## 1. Introduction

Differential radiative cooling over Earth’s poles gives rise to the strong circumpolar vortices that form in the winter stratosphere. These are dynamically perturbed by planetary-scale Rossby waves, producing intermittent vacillations in the strength of the vortices as a result of competition between radiative cooling and wave–mean flow interactions. These vacillations are most severe in the Arctic vortex, which undergoes stratospheric sudden warmings in which the climatological westerly flow completely reverses roughly two out of every three winters (Charlton and Polvani 2007). They can occur at any time throughout the extended winter season, and exert a significant downward influence on the troposphere (Baldwin and Dunkerton 2001; Hitchcock and Simpson 2014). Because they are not tied to the seasonal cycle, and because the associated anomalies can persist for considerably longer than typical tropospheric midlatitude weather, their contribution to subseasonal and seasonal forecasting is increasingly valued (Sigmond et al. 2013; Butler et al. 2019; Domeisen et al. 2019a,b).

The central role of planetary-wave-induced deceleration of the vortex in stratospheric vacillations was first demonstrated by Matsuno (1971). Matsuno described the effects on the mean flow of a conservative Rossby wave train propagating upward into the climatological westerlies. This wave train acts to decelerate the winds, ultimately generating a critical line as the winds reverse. Nonlinearity at this critical line produces irreversible absorption of the wave activity, ultimately leading to the breakdown of the westerly vortex.

In Matsuno’s model, the amplification of planetary waves is explained as a result of tropospheric processes. This ties the intermittency of stratospheric vacillations to tropospheric variability. An alternative explanation for the amplification is that the planetary waves grow when a free mode of the vortex comes into resonance with the quasi-stationary forcing (Clark 1974; Tung and Lindzen 1979; Plumb 1981). In contrast to Matsuno’s theory, this places more emphasis on the role of the stratosphere itself in generating the intermittency of these events. The validity of these two perspectives in explaining observed variability continues to be debated (e.g., Albers and Birner 2014; Butler et al. 2019).

The theory has been further developed in the context of “contour dynamics,” in which the potential vorticity distribution is assumed to be piecewise constant, allowing for a description of the system in terms of the boundaries between regions with different values of potential vorticity (PV). The polar vortex is idealized as a patch of constant potential vorticity, separated by a discontinuous jump from the “surf zone,” a region of lower potential vorticity. Under adiabatic, inviscid conditions, the evolution of the vortex can be fully described in terms of the contour separating the two regions. In this context, the existence and importance of a barotropic mode was pointed out by Esler and Scott (2005). The role of this barotropic mode in the resonance-based theory of stratospheric variability has been further discussed by Matthewman and Esler (2011) and by Esler and Matthewman (2011).

These essentially adiabatic theories focus on the dynamics of the wind reversals themselves, ignoring the role of the slower diabatic processes that restore the westerly winds. In contrast, the competition between these two processes lies at the heart of the “Holton–Mass” model first introduced by Holton and Mass (1976). They considered a continuously stratified quasigeostrophic flow in a beta channel, reducing the flow to a vertically varying profile of zonal mean winds and a single planetary wave mode, both with a fixed meridional structure. The wave propagates quasi linearly from a source at the lower boundary, and influences the mean flow through the eddy fluxes of PV that it induces. Both the wave and the mean flow are subject to a simple parameterization of radiative damping, which relaxes the system toward a zonally symmetric, westerly vortex.

Despite the highly simplified nature of the Holton–Mass model and the absence of an explicitly varying wave source, it exhibits intermittent “vacillations” that resemble the dynamics of observed stratospheric variability in meaningful ways (Yoden 1987b). The model exhibits multiple steady states (Yoden 1987a), quasi periodicity, and chaos (Christiansen 2000). It has been highly influential as a conceptual model of stratospheric variability (Yoden et al. 1999), providing insight into how the stratosphere can produce its own dynamical variability (Yoden 1987c; Sjoberg and Birner 2014; de la Cámara et al. 2019).

The purpose of this paper is to introduce a new, minimal model of stratospheric vacillations, based on contour dynamics but incorporating diabatic effects. Like the Holton–Mass model, it includes a representation of the strength of the zonal mean vortex and a single planetary-wave mode. The model is developed within the framework of contour dynamics, which arguably provides a more natural reduction of the meridional and vertical structure of the flow than the beta channel of the Holton–Mass model. The derivation presented here begins with the quasigeostrophic shallow-water equations (Polvani and Plumb 1992; Matthewman and Esler 2011), emphasizing barotropic, horizontal dynamics of the vortex over the vertical propagation of waves (see also Liu and Scott 2015; Scott 2016). However, the equations can also be derived from the continuously stratified linear theory of Esler and Matthewman (2011). The essential new ingredient is to introduce a simple parameterization of irreversible wave–mean flow interactions between the vortex and an edge mode. A simplified representation of radiative dissipation of the waves and restoration of the vortex PV gradient is also invoked.

