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Acoustic waves in a stratified atmosphere


Abstract and Figures

In a gravitationally stratified atmosphere, small temperature variations distort the paths of acoustic waves from the rectilinear paths in an isothermal atmosphere. For temperature increasing upward, low-frequency waves near the acoustic cutoff frequency propagating at a given polar angle are refracted towards the vertical direction (focused) and high-frequency waves, away from the vertical (defocused). Similarly, for temperature increasing towards the axis of a vertical cylinder, low-frequency waves are focused and high-frequency waves are defocused. This effect of temperature inhomogeneities may be important for wave propagation in the chromospheric K$_{\rm 2v}$ bright point phenomenon.
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Waves & Oscillations in the Solar Atmosphere:
Heating and Magneto-Seismology
Proceedings IAU Symposium No. 247, 2008
R. Erd´elyi & C. A. Mendoza-Brice˜no, eds.
°2008 International Astronomical Union
DOI: 00.0000/X000000000000000X
Resonant acoustic waves in
a stratified atmosphere
Y. Taroyan and R. Erd´elyi
SP2RC, Department of Applied Mathematics, University of Sheffield, Sheffield S3 7RH, UK
email: y.taroyan;
Abstract. The upward propagation of linear acoustic waves in a gravitationally stratified atmo-
sphere is studied. The wave motion is governed by the Klein-Gordon equation which contains
a cut-off frequency introduced by stratification. The acoustic cut-off may act as a potential
barrier when the temperature decreases with height. It is shown that waves trapped below the
barrier could be subject to a resonance which extends into the entire unbounded atmosphere.
The parameter space characterizing the resonance is explored.
Keywords. waves, hydrodynamics, Sun: atmosphere, Sun: photosphere, Sun: chromosphere,
Sun: Corona, Sun: oscillations
1. Introduction
Acoustic waves have often been invoked as possible candidates for the heating of solar
and stellar chromospheres and coronae. Until recently it was thought that high frequency
waves could be responsible for the heating of the non-magnetic chromosphere of the Sun
as they develop into shocks. On the other hand, low frequency waves were believed to
play little role as far as the dynamics and energetics of the atmosphere are concerned due
to reflection from regions with steep temperature gradients. Recent works have changed
these views. It was established that the power of the observed high frequency propa-
gating (>5 mHz) acoustic waves is not enough to balance the radiative losses in the
chromosphere (Fossum & Carlsson 2005). On the other hand, new observations have
shown that the energy flux carried by the low frequency (<5 mHz) acoustic waves into
the chromosphere is about a factor of 4 greater than that carried by high frequency waves
(Jefferies et al. 2006). It was argued that these low frequency waves could propagate and
carry their energy into the higher layers of the atmosphere through portals formed by the
inclined magnetic field lines. Further, it was demonstrated that dynamic features such
as solar chromospheric spicules or fibrils could be associated with the leakage of global
p-modes into the atmosphere along inclined field lines (De Pontieu et al. 2004). A strong
correlation was also found between propagating intensity oscillations in the corona and
p-modes (De Pontieu et al. 2005). These and other results have prompted renewed strong
interest in the theory of low frequency acoustic wave propagation in stratified media.
The present paper deals with a two-layer model (Fig. 1) to study the vertical propaga-
tion of acoustic waves in a stratified atmosphere (either plasma or gaseous). The waves
are described by the Klein-Gordon (KG) equation. The main result here is the discovery
of a resonance occurring at low frequencies which extends into the entire unbounded
atmosphere. This previously unknown resonance may be responsible for the transfer of
wave energy which could have dynamic consequences and heat the higher atmospheric
layers. The KG equation is widely used in a range of fields such as atmospheric physics,
cosmology, quantum field theory, solid state physics, solar/stellar physics. Therefore, the
2 Y. Taroyan & R. Erd´elyi
presented results may have wider applicability in distinct areas of physics and astro-
Figure 1. Two-layer model depicting a stratified solar atmosphere. The lower part of the at-
mosphere (index 1) is separated from the upper part (index 2) by a density and temperature
discontinuity at z=L. Waves are launched at z= 0 and propagate in the vertical z-direction.
2. Model and Governing Equations
The proposed one dimensional model is shown in Fig. 1. The atmosphere is stratified
under gravity in the zdirection. The temperature T=T(z) linearly decreases in the
lower part and remains constant in the upper part of the atmosphere:
T0(1 az),0< z < L,
T2, z > L,
where the constant a(aL = 1 TL/T0>0) characterizes the steepness of temperature
decrease from T0=T(0) to TL=T(L), and Lis the thickness of the nonuniform layer.
