Introduction

This chapter deals with the search for the oscillatory signatures of propagating waves in the chromosphere by extending the work in the previous chapter into two further dimensions (namely height in atmosphere and time). This study uses the fact that both longitudinal and transverse travelling waves are observable as intensity changes in carefully chosen wavelength bandpasses. A propagating longitudinal wave results in a periodic compression and hence heating of the surrounding plasma. This leads to increased radiation, which is then viewable as a periodic intensity change in the filter bandpass. Transverse waves propagating through a plasma result in a Doppler shift of the line profile of the emitted radiation. When the wave is not travelling along the observers line-of-sight ,this results in a periodic change in intensity in the filtergram as the emitted line profile moves in and out of the chosen bandpass. Detailed MHD simulations predict various types of oscillatory signatures for different wave modes at different heights in the chromosphere. The search for these oscillatory signatures is therefore vital in providing observational evidence for MHD waves.

Simulations of MHD waves are commonly modelled using the thin flux tube approximation, whereby the flux tube radius is much smaller than the characteristic wavelength. This approximation will break down in the mid chromosphere, where the flux tube will expand radially due to decreased density and hence its diameter will exceed the pressure scale height (Spruit 1981a). Further discussion of the validity of the thin flux tube approximation in modelling the network can be found in Hasan et al. (2003).

In the thin flux tube approximation a magnetic flux tube in an isothermal atmosphere supports three types of waves (Spruit & Roberts 1983), as discussed in Chapter 2. To summarise, the pure Alfvén torsional wave is non-dispersive and is propagating for any frequency, which negates it as a possible heating mechanism. In a flux tube with a strong magnetic field (as occurs in NBPs), the other types are the transverse, kink-mode wave and the longitudinal, sausage-mode wave. The corresponding tube speeds are given in Kalkofen (1997) in terms of the acoustic speed, c_s (which is around 9 km s in the low chromosphere). Both wave modes are dispersive and propagate at frequencies above their respective cut-offs; otherwise they are evanescent. The wave will propagate at the appropriate tube speed, followed by a wake oscillating at the corresponding cut-off frequency. The longitudinal cut-off frequency, ν_l, and the transverse cut-off frequency, ν_k, are defined by Kalkofen (1997) in terms of the acoustic cut-off frequency, ν_ac (typically around 5.5 mHz in the chromospheric internetwork K_2v bright points, Liu 1974) and given in Chapter 2. For ease of reference they are reproduced as,

With a maximum plasma β value of 0.5 (Kneer, Hasan, & Kalkofen 1996), this gives a minimum ν_k=2.2 mHz and ν_l=5.7 mHz. However, the local conditions (e.g., density, magnetic field) in the network can be quite different from the internetwork such that although ν_l will not vary by more than 2% from ν_ac (Kalkofen 1997), ν_k may be much lower than 2.2 mHz. Roberts (1983) also discuss the possibility of non-adibatic conditions in the network, which will drastically alter ν_l. The exact value of ν_ac in the magnetised atmosphere may also differ from the internetwork.

Transverse waves may be generated at the photosphere by granular buffeting of network flux tubes at a frequency above the transverse cut-off. Muller & Roudier (1992) and Muller et al. (1994) have discovered that NBPs do possess a rapid, intermittent motion which Choudhuri, Auffret, & Priest (1993) and Choudhuri, Dikpati, & Banerjee (1993) modelled as the creation of transverse mode waves at the photospheric level. These waves can then propagate up along the field lines in a NBP at the corresponding tube speed. Their speed amplitude will increase due to density stratification, so that when it becomes comparable to the tube speed, they enter the non-linear range. At this point mode-transformation can occur (Ulmscheider et al. 1991), and the waves can couple (and hence transfer power) to longitudinal waves. This mode-coupling occurs preferentially for transverse waves at a frequency, ν, which can transfer most of their power to longitudinal waves at a frequency 2ν, with a remnant of power at the original frequency. The interaction is greatest when the two tube speeds are equal (i.e., c_k = c_l). From Eqn. 61 and Eqn. 59 it turns out this corresponds to β = 0.2, which gives a speed of maximum interaction of around 0.93 c_s. The longitudinal waves can then shock and heat the surrounding plasma (Zhugzhda, Bromm, & Ulmscheider 1995).

In order to identify such a mechanism in the chromosphere, a number of tests can be carried out.

• Test 1: Power at a frequency ν in the lower chromosphere should also be identifiable at much reduced power in the upper chromosphere.
  
• Test 2: Power at frequency 2ν should appear in the upper chromosphere.
  
• Test 3: Waves at frequencies above the transverse cut-off should propagate at a speed close to c_s.
  
• Test 4: In the upper chromosphere there may be evidence for waves above the longitudinal cut-off frequency although to a lesser extent as they should shock in this region.
  
• Test 5: Oscillations should be quasi-periodic, resulting from the nature of the granular buffeting.

In this chapter I search for observational evidence of MHD waves by studying intensity changes in NBPs at several wavelengths, corresponding to a range of heights from the photosphere to the upper chromosphere. Section 5.2 provides a summary of the observations and data analysis. Both Fast Fourier Transforms (FFTs) and wavelet transforms are used to study NBP oscillations in space and time. The wavelet transforms are also cross-correlated at all frequencies to check for the signature of possible travelling waves. In Section 5.3 I present and discuss the results from each NBP individually and collectively. These results are then further discussed in Section 5.4 in relation to the observational tests mentioned above.