The result is a system of three coupled nonlinear ordinary differential equations, representing the amplitude and phase of the planetary wave, as well as the PV jump at the vortex edge. Despite its extreme simplicity, it exhibits many characteristics in common with the Holton–Mass model. The dependence of the model’s variability on external parameters is made much more transparent. This clarifies, for instance, that in the more dynamically active regions of parameter space, the time-mean state of the vortex is determined more by the nature of the linear resonance than by the strength of the wave driving.

This new low-dimensional system of equations is similar to a previously derived set due to Ruzmaikin et al. (2003), based on a severe truncation of the finite-difference form of the Holton–Mass model (see also Birner and Williams 2008; Esler and Mester 2019). One advantage of the present system is that the approximations made are arguably more natural and transparent. This allows for a clear physical interpretation of the external parameters. The system is also similar to the highly truncated barotropic model of blocking due to Charney and DeVore (1979). Comparisons to these models are discussed further in the main text.

The paper is organized as follows. In the next section the model equations are derived, reviewing the formalism of contour dynamics and introducing the diabatic parameterization. In section 3, the behavior of the equations is described, demonstrating multiple steady states, periodic and chaotic trajectories. The implications of the model are discussed in section 4, and a summary and conclusions are given in section 5. A glossary of the most relevant symbols is provided (Table 1). Finally, some analytical results regarding the steady states of the equations and their stability are presented in two appendixes.

Definitions of symbols directly relevant to the three-parameter model and relevant defining equations. Nondimensionalized quantities are indicated by (ND).

## 2. Deriving the model equations

### a. Inviscid, adiabatic flow

The starting point is the evolution of a linear perturbation to a quasigeostrophic shallow-water vortex on a polar *f* plane (Polvani and Plumb 1992; Matthewman and Esler 2011). The vortex is modeled as a near-circular patch of constant potential vorticity. The inviscid, adiabatic dynamics of this perturbation are reviewed here in detail because they form the basis for the diabatic model.

*ψ*is the streamfunction for the geostrophic flow

**u**

_{g}; thus in polar coordinates (

*r*,

*θ*) the azimuthal and radial velocities are given by

*u*

_{g}= ∂

_{r}

*ψ*and

*θ*will be referred to as the longitude. The deformation radius

*f*

_{0}is the Coriolis parameter,

*D*is the reference depth, and

*g*is the acceleration due to gravity. Topographic forcing is included by specifying the bottom topography

*h*

_{b}.

*R*and potential vorticity jump

In the following *ϵ* is taken to be small so that the dynamics remain linear in *r* → ∞, with the exception that a zonally symmetric flow is included to allow for a background flow independent of the PV jump

*I*

_{n}(

*r*) and

*K*

_{n}(

*r*) are the modified Bessel functions of the first and second kinds, of order

*n*. The flow consists of a component proportional to the PV jump itself, and a background flow

*R*Ω

_{b}unrelated to

*ψ*

_{Q}is associated with the anomaly

*k*:

*ψ*

_{Q}has a discontinuity in its meridional derivative at the vortex edge that is related to

*r*=

*R*. Away from the vortex edge the perturbation PV vanishes. These conditions determine

*ψ*

_{Q}, producing a meridional flow at the vortex edge:

*ψ*

_{T}relatively simple: setting

*h*

_{b}(

*r*,

*θ*) =

*h*

_{0}

*J*

_{k}(

*λr*)cos

*kθ*leads to an additional meridional flow at the vortex edge:

*J*

_{n}(

*r*) is the Bessel function of the first kind of order

*n*,

*k*is the zonal wavenumber, and

*λ*sets the meridional scale of the topography.

*ϵ*, perturbations to the circular vortex are advected by the flow (5) at the vortex edge:

*S*

_{k}(

*R*). Multiplied by the PV jump, this corresponds to the intrinsic phase speed of the wave. Notably, this is always nonnegative, meaning that the intrinsic phase speed of the wave becomes more easterly as the PV gradient sharpens. The dependence of the phase speed of the wave on the geometry

*R*and strength

### b. Diabatic parameterizations

Taking into account diabatic effects such as radiative cooling or irreversible mixing in the context of contour dynamics is problematic, because a realistic account of these effects will tend to generate a smooth distribution of potential vorticity. This destroys the piecewise constant character of the potential vorticity distribution that permits the contour dynamics approach in the first place (Dritschel and Ambaum 2006). However, the action of breaking Rossby waves tends to sharpen them (Dritschel and McIntyre 2008), justifying the use of a constant PV jump in the first place.

The approach adopted here is to assume that this tendency to sharpen PV gradients is sufficiently strong that the contour dynamics description remains valid. To form a closed, forced-dissipative dynamical system, the diabatic effects of radiative heating and small-scale mixing are parameterized in the context of the adiabatic system just described in a manner that retains the piecewise constant character of the PV distribution.

The basic, zonal mean flow is specified by the vortex radius *R* and PV jump *R* or *R* through *S*_{k}(*R*) is nonlinear. Thus for simplicity the PV jump

The evolution of _{E} on the time scale *τ*_{V}. Since this diabatic equilibrium includes the gradient-sharpening effects of the breaking waves, it is not a purely radiative equilibrium.

