Nonlinear Effects in the Shock-Associated
EUV Wave Propagation
Andrey Afanasyev
Institute of Solar-Terrestrial Physics
The Siberian Branch of the Russian Academy of Sciences
Irkutsk, Russia
Methodologies adopted in EUV-­‐wave research have direct impact on our capabilities to achieve the best science from observations and to probe their underlying physics. Wei Liu and Leon Ofman, Solar Phys. 289, 3233 (2014) “Advances in Observing Various Coronal EUV Waves in the SDO Era and Their Seismological ApplicaNons (Invited Review)” Outline
•
Global waves on the Sun
•
Method of nonlinear geometrical acoustics
•
Modelling of the propagation of Moreton
waves and EUV waves
•
Modelling of the wave propagation near a
magnetic null point
•
Conclusions
Moreton waves
•  are large-scale single-pulse wave disturbances
propagating along the solar surface
•  were discovered in 1960 (Moreton, and
Moreton&Ramsey)
•  are observed in the chromospheric H-alfa line
•  were described by Uchida in 1968 as
chromospheric imprints of global coronal fastmode waves
Uchida, 1968
06-­‐12-­‐2006 Hα MLSO Moreton waves
17-01-2005 02-05-1998 Warmuth et al., 2001 Veronig et al., 2006 19-May-2007
EUV waves
17-January-2010
EUV waves
•  are single-pulse wave disturbances propagating
over large distances along the solar surface
•  were discovered in 1996 with SOHO/EIT (‘EIT
waves’)
•  are observed in Extreme Ultraviolet lines
•  Nature of EUV waves has been debated very
intensively starting from their discovery
Physical nature of EUV waves
•  Signature of a coronal fast-mode wave in the
lower corona (e.g., Thompson et al., 1999;
Warmuth et al., 2001) – like a Moreton wave
‘a coronal counterpart of a Moreton wave’
•  Plasma compression in bases of coronal loops
in their successive stretching by a CME/eruptive
filament (e.g., Delannée, Aulanier, 1999; Chen,
Fang, Shibata, 2005)
Chen, Fang, Shibata, 2005
Physical nature of EUV waves
•  Other mechanisms exist (e.g. current dissipation)
•  Both mechanisms considered are valid –
bimodal, or 2-component nature
•  Some of EUV waves are signatures of coronal
fast-mode MHD waves in the lower corona
•  We consider just this class of global EUV
disturbances
Warmuth & Mann, A&A, 2011
Observed properties of global waves
deceleration
17-01-2005 – Moreton wave
Veronig et al., 2006; Temmer et al., 2009
deceleration
19-05-2007 – EUV wave (STEREO/EUVI)
Veronig, Temmer, Vršnak, 2008
02-05-1998 Moreton wave
Warmuth et al., 2001; Warmuth et al., 2004a
deceleration
deceleration
lengthening
weakening
Modelling of the propagation of Moreton
waves and EUV waves
•
•
•
•
Uchida 1968, 1974
Uchida et al., 1973
Wang, 2000
Patsourakos et al., 2009
… used the LINEAR ray-tracing method
However, the linear method cannot explain wave
deceleration and its lengthening when it
propagates in the quiet Sun regions
Uchida, 1968
acceleration
Linear wave keeps its
length unchanged if
average plasma
parameters are constant
along the solar surface
(quiet Sun regions)
Modelling the propagation of Moreton
waves and EUV waves
•  We should take into account nonlinear properties
of propagating disturbances
•  We apply the method of nonlinear geometrical
acoustics
•  This method is based on the linear geometrical
acoustics method (ray-tracing method, WKB
approximation)
Method of LINEAR geometrical
acoustics
•  allows one to calculate the propagation of linear waves
in smoothly inhomogeneous media
•  takes into account the wave refraction as well as the
wave amplitude variation owing to the inhomogeneity of
a medium
i ( ωt − kr )
A
e
Plane wave in a homogeneous medium:
In an inhomogeneous medium:
∂Ψ
+ a gradΨ = 0
∂t
Eikonal Equation:
Ray Equations:
A(r, t )eiΨ (r , t )
à
a – fast-mode
speed
Ray equations for linear waves
dr
k
∂a
=a +k
,
dt
k
∂k
dk
= − k ∇a,
dt
in Cartesian coordinates
Calculation of wave amplitude
In linear geometrical acoustics, the flux of the energy of a
disturbance travelling at group velocity q is directed along the
rays, and its magnitude is conserved within a ray tube:
u – wave amplitude –
longitudinal component
of plasma velocity,
v – transverse
component of plasma
velocity
div (Δε q) = 0.
Δε = ρ (u 2 + v 2 )
dS qρ u2 (1 + µ 2 ) = const.
