Editorial Type:
Article
Article Type:
Research Article

Stratospheric Vacillations in a Minimal Contour Dynamics Model

Peter Hitchcock aDepartment of Earth and Atmospheric Sciences, Cornell University, Ithaca, New York

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https://orcid.org/0000-0001-8993-3808
Online Publication:
28 Jun 2021
Print Publication:
01 Jul 2021
DOI:
https://doi.org/10.1175/JAS-D-20-0296.1
Page(s):
2177–2193
Free access

Abstract

A low-dimensional dynamical system that describes dynamical variability of the stratospheric polar vortex is presented. The derivation is based on a linearized, contour-dynamics representation of quasigeostrophic shallow-water flow on a polar f plane. The model consists of a single linear wave mode propagating on a near-circular patch of constant potential vorticity (PV). The PV jump at the vortex edge serves as an additional degree of freedom. The wave is forced by surface topography, and interacts with the vortex through a simplified parameterization of diabatic wave–mean flow interaction. The approach can be generalized to other geometries. The resulting three-component system depends on four nondimensional parameters, and the structure of the steady-state solutions can be determined analytically in some detail. Despite its extreme simplification, the model exhibits variability that is closely analogous to the Holton–Mass model, a well-known and more complex dynamical model of stratospheric variability. The present model exhibits two stable steady solutions, one consisting of a strong vortex with a small amplitude wave and the second consisting of a weak vortex with a large amplitude wave. Periodic and aperiodic limit cycles are also identified, analogous to similar solutions in the Holton–Mass model. Model trajectories also exhibit a number of behaviors that have been identified in observations. A key insight is that the time-mean state of the vortex is predominantly controlled by the properties of the linear mode, while the strength of the topographic forcing plays a far weaker role away from bifurcations.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Peter Hitchcock, aph28@cornell.edu

Abstract

A low-dimensional dynamical system that describes dynamical variability of the stratospheric polar vortex is presented. The derivation is based on a linearized, contour-dynamics representation of quasigeostrophic shallow-water flow on a polar f plane. The model consists of a single linear wave mode propagating on a near-circular patch of constant potential vorticity (PV). The PV jump at the vortex edge serves as an additional degree of freedom. The wave is forced by surface topography, and interacts with the vortex through a simplified parameterization of diabatic wave–mean flow interaction. The approach can be generalized to other geometries. The resulting three-component system depends on four nondimensional parameters, and the structure of the steady-state solutions can be determined analytically in some detail. Despite its extreme simplification, the model exhibits variability that is closely analogous to the Holton–Mass model, a well-known and more complex dynamical model of stratospheric variability. The present model exhibits two stable steady solutions, one consisting of a strong vortex with a small amplitude wave and the second consisting of a weak vortex with a large amplitude wave. Periodic and aperiodic limit cycles are also identified, analogous to similar solutions in the Holton–Mass model. Model trajectories also exhibit a number of behaviors that have been identified in observations. A key insight is that the time-mean state of the vortex is predominantly controlled by the properties of the linear mode, while the strength of the topographic forcing plays a far weaker role away from bifurcations.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Peter Hitchcock, aph28@cornell.edu
Keywords: Nonlinear dynamics; Rossby waves; Shallow-water equations; Stratospheric circulation; Quasigeostrophic models

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.

Table 1.

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

Table 1.

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.

Under appropriate scaling assumptions, the shallow-water quasigeostrophic potential vorticity
q=f0+∇2ψ−Ld−2ψ+f0Dhb
is materially conserved:
∂tq+ug⋅∇q=0.
Here ψ is the streamfunction for the geostrophic flow ug; thus in polar coordinates (r, θ) the azimuthal and radial velocities are given by ug = ∂rψ and υg=−(1/r)∂θψ, respectively. With the geometry of the stratospheric polar vortex in mind, the term “zonal” will henceforth be used instead of “azimuthal,” the term “meridional” will be used instead of “radial,” and the angle θ will be referred to as the longitude. The deformation radius Ld=gD/f0 is chosen as the scale for the horizontal coordinate, in which f0 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 hb.
The flow under consideration consists of a single, near-circular patch of elevated potential vorticity (the polar vortex) within a region of lower potential vorticity (the “surf zone”):
q={Q0+Δ˜ifr<R+ϵη˜⁡(θ,t),Q0otherwise.
The basic state is determined by the radius R and potential vorticity jump Δ˜, while perturbations to the vortex edge are specified by η˜. The tildes anticipate further rescaling discussed below.

In the following ϵ is taken to be small so that the dynamics remain linear in η˜. This fully specifies the potential vorticity distribution. The streamfunction is required to vanish as r → ∞, with the exception that a zonally symmetric flow is included to allow for a background flow independent of the PV jump Δ˜.