*R*and

*k*is absorbed into the “effective diffusivity” coefficient

*κ*.

*τ*

_{W}:

Equations (14) and (15) form a closed, forced-dissipative dynamical system. The zonal mean vortex strength

The nature of the wave–mean flow interaction in these equations warrants some further discussion. The aspect that has been included is dissipative and irreversible in nature, and does not include the effects of wave transience (Holton and Dunkerton 1978). The latter are central to the theories of self-tuned resonance (Plumb 1981; Matthewman and Esler 2011; Esler and Matthewman 2011), which focus on the transient evolution of the vortex around the onset of major sudden warmings. They arise to leading order in this context to third order in *ϵ* as an additional term proportional to

## 3. A three-parameter model of stratospheric vacillations

### a. Nondimensionalization

*τ*

_{W}. To simplify notation,

*t*from this section forward refers to the nondimensional time.

*k*Δ

_{E}

*S*

_{k}(

*R*) to the damping rate of the wave

_{b}to the intrinsic phase speed of the wave at diabatic equilibrium Δ

_{E}

*S*

_{k}(

*R*). Critically, when the nondimensional PV jump Δ is equal to

The third parameter, *τ*_{W} to the time scale of the diabatic restoration of the vortex *τ*_{V}.

The dynamics of the wave mode (18a) are that of a forced, damped, harmonic oscillator. Under the chosen nondimensionalization, the forcing and dissipation are fixed but the frequency of the oscillation

The dynamics of the nondimensional PV jump are a balance between the radiative restoration of the vortex toward an equilibrium at Δ = 1, and the wave driving term

The three real degrees of freedom in (20) are illustrated schematically in Fig. 1 for a zonal wavenumber two perturbation. The sign of *φ* is chosen so that a crest of the wave can be found at longitude *θ* = *ϕ*/*k*, as can be confirmed by substituting back into (6). The topographic flow is most effective at amplifying the wave when the phase *ϕ* = *π*/2 + 2*mπ* for integer *m*.

*x*=

*a*cos

*ϕ*and

*y*=

*a*sin

*ϕ*:

### b. Some limiting behavior

If the initial wave amplitude is not greater than its resonant value, the amplitude *a* remains between 0 and 1. Likewise, the tendency of the PV jump is zero or negative when Δ is at its equilibrium value of 1, hence if the initial value of Δ lies between 0 and 1 it remains so. Transient behavior from initial values outside these ranges will not be further considered here. The system is invariant under the transformation

The nondimensional parameter

#### 1) Slow restoration of the vortex

*τ*and

*T*can be defined such that

*a*and

*φ*to be functions of

*τ*and

*T*, while Δ is only a function of the slow time scale

*T*. At leading order, the evolution of the wave components on the fast time scale sees a constant PV jump Δ(

*T*). In this limit the wave dynamics are that of a linear, forced, damped oscillator. On the fast time scale

*τ*, the wave (18a) relaxes toward a quasi-steady state given by

Tendency of vortex PV jump, (23), as a function of the PV jump itself for (a) constant

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Tendency of vortex PV jump, (23), as a function of the PV jump itself for (a) constant

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Tendency of vortex PV jump, (23), as a function of the PV jump itself for (a) constant

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

The weakening of the PV jump produced by wave–mean flow interactions is strongest when the wave amplitude is large, that is, when the phase speed of the wave is at resonance. Since Δ lies between 0 and 1, this resonance is directly accessible for values of

Finally,

Since the value of

When (23) holds, one can expect that the system will ultimately approach one or the other of the two steady states. There are nonetheless transient trajectories in this limit that bear some resemblance to stratospheric sudden warmings. For instance, consider an initial state with a PV jump just less than the unstable intermediate steady state. An example of such a trajectory is given in Fig. 3, which shows the evolution of the wave amplitude and the PV jump for both the full set of equations, (20), and for the asymptotic system.

Time series of wave amplitude *a* and PV jump Δ for parameters given in panel title. Solid curves show the evolution of the full system, while dashed lines show the asymptotic form, (23).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Time series of wave amplitude *a* and PV jump Δ for parameters given in panel title. Solid curves show the evolution of the full system, while dashed lines show the asymptotic form, (23).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Time series of wave amplitude *a* and PV jump Δ for parameters given in panel title. Solid curves show the evolution of the full system, while dashed lines show the asymptotic form, (23).

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Initially the decay of the PV jump is weak since the system remains far from resonance and the wave amplitude is small. However, as the PV jump approaches the resonance, the wave will begin to be more efficiently forced by the topographic flow, leading to rapid amplification (relative to the slow time scale *T*). This in turn will accelerate the depletion of the PV jump. Once the jump drops below the resonance the wave will weaken again until a steady state is achieved. The asymptotic solution agrees well with the full solution over the initial slow depletion of the vortex. However, the quasi-steady assumption for the wave fails as the vortex approaches the resonant point. In the full system, the wave takes a finite time to amplify, leading to large amplitude transients in the aftermath of the period of rapid weakening. As these transients die out, the full system relaxes again toward the asymptotic solution.

