Analytical Description of Nonlinear Acoustic Waves in the Solar Chromosphere

A&A 599, A15 (2017) Astronomy DOI: 10.1051/0004-6361/201629568 & c ESO 2017 Astrophysics

Analytical description of nonlinear acoustic waves in the solar chromosphere

Yuri E. Litvinenko1 and Jongchul Chae2

1 Department of Mathematics, University of Waikato, PO Box 3105, 3240 Hamilton, New Zealand e-mail: [email protected] 2 Astronomy Program, Department of Physics and Astronomy, Seoul National University, 151-742 Seoul, Korea

Received 23 August 2016 / Accepted 22 December 2016

ABSTRACT

Aims. Vertical propagation of acoustic waves of finite amplitude in an isothermal, gravitationally stratified atmosphere is considered. Methods. Methods of nonlinear acoustics are used to derive a dispersive solution, which is valid in a long-wavelength limit, and a non-dispersive solution, which is valid in a short-wavelength limit. The influence of the gravitational field on wave-front breaking and shock formation is described. The generation of a second harmonic at twice the driving wave frequency, previously detected in numerical simulations, is demonstrated analytically. Results. Application of the results to three-minute chromospheric oscillations, driven by velocity perturbations at the base of the solar atmosphere, is discussed. Numerical estimates suggest that the second harmonic signal should be detectable in an upper chromosphere by an instrument such as the Fast Imaging Solar Spectrograph installed at the 1.6-m New Solar Telescope of the Big Bear Observatory. Key words. hydrodynamics – Sun: atmosphere – Sun: chromosphere – Sun: oscillations – waves

1. Introduction lower boundary (e.g., Sutmann et al. 1998; Petukhov & Petukhov 2001). Excitation by a single pulse or by a periodic, low- Fleck & Schmitz(1991) were the first to point out that three- frequency driver generally leads to a chromospheric response minute chromospheric oscillations naturally arise as a response at the cutoff frequency, which decays with time. The observed of the solar atmosphere to a general velocity disturbance at its persistent three-minute oscillations are probably excited by a base. Kalkofen et al.(1994) presented solutions for both impul- more complicated time-dependent driving, for instance by a sive and continuous driving at the photospheric boundary, which sequence of random pulses (Sutmann et al. 1998). Recently, illustrated how a velocity disturbance at the base of the chro- Chae & Goode(2015) analyzed vertical propagation of a spec- mosphere generates free atmospheric oscillations at the acoustic trum of acoustic waves with frequencies close to the cutoff fre- cutoff frequency. An important feature of the solutions is that quency and argued that the power of the observed chromospheric the growth rate with height at the driving frequency, ω, is less oscillations at frequencies ω ≥ ω0 is consistent with the acoustic than the growth rate of the oscillations excited at the acoustic energy flux in the photosphere, produced by a series of impulsive cutoff frequency, ω0, as long as ω < ω0. Kalkofen et al.(1994) disturbances. termed the former solutions evanescent because they grow with Traditionally, analytical studies of acoustic wave propaga- height more slowly than the free atmospheric oscillations. Con- tion in a gravitationally stratified atmosphere used linearized sequently, the atmospheric response to a low-frequency driving equations to describe the evolution of small perturbations. Here is dominated at sufficiently large heights by an oscillation at the we use methods of nonlinear acoustics (e.g., Naugolnykh & cutoff frequency, ω0, corresponding to wave periods of about Ostrovsky Naugolnykh & Ostrovsky 1998) to derive an ana- three minutes. (Non-evanescent waves at both frequencies are lytical solution for dispersive nonlinear waves in the solar present if ω > ω0.) chromosphere. Independent studies of chromospheric oscillations indicated The theoretical study is motivated by new opportunities for the presence of a strong three-minute peak in the data (e.g., observing the three-minute oscillations, provided by instruments O’Shea et al. 2002; Jefferies et al. 2006; Centeno et al. 2009). such as the HELioseismological Large Regions Interferomet- The excitation of the cutoff frequency mode provides a gener- ric DEvice (HELLRIDE) operating on the Vacuum Tower Tele- ally accepted explanation of the three-minute oscillations (e.g., scope located at Tenerife (Wisniewska´ et al. 2016) and the Fast Felipe et al. 2010; Chae & Goode 2015), although the oscilla- Imaging Solar Spectrograph (FISS) installed at the 1.6-m New tions were also modeled as standing acoustic waves trapped in a Solar Telescope (NST) of the Big Bear Observatory (Kwak et al. chromospheric cavity (e.g., Cally & Bogdan 1993), and as waves 2016). The theoretical results may also contribute to the anal- escaping a leaky resonator and propagating through a boundary ysis of the data from the Interferometric BIdimensional Spec- (e.g., Zhugzhda 2008). tropolarimeter (IBIS) installed at the Dunn Solar Telescope (e.g., Theoretical studies investigated the atmospheric response Stangalini et al. 2011), the CRisp Imaging SpectroPolarimeter to various types of continuous and pulsed driving at the (CRISP) installed at the 1-m Swedish Solar Telescope (e.g.,

