Weak Turbulence Theory for Reactive Instabilities Peter H. Yoon 1 - - PowerPoint PPT Presentation

Laboratory Space and Astrophysical Plasmas Pohang, Korea, June, 2008

Weak Turbulence Theory for Reactive Instabilities

Peter H. Yoon

1

• Kinetic instabilities

Re ǫ(ω, k) = 0, γ = − Im ǫ(ω, k) ∂Re ǫ(ω, k)/∂ω

• Reactive instabilities

Re ǫ(ω + iγ, k) + i Im ǫ(ω + iγ, k) = 0

• Weak turbulence theory available in the literature is valid only

for kinetic instabilities

• Most plasma instabilities that lead to turbulence is reactive

Review of Textbook Weak Turbulence Theory

• ∂

∂t + v ∂ ∂x + eaE ma ∂ ∂v

• fa = 0,

∂E ∂x = 4πˆ n

• a

ea

• dv fa
• fa(x, v, t) = Fa(v, t) + δfa(x, v, t),

E(x, t) = δE(x, t)

• ∂

∂t + ea ma δE ∂ ∂v

• Fa +

∂

∂t + v ∂ ∂x + ea ma δE ∂ ∂v

• δfa = 0

∂ ∂x δE = 4πˆ n

• a

ea

• dv δfa

Averaging over phase

• ∂ Fa

∂t = − ea ma ∂ ∂v δfa δE Insert back to the original equation

• ∂

∂t + v ∂ ∂x

• δfa = − ea

ma δE ∂Fa ∂v − ea ma ∂ ∂v (δfa δE− < δfa δE >) Two-time scales (slow and fast)

• δfa(x, v, t) =
• dk
• dω δfa

kω(v, t) eikx−iωt

⇑ ⇑ slow fast

• ω − kv + i ∂

∂t

• δfa

kω = −i ea

ma δEkω ∂Fa ∂v − i ea ma

• dk′
• dω′ ∂

∂v

• δEk′ω′ δfa

k−k′ ω−ω′− < δEk′ω′ δfa k−k′,ω−ω′ >

• ω − kv + i ∂

∂t

• δfa

kω = −i ea

ma δEkω ∂Fa ∂v − i ea ma

• dk′
• dω′ ∂

∂v

• δEk′ω′ δfa

k−k′ ω−ω′− < δEk′ω′ δfa k−k′,ω−ω′ >

• ω → ω + i ∂/∂t
• K = (k, ω),

gK = −i ea ma 1 ω − kv + i0 ∂ ∂v

• fK = gK F EK +
• dK′ gK (EK′ fK−K′− < EK′ fK−K′ >)

• fK = gK F EK +
• dK′ gK
• EK′ fK−K′− < EK′ fK−K′ >
• fK = f(1)

K

+ f(2)

K

+ · · ·

• fK = gK F EK+
• dK′ gK gK−K′ F
• EK′EK−K′− < EK′EK−K′ >
• Insert fK to Poisson equation
• EK = −i
• a

4πˆ nea k

• dv fK

ǫ(K): linear dielectric response ⇓

• 0 =
• 1 + i
• a

4πeaˆ n k

• dv gK F
• EK

+

• dK′

a

4πeaˆ n i k

• dv gK gK−K′ F
• EK′EK−K′− < EK′EK−K′ >
• ⇑

χ(2)(K′|K−K′): (second-order) nonlinear response

• 0 = ǫ(K) EK+
• dK′χ(2)(K′|K − K′)
• EK′EK−K′− < EK′EK−K′ >

• 0 = ǫ(K) < EKE−K > +
• dK′χ(2)(K′|K − K′) < E−KEK′EK−K′ >

At this point, reintroduce slow-time derivative ω → ω + i∂/∂t

• ǫ(k, ω) < E2 >kω→ ǫ
• k, ω + i ∂

∂t

• < E2 >kω

→

• ǫ(k, ω) + i

2 ∂ǫ(k, ω) ∂ω ∂ ∂t

• < E2 >kω
• 0 = i

2 ∂ǫ(K) ∂ω ∂ ∂t < E2 >K +Re ǫ(K) < E2 >K +i Im ǫ(K) < E2 >K +

• dK′ χ(2)(K′|K − K′) < E−KEK′EK−K′ >

Summary of weak turbulence theory for kinetic instabilities

• Re ǫ(K) < E2 >K= 0

dispersion relation

• ∂

∂t < E2 >K= 2 −Im ǫ(K) ∂Re ǫ(K)/∂ω

• < E2 >K

⇑ γ growth rate + Im 2i ∂Re ǫ(K)/∂ω

• dK′ χ(2)(K′|K − K′) < E−KEK′EK−K′ >

Weak Turbulence Theory for Reactive Instabilities

• ∂

∂t + v ∂ ∂x

• δfa(x, v, t) = − ea

ma δE(x, t) ∂Fa(v, t) ∂v − ea ma ∂ ∂v [ δE(x, t) δfa(x, v, t) − δE(x, t) δfa(x, v, t) ] Fourier transformation in space