#### 2) Fast restoration of the vortex

### c. Steady states and their stability

_{1}is real for

Further analytical results regarding the location of transitions from one to three steady states are given in appendix A.

Figure 4a shows the steady-state values of Δ as a function of the nondimensional forcing

(a) Vortex PV jump Δ of the steady-state solutions as a function of *κ* close to the emergence of the weak and intermediate vortex states.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

(a) Vortex PV jump Δ of the steady-state solutions as a function of *κ* close to the emergence of the weak and intermediate vortex states.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

(a) Vortex PV jump Δ of the steady-state solutions as a function of *κ* close to the emergence of the weak and intermediate vortex states.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

The stability of these steady states can be studied through the Jacobian matrix of (20); see appendix B for details. The real and imaginary components of the eigenvalues of this matrix at the weak vortex state are shown in Figs. 4b and 4c. As

The behavior of the full system is summarized for a broader range of parameter space in Fig. 5. Figures 5a–c show the

Phase portraits of the three-parameter dynamical system in the

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Phase portraits of the three-parameter dynamical system in the

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Phase portraits of the three-parameter dynamical system in the

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

The solid black curves in Fig. 5 indicate the transitions between one and three steady states. Within the wedges there are three steady states, outside there is only one. For instance, the parameter values in Fig. 4a correspond to a vertical line at

The filled contours in Fig. 5 show the time-mean value of Δ for representative trajectories. The trajectories are initialized in one direction along the unstable manifold of the intermediate vortex state where it exists; where it does not, the real part of the least stable eigenvector of the sole steady state is used instead.

Figure 5 also indicates the linear stability of the weak vortex state. Along the dashed blue curves, the largest real component of the eigenvalues of the Jacobian matrix at the steady state with the lowest value of Δ. To the right of this contour the weak vortex state is unstable.

Consider first the behavior of the system at fixed

For larger values of

Where the weak vortex state is unstable, trajectories generally tend toward nonsteady limit cycles as will be explored further in the next section. An exception to this is the series of lobe-like regions in which the trajectories again tend toward the strong vortex state. With the exception of these regions, the time mean value of Δ in the nonsteady trajectories remains close to

Considering instead the behavior as

The nature of the dependence of the time-mean vortex jump on the external parameters is one of the central results of this work: above the critical point where the weak vortex state emerges discontinuously, the strength of the vortex in these steady states is primarily determined by the location of the resonance

### d. Unsteady trajectories

The character of stable orbits is further explored in Fig. 6. Each panel indicates local extrema of Δ as the forcing magnitude

Bifurcation diagrams showing local extrema of Δ (black points) in trajectories as a function of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Bifurcation diagrams showing local extrema of Δ (black points) in trajectories as a function of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Bifurcation diagrams showing local extrema of Δ (black points) in trajectories as a function of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

At

At

As

(a) Trajectories along unstable manifold of intermediate fixed point for three values of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

(a) Trajectories along unstable manifold of intermediate fixed point for three values of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

(a) Trajectories along unstable manifold of intermediate fixed point for three values of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Consistent with Fig. 5b, there exists a range of values of

A further period doubling bifurcation at

At

More complex cases can also be found in which multiple stable attractors coexist. For instance, Fig. 7b shows three trajectories with different initial conditions for

### e. Dynamics of the vacillations

Figure 8 shows the evolution of the amplitude and phase of the wave and the PV jump for an example trajectory in the chaotic regime highlighted in Fig. 6d.

Time series of (a) PV jump Δ and wave amplitude *a* and (b) wave phase *ϕ* for a chaotic trajectory. Vertical gray lines are placed for reference at local maxima in the tendency of *ϕ*.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Time series of (a) PV jump Δ and wave amplitude *a* and (b) wave phase *ϕ* for a chaotic trajectory. Vertical gray lines are placed for reference at local maxima in the tendency of *ϕ*.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Time series of (a) PV jump Δ and wave amplitude *a* and (b) wave phase *ϕ* for a chaotic trajectory. Vertical gray lines are placed for reference at local maxima in the tendency of *ϕ*.

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

The trajectory is characterized by vacillations in both the strength of the PV jump and the amplitude of the wave that occur on the radiative time scale, or equivalently, since

The PV jump vacillates between roughly 30% and 50% of the equilibrium value. For the majority of the time it lies somewhat below the location of the resonance at

Wave growth occurs when *ϕ* lies between 0 and *π*; that is, when the topographic flow is aligned favorably with the orientation of the vortex edge perturbation (see Fig. 1). Thus the most sustained periods of wave growth occur when the phase speed of the wave slows down or even reverses within this range. Examples of this are highlighted by vertical lines in Fig. 8, occurring near *t* = 51.1, 54, and 58.2. This requires the PV jump to bring the intrinsic frequency near to the background wind speed. These periods are initiated when the PV jump grows close to the location of the resonance at Δ = 0.5.

At this point, the westward progression of the wave comes to a near halt. This occurs at a consistent phase relative to the topographic forcing favorable for wave amplification. For a wave two perturbation this is roughly consistent with the orientation shown in Fig. 1. The wave amplitude, initially near a global minimum, then grows in response to the topographic forcing until it approaches a global maximum value. This amplification is associated with an increase in wave activity matching the barotropic structure of the wave.

