Journal of Geophysical Research: Atmospheres

Interaction of the 2-day wave with solar tides

Authors


Abstract

[1] Nonlinear integrations with a three-dimensional model of the middle and upper atmosphere are used to study how the 2-day wave interacts with solar tides. At small 2-day wave amplitude, the modulation by tides is linear and, therefore, largely reversible. At large 2-day wave amplitude, however, the modulation leads to a nonlinear interaction that is irreversible. Local reinforcement, chiefly by the diurnal tide, introduces local instability and wave breaking. This nonlinear behavior results in a cascade of variance to small scales, at the expense of large-scale wave activity. By accelerating wave damping, this process yields slower amplification and the 2-day wave being limited to amplitudes at which nonlinear interaction prevails. At altitudes of strong tidal amplitude, regular propagation is disrupted, replaced by broadband behavior associated with secondary scales that are generated nonlinearly through eddy mixing. The same process influences solar tides. However, because they are continually forced, tidal amplitudes are reduced only modestly, by 10–20%. This contrasts with the 2-day wave, which, for sufficiently strong tides, is limited to small amplitude.

1. Introduction

[2] The 2-day wave is a prominent feature of the middle and upper atmosphere. Dominated by zonal wave number 3, this global disturbance propagates westward with a period of 2.1–2.3 days [Müller, 1972; Müller and Kingsley, 1974; Glass et al., 1975; Müller and Nelson, 1978; Rodgers and Prata, 1981; Wu et al., 1993, 1996]. A peak at the same period is evident in spectra of sea level pressure [Hamilton and Garcia, 1996]. In the middle atmosphere, the 2-day wave introduces perturbations in the distributions of trace species [Limpasuvan and Wu, 2003; Limpasuvan et al., 2005]. Analogous behavior is observed in the thermosphere, where it is associated with geomagnetic variations in the ionospheric E layer and a dynamo current [Voiculescu et al., 2000; Yamada, 2002].

[3] The 2-day wave is characterized by eddy (perturbation) meridional velocity v′ that is in phase between the hemispheres and eddy zonal velocity u′ that is out of phase between the hemispheres. Jointly, those perturbations form a wave number 3 pattern of equatorial gyres that achieve strong cross-equatorial motion and propagate westward [Wu et al., 1993; Salby and Callaghan, 2000].

[4] The 2-day wave has been observed during much of the year, especially at low latitude where velocity perturbations are strong [e.g., Salby and Roper, 1980; Harris and Vincent, 1993; Wu et al., 1996; Thayaparan et al., 1997; Pancheva et al., 2004; Chshyolkova et al., 2005]. However, it amplifies sharply around solstice, contemporaneously in both hemispheres [Müller and Nelson, 1978; Craig et al., 1983; Tsuda et al., 1988].

[5] The robust nature of its period led to the 2-day wave being identified with the gravest planetary normal mode of wave number 3, the “Rossby-gravity mode” [Salby and Roper, 1980]. The Rossby-gravity mode propagates westward with a period of 2.1 days. It has eddy motion very similar to that observed of the 2-day wave, even when calculated in the presence of realistic wind shear [Salby, 1981a, 1981b; Hagan et al., 1993]. The normal mode is excited preferentially from broadband forcing, as characterizes baroclinic weather systems that punctuate the tropospheric circulation.

[6] On the other hand, the 2-day wave's semiannual amplification around solstice suggests auxiliary forcing, in association with instability of the summer jet [Plumb, 1983; Pfister, 1985]. This interpretation is supported by the zonal mean gradient of potential vorticity (PV). It reverses semiannually in the summer hemisphere, where it marks dynamical instability [Randel, 1994; Limpasuvan et al., 2000]. Neighboring the reversed gradient of PV is a divergence of EP flux, which represents wave overreflection or the emission of wave activity [Lieberman, 1999, 2002; Salby and Callaghan, 2000]. There, wave number 3 is attended by other wave numbers of similar phase speed [see also Rodgers and Prata, 1981; Garcia et al., 2005]. This is especially true in temperature, which is characteristic of eddy heat flux and, hence, of EP flux. During southern summer, when the 2-day wave is strongest, wave number 3 is dominant [Rodgers and Prata, 1981; Garcia et al., 2005]. During northern summer, it is attended by adjacent wave numbers, particularly wave number 4. Analogous behavior appears in a general circulation model [Norton and Thuburn, 1996]. Even there, however, wave structure in the presence of the reversed PV gradient is quite similar to that of the normal mode [also Hunt, 1981; Manzini and Hamilton, 1993; Palo et al., 1998].

