ULF FIELD-LINE RESONANCES IN THE EARTH’S MAGNETOSPHERE Massimo Vellante Physics Department, University of L’Aquila, Italy BG - URSI School on Waves and Turbulence Phenomena in Space Plasmas 1-9 July, 2006, Kiten, Bulgaria 1

ULF FIELD-LINE RESONANCES IN THE EARTH’S MAGNETOSPHERE 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. 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 Basic MHD equations for small amplitude perturbations (b, e, v, j) in a cold, highly conducting, collisionless, magnetized fluid: 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 2e/t2 = VA  VA   e VA = B/(o)1/2 = Alfvén velocity 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 5

MHD WAVE MODES IN A UNIFORM, COLD PLASMA ALFVEN (TRANSVERSE) MODE Non-compressional, guided along the ambient field v,b b  B Poynting vector S // B  = k// VA VA = B/(o)1/2 e   k j B 2) FAST (COMPRESSIONAL) MODE No preferencial guidance: isotropic mode j,e b//  0 S  0  = k VA v   k b B 6

Thus, typical wavelengths are in the range 1 -100 RE: 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 h2 e = h3 b 1)     2) h1 e =  h3 b (h3 b ) = (h1 e )  (h2 e ) 3) g1, g2, h1, h2, h3: dipole metric functions Transverse and compressional modes are coupled 8

Polarization: e , b , b , v Axial symmetry: / = 0 TOROIDAL MODE POLOIDAL MODE h2 e = 0 h1 e = 0 Polarization: e , b , v Polarization: e , b , 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. Plasma and field line oscillations in the meridian plane. Propagation across the magnetic field. b = b//  0 compressional mode 9

Localised mode: /   GUIDED POLOIDAL MODE h1 e = 0 Polarization: e , b , v S b,v e 10

SCHEMATIC PLOT OF THE GUIDED STANDING OSCILLATIONS IN THE MAGNETOSPHERE POLOIDAL TOROIDAL 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. 11

CONJUGATE OBSERVATIONS H  North D  East (a), odd mode (b), even mode 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). 12

Temporal variations: exp(-it) GUIDED EIGENMODES CALCULATION Temporal variations: exp(-it) LOCALISED POLOIDAL MODE AXISYMMETRIC TOROIDAL MODE z = cos  VA = B/(o)1/2 = Alfvén velocity 1) specifying a model for the plasma density distribution  (z); 2) using boundary conditions for the electric field  ,  = 0 at r  RE (ionospheric level);  r RE req 3) equations can be solved numerically to find the eigenfrequencies . r = req sin2 13

WKB / TIME-OF-FLIGHT APPROXIMATION field-line eigenperiods  time-of-flight : P1   n = 1,2,3,… VA(s) = B(s) / [o (s)]1/2 = field-aligned Alfvén velocity  P2  14

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 16

(Hattingh and Sutcliffe, 1987) 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

167- 200 s 117- 133 s 87- 105 s 69 - 80 s L 3.83 3.28 3.04 2.75 2.44 167- 200 s 117- 133 s 3.83 3.28 3.04 2.75 2.44 87- 105 s 69 - 80 s 20

Kelvin-Helmholtz instability Diurnal and latitudinal polarization pattern Polarization reversals across: - noon meridian - latitude of maximum amplitude Kelvin-Helmholtz instability 21

Field line resonance model (Southwood; Chen and Hasegawa, 1974) z   IONOSPHERE l B IONOSPHERE EARTH MAGNETOPAUSE x    -H y    D eigenfrequency R(x) = ( π / l ) VA (x) = = ( π / l ) B [o  (x)]-1/2 Plasma density  =  (x) 22

R= R(x) resonance: R=  IONOSPHERE z assuming magnetic perturbations of the form: b = b (x) exp [i (k// z + my - t)], l monochromatic wave: e - i  t near the resonance: IONOSPHERE x xR y scale length of the inhomogeneity R= R(x) resonance: R=  singularity at x = xR because of dissipative effects:    - i  By expanding R(x) around R(xR):  = 2γδ/ω = resonance width 23

At the resonant latitude xR : AMPLITUDE (by)  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) /  xR x PHASE (by) π/2 -π/2 xR x 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; 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. 24

Theory Observations CW CCW CW 25

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

Magnetospheric signal Ionospheric effect on the resonance structure Magnetospheric signal Ground signal 800 400 0 - 400 - 800 800 400 0 - 400 - 800 Hughes and Southwood, 1976 LATITUDE PROFILE (km) resonance 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. 27

Resonance effects can be easily masked by: 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

1 < 2 < 3 2  S - N 2  Gradient method for detecting field line resonances from ground-based ULF measurements (Baransky et al., 1985) N S EARTH Higher latitude field line → Lower resonant frequency ( N ) Separation: 1°- 3° Lower latitude field line → Higher resonant frequency ( S ) 1 < 2 < 3 AMPLITUDE RATIO A N /A S AMPLITUDE RESPONSE 1 Δx/ SOUTH x1 x2 x3 2  N S CROSS-PHASE S - N 2 tan-1(Δx/2) PHASE RESPONSE SOUTH x1 x2 x3 N S 2  30

An example of gradient measurements (Green et al., 1993) Power spectra at TM, L = 1.59 Δx ~ 240 km Power spectra at AK, L = 1.51 H/D Amplitude ratio fR = 78 mHz Cross-phase 31

Ground-satellite comparative study Event of July 6, 2002 (Vellante et al., 2004) CHAMP trajectory SEGMA array 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°. 32

N C R A 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. 33

Spectral analysis CHAMP azimuthal comp: peak at ~65 mHz NCK - CST, L=1.82 CST - RNC, L=1.70 RNC - AQU, L=1.60 CHAMP compress. NCK CST RNC AQU SEGMA - H 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 NCK CST RNC AQU SEGMA - D To be explained the higher frequency of the azimuthal comp. Source frequency: 55 mHz Resonance at L=1.60 34

Simulation of the signals observed by the CHAMP satellite At the resonant point : Vphase = 2π f  ~ 30 km/s Resonance structure,  = 80 km, VSAT = 7.6 km/s Amplitude Phase resonance VSAT VSAT = 7.6 km/s equatorward Vphase = 30 km/s poleward signal driving signal (f = 55 mHz) forced signal 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%). power spectra 35

Dynamic cross-phase analysis at SEGMA array An example of diurnal variation of the resonant frequency fR (midday) 53 mHz 64 mHz 69 mHz 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 39

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) Vellante et al. (1996) ρmax ≈ 2 ρmin Assuming 41

Plasmapause identification 3 4 5 6 7 8 9 3 4 5 6 7 8 9 L value 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

FLRs – cavity modes coupling 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). 44