Full-Wave Modelling of VLF Wave Scattering and Propagation...

STANFORDE L E C T R I C A LE N G I N E E R I N G

Full-Wave Modelling of VLF Wave Scattering andPropagation in Curvilinear Stratified Ionosphere

Nikolai G. Lehtinen, Timothy F. Bell, Linhai Qiu, Morris B. Cohenand Umran S. Inan

EE Department (STAR Laboratory), Stanford University, Stanford, CA, U.S.A.ICEAA ’12, Cape Town, South Africa

September 3, 2012

STANFORDE L E C T R I C A LE N G I N E E R I N G

N. Lehtinen (Stanford) EM Waves in Ionosphere September 3, 2012 1

StanfordFWM code STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline

1 Stanford Full-Wave Method (StanfordFWM) codeDescriptionExample: VLF transmitter radiation

2 Scattering calculations with the method of moments (MoM)HF heating disturbanceMethod of moments descriptionResults

3 StanfordFWM in orthogonal curvilinear coordinatesIntroductionValidation in cylindrical coordinatesEarth-ionosphere waveguide modes

4 Conclusions


StanfordFWM code Description STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



Capabilities and applications

Capabilities:Arbitrary plane stratified medium, e.g., a horizontally-stratifiedmagnetized plasma with an arbitrary direction of geomagnetic field(such as ionosphere)Arbitrary configuration of harmonically varying currentsProvides full wave 3D solution of both whistler waves launchedinto ionosphere and VLF waves launched into Earth-ionospherewaveguideStable against the “swamping” instability by evanescent wavesEfficient use of the computer resources, easily parallelized

Applications:Trans-ionospheric propagationEarth-ionosphere waveguide propagationScattering on D-region disturbances



Algorithm

z

xy

Ne

B

z2

zk

zk+1

zM

z1=0

εk

We work in Fourier (horizontal wave vector k⊥) domain:1 For each k⊥ = const (Snell’s law) =⇒ find kz , E and H in each

layer for each of 4 plane wave modes: 2 up (u) and 2 down (d)2 Use continuity of E⊥ and H⊥ between layers to find reflection

coefficients Ru,d and mode amplitudes u, dRecursion order Ru

k+1 → Ruk and uk → uk+1 provides stability

against “swamping” of solution by evanescent wavesRepresent source currents as boundary conditions on E⊥ and H⊥between layers

3 Inverse Fourier transform from k⊥ to r⊥N. Lehtinen (Stanford) EM Waves in Ionosphere September 3, 2012 5


Booker equation: kz and {E,H} in each layer

From Maxwell’s equations, obtain the propagation equation alongz for “continuous” components E⊥, H⊥ [Clemmow and Heading,1954, doi:10.1017/S030500410002939X]:

∂z

ik0f = Lf, f =

(E⊥

Z0H⊥

), L(k⊥, K) is a 4× 4 matrix

where k0 = ω/c and K = K(ω) is the dimensionless constitutivetensor, connecting (D/ε0, cB)T with (E,Z0H)T

Find q = kz/k0 and (E⊥,Z0H⊥)T as eigenvalues and eigenvectorsof the above equationExpress the remaining components in terms of the continuouscomponents:(

EzZ0Hz

)= Lf, L(k⊥, K) is a 2× 4 matrix


StanfordFWM code Example: VLF transmitter radiation STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



Configuration



Upward flux in k⊥-space

Earth-ionosphere waveguide modes manifest as maxima in k⊥-space

k⊥ = {kx , ky }

r⊥ = {x , y }k⊥ ↔ r⊥

by Fourier transform.

Pup =

∫∫Sz(k⊥)

d2k⊥(2π)2

– gives a moreaccurate result than∫∫

Sz(r⊥) d2r⊥:

Pup

P= 13%

nx=k

x/k

0 (k

0=w/c)

n y=k y/k

0

NPM: iri 00h 22−Jun−2010Upward power flux at h=137.5 km in k⊥ −space

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

log10

Sz, W

7

8

9

10

11

12

13

14



EM field in position (r⊥) space (ground and space)Features of VLF radiation from NPM (f = 21.4 kHz, P = 424 kW, B0 = 34 µT, d = 38.4◦):

Mode interference (both on the ground and in space)

Higher attenuation westward on the ground

Radiation higher along B0 into space

West-East asymmetry for radiation into space

x, km (← geomag. W-E→)

y,km

(←geom

ag.S-N→)

