a. Periodicity recovery for increasing attenuation length scale

In the following, we will explore the parameter space Re = 0–100 and $\widehat{h}=0.1\text{–}1$. While the range in Re is well justified by estimates, values of $\widehat{h}$ exceeding 0.5 are not, as this is equivalent to imposing a lid at the top of the domain, which has no physical analog in the stratosphere. However, exploring a wide range of possible attenuation length scales is crucial to understand the synchronization mechanism discussed later in the multimodal section of the manuscript, as the mean flow of a given wave type will increase the Doppler-shifted attenuation length scale of the other type, thus drawing an analogy to cases with larger $\widehat{h}$.

Figure 1b presents the bifurcation diagram obtained by varying both dimensionless control parameters Re and $\widehat{h}$. Bifurcation diagrams are constructed using Poincaré maps, where values of $\overline{u}$ near the top of the domain that intersect with $\overline{u}=0$ near the bottom are grouped into bins. The shading in the figure corresponds to the total number of bins populated in a given experiment (for more details, see Renaud et al. 2019). Experiments with only one bin indicate a stable resting state; those with two bins represent a QBO-like periodic oscillation, while more than two bins signify the progression toward aperiodicity and chaotic behavior (Renaud et al. 2019). The transition from a resting state to a QBO-like oscillation is referred to as the first bifurcation, whereas the transition from a QBO-like oscillation to higher oscillatory modes is termed the second bifurcation.

View Full Size

Bifurcation diagram for a monochromatic standing wave forcing. (a) Growth rate of the linearized system for $\widehat{h}=1/3.5$. Black lines with starred indices: oscillatory modes [Im(σ) ≠ 0]. Orange lines: nonoscillatory modes [Im(σ) = 0]. Indices refer to the vertical structure of the modes (see Fig. 2). (b) Dynamical regimes of the quasilinear model in $(\widehat{h},\text{Re})$ space, using the Poincaré map metric. The shading corresponds to the total number of bins populated in a Poincaré section of the mean flow $\overline{u}$ at z = 0.75H intersecting $\overline{u}=0$ near the bottom at z = 0.1H. Black lines indicate the appearance of an unstable oscillatory mode in the linearized system; orange lines correspond to the nonoscillatory ones. Green dots correspond to those in (a).

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0220.1

• Download Figure
• Download figure as PowerPoint slide

In standard textbook treatments, the assumption is often made of a semi-infinite stratosphere, where $\widehat{h}\to 0$, corresponding to the left of Fig. 1b. Remarkably, for any fixed value of the forcing parameter Re, the system transitions toward more periodic states as $\widehat{h}$ becomes sufficiently large. This implies that, for a finite domain depth H, modulations of wind reversal regimes can be achieved solely by varying the attenuation length scale h, which is determined by the wave properties originating in the upper troposphere, regardless of their forcing amplitude.

Focusing on the structure of bifurcation diagrams and considering the entire range of Re as $\widehat{h}$ increases, a new dynamical regime emerges, which is not present in the semi-infinite assumption ($\widehat{h}\to 0$). This regime is characterized by bistability, represented by the light gray shading in Fig. 1b. Within this bistable regime, two steady nonzero zonal wind solutions coexist, effectively breaking the east–west system symmetry imposed by the wave generator. The introduction of this new dynamical regime has a significant impact on the behavior of the first bifurcation, which occurs between resting states and QBO-like oscillations. This transition is marked by a shift from a supercritical Hopf bifurcation to a subcritical one as $\widehat{h}$ surpasses a critical threshold, denoted as ${\widehat{h}}_c$. For values of $\widehat{h}$ below ${\widehat{h}}_c$, the system undergoes a first bifurcation from a state of rest to a QBO-like oscillation. However, when $\widehat{h}$ exceeds ${\widehat{h}}_c$, the system first transitions to a steady nonzero zonal wind profile before generating QBO-like regimes. The system exhibits hysteresis behavior as Re is varied continuously across this regime. Similar transitions from supercritical to subcritical Hopf bifurcations have been previously reported by Semin et al. (2018) when varying other parameters of the quasilinear model, such as the strength of an additional linear drag; see also Yoden and Holton (1988) in the case of asymmetric wave forcing.