⇒ We have to calculate the cross-section dS of ray tubes
Calculation of wave amplitude
We calculate the cross section, using the Jacobian of the
transformation from the used coordinates to ray ones
(Kravtsov&Orlov, 1990)
η – ray coordinate
σ – length of the
ray tube
2D case à
D(t ) ρ u2 (1 + µ 2 ) = const
We determine the derivatives by integrating the so called
adjoint set of equations
Adjoint set of equations
2D case, Cartesian coordinates
Set of ray equations was:
dr
k
∂a
=a +k
,
dt
k
∂k
dk
= − k ∇a,
dt
3D case,
Spherical
coordinates
Method of NONLINEAR geometrical
acoustics
q allows one to consider weak nonlinear disturbances in a
smoothly inhomogeneous medium
q is based on the linear method
q takes into account:
•  Wave refraction
•  Nonlinear increase in the wave propagation speed
•  Nonlinear damping of the wave amplitude in addition
to its linear variation
Properties of shock waves
Linear disturbance
u
U =a
Nonlinear disturbance
u
U = a +κ u
Shock front
All points of the wave
profile move at the same
speed equal to the fastmode speed a
Speed of a point of the wave
profile depends on the
perturbation value at this point
Damping of shock waves
In inhomogeneous media:
u
a+
u
κ
2
u sh
aT*
Ush
aTsh
Uralov, 1982
u1
⎛ τ 1 ⎞
U sh = u1 ⎜⎜1 + ⎟⎟
⎝ T* ⎠
⎛ τ 1 ⎞
Tsh = T* ⎜⎜1 + ⎟⎟
⎝ T* ⎠
dτ 1 κ u1
=
dt
a
−
1
2
1
2
u1 is the amplitude of the
linear wave in an
inhomogeneous medium
Set of ‘nonlinear’ ray equations
2D, Cartesian coordinates
•  Calculation of the wave amplitude is necessary now
Thus we have 19 ODEs in the 3D case and 9 ODEs in the
2D case
•  Numerical solving these ODEs
3D case,
Spherical
coordinates
κ U sh ⎞ k r
dr
∂a
⎛
= Vr + ⎜ a +
+
k
,
⎟
dt
2 ⎠ k
∂k r
⎝
κ U sh ⎞ kθ
dθ ⎛
∂a
r
= ⎜ a +
,
⎟ + k
dt ⎝
2 ⎠ k
∂kθ
κ U sh ⎞ kϕ
dϕ ⎛
∂a
r sin θ
= ⎜ a +
,
⎟ + k
dt ⎝
2 ⎠ k
∂kϕ
dk r
∂V
∂a
a 2
= − r kr −
k+
kθ + kϕ2 ,
dt
∂r
∂r
kr
dk
∂V
∂a
a
dr
r θ = − r kr −
k + kϕ2 cot θ − kθ ,
dt
∂θ
∂θ
k
dt
dkϕ
∂V
∂a
dr
dθ
r sin θ
= − r kr −
k − sin θ kϕ
− kϕ r cosθ
.
dt
∂ϕ
∂ϕ
dt
dt
(
)
Model of the solar corona
⎛ Rs ⎛ Rs
⎞ ⎞
ρ (r ) = ρ 0 exp ⎜⎜ ⎜ − 1⎟ ⎟⎟,
⎠ ⎠
⎝ H ⎝ r
2
⎛ Rs ⎞
Br = ± B0 ⎜ ⎟ ,
⎝ r ⎠
‘Quiet Sun’
•  Alfven speed increases with height up to ~ 2.4
solar radii
•  Rays tend to the low Alfven speed regions
Results of modelling
Vsound=180 km/s
At the base of the
corona:
VAlfven = 285 km/s,
n = 3×108 cm-3,
B0 = 2.3 G
DeceleraNon Lengthening Nonlinear wave decelerates linear Wave
velocities →
linear
nonlinear
Amplitude evolution
←
These results have the qualitative agreement with the
observations discussed à quantitatively? à
EUV wave in the 17-01-2010 Event
STEREO-B EUVI 195
Grechnev, Afanasyev, Uralov et al. (2011)
Kinematics of the EUV wave
Model of the solar corona Taking into account the nonlinear
properties of the EUV wave we
can fit the initial part of the plot
Front of the coronal shock
wave
Magnetic null points in the solar corona
Alfven speed in
the solar corona
in the model with
an AR dipole à
Null point
q  It is interesting to analyse the interaction of fast-mode MHD
waves with null points because it can result in the wave energy
dissipation, which is important for the coronal heating problem
Magnetic null points in the solar corona
q  Behaviour of waves in the neighbourhood of a
null point has been investigated by many authors:
Syrovatskii, 1966;
Craig and McClymont, 1991;
McLaughlin and Hood, 2004;
McLaughlin and Hood, 2006;
McLaughlin, Ferguson and Hood, 2008;
McLaughlin et al., 2009;
Gruszecki et al., 2011;
Review article by McLaughlin, Hood, and De Moortel, 2011
2D magnetic null point
•  2D magnetic null point: B=(x,0,-z).
•  The plasma density and temperature are assumed to be constant.
