Reflection and transmission of atmospheric gravity waves in a stably sheared horizontal wind field

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi: /2009jd012687, 2010 Reflection and transmission of atmospheric gravity waves in a stably sheared horizontal wind field Kai Ming Huang, 1,2,3,4 Shao Dong Zhang, 1,3,4 and Fan Yi 1,3,4 Received 18 June 2009; revised 23 March 2010; accepted 7 April 2010; published 18 August [1] Applying a second order fully nonlinear numerical scheme, we have investigated the characteristics of reflection and transmission of atmospheric gravity wave packets in a vertically sheared horizontal wind. When the leading edge of incident wave arrives at the reflecting level predicted by the linear theory, the wave reflection begins to occur. In the reflection process, the reflection and incident waves are superposed with obvious phase staggering, which is different from the wave reflection in a meridionally sheared horizontal wind. In the evanescent region, the wave phase has only weak variation; the wave amplitude decays with the sheared wind growth and vice versa, which is in good agreement with the evanescent wave configuration predicted by the linear theory. Some spectral components of the incident wave can penetrate through the evanescent region and produce a transmitted wave. Both the reflection and transmission coefficients slightly decrease with the moderate increase of the initial amplitude of the incident wave, which is because a large amplitude wave can induce a strong mean flow; moreover, the waveinduced mean flow plays a more significant role in transferring the energy from the waves to the background flow than in enhancing the transmission of the waves. This is distinguished from the wave reflection in a sheared flow under the wave propagation distance smaller than the density scale height, in which the mean flow induced by a largeramplitude wave significantly strengthens the transmission of the wave. The simulation shows that the reflection loop predicted by the linear theory is not a common phenomenon in the wave reflection. Several groups of simulated cases indicate that the reflection and transmission coefficients depend on not only the amplitude, frequency, and wavenumbers of the incident wave but also the strength and thickness of the evanescent region. The reflection coefficient increases but the transmission coefficient decreases with the relative evanescent thickness growth, and once the strength and thickness of the evanescent region are large enough, the wave hardly penetrates through the sheared wind zone, and the reflection coefficient approaches a constant value, too. Except for the disappearance of the evanescent region, the total sum of the reflection and transmission coefficients of the wave pseudoenergy flux is slightly <1 because of the interaction between the wave and flow. These results suggest that the effects of wave reflection and transmission should be correctly included in the parameterization of gravity waves to attain more realistic middle atmospheric climatology from general circulation models. Citation: Huang, K. M., S. D. Zhang, and F. Yi (2010), Reflection and transmission of atmospheric gravity waves in a stably sheared horizontal wind field, J. Geophys. Res., 115,, doi: /2009jd Introduction [2] It is generally accepted that gravity waves play a crucial role in determining large scale circulation, thermal 1 School of Electronic Information, Wuhan University, Wuhan, China. 2 Also at State Key Laboratory of Space Weather, Chinese Academy of Sciences, Beijing, China. 3 Also at Key Laboratory of Geospace Environment and Geodesy, Ministry of Education, Wuhan, China. 4 Also at State Observatory for Atmospheric Remote Sensing, Wuhan, China. Copyright 2010 by the American Geophysical Union /10/2009JD states, and dynamics of the middle atmosphere because of their inherent ability to transport momentum from one level to another and deposit energy at the heights where these waves undergo kinds of dissipation [Lindzen, 1981; Dunkerton, 1982; Holton, 1982, 1983; Vincent and Reid, 1983; Fritts and Dunkerton, 1985]. Convection, topography, wind shear, and geostrophic adjustment are generally viewed as the dominant sources of gravity waves in the lower atmosphere [Fritts and Alexander, 2003; Fritts et al., 2006], and additional sources, including body forcing accompanying localized wave dissipation [Fritts et al., 2002; Vadas and Fritts, 2002; Vadas et al., 2003], auroral heating, and wavewave interactions [Sofko and Huang, 2000; Zhang and Yi, 1of17

2 2004; Huang et al., 2007, 2009], may be of significance in the middle and upper atmosphere [Fritts and Alexander, 2003; Fritts et al., 2006]. On account of the diversity and variability of wave sources, generated gravity waves have different spatial and temporal scales. Because the propagation of gravity waves is intensely affected by the background atmosphere, gravity waves with different spatial and temporal scales have different propagation characteristics. When a gravity wave propagates in a vertically sheared wind field, the linear theory deals with this case by assuming that the background atmosphere is stratified horizontally; thus the horizontal wavenumber k x is constant, and then the vertical wavenumber k z can be calculated from the dispersion equation of gravity waves, which is written in the following form [Hines, 1960; Einaudi and Hines, 1971]: kz 2 ¼ k2 x W 2 N 2 W 2 W 2! 