Article published by EDP Sciences A15, page 1 of6 A&A 599, A15 (2017)

Stangalini et al. 2015), and the Interface Region Imaging Spec- As an illustration of distinct parameter regimes, consider the trograph (IRIS; e.g., Tian et al. 2014). These instruments can ac- plasma density n ' 109 cm−3 and temperature T ' 5 × 105 K, quire high-cadence data in the upper solar chromosphere where, corresponding to the coronal values at height h ' 2500 km as we argue below, the nonlinear effects lead to observable con- above the photosphere (e.g., level 1 of model C in Vernazza et al. sequences. The new analytical results may also help to inter- 1981). Neglecting the neutral component of the gas at that tem- pret and guide numerical studies (e.g., Kalkofen et al. 2010; perature, we see that the plasma beta β = 8πknT/B2 < 1 if the Fawzy & Musielak 2012). magnetic field B exceeds a couple of Gauss. Conversely, in a much cooler and denser chromosphere, we expect our nonlinear solution – and its key prediction of an observable second har- 2. Basic equations monic at twice the driving wave frequency in an upper chromo- The vertical propagation of plane adiabatic acoustic waves in the sphere – to be sufficiently reliable to justify the neglect of those presence of gravity is governed by the momentum equation, additional effects. ! On differentiating Eq. (1) with respect to time and simplify- dv ∂v ∂v ∂p ρ = ρ + v = − − ρg, (1) ing the result using Eqs. (2) and (3), we get dt ∂t ∂z ∂z d2v ∂2v ∂v ∂v dv = c2 − γg − γ · (7) the continuity equation, dt2 ∂z2 ∂z ∂z dt dρ dρ dv ∂v = = −ρ , (2) It is straightforward to verify that Eq. (7) contains two familiar dt dv dt ∂z limiting cases. First, in the small perturbation theory, lineariza- and the entropy equation, tion yields dp dρ ∂2v ∂2v ∂v = c2 · (3) = c2 − γg · (8) dt dt ∂t2 0 ∂z2 ∂z Here t is the time, z is the vertical coordinate, v(z, t) is the veloc- The well-known solution (in complex form) for a small- ity along the z axis, ρ = ρ(v) and p = p(v) are the fluid density p amplitude, vertically propagating wave is given by and pressure in the wave, c = γp/ρ is the local sound speed, z and γ is the ratio of specific heats. The vector gravitational ac- v(z, t) ∼ exp + ikz − iωt . (9) celeration g = (0, 0, −g) is directed downward. 2H We consider nonlinear waves in an isothermal atmosphere. Here the frequency ω and wave number k are connected by the The background density ρ0(z) and pressure p0(z) are related by dispersion relation the hydrostatic equation, 2 2 2 2 ω = c0k + ω0, (10) dp0 + ρ0g = 0, (4) dz and the lower cutoff frequency is defined as follows: and p0/ρ0 = const. Consequently, c0 ω0 = · (11) dρ 1 2H 0 = − ρ , (5) dz H 0 Second, in the absence of gravity (g = 0), a simple (or Riemann) where the gravitational scale height wave p c2 v(z, t) = F(z − ut) (12) H = 0 = 0 = const. (6) ρ0g γg is an exact solution of the nonlinear Eq. (7). Here F is an arbi- p trary function and and c0 = γp0/ρ0. In the remainder of the paper, we develop an analytical de- γ + 1 u = ±c + v = ±c + v (13) scription of nonlinear acoustic waves, which should apply to the 0 2 observed three-minute oscillations, caused by slow magnetoacoustic waves propagating along the chromospheric magnetic is the speed of a point in the wave profile (e.g., Lighthill 1978). field (O’Shea et al. 2002). We focus on the propagation of finite- The two signs correspond to waves propagating in the positive amplitude acoustic waves and neglect the effects caused by a and negative directions along the z axis. temperature gradient (Routh & Musielak 2014) and a nonuniform background magnetic field (Afanasyev & Nakariakov 3. Nondispersive short-wavelength waves 2015). Our approach is justified as long as the background temperature gradient and magnetic field are weak enough. Quanti- A simple-wave solution v(z, t) = F(z − ut) cannot exactly sat- tatively, our model remains accurate as long as its characteris- isfy Eq. (7) if the gravitational acceleration g , 0. To understand tic length scale, that is, the height of wave breaking and shock how acoustic waves of finite amplitude are modified by gravity, formation, given by Eq. (22), is small in comparison with the it is instructive to use a heuristic argument (Lighthill 1978) first; characteristic lengths of a more complete solution incorporating this argument is valid if an effective wavelength λ ∼ |v/(∂v/∂z)| the temperature gradient and magnetic field effects. Analysis of is short compared with a distance over which ρ0(z) varies sig- the linearized equations suggest that those effects become significantly, that is, if λ/H 1. We break down the problem nificant in a low-beta plasma above the transition region sepa- of wave propagation through a stratified medium into succes- rating the chromosphere and corona (Routh & Musielak 2014; sive small sections of length ∆z H. Within each section, Afanasyev & Nakariakov 2015). the change in the background density ρ0 is small, and the wave

