In Situ Generation of Planetary Waves in the Mesosphere by Zonally Asymmetric Gravity Wave Drag: A Revisit

a. Model description and configuration

The present study employs SWAD, which is developed for exploring the atmospheric dynamical processes from the troposphere to the lower thermosphere. This modeling system adopts the semi-implicit, semi-Lagrangian dynamical core (Williamson and Olson 1994) implemented in the Community Atmosphere Model, version 3 (CAM3; Collins et al. 2004), of the National Centre for Atmospheric Research (NCAR). The spherical harmonic (SPH) expansion, on which the semi-implicit, semi-Lagrangian dynamical core is based, allows for preserving the zonal-mean field while enabling the wave components to evolve freely. This model solves the dry and hydrostatic primitive equations in a vorticity–divergence form in the hybrid vertical coordinate using the SPH transform method for horizontal discretization, a two-time-level semi-implicit scheme for time integration, and a semi-Lagrangian transport scheme for advection (see section 3.2 of the CAM3 documentation for details). The horizontal resolution is T42, which can resolve up to ZWN 42 on the Gaussian latitudinal grid at an approximate 2.8° interval, and 66 vertical levels are employed from the surface to an altitude of approximately 130 km. The modeling system is driven by radiative heating and cooling computed by relaxing temperature toward a Held–Suarez-type radiative equilibrium temperature which is extended into the lower thermosphere (Held and Suarez 1994; Polvani and Kushner 2002). This model incorporates a GWD parameterization identical to that used in NASA’s GEOS-5 and GEOS-6 models (Molod et al. 2012, 2015). Vertical diffusion processes are also implemented for the planetary boundary layer (Held and Suarez 1994) below about 700 hPa and molecular diffusion (Banks and Kockarts 1973) above 0.5 hPa. Perpetual January and July runs exhibit the typical structure of the zonal-mean fields and meridional circulations. Moreover, middle atmospheric variabilities, such as SSWs and quasi-biennial oscillations, are simulated reasonably well, although SSW frequency and periods of quasi-biennial oscillations depend on some tunable factors in the upper troposphere of the winter polar and equatorial regions, respectively (Song 2023).

The primary aim of this study is to investigate the characteristics of PWs generated by zonally asymmetric GWD in the absence of any other sources. To eliminate major stationary sources of PWs (e.g., topography and land–sea thermal contrast), the lower boundary of the modeling system is set as a flat bottom with zero geopotential height. In addition, during integration, another possible in situ source, BT/BC instability, is removed from the zonal-mean state of the entire atmosphere by employing a nudging approach elaborated on in the subsequent section. In an idealized framework, henceforth, the aforementioned physical processes (radiation scheme, GWD, and vertical diffusion parameterizations) are disabled in our simulations. To avoid potentially harmful impacts from the top boundary, the analysis is restricted to the middle atmosphere below an altitude of 90 km. In each simulation, the model is integrated for 30 days with a time step of 600 s and daily averaged outputs are used for analysis.

b. Nudging of the zonal-mean field without BT/BC instability

To prevent the emergence of BT/BC instability during integration, we first establish a barotropically and baroclinically stable NH winter zonal-mean flow by using the zonally averaged global ground-to-space (G2S) data on 1 January 2020, considered representative of the NH winter condition. Figure 1 describes the sequential stages of the process. Vertically continuous G2S atmospheric profiles are constructed by fitting B-spline curves to reanalysis data, including NASA’s Modern-Era Retrospective Analysis for Research and Applications (MERRA-2; Gelaro et al. 2017) and the fifth major global reanalysis produced by ECMWF (ERA5; Hersbach et al. 2020) for below the lower mesosphere and empirical model results for the upper atmosphere; see Song et al. (2018) for details.

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Latitude–altitude cross sections of zonally averaged G2S data for (a) zonal wind U and (b) temperature T. (c) The meridional gradient of the QGPV ${\overline{q}}_y$, calculated using the G2S data. (d) The zonal-mean stable U and (e) ${\overline{q}}_y$ calculated using the stable U and G2S T. (f) The difference between G2S U and stable U (G2S U − stable U).

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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Figure 1d depicts the stable $\overline{u}$ (${\overline{u}}_{\text{stable}}$). Despite being solved with the modified ${\overline{q}}_y$, ${\overline{u}}_{\text{stable}}$ preserves typical zonal wind structures seen during the boreal winter. The ${\overline{q}}_y$ calculated with this ${\overline{u}}_{\text{stable}}$ and G2S temperature, shown in Fig. 1e, verifies that the potential instability regions seen in Fig. 1c virtually disappear, with only a few insignificantly small negative values in the lower troposphere. The difference between the original $\overline{u}$ and stable $\overline{u}$ is negative overall (Fig. 1f), indicating acceleration of westerlies and deceleration of easterlies in $\overline{u}$. The most remarkable differences appear in higher southern latitudes. This discrepancy arises from the disappearance of the secondary easterlies peak around 60°S at a 0.05-hPa altitude, coupled with an intensification of westerlies in the lower mesosphere above the secondary easterly peak.

