Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
Dissipation of acoustic-gravity waves: An asymptotic approach Oleg A. Godina) Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado 80309 [email protected]
Abstract: Acoustic-gravity waves in the middle and upper atmosphere and long-range propagation of infrasound are strongly affected by air viscosity and thermal conductivity. To characterize the wave dissipation, it is typical to consider idealized environments, which admit planewave solutions. Here, an asymptotic approach is developed that relies instead on the assumption that spatial variations of environmental parameters are gradual. It is found that realistic assumptions about the atmosphere lead to rather different predictions for wave damping than do the plane-wave solutions. A modification to the Sutherland-Bass model of infrasound absorption is proposed. C 2014 Acoustical Society of America V
PACS numbers: 43.28.Dm, 43.20.Hq, 43.28.Py, 43.20.Bi [VO] Date Received: September 4, 2014 Date Accepted: November 8, 2014
1. Introduction Wave energy dissipation through irreversible thermodynamic processes is a major factor influencing propagation of infrasound and acoustic-gravity waves (AGWs) in the atmosphere. Accurate modeling of wave dissipation is important in a wide range of problems from understanding the momentum and energy transport by waves into the upper atmosphere1 to predicting long-range propagation of infrasound2–4 to the acoustic remote sensing of mesospheric and thermospheric winds.5,6 Variations with height of the mass density, kinematic viscosity, and other physical parameters of the atmosphere have a profound effect on the wave dissipation and its frequency dependence. To characterize the wave dissipation, it is typical to consider an idealized environment, which allows for plane-wave solutions. For instance, kinematic viscosity is often assumed to be constant in derivations of dispersion equations of atmospheric waves.7–11 It is a dubious approach since the kinematic viscosity changes by 5 orders of magnitude between ground level and 200 km altitude. While the assumption of constant shear viscosity would be much more realistic, it does not lead to plane-wave solutions.7,8,10 In this paper, we present an asymptotic approach to the derivation of dispersion equations of AGWs in dissipative fluids. The approach does not presuppose the existence of any plane-wave solutions and relies instead on the assumption that spatial variations of environmental parameters are gradual. The atmosphere is modeled as a three-dimensionally inhomogeneous, moving fluid of variable composition. 2. Derivation of AGW dispersion relations Motion of viscous, thermally conducting fluids is described by the Euler, continuity, and general heat transfer equations:12 @vj @vj @p @ @vn @vj @ 2 þ vn ¼ qgj q þ g þ f g r v ; (1) þ @t @xn @xj @xn @xj @xn @xj 3
a)
Author to whom correspondence should be addressed. Also at: NOAA Earth System Research Laboratory, Physical Sciences Division, Boulder, CO 80305-3328.
J. Acoust. Soc. Am. 136 (6), December 2014
C 2014 Acoustical Society of America EL411 V
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
@q=@t þ r ðqvÞ ¼ 0; @s g @vn @vj 2 2 2 qT þ v rs ¼ r ðjrT Þ þ þ þ f g ðr vÞ ; @t 2 @xj @xn 3
(2) (3)
as well as an equation of state, which can be written as p ¼ p(q, s) or p ¼ p(q, T). Here summation over repeated indices j, n ¼ 1, 2, 3 is implied; t, xj, and g are time, Cartesian spatial coordinates, and acceleration due to gravity; p, q, s, T, v, g, f, and j are pressure, mass density, entropy density, temperature, velocity, shear viscosity, bulk viscosity, and thermal conductivity of the fluid. We are interested in linear waves propagating in a stationary, horizontal background flow of a fluid (not necessarily an ideal gas), in which flow velocity, temperature, and fluid composition change gradually, with the representative spatial scale L being much larger than the wavelength. Let subscripts “0” and “1” designate background characteristics of the fluid and their wave-induced perturbations. Then p ¼ p0 þ p1, vj ¼ v0,j þ v1,j, etc. Neglecting terms O(L1), in the absence of waves, from Eqs. (1)–(3) it follows that ðz c2 dz hð0Þp0 ðzÞ ; h¼ 0; (4) p0 ðzÞ ¼ p0 ð0Þexp ; q0 ðzÞ ¼ q0 ð0Þ cg hðzÞp0 ð0Þ 0 h in a Cartesian coordinate system with horizontal coordinates x and y and the vertical coordinate z increasing upward. Here, c2 ¼ ð@p=@qÞs and c ¼ ð@p=@qÞs =ð@p=@qÞT ¼ ð@s=@TÞp =ð@s=@TÞq are the sound speed squared and the ratio of specific heats at constant pressure and constant volume. Background pressure and density profiles [Eq. (4)] are the same as in a motionless atmosphere.13 Under the assumptions made above, variations of the sound speed c0 and the scale height h have the same representative spatial scale L as the temperature T0. The scale height can be much smaller than L; no assumptions are made about h being large or small compared to the wavelength. In the ray approximation for continuous AGWs,10 q01/2p1, q01/2v1, q01/2q1, are given by the product of a slowly varying complex amplitude and a rapidly varying factor exp(iu ixt), where u and x are the eikonal and wave frequency; ru ¼ (k, m), and k and m are the horizontal and vertical components of the wave vector. The complex amplitudes are distinct for different variables but share the representative spatial scale L of their variation. Governing equations for linear waves are obtained by linearizing Eqs. (1)–(3) with respect to the wave-induced perturbations. Using Eq. (4) and again neglecting terms O(L1), we find " 2 # 1 1 g0 1 2 im þ g0 v1;3 þ f0 þ im p1 þ gq1 ¼ ixd q0 k g0 þ im þ r v1 ; 3 2h 2h 2h