NPM: iri 00h 22−Jun−2010Field on the ground

−600 −400 −200 0 200 400 600−600

−400

−200

0

200

400

600

log10

B⊥ , T

−11

−10.5

−10

−9.5

−9

−8.5

x, km (← geomag. W-E→)

y,km

(←geom

ag.S-N→)

NPM: iri 00h 22−Jun−2010Upward power flux at h=137.5 km

−600 −400 −200 0 200 400 600−600

−400

−200

0

200

400

600

log10

Sz, W/m2

−10.5

−10

−9.5

−9

−8.5

−8

−7.5

−7

−6.5


Scattering with MoM STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions


Scattering with MoM HF heating disturbance STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



VLF scattering on D-region disturbancesThe VLF perturbations are caused by D-region disturbances due to HF heaters and can becalculated using Earth-ionosphere waveguide mode theory:

with WKB and Born approximations [Barr et al, 1985; Demirkol, Ph. D thesis, 1999].with Born but no WKB [Lehtinen et al, 2011]neither Born nor WKB [Foust et al, 2011; present work]

We use StanfordFWM together with the method of moments (MoM), which uses less computerresources than discontinuous Galerkin (DG) finite element method [Foust et al, 2011].



VLF scattering by an HF heater: NLK/HAARP

NLK

HAARP

150 ° W 140° W 130° W

120° W

50 ° N

60 ° N

70 ° N

NLK VLF transmitter:Modelled as a ground-based verticaldipolef = 24.8 kHzP = 250 kW

HAARP HF heater:fHF = 5 MHzERP = 1 GWBeam width ∼23 km [Payne et al, 2007],we assume Gaussian horizontal shape∆Te and ∆νe are found using kineticequations



Incident VLF wave: Strongest waveguide modesAt R0 = 2000 km (location of the disturbance)

Modes are calculated using StanfordFWM using night-time ionosphereAttenuation is due to both absorption and radiation into ionosphere

10−5

10−4

10−3

10−2

0

20

40

60

80

100

120

E, V/m, at 2000 km from transmitter (P=250 kW)

h, k

m

QTM1QTM2QTM3QTM4



Change in νe due to heatingKinetic model results for steady heating starting at t = 0

106

107

108

109

1010

80

100

120

Ambient Ne, m−3

h, k

mfHF

=5 MHz, fmod

=0 Hz (T=Inf ms)

102

103

104

105

106

107

80

100

120

νe, s−1

h, k

m

backgroundt=0 mst=6 mst=12 mst=18 mst=24 mst=30 ms


Scattering with MoM MoM description STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



Previously used Born approximation

Neglect the scattered field Es compared to the incident field E0inside the perturbed regionE0 acting together with the perturbation ∆σ creates currents whichradiate Es

What if Es is comparable to E0?



Motivation: ∆σ may be large

0 0.5 1 1.5 2 2.5 3 3.5

x 10−4

80

90

100

110

120

130

140h

, km

max|σeig

|, S/m

Background

Heated

∆σ



Summary of the method of moments (MoM)

Green’s function is a 3× 3 matrix G with components

Gij(ro, rs) ≡ Ei(ro) created by current J(r) = xjδ(r − rs)

rs — source positionro — observer position

In our case, Green’s function is in the stratified medium, andcurrents J = ∆σE are due to conductivity perturbation.We have an integral equation for the scattered field Es:

Es(r) =∫

G(r, r ′)∆σ(r ′)[E0(r ′) + Es(r ′)

]d3r ′

where the integration is over the perturbed region (∆σ 6= 0).MoM makes use of discretisation of J, E. Then the integralequation is solved numerically, and involves an inversion of a largematrix.


Scattering with MoM Results STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



3D calculation of Ez from scattering of QTM1 modeThe field is significantly modified inside the lower part of the perturbation region (outlined by thin yellow line)