Nudging for certain parts or components of the general circulation has been employed in several contexts to examine its effects on the other regions or components (Hoskins et al. 2012; Simpson et al. 2011). Our approach is similar to that of Simpson et al. (2011), which relaxed the zonal-mean component of stratospheric circulation toward the seasonally varying model climatology in the stratosphere and above to quantify the influence of stratospheric variability on tropospheric annular-mode time scales. However, the nudging technique of the present study, used to constrain the zonal-mean state without BT/BC instability, differs from that of Simpson et al. (2011) in certain aspects. First, we apply relaxation throughout the atmosphere, not limiting it to the stratosphere and above as in Simpson et al. (2011). Accordingly, second, our approach does not necessitate additional physical processes to construct the general circulation, while in Simpson et al. (2011), the comprehensive GCM was integrated with the seasonally varying climatological sea surface temperatures, sea ice, and greenhouse gases fixed at 1990s values. Simpson’s method can induce spurious responses in waves through anomalous meridional circulation and artificial sponge-layer feedback (Hitchcock and Haynes 2014). Nudging the entire atmosphere without any longitudinal disturbances in the present study eliminates the likelihood of such spurious responses. Another distinction in our approach is nudging the zonal-mean temperature toward T₀(z) without latitudinal variation.

Before introducing the zonally asymmetric GWD, a simulation controlled exclusively by nudging without GWD is conducted to verify its performance. We refer to this experiment as “stable $\overline{u}$ nudging without GWD” (StaNOGWD) (see Table 1). Figure 2 presents the zonal-mean zonal wind, its deviation from ${\overline{u}}_{\text{stable}}$ ($\overline{u}-{\overline{u}}_{\text{stable}}$), and the amplitude of geopotential height perturbation (GHP), indicative of the presence of waves, at integration times of 5, 15, and 25 days in the StaNOGWD simulation. The $\overline{u}$ (Fig. 2a) is close to ${\overline{u}}_{\text{stable}}$ except for an overestimation by about 10% of westerlies and easterlies just above and below 0.01 hPa, respectively, in the Southern Hemisphere (SH) mesosphere at day 5 (Fig. 2b). However, these discrepancies diminish to under 1% by day 15. Underestimations of about 1–2 m s⁻¹ in the westerlies of the SH polar troposphere are also reduced with time. This indicates that the zonal-mean field is driven toward the stable conditions reasonably well after day 15, as demonstrated by the sparse regions of negative ${\overline{q}}_y$ (green thick lines in Fig. 2b). While some regions with negligibly small negative ${\overline{q}}_y$ appear near the surface, the substantial distances of these instabilities from the middle atmosphere make them incapable of influencing in situ PW excitation by GWD.

Table 1.

Summary of the configurations for the different experimental simulations.

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Latitude–altitude cross sections of (a) the zonal-mean U, (b) its deviation from the stable U, and (c) the GHP amplitude in the StaNOGWD simulation at the days (left) 5, (center) 15, and (right) 25 of integration times. Green thick lines overlaid on (b) represent negative ${\overline{q}}_y$ areas, while black lines denote zonal-mean U. Solid, thick solid, and dashed lines indicate positive, zero, and negative winds, respectively.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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The complete absence of any longitudinally varying forcings results in the absence of waves, producing a GHP amplitude of zero (Fig. 2c). However, it is noteworthy that this waveless state persists only up to approximately 60 days of integration (not shown). Beyond this time, small-scale waves emerge from the tropospheric jets and propagate upward and downward. The excitation of small-scale waves from the tropospheric jet stream consistently occurs across multiple simulations with a number of different setups (e.g., horizontal diffusion, Rayleigh damping, and nudging time scale), featuring variations solely in the timing of the appearance. However, given that nudging effectively constrains the background state from day 15, long-time integration is considered unnecessary. Henceforth, experiments are carried out for only 30 days.

c. Prescribed GWD

The absence of waves in StaNOGWD confirms that the nudging approach does not initiate any zonal disturbances during the selected analysis period (30 days). Therefore, in subsequent simulations where zonally varying GWD is included, the in situ generation of waves is solely attributed to the inherent zonal asymmetries in GWD.

To explicitly examine the influence of varying configurations of zonally asymmetric GWD, an idealized three-dimensional zonal component of GWD (XGWD) is considered. Spatial variations in GWD can arise from the (i) temporal and/or geographical variabilities in GW sources, (ii) filtering of GWs by PWs in the lower atmosphere, and (iii) nonlinear influence of background wind on GWD (Smith 1996, 1997, and references therein). Smith (1996) acknowledged the predominance of the second mechanism in shaping zonal variations in the mesospheric GWD, whereas Šácha et al. (2016) reported the significance of the first mechanism, particularly for OGWs in the stratosphere. Although the detailed zonal distribution of GWD may vary depending on the cause, we represent the large-scale zonal variability of GWD simply by adopting a zonally sinusoidal structure:

$\text{XGWD}(\lambda ,\varphi ,z)=\text{XGWD}(\varphi ,z)X_0\sin \hspace{0.17em}n\lambda ,$(4)

where λ is longitude, n is the ZWN, and X₀ is the magnitude of peak GWD. This structure, characterized by a zonally asymmetric component with a zero zonal-mean value, is suitable for making longitudinal variability without altering the stable zonal-mean flow constrained by the nudging processes. This GWD is introduced in the midlatitudes (30°–60°N) and 50–80-km altitude having a shape defined by the equation:

$\text{XGWD}(\varphi ,z)=\sin (6\varphi )\{\frac{1}{2}\left[\tanh \left(\frac{z-58}{3}\right)+\tanh \left(\frac{z+58}{3}\right)\right]-\frac{1}{2}\left[\tanh \left(\frac{z-68}{3}\right)+\tanh \left(\frac{z+68}{3}\right)\right]\}.$(5)

In determining this spatial distribution, we reflect the parameterized total GWD (the sum of orographic and nonorographic GWD) under the stable zonal-mean field (Fig. 1d), which is obtained through the GWD parameterization schemes implemented in this model (not shown). Accordingly, referring to the magnitude of large-scale (ZWN 1–2) asymmetries in the parameterized GWD, X₀ is set to 100 m s⁻¹ day⁻¹. Figure 3a presents the location and corresponding amplitude of sinusoidal XGWD in a latitude–height cross section. XGWD exhibits the maximum amplitude at around 45°N and 0.2 hPa (around 63 km in altitude) and decreases with distance from the peak region.