Predicting bifurcations with the linearized system

To delve into the dynamics elucidating the bifurcation diagram presented in Fig. 1b, we linearize the model around a state of rest, as in Plumb (1977), and compute eigenmodes of the linear operator, following the approach of Semin et al. (2018). Using nondimensional parameters,

$U=\frac{\overline{u}}{c},Z=\frac{z}{H},T=\frac{tF}{ch},$(6)

and assuming U ≪ 1, we obtain, at the lowest order, the rescaled linear system as follows:

${\partial }_{T}U=\frac{{\widehat{h}}^2}{\text{Re}}{\partial }_{ZZ}U+4e^{-Z/\widehat{h}}\left(U-\frac{1}{\widehat{h}}{\displaystyle {\int }_0^{Z}U}d{Z}^{\prime }\right),$(7)

with boundary conditions

$U=0\hspace{0.17em}\text{at}\hspace{0.17em}Z=0,{\partial }_{Z}U=0\hspace{0.17em}\text{at}\hspace{0.17em}Z=1.$(8)

This system can be turned into an eigenvalue problem $\sigma \widehat{U}=L\widehat{U}$ by writing $U(z,t)=\widehat{U}{e}^{\sigma t}$. The eigenspectrum is spanned by solving numerically the integrodifferential operator L, corresponding to the right-hand side of Eq. (7) (more details are provided in the supplemental material). We first focus on the classical scenario of a semi-infinite stratosphere, characterized by $\widehat{h}\ll 1$, for which the real part of the eigenvalue σ is plotted in Fig. 1a as a function of Re. In this figure, black signifies that a mode is oscillatory [Im(σ) ≠ 0], while orange signifies a mode that is not oscillatory [Im(σ) = 0]. Vertical modes are numbered based on the maximum number of zeros or nodes, excluding boundary conditions (see Fig. 2).

View Full Size

Indexation of eigenmodes by the number of nodes (colored dots). The real part of $\widehat{U}$ for oscillatory modes indexed $1^{\star }$, $3^{\star }$, and $5^{\star }$ in Fig. 1b, presenting, respectively, 1, 3, and 5 nodes is depicted. The same indexation applies for stationary modes.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0220.1

• Download Figure
• Download figure as PowerPoint slide

The point at which an eigenvalue changes sign marks the onset of unstable exponential growth (green dots in Fig. 1a). Within this semi-infinite domain context, the threshold of the first bifurcation in Fig. 1b corresponds to the instability of the first oscillating mode, denoted as $1^{\star }$. Remarkably, the threshold of the second and third bifurcations, as predicted in the quasilinear simulations, closely aligns with the instability of the third and fifth oscillating modes, referred to as $5^{\star }$ and $9^{\star }$. We conclude that the transition to chaos corresponds to the excitation of unstable modes with an increasing number of alternating horizontal jets in the vertical direction as the Reynolds number is increased. Interestingly, in the infinite Reynolds limit, the eigenmode analysis conceals a more streamlined analytical solution detailed in Plumb (1977), where the growth rate gradually diminishes from the bottom to the top in an exponential fashion. The key outcome of both analyses is the existence of a bottom-intensified amplification mechanism, on the order of F/(ch) in dimensional units.

