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Acoustic waves in a stratified atmosphere

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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.
c
°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;robertus@sheffield.ac.uk
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
1
2 Y. Taroyan & R. Erd´elyi
presented results may have wider applicability in distinct areas of physics and astro-
physics.
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:
T=
T0(1 az),0< z < L,
T2, z > L,
(2.1)
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:
2Q
∂t2c2(z)2Q
∂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
system:
2=c2
2µ1+2dΛ(z)
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:
lim
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
lim
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
c0a
1az+B1Yνµ2ω
c0a
1az¶¸,(3.1)
where Jνand Yνare the Bessel functions of the first and second kind, respectively and
ν=γg/(ac2
0)1 is the polytropic index. In the upper layer z > L, the solution has the
form
Q(z, t) = Aexp(ikz t),where k=
i
p2
2ω2
c2
, ω < 2,
pω22
2
c2
, ω > 2,
(3.2)
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)
1
L
2
+L
p2
2ω2
c2
,
(3.3)
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
ω=c2
LrL
Λ21,(3.4)
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.
Acknowledgements
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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  • H C Spruit
  • B Roberts
Spruit, H.C. & Roberts, B. 1983, Nature, 304, 401