Physics Department, University of L’Aquila, Italy

ULF FIELD-LINE RESONANCES

IN THE EARTH’S MAGNETOSPHERE

Massimo Vellante

BG - URSI School onWaves and Turbulence Phenomena in Space Plasmas1-9 July, 2006, Kiten, Bulgaria

1

OUTLINE

• MHD wave modes in a uniform, cold plasma;

• MHD wave modes in a dipole field;

• Uncoupled toroidal and poloidal modes, eigenfunctions

calculation for the Earth’s magnetosphere;

• Field line resonance (FLR): basic theoretical characteristics

(Southwood’s box model);

• Effect of the ionosphere;

• Methods for detecting FLRs: gradient technique;

• Experimental observations of FLRs in space and on the ground;

• Monitoring the magnetospheric dynamics by FLRs.

ULF FIELD-LINE RESONANCESIN THE EARTH’S MAGNETOSPHERE

2

The Magnetosphere

3

Geomagnetic pulsations

ULF (1 mHz – 1 Hz) oscillations of the geomagnetic field observed

both in space and on the ground.

In 1954, Dungey suggested that the common occurrence of regular

geomagnetic pulsations with distinct periods could be the signature of

wave resonances in the magnetospheric plasma.

Magnetospheric waves in the ULF frequency range can be described

using the MHD approximation: frequency lower than the ion girofrequency.

4

1) e = b/t Faraday’s law

2) b = o j Ampère’s law

3) e = v B Ohm’s law (frozen field)

4) v/t = j B Momentum equation

Basic MHD equations for small amplitude perturbations (b, e, v, j)

in a cold, highly conducting, collisionless, magnetized fluid:

In the ionosphere, where collisions are

important:

3’) j = o e// + P e + H e (B/B)

o: parallel conductivity

P: Pedersen conductivity

H: Hall conductivity

2e/t2 = VA VA e

VA = B/(o)1/2 = Alfvén velocity

5

MHD WAVE MODES IN A UNIFORM, COLD PLASMA

1) ALFVEN (TRANSVERSE) MODENon-compressional, guided along the ambient field

2) FAST (COMPRESSIONAL) MODENo preferencial guidance: isotropic mode

b B

Poynting vector S // B

= k// VA VA = B/(o)1/2

b// 0

S 0

= k VA

v,b

e

jk

B

B

kv

b

j,e

6

A typical Alfvén velocity in the magnetosphere is 103 km/s,

while typical periods of ULF waves (geomagnetic pulsations)

are in the range 10 – 500 s.

Thus, typical wavelengths are in the range 1 -100 RE:

comparable or even greater than the size of the magnetosphere.

Therefore, the uniform cold plasma approximation is not

appropriate.

7

MHD WAVE MODES IN A DIPOLE FIELD

)1

( 2

2

221tV

ggA

h2 e = )( 1 t

g

h3 b

)1

(2

2

212tV

ggA

h1 e = )( 2 t

g

h3 b

t

(h3 b ) = (h1 e ) (h2 e )

g1, g2, h1, h2, h3: dipole metric functions

Transverse and compressional modes are coupled

1)

2)

3)

8

Axial symmetry: / = 0

)1

(2

2

221tV

ggA

h2 e = 0

TOROIDAL MODE

Polarization: e , b , v

Azimuthal oscillations of plasma and field lines.

Independent, in-phase torsional oscillation of

individual magnetic shells.

Poynting vector S e b along B

Wave guided along the field line.

h1 e = 0

POLOIDAL MODE

2

2

22

2

2121

tVggg

A

Polarization: e , b , b , v

Plasma and field line oscillations in the

meridian plane.

Propagation across the magnetic field.

b = b// 0 compressional mode

9

Localised mode: /

Polarization: e , b , v

GUIDED POLOIDAL MODE

)1

(2

2

212tV

ggA

h1 e = 0

b,vS

e

10

SCHEMATIC PLOT OF THE GUIDED STANDING

OSCILLATIONS IN THE MAGNETOSPHERE

Magnetic field lines have fixed ends in the ionosphere, assumed as a perfect conductor.

Multiple reflections of the guided Alfvén wave generate a standing structure.

POLOIDAL TOROIDAL

11

Symmetry relations of wave hodograms at conjugate stations allow to determine

if the mode is odd (case a) or even (case b). Lanzerotti et al. (1972).