"
2 #
1 g0 v1;3 2h 2 # 4g 1 ixd q0 k2 f0 þ 0 þ im þ g0 r v1 ; 3 2h " # 2 1 k 2 T1 ; ixd q1 ¼ q0 r v1 q0 h1 v1;3 ; ixd q0 T0 s1 ¼ j0 im þ 2h @p0 @T0 @T0 s1 ; T1 ¼ q1 þ s1 : p1 ¼ c20 q1 þ @s0 q0 @q0 s0 @s0 q0 k 2 p1 ¼
1 im þ 2h "
ixd q0 k2 g0 þ im þ
(5)
(6)
(7) (8)
Here, the intrinsic frequency xd ¼ x k v0 has the meaning of wave frequency in a reference frame moving with the local velocity of the background flow.10,14
EL412 J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
Equations (5) and (6) follow from the Euler Eq. (1), Eq. (7)—from the continuity and heat transfer Eqs. (2) and (3), and Eq. (8)—from the equation of state. The Eucken expression for thermal conductivity15 and the Sutherland equation for shear viscosity15 indicate that spatial scales of variation of viscosity and thermal conductivity of dry air are the same as for T and c. Therefore, in derivation of Eqs. (5)–(8) we assumed that the gradients of g0, f0, and j0 are small quantities O(L1). Equations (5)–(8) comprise a set of six linear, homogeneous, algebraic equations in six unknowns: p1, v1,3, r v1, q1, s1, and T1. The set has a nontrivial solution only when its determinant equals zero. A direct but cumbersome calculation of the determinant renders this condition in the form " # ! x2d gk2 1 g i ðc 1Þj0 1 þ þ 2 B1 g0 þ B2 f0 þ B3 (9) ¼ k 2 þ m2 þ 2 ; Cp xd q 0 4h c20 xd h c20 where
!" 2 # 7x2d 1 i 2 2 2 B1 ¼ k m 2 k þ m ; 4h 2h 3c20 " ! 2 # x2d 2 i g2 k 2 ; B3 ¼ 1 4 B2 : B2 ¼ 2 k þ m 2h c0 xd
(10)
Equation (9) applies to the solutions of Eqs. (5)–(8), which remain nontrivial in the ideal fluid limit, i.e., when g0, f0, and j0 vanish. In Eq. (10), Cp stands for the specific heat at constant pressure. We have used the thermodynamic identity ð@p=@TÞq ð@T=@qÞs ¼ ðc 1Þc2 to simplify Eq. (10). This identity can be derived by substituting dT ¼ ð@T=@qÞs dq þ ð@T=@sÞq ds into the equation dp ¼ ð@p=@sÞq ds þ c2 dq ¼ ð@p=@TÞq dT þ ð@p=@qÞT dq: Only linear terms in g0, f0, and j0 are retained in Eq. (9) for simplicity. Physically, this corresponds to the assumption that wave attenuation per wavelength is small. Equation (9) relates the wave frequency and wave vector to the material parameters in the background state of the atmosphere and has the meaning of the dispersion equation of AGWs. In alternative derivations7–11 of the AGW dispersion equations, the kinematic viscosity g0/q0 and the ratios f0/q0 and j0/q0 are assumed to be constant. To reveal the consequences of such an assumption, we now suppose that g0/q0, f0/q0, and j0/q0 are slowly varying in the sense that the gradients of these quantities are O(L1). Then, similar to derivation of Eqs. (5)–(8), one obtains from Eqs. (1)–(4) a set of linear algebraic equations for p1, v1,3, r v1, q1, s1, and T1, which differs from Eqs. (5)–(8) by the terms containing g0/h, f0/h, and j0/h. The condition of the existence of nontrivial wave solutions is again given by Eq. (9), where now ! 7x2d 1 1 5gk 2 k2 2 2 2 2 k m 2 ; B1 ¼ k þm þ 2 þ 2 4h 4h 3c0 3hc20 h2 ! x4d g2 k 2 gc20 k 2 g2 k2 ; B3 ¼ 1 þ 4 4 B2 : (11) B2 ¼ c40 xd h xd Consider the special cases of (1) an isothermal atmosphere10,13 and (2) an incompressible wave motion,16,17 which admit exact analytic solutions of the problem. In the absence of dissipation, Eq. (9) reduces to the textbook dispersion equation10,13 of AGWs in an isothermal atmosphere. Equation (11) agrees with earlier results on AGW damping,7,8,10 which were obtained for a windless atmosphere assuming that the kinematic viscosities g0/q0 and f0/q0 and the thermal diffusivity j0/Cpq0 are constant. Uniform background flows of arbitrarily stratified viscous, thermally conducting fluid with g0 ¼ const and arbitrary f0 and j0 support linear AGWs with
J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves EL413
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
2 v1;3 ðrÞ ¼ A expðik r þ kz ixtÞ; r v1 ¼ 0; and p1 ðrÞ ¼ ix1 d gq0 ðzÞv1;3 ðrÞ; where xd 16,17 ¼ kg and A is a constant. This exact solution can be used to check asymptotic results. To be consistent with the exact solution, the dispersion equations [Eqs. (9) and (10)] should be satisfied, when x2d ¼ kg and im þ 1=2h ¼ k; B2 and B3 in Eq. (11) should vanish under these conditions. An inspection shows that Eqs. (9)–(11) satisfy these requirements. In the general case, there are substantial quantitative and qualitative differences between predictions of Eqs. (10) and (11), unless jmjh 1 (Fig. 1). For propagating waves having real-valued x and the wave vector (k, m) in the absence of dissipation, dissipation-induced wave damping and phase corrections are described by real and imaginary parts of the coefficients Bj, respectively. Equation (10) shows that AGWs experience both phase and amplitude changes due to the viscosity and thermal conductivity. Equation (11) predicts only damping [at a different rate than does Eq. (10)] of propagating waves. Figure 1 illustrates the large differences between the damping rates obtained under realistic [Eq. (10)] and traditional [Eq. (11)] assumptions about the atmosphere. Thermal conductivity and shear viscosity are closely related in the ideal gas. The Prandtl number Pr ¼ g0 Cp =j0 is nearly constant in the atmosphere;7,15 to a good accuracy Pr ¼ 4c=ð9c 5Þ.15 Therefore, it was concluded8,10,11 that, in various special cases, contributions of shear viscosity and thermal conductivity to AGW dissipation rate are identical up to a numerical constant O(1). In fact, these contributions can differ rather substantially in the general case (Fig. 1).
