1. Coronal Loop Oscillations observed by TRACE
(Today’s reference is Ofman, ApJ, 568, L135)
（Newton, 2002年10月号）

2. Coronal Loop Oscillations observed by TRACE
Nakariakov et al. 1999, Science, 285, 862
Aschwanden et al.2002, Sol.Phys, 206,99
Aschwanden et al. 1999, ApJ, 520, 880

3. Nakariakov et al. 1999, Science, 285, 862
Aschwanden et al. 1999, ApJ, 520, 880
coronal loop oscillations associated with flares
- loop length : km
- period : about 4 min
- decay time : about 870 s
What is the mechanism of enhanced dissipation?
- enhanced large Viscosity or Reynolds number (Nakariokov, Ofman,
Aschwanden)  R=10^(5-6) ! classical value : 10^(14)
- phase mixing  large R number (Ofman and Aschwanden, 2002)
- leakage under chromosphere (De Pontieu et al. 2001)
- fast mode propagation (Miyagoshi & Yokoyama, in prep.)
The analysis of loop oscillations and its damping
mechanism is ...
- coronal plasma characteristics
- an important impact on coronal heating theories
- wave heating
- reconnection theories

4. Ofman 2002, ApJ, 568, L135 1.5 dimensional MHD Three components of V &
B are included
(the thermal pressure is neglected compared with the magnetic pressure)
X : along the loop
Y : transverse to the loop
Z : vertical

5. Initial Condition Normalize Unit : L = 10000 km n = 10^14 V = 10km/s
(Loop length : km)
(Typical case)
(high freq.)
Oscillation amplitude 20km <<
Coronal Va 1000km :
the oscillations are nearly linear

6. Previous Work (Linear Theory)
The linear theory predicts that the leakage time of the
Alfvén waves is given by (e.g., Berghmans & De Bruyne 1995)
R : wave reflection
coefficient
When the Alfvén speed jumps discontinuously from the
photospheric value to the coronal value, the reflection
coefficient can be approximated by (e.g., Davila 1991)
here, Va0=10km/s,
Va=1000km/s,
L = km so R=0.980
and Td = 6500 s
(observation : 870s)

7. Previous Work (Linear Theory)
De Pontieu et al. (2001) use the following expression for the
timescale of the leakage through the ends of the loop
(with Chromosphere model):
De Pontieu et al. (2001) made
a mistake. The correct value is,
Td=3179s
(Re,spot = 0.90, Re, plage = 0.94)
The energy reflection coefficient of Alfvén waves was obtained
by Leer et al. (1982) for Alfvén waves in the solar wind
(their eqs. [44] and [46]) and used by Hollweg
(1984) and De Pontieu et al. (2001) in coronal loops.
Td = 4127 s (typical case)
Td = s (short wavelength)

8. Numerical Results
The linear analytical calculation of the leakage time is sensitive
to the estimated theoretical value of the reflection coefficient.
In the numerical model, the leakage time is calculated from the decrease of the wave amplitude in the loop as a function of time, without the need to estimate the reflection coefficient.
Without Choromosphere:
Td = 6453 s
 excellent agreement with
the theoretical value
With Choromosphere:
Td = 4406 s
 good agreement
(7% longer)

9. Numerical Results (t = 939s) Vx component of the velocity is
produced by the nonlinear coupling
of the Alfvén wave to the fast wave. Vy Vx Va
However, since the amplitude of the
wave is small (2%) compared with
the coronal Alfvén speed, the
nonlinearity does not affect the
dynamics or the leakage time
significantly.
Short wavelength case (k=5):
Td = 397 s
 good agreement
(16% longer)

10. Discussions and Conclusions
They find that the leakage time in this loop is 4406 s, which is 5
times longer than the observed damping time.
Based on this model, they conclude that the chromospheric
footpoint leakage cannot explain the observed rapid oscillation
damping, in agreement with the conclusion of Nakariakov et al.
(1999). I find that short-wavelength Alfvén waves will leak on a
much shorter timescale.

11. Discussions and Conclusions
The leakage time depends on several observed and
assumed parameters.
-An increased or decreased chromospheric scale height will
leaded to a faster or slower wave leakage,
-the values of photospheric and coronal Alfvén speeds, the loop length,
and the wavelength can vary from one
location to another in an active region and may be different in various
active regions and in various loops
The dynamic variations and shocks in the chromosphere
were not included in the model.
In the present 1.5-dimensional MHD coronal loop model, the
geometric divergence of the magnetic field near the footpoints
is not included. To account for the magnetic divergence
requires multidimensional modeling of the coronal loop.

12. Miyagoshi & Yokoyama (2003, in prep)
Three – dimensional MHD numerical simulations
Vy of typical case is 80 % of the local sound speed
(strong nonlinear effect)

13. Numerical Simulation Results
It can be explained…
Energy loss by fast mode
propagation around

14. Numerical Simulation Results
For this mechanisim, three dimensional effect is essential.
Comparison between TRACE observation and our model…
Our Model
(Miyagoshi and Yokoyama)
Period : 260 s
Decay time : 750 s
For loop length km,
Coronal sound speed 100km/s
Observation
(Nakariakov et al. 1999)
Period : about 240 s (4 min)
Decay time : about 870 s
For loop length km