Content uploaded by S. Massaglia

Author content

All content in this area was uploaded by S. Massaglia on Oct 06, 2014

Content may be subject to copyright.

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

∂t2−c2(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:

Ω2=c2

4Λ2µ1+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

z→0Q(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

z→0Q(z, t) = I(ω)e−iωt .(2.5)

3. Results and Discussion

We seek steady state solutions of the form Q(z, t)∼exp(−iωt). 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(−iωt)√1−az ·A1Jνµ2ω

c0a

√1−az¶+B1Yνµ2ω

c0a

√1−az¶¸,(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 −iωt),where k=

i

pΩ2

2−ω2

c2

, ω < Ω2,

pω2−Ω2

2

c2

, ω > Ω2,

(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

2Λ2

+L

pΩ2

2−ω2

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/2Λ2>1. The waves are resonantly ampliﬁed when

ω=c2

LrL

Λ2−1,(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 < Ω2driven 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.

K´eray for patient encouragement and NSF, Hungary (OTKA, ref. no. TO43741).

References

H. Lamb, Hydrodynamics, Cambridge University Press, Cambridge, 1953

Fossum, A. & Carlsson, M. 2005, Nature, 435, 919

Jeﬀeries, S.M., McIntosh, S.W., Armstrong, J.D., Bogdan, T.J., Cacciani, A. et al. 2006, ApJ,

648, L151

De Pontieu, B., Erd´elyi, R., & James, S.P. 2004 Nature, 430, 546

De Pontieu, D., Erd´elyi, R. & De Moortel, I. 2005, ApJ, 624, L61

Roberts, B. 2004, in: R. Erd´elyi, J.L. Ballester & B. Fleck (sci eds.) SOHO 13 Waves, Oscillations

and Small-Scale Transients Events in the Solar Atmosphere: Joint View from SOHO and

TRACE, ESA-SP, 547, 1

Landau, L.D. & Lifshitz, E.M. Quantum Mechanics, Pergamon Press, Oxford, 1977

Spruit, H.C. & Roberts, B. 1983, Nature, 304, 401