Alfvn waves Troy Carter Dept. of Physics and Astronomy, UCLA - PowerPoint PPT Presentation

Alfvén waves Troy Carter Dept. of Physics and Astronomy, UCLA

Importance of plasma waves • Along with single particle motion, understanding of linear waves are foundation for physical intuition for behavior of plasmas • Waves play direct role in important physical processes: RF heating in fusion plasmas, particle acceleration by waves in space plasmas, plasma turbulence in astrophysical objects

Importance of plasma waves • Along with single particle motion, understanding of linear waves are foundation for physical intuition for behavior of plasmas • Waves play direct role in important physical processes: RF heating in fusion plasmas, particle acceleration by waves in space plasmas, plasma turbulence in astrophysical objects • Wave is collective response of plasma to perturbation, however, intuition for waves starts with considering single particle response to electric/magnetic fields that make up the wave • Focus on magnetized plasmas: particle response is anisotropic, orientation of wave E-field wrt background magnetic field is essential in determining response

Wave equation, plasma dielectric model for linear waves • Treat plasma as conducting medium; will lead to dielectric description (but start by treating plasma charge and currents as free) ∂ E ⇤ ⇥ E = � ∂ B ⇥ � B = µ o j + 1 c 2 ∂ t ∂ t ∂ 2 E ∂ j ⇥ � ⇥ � E + 1 ∂ t 2 + µ o ∂ t = 0 c 2 • Plasma effects buried in current, need model to relate current to E • Model plasma as cold fluid, will find a linear, tensor conductivity j = σ · E

Important intuition: Single particle response to wave fields • Conductivity tensor tells us plasma response to applied electric field; useful to think about single particle orbits • In particular for magnetized plasmas and wave electric fields that are perpendicular to B • Two drifts matter (in uniform plasma): ExB drift and polarization drift • ExB drift is the dominant particle response for low frequency wave fields ω < Ω c • Polarization drift is dominant at higher frequencies

ExB and Polarization Drifts v drift B B E E Polarization drift, ExB drift, DC E Field ExB drift removed ∂ v E = E × B v p = 1 E ⊥ B 2 Ω ∂ t B • No currents from ExB at low freq (ions and electrons drift the same); above ion cyclotron freq, ions primarily polarize, no ExB, can get ExB current from electrons

Model for plasma conductivity • Use cold, two-fluid model; formally cold means: v φ � v th , e , v th , i d v s = n s q s ( E + v s × B ) n s m s dt � = j n s q s v s ≡ σ · E s • Assume plane wave solution (uniform plasma), linearize the equations: f ( r , t ) = f exp ( i k · r − i ω t ) f = f 0 + f 1 + . . . f 1 � f o ; • Ignore terms higher than first order: arrive at equation that is the same for motion of a single particle (importance of understanding drifts!)

Plasma model, cont. Choose B = B 0 ˆ z , E = E 1 = E x ˆ x + E z ˆ z Ion momentum equation becomes: eE x − i ω v x − Ω i v y Ω i = eB = m i m i Ω i v x − i ω v y = 0 Solve for v x , v y : − i ω e v x = E x (polarization) Ω 2 i − ω 2 m i − Ω i e v y = E x (E × B) Ω 2 i − ω 2 m i ie For the parallel response: v z = E z (inertia-limited response) ω m i

Plasma model, cont. Back to the wave equation, rewrite with plane wave assumption: − k × k × E − ω 2 c 2 E − i ω µ o σ · E = 0 Can rewrite in the following way: M · E = 0 n 2 = c 2 k 2 k − I ) n 2 + � M = (ˆ k ˆ index of refraction ω 2 � = I + i ⇥ � o ⇤ dielectric tensor unit tensor

Cold plasma dispersion relation Using the cold two-fluid model for σ , the dielectric tensor becomes: ω 2 ω 2 pi pe S = 1 − (polarization) − ω 2 − Ω 2 ω 2 − Ω 2   S − iD 0 e i Ω i ω 2 Ω e ω 2 pi pe iD S 0 � = D = (E × B response) i ) −   ω ( ω 2 − Ω 2 ω ( ω 2 − Ω 2 e ) P 0 0 P = 1 − ω 2 ω 2 − ω 2 pi pe (inertial response) ω 2 Defining θ to be the angle between k and B o , the wave equation becomes: S − n 2 cos 2 θ n 2 sin θ cos θ     − iD E x  = 0 S − n 2 iD E y 0    P − n 2 sin 2 θ n 2 sin θ cos θ E z 0 det M = 0 provides dispersion relation for waves – allowable combinations of ω and k

Low frequency waves: Alfvén waves • For freq. much less than ion cyclotron frequency, primary waves are Alfvén waves k Magnetic field lines Shear Alfvén wave • Primary motion: ExB motion of electrons and ions together (D → 0) • To pull this out of our cold plasma model: k = k z ˆ z ( θ = 0)

Shear wave in cold plasma model  S − n 2    E x 0 0  = 0 S − n 2 E y 0 0    P E z 0 0 ω 2 ω 2 ≈ ω 2 = c 2 n 2 = S = 1 − pi pe pi − ω 2 − Ω 2 Ω 2 v 2 ω 2 − Ω 2 e i i A B 2 ω 2 = k 2 � v 2 v 2 ; A = A µ o m i n i • Like wave on string: magnetic field plays role of tension, plasma mass → string mass

