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**Magnetic Structures in Electron-scale Reconnection Domain**

Dynamical Processes in Space Plasmas Eyn Bokkek, Israel, April 2010 Ilan Roth Space Sciences UC Berkeley, CA Thanks: Forrest Mozer Phil Pritchett Electron compressibility n(x); far from answering all these questions

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**Fundamental plasma processes with global **

implications may occur in a narrow layer Magnetic Reconnection Magnetic shears - electron dominated region What can we learn about electron scale structures without (full) simulations?

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**Symmetric Configuration à la texbook cartoon**

Nonsymmetric!

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**Classical Symmetric Crossing à la observations- Mozer, 2002**

Hall reconnect Hall Classical , large scale; Ey – inductive field; reconnect

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**Non-symmetric crossing**

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**Main purpose: assessing the non ideal effects of Ohms Law **

Environment: electron (current) velocity >> mass velocity Collective plasma scales determine the different (nested) layers: Outer: Hall effect – ions decouple from B Intermediate: e- inertia (pressure) Inner: break(s) the e- Innermost: frozen-in condition Electron diffusion region controls the global structure. Note: resistive time absent NOTE: density fluctuations <d_e

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**Two Fluid: coupling (B,v)**

“Ion” fluid Electron fluid Two approaches: simplify electron(ion) fluid and concentrate on ions(electrons)

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**Sheared field, Inhomogeneous Plasma General coupling between**

Shear Alfven Compressional Alfven Slow Magneto-Acoustic modified on short scales by (mainly) electron effects Two approaches: simplify electron(ion) fluid and concentrate on ions(electrons) 9

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**Two (extreme) approaches**

Lowest approximation of the electron dynamics + follow ion dynamics Lowest approximation of the ion dynamics + follow electron dynamics

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**A. Faraday and Ohm’s law couple magnetic and velocity fields**

MHD: Magnetic field is frozen in the fluid drift Alfven vs Whistler(Helicon); few manipulations

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Magnetic field – fictitious diagram of lines in R3 satisfying specific rules. MHD – approximate description of magnetic field motion in a plasma fluid. Knot - closed loop of a non-self-intersecting curve, transformed via continuous deformation of R3 upon itself, following laws of knot topology - pushed smoothly in the surrounding viscous fluid, without intersecting itself (stretching or bending). MHD field evolves as a topological transformation of a knot. MHD dynamics forms equivalent knot configurations with a set of knot invariants. Invatriant of link - + or- .NOTE: MHD Turbulence is a great example for knots

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All KNOT deformations can be reduced to a sequence of Reidemeister “moves”: (I) twist (II) poke , and (III) slide. Type 3 Type 1 Type 2 Knot topology described through knot diagrams

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Reidemeister moves Invariants: assign to each crossing specific value, o 1 –1, t and form determinant polynomial

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**MHD invariants: (cross) helicity, generalized vorticity, Ertel,…**

Reidemeister moves preserve several invariants of the knot or link represented by their diagram - topological information. MHD invariants: (cross) helicity, generalized vorticity, Ertel,… Every knot can be uniquely decomposed as a knot sum of prime knots, which cannot themselves be further decomposed - Schubert (1949) Diagram- projection of 3D 0n 2D. Link – set of knots. Franz Peter Schubert – composer; Ertel – curlv . Grad s

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Prime knots Characterization based on crossing number – Tait 1877

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**Flux-rope is a KNOT MHD Turbulence forms a LINK-**

HELIOSPHERE Flux-rope is a KNOT MHD Turbulence forms a LINK- Collection of knots Reconnection is NOT a KNOT: it forms a KNOT SUM Schubert (1949) showed that every knot can be uniquely decomposed (up to the order in which the decomposition is performed) as a knot sum of a class of knots known as prime knots, which cannot themselves be further decomposed. Knots that are the sums of prime knots are known as composite knots. 17

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**MHD (KNOT) can be broken via several physical processes**

Various physical regions Reconnection: topological transition Diffusion: violation of frozen–in condition Dissipation: conversion of em energy (no consensus on definitions) Topological transition: fields from different regions encounter themselves. Dissipation : < ion skin; diffusion – filamentary current, dens gradients…Interesting physics NOT in reconnection;most energy conversion – to electrons. Important: parts of electron diffusion has also strong currents.

