Kinetic Equilibrium of Dipolarization Fronts*

Guru Ganguli, Chris Crabtree, Erik Tejero, Alex Fletcher, and David Malaspina1

Plasma Physics Division, Naval Research Laboratory, Washington DC 1 Laboratory for Atmospheric and Space Physics, University of Colorado, Boulder CO

*Work Supported by NRL base funds and NASA

March 4 – 9, 2018

Particle Dynamics in the Earth’s Radiation BeltsAGU Chapman Conference, Cascais, Portugal

foa (Xg, Ha (x)) =Na

(pvta

2 )2Qa (Xg )e

-Ha (x )

kTa ,

Qa (Xg ) =

Ra

Ra + (Sa

Sa

ì

íï

îïï

- Ra )Xg - Xg1

Xg2 - Xg1

æ

èç

ö

ø÷

, Xg < Xg1

, Xg1 < Xg < Xg2

, Xg > Xg2

< f0a >º f0a (v,Y(x))dv = n0a (ò Y(x))

n0a (Y(x)) = 0a

å

Ha = mv2 / 2 + eY(x), Xg = x + vy / W

Qa (Xg )

Distribution function

Guiding center distribution

Density:

Quasi-neutrality determines the electrostatic potential

With Ψ(x) determined, the distribution function is fully specified

[Romero et al., GRL, 1990]

Generic Configuration: Plasma With Spatial Gradient

TransitionRegion I Region II

PlasmaParameters (I)

PlasmaParameters (II)

Ra

Sa

Xg1Xg2

B0

Constants of motion:

3

Dipolarization Fronts are Pressure Gradient Layers

In the aftermath of reconnection plasma is dragged by the

magnetic field lines to create a pressure gradient layer with

scale size ~ i or less

(a, b) Low-frequency magnetic field data (128 Samples/s) in GSE coordinates.

(c) Electron and proton density.

(d) Electron temperature parallel and perpendicular to the background magnetic field.

(e) Parallel and perpendicular proton temperatures

(f) Proton velocity components, in GSE coordinates.

(g) Electric field wave power spectral density.

The temperature ratio Te0/Ti0 ~ 0.16 Earthward of DF

Steady state can be modeled by

- Xg1 = -1.0 i , Xg2 = -1,5 i : Ions

- Xg1 = -0.5 i , Xg2 = -1.2 i : Electrons

- R = 1.0, S = 0.75 for both ions and electrons

- i evaluated in the Earthward region outside DF

Global Compression Surrogate:

- R, S, determine magnitude

- ∆Xg the pressure gradient scale size

Obtained from observation or global model

θ

X

Z

X’

Z’

X

Z

P

P’

B(s)

Qa (Xg )

ri ³ L > re

Ra

Sa

Xg1Xg2

B0

DF Geometry

3i-3i 0

4

Generation of Ambipolar Potential and Density

Ambipolar Electrostatic Potential Self-Consistent Density

Separation of electron and ion scales evidentNot possible in a MHD/Fluid model

Electron scale variations

na º< f0a >

5

Formation of Distinct Electron and Ion Layers

Electric field magnitude and gradient is stronger in electron layerKinetic property unlike in MHD/Fluid models

Ambipolar electric field

saturates for L ≲ ρi

In MHD/Fluid models

E ∝ (▽Pi)/n blows up for

L → 0

Separate flow layers

seen in ISEE data [Parks

et al. JGR, 1979]Electron scale variations

Perpendicular Electric Field Self-Consistent Flows

∇P(x)

DF

We >dVE

dx> Wi

E(x) = -dY

dx

Electric Field Gradient Modifies Particle Orbits

L E(x)

B0

vE(x)x

-y

zVelocity shear becomes an important factor

- for > 0 orbit oscillatory but for < 0 exponential

Kinetic property unlike in MHD/Fluid models

b =1/ vth

2

Affects zeroth-order plasma dynamics- Particles can move across magnetic field lines

- Unique plasma distribution is created withtemperature anisotropy in x and y directions

The ions experience strong velocity shear while electrons experience weak shear

x = x + (vy -VE (x)),

VE (x) = -cE(x) / B0,

- where > 0 and L > e

[Ganguli et al., Phys Fluids. 1988]

We >dVE

dx> Wi

7

Electric Field Gradient Breaks Gyrotropy and Introduces Temperature Anisotropy

Non-gyrotropy and temperature anisotropy small for electrons but large for ions

because velocity shear dVE/dx > ι but < e

Temperature Anisotropy

8

Self-Consistent Currents and Magnetic Flux Pileup

Unequal electron and Ion flows lead to net perpendicular

current and consequent magnetic flux pileup

Electron scale variations

9

Parallel Electric field and Pressure Anisotropy

θ X

Z

X’Z’

X

Z

P

P’

B(s)

L|| ~ (¶ln B(s) / ¶s)-1E||(s) º -¶y (B(s)) / ¶s = (x / L||)Ex(x)

The x-z component of the magnetic field rotates along the DF field lines

The es potential Ψ(B(s)) changes along the field lines

generating a parallel electric field

Existence of parallel electric field implies non-zero off-diagonal terms of the

pressure tensor for parallel force balance;