2 þ a ; ð1þ where N is the buoyancy frequency, v a is the acoustic speed, w a is the acoustic cutoff frequency, and W is the intrinsic frequency observed in a coordinate system moving with the background atmosphere, which varies with height dependent wind due to the Doppler shift. According to the Doppler shift, W is given by the following equation: W ¼! k x u 0 ; where w is a constant ground based wave frequency and u 0 is the horizontal background wind velocity, which is a function of the height in a vertically sheared flow. [3] For a gravity wave propagating along a vertically sheared background wind, it can be deduced from equations (1) and (2) that when it reaches a level at which the horizontal phase speed of a wave equals the velocity of the background wind, the Doppler shifted intrinsic frequency decreases to zero and the vertical wavenumber approaches infinity. Thus the wave meets a critical level, which leads to the wave momentum deposition in the background flow due to the critical level absorption and wave action dissipation. The research on the interaction between the critical level and gravity waves is extensive and detailed by a great deal of the theoretical and experimental efforts in the past several decades [Bretherton, 1966; Booker and Bretherton, 1967; Hines, 1968; Breeding, 1971; Thorpe, 1973, 1981; Hartman, 1975; Fritts, 1978; Holton, 1982, 1983; Dunkerton and Fritts, 1984; Winters and D Asaro, 1989; Yi et al., 1991, 1992; Huang et al., 1998; Walterscheid, 2000; Wang et al., 2006]. Moreover, the critical level filtering is regarded as a major mechanism to explain the anisotropy of the observed gravity wave propagation directions in the middle and upper atmospheres. [4] In contrast, a gravity wave propagating against the sheared flow may be Doppler shifted to a higher frequency so that the vertical wavenumber is close to zero, indicating that the wave encounters a reflecting level. If the background wind across the reflecting level becomes stronger, the square of the vertical wavenumber is less than zero; thus an evanescent region exists above the reflecting level. Waves penetrating into the evanescent region become vertically evanescent waves without vertical phase variation in contrast with an internal gravity wave with phase propagation, and the amplitude decays away from the reflecting v 2 a ð2þ level with the increasing sheared wind. On the contrary, when the background wind turns to decrease with height, even with direction reversion, an evanescent wave has its frequency Doppler shifted to a lower frequency and may turn into a freely propagating internal gravity wave. According to equations (1) and (2), waves with high frequencies and small horizontal wavelengths are more likely to be reflected in the atmospheric wind field. For some high frequency waves, there may be two reflecting levels at different heights in the atmospheric wind field, which forms a Doppler ducting [Pitteway and Hines, 1965; Hines and Reddy, 1967; Francis, 1973; Chimonas and Hines, 1986; Fritts and Yuan, 1989; Monserrat and Thorpe, 1996; Isler et al., 1997; Walterscheid et al., 1999; Hecht et al., 2001; Hickey, 2001; Nappo, 2002; Yu and Hickey, 2007; Snively et al., 2007]; thus these waves are trapped between two reflecting levels (or two evanescent regions), leading to the energies and momenta of waves being confined to a vertical region, and then could be transported over a considerable horizontal distance. The other case is that gravity waves may be ducted between the ground and some high regions of the atmosphere in which waves become locally evanescent [Francis, 1973; Yeh and Liu, 1974; Richmond, 1978; Tuan and Tadic, 1982; Mayr et al., 1984; Wang and Tuan, 1988; Munasinghe et al., 1998]. The wave reflection events were reported from imager and foil chaff measurements [Schubert et al., 1999; Ejiri et al., 2001; Wüst and Bittner, 2008], even including the reflection of a long period gravity wave [Walterscheid et al., 2000], and the evidence of ducted gravity waves in the mesopause region was revealed by many imager observations [e.g., Taylor et al., 1995; Isler et al., 1997; Walterscheid et al., 1999; Hechtetal., 2001]. Moreover, Hines and Tarasick [1994] and Makhlouf et al. [1995, 1998] suggested that an out of phase relation between airglow brightness and temperature was attributed to the strong wave reflection, which results in the establishment of vertically standing waves. [5] Previous work on gravity wave reflection was generally based on the linear theory of gravity waves. Combining the linearized motion equations and the Doppler shift equation, Hines and Reddy [1967] and Blumen [1985] deduced complex reflection and transmission coefficients, which were expressed by the incident wave wavenumbers and frequency, buoyancy frequency, and sheared wind of the background atmosphere, and according to the deduced complex reflection coefficient, the variation of the energy reflection coefficient with the horizontal wavenumber of incident wave was discussed [McKenzie, 1972]. Cowling et al. [1971] and Broutman and Rottman [2004] brought forward that when gravity waves were reflected by a sheared background wind, a reflection loop might occur in their ray paths derived from the ray equations. By modifying the wave action continuity equation presented by Dunkerton [1981], Robinson [1997] explored the nonlinear effects of wave induced mean flow on the reflection properties of transient internal gravity waves in thermospheric wind. The configuration of gravity waves at the evanescent region was given by the linear theory in terms of an Airy function [Lighthill, 1978], and Walterscheid and Hecht [2003] reexamined the special properties of evanescent acoustic gravity waves in a linear regime and their significant role in the upper mesosphere and lower thermosphere. On the basis of the Taylor Goldstein equation, many scientists [Chiminas and Hines, 1986; Wang 2of17

3 and Tuan, 1988; Fritts and Yuan, 1989; Isler et al., 1997; Nappo, 2002] analyzed the possible ducted propagation of gravity waves in the atmosphere and pointed out the probable ducting regions for specific high frequency waves. Sutherland and Yewchuk [2004] and Brown and Sutherland [2007] further derived analytic formulae of wave transmission across idealized piecewise linear profiles of the buoyancy frequency and shear layer and termed this phenomenon internal wave tunneling. The studies on waves at critical levels also showed that the wave reflection and transmission vary with the gradient Richardson number and wave frequency and wavenumber [Jones, 1968; Breeding, 1971; Bowman et al., 1980; Nault and Sutherland, 2007]. Huang et al. [2008] further compared the wave reflection in a meridionally sheared wind within a vertically sheared flow by applying the dispersion equation and Doppler shift equation, which revealed some different characteristics of the wave reflection between the two sheared winds; for example, low frequency rather than high frequency waves are likely to encounter the reflecting level in a meridionally sheared wind. Although these theoretical studies on the wave reflection were highly idealized in a linear regime, they provided an insight into the essential features of the gravity wave reflection. Sutherland [1996, 1999, 2000] numerically investigated the reflection and transmission of internal