Horizontal and vertical propagation and dissipation of gravity waves in the thermosphere from lower atmospheric and thermospheric sources

Authors


Abstract

[1] The dissipation of high-frequency gravity waves (GWs) in the thermosphere is primarily due to kinematic viscosity and thermal diffusivity. Recently, an anelastic GW dispersion relation was derived which includes the damping effects of kinematic viscosity and thermal diffusivity in the thermosphere and which is valid before and during dissipation. Using a ray trace model which incorporates this new dispersion relation, we explore many GW properties that result from this dispersion relation for a wide range of thermospheric temperatures. We calculate the dissipation altitudes, horizontal distances traveled, times taken, and maximum vertical wavelengths prior to dissipation in the thermosphere for a wide range of upward-propagating GWs that originate in the lower atmosphere and at several altitudes in the thermosphere. We show that the vertical wavelengths of dissipating GWs, λz(zdiss), increases exponentially with altitude, although with a smaller slope for z > 200 km. We also show how the horizontal wavelength, λH, and wave period spectra change with altitude for dissipating GWs. We find that a new dissipation condition can predict our results for λz(zdiss) very well up to altitudes of ∼500 km. We also find that a GW spectrum excited from convection shifts to increasingly larger λz and λH with altitude in the thermosphere that are not characteristic of the initial convective scales. Additionally, a lower thermospheric shear shifts this spectrum to even larger λz, consistent with observations. Finally, we show that our results agree well with observations.

1. Introduction

[2] Gravity waves (GWs) have been observed in the thermosphere for many decades as traveling ionospheric disturbances (TIDs) [e.g., Hocke and Schlegel, 1996; Hocke et al., 1996; Oliver et al., 1997; Djuth et al., 1997, 2004]. Many of these observed and modeled GWs are from auroral sources in the lower thermosphere at high latitudes [Hocke and Schlegel, 1996; Richmond, 1978; Hickey and Cole, 1988], although medium scale GWs which are thought to arise from lower atmospheric sources have also been detected in the thermosphere with periods less than an hour and phase speeds less than 250 m s−1 [Georges, 1968; Waldock and Jones, 1986; Crowley et al., 1987; Ogawa et al., 1987; Hocke and Schlegel, 1996]. GWs from severe storms have been observed as concentric rings in airglow layers near the mesopause, just below the thermosphere [Taylor and Hapgood, 1988; Dewan et al., 1998; Sentman et al., 2003]. Reverse ray tracing has shown that the origin of a very large number of medium-scale TIDs (MSTIDs) is the troposphere [Waldock and Jones, 1987]. Additionally, enhanced GW occurrence has been found to occur when convection occurs [Röttger, 1977; Hocke and Tsuda, 2001], and strong ionospheric activity has resulted from hurricanes and tornados [Bauer, 1958; Hung et al., 1978; Hung and Kuo, 1978; Hung and Smith, 1978; Bishop et al., 2006]. Finally, recent theoretical efforts have shown that GWs from convective sources in the troposphere can propagate well into the thermosphere before dissipating, and the momentum flux divergence which occurs where GWs dissipate in the thermosphere likely results in the excitation of large-scale TIDs (LSTIDs) and MSTIDs and the creation of neutral winds [Vadas and Fritts, 2006, hereafter VF2006].

[3] Early theoretical work estimated the effect individual GWs have on the thermosphere [Hines, 1960, 1967; Francis, 1973]. Until recently, GW propagation in the thermosphere with dissipation was mainly accomplished with multilayer techniques [Midgley and Liemohn, 1966; Volland, 1969; Klostermeyer, 1972; Hickey and Cole, 1988], numerical simulations [Richmond, 1978; Francis, 1973; Zhang and Yi, 2002), and approximate or numerical solutions to complex dispersion relations [Pitteway and Hines, 1963; Yeh et al., 1975; Hickey and Cole, 1987]. Recently, an exact analytic anelastic GW dispersion relation was derived which includes kinematic viscosity and thermal conductivity [Vadas and Fritts, 2005, hereafter VF2005]. This dispersion relation is real and was derived from the more general complex dispersion relation by assuming a complex intrinsic frequency and real vertical wave number. This dispersion relation allows for the accurate ray tracing of GWs in the atmosphere when other dissipative mechanism such as ion drag can be neglected.

[4] Although this new dispersion relation has been used to study the thermospheric response to GWs from tropospheric convection [Vadas and Fritts, 2004; VF2006], many GW properties have not yet been explored. The purpose of this paper is to further explore these properties. Our paper is structured as follows. Section 2 contains a brief description of the GW dispersion relation and ray trace model. Section 3 shows GW dissipation altitudes, horizontal distances traveled, and total time taken to dissipate for a wide variety of individual GWs, thermospheric temperatures, and launch altitudes. Section 4 displays vertical wavelength, horizontal wavelength, and intrinsic wave period spectra as a function of altitude for dissipating GWs and compares these results with observational results. Section 5 shows the thermospheric GW spectra that result from a single deep convective plume in the troposphere, with and without a thermospheric shear. Our conclusions are provided in section 6. An Appendix follows.

2. Model Review

[5] Although the average temperature in the lower atmosphere is equation image ≃ 250 K, the temperature increases rapidly in the lower thermosphere. During extreme solar minimum, the thermosphere is relatively cold, equation image ≃ 600 K. During active solar conditions however, the temperature in the thermosphere can be equation image ≃ 2000 K [Banks and Kockarts, 1973]. Figure 1 shows the four canonical temperature profiles we use in this paper. The parameters used to generate these profiles are listed in Table 1 of the work by VF2006, with equation image0 = 237 K. Temperature profiles II and V approximate extreme solar minimum and very active solar conditions, respectively. From a given temperature profile equation image(z), we determine the pressure, equation image, using the hydrostatic balance equation dequation image/dz = -gequation image and the ideal gas law equation image = Requation imageequation image:

equation image

where equation image is the mean density, g is the acceleration due to gravity, p0 = equation image(z = 0), R = 8314.5/XMW m2 s−2 K−1, and XMW is the mean molecular weight of the particle in the gas. In addition, equation image = equation image/(Requation image), the density scale height is H = −equation image(dequation image/dz)−1, the potential temperature is equation image = equation image(po/equation image)R/Cp, the buoyancy frequency is N = equation image, Cp = γR/(γ − 1), and γ = Cp/Cv. Here Cp and Cv are the mean specific heats at constant pressure and volume, respectively. The density at the ground is set to equation image(z = 0) = 1 × 103 gm m−3. This yields densities at z ∼ 125 km which agree with the thermosphere-ionosphere-mesosphere-electrodynamics general circulation model (TIME-GCM) (see Appendix A). The coefficient of molecular viscosity is

equation image

[Dalgarno and Smith, 1962]. The kinematic viscosity is ν = μ/equation image and the thermal diffusivity is κ = ν/Pr. Here we set the Prandtl number to be Pr = 0.7 [Kundu, 1990] and thus ignore its slight variations with temperature [Yeh et al., 1975]. We also set the mean molecular weight and ratio of mean specific heat capacities to be

equation image
equation image

respectively, where s = −ln(equation image) and equation image has units of gm m−3. Here ln is the natural logarithm, image = 28.9, image = 16, a = 14.9, Δa = 4.2, γ0 = 1.4, γ1 = 1.67, b = 15.1, and Δb = 4.0 These parameters represent the best fit for a month of 2004 TIME-GCM data (see Appendix A). The decrease of XMW and increase of γ with altitude represent the change in composition from primarily diatomic N2 and O2 to monotomic O. Figure 2a shows XMW and γ using equations (3) and (4). We also show the corresponding altitudes using temperature profile III; in this case, the change from diatomic to monotomic occurs from z ∼ 150–300 km. We also show the local speed of sound, csequation image in Figure 2b. Figure 2c shows the density scale height. Because H = Requation image/g in an isothermal atmosphere and because R increases with altitude, H is twice as large in the thermosphere than if XMW and γ were constant.

Figure 1.

Model temperature profiles II, III, IV, and V as solid, dash, dash-dot, and dash-dot-dot-dot lines, respectively. Temperature profiles II, III, IV, and V have asymptotic temperatures (for example, for z > 300 km) of equation image = 600 K, 1000 K, 1500 K, and 2000 K, respectively.

Figure 2.

(a) Mean molecular weight (solid line) and ratio of mean specific heats (dash line) as a function of −ln(equation image), where equation image has units of gm m−3. The corresponding altitudes in a thermosphere with temperature profile III is shown on the right-hand y axis. (b) Sound speed profiles as solid, dash, dash-dot, and dash-dot-dot-dot lines for temperature profiles II, III, IV, and V, respectively. (c) Same as Figure 2b but for the density scale height.

