Warwick PX420 Solar MHD 2014-2015: Problems
1
Problems
See also the module’s web-site:
http://www2.warwick.ac.uk/fac/sci/physics/research/cfsa/people/valery/teaching/px420/
Problem 1 Estimate in km the distance on the Sun corresponding to one
arcsec when the Earth is in perihelion (about 4th of January each year) and
is in 147.5 million km from the Sun and when it is in aphelion (about 4th of
July), in 152.6 million km from the Sun.
Problem 2 Show that the adiabatic equation can be expressed as
∂p
+ V · ∇p = −γp∇ · V.
∂t
(1)
How would the expression change in the incompressible case?
Problem 3 Show that in the case of constant pressure, the energy equation
ργ d
γ − 1 dt
p
ργ
!
= −L,
(2)
can be rewritten as
dT
= −L,
dt
where cp is specific heat at constant pressure,
ρcp
cp =
(3)
γ kB
.
γ−1 m
Problem 4 There is a dimensionless number, Rm , called the magnetic Reynolds
number.
LV
Rm =
,
(4)
η
where V is a typical plasma velocity, L is a typical scale and η is the magnetic
diffusivity. The magnetic diffusivity is defined as
η=
1
,
µσ
(5)
Warwick PX420 Solar MHD 2014-2015: Problems
2
where σ is the electrical conductivity in ohm m−1 .
According to Braginski, in a strongly magnetised plasma η can be estimated as
η = 109 T −3/2 m2 s−1
Estimate the magnetic Reynolds number in the corona, in sunspots and
in the solar wind, making appropriate assumptions about the typical spatial
scales and times.
Problem 5 If the radius of a sunspot is l = 107 m and η = 103 m2 s−1 , estimate the time for the sunspot magnetic field to diffuse away.
Problem 6 A chromospheric magnetic element has a magnetic field of 1000 G,
and a radius 100 km. Determine the radius of the coronal loop which is a
continuation of this element if the field is estimated to be 15 G.
Problem 7 A neutral slab of a non-magnetized plasma of width l is confined
by two regions of a magnetized plasma penetrated by the magnetic field of
strength B0 , parallel to the slab boundaries. The field reverses its direction
from one side of the slab to the other. Neglecting the magnetic diffusivity η,
show that the slab can be in equilibrium. What is the thermodynamic pressure
in the slab?
Problem 8 Sketch the magnetic field lines for B = 1ex + 2xey , and determine the j × B force at the point x = 0, y = 1.
Problem 9 Investigate the polytropic model of the solar wind. Consider a
steady spherically symmetric outflow from the surface of the Sun. Show that
the MHD equations in spherical coordinates reduce to
1 d 2
(r ρv) = 0,
r2 dr
ρv
d p GM ρ
dv
=−
−
,
dr
dr
r2
d
v
dr
p
ργ
(6)
(7)
!
= 0.
(8)
Warwick PX420 Solar MHD 2014-2015: Problems
3
Show that the Euler equation may be rewritten as
dv
γ
v
=−
dr
γ−1
p
ργ
!
d γ−1 GM
ρ
−
,
dr
r2
(9)
and deduce the expression
Cs2
GM
v2
+
−
= const.
2
γ−1
r
(10)
Problem 10 Observations by GOES spacecraft show that the integrated soft
X-ray flux of a solar flare varies with time as F (t) ∝ exp(−t2 /t2f ), where tf
is about 500 s. Determine the expected time dependence of the hard X-ray
and microwave fluxes generated by the flare.
Problem 11 The vertical dependence of the absolute value of the magnetic
field in an active region is estimated as B = B0 (z/a)−4 , where a and B0 are
constant. Take that the density profile is given by the hydrostatic equilibrium,
ρ = ρ0 exp(−z/Λ), where ρ0 and Λ are constant. Determine the height of the
minimum of the Alfvén speed.
Problem 12 The Extreme Ultraviolet Imager onboard SOHO spacecraft observes an off-limb loop. The emission intensity in the loop is measured to
have the scale height two times larger than outside it. Under the assumption
that the loop cross-section does not change with height, determine the ratio
of the plasma temperatures inside and outside the loop.
Problem 13 Spectroscopic measurements show that outside the loop the plasma
is cooler and less dense than inside. What information it gives you about the
ratio of the magnetic field inside and outside the loop?
Problem 14 Imaging observations in EUV show that the global kink mode
of a coronal loop has the phase speed of 1000 km/s. The loop appears to
be 100 times brighter than the background plasma. Show
that in the low-β
q
plasma the kink speed can be estimated as Ck ≈ CAi 2/(1 + ρe /ρi ). Using
this expression or otherwise estimate the Alfvén speed in the loop. (Here the
indices i and e refer to the physical parameters inside and outside in the loop,
respectively.)
Warwick PX420 Solar MHD 2014-2015: Problems
4
Problem 15 The SOHO/SUMER spectral instrument discovered quasi-periodic
oscillations of the intensity and Doppler shift in the coronal emission line
Fexix . The period of the oscillations is in the range of 5-15 min. It is suggested that these oscillations can be interpreted as standing sound waves in
coronal loops.
(a) Consider linear sound waves propagating strictly along the magnetic
field, parallel to the z-axis, neglect dissipation and gravity, and show that the
Euler equation and continuity equations may be reduced to the equations,
ρ0
∂ p̃
∂ Ṽ
=− ,
∂t
∂z
∂ ρ̃
∂ Ṽ
+ ρ0
∂t
∂z
where ρ0 is the constant equilibrium density, and the tilde denoted perturbations of the physical value.
(b) Show that with the use of the expression p̃ = Cs2 ρ̃ where Cs is the
sound speed, the above equations combine to the wave equation
2
∂ 2 Ṽ
2∂ V
−
C
= 0.
s
∂t2
∂z 2
(c) Apply the “rigid wall” boundary conditions (V = 0) at the loop footpoints (z = 0 and z = L) and determine the resonant frequency of the global
acoustic (also called “longitudinal”) mode of the loop.
Problem 16 A global coronal wave is observed to propagate in a coronal
hole in the horizontal direction. The speed of the wave at the height 50 Mm
is estimated to be 2 times higher than near the bottom of the corona. Consider
the atmosphere to be plane, the magnetic field vertical and uniform, and the
temperature constant, and estimate the temperature.
Problem 17 The temperature at the centre of the Sun is is 15million K.
What is the sound speed in cm s−1 . Take the adiabatic index γ = 5/3 and the
mean molecular weight to be µ = 0.6.
Problem 18 Determine the acoustic cut-off frequency at R = R taking
the temperature to be 5800 K and using the equation of state for an ideal gas.
Discuss why this demonstrates that a different equation of state is required.
Problem 19 The ` = 1, n = 21 has typical damping time of approx.
10 days. In Sun-as-a-star data ` = 1, m = +1 and ` = 1, m = −1 modes are
Warwick PX420 Solar MHD 2014-2015: Problems
5
separated by 0.8 µHz because of the Sun’s rotation. Which is greater the half
width at half maximum of the modes or the 0.8 µHz separation and what are
the implication of this for observations? If an data point is recorded every
60 s for 365 d there will be a data point at, based on a Lorenztian mode profile
in the power spectrum, how many data points will span the full width at half
maximum of the mode.