Alfvén waves from MHD ∂ρ Continuity ∂ t + ∇ · ( ρ ~ v ) = 0 ✓ ∂ ~ ◆ v Momentum = − ∇ p + ~ j × ~ ρ v · ∇ ~ ∂ t + ~ v B Ohm’s Law ~ v × ~ E + ~ B = 0 (electron momentum) ✓ p ◆ d Pressure closure (adiabatic) = 0 dt ρ γ + Maxwell’s Equations • Linearizing this system reveals four waves: fast and slow magnetosonic waves, the shear Alfvén wave, and the entropy wave

MHD Waves • For freq. much less than ion cyclotron frequency, primary waves are Alfvén waves Magnetic field lines Compressional Slow magnetosonic Shear Alfvén wave Alfvén wave (fast magnetosonic) ω 2 = k 2 k v 2 A ω 2 = k 2 ✓ ◆ q c 2 s + v 2 s + v 4 s v 2 c 4 A − 2 c 2 A cos2 θ A ± 2 sound wave response (in fast/slow modes) not in our cold two-fluid model

Shear wave dispersion derivation B · ∇ ~ ~ p + B 2 ✓ ∂ ~ ◆ ✓ ◆ v B = − ∇ p + ~ j × ~ ρ v · ∇ ~ B = − ∇ ∂ t + ~ v + 2 µ o µ o magnetic pressure magnetic tension ∂ ~ B ∂ t = − ∇ × ~ v × ~ E = ∇ × ~ B • We are looking for the shear wave, so we’ll make appropriate assumptions: ~ incompressible motion k · δ ~ v = 0 ~ no field line compression, linearly polarized z + δ B ˆ B = B o ˆ x ~ plane waves δ B , δ v ∝ exp ( i r − i ω t ) k · ~ δ p = 0 follows from the first assumption, adiabatic assumption

Shear wave dispersion derivation, cont ~ B · ∇ ~ p + B 2 ✓ ∂ ~ ◆ ✓ ◆ v B ρ v · ∇ ~ = − ∇ ∂ t + ~ v + 2 µ o µ o v = ik k B o δ B � i ωρδ ~ x ˆ µ o ∂ ~ B v × ~ ∂ t = ∇ × ~ B � i ωδ B = ik k δ vB o • Combine these two to get: B 2 ω 2 = k 2 µ o ρ = k 2 k v 2 k A

Currents in MHD AW j = 1 B = 1 ~ ~ ∇ × ~ k × ( δ B ˆ x ) δ ~ i E = − δ ~ z = − δ vB o ˆ v × B o ˆ y µ o µ o E = k k B 2 o δ ~ δ B ˆ y ωρ µ o δ ~ j = − i ω ne − ik y δ B E ~ z ˆ Ω i B o µ o Polarization current • Current in k ⊥ =0 AW is entirely due to ion polarization current: no field aligned current • As k ⊥ is introduced, current closes along the field (inductively driven)

Finite k ⊥ introduces parallel current, electric field S − n 2     n k n ? 0 E x k  = 0 S − n 2 E y 0 0    � zz − n 2 E z n k n ? 0 ? • Shear Alfvén wave currents without k ⊥ are purely due to ion polarization and are cross-field • With finite k ⊥ , wave currents must close along the field: introduce parallel electric field and parallel particle response (easy to find departures from MHD…) • Ions carry current across field, electrons carry parallel current (ion parallel response important at higher β ) • Electron parallel response introduces dispersion and damping to Alfvén wave

Kinetic and Inertial Alfvén waves: introduce dispersion and damping at finite k ⊥ • At finite k ⊥ , wave obtains parallel electric field • In low β plasma, key additional physics is parallel electron response v 2 v 2 = B 2 m i m e = 1 = 1 A A v 2 v 2 µ o ρ T i β β m i th , i th , e • Use kinetic electron response in parallel direction (ignore ion response) 1 ⌅ � zz ≈ 1 + (1 + ⇥ e Z ( ⇥ e )) ; ⇥ e = √ k 2 k ⇤ 2 2 k k v th , e D • For simplicity, assume cold ions, k ⊥ ρ e << 1; use cold perpendicular response

Kinetic and Inertial Alfvén waves S − n 2     n k n ? 0 E x k  = 0 S − n 2 E y 0 0    � zz − n 2 E z n k n ? 0 ? • Shear wave: k ⊥ k E ⊥ , E y = 0 k − ⇥ 2 ✓ ◆ k 2 = − k 2 c 2 S ? S � zz S ≈ c 2 v 2 A k − ⇥ 2 c 2 ✓ ◆ k 2 = − k 2 � zz ? v 2 v 2 A A

Kinetic and Inertial Alfvén waves • Inertial Alfvén wave: cold electron response, v A >> v th,e + i √ ⌅⇥ e � zz ≈ 1 − ⇧ 2 − ⇥ 2 � � ( ζ e � 1) exp e k 2 k ⇤ 2 ⇧ 2 pe D k 2 k v 2 c A ω 2 r = δ e = , (1 + k 2 ? δ 2 e ) ω pe • Kinetic Alfvén wave: hot electron response, v A << v th,e + i √ ⌅⇥ e 1 − ⇥ 2 � � � zz ≈ 1 + exp ( ζ e ⌧ 1) e k 2 k ⇤ 2 k 2 k ⇤ 2 D D ρ s = C s ω 2 r = k 2 k v 2 1 + k 2 ? ρ 2 � � , A s Ω i