[6] Our ray trace model follows the formalism of Lighthill [1978]. The GW dispersion relation we use here includes the primary damping mechanisms for high-frequency GWs with large vertical wavelengths, kinematic viscosity and thermal diffusivity. It is nonhydrostatic and compressible but excludes acoustic waves similar to Marks and Eckermann [1995]. This new anelastic GW dispersion relation can be written as [equation (26) from the work of VF2005, rearranged]

equation image

where k, l, and m are the zonal, meridional, and vertical wave number components of the GW, respectively, kH2 = k2 + l2, k2 = kH2 +m2, ωIr is the intrinsic frequency of the GW, δ = νm/Ir, δ+ = δ(1 + Pr−1), and ν+ = ν(1 + Pr−1). Note that δ is negative for an upward-propagating GW, because m is negative. This dissipative dispersion relation yields the usual GW anelastic dispersion relation when dissipation is negligible [Gossard and Hooke, 1975]:

equation image

Note that the dispersion relation we use here neglects other forms of dissipation such as ion drag and wave-induced diffusion. Ion drag is unimportant during the night for GWs with periods less than a few hours and for GWs with periods less than an hour during the day [Hines and Hooke, 1970; Francis, 1973; Hocke and Schlegel, 1996; Gossard and Hooke, 1975]. Wave-induced diffusion is important for z ∼ 140–220 km [Del Genio and Schubert, 1979], whereby it may significantly decrease GW amplitudes with periods longer than an hour or two.

[7] The inverse decay rate in time for a dissipating GW is [equation (25) from the work of VF2005]

equation image

Therefore a GW's momentum flux (per unit mass) when launched from z = zi is

equation image

where we put an absolute value around ωIi to ensure that a GW decays in time even when k2 < 1/4H2. These expressions, the GW anelastic dispersion relation and decay rate, were derived under the assumption that acoustic waves can be neglected. When dissipation is unimportant, this assumption is (ωIr/cs)2equation image (k2 + 1/4H2) [Vadas and Fritts, 2005]. Since a GW propagates at the group velocity, the ray-tracing condition we adopt here is that each GW propagates slower than the speed of sound:

equation image

where the factor 0.9 is arbitrarily chosen. Here cg = equation image is the group velocity in the direction of propagation, and image image and image are given by equations (C1), (C2), and (C3) in the work of VF2005. If a GW violates equation (9), it is removed from the spectrum.

[8] In Figure 3, we show the vertical wavelengths, λz ≡ 2π/|m|, for GWs launched from zi = 0 (Figure 3a) and zi = 120 km (Figure 3b and 3c) through zero background winds. As a GW propagates upwards in the thermosphere, its raypath bends toward the vertical because its vertical wavelength λz ≡ 2π/|m| increases, with a larger increase when the thermosphere is hot than when it is cold [Richmond, 1978; VF2006]. We also show the dissipation altitudes, zdiss, which are the altitudes where each GW's momentum flux (per unit max) is maximum. For GWs launched from zi = 0, λz decreases in the lower thermosphere because equation image increases rapidly. Where equation image increases less rapidly (i.e., for zequation image120 km), λz increases. If a GW dissipates in a region of the thermosphere where the temperature is approximately constant, its raypath bends rapidly toward the horizontal because λz decreases rapidly [VF2005; Zhang and Yi, 2002]. However, if a GW instead dissipates in a region of the thermosphere where the temperature increases, its raypath continues to bend toward the vertical because λz continues to increase above its dissipation altitude.

Figure 3.

(a) GW vertical wavelengths as a function of altitude. GWs propagate upwards from zi = 0 with λH = 160 km and λz(zi) = 80 km. The solid, dash, dash-dot, and dash-dot-dot-dot lines are for GWs propagating in temperature profiles II, III, IV, and V, respectively. The squares, diamonds, triangles, and Xs show the altitudes where the GW momentum fluxes (per unit mass) are maximum (i.e., zdiss) for profiles II, III, IV, and V, respectively. The solutions between the black circles denotes where R2 ≥ 1 using equation (11). (b) Same as in Figure 3a but for GWs which propagate upwards from zi = 120 km and have λH = 300 km and λz(zi) = 100 km. (c) Same as in Figure 3a but for GWs which propagate upwards from zi = 120 km and have λH = 400 km and λz(zi) = 60 km. (d) Momentum fluxes (per unit mass) for the GWs shown in Figure 3a with the same linetypes and symbols. (e) Same as in Figure 3d but for the GWs shown in Figure 3b. (f) Same as in Figure 3d but for the GWs shown in Figure 3c. Note that the altitudes displayed in Figures 3d–3f differ from those in Figures 3a–3c.

[9] A GW's dissipation altitude, zdiss, increases as equation image increases. The result that GWs dissipate at higher altitudes during active solar (and day time) conditions than during extreme solar minimum (and nighttime) conditions has been studied previously [Pitteway and Hines, 1963; Francis, 1973; Yeh et al., 1975; Richmond, 1978; Cole and Hickey, 1981; VF2006] and occurs because of the substantial increase of λz with equation image. This effect is enhanced for z > 250 km, where ν increases less rapidly with altitude for hotter as compared with cooler thermospheres.

[10] In Figure 3d–3f, we show the GW momentum fluxes (per unit mass), equation image, for the GWs from Figure 3a–3c, respectively, using equation (8). Because equation image increases exponentially with z, equation image also increases exponentially with z. Thus, equation image increases by ∼108 for GWs propagating from the troposphere to the lower thermosphere. After a GW reaches zdiss, equation image decreases rapidly if λz decreases and decreases less rapidly if λz continues to increase.

[11] The GWs shown in Figure 3c have λH = 400 km and intrinsic periods of τIr = 2π/ωIr ∼ 35 min, which are typical of the aurorally generated GWs observed by Bristow et al. [1996]. For equation image = 1000–1500 K, Figure 3c shows that this GW dissipates at zdissequation image 200–210 km, with λzequation image 80–100 km at zdiss. However, this GW is likely to be observed up to z ∼ 250 km because equation image is still reasonably large there. Bristow et al. [1996] examined this GW using the nondissipative and dissipative dispersion relations of Hines [1960] and Francis [1973], respectively with equation image = 1200. They found that at z ∼ 210 km, λz ≃ 150 km and 160 km, respectively (see Figures 7 and 8 from Bristow et al. [1996]). In addition, the attenuation distance decreased rapidly only above z ≃ 230 km. Thus, Francis' dissipative dispersion relation may result in a larger λz and a larger zdiss than the dissipative dispersion relation used here.

[12] In deriving this dispersion relation, we made several assumptions. The first is that equation image, the horizontal winds, and ν change “slowly enough” (equation (8) of the work by VF2006). We showed in the work of VF2006 that for most of the GWs originating in the lower atmosphere, the temperature profiles employed here do change slowly enough. We also found that even for very steep wind gradients which do not satisfy the “slowly enough” condition, the ray-trace solutions agree with the exact solutions. Therefore the “slowly enough” criteria may be overly restrictive. The slowly enough condition for ν was estimated to be λzequation image 2π[(dν/dz)/ν]−1equation image 2π (0.71equation image−1dequation image/dz + H−1)−1. However, the ray-trace solutions displayed in Figure 2c in the work of VF2005 showed that this condition is overly restrictive. Therefore the slowly enough condition for ν we adopt here is

equation image

although this condition may also be overly restrictive. Here λz(zdiss) is the vertical wavelength of a GW at zdiss. For the GWs shown in Figure 3a in temperature profiles II and V, 2π[(dν / dz) /ν]−1 at zdiss is ∼130 and ∼285 km, respectively, while λz(zdiss) ∼ 65 and 100 km, respectively. Therefore ν changes slowly enough for these GWs.

[13] The second assumption we made is that the WKB approximation is satisfied while ray tracing. Einaudi and Hines [1970] showed that the WKB approximation is valid as long as the residue is much less than one, where the residue is defined in our notation as

equation image

When R2 > 1, the WKB approximation fails because the solution cannot be written as a single upgoing or downgoing GW. This occurs if dissipation causes an upward propagating GW to partially reflect downward [Midgley and Liemohn, 1966; Yanowitch, 1967; Volland, 1969]. Those portions of the ray solutions where R2 ≥ 1 are shown in Figure 3 between the small black circles. For these GWs, the solutions fail well after they dissipate.

3. Properties of Gravity Wave Propagation and Dissipation

[14] In this section, we show many properties of dissipating GWs, such as dissipation altitudes, range of allowed vertical wavelengths, horizontal distances traveled prior to dissipation, and total time taken to travel these distances. These GWs propagate upwards from launch altitudes of zi = 0 (i.e., the lower atmosphere), zi = 120 km (approximate auroral excitation altitude), zi = 150 km, and zi = 180 km (approximate thermospheric body force altitude). The launch altitude zi = 180 km is utilized because it is the approximate altitude where convectively generated GWs dissipate in the thermosphere and create thermospheric body forces (VF2006). These thermospheric body forces likely generate medium and large-scale secondary GWs, and thus may be a new source of MSTIDs and LTIDs which occur during geomagnetically quiet and active conditions.