waves under the wave propagation distance smaller than the density scale height by using a set of nonlinear 2 D motion equations which showed that if the amplitude of wave packet was large, nonlinear effects could significantly enhanced the transmission of wave packet across the reflecting level as a result of the effective mean flow induced by the wave. For finite amplitude waves in a nonuniformly stratified fluid, the simulation showed that the transmission coefficient exhibits a monotonic increase as a function of the incident wave amplitude but little dependence on the wavepacket extent [Brown et al., 2008]. However, the nonlinear physical process of atmospheric gravity wave reflection in the atmospheric jets is not quite clear. [6] If the amplitudes for upgoing gravity waves increase large enough to exceed their instability threshold due to the exponential decrease of the atmospheric density with height, waves will break and deposit their energies and momenta into the background atmosphere [Breeding, 1972; Lindzen, 1981; Holton, 1982, 1983; Fritts, 1984; Fritts et al., 1996]. By assuming that each wave propagates independently and transfers its momentum into the background atmosphere due to the production of turbulence at its breaking level, Lindzen [1981] presented a parameterization scheme of gravity waves. Alexander and Dunkerton [1999] put forward a variant of Lindzen s scheme in which the reflected component of spectrum is eliminated at the reflecting level. According to Alexander and Dunkerton s parameterization scheme, the reflection effect had a considerable influence on the parameterized wave driven force because the spectra of gravity waves generated by convections are especially sensitive to the vertically sheared wind, owing to the decrease in the amount of spectral components propagating into the upper atmosphere through the wave reflection [Marks and Eckermann, 1995; Alexander and Dunkerton, 1999; Beres et al., 2002, 2004; Alexander et al., 2006]. This indicates the necessity of the parameterized reflection effect in considering the drag from a full wave spectrum. In the linear theory, the wave is regarded as a complete unit to be reflected at its reflecting level; however, in fact, the wave reflection may be a long lasting process in which the nonlinearity and the work due to Reynolds stress may have a significant influence on the wave energy reflection. Therefore, if we can understand the nonlinear reflection and transmission process of gravity waves in detail, in particular the reflection and transmission coefficients in the sheared flow, this would be helpful in further improving the parameterization of the reflection and transmission effects. [7] The main purpose of this paper is to numerically study the features of the reflection and transmission of gravity waves in a vertically sheared horizontal wind by using a full nonlinear model. We begin in section 2 with a brief description of the numerical model. In section 3, we analyze quantitatively the computational results, including the effect of the wave induced mean flow. Several groups of cases in section 4 are examined to explore the relations among the reflection and transmission coefficients, the incident wave parameters, and the sheared flow configurations. In section 5, the reflection loop predicted by the linear theory is discussed, and the summary and discussion are presented in section Numerical Model 2.1. Governing Equations [8] Starting from a set of hydrodynamic equations in an adiabatic, inviscid, and compressible 2 D atmosphere, the governing equations can be written as þ þ ; þ g þ >: ¼ where x and z are the horizontal and vertical (positive upward) coordinates, respectively; u and w are the horizontal and vertical components of total wind velocities, respectively; r and T are density and temperature, respectively; R = 287 Jkg 1 k 1 is the universal gas constant; g = c p /c v (c p = 1005 Jkg 1 k 1 and c v = 718 Jkg 1 K 1 are specific heats at constant pressure and volume, respectively); and g = ms 2 is the gravitational acceleration. [9] Usually, propagation of gravity waves in the atmosphere is a long lasting and slow varying process. For the sake of precisely simulating the propagation of gravity waves, a numerical scheme should be of high accuracy and fine stability. Here, the operator splitting scheme, Crank Nicholson temporal difference scheme, and center spatial difference scheme are combined into one composite scheme that has a second order precision for both the time and space differences. In addition, a uniform Eulerian mesh with staggered grids is applied to avoid the checkerboard error. The difference equations are given in detail by Huang et al. [2007]. 3of17

4 context, the spatial step lengths are set to be Dx = 1.5 km and Dz = 0.3 km, respectively. The mesh is chosen to be , which corresponds to horizontal and vertical domains of 0 km x 750 km and 0 km z 150 km, respectively. [13] For avoiding the boundary reflection, here the horizontal boundaries are set to be periodical boundaries, and projected characteristic line boundaries [Hu and Wu, 1984; Zhang and Yi, 1999] are employed on vertical boundaries. [14] Considering that an explicit scheme is applied in the projected characteristic line boundaries, the time step Dt should be restricted by the Courant condition: 1 Dt < Dt c ¼ v a þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p u 2 þ w 2 1 Dx 2 þ 1 1 ð4þ 2 Dz 2 Figure 1. Profile of the sheared background wind in case 1. The height between two red dashed dotted lines is the evanescent region, and the lower red dashed dotted line stands for the reflecting level. The wind speeds at the heights marked by two orange dashed dotted lines is only 0.20 ms 1, and the green dashed dotted line denotes the position of the maximum wind speed Initial Background and Perturbation [10] To study the characteristic of the gravity wave reflected only by a sheared wind field, we should isolate the influences of inhomogeneous temperature. So, assuming that the initial background atmosphere is isothermal in hydrostatic equilibrium with a sheared horizontal wind field, the values for the background atmosphere are taken to be T 0 ¼ 290 K; gz RT0 0 ¼ c e = ; w 0 ¼ 0ms 1 ; where r c = 1.2 kg m 3 is the density on the ground, and the vertically sheared horizontal wind u 0 will be given in the following simulation cases. [11] At the beginning, an