The tendency for the wave to slow down prior to amplification is consistent with similar adiabatic models (Matthewman and Esler 2011) and with behavior observed prior to stratospheric sudden warmings (Matthewman and Esler 2011).

## 4. Discussion

The primary benefit of such a highly simplified model is to clarify how the model’s behavior depends on its external parameters. The most important insight from the analysis above is the central role played by the dependence of the Rossby wave’s phase speed on the strength and structure of the polar vortex.

For the shallow-water, barotropic case considered here, the phase speed depends linearly on the PV jump at the vortex edge, (12), and is determined by the nondimensional parameters

On the other hand, if

The critical point described in Fig. A1 (and discussed in more detail in appendix A) provides one quantitative indication of the transition between these two regimes. Above the critical values of

A second benefit of such a simple model is that the relevant dynamical processes are more easily identified than in more complex models. In the present model, by construction, there is no variability in the wave source. The vacillations exhibited arise from the interaction between the wave forcing and the vortex. As was emphasized by Holton and Mass (1976), it demonstrates that stratospheric variability can arise internally, even in the absence of variability in the wave source. Indeed, the general lack of sensitivity of the vortex to the wave driving in the present model just discussed might suggest a reason why the observed Arctic polar vortex in fact does not respond to many episodes of enhanced tropospheric wave driving (Birner and Albers 2017). The barotropic nature of the present model demonstrates that vertical propagation is not necessary for this internal variability. This is consistent with Liu and Scott (2015) and Scott (2016) who demonstrate sudden-warming-like vacillations in a fully nonlinear shallow-water model on the sphere.

The chaotic trajectory in Fig. 8 also presents a particularly clear example of the relevance of “preconditioning” to sudden stratospheric warmings (see Liu and Scott 2015, and references therein). The PV jump grows just prior to periods of sustained wave growth, preconditioning the vortex for its own destruction as the wave driving amplifies. The role this preconditioning plays is made particularly clear in this context, but it seems likely that any other aspect of the vortex geometry that affects the phase speed of free modes could play a similar role in controlling the wave response if it were also allowed to vary.

While this model is intended to be paradigmatic, not quantitative, it is nonetheless worth briefly considering what values of *S*_{k}(*R*) plays a central role: for a wave-two mode on a shallow-water vortex with a radius several times the deformation radius, *τ*_{W} of 20 days, this implies an equilibrium PV jump of close to 2*f*_{0}. Previous research has taken Δ = *f*_{0} as a characteristic value of the observed PV jump in the Arctic vortex (Matthewman and Esler 2011). This would then require that the equilibrium vortex be stronger than the observed state by a factor of 2. This is plausible, given that radiative equilibrium temperatures are also estimated to be considerably colder than those observed (Fels 1985). Finally, in order for *f* plane (Polvani and Plumb 1992; Matthewman and Esler 2011).

The restriction to a single wave mode is of course highly unrealistic; in the real stratosphere there exists a whole spectrum of wave modes, including the vertically propagating continuum (Esler and Scott 2005; Esler and Matthewman 2011). Nonetheless, the behavior of the present model is in many ways similar to the Holton–Mass model, which includes a much broader spectrum of waves and a more complete depiction of their dependence on the vortex state. The results presented here suggest that understanding of the phase speeds of relevant barotropic modes of the polar vortex and their dependence on the vortex strength and shape could improve not only our understanding of the dynamics of stratospheric sudden warmings themselves, but also our understanding of the climatological strength of the polar vortices.

## 5. Summary and conclusions

A minimal model of stratospheric vacillations has been developed based on quasigeostrophic contour dynamics and a parameterization of several key diabatic effects. The model consists of a set of three coupled ordinary differential equations, (20), representing the amplitude and phase of single Rossby wave mode, and the strength of potential vorticity (PV) jump at the edge of the polar vortex on which the wave propagates. Notably this model explicitly captures the competition between wave driving and radiative restoration at the core of polar vortex variability.

The behavior of the model is controlled by four nondimensional parameters. Two parameters determine the linear relationship between the phase speed of the wave and the PV jump:

The model exhibits multiple stable limiting behaviors: a strong, stable vortex state close to equilibrium, and a weak vortex state at which the PV jump at the vortex edge lies just below the resonant value

The model exhibits a range of behaviors that share key features with the Holton–Mass model (Holton and Mass 1976), another simplified model of stratospheric vacillations that has been influential in forming the conceptual understanding of polar stratospheric variability. The strong and weak states can be identified with similar steady states in the Holton–Mass model (Yoden 1987a), as can the instability of the weak vortex state to periodic and aperiodic vacillations (Yoden 1987a; Christiansen 2000).

The trajectories in this dynamically active regime share some characteristics with observed events, including preconditioning of the vortex and the alignment of phase with the topographic forcing (Matthewman et al. 2009) just prior to the rapid amplification of the wave and depletion of the PV jump at the vortex edge.

