## 1. Introduction

Stratospheric polar vortex in the Arctic winter is significantly more volatile than its Antarctic counterpart, the culmination of which being stratospheric sudden warmings (SSWs). Recurring every 1.5–2 winters on average, SSWs involve a rapid transformation of the polar vortex, reversal of the circumpolar winds and a sudden increase in the polar temperatures (Scherhag 1952; Quiroz 1975; McIntyre and Palmer 1983, 1984; Limpasuvan et al. 2004; Charlton and Polvani 2007; Butler et al. 2015). SSWs not only disrupt stratospheric circulation but also affect tropospheric circulation of the extratropics in weeks to follow (Kodera and Chiba 1995; Baldwin and Dunkerton 1999, 2001; Kushner and Polvani 2004; Polvani and Waugh 2004; Gerber et al. 2009; Hitchcock and Simpson 2014).

SSWs are arguably the most dramatic manifestation of wave–mean flow interaction in Earth’s atmosphere, for which transience in the planetary-scale Rossby waves plays a central role (McIntyre 1982; Limpasuvan et al. 2004). In the first successful modeling of SSW, Matsuno (1971) assumes that wave transience is a forced response of the stratosphere to tropospheric processes such as atmospheric blocking, a paradigm adopted by many subsequent studies (e.g., Holton 1976; Polvani and Saravanan 2000; Scott and Haynes 2000; Sjoberg and Birner 2012), with substantial observational support (Randel and Boville 1987; Polvani and Waugh 2004; Martius et al. 2009; Woollings et al. 2010). An impulsive wave forcing at the tropopause produces a vertically propagating packet of quasi-stationary Rossby waves, and the converging Eliassen–Palm (E–P) flux at the leading edge of the wave packet decelerates the zonal-mean zonal wind (Matsuno 1971).

An alternative paradigm of SSW asserts that wave transience arises from the internal dynamics of the stratosphere. Holton and Mass (1976) demonstrate with an idealized model that even a steady wave forcing at the tropopause generates stratospheric vacillation in wave activity and the zonal-mean zonal wind when a critical forcing amplitude is exceeded. Plumb (1981) proposes instability due to interaction between the steady wave forcing and near-resonant free waves in the stratosphere as a cause of wave transience (see also Geisler 1974; Tung and Lindzen 1979; Smith 1989). Unlike the transient forcing paradigm, the resonance theory predicts the amplification of the stationary wave as part of the solution. Resonance and wave–mean flow interaction also produce multiple equilibria (cf. Charney and DeVore 1979; Held 1983), prompting some to characterize SSW as a transition from a “high-index” (nearly zonal) state to a “low-index” (perpetual resonance) state of the polar vortex (Chao 1985; Yoden 1987, 1990; Ruzmaikin et al. 2003; Birner and Williams 2008; Yasuda et al. 2017). The resonance idea has been applied to nearly barotropic, split-type SSWs (Esler and Scott 2005; Matthewman and Esler 2011; Liu and Scott 2015).

Whether one adopts the first paradigm or the second, modeling studies have long shown that the properties of wave transience are sensitive to the initial profile of the stratospheric winds and their vertical shear (Holton and Dunkerton 1978; Chen and Robinson 1992; Scott and Polvani 2006). This suggests that the condition of the stratosphere has a strong filtering effect on the forcing from below (Hitchcock and Haynes 2016). Generally, weak (but not too weak) zonal winds are conducive to SSWs, but such “preconditioning” of stratospheric winds in turn depends on prior wave events (or lack thereof) (Polvani and Waugh 2004; Jucker 2016; de la Cámara et al. 2017).

Although the short-term predictability of SSWs has improved over time (Mukougawa and Hirooka 2004; Stan and Straus 2009; Tripathi et al. 2016; Taguchi 2016; Rao et al. 2018), there remains considerable uncertainty in GCMs’ ability to predict the frequency of SSWs and its trend under a changing climate (Labitzke and Naujokat 2000; Charlton-Perez et al. 2008; McLandress and Shepherd 2009; Ayarzagüena et al. 2018). This may be due partly to variations in the definition of SSW (Butler et al. 2015), but more fundamentally, our understanding of SSW’s onset is still incomplete, limiting our ability to interpret the disparate model results. Stationary waves in the stratosphere sometimes attain large amplitude and yet do not produce a full-blown SSW; other times they do (Solomon 2014). When an SSW does occur, deceleration and reversal of the zonal-mean zonal wind proceed swiftly, exhibiting a sense of “suddenness.” What separates SSW and non-SSW conditions and why is vortex breakdown abrupt? The resonance paradigm partially addresses this question but supportive evidence from observation is still circumstantial at best (e.g., Smith 1989; Esler et al. 2006).