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(a) Latitude–altitude cross section of the amplitude of the idealized zonal component of GWD in a sinusoidal zonal shape. Longitude–latitude cross sections of GWD having ZWNs of (b) 1 and (c) 2 at the 0.2-hPa altitude. Black lines overlaid on (a) denote zonal-mean U, where solid, thick solid, and dashed lines indicate positive, zero, and negative winds, respectively.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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To elucidate the characteristics of PWs depending on various GWD configurations, a series of experiments with different GWD configurations, which is listed in Table 1, is conducted. As illustrated in the longitude–latitude cross section of XGWD at the peak altitude (0.2 hPa) in Fig. 3b, XGWD is set to default with a ZWN 1 structure (n = 1). Therefore, the control simulation (“Sta1GWD100”) is run with XGWD having a ZWN 1 structure with X₀ = 100 m s⁻¹ day⁻¹ under the stable background state, where the naming convention is given by “Nudging field + ZWN + GWD + X₀.” For “Sta1GWD200,” X₀ is double that of ZWN 1, while “Sta2GWD100” is a run with ZWN 2 (n = 2) and the default X₀ (Fig. 3c). To examine PW generation based on the linearized disturbance QGPV equation, an additional run with 1/100 of X₀ is conducted using the GWD of ZWN 1, and then, the results are multiplied by 100. This approach, so-called quasi-linear simulation, is necessary to examine the effects of nonlinearity in the results of fully nonlinear numerical simulations. Finally, a simulation (Uns1GWD100) where the zonal-mean zonal wind is nudged toward the unstable condition, which is shown in Fig. 1a, is conducted.

In all the experiments, prescribed XGWD is initiated on day 15, coinciding with the successful nudging of the background field toward the stable state. To minimize noise induced by the initial adjustment process, XGWD, which is initially zero, increases exponentially for 7 days, attaining its ultimate steady value on day 22.

d. Analysis of PWs

a. PW amplitude

Analyzing the characteristics of PWs begins with their GHP amplitude. This is conducted by decomposing the resolved waves into ZWN 1, ZWN 2, and ZWN > 2 components at days 15, 22, and 27 (Fig. 5). Unlike in StaNOGWD, PWs manifest in Sta1GWD100 above the region where the idealized GWD is introduced. The PWs exhibit predominant ZWN 1 structure, equivalent to that of GWD. We also observe weak amplifications in waves having ZWNs greater than 1. This is likely associated with nonlinear wave–wave interactions, and the verification of this will be explored in section 3d. Throughout the period of increasing GWD (days 15–22), PWs are not only reinforced but also extend into the upper mesosphere of the SH. Once GWD reaches a peak (from day 22), PWs attain a maximum GHP amplitude of about 700 m and settle into a quasi-steady state with minor fluctuations (please refer to the Hovmöller diagram in Fig. S2). It is interesting to see that PWs are also found in the SH upper mesosphere. This is likely due to equatorward propagation of these waves into the SH, which will be discussed further in sections 3c and 4c.

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Latitude–altitude cross sections of the GHP amplitude of resolved waves having ZWNs of (a) 1, (b) 2, and (c) greater than 2 at days (left) 15, (center) 22, and (right) 27 in the Sta1GWD100 simulation.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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b. Linearized disturbance QGPV arguments

Even within the GWD area, the amplification of PWs occurs at two distinct centers located at 30° and 60°N (Fig. 5). These regions do not align with the maximum GWD regions located at 45°N but correspond to the meridional boundaries of the peak. This is associated with the nonconservative GWD forcing Z′, taking the form of the meridional gradient of the idealized XGWD. Figure 6 shows the amplitude of Z′ and EP fluxes along with EPFD of PWs on latitude–height cross sections. The XGWD (=X′) with a maximum at 45°N results in two peaks of Z′ at the areas of its most significant latitudinal gradient, specifically 30°–40°N and 50°–60°N (Fig. 6a). These regions coincide with the two major PW GHP amplification locations observed in Fig. 5. In addition, due to the tangential relation of Z′ with respect to latitudes [Eq. (8): $Z^{\prime }=–{(X^{\prime }\text{cos}\varphi )}_{\varphi }/a\text{cos}\varphi =(–X_{\varphi }^{\prime }+X^{\prime }\text{tan}\varphi )/a$], Z′ peak at the higher latitude is larger than that at the lower latitude for a given X′. This latitudinal dependency of Z′ results in that of GHP, along with the contribution by the relation of f q′ ∝ ∇² Φ′ outlined by Eq. (7). At the maximum Z′ regions, EP fluxes emanate away from positive EPFD (Fig. 6b), confirming the in situ generation of PWs caused by Z′. This is also evidenced by wave activity density [$(1/2)a\cos \varphi {\rho }_0\overline{{q^{\prime }}^2}/{\overline{q}}_y$; Palmer 1982] that arises from the peak Z′ regions (Fig. S3). With intensifying Z′ (Fig. 6a), the initially weak EP fluxes and EPFD at the higher-latitude peak of Z′ (50°–60°N) seen on day 15 become strong by day 22 (Fig. 6b) with a positive EPFD (∼5 m s⁻¹ day⁻¹) that is relatively weaker than the lower-latitude counterpart (greater than 20 m s⁻¹ day⁻¹).

