VLADIMIR SOTNIKOV

Sensors Directorate Air Force Research Laboratory

29 July, 2011

SCATTERING OF ELECTROMAGNETIC WAVES ON TURBULENT PULSATIONS INSIDE A PLASMA SHEATH

In collaboration with

S. Mudaliar Air Force Research Laboratory, Sensors Directorate, Hanscom AFB, MA 01731, USA

J.N. Leboeuf JNL Scientific, Casa Grande, AZ 85194, USA

T. Genoni and D. Rose Voss Scientific, Albuquerque, NM 85294, USA

B.V. Oliver Sandia National Laboratories, NM 87185, USA

T.A. Mehlhorn Naval Research Laboratory, Washington, DC 20375, USA

! Introduction

Shear flow instability in a partially ionized compressible plasma sheath around a fast moving vehicle

Scattering and transformation of electromagnetic waves on excited density perturbations

Conclusions

Overview

Introduction

The plasma formed around hypersonic or reentry vehicles can interfere with antenna performance at microwave frequencies even at electron densities below the overdense level

Turbulence excited inside a plasma sheath leads to appearance of EM fields modulated in amplitude and phase and can have significant impact on phase-locking-in processes, in particular for GPS signals

Currently no existing computational fluid dynamic (CFD) code can be used to predict the plasma turbulence around the vehicle

We analyze growth rates of excited low frequency ion-acoustic type oscillations in a compressible plasma flow with velocity shear in the presence of neutral particles

We examine influence of such turbulent pulsations on scattering and transformation of high frequency electromagnetic waves used for communication purposes

Instability Of Plasma Flow Around A Fast Moving Vehicle

GPS signal

We are interested in the growth rates of instability of a flow with velocity

shear for different flow profiles in the presence of neutrals

Linearized Equations For Compressible Plasma Flow With Velocity Shear

Linearized momentum equation for the ions

Linearized momentum equation for the electrons

Linearized momentum equation for the neutrals

Basic Equations For Compressible Plasma Flow With Velocity Shear

Poisson equation for electrostatic potential in oscillations

Flow velocity can be presented in the following form

Δϕ = 4πe (ne − Zni )

vi =V0y (x)

ey + δ

vi(x,y,t)

Linearized mass conservation equation for ions, electrons and neutrals

∂n1α

∂t+ u0αy

∂n1α

∂y+ N0α∇ ⋅u1α = 0 j = i,e,n

Dimensionless Variables

Dimensionless electrostatic potential in IA wave

Φ(x,y)=Miωpi

2 Δ i2

Z eΦ(x,y)

Plasma And EM Wave Parameters

Ion Thermal speed Vehicle speed

Doppler shift Electron-neutral collision frequency

Electron plasma frequency Ion plasma frequency Frequency of GPS signal

Electron density Neutral density Electron temperature Debye radius

Atomic number of Potassium ions

Wave number of incident EM wave

Minimum wave number in IAW

Equation For Eigenfunctions F(x,y ,t) ~ f (x)exp(iky y − iωt)

Dependence of perturbed quantities on time and coordinates

Le

d 2φdx2 − k 2φ − φ + ni

⎡

⎣⎢

⎤

⎦⎥ = mΩΩe

d 2φdx2 − k 2φ + ni

⎡

⎣⎢

⎤

⎦⎥

mΩΩe

k 2 1,

Le

d 2φdx2 − k 2φ − φ + ni

⎡

⎣⎢

⎤

⎦⎥ = 0

Le ≡

d 2

dx2 +2kΩe

dV0

dxddx

− k 2.

Decoupling of electron and ion dynamics

Poisson Equation

Equation For Eigenfunctions

D2

∂2

∂x2 Φ + D1∂∂x Φ + D0Φ = 0

D2 = ε i = 1−

1ΩΩi

D1 =−2k

dV0

dxΩΩi

2 1−12ν inνni

Ωn2

⎛

⎝⎜

⎞

⎠⎟

D0 = − k 2ε i +1( )

Ω ≡ω − kV0 x( ) Ωi ≡ ω − kV0 x( ) + iν in 1−

iνni

Ωn

⎛

⎝⎜⎞

⎠⎟ Ωn ≡ ω − kV0 x( ) + iνni .

