1. Electrostatic fluctuations at short scales in the solar-wind turbulent cascade. Francesco Valentini Dipartimento di Fisica and CNISM, Università della Calabria, Italy francesco.valentini@fis.unical.it In collaboration with: D. Perrone and P. Veltri, Università della Calabria F. Califano and F. Pegoraro, Università di Pisa

2. Vienna WPI - Valentini Outline Vlasov simulations of solar wind turbulence: electrostatic activity at short spatial lengths, consisting of waves with phase speed close to the thermal velocity of protons (ion-bulk waves) The ion-bulk waves, electrostatic oscillations driven by particle trapping effects: the Vlasov-Yukawa model Dispersion relation of the ion-bulk waves: analogies with non-neutral plasmas Numerical Vlasov-Yukawa simulations: the excitation of the ion-bulk waves can be obtained even at low values of the electron to ion temperature ratio (relevant for space plasmas) The late time evolution: generation of long lived soliton-like waveform Conclusions

3. Vienna WPI - Valentini Electrostatic activity at short wavelengths Ion-Acoustic waves New branch of waves The proton distribution function Diffusive longitudinal plateau: - F. Kennel and F. Engelmann Phys. Fluids 9, 2377 (1966) - M. Heuer and E. Marsch J. Geophys. Res. 112, A03102 (2007) Ion-Bulk waves at phase speed close to Fourier spectrum of the electric energy Valentini et al., PRL 2008; Valentini and Veltri, PRL 2009

4. Vienna WPI - Valentini The ion-bulk waves Boltzmanian electrons Debye length Vlasov – Yukawa (VY) model Proton distribution function Valentini et al., in press in PRL

5. Vienna WPI - Valentini Plasma dielectric function Weak damping or amplification: Ion-acoustic waves We assume a plateau of vanishing velocity width in the proton velocity distribution that suppress wave damping See, for example, Valentini et al., PoP 2006 and Johnston et al., PoP 2009

6. Ion-bulk wave dispersion relation From thumb to tear drop for an non-neutral electron plasma: No solutions are recovered for

7. Vienna WPI - Valentini Numerical simulations 10.00 External driver electric field (applied ONLY to the Vlasov equation for protons) The system dynamics is investigated in terms of the electron to proton temperature ratio See, for example, Valentini et al., PoP 2006 and Johnston et al., PoP 2009

8. Vienna WPI - Valentini Numerical results Time evolution of three electric field Fourier components m=1,4,6 Resonance curve IBk IA

9. Vienna WPI - Valentini Proton distribution function During the driving process we observe the generation of trapped proton populations and of unstable regions with positive velocity slope

10. Vienna WPI - Valentini Proton distribution function Generation of wavepackets locked in the phase space votices, in the spatial region of positive velocity slope LOCAL SECONDARY INSTABILITY

11. Vienna WPI - Valentini Bump-on-tail like instability The bump is not on the tail of the distribution but the theory works very well We evaluate the imaginary part of the wave frequency using the distribution function from the simulations

12. Late time evolution Secondary vortex merging and very long lived soliton-like waveform Mangeney et al., 1999: Isolated electrostatic structures (data from WIND)

13. Vienna WPI - Valentini Summary and conclusions We have demonstrated the existence of a new branch of electrostatic waves with acoutic-like dispersion and sustained by trapping effects These waves have phase speed comparable to the proton thermal velocity and can be excited even at low values of the electron to ion temperature ratio Our numerical results can represent a new interpretation of the electrostatic noise recovered in the high frequency region of the turbulent spectra in the solar wind THANK YOU!