3. Infrasound absorption In the acoustic limit, where g ! 0 and h1 ! 0, Eqs. (9)–(11) reduce to " !# pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xd ixd 4 ðc 1Þj0 2 2 : 1þ f0 þ g0 þ k þm ¼ c0 Cp 3 2q0 c20
(12)
This result is consistent with the well-known dispersion equation of sound in homogeneous, quiescent, viscous, and thermally conductive fluids.12,14,15 Note that the wave frequency x and the background flow (wind) velocity v0 enter Eq. (12) as well as the more general Eqs. (9)–(11) through the intrinsic frequency xd only. On the right side of Eq. (12), the terms with the shear viscosity g0 and thermal conductivity j0 describe the classical absorption; the term with the bulk viscosity f0 represents absorption due to various relaxation processes, including molecular vibrational and rotational
Fig. 1. (Color online) Effect of dissipative processes on the dispersion equation of AGWs. Real parts of the coefficients B1, B2, and B3 defined by Eq. (10) are shown at x2d =k2 c20 ¼ 1:2 by solid lines 1–3, respectively. The coefficients B1, B2, and B3 defined by Eq. (11) are shown by dashed lines. All the coefficients Bj are multiplied 4 by the factor x4 d c0 to arrive at dimensionless quantities; the ratio of specific heats at constant pressure and constant volume c ¼ 1.4.
EL414 J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
relaxations. The relaxation processes are rather important in the troposphere and stratosphere, but the total infrasound absorption is closely approximated by the classical absorption for waves propagating to mesospheric and thermospheric heights.15 Wave damping by dissipative processes is given by an integral along a ray of the imaginary part of the wave vector. In horizontally stratified (layered) media, k ¼ const on the ray, and the integral along the ray readily reduces to an integral over z. For infrasound absorption losses Q at propagation between heights z1 and z2, z1 < z2, from Eq. (12) we find " !# ð z2 ð z2 x3 g0 dz qdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: (13) Imm dz ¼ 10:86 Q ¼ 20 log exp 2 k 2 c2 z1 z1 q c3 x 0 0 0 d Here, we neglect the bulk viscosity and assume that c ¼ 1.4 and Pr ¼ 4c=ð9c 5Þ.15 Wind has a significant effect on infrasound absorption in the middle and upper atmosphere [Fig. 2(a)], where the ratio v0/c0 is not small. The ratio A of the absorption rates (in dB per 1 m of height) in moving and quiescent media for a wave with a given wave vector varies from zero to infinity depending on the wave propagation direction. In Fig. 2(a), A ! 0 at h ! p/2, when the wave propagates horizontally in the quiescent medium but at a finite grazing angle in the moving one. Conversely, A ! 1, when the wave propagates horizontally in the moving medium but at a finite grazing angle in the quiescent one. The factor A ¼ 1 in Fig. 2(a) and the wind has no effect on the absorption, when either h ¼ 0 or u ¼ p/2, i.e., the wave vector is orthogonal to the wind. Wind affects the absorption by changing, first, the ray geometry and, second, the effective frequency of the wave. The contributions of these two mechanisms are illustrated in Figs. 2(b) and 2(c) in the simple case of an isothermal atmosphere. Exponential decrease of density with height makes the absorption loss Q [Eq. (13)] particularly sensitive to the wind velocity in the vicinity of the upper point of the ray trajectory. For the wind profile assumed in Figs. 2(b) and 2(c), ray turning points are located at the heights of 150–170 km, when h 42 , and, therefore, the contributions of both mechanisms change most rapidly with h in the vicinity of this angle. In the Sutherland-Bass infrasound absorption model, x takes the place of xd in Eq. (12), and the absorption rate is isotropic.15,18 Equations (12) and (13) show that deviations of the intrinsic frequency xd from x make the absorption anisotropic; for given x and k
Fig. 2. (Color online) Impact of winds on infrasound absorption. (a) Local effect of the wind. The absorption rate (in dB per 1 m of height) in quiescent atmosphere is to be multiplied by the correction factor A shown in the figure. u is the angle between the horizontal wave vector k and the wind velocity v0; h ¼ arcsin(c0k/x) characterizes direction of the wave vector (k, m) in the vertical plane and, when v0 ¼ 0, is the acute angle between the wave vector and the vertical. In the figure, v0/c0 ¼ 0.5. (b) Absorption losses at infrasound propagation from the ground level z ¼ 0 to z ¼ 160 km in an isothermal atmosphere with a linear wind velocity profile. The losses are calculated with (1) and without (2) accounting for variations with height of the intrinsic frequency xd. For comparison, absorption losses in quiescent atmosphere (3) are also shown. (c) Effect of the absorption anisotropy on one-way absorption losses for infrasound propagation from ground level to a height of 150 km (1), 160 km (2), and 170 km (3). Wave frequency f ¼ 0.2 Hz, k ¼ xc01 (sin h, 0, 0), v0 ¼ (zc0/H, 0, 0), and H ¼ 320 km in (b) and (c).