The onset of instability of each mode is illustrated as black and orange lines in the bifurcation diagram of Fig. 1b. As $\widehat{h}$ increases, all unstable eigenvalues become purely real, as the wave operator L progressively becomes proportional to a Laplacian. This is why any black branch (oscillatory modes) split into two orange branches (nonoscillatory) modes as $\widehat{h}$ is increased in Fig. 1b. Consequently, the first QBO bifurcation at large $\widehat{h}$ is different from that in the semi-infinite case ($\widehat{h}\ll 1$). For values of $\widehat{h}$ larger than a critical threshold denoted as ${\widehat{h}}_c$, the emergence of the first unstable eigenvalue corresponds to a pitchfork bifurcation of fixed points (from a state of rest to bistability). The threshold of this first bifurcation corresponds to the instability of the first stationary mode, denoted as 0 in Fig. 1b. This bistable configuration becomes unstable at higher Re, with the emergence of a limit cycle around the two fixed points. The transition occurs before the second stationary unstable eigenvalue and is not captured by the linearized dynamics around a state of rest, since this would instead require a linearization around the fixed points. Surprisingly, the threshold of the second bifurcation, originally captured by the instability of the oscillatory mode $5^{\star }$ at $\widehat{h}\ll 1$, coincides with the instability of the stationary mode 4 as $\widehat{h}$ increases.

It is intriguing that the instability of the eigenmodes of the linearized system is capable of capturing the emergence of successive bifurcations beyond the first one given that they are computed around a state of rest. Additionally, the connection between the instability of certain modes and their impact on bifurcations, while others have a limited effect, is also somewhat puzzling. To fully understand these selection rules and explain the predictive ability of the linearized dynamics around a state of rest for subsequent bifurcations, a comprehensive Floquet analysis would be required. However, conducting such an analysis falls beyond the scope of this paper.

Dimensional analysis helps explain why increasing $\widehat{h}$ promotes periodicity. In a semi-infinite stratosphere where $\widehat{h}\to 0$, Re is the sole dimensionless parameter governing the system’s dynamics (Vallis 2017). When $\widehat{h}$ is large, a direct inspection of Eq. (7) shows that the appearance of new unstable eigenmodes is controlled by $\text{Re}/{\widehat{h}}^2$. This amounts to defining a new Reynolds number based on the domain height H rather than on the attenuation length scale h and on the reduced forcing $F/\widehat{h}$ rather than F. In the limit $\widehat{h}\gg 1$, bifurcations occur at finite values of $\text{Re}/{\widehat{h}}^2$, which causes the range of Re between each bifurcation to widen as $\widehat{h}$ increases, as observed in Fig. 1b.

We have seen that the emergence of aperiodic states involves the excitation of higher vertical modes with multiple jets. During their nonlinear evolution, some of these jets grow until reaching the maximum velocity set by the critical layers $\overline{u}=c$. In the case of a semi-infinite domain, the altitude at which these critical layers form scales roughly with h. As $\widehat{h}$ becomes comparable to 1 or larger, this altitude scales with the stratosphere height H. More generally, for a given Re, increasing h implies reduced wave damping, higher altitude wave propagation, and increased vertical homogeneity, concomitantly with periodicity recovery. We hypothesize that this vertical homogeneity in wave forcing promotes periodicity as it hinders the development of higher vertical modes. This hypothesis will be further substantiated in the case of multimodal forcing in section 3c.

b. A new metric to characterize dynamical regimes using vertical modes

As noted above, the onset of aperiodicity is inherently associated with the instability of higher vertical modes of the linearized system, with the number of alternating jets increasing as the forcing parameter grows. This excitation of higher vertical modes with increasing forcing is manifested in the quasilinear model, as illustrated using the Hovmöller plots of the zonal winds in Figs. 3a–d. Periodic oscillations are characterized by a maximum of one downward propagating flow reversal in the vertical at any instant (or 1-node).² In contrast, strongly forced aperiodic regimes exhibit time periods with multiple downward propagating nodes. Figure 3e depicts a bifurcation diagram of the histograms employing Poincaré maps, for which the information is condensed as a number of bins in Fig. 1b. This is contrasted with Fig. 3f, where the ratio of the number of flow reversals compared to that of the topmost grid point is plotted as a function of altitude. As the forcing is increased, faster flow reversals are observed at the bottom of the stratosphere compared to the top. These higher vertical structures can be interpreted as an entanglement of multiple oscillators with increasing frequency toward the ground (see Fig. 3d). Importantly, each major transition in the bifurcation diagram of Fig. 3e is closely linked to a regime transition in the vertical modes in Fig. 3f.