The model is closely related to similar sets of equations derived by Charney and DeVore (1979) and Ruzmaikin et al. (2003). These also consist of a wave interacting with a mean flow, but the wave driving occurs through a mountain torque effect instead of the parameterized wave breaking considered here. They also exhibit multiple steady states; the explanation given by Held (1983) depends on the same resonant character of the wave forcing (their Fig. 6 can be compared with Fig. 2 above). However, to the author’s knowledge, periodic and aperiodic trajectories have not been reported in either system. It may be that the form of the wave–mean flow interaction term is important for the nature of the unsteady trajectories, but this speculation has not been confirmed.

The primary merit to the present model is that the dependence of the behavior on external parameters is made particularly clear. Crucially, the behavior is strongly determined by the linear relationship between the phase speed of the free mode and the PV jump at the vortex edge. In particular, in much of parameter space the time-mean strength of the vortex in this minimal model tends to lie just below the value at which the free mode is in resonance with the topographic forcing. It is only weakly sensitive to the strength of the forcing, with the exception of bifurcation points at which the character of the trajectories change discontinuously. This may provide an explanation for why the observed Arctic vortex frequently does not respond to episodes of enhanced tropospheric wave driving (e.g., Birner and Albers 2017). It also points toward the importance of better understanding the free modes of the observed vortex, including their dependence on properties of the vortex itself.

This model also exhibits two dynamical behaviors that have not previously been reported in similar models. First, in certain regions of parameter space, the weak vortex state exists, but there appear to be no stable dynamically active trajectories. Instead, the typical limiting behavior is the strong vortex state. It should be noted that this claim is based on finite numerical explorations; other stable limit cycles may exist that were not found by the present investigations. Second, at larger values of

Finally, the simple diabatic parameterizations proposed here can easily be applied to more general configurations of potential vorticity contours, allowing for the construction of other low-dimensional models.

## Acknowledgments

The author gratefully acknowledges support from Cornell University. Conversations with Gavin Esler, Peter Haynes, and Richard Scott were helpful in developing the ideas underlying the diabatic parameterization. Insightful reviews from Shigeo Yoden and two anonymous reviewers and editing by Hannah Moland substantially improved the manuscript.

## APPENDIX A

### Transitions One to Three Steady States

*D*of (27). After some algebra, this can be shown to be the following polynomial in

*D*is positive (27) has three roots; when it is negative (27) has only one. The sign of

*D*is determined by the quadratic in

Considering first the limit *D* will be dominated by the polynomial *p*_{3}. Setting *p*_{3} = 0 again confirms the asymptotic form (24) for the saddle-node bifurcation at which the weak and intermediate states emerge. In Figs. 5a–c this corresponds to the lower boundaries of the wedges indicated by the black curves, which approach the constant values of

*O*(1) terms in

*D*yields the quadratic

*β*≈ 0.069, 0.023, and 0.0013, respectively. Hence the strong vortex state survives to much larger forcing strengths

The upper boundary in Fig. 5d also indicates the disappearance of the strong vortex state. However, in this plane, as

In both planes, the two boundaries meet at a critical point below which the strong vortex state can transition smoothly to the weak vortex state.

Some further understanding of how these phase portraits differ in other parameter regimes can be inferred from the asymptotic forms (24) and (A4), and from the location of the critical point where these two boundaries intersect. Figure A1 shows the roots of (A4) as a function of *β* grows as the resonance moves to lower values of the PV jump, indicating that the strong vortex state survives to larger magnitude forcings the further away the resonance is from the equilibrium value of the PV jump. At

(a) Positive roots of (A4) giving the coefficient *β* of the asymptotic boundary at which the strong vortex state ceases to exist. (b) Values of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

(a) Positive roots of (A4) giving the coefficient *β* of the asymptotic boundary at which the strong vortex state ceases to exist. (b) Values of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

(a) Positive roots of (A4) giving the coefficient *β* of the asymptotic boundary at which the strong vortex state ceases to exist. (b) Values of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

The critical point occurs when the discriminant

## APPENDIX B

### Stability of the Weak Vortex State

*a*

_{0}= sin

*ϕ*

_{0}and

*λ*of this matrix are given by the roots of its characteristic polynomial:

*λ*are purely imaginary complex conjugates. This requires that

*a*

_{0}= Δ

_{1}, which does not depend on

Value of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Value of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

Value of

Citation: Journal of the Atmospheric Sciences 78, 7; 10.1175/JAS-D-20-0296.1

In contrast *τ*_{V} is longer than the wave damping time scale *τ*_{W} (*τ*_{W} is longer than *τ*_{V} (

## REFERENCES

Albers, J. R., and T. Birner, 2014: Vortex preconditioning due to planetary and gravity waves prior to sudden stratospheric warmings.

,*J. Atmos. Sci.***71**, 4028–4054, https://doi.org/10.1175/JAS-D-14-0026.1.Baldwin, M. P., and T. J. Dunkerton, 2001: Stratospheric harbingers of anomalous weather regimes.

