1. MHD wave propagation in the neighbourhood of a two-dimensional null point James McLaughlin Cambridge 9 August 2004

2. Introduction Coronal heating remains a key unsolved problem in solar physics. Rival theories; reconnection models (current sheets or X-point collapse), wave heating (small length scale wave motions). Theory predicts existence of null points. Important because Alfv é n speed is zero at that point. Potential field extrapolations suggest there are null points in the corona.

3. Basic Equations The usual MHD equations for the solar corona are used: Where we consider a zero β plasma and ideal MHD.

4. Basic Equations II We take the basic equilibrium magnetic field as a simple 2D X-type neutral point;

5. Basic Equations III We now perform a perturbation analysis and consider the linearised perturbation equations; Assume ρ 0 uniform and non-dimensionalise. Under this process, t=1 refers to time in units of.

6. Basic Equations IV These ideal linearised MHD equations naturally decouple into two equations for the fast magnetoacoustic wave and for the Alfv é n wave. The linearised equations for the fast wave are: Variable V is related to the perpendicular velocity; and Linearised Alfv é n wave equations are:

7. Basic Equations V The fast wave equations can be reduced to a single wave equation: Where (non-dimensionalised). The Alfv é n wave equations reduce to:

8. Fast wave - Numerical Work For the fast wave solve the linearised perturbation equations numerically using a two-step Lax-Wendroff scheme. The numerical scheme is run in a box with -6 ≤ x ≤ 6 and -6 ≤ z ≤ 6. Boundary conditions:

9. Fast wave - Results I The linear, fast magnetoacoustic wave travels towards the vicinity of the X-point and bends around it. Since the Alfv é n speed is spatially varying, it travels faster the further it is away from the origin. The wave demonstrates refraction. It is this refraction effect that wraps the wave around the null point and this is the key feature of fast wave propagation.

10. Fast wave - Results II

11. Fast wave - Current I Alfv é n speed zero at the null point, so wave never reaches there, but the length scales rapidly decrease, indicating that gradients will increase. Changing perturbed magnetic field whose gradients are increasing in time means we have a build up of current density. Simulations show that there is a large current accumulation at the neutral point.

12. Current III

13. Fast wave : Analytical IV

14. Alfvén wave - Numerical Work For the Alfv é n wave solve the linearised perturbation equations numerically using a two-step Lax- Wendroff scheme. The numerical scheme is run in a box with 0 ≤ x ≤ 6 and 0 ≤ z ≤ 6. Boundary conditions:

15. Alfvén wave - Results It is found that the linear Alfv é n wave travels down from the top boundary and begins to spread out, following the field lines. As the wave approaches the separatrix, it thins but keeps its original amplitude. The wave eventually accumulates very near the separatrix (x axis).

16. Alfvén wave : Results II

17. Alfvén wave – Current I We have a varying perturbed magnetic field and so current is forming, in the Alfv é n case only and are present. j x spreads out along the field and accumulates at the separatrix and increases in time. From analytical work, we see j x grows like e t. j z spreads out along the field lines and decays in amplitude as it approaches the separatrix. From analytical work, we see j z decays like e -t.

18. Alfvén wave : j x

19. Alfvén wave : j z

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

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

22. Conclusions III Numerical and analytical work here was ideal. However, for the fast wave all the current density accumulates at the null point and appears to form a null line. Hence, no matter how small the value of the resistivity is, if we include it, then eventually the term will become non-negligible and dissipation will become important. Similarly for the Alfv é n wave, with, is found to decay away but increases (exponentially). Thus, if results transfer then null points and separatrices will be the locations of wave energy deposition and preferential heating.

23. Summary Fast wave is attracted to coronal null points. Wave is trapped by null point and wraps around it. Current density accumulates at the null. Ohmic dissipation will extract the energy in the wave at this point. So coronal nulls play an important role in the rapid dissipation of fast magnetoacoustic waves. Their dissipation will contribute to the overall energy budget of the corona. McLaughlin, J.A. & Hood, A.W., (2004), “MHD wave propagation in the neighbourhood of a two-dimensional null point”, A&A, 420, 1129-1140