J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves EL415
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
Fig. 3. (Color online) Absorption losses in a real atmosphere. (a) Profiles of the sound speed c0 and the zonal v0,1 (2) and meridional v0,2 (3) components of the wind velocity predicted by the Whole Atmosphere Model (Refs. 19 and 20). (b) Absorption losses at infrasound propagation from ground level to 90 km at frequency f ¼ 4 Hz (1), 120 km at f ¼ 0.4 Hz (2), and 150 km at f ¼ 0.1 Hz (3). The losses are shown for different azimuthal angles u between the horizontal wave vector k and the zonal direction; k ¼ x/C, C ¼ 570 m/s. Absorption losses are calculated with (solid lines) and without (dashed lines) accounting for wind-induced absorption anisotropy.
the absorption per unit ray length depends on the angle between k and the wind velocity v0. The Sutherland-Bass model underestimates infrasound damping at upwind and overestimates it at downwind propagation [respectively, at h < 0 and h > 0 in Figs. 2(b) and 2(c)]. Effects of the absorption anisotropy become particularly strong in the vicinity of turning points [Fig. 2(c)] and can be in tens of dBs for thermospheric returns at ground level [Figs. 2(c) and 3]. Since turning points typically occur when k v0 > 0, the absorption anisotropy can potentially help to explain the previously reported3 overestimation of damping of thermospheric returns by the Sutherland-Bass model. At infrasound propagation to different heights in a real atmosphere, absorption minima occur at different azimuthal directions of the wave vector (Fig. 3) because of the wind direction changes with height. Lines 1 and 3 in Fig. 3(b) illustrate that minima of the azimuthal dependence of absorption can be incorrectly predicted as maxima, when the absorption anisotropy is not taken into account. 4. Conclusion Unlike the traditional approach that requires plane-wave solutions, the asymptotic approach presented in this paper allows one to derive AGW dispersion equations in a consistent manner for a wide range of scenarios and to describe the wave attenuation more realistically. Realistic assumptions about the atmosphere are found to lead to rather different predictions for AGW damping than the plane-wave solutions. Wind speed and wave frequency appear only through the intrinsic wave frequency in the AGW dispersion equation, including the terms that describe the effects of viscosity and thermal conductivity. The Sutherland–Bass model of infrasound absorption15 should be modified to account for wind-induced absorption anisotropy, which results from deviation of the intrinsic frequency from the wave frequency in a stationary reference frame. The anisotropy is expected to result in a significant decrease in the predicted attenuation of thermospheric returns. Further research is necessary regarding the effects of the absorption anisotropy on long-range propagation of infrasound.
EL416 J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
Acknowledgments Stimulating discussions with D. P. Drob and L. G. Evers are gratefully acknowledged. The author thanks V. E. Ostashev for insightful comments, which helped to improve the presentation. This work was supported, in part, by the Office of Naval Research, Grant No. N00014-13-1-0348. References and links 1
D. C. Fritts and M. J. Alexander, “Gravity wave dynamics and effects in the middle atmosphere,” Rev. Geophys. 41, Art. 1003, doi:10.1029/2001RG000106 (2003). 2 L. G. Evers and H. W. Haak, “Infrasonic forerunners: Exceptionally fast acoustic phases,” Geophys. Res. Lett. 34, L10806, doi:10.1029/2007GL029353 (2007). 3 C. de Groot-Hedlin, M. A. Hedlin, and K. Walker, “Finite difference synthesis of infrasound propagation through a windy, viscous atmosphere: Application to a bolide explosion detected by seismic networks,” Geophys. J. Int. 185, 305–320 (2011). 4 I. Chunchuzov, S. Kulichkov, O. Popov, and M. Hedlin, “Modeling propagation of infrasound signals observed by a dense seismic network,” J. Acoust. Soc. Am. 135, 38–48 (2014). 5 D. P. Drob, R. R. Meier, J. M. Picone, and M. Garces, “Inversion of infrasound signals for passive atmospheric remote sensing,” in Infrasound Monitoring for Atmospheric Studies, edited by A. Le Pichon, E. Blanc, and A. Hauchecorne (Springer, New York, 2010), pp. 701–732. 6 J. D. Assink, R. Waxler, W. G. Frazier, and J. Lonzaga, “The estimation of upper atmospheric wind model updates from infrasound data,” J. Geophys. Res.: Atmos. 118, 10707–10724 (2013). 7 M. L. V. Pitteway and C. O. Hines, “The viscous damping of atmospheric gravity waves,” Can. J. Phys. 41, 1935–1948 (1963). 8 G. S. Golitsyn, “Damping of small oscillations in the atmosphere due to viscosity and thermal conductivity,” Izv., Acad. Sci., USSR, Atmos. Oceanic Phys. (Engl. Transl.) 1, 82–89 (1965). 