,*Science***294**, 581–584, https://doi.org/10.1126/science.1063315.Birner, T., and P. D. Williams, 2008: Sudden stratospheric warmings as noise-induced transitions.

,*J. Atmos. Sci.***65**, 3337–3343, https://doi.org/10.1175/2008JAS2770.1.Birner, T., and J. R. Albers, 2017: Sudden stratospheric warmings and anomalous upward wave activity flux.

,*SOLA***13A**, 8–12, https://doi.org/10.2151/sola.13A-002.Butler, A., and Coauthors, 2019: Sub-seasonal predictability and the stratosphere.

*Sub-Seasonal to Seasonal Prediction*, A. W. Robertson, and F. Vitart, Eds., Elsevier, 223–241, https://doi.org/10.1016/B978-0-12-811714-9.00011-5.Charlton, A. J., and L. M. Polvani, 2007: A new look at stratospheric sudden warmings. Part I: Climatology and modeling benchmarks.

,*J. Climate***20**, 449–469, https://doi.org/10.1175/JCLI3996.1.Charney, J. G., and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking.

,*J. Atmos. Sci.***36**, 1205–1216, https://doi.org/10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2.Christiansen, B., 2000: Chaos, quasiperiodicity, and interannual variability: Studies of a stratospheric vacillation model.

,*J. Atmos. Sci.***57**, 3161–3173, https://doi.org/10.1175/1520-0469(2000)057<3161:CQAIVS>2.0.CO;2.Clark, J. H. E., 1974: Atmospheric response to the quasi-resonant growth of forced planetary waves.

,*J. Meteor. Soc. Japan***52**, 143–163, https://doi.org/10.2151/jmsj1965.52.2_143.de la Cámara, A., M. Abalos, and P. Hitchcock, 2018: Changes in stratospheric transport and mixing during sudden stratospheric warmings.

,*J. Geophys. Res. Atmos.***123**, 3356–3373, https://doi.org/10.1002/2017JD028007.de la Cámara, A., T. Birner, and J. R. Albers, 2019: Are sudden stratospheric warmings preceded by anomalous tropospheric wave activity?

,*J. Climate***32**, 7173–7189, https://doi.org/10.1175/JCLI-D-19-0269.1.Domeisen, D. I. V., and Coauthors, 2019a: The role of the stratosphere in subseasonal to seasonal prediction: 1. Predictability of the stratosphere.

,*J. Geophys. Res. Atmos.***125**, e2019JD030920, https://doi.org/10.1029/2019JD030920.Domeisen, D. I. V., and Coauthors, 2019b: The role of the stratosphere in subseasonal to seasonal prediction: 2. Predictability arising from stratosphere-troposphere coupling.

,*J. Geophys. Res. Atmos.***125**, e2019JD030923, https://doi.org/10.1029/2019JD030923.Dritschel, D. G., and M. H. P. Ambaum, 2006: The diabatic contour advective semi-Lagrangian model.

,*Mon. Wea. Rev.***134**, 2503–2514, https://doi.org/10.1175/MWR3202.1.Dritschel, D. G., and M. E. McIntyre, 2008: Multiple jets as PV staircases: The Phillips effect and the resilience of eddy-transport barriers.

,*J. Atmos. Sci.***65**, 855–874, https://doi.org/10.1175/2007JAS2227.1.Esler, J. G., and R. K. Scott, 2005: Excitation of transient Rossby waves on the stratospheric polar vortex and the barotropic sudden warming.

,*J. Atmos. Sci.***62**, 3661–3682, https://doi.org/10.1175/JAS3557.1.Esler, J. G., and N. J. Matthewman, 2011: Stratospheric sudden warmings as self-tuning resonances. Part II: Vortex displacement events.

,*J. Atmos. Sci.***68**, 2505–2523, https://doi.org/10.1175/JAS-D-11-08.1.Esler, J. G., and M. Mester, 2019: Noise-induced vortex-splitting stratospheric sudden warmings.

,*Quart. J. Roy. Meteor. Soc.***145**, 476–494, https://doi.org/10.1002/qj.3443.Fels, S. B., 1985: Radiative-dynamical interactions in the middle atmosphere.

,*Adv. Geophys.***28**, 277–300, https://doi.org/10.1016/S0065-2687(08)60227-7.Held, I. M., 1983: Stationary and quasi-stationary eddies in the extratropical troposphere: Theory.

*Large-Scale Dynamical Processes in the Atmosphere*, B. J. Hoskins and R. P. Pearce, Eds., Academic Press, 127–168.Hitchcock, P., and I. R. Simpson, 2014: The downward influence of stratospheric sudden warmings.

,*J. Atmos. Sci.***71**, 3856–3876, https://doi.org/10.1175/JAS-D-14-0012.1.Holton, J. R., and C. Mass, 1976: Stratospheric vacillation cycles.

,*J. Atmos. Sci.***33**, 2218–2225, https://doi.org/10.1175/1520-0469(1976)033<2218:SVC>2.0.CO;2.Holton, J. R., and T. Dunkerton, 1978: On the role of wave transience and dissipation in stratospheric mean flow vacillations.