[7] Although its amplification has the same seasonality as the reversal of PV gradient, the 2-day wave's appearance is not always contemporaneous with such reversals. The 2-day wave has been observed during other seasons, when the PV gradient remains positive. Even during solstice, the 2-day wave can appear when the PV gradient reverses only briefly, sometimes, not at all [e.g., Randel, 1994]. Indeed, 2-day wave amplifications at this time of year operate coherently with sporadic amplifications of planetary waves in the winter hemisphere [Wu et al., 1996]. A similar relationship is apparent in GCM integrations [Liu et al., 2004]. Such amplifications develop through upward propagation of wave activity, originating in the winter troposphere.

[8] Calculations of the normal mode in the presence of instability are consistent with such forcing and with auxiliary reinforcement by instability in the middle atmosphere [Salby and Callaghan, 2000]. Those calculations reveal a global-scale disturbance which has structure of the Rossby-gravity mode but which amplifies twice yearly around solstice, when the gradient of PV reverses near the normal mode's critical line. The Rossby-gravity mode then amplifies through sympathetic interaction with the mean flow. Wave activity generated at the unstable region disperses globally, assuming the structure and propagation characteristics of the Rossby-gravity mode. Such behavior appears in satellite observations [Wu et al., 1993; Limpasuvan et al., 2005; Garcia et al., 2005]. Integrations under differing zonal mean conditions indicate that these features of the Rossby-gravity mode are robust. While its amplification is sensitive to the mean flow, the mode's structure and period are not.

[9] Much the same behavior develops in nonlinear integrations [Salby and Callaghan, 2003]. However, upon attaining amplitudes comparable to zonal mean wind inside the region of instability, the 2-day wave breaks: It overturns the distribution of PV, resulting in eddy mixing. Such mixing homogenizes PV, which, in turn, destroys the reversed gradient of PV and instability, limiting further wave amplification.

[10] The 2-day wave routinely achieves amplitudes of several tens of m/s [e.g., Craig et al., 1981; Wu et al., 1993; Limpasuvan et al., 2005]. It then rivals the amplitudes of solar tides, which prevail in the MLT region at other times. The substantial amplitude of both implies a nonlinear (and, hence, irreversible) interaction between these global disturbances. Such interaction has been suggested in GCMs, albeit through different mechanisms [Palo et al., 1998; Norton and Thuburn, 1999; McLandress, 2002].

[11] Here, we describe nonlinear integrations of the 2-day wave in the presence of solar tides. Performed with a three-dimensional (3-D) model of the middle and upper atmosphere, the integrations explore how these global disturbances interact as well as the concomitant impact upon them. Following an overview of the model, section 3 describes the amplification, propagation, and structure of the 2-day wave in isolation, namely, in the absence of tides. Section 4 then investigates how those features of the 2-day wave are modified in the presence of tides. Conclusions are drawn in section 5.

2. Three-Dimensional Model

2.1. Numerical Formulation and Stochastic Forcing

[12] The nonlinear model is same one used to study the 2-day wave earlier [Salby and Callaghan, 2003]. Solving the global primitive equations in isentropic coordinates, it was developed from the spectral model described by Callaghan et al. [1999].

[13] The domain is three-dimensional, extending upward from an isentropic surface near the tropopause. The upper boundary is removed to 185 km, buried inside a deep sponge layer. There, thermal dissipation ramps up sharply above 142 km to absorb upward propagating wave activity and enforce the radiation condition. The useful domain thus extends from the tropopause to 142 km.

[14] For the integrations presented here, the model is configured with vertical resolution of 3–4 km and horizontal resolution comparable to T20. These are more than adequate to resolve scales of the 2-day wave and solar tides. The model is forced stochastically at its lower boundary by fluctuations representative of tropospheric wave structure. Individual wave numbers have complex amplitudes that are defined as a second-order stochastic process. Associated covariance is specified, with latitudinal structure that simply concentrates forcing in the winter hemisphere. The resulting fluctuations correspond to a red power spectrum (Figure 1), with a correlation time of ∼5 days. A more detailed description of the 3-D model and its forcing by stochastic fluctuations is available in the foregoing references.

