A Self-consistent Model of Alfv é n Wave Phase Mixing

1. A Self-consistent Model of Alfvén Wave Phase Mixing G.kiddie, i. de moortel, p.Cargill & A.hood

2. Phase Mixing • Occurs when Alfvén waves are travelling in an inhomogeneous plasma • This leads to the creation of strong gradients which enhance the visco-resistive damping • First suggested by Heyvaerts & Priest (1982) • Require weak damping and “strong phase mixing” • Heating is focused at edges of loops

3. Coupling of Corona and Chromosphere • We consider the effect of including the coupling of the corona and chromosphere in a phase mixing experiment • Most models don’t consider relationship between corona, chromosphere and heating • The dependence of heating on density is a direct consequence of the dynamic coupling Heating

4. Ofmanet al. (1998) • Considered the coupling of the corona and chromosphere with a resonant absorption experiment • This interaction moves the resonance layer around, leads to spatially bursty heating

5. Model z y • 2.5D model • Phase mixing is introduced through a density inhomogeneity • Background magnetic field in y-direction • We solve the linear phase mixing equations for a standing Alfvén wave x

6. Scaling Laws • Use a full energy equations to derive scaling laws linking the temperature, density and heating ( Rosner et al. 1978, Hood & Priest 1979, Craig et al. 1978) • d and r come from the non-dimensionalisation • 1st scaling law comes from the comparison between the conduction and heating terms • 2nd scaling laws comes from the comparison between the conduction and radiation terms • Set

7. Introduction of Density Feedback • Have a background field of , initial density approx. ) • Additional peaks form on the density profile • Spreading of the heating layers

8. Introduction of Density Feedback

9. Damping Timescale • Damping timescale

10. Lower density • The effect is small, simplest method is to enhance the effect is to decrease the density • Now enough energy in the wave to power the background atmosphere

11. Lower density • Density is now greatly enhanced via feedback mechanism • Resolution is lost before the wave has fully damped

12. Driven Case • All the energy in previous experiment in initial velocity profile • We introduce more energy into the system by continuously driving the bottom boundary • To continue to use our scaling laws we need a closed system • We set up reflective boundary conditions at the top boundary • Turn off the driver before the wave reaches the bottom boundary and bottom boundary becomes reflective

13. Ideal Case • Initially test with resistivity switched off • Complex chequered pattern where reflected wave interferes with outward propagating wave

14. Ideal Case • Additional gradients created by reflecting wave

15. Current • Investigate how the current changes with this reflection • Current becomes very difficult to resolve

16. Conclusions • Alfvén waves deposit heat in the solar atmosphere • By considering the mass exchange between corona and chromosphere this heat can be spread • The effect is not big using coronal parameters • Situation even more complex when considering driven waves