Mode conversion occurs when a wave passes through a region where the sound
and Alfven speeds are equal. At this point there is a resonance, which allows
some of the incident wave to be converted into a different mode. We study this
phenomenon in the vicinity of a two-dimensional, coronal null point. As a wave
approaches the null it passes from low- to high-beta plasma, allowing
conversion to take place. We simulate this numerically by sending in a slow
magnetoacoustic wave from the upper boundary; as this passes through the
conversion layer a fast wave can clearly be seen propagating ahead. Numerical
simulations combined with an analytical WKB investigation allow us to determine
and track both the incident and converted waves throughout the domain.Comment: 4 pages, 2 figure

A. M. D. McDougall

A. W. Hood

Eric Stempels

English

arXiv

Crossref

MHD Mode Conversion around a 2D Magnetic Null Point

ar
X
iv
:0
90
7.
15
41
v1
  [
as
tr
o-
ph
.S
R
]  
9 
Ju
l 2
00
9
MHD Mode Conversion
around a 2D Magnetic Null Point
A. M. D. McDougall and A. W. Hood
School of Mathematics and Statistics, University of St Andrews, St Andrews, KY16 9SS, UK
Abstract. Mode conversion occurs when a wave passes through a region where t e sound and
Alfvén speeds are equal. At this point there is a resonance, which allows some of the incident
wave to be converted into a different mode. We study this phenomenon in the vicinity of a two-
dimensional, coronal null point. As a wave approaches the null it passes from low- to high-β plasma,
allowing conversion to take place. We simulate this numerically by sending in a slow magnetoacous-
tic wave from the upper boundary; as this passes through the conversion layer a fast wave can clearly
be seen propagating ahead. Numerical simulations combinedwith an analytical WKB investigation
allow us to determine and track both the incident and converted waves throughout the domain.
Keywords: Magnetohydrodynamics, Sun: corona, Sun: oscillations.
PACS: 52.35.Bj, 95.30.Qd
INTRODUCTION
At the conversion region the plasmaβ , the ratio of the gas pressure to the magnetic
pressure, is approximately equal to unity. This layer generally lies low in the solar
atmosphere, in the chromosphere. However, if a coronal nullpoint is considered, where
the magnetic fieldB = 0, there will be an area surrounding the null whereβ ≈ 1. Thus, it
is reasonable to expect mode conversion to occur as a wave approaches the null, passing
from low- to high-β plasma.
Extensive work has been done in examining mode conversion inone-dimension.
Zhugzhda and Dzhalilov [1] looked at the conversion of fast and slow magnetoacoustic
waves propagating from high- to low-β , in a gravitationally stratified, isothermal atmo-
sphere. Transmission and conversion coefficients were found, sing an exact solution.
McDougall and Hood [2] studied conversion of slow to fast magnetoacoustic waves
propagating downwards from low- to high-β . Using a method developed by Cairns and
Lashmore-Davies [3] transmission and conversion coefficients were found. This method
has the advantage that an exact analytical solution need notbe known. This was extended
to include a non-isothermal atmosphere [4], where it was found that the same conversion
and transmission coefficients apply. Mode conversion around a two-dimensional, mag-
netic null point has also been investigated [5]. This lookedat a fast wave propagating
towards the null point from above, and focused on how the proximity of theβ ≈ 1 layer
affects the competing refraction and conversion effects.
We look at driving a slow wave pulse along the field lines from above. This travels
from low- to high-β towards the null point, and conversion is observed as it crosses the
cs = vA layer. We use the WKB method to track the incoming slow wave, and as this hits
the conversion region we also track the fast wave which propagates ahead.
FIGURE 1. Equilibrium magnetic field with a magnetic null point situated at the origin, denoted by a
cross. The circle shows where the sound and Alfvén speeds areequal.
MODEL
We use the ideal MHD equations with gravity neglected. The equilibrium magnetic field
is given byB0 = B0(x,0,−z)/L, and is shown in Figure 1. The squared sound and
Alfvén speeds are given byc2s = γ p0/ρ0 andv2A = B
2
0
(
x2+ z2
)
/
(
µρ0L2
)
respectively,
whereγ is the ratio of specific heats,p0 andρ0 are the constant equilibrium pressure and
density,µ is the magnetic permeability, andL is a typical coronal length scale.
The ideal MHD equations may be linearised and combined to give a pair of linear
wave equations:
∂ 2vx
∂ t2
= c2s
(
∂ 2vx
∂x2
+
∂ 2vz
∂x∂ z
)
+
B20z
µρ0L2
[
z
(
∂ 2vx
∂x2
+
∂ 2vx
∂ z2
)
+ x
(
∂ 2vz
∂x2
+
∂ 2vz
∂ z2
)]
, (1)
∂ 2vz
∂ t2
= c2s
(
∂ 2vx
∂x∂ z
+
∂ 2vz
∂ z2
)
+
B20x
µρ0L2
[
x
(
∂ 2vz
∂x2
+
∂ 2vz
∂ z2
)
+ z
(
∂ 2vx
∂x2
+
∂ 2vx
∂ z2
)]
. (2)
We define the velocity parallel and perpendicular to the magnetic field as
v‖ = (xvx − zvz)/
√
x2+ z2 and v⊥ = (zvx + xvz)/
√
x2+ z2 respectively. We may then