Taking advantage of the linearity of (1), the streamfunction is decomposed into three components:
ψ=ΨQ+ψQ+ψT.
The first component accounts for the potential vorticity of the basic flow, the second for the potential vorticity perturbation, and the third for the effects of topography.
The basic state streamfunction ΨQ gives rise to a zonal flow at the vortex edge given by (see, e.g., Polvani and Plumb 1992)
U⁡(R)=∂rΨQ=RΩb+RΔ˜I1⁡(R)K1⁡(R).
Here In(r) and Kn(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 RΩb unrelated to Δ˜.
The perturbation streamfunction ψQ is associated with the anomaly η˜. The latter is taken to consist of a single sinusoid with wavenumber k:
η˜=ℜ{η˜k⁡(t)eikθ}.
The streamfunction ψQ has a discontinuity in its meridional derivative at the vortex edge that is related to η˜ (e.g., Swanson et al. 1997):
η˜Δ˜=−[∂rψQ]R,
where the right-hand side denotes the jump in the derivative at r = R. Away from the vortex edge the perturbation PV vanishes. These conditions determine ψQ, producing a meridional flow at the vortex edge:
υQ⁡(R,t)=ℜ{ikη˜k⁡(t)Δ˜Ik⁡(R)Kk⁡(R)eikθ}.
The topography is chosen to make the form of ψT relatively simple: setting hb(r, θ) = h0Jk(λr)coskθ leads to an additional meridional flow at the vortex edge:
υT=ℜ{−ikFkeikθ},Fk=f0λ2+1h0D1RJk⁡(λR).
Here Jn(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.
Finally, kinematic conditions at the edge of the vortex relate the material derivative of η˜ to the flow normal to the contour, (8) and (9). To linear order in ϵ, perturbations to the circular vortex are advected by the flow (5) at the vortex edge:
∂tη˜+U⁡(R)R∂θη˜=υQ⁡(R)+υT⁡(R).
Substituting in (5), (8), and (9) into (10) gives an ordinary differential equation for the complex amplitude of the wave:
dη˜kdt+ikcpη˜k=−ikFk.
The angular phase speed of the unforced wave is given by
cp=Ωb+Δ˜[I1⁡(R)K1⁡(R)−Ik⁡(R)Kk⁡(R)].
Physically, the first term is the advection of the wave by the background solid body rotation of the surf zone (5), the second term is the advection by the zonal mean winds associated with the PV jump (5), and the third term is the counter propagation characteristic of Rossby waves induced by the meridional flow associated with the perturbation, (8). For later convenience, the term in square brackets will be denoted Sk(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 Δ˜ of the vortex plays a central role in the following.

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 Δ˜. To represent the dynamical evolution of the vortex itself, an additional dynamical degree of freedom is needed. In principal either R or Δ˜ (or both) could play this role. However, the angular phase speed (12) of the wave depends linearly on Δ˜ while its dependence on R through Sk(R) is nonlinear. Thus for simplicity the PV jump Δ˜ is chosen to represent the strength of the vortex.

The evolution of Δ˜ is assumed to be driven by a competition between diabatic mixing across PV contours that acts to reduce the potential vorticity jump, and radiative relaxation that, when combined with the gradient-sharpening tendency discussed above, acts to strengthen it. The latter is parameterized as simple linear relaxation toward an equilibrium value ΔE on the time scale τV. Since this diabatic equilibrium includes the gradient-sharpening effects of the breaking waves, it is not a purely radiative equilibrium.

The effects of small-scale mixing are parameterized as an anomalous diffusive flux of potential vorticity across the contour. Within the conceptual framework of effective diffusivity (Shuckburgh et al. 2001; Nakamura and Zhu 2010; de la Cámara et al. 2018), this flux is assumed to be proportional to the PV jump Δ˜ and to the length of the contour:
∫02πR{1+12ϵ2[∂θη˜⁡(t,θ)R]2+O⁡(ϵ3)}dθ.
To leading order, the PV flux is proportional to the amplitude of the wave squared. This leading-order term is used to model the wave-driven depletion of the PV jump, drastically simplifying the highly nonlinear process of filamentation and transport across the vortex edge (e.g., Polvani and Plumb 1992).
Combining these two processes gives rise to the following evolution equation for the potential vorticity jump:
dΔ˜dt=−τV−1⁡(Δ˜−ΔE)−κη˜k†η˜kΔ˜,
where η˜k† is the complex conjugate of η˜k. The dependence of the diffusive flux on R and k is absorbed into the “effective diffusivity” coefficient κ.
Finally, dissipation of the wave itself is also included by adding a linear damping term to the right-hand side of (11) with a time scale τW:
dη˜kdt+ikcpη˜k=−ikFk−τW−1η˜k.

Equations (14) and (15) form a closed, forced-dissipative dynamical system. The zonal mean vortex strength Δ˜ influences the evolution of the wave by modifying the mean flow on which it is propagating, and by modulating the influence of remote components of the wave. The wave acts to acts to modify the zonal mean vortex strength by increasing the diffusive flux across the contour.

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 ikη˜|η˜2| in the wave evolution equation. They are omitted here for simplicity; as will be seen below, the irreversible, dissipative effects that are included here produce some similar effects. The consequences of including both effects are left for future study.

3. A three-parameter model of stratospheric vacillations

a. Nondimensionalization

Thus far the only nondimensionalization that has been discussed is that of the horizontal coordinate system. To nondimensionalize these equations, the wave amplitude is rescaled by its steady-state value when the mode is in resonance
η˜k=kτWFkη,
the PV jump by its equilibrium value
Δ˜=ΔEΔ,
and time by the radiative time scale of the wave τW. To simplify notation, t from this section forward refers to the nondimensional time.
The resulting two nondimensional, first-order, ordinary differential equations, one complex and one real valued,
dηdt=−iS^⁡(Δ−δ^)η−η−i,
dΔdt=γ^⁡(1−Δ−κ^η†ηΔ),
are determined by four nondimensional external parameters (indicated by hats):
S^=kτWΔESk⁡(R),
δ^=−ΩBΔESk⁡(R),
κ^=κτVk2τW2Fk†Fk,
γ^=τWτV.
These nondimensional parameters can be physically interpreted as follows. The first, S^, is the ratio of the intrinsic frequency of the wave when the vortex is at equilibrium kΔESk(R) to the damping rate of the wave τW−1. It is a nondimensional linear sensitivity of the wave’s phase speed to the potential vorticity jump [cf. (12)]. The second, δ^, is (minus) the ratio of the background surf-zone rotation Ωb to the intrinsic phase speed of the wave at diabatic equilibrium ΔESk(R). Critically, when the nondimensional PV jump Δ is equal to δ^, the intrinsic phase speed cancels the advection by the background surf-zone rotation, and the wave is stationary in the frame of the topography. Together, these two parameters specify the dependence of the phase speed of the wave on the vortex PV jump.

