SCATTERING OF ELECTROMAGNETIC WAVES ON TURBULENT PULSATIONS INSIDE A PLASMA SHEATH

Transcription

1 SCATTERING OF ELECTROMAGNETIC WAVES ON TURBULENT PULSATIONS INSIDE A PLASMA SHEATH 29 July, 2011 VLADIMIR SOTNIKOV Sensors Directorate Air Force Research Laboratory

2 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

3 Overview! 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

4 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

5 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

6 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

7 Basic Equations For Compressible Plasma Flow With Velocity Shear Linearized mass conservation equation for ions, electrons and neutrals n 1α t + u 0α y n 1α y + N 0α u 1α = 0 j = i,e,n Poisson equation for electrostatic potential in oscillations Δϕ = 4πe (n e Zn i ) Flow velocity can be presented in the following form v i = V 0y (x) e y + δ v i (x,y,t)

8 Dimensionless Variables Dimensionless electrostatic potential in IA wave 2 M i ω 2 Δi pi Φ( x, y)= Φ(x,y) Z e

9 Plasma And EM Wave Parameters Electron density Neutral density Electron temperature Debye radius Electron plasma frequency Ion plasma frequency Frequency of GPS signal Atomic number of Potassium ions Wave number of incident EM wave Minimum wave number in IAW Ion Thermal speed Vehicle speed Doppler shift Electron-neutral collision frequency

10 Poisson Equation Dependence of perturbed quantities on time and coordinates F(x,y,t) ~ f (x)exp(ik y y iωt) L e Equation For Eigenfunctions d 2 φ dx k 2 φ φ + n 2 i = mωω e d 2 φ dx k 2 φ + n 2 i L e d 2 dx + 2k dv 0 2 Ω e dx d dx k 2. m ΩΩ e k 2 1, Decoupling of electron and ion dynamics L e d 2 φ dx k 2 φ φ + n 2 i = 0

11 Equation For Eigenfunctions D 2 2 x 2 Φ + D 1 x Φ + D 0 Φ = 0 D 2 = ε i = 1 1 ΩΩ i D 1 = 2k dv 0 dx 1 1 ν in ν ni 2 2 ΩΩ i 2 Ω n D 0 = k 2 ε i + 1 ( ) Ω ω kv 0 ( x) Ω i ω kv 0 ( x) + iν in 1 iν ni Ω n Ω ω kv x n 0 ( ) + iν ni.

12 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 dφ dx + kφ x= L = 0

13 Linear Velocity Profile Normalized to ion sound speed flow velocity Normalized to ion Debye length coordinate across plasma sheath

14 Growth Rate Of Instability With Linear Velocity Profile Frequency and growth rate normalized to ion plasma frequency a) Real part of the frequency ω b) The growth rate γ as a function of kl

15 Growth Rate Of Instability With Linear Velocity Profile Growth rate normalized to ion plasma frequency Peak growth rate as a function of neutral density for linear velocity shear with c 0 = 0.5 / L and c 0 = 1.0 / L in the collisionless limit

16 Gaussian Velocity Profile Normalized to ion sound speed flow velocity Normalized to ion Debye length coordinate across plasma sheath

17 Growth Rate Of Instability With Gaussian Velocity Profile Growth rate γ as a function of kl Growth rate normalized to ion plasma frequency Normalized wave vector

18 Velocity Profiles Produced By CFD Code Vulkan V 0 (x) = u 0 y (x) / V s = tanh(x / α) tanh(l / α) Normalized to ion sound speed flow velocity Normalized to ion Debye length coordinate across plasma sheath

19 Growth Rate Of Instability With Velocity Shear Profiles From Vulkan Simulations Growth rate normalized to ion plasma frequency Normalized wave vector

20 Growth Rate Of Instability With Velocity Profiles From Vulkan Simulations Growth rate normalized to ion plasma frequency Peak growth rate as a function of neutral density for the velocity shear profiles obtained from CFD Vulcan simulations

21 Air Density From G. Abell, Exploration of the Universe, 1982

22 Nonlinear Equations For Compressible Plasma Flow With Velocity Shear Equation of motion for ions m i n i [ v i t + ( v i ) v i ] = Zen i ϕ Pi + m i n i ν i Δ v i µ ia n i ν ia ( v i v a ) Equation for adiabatic electrons T e ne = en e ϕ Mass conservation equation for ions n i t + div(n iv i ) = 0