Figure 1.

Power spectrum of stochastic forcing in wave number 3.

[15] When forced by tropospheric wave structure, the simulations recover dynamical structure representative of that observed. The latter reflects diabatic cooling over the winter pole that is offset by dynamical heating associated with downwelling of the Brewer-Dobson circulation and with isentropic mixing by planetary waves. At thermospheric altitudes, strong molecular diffusion prevails, accompanied by ion drag. Jointly, these features yield zonal mean structure that reproduces major features of the circulation in the middle and upper atmosphere [Callaghan et al., 1999; Francis and Salby, 2001; Callaghan and Salby, 2002; Salby and Callaghan, 2003]. They also reproduce salient features of the 2-day wave.

2.2. Tidal Forcing

[16] The present integrations include solar tides. They are forced thermally via water vapor and ozone heating. Ozone heating is imposed analytically in terms of time and solar zenith angle to represent structure similar to that given by Chapman and Lindzen [1970]. Water vapor heating, which is concentrated beneath the tropopause, is absorbed into the lower boundary condition to account for forcing at tropospheric levels. Imposed at the lowest isentropic surface is a diurnal variation in Montgomery stream function. Through the hydrostatic relation in isentropic coordinates [e.g., Salby, 1996], it reflects anomalous temperature below integrated vertically. Like ozone heating, the perturbation in Montgomery stream function is specified in terms of time and solar zenith angle. It is then multiplied by the latitudinal structure of water vapor mixing ratio, which also emphasizes low latitudes. These forms of heating are imposed with a prescribed amplitude, scaled by the maximum ozone heating rate in the stratosphere equation image. They produce migrating tides representative of those observed.

[17] Figure 2 plots, for equation image = 6 K/d, the amplitude (solid) and phase (dashed) of the migrating diurnal tide. Its structure has been evaluated by normalizing the complex amplitude of wave number 1 on individual days by its equatorial value at a reference level and then averaging the normalized structure over successive days. As developed by Salby and Callaghan [2003], this procedure is tantamount to averaging structure relative to a frame moving with wave activity at the reference site (e.g., with the tide). It therefore averages structure with nearly identical phase, averting sporadic phase shifts introduced by the stochastic forcing.

Figure 2.

Average amplitude (solid) and phase (dashed) of migrating diurnal tide for (a) temperature, (b) zonal wind, and (c) meridional wind. Stochastic forcing at lower boundary set to zero. Phase increment equals 45°.

[18] Perturbation temperature (Figure 2a) exhibits modal structure in latitude: A maximum over the equator, flanked in the subtropics by phase reversals and secondary maxima. In altitude, T1 oscillates with a vertical wavelength of order 25 km. It amplifies upward, exceeding 10 K above the mesopause.

[19] Perturbation zonal wind (Figure 2b) likewise exhibits modal structure in latitude. It is characterized by maxima in the subtropics that are symmetric (in phase) about the equator. Like perturbation temperature, u1 amplifies upward, exceeding 20 m/s in the lower thermosphere.

[20] Perturbation meridional wind (Figure 2c) also has modal structure. However, maxima of v1 in the subtropics are antisymmetric (out of phase) about the equator, separated by a phase jump of 180°. They too amplify upward, exceeding 30 m/s in the lower thermosphere.

[21] Figure 3 plots analogous characteristics of the semidiurnal tide. Perturbation temperature (Figure 3a) exhibits modal structure in latitude. It is symmetric about the equator, but broader than that of the diurnal tide. For the semidiurnal tide, T2 possesses a single maximum over the equator, with phase invariance in latitude. In altitude, perturbation temperature is nearly barotropic. A weak phase tilt develops above the turbopause, at altitudes of strong molecular diffusion. As for the diurnal tide, T2 amplifies upward, approaching 30 K in the lower thermosphere.

Figure 3.

As in Figure 2 but for migrating semidiurnal tide.

[22] Perturbation zonal wind (Figure 3b) is noticeably broader than for the diurnal tide. Assuming global structure, u2 is symmetric about the equator, with phase invariance. Above the turbopause, it develops weak vertical phase tilt. Like temperature, u2 amplifies upward, exceeding 40 m/s in the lower thermosphere.