The wave motion is governed by the KG equation:
∂z2+ Ω2(z)Q= 0 (2.2)
where Qis the scaled velocity (Roberts 2004) and c= (γRT )1/2is the sound speed. The
quantity Ω represents the acoustic cut-off frequency, which imposes a time-scale on the
dz ,(2.3)
where Λ is the pressure scale-height proportional to the temperature. An extensive review
on solar applications of the KG equation is presented by Roberts (2004). In the present
work, the KG equation (2.2) is applied to the study of waves driven at a boundary of a
semi-infinite non-isothermal atmosphere:
z0Q(z, t) = I(ω) cos(ωt),(2.4)
where ωis the driver frequency and I=I(ω) is the frequency dependent amplitude
of the driver. For simplicity, we assume that Qis a complex variable. The boundary
JD 11. Resonant acoustic waves 3
condition (2.4) is then replaced with
z0Q(z, t) = I(ω)eiωt .(2.5)
3. Results and Discussion
We seek steady state solutions of the form Q(z, t)exp(t). In the lower part of
the atmosphere, Eq. (2.2) can be transformed to a Bessel equation. It possesses solutions
of the form
Q(z, t) = exp(t)1az ·A1Jνµ2ω
where Jνand Yνare the Bessel functions of the first and second kind, respectively and
0)1 is the polytropic index. In the upper layer z > L, the solution has the
Q(z, t) = Aexp(ikz t),where k=
, ω < 2,
, ω > 2,
with c2and Ω2being the constant sound speed and cut-off frequency in the upper layer.
The coefficient Ais uniquely determined by appropriately matching the solutions across
z=L. It determines the wave amplitude in the region above z=L. In the WKB limit
the coefficient Ais reduced to
A=I(ω) exp(ikL)
Eq. (3.3) is valid when the lower layer is thin. It shows that the wave amplitude may be-
come infinite in the adopted linear approximation. A necessary condition for the existence
of a resonance is L/2>1. The waves are resonantly amplified when
In Fig. 2, the scaled resonant frequency ωis plotted against the scaled thickness L. The
temperature T2is fixed (and so are the cut-off frequency Ω2and the scale height Λ2).
Four different cases with different temperature ratios (T0/T2= 2,5,8,15) are shown. In
all four cases, TL=T2is set. The resonant frequencies consecutively appear and decrease
as the thickness of the lower layer Lincreases. In general, the higher the length ratio
L/Λ the lower the temperature ratio T0/T2>1 required for the existence of a resonance.
The physical mechanism responsible for wave amplification is the following: the de-
creasing temperature results in an increasing acoustic cut-off frequency Ω = Ω(z) which
forms a potential barrier similar to the one in quantum mechanics (Landau & Lifshitz
1977). Low frequency waves with Ω0< ω < 2driven at z= 0 are reflected back from
the barrier and trapped in the lower layer 1. When the driver frequency matches the nat-
ural frequency of the cavity where the waves are trapped a standing wave is set up and
amplified resonantly. In the case of a thin layer, only the fundamental mode is present
with a frequency given by Eq. (3.4). The frequency of the fundamental mode decreases
and higher harmonics appear as the thickness Lincreases. The resonance affects the
evanescent tail of the waves in the upper atmosphere leading to a global resonance.
4 Y. Taroyan & R. Erd´elyi
Figure 2. Scaled resonant frequency ωas a function of the length of the non-uniform layer
L. Four different cases are shown: T0/T2= 2 (solid line), T0/T2= 5 (dashed line), T0/T2= 8
(dotted line) and T0/T2= 15 (dash-dotted line). In all four cases, TL=T2is set.
The nonlinear development and dissipation of such waves generated in various physical
systems and their energetic implications must be treated separately. These may include
the generation of spicules or the heating of the lower atmosphere. An important extension
of the present work is the treatment of the problem in 2D and the consideration of a
non-monochromatic driver. The resonant waves presented here arise due to the variation
of the cut-off frequency introduced by stratification. However, such waves could operate
in any other systems with varying cut-off frequencies. The cut-off frequencies of waves,
such as Alfv´en, kink and slow waves, in thin magnetic flux tubes vary due to, e.g., cross-
section expansion of the flux tube or the variations in the magnetic field strength and
density (Spruit & Roberts 1983). Therefore, such waves could readily become subject to
a resonance in solar/stellar structures. The applicability of the presented mechanism in
other areas of physics and astrophysics is open for discussion.
Y.T. is grateful to the Leverhulme Trust for financial support. R.E. acknowledges M.
eray for patient encouragement and NSF, Hungary (OTKA, ref. no. TO43741).
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... Most of the wave-like oscillations in the atmosphere can be described/parametrized using basic acoustic-gravity wave theory in the atmosphere. Details can be found, for instance, in works of Davies (1990), Bodo et al. (2001), Hargreaves (1982), Yeh & Liu (1974) among others. Here, we show brief derivation of the dispersion relation that any wave motion of the AGW type must satisfy. ...