9 S. H. Francis, “Acoustic-gravity modes and large-scale traveling ionospheric disturbances of a realistic, dissipative atmosphere,” J. Geophys. Res. 78, 2278–2301, doi:10.1029/JA078i013p02278 (1973). 10 E. E. Gossard and W. H. Hooke, Waves in the Atmosphere (Elsevier, Amsterdam, 1975), pp. 168–170, 215–250. 11 S. L. Vadas and M. J. Nicolls, “The phases and amplitudes of gravity waves propagating and dissipating in the thermosphere: Theory,” J. Geophys. Res. 117, A05322, doi:10.1029/2011JA017426 (2012). 12 L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics, Vol. 6, Fluid Mechanics, 2nd ed. (Elsevier, Oxford, 2004), pp. 1–2, 44–51, 192–196, 300–305. 13 A. E. Gill, Atmosphere–Ocean Dynamics (Academic Press, New York, 1982), pp. 45–50, 169–175. 14 L. M. Brekhovskikh and O. A. Godin, Acoustics of Layered Medium. 1: Plane and Quasi-Plane Waves, 2nd ed. (Springer, Berlin, 1998), pp. 144–147, 192–204. 15 L. C. Sutherland and H. E. Bass, “Atmospheric absorption in the atmosphere up to 160 km,” J. Acoust. Soc. Am. 115, 1012–1032 (2004). 16 O. A. Godin, “Incompressible wave motion of compressible fluids,” Phys. Rev. Lett. 108, 194501 (2012). 17 O. A. Godin, “Shear waves in inhomogeneous, compressible fluids in a gravity field,” J. Acoust. Soc. Am. 135, 1071–1082 (2014). 18 H. E. Bass, C. H. Hetzer, and R. Raspet, “On the speed of sound in the atmosphere as a function of altitude and frequency,” J. Geophys. Res. 112, D15110, doi:10.1029/2006JD007806 (2007). 19 R. A. Akmaev, “Whole atmosphere modeling: Connecting terrestrial and space weather,” Rev. Geophys. 49, RG4004, doi:10.1029/2011RG000364 (2011). 20 N. A. Zabotin, O. A. Godin, P. C. Sava, and L. Y. Zabotina, “Tracing three-dimensional acoustic wavefronts in inhomogeneous, moving media,” J. Comput. Acoust. 25, Art. 1450002 (2014).
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Dissipation of acoustic-gravity waves: An asymptotic approach Oleg A. Godina) Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado 80309 [email protected]
Abstract: Acoustic-gravity waves in the middle and upper atmosphere and long-range propagation of infrasound are strongly affected by air viscosity and thermal conductivity. To characterize the wave dissipation, it is typical to consider idealized environments, which admit planewave solutions. Here, an asymptotic approach is developed that relies instead on the assumption that spatial variations of environmental parameters are gradual. It is found that realistic assumptions about the atmosphere lead to rather different predictions for wave damping than do the plane-wave solutions. A modification to the Sutherland-Bass model of infrasound absorption is proposed. C 2014 Acoustical Society of America V
PACS numbers: 43.28.Dm, 43.20.Hq, 43.28.Py, 43.20.Bi [VO] Date Received: September 4, 2014 Date Accepted: November 8, 2014
1. Introduction Wave energy dissipation through irreversible thermodynamic processes is a major factor influencing propagation of infrasound and acoustic-gravity waves (AGWs) in the atmosphere. Accurate modeling of wave dissipation is important in a wide range of problems from understanding the momentum and energy transport by waves into the upper atmosphere1 to predicting long-range propagation of infrasound2–4 to the acoustic remote sensing of mesospheric and thermospheric winds.5,6 Variations with height of the mass density, kinematic viscosity, and other physical parameters of the atmosphere have a profound effect on the wave dissipation and its frequency dependence. To characterize the wave dissipation, it is typical to consider an idealized environment, which allows for plane-wave solutions. For instance, kinematic viscosity is often assumed to be constant in derivations of dispersion equations of atmospheric waves.7–11 It is a dubious approach since the kinematic viscosity changes by 5 orders of magnitude between ground level and 200 km altitude. While the assumption of constant shear viscosity would be much more realistic, it does not lead to plane-wave solutions.7,8,10 In this paper, we present an asymptotic approach to the derivation of dispersion equations of AGWs in dissipative fluids. The approach does not presuppose the existence of any plane-wave solutions and relies instead on the assumption that spatial variations of environmental parameters are gradual. The atmosphere is modeled as a three-dimensionally inhomogeneous, moving fluid of variable composition. 2. Derivation of AGW dispersion relations Motion of viscous, thermally conducting fluids is described by the Euler, continuity, and general heat transfer equations:12 @vj @vj @p @ @vn @vj @ 2 þ vn ¼ qgj q þ g þ f g r v ; (1) þ @t @xn @xj @xn @xj @xn @xj 3
a)
Author to whom correspondence should be addressed. Also at: NOAA Earth System Research Laboratory, Physical Sciences Division, Boulder, CO 80305-3328.