[15] Because the intrinsic properties of a GW determines its dissipation altitude (VF2005), and because we are only interested in exploring general properties of GW dissipation here, we do not include background winds. However, many of our results are valid when backgrounds winds are included, such as the dissipation altitudes and maximum vertical wavelengths achieved prior to dissipation, as long as the intrinsic GW properties at and somewhat below the dissipation altitudes are utilized.

[16] In Figure 4, as functions of the horizontal wavelength λH ≡ |2π/kH| and the initial vertical wavelength λz(zi), we show the dissipation altitudes, zdiss, for upward-propagating GWs as pink dash lines, and the maximum GW vertical wavelengths prior to dissipation, λz(zmax), as blue solid lines. Green dot lines show the intrinsic GW periods (at and near the dissipation altitudes), τIr = 2π/ωIr. Each GW's horizontal wavelength is constant with altitude here because we are not allowing for horizontal variations in the background densities, temperature, etc. [Lighthill, 1978]. GWs launched from zi = 0 are shown in the left column, while those GWs launched from zi = 120 km are shown in the right column. The top to bottom rows correspond to temperature profiles II, III, IV, and V, respectively. We also show those GWs which violate equation (9) as dark blue shading, those GWs with 2π[(dν/dz)/ν]−1 < λz(zdiss) < 4π[(dν/dz)/ν]−1 with aqua shading, and those GWs with λz(zdiss) > 4π[(dν/dz)/ν]−1 as dark green triangular-shaped shading on the left-hand side of each plot. These latter GWs have very large ωIr, causing them to reflect in the hotter thermosphere; however, they dissipate prior to reflecting, so that λz(zdiss) is very large. We also show those GWs with R2 > 1 after they reach zdiss but before their momentum flux amplitudes decrease by a factor of 2 with light pink-grey shading. Because a GW's momentum flux decreases above zdiss, the results in the light pink-grey shaded regions are likely reasonably accurate. However, we discard the results in the dark blue and dark green shaded regions.

Figure 4.

Dissipation properties of GWs when launched from zi = 0 (left column) and zi = 120 km (right column) as functions of λH and λz(zi). Pink dash lines show GW dissipation altitudes, zdiss, in 25-km intervals. Blue solid lines show maximum vertical wavelengths prior to dissipation, λz(zmax), in 50-km intervals. Green dot lines show the GW intrinsic periods, τIr, at 10, 20, 30, 40, 50, 60, 80, 100, and 180 min.

[17] General features of Figure 4 include (1) GWs launched from zi = 0 with λz(zi) > 50 km and λH ∼ 100–400 km dissipate at the highest altitudes, (2) GWs launched from zi = 120 km with λz(zi) > 100 km and λH ∼ 150–600 km dissipate at the highest altitudes, and (3) GWs penetrate to higher altitudes during active solar conditions than during extreme solar minimum. We show how to utilize the results from Figure 4 by employing an example. Imagine observing an upward-propagating GW at z ∼ 130 km that has λH = 200 km and an apparent (ground-based) horizontal phase speed of cH = 83 m s−1. Using the relation cH = ωr/kH, where ωr is the ground-based frequency, the ground-based period τr = 2π/ωr is calculated to be 40 min. If we neglect background winds, then the intrinsic period is τIr = 40 min. If the thermospheric temperature is 600 K, we use temperature profile II. Assuming that this GW originates in the lower atmosphere, we view Figure 4a. Locating the “λH = 200” tick mark on the x axis and drawing an imaginary vertical line, this line intersects the 40-min intrinsic period green dot line when λz(zi) equation image 25–30 km, the dissipation altitude is zdiss ∼ 150 km, and the maximum vertical wavelength prior to dissipation, λz(zmax), is λz(zmax) ∼ 30 km. A similar result is obtained if this GW was excited at zi = 120 km instead (see Figure 4b), and similar results are obtained if equation image = 1000, 1500 or 2000 K (see Figures 4c–4h). For all temperature profiles, zdiss ∼ 140–150 km and λz(zmax) ∼ 25–30 km for this GW. Because λz is small, zdiss does not depend sensitively on the temperature profile because this GW dissipates in the lower thermosphere where the different temperature profiles have similar values. Additionally, λz(zmax) is not very different from λz(zi) because λz(zi) is relatively small and τIr > 15–20 min.

[18] We now show how to utilize the results from Figure 4 if the background winds are known. The intrinsic frequency of a GW is

equation image

where U and V are the background zonal and meridional wind components, respectively, and ωr is the ground-based GW frequency. If eastward and westward propagating GWs with λx = 200 km and ground-based periods of τr = 2π/ωr = 40 min are observed propagating at z ∼ 100 km in an eastward zonal wind of U = 50 m s−1, the eastward propagating GW is Doppler shifted to a smaller intrinsic frequency while the westward propagating GW is Doppler shifted to a larger intrinsic frequency. Using equation (12) and k = ±2π/λx = ±3.14 × 10−5 m−1, the intrinsic frequencies of the eastward and westward propagating GWs are ωIr = ωrkU = 2π/(2400 s) ∓ 0.0016 s−1 = 0.001 and 0.0042 s−1, which implies intrinsic periods of τIr = 2π/ωIrequation image 100 and 25 min, respectively. Assuming a thermospheric temperature of equation image = 1000 K and a constant eastward wind of U = 50 m s−1 above zequation image 100 km, we use Figure 4c to estimate the dissipative properties of these GWs. Locating the λH = 200-km tick mark on the x axis and drawing a vertical line, the intersection of this line with the green dot “100”-min and (interpolated) “25”-min lines yield zdiss ∼ 110–115 km and λz(zmax) ∼ 10–15 km for the eastward-propagating GWs, and zdiss ∼ 180–190 km and λz(zmax) ∼ 55–65 km for the westward-propagating GWs, respectively. (Note that when background winds are included, the initial vertical wavelengths on the y axis are not correct and should not be utilized.) Therefore the influence of background winds can be large, allowing much deeper penetration for those GWs moving against the wind with larger intrinsic frequencies (but with ωIr < N to avoid reflection) than those GWs moving into the wind with smaller intrinsic frequencies. In this example, the winds above the observing altitude were assumed constant. If the winds decrease above z ≥ 100 km however, the intrinsic periods of the westward-propagating GWs will increase, thereby decreasing their dissipation altitude. Therefore, in order to accurately calculate dissipation altitudes using Figure 4, the intrinsic period near or at the dissipation altitude must be known.

[19] In addition to displaying zdiss and λz(zmax), Figure 4 displays the range of vertical wavelengths each GW has for zzdiss (excepting the decrease of λz in the mesosphere and lower thermosphere). For example, an upward-propagating GW excited by auroral heating at zi = 120 km in temperature profile V with λH = 600 km, λz(zi) = 80 km, and τIr ∼ 40 min has a range of vertical wavelengths of λz(z) ≃ 80–125 km along its raypath up to z = zdiss, using Figure 4h.

[20] In addition to vertical aspects of GW dissipation in the thermosphere, there are horizontal aspects as well because these GWs travel horizontally and vertically simultaneously. In Figure 5, we show the horizontal distances traveled from the launch location until the GWs dissipate, Xdiss, as reddish-brown dash lines, the intrinsic horizontal phase speeds, cIHωIr/kH, as yellow solid lines, and the total times taken to reach zdiss from the launch location, tdiss, as blue dot lines for the same internal, upward-propagating GWs shown in Figure 4. The columns, rows, and shading are the same as in Figure 4. The launch altitude zi = 120 km can either be utilized as the altitude where the GWs are created or as the altitude where the GWs (or TIDs) are observed propagating within the thermosphere. As an example of the latter usage, if an upward propagating GW (excited at any altitude below 120 km) is observed at z ∼ 120 km with λH = 400 km and cIHequation image 200 m s−1 in a thermosphere with equation image = 1000 K, then from Figure 5d, the time taken to dissipate from that altitude is tdiss ∼ 1 hour, and the horizontal distance traveled during that time is Xdiss ∼ 500 km. We can determine the dissipation altitude as well by locating the corresponding value of λz(zi) on the y axis in Figure 5d, locating this same value in Figure 4d, then finding where it intersects the λH = 400 km vertical line in Figure 4d. In this case, λz(zi) ≃ 60 km, so this GW dissipates at zdiss ≃ 200 km.

Figure 5.

Dissipation properties for the same GWs shown in Figure 4. Reddish-brown dash lines show the horizontal distances traveled while these GWs propagated from z = zi to z = zdiss, Xdiss, in 1000-km intervals up to 6000 km (left column) and 250-km intervals (right column), as labeled. Blue dot lines show the times taken to dissipate, tdiss, at 2, 4, 6, 8, 10, and 20 hours (left column) and in 0.5 hour intervals (right column). Yellow solid lines show the intrinsic horizontal phase speeds, cIH, in 50 m s−1 intervals.

