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 Im ǫ ( ω, k ) Re ǫ ( ω, k ) = 0 , γ = − ∂ 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 + e a E ∂ ∂E � • f a = 0 , ∂x = 4 π ˆ n e a dv f a m a ∂v a f a ( x, v, t ) = F a ( v, t ) + δf a ( x, v, t ) , E ( x, t ) = δE ( x, t ) • � ∂ � ∂ � � ∂t + e a δE ∂ ∂t + v ∂ ∂x + e a δE ∂ F a + δf a = 0 • m a ∂v m a ∂v � ∂ � ∂x δE = 4 π ˆ n e a dv δf a a

Averaging over phase ∂ F a = − e a ∂ • ∂v � δf a δE � ∂t m a Insert back to the original equation � ∂ � ∂t + v ∂ δf a = − e a δE ∂F a ∂v − e a ∂ • ∂v ( δf a δE − < δf a δE > ) ∂x m a m a Two-time scales (slow and fast) � � dω δf a kω ( v, t ) e ikx − iωt • δf a ( x, v, t ) = dk ⇑ ⇑ fast slow � � ω − kv + i ∂ kω = − i e a ∂F a δf a • δE kω ∂t m a ∂v � � − i e a dω ′ ∂ � � dk ′ δE k ′ ω ′ δf a k − k ′ ω − ω ′ − < δE k ′ ω ′ δf a k − k ′ ,ω − ω ′ > m a ∂v

� � ω − kv + i ∂ kω = − i e a ∂F a δf a • δE kω ∂t m a ∂v � � − i e a dω ′ ∂ � � dk ′ δE k ′ ω ′ δf a k − k ′ ω − ω ′ − < δE k ′ ω ′ δf a k − k ′ ,ω − ω ′ > m a ∂v • ω → ω + i ∂/∂t 1 g K = − i e a ∂ K = ( k, ω ) , • m a ω − kv + i 0 ∂v � dK ′ g K ( E K ′ f K − K ′ − < E K ′ f K − K ′ > ) • f K = g K F E K +

� � � dK ′ g K • f K = g K F E K + E K ′ f K − K ′ − < E K ′ f K − K ′ > f K = f (1) + f (2) • + · · · K K � � � dK ′ g K g K − K ′ F • f K = g K F E K + E K ′ E K − K ′ − < E K ′ E K − K ′ > Insert f K to Poisson equation � 4 π ˆ ne a � • E K = − i dv f K k a

ǫ ( K ): linear dielectric response ⇓ � �� � � � � 4 πe a ˆ n � • 0 = 1 + i dv g K F E K k a � � 4 πe a ˆ n i � � dK ′ � + dv g K g K − K ′ F E K ′ E K − K ′ − < E K ′ E K − K ′ > k a � �� � ⇑ χ (2) ( K ′ | K − K ′ ): (second-order) nonlinear response � � � dK ′ χ (2) ( K ′ | K − K ′ ) 0 = ǫ ( K ) E K + • E K ′ E K − K ′ − < E K ′ E K − K ′ >

� dK ′ χ (2) ( K ′ | K − K ′ ) < E − K E K ′ E K − K ′ > 0 = ǫ ( K ) < E K E − K > + • At this point, reintroduce slow-time derivative ω → ω + i∂/∂t � � k, ω + i ∂ ǫ ( k, ω ) < E 2 > kω → ǫ < E 2 > kω • ∂t � � ǫ ( k, ω ) + i ∂ǫ ( k, ω ) ∂ < E 2 > kω → 2 ∂ω ∂t 0 = i ∂ǫ ( K ) ∂ ∂t < E 2 > K +Re ǫ ( K ) < E 2 > K + i Im ǫ ( K ) < E 2 > K • 2 ∂ω � dK ′ χ (2) ( K ′ | K − K ′ ) < E − K E K ′ E K − K ′ > +

Summary of weak turbulence theory for kinetic instabilities Re ǫ ( K ) < E 2 > K = 0 • dispersion relation − Im ǫ ( K ) ∂ ∂t < E 2 > K = 2 < E 2 > K • ∂ Re ǫ ( K ) /∂ω � �� � ⇑ γ growth rate � 2 i dK ′ χ (2) ( K ′ | K − K ′ ) < E − K E K ′ E K − K ′ > + Im ∂ Re ǫ ( K ) /∂ω

Weak Turbulence Theory for Reactive Instabilities � ∂ � ∂t + v ∂ δf a ( x, v, t ) = − e a δE ( x, t ) ∂F a ( v, t ) • ∂x m a ∂v − e a ∂ ∂v [ δE ( x, t ) δf a ( x, v, t ) − � δE ( x, t ) δf a ( x, v, t ) � ] m a Fourier transformation in space � dk δf a k ( v, t ) e ikx • δf a ( x, v, t ) = � ∂ � k ( v, t ) = − e a δE k ( t ) ∂F a ( v, t ) δf a • ∂t + ikv m a ∂v � − e a ∂ dk ′ [ δE k ′ ( t ) δf a k − k ′ ( v, t ) − � δE k ′ ( t ) δf a k − k ′ ( v, t ) � ] m a ∂v

