Modeling of Whistler Amplification

Anatoly V. StreltsovAnatoly V. Streltsov

Thayer School of Engineering, Dartmouth College, Hanover, NH 03755Thayer School of Engineering, Dartmouth College, Hanover, NH 03755

MURI meeting March 3-6, 2008 Lake Arrowhead, CA

Our goal is to model and explain VLF triggering events Our goal is to model and explain VLF triggering events measured on Siplemeasured on Siple

(this example is courtesy of E. Mishin and A. Gibby)(this example is courtesy of E. Mishin and A. Gibby)

Hybrid Fluid-Particle Model

cold fluid electron momentum equation:

Maxwell’s equations:

e

e

t m

u

u u E+ u×B

t

B

×E

0 B

0 hc ×B j j

iit

x

v hot electrons equations: ; ii

e

e

t m

pE v ×B

2 2 1/2 , 1 /i i i i i c

p v v

ih hi

en j v

c cenj u

Siple Simulation

Geometry Background

0 zz x y z y z z xE v B v B v v

0 zz y x z x z z yE v B v B v v

t x z yB E

t y z xB E

y z xv B

1D FDTD electron MHD model in Cartesian geometry

x z yv B

0ne / ee m

z is pointing along B0

0 1

x z z xy

xy z x z

y z y z z x zv B v B v vh

h hh h

Eh h

E

t yx y y zzB Eh h h

1D electron MHD model in Dipole geometry

yx y zz yv Bh h h

t y x zz x xB Eh h h

xy x zz xv Bh h h

0 1

y z z yx

y zx z y z

y z y z z xv B v B v vh

h hh h

Eh h

E

2 2 3 2 2sin 1 3 cos , sin , 1 3 cosx E y z Eh h r h r Rr R

r

1 KHz and 3 KHz signals

Str

elts

ov

et

al.

[200

7]

1 KHz, 1.5 KHz and 3 KHz signals

Dipole Magnetic Field

Dipole Magnetic Field

||

( ) B

A

g

dT f

l

v1

)( () ( )T ff fT T

2 2 2

||1 /

ce pec k

T

2

||

|| 22 2 2

|| ||

2

/

ce pe

g

pe

k c

k k c

v

1D Whistler Dispersion relation:

Parallel group velocity:

Traveling time:

VLF wave examples from SipleVLF wave examples from Siple(courtesy E. Mishin and A. Gibby)(courtesy E. Mishin and A. Gibby)

Fluid + PIC CodeBefore we run simulations on 512 grid cells with 500 particles per cell or 256,000 particles total.

Simulations on the grid with 5000 cells and with 500 particles per cell (2,500,000 particles total) let us to resolve about 10% of L=4.2 magnetic field line.

Test of a new Fluid + PIC Code (time dynamic of Ey)

Summary

Our Results Our Goal