drive a slow wave pulse exactly along the field lines by setting v⊥ = 0 andv‖ = sinωt,
whereω is the driving frequency. As the pulse is curved compared to our horizontal
boundary, we calculate when each point hits the boundary, sothe pulse is released along
the boundary at the correct time.
WKB METHOD AND SIMULATIONS
Starting with the wave equations (1) and (2), we writevx = aexp(iφ (x, z, t)) and
vz = bexp(iφ (x, z, t)) under the assumption thatφ ≫ 1. We then obtain the quadratic
F =
1
2
[
ω4−
(
c2s + v
2
A
)(
p2+q2
)
ω2+ c2s
(
p2+q2
)
(xp− zq)2
]
= 0,
which we have set equal toF , wherep = ∂φ/∂x andq = ∂φ/∂ z. The roots of this
FIGURE 2. Contour plots of the parallel velocity from the numerical simulation. The circle denotes
wherecs = vA, and the cross is the location of the null point. The lines show the front, middle and back of
the slow wave and fast wave as predicted by the WKB method.
quadratic are
2ω2 =
(
c2s + v
2
A
)(
p2+q2
)
±
√
(
c2s + v
2
A
)2
(p2+q2)2−4c2s (p2+q2)(xp− zq)2,
where the plus sign gives the fast wave solution, and the minus sign the slow wave.
Using Charpit’s relations we may then obtain a system of ODEsgoverning the fast
and slow magnetoacoustic waves:
dp
dt
=
x
(
p2+q2
)
2ω
±
[
x
(
p2+q2
)2(
c2s + v
2
A
)
−2pc2s
(
p2+q2
)
(xp− zq)
]
2ω
√
(
c2s + v
2
A
)2
(p2+q2)2−4c2s (p2+q2)(xp− zq)2
,
dq
dt
=
z
(
p2+q2
)
2ω
±
[
z
(
p2+q2
)2(
c2s + v
2
A
)
+2qc2s
(
p2+q2
)
(xp− zq)
]
2ω
√
(
c2s + v
2
A
)2
(p2+q2)2−4c2s (p2+q2)(xp− zq)2
,
dx
dt
= − p
(
c2s + v
2
A
)
2ω
∓
∓
[
p
(
p2+q2
)(
c2s + v
2
A
)2−2pc2s (xp− zq)2−2xc2s
(
p2+q2
)
(xp− zq)
]
2ω
√
(
c2s + v
2
A
)2
(p2+q2)2−4c2s (p2+q2)(xp− zq)2
,
dz
dt
= −q
(
c2s + v
2
A
)
2ω
∓
∓
[
q
(
p2+q2
)(
c2s + v
2
A
)2−2qc2s (xp− zq)2+2zc2s
(
p2+q2
)
(xp− zq)
]
2ω
√
(
c2s + v
2
A
)2
(p2+q2)2−4c2s (p2+q2)(xp− zq)2
.
We solve these using a fourth order Runge-Kutta scheme, allowing the position of the
wave pulses to be tracked throughout the domain, as shown in Figure 2.
The slow wave is tracked as it enters the domain and approaches the conversion layer.
Once the front of the pulse reaches this point, we begin to foll w the fast wave pulse
that is created. This is only tracked for points which pass through the conversion layer,
hence a much smaller portion of the wave front is followed.
CONCLUSIONS
We have studied a two-dimensional mode conversion layer situated around a magnetic
null point. A slow wave pulse is driven along the field lines onthe upper boundary,
and as it hits the conversion region, wherecs = vA, some of its energy is transferred to
the fast mode, which we see propagating out ahead of the slow mode pulse. The WKB
method is then used to predict the position of both wave fronts as they travel through
the domain. These predicted positions are in excellent agreement with the simulations,
showing the slow wave stretching out and slowing as it approaches the null, while the
fast wave propagates out in front.
In future we plan to extend the Cairns and Lashmore-Davies [3] method to two
dimensions, allowing the quantity of conversion and transmis ion to be calculated, as
has been done for an isothermal [2], and non-isothermal [4],one-dimensional model.
REFERENCES
1. Y. D. Zhugzhda, and N. S. Dzhalilov,A&A 112, 16 – 23 (1982).
2. A. M. D. McDougall, and A. W. Hood,Sol. Phys. 246, 259 – 271 (2007).
3. R. A. Cairns, and C. N. Lashmore-Davies,Phys. Fluids 26, 1268 – 1274 (1983).
4. A. M. D. McDougall, and A. W. Hood, “MHD Mode Conversion in aStratified Atmosphere,” in
Proceedings of the International Astronomical Union, Waves & Oscillations in the Solar Atmosphere:
Heating and Magneto-Seismology, edited by R. Erdélyi, and C. A. Mendoza-Briceño, 2008, vol.247,
pp. 296 – 302.
5. J. A. McLaughlin, and A. W. Hood,A&A 459, 641 – 649 (2006).


Mode conversion occurs when a wave passes through a region where the sound and Alfvén speeds are equal. At this point there is a resonance, which allows some of the incident wave to be converted into a different mode. We study this phenomenon in the vicinity of a twodimensional, coronal null point. As a wave approaches the null it passes from low- to high-β plasma, allowing conversion to take place. We simulate this numerically by sending in a slow magnetoacoustic wave from the upper boundary; as this passes through the conversion layer a fast wave can clearly be seen propagating ahead. Numerical simulations combined with an analytical WKB investigation allow us to determine and track both the incident and converted waves throughout the domain.</p

Mcdougall, A. M.D.

Hood, A. W.

University of St. Andrews - Pure

Mhd Mode conversion around a 2d magnetic null point

https://research-portal.st-andrews.ac.uk/en/researchoutput/mhd-mode-conversion-around-a-2d-magnetic-null-point(05d4f62d-218f-4b1e-9150-d1720ac1e509).html

MHD Mode Conversion around a 2D Magnetic Null Point

Abstract

Similar works

Full text

Available Versions

Crossref

Crossref

Crossref

University of St. Andrews - Pure

Crossref

Crossref

University of St. Andrews - Pure