[21] What limits a GW's ability to propagate large distances horizontally is the duration of time it propagates vertically before dissipating, which is partially determined by its vertical group velocity cg,z = ∂ωIr/∂m. If the same GW in the previous example has twice the horizontal wavelength instead (i.e., λH = 800 km), then the time taken to dissipate from zi = 120 km is longer, tdiss ∼ 100 min, and the horizontal distance traveled during that time is also longer, Xdiss ∼ 850 km, even though the dissipation altitude is lower, zdiss ≃ 180 km. This occurs because the former GW has a larger ωIr than that of the latter GW and therefore propagates upwards more quickly because of its faster vertical group velocity cg,zλzωIr/2π. Here λz(zi) is approximately the same for both GWs. The larger cg,z for the former GW shortens tdiss and therefore shortens Xdiss, since the horizontal group velocity, cg,H = ∂ωIr/∂kHλzN/2π, is nearly the same for both GWs.

[22] Therefore upward-propagating, aurorally generated GWs which travel large horizontal distances prior to dissipating have small cg,z, large λH, and sufficiently large λz (in order that they penetrate into the lower thermosphere before dissipating). This effect is observed in the right-hand column of Figure 5; for λz(zi) > 40 km, cIH ≥ 150 m s−1, and λH ≥ 100 km, Xdiss is approximately proportional to λH (that is, if λH is larger, then Xdiss is larger). Additionally, for those GWs with large λH, those with the largest values of λz(zi) travel the largest horizontal distances. Comparing Figures 5 and 4, large λH generally corresponds to large τIr. Thus, GWs with large λH and τIr tend to travel large horizontal distances prior to dissipating. The result that long period, high phase speed GWs travel large horizontal distances before dissipating was shown previously [Richmond, 1978; Walker et al., 1988; Hocke and Schlegel, 1996]. For λH ≃ 1200 km, the largest horizontal distance traveled is Xdiss ≃ 1250 km for GWs launched from zi ≃ 120 km, using Figure 5. Extending these results out to λH ≃ 3000 km, the largest horizontal distances traveled are instead double, Xdiss ≃ 2500 km (not shown). These results are consistent with the observational result that TIDs tend to propagate less than a quarter of the way around the Earth (which is about 10,000 km). However, downward-propagating GWs excited in the thermosphere may reflect from the Earth and propagate upwards into the thermosphere prior to dissipating, therefore traveling much longer horizontal distances than those shown here. Although we are not considering Earth-reflected GWs here, they can be important (see below).

[23] The left columns of Figures 4 and 5 show that small- and medium-scale GWs with λH < 500 km, cIH ≥ 100 m s−1, and τIr < 60 min propagate less than ∼2000 km horizontally from the ground prior to dissipating at zdiss ∼ 150–250 km. This suggests that if GWs from a tropospheric source propagate into the thermosphere, the corresponding MSTIDs will not be observed more than ∼2000 km from this source. This result is consistent with those of Waldock and Jones [1987], who showed that most MSTIDs could be reverse ray-traced back to tropospheric altitudes within 250–1500 km horizontally of the observation location. Our result for zdiss is also consistent with the mostly daytime observations of Waldock and Jones [1986], who observed medium-scale GWs in the F region with horizontal phase speeds of 100–200 m s−1 and periods of 10–30 min.

[24] GWs excited from the aurora and observed by oblique HF radar have ground-based periods of τr ≃ 20–50 min, λH ≃ 200–450 km, and cH = 100–200 m s−1 and are often observed propagating more than 1000 km from their source region 2000 km away [Bristow et al., 1996]. Many of these GWs are believed to be Earth-reflected [Samson et al., 1989, 1990; Bristow et al., 1994]. The right columns of Figures 4 and 5 show that if these medium-scale GWs are upward-propagating, they would not propagate more than 600 km from the source at zi = 120 km. However, if these GWs instead propagated downward, reflected at the Earth’s surface, then propagated upwards, they would travel horizontally ∼2 × (1000–2000) km ∼ 2000–4000 km prior to dissipating in the thermosphere (see the left column of Figure 5). Therefore our estimate based on this new dispersion relation is consistent with these observational results.

[25] As discussed previously, we also consider the propagative and dissipative properties of upward-propagating GWs launched from higher altitudes within the thermosphere. For upward-propagating GWs that are launched from zi = 150 km (left column) and zi = 180 km (right column), Figure 6 shows the dissipation altitudes and maximum vertical wavelengths, while Figure 7 shows the horizontal distances traveled and the total time taken to reach zdiss from zi. We use the same linetypes, colors, and shading as in Figures 4 and 5. As before, the top to bottom rows correspond to temperature profiles II, III, IV, and V, respectively. Note that Volland [1969] showed that reflection is negligible for a GW with λx ≃ 300 km and wave period of 21 min. This agrees with our results, as this GW is outside the light pink-grey shaded regions in Figure 6.

Figure 6.

Same as in Figure 4 but for GWs launched from zi = 150 km (left column) and zi = 180 km (right column). Pink dash lines show zdiss in 25-km intervals, blue solid lines show λz(zmax) in 50-km intervals, and green dot lines show τIr in minutes with the same values as in Figure 4.

Figure 7.

Same as in Figure 5 but for GWs launched from zi = 150 km (left column) and zi = 180 km (right column). (a, d) Reddish-brown dash lines show Xdiss in 200-km intervals, (b) 100-km intervals, (c, e–h) and 250-km intervals, as labeled. Blue dot lines show tdiss in (a, c–h) 0.25-hour intervals and (b) 0.1-hour intervals, as labeled. Yellow solid lines show cIH in 100 m s−1 intervals.

[26] Figures 6 and 7 are utilized the same way as Figures 4 and 5 to obtain zdiss, λz(zmax), Xdiss, and tdiss. As before, the launch altitudes zi can either be utilized as the altitudes where the GWs are created or as the altitudes where upward-propagating GWs are observed. As an example, consider an upward-propagating GW with λH = 1000 km and τIr = 60 min, which is observed at z = 150 km in a thermosphere with equation image = 1000 K. Using Figures 6c and 7c, this GW dissipates at zdiss ∼ 210 km, has a range of vertical wavelengths λz = 100–120 km for 150 km ≤ z ≤ 210 km, has an intrinsic horizontal phase speed cIH ∼ 275 m s−1, travels horizontally Xdiss ∼ 600–650 km before dissipating, and takes tdiss ∼ 1 hour to dissipate from zi = 150 km.

[27] Figure 6b shows that many GWs launched from zi = 180 km dissipate within a scale height during extreme solar minimum, with zdiss ≃ 250 km being the maximum attainable altitude. During very active solar conditions, however, the maximum attainable altitude from the same launch altitude is much higher, up to zdiss ∼ 450 km (see Figure 6h). However, even during very active solar conditions, those GWs with λz(zi) equation image 100 km dissipate just above zi = 180 km. For example, a GW with λH ∼ 900 km and τIr ≃ 60 min dissipates at zdiss ∼ 200 km regardless of its launch altitude (see Figures 4h, 6g, and 6h).

[28] For GWs launched from the thermosphere, those with large λH and λ(zi) travel the largest horizontal distances. As before, GWs with larger τIr tend to propagate larger horizontal distances Xdiss than GWs with smaller τIr. Therefore, because of wave dissipation in the vertical direction, as a GW packet propagates away from an auroral source, GWs with smaller τIr tend to be spectrally filtered out, causing the dominant wave period of the wave packet to increase with time and distance away from the source. This result is consistent with observational and theoretical results of Earth-reflected GWs [Bristow and Greenwald, 1996].

[29] One of the striking features of Figures 4, 5, 6, and 7 is that lines of constant τIr are oriented in a similar manner to lines of constant tdiss for GWs with large enough cIH. For GWs launched from zi = 120 km with τIr ∼ 20–100 min and cIH > 100 m s−1, tdiss/τIr ∼ 1–2. For GWs launched from zi = 180 km with τIr ∼ 20–60 min and cIH > 300 m s−1, tdiss/τIr ∼ 0.5–1.5. This implies that upward-propagating GWs generated in the thermosphere will typically only cycle through one half to two wave cycles prior to dissipating. Thus, upward-propagating GWs generated in the thermosphere may appear quasiperiodic, with significantly decreasing amplitudes over a wave cycle or two.