CONJUGATE OBSERVATIONS

(a), odd mode (b), even mode

H North

D East

12

GUIDED EIGENMODES CALCULATION

AXISYMMETRIC TOROIDAL MODE LOCALISED POLOIDAL MODE

0)(

)31(2

222

2

2

νν εε

zVzr

dz

d

Aeq

0

)()31(

31

62

222

22

2

εεε

zVzr

dz

d

z

z

dz

d

Aeq

r

req

r = req sin2

RE

z = cos

VA = B/(o)1/2 = Alfvén velocity1) specifying a model for the plasma

density distribution (z);

2) using boundary conditions for the electric field

, = 0 at r RE (ionospheric level);

3) equations can be solved numerically

to find the eigenfrequencies .

Temporal variations: exp(-it)

13

WKB / TIME-OF-FLIGHT APPROXIMATION

P1

P2

n = 1,2,3,…

field-line eigenperiods time-of-flight :

2

1 )(

2 P

P sV

dsn

AnT

2

1

2/1

2/1

)(

)(2 P

Pods

sB

sn

nT

VA(s) = B(s) / [o (s)]1/2 = field-aligned Alfvén velocity

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f5 / f1 = 6.09 8.33

f4 / f1 = 4.82 6.59

f3 / f1 = 3.55 4.84

f2 / f1 = 2.28 3.08

TOR POL

15

Latitudinal variation of the fundamental field-line eigen-period

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EIGEN-OSCILLATIONS OF LOW-LATITUDE FIELD LINES

only H+

both H+ and O+

South Africa array

(Hattingh and Sutcliffe, 1987)

17

Space observations of field-line eigen-scillations

Anderson et al., 1989

18

An example of a pulsation event with latitude-dependent period

Φ = 52°

Φ = 47°

Φ = 41° Miletits et al., 1990

19

L

3.83

3.28

3.04

2.75

2.44

3.83

3.28

3.04

2.75

2.44

167- 200 s 117- 133 s 87- 105 s 69 - 80 s

167- 200 s 117- 133 s

87- 105 s 69 - 80 s

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Diurnal and latitudinal polarization pattern

Polarization reversals across:

- noon meridian

- latitude of maximum amplitude

Kelvin-Helmholtz instability

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Field line resonance model

(Southwood; Chen and Hasegawa, 1974)

IONOSPHERE

IONOSPHERE

l

EARTH

x -H

z

MAGNETOPAUSE

y D

Plasma density = (x)

eigenfrequency

R(x) = ( π / l ) VA (x) =

= ( π / l ) B [o (x)]-1/2

B

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IONOSPHERE

IONOSPHERE

monochromatic wave: e - i t

l

x xR

z

y

R= R(x) resonance: R=

assuming magnetic perturbations of the form:

b = b (x) exp [i (k// z + my - t)],

near the resonance:

0)]([

222

2

2

2

xx

R

x bmdx

db

xdx

bd

1

)( 1

dx

d

scale length of the inhomogeneity

singularity at x = xRbecause of dissipative effects: - i

By expanding R(x) around R(xR):

01 2

2

2

xx

R

x bmdx

db

ixxdx

bd

= 2γδ/ω = resonance width

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01 2

2

2

xx

R

x bmdx

db

ixxdx

bd

Solutions are given by the modified Bessel functions,

bx bo log [m (x – xR – i ) ]

close to xR : i bo by (xR) by = m (x – xR - i) 1+ i (x - xR) /

At the resonant latitude xR :

a) More pronounced peak and sharper phase change for by, i.e. in the azimuthal direction:

Toroidal mode dominates;

b) The horizontal polarization is predicted to reverse the sense across the resonance where

becomes almost linear;

c) The horizontal polarization changes sense according to the sign of the azimuthal wave

number m: if the driving source is due to the Kelvin Helmholtz instability polarization reversal

across the noon meridian.

AMPLITUDE (by)

PHASE (by)

x

x

xR

xR

-π/2

π/2

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CW CCW CW

Theory Observations

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IONOSPHERIC EFFECT (Hughes, Southwood, 1976)

The Pedersen current shields the incident signal. On the ground, we observe the signal

generated in the E-region (altitude ~120 km) by the Hall current.26

On the ground, the signal is rotated through 90° and smoothed: features with

scale lengths less than ~120 km (E-region altitude) are strongly attenuated.