The third parameter, κ^, is the ratio of the wave-driven PV flux across the contour when the wave is at its steady-state resonant amplitude and the vortex is at equilibrium κ|kτWFk|2ΔE to the diabatic PV flux restoring the vortex when the vortex PV jump is zero ΔEτV−1. Thus this parameter characterizes the strength of the wave forcing relative to the strength of the diabatic restoration of the vortex. The fourth parameter, γ^, is the ratio of the damping time scale of the wave τW to the time scale of the diabatic restoration of the vortex τV.

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 S^⁡(Δ−δ^) is linearly related to the PV jump and depends on S^ and δ^.

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 κ^η†ηΔ. The efficiency of the latter is determined by κ^. Finally, the time scale of the vortex’s evolution relative to that of the wave is determined by γ^. Further interpretation of the degrees of freedom and of the external parameters is given below.

Writing η⁡(t)=a⁡(t)e−iϕ⁡(t), (18) can be rewritten as three real-valued equations:
dadt=−a+sinϕ,
dϕdt=S^⁡(Δ−δ^)+cosϕa,
dΔdt=γ^⁡(1−Δ−κ^a2Δ).
The amplitude of the wave is affected only by the damping and the topographic forcing. The latter can either amplify or dampen the wave depending on the phase of the wave relative to the topography. The phase of the wave, in turn, is dictated by the internal frequency of the oscillation, but is also directly affected by the forcing.

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 + 2mπ for integer m.

Fig. 1.
Fig. 1.

A schematic of the minimal model. The vortex edge (thick black line) is perturbed from the circular basic state (gray circle) by a single sinusoidal wave (in this case k = 2). The wave is forced by idealized topography; the thin orange contours can be regarded either as level sets of hb or as streamlines of the topographic flow. The meridional flow at the vortex edge is indicated by arrows. The 3 degrees of freedom in the model are the amplitude of the wave a, the phase ϕ, and the strength of the PV jump at the vortex edge Δ.

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

These first two forms (18) and (20) are used directly in the discussion below. A third equivalent form can be derived in terms of Cartesian components of the complex wave amplitude x = acosϕ and y = asinϕ:
dxdt=−S^⁡(Δ−δ^)y−x,
dydt=S^⁡(Δ−δ^)x−y+1,
dΔdt=γ^[1−Δ−κ^⁡(x2+y2)Δ].
This will be used for plotting certain trajectories. It can also be directly compared with the sets of equations derived by Charney and DeVore (1979) and Ruzmaikin et al. (2003). These two models also describe the real and imaginary components of a single wave mode interacting with a zonal mean flow, although the zonal mean zonal wind is used instead of the PV jump. The same linear dependence of the phase speed of the wave on the zonal mean flow is present in all three sets. However, the wave–mean flow interaction terms arise from mountain torque effects and have a slightly different, zonally asymmetric form.

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 S^→−S^, ϕ→−ϕ; as such S^≥0 can be considered without loss of generality.

The nondimensional parameter γ^ is the ratio of the damping time scales for the vortex and the wave. It is helpful to build some physical intuition by considering the limiting behavior of this three-component model when γ^ is far from unity. This introduces a time-scale separation between the dynamics of these two components.

1) Slow restoration of the vortex

Somewhat more formally this can be seen through a multiple-time-scale analysis. Taking first the case where γ^≪1, a fast and a slow time, τ and T can be defined such that T=γ^τ. One then considers 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
η=−i1+iS^⁡(Δ−δ^).
The largest amplitude response occurs when the phase speed of the wave is zero, and is resonantly amplified by the topographic forcing. The damping prevents the response from being singular; under the present scaling the nondimensional amplitude at this resonant maximum is 1.
Heuristically, on the slow time scale, the dynamics of the PV jump are determined by the quasi-steady amplitude of the wave:
dΔdT=1−Δ−κ^Δ1+S^2⁡(Δ−δ^)2.
The validity of this approximation will not be considered in detail here. The tendency of Δ on the right-hand side of (23) is plotted in Fig. 2 for various values of δ^, S^, and κ^.
Fig. 2.
Fig. 2.

Tendency of vortex PV jump, (23), as a function of the PV jump itself for (a) constant S^=100 and several values of δ^ and κ^ and (b) constant δ^=0.5 and κ^=1.5 and several values of S^.

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 δ^ from 0 to 1 (e.g., Fig. 2a). The sharpness of this resonance is controlled by S^; specifically, S^−1 corresponds roughly to the half-width at half-minimum of the minima in Fig. 2b. When S^ is large, the strength of the interactions between the wave and the vortex are sensitively dependent on the vortex being close to the resonant value δ^.

Finally, κ^ determines the overall efficiency with which the wave depletes the background PV gradient (e.g., Fig. 2a); it will be referred to as the forcing strength. Neglecting the radiative restoration term in (23), the PV jump tendency has a minimum at resonance of 1−δ^⁡(κ^+1). The efficiency of this depletion at resonance thus depends on the location of the resonance as well as the forcing strength. For fixed κ^, resonances closer to diabatic equilibrium will experience more efficient wave driving than those occurring at weaker values of the PV jump.

There are either one or three steady states present in this system. For sufficiently small values of κ^, such as the κ^=2 curve in Fig. 2a, the wave driving is never strong enough to compete against the radiative restoration of the vortex. In this case, there is a single, stable steady state with Δ close to one, corresponding to a strong vortex near diabatic equilibrium. For larger values of κ^, the value of the PV jump tendency at resonance drops below zero (here again neglecting the radiative restoration term), at
κ^=1−δ^δ^.
Above this threshold the wave–mean flow interaction tends to be sufficiently strong that two additional steady states can exist above and below the location of the resonance. Note that this is not a sufficient condition for there to be three steady states. The minimum tendency in fact occurs at values of Δ slightly greater than δ^ due to the radiative restoration term. In addition, at sufficiently large values of κ^, the strong vortex state can disappear.

