Connections in Plasmas Felipe A. Asenjo 1 Universidad Adolfo Ib a - PowerPoint PPT Presentation

ICTP-IAEA College on Plasma Physics, 2016 Connections in Plasmas Felipe A. Asenjo 1 Universidad Adolfo Ib´ a˜ nez, Chile 1 felipe.asenjo@uai.cl

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Newcomb’s Theorem

In 1958, Newcomb showed that in a plasma that satisfies the ideal Ohms law, two plasma elements connected by a magnetic field line at a given time will remain connected by a field line for all subsequent times. This occurs because the plasma moves with a transport velocity that preserves the magnetic connections between plasma elements. This is one of the most fundamental and relevant ideas in plasma physics.

Proof: d / dt is the convective derivative Ohm’s law � v × � E + � B = 0 implies B ) = d � B ∂ t � v × � v · ∇ ) � B = ∇ × ( � dt − ( � B

Proof: d / dt is the convective derivative Ohm’s law � v × � E + � B = 0 implies B ) = d � B ∂ t � v × � v · ∇ ) � B = ∇ × ( � dt − ( � B x ′ − � � l = � Be d x the 3D vector connecting two infinitesimally close fluid elements. d � � � x ′ ) − � l = � v ( � v ( � x ) = � v ( � x + d l ) − � v ( � x ) = ( d l · ∇ ) � dtd v

Proof: d / dt is the convective derivative Ohm’s law � v × � E + � B = 0 implies B ) = d � B ∂ t � v × � v · ∇ ) � B = ∇ × ( � dt − ( � B x ′ − � � l = � Be d x the 3D vector connecting two infinitesimally close fluid elements. d � � � x ′ ) − � l = � v ( � v ( � x ) = � v ( � x + d l ) − � v ( � x ) = ( d l · ∇ ) � dtd v Then d � � � l × � � l × � � l × � dt ( d B ) = − ( d B )( ∇ · � v ) − ( d B ) × ∇ × � v � l × � B = 0, it always remains null Wich means that if d

Pegoraro’s generalization

Ohm’s law u µ = dx µ d τ F µν u ν = 0

Ohm’s law u µ = dx µ d τ F µν u ν = 0 dF µν = ( ∂ µ u α ) F να − ( ∂ ν u α ) F µα d τ d / d τ = u µ ∂ µ

Ohm’s law u µ = dx µ d τ F µν u ν = 0 dF µν = ( ∂ µ u α ) F να − ( ∂ ν u α ) F µα d τ d / d τ = u µ ∂ µ d d τ dl µ = dl α ∂ α u µ where dl µ is the 4D displacement of a plasma fluid element.

Ohm’s law u µ = dx µ d τ F µν u ν = 0 dF µν = ( ∂ µ u α ) F να − ( ∂ ν u α ) F µα d τ d / d τ = u µ ∂ µ d d τ dl µ = dl α ∂ α u µ where dl µ is the 4D displacement of a plasma fluid element. d d τ ( dl µ F µν ) = − ( ∂ ν u β ) dl α F αβ This means that if dl µ F µν = 0, it always remains null

Relativistic Plasma We extend the connection concept beyond

A plasma governed by generalized relativistic MHD equations. Effects such as thermal-inertial effects, thermal electromotive effects, current inertia effects and Hall effects. Minkowski metric tensor η µν = diag ( − 1 , 1 , 1 , 1 ) , and an electron-ion plasma with density n , charge density q = ne , normalized four-velocity U µ ( U µ U µ = − 1) and four-current density J µ continuity ∂ µ ( qU µ ) = 0 generalized momentum equation � hU µ U ν + µ h � = − ∂ µ p + J ν F µν , q 2 J µ J ν ∂ ν generalized Ohm’s law � µ h q ( U µ J ν + J µ U ν ) − µ ∆ µ h � = 1 2 ∂ µ Π + qU ν F µν − ∆ µ J ν F µν + qR µ . J µ J ν ∂ ν q 2 h denotes the MHD enthalpy density, Π = p ∆ µ − ∆ p , p = p + + p − and ∆ p = p + − p − , µ = m + m − / m 2 , m = m + + m − , ∆ µ = ( m + − m − ) / m . The frictional four-force density between the fluids is R µ = − η [ J µ + Q ( 1 + Θ) U µ ] , where Θ is the thermal energy exchange rate from the negatively to the positively charged fluid, η is the plasma resistivity, and Q = U µ J µ .