J. Acoust. Soc. Am. 136 (6), December 2014
C 2014 Acoustical Society of America EL411 V
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 205.208.120.206 On: Fri, 26 Dec 2014 20:06:58
Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
@q=@t þ r ðqvÞ ¼ 0; @s g @vn @vj 2 2 2 qT þ v rs ¼ r ðjrT Þ þ þ þ f g ðr vÞ ; @t 2 @xj @xn 3
(2) (3)
as well as an equation of state, which can be written as p ¼ p(q, s) or p ¼ p(q, T). Here summation over repeated indices j, n ¼ 1, 2, 3 is implied; t, xj, and g are time, Cartesian spatial coordinates, and acceleration due to gravity; p, q, s, T, v, g, f, and j are pressure, mass density, entropy density, temperature, velocity, shear viscosity, bulk viscosity, and thermal conductivity of the fluid. We are interested in linear waves propagating in a stationary, horizontal background flow of a fluid (not necessarily an ideal gas), in which flow velocity, temperature, and fluid composition change gradually, with the representative spatial scale L being much larger than the wavelength. Let subscripts “0” and “1” designate background characteristics of the fluid and their wave-induced perturbations. Then p ¼ p0 þ p1, vj ¼ v0,j þ v1,j, etc. Neglecting terms O(L1), in the absence of waves, from Eqs. (1)–(3) it follows that ðz c2 dz hð0Þp0 ðzÞ ; h¼ 0; (4) p0 ðzÞ ¼ p0 ð0Þexp ; q0 ðzÞ ¼ q0 ð0Þ cg hðzÞp0 ð0Þ 0 h in a Cartesian coordinate system with horizontal coordinates x and y and the vertical coordinate z increasing upward. Here, c2 ¼ ð@p=@qÞs and c ¼ ð@p=@qÞs =ð@p=@qÞT ¼ ð@s=@TÞp =ð@s=@TÞq are the sound speed squared and the ratio of specific heats at constant pressure and constant volume. Background pressure and density profiles [Eq. (4)] are the same as in a motionless atmosphere.13 Under the assumptions made above, variations of the sound speed c0 and the scale height h have the same representative spatial scale L as the temperature T0. The scale height can be much smaller than L; no assumptions are made about h being large or small compared to the wavelength. In the ray approximation for continuous AGWs,10 q01/2p1, q01/2v1, q01/2q1, are given by the product of a slowly varying complex amplitude and a rapidly varying factor exp(iu ixt), where u and x are the eikonal and wave frequency; ru ¼ (k, m), and k and m are the horizontal and vertical components of the wave vector. The complex amplitudes are distinct for different variables but share the representative spatial scale L of their variation. Governing equations for linear waves are obtained by linearizing Eqs. (1)–(3) with respect to the wave-induced perturbations. Using Eq. (4) and again neglecting terms O(L1), we find " 2 # 1 1 g0 1 2 im þ g0 v1;3 þ f0 þ im p1 þ gq1 ¼ ixd q0 k g0 þ im þ r v1 ; 3 2h 2h 2h
"
2 #
1 g0 v1;3 2h 2 # 4g 1 ixd q0 k2 f0 þ 0 þ im þ g0 r v1 ; 3 2h " # 2 1 k 2 T1 ; ixd q1 ¼ q0 r v1 q0 h1 v1;3 ; ixd q0 T0 s1 ¼ j0 im þ 2h @p0 @T0 @T0 s1 ; T1 ¼ q1 þ s1 : p1 ¼ c20 q1 þ @s0 q0 @q0 s0 @s0 q0 k 2 p1 ¼
1 im þ 2h "
ixd q0 k2 g0 þ im þ
(5)
(6)
(7) (8)
Here, the intrinsic frequency xd ¼ x k v0 has the meaning of wave frequency in a reference frame moving with the local velocity of the background flow.10,14
EL412 J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves
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Oleg A. Godin: JASA Express Letters
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Equations (5) and (6) follow from the Euler Eq. (1), Eq. (7)—from the continuity and heat transfer Eqs. (2) and (3), and Eq. (8)—from the equation of state. The Eucken expression for thermal conductivity15 and the Sutherland equation for shear viscosity15 indicate that spatial scales of variation of viscosity and thermal conductivity of dry air are the same as for T and c. Therefore, in derivation of Eqs. (5)–(8) we assumed that the gradients of g0, f0, and j0 are small quantities O(L1). Equations (5)–(8) comprise a set of six linear, homogeneous, algebraic equations in six unknowns: p1, v1,3, r v1, q1, s1, and T1. The set has a nontrivial solution only when its determinant equals zero. A direct but cumbersome calculation of the determinant renders this condition in the form " # ! x2d gk2 1 g i ðc 1Þj0 1 þ þ 2 B1 g0 þ B2 f0 þ B3 (9) ¼ k 2 þ m2 þ 2 ; Cp xd q 0 4h c20 xd h c20 where
!" 2 # 7x2d 1 i 2 2 2 B1 ¼ k m 2 k þ m ; 4h 2h 3c20 " ! 2 # x2d 2 i g2 k 2 ; B3 ¼ 1 4 B2 : B2 ¼ 2 k þ m 2h c0 xd
(10)