Vertical slice

Interpolated |Ez|

x, km

h, km

−60 −40 −20 0 20 40 60

90

100

110

Calculated |Ez|

x, km

h, km

−60 −40 −20 0 20 40 60

90

100

110

Interpolated Ez, φ=0

°

x, km

h, km

−60 −40 −20 0 20 40 60

90

100

110

Calculated Ez, φ=0

°

x, km

h, km

−60 −40 −20 0 20 40 60

90

100

110

0.010.020.030.040.05

0.010.020.030.040.05

−0.05

0

0.05

−0.05

0

0.05

Horizontal slice

Interpolated |Ez|, h=90 km

x, km

y,

km

−50 0 50−60

−40

−20

0

20

40

60

0

0.005

0.01

0.015

Calculated |Ez|, h=90 km

x, km

y,

km

−50 0 50−60

−40

−20

0

20

40

60

0

0.005

0.01

0.015

Interpolated Ez, h=90 km, φ=0

°

x, km

y,

km

−50 0 50−60

−40

−20

0

20

40

60

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

Calculated Ez, h=90 km, φ=0

°

x, km

y,

km

−50 0 50−60

−40

−20

0

20

40

60

−0.015

−0.01

−0.005

0

0.005

0.01

0.015



Error in ∆J = ∆σE due to Born approximationThe Born approximation, e.g., overestimates Jz at higher altitudes. MoM shows that the current is smaller, possibly due to field

being decreased by the perturbation.

0 0.5 1 1.5 2 2.5 3 3.5

x 10−5

80

85

90

95

100

105

110

115

120

∆J, A/m2

h,

km

∆J

x

∆Jy

∆Jz

∆Jx (Born appr)

∆Jy (Born appr)

∆Jz (Born appr)



Amplitude change on the groundMoM gives a lower scattered field than Born approximation. The pattern of ∆A on the ground is also modified, especially close to

the scatterer (shown by purple circle).

→VLF propagation→Method of Moments, ∆A∈[−0.00632,0.00624] dB

y, km

−200 0 200 400 600 800 1000−200

−100

0

100

200∆A, dB

−6

−4

−2

0

2

4

6x 10

−3

Born approximation, ∆A∈[−0.0202,0.0213] dB

x, km

y, km

−200 0 200 400 600 800 1000−200

−100

0

100

200

−0.02

−0.01

0

0.01

0.02



Upward flux change at 137.5 kmAgain, Born approximation overestimates the scattering strength.

Method of Moments, ∆Sz∈[−3.59,0.0813] dB

y, km

−200 0 200 400 600 800 1000−200

−100

0

100

200

∆Sz, dB

−3

−2

−1

0

Born approximation, ∆Sz∈[−11.8,0.067] dB

x, km

y, km

−200 0 200 400 600 800 1000−200

−100

0

100

200

−10

−8

−6

−4

−2

0


Curvilinear StanfordFWM STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions


Curvilinear StanfordFWM Introduction STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



Motivation

The mode height gains and therefore the attenuation coefficients may bedifferent if we include Earth’s curvatureThe previous models (LWPC) only included curvature in the isotropic part of theEarth-ionosphere waveguide and used cylindrical geometryThe curved stratification may be used for other problems, e.g., ducting ofwhistlers by a curved boundary of plasmasphere.

We would like to include curvatures in both directions (x1, x2) in a medium which istranslationally-symmetric in these direcions:



Modifications to StanfordFWM

Operator L transports the transverse (ξ1, ξ2)-components of the E/Mfield ∝ e−iωt in ξ3 ≡ Z -direction:

Lf =1

ik0

∂f∂Z

, f =(

E⊥H⊥

), k0 =

ω

c

Consider a curvilinear orthogonal coordinate system {ξ1, ξ2, ξ3} withscale factors {h1,h2,h3}, such that

h1,2 = 1 + α1,2ξ3, h3 = 1, ξ3 ≡ Z

where α1,2 = const are curvatures. Examples:cylindrical: Z = ρ− R, α1 = 1/R, α2 = 0spherical: Z = r − R, α1 = 1/R, α2 = 1/R

When hi = 1 (i.e., at Z = 0), we have

L = Lflat −α1

ik0I1 −

α2

ik0I2

where I1,2 are projection operators on ξ1,2.N. Lehtinen (Stanford) EM Waves in Ionosphere September 3, 2012 29

Curvilinear StanfordFWM Validation in cylindrical coordinates STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



Comparison with analytic solutionPerfect conductor at r = a = 2, the source current ||z is at r = b = 2.5

2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

1.2

1.4

r

E,H

Analytical results (solid: flat, dashed: cylindrical)ie

z=1 and i

m

φ=0 at r=b=2.5, k

0=4, k

z=0, k

r=4

|Er|

|Ez|

|Hφ|

2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

1.2

1.4

r

E,H

Cylindrical results (solid: FWM, dashed: analytic)ie

z=1 and i

m

φ=0 at r=b=2.5, k

0=4, k

z=0, k

r=4

|Er|

|Ez|

|Hφ|

These FWM results coincide with analytic in the right panel. The × signs show thechosen stratification boundaries (i.e., in this case ∆r = 0.1). Relative error < 2× 10−3

and is due to imperfect radiation condition at r = 3 (currently using flat geometry, incurvilinear this translates into a presence of a small reflected wave).



