ULF MHD perturbations in the inner magnetosphere of the Earth: a theoretical study

1. ULF MHD perturbations inthe inner magnetosphere of the Earth:a theoretical study Oleg Cheremnykh Aleksei Parnowski Space Research Institute, Kyiv, Ukraine Tamao workshop, Kyushu U, Fukuoka, Japan

2. Tamao workshop, Kyushu U, Fukuoka, Japan Space vs. fusion plasma • largely independent fields, “withering” on their own • few examples of collaboration, but very successful • fusion people move to space, but not vice versa • lots of synergies and great prospects for cooperation New fusion Old fusion

3. Tamao workshop, Kyushu U, Fukuoka, Japan Magnetic surfaces • a is called a label of a magnetic surface • The geometrical relations do not depend on the equation of equilibrium • It is not necessary to know the value of a to derive the equations of small perturbations The magnetosphere is both a tokamak and a stellarator

4. Tamao workshop, Kyushu U, Fukuoka, Japan Displacement vector • Small perturbations approach: • all perturbed quantities are expressed • through a displacement vector x

5. Elsasser (1946) Modern scaloidal poloidal toroidal toroidal poloidal longitudinal / field-aligned Tamao workshop, Kyushu U, Fukuoka, Japan Local geometry • Orthogonal basis:

6. Tamao workshop, Kyushu U, Fukuoka, Japan General equations of small perturbations in arbitrary geometry • Cheremnykh (2010) • alternative form • Cheng and Chance (1986)

7. Tamao workshop, Kyushu U, Fukuoka, Japan Transversally small-scale (ballooning) modes • Large wavelength (l|| ~ RE) • ULF frequencies (10–4 < f < 0.1 Hz) • Ballooning approximation (k^ >> L–1, k||): perturbations are localized at the magnetic field lines • Furth et al. (1966) • Kruskal and Schwarzschild (1954) Ballooning Flute (interchange)

8. Tamao workshop, Kyushu U, Fukuoka, Japan Eikonal approximation • Let • As usual, the Re sign in the right-hand part is assumed, so both and S are purely real. • We denote • This yields • The applicability criteria are: • and

9. Tamao workshop, Kyushu U, Fukuoka, Japan Equations of small ballooning perturbations • Dewar and Glasser (1983) – variation methods • Pustovitov and Shafranov (1987) – differential operators • Cheremnykh et al. (2004) – small transverse scale

10. Tamao workshop, Kyushu U, Fukuoka, Japan Dipolar geometry • Nested magnetic surfaces • Axial symmetry (d/dj = 0) • Dipolar magnetic field • B = [y×j], y = Mcos2q/r • No convection (v0 = 0) • Low beta (b = gp/B02 << 1) • No shear or FAC (s = gs = 0) • This is the simplest 3D model of the geomagnetic field

11. Tamao workshop, Kyushu U, Fukuoka, Japan Eigenmodes in dipolar geometry poloidal even odd toroidal

12. Tamao workshop, Kyushu U, Fukuoka, Japan Toroidal Alfvén eigenmodes • Cheng et al. (1993)

13. Tamao workshop, Kyushu U, Fukuoka, Japan Poloidal Alfvén eigenmodes • Cheremnykh et al. (2004) • Poloidal Alfvén modes are coupled with compression modes through the radial curvature

14. Tamao workshop, Kyushu U, Fukuoka, Japan Does FLR exist in reduced MHD? • When approaching a resonant surface, the amplitudes of the perturbations grow according to the logarithmic or inverse power law • However, in the vicinity of the resonant surface the amplitude of the perturbed quantities cannot be considered small and the linearised equations do not apply. Thus, to correctly analyze FLR or any other resonance in plasma, one should consider the full non-reduced set of MHD equations.