Ionospheric effect on the resonance structure

Magnetospheric signal

Ground signal

Hughes and Southwood, 1976800 400 0 - 400 - 800 800 400 0 - 400 - 800

LATITUDE PROFILE (km)resonance

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Observed spectrum

Source spectrum

Resonance response

Resonance effects can be easily masked by:

- peculiarities of the source spectrum

- ionospheric smoothing 28

Detection of different harmonics at L = 2.3 by the H/D technique(Vellante et al., 1993)

29

Gradient method for detecting field line resonances from

ground-based ULF measurements (Baransky et al., 1985)

N

S

Higher latitude field line → Lower resonant frequency ( N )

Lower latitude field line → Higher resonant frequency ( S )

EARTH

Separation: 1°- 3°

1 < 2 < 3

x2

N S

AM

PL

ITU

DE

RE

SP

ON

SE

SOUTH

SOUTH

PH

AS

E R

ES

PO

NS

E

x3x1

x2x1 x3

N S

A N

/A S

1

2

Δx/

2

S -

N2 tan-1(Δx/2)

AMPLITUDE RATIO

CROSS-PHASE

30

An example of gradient measurements (Green et al., 1993)

Power spectra at TM, L = 1.59

Power spectra at AK, L = 1.51

H/D

Amplitude ratio

Cross-phase

Δx ~ 240 km

fR = 78 mHz

31

CHAMP trajectory and SEGMA lines of force in a meridional plane. CHAMP spends about

1.5 min to cover the latitudinal range of the SEGMA array. The longitudinal difference between

CHAMP and SEGMA is less than 4°.

Ground-satellite comparative studyEvent of July 6, 2002 (Vellante et al., 2004)

CHAMP trajectory

SEGMA array

32

Magnetic field data from CHAMP and SEGMA array. The data are filtered in the frequency band 20–100 mHz.

The gray region indicates the time interval of the conjunction. The stars indicate the conjunction to each station.

N C R A

33

CHAMP azimuthal

compress.

SEGMA - H

SEGMA - D

NCK

CST

RNC

AQU

NCK

CST

RNC

AQU

Spectral analysis

NCK - CST, L=1.82

CST - RNC, L=1.70

RNC - AQU, L=1.60

azimuthal comp: peak at ~65 mHz

compressional comp: peak at ~55 mHz

ground stations: peak at ~55 mHz

FLR frequency at L=1.60: ~55 mHz

Source frequency: 55 mHz

Resonance at L=1.60

To be explained the

higher frequency of

the azimuthal comp.

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Simulation of the signals observed by the CHAMP satellite

Resonance structure, = 80 km, VSAT = 7.6 km/s

Amplitude

Phase

driving signal

(f = 55 mHz)

forced signal

signal

power spectra

resonance

At the resonant point :

Vphase = 2π f ~ 30 km/s

VSAT = 7.6 km/s

equatorward

Vphase = 30 km/s

poleward

Resonant signal shifted in frequency

in the satellite frame of reference.

The shift expected from the theoretical

FLR structure agrees with that

experimentally observed (~ +20%).

VSAT

35

fR (midday)

53 mHz

64 mHz

69 mHz

Dynamic cross-phase analysis at SEGMA array

An example of diurnal variation of the resonant frequency

36

An example of harmonics detection

fR (midday) f2 / f1

40 mHz ~ 2.1

52 mHz ~ 2.0

66 mHz ~ 1.8

f2 / f1 < 2 at L = 1.6 in agreement with theoretical expectations (Poulter et al, 1988)37

Monitoring the plasma dynamics during geomagnetic storms

detection of plasma depletion (Villante et al., 2005)

pre-storm recovery phase

38

ANNUAL VARIATION OF THE FLR FREQUENCY AT L = 1.61, YEAR 2003

DAILY AVERAGES (0900 – 1600 LT)

27 Days

a nearly 27-days modulation appears which must be

connected to the recurrence of active regions of the Sun

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SOLAR IRRADIANCE DEPENDENCE OF THE FLR FREQUENCY (L = 1.61)

An increase of the solar EUV/X-ray radiation increases the ionization rate in the ionosphere.

This influences the whole distribution of the plasma along the low-latitude field line.

Vellante et al., 2006

40

Solar Cycle Variation of the Field Line Resonant Frequency at L’Aquila (L = 1.56)

Assuming ρmax ≈ 2 ρminfr

2

Vellante et al. (1996)

41

Plasmapause identification

3 4 5 6 7 8 9

L value

3 4 5 6 7 8 9

L value

Menk et al., 2004 42

Mid-continent MAgnetoseismic Chain (McMAC):

A Meridional Magnetometer Chain for Magnetospheric Sounding

1.3 < L < 11.7

43

A schematic representation of field line

resonances driven by resonant magnetospheric

cavity modes. The top panel shows the periods

of three harmonics of cavity resonance, which

do not vary with L-shell, and the variation of the

fundamental field line eigenperiod with L shell.

There are three L shells where the fundamental

field line eigenperiod matches one of the cavity

mode eigenperiods. In the lower panel the

variation of wave amplitude at each of the three

cavity eigenperiods with L shell is drawn.

Note how a field line resonance is driven each

time a field line eigenperiod match.

(Hughes, 1994).

FLRs – cavity modes coupling

44