23 Nonlinear Equations For Compressible Plasma Flow With Velocity Shear Poisson equation for electrostatic potential in oscillations Δϕ = 4πe(n e n i ) Flow velocity can be presented in the following form v i = V 0 y (x) e y + δ v i (x, y,t)

24 Gaussian Velocity Profile V 0 y (x) = exp[ (x x mid )2 ] 2 2x wid

25 Nonlinear Stage Of Instability Potential at t = 500 Potential at t = 4000

26 Density Spectrum Of Excited IAW Turbulence Spectrum of density perturbations Normalized wave vector Sotnikov et al., TPS IEEE, 2010

27 Electromagnetic Wave Scattering Frequency Wave vector Dispersion of waves inside a plasma sheath 1 Electromagnetic wave (EM) 2 Langmuir wave (L) 3 Ion acoustic wave (IA)

28 Electromagnetic Wave Scattering EM wave scattering Nonresonant ε t (k,ω) 0 EM wave transformation Resonant ε L (k,ω) = 0

29 Scattering Of EM Wave On IA Waves Interaction of EM wave with the excited spectrum of IAW results in appearance of scattered EM waves Incident EM wave E 0 (r,t) = E k0 cos(k 0 r ω k0 t) IA Stokes component Antistokes component Nonresonant scattering ε t (k ±,ω ± ) 0 Scattered EM waves in frequency space

30 Scattering Of EM Wave On IA Waves Scattered EM waves in k - space Incident EM wave Nonresonant scattering ε t (k ±,ω ± ) 0

31 Nonlinear Coupling Model Nonlinear coupling of the GPS signal and IAW causes a beat-wave field at the combination frequencies: ω ± = ω k1 ± ω k2 The scattered wave numbers are matched according to: k ± = k 1 ± k 2 Eigenmode frequency matching condition is not necessarily satisfied ω ± ω k1 ±k 2

32 Generation Of Scattered Wave Fields ε t ( ω ±,k ± ) E t k ± = i 4π NL j ω k± ± Spectral component of the scattered EM wave field

33 Scattered Power Power going into the scattered EM waves δq = Re < δ j N ( r,t) E * ( r,t) > d 3 r δq = π 3 α 2 4 ω pe n 2 0 E 0 2 Re i d k dω ε t * k ',ω ' ω 3 δn k0 k, ω 0 ω 2 Re( i ε t* k,ω ) = ε t 2,k,ω ε 2 ( ε t ) 2 1,k,ω + ( ε t ) 2 t ε 1 2,k,ω t 2 + ε 2 t π δ( ω ω ) k 2 [ ε t 1 ω ] ω = ω ( k )

34 Scattering Cross Section σ = scattered power energy flux of the incident wave = δq c E 2 0 4π σ ~ dk Ax ε dk Ay dω 2 A ε t 1 2 t + ε 2 t 2 1 ( ω 0 ω A ) 3 δn k Ax,k Ay,ω A 2 Spectrum of the excited IA waves

35 Amplitudes Of Scattered Waves Density inside a plasma sheath 10 6 < n 0 < cm 3 Amplitudes of scattered waves Width of a plasma sheath 3 < L x < 15 cm Scattering cross section 10 5 < E ± E 0 < < σ ± < 10 8 cm 2 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

36 Resonance Coupling Model Nonlinear coupling of the GPS signal and IAW causes a beat-wave field at the combination frequencies: ω ± = ω k1 ± ω k2 The scattered wave numbers are matched according to: k ± = k 1 ± k 2 For Langmuir waves eigenmode frequency matching condition is satisfied ω ± = ω k1 ±k 2

37 Amplitudes Of Scattered Waves Density inside a plasma sheath n 0 ~ cm 3 Width of a plasma sheath 3 < L x < 15 cm Amplitudes of scattered Langmuir waves 10 2 < E L ± E 0 < 1 Scattering cross section 10 5 < σ L ± < 10 3 cm 2 Phase changes in scattered waves can impact phase-locking-in process for signals from GPS satellites

38 Electron Beam Injection Experiment CHARGE 2B Modulated electron beam VLF wave excited by the modulated electron beam with frequency ~ 18 khz rocket Wave instrument mounted on a deployable free flying diagnostic package (DFF) ELF wave radiated by the Mother payload with frequency ~ 5 khz 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.