An example with shorter wavelength

2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

1.2

1.4

r

E,H

Analytical results (solid: flat, dashed: cylindrical)ie

z=1 and i

m

φ=0 at r=b=2.5, k

0=20, k

z=10, k

r=17.321

|Er|

|Ez|

|Hφ|

2 2.2 2.4 2.6 2.8 30

0.2

0.4

0.6

0.8

1

1.2

1.4

r

E,H

Cylindrical results (solid: FWM, dashed: analytic)ie

z=1 and i

m

φ=0 at r=b=2.5, k

0=20, k

z=10, k

r=17.321

|Er|

|Ez|

|Hφ|

Again, FWM is visibly different from the flat geometry case and gives good agreementwith the analytic solution. Relative error < 1× 10−4.


Curvilinear StanfordFWM Earth-ionosphere waveguide modes STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions


Curvilinear StanfordFWM Earth-ionosphere waveguide modes STANFORDE L E C T R I C A LE N G I N E E R I N G

Earth-ionosphere waveguide modesPropagation in x-direction

Curvature in y -direction did not change the results appreciably (i.e.,cylindrical Earth assumption gave correct results.

0.95 1 1.05 1.1

0

2

4

6

8

10

vph

/c

Attenuation, dB

/Mm

No curvature

With curvature


Conclusions STANFORDE L E C T R I C A LE N G I N E E R I N G

Outline




4 Conclusions



Summary

We have developed and used StanfordFWM (Stanford full-wavemethod) code which

calculates VLF propagation in ionosphere and theEarth-ionosphere waveguidecan be used to calculate scattering on D-region disturbanceshas been generalized to curvilinear coordinates

Calculation results indicate thatthe satellite-observed waves have been scattered on irregularitiesuse of Born approximation overestimates scattering on strongdisturbancesthe Earth’s curvature contributes significantly to attenuation ofmodes



Acknowledgments

This work was supported by the following grants to Stanford University:

AFRL FA9453-11-C0011DARPA HR0011-10-1-0058DTRA HDTRA1-10-1-0115



DEMETER pass over NWC transmitter on October 24, 2006, starting at 14:50:40 UT: (a) Ne ; (b) VLF energy flux at 700 km;

(c) data and StanfordFWM results for two profiles of electron-neutral collision rate νe [Lehtinen and Inan, 2009].

Ew

spectrogram

f, k

Hz (c)

19.2

19.4

19.6

19.8

102

103

104

105

Ew

, µ

V/m

Ew

data

FWM, νen

[V02]

FWM, νen

[H65]

Bw

spectrogram

f, k

Hz

19.2

19.4

19.6

19.8

−24 −22 −20 −18 −16 −14 −12 −10 −810

0

101

102

103

Latitude, deg

Bw

, p

T

Bw

data

FWM, νen

[V02]

FWM, νen

[H65]

100

101

102

103

104

80

85

90

95

100

105

110

Ne, cm

−3

h,

km

(a)

(b)

Longitude, deg

La

titu

de

, d

eg

DEMETER passes over NWC

110 112 114 116 118

−24

−23

−22

−21

−20

−19

−18

−17

dB

re

W/m

2

−95

−90

−85

−80

−75

−70

−65

−60

At the peak: Calculations overshoot observationsOff the peak: Structures in E field (absent in calculations)

Starks et al [2008]: models overestimate VLF field by ∼10 dB (day) to ∼20 dB (night) for L > 1.5.

The deficit of measured VLF energy points to scattering on irregularities in the ionosphere.



0+ whistlers vs QEMW

0+ whistler mode waves:Small values of k⊥ (in respect to B)Circularly polarizedElectromagnetic, i.e. cB/E ∼ n with the refractive index n ∼ 10–12for presented observations

QEMW:High values of k⊥ (in respect to B)Longitudinal polarization i.e. E ‖ kQuasi-electrostatic, i.e. cB/E � 1


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Full-Wave Modelling of VLF Wave Scattering and Propagation...

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