Acoustic Events in the Solar Atmosphere from Hinode/SOT NFI Observations

Abstract

We investigate the properties of acoustic events (AEs), defined as spatially concentrated and short duration energy flux, in the quiet Sun, using observations of a 2D field of view (FOV) with high spatial and temporal resolution provided by the Solar Optical Telescope (SOT) onboard Hinode. Line profiles of Fe i 557.6 nm were recorded by the Narrow-band Filter Imager (NFI) on a 82″×82″ FOV during 75 min with a time step of 28.75 s and 0.08″ pixel size. Vertical velocities were computed at three atmospheric levels (80, 130, and 180 km) using the bisector technique, allowing the determination of energy flux to be made in the range 3 – 10 mHz using two complementary methods (Hilbert transform and Fourier power spectrum). Horizontal velocities were computed using local correlation tracking (LCT) of continuum intensities providing divergences. We found that the net energy flux is upward. In the range 3 – 10 mHz, a full FOV space and time averaged flux of 2700 W m⁻² (lower layer 80 – 130 km) and 2000 W m⁻² (upper layer 130 – 180 km) is concentrated in less than 1 % of the solar surface in the form of narrow (0.3″) AE. Their total duration (including rise and decay) is of the order of 10³ s. Inside each AE, the mean flux is 1.6×10⁵ W m⁻² (lower layer) and 1.2×10⁵ W m⁻² (upper). Each event carries an average energy (flux integrated over space and time) of 2.5×10¹⁹ J (lower layer) to 1.9×10¹⁹ J (upper). More than 10⁶ events could exist permanently on the Sun, with a birth and decay rate of 3500 s⁻¹. Most events occur in intergranular lanes, downward velocity regions, and areas of converging motions.

Appendix

In order to compute the complex signal $V(t) = v(t) + \mathrm{i}\underline{v}(t)$, we used the temporal FFT together with the following properties of the Hilbert transform (HT). Given an observed signal v(t), the HT $\underline{v}(t)$ is given by

$$\underline{v}(t) = h(t) \star v(t),$$

where ⋆ denotes the convolution product and $h(t)= \frac{1}{\pi t}$.

For a sinusoidal signal of the form v(t)∝cos(ωt+φ), the HT is simply $\underline{v}(t) \propto\sin(\omega t + \varphi)$, so that in this case V(t)∝exp[i(ωt+φ)].

The easiest way to get $\underline{v}(t)$ is to use the Fourier transform (FT with u denoting the temporal frequency):

$$\mathrm{FT}_{\underline{v}}(u) = \mathrm{FT}_{h}(u)\mathrm{FT}_{v}(u).$$

It can easily be shown that

$$\mathrm{FT}_{h}(u) = - \mathrm{i} \operatorname{sgn}(u),$$

where $\operatorname{sgn}(u)$ is the sign function, +1 for u>0, 0 for u=0 and −1 for u<0.

As a consequence, the FT of the complex signal $V(t) = v(t) + \mathrm{i}\underline{v}(t)$ is given by

$$\mathrm{FT}_{V}(u) = \mathrm{FT}_{v}(u) + \mathrm{i}\mathrm{FT}_{h}(u) \mathrm{FT}_{v}(u) =\mathrm{FT}_{v}(u) \bigl( 1 + \operatorname{sgn}(u) \bigr) .$$

This expression reduces simply to

• i) u<0,FT _V (u)=0,

• ii) u=0,FT _V (u)=FT _v (u),

• iii) u>0,FT _V (u)=2FT _v (u).

We first computed the temporal FT of the observed signal v(t); in order to get the FT of the complex velocity $V(t) = v(t) + \mathrm{i}\underline{v}(t)$, values corresponding to negative frequencies were eliminated and values corresponding to positive frequencies were doubled. Finally, the complex velocity $V(t) = v(t) + \mathrm{i}\underline{v}(t)$ was derived from the inverse FT.