,*J. Atmos. Sci.***35**, 740–744, https://doi.org/10.1175/1520-0469(1978)035<0740:OTROWT>2.0.CO;2.Liu, Y. S., and R. K. Scott, 2015: The onset of the barotropic sudden warming in a global model.

,*Quart. J. Roy. Meteor. Soc.***141**, 2944–2955, https://doi.org/10.1002/qj.2580.Matsuno, T., 1971: A dynamical model of the stratospheric sudden warming.

,*J. Atmos. Sci.***28**, 1479–1494, https://doi.org/10.1175/1520-0469(1971)028<1479:ADMOTS>2.0.CO;2.Matthewman, N. J., and J. G. Esler, 2011: Stratospheric sudden warmings as self-tuning resonances. Part I: Vortex splitting events.

,*J. Atmos. Sci.***68**, 2481–2504, https://doi.org/10.1175/JAS-D-11-07.1.Matthewman, N. J., J. G. Esler, A. J. Charlton-Perez, and L. M. Polvani, 2009: A new look at stratospheric sudden warmings. Part III: Polar vortex evolution and vertical structure.

,*J. Climate***22**, 1566–1585, https://doi.org/10.1175/2008JCLI2365.1.Nakamura, N., and D. Zhu, 2010: Finite-amplitude wave activity and diffusive flux of potential vorticity in eddy–mean flow interaction.

,*J. Atmos. Sci.***67**, 2701–2716, https://doi.org/10.1175/2010JAS3432.1.Plumb, R. A., 1981: Instability of the distorted polar night vortex: A theory of stratospheric warmings.

,*J. Atmos. Sci.***38**, 2514–2531, https://doi.org/10.1175/1520-0469(1981)038<2514:IOTDPN>2.0.CO;2.Polvani, L. M., and R. A. Plumb, 1992: Rossby wave breaking, microbreaking, filamentation, and secondary vortex formation: The dynamics of a perturbed vortex.

,*J. Atmos. Sci.***49**, 462–476, https://doi.org/10.1175/1520-0469(1992)049<0462:RWBMFA>2.0.CO;2.Ruzmaikin, A., J. Lawrence, and C. Cadavid, 2003: A simple model of stratospheric dynamics including solar variability.

,*J. Climate***16**, 1593–1600, https://doi.org/10.1175/1520-0442-16.10.1593.Scott, R. K., 2016: A new class of vacillations of the stratospheric polar vortex.

,*Quart. J. Roy. Meteor. Soc.***142**, 1948–1957, https://doi.org/10.1002/qj.2788.Shuckburgh, E., W. Norton, A. Iwi, and P. Haynes, 2001: Influence of the quasi-biennial oscillation on isentropic transport and mixing in the tropics and subtropics.

,*J. Geophys. Res.***106**, 14 327–14 337, https://doi.org/10.1029/2000JD900664.Sigmond, M., J. F. Scinocca, V. V. Kharin, and T. G. Shepherd, 2013: Enhanced seasonal forecast skill following stratospheric sudden warmings.

,*Nat. Geosci.***6**, 98–102, https://doi.org/10.1038/ngeo1698.Sjoberg, J. P., and T. Birner, 2014: Stratospheric wave-mean flow feedbacks and sudden stratospheric warmings in a simple model forced by upward wave activity flux.

,*J. Atmos. Sci.***71**, 4055–4071, https://doi.org/10.1175/JAS-D-14-0113.1.Swanson, K. L., P. J. Kushner, and I. M. Held, 1997: Dynamics of barotropic storm tracks.

,*J. Atmos. Sci.***54**, 791–810, https://doi.org/10.1175/1520-0469(1997)054<0791:DOBST>2.0.CO;2.Tung, K. K., and R. S. Lindzen, 1979: A theory of stationary long waves. Part I: A simple theory of blocking.

,*Mon. Wea. Rev.***107**, 714–734, https://doi.org/10.1175/1520-0493(1979)107<0714:ATOSLW>2.0.CO;2.Yoden, S., 1987a: Bifurcation properties of a stratospheric vacillation model.

,*J. Atmos. Sci.***44**, 1723–1733, https://doi.org/10.1175/1520-0469(1987)044<1723:BPOASV>2.0.CO;2.Yoden, S., 1987b: Dynamical aspects of stratospheric vacillations in a highly truncated model.

,*J. Atmos. Sci.***44**, 3683–3695, https://doi.org/10.1175/1520-0469(1987)044<3683:DAOSVI>2.0.CO;2.Yoden, S., 1987c: A new class of stratospheric vacillations in a highly truncated model due to wave interference.

,*J. Atmos. Sci.***44**, 3696–3709, https://doi.org/10.1175/1520-0469(1987)044<3696:ANCOSV>2.0.CO;2.Yoden, S., T. Yamaga, S. Pawson, and U. Langematz, 1999: A composite analysis of the stratospheric sudden warmings simulated in a perpetual January integration of the Berlin TSM GCM.

,*J. Meteor. Soc. Japan***77**, 431–445, https://doi.org/10.2151/jmsj1965.77.2_431.