Equation (9) applies to the solutions of Eqs. (5)–(8), which remain nontrivial in the ideal fluid limit, i.e., when g0, f0, and j0 vanish. In Eq. (10), Cp stands for the specific heat at constant pressure. We have used the thermodynamic identity ð@p=@TÞq ð@T=@qÞs ¼ ðc 1Þc2 to simplify Eq. (10). This identity can be derived by substituting dT ¼ ð@T=@qÞs dq þ ð@T=@sÞq ds into the equation dp ¼ ð@p=@sÞq ds þ c2 dq ¼ ð@p=@TÞq dT þ ð@p=@qÞT dq: Only linear terms in g0, f0, and j0 are retained in Eq. (9) for simplicity. Physically, this corresponds to the assumption that wave attenuation per wavelength is small. Equation (9) relates the wave frequency and wave vector to the material parameters in the background state of the atmosphere and has the meaning of the dispersion equation of AGWs. In alternative derivations7–11 of the AGW dispersion equations, the kinematic viscosity g0/q0 and the ratios f0/q0 and j0/q0 are assumed to be constant. To reveal the consequences of such an assumption, we now suppose that g0/q0, f0/q0, and j0/q0 are slowly varying in the sense that the gradients of these quantities are O(L1). Then, similar to derivation of Eqs. (5)–(8), one obtains from Eqs. (1)–(4) a set of linear algebraic equations for p1, v1,3, r v1, q1, s1, and T1, which differs from Eqs. (5)–(8) by the terms containing g0/h, f0/h, and j0/h. The condition of the existence of nontrivial wave solutions is again given by Eq. (9), where now ! 7x2d 1 1 5gk 2 k2 2 2 2 2 k m 2 ; B1 ¼ k þm þ 2 þ 2 4h 4h 3c0 3hc20 h2 ! x4d g2 k 2 gc20 k 2 g2 k2 ; B3 ¼ 1 þ 4 4 B2 : (11) B2 ¼ c40 xd h xd Consider the special cases of (1) an isothermal atmosphere10,13 and (2) an incompressible wave motion,16,17 which admit exact analytic solutions of the problem. In the absence of dissipation, Eq. (9) reduces to the textbook dispersion equation10,13 of AGWs in an isothermal atmosphere. Equation (11) agrees with earlier results on AGW damping,7,8,10 which were obtained for a windless atmosphere assuming that the kinematic viscosities g0/q0 and f0/q0 and the thermal diffusivity j0/Cpq0 are constant. Uniform background flows of arbitrarily stratified viscous, thermally conducting fluid with g0 ¼ const and arbitrary f0 and j0 support linear AGWs with
J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves EL413
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2 v1;3 ðrÞ ¼ A expðik r þ kz ixtÞ; r v1 ¼ 0; and p1 ðrÞ ¼ ix1 d gq0 ðzÞv1;3 ðrÞ; where xd 16,17 ¼ kg and A is a constant. This exact solution can be used to check asymptotic results. To be consistent with the exact solution, the dispersion equations [Eqs. (9) and (10)] should be satisfied, when x2d ¼ kg and im þ 1=2h ¼ k; B2 and B3 in Eq. (11) should vanish under these conditions. An inspection shows that Eqs. (9)–(11) satisfy these requirements. In the general case, there are substantial quantitative and qualitative differences between predictions of Eqs. (10) and (11), unless jmjh 1 (Fig. 1). For propagating waves having real-valued x and the wave vector (k, m) in the absence of dissipation, dissipation-induced wave damping and phase corrections are described by real and imaginary parts of the coefficients Bj, respectively. Equation (10) shows that AGWs experience both phase and amplitude changes due to the viscosity and thermal conductivity. Equation (11) predicts only damping [at a different rate than does Eq. (10)] of propagating waves. Figure 1 illustrates the large differences between the damping rates obtained under realistic [Eq. (10)] and traditional [Eq. (11)] assumptions about the atmosphere. Thermal conductivity and shear viscosity are closely related in the ideal gas. The Prandtl number Pr ¼ g0 Cp =j0 is nearly constant in the atmosphere;7,15 to a good accuracy Pr ¼ 4c=ð9c 5Þ.15 Therefore, it was concluded8,10,11 that, in various special cases, contributions of shear viscosity and thermal conductivity to AGW dissipation rate are identical up to a numerical constant O(1). In fact, these contributions can differ rather substantially in the general case (Fig. 1).
3. Infrasound absorption In the acoustic limit, where g ! 0 and h1 ! 0, Eqs. (9)–(11) reduce to " !# pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xd ixd 4 ðc 1Þj0 2 2 : 1þ f0 þ g0 þ k þm ¼ c0 Cp 3 2q0 c20
(12)
This result is consistent with the well-known dispersion equation of sound in homogeneous, quiescent, viscous, and thermally conductive fluids.12,14,15 Note that the wave frequency x and the background flow (wind) velocity v0 enter Eq. (12) as well as the more general Eqs. (9)–(11) through the intrinsic frequency xd only. On the right side of Eq. (12), the terms with the shear viscosity g0 and thermal conductivity j0 describe the classical absorption; the term with the bulk viscosity f0 represents absorption due to various relaxation processes, including molecular vibrational and rotational
Fig. 1. (Color online) Effect of dissipative processes on the dispersion equation of AGWs. Real parts of the coefficients B1, B2, and B3 defined by Eq. (10) are shown at x2d =k2 c20 ¼ 1:2 by solid lines 1–3, respectively. The coefficients B1, B2, and B3 defined by Eq. (11) are shown by dashed lines. All the coefficients Bj are multiplied 4 by the factor x4 d c0 to arrive at dimensionless quantities; the ratio of specific heats at constant pressure and constant volume c ¼ 1.4.