In what follows, we propose a view that the timing and suddenness of SSWs are determined, on average, by a threshold behavior of Rossby waves after they attain finite-amplitude wave activity flux in the vertical, regardless of how they attain it. Based on the observed finite-amplitude wave activity [FAWA; Nakamura and Zhu (2010), hereafter NZ10, Nakamura and Solomon (2010), hereafter NS10] and its vertical flux, we present evidence that there is an upper bound on the upward E–P flux (“transmission capacity”) of a stationary Rossby wave for a given altitude and flow condition. A rapid, spontaneous vortex breakdown occurs once the incident E–P flux from below reaches this capacity. As we will see, this happens when the zonal-mean zonal wind *u*_{REF} (to be defined below). We thus propose the ratio

The dynamics underlying the threshold behavior is akin to that discussed by Wang and Fyfe (2000), who use a Boussinesq contour dynamics model of the polar vortex to show that the onset of wave breaking occurs once the zonal-mean zonal wind drops below about one-half of the initial value. However, unlike Wang and Fyfe, our diagnostic applies to instantaneous data without a need for solving an initial-value problem. We will also construct an idealized 1D model of SSWs that encapsulates the essence of the threshold behavior. Mathematically similar to the “traffic jam” model of Nakamura and Huang (2018) for atmospheric blocking, this model predicts salient features of SSWs including a rapid vortex breakdown and a downward migration of wave activity/zonal wind anomalies that follows.

In the next section we will review the diagnostic formalism and describe wave–mean flow interaction observed during the life cycles of SSWs in terms of FAWA, *u*_{REF}. Section 3 explains how wave–mean flow interaction makes the vertical component of E–P flux a nonlinear function of FAWA and how this leads to a threshold behavior. We will demonstrate the threshold behavior in response to transient wave forcing with an idealized, semiempirical 1D model of SSW based on the WKB theory. In section 4 we will examine the effects of radiative damping and mixing on the threshold behavior. In section 5 we present an observational support for the theory and demonstrate the utility of

## 2. Wave–mean flow interaction during SSW life cycles

### a. FAWA and reference state

*ϕ*

_{e}[e.g., Eq. (4) of NS10]:

*q*(

*λ*,

*ϕ*,

*z*,

*t*) is PV (see Table 1 for the list of symbols) and

*dS*=

*a*

^{2}cos

*ϕdλdϕ*. Domains

*D*

_{1}and

*D*

_{2}and equivalent latitude

*ϕ*

_{e}are defined by

*Q*(

*ϕ*

_{e},

*z*,

*t*) specifies the PV contour that encloses the same area as the polar cap bounded by

*ϕ*=

*ϕ*

_{e}, that is, the areas of

*D*

_{1}and

*D*

_{2}are the same [see Fig. 1 of NS10]. By definition FAWA

*A*(

*ϕ*

_{e},

*z*,

*t*) is nonnegative and vanishes only when the PV contour is zonally symmetric.

List of symbols.

*A*(

*ϕ*

_{e},

*z*,

*t*) are related through the following relation, assuming

*ϕ*

_{e}=

*ϕ*[e.g., Eq. (10) of NS10 and Eq. (3) of Nakamura and Solomon (2011), hereafter NS11]:

*μ*≡ sin

*ϕ*and

*S*,

*A*,

*A*/∂

*t*= 0),

*reference-state wind u*

_{REF}(

*ϕ*,

*z*,

*t*) that satisfies the following:

*u*

_{REF}is related to the reference-state PV

*q*

_{REF}(

*ϕ*,

*z*,

*t*), which is obtained by zonalizing the instantaneous PV (

*q*) through area-preserving map on each

*z*surface (NZ10, NS10):

^{1}Since

*q*

_{REF}changes only in response to nonconservative processes,

*u*

_{REF}is invariant under conservative dynamics [Eq. (6), see Methven and Berrisford (2015) for a related discussion]. In what follows,

*u*

_{REF}is inverted hemispherically from

*q*

_{REF}with Eq. (7) with no-slip lower boundary condition [see supplementary material of Nakamura and Huang (2018)].