The parallel electric can accelerate the non-thermal particles to form parallel

beams

Hierarchy of Velocity Shear Driven Waves Possible

Velocity shear, generated by global compression,

is the natural source of free energy in DF

Inhomogeneous Flow

Electrostatic

Oscillations

Perpendicular Flow

KH

IEDDI*

EIH*

Parallel Flow

D’Angelo, SMIA *

SMIC*

SMLH

Electromag

Oscillations

Perpendicular Flow

EM-KH

EM-IEDDI*

EM-EIH

(Whistler)

Parallel Flow

EM-SMIA

EM-D’Angelo

TBD

TBD

L > ri ®w < Wi

L > ri ®w < Wi

L ~ ri ®w ~ Wi

L < ri ®w > Wi

L ~ ri ®w ~ Wi

L < ri ®w > Wi

L > ri ®w < Wi L > ri ®w < Wi

L ~ ri ®w ~ Wi L ~ ri ®w ~ Wi

L < ri ®w > Wi L < ri ®w > Wi

* Experiments in

- Iowa U

- WVU

- NRL

- Auburn U

- Japan

- S. Africa

- India

3i-3i 0

• Dispersion relation for waves with k|| ~ 0 and both density and velocity gradient is,

– For VE →0 reduces to the equation for the Lower Hybrid Drift Instability (LHDI) [Mikhailvoski and Tsypin, JETP, 1963; Krall and Liewer, Phys Rev A, 1971]

– For Ln➝∞ reduces to the Electron Ion Hybrid (EIH) Instability [Ganguli et al., Phys Fluids, 1988]

11

Velocity Gradient (not the Density Gradient) is the Source for

Lower Hybrid Waves in Dipolarization Fronts

1

Ln

º1

n

dn

dx, G(w ) =

w pe

2

w pe

2 + We

2

æ

èç

ö

ø÷

w 2

w 2 -w LH

2

æ

èçö

ø÷, w LH

2 =w pi

2 We

2

We

2 +w pe

2

Qa (Xg )

ri ³ L > re

Ra

Sa

Xg1 Xg2

B0

S/R

Conclusions

Kinetic equilibrium model for a dipolarization front shows– A strong ambi-polar electric field across the magnetic field as a result of global compression

– Spatial variation in both electron and ion scales

- Ion and electron orbits are affected differently resulting in unique particle distribution

– Spatial gradient in the electric field that causes anisotropy and non-gyrotropy in the distribution function

– MHD/fluid descriptions become inadequate for L ≲ 2ρi

Analytical model validated in a 1D PIC simulation

Magnetic field curvature leads to parallel electric field– Parallel electric field can accelerate non thermal particles to form parallel beams

- Provides a non-reconnection basis for their existence

– Existence of parallel electric field implies anisotropic pressure tensor

Plasma compression due to dipolarization of field lines generates velocity shear – Velocity shear generates a hierarchy of instabilities in a broad frequency and wave vector band

– In the collisionless plasma environment these waves lead to relaxation of stress in a DF

12

13

PIC Simulation of Broadband Noise in Compressed Layers

21/2 D PIC Simulation

Shear stronger in the electron layer where E X B

drift is larger than the diamagnetic drift

Signatures consistent with the EIH instability

Contours of electrostatic potentialBroadband emission

Relaxation of initial flow

[Romero and Ganguli, Phys. Fluids, 1993; Romero and Ganguli, GRL, 1994]

• KyL ~ O(1)

• L becomes larger with

time

• This can successively

trigger lower frequency

waves

L > ri Þw < Wi

L ~ ri Þw ~ Wi

L < ri Þw ~ wLH > Wi

0.1 1 10

pe

14

1D PIC Simulation Validates the Analytical Model

15

Electric Field Gradient Breaks Gyrotropy and Introduces Temperature Anisotropy

Non-gyrotropy and temperature anisotropy small for electrons but large for ions

because velocity shear dVE/dx > ι but < e

Analytical Model 1D PIC Simulation

Transverse Velocity Shear Can Drive Broadband Waves

d2

dx2- ky

2 +kyVE

"

w - kyVE

æ

èç

ö

ø÷f1(x) = 0

Kelvin-Helmholtz (w - kyVE ) << Wi, kyri <1

U º¶w D

¶wµw (w - k

yV

E)

Doppler-ShiftedFrequency

gUILA

y ,z

Energy Creationin Re gion-I

= - VgU

IIA

y ,z

Energy ConvectionFrom Re gion-I

/II IU U

IEDDI (w - kyVE ) ~ nWi , kyri ³1

x

III

E(x)

L/2-L/2

IIVg

y

z

Vg

(w - k

yV

E) < 0

U º¶w D

¶wµw (w - k

yV

E)

Doppler-ShiftedFrequency

gUILA

y ,z

Energy Creationin Re gion-I

= - VgU

IIA

y ,z

Energy ConvectionFrom Re gion-I

/II IU U

Electron IEDDI (w - kyVE ) ~ nWe, kyre ~1

x

III

E(x)

L/2-L/2

IIVg

y

z

Vg

(w - k

yV

E) < 0

Weak Shear L > ri , dVE / dx < WiStrong Shear L < ri , dVE / dx > Wi

Electron-Ion Hybrid (EIH) Wi <w < We, kyri >1

1+w pe

2

We

2-

w pi

2

w 2

æ

èç

ö

ø÷ Ñ

2f1(x)+w pe

2

We

2

æ

èç

ö

ø÷

kyVE

"

w - kyVE

f1(x) = 0