• δfa(x, v, t) =
• dk δfa

k(v, t) eikx

• ∂

∂t + ikv

• δfa

k(v, t) = − ea

ma δEk(t) ∂Fa(v, t) ∂v − ea ma ∂ ∂v

• dk′ [ δEk′(t) δfa

k−k′(v, t) − δEk′(t) δfa k−k′(v, t) ]

Quasilinear Theory Temporal dependence

• δfa

k(v, t) =

• dω δfa

kΩ(v) e−iΩt

Ω = ω + iγ

• − i (Ω − kv) δfa

kΩ(v) = − ea

ma δEkΩ ∂Fa ∂v Inserting the above to Poisson equation we have

• 0 = 1+
• a

ω2

pa

k

• dv

1 Ω − kv ∂Fa ∂v = Re ǫ(ω+iγ, k)+i Im ǫ(ω+iγ, k)

• ∂

∂t < δE2 >kΩ e2γt = 2γ < δE2 >kΩ e2γt

Weak Turbulence Theory Temporal dependence

• δfa

k(v, t) =

• dω δfa

kΩ(v, t) e−iΩt,

δEa

k(t) =

• dω δEa

kΩ(t) e−iΩt

Ω = ω + iγ ⇑ slow time

• ∂

∂t < δE2 >kΩ e2γt = 2γ < δE2 >kΩ e2γt + ∂ < δE2 >kΩ ∂t e2γt The extra factor ∂ < δE2 >kΩ ∂t is determined by nonlinear wave kinetic equation

Nonlinear theory

• Ω → Ω + i ∂

∂t

• − i (Ω − kv) δfa

kΩ = − ea

ma δEkΩ ∂Fa ∂v − ea ma ∂ ∂v

• dk′
• dΩ′

δEk′Ω′ δfa

k−k′,ω−Ω′ −

• δEk′Ω′ δfa

k−k′,Ω−Ω′

• 0 = ǫ(k, Ω) < δEkΩ δE∗

kΩ >

+

• dk′
• dΩ′ χ(k′, Ω′|k − k′, Ω − Ω′) < δE∗

kΩ δEk′Ω′ δEk−k′,Ω−Ω′ >

Re-introduce slow time derivative Ω → Ω + i ∂/∂t

• 0 = ǫ(k, Ω) < δEkΩ δE∗

kΩ > + i

2 ∂ǫ(k, Ω) ∂Ω ∂ ∂t < δEkΩ δE∗

kΩ >

+

• dk′
• dΩ′ χ(k′, Ω′|k − k′, Ω − Ω′) < δE∗

kΩ δEk′Ω′ δEk−k′,Ω−Ω′ >

Dispersion relation

• 0 = ǫ(k, ω + iγ) = Re ǫ(k, ω + iγ) + i Im ǫ(k, ω + iγ)

Wave kinetic equation

• ∂

∂t < δEkΩ δE∗

kΩ >=

2i ∂ǫ(k, Ω)/∂Ω

• dk′
• dΩ′ χ(k′, Ω′|k−k′, Ω−Ω′)

× < δE∗

kΩ δEk′Ω′ δEk−k′,Ω−Ω′ >

Making use of

• ∂

∂t < δE2 >kΩ e2γt = 2γ < δE2 >kΩ e2γt + ∂ < δE2 >kΩ ∂t e2γt and

• ∂ < δE2 >kΩ

∂t = 2i ∂ǫ(k, Ω)/∂Ω

• dk′
• dΩ′ χ(k′, Ω′|k − k′, Ω − Ω′)

× < δE∗

kΩ δEk′Ω′ δEk−k′,Ω−Ω′ >

We finally arrive at

• ∂

∂t < δE2 >kΩ e2γt = 2γ < δE2 >kΩ e2γt + 2i ∂ǫ(k, Ω)/∂Ω

• dk′
• dΩ′ χ(k′, Ω′|k − k′, Ω − Ω′)

× < δE∗

kΩ δEk′Ω′ δEk−k′,Ω−Ω′ > e2γt

Weak turbulence theory for kinetic vs reactive instabilities: Dispersion relation

• Re ǫ(k, ω) = 0

kinetic

• Re ǫ(k, ω + iγ) + i Im ǫ(k, ω + iγ) = 0

reactive

Wave kinetic equation

• ∂

∂t < δE2 >kω= − 2 Im ǫ(k, ω) ∂Re ǫ(k, ω)/∂ω < δE2 >kω + Im 2i ∂Re ǫ(k, ω)/∂ω

• dK′ χ(2)(k′, ω′|k − k′, ω − ω′)

× < δE−k,−ω δEk′,ω′ δEk−k′,ω−ω′ > kinetic

• ∂

∂t < δE2 >k,ω+iγ e2γt = 2γ < δE2 >k,ω+iγ e2γt + 2i ∂ǫ(k, ω + iγ)/∂(ω + iγ)

• dk′
• d(ω + iγ)′

×χ(k′, ω′ + iγ′|k − k′, ω − ω′ + iγ − iγ′) × < δE∗

k,ω+iγ δEk′,ω′+iγ′ δEk−k′,ω−ω′+iγ−iγ′ > e2γt

reactive