Since the value of γ^ does not affect the steady-state solutions, these steady states correspond to those of the full system, as will be discussed further below. When the quasi-steady-state description holds, their stability is also evident. Small perturbations in either direction from the intermediate steady state will tend to grow, and the system will approach either the strong-vortex, weak-wave state, or the weak-vortex, strong-wave state. Physically these stability properties can be understood as follows. Assuming that the resonance is well separated from the strong vortex state, the stable dynamics of radiative restoration will dominate the system near the latter. For the intermediate state, small positive perturbations in the PV jump will tune the vortex away from resonance leading to further strengthening of the jump, while negative perturbations lead to further wave growth. In contrast, at the weak vortex state below the resonance, the wave dynamics act to stabilize the system.

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.

Fig. 3.
Fig. 3.

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

Alternatively, the limit γ^≫1 can also be considered. In this case the vortex is restored much more rapidly than the wave is damped. The wave parameters are taken to be functions only of the slow time T=γ^−1τ, and the PV jump Δ relaxes on the fast time scale toward a steady state given by
Δ=11+κ^η†η.
On the slow time scale, the wave dynamics become that of a nonlinear oscillator:
dηdT=−iS^⁡(11+κ^η†η−δ^)η−η−i.
For small values of κ^ this recovers the cubic nonlinearity discussed by Matthewman and Esler (2011) and Esler and Matthewman (2011). In this limit the irreversible wave–mean flow interaction arising from dissipation occurs rapidly enough that it can play a role similar to that of the transience effects at the heart of their discussion.

c. Steady states and their stability

Steady states of (20) correspond to zeroes of the right-hand side of (23). Since the denominator of the wave-forcing term is always positive, these correspond to the roots of the cubic polynomial:
⁡(Δ−1)⁡(Δ−δ^)2+S^−2[⁡(1+κ^)Δ−1]=0.
Consistent with the symmetry identified earlier, the steady states for Δ depend only on the absolute value of S^. As inferred above, the system has either one or three steady-state solutions, corresponding to the real roots of this cubic polynomial.
Motivated by the form of (27) and by Fig. 2, an approximate expression can be found for the pair of roots near δ^, when they exist. Expanding Δ in S^−1,
Δ=δ^+S^−1Δ1+O⁡(S^−2),
and substituting into (27), the O⁡(1) and O⁡(S^−1) terms vanish. The O⁡(S^−2) terms yield
Δ12=δ^⁡(1+κ^)−1δ^−1.
The correction Δ1 is real for κ^>δ^−1−1, in agreement with the heuristic condition (24) inferred above. The two roots lie near
Δ=δ^±S^−1δ^⁡(1+κ^)−11−δ^
in this limit. The positive root in the expression corresponds to the intermediate state, while the negative root corresponds to the weak vortex state.
A similar expression can be found for the strong vortex state that lies near
Δ=1−κ^⁡(1−δ^)2S^2
when it exists in the limit of small S^−2.

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 κ^ for fixed values of the other three nondimensional parameters. For small forcing amplitudes only the strong vortex state exists; at κ^≈0.99, the two additional states emerge in a saddle-node bifurcation very close to the value of κ^=1 predicted by (24). Near κ^≈9.25 the strong vortex state merges with the intermediate state in another saddle-node bifurcation. Above this value, only the weak vortex state is present. Also shown are the approximate expressions (30) and (31) for comparison with the exact values. The expressions become less accurate for larger values of the forcing, with (31) overestimating the strength of the vortex, and (30) underestimating Δ for both the intermediate and weak vortex state as κ^ increases.

Fig. 4.
Fig. 4.

(a) Vortex PV jump Δ of the steady-state solutions as a function of κ^ for fixed δ^=0.5, S^=20, and γ^=1. The thick solid black curve corresponds to the strong vortex state, the thick dashed curve to the unstable intermediate state, and the thick solid red curve to the weak vortex state. The thin lines correspond to the asymptotic forms (31) and (30), respectively. (b) Real and (c) imaginary components of the three eigenvalues of the weak vortex state [thick red curve in (a)]. The vertical black line in each panel indicates the value of κ^ at which the weak vortex state becomes unstable. The inset in (b) shows real and imaginary components of the eigenvalues for a narrow range 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 κ^ increases after the weak vortex state emerges, the eigenvalues rapidly transition from a pair of complex conjugates and a negative real value, to three negative real values, then back to a pair of complex conjugates and a negative real value (see inset in Fig. 4b). This corresponds to a transition from a stable spiral to a stable node and back. However, the weak vortex state remains stable to small amplitude perturbations up to a value of κ^≈2.58, at which point the real component of the complex conjugate pair becomes positive. This indicates that the state is transitioning from a stable to an unstable spiral node, corresponding to a Hopf bifurcation. The character of the trajectories above this bifurcation will be further discussed in the next section. Similar plots for the other two steady states confirm the implications of Fig. 2 that the strong vortex state remains stable where it exists and the intermediate vortex is unstable in one direction (not shown).

The behavior of the full system is summarized for a broader range of parameter space in Fig. 5. Figures 5a–c show the S^,κ^ plane for values of δ^=0.3, 0.5,and0.8, and Fig. 5d shows the S^,δ^ plane for κ^=10. In all cases γ^=1.

Fig. 5.
Fig. 5.