View Full Size

(a)–(d) Hovmöller diagrams for the wind speed profile at different Reynolds numbers for $\widehat{h}=1/3.5$. (e) Corresponding bifurcation diagram computed with the Poincaré metric. (f) Same bifurcation diagram computed with the new metric $N_0$, defined as the ratio between the number of flow reversals on every level to that of the upper one.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0220.1

• Download Figure
• Download figure as PowerPoint slide

Based on this observation, we propose a new metric for diagnosing QBO-like regimes that focuses on the vertical structure of the oscillations, rather than employing histograms of Poincaré maps (e.g., Renaud et al. 2019; Léard et al. 2020). For a given total simulated time, this new metric computes the ratio of the number of zeros counted in the bottom boundary layer (gray region in Fig. 3f) compared to that counted near the top. This ratio is loosely related to a vertical mode as it yields a time-averaged number of nodes in the column, excluding periods without zonal wind reversal from this average. The numbering of each vertical mode, depicted in Fig. 2, is based on the maximum number of zeros in the column, excluding boundary conditions. For oscillatory modes, we consider the maximum through a complete oscillation.

Hereafter, states with a ratio of 1, meaning a single node or mode-1, will be referred to as QBO-like states. Notably, this metric is better suited to describe the actual Earth QBO, which exhibits fluctuations around its average period while remaining trapped in a mode-1 state prior to 2016. Yet, under this new metric, both periodic and quasiperiodic solutions with mode-1 vertical structure are considered QBO-like. This results in a slight shift in the value of the second bifurcation compared to the Poincaré metric, corresponding to the quasiperiodic region (Figs. 3e,f).

This minor limitation is outweighed by a major advantage compared to Poincaré sections: It exhibits a monotonic increase with both Re and h. For example, quasiperiodic regimes are often wedged between periodic and frequency-locked regimes when increasing Re (Fig. 3e). This is diagnosed as a local maximum using Poincaré sections, whereas a monotonic increase is observed using vertical modes. The monotonicity of the metric greatly helps in interpreting the response of the system to a given change in parameter. In particular, if increasing a parameter reduces the number of nodes, it can be interpreted as favoring periodicity. Conversely, if the number of nodes increases with an increasing parameter, it suggests a tendency toward aperiodicity or more complex dynamics. This idea will greatly facilitate the interpretation of the results in the following section, which involve perturbations to the background QBO.

c. Periodicity recovery by perturbations to an aperiodic background state

To disentangle the role of the perturbation from that of the base state, we select the parameters of the background waves corresponding to the mode-3 oscillation shown in Fig. 3c, beyond the second bifurcation. We then conduct a series of experiments where we incrementally increase the perturbation F_p while keeping F_b constant—thus increasing the total forcing—for a wide range of phase speed ratios c_p/c_b.³ One might intuitively anticipate that as the total forcing increases, the system would transition toward higher vertical modes and chaos, as noted by Renaud et al. (2019). Paradoxically, our results, depicted in Fig. 4a, reveal a parameter space featuring instead two prominent regions where the flow transitions back to a periodic QBO-like oscillation: one for c_p < c_b, denoted as QBO₍₋₎, and another for c_p > c_b, denoted as QBO₍₊₎. Apart from a small region in the upper left corner of Fig. 4a (c_p ≪ c_b), where higher vertical modes with chaotic reversals are excited, no significant changes are observed elsewhere.