upgoing Gaussian gravity wave packet is introduced as the initial wave perturbations. The horizontal perturbation velocity has the following form: " # " # u 0 ð ðx; z; 0Þ ¼ u c exp x x cþ 2 ð exp z z cþ x sin½k x ðx x c Þþk z ðz z c ÞŠ; where u c =1ms 1 is the initial maximum amplitude of the wave packet; x c = 60 km and z c = 35 km are initial geometry center position of the wave packets in x and z directions, respectively; and s x and s z represent, respectively, the halfwidths of the wave packet in x and z directions, which are set to be s x = l x, and s z = l z (l x and l z are the horizontal and vertical wavelengths, respectively). The other initial perturbation quantities (v, w, r, T ) are derived from the polarization equations of the gravity wave Boundary Conditions and Step Lengths [12] In the numerical computation, considering the scale of the gravity wave which is presented in the following 2 2 z In this paper we select Dt = 0.5Dt c. 3. Computational Results [15] First, we simulate case 1, in which the sheared background wind is chosen to be the following Gaussian form: " # ð u 0 ðþ¼ u z 00 exp z z bþ 2 ; where the maximum value position and half width of the sheared background wind are set to be z b = 60 km and s b = 4 km, respectively; the maximum wind speed u 00 is set to be 60 ms 1 ; and the negative sign in the expression of u 0 denotes a westward background wind. The profile of the sheared wind is shown in Figure 1. It is known that the similar sheared wind form is common in the middle and upper atmosphere, such as the various jets at different altitudes and latitudes. For instance, the modeling result indicated that the maximum of the mesospheric summer jet in a latitude band near the height of 60 km has the typical westward wind speeds of 60 ms 1 in July and 70 ms 1 in January, whereas the eastward mesospheric winter jets arrive at the wind speed of about 60 ms 1 during January and 110 ms 1 during July [Berger, 2008]. The observational data of radar and rocket and satellite showed that the wind field has usually an extremum of >100 ms 1 in the vicinity of the mesopause in the middle latitude [Burrage et al., 1993; McLandress et al., 1996; Wüst and Bittner, 2008]. [16] In the respect that the higher frequency gravity wave is more likely to encounter the reflecting level in the atmospheric sheared wind field, we select a gravity wave with ground based frequency of w = Rads 1 (the corresponding period of 29.4 min) and horizontal wavelength of l x = 15 km. Then its vertical wavelength can be calculated to be l z = 3 based on the dispersion equation of gravity waves because the whole wave packet is designed to be out of the background sheared wind; here the negative sign of the vertical wavelength denotes the downward phase propagation. According to equations (1) and (2), when such a gravity wave propagating eastward and upward arrives at the height of 55.8 km, where the westward background wind speed increases to ms 1, its vertical wavenumber falls to zero, indicating that the gravity wave 2 2 b 4of17

5 encounters the reflecting level predicted by the linear theory. In Figure 1, the lower red dashed dotted line at the altitude of 55.8 km represents the reflecting level, and at the position of the upper red dashed dotted line (z = 64.2 km), the background wind speed also equals ms 1 ; thus the height between these two red dashed dotted lines is the wave evanescent region. At the position of two orange dashed dotted lines (z = 46.5 and 73.5 km), the sheared wind velocity is 0.20 ms 1, which means that the sheared wind is rather weak, and then outside these two orange dashed dotted lines, the effect of the sheared background wind on the wave packet may be negligible. The green dashed dotted line at z = 60 km represents the position of the maximum background wind velocity Evolution of Disturbance Velocities [17] Figure 2 displays the propagation and reflection process of the gravity wave in square root density weighted horizontal disturbance velocities, which are calculated from the expression of u (z) =[r 0 (z)/r 0r ] 1/2 u (z), where r 0r is the background density at a reference level of z = 36 km. From Figure 2, one can see that the region occupied by the wave packet tends to become larger in the propagation and reflection process. At the beginning, the phase tilt of the incident wave is toward the right up direction, which indicates that the upgoing wave packet propagates against the sheared horizontal wind field. At the time of 1.5 h, part of the wave packet crosses the reflecting level and penetrates into the evanescent region, of which the phase front becomes nearly perpendicular due to the effect of the sheared wind, implying that the vertical wavelength of this part wave approaches infinity. After 2.5 h of propagation, a new wave perturbation which staggers with the incident wave appears, and the tilt of its phase indicates that the new wave is a reflected wave and propagates downward and eastward; moreover, the part of the incident wave penetrating into the evanescent region can travel through this region and brings forth a transmitted wave. From 3 to 3.5 h, the reflected and transmitted waves become stronger and stronger due to the effect of the reflection and transmission in the sheared wind field, whereas the incident wave becomes weaker and weaker. At the time of 4.5 h, the incident wave reduces to a rather weak level because the energy is continuously transferred to the other two waves. And from 6 to 7.5 hours, the reflected and transmitted waves move obviously downward and upward, respectively. [18] To more clearly exhibit the wave configurations in the reflection and transmission process, we present Figure 3, which shows the profile of u (z) at the position of the maximum wave amplitude at several selected times. In Figure 3, the colored dashed dotted lines have the same meaning as those in Figure 1; the first and fourth columns are the total disturbance winds; the second and fifth columns are the incident and transmitted waves; and the third and sixth columns are the reflected waves. The incident and transmitted waves are calculated from the total winds by applying a band pass filter with high and low cutoff wavelengths of 37.5 km and 9.0 km in the horizontal direction and 75 km and 1.5 km in the vertical directions, respectively, where the high cutoff vertical wavelength is chosen to be a very large