Phase portraits of the three-parameter dynamical system in the S^,κ^ plane for fixed (a) δ^=0.3, (b) δ^=0.5, and (c) δ^=0.8, and in the S^,δ^ plane for fixed (d) κ^=10. The black contour shows where the system transitions from having one steady state to three steady states. Filled contours show the time-mean value of Δ for representative trajectories. Dashed blue lines show the zero contour of the largest real component of the eigenvalues of the Jacobian matrix at the weak vortex state. The thin vertical black lines in (b) correspond to parameter ranges shown in Fig. 6.

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 S^=20 in Fig. 5b; the saddle-node bifurcations illustrated there correspond to values of κ^ at which the black curves cross S^=20. The region of the S^,κ^ plane with multiple steady states becomes narrower as the location of the resonance δ^ moves closer to the equilibrium PV jump.

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 δ^ and S^ (Figs. 5a–c). For small values of S^ there is only ever a single, stable steady state that transitions smoothly from a strong PV jump (Δ ≈ 1) to a weak PV jump (Δ ≈ 0) as κ^ increases. Above a critical value of S^, however, there emerges a finite range of forcing strengths κ^ for which three steady states exist. Above this critical point, the transition from strong to weak vortex states is no longer continuous, with the emergence of the weak vortex near the forcing strength given by (24). For intermediate values of S^, the weak vortex state remains stable above this saddle-node bifurcation (to the left of the dashed blue line, above the lower black curve). In this region most of the representative trajectories tend toward the weak vortex state, although some small regions exist in which they tend toward the strong vortex instead. The PV jump at the weak vortex state lies close to the resonant value δ^, responding only weakly to further increases in the forcing strength κ^.

For larger values of S^, the weak vortex state ultimately becomes unstable (to the right of the dashed blue line). This may occur where the weak vortex state is the only steady state present (above the upper black curve), or while all three steady states exist. At even larger values of S^ this occurs for forcing strengths κ^ only slightly larger than those at which the weak vortex state first emerges.

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 δ^, varying only weakly with κ^ and S^.

Considering instead the behavior as S^ increases at fixed κ^=10 (Fig. 5d), the weak state becomes unstable at lower values of S^ for resonances closer to, but not directly at, the equilibrium PV jump. The importance of δ^ for determining the time-mean value of Δ for larger values of S^ is again apparent.

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 δ^ and is only weakly affected by the strength of the forcing κ^ away from discontinuous bifurcations. This inference, which is made explicit for the steady states by (30), holds true for the time mean of the unsteady states as well. The location of the critical point is discussed further in appendix A.

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 κ^ is varied for δ^=0.5 and γ^=1, for four increasingly sharp resonances corresponding to increasing values of S^. The trajectories are initialized in both directions along the most unstable eigenvector of the intermediate steady state. The steady states are also indicated in gray.

Fig. 6.
Fig. 6.

Bifurcation diagrams showing local extrema of Δ (black points) in trajectories as a function of κ^ for δ^=0.5 and (a) S^=10, (b) S^=20, (c) S^=40, and (d) S^=85. For each value of κ^, two trajectories are initialized, in each direction along the most unstable eigenvector of the unstable steady state. Also shown (gray points) are the steady states.

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

At S^=10 (Fig. 6a), the strong vortex state exists up to a value of κ^=2.5. At this location of the resonance, β=0.225 and the asymptotic form (A4) gives a value of κ^=2.25 for this transition. The weak vortex state emerges near κ^=0.96, just below κ^=1 suggested by (24). Both steady states are stable where they exist.

At S^=20 (Fig. 6b), the strong vortex state persists until κ^≈9.25; (A4) predicts 9.02. The weak state emerges very close to κ^=1. The weak vortex state then loses its stability to a periodic limit cycle in a Hopf bifurcation just above κ^=2.5. These values are consistent with the linear stability calculations shown in Fig. 4. This behavior closely resembles that of the Holton–Mass model as discussed by Yoden (1987a, compare with their Fig. 2a). For sufficiently large κ^ this limit cycle eventually disappears, and the weak vortex state becomes stable again (not shown).

As κ^ increases when S^=40 (Fig. 6c), the weak vortex state becomes unstable just above κ^=1.2, very shortly after it emerges at κ^=1. The limit cycle itself then loses its stability near κ^=2.1. This homoclinic bifurcation is further illustrated in Fig. 7a that shows trajectories for three values of κ^ near the bifurcation. As κ^ increases, the limit cycle (blue curve) merges with the intermediate steady state, becoming a homoclinic orbit at the bifurcation (orange curve). Above this point the trajectory tends toward the strong vortex state (green curve).

Fig. 7.
Fig. 7.

(a) Trajectories along unstable manifold of intermediate fixed point for three values of κ^ (2, 2.128 775 7, and 2.25) near a homoclinic bifurcation. The three trajectories are integrated for 9.8, 54, and 7.1 nondimensional time units, respectively. Colored circles indicate start points and squares indicate end points. These nearly coincide for the second (orange) trajectory. (b) Three trajectories for the parameter values given earlier in the figure caption and in the legend in (a), differing only in the initial conditions. In this case initial transients of the periodic and chaotic orbits are omitted to emphasize their limiting behavior. Fixed points are indicated by black circles. Note that periodic trajectory is overlaid on top of the chaotic trajectory, despite the fact that it passes behind the chaotic trajectory in some cases.

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 κ^ for which the strong vortex state again appears to be the only stable steady state, above the threshold (24) for the existence of the weak vortex state. Near κ^=6.6 the stable limit cycle reemerges in another homoclinic bifurcation.

A further period doubling bifurcation at κ^=9.8 leads to a range of values for which a period-four limit cycle is stable, before the period-two limit cycle returns near κ^=22. It is worth noting that the amplitude of these cycles is very insensitive to the magnitude of κ^, tending to decrease weakly as S^ and κ^ increase, with the exception of the strong transitions near bifurcations. The strong vortex persists until near κ^=36.