### b. Reversible and irreversible components of wave–mean flow interaction

*A*= 0) one obtains

*u*

_{REF},

*A*. Their relationship at 60°N and

*z*= 32 km (10.34 hPa) during SSW’s life cycle is demonstrated in Fig. 1a for the 2009 event (Harada et al. 2010) [data source: ERA-Interim (Dee et al. 2011)]. Vortex conditions at four stages of this splitting event, labeled A–D, are illustrated in Fig. 2. As the vortex is distorted,

*A*increases and Δ

*u*decreases, following a nearly linear relationship (Fig. 1a, red dots). Note that Δ

*u*is always negative, meaning that

*u*

_{REF}and it becomes even more so as the event matures. As the flow recovers after the event (blue dots),

*A*decreases and Δ

*u*increases, tracing closely the linear path of the lead. Both

*A*and Δ

*u*almost return to their original values. Figure 1b shows the same diagram for a composite of 18 SSW events between 1979 and 2016. (To obtain a typical picture of SSWs, we have excluded a few late-season events and those that are preceded by another significant event during the 30-day lead time. See Table 2 for the list of the events.) Despite the broad type variations from one event to another (Table 2), and despite the visually evident irreversibility in Fig. 2, the reversible, linear relationship between

*A*and Δ

*u*is robust through the life cycles of SSWs.

List of SSWs included and excluded in Figs. 1a, 1b, and 11. SSW type categories are based on Charlton and Polvani (2007) and Kodera et al. (2016).

*A*/

*u*

_{REF}during 60-day leads of SSW events on different pressure levels, using both ERA-Interim and MERRA-2 (Gelaro et al. 2017). Data points in both panels cluster around a linear relationship expressed approximately

^{2}

*α*≈ 0.4 means that about 40% of FAWA is used to drive

*u*

_{REF}, whereas the remainder is used to drive the residual circulation (NS10). Although the local relation [Eq. (9)] cannot be expected directly from Eq. (8) (except for barotropic case in which

*α*= 1), given the robust observational support, we will use Eq. (9) as a key empirical constraint in the subsequent theoretical development.

In contrast to the reversible nature of Δ*u*, the behavior of *u*_{REF} during SSW is distinctly irreversible. Figures 1c and 1d plot *u*_{REF} cos*ϕ* against *A* cos*ϕ* for the corresponding periods in Figs. 1a and 1b at 60°N and *z* = 32 km. During the 30-day lead, *u*_{REF} remains more or less steady for small values of *A*. For larger values of *A*, *u*_{REF} starts to decrease steadily and it does not recover even after *A* reverses its course. The steady decline of *u*_{REF} is driven by nonconservative processes [Eq. (6)], and although part of this driving is seasonal radiative forcing, the majority of it comes from mixing (Lubis et al. 2018a,b; Martineau and Son 2015). The irreversible loss of *u*_{REF} through mixing delays the vortex recovery in some SSW events (Fig. 9 of Lubis et al. 2018a).

## 3. Theory of threshold behavior

In the previous section we described how finite-amplitude Rossby waves modify the zonal-mean zonal wind during the life cycles of SSWs. In this section we discuss how this affects the propagation of Rossby waves and gives rise to a threshold behavior, using a semiempirical theory.^{3}

### a. Vertical group velocity of a stationary Rossby wave

^{4}The same theory also governs FAWA at small amplitude under conservative dynamics (NZ10). To maximize the affinity of quasigeostrophic theory for data, we formulate the dispersion relation for a near plane wave in the Mercator coordinate (e.g., section 5b of Hoskins and Karoly 1981). This leads to the following expression for the vertical group velocity for the stationary Rossby wave (see appendix for derivation):

*k*=

*n*/

*a*is equatorial wavenumber and

*n*> 0 is the integer zonal wavenumber, whereas

*m*is vertical wavenumber given by Eq. (A9). We assume a positive root of

*m*and

*N*nearly constant with height in the stratosphere (≈2.1 × 10

^{−2}s

^{−1}). The factor

*N*, this is the only other empirical constraint we use to simplify the theory.

