy +
[H0 , dh]zs −
K(dh) = C(df )
2
∂t
B
B
2e
where U is relabelled as z, and
dg = df + F
M
ez
J0 dAk
T c
dh = df + F
M
e
J0 dφ
T
z
ψ = J0 dφ − J0 dAk
c
• polarisation
XZ
sp
2
e
dW e(J0 dg) + F M (J02 − 1)dφ = 0
T
• induction
Z
2 2
X
4π
Me z
2
dW ez(J0 dg) − F
J02 dAk = 0
∇⊥ dAk +
c sp
cT
Free Energy – why it works
• brackets and curvature operator in the equations are all with dh
free energy as second order Casimir: quadratic functional of dependent variables
multiply the equation by dh and integrate over phase space
XZ
sp
dΛ
xy
T
∂dg
cF
dh
=
−
T
2
FM
∂t
B
sp
X
Z
dΛ dh[ψ, log f0 ]xy =
XZ
dΛ
sp
T
dhC(df )
M
F
• the nonlinearity, parallel compressions and curvature all conserve this free energy
• free energy source due to the background gradient (∂ log f0 /∂x)
flux down the gradient given by dh times (−cF xy /B 2 )∂ψ/∂y
includes both ExB advection and magnetic flutter (ψ contains both dφ and dAk )
• dissipative sink due to the collision operator C (details of this on Day 8)
gradient driven dynamics as a driven, dissipative system
Free Energy Components
• the time derivative of the free energy is given by
XZ
∂E
T
∂dg
=
dΛ M dh
∂t
F
∂t
sp
• this is suggestive of the quadratic functional
XZ
T
E=
dΛ
(dh dg)
M
2F
sp
• this splits into three pieces (the fourth, with dφ and dAk , vanishes)
we will show that the latter two are ExB and magnetic energy, respectively
XZ
XZ
XZ
1
1 z
T
2
(df ) +
dΛ e(J0 dφ)(df ) +
dΛ e (J0 dAk )(df )
E=
dΛ
M
2F
2
2 c
sp
sp
sp
role of polarisation
• the first field piece in the free energy is due to dφ
XZ
1
EE =
dΛ e(J0 dφ)(df )
2
sp
• use the Hermicity of J0 to get (dφ)(eJ0 df )
• the polarisation equation is
XZ
dW (eJ0 df ) =
sp
XZ
sp
dW F
M
e2
(1 − J02 )dφ
T
• therefore the field energy due to dφ is the ExB energy
EE =
XZ
sp
2
e
(1 − J02 )(dφ)2
dΛ F M
2T
role of induction
• the second field piece in the free energy is due to dAk
EM =
XZ
sp
1 z
dΛ e (J0 dAk )(df )
2 c
• use the Hermicity of J0 to get (dAk )(eJ0 df )
• the induction equation is
XZ
sp
z
dW 4π (eJ0 df ) = −∇2⊥ dAk
c
• therefore the field energy due to dAk is the magnetic energy
EM =
Z
2
1 ∇⊥ dAk dV
8π
time derivatives of free energy pieces
• ExB and magnetic energies are quadratic in the fields, so that
∂EE
=
∂t
sp
XZ
∂EM
=
∂t
Z
2
e
∂
F M (1 − J02 )dφ
dΛ dφ
∂t
T
∂dAk
1
2
dV
(−∇⊥ dAk ) ·
4π
∂t
• reverse the preceding manipulations to find
∂EE
=
∂t
sp
XZ
∂df
dΛ e(J0 dφ)
∂t
∂EM
=
∂t
sp
XZ
put these together with the (df )2 piece
XZ
T
∂df
∂EF
dΛ M (df )
=
∂t
F
∂t
sp
∂dAk
z
dΛ e (J0 df )
c
∂t
time derivatives of the total free energy
• we have the three pieces
∂EE
=
∂t
sp
XZ
∂df
dΛ e(J0 dφ)
∂t
∂EF
=
∂t
sp
XZ
∂EM
=
∂t
sp
XZ
dΛ
∂dAk
z
dΛ e (J0 df )
c
∂t
∂df
T
(df
)
FM
∂t
• with the fourth involving zdAk and dφ vanishing by symmetry, we have
∂E
∂
(EF + EE + EM ) =
=
∂t
∂t
sp
XZ
dΛ
T
∂dg
dh
FM
∂t
• similarly, by adding the free energy pieces we have
XZ
T
(dh dg)
EF + EE + EM = E =
dΛ
M
2F
sp
Delta-f Gyrokinetic Energy
• ExB and magnetic energy (same as in total-f model as found)
1X
EE =
2 sp
Z
2
e
dΛ F M (1 − J02 )dφ2
T
EM
1
=
8π
Z
2
dV k⊥
dA2k
• time derivatives (reformulate using polarisation/induction as shown)
XZ
∂df
∂EE
=
dΛ e(J0 dφ)
∂t
∂t
sp
XZ
∂EM
e
∂
=
dΛ z df (J0 dAk )
∂t
c
∂t
sp
• thermal free energy and time derivative (the remaining piece)
EF =
Z
X
1
2 sp
dΛ
T
2
df
FM
∂EF
=
∂t
sp
XZ
dΛ
T
∂df
df
FM
∂t
Gyrokinetic Energy Theorem
• combined time derivative for E = EE + EM + EF
Z
XZ
∂E
T
∂dg X
cF xy ∂ψ
dh
∂ log f0
=
dΛ M dh
=
T dΛ
dh
+ M C(df )
2
∂t
F
∂t
∂x
B ∂y
F
sp
sp
• conservation by brackets, curvature, leaving linear drive, and collision damping
• linear drive: radial flux down the background gradient (x-direction)
• collisions: diffusion across H0 -lines in zw-space (velocity)
also in perpendicular space for guiding centers (treated on Day 8)
• turbulence cascade carries EF to small scales in all four {x, z} coordinates,
requires numerical dissipation (treated on Day 7)
• use energy theorem pieces to diagnose energy transfer, (afternoon sessions)
Main Points
• delta-f model is derived from total-f model at same consistency level
• delta-f and total-f models solve different equations
• total-f and delta-f models conserve different quantities
the (df )2 piece is related to f log f in the total-f model
• instead of global conservation, the delta-f model has a drive/dissipation theorem
if the background gradient variation is incorporated into the dependent variable,
the drive becomes a source to maintain the gradient, as in total-f
• collisions represent a sink for delta-f free energy
similar to their effect on entropy in the total-f model