•  We calculate the propagation of waves in cold and warm plasmas.
c = 0: cold plasma
c ≠ 0: warm
plasma
VA=c layer
Linear wave in a cold plasma
The rays along which a linear wave travels towards the null point in a
cold plasma were obtained by McLaughlin and Hood (2004):
x=e
−
z0
t
r0
⎛
x
x ⎞
⎜⎜ x0 cos 0 t + z0 sin 0 t ⎟⎟, z = e
r0
r0 ⎠
⎝
−
z0
t
r0
⎛
x
x ⎞
⎜⎜ z0 cos 0 t − x0 sin 0 t ⎟⎟
r0
r0 ⎠
⎝
x0 and z0 are the initial values of a ray trajectory and r0 =
Wave front
rays
x02 + z02
Linear wave in a cold plasma
• The wave is captured
by the null point
• The wave energy
accumulates at the
null point
• Propagation speed at
the null point drops to
zero
Linear wave in a cold plasma
Geometrical acoustics allows the wave amplitude A to be calculated:
A(t ) / A0 =
r0
e
z0t + r0
z0
t
r0
if we assume the wave amplitude A to be the
magnitude of the plasma velocity vector, A = u 2 + υ 2
x0
z0
x0
A(t ) / A0 = cos 2 t + sin 2 t
r0
x0
r0
The plasma velocity component u
oscillates and drops to zero from
time to time, while the magnitude of
the plasma velocity vector grows
exponentially.
Such a behavior of the component
u is due to changing the direction
of the wave propagation.
r0
e
z0t + r0
z0
t
r0
if we assume A to be the
plasma velocity
component u along the
normal to the wave front
Linear wave in a warm plasma
VA=c layer
caustic
wave front
Wave passes through the null point, which results in
considerable distortion of the wave front as well as
formation of a caustic.
Linear wave in a warm plasma
Linear wave in a warm plasma
Ray touches the
caustic
In this case heating is
distributed in space
The wave amplitude tends to infinity at a caustic. Caustics in
the ray pattern represent places of the most efficient plasma
heating
Shock wave in a cold plasma
Any initial linear wave transforms into a shock wave, which
results in rapid conversion of the wave energy into heat
Shock wave passes through the null point
Shock wave in a cold plasma
A significant part of the shock wave energy is spent on
plasma heating.
The amplitude of a shock wave does not increase
Shock wave in a warm plasma
The ray pattern as well
as the form of the caustic
are similar to those in the
‘linear warm’ case, but it
is ‘shifted’ in the direction
of the wave incidence.
Wave amplitude→
Summary
•
•
•
For a wave approaching the null point, the amplitude
growth has been calculated analytically. The amplitude
growth of a linear wave results in its transformation into a
shock wave. The amplitude of a shock wave does not
increase.
A complex caustic is formed around the null point, with
the wave amplitude increasing significantly at the caustic.
Plasma heating associated with the caustic is distributed
in space.
A shock wave is able to pass through the null point even
in a cold plasma. Along with the heating at a caustic, a
substantial heating of plasma due to the nonlinear energy
dissipation in the shock front occurs in the neighbourhood
of the null point.
Summary
•
Taking into account the nonlinear nature of Moreton
waves and EUV waves associated with fast-mode
coronal shock waves, we explained the observed
deceleration and lengthening of these large-scale
coronal disturbances in the quiet Sun regions.
•
Modelling the EUV wave propagation in the 17Jan-2010 event showed the quantitative agreement
with observations.
Concluding remarks
•
Nonlinearity of wave disturbances in the solar corona is
crucial for their propagation and evolution and should be
taken into account.
•
Method of nonlinear geometrical acoustics allows one to
take into account nonlinear properties of fast-mode shock
waves, and calculate their propagation and amplitude
evolution in various inhomogeneous media in the ray
approximation. The method is quite effective and can be
applied in other problems.
References 1.
2.
3.
4.
5.
Afanasyev A.N., Uralov A.M. Coronal shock waves, EUV waves, and their rela7on to CMEs. II. Modeling MHD shock wave propaga7on along the solar surface, using nonlinear geometrical acous7cs. Solar Phys. 2011. V. 273. P. 479-­‐491. Afanasyev A.N., Uralov A.M. Modelling the propaga7on of a weak fast-­‐mode MHD shock wave near a 2D magne7c null point, using nonlinear geometrical acous7cs. Solar Phys. 2012. V. 280. P. 561-­‐574. Grechnev V.V., Afanasyev A.N., Uralov A.M., Chertok I.M., Eselevich M.V., Eselevich V.G., Rudenko G.V., Kubo Y. Coronal shock waves, EUV waves, and their rela7on to CMEs. III. Shock-­‐associated CME/EUV wave in an event with a two-­‐component EUV transient. Solar Phys. 2011. V. 273. P. 461-­‐477. Afanasyev A.N., Uralov A.M., Grechnev V.V. Using the nonlinear geometrical acous7cs method in the problem of Moreton and EUV wave propaga7on in the solar corona. Geomag. Aeronomy. 2011. V. 51. P. 1015-­‐1023. Grechnev V.V., Uralov A.M., Chertok I.M., Kuzmenko I.V., Afanasyev A.N., Meshalkina N.S., Kalashnikov S.S., Kubo Y. Coronal shock waves, EUV waves, and their rela7on to CMEs. I. Reconcilia7on of “EIT waves”, type II radio bursts, and leading edges of CMEs. Solar Phys. 2011. V. 273. P. 433-­‐460. Thank you!