EL414 J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves
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Oleg A. Godin: JASA Express Letters
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Published Online 24 November 2014
relaxations. The relaxation processes are rather important in the troposphere and stratosphere, but the total infrasound absorption is closely approximated by the classical absorption for waves propagating to mesospheric and thermospheric heights.15 Wave damping by dissipative processes is given by an integral along a ray of the imaginary part of the wave vector. In horizontally stratified (layered) media, k ¼ const on the ray, and the integral along the ray readily reduces to an integral over z. For infrasound absorption losses Q at propagation between heights z1 and z2, z1 < z2, from Eq. (12) we find " !# ð z2 ð z2 x3 g0 dz qdffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi: (13) Imm dz ¼ 10:86 Q ¼ 20 log exp 2 k 2 c2 z1 z1 q c3 x 0 0 0 d Here, we neglect the bulk viscosity and assume that c ¼ 1.4 and Pr ¼ 4c=ð9c 5Þ.15 Wind has a significant effect on infrasound absorption in the middle and upper atmosphere [Fig. 2(a)], where the ratio v0/c0 is not small. The ratio A of the absorption rates (in dB per 1 m of height) in moving and quiescent media for a wave with a given wave vector varies from zero to infinity depending on the wave propagation direction. In Fig. 2(a), A ! 0 at h ! p/2, when the wave propagates horizontally in the quiescent medium but at a finite grazing angle in the moving one. Conversely, A ! 1, when the wave propagates horizontally in the moving medium but at a finite grazing angle in the quiescent one. The factor A ¼ 1 in Fig. 2(a) and the wind has no effect on the absorption, when either h ¼ 0 or u ¼ p/2, i.e., the wave vector is orthogonal to the wind. Wind affects the absorption by changing, first, the ray geometry and, second, the effective frequency of the wave. The contributions of these two mechanisms are illustrated in Figs. 2(b) and 2(c) in the simple case of an isothermal atmosphere. Exponential decrease of density with height makes the absorption loss Q [Eq. (13)] particularly sensitive to the wind velocity in the vicinity of the upper point of the ray trajectory. For the wind profile assumed in Figs. 2(b) and 2(c), ray turning points are located at the heights of 150–170 km, when h 42 , and, therefore, the contributions of both mechanisms change most rapidly with h in the vicinity of this angle. In the Sutherland-Bass infrasound absorption model, x takes the place of xd in Eq. (12), and the absorption rate is isotropic.15,18 Equations (12) and (13) show that deviations of the intrinsic frequency xd from x make the absorption anisotropic; for given x and k
Fig. 2. (Color online) Impact of winds on infrasound absorption. (a) Local effect of the wind. The absorption rate (in dB per 1 m of height) in quiescent atmosphere is to be multiplied by the correction factor A shown in the figure. u is the angle between the horizontal wave vector k and the wind velocity v0; h ¼ arcsin(c0k/x) characterizes direction of the wave vector (k, m) in the vertical plane and, when v0 ¼ 0, is the acute angle between the wave vector and the vertical. In the figure, v0/c0 ¼ 0.5. (b) Absorption losses at infrasound propagation from the ground level z ¼ 0 to z ¼ 160 km in an isothermal atmosphere with a linear wind velocity profile. The losses are calculated with (1) and without (2) accounting for variations with height of the intrinsic frequency xd. For comparison, absorption losses in quiescent atmosphere (3) are also shown. (c) Effect of the absorption anisotropy on one-way absorption losses for infrasound propagation from ground level to a height of 150 km (1), 160 km (2), and 170 km (3). Wave frequency f ¼ 0.2 Hz, k ¼ xc01 (sin h, 0, 0), v0 ¼ (zc0/H, 0, 0), and H ¼ 320 km in (b) and (c).
J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves EL415
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Published Online 24 November 2014
Fig. 3. (Color online) Absorption losses in a real atmosphere. (a) Profiles of the sound speed c0 and the zonal v0,1 (2) and meridional v0,2 (3) components of the wind velocity predicted by the Whole Atmosphere Model (Refs. 19 and 20). (b) Absorption losses at infrasound propagation from ground level to 90 km at frequency f ¼ 4 Hz (1), 120 km at f ¼ 0.4 Hz (2), and 150 km at f ¼ 0.1 Hz (3). The losses are shown for different azimuthal angles u between the horizontal wave vector k and the zonal direction; k ¼ x/C, C ¼ 570 m/s. Absorption losses are calculated with (solid lines) and without (dashed lines) accounting for wind-induced absorption anisotropy.
the absorption per unit ray length depends on the angle between k and the wind velocity v0. The Sutherland-Bass model underestimates infrasound damping at upwind and overestimates it at downwind propagation [respectively, at h < 0 and h > 0 in Figs. 2(b) and 2(c)]. Effects of the absorption anisotropy become particularly strong in the vicinity of turning points [Fig. 2(c)] and can be in tens of dBs for thermospheric returns at ground level [Figs. 2(c) and 3]. Since turning points typically occur when k v0 > 0, the absorption anisotropy can potentially help to explain the previously reported3 overestimation of damping of thermospheric returns by the Sutherland-Bass model. At infrasound propagation to different heights in a real atmosphere, absorption minima occur at different azimuthal directions of the wave vector (Fig. 3) because of the wind direction changes with height. Lines 1 and 3 in Fig. 3(b) illustrate that minima of the azimuthal dependence of absorption can be incorrectly predicted as maxima, when the absorption anisotropy is not taken into account. 4. Conclusion Unlike the traditional approach that requires plane-wave solutions, the asymptotic approach presented in this paper allows one to derive AGW dispersion equations in a consistent manner for a wide range of scenarios and to describe the wave attenuation more realistically. Realistic assumptions about the atmosphere are found to lead to rather different predictions for AGW damping than the plane-wave solutions. Wind speed and wave frequency appear only through the intrinsic wave frequency in the AGW dispersion equation, including the terms that describe the effects of viscosity and thermal conductivity. The Sutherland–Bass model of infrasound absorption15 should be modified to account for wind-induced absorption anisotropy, which results from deviation of the intrinsic frequency from the wave frequency in a stationary reference frame. The anisotropy is expected to result in a significant decrease in the predicted attenuation of thermospheric returns. Further research is necessary regarding the effects of the absorption anisotropy on long-range propagation of infrasound.
EL416 J. Acoust. Soc. Am. 136 (6), December 2014
Oleg A. Godin: Dissipation of atmospheric waves
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Oleg A. Godin: JASA Express Letters
[http://dx.doi.org/10.1121/1.4902426]
Published Online 24 November 2014
Acknowledgments Stimulating discussions with D. P. Drob and L. G. Evers are gratefully acknowledged. The author thanks V. E. Ostashev for insightful comments, which helped to improve the presentation. This work was supported, in part, by the Office of Naval Research, Grant No. N00014-13-1-0348. References and links 1
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Oleg A. Godin: Dissipation of atmospheric waves EL417
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