[23] Perturbation meridional wind (Figure 3c) also has modal structure, but with maxima in the subtropics that are antisymmetric about the equator. Like other properties of the semidiurnal tide, v2 has barotropic structure, except in the lower thermosphere where weak vertical phase tilt develops. Amplifying upward, it exceeds 50 m/s in the lower thermosphere.

[24] The narrow meridional and vertical structure of the diurnal tide (Figure 2) corresponds to a short equivalent depth. Conversely, the broad meridional and vertical structure of the semidiurnal tide (Figure 3), corresponds to tall equivalent depth. Both are consistent with the structures of classical tidal theory and with observed amplitudes [Chapman and Lindzen, 1970; Forbes, 1984a, 1984b].

3. Two-Day Wave in Isolation

[25] The model is integrated under January conditions, when strong easterlies lead to a reversal of PV gradient in the summer mesosphere. For equation image = 0, solar tides are absent. Stochastic forcing at the model's lower boundary then excites the Rossby-gravity mode, which amplifies through sympathetic interaction with the mean flow.

[26] Figure 4 plots, as a function of time, the amplitude of wave number 3 meridional velocity over the equator. Reflecting cross-equatorial motion, v3EQ has been averaged vertically between the tropopause and mesopause (solid). 〈v3EQ〉 measures the strength of equatorial gyres that characterize the 2-day wave. Initially, 〈v3EQ〉 fluctuates randomly about a constant value, reflecting sporadic reinforcement and cancelation through interference with the stochastic forcing. This eventually gives way to steady amplification, albeit punctuated by sporadic increases and decreases. By day 80, vertically averaged cross-equatorial motion exceeds 10 m/s.

Figure 4.

Evolution of vertically averaged cross-equatorial wind in wave number 3 in the absence of tides (solid), in the presence of tides (blue-dashed), and in the presence of amplified tides (red-dotted).

[27] Figure 5 plots the same information, but at that level where cross-equatorial motion maximizes (solid). Located near the mesopause, ∣v3EQmax undergoes a similar evolution. However, it achieves amplitudes much stronger than the vertical average. By day 80, ∣v3EQmax approaches 70 m/s.

Figure 5.

As in Figure 4 but for maximum cross-equatorial wind (found near the mesopause).

[28] Figure 6 plots the power spectrum of 〈v3EQ〉, which describes the propagation characteristics of the unsteady response. 〈v3EQ〉 power (solid) is concentrated at a westward period of ∼2.1 days. Secondary power is visible out to periods of 2.5 days. However, it is overshadowed by the discrete peak at 2.1 days.

Figure 6.

Power spectrum of vertically averaged cross-equatorial wind in wave number 3 in the absence of tides (solid), in the presence of tides (blue-dashed), and in the presence of amplified tides (red-dotted). Spectral resolution ≅ 0.01 cpd.

[29] The average amplitude and phase structure of the unsteady response is evaluated in the same manner as that of tides: by averaging structure relative to a frame moving with the wave number 3 disturbance at a reference level over the equator (section 2). Figure 7 plots the average amplitude and phase structure of wave number 3 meridional velocity. Symmetric in latitude, v3 maximizes over the equator. It reflects equatorial gyres that achieve strong cross-equatorial velocity. v3 amplifies with altitude, exceeding 25 m/s in the lower thermosphere.

Figure 7.

Average amplitude (solid) and phase (dashed) of wave number 3 meridional wind, in the absence of tides.

[30] Figure 8 plots the corresponding horizontal structure, in eddy velocity and Montgomery stream function Ψ (the analogue, in isentropic coordinates, of geopotential height). As for amplitude and phase structure (Figure 7), the behavior in Figure 8 represents the average structure in a frame moving with the disturbance. At 72 km (Figure 8a), Ψ (contoured) forms a wave number 3 pattern that maximizes in the subtropics of each hemisphere and is antisymmetric about the equator. Anomalies are strongest in the summer hemisphere, adjacent to the reversed gradient of PV (not shown). Eddy velocity (vectors) follows contours of Ψ outside the tropics. However, in a neighborhood of the equator, v deviates from Ψ, forming strong cross-equatorial motion. It comprises a wave number 3 pattern of equatorial gyres that propagates westward with a period of ∼2.1 days (Figure 6). Together, the distributions of Ψ and v embody the salient features of the Rossby-gravity mode.