Boundary Conditions

At the conducting surface of the vehicle we require

φ x = 0( ) = 0

In the free space beyond the sheath edge at x = L the potential

φ x( ) = φ L( )exp −k x − L( )⎡⎣ ⎤⎦

Boundary conditions on the sheath edge

dφdx

+ kφ⎡

⎣⎢

⎤

⎦⎥

x= L

= 0

Linear Velocity Profile

Nor

mal

ized

to io

n so

und

sp

eed

flow

vel

ocity

Normalized to ion Debye length coordinate across plasma sheath

Growth Rate Of Instability With Linear Velocity Profile

Freq

uenc

y an

d gr

owth

rate

nor

mal

ized

to

io

n pl

asm

a fre

quen

cy

a) Real part of the frequency ω b) The growth rate γ as a function of kL

Growth Rate Of Instability With Linear Velocity Profile

Peak growth rate as a function of neutral density for linear velocity shear with c0 = 0.5 / L and c0 = 1.0 / L in the collisionless limit

Gro

wth

rate

nor

mal

ized

to

io

n pl

asm

a fre

quen

cy

Gaussian Velocity Profile

Nor

mal

ized

to io

n so

und

sp

eed

flow

vel

ocity

Normalized to ion Debye length coordinate across plasma sheath

Growth Rate Of Instability With Gaussian Velocity Profile

Gro

wth

rate

nor

mal

ized

to

io

n pl

asm

a fr

eque

ncy

Normalized wave vector

Growth rate γ as a function of kL

Velocity Profiles Produced By CFD Code Vulkan

V0 (x) = u0 y (x) / Vs =

tanh(x /α )tanh(L /α )

Nor

mal

ized

to io

n so

und

sp

eed

flow

vel

ocity

Normalized to ion Debye length coordinate across plasma sheath

Growth Rate Of Instability With Velocity Shear Profiles From Vulkan Simulations

Gro

wth

rate

nor

mal

ized

to

io

n pl

asm

a fre

quen

cy

Normalized wave vector

Growth Rate Of Instability With Velocity Profiles From Vulkan Simulations

Gro

wth

rate

nor

mal

ized

to

io

n pl

asm

a fr

eque

ncy

Peak growth rate as a function of neutral density for the velocity shear profiles obtained from CFD Vulcan simulations

Air Density

From G. Abell, Exploration of the Universe, 1982

Nonlinear Equations For Compressible Plasma Flow With Velocity Shear

mini[

∂vi∂t

+ (vi ⋅∇)vi ] = −Zeni

∇ϕ −

∇Pi + miniν iΔ

vi − µianiν ia (vi −va )

Equation of motion for ions

Te∇ne = ene

∇ϕ

Equation for adiabatic electrons

∂ni∂t

+ div(nivi ) = 0

Mass conservation equation for ions

Nonlinear Equations For Compressible Plasma Flow With Velocity Shear

Poisson equation for electrostatic potential in oscillations

Flow velocity can be presented in the following form

Δϕ = 4πe(ne − ni )

vi = V0y (x)

ey + δvi(x, y,t)

Gaussian Velocity Profile

V0y (x) = exp[−(x − xmid )

2

2xwid2 ]

Nonlinear Stage Of Instability

Potential at t = 500 Potential at t = 4000

Density Spectrum Of Excited IAW Turbulence

Sotnikov et al., TPS IEEE, 2010

Normalized wave vector

Spe

ctru

m o

f den

sity

per

turb

atio

ns

Electromagnetic Wave Scattering

Dispersion of waves inside a plasma sheath

1 – Electromagnetic wave (EM) 2 – Langmuir wave (L) 3 – Ion acoustic wave (IA)

Wave vector

Freq

uenc

y

Electromagnetic Wave Scattering

EM wave transformation

Nonresonant ε t (k,ω ) ≠ 0

Resonant εL(k,ω ) = 0

EM wave scattering

Scattering Of EM Wave On IA Waves

Stokes component Antistokes component

IA

Incident EM wave

Interaction of EM wave with the excited spectrum of IAW results in appearance of scattered EM waves

E0(r,t) = Ek0 cos(k0r −ωk0

t)

Scattered EM waves in frequency space

Nonresonant scattering ε t (k±,ω ± ) ≠ 0

Scattering Of EM Wave On IA Waves

Incident EM wave

Scattered EM waves in k - space

Nonresonant scattering ε t (k±,ω ± ) ≠ 0

Nonlinear Coupling Model

Nonlinear coupling of the GPS signal and IAW causes a beat-wave field at the combination frequencies:

ω ± =ωk1 ±ωk2The scattered wave numbers are matched according to:

k± = k1 ± k2Eigenmode frequency matching condition is not necessarily satisfied

ω ± ≠ ωk1 ±k2

Generation Of Scattered Wave Fields

ε t ( ω ±,k± ) E

tk±

= −i4πω ±

jk±NL

Spectral component of the scattered EM wave field

Scattered Power

δQ− = −

π 3α 2ω pe4

n20

E02 Re d k−∫ dω−

iε t *k '− ,ω '