[30] Shiokawa et al. [2006] observed quasiperiodic southward-moving waves in their OI 630-nm airglow images at Kototabang, Indonesia (altitude range 200–300 km), which they argued may have been GWs. Typical waves had τr ≃ 40 min and cH ≃ 310 m s−1. From Figures 4 and 5, a GW with cH ≃ 310 m s−1 and τIr ∼ 40–50 min cannot originate in the lower atmosphere unless the horizontal winds are >60 m s−1, since cIH = cHUH from equation (12). Here UH is the component of the horizontal background wind along the propagation direction of the GW. Alternatively, GWs with τIr ≃ 40 min and cIH ≃ 310 m s−1 can be excited in the thermosphere. Since the observed GWs do not correlate with the Kp index, it is unlikely that these waves were excited by geomagnetic processes. Because the waves observed in this study were most frequently observed in May to July, which is the Asian monsoon season, and because the waves are medium to large scale, it is possible that during this time, small-scale GWs excited from the monsoon dissipated in the thermosphere, creating thermospheric body forces at z ∼ 150–250 km in a manner similar to that discussed in the work of VF2006. These body forces then would have excited medium and large-scale secondary GWs with periods and wavelengths similar to those observed here. Regardless of the source, if this GW was excited at zi = 150–180 km, and if we assume that the background winds were small, then τIr ≃ 40 min and cIH ∼ 310 m s−1. Then using Figures 6 and 7, λH ∼ 700 km, λz(zi) ∼ 100–120 km, λz(zmax) ≃ 110–160 km, and zdiss ∼ 200–250 km. A larger (smaller) intrinsic frequency (because of background winds) results in higher (lower) dissipation altitudes. Because this GW's cIH is large, only background winds ≥50 m s−1 can appreciably change zdiss. This is possible, however, because horizontal thermospheric winds up to 100–200 m s−1 can occur daily. Note that these dissipation altitudes are consistent with the observation that these GWs were not observed equatorward of the anomaly (i.e., at higher altitudes) [Shiokawa et al., 2006].

4. Spectra of Dissipating Gravity Waves With Altitude

[31] In the last section, we presented key dissipation parameters for a wide variety of individual, upward-propagating GWs in the thermosphere. Because GWs dissipate at differing altitudes in the thermosphere, with GWs having smaller (larger) λz dissipating at lower (higher) altitudes, a GW packet will be dissipatively filtered as it travels upwards in the thermosphere, shifting to larger λz as it propagates. We combine the results of individual GWs in this section, in order to determine if there are general trends for λz, λH, and τIr with altitude for the dissipating GWs. This also allows for comparison with observations.

[32] Figure 8 shows λz(zdiss) as a function of the dissipation altitudes, zdiss, using the results for all of the launch altitudes from Figures 4 and 6 for GWs with λH = 20, 50, 100, 200, 300, 400, 1000, 1500, and 2000 km. The squares, diamonds, triangles, and Xs denote GWs which propagate within temperature profiles II, III, IV, and V, respectively. We do not include those GWs which violate equations (9) and (10), dissipate within H/2 of the launch altitude or have intrinsic periods τIr > 6 hours. This last condition eliminates large-scale, shallow GWs which dissipate near the turbopause (for example, dissipation altitudes less than z < 115 km when λH = 2000 km in Figure 8i). We see that λz(zdiss) increases linearly with zdiss for a fixed value of λH. This increase occurs as ωIr and λz(zi) increase for differing GWs in the spectrum. If ωIr is near N (as occurs for some GWs with λH ≤ 400 km and large zdiss), then λz(zdiss) increases rapidly with zdiss and appears to reach “altitudinal ceilings” that are different for each value of λH. This latter behavior does not occur for GWs with λH ≥ 1000 km because ωIrN (see Figures 4 and 6). Note that this rapid increase of λz with z occurs mainly within the aqua regions in Figures 4 and 6.

Figure 8.

GW vertical wavelengths at the dissipation altitudes, λz(zdiss), as a function of the GW dissipation altitudes, zdiss, for GWs with λH = (a) 20, (b) 50, (c) 100, (d) 200, (e) 300, (f) 400, (g) 1000, (h) 1500, and (i) 2000 km. Each plot includes all of the launch altitudes zi = 0, 120, 150, and 180 km. Results for temperature profiles II, III, IV, and V are shown as squares, diamonds, triangles, and Xs, respectively. The solid lines show the solutions of the dissipation condition, equation (13). These lines are not visible in Figures 8a–8c because they overlap with the symbols.

[33] Although the curves for differing thermospheric temperatures are similar for fixed λH, they are not identical; when the thermosphere is hotter, GWs with the same values of λH and λz(zdiss) typically dissipate at somewhat higher altitudes than when the thermosphere is cooler, especially when ωIr nears N. For example, GWs with λH = 300 km and λz(zdiss) equation image 200 km dissipate at z ∼ 250 km during extreme solar minimum and at z ∼ 300 km during active solar conditions (see Figure 8e). This behavior can be explained theoretically. In the work of VF2006, we derived an approximate expression for GWs which dissipate from kinematic viscosity and thermal conductivity. This dissipation condition is

equation image

which is solved iteratively for the absolute value of the vertical wave number ma = 2π/λz:

equation image

where the first guess for ma on the right-hand side of equation (14) is

equation image

and succeeding results are substituted in as guesses until convergence in ma is obtained (equations (23), (24), and (25) from the work of VF2006). These results are shown in Figure 8 as solid lines. The dissipation condition is seen to agree with the ray-trace results very well; the increase of λz with z in the linear and rapidly increasing regimes are reproduced, as well as the increase of zdiss for increasing equation image when λH and λz(zdiss) are fixed. From equation (13), the dissipation condition implies that νH/N is approximately constant for GWs with the same m and kH and which satisfy m2k2 + 1/4H2. Because of the exponential dependence of ν on z/H, the variation of ν is more important than the variations of H or N. Since ν is much smaller in a hot than in a cool thermosphere at the same altitude, we estimate that GWs with the same kH and λz(zdiss) dissipate at somewhat higher altitudes in hot than in cool thermospheres. This is the observed behavior in Figure 8.

[34] For all of the small, medium, and large-scale GWs displayed in Figure 8, we show λz(zdiss), λH, and τIr for these dissipating GWs as functions of zdiss in Figure 9. Figure 9a shows that the λz(zdiss) versus zdiss curves are quite similar for GWs with differing values of λH and ωIrN. This implies a general relationship between λz(zdiss) and zdiss for GWs with ωIrN, regardless of λH (note that GWs with ωIrN have much larger λz(zdiss) at any given altitude). We also see that overall, λz(zdiss) increases exponentially with altitude, although with a smaller slope for z > 200 km. Figure 9b–9c shows that the highest dissipation altitudes of z ∼ 400–500 km are achieved only for GWs with horizontal wavelengths of λH ∼ 400–2000 km and intrinsic periods of τIr ∼ 10–50 min. Additionally, GWs with small horizontal wavelengths of λH ≲≲ 20 km do not dissipate above z ≃ 150 km. Finally, there is a clear filtering with respect to intrinsic wave period with altitude; GWs with intrinsic periods of τIr ≤ 7 min and τIr ≥ 300 min do not dissipate above z ≃ 150 km because of reflection and dissipative filtering, respectively.

Figure 9.

(a) Vertical wavelengths λz(zdiss) as a function of zdiss for all of the dissipating GWs shown in Figure 8 at all launch altitudes. (b) Horizontal wavelengths λH as a function of zdiss for the dissipating GWs in Figure 9a. (c) Intrinsic periods as a function of zdiss for the dissipating GWs in Figure 9a. The symbols are the same as in Figure 8.

[35] The exponential increase of λz as a function of altitude seen in Figure 9a has been noted up to altitudes of 240 km [Oliver et al., 1997]. In Figure 10, we reprint Figure 10 from the work of Oliver et al. [1997], which shows observed daytime GW vertical wavelengths as a function of altitude measured by the MU radar system in Japan. Note the logarithmic x axis and linear y axis (as in our Figure 9a).

Figure 10.

Figure 10 from the work of Oliver et al. [1997] showing the observed GW vertical wavelengths as a function of altitude for the MU radar in Japan. Reproduced by permission of the American Geophysical Union.

[36] In Figure 11, we show the GW dissipation altitudes, zdiss, binned as a function of the vertical wavelengths, λz(zdiss), for the same GWs shown in Figure 8. Here we show the results for all four temperature profiles separately, with each plot including all four launch altitudes. As discussed previously, higher penetration altitudes are achievable for GWs propagating in hotter thermospheres. For temperature profiles II, III, IV, and V, the highest dissipation altitudes are 275, 350, 400, and 500 km, respectively. These highest altitudes are only obtained for GWs launched from zi = 180 km rather than from zi = 0 (see Figures 4 and 6). We also show the binned and averaged values of (zequation image/2zdiss)/H in Figure 11 with grey-scale shading, where white (dark grey) bins denote values of zero (one). Here zequation image/2 is the altitude above zdiss where equation image decreases by a factor of two from its maximum at zdiss. Because most of the bins show values of (zequation image/2zdiss)/Hequation image 0.5–1, most GWs reach zequation image/2 approximately equation image − 1 scale heights above zdiss. Therefore we expect most GWs to be observable up to ∼1–2 scale heights above zdiss, i.e., to altitudes of zzdiss + (1–2)H. For example, an aurorally generated GW with λH = 400 km and τIr = 35 min in temperature profile III has a maximum momentum flux (i.e., dissipates) at zdiss = 200 km (see Figure 3c). Because H ≃ 30 km at that altitude (see Figure 2c), this GW is expected to be observable up to altitudes of z ∼ 240–280 km. This is verified for this GW in Figure 3f.

Figure 11.

