We have carried out a linear analysis of the three-dimensional wave propagation in a stratified, isothermal atmosphere. The motivation was to study the limitations of the generally assumed plane waves. Although the analysis was based on linearized equations and idealized medium our results may suggest the sense in which 3D and 1D waves differ from one another in the nonlinear regime. A pulse generated in the `photosphere' by a `point source' propagates upward with exponential amplification. The source region has a Gaussian pressure perturbation with a full half-width of , where H is the pressure scale height, for a diameter of 40 km. An initial perturbation of strength becomes nonlinear at a height of . So this strength of the source is near the minimum needed for bright points.
At 1 Mm above the source, which corresponds to the height of formation of the H and K lines of Ca II in the solar chromosphere, the pressure and velocity perturbations reach large values over a region with a horizontal extent of 1 Mm. This is comparable to the size of Ca bright points.
The initial pulse is followed by a wake in the upward direction at the acoustic cutoff period, approximately three minutes in the solar atmosphere. There is a wake also in the horizontal direction, with the same period but much lower amplitude.
The energy of the wave is concentrated in the vertical direction: One quarter of the upward-propagating energy is contained within a cone with a half-angle of 30o about the vertical axis; that cone constitutes only about 13% of the volume of the hemisphere. The energy is concentrated also in a narrow layer behind the initial pulse: When the wave has reached a height of above the source, 60% of the energy on the vertical axis is contained within the first two scale heights behind the apex.
The height where a given magnitude of the perturbation is reached increases with the size of the region. Assuming that nonlinear conditions are reached on the vertical axis at a height of , a `bright point' with a diameter of 1 Mm requires that the wave at the edge travel another , which requires an additional 21 s (at a sound speed of 7 km/s), and for a diameter of 2 Mm, the additinal travel distance and time are and 45 s. Thus the `bright point' grows from the center outward and upward. For the larger diameter and height, correspondingly higher layers in the chromosphere would form the spectrum.
The amplitude of the oscillations behind the pulse weakens in strength and shrinks in size. As a consequence, the maximal intensities associated with bright points in the wake are weaker and smaller than those in the initial pulse.
The maxima in the wake decay with time as , depending on the geometry. For a plane wave in a one-dimensional medium, , the velocity decays as ; it matches the plane wave solution of KRBM; for a line source in a two-dimensional medium, the decay is as ; and for a point source in 3D, it is as . The energy flux decays as . Thus the energy contained within the spherical volume of the wave decays in step with the increase in volume and hence the energy within is conserved.
For comparison with observations of chromospheric oscillations it needs to be borne in mind that the initial atmosphere in our analytic solution is at rest. The numerical simulations by Carlsson & Stein (1997) compare well with observations only when the waves are launched into a disturbed atmosphere.
© European Southern Observatory (ESO) 2000
Online publication: January 31, 2000