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.