39 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

40 Areol 3 Satellite Data Sotnikov et al., J. Geophys. Res., 1991

41 Analysis of Electron Acoustic Wave Propagation and Transformation in a Two-Electron-Temperature Plasma Layer

42 Excita'on Of Waves By Dipole Antenna In Isotropic Plasma ω ( k) 2ω pe ω = kc GHz n ~ cm 3 f pe ~ 9 *10 9 Hz ω pe 9 GHz 3 k 1 - transverse electromagne1c wave ω k = ω 2 pe + k 2 c Langmuir wave ω k = ω pe ( k 2 r de 2 ) 3 - electron acous1c wave ( T eh >> T ec ) ω s (k )=ω pc kr dh 1+ k 2 2 r dh Ra1o of the energy in transverse electromagne1c and Langmuir waves for ω ω pe W EM W L ~ 3.7 V 3 te c << 1 3

43 Dispersion Equa'on D(ζ ) = 1+ k 2 dc k [1+ζ cz(ζ 2 c )] + k 2 dh k [1+ζ hz(ζ 2 h )] + k 2 db k [1+ζ bz(ζ 2 b )] = 0 ζ c = ω 2kV Tc ; ζ h = V Tc V Th ζ c ; ζ b = V Tc V Tb ζ c 1 2 V 0b V Tb cosθ Z(ζ ) = 1 π C exp( y 2 ) dy ζ y

44 Plasma And Beam Parameters Inside The Sheath n 0c = cm 3 n 0h = cm 3 n 0b = cm 3 T ec = 0.2 ev T eh = 20 ev T eb = 0.1 ev V 0b = cm / sec ω pc = rad / sec ω ph = rad / sec

45 Damping Of Electron Acous'c Waves

46 Influence Of Inhomogeneity On The Damping Of Electron Acous'c Waves n 0e (x) Plasma sheath Atmosphere ω k ~ ω pc (x)kr Dh = n c n h kv Th k(x) ~ n h (x) n c (x) ~ const V f = ω k

47 Influence Of Collisions On The Damping Of Electron Acous'c Waves 1+ δε eh + δε ec + δε eb = 0 δε ec = ω 2 pc ω + i π ωω pc k 3 V exp{ ω 2 3 Tc 2k 2 V } + i ω 2 pcν en 2 Tc ω 3 δε eh = 1 k 2 2 r Dh γ col = Imε col Reε ω = ν en 2 κ col = γ col V g ~ 1 2 n h n c E(x) ~ E 0 exp( κ x) ν en V Th ν en < 2 n c n h V Th Δx ν en < 2 *10 9 sec -1

48 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 (E x, E z, H y ).

49 Basic Equations d 2 H y + ( ω 2 dx 2 c ε k 2 2 c tz)h y = 0 2 γ ε 0 V Th 2 E x c c(ω + iν eh ) x 2 + ω c [ε c 2 ω ph ω(ω + iν eh ) ](1 γ 0V 2 2 Th k tz ω(ω + iν eh ) )E x = (1 γ 0 V 2 2 Th ω ph c 2 (ω + iν eh ) )k tzh 2 y 2 ω ph 2 [ ω(ω + iν eh ) ε γ 0 V Th c] (ω + iν eh ) 2 E z x 2 + [ 2 ω ph 2 k tz ω ph (ω + iν eh ) ε c(ω γ 0V Th (ω + iν eh ) ]E γ 0 V Th E z ik x tz ω(ω + iν eh ) 2 x = ic H y x ε c = 1 ω pc 2 ω 2

50 Solutions Of Wave Equations H 1y = ω ' ε ck tot GE 1 exp( κ 1tx x) 1zt ' E 1x = E 1 exp( iq 1 x) GE 1 exp( iκ 1tx x) E 1z = E 1 k 1sz q 1 exp( iq 1 x) + ig κ ' 1tx ' E k 1 exp( iκ 1tx x) 1tz H 2y = ω ' ε ck tot GE 2 exp( κ 1tx x) 1zt ' E 1x = E 21 exp(iq 1 x) GE 2 exp( iκ 1tx x) E 2z = E 2 k 1sz q 1 exp(iq 1 x) + ig κ ' 1tx ' E k 1 exp( iκ 1tx x) 1tz

51 Transformation Coefficient 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: S EM = c E 2 3 4π cosθ 3 to the normal component of the energy density flux of an electron acoustic wave: S EAW = 1 k s r Dh ω pc k s 2 E s 4π Transformation coefficient W T W T = 16π ε c V ω Teh pc cosθ 3 sin 2 θ 3 c ω ph ( ω 1 cq 1 G sin2 θ 3 ε tot cosθ 3 ) 2 + ( ε c +sin 2 θ 3 )

52 Coefficient Of Transformation On The Sheath Boundary

53 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: N n ~ cm 3 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.

54 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.

55 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- AC04-94AL 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, , 2011.