At S^=85 (Fig. 6d), the orbits near the weak vortex state undergo a sequence of period-doubling bifurcations, leading to a range of κ^ for which the orbits are chaotic. Again this occurs at forcing magnitudes for which the strong vortex still exists and is stable; at this value of S^ it persists until κ^=162.8.

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 S^=100, δ^=0.5, κ^=25, and γ^=1. At these parameter values, a stable fixed point (the strong vortex state), a stable periodic limit cycle, and a chaotic attractor all coexist with different basins of attraction. The coexistence of periodic and chaotic trajectories has not been reported in the Holton–Mass model, but given other similarities in the character of the two models, such cases may exist.

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.

Fig. 8.
Fig. 8.

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 γ^=1, the time scale of the radiative restoration of the PV jump.

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 δ^=0.5. The wave amplitude varies from nearly 5% to 45% of its maximum amplitude. Consistent with the fact that the PV jump tends to lie below the resonance, the phase propagation is nearly always negative (or westward).

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 S^ and δ^. Because the topographic forcing is stationary, it is much more effective at perturbing the vortex when the phase speed of the free mode is near zero, that is, when Δ≈δ^. If S^ is large, this occurs only for a narrow range of PV jumps near δ^. As a result, the time mean of the dynamically active vortex trajectories are only weakly dependent on the nondimensional amplitude of the forcing κ^, with the exception of highly nonlinear transitions that occur where the character of stable solutions change (Figs. 5, 6). Because the range of Δ that allows for significant wave forcing narrows as S^ increases, the dynamically active states remain close to Δ=δ^. This lack of sensitivity to κ^ is made explicit by the asymptotic location of the steady states, (30), and by Fig. 6. The strong vortex state near the equilibrium point is also only weakly affected by the forcing strength since the wave mode is far from resonance at that point.

On the other hand, if S^ is small, the phase speed of the mode is relatively insensitive to the value of Δ, and the system is susceptible to the topographic forcing regardless of the strength of the PV jump. In this regime, the vortex state is much more directly sensitive to κ^, although since the wave-driving term in (18b) is proportional to Δ, the response of the system still ultimately saturates.

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 S^ and κ^ the system develops a nonlinear transition between the strong and weak vortex states.

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 δ^ and S^ are plausible for barotropic modes relevant to the Arctic polar vortex. To be consistent with the dynamical variability present in the real system, the value of S^ must be at least 20 or so. The dimensional parameter Sk(R) plays a central role: for a wave-two mode on a shallow-water vortex with a radius several times the deformation radius, Sk⁡(R)≈0.02. Assuming a target value of S^=20 and a radiative dissipation time scale τW of 20 days, this implies an equilibrium PV jump of close to 2f0. Previous research has taken Δ = f0 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 δ^ to lie in the range 0 to 1, the background rotation must be weak and westward. This is consistent with other dynamical models based on a shallow-water 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: δ^ is the nondimensional PV jump at which the wave mode is stationary, and S^ is the linear coefficient determining the sensitivity of the phase speed to the PV jump. A third parameter, κ^, is a nondimensional measure of the strength of the wave forcing relative. Finally, γ^ is the ratio of diabatic time scales acting on the wave and the vortex.

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 Δ=δ^. In some parameter regimes the latter becomes unstable to periodic and chaotic vacillations. Closed-form, asymptotic expressions are obtained for the system’s steady states (section 3), the location of saddle-node bifurcations where the system transitions from one to three steady states (appendix A), and the location of Hopf bifurcation at which the weak vortex state becomes unstable (appendix B).

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 S^, regions exist where both periodic and aperiodic stable orbits coexist with the strong vortex state. It is not clear whether either behavior can be identified in more complex models.

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

The location of the saddle-node bifurcations between one and three states can be found explicitly by considering the discriminant D of (27). After some algebra, this can be shown to be the following polynomial in S^−2:
D=−4S^−2⁡(p1S^−4+p2S^−2+p3),
where
p1=⁡(κ^+1)3,
p2=2⁡(κ^+1)2δ^2+(5κ^2+κ^−4)δ^−14κ^2−5κ^+2,
p3=⁡(1−δ^)3⁡(1−δ^−δ^κ^).
When 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 S^−2 on the right-hand side of (A1) in parentheses. Since the quadratic has at most two real roots, for fixed κ^ and δ^ there will be either zero, one, or two transitions between one and three steady states.

Considering first the limit S^−1→0 for fixed κ^, D will be dominated by the polynomial p3. Setting p3 = 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 κ^=7/3, 1,and1/4, respectively. In Fig. 5d this also corresponds to the lower boundary, which approaches the value δ^=1/11.

Alternatively, (A1) has a further root asymptotic to
κ^=βS^2+O⁡(S^).
Retaining only O(1) terms in D yields the quadratic
β2+(2δ^2+5δ^−14)β^+δ^⁡(δ^−1)3=0.
This corresponds to the saddle-node bifurcation at which the strong vortex state merges with the intermediate state. In Figs. 5a–c this corresponds to the upper boundary that asymptotes to the curve (A3) with β ≈ 0.069, 0.023, and 0.0013, respectively. Hence the strong vortex state survives to much larger forcing strengths κ^ if the resonance is further from the diabatic equilibrium PV jump. Since κ^ is quadratic in the height of the topography, the height required to eliminate the strong vortex state in this limit is linear in S^.

The upper boundary in Fig. 5d also indicates the disappearance of the strong vortex state. However, in this plane, as S^−1→0 this curve asymptotes to the constant δ^=1, though it approaches this limit much more slowly than the other limiting forms considered.