Figure 8.

Average horizontal structure in the absence of tides, in a frame moving with the propagating response, of Montgomery stream function (contoured) and wind (vectors) at (a) 72 km, (b) 83 km, and (c) 110 km.

[31] Near 85 km (Figure 8b), the disturbance has similar structure, but with meridional phase tilt in the summer hemisphere. At 110 km (Figure 8c), the disturbance has assumed virtually modal structure: Anomalies appear to either side of the equator, with comparable amplitude but opposite sign. Characteristic of the Rossby-gravity mode, these features have the same for as eddy structure observed by HRDI [Wu et al., 1993].

[32] Plotted in Figure 9 is the accompanying EP flux F (vectors) and its divergence (contoured). In the subtropics of the summer mesosphere is positive EP flux divergence. The region of ∇ · F > 0 near 65 km coincides with the reversed gradient of PV and zonal mean instability [cf. Salby and Callaghan, 2003]. It represents the emission of wave activity through interaction with the mean flow. EP flux is seen to radiate away from this region, upward and northward. Overhead and to the north are regions of negative EP flux divergence, where wave activity is absorbed. (Notice: The emission of wave activity (∇ · F > 0) is sharply confined to the summer subtropics. Yet, the wave itself (e.g., in Ψ: Figure 8) is globally extensive, occupying both hemispheres.)

Figure 9.

Average EP flux (normalized) and its divergence (m/s · d) resulting from the unsteady response in the absence of tides. F scaled by (ps/p) to elucidate EP flux at upper levels.

[33] Plotted in Figure 10 is the distribution of PV on day 90. The level 72 km (Figure 10a) intersects the critical region of the disturbance, where eddy velocity is comparable to the mean flow. There, PV has wrapped up into a series of closed vortices that are positioned just south of the equator. Those eddies mix air horizontally along isentropic surfaces, homogenizing long-lived species, like PV.

Figure 10.

Distribution of PV on day 90, in the absence of tides, at (a) 72 km and (b) 84 km.

[34] Near 85 km (Figure 10b), the disturbance is wavelike. However, at this time, undulations have amplified sufficiently to just overturn the PV distribution. Low-PV air is then folded north of high-PV air, reversing the PV gradient locally. This introduces local instability, through which anomalies amplify. Near Australia, the PV gradient is reversed between ∼15S and 40S. At subsequent times, such features wrap up to form closed vortices like those in Figure 10a. Reflecting an upward expansion of the critical region, they lead to an irreversible rearrangement of air along isentropic surfaces. The attendant mixing dissipates large-scale wave activity, as comprises the 2-day wave. Such dissipation is manifested in Figure 9 as the convergence of EP flux, which measures the absorption of wave activity. Notice that, at the level near 85 km, the local reversal of PV gradient between 15S and 40S (Figure 10) coincides with the region of strong ∇ · F < 0 (Figure 9).

4. Interaction With Tides

[35] We consider now the same diagnostics, but in integrations that include solar tides (equation image > 0). In those integrations, stochastic forcing is identical to that in the integration exclusive of tides (equation image = 0). The resulting evolutions can therefore be compared directly.

4.1. Nominal Solar Tides

[36] Solar heating of equation image = 6 K/d produces tidal amplitudes representative of annual mean behavior (Figures 2 and 3). Superimposed on Figure 4 is the accompanying record of 〈v3EQ〉 (blue-dashed). Vertically averaged cross-equatorial motion undergoes an evolution similar to that in the absence of tides. This is especially true during the first 30 days, when 〈v3EQ〉 is weak. The two evolutions then track one another closely. Their resemblance reflects linear conditions and tidal modulation of the 2-day wave that is largely reversible. Thereafter, however, the growth rate in the presence of tides is noticeably reduced. At day 80, when 〈v3EQ〉 in the absence of tides exceeds 10 m/s, 〈v3EQ〉 in their presence is only 6 m/s.