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Latitude–altitude cross sections of (a) the amplitude of Z′ at days 15 and 22 and (b) EP fluxes (vectors) overlaid on its divergence (EPFD, shading) at days 15, 22, and 27 in the Sta1GWD100 simulation. The EP fluxes at day 15 are multiplied by 5 for better representation. The black contours in (b) represent the zonal-mean zonal wind: Solid, dashed, and thick solid lines indicate positive, negative, and zero winds, respectively, while thick green lines represent areas of instability (${\overline{q}}_y<0$).

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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Figures 7a–i present a series of polar stereographic projections depicting q′ development over 6 h from day 15 and the contributions by each term in Eq. (11) calculated from the results of the Sta1GWD100 simulation. The initially unperturbed q′ (Fig. 7a) manifests an obvious ZWN 1 pattern after 6 h (Fig. 7b). During this period, both $-[\overline{u}/(a\cos \varphi )](\partial q^{\prime }/\partial \lambda )$ (Fig. 7d) and $-({\upsilon }^{\prime }/a)(\partial \overline{q}/\partial \varphi )$ (Fig. 7e) and thus the sum of the zonal and meridional advection terms (referred to “Advection” in Fig. 7f) are weak and contribute little to $\partial q^{\prime }/\partial t$ (Fig. 7c). Rather, Z′ (Fig. 7g) almost entirely accounts for $\partial q^{\prime }/\partial t$, dominating the overall advection (Fig. 7h). This indicates that Z′ induces q′, determining its overall tendency. Accordingly, q′ attributed to the lower-latitude Z′ (30°–45°N) has an opposite phase to that associated with the higher-latitude Z′ (45°–60°N). Following the theoretical relationship outlined by Eq. (7) (q′ ∝ ∇²Φ′ ∝ − Φ′), the enhanced q′ leads to the horizontally broadened amplification of GHP above both the lower and higher peaks of Z′ with opposite phases (Fig. 5), corresponding to an out-of-phase relationship with the distribution of Z′ (not shown). Despite the negligible contribution of advection terms even smaller than that of Residual, it is crucial to acknowledge their phase with respect to that of Z′. As they arise from q′, which is in phase with Z′, the zonal advection, $[\overline{u}/(a\cos \varphi )](\partial q^{\prime }/\partial \lambda )\propto -i{q}^{\prime }$, is shifted eastward by 90° relative to Z′. Similarly, given that ${\upsilon }^{\prime }=[1/(fa\cos \varphi )](\partial {\text{Φ}}^{\prime }/\partial \lambda )\propto i{\text{Φ}}^{\prime }\propto -i{q}^{\prime }$ [Eq. (7)], the meridional advection, $-({\upsilon }^{\prime }/a)(\partial \overline{q}/\partial \varphi )\propto i{q}^{\prime }$, is shifted westward by 90° relative to Z′. The zonal advection prevailing over the meridional one shapes the total advection, despite being partially counteracted by the out-of-phase relationship between the two advection terms.

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Polar stereographic projections of q′ at (a) 0 h and (b) 6 h, (c) the tendency of q′, (d) the zonal advection of q′, (e) the meridional advection of $\overline{q}$, (f) the sum of the two advection terms (denoted as Advection), (g) nonconservative GWD forcing Z′, (h) the sum of Advection and Z′, and (i) the residual [Residual, defined as $(\partial q^{\prime }/\partial t)-(\mathit{\text{Advection}}+Z^{\prime })$] on day 15 in the Sta1GWD100 simulation. The unit of q′ is per second with a scale factor of 0.01, while that of others is per second per hour. (j)–(r) As in (a)–(i), but calculated on day 22 and using a scale factor of 1/30 for q′.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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The initial behavior of the q′ evolution undergoes a notable change with time. Figures 7j–r illustrate q′ development at day 22. A remarkable distinction is the eastward movement of q′ (Fig. 7j) by approximately 90° from its state on day 15 (Fig. 7b). This is associated with the zonal advection of q′. Tied to the amplifying $q^{\prime },-[\overline{u}/(a\cos \varphi )](\partial q^{\prime }/\partial \lambda )$ is intensified and gradually moves eastward due to the zonal-mean westerlies. Despite being partially counteracted by the meridional advection, which also strengthens and shifts eastward, the zonal advection drives the eastward propagation of q′. This, in turn, triggers further eastward propagation of the advection terms. Through this positive feedback, both advection terms undergo a 90° eastward shift from their original states on day 15 (Figs. 7m–o). The enhanced Advection (Fig. 7o) significantly offsets Z′ (Fig. 7p) due to their out-of-phase relationship and comparable magnitude (Fig. 7q), resulting in a markedly reduced $\partial q^{\prime }/\partial t$ (Fig. 7l). Therefore, the in situ generated PWs attain a quasi-steady and stationary state with minimal change in q′ (Figs. 7j,k).

