Vorticities in relativistic plasmas: from waves to reconnection - PowerPoint PPT Presentation

ICTP-IAEA College on Plasma Physics, 2018 Vorticities in relativistic plasmas: from waves to reconnection Felipe A. Asenjo 1 Universidad Adolfo Ib´ a˜ nez, Chile ◮ Part I: Waves in relativistic plasmas ◮ Part II: Electro–Vortical formulation ◮ Part III: Generalized Connetion and Reconnection 1 felipe.asenjo@uai.cl; felipe.asenjo@gmail.com

ICTP-IAEA College on Plasma Physics, 2018 Part I: VORTICITY AND WAVES IN RELATIVISTIC PLASMAS ◮ Vortical model for relativistic plasmas ◮ Circular polarized waves

Relativistic Plasma equations ◮ the rest-frame density of the fluid n . ◮ the energy density ǫ , pressure p , enthalpy density h = ǫ + p , and temperature T . ◮ relativistic velocities and the Lorentz factor γ = ( 1 − v 2 ) − 1 / 2 . ◮ coupled to Maxwell equations via the current density n γ v . Plasma fluid equation for specie j � ∂ � ( f j γ j v j ) = q j γ j ( E + v j × B ) − 1 m j γ j ∂ t + v j · ∇ ∇ p j n j Continuity equation ∂ ( γ j n j ) + ∇ · ( γ j n j v j ) = 0 ∂ t f ≡ h mn = f ( T ) And an equation of state for pressure and density.

We re-write the fluid equation as... Let us assume constant rest-frame density n and constant temperature ∂ ( γ j v ) − m j f j v j × ∇ × ( γ j v j ) = q j ( E + v j × B ) − 1 2 ∇ ( v j · v j ) m j f j ∂ t where we have used a × ( ∇ × b ) = ( ∇ b ) · a − ( a · ∇ ) b Now, we notice ∂ ( γ j v j ) � � �� B + m j f j − 1 = q j E + v j × ∇ × ( γ j v j ) 2 ∇ ( v j · v j ) m j f j ∂ t q j it appears the interesting field Ω j = B + m j f j ∇ × ( γ j v j ) = ∇ × P j q j that will be a generalized vorticity with the potential [the canonical momentum] P j = A + m j f j γ j v j q j

Generalized vorticity equation Taking the curl of the previous equation ∂ ∇ × ( γ j v j ) m j f j = ∇ × E + ∇ × ( v j × Ω J ) ∂ t q j and remembering that ∇ × E = − ∂ t B we obtain ∂ Ω j ∂ t = ∇ × ( v j × Ω j ) The plasma dynamics becomes simplified in terms of the Generalized vorticity! Ω j = B + m j f j ∇ × ( γ j v j ) q j

Maxwell equations E , B electric and magnetic fields ∂ B = −∇ × E ∂ t ∂ E � ∂ t + q i n i γ i v i = ∇ × B i ∇ · B = 0 � = q i n i γ i ∇ · E i

From Maxwell equations we obtain... ∂ B = −∇ × E ∂ t ∂ E � ∂ t + q i n i γ i v i = ∇ × B i ∇ · B = 0 � = q i n i γ i ∇ · E i ———————————————————————————– ∇ × ( ∇ × B ) + ∂ 2 B � ∂ t 2 = q i ∇ × ( n i γ i v i ) i

Vorticity and helicity The vorticity field is any psedovector that is the rotational (curl) of a vector field (potential). The vorticity field has associated a quantity called helicity For example, the magnetic helicity is � A · B d 3 x h = such that � ∂ A ∂ h A · ∂ B � ∂ t · B d 3 x + ∂ t d 3 x = ∂ t � � ( − E − ∇ φ ) · B d 3 x − A · ∇ × E d 3 x = � � E · B d 3 x − ( φ B + E × A ) · d 2 x ≡ − 2 � E · B d 3 x ≡ − 2 is not always conserved!

Plasma fluid generalized helicity The helicity associated to the relativistic plasma fluid (for constant density and pressure) is � P · Ω d 3 x h = which satisfies � ∂ P ∂ h � P · ∂ Ω ∂ t · Ω d 3 x + ∂ t d 3 x = ∂ t � � ( v × Ω) · Ω d 3 x + P · [ ∇ × ( v × Ω)] d 3 x = ≡ 0 the Generalized Helicity is conserved 2 2 Mahajan & Yoshida, Phys. Plasmas 18 , 055701 (2011).