[37] The difference is even more pronounced at the level of maximum cross-equatorial motion, near the mesopause. Superimposed in Figure 5 is analogous information for ∣v3EQmax (blue-dashed). During the first 50 days, the evolution remains close to that in the absence of tides. This is especially true during the first 20 days, when ∣v3EQmax is weak and the two curves fall on top of one another. Even at day 50, ∣v3EQmax approaches 25 m/s in the presence of tides, whereas it is only a couple of m/s stronger in isolation. Much the same applies at subsequent times, so long as ∣v3EQmax remains less that 30 m/s. Beyond day 70, however, the unsteady response attains stronger amplitude. The integrations then diverge. In the absence of tides (solid), ∣v3EQmax amplifies sharply, approaching 70 m/s near day 80. In the presence of tides, that amplification collapses once ∣v3EQmax exceeds 30 m/s.

[38] Superimposed in Figure 6 is the corresponding power spectrum of 〈v3EQ〉 (dashed). It has form similar to that in the absence of tides. However, the peak at 2.1 days no longer dominates. Instead, the spectrum has significant support from periods near 2.5 days. Some evidence of this period appears in radar observations of near-equatorial v, especially at levels where tides are amplified [Pancheva et al., 2004].

[39] Plotted in Figure 11 is the average amplitude and phase structure of wave number 3 meridional velocity. In the presence of tides, v3 has much the same form as it does in their absence (Figure 7). However, its maximum, found in the equatorial lower thermosphere, is reduced from 25 m/s to 18 m/s.

Figure 11.

As in Figure 7 but in the presence of tides.

[40] Figure 12 plots the horizontal structure. At 72 km (Figure 12a), the structure is similar to that in the absence of tides (Figure 8a). However, the strength of anomalies now varies with longitude. The dependence on longitude is conspicuous north of the equator. It reflects a modulation of the 2-day wave by solar tides: a wave number 1 modulation by the diurnal tide and a wave number 2 modulation by the semidiurnal tide.

Figure 12.

As in Figure 8 but in the presence of tides.

[41] At 110 km (Figure 12b), where tides are amplified, the regular modal structure evident in the absence of tides (Figure 8c) is disrupted. Replacing it is a complex pattern, which implies 2-day wave fluctuations that are less regular and involve more scales than develop in the absence of tides. This is consistent with the broader nature of the power spectrum, evident even at lower levels (Figure 6).

[42] Plotted in Figure 13a is EP flux and its divergence. Differing conspicuously from that in the absence of tides (Figure 9) is strong EP flux convergence in the thermosphere. That forcing of the zonal mean flow represents the absorption of tides that have propagated upward from below. At lower levels, the pattern is similar to ∇ · F in the absence of tides. Still visible is the region of EP flux divergence in the summer mesosphere (wave emission), flanked above and to the north by regions of EP flux convergence (wave absorption). However, the region of absorption (∇ · F < 0) is now more extensive, prevailing overhead and to either side of the equator. It is also stronger. Some of the additional absorption is contributed by tides, but the contribution from the 2-day wave is also stronger. Evaluating ∇ · F from wave numbers 3 and greater excludes tidal contributions. Plotted in Figure 13b, it recovers a pattern similar to that in the absence of tides (Figure 9). However, relative to ∇ · F > 0 near 65 km (wave emission), ∇ · F < 0 overhead (wave absorption) is magnified by ∼30%. This is, in fact, close to the reduction of average 2-day wave amplitude (Figure 7 versus Figure 11), which in turn reflects ∼30% slower amplification.

Figure 13.

(a) As in Figure 9 but in the presence of tides. (b) As in Figure 13a but from wave numbers of 3 and greater.

[43] Relative to behavior in the absence of tides, magnified ∇ · F in the presence of tides represents accelerated absorption of large-scale wave activity. Some insight into the accelerated absorption comes from the horizontal distribution of PV, shown in Figure 14 on day 80. At 72 km (Figure 14a), closed vortices that comprise the critical region are still apparent. However, they are modulated zonally: strongest near the dateline. The dependence on longitude is analogous to that observed in Montgomery stream function (Figure 12). It reflects a wave number 1 modulation of the PV distribution, namely, by the diurnal tide.

Figure 14.

As in Figure 10 but on day 80 and in the presence of tides.

