MHD Mode Conversion in a Stratified Atmosphere

1
MHD Mode Conversion in a Stratified Atmosphere

• Dee McDougall and Alan Hood
• University of St Andrews, UKIAUS 247 Venezuela
• 17 22 September 2007

2

• Zhugzhda (1979) Zhugzhda and Dzhalilov (1981,
  1982) Cally (2001).
• Investigate propagation from high- to low-ß
  plasma.
• Solutions found for harmonic waves.

• We examine propagation from low- to high-ß
  plasma.
• We drive on the upper boundary.
• Predominantly a slow wave.
• Fast wave is evanescent.

3

• We derive wave equations from the Ideal,
  linearised MHD equations.
• Fast and slow modes coupled via the horizontal
  wave number
• If there is no mode conversion.
• Equations are solved numerically using the
  MacCormack scheme.

4
Numerical Simulations Isothermal

• Strong exponential behaviour.
• Masks what is happening around conversion region.

• Now clear that mode conversion is occurring where
  the sound and Alfvén speeds are equal.

5

• In high ß we see the fast wave propagating ahead
  (transmitted wave).
• Slow wave lags behind and is visible as
  interference (converted wave).
• Amplitude of transmitted wave may be compared to
  the incident wave to quantify the transmission.

6

• Conversion occurs at where
• Single wave mode will have the same properties in
  both high and low ß.
• When we drive a slow wave in low ß
• Transmitted wave is the fast wave.
• Converted wave is the slow wave.

7

• Using WKB method we may expand and in
  inverse powers of assuming that
• Substituting these expansions into the wave
  equations gives solutions valid away from the
  mode-conversion region.
• and are the conversion and transmission
  coefficients respectively.

8

• To solve our wave equations around the
  mode-conversion region we follow the method
  developed by Cairns and Lashmore-Davies (1983).
• Method is valid under the assumptions
  ,
• Writing the wave equations in the form
• the transmission coefficient is given by
• the conversion coefficient is given by

9

• Under the assumptions, if we make the
  transformation and expand about
  the conversion region the wave equations become
• So the conversion and transmission coefficients
  are given by

• On comparing with the results of the
  numerical simulations we find excellent
  agreement.
• Work to be published in Solar Physics, McDougall
  and Hood (2007, in press).

10
Non-Isothermal Atmosphere

• We may follow exactly the same analysis process
  for a non-isothermal atmosphere.

• We choose a tanh profile to model the steep
  temperature gradient at the transition region.
• .
• We set in the centre of the
  temperature gradient.

11

• Using a WKB analysis the behaviour away from the
  mode-conversion region is given as
• This allows us to transform the velocity so that
  the incident amplitude is constant

12

• Following Cairns and Lashmore-Davies (1983)
  around the mode-conversion region, we find
  conversion and transmission coefficients

• As before, we can compare the transmission
  predicted to the numerical simulations.

13
Conclusions and Future Work

• Using various analytical techniques we have found
  conversion and transmission coefficients which
  accurately quantify mode conversion.
• It turns out that the conversion and transmission
  coefficients are the same regardless of the
  chosen temperature profile.
• We plan to investigate the same phenomenon in
  more complex magnetic topologies, such as around
  2D null points, extending on work done by
  McLaughlin and Hood (2006).