GW dissipation altitudes, zdiss, binned as a function of the vertical wavelengths, λz(zdiss), for the GWs shown in Figure 8. GWs in temperature profiles (a) II, (b) III, (c) IV, and (d) V. The grey-scale shading indicates the average value of the quantity, (zequation image/2zdiss)/H, binned in intervals of 0.125 from 0 to 1.0, as indicated by the color bar in the lower right hand corner of each plot.

[37] We overlay in Figure 11 the observational results of Oliver et al. [1997] for z ≥ 80 km, boxes D and E from Figure 10, as dash boxes. We also overlay the observational results of Djuth et al. [1997, 2004], who observed GWs with λzequation image 4–50 km for z = 115–160 km and λz ∼ 100–300 km for z = 170–500 km at Arecibo Observatory. We plot these results as dash-dot boxes. The observational results agree with the theoretical results fairly well for thermospheric temperatures of equation image ≃ 600–1000 K. Finally, we overlay the solid line in Figure 10 (the “fit” line from the work of Oliver et al. [1997]) as long dash lines in Figure 11. Although this line matches the slope well for GWs with λz ≤ 100 km and zdiss < 200 km, it does not match the slope well for GWs with larger vertical wavelengths when z > 200 km, because the growth of λz with z when z > 200 km is slower than this line implies. Indeed, the slower increase of λz with z can even be seen in Figure 10 because this line cuts through only the lower right-hand portion of box E.

[38] In Figure 12, we show λz(zdiss) as a function of zdiss binned into shaded boxes for the GWs from Figure 11b. The grey-scale shading shows the average horizontal phase speed for the GWs in each bin. We also overlay the observational results of Oliver et al. [1997] and Djuth et al. [1997, 2004] as dash and dash-dot boxes, respectively.

Figure 12.

Binned vertical wavelengths at the dissipation altitudes, λz(zdiss), as a function of the binned dissipation altitudes, zdiss, for GWs propagating in temperature profile III, as shaded boxes. (a) GWs with λH = 20, 50, 100, 200, 300, 400, 1000, 1500, and 2000 km launched from zi = 0, 120, 150, and 180 km. (b) GWs with λH = 20, 50, 100, 200, 300, and 400 km launched from zi = 0 km. (c) GWs with λH = 1000, 1500, and 2000 km launched from zi = 120 km. (d) Same as Figure 12c but launched from zi = 150 km. (e) Same as Figure 12c but launched from zi = 180 km. (f) GWs with λH = 20, 50, 100, 200, 300, 400, 1000, 1500, and 2000 km launched from zi = 180 km. The grey-scale shading of each box shows the average value of |cIH| for this bin on a linear scale, with a maximum value of 600, 300, 400, 600, 600, and 600 in Figures 12a–12f, respectively.

[39] Figure 12a shows the results for GWs with all values of λH and for all four launch altitudes. The agreement between observation and theory is generally very good, as mentioned previously. At z ∼ 80–120 km, the agreement is very good except for those GWs with λz ∼ 20–80 km, which were observed but which do not appear in the theoretical boxes. This deficit occurs for several reasons. The first is that we limited our results to those GWs with intrinsic periods smaller than 6 hours; however, large-scale GWs with λH ∼ 1000 km, τIr > 6 hours, and λzequation image 50 km, for example, likely dissipate at z ∼ 90–110 km (see Figure 8g). The second reason is that the theoretical boxes in Figure 12 only display GWs which are dissipating. GWs with vertical wavelengths larger than 20 km (but that dissipate at higher altitudes) can be observed at z ∼ 100–120 km if their amplitudes are large enough. Using a GW spectrum from a convective plume model, we show in the next section that GWs up to λz ∼ 70 km can be observed at z ∼ 80–100 km.

[40] For z = 115–160 km, the theoretical and observational results also agree very well in Figure 12a. For z = 170–200 km, the agreement between theory and both sets of observations is good for most GWs. However, the observations do not include GWs with λz = 30–100 km for z = 170–200 km that are predicted to be dissipating at these altitudes. This likely occurs because of spectral Doppler-shifting because of the presence of large thermospheric winds, as we show in the next section. Finally, for the z = 200–325 km altitude range, theory and observation agree quite well. Note that the GWs with the largest horizontal phase speeds of cH ≃ 300–600 m s−1 dissipate at the highest altitudes of z ≥ 200 km. Additionally, those GWs with the smallest horizontal phase speeds of cH < 50 m s−1 dissipate at z < 160 km with small vertical wavelengths of λz(zdiss) < 10 m s−1.

[41] Figure 12b shows the theoretical results for GWs with λH = 20–400 km launched from the lower atmosphere (note that this plot looks the same if all GWs with λH = 20–2000 km are included instead (not shown)). These small- and medium-scale GWs overlap reasonably well with observations at z ≃ 80–160. However, many of these GWs dissipate at z ∼ 170–200 km with λz = 30–100 km, seemingly inconsistent with observations. However, the background winds were assumed zero here. In a zero-wind environment, the vertical wavelengths of most GWs launched from the lower atmosphere do not become very large in the thermosphere because the increase of λz in the lower thermosphere mostly offsets the decrease of λz near the mesopause (however, λz for GWs with ωIrN increases significantly just prior to dissipation and reflection). Horizontal winds, however, can substantially alter λz. For GWs with m2 > kH2 + 1/4H2 and where dissipation is unimportant, the GW dispersion relation becomes

equation image

using equation (6). As −(kU + lV) increases (decreases), the intrinsic frequency increases (decreases), and λz increases (decreases). Additionally, larger (smaller) λz leads to higher (lower) dissipation altitudes (see Figures 4 and 6). We show how thermospheric horizontal winds can alter GW spectra in the next section using a simple convection model.

[42] We also mention another possibility as to why theory and observation can differ when observations infer λz from GW phase speeds. Using equation (16), the intrinsic phase speed of a GW is

equation image

If cH is used in equation (17) instead of cIH because the background winds are not known, then λz is calculated to be smaller (larger) than the GW's real vertical wavelength when the background wind is against (in) the GW's direction of propagation. In the lower thermosphere, horizontal winds are generally present and vary daily, with speeds up to 100–200 m s−1. Such large winds can substantially alter the intrinsic properties of observed GWs. As another example, GWs from convection dissipate in the thermosphere in the altitude range, z ∼ 180–200, creating body forces and large horizontal winds (VF2006). Because GWs which have not yet dissipated propagate through this region in the same direction as the induced horizontal winds, their intrinsic phase speeds are smaller than they would otherwise be. (GWs propagating in the opposite direction dissipate at lower altitudes). If these induced horizontal winds are neglected, then λz is calculated from equation (17) to be larger than the GW's true vertical wavelength.

[43] Figures 12c, 12d, and 12e shows the theoretical results for only those GWs with λH = 1000–2000 km launched from zi = 120, 150, and 180 km, respectively. The agreement with observations at z = 170–330 km is excellent for GWs launched from zi ∼ 150–180 km because GWs with λz(zi) < 100 km dissipate within H/2 of the launch altitude (see Figure 6) and thus are not included here. Because Figure 12e only includes large-scale GWs, and because thermospheric body forcings at z ∼ 180 km likely generate medium-scale GWs as well (VF2006), we show in Figure 12f the same results as in Figure 12e but for all GWs (small, medium, and large) launched from zi = 180 km. Again, the theoretical predictions agree very well with observations because the small and medium-scale GWs capable of propagating and dissipating away from the launch site have λz > 100 km, consistent with observations.

[44] Finally, we overlay the solid line in Figure 10 (the “fit” line from the work of Oliver et al. [1997]) as long dash lines in Figure 12. Although it matches the slope well for GWs with small to medium λz, it does not match the slope for GWs with large λz when z > 200 km. Excellent fits are obtained instead by the curved solid lines, which are the iterative solutions of our dissipation condition, equation (13), using temperature profile III and λH = 1500 km for all except Figure 12b where λH = 200 km is used instead. We also overlay the quenching criteria of Hines [1964] (dotted line),

equation image

Hines' quenching criteria underestimates the vertical wavelengths for zequation image 300 km because Hines' derivation is based on the value of λz when dissipation just starts to affect a GW and does not take into account the growth of λz above that altitude.

[45] Figure 13 shows the iterative solutions of the dissipation condition for GWs with λH = 20–2000 during active solar conditions. Although each solution is distinct because of the altitudinal “ceiling” which prevents further vertical penetration for GWs with a fixed value of λH, the upward trends of the solutions are similar, with only small altitudinal differences. This explains why the λH = 1500 km dissipation condition shown in Figure 12a, for example, is an excellent fit even in the lower thermosphere where small- and medium-scale GWs likely dominate. We see that λz(zdiss) grows exponentially with altitude, although with a smaller slope for z > 200 km. We also overlay Hines' quenching criteria in Figure 13.

Figure 13.

Dissipation condition, equation (13), shown as solid lines using temperature profile IV for GWs with horizontal wavelengths of λH = 20, 50, 100, 200, 300, 400, 1000, 1500, 2000 km. The dot line shows the quenching criteria of Hines [1964], equation (18), using the same temperature profile.

