MHD wave propagation in the neighbourhood of a

two-dimensional null point

James McLaughlinCambridge

9 August 2004

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 Alfvén speed is zero at that point. Potential field extrapolations suggest there are null points in the corona.

Basic Equations

The usual MHD equations for the solar corona are used:

Where we consider a zero β plasma and ideal MHD.

0.

1.

2

v

BBvB

BBvvv

t

t

pt

Basic Equations II

We take the basic equilibrium magnetic field as a simple 2D X-type neutral point; ( )0 ,0,B x z

a= -B

Basic Equations III

We now perform a perturbation analysis and consider the linearised perturbation equations;

Assume ρ0 uniform and non-dimensionalise.

Under this process, t=1 refers to time in units of .A

av

0.

1

101

111

011

0

v

BvB

BBv

t

t

t

Basic Equations IV

These ideal linearised MHD equations naturally decouple into two equations for the fast magnetoacoustic wave and for the Alfvén wave.

The linearised equations for the fast wave are:

Variable V is related to the perpendicular velocity; and

Linearised Alfvén wave equations are:

( )1 , ,x y zb b b=B yV eBv ˆ.01

xV

tb

zV

tb

zb

xbzx

tV zxxz

,,22

zv

zxv

xtb

zbz

xbx

tv yyyzxy

,

Basic Equations V

The fast wave equations can be reduced to a single wave equation:

Where (non-dimensionalised).

The Alfvén wave equations reduce to:

2 2Av x z= +

2

2

2

22

2

2

,zV

xVzxv

tV

A

yy v

zz

xx

tv 2

2

2

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:

0,0,0

otherwise00sin6,

666

zxx zV

xV

xV

ttxV

Fast wave - Results I

The linear, fast magnetoacoustic wave travels towards the vicinity of the X-point and bends around it.

Since the Alfvén 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.

Fast wave - Results II

Fast wave - Current I

Alfvén 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.

Current III

Fast wave : Analytical IV

Alfvén wave - Numerical Work

For the Alfvén 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:

0,0,0

otherwise0

300

3cos1sin6,

060

6

z

y

x

y

x

y

z

y

y

zv

xv

xv

zv

xtxtxv

Alfvén wave - Results

It is found that the linear Alfvén 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).

Alfvén wave : Results II

Alfvén wave – Current I

We have a varying perturbed magnetic field and so current is forming, in the Alfvén case only and are present.

jx spreads out along the field and accumulates at the separatrix and increases in time. From analytical work, we see jx grows like et .

jz spreads out along the field lines and decays in amplitude as it approaches the separatrix. From analytical work, we see jz decays like e-t .

zb

j yx

xb

j yz

Alfvén wave : jx

Alfvén wave : jz

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.

Conclusions II – Alfvén wave

For the Alfvén 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 jx increases and accumulates along the separatrix, whilst jz decays away.

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 Alfvén 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.

xjz

zjx

12B

zj

xj xz

12B

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