Space and Astrophysics Generation of quasi- periodic pulsations in solar flares by MHD waves Valery M. Nakariakov University of Warwick United Kingdom Nobeyama, Japan 16/03/2006

Space and Astrophysics A hypothesis or theory is clear, decisive, and positive but it is believed by none but the man who created it. Experimental findings, on the other hand are messy, inexact things which are believed by everyone except the man who did the work. Harlow Shapley

Space and Astrophysics Observational evidence of quasi-periodic pulsations in solar flares is abundant. (Quasi) Periodicity: Resonance (characteristic spatial scales) Dispersion Nonlinearity / self-organisation Characteristic scales: 1 Mm-100 Mm, Alfvén speed 1-2 Mm/s, sound speed Mm/s → periods 1 s – several min - MHD waves Seismological information

Space and Astrophysics Dispersion relations of MHD modes of a magnetic flux tube: Magnetohydrodynamic (MHD) equations  Equilibrium  Linearisation  Boundary conditions Zaitsev & Stepanov, B. Roberts and colleagues, Standard theoretical model:

Space and Astrophysics Main MHD modes in observed in the corona: sausage (|B|,  ) kink (weakly compressible) torsional (incompressible) acoustic ( , V) ballooning (|B|,  ) Dispersion curves of coronal loop:

Space and Astrophysics 1.Kink oscillations of coronal loops (Aschwanden et al. 1999, Nakariakov et al. 1999) 2.Propagating longitudinal waves in polar plumes and near loop footpoints (Berghmans & Clette, 1999; Nakariakov et al. 2000, De Moortel et al ) 3.Standing longitudinal waves in coronal loops (Kliem at al. 2002; Wang & Ofman 2002; Nakariakov et al. 2004) 4.Global sausage mode (Nakariakov et al. 2003) 5.Propagating fast wave trains. (Williams et al. 2001, 2002; Cooper et al. 2003; Katsiyannis et al. 2003; Nakariakov et al. 2004, Verwichte et al. 2005) Observed wave phenomena (to 2006):

Space and Astrophysics Kink mode: θ(t)  modulation I f (t) Sausage mode: B(t)  modulation I f (t) Longitudinal mode can also modulate the GS emission E.g., the optically thin gyrosynchrotron emission intensity I f at a frequency f can be estimated as In general, MHD waves should be well seen in microwave:

Space and Astrophysics

But, often QPP are seen in both microwave (GS) and hard X-ray : e.g. Asai et al. (2001)

Space and Astrophysics Suppose that QPP are connected with some MHD oscillations. The model has to explain: the modulation of both microwave and hard X-ray (and possibly WL) emission simultaneously and in phase; the modulation depth (> 50% in some cases, while the amplitudes of known coronal MHD waves are usually just a few percent); the mechanism responsible for the periodicity; the “2D” structure of the pulsations.

Space and Astrophysics Electron acceleration Nonthermal electron dynamics Emission in WL, microwave, X-rays MHD waves In general, MHD waves can affect the whole chain responsible for the emission:

Space and Astrophysics Flaring loop Faint cool loop At the reconnection site: B ~ cos(Ωt)  N e ~ cos(Ωt) Kink oscillation Quadrupolar magnetic configuration What if the modulating wave is magnetically disconnected with the flaring loop? 1. Long period pulsations (> 60 s)

Space and Astrophysics MHD oscillation in the external loop (very small amplitude) Fast wave perpendicular to B approaches X-point Electric currents build up (time variant) Current driven micro-instabilities Anomalous resistivity Triggers fast reconnection Acceleration of non-thermal electrons (Nakariakov et al. A&A, 2006, in press)

Space and Astrophysics Full MHD 2.5D simulations of the interaction of a periodic fast wave with a magnetic X-point. The fast wave energy is accumulated near the separatrix. The current density near the X-point experiences periodic building up. The absolute value of the velocity perturbation. run The electric current density. run The electric current density, side view run

Space and Astrophysics Thus, the electric current at the x-point varies periodically in time: The amplitude of the source fast wave is just 1%.

Space and Astrophysics Current-driven plasma microinstabilities were suggested as a triggering mechanism for fast reconnection: Periodic variation of the current density causes periodic triggering of fast reconnection

Space and Astrophysics There is some observational evidence: (Foullon et al. 2005) Unseen kink oscillations of the faint trans-equatorial EUV loop cause modulation of the hard X-ray emission near the magnetically conjugate points.

Space and Astrophysics 2. Medium period case (10-60 s) Sausage modes: -for trapped modes here L is the loop length The commonly used expression is incorrect, (here a is the loop minor radius) (Nakariakov et al., 2003; Aschwanden et al., 2004)

Space and Astrophysics What about leaky modes? Full MHD simulations: Thus, the period of leaky sausage modes is also determined by the loop length, not by the loop minor radius. (Pascoe et al., 2006) trapped leaky

Space and Astrophysics 3. Short period case (< 10 s) Higher spatial harmonics (but the problem is the selectivity of the excitation); Fast wave trains formed by dispersion: (But, they should have period modulation - the “crazy tadpole wavelet spectra …)

Space and Astrophysics Conclusions: There are simple mechanisms for modulation of microwave and hard x-ray emission by the modes. The longer periodicities can be connected with small-amplitude oscillations of an external cool loop. In this case the cool loop acts as a resonator and determines the period of oscillations. Sausage mode periods are prescribed by the flaring loop length, even in the leaky mode regime. Shorter periodicites can be connected with dispersion. However, there should be period modulation. In general, shorter periodicities (1-5 s) are still without appropriate interpretation (the “fast mode formula” does not work).