A15, page 2 of6 Y. E. Litvinenko and J. Chae: Nonlinear acoustic waves in the solar chromosphere motion is governed by the Riemann solution. Consequently, we can describe only a shortwave regime λ 2πc0/ω0 = 4πH, cor- have v ∼ F(z − ut) within each section, where the speed u is responding to the wave frequencies significantly exceeding the given by Eq. (13), and a proportionality constant is determined cutoff frequency ω0. In the solar chromosphere, H ≈ 100 km, by the entry conditions. Neglecting errors that might accumulate and so we expect the analytical solution to provide a reasonably after a number of successive sections, we approximately take accurate description for wavelengths up to a few hundred kilo- v(z, t) = φ(z)F(z − ut) to describe a disturbance moving in the metres. positive direction along the z axis. Here φ(z) is a function, de- We can use the approximate wave solution for v(z, t) to il- fined by the density profile ρ0(z), which varies on a scale ∆z. lustrate how the gravitational field influences the formation of a Consistency with Eq. (9) of the small perturbation theory dic- shock. Wave breaking occurs at a height zwb, defined by tates that φ(z) = exp(z/2H). It follows that ∂τ ∂ξ ∂2τ ∂2ξ   0 = = = = (21)  z γ + 1 vz  ∂u ∂u ∂u2 ∂u2 v(z, t) ≈ ez/2H F t − +  (14)  c 2  0 2 c0 (Ostrovskii 1963). On using Eqs. (19) and (20), we get ! for an upward-propagating wave, where we redefined the arbi- γ g trary function F and expanded c /u, assuming v/c to be suffi- zwb = 2H ln 1 + (22) 0 0 γ + 1 ωv0 ciently small. To obtain a more accurate solution for finite values of the pa- or alternatively rameter λ/H, we use the results from nonlinear acoustics (e.g., zwb,0 Naugolnykh & Ostrovsky 1998). The method of multiple scales zwb = 2H ln 1 + , (23) yields a first-order uniform expansion for finite-amplitude plane 2H waves propagating in an inhomogeneous medium. The resulting where expression, given by Eq. (32) in Nayfeh(1975), describes non- c2 linear adiabatic waves in an inhomogeneous medium in a hy- 2 0 zwb,0 = (24) drostatic, but not necessarily isothermal, equilibrium for which γ + 1 ωv0 ρ (z) and c (z) are assumed to be known. On introducing the new 0 0 is the height of wave breaking in the absence of gravity (H = ∞). variables Clearly gravity acts to lower the height of shock formation. Phys- Z dz ically, gravity modifies the background density profile, causing τ = t − , (15) an increase of the amplitude of an upward-propagating wave, c0 which in turn leads to a stronger deformation of an initial wave Z dz profile. Notably, zwb → 0 as H → 0, confirming that the ana- ξ = √ , (16) lytical multiple-scale solution for v(z, t) leads to a qualitatively c2 c ρ 0 0 0 correct description of nonlinear waves even if λ/H > 1. √ Nonlinear oscillating systems generally exhibit oscillations w = c0ρ0v, (17) with combination frequencies that are superposed on the nor- the solution is expressed in our notation as follows: mal oscillations of the system (Landau & Lifshitz 1969). Here we consider the basic nonlinear effect of a second harmonic ex- " # γ + 1 citation by nonlinear acoustic waves in the solar atmosphere, w(ξ, τ) ≈ F τ + ξw . (18) which may produce observable features at double frequency 2ω. 2 Kalkofen et al.(1994) appear to have identified this e ffect in In the particular case of an isothermal background, we have their numerical simulations of chromospheric oscillations. 2 The second harmonic generation is demonstrated most easily ρ0(z) = ρ0(0) exp(−z/H), c0 = const and H = c0/γg = const. It is then straightforward to rewrite Eq. (18) in terms of the orig- for moderately nonlinear, short-wavelength waves, in which case inal variables. We obtain we can replace v(z, t) on the right-hand side of Eq. (19) by a   linear solution from Eq. (9) in the limit ω ω0, z/2H  z vH −z/2H " v(z, t) ≈ e F t − + (γ + 1) 1 − e  , (19) ωz  c 2  z/2H 0 c0 v(z, t) ≈ e v0 sin ωt − c0 which clearly reduces to Eq. (14) in the limit z/H 1. Below we (25) ! model an evolving wave profile by assuming a simple harmonic ωHv0 ωz  +(γ + 1) ez/2H − 1 sin ωt −  · perturbation at the boundary: 2 c  c0 0 v(0, t) = v0 sin(ωt), (20) Now v(z, t) can be written as its Fourier series ∞ where ω > ω0 for a nonevanescent propagating wave. The ! z/2H X ωz boundary condition yields F(t) = v sin(ωt). v(z, t) ≈ e v0 bn sin n ωt − , (26) 0 c It is worth stressing that, although the analytical multiple- n=1 0 scale solution formally requires the wavelength λ to be a small where parameter, in practice the solution remains qualitatively correct 2 3 even for λ/H > 1. This advantageous feature is common to uni- b = J (a) − 2J (a) = 1 − a2 + ..., (27) form expansions derived by the method of multiple scales. The 1 a 1 2 8 solution, however, neglects the dispersive nature of the acous- 4 1 1 tic waves because it implies that ω ≈ c0k and thus ω ω0 in b = J (a) = a − a3 + ..., (28) a small-amplitude limit. Therefore, the multiple-scale solution 2 a 2 2 24