value by considering the increase of the vertical wavelength when the wave approaches the reflecting level. The reflected wave is extracted by applying a band pass filter with high and low cutoff wavelengths of 37.5 km and 9.0 km in the horizontal direction and 75 km and 1.5 km in the vertical direction, respectively. At t = 0 h, below the sheared background wind, the peak value of u (z) for the incident wave is 1.0 ms 1. After 1.5 h of propagation, the vertical wavelength for the wave entering into the sheared wind field becomes evidently longer, and a reflected wave perturbation appears under the reflecting level. At the time of 2.5 h, an obvious transmitted wave is above the evanescent region, which has an amplitude of 0.14 ms 1. Although part of the incident wave penetrates through the evanescent region, the maximum value of the total disturbance velocity arrives at 1.06 ms 1 owing to the superposition of the incident and reflected waves, of which the peak amplitudes are 0.50 ms 1 and 0.60 ms 1, respectively. From 3 h to 4.5 h, continuing to come upward into the sheared wind region and encountering the reflecting level, the incident wave lasts to undergo the reflection and transmission in the sheared flow, and the reflected and transmitted waves propagate downward and upward, respectively. Moreover, we note that in the duration from 2.5 h to 4.5 h, the perturbation configuration in the evanescent region has three extremums owing to being modulated by the sheared wind stronger than that at reflecting level, displaying clear evanescent wave form. The first maximum value is just at the reflecting level; beyond the reflecting level, the disturbance velocity decreases with the increase of the sheared wind speed and arrives at a minimum value at the position of the strongest background wind. Thereafter, when the sheared wind becomes weak, the disturbance wind grows and achieves another maximum value at the upper boundary of the evanescent region, and once having propagated across this level, the perturbation exhibits a wave packet structure again. At the time of 6 h, the incident wave almost disappears, and both the reflected and transmitted waves exhibit a Gaussian wave packet structure with vertical extent larger than the initial wave. After 7.5 h of propagation, the reflected and transmitted waves depart from the sheared background flow, and their amplitudes decrease to 0.29 and 0.12 ms 1 from the maximum values of 0.51 ms 1 and 0.18 ms 1 at t = 3.5 h, respectively, which results mainly from the negative work due to Reynolds stress in the sheared wind and the larger spatial coverage. [19] The wave number spectra of these three waves were obtained by separately making a discrete Fourier transform on the square root density weighted horizontal disturbance velocities for the computational domain below and above the height of 60 km. The ratios of the calculated wave number spectra to the spectral peak value k 0m at the beginning time are named as the normalized relative wavenumber spectra, which are shown in Figure 4 for four selected times. In Figure 4, the blue lines represent the normalized relative wavenumber spectra of the incident and reflected waves, the red lines represent the spectra of the transmitted wave, and the positive and negative values of k z /k 0m mean the upward and downward propagations of the waves, respectively. As shown in Figure 4, one spectral center of the incident wave at the beginning time evolves gradually into two spectral centers at t = 6 h, of which one with negative value is the spectrum of the reflected wave and the other one with positive value is the spectrum of the transmitted wave. At 5of17

6 Figure 2. Reflection and transmission process of gravity wave in square root density weighted horizontal disturbance velocities. the time of 6 h, the reflected and transmitted waves propagate outside the sheared wind zone, and their spectral peaks show that the reflected and transmitted waves have almost the same vertical dominant scale as the initial incident wave, whereas the dominant horizontal wavelength of the transmitted wave is obviously larger than that of the reflected wave. This implies that the spectral components with larger horizontal wavenumber are more likely to be reflected, whereas the spectral components having smaller horizontal wavenumber are more likely to be transmitted in the evanescent region, which is consistent with the result derived from equations (1) and (2) that the components of the incident wave with smaller horizontal wavenumber is reflected under the condition of a stronger sheared wind Reflection and Transmission of Pseudoenergy Flux [20] Traditionally, the transmission (reflection) coefficient is defined as the ratio of transmitted (reflected) to incident 6 of 17

7 Figure 3. Profile of square root density weighted horizontal disturbance velocities at the horizontal position of maximum wave amplitude. The first and fourth columns are the total disturbance winds, the second and fifth columns are the incident and transmitted waves, and the third and sixth columns are the reflected waves. The colored dashed dotted lines are the same as those in Figure 1. energy. When a gravity wave propagates in a sheared wind, the work due to Reynolds stress causes an energy exchange between the wave and flow, which means that the wave energy E is no longer conserved. According to the linear wave theory, for small amplitude waves, the corresponding conserved quantity is wave action, A = E/W. Then the pseudoenergy E = E w/w, with the same units as energy, also is a conserved quantity. So, Brown and Sutherland [2007] introduced an appropriate definition of the transmission coefficient to be the ratio of the transmitted to incident vertical flux of pseudoenergy; here, the vertical flux of pseudoenergy, F = v gz Ew/W, is the pseudoenergy 7of17