15. Tamao workshop, Kyushu U, Fukuoka, Japan Ionospheric boundary conditions • Hameiri and Kivelson (1991) • Hameiri (1999) • Cheremnykh and Parnowski (2004) • Small parameters: • a/l|| ~ 10–1, s/s|| ~ 10–4 • Closure of magnetospheric current in the ionosphere: • No perturbations in the atmosphere: Indices: M – magnetospheric, b – boundary, S - surface

16. Tamao workshop, Kyushu U, Fukuoka, Japan Dimensionless equations • Scaling: • Equations: • Boundary conditions:

17. Tamao workshop, Kyushu U, Fukuoka, Japan Stability boundaries Ballooning Flute perturbations define the overall MHD stability of magnetospheric plasmas for any ionospheric conductivity Instability Flute

18. 3rd Alfvén mode (even) 2nd Alfvén mode (odd) 1st Alfvén mode (even) Flute modes (aperiodic) Tamao workshop, Kyushu U, Fukuoka, Japan Flute instability When the ionospheric conductivity decreases, so do the eigenmode frequencies. The frequency of the first Alfvén eigenmode reaches zero and this mode transforms into flute modes with zero frequency. Flute Alfvén

19. Tamao workshop, Kyushu U, Fukuoka, Japan Flute instability (noon) • Colour code: • Black = frequency • Blue = growth/decay rate with ω≠0 • Red = growth/decay rate with ω = 0 • Note that an unstable flute mode always exists, but it can be neglected for β < 0.14

20. Tamao workshop, Kyushu U, Fukuoka, Japan Flute instability (dawn/dusk) • Colour code: • Black = frequency • Blue = growth/decay rate with ω≠0 • Red = growth/decay rate with ω = 0 • The value a~3.3 is critical and separates two different solutions

21. Tamao workshop, Kyushu U, Fukuoka, Japan Flute instability (midnight) • Colour code: • Black = frequency • Blue = growth/decay rate with ω≠0 • Red = growth/decay rate with ω = 0 • Note that at a > 4 the frequency is always zero. This is due to the same reason as above

22. Tamao workshop, Kyushu U, Fukuoka, Japan Flute instability • Growth rates: • Day (high conductivity): GD ~ d ~ SP-1 • Night (low conductivity): GN ~ d-1 ~ SP, GD ~ GN • Weak wave activity • Dawn/Dusk (intermediate conductivity): |G| >> |GD|, |GN| • Strong wave activity • Amplitude of flute perturbations:

23. Tamao workshop, Kyushu U, Fukuoka, Japan Effect of TEC on mode coupling

24. Tamao workshop, Kyushu U, Fukuoka, Japan Waveforms 100 1.5 b = 10-2

25. Tamao workshop, Kyushu U, Fukuoka, Japan Waveforms 1 60000 b = 10-2

26. Tamao workshop, Kyushu U, Fukuoka, Japan Waveforms: asymptotes • Near the ionosphere when e = 1 – x << 1 the equation for longitudinal perturbations can be reduced to the Bessel equation e2t" + 9et' + (12 + e2F2)t = ¼(x' + x/e) • Its solution is t = t1 + t2 + t3, where • F = 2b–1W, e = 1 – x, e0 = 1 – x0, J2and Y2 are Bessel functions, and C is an integration constant.

27. Tamao workshop, Kyushu U, Fukuoka, Japan Waveforms: observational manifestations • In the indicated area strong longitudinal oscillations with different frequencies meet together • As a result, one should expect generation of non-linear structures • Possibly, they were observed on Freja S/C by Stasiewicz et al. (1997) Maximum longitudinal displacement Maximum compression Non-linear effects

28. Tamao workshop, Kyushu U, Fukuoka, Japan Summary • The displacement vector approach allowed to derive a set of small perturbation equations, which can be put down in a convenient form • These equations can be treated in two different ways: by applying the ballooning approximation or by looking for solutions in a specific form • Following the first approach, we managed to derive the stability criteria for flute and ballooning modes and to study their spectra and waveforms

29. Tamao workshop, Kyushu U, Fukuoka, Japan Summary (continued) • The second approach allowed us to demonstrate that in dipolar geometry Alfvén modes can have only toroidal and poloidal polarisation • Toroidal modes are always stable in this case and poloidal modes can become unstable and are described in general case by the Spies criterion • Boundary conditions significantly affect the spectrum of the perturbations and their stability • There is a qualitative agreement between the results of this approach and the observations

30. Tamao workshop, Kyushu U, Fukuoka, Japan Acknowledgements • We would like to express our extreme gratitude to Oleksiy Agapitov for his valuable input in this research. • We would like to thank the PIs and data providers of the following projects: INTERMAGNET, AMPTE/CCE, AMPTE/IRM for keeping their data freely available and for maintaining high data quality standards. • And, of course, we would like to • Thank you for your kind attention!