5. Spectral Evolution of GWs in the Thermosphere From Convection

[46] In section 4, we displayed GW dissipation curves with altitude for a wide range of GWs. However, we did not consider how GW spectra from specific sources might evolve with altitude. In particular, GWs may be observed at altitudes lower or higher than zdiss if their amplitudes are large enough. Additionally, background winds can alter the observed spectra significantly. In this section, we estimate the evolution of a GW spectrum which is generated from a single, deep plume in a local convection model, as described in the works of Vadas and Fritts [2004, 2006]. This plume (plume 8) is created from a vertical body force with full duration 15 min, full diameter ≃ 18 km, and full depth ≃ 12 km. This yields a convective plume with a maximum vertical updraft velocity of ∼6 m s−1. The GW momentum flux spectrum in flux content form is shown in Figure 14.

Figure 14.

GW momentum flux spectrum in flux content form modeled after a single, deep, convective plume (solid lines). The contours show the amplitudes in intervals of 1/7 of the maximum value.

[47] We now estimate how this GW spectrum evolves with altitude in a simple zero-wind environment using the ray-trace results from Figure 4. The spectral filtering is assumed to be kinematic viscosity and thermal diffusivity only; thus we are neglecting the filtering effects from eddy viscosity, ion drag, wave saturation, and wave breaking. Because equation image and m are only outputted at zi, zmax, zdiss, and zequation image/2 for these runs, some assumptions are needed in order to estimate equation image and m at all altitudes for each GW. Here zmax is the altitude below or at zdiss where λz is maximum and equals λz(zmax).

[48] 1. For zzmaxH, equation image is assumed to grow as equation image(zi)/equation image, and m is calculated from the nondissipative GW anelastic dispersion relation given by equation (6);

[49] 2. For each of the altitude ranges zmaxHzzmax, zmaxzzdiss, and zdisszzequation image/2, equation image and λz are linearly interpolated;

[50] 3. For zequation image/2zzequation image/2 + H, equation image is driven exponentially to 0 using the arbitrarily chosen function exp(−5(zzequation image/2)/H). Additionally, λz grows linearly with altitude with the same slope if λz(zequation image/2) > λz (zdiss); otherwise, λz is linearly interpolated to λz(zi);

[51] 4. For zzequation image/2 + H, equation image = 0 and λz = λz(zequation image/2 + H).

[52] The GWs in this spectrum have horizontal wavelengths from 10–3020 km in 10-km increments and vertical wavelengths from 5–309 km in 4-km increments and are launched from zi = 0 in the troposphere. We do not include those GWs which violate equations (9) and (10) or have intrinsic periods τIr > 6 hours. We show the resulting vertical wavelength (first row), horizontal wavelength (second row), intrinsic wave period (third row), and intrinsic horizontal phase speed (forth row) spectra as functions of altitude in Figure 15 for thermospheric temperatures 600 K (left column), 1000 K (middle column), and 1500 K (right column). The shaded boxes in Figure 15 show those GWs with amplitudes that are at least 90% of the maximum GW amplitude at that altitude, while the extended rectangular boxes with no shading show those GWs with amplitudes that are at least 25% of the maximum GW amplitude at that altitude. The grey-scale color of the shading indicates the value of β(z) = log10 of the maximum momentum flux amplitude at that altitude, divided by the largest value of β for all altitudes. Below z ∼ 125 km, GWs with amplitudes that are >90%, the maximum have λzequation image 20 km, while those GWs with amplitudes that are >25%, the maximum have λzequation image 70 km. Similar sensitivities occur in the horizontal wavelength and intrinsic period spectra. Therefore the particular GWs observed at a given altitude depends on the sensitivity of the observations, with more sensitive observations seeing a proportionately larger portion of the convective GW spectrum.

Figure 15.

Estimated GW spectra as a function of altitude for the GWs in Figure 14 launched from zi = 0 in temperature profiles II, III, and IV in the left, middle, and right columns, respectively. (a–c) Vertical wavelength spectra. (d–f) Horizontal wavelength spectra. (g–i) Intrinsic wave period spectra. (j–l) Intrinsic horizontal phase speed spectra. The values of the grey-scale shading are described in the text. The light to dark shading indicates the values 0–1.0 in intervals of 0.125, as shown by the color bar in the upper left-hand corner of each plot.

[53] In Figure 15, λz increases exponentially with altitude above z ∼ 125 km, although with a smaller slope for z > 200 km. This is because GWs with initially “undetectable,” small amplitudes are eventually detectable in the thermosphere, since their amplitudes grow exponentially with altitude and those GWs with larger initial amplitudes (but with smaller λz) are dissipatively filtered out of the spectrum. The exponential increase also occurs because each GW's λz increases with z as the temperature increases. We also see that λH increases rapidly with altitude as well for z > 125 km. The intrinsic periods, however, do not change appreciably with altitude. This is likely because typical GW periods from convection lie in the range of wave periods for GWs which can propagate to the highest altitudes (see Figure 9c). As expected, the GWs with significant amplitudes penetrate to the highest altitudes when the thermosphere is the hottest, z ∼ 225, 300, and 325 km in temperature profiles II, III, and IV, respectively. As a function of thermospheric temperature, the largest variations occur in the vertical wavelength spectra; from extreme solar minimum to solar maximum, the peak vertical wavelength of GWs with reasonably significant amplitudes doubles from λz ∼ 60 km to λz ∼ 120 km. The GWs which penetrate to the highest altitudes of z ∼ 300 km with amplitudes that are >25% of the maximum amplitude at that altitude in a equation image = 1000 K thermosphere, for example, are medium scale and have λz ∼ 50–300 km, λH ∼ 100–300 km, τIr ∼ 10–30 min, and cIH ∼ 100–250 m s−1. These large vertical scales, horizontal scales, and intrinsic phase speeds are not characteristic of the dominant convective scales in the initial GW spectrum. In contrast, those GWs at the peak of the initial convective spectrum have λz(zi) ∼ 15 km, λH ∼ 50 km, and cIH ∼ 50 m s−1 and dissipate at zdiss ∼ 130 km from Figure 4c.

[54] We overlay the dissipation condition, equation (13), for λH = 200 km for each temperature profile as solid lines in Figure 15a–15c. Because the dissipation condition predicts the altitudes at which GWs are dissipating, the actual GW spectra (consisting of dissipating and not-yet-dissipating GWs) are centered at smaller vertical wavelengths for a given altitude, especially at the highest altitudes. We also overlay the observational results of Oliver et al. [1997] and Djuth et al. [1997, 2004] as dash and dash-dot boxes, respectively. The observational results agree very well with this model spectrum for z = 80–170 km. For z = 170–250 km, however, many of these GWs do not have large enough λz to agree with these observations.

[55] Although GWs from convection in a zero-wind environment do not appear to have large enough λz at zequation image 170–250 km to agree with observations, strong background winds can substantially lengthen GW vertical wavelengths to better agree with observations. Here we consider the same convective GW spectrum shown in Figure 14 but with a sudden westward wind of U = −100 m s−1 above z ≥ 120. Using equations (6) and (12), we recalculate the vertical wavelengths of GWs at z = 120 km that have zmaxH > 120 km (i.e., that are not yet dissipating). Then we use the zi = 120 km launch solutions with the closest values of λz (zi) in order to determine the values of equation image and m at the new values of zmax, zdiss, and zequation image/2. Finally, we use the same approximate altitude ranges as before to estimate equation image and m as a function of altitude above z ≥ 120 km. The initial GW spectrum is approximated to be 2D, and we assume that it contains equal amounts of eastward and westward-propagating GWs.

[56] In Figure 16, we show the estimated vertical and horizontal wavelength, ground-based wave period, and ground-based horizontal phase speed spectra for GWs propagating in temperature profile III. The boxes and shading are the same as in Figure 15, with the shaded (unshaded) rectangular boxes showing the GWs with amplitudes that are 90% (25%) of the maximum amplitude at that altitude. Additionally, Figure 16a shows the dissipation condition with λH = 200 km (solid line) and the same Oliver and Djuth results shown in Figure 15a–15c. We see that the vertical wavelengths of GWs for z > 170 km are much larger here than in the zero-wind example because of the Doppler shifting of the eastward-moving GWs; in particular, there are no longer GWs with λz ∼ 20–100 km in the z ≃ 170–250 km altitude range. However, the horizontal wavelengths, ground-based periods, and horizontal phase speeds for the GWs reaching the highest altitudes of z ∼ 325 km (with significant amplitudes) at the 25% detection level are not very different from when there are no thermospheric winds, λH ≃ 100–300 km, τr ∼ 10–40 min, and cH ∼ 100–250 m s−1. The GW spectra at the highest altitudes are composed nearly entirely of eastward-propagating GWs in this example; those GWs propagating in the direction of the wind negligibly affect the spectra because they dissipate at lower altitudes where their amplitudes are smaller. Additionally, GWs moving perpendicular to the background wind (northward and southward in this example) would also negligibly affect the spectra if included because their vertical wavelengths would not change from the wind, so they would dissipate at lower altitudes with smaller amplitudes as well (VF2006).