A15, page 3 of6 A&A 599, A15 (2017) and we used the Taylor expansions of the Bessel functions J1 or in complex form, and J2 in terms of a dimensionless velocity amplitude " q !# z/2H 2 2 z ωHv vlin(z, t) = v0e Im exp iωt − i ω − ω · (35) a γ 0 z/2H − . 0 c = ( + 1) 2 e 1 (29) 0 c0 In the next iteration, we solve Eq. (31) to obtain a correction δv The wave-breaking condition, specified by Eq. (22), is equiva- to vlin(z, t), such that v = vlin +δv is correct to a second order in v0. lent to a = 1. 2 Equations (32) and (33) yield the first-order density and pressure For a small nonlinearity parameter a, the terms of order a perturbations, and higher can be neglected, and so the first two terms in the Fourier series provide an accurate approximation of the complete  q  2 2 ω0 + i ω − ω  solution, ρ1(z, t)  0  = Im  vlin , (36) " ! !# ρ0(z)  iωc0  z/2H ωz a ωz   v(z, t) ≈ e v0 sin ωt − + sin 2 ωt − , (30) c0 2 c0 which is just the sum of the linear solution and a second- " # p1(z, t) ρ1(z, t) 2(1 − γ)ω0 harmonic correction. This expression also follows directly from = γ + Im vlin , (37) Eq. (25) on replacing its right-hand side by a linear Taylor ap- p0(z) ρ0(z) iωc0 proximation with respect to the dimensionless amplitude a. where the complex form is used for compactness. On substituting vlin, ρ1, and p1 into the right-hand side of Eq. (31), we obtain 4. Dispersive long-wavelength waves a linear equation for δv. The general solution is a sum of solutions of the corresponding homogeneous and inhomogeneous As noted above, a nondispersive solution for acoustic wave prop- equations. We select a solution of the former equation, which de- agation in the presence of gravity can only be justified in a limit- scribes an upward-propagating wave, whose amplitude increases ing case of short wavelengths. In practice, the dispersive nature with height as exp(z/2H), which is the same as the growth rate of the waves is essential in a long-wave regime λ > 2πc0/ω0, of the linear wave. The solution of the latter equation describes a which corresponds to 0 < ω/ω0 − 1 1. As an illustration, wave, whose amplitude increases with height as exp(z/H). Given suppose that the gravitational scale height in a lower chromo- the weak dependence of Eq. (30) on γ, we assume that disper- −2 −1 sphere is H = 100 km, g = 0.27 km s , and so c0 ≈ 6.7 km s . sive solutions also depend weakly on γ and simplify the algebra On calculating the cutoff frequency ω0 from Eq. (11) and using by performing the analysis for isothermal perturbations. Then ω = 2π/T with T = 180 s, we get ω/ω0 ≈ 1.1. Clearly Eq. (30) Eq. (31) yields is not applicable in this case. 2 " In order to derive a dispersive-wave solution, we begin by ∂2δv ∂2δv ∂δv 2ωv q − c2 γg 0 ω2 − ω2 − ω noting that moderately nonlinear waves can be described ana- 2 0 2 + = Im i 0 0 ∂t ∂z ∂z c0 lytically if v/c0 is sufficiently small. We obtain a second-order (38) q !