If pressure is not constant... ∂ Ω ∂ t = ∇ × ( v × Ω) + 1 n 2 ∇ n × ∇ p the last term is so-called Biermann battery. It can generate vorticity from plasma thermodynamical inhomogenities.

◮ The conservation of helicity establishes topological constraints. It can forbid the creation (destruction) of vorticity in plasmas. ◮ We can see that the generalized helicity remains unchanged in ideal dynamics. This conservation implies serious contraints on the origin and dynamics of magnetic fields. ◮ Otherwise, the nonideal effects can change the helicity. For example, if gradients of pressure and temperature have different directions [Biermann battery]. ◮ An anisotropic pressure tensor may also generate vorticity.

Dimensionless system. Positive ( q = e ) and negative ( q = − e ), two-fluids plasma Magnetic fields are normalized to background magnetic field B 0 (measured in rest frame), time to Ω 0 , distance to Ω − 1 0 , with the generalized cyclotron eB 0 Ω 0 = ( m + f + + m − f − ) c The equations are now ∂ Ω ± = ∇ × ( v ± × Ω ± ) ∂ t m ± f ± Ω ± = B ± µ ± ∇ × ( γ ± v ± ) ; µ ± = m + f + + m − f − ∇ × ( ∇ × B ) + ∂ 2 B 1 ∂ t 2 = ∇ × ( γ + v + − γ − v − ) U 2 A 0 � with the normalzied Alfven speed U A 0 = B 0 / 4 π n 0 ( m + f + + m − f − )

Exact propagation circularly polarized waves No background flow, background magnetic field in ˆ z . Transverse waves propagating in ˆ z direction, with constant frequency and constant wavevector. Hence, v · ˆ z = 0 and B · ˆ z = 0 v ± = v ± B = B y ) e ikz − i ω t + c . c . y ) e ikz − i ω t + c . c . � � � � (ˆ x + i ˆ ; (ˆ x + i ˆ 2 2 where v ± and B are constant amplitudes. Notice that 1 1 γ = = √ 1 − v ± · v ± � 1 − v 2 ± is now constant. The system is reduced to ω B + ω k µ + γ + v + = kv + ω B − ω k µ − γ − v − = kv − k ( k 2 − ω 2 ) B = ( γ + v + − γ − v − ) U 2 A 0

Dispersion relation for pair plasmas 3 Consider m + = m − = m and f + = f − = f . Then µ + = µ − = 1 / 2. � 1 / 2 �� k 2 � 2 ± = k 2 k 2 2 2 1 1 ω 2 2 + + 4 + + ± 2 − U 2 γ 2 + γ 2 U 2 γ 2 + γ 2 γ 2 + γ 2 A 0 − A 0 − − High–frequency modes in physical units + ≈ c 2 k 2 + ω 2 Ω 2 p c ω 2 + f 2 γ 2 + γ 2 f − Low–frequency modes in physical units � − 1 � − ≈ V 2 A k 2 1 + c 2 k 2 V 2 ω 2 A + f γ 2 + γ 2 ω 2 c 2 f γ 2 + γ 2 p − − 8 π n 0 e 2 / m , Ω c = eB 0 / ( mc ) , and V A = B 0 / √ 8 π n 0 m . � with ω p = 3 Mahajan & Lingam, Phys. Plasmas 25 , 072112 (2018).

Amplitude–dependent dispersion relation High–frequency wave cut–off c 2 k 2 = ω 2 − ω 2 cut − off � � cut − off = ω 2 Ω 2 p ω 2 c 1 + f ω 2 p γ 2 + γ 2 f − ⇒ c 2 k 2 = ω 2 − ω 2 p if γ + γ − ≫ 1 = approaches to a light wave in a plasma! f For high–amplitude, the plasma wave behaves as if the plasma were unmagnetized. Simiarly, the Alfven mode frequency decreases

Estimations For a pair plasma with n 0 ≈ 10 8 cm − 3 , and then ω p ≈ 3 × 10 8 s − 1 , in a magnetosphere in a pulsar with magnetic field B 0 ≈ 10 10 G, then Ω c ≈ 2 × 10 17 s − 1 . For high temperatures, f ≈ 4 k B T / ( mc 2 ) . For T ∼ 10 11 K , then f ≈ 100. The cut–off � 1 + 10 16 ω cut − off ≈ ω p √ f γ 2 + γ 2 − Then if γ + ∼ γ − ∼ 10 5 , the wave behaves as a light wave with ω cut − off ≈ ω p / √ f .

That’s all (for now). Thanks!