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 δ^. For δ^≥1 there are no positive roots; in this regime only one steady state is accessible. The coefficient 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 δ^=0 a second root becomes positive, at which point the condition (24) also ceases to be valid. The smaller root gives the boundary where the weak vortex state emerges; in this regime there is a narrow interval of κ^ for which the three states coexists. This regime moves to increasingly large forcing magnitudes until the two roots merge at δ^=−1/8. Below this value only one steady state is present.

Fig. A1.
Fig. A1.

(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 κ^ and S^ at the critical point. See text for discussion.

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

The critical point occurs when the discriminant p22−4p1p3 of the quadratic in (A1) vanishes. The values of κ^ and S^ at this point are shown in Fig. A1b. Below and to the left of this point in the S^,κ^ plane the strong vortex state transitions smoothly to the weak vortex state. The critical κ^ diverges at δ^=−1/8 and rapidly decreases as the resonance becomes directly accessible and the wave driving can more easily maintain a weak-vortex state. For much of the interval 0<δ^<1, the critical κ^ is order unity or less. In the same interval, the critical value of S^ is just less than 5 until δ^ approaches the equilibrium value of the PV jump. In this limit the resonance must be increasingly narrow (S^→∞) to be well separated from the strong vortex state.

APPENDIX B

Stability of the Weak Vortex State

The linear stability of the weak vortex state is determined by the eigenvalues of the Jacobian matrix of (20). At the steady weak vortex state, this is given by
J=[−1a0 cosϕ00−a0−2 cosϕ0−1S^−2γ^κ^a01+κ^a020−γ^⁡(1+κ^a02)],
where subscripts indicate the steady-state values and a0 = sinϕ0 and Δ0=⁡(1+κ^a02)−1 have been used. Eigenvalues λ of this matrix are given by the roots of its characteristic polynomial:
⁡(λ+q1)⁡(λ2+2λ+q2)+q3=0,
q1=γ^⁡(1+κ^a02),
q2=a0−2,
q3=2S^γ^κ^a01+κ^a021−a02.
As shown in Fig. 4, the weak vortex loses its stability in a Hopf bifurcation, at which two of the eigenvalues λ are purely imaginary complex conjugates. This requires that
q1q2+q3=(2+q1)⁡(2q1+q2),
from which the following implicit equation for the location of the bifurcation can be found:
S^H=1+κ^a02γ^κ^a01−a02[γ^2⁡(1+κ^a02)2+2γ^⁡(1+κ^a02)+a0−2].
This can be solved iteratively to find S^H, given the other three parameters.
In the limit that (30) is valid, the steady-state wave amplitude a0 = Δ1, which does not depend on S^. This can be used to find a closed form approximate expression:
S^H=1+κ^−δ^−1+⁡(1+γ^δ^)2γ^⁡(1−δ^)⁡(δ^⁡(1+κ^)−1).
While the location of the steady states do not depend on γ^, their stability does. Figure B1a shows the value S^H as a function of κ^ for fixed δ^=0.5 and three values of γ^. After the weak vortex state emerges, the value S^H at which it destabilizes drops rapidly with the strength of the forcing κ^. However, beyond this initial rapid adjustment, S^H depends only weakly on κ^. The closed asymptotic form (B8) is a better approximation in the rapid transition where S^H is large and κ^ is small, losing accuracy at higher values of the forcing.
Fig. B1.
Fig. B1.

Value of S^H at which the weak vortex loses stability for (a) δ^, (b) κ^, and (c) γ^ as indicated in the labels. In (a) the thin lines show the asymptotic form, (B8). Note the logarithmic scale in (c).

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

In contrast S^H depends strongly on the location of the resonance δ^ and the ratio of diabatic time scales γ^ (Figs. B1b,c). For fixed κ^ and γ^, S^H has a minimum as a function of δ^; that is, there is a particular location of the resonance at which the weak vortex loses its stability earliest. This shifts to lower values of δ^ when vortex restoration time scale τV is longer than the wave damping time scale τW (γ^<1) and to higher values of δ^ when τW is longer than τV (γ^>1). The minimum values of S^H also shift lower as γ^ increases. Notably, in the latter case the weak vortex state can destabilize even when the linear resonance lies outside the range of PV jumps accessible to the vortex (δ^>1).

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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation
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Journal of the Atmospheric Sciences

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation

  • 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.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    A schematic of the minimal model. The vortex edge (thick black line) is perturbed from the circular basic state (gray circle) by a single sinusoidal wave (in this case k = 2). The wave is forced by idealized topography; the thin orange contours can be regarded either as level sets of hb or as streamlines of the topographic flow. The meridional flow at the vortex edge is indicated by arrows. The 3 degrees of freedom in the model are the amplitude of the wave a, the phase ϕ, and the strength of the PV jump at the vortex edge Δ.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    Tendency of vortex PV jump, (23), as a function of the PV jump itself for (a) constant S^=100 and several values of δ^ and κ^ and (b) constant δ^=0.5 and κ^=1.5 and several values of S^.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    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).

  • <sc>Fig</sc>. 4.
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    Fig. 4.

    (a) Vortex PV jump Δ of the steady-state solutions as a function of κ^ for fixed δ^=0.5, S^=20, and γ^=1. The thick solid black curve corresponds to the strong vortex state, the thick dashed curve to the unstable intermediate state, and the thick solid red curve to the weak vortex state. The thin lines correspond to the asymptotic forms (31) and (30), respectively. (b) Real and (c) imaginary components of the three eigenvalues of the weak vortex state [thick red curve in (a)]. The vertical black line in each panel indicates the value of κ^ at which the weak vortex state becomes unstable. The inset in (b) shows real and imaginary components of the eigenvalues for a narrow range of κ close to the emergence of the weak and intermediate vortex states.

  • <sc>Fig</sc>. 5.
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    Fig. 5.