[44] The impact of such modulation is evident near 85 km (Figure 14b). On day 80, the PV distribution at this level is on the verge of overturning. Modulation by the diurnal tide has steepened the wave near 30 E. There, contours of PV have folded, crossing the threshold for local instability and wave breaking: Introduced is a local reversal of PV gradient and, hence, instability. The wave then breaks. Vortex wrap up, with the attendant generation of secondary scales, accelerates wave dissipation near 30 E. (This process reflects a cascade of enstrophy (PV variance) from the large scale of organized wave motion to small scales that are generated nonlinearly through eddy mixing.) Introduced through modulation by the tide, the vortex wrap up near 30 E represents a local expansion of the critical region, where wave activity is absorbed nonlinearly through irreversible mixing of PV.

[45] At other longitudes, modulation by the diurnal tide only weakens anomalies. Contours of PV then remain wavelike and, hence, short of the threshold for instability and wave breaking. Wave absorption at those longitudes is therefore about the same as it is in the absence of tides. Consequently, tidal modulation, which magnifies irreversibility locally, leads to a net acceleration of wave absorption.

[46] At altitudes of the 2-day wave's critical region, the diurnal tide is the prevailing tidal component (Figures 2 and 3). Its interaction with the Rossby-gravity mode intensifies 3 times per day: Each time the tide comes into phase with one of the 3 undulations in PV, that undulation is steepened. If the 2-day wave is sufficiently strong, reinforcement by the tide drives PV structure at those longitudes beyond the threshold for wave breaking. Vortex wrap up then leads to eddy mixing of PV and the absorption of large-scale wave activity. This nonlinear interaction leads to an overall acceleration of wave absorption, because other eddies are left wavelike and, hence, accomplish wave absorption at about the same rate as they do in the absence of tides.

[47] The accelerated absorption of wave activity damps the 2-day wave. Amplification of the 2-day wave must therefore proceed at a slower rate because the generation of wave activity must offset the accelerated absorption of wave activity that is introduced through interaction with the tide. Nonlinear interaction with the tide should be strongest when the Rossby-gravity mode is amplified. Support for this inference comes from Figure 5. During the first 20 days, when 2-day wave amplitude is small, the integrations with and without tides are nearly identical. However, after maximum cross-equatorial motion exceeds 30 m/s, the integrations diverge sharply.

4.2. Amplified Solar Tides

[48] We consider next an integration in which solar tides are amplified. equation image = 10 K/d produces tides that are about 50% stronger than in the foregoing integration. The resulting amplitudes are typical of equinox, when tides are stronger [e.g., Hagan et al., 1997; McLandress, 2002].

[49] Superimposed in Figure 4 is the accompanying record of 〈v3EQ〉 (red-dotted). Vertically averaged cross-equatorial motion now diverges from 〈v3EQ〉 in the integration without tides (solid) early on, beyond day 20, when 〈v3EQ〉 is only 1–2 m/s. The steady amplification that developed in the absence of tides has been virtually eliminated. On day 80, 〈v3EQ〉 has values of only 2–3 m/s. By contrast, in the absence of tides, it exceeds 10 m/s.

[50] Superimposed in Figure 5 is analogous information at the level of maximum cross-equatorial motion. The evolution of ∣v3EQmax (red-dotted) is similar to that in the absence of tides (solid). (It is imprinted on each evolution by fluctuations of stochastic forcing.) However, the magnitude of ∣v3EQmax is now less than half of that when the Rossby-gravity mode amplifies in isolation. Near day 80, ∣v3EQmax reaches only 23 m/s. In the absence of tides, it approaches 70 m/s.

[51] The corresponding power spectrum of 〈v3EQ〉 is superimposed in Figure 6 (dotted). Wave number 3 power is noticeably reduced at all periods. Still visible is the peak at 2.1 days. However, that discrete feature is now no larger than the other peak near 2.5 days. Consequently, amplified tides typical of equinox lead to 〈v3EQ〉 variance that is not only weaker, but distributed broadly between 2.0 and 2.5 days.

[52] The average amplitude and phase structure of v3 (Figure 15) has the salient form seen earlier. However, the maximum over the equator is found distinctly lower, not far above the mesopause. There, v3 does not exceed 10 m/s. (Notice: With the reduced contour interval, modal structure of v3 is seen to extend downward to the tropopause.) At higher levels, where tides are amplified, regular propagation is disrupted, causing v3 to decrease sharply.