Quasilinear Theory Temporal dependence � k Ω ( v ) e − i Ω t δf a dω δf a • k ( v, t ) = Ω = ω + iγ k Ω ( v ) = − e a ∂F a − i (Ω − kv ) δf a • δE k Ω m a ∂v Inserting the above to Poisson equation we have ω 2 � 1 ∂F a � pa • 0 = 1+ dv ∂v = Re ǫ ( ω + iγ, k )+ i Im ǫ ( ω + iγ, k ) Ω − kv k a ∂ ∂t < δE 2 > k Ω e 2 γt = 2 γ < δE 2 > k Ω e 2 γt •

Weak Turbulence Theory Temporal dependence � � δf a dω δf a k Ω ( v, t ) e − i Ω t , δE a dω δE a k Ω ( t ) e − i Ω t k ( v, t ) = k ( t ) = • ⇑ Ω = ω + iγ slow time ∂t < δE 2 > k Ω e 2 γt = 2 γ < δE 2 > k Ω e 2 γt + ∂ < δE 2 > k Ω ∂ e 2 γt • ∂t The extra factor ∂ < δE 2 > k Ω ∂t is determined by nonlinear wave kinetic equation

Nonlinear theory Ω → Ω + i ∂ • ∂t k Ω = − e a ∂F a − i (Ω − kv ) δf a • δE k Ω m a ∂v � � − e a ∂ d Ω ′ � � �� dk ′ δE k ′ Ω ′ δf a δE k ′ Ω ′ δf a k − k ′ ,ω − Ω ′ − k − k ′ , Ω − Ω ′ m a ∂v 0 = ǫ ( k, Ω) < δE k Ω δE ∗ • k Ω > � � d Ω ′ χ ( k ′ , Ω ′ | k − k ′ , Ω − Ω ′ ) < δE ∗ dk ′ + k Ω δE k ′ Ω ′ δE k − k ′ , Ω − Ω ′ >

Re-introduce slow time derivative Ω → Ω + i ∂/∂t k Ω > + i ∂ǫ ( k, Ω) ∂ 0 = ǫ ( k, Ω) < δE k Ω δE ∗ ∂t < δE k Ω δE ∗ • k Ω > 2 ∂ Ω � � d Ω ′ χ ( k ′ , Ω ′ | k − k ′ , Ω − Ω ′ ) < δE ∗ dk ′ + k Ω δE k ′ Ω ′ δE k − k ′ , Ω − Ω ′ > Dispersion relation 0 = ǫ ( k, ω + iγ ) = Re ǫ ( k, ω + iγ ) + i Im ǫ ( k, ω + iγ ) • Wave kinetic equation � � ∂ 2 i d Ω ′ χ ( k ′ , Ω ′ | k − k ′ , Ω − Ω ′ ) ∂t < δE k Ω δE ∗ dk ′ k Ω > = • ∂ǫ ( k, Ω) /∂ Ω × < δE ∗ k Ω δE k ′ Ω ′ δE k − k ′ , Ω − Ω ′ >

Making use of ∂t < δE 2 > k Ω e 2 γt = 2 γ < δE 2 > k Ω e 2 γt + ∂ < δE 2 > k Ω ∂ e 2 γt • ∂t and ∂ < δE 2 > k Ω � � 2 i d Ω ′ χ ( k ′ , Ω ′ | k − k ′ , Ω − Ω ′ ) dk ′ • = ∂t ∂ǫ ( k, Ω) /∂ Ω × < δE ∗ k Ω δE k ′ Ω ′ δE k − k ′ , Ω − Ω ′ > We finally arrive at ∂ ∂t < δE 2 > k Ω e 2 γt = 2 γ < δE 2 > k Ω e 2 γt • � � 2 i d Ω ′ χ ( k ′ , Ω ′ | k − k ′ , Ω − Ω ′ ) dk ′ + ∂ǫ ( k, Ω) /∂ Ω × < δE ∗ k Ω δE k ′ Ω ′ δE k − k ′ , Ω − Ω ′ > e 2 γt

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 ∂ 2 Im ǫ ( k, ω ) ∂t < δE 2 > kω = − ∂ Re ǫ ( k, ω ) /∂ω < δE 2 > kω • � 2 i dK ′ χ (2) ( k ′ , ω ′ | k − k ′ , ω − ω ′ ) + Im ∂ Re ǫ ( k, ω ) /∂ω × < δE − k, − ω δE k ′ ,ω ′ δE k − k ′ ,ω − ω ′ > kinetic ∂ ∂t < δE 2 > k,ω + iγ e 2 γt = 2 γ < δE 2 > k,ω + iγ e 2 γt • � � 2 i dk ′ d ( ω + iγ ) ′ + ∂ǫ ( k, ω + iγ ) /∂ ( ω + iγ ) × χ ( k ′ , ω ′ + iγ ′ | k − k ′ , ω − ω ′ + iγ − iγ ′ ) × < δE ∗ k,ω + iγ δE k ′ ,ω ′ + iγ ′ δE k − k ′ ,ω − ω ′ + iγ − iγ ′ > e 2 γt reactive