*l*[and through Eq. (A9),

*m*] as follows. Invariance of Eq. (11) through wave–mean flow interaction suggests

*aμ*),

^{5}the lhs of Eq. (12) may be rewritten as

*A*with respect to latitude:

*α*= 0.4 in the last approximation. This gives the meridional scale of FAWA,

*L*≈ 1.45 × 10

^{6}m, at 60°N. Then assuming that

*q*′ is modal in latitude and proportional to sin

*ly*[

*y*is Mercator latitude; Eq. (A1)]:

^{6}

*m*= 2.36 × 10

^{−4}m

^{−1}for

*n*= 1 and 2.18 × 10

^{−4}m

^{−1}for

*n*= 2. Substituting Eq. (15) into Eq. (10), we obtain the vertical group velocity of the stationary Rossby wave as

*C*is the ratio of

*c*

_{gz}to

*C*is nearly independent of height and

*C*≈ 2.23 × 10

^{−3}for

*n*= 1 and ≈4.11 × 10

^{−3}for

*n*= 2 at 60°N. Equation (17) highlights the proportionality between the group velocity and

*A*according to Eq. (9),

### b. Wave activity budget and threshold

^{7}

*c*

_{gz}

*Ae*

^{−z/H}≡

*F*

_{z}is the vertical component of wave activity (E–P) flux density. Here the convergence of the meridional wave activity flux and nonconservative terms are ignored. These omissions would make Eq. (19) a poor approximation for describing the full life cycle of SSW (e.g., O’Neill and Taylor 1979; Lubis et al. 2018a,b). Yet there is mounting evidence that early development of SSW is dictated by the upward E–P flux and its convergence (Polvani and Waugh 2004; Jucker 2016), so we take Eq. (19) as a model for the SSW’s onset. [Alternatively, Eq. (19) may be thought of as the budget of meridionally integrated wave activity (e.g., Plumb and Semeniuk 2003).] With Eq. (18) we have

*c*

_{gz}is a linear function of

*A*,

*F*

_{z}is a quadratic function of

*A*. We will compare this theoretical prediction of

*F*

_{z}with observation in section 5. For now we demonstrate how the nonlinear flux of FAWA brings about a threshold behavior.

*F*

_{0}> 0 for upward propagation):

*F*

_{0}may be determined, for example, by a steady wave forcing at the lower boundary. Solving Eq. (22) for a steady-state

*ξ*:

*ξ*is unstable and hence unrealizable (we will see why shortly). The discriminant condition in Eq. (23) means that there is a height-dependent upper bound on the wave activity flux that a steady stationary wave can transmit. This places a “transmission capacity” for finite-amplitude Rossby waves, whose vertical transmission is guaranteed at small amplitude (Charney and Drazin 1961). For a given

*F*

_{0}and Γ(

*z*), Eq. (23) predicts the altitude at which this upper bound is first reached. Because of the exponential factor, Γ(

*z*) is a decreasing function of

*z*at sufficiently high altitude [Eq. (21)], and therefore even if

*F*

_{0}< Γ(

*z*)/4 at low altitudes,

*F*

_{0}eventually equals Γ/4 at some

*z*. This altitude

*z*

_{c}is lower for larger

*F*

_{0}. At this altitude, the following conditions are met:

*u*

_{REF}at

*z*=

*z*

_{c}, and this criterion does not depend on the value of

*α*as long as it is positive. The above conditions correspond to the vertex of the quadratic curve in Fig. 5 (point “c”) touching the line

*F*

_{z}=

*F*

_{0}. For

*z*>

*z*

_{c}, the vertex will be lower than

*F*

_{0}, so Eq. (22) has no real root. To ensure a real value for

*ξ*,

*F*

_{z}must be capped at the transmission capacity:

*F*

_{z}/∂

*z*< 0 above

*z*

_{c}(see the left panel of Fig. 5b), which is incompatible with the steady-state assumption:

*A*(and hence

*ξ*) must increase with time according to Eq. (19). From Eqs. (19), (25), and (21), the growth rate of wave activity is estimated to be

*e*-folding time of 1.5–3 days, consistent with the suddenness of SSW. The growth of