Considering the nonnegligible magnitude of the Residual term in both the initial (Fig. 7i) and quasi-steady (Fig. 7r) states, its potential sources are worth noting. First, the linearized disturbance QGPV equation [Eq. (7)], despite being modified to include an ageostrophic term (the isallobaric contribution to υ′, which is ${\upsilon }_I^{\prime }{}_{}=(1/f)\{(\partial /\partial t)+\overline{u}[\partial /(a\cos \varphi \partial \lambda )]\}{u}^{\prime }$ in Palmer 1982), cannot fully explain the dynamics of the atmosphere governed by the primitive equation model. We identify that the geostrophic zonal wind, which is somewhat weaker than the stable zonal wind nudged in this model, contributes to the Residual in the polar latitudes (not shown). Nonlinear interactions among waves stand out as the important cause of the Residual. As evidenced in Fig. 7r, the Residual term exhibits a ZWN 2 pattern, aligning with the amplification of ZWN 2 in Fig. 5a, supporting our speculation that the nonlinear interaction induces the waves of higher ZWN than that of the idealized GWD. This will be confirmed with Sta1GWD1, the quasi-linear simulation in section 3d. Numerical diffusion is another possible factor, although its influence on large-scale PWs is minimal. Despite the presence of the Residual, our theoretical inspection clarifies the development of PWs generated in situ by Z′ until achieving the quasi-steady state.

c. Influences of in situ generated PWs

The analysis of EP flux and EPFD in Fig. 6 also provides insights into the propagation and influences of the GWD-induced PWs. Poleward and equatorward EP fluxes propagating away from the equatorward and poleward peaks of Z′, respectively, converge above the maximum GWD area, depositing a localized negative EPFD of about 10 m s⁻¹ day⁻¹. Prominent positive EPFD at the equatorward peak of Z′ and negative EPFD above the maximum GWD are responsible for the acceleration and deceleration of the zonal-mean zonal wind, respectively, in Fig. 4b, though the magnitude is significantly diminished due to the mitigating effect of nudging. The influence of the in situ PWs is not confined to their origin. Following westerlies extended into the SH upper mesosphere, these PWs propagate across the equator. A similar feature was previously identified in some satellite observation studies (e.g., Garcia et al. 2005; Forbes et al. 2002) as well as in the numerical study by Smith (2003), where SPWs from the winter hemisphere propagate toward the summer hemisphere along westerlies in the mesosphere tropics. Along the path toward the SH mesosphere above the zero-wind line, these waves not only initiate zonal disturbances (Fig. 4) but also induce negative EPFD due to their decreasing amplitude with distance from the source region. This interhemispheric propagation property aligns with the theoretically inferred characteristic of quasi-stationary PWs which propagate under westerlies weaker than the critical Rossby speed (Charney and Drazin 1961). Although interhemispheric propagation could be contingent on the prevailing zonal wind condition, this feature indicates a possible contribution of the GWD-generated PWs in the NH to PWs in the SH mesosphere.

d. Characteristics of the in situ PWs induced by different GWD configurations

This section delves into a comparative analysis of the characteristics of PWs dependent on the properties of zonally asymmetric GWD. Figure 8 illustrates the amplitude of PWs across the simulations with varying zonal asymmetries in GWD. In the simulation Sta1GWD200 (Fig. 8b), PW amplitude is intensified to about twice the levels seen in Sta1GWD100, in concordance with the doubled peak value of GWD. This indicates that the GWD-induced PWs have magnitudes in proportion to the peak magnitude of asymmetries in GWD. The amplification is not restricted to PWs of ZWN 1 but extends into the higher ZWN ones arising from the nonlinear process. Regarding this, we speculate whether the degree of the nonlinear interaction increases with the magnitude of zonal asymmetries in GWD. To verify this, an additional experiment, Sta1GWD1, in which the peak magnitude of GWD is reduced by a factor of 100 compared to that of Sta1GWD100, is conducted. In this simulation, the PW amplitude is recovered by multiplying by 100 from the model result. As evidenced in Fig. 8c, the amplitude of PWs with ZWNs larger than 1, indicative of the nonlinear interaction, almost disappears. This result confirms our speculation that as the peak magnitude of the GWD increases, the amplitude of shorter wavelength PWs is enhanced by nonlinear interactions.

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Latitude–altitude cross sections of the GHP amplitude of the resolved waves with ZWNs of (left) 1, (center) 2, and (right) greater than 2 at day 22 in the (a) Sta1GWD100, (b) Sta1GWD200, (c) Sta1GWD1, and (d) Sta2GWD100 simulations. The GHP amplitude in (c) is multiplied by 100.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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In the simulation with ZWN 2 GWD (Sta2GWD100; Fig. 8d), PWs of ZWN 2 (PW2) are excited as expected. However, although the GWD magnitude is identical to that in Sta1GWD100, the amplitude of PW2 is reduced to approximately a quarter that of the ZWN 1 PWs seen in Sta1GWD100, with a maximum of only ∼150 m. The decrease in PW amplitude with increasing ZWN follows the relationship q′ ∝ ∇²Φ′ ∝ − k²Φ′, implying that the Z′-enhanced q′ induces PWs with GHP amplitudes inversely proportional to the square of their ZWN. This indicates that larger-scale asymmetries in GWD tend to generate PWs with more significant amplitude and influence.