−ω 3

−

| δnk0 − k− , ω0 −ω−|2

Power going into the scattered EM waves

δQ− = −Re < δ jN

− ( r,t) ⋅E−*( r,t) > d 3r∫

Re( i

ε t*k,ω

) = −ε t

2,k,ω

( ε t1,k,ω ) 2 + ( ε t

2,k,ω ) 2

ε2t

| ε1t |2 + | ε2

t |2≠ π

δ ( ω− − ωk−)

[ ∂ε1t

∂ ω−ω− = ω ( k− ) ]

Scattering Cross Section

σ − =scattered power

energy flux of the incident wave=δQ−

cE02

4π

σ − ~ dkAx∫ dkAydωA

ε2t

| ε1t |2 + | ε2

t |21

( ω0 −ωA ) 3 |δnkAx ,kAy ,ωA|2

Spectrum of the excited IA waves

Amplitudes Of Scattered Waves

10−17 < σ ± < 10−8 cm2

106 < n0 < 1014 cm−3

10−5 < E±

E0

< 10−1

3 < Lx < 15 cmDensity inside a plasma sheath Width of a plasma sheath

Scattering cross section Amplitudes of scattered waves

Phase changes in scattered waves can impact phase-locking-in process for signals from GPS satellites

Phase modulation of a data signal onto a carrier

Resonance Coupling Model

ω ± =ωk1 ±ωk2

The scattered wave numbers are matched according to:

k± = k1 ± k2For Langmuir waves eigenmode frequency matching condition is satisfied

ω ± =ωk1 ±k2

Nonlinear coupling of the GPS signal and IAW causes a beat-wave field at the combination frequencies:

Amplitudes Of Scattered Waves

10−5 < σL± < 10−3 cm2

n0 ~ 1010 cm−3

10−2 < EL

±

E0

< 1

3 < Lx < 15 cm

Density inside a plasma sheath Width of a plasma sheath

Scattering cross section Amplitudes of scattered Langmuir waves

Phase changes in scattered waves can impact phase-locking-in process for signals from GPS satellites

Symmetric very low frequency (VLF) scattered emissions were observed by wave instrument near the rocket. They can be explained by nonlinear coupling mechanism due to interaction between the VLF and ELF waves.

Modulated electron beam

rocket

Electron Beam Injection Experiment CHARGE 2B

Wave instrument mounted on a deployable free flying diagnostic package (DFF)

VLF wave excited by the modulated electron beam with frequency ~ 18 kHz

ELF wave radiated by the Mother payload with frequency ~ 5 kHz

Experimental Detection Of Scattered Waves CHARGE 2B Mission

A wave power spectrum diagram from the VLF wave instrument mounted aboard the deployable free flyer (DFF) with magnetic field shown versus frequency.

Sotnikov et al., J. Geophys. Res, 1994

Areol 3 Satellite Data

Sotnikov et al., J. Geophys. Res., 1991

Analysis of Electron Acoustic Wave

Propagation and Transformation in a

Two-Electron-Temperature Plasma Layer

ω = kc

ω pe

k

( )kω

1   2  

n ~ 1012 cm−3 fpe ~ 9 *109Hz

Excita'on  Of  Waves  By  Dipole                      Antenna  In  Isotropic  Plasma  

9 GHz

18 GHz

1  -­‐  transverse  electromagne1c  wave  

2  -­‐  Langmuir  wave    

3  -­‐  electron  acous1c  wave  (  Teh  >>  Tec  )  

2ω pe

ω k = ω pe2 + k2c2

ω k =ω pe(1+32k2rde

2 )

Ra1o  of  the  energy  in  transverse  electromagne1c  and  Langmuir  waves  

for ω ≥ω pe WEM

WL

~ 3.7Vte3

c3 << 1

3

ω s (k )=ω pckrdh1+ k2rdh

2

D(ζ ) = 1+ kdc2

k2[1+ζcZ(ζc )]+

kdh2

k2[1+ζhZ(ζh )]+

kdb2

k2[1+ζbZ(ζb )] = 0

Dispersion  Equa'on    

ζc =ω

2kVTc; ζh =

VTcVTh

ζc ; ζb =VTcVTb

ζc −12V0b

VTbcosθ

Z(ζ ) = 1π

exp(−y2 )ζ − yC

∫ dy

Plasma  And  Beam  Parameters  Inside  The  Sheath  

n0c = 0.4 ×1011 cm−3 n0h = 0.05 ×1011 cm−3 n0b =0.09 ×1011 cm−3

Tec = 0.2 eV Teh = 20 eV Teb = 0.1 eV

V0b = 6.5 ×108 cm / sec

ωpc = 5.64 ×1010 rad / sec ωph = 1.7 ×1010 rad / sec

Damping  Of  Electron  Acous'c  Waves  

Influence  Of  Inhomogeneity  On  The  Damping  Of  Electron  Acous'c  Waves  

ω k ~ω pc (x)krDh =ncnhkVTh

k(x) ~ nh (x)nc (x)

