Using the Butterfly Effect to Probe How the Sun Generates Acoustic Noise

Abstract

A major encumbrance to recognition of individual episodes of noise emission is the accumulation over hours of other noise emitted long before. This is true in simulations just as it is in the solar environment itself. The composite seismic signature of acoustic radiation accumulated over preceding hours drowns out the signature of newly emitted “acoustic pings.” This problem could be alleviated in simulations by periodically damping the accumulated acoustic radiation – if this can be done benignly, i.e. in such a way that the onset transient of the damping (and its subsequent termination) does not emit its own acoustic noise. We introduce a way of doing this based upon a study of the butterfly effect in compressible radiative MHD simulations of convection that excites p-modes. This gives us an encouraging preview of what further development of this utility offers for an understanding of the character of p-mode generation in convective atmospheres.

Appendix

The spatial domain of the simulations in this study was a square slab whose horizontal dimensions were $98~\mbox{Mm}\times 98~\mbox{Mm}$ ($768\times 768$ pixels), with a depth of 18.4 Mm (384 pixels), noting that the horizontal separation between neighboring pixels, 128 km, is 8/3 the vertical pixel separation of 48 km. The simulation domain is evolved with a cadence of 0.5 seconds, but it is saved only every 22.5 seconds.

The disturbance injected into the butterfly version of the simulation was a pseudo-random perturbation in the rms range of ${\pm}\,3~\mbox{m}\,\mbox{s}^{-1}$ applied independently to the vertical velocity of each pixel extending downward from the base of the photosphere to a depth of 1 Mm. This disturbance was applied in a single 3D frame of the butterfly version of the simulation, the computation proceeding from that moment with no further intervention from the authors.

In standard convective simulations run in the past, the bottom boundary has been, like the top, an acoustic reflector hence the strong accumulation of p-modes discussed in Sections 1, 4, and 5, and the sharp resonances these show in $\ell$–$\nu$ diagrams computed from standard compressible simulations of convection (Stein and Nordlund, 2001). For the simulations run in this study, we replaced the reflecting lower boundary by one that efficiently absorbs low-$\ell$ modes that penetrate to it. This is devised by a boundary condition that prescribes a pressure perturbation $[\delta p]$ at the lower boundary that resists vertical motion by the rule

$$ \delta p = -c\rho v_{z}, $$

(2)

a simple adaptation of the standard practice in classical electronics of terminating a transmission line with a resistor that matches its characteristic impedance to suppress a reflection back to the source that would occur in one that was terminated by a short or left simply open.

By this contrivance, the back-reflection of the pulse that delivers the butterfly perturbation to the granulation passes through the bottom boundary and disappears, leaving $\Delta ^{\mathrm{lep}}v_{z}$ in the purely acoustic region undisturbed until the first acoustic disturbances from the growing butterfly-difference sources arrive therein. This innovation, then, represents an early step in the development of the acoustic mop anticipated in Section 5 and reiterated in item vi) of the Summary (Section 7).