The ZWN dependency is also identified in the time required to achieve the stationary state: In Sta2GWD100, the stationary state is attained in roughly half the time (3 days) taken in Sta1GWD100 (6 days). Close inspection of the q′ evolution provides insight into the underlying cause. As discussed above with Fig. 7, Z′ initially causes in-phase changes in q′, thereby inducing the two advection terms shifted 90° relative to that of Z′. Phase difference between Z′ and subsequent Advection (the sum of two advection terms), which is dominated by the zonal advection, triggers the positive feedback, which moves the resultant q′ and Advection eastward until when Advection balances with Z′, achieving the stationary state. Concerning this relation, we compare the phase relationship between Advection and Z′ along with their combined effect in Sta2GWD100 to those in Sta1GWD100, as shown in Fig. 9. The initial phase difference and resultant positive feedback occur consistently in Sta2GWD100. However, aligning with the normal mode method, the doubled ZWN in Sta2GDW100 induces a phase difference of only 45° at the initial state (Fig. 9c). Accordingly, under the same zonal-mean zonal wind condition, Advection reaches the phase opposite that of Z′ in about 3 days, faster than the approximate 6 days required in Sta1GWD100.

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Polar stereographic projections of (left) Advection, (center) Z′, and (right) the sum of Advection and Z′ accumulated for 6 h, from 0000 to 0600 UTC, on (a),(c) day 15 and (b),(d) day 22 in the (a),(b) StaGWD100 and (c),(d) Sta2GWD100 simulations.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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Additional experiments varying the meridional location and width of the zonally asymmetric GWD demonstrate the dependence of in situ excited PWs on these factors (Text S2, Table S1, and Figs. S4 and S5). As GWD shifts toward lower latitudes, the opposite-phase PWs originating from the lower- and higher-latitude Z′ peaks exhibit an increasing difference in their GHP amplitude (Fig. S4c). The degree of cancellation between these two opposite-phase PWs decreases with decreasing latitudes. Consequently, PWs at higher-latitude Z′ peak tend to strengthen with decreasing latitude. This phenomenon also appears in simulations varying meridional width of GWD (Fig. S5). With the widening meridional range of GWD, the lower and higher Z′ locations become more distant from each other (Fig. S5b). Furthermore, the difference in GHP amplitude between PWs at these two Z′ peaks also increases. Attributed to both factors, the offsetting of opposite-phase PWs tends to decrease as the meridional width increases. Therefore, PWs at higher-latitude peak intensify with increasing meridional width. Note that the steady-state balance in the linearized disturbance QGPV equations remains consistent across the different meridional locations and width of GWD, thereby not accounting for the latitudinal dependence in GHP amplitude (not shown).

e. Influences of instability on the in situ PW generation by GWD

Focusing on the exclusive role of GWD in exciting PWs without another in situ source, BT/BC instability, may prompt the question “How does in situ PW excitation by GWD alter in the presence of instability?” This question is addressed by performing an additional experiment, Uns1GWD100, in which the zonal-mean atmosphere is constrained toward the G2S wind and temperature data including BT/BC instability (Figs. 1a,b). Figure 10 compares the nudged zonal-mean zonal wind and properties of PWs in the Sta1GWD100 and Uns1GWD100 simulations. In the simulation with an unstable background state, PW amplitudes are notably enhanced. While quantifying the contributions of GWD and instability to the overall increase in PW amplitudes seen in the Uns1GWD100 is not straightforward, we can qualify the influence of instability when PWs are forced by the zonally asymmetric GWD. At the higher-latitude GWD source region (60°N at a 65–70 km altitude), instabilities are concurrent with the enhanced PW amplitudes, suggesting a role of instability in amplifying PWs initiated by GWD. In addition, outside the GWD region in the polar mesosphere (60°–90°N at altitudes of 70–90 km; Fig. 10e), the intensification of PWs with ZWNs larger than 1 is remarkable (Figs. 10c,d) and is accompanied by substantial positive EPFD (Fig. 10e). This suggests an additional but important effect of instability: the strengthening of nonlinear wave–wave interactions. We hypothesize that this nonlinear interaction likely aligns with the findings of Hartmann (1983), which showed that barotropic instability coupled with the preexisting ZWN 1 PWs destabilizes waves having shorter wavelength. To confirm this, an analysis examining unstable wave growth rates under corresponding background conditions is necessary. However, this is beyond the scope of this study.

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Latitude–height cross sections of (a) zonal-mean U, the GHP amplitude of the resolved waves having ZWNs of (b) 1, (c) 2, and (d) greater than 2, and (e) EP flux (vectors) overlaid on EPFD (shading) under the (left) stable and (right) unstable nudging fields in the Sta1GWD100 and Uns1GWD100 simulations, respectively. Specifications of overlaid lines on (e) are the same as in Fig. 6b.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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a. PW amplitude

Regarding the increase in perturbations with height due to decreasing air density, we compare PW amplitudes taking note of their altitudinal region. Furthermore, we also make an effort to note the magnitude of the zonal asymmetries and meridional width of GWD depicted in previous studies, which are crucial determinants of nonconservative GWD forcing. Quasi-steady-state PWs in this study have a GHP amplitude in a range of 500–700 m above their generation altitude (60 km). This magnitude is comparable to that of the OGWD-induced PWs at the 71-km altitude found by Holton (1984). This is likely attributed to three reasons: (i) Although Holton (1984) parameterized OGWs forced by the idealized ZWN 1 topography, OGWD had similar characteristics to our idealized GWD, showing zonal asymmetries of 100 m s⁻¹ day⁻¹ between 30° and 70°N at the 71-km altitude. Moreover, (ii) the absence of PWs originating from the lower boundary sources and (iii) the extinction of barotropic and inertial instability in Holton (1983), excluding other PW sources, are consistent with our approach. MM93 had put forth a similar argument that the concurrent presence of PWs forced from below and the GWD-induced PWs in their simulation led to larger amplitude mesospheric PWs than those seen in Holton (1984). While the GWD-induced PWs in Smith (2003) exhibited an amplitude of 200–300 m within an altitude of 75–85 km, about half of that in the present study, this can likely be attributed to a smaller GWD asymmetry (∼25 m s⁻¹ day⁻¹) in Smith (2003).