# approximation by retaining only linear and quadratic terms with 2 2 z z × exp i2ωt − i2 ω − ω0 + , respect to v in Eq. (7) as follows: c0 H ! ∂2v ∂2v ∂v p ρ ∂2v ∂2v ∂v ∂v and the solution for δv is given by −c2 +γg = c2 1 − 1 −2v −(1+γ) , ∂t2 0 ∂z2 ∂z 0 p ρ ∂z2 ∂z∂t ∂z ∂t 0 0 2 " ! ωHv q z z (31) δv = 0 Im exp i2ωt − i2 ω2 − ω2 + 2 0 c H c0 0 !# (39) where the quadratic terms are collected on the right-hand side of q z z the equation (see also Petukhov & Petukhov 2001 and references − exp i2ωt − i 4ω2 − ω2 + , 0 c 2H therein). Because ρ1 and p1 are multiplied by a derivative of v, in 0 this second-order approximation the density and pressure perturbations should be calculated from the linearized Eqs. (2) and (3), where integration constants in the solution of the homogeneous i.e., equation are determined from the boundary condition ! ∂ρ v ∂v δv(0, t) = 0, (40) 1 = ρ − , (32) ∂t 0 H ∂z which describes the absence of a second harmonic at z = 0. Of ∂p1 2 ∂ρ1 course the factor exp(z/H) in the solution indicates the break- = c0 + (1 − γ)gρ0v. (33) ∂t ∂t down of the formal expansion in the small parameter v0/c0 at As a concrete example, consider the chromospheric response to a large heights. Next we use a simple matching argument to correct the source of monochromatic acoustic waves with frequency ω > ω0 at the lower boundary z = 0. The boundary condition is given by nondispersive nonlinear solution given by Eq. (30). We mod- Eq. (20). On neglecting the right-hand side of Eq. (31), we obtain ify Eq. (30), so that it reduces to the dispersive solution given a linear equation for small-amplitude waves, which yields the by Eqs. (34) and (39) when ω ≈ ω0 and γ = 1. The resulting familiar linear solution approximate solution for an upward-propagating wave incorpo- rates both the nonlinear and dispersive effects q ! z/2H 2 2 z vlin(z, t) = v0e sin ωt − ω − ω0 , (34) c0 v(z, t) = vlin + δv, (41) A15, page 4 of6 Y. E. Litvinenko and J. Chae: Nonlinear acoustic waves in the solar chromosphere where vlin is defined by Eq. (34), and the second-order correction 5. Discussion δv is given by The observed three-minute oscillations in the solar chromo- 2 " ! sphere, caused by slow magnetoacoustic waves, can be modeled ωHv q z δv = (γ + 1) 0 ez/H sin 2 ωt − ω2 − ω2 under certain simplifying assumptions as vertically propagating 2 0 c acoustic waves. We used methods of nonlinear acoustics to de- 2c0 0 !# (42) scribe vertical propagation of finite-amplitude acoustic waves in q z z/2H 2 2 a gravitationally stratified atmosphere. Since dispersive effects −e sin 2 ωt − 4ω − ω0 · 2c0 are essential in the propagation of the low-frequency waves re- sponsible for the chromospheric oscillations, we derived a non- The solution generalizes Eqs. (30) and (39) and therefore linear dispersive solution that is valid in a long-wavelength limit. describes the second harmonic excitation in both the short- We also matched the solution to a nondispersive solution that is wavelength and long-wavelength limits. valid in a