Figure 16.

Estimated GW spectra as a function of altitude for convective GWs launched from zi = 0 in temperature profile III. A sudden shear of −100 m s−1 is assumed at z = 120 km. (a) Vertical wavelength spectrum. (b) Horizontal wavelength spectrum. (c) Ground-based wave period spectrum, τr. (d) Ground-based horizontal phase speed spectrum, cH. The shading is the same as in Figure 15.

[57] Figure 16a shows that the Doppler-shifted thermospheric GW spectrum agrees very well with observations for z ≥ 170 km. The result that modeled GW spectra from convection agree well with observations for z ≥ 170 km when strong thermospheric winds are present (and does not agree well for zero background winds) should not be surprising, as horizontal winds in the thermosphere tend to be strong because of diurnal and semidiurnal tides, with magnitudes of order ∼100–200 m s−1 [e.g., Roble and Ridley, 1994; Larsen, 2002; Larsen et al., 2003]. Therefore it is likely in general that lower atmospheric GW spectra are Doppler-shifted to larger λz in the direction opposite to prevailing background winds in the thermosphere. We emphasize that this model GW spectrum is a simple example that we assumed in order to gain a better understanding of the relationship between observed GW scales and theoretical predictions using this new anelastic dispersion relation. Further work using ray tracing with realistic temporally and spatially variable GW convective spectra and realistic background winds are needed for better comparison with observations.

6. Conclusions

[58] In this paper, we explored many properties of a new anelastic, GW dispersion relation which includes kinematic viscosity and thermal diffusivity in the thermosphere. We calculated the dissipation altitudes, range of vertical wavelengths, horizontal distances traveled, and time taken for GWs to travel until they dissipate for GWs with horizontal wavelengths of 10–3020 km and vertical wavelengths of 5–400 km, for four different temperature profiles from extreme solar minimum to very active solar conditions and for four different launch altitudes in the lower atmosphere and thermosphere. These results were shown and described in Figures 47. Because of the complexity of the results and because the dissipation altitudes and maximum vertical wavelengths only depend on the intrinsic wave properties, our calculations did not include background winds. However, we explained how background winds can be included to obtain GW dissipation altitudes and maximum vertical wavelengths when background winds are known. Therefore Figures 47 can be used as look-up figures if the approximate intrinsic GW properties at or near the dissipation altitudes are known or can be estimated. Note that the dissipation altitude, zdiss is defined as the altitude where a GW's momentum flux is maximum and is therefore not the maximum altitude attainable by a GW. Instead, we showed that GWs can be observed one to two density scale heights above zdiss. We also found that GWs generated in the thermosphere with large enough horizontal phase speeds will appear quasiperiodic because they typically only oscillate through one-half to two wave cycles before dissipating, depending on the launch altitude and thermospheric temperature.

[59] For GWs with the same horizontal wavelength λH, we found that λz at zdiss (i.e., λz(zdiss)) increases approximately linearly with altitude because of the increase of the initial vertical wavelengths λz(zi) in the GW spectra. However, if a GW's intrinsic frequency nears the thermospheric buoyancy frequency, then λz(zdiss) increases much more rapidly with zdiss. We combined our ray-trace results for GWs with differing λH and λz(zi) and found that as a whole, λz(zdiss) increases exponentially with zdiss, although with a smaller slope for z > 200 km. These results agree well with observational data of Oliver et al. [1997] and Djuth et al. [1997, 2004]. We found that GWs dissipating at the highest altitudes of z ∼ 400–500 km have horizontal scales λH ≃ 400–2000 km and intrinsic wave periods of τIr ≃ 10–50 min. Additionally, GWs with λH ≤ 20 km (commonly observed in airglow images near the mesopause) do not dissipate above z ∼ 150 km, and GWs with τIr ≤ 7 min and τIr ≥ 300 min do not dissipate above z ∼ 150 km as well. We also found that our dissipation condition, which calculates the estimated vertical wavelengths of dissipating GWs as a function of altitude [given by equation (13)], agrees with the ray trace results very well (solid lines in Figures 8, 12, 13, 15, and 16).

[60] Last, we employed a simple GW spectrum modeled after a deep plume in tropospheric convection to estimate the horizontal wavelength, vertical wavelength, and wave period spectra with altitude. We found that dissipative filtering caused the GW spectra to shift to increasingly larger horizontal and vertical scales while propagating upwards in the thermosphere, with differing portions of the initial GW spectra being important at differing altitudes within the thermosphere. The wave period spectrum, however, did not alter appreciably with altitude. This is likely because typical GW periods from convection lie in the range of wave periods for GWs which can propagate to the highest altitudes. At altitudes above z ∼ 135 km, the horizontal and vertical scales which dominate the GW spectra are not characteristic of the typical scales in the initial convective GW spectrum. This may be one of the reasons it has been difficult to trace GWs in the thermosphere back to specific small-scale tropospheric convective regions. The GWs with the largest amplitudes which penetrate to the highest altitudes of z ∼ 300 km in a equation image = 1000 K thermosphere, for example, have λz ∼ 100 km, λH ∼ 100–300 km, intrinsic wave periods of τIr ≃ 10–20 min, and intrinsic horizontal phase speeds of cIH ∼ 200 m s−1. We also estimated the GW spectra in the thermosphere for GWs excited from the same convective plume but which propagated through a shear of −100 m s−1 in the lower thermosphere. We found that the vertical wavelength spectra shifted to much larger λz ∼ 100–300 km for z ≥ 170 km, thereby agreeing well with observational results in this altitude range. We also found that those GWs penetrating to the highest altitudes have horizontal wavelengths and ground-based phase speeds that are similar to those when background winds are zero. Because convection may generate medium-scale GWs in the F region with horizontal phase speeds of cH ∼ 100–250 m s−1, because medium-scale TIDs with these characteristics are ubiquitous in the ionosphere, and because our results show that these GWs may travel up to 2000 km horizontally from their source prior to dissipating, our results suggest that some of these observed TIDs may be a direct result of convection.

Appendix A:: Molecular Weight and Ratio of Specific Heat Capacities

[61] The TIME-GCM model data used to calculate the “best fit” functions for XMW and γ as a function of the mean density used in this paper encompass 28 September 2004 to 27 October 2004 over Brazil, with latitudes of −22.5° to −2.5° and longitudes of −60° to −40° on a 5° grid. The TIME-GCM is a global mesospheric and thermospheric dynamics and chemistry model [e.g., Roble and Ridley, 1994]. Our analysis only utilized the model data every 3 hours. During this period, the minimum and maximum thermospheric temperatures were 740 and 1100 K, respectively. We show these temperature profiles in Figure A1a along with the temperature profiles we use in this paper. Our temperatures are consistent with the TIME-GCM temperatures in the lower thermosphere above the cold mesopause. Figure A1b shows the mean XMW for these minimum and maximum TIME-GCM temperature profiles, along with our “best fit” profile [equation (3)]. Figure A1c shows the best fit profile as well as all of the mean XMW profiles for this month. The best fit analytic function for XMW fits the model data very well. Figure A1d shows the mean density profiles for the minimum and maximum TIME-GCM temperature profiles shown in Figure A1a, as well as the density profiles we use in this paper. Our densities are consistent with the TIME-GCM mean densities in the lower thermosphere (at z ∼ 125 km). Figure A1e shows the ratio of the mean Cp to mean Cv profiles for the minimum and maximum TIME-GCM temperature profiles shown in Figure A1a, along with our “best fit” profile [equation (4)]. Figure A1f shows the best fit profile as well as all of the mean γ profiles for this month. Again, the best fit analytic function for γ fits the TIME-GCM model data very well.

Figure A1.

(a) Minimum and maximum TIME-GCM temperature profiles (dash and dash-dot lines, respectively). Boxes, diamonds, triangles, and Xs show our temperature profiles II, III, IV, and V, respectively. (b) Mean molecular weight profiles, XMW, for the minimum and maximum TIME-GCM thermospheric temperature profiles shown in Figure A1a as dash and dash-dot lines, respectively. The solid line shows the “best fit” distribution given by equation (3). (c) Mean molecular weight profiles, XMW, for all TIME-GCM model data (dash lines). Solid line same as in Figure A1b. (d) Same as in Figure A1a but showing the density profiles. (e) Ratio of mean specific heat capacities, γ, for the minimum and maximum TIME-GCM thermospheric temperature profiles shown in Figure A1a as dash and dash-dot lines, respectively. The solid line shows the “best fit” distribution given by equation (4). (f) Profiles of mean γ for all TIME-GCM model data (dash lines). Solid line same as in Figure A1e.

Acknowledgments

[62] This research was supported by the National Science Foundation under grants ATM-0307910 and ATM-0537311. The author would like to thank Ray Roble and Han-Li Liu for the TIME-GCM data and Frank Djuth for helpful comments regarding this manuscript.

[63] Zuyin Pu thanks Michael Hickey and Frank Djuth for their assistance in evaluating this paper.

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