Another important finding in Smith (2003) is that GWD-induced PWs in the upper mesosphere (around 80 km altitude) take precedence over the upward-propagating Rossby waves due to the presence of zonal wind reversals from westerlies to easterlies. To examine this possibility, we conducted an additional experiment (not shown) by prescribing the idealized topography devised by Gerber and Polvani (2009) as a source for stationary PWs within 30°–60°N at a lower boundary, wherein the same stable zonal wind nudging is performed. Although the elevation of the topography is lowered to 1.5 km (from the original 3 km) in this simulation, the orographically induced PWs have amplitudes (700–1000 m) greater than those of the GWD-induced waves. This is likely due to the westerlies extending up to 90 km without reversal, enabling the continuous amplification of the stationary waves propagating upward without being filtered.

b. Wind responses to the zonally asymmetric GWD

By analyzing the linearized disturbance QGPV equation, this study elucidates the initial excitation and subsequent growth of the GWD-induced PWs toward a steady state: Initially, nonconservative GWD forcing generates PWs by determining the q′ tendency. Within 7 days, a quasi-steady state is achieved as the nonconservative GWD forcing balances with the dominant zonal advection of q′, which has an opposite phase but comparable magnitude, following its eastward movement driven by the zonal-mean westerlies. The initial feature is consistent with the generation of eastward-propagating mesospheric PWs seen on 18 January 2009 by Song et al. (2020) using the MERRA-2 reanalysis data, which was attributed to the q′ tendency determined by nonconservative GWD forcing (see Fig. 8 in their study). The steady-state balance, on the other hand, aligns with the findings in MM93 and Smith (2003) that the nonconservative GWD forcing was primarily balanced by the zonal advection of q′. Even in a steady state, in Holton (1984), the curl of GWD was balanced by the meridional advection of planetary vorticity.

Despite varying factors balancing the nonconservative GWD forcing in these studies, the mesospheric wind response to a localized GWD showed a consistent downstream shift of the wind minimum relative to the strongest negative drag position (here, eastward). Holton (1984) observed the minimum zonal wind downstream (to the east) of the strongest OGWD by 90°, while a smaller shift, 30°–40°, was noted by MM93. Additional details regarding the wind response were provided by Smith (1996) using a simple barotropic model. Smith (1996) reported that during and immediately after a strong GWD pulse, the minimum wind coincides with the strongest drag, whereas it shifts downstream and persists there until another pulse or the decay of longitudinal variability. Considering the prominent eastward advection of q′ in the present study, similar behavior is expected. This is confirmed in Fig. 11, which depicts the zonally varying GWD and response of $u^{\prime }(u-\overline{u})$ at an initial state and a steady state. Showing consistency with Smith (1996), X′ and u′ are initially in phase, but at the steady state, u′ shifts 90° to the east from X′. However, as summarized by Smith (1996), the downstream displacement of the wind response is not always guaranteed but varies in response to the zonal-mean state (particularly $\overline{u}$ and ${\overline{q}}_y$) and, thus, the relative importance of zonal and meridional vorticity advection. Furthermore, the intermittency of GWD over time also affects the longitudinal displacement in the real atmosphere.

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Polar stereography of (left) the zonal perturbation of the idealized GWD X′ (m s⁻¹ day⁻¹) and (right) the zonal perturbation of the zonal wind u′ at 0.2 hPa (m s⁻¹) at days (a) 15 and (b) 22. The terms X′ and u′ at day 15 are multiplied by a factor of 3 and 30, respectively, and u′ at day 22 is scaled by 3/2 for better representation with the same label bar.

Citation: Journal of the Atmospheric Sciences 81, 9; 10.1175/JAS-D-24-0026.1

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Meanwhile, Lieberman et al. (2013) suggested that the zonal and meridional GWD perturbations induce the meridional ${\upsilon }_{\text{ag}}^{\prime }$ and zonal $u_{\text{ag}}^{\prime }$ ageostrophic wind perturbations following $-f{\upsilon }_{\text{ag}}^{\prime }=X^{\prime }$ and $f{u}_{\text{ag}}^{\prime }=Y^{\prime }$ [Eqs. (5) and (6) in their study], respectively. The resulting divergence/convergence of the ageostrophic wind perturbations serves as an in situ source of PWs in the mesosphere. We briefly examine whether such relationships also appear in our study by investigating the geopotential–wind relationship (not shown). Equatorward and poleward ${\upsilon }_{\text{ag}}^{\prime }$ emerging above positive and negative X′ regions, respectively, are consistent with Lieberman et al. (2013), although the magnitude of ${\upsilon }_{\text{ag}}^{\prime }$ is about half of that attributed to X′ following $-f{\upsilon }_{\text{ag}}^{\prime }=X^{\prime }$. Also, $u_{\text{ag}}^{\prime }$ is obtained in the present study without considering Y′. These differences stem mainly from the zonal advection of q′ that was disregarded in Lieberman et al. (2013). In the present study, the zonal advection of q′ primarily balances the nonconservative GWD forcing.

