1. 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

2. 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?

3. Symmetric Configuration à la texbook cartoon
Nonsymmetric!

4. Classical Symmetric Crossing à la observations- Mozer, 2002
Hall
reconnect
Hall
Classical , large scale; Ey – inductive field;
reconnect

6. Non-symmetric crossing

7. 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

8. Two Fluid: coupling (B,v)
“Ion” fluid
Electron fluid
Two approaches: simplify electron(ion) fluid and concentrate on ions(electrons)

9. 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

10. Two (extreme) approaches
Lowest approximation of the electron dynamics + follow ion dynamics
Lowest approximation of the ion dynamics + follow electron dynamics

11. 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

12. 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

13. 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

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

15. 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

16. Prime knots
Characterization based on crossing number – Tait 1877

17. 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

18. 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.

19. Parallel electric field is observed in tandem with density gradients
Mozer +, 2005
Localized electric field over
scale ≤ de=c/ωe – electron inertia effect?

20. 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.

21. 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?

22. Asymmetric Simulation – Pritchett, 2009
Violation of electron frozen-in condition
Elongated Electron Diffusion regions

23. Asymmetric PIC simulation – Pritchett, 2009

24. Magnetic field vs Electron Vorticity vs dissipation24

25. Electron Diffusion vs Dissipation25

26. 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

27. MHD: “Ion” fluid eMHD: Electron fluid:
Alfven vs Whistler(Helicon); few manipulations

28. 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

29. Inhomogeneous electron fluid

30. 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

31. The eigenmodes evolve from linear perturbations

32. 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

33. 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

34. Eigenmodes: two components of the magnetic fieldbx
By=tanh(x/L)
de/L=1 bz
Califano, 1999
Unstable mode in a whistler regime

35. B. Compressible Homogeneous Plasma

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

37. B. Compressible Homogeneous Plasma
Increase in the effective electron skin depth
Compressibility - “Guiding” field: enhance the electron inertia effect 37

38. C. Inhomogeneous, compressible plasma
Density dips enhance the electron inertia effect

39. D. Inhomogeneous, compressible plasma – generalized configuration
3D structure may enhance the electron inertia effect

40. E. Kinetic, incompressible, inhomogeneous plasma

41. 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.

42. For kz=0, n=0
Resonances

43. E. Kinetic, incompressible, inhomogeneous plasma
Attico +, 2002

44. 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.