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**Parallel electric field is observed in tandem with density gradients**

Mozer +, 2005 Localized electric field over scale ≤ de=c/ωe – electron inertia effect?

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**Electron diffusion region:**

filamentary currents on scale ≤ de=c/ωe – dissipation region due to electron inertia effect? ELECTRON PHYSICS COVERS LARGE SPATIAL SCALES. Tangential E field – reconnection rate5. Note” pressure terms scales - ion skin; el inertia – electron skin depth.

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**Electron diffusion found NOT at the null of magnetic field: β<1**

Large localized electric field spikes – electron inertia? Small fluctuations in B – large in E; U different from ExB Localized electric field – scale – electron inertia effect?

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**Asymmetric Simulation – Pritchett, 2009**

Violation of electron frozen-in condition Elongated Electron Diffusion regions

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**Asymmetric PIC simulation – Pritchett, 2009**

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**Magnetic field vs Electron Vorticity vs dissipation**

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**Electron Diffusion vs Dissipation**

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**B. Faraday and Ohm’s law couple magnetic and velocity fields**

eMHD: Generalized vorticity field is frozen in the electron fluid drift vorticity Alfven vs Whistler(Helicon); few manipulations 26

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**MHD: “Ion” fluid eMHD: Electron fluid:**

Alfven vs Whistler(Helicon); few manipulations

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**Homogeneous, incompressible electron fluid**

Magnetic field slips with respect to the electron fluid Generalized vorticity G is frozen in the electron drift u Electrons can slip with respect to the mag field; the role of Alfven helicon wave Electron inertia Hall

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**Inhomogeneous electron fluid**

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**Generalized Vorticity – Inhomogeneous fluid**

Electrons can slip with respect to the mag field; the role of Alfven helicon wave; can add resistivity and viscosity Linear homogeneous infinite plasma waves Whistler branch

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**The eigenmodes evolve from linear perturbations**

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**A. Incompressible Homogeneous Plasma; [n(x)=no]**

Electron inertia effect is manifested on the small spatial scale Express the perturbed velocity with the help of the magnetic field components

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**Inclusion of ion dynamics in the limit**

Coupling of shear Alfven and compressional Alfven Mirnov+, 2004 eMHD limit: Express the perturbed velocity with the help of the magnetic field components 33

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**Eigenmodes: two components of the magnetic field**

bx By=tanh(x/L) de/L=1 bz Califano, 1999 Unstable mode in a whistler regime

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**B. Compressible Homogeneous Plasma**

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Lowest order : Increase in the effective electron skin depth Compressibility - “Guiding” field: enhance the electron inertia effect

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**B. Compressible Homogeneous Plasma**

Increase in the effective electron skin depth Compressibility - “Guiding” field: enhance the electron inertia effect 37

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**C. Inhomogeneous, compressible plasma**

Density dips enhance the electron inertia effect

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**D. Inhomogeneous, compressible plasma – generalized configuration**

3D structure may enhance the electron inertia effect

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**E. Kinetic, incompressible, inhomogeneous plasma**

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Calculation of σ+ , σ- requires integration over electron trajectories in an inhomogeneous magnetic field; choose the magnetic field in y (parallel) direction and include both ky and kz.

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For kz=0, n=0 Resonances

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**E. Kinetic, incompressible, inhomogeneous plasma**

Attico +, 2002

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SUMMARY A. MHD satisfies the axioms of knot theory – both evolve preserving various invariants. Knot sum is equivalent to violation of frozen-in condition. B. Density gradients/dips, compressibility, and thermal effects may have a significant effect on the electron vorticity, which determines the slipping of the magnetic field with respect to the electrons. These effects modify the structure of the magnetic field on the short-scale, forming current filaments, parallel electric fields, which violate the frozen-in condition and contribute to electron heating. These regions are ubiquitous and are observed outside of the x-points in the reconnection domain.

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