We analyzed chromospheric events and their connection to oscillation phenomena and photospheric dynamics. The observations were done with the New Solar Telescope of Big Bear Solar Observatory using a broad-band imager at the wavelength of a TiO band and FISS spectrograph scanning Ca ii and Hα spectral lines. The event in Ca ii showed strong plasma flows and propagating waves in the chromosphere. The movement of the footpoints of flux tubes in the photosphere indicated flux tube entanglement and magnetic reconnection as a possible cause of the observed brightening and waves propagating in the chromosphere. An upward propagating train of waves was observed at the site of the downflow event in Hα. There was no clear relationship between photospheric waves and the Ca ii and Hα events. Our observations indicate that chromospheric waves that were previously thought to originate from the photosphere may be generated by some events in the chromosphere as well.
The efficiency of non-modal self-heating by acoustic wave perturbations is examined. Considering different kinds of kinematically complex velocity patterns, we show that non-modal instabilities arising in these inhomogeneous flows may lead to significant amplification of acoustic waves. Subsequently, the presence of viscous dissipation damps these amplified waves and causes the energy transfer back to the background flow in the form of heat; viz. closes the ‘self-heating’ cycle and contributes to the net heating of the flow patterns and the chromospheric network as a whole. The acoustic self-heating depends only on the presence of kinematically complex flows and dissipation. It is argued that together with other mechanisms of non-linear nature the self-heating may be a probable additional mechanism of non-magnetic chromospheric heating in the Sun and other solar-type stars with slow rotation and extended convective regions.
I report observations of unusually strong photospheric and chromospheric velocity oscillations in and near the leading sunspot of NOAA 10781 on 3 July 2005. I investigate an impinging wave as a possible origin of the velocity pattern and the changes of the wave after the passage through the magnetic fields of the sunspot. The wave pattern found consists of a wave with about 3 Mm apparent wavelength, which propagates towards the sunspot. This wave seems to trigger oscillations inside the sunspot’s umbra, which originate from a location inside the penumbra on the side of the impinging wave. The wavelength decreases and the velocity amplitude increases by an order of magnitude in the chromospheric layers inside the sunspot. On the side of the sunspot opposite to the impinging plane wave, circular wave fronts centered on the umbra are seen propagating away from the sunspot outside its outer white-light boundary. They lead to a peculiar ring structure around the sunspot, which is visible in both velocity and intensity maps. The fact that only weak photospheric velocity oscillations are seen in the umbra – contrary to the chromosphere where they peak – highlights the necessity to include the upper solar atmosphere in calculations of wave propagation through spatially and vertically extended magnetic field concentrations such as sunspots.
Full-text available
Context. The quiet solar chromosphere in the interior of supergranulation cells is believed to be heated by the dissipation of acoustic waves that originate with a typical period of 3 min in the photosphere. Aims: We investigate how the horizontal expansion with height of acoustic waves traveling upward into an isothermal, gravitationally stratified atmosphere depends on the size of the source region. Methods: We have solved the three-dimensional, nonlinear, time-dependent hydrodynamic equations for impulsively-generated, upward-propagating acoustic waves, assuming cylindrical symmetry. Results: When the diameter of the source of acoustic waves is small, the pattern of the upward-propagating waves is that of a point source, for which the energy travels upward in a vertical cone, qualitatively matching the observed pattern of bright-point expansion with height. For the largest plausible size of a source region, i.e., with granular size of 1 Mm, wave propagation in the low chromosphere is approximately that of plane waves, but in the middle and upper chromosphere it is also that of a point source. The assumption of plane-wave propagation is not a good approximation in the solar chromosphere. The upward-directed energy flux is larger than that of the solar chromosphere, at least in the middle and upper chromosphere, and probably throughout. Conclusions: Simulations of impulsively generated acoustic waves emitted from source regions with diameters that are small compared to the pressure scale height of the atmosphere qualitatively reproduce the upward expansion observed in chromospheric bright points. The emission features in the cores of the H and K lines are predicted to be blueshifted for a pulse and redshifted for the waves in its wake. The contribution of internal gravity waves to the upward energy flux is small and decreases with increasing size of the source region.
The raditation loss of the solar chromosphere is evaluated on the basis of the Harvard Smithsonian Reference Atmosphere. The total radiative flux is found to be between 2.5 and 3.3 E6 erg cm–2 s–1. A discussion of possible heating mechanisms shows that the short period acoustic wave theory is the only one able to balance the chromospheric radiation loss and is consistent with observation.
  • H C Spruit
  • B Roberts
Spruit, H.C. & Roberts, B. 1983, Nature, 304, 401