8 Figure 4. Normalized relative wavenumber spectra. The blue contours represent the spectra of the incident and reflected waves, and the red contours represent the spectra of the transmitted waves. The values of contours are 0.1, 0.2, 0.3, 0.5, 0.7, and 0.9, respectively. multiplied by the vertical group velocity v gz. In the absence of the background shear, this definition of the transmission coefficient is equivalent to the ratio of transmitted to incident wave energy. Brown et al. [2008] further compared the transmission coefficient T E = E trans /E total with T F = F trans /F total, and found that two different measures gave consistent results. [21] Considering the energy transfer between the wave kinetic and potential energies when waves propagate in a shear flow [Huang et al., 2008], the wave energy density " is calculated on the basis of the following expression [Yeh and Liu, 1974]: " ¼ u 02 þ w 02 p 0 þ 2 0 v 2 a þ p0 0 v 2 a 2 0 ð 1Þv 2 ; a where p is the perturbation pressure. We can extract the perturbation quantities of the different spectral components by applying a band pass filter; thus the wave energy density " w of the different spectral components is calculated. The vertical flux density w of the pseudoenergy can be written as follows: w ¼ v gz" w! W ; where W and v gz can be expressed as the functions of the wavenumbers on the basis of the dispersion equation. Therefore, the vertical flux densities w of the pseudoenergy for the different spectral components can be calculated. The densities of the vertical flux of the pseudoenergy for the incident and transmitted waves are calculated by integrating w from 9.0 km to 37.5 km of the corresponding horizontal wavelengths and from 75 km to 1.5 km of the corresponding vertical wavelengths, and the density of the vertical flux of the reflected wave pseudoenergy is obtained by integrating w from 9.0 km to 37.5 km of the horizontal wavelengths and from 1.5 km to 75 km of the vertical wavelengths. Finally, the vertical fluxes F of the wave pseudoenergies are obtained by integrating the densities in the horizontal direction and then in time at the specified vertical levels. The vertical pseudoenergy flux of the transmitted (reflected) wave is the flux that moves upward (downward) through the level of z =60km(z = 55.8 km) represented by the green (lower red) dashed dotted line in Figure 1, which is used to compute the transmission (reflection) coefficient in the following text. Figure 5a shows the variation of the absolute vertical flux of three wave pseudoenergies with time, in which the solid line represents the total vertical flux of these three wave pseudoenergies. At the beginning 1.5 h, the vertical flux of the incident wave pseudoenergy holds nearly the initial value of Jms 1. With the wave reflection and transmission in the shear zone, from 1.5 h to 3.5 h, the vertical flux of the incident wave pseudoenergy rapidly decreases while the magnitude of the vertical fluxes of the reflected and transmitted wave pseudoenergies remarkably increases. At t = 5 h, the vertical flux of the incident wave pseudoenergy is close to zero, and the reflected and transmitted waves have the maximum vertical fluxes ( Jms 1 and Jms 1, respectively) of the pseudoenergy. After this, the vertical fluxes of the reflected and transmitted wave pseudoenergies almost maintain constant values with a rather slight decrease. In the whole process, the total vertical flux of these three wave pseudoenergies falls slightly, and at t = 7.5 h, the total value is 90.72% of the initial vertical flux of the incident wave pseudoenergy. Besides the dissipation in the numerical computation and filtering, a little wave energy transfers to the background flow due to the occurrence of the wave induced flow in the long lasting nonlinear propagation; hence, in the nonlinear environment, the wave pseudoenergy flux may be only approximately conserved, which is not completely the same as the prediction by the linear wave theory. [22] According to the definitions of the reflection coefficient R = F ref /F initial and the transmission coefficient T = F trans /F initial, which is equivalent to the ratios of the fluxes of the wave action [Brown and Sutherland, 2007], we calculate R(t) and T(t). Figures 5b and 5c show the temporal evolution of the reflection and transmission coefficients. The reflection and transmission coefficients arrive at their maximums of and at t = 5 h and then slightly fall to and at t = 7.5 h, respectively, which may be caused by the tiny energy transfer from the waves to the background field. In this case 1, the reflection and transmission coefficients are selected to be R=0.806 and T=0.101, respectively Propagation Path of Wave Energy [23] We calculate the propagation path of energy center of the incident wave in the duration of the first 5.5 h and the 8of17

9 paths of energy centers of the reflected and transmitted waves from 1.5 to 7.5 h, respectively, which are shown in Figure 6. At the beginning time, the energy center of the incident wave is at the position of x = 60.0 km and z = 35.4 km and arrives at x = km and z = 52.2 km after 5.5 h of propagation. By ignoring little residual energy after 5.5 h, z = 52.2 km is almost the maximum height that the incident wave energy can reach, which is lower than the reflecting level of z = 55.8 km predicted by the linear theory. When the reflected wave appears at t = 1.5 h, its energy center locates at x = km and z = 51.3 km, and arrives at the position of x = km and z = 21.0 km at t = 7.5 h; during this period, the energy center of the transmitted wave propagates from x = km and z = 65.7 km to x = km and z = 93.0 km. Since most of the incident wave has not propagated into the sheared wind zone yet at the onset of the reflected wave at t = 1.5 h, the energy center of the reflected wave is slightly higher than that of the incident wave, as shown in Figure 6. Additionally, it can be noted that the positions of the incident wave in each half hour tend to become gradually nearer, whereas the positions of the reflected and transmitted waves have an opposite variational tendency. This is in agreement with the result in the linear theory that the propagation velocity of wave energy equals the sum of the background wind velocity and the intrinsic group velocity [Bretherton, 1966; Cowling et al., 1971; Yeh and Liu, 1974]. Figure 5. Evolution of absolute vertical flux of three wave (a) pseudoenergies, (b) reflection, and (c) transmission coefficients. In Figure 5a, the dashed dotted, dashed, and dotted curves denote the magnitude of the pseudoenergy fluxes of the incident, reflected, and transmitted waves, respectively, and the solid curve denotes their sum Reflection of Large Amplitude Wave Packet [24] An additional simulation (case 2) was performed to investigate the properties of a wave packet with large amplitude of u c = 3.5 ms 1 in the reflection and transmission process, which is about four tenths of the horizontal phase speed (8.5 ms 1 ) of the wave. Except for the initial amplitude, the other parameters in case 2 are taken to be the same as those in case 1. Figure 7a shows the evolution of the magnitude of the vertical fluxes of three wave pseudoenergies and their sum in case 2, in which the variational tendencies of the pseudoenergy fluxes are similar to those in case 1 shown in Figure 5a. However, there are several slight distinctions between Figures 7a and 5a. As shown in Figure 7a, at t = 5 h, the reflected wave has the maximum pseudoenergy flux ( Jms 1 ), whereas the pseudoenergy flux of the transmitted wave arrives at the maximum value of Jms 1 at t = 4 h; after these, the pseudoenergy fluxes of the transmitted wave decreases faster than that in Figure 5a; at t = 7.5 h, the total pseudoenergy flux ( Jms 1 ) is only 82.53% of the initial value ( Jms 1 ), which is <90.72% in case 1. The variation of the reflection and transmission coefficients with time is exhibited in Figures 7b and 7c. At t = 7.5 h, the reflection and transmission coefficients are R = and T = 0.081, respectively, and both coefficients are less than those in case 1 (R =0.806 and T=0.101). [25] It is known that a gravity wave with large amplitude can yield a significant wave induced mean flow [Dunkerton Figure 6. Energy propagation paths of three wave packets. The dashed lines with asterisks, triangles and squares denote the paths of the incident, transmitted, and reflected waves, respectively. The asterisks, triangles, and squares stand for the positions at each half hour. 9of17

10 result is regarded as the wave induced mean flow. Figure 9 shows the profile of the wave induced mean flow at three selected integral hours. Although the major wave perturbations propagate out of the sheared wind zone, a strong wave induced mean flow persists in the sheared wind zone. In the wave transmission process, the wave induced local mean flow weakens the sheared background field; thus the transmission effect can be enhanced. Simultaneously, a large wave induced mean flow implies that much energy of the waves transfers into the background field. Hence, the transmission coefficient has the maximum value of at t = 4 h, which is nearly the same as in case 1; however, at t = 5 h, of the reflection coefficient is evidently <0.824 in case 1. It can be noted that compared with the initial background wind, the wave induced mean flow is very small; therefore, the wave induced mean flow plays a more significant role in transferring the energy from the waves to the background field than in enhancing the wave transmission. Additionally, because the amplitude of the upgoing transmitted wave grows up with height, the transmitted wave can induce a much stronger mean flow than the reflected wave as shown in Figure 9. This causes the relatively obvious decrease of the transmission coefficient with time after 4 h. At t = 7.5 h, not only the reflection coefficient but also the transmission coefficient in case 2 are slightly smaller than those in case 1. Some different results were presented in previous numerical studies on the reflection of internal waves in a sheared fluid on the basis of a 2 D model with Boussinesq approximation [Sutherland, 1996, 1999, 2000]. Under the propagation distance smaller than the density scale height, for a large amplitude wave, the strong wave induced mean flow due to the superposition of the incident and reflected waves acts to greatly enhance the transmission of the wave [Sutherland, 1999, 2000]. Figure 7. Same legend as Figure 5a, but for case Reflection and Transmission Coefficients [28] The coefficients of reflection and transmission are an attracting aspect in the propagation process. In this section, and Fritts, 1984; Fritts and Dunkerton, 1984; Sutherland, 1996, 1999; Zhang and Yi, 1999, 2002]. [26] To clearly analyze the wave induced mean flow, by removing the initial background wind from the total wind, the horizontal disturbance velocities are obtained, and then we make a discrete Fourier transform on the square root densityweighted horizontal disturbance velocities. Figure 8 shows the normalized relative wavenumber spectra at the time of t = 6 h. In comparison with Figure 4, there are two different characteristics in Figure 8. One is that the Gaussian spectral shape is destroyed to some extent, especially for the reflected wave, and the other is that some small wavenumber spectra components arise, which correspond to large scale perturbation and can be named as the wave induced mean flow. The occurrence of the strong wave induced mean flow is responsible for the obvious decrease of the total pseudoenergy flux. [27] By adopting the same filter as that in case 1, the wave perturbation is extracted from the horizontal disturbance velocities, and then we average the residual disturbance field over one horizontal wavelength coverage. So, the calculated Figure 8. Same legend as Figure 4, but for case 2 at the time of 6 h. 10 of 17

11 Figure 9. Profile of the wave induced mean flow at several selected times. we investigate the influence of the characteristics of the wave and background wind on the reflection and transmission coefficients by several groups of the numerical experiments. In the following computation, the presented reflection (transmission) coefficient is approximately the same as that at the next half hour with change <0.02 (0.005) in each case. [29] In the previous section, we studied the effect of the wave induced mean flow on the reflection and transmission. Here, more cases are studied to further examine the variation of the reflection and transmission coefficients with the amplitude of the incident wave. In cases 3 8, the amplitudes of the incident waves are selected to be 0.5, 1.5, 2.0, 2.5, 3.0, and 4.0 ms 1, respectively, and the other parameters are the same as those in case 1. Here, for the sake of avoiding the wave breaking in the propagation [Fritts, 1984; Sutherland, 2001], more than 4.0 ms 1 (about one half of the horizontal phase speed) is not chosen. It can be noted that in this first group of cases, the reflecting level is unchanged, as shown in Figure 1. The computational results are displayed in Figure 10. One can clearly see from Figure 10 that the coefficients of reflection and transmission show the same variational tendency. When the amplitude of the incident wave is <3.0 ms 1, the wave induced mean flow slowly increases with the initial amplitude of the incident wave, which leads to the reflected and transmission coefficients slowly decreasing by considering that the wave induced mean flow plays a more significant role in transferring energy from the waves to the background field than in enhancing the transmission effect. Once the amplitude of the incident wave exceeds 3.0 ms 1, the wave induced mean flow rapidly increases; hence, the reflection and transmission coefficients markedly reduce. In case 8 with the incident wave amplitude of 4.0 ms 1, the reflection and transmission coefficients fall to and 0.070, respectively; thus the vertical flux of the wave pseudoenergy decreases 21%, which implies that the significant wave energy transfers into the background field due to the occurrence of the strong wave induced mean flow. [30] It is known that the high frequency gravity waves might encounter the reflecting level in the atmospheric wind field. In fact, the gravity waves with an identical frequency may also have different spatial scales. By altering the horizontal wavelengths, we used the second group of cases (cases 9 13) to explore the reflection and transmission of the waves with a constant frequency of w = Rads 1. In cases 9 13, the horizontal wavelengths of the incident waves are selected to be 10, 12, 14, 16, and 18 km (the corresponding vertical wavelengths are 2.00, 2.40, 2.80, 3.20, and 3.60 km at the center position of the wave packets), respectively. The other parameters maintain the same values as in case 1. Here, we introduce a relative evanescent thickness s e, which is defined to be the ratio of the thickness of the evanescent region to the vertical wavelength of the incident wave. In cases 9 13, the westward wind velocities at the reflecting level are 23.21, 27.84, 32.47, 37.08, and ms 1 derived from equations (1) and (2), respectively, and then the reflecting levels are calculated to be at the heights of 54.5, 55.0, 55.6, 56.1, and 56.6 km, respectively; thus the relative evanescent thicknesses are s e = 5.51, 4.13, 3.17, 2.45, and 1.90, respectively. Figure 11 shows the evolution of the reflection and transmission coefficients with s e. The growth of the relative evanescent thickness means that the waves undergo a stronger obstructive effect in the evanescent region, which leads to the increase of the reflection coefficient and the decrease of the transmission coefficient, as shown in Figure 11. In case 9, with l x =10km and s e = 5.51, the reflection and transmission coefficients are and only 0.001, respectively. Case 13 has the horizontal wavelength of 18 km and the relative evanescent Figure 10. Coefficients of (a) reflection and (b) transmission for the first group of cases. 11 of 17

12 Figure 11. Coefficients of (a) reflection and (b) transmission for the second case group (solid curve) and the third case group (dashed curve). thickness of 1.90, and then the reflection coefficient reduces to but the transmission coefficient increases to This indicates that even though the incident waves have a fixed frequency, the reflection and transmission coefficients depend closely on the wavelengths of the incident waves due to the variation of the relative evanescent thickness. [31] Next, we study the effect of the sheared background wind on the reflection and transmission of waves under the condition of an incident wave identical to that presented in case 1. By changing the half width of the Gaussian sheared background wind, we simulated the third group of cases (cases 14 18). In comparison with case 1, the half widths of the sheared wind are replaced with 2.0, 2.5, 3.0, 3.5, and 4.5 km in cases 14 18, respectively, and the relative evanescent thicknesses are calculated to be s e = 1.93, 1.74, 2.09, 2.44, and 3.13, respectively. The results are shown in Figure 11. Although the peak value of the westward background winds is fixed to be 60 ms 1 at the height of 60 km, the wave components getting across the reflecting level are likely to be reflected back at higher height under a larger relative evanescent thickness. With the growth of the relative evanescent thickness, the reflection coefficient increases from in case 14 to in case 18, and the transmission coefficient decreases from to Therefore, despite the fact that the relative evanescent thickness variation results from the alterations of the sheared background wind or the wavelengths of the incident wave, the second and third case groups show a similar evolution of wave reflection and transmission with the relative evanescent thickness as shown Figure 11. In the fourth group of cases (cases 19 23), we researched the relation between the reflection and transmission of wave and the sheared background wind magnitude on the condition of a same thickness of the evanescent region. In cases 19 23, besides the fact that the westward sheared background winds have the maximum values of 45, 75, 90, 100, and 120 ms 1 at z = 60 km, respectively, the half widths of the sheared winds are respectively adjusted to be 5.82, 3.37, 3.03, 2.87, and 2.65 km, so that the evanescent region (from 55.8 to 64.2 km) and the relative evanescent thicknesses (s e = 2.79) are consistent with those in case 1. Figure 12 illustrates the evolution tendencies of the reflection and transmission coefficients with the sheared wind magnitude. Despite the fact that the relative evanescent thicknesses is fixed, less wave energy can penetrate through the evanescent region when the sheared background flow in the evanescent region becomes stronger, which implies the decrease of the transmission coefficient but the increase of the reflection coefficient. Especially under the condition that the maximum velocity of the background wind is <75 ms 1, the reflection and transmission coefficients dramatically vary with the sheared wind magnitude. When the background wind peak grows up to 90 ms 1, the reflection and transmission coefficients are and 0.021, respectively, which indicates that the most energy of the incident wave is reflected; thus if the maximum velocity of the background wind exceeds 90 ms 1, the energies of the reflected and transmitted waves cannot remarkably vary any more with the sheared wind magnitude. Hence, the results in the third and fourth groups of cases indicate that the half width and magnitude varia- Figure 12. Coefficients of (a) reflection and (b) transmission for the fourth group of cases. 12 of 17