c. Influences of the in situ PWs

The GWD-induced PWs exhibit divergent fluxes away from their source regions and subsequent dissipation along their propagation, providing positive and negative EPFD, respectively (Fig. 6). A similar divergent feature of EP fluxes, directed upward and downward away from the source level (71 km altitude), was noted by Holton (1984). As previously mentioned, the clear discernment of local PW generation by GWD in Holton (1984) was attributable to the absence of upward-propagating PWs forced from below in his ideal simulation. Similar features also manifested in the numerical simulation of MM93 (refer to Figs. 7 and 8 of MM93, which compare EPFD between cases A and B), where PWs forced from the tropopause existed simultaneously. In Case A, which incorporated the curl of GWD, positive EPFD emerged above the significant nonconservative GWD forcing area (40°–50°N), accompanied by negative EPFD in the adjacent higher latitude. While not explicitly stated in MM93, it is reasonable to attribute the positive EPFD to the in situ PW generation. In Case B, excluding the curl of GWD led to a marked reduction in the midlatitude positive EPFD and higher-latitude negative EPFD. This demonstrated that the GWD-induced PWs emanating away from the source region (40°–50°N) contributed to a substantial portion of the negative EPFD along with the upward-propagating PWs. These findings emphasize the ability of GWD-induced PWs to not only deposit positive forcing locally at their origin but also exert negative forcing over a wide area through dissipation.

Related to changes in the PW forcing driven by the curl of GWD, MM93 also observed an interesting phenomenon: compensation between GW and PW forcings. In the framework enabling the interaction among GWs, PWs, and zonal-mean flow in MM93, reduced negative EPFD accelerates the zonal-mean westerlies, thereby negatively enhancing the parameterized OGWD [see Eqs. (1) and (2) in MM93]. Although the value of the negative ΔOGWD equivalent to that of the positive ΔEPFD remains elusive, the compensation resulted in the sum of the two forcings being nearly identical in cases A and B, as depicted in Fig. 10 of MM93. Our EPFD results, which are consistent with those in MM93, support a potential for compensation, suggesting the mechanism of in situ PW generation by zonally asymmetric GWD as an addition to the three compensation mechanisms between parameterized GWs and resolved waves outlined in Cohen et al. (2014): (i) a stability constraint, (ii) a potential vorticity mixing constraint, and (iii) refractive index modification. It is also conceivable that the Cohen et al. (2014) compensation interactions driven by zonal-mean GWD could be influenced by the changes in EPFD attributed to the longitudinal variability in GWD (Šácha et al. 2016). While Cohen et al. (2013) showed that the impact of zonal asymmetries in GWD on the compensation associated with instability is insignificant in the NH winter stratosphere, its importance is expected to rise notably in the mesosphere, where GWD magnitudes and asymmetries increase.

The influence of GWD-induced PWs originating in the NH is not confined to that hemisphere but extends into the SH as the waves propagate across the equator in the westerly waveguide (Fig. 4). This is a consistent phenomenon identified in previous observational and numerical studies (e.g., Forbes et al. 2002; Garcia et al. 2005; Smith 2003), suggesting that the interhemispheric propagation of GWD-induced SPWs from the winter hemisphere is a plausible explanation for SPWs observed in the summer mesosphere (Garcia et al. 2005; Wang et al. 2000). These waves also deposit the negative EPFD in the SH MLT, where positive GWD predominates in the real atmosphere (see Fig. 2 of Yasui et al. 2021). The introduction of a negative momentum anomaly in the SH summer MLT is worth discussing in the context of the interhemispheric coupling of the middle atmospheric circulation (Becker et al. 2004). Körnich and Becker (2010) proposed a plausible mechanism for interhemispheric coupling in association with GW forcing: An increase in negative Rossby wave forcing in the winter stratosphere decelerates the westerlies, which, in turn, facilitates the propagation of eastward-propagating GWs into the winter mesosphere, thereby reducing the dominant negative GWD in that region. The positive GWD anomaly weakens the meridional circulation toward the winter pole, leading to a warm anomaly in the tropical mesosphere. Responding to this tropical warm anomaly via thermal wind relation, the easterlies in the summer upper stratosphere and lower mesosphere are weakened, causing a downward shift in the eastward GWD. A westward GWD anomaly at the altitude of dominant eastward GWD in the summer MLT weakens the meridional circulation toward the equator, leading to a warm anomaly in the summer polar mesosphere. Recently, Yasui et al. (2021), based on growing evidence for a significant influence of in situ excited waves in the MLT, demonstrated that quasi-2-day PWs and secondary GWs, spontaneously generated by destabilized easterlies in the summer stratosphere, propagate upward and deposit westward momentum in the summer MLT, thereby inducing a warm anomaly in the summer polar MLT. The instability-associated quasi-2-day waves were also identified in Lieberman et al. (2021) based on high-altitude observations and observation-driven modeling, while the destabilization of the summer easterly jet was induced by the inertial instability in the lower latitude, resulting from anomalously strong PW breaking in the winter stratosphere. Considering earlier findings of in situ PW generation resulting from enhanced zonal asymmetries in GW fluxes, as filtered by intensified stratospheric PWs during the onset of SSW events (Song et al. 2020) coupled with the interhemispheric propagation toward the SH MLT observed in this study, there is an apparent possibility that GWD-induced PWs in the NH propagate across the equator and deposit negative momentum in the SH, thereby contributing to the warm anomaly. This hypothesis is worth further exploration in elucidating the mechanism of interhemispheric coupling.