View Full Size

(a) Bifurcation diagram for the case of multiple wave forcing. See the text and Fig. 3 for a definition of the metric $N_0$. (b) Maximal wind speed recorded in each regime. (c) Frequency of wind reversals computed from a time series of the fastest oscillations near the lower boundary in each regime. In each panel, thick black contours indicate isolines of τ_p/τ_b.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0220.1

• Download Figure
• Download figure as PowerPoint slide

To understand the primary drivers in these regions, we consider the maximum velocity of the mean flow, displayed in Fig. 4b. When the background wave with phase speed c_b governs the system, the maximum speed of the mean flow is ${\overline{u}}_{\text{max}}\approx c_b$. In most of the parameter space, regimes are dominated by the background wave. There are two notable exceptions. First, the system undergoes an abrupt transition when c_p ≪ c_b, where the mean flow decelerates suddenly (Fig. 4b) as it becomes governed by the perturbation associated with chaotic high vertical modes (Fig. 4a). Second, a smooth transition occurs for c_p > c_b, where the mean flow gradually accelerates as it increasingly falls under the influence of the perturbation. This smooth transition for c_p > c_b aligns with a decrease in the frequency of wind reversals (as indicated in Fig. 4c) and is closely linked to the QBO₍₊₎ region of periodicity recovery observed in Fig. 4a. This contrasts with the region QBO₍₋₎, where the frequency of wind reversals increases while no change in mean flow is observed. Below, we focus individually on each of the periodic regions.

In the QBO₍₋₎ region, where the background wave governs the system, two types of critical levels can emerge: (i) critical levels associated with the background waves, which occur when the maximum wind speed approaches the phase speed of the background waves (${\overline{u}}_{\max }\approx \pm c_b$), and (ii) critical levels linked to the perturbation waves, which occur when the mean wind speed matches the horizontal phase speed of the perturbation waves ($\overline{u}=\pm c_p$).

To gain insight into the role of each type of critical level, we conduct an experiment depicted in Fig. 5, where a perturbation is introduced at t = 0 from the reference experiment using monochromatic forcing. After just a few oscillation cycles following the introduction of the perturbation wave, a QBO-like cycle is recovered (Fig. 5a), corresponding to the experiment marked with an orange star in Fig. 4. In both the monochromatic and multimodal cases, we analyze two snapshots typical of the westward phase of the mean winds ($\overline{u}<0$), as well as their associated wave forcing. The first snapshot is taken soon after the previous wind reversal, while the second snapshot is right before the next wind reversal (as shown in Fig. 5).

View Full Size

Transition from mode-3 to QBO-like oscillations with the addition of a perturbation wave at t = 0, corresponding to the orange star in Fig. 4. (a) Hovmöller diagram. Black and yellow contours trace the downward propagation of critical layers of the perturbation and the background waves, respectively. (b),(c) Instantaneous vertical profiles of the mean flow (green line) under monochromatic wave forcing, with blue and red dots representing the westward and eastward Reynolds stress derivatives, respectively (dot size is proportional to amplitude). (d),(e) As in (b) and (c), but for multiple wave forcing, where Reynolds stresses are decomposed into background wave (at $\overline{u}=\pm c_b$) and perturbation wave (at $\overline{u}=\pm c_p$) contributions.

Citation: Journal of the Atmospheric Sciences 81, 7; 10.1175/JAS-D-23-0220.1

• Download Figure
• Download figure as PowerPoint slide

In both cases, the forcing from the background waves is similar at both times (Figs. 5b–e). However, in the multimodal case, an additional eastward forcing from the perturbation is initially introduced in the upper part of the stratosphere (see t₃ in Fig. 5d). This high-altitude eastward forcing is a consequence of the Doppler shift’s impact on the local attenuation length scale,⁴ which allows the perturbation wave with an eastward phase speed to propagate upward with minimal attenuation until it reaches the upper part of the stratosphere, where the winds are much weaker. This efficient transport leads to high-altitude deposition of eastward momentum from the perturbation and to the development of a fast downward propagating critical layer that closely follows the zero-wind line (Figs. 5d,e). This fast-moving critical layer prevents the excitation of secondary westward jets or higher vertical modes responsible for aperiodicity, by smoothing the vertical wind profile (cf. Figs. 5c,e). This interaction between the perturbation and the mean flow is shown in greater detail in the online supplemental material videos, displaying complete mean-flow cycles prior to (t < 0) and after the injection of the perturbation (t ≥ 0) (see Videos 1 and 2, respectively).