As usual, F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor ( A µ is the four-vector potential), which obeys Maxwell’s equations ∂ ν F µν = 4 π J µ , ∂ ν F ∗ µν = 0 . Of course, F ∗ µν = ( 1 / 2 ) ǫ µναβ F αβ is the dual of F µν , and ǫ µναβ indicates the Levi-Civita symbol.

Ohm’s law Σ µ = U ν M µν + R µ , U µ = U µ − ∆ µ M µν = F µν − µ q J µ , ∆ µ W µν , � h � � h � W µν = S µν − ∆ µ Λ µν = ∂ µ q U ν − ∂ ν q U µ , � h � � h � S µν ∂ µ qU ν − ∂ ν qU µ = , � h � h � � Λ µν ∂ µ q 2 J ν − ∂ ν q 2 J µ = . and Σ µ = ∂ µ � µ hQ / q 2 + µ h / ( q ∆ µ ) � + ( µ/ ∆ µ ) χ µ , with � h � � h � + ∆ µ Q − ∆ µ χ µ = U ν ∂ ν qU µ ∂ µ 2 µ q ∂ µ Π . q q

Curl of the Ohm’s law d M λφ = ∂ λ U ν M φν − ∂ φ U ν M λν − µ ∆ µ Z λφ + ∂ λ R φ − ∂ φ R λ , d τ with d / d τ = U ν ∂ ν , and Z λφ = Z λφ + Z λφ + Z λφ + Z λφ , p c h H where � � Q � � h � � Q � � h �� Z λφ ∂ λ ∂ φ − ∂ φ ∂ λ = ∆ µ , h q q q q ∂ λ q − ∂ φ q � p + ∆ µ � � p + ∆ µ � Z λφ q 2 ∂ φ q 2 ∂ λ = 2 µ Π 2 µ Π , p � 1 � � 1 � Z λφ ∂ λ qJ ν F φν − ∂ φ qJ ν F λν = , H � h � h � µ �� � µ �� Z λφ − ∂ λ q J α ∂ α q 2 J φ + ∂ φ q J α ∂ α q 2 J λ = . c Z λφ and Z λφ are due to the thermal-inertial and thermal electromotive effects. The p h contributions coming from the Hall effect in the generalized Ohm’s law are instead retained by the tensor Z λφ H , while Z λφ appears owing to current inertia effects. c

Displacement of a plasma element Define a general displacement four-vector ∆ x µ of a general element that is transported by the general four-velocity ∆ x µ ∆ τ = U µ + µ ∆ µ D µ , where ∆ τ is the variation of the proper time and D µ is a four-vector field which satisfies the equation M νφ ∂ λ D ν − M νλ ∂ φ D ν = Z λφ . The four-vector D µ contains all the (inertial-thermal-current-Hall) information of Z µν . We introduce the event-separation four-vector dl µ = x ′ µ − x µ between two different elements. Then ( d / d τ ) dl µ = U ′ µ + ( µ/ ∆ µ ) D ′ µ − U µ − ( µ/ ∆ µ ) D µ = U µ ( x α + dl α ) + ( µ/ ∆ µ ) D µ ( x α + dl α ) − U µ ( x α ) − ( µ/ ∆ µ ) D µ ( x α ) . Therefore, the four-vector dl µ fulfills � U µ + µ � d d τ dl µ = dl α ∂ α ∆ µ D µ .

Connections when resistivity is neglected! Finally we find � � d U ν + µ � dl λ M λφ � � dl λ M λν � ∂ φ = − . ∆ µ D ν d τ This equation reveals the existence of generalized magnetofluid connections that are preserved during the plasma dynamics. Indeed, from this equation it follows that if dl λ M λφ = 0 initially, then d / d τ ( dl λ M λφ ) = 0 for every time, and so dl λ M λφ will remain null at all times. The “magnetofluid connection equation” all previous results for a relativistic electron-ion MHD plasma with thermal-inertial, Hall, thermal electromotive and current inertia effects dl λ M λφ = dl λ F λφ − µ ∆ µ dl λ W λφ ,

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