Figure 15.

As in Figure 7 but in the presence of amplified tides.

[53] In horizontal structure (not shown), the modulation with longitude is now visible as low as 50 km. The modal structure is disrupted as low as 85 km. This is analogous to the disruption seen earlier at 110 km (Figure 12b), but consistent with the lower maximum in v3 (Figure 15).

[54] Plotted in Figure 16 is EP flux and its divergence. The pattern of ∇ · F is now dominated by EP flux convergence in the thermosphere, associated with tidal absorption. Its broad meridional structure indicates that, at those levels, a major contribution to ∇ · F comes from the semidiurnal tide (see Figures 2 and 3). At lower levels, the EP flux divergence (wave emission) seen earlier in relation to amplification of the Rossby-gravity mode is overshadowed by an array of anomalies that straddle the equator. Their symmetry and narrow meridional structure reflect the diurnal tide, which prevails at these altitudes. Generation of wave activity through instability has been virtually eclipsed by strong absorption of wave activity associated with the diurnal tide.

Figure 16.

As in Figure 9 but in the presence of amplified tides.

[55] The corresponding distribution of PV (not shown) exhibits features similar to that in Figure 14. Although wavelike at most longitudes, one of the PV undulations is steepened sufficiently to drive PV contours north-south. This eliminates the local meridional gradient of PV and, hence, dynamical stability. Eddy mixing then develops nonlinearly, absorbing large-scale wave activity and limiting subsequent amplification of the Rossby-gravity mode. Thus, for amplified tides, nonlinear absorption of wave activity ensues at considerably weaker 2-day wave amplitude (Figure 4). By damping the Rossby-gravity mode, such interaction prevents the 2-day wave from amplifying further.

5. Conclusions

[56] Solar tides influence the amplification of the 2-day wave. At small 2-day wave amplitude, tidal modulation is linear and, hence, largely reversible. This is evident from the initial amplification of the 2-day wave, which is virtually independent of tidal amplitude. At large 2-day wave amplitude, however, the modulation introduces a nonlinear interaction that is irreversible. Local reinforcement, chiefly by the diurnal tide, steepens individual undulations of the 2-day wave, driving anomalies of PV across the threshold for local instability and wave breaking. The resulting cascade of enstrophy to smaller scales magnifies eddy mixing, which homogenizes PV at the expense of large-scale wave activity.

[57] Derived through nonlinear interaction between the 2-day wave and tides, this process accelerates the absorption of large-scale wave activity. The accelerated absorption must be offset by the generation of wave activity. This results in slower amplification and the 2-day wave being limited to amplitudes at which nonlinear interaction prevails.

[58] In the presence of strong solar tides, typical of equinox, nonlinear interaction ensues even at modest 2-day wave amplitude. This quickly arrests amplification, limiting the 2-day wave to smaller amplitude. Regular propagation, which characterizes the 2-day wave in the presence of weaker tides, is then disrupted. Replacing it is complex structure and propagation, which are mirrored in the power spectrum. Discrete spectral behavior is transformed into broadband behavior, reflecting a cascade of variance to other scales and eddy mixing.

[59] The seasonality of tides is relevant. Tidal amplitudes are weaker by 25–50% during solstice [e.g., Hagan et al., 1997], when the 2-day wave can amplify through sympathetic interaction with the mean flow. Weaker tidal amplitudes then enable the 2-day wave to attain amplitudes that would otherwise be impossible. By the same token, stronger tides during equinox may limit the 2-day wave from amplitudes that would otherwise be attained.

[60] An analogous process influences solar tides. Like the 2-day wave, tidal amplitudes are reduced from those in isolation. However, because tides are continually forced diabatically, they depend only weakly upon PV and its irreversible rearrangement through interaction with the 2-day wave. Consequently, their interaction with the 2-day wave reduces tidal amplitudes only modestly, by 10–20%. This is consistent with the GCM behavior reported by McLandress [2002], wherein the observed seasonality of tides is recovered even exclusive of interaction with the 2-day wave. By contrast, such interaction strongly influences the 2-day wave, which, for sufficiently strong tides, is limited to small amplitude.

Acknowledgments

[61] The authors are grateful for constructive comments provided during review. This work was supported by NSF grant ATM-0120512.

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