~ const Vf =ωk

Plasma  sheath   Atmosphere  

n0e (x)

         Influence  Of  Collisions  On  The  Damping  Of  Electron  Acous'c  Waves  

δεeh =1

k2rDh2δεec = −

ω pc2

ω 2 + i π2ωω pc

2

k 3VTc3 exp{−

ω 2

2k2VTc2 } + i

ω pc2 νen

ω 3

γ col = −Imεcol∂Reε∂ω

= −νen

2 κ col =γ col

Vg~ − 1

2nhnc

νen

VTh

νen < 2ncnh

VThΔx νen < 2 *109 sec-1

1+δεeh +δεec +δεeb = 0

E(x) ~ E0 exp(−κ x)

Wave  Transforma'on  On  A  Sheath  Boundary  

Incident  EA  wave  

Incident and reflected EA waves (in blue with index s) and incident, transformed and reflected EM waves (in red with index t) of P-polarization (Ex, Ez, Hy).

�

d2Hy

dx 2+ (ω

2

c 2εc − ktz

2 )Hy = 0

Basic Equations

�

εcγ 0VTh

2

c(ω + iνeh )∂ 2Ex

∂x 2+ωc[εc −

ω ph2

ω(ω + iνeh )](1−

γ 0VTh2 ktz

2

ω(ω + iνeh ))Ex = (1−

γ 0VTh2ω ph

2

c 2(ω + iνeh )2 )ktzHy

�

[ω ph2

ω(ω + iνeh )− εc ]

γ 0VTh2

(ω + iνeh )∂ 2Ez

∂x 2+ [

ω ph2

(ω + iνeh )− εc (ω −

γ 0VTh2 ktz

2

(ω + iνeh )]Ez − iktz

γ 0VTh2ω ph

2

ω(ω + iνeh )2∂Ex

∂x= −ic

∂Hy

∂x

�

εc = 1−ω pc2

ω 2

Solutions Of Wave Equations

�

H1y = −ωck1zt

ε totGE1 exp(−κ1tx' x)

�

E1x = E1 exp(−iq1x) −GE1 exp(−iκ1tx' x)

�

E1z = −E1k1szq1exp(−iq1x) + iGκ1tx

'

k1tzE1 exp(−iκ1tx

' x)

�

E2z = E2k1szq1exp(iq1x) + iGκ1tx

'

k1tzE1 exp(−iκ1tx

' x)�

H2y = −ωck1zt

ε totGE2 exp(−κ1tx' x)

�

E1x = E21 exp(iq1x) −GE2 exp(−iκ1tx' x)

WT = 16π | εc |VTehc

ω pc

ω ph

cosθ3 sin2θ3

( ωcq1

1Gsin2θ3− | εtot | cosθ3)

2 + (| εc | + sin2θ3)

Coefficient of transformation of electron acoustic wave into an electromagnetic wave can be defined as the ratio of the of the normal component of the energy density flux of an electromagnetic wave:

Transformation Coefficient

SEM = c E32

4πcosθ3

to the normal component of the energy density flux of an electron acoustic wave:

SEAW =1

ksrDh

ω pc

ks

Es2

4π

Transformation coefficient WT

Coefficient Of Transformation On The Sheath Boundary

Conclusions

  In the plasma sheath around a hypersonic vehicle flow with velocity shear can excite intense ion-acoustic type perturbations.

  Presence of neutrals can significantly influence the growth rate. Suppression of the instability with different velocity profiles takes place at neutral densities:

  EM scattering on density perturbations excited by the flow leads to appearance of frequency wave spectra shifted above and below the frequency of the incident wave.

  Integrated phase shift and scattering cross section can be significantly affected due to interaction with ion acoustic density perturbations.

Nn ~ 1013 ÷1014 cm−3

Conclusions  

• Damping of electron acoustic waves propagating inside the plasma sheath in the presence of an electron beam is small.

• Presence of inhomogeneity does not increase EAW damping inside the plasma sheath.

• Damping of EAW due to electron-neutral collisions for the plasma parameters inside the sheath is small.

• EAW transformation into EM wave on the sheath boundary can lead to significant attenuation of the EM wave amplitude.

Acknowledgements  

Work supported by the Air Force Research Laboratory and Air Force Office of Scientific Research.

Sandia is a multiprogram laboratory operated by Sandia Corporation, A Lockheed Martin Company, for the United States Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.

References

V. I. Sotnikov, J. N. Leboeuf, and S. Mudaliar, IEEE Trans. Plasma Sci., 2010, 9, 38. V.

V. Sotnikov, S. Mudaliar, T. Genoni, D. Rose, B.V. Oliver, T.A. Mehlhorn, Phys. Plasmas, 18, 016104, 2011.