    Phase portraits of the three-parameter dynamical system in the S^,κ^ plane for fixed (a) δ^=0.3, (b) δ^=0.5, and (c) δ^=0.8, and in the S^,δ^ plane for fixed (d) κ^=10. The black contour shows where the system transitions from having one steady state to three steady states. Filled contours show the time-mean value of Δ for representative trajectories. Dashed blue lines show the zero contour of the largest real component of the eigenvalues of the Jacobian matrix at the weak vortex state. The thin vertical black lines in (b) correspond to parameter ranges shown in Fig. 6.

  • <sc>Fig</sc>. 6.
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    Fig. 6.

    Bifurcation diagrams showing local extrema of Δ (black points) in trajectories as a function of κ^ for δ^=0.5 and (a) S^=10, (b) S^=20, (c) S^=40, and (d) S^=85. For each value of κ^, two trajectories are initialized, in each direction along the most unstable eigenvector of the unstable steady state. Also shown (gray points) are the steady states.

  • <sc>Fig</sc>. 7.
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    Fig. 7.

    (a) Trajectories along unstable manifold of intermediate fixed point for three values of κ^ (2, 2.128 775 7, and 2.25) near a homoclinic bifurcation. The three trajectories are integrated for 9.8, 54, and 7.1 nondimensional time units, respectively. Colored circles indicate start points and squares indicate end points. These nearly coincide for the second (orange) trajectory. (b) Three trajectories for the parameter values given earlier in the figure caption and in the legend in (a), differing only in the initial conditions. In this case initial transients of the periodic and chaotic orbits are omitted to emphasize their limiting behavior. Fixed points are indicated by black circles. Note that periodic trajectory is overlaid on top of the chaotic trajectory, despite the fact that it passes behind the chaotic trajectory in some cases.

  • <sc>Fig</sc>. 8.
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    Fig. 8.

    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.

  • <sc>Fig</sc>. A1.
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    Fig. A1.

    (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 κ^ and S^ at the critical point. See text for discussion.

  • <sc>Fig</sc>. B1.
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    Fig. B1.

    Value of S^H at which the weak vortex loses stability for (a) δ^, (b) κ^, and (c) γ^ as indicated in the labels. In (a) the thin lines show the asymptotic form, (B8). Note the logarithmic scale in (c).

Fig. 1.

A schematic of the minimal model. The vortex edge (thick black line) is perturbed from the circular basic state (gray circle) by a single sinusoidal wave (in this case k = 2). The wave is forced by idealized topography; the thin orange contours can be regarded either as level sets of hb or as streamlines of the topographic flow. The meridional flow at the vortex edge is indicated by arrows. The 3 degrees of freedom in the model are the amplitude of the wave a, the phase ϕ, and the strength of the PV jump at the vortex edge Δ.
Fig. 2.

Tendency of vortex PV jump, (23), as a function of the PV jump itself for (a) constant S^=100 and several values of δ^ and κ^ and (b) constant δ^=0.5 and κ^=1.5 and several values of S^.
Fig. 3.

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).
Fig. 4.

(a) Vortex PV jump Δ of the steady-state solutions as a function of κ^ for fixed δ^=0.5, S^=20, and γ^=1. The thick solid black curve corresponds to the strong vortex state, the thick dashed curve to the unstable intermediate state, and the thick solid red curve to the weak vortex state. The thin lines correspond to the asymptotic forms (31) and (30), respectively. (b) Real and (c) imaginary components of the three eigenvalues of the weak vortex state [thick red curve in (a)]. The vertical black line in each panel indicates the value of κ^ at which the weak vortex state becomes unstable. The inset in (b) shows real and imaginary components of the eigenvalues for a narrow range of κ close to the emergence of the weak and intermediate vortex states.
Fig. 5.

Phase portraits of the three-parameter dynamical system in the S^,κ^ plane for fixed (a) δ^=0.3, (b) δ^=0.5, and (c) δ^=0.8, and in the S^,δ^ plane for fixed (d) κ^=10. The black contour shows where the system transitions from having one steady state to three steady states. Filled contours show the time-mean value of Δ for representative trajectories. Dashed blue lines show the zero contour of the largest real component of the eigenvalues of the Jacobian matrix at the weak vortex state. The thin vertical black lines in (b) correspond to parameter ranges shown in Fig. 6.
Fig. 6.

Bifurcation diagrams showing local extrema of Δ (black points) in trajectories as a function of κ^ for δ^=0.5 and (a) S^=10, (b) S^=20, (c) S^=40, and (d) S^=85. For each value of κ^, two trajectories are initialized, in each direction along the most unstable eigenvector of the unstable steady state. Also shown (gray points) are the steady states.
Fig. 7.

(a) Trajectories along unstable manifold of intermediate fixed point for three values of κ^ (2, 2.128 775 7, and 2.25) near a homoclinic bifurcation. The three trajectories are integrated for 9.8, 54, and 7.1 nondimensional time units, respectively. Colored circles indicate start points and squares indicate end points. These nearly coincide for the second (orange) trajectory. (b) Three trajectories for the parameter values given earlier in the figure caption and in the legend in (a), differing only in the initial conditions. In this case initial transients of the periodic and chaotic orbits are omitted to emphasize their limiting behavior. Fixed points are indicated by black circles. Note that periodic trajectory is overlaid on top of the chaotic trajectory, despite the fact that it passes behind the chaotic trajectory in some cases.
Fig. 8.

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.
Fig. A1.

(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 κ^ and S^ at the critical point. See text for discussion.
Fig. B1.

Value of S^H at which the weak vortex loses stability for (a) δ^, (b) κ^, and (c) γ^ as indicated in the labels. In (a) the thin lines show the asymptotic form, (B8). Note the logarithmic scale in (c).
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