short-wavelength limit. Therefore, some information In the limit ω → ω0, the matching procedure that yields can be obtained in an intermediate case by interpolating between Eq. (42) leads to an error on the order of (γ − 1)/2. Consider the two extreme cases. a second-order approximation in the limiting case ω = ω0 for an A key feature of the analytical solution is that it yields arbitrary γ. Then Eq. (31) yields a simple description of the basic nonlinear effect of the second harmonic generation at frequency 2ω, where the wave 2 2 2 driving frequency ω only slightly exceeds the chromospheric ∂ δv ∂ δv ∂δv ω0v − c2 γg − 0 ( ω t) z/H, ≈ −1 2 0 2 + = sin 2 0 e (43) cutoff frequency ω0 0.03 s . It appears that this ef- ∂t ∂z ∂z H fect was previously noted in a numerical study of chromospheric oscillations (Kalkofen et al. 1994). More general solu- and the solution for δv is given by tions incorporating the effects caused by a temperature gradient √ (Routh & Musielak 2014) or a nonuniform background mag- v2    0  z/H z/2H  3 z  netic field (Afanasyev & Nakariakov 2015) would be required to δv = e sin (2ω0t) − e sin 2ω0t −  , (44) 2c0 2 H achieve a more accurate description of the waves in the transition region where the plasma beta decreases rapidly with height. which depends on γ only through the dependence on c . A more It is worth stressing that the nonlinearity of the system con- 0 trols the evolution of a large-amplitude acoustic wave, and the detailed matching of the solutions with ω ≈ ω0 and ω ω0 could be performed but would not lead to qualitatively different strength of the second harmonic is quantified by the dimension- results in the second-order approximation. less nonlinearity parameter a, defined by Eq. (29), rather than just by a dimensionless wave amplitude v /c . Equations (22) Finally, although the nonlinear solutions above describe 0 0 and (29) show that wave breaking occurs at a height where a = 1. upward-propagating waves, Eq. (31) with different initial con- It follows from Eq. (30) that wave breaking occurs when the ditions also admits both downward-propagating and standing- second-harmonic amplitude reaches a half of the wave ampli- wave solutions, which may be of physical interest. In practice, tude at the driving frequency ω. the initial and boundary conditions are hard to specify for re- Our results suggest that the nonlinearity of acoustic waves alistic photospheric driving. For instance, the simplest assump- is strong enough in the solar chromosphere to allow the experi- tion that the atmosphere be initially at rest and unperturbed, mental detection of a second harmonic signal. On setting a = 1 v(z, 0) = ρ (z, 0) = p (z, 0) = 0, leads to a standing-wave solu- 1 1 in Eq. (29), we arrive at a rough estimate of the height, where tion. As an illustration, consider the low-frequency limit ω = ω . 0 the strongest second harmonic signal should be expected, as The linear standing wave is described by follows:

z/2H vlin = v0e sin (ω0t) . (45) z ≈ 2H ln(c0/v0). (49)

As previously, we ﬁnd a weakly nonlinear solution of Eq. (31) This simple analytical expression, derived in the nondispersive by substituting Eq. (45) into the right-hand side of Eq. (31) and limit, remain essentially unaltered for the dispersive solution solving the resulting linear equation, which leads to given by Eq. (42). If v0 is assumed to be comparable with the photospheric convection speed, then c0/v0 ≈ 100, and so z ≈ 9H ≈ 900 km, implying that the second harmonic signal v(z, t) = vlin + δv1 + δv2. (46) in chromospheric oscillations should be detectable in an upper chromosphere by an instrument such as the FISS NST at the Big Here δv1 is a second-order correction to the amplitude of the os- Bear Observatory (Chae et al. 2013). cillation at the driving frequency ω0, and δv2 is the second har- Work is currently underway to isolate the second harmonic monic, which is of primary interest to us. On employing the zero signal in the FISS data (J. Chae et al., in prep.). initial and boundary conditions, after some algebra we obtain Acknowledgements. Y.L. is grateful to the hospitality of Seoul National Univer- sity where this work began. The authors acknowledge the comments and sugges- v2 z z δv = (γ − 1) 0 exp − exp sin(ω t), (47) tions by the referee, Marco Stangalini. 1 c H 2H 0 0 √ v2    0  z z  3 z  δv2 = exp − exp cos   sin(2ω0t). (48) 2c0 H 2H 2 H References Afanasyev, A. N., & Nakariakov, V. M. 2015, A&A, 582, A57 While both terms in Eq. (48) are nonevanescent, the spatially Cally, P. S., & Bogdan, T. J. 1993, ApJ, 402, 721 nonoscillating term dominates at larger heights. Centeno, R., Collados, M., & Trujillo Bueno, J. 2009, ApJ, 692, 1211

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