1. Wave propagation in a non-uniform, magnetised plasma: Finite beta James McLaughlin Leiden March 2005

2. Introduction Models of wave motions that should occur in the neighbourhood of 2D X-point. There are three types of wave motions that should occur; slow MA, fast MA and Alfvén. Wave motions recently observed in the corona. Research has also illustrated importance of magnetic topology. Extrapolations predict null points; Alfvén speed zero. Aim - Bring these areas together. Show how plasma waves behave when they travel into magnetic structures (inhomogeneous medium).

3. Model Perform perturbation about equilibrium magnetic field – simple 2D X-point.

4. Model Perform perturbation about equilibrium magnetic field – simple 2D X-point. Idea is to send in wave from top boundary, and see (and explain!) what happens.

5. Model II Perturbation analysis on ideal MHD equations- linearised perturbation equations Take velocity of the form:

6. Model III Equations decouple – fast and slow MA wave & Alfvén wave. MA waves: Alfvén wave -

7. Fast beta=0 & Alfvén wave Linear, fast magnetoacoustic wave travels towards the vicinity of the X-point and bends around it. Alfvén speed spatially varying - travels faster further away from the origin. Wave demonstrates refraction – wraps wave around null Key feature of fast wave propagation. Alfvén wave travels down from top boundary and begins to spread out, following the field lines. As the wave approaches the separatrix, thins but keeps its original amplitude. The wave eventually accumulates very near the separatrix (x axis).

8. Fast MA wave I Fast wave case reduces to single wave equation: Where (non-dimensionalised. Spatially dependent). Fast wave solved numerically - two-step Lax-Wendroff scheme. -6 ≤ x ≤ 6 and -6 ≤ z ≤ 6. Boundary conditions

9. Fast MA wave III

10. Fast MA wave II Linear, fast magnetoacoustic wave travels towards the vicinity of the X-point and bends around it. Alfvén speed spatially varying - travels faster further away from the origin. Wave demonstrates refraction – wraps wave around null Key feature of fast wave propagation.

11. Alfvén wave I Alfvén wave solved numerically - two-step Lax- Wendroff scheme. 0 ≤ x ≤ 6 and 0 ≤ z ≤ 6 (first quadrant) Boundary conditions:

12. Alfvén wave III

13. Alfvén wave II Alfvén wave travels down from top boundary and begins to spread out, following the field lines. As the wave approaches the separatrix, thins but keeps its original amplitude. The wave eventually accumulates very near the separatrix (x axis).

14. Fast and slow MA waves How does this simple model change with the inclusion of a finite term? Main questions: 1.How does the slow wave couple to the (driven) fast wave? 2.How does the nature of the coupling change with ? 3.What is the effect of pressure at the origin?

15. Fast and slow I Now consider finite case. Recall equations: First thing; much more complicated! We now have a coupled system: driving the system with a fast wave will generate a slow wave, which in turn will act back on the system. Difficult to think about: pure fast and pure slow wave concepts no longer valid.

16. Fast and slow II Also, now concerned that so (even though the fast wave may be able to pass through the origin. Fast wave solved numerically - two-step Lax-Wendroff scheme. -4 ≤ x ≤ 4 and -4 ≤ z ≤ 4. Boundary conditions We find that the fast wave (driven on the upper boundary) travels into the box (across fieldlines) mainly as before, with some small additions: ),

17. Movies I Fast zero beta : perpendicular velocity

18. Movies II Perpendicular velocity component

19. Movies III Parallel velocity component

20. Fast and slow Main questions: 1.How does the slow wave couple to the (driven) fast wave? 2.How does the nature of the coupling change with ? 3.What is the effect of pressure at the origin?

21. How does the slow wave couple to the (driven) fast wave? I There is now coupling to the parallel velocity component, such that two waves are produced:  firstly, an aspect travelling at the same speed and frequency as the driven (fast) wave; this wraps around the null in a similar way to the zero beta fast wave.  secondly, an aspect that trails behind the first, travelling at a much slower speed ( ). This is the slow wave.

22. How does the slow wave couple to the (driven) fast wave? II Parallel velocity component

23. How does the slow wave couple to the (driven) fast wave? II Parallel velocity component

24. How does the slow wave couple to the (driven) fast wave? II slow wave generated by the system / homogeneous solution (driven) perpendicular wave showing up in parallel velocity component / inhomogeneous solution Parallel velocity component

25. Fast and slow Main questions: 1.How does the slow wave couple to the (driven) fast wave? 2.How does the nature of the coupling change with ? 3.What is the effect of pressure at the origin?

26. How does the nature of the coupling change with ? The size of the used will affect the magnitude of the coupling and determine how much the pressure and parallel velocity feedback to affect the (driven) perpendicular velocity wave. In the corona, typical values of are often cited.

27. Position reached by (homogenous solution) slow wave after time t=1.5 along x=1.5 ) Linear plot Gradient = -0.989 Linear plot makes sense if

28. Time taken by (homogenous solution) slow wave to reach z=3.0 along x=1.5 Linear plot Gradient = 1.223 Linear plot makes sense if

29. Amplitude of (homogenous solution) slow wave at (x, z)=(1.5,3.0) ) e Gradient = 1.599 Amplitude ~ So perhaps negligible For coronal vales of ?

30. Amplitude of (inhomogenous solution) wave at (x, z) = (0.5,1.0) at t=1.67 Linear plot Gradient = 0.8702 Amplitude increases Also, position of max always occurs at same point in space, i.e. position of coupling independent of

31. Fast and slow III Main questions: 1.How does the slow wave couple to the (driven) fast wave? 2.How does the nature of the coupling change with ? 3.What is the effect of pressure at the origin?

32. What is the effect of pressure at the origin? I Idea is that, as opposed to case, when all wave is packed around null, pressure may now critically change the results; i.e. either expel the wave away from the null or in some other way impede its propagation (because of the term). However, we do not find that the inclusion of a pressure term significantly changes the system. Yes, the pressure is increasing as we approach the null ( at null), and yet fast wave still wraps around the null in a very similar way to before, and so wave will still dissipate close to the origin. We also find no evidence of the fast wave travelling through the null (perhaps is still true, but may be tiny (so dissipates).

33. Conclusions I – Fast wave When fast magnetoacoustic wave propagates near X-type neutral point, wave wraps itself around due to refraction (at least in 2D). Large current density accumulation at the null. Build up exponential in time. Refraction of the wave focuses energy of the incident wave towards null point - wave continues to wrap itself around null point, again and again.

34. Conclusions II – Alfvén wave Alfvén wave - wave propagates along field lines, accumulating on separatrix (along separatrices due to symmetry). Wave thins and stretches. The current j x increases and accumulates along the separatrix, whilst j z decays away.

35. Conclusions III – wave coupling Perpendicular and parallel velocity components now (inevitably) coupled with inclusion of. Driven perpendicular ( fast ) wave generates waves in parallel velocity component (one of which is slow wave). Magnitude of the coupling dependent on, especially on amplitude of parallel velocity waves. Hence in a low corona, effect is very small, i.e. preferential heating still occurs at the origin for the fast MA wave. Pressure does have an effect on the system, but only causes a very small impediment; fast wave still refracts around origin, wrapping around again and again. Hence, inclusion of finite term does not significantly change the nature of the system.