Periodicity recovery in the regime QBO₍₋₎ results from a synchronization mechanism between the fast descent of the perturbation-driven zero-wind line and the background-driven mean-wind oscillation. Synchronization occurs when the downward propagation of the perturbation-driven critical layers is sufficiently faster than the growth of the higher unstable modes of the background flow. Assuming that all the momentum of the perturbation wave is efficiently used to move downward to the location where $\overline{u}=c_p$, the typical descent time scale of this critical layer is τ_p = c_ph_p/F_p. Meanwhile, following the linear analysis of Plumb (1977) in the large Re_b limit, the growth of higher vertical modes of the background flow occurs over a time scale τ_b = c_bh_b/F_b. A periodic state is thus expected when τ_p < τ_b.

In each panel of Fig. 4, lines representing the ratio τ_p/τ_b = 0.1, 1, 10 are displayed. The region QBO₍₋₎ is well represented by the area between τ_p/τ_b = 0.1 and 1, as expected from the above analysis. In this region, a periodic behavior with a QBO-like vertical structure emerges from the combination of perturbed and background wave forcing, which would independently lead to nonperiodic behavior. This synchronization mechanism ceases to be effective when the critical levels associated with the perturbation are fast enough to govern the oscillation itself, which becomes chaotic at τ_p/τ_b ≈ 0.1.

The occurrence of the region QBO₍₊₎ coinciding with the line of τ_p/τ_b = 10 suggests that a similar synchronization mechanism is at play in this second regime of periodicity recovery. However, in this case, the roles of the background and perturbation are somewhat reversed: Critical levels associated with the background forcing occur progressively at higher altitudes as the maximum mean flow gradually accelerates when the oscillation falls under the influence of the perturbation (see Fig. S2 and Video 3). Additionally, the transfer of momentum from the perturbation to the background state is much less efficient than in the QBO₍₋₎ region due to a compensation effect between the wave momentum flux induced by the eastward-moving and westward-moving perturbations, which prevents these perturbations from reaching critical levels. An extreme case is observed when c_p/c_b ≫ 1, where the perturbation has virtually no effect on the wind reversals. This is evident in Fig. 4, where QBO properties remain unchanged beyond c_p = 10c_b. This occurs because the wave momentum flux induced by the eastward-moving perturbation becomes nearly equal to the contribution from the westward-moving perturbation when $\overline{u}/c_p\ll 1$, as described by Eq. (2). For finite values of c_p/c_b > 1, the primary effect of the perturbation is to shift the maxima of the wind profile to values greater than the critical level c_b. Assuming that most of the descent of these maxima remains dominated by contributions from the background wave forcing, the period of the QBO can be estimated as ${\overline{u}}_{\text{max}}{h}_b/F_b$, with ${\overline{u}}_{\text{max}}$ being an increasing function of F_p. This explains the decrease in frequency associated with the QBO₍₊₎ region in Fig. 4c.

In summary, perturbation waves with phase speeds c_p < c_b lead to a faster descent of the QBO front to the bottom of the stratosphere, resulting in a shorter period of wind reversals. On the other hand, waves with c_p > c_b amplify the amplitude of the QBO front, leading to an extended period of reversals. In both cases, the presence of a wave with a slower phase speed prevents the growth of unstable modes of the larger wave due to resonance between two time scales: any local extremum in the velocity profile is shifted downward before its amplitude reaches a critical level. This effectively prevents the excitation of higher vertical modes responsible for aperiodicity and explains the observed periodicity recovery regions characterized by a vertical mode-1 akin to the QBO. This phenomenon is all the more surprising as it emerges from the combination of perturbed and background wave forcing that independently would lead to nonperiodic reversals.