*ξ*beyond

*ξ*

_{c}causes

*F*

_{z}to decrease further, since

*F*

_{z}is a decreasing function of

*ξ*for

*ξ*>

*ξ*

_{c}[Eq. (21), Fig. 5a]. This sets up a positive feedback between wave activity and its flux: the decreasing flux above the threshold altitude

*z*

_{c}reinforces the flux convergence and further increase in

*A*, which then promotes further reduction in

*F*

_{z}, provided that there is a continued supply of incident flux from below. Therefore, above

*z*=

*z*

_{c}where the threshold (24) is first met, wave activity increases rapidly until it reaches

*ξ*>

*ξ*

_{c}and corresponds to “vortex breakdown.”

^{8}The threshold behavior and spontaneous breakdown of the vortex arises only because the E–P flux is a nonlinear function of

*ξ*and unimodal.

Note that vortex breakdown starts when

### c. Numerical experiments

*A*and

*z*≤ 110 km and discretized with grid spacing of 100 m. A small numerical diffusion with a diffusion coefficient of 10 m

^{2}s

^{−1}is added to the rhs to smooth out noise. We assume

*C*= 3 × 10

^{−3},

*H*= 7 km, and

*A*as

*α*= 0.4. The initial condition is

*u*

_{REF}increases linearly from 20 m s

^{−1}at

*z*= 10 km to 80 m s

^{−1}at

*z*= 50 km and remains 80 m s

^{−1}above 50 km. This is broadly consistent with the climatology of

*u*

_{REF}at the beginning of the winter (see Fig. 12 below). At the lower boundary (

*z*= 10 km), wave forcing is prescribed as

*z*= 110 km) a radiation condition is assumed.

In the first experiment, we turn on the wave forcing around day 10 (*t*_{on} = 10 days) relatively quickly (*t*_{0} = 6 h) and leave it on. The result is shown in the top row of Fig. 6 in height–time cross sections for *z* ≤ 50 km. After forcing is turned on, a trace of wave activity moves up and quickly amplifies toward the top of the stratosphere (Fig. 6a). During days 12–15, *A* undergoes a rapid increase in the upper stratosphere, whereas *A* and zero

The progression of vortex breakdown is understood as follows. Soon after the wave forcing is turned on, *F*_{z} reaches a steady value of *F*_{0} = 7.87 × 10^{−2} m^{2} s^{−2} in the lower stratosphere. For this value of *F*_{0}, the threshold (24) is reached at *z*_{c} = 27.5 km. Therefore, a runaway flux convergence and vortex breakdown occur above this altitude first, shifting the vortex state from “c” to “d” in Fig. 5a. Subsequently the region of vortex breakdown (a stagnant reservoir of FAWA like critical layer) expands downward by absorbing the incident wave activity flux from below, which causes the vortex terminus to move down (right panel of Fig. 5b). This is akin to a traffic jam on a highway, where the jam expands backward by absorbing the traffic from behind (Lighthill and Whitham 1955; Richards 1956). Nakamura and Huang (2017, 2018) and Paradise et al. (2019) demonstrate that a similar dynamics is at play in the formation of atmospheric blocking.

*A*and define its altitude as

*z*

_{b}(

*t*), we can estimate its migration speed

*dz*

_{b}(

*t*)/

*dt*using the Rankine–Hugoniot relation [LeVeque 2002, chapter 3]. Just above

*z*

_{b}, from Eq. (27):

*z*

_{b}, from Eq. (23):

*z*

_{b}=

*z*

_{c}where the shock first forms,

*dz*

_{b}/

*dt*= −

*Cu*

_{REF}(

*z*

_{c})/2. This corresponds to the slope of the line segment

*z*

_{c}. We have computed

*z*

_{b}(

*t*) by numerically integrating Eq. (33) and show it in Fig. 6b. It captures the descent of the vortex boundary very well below

*z*

_{c}= 27.5 km.

When the wave forcing is switched on more slowly (*t*_{0} = 96 h, *t*_{on} = 10 days), the result is qualitatively similar but the vortex breakdown occurs earlier and the descent of the vortex terminus is noticeably slower in the upper stratosphere (Figs. 6c,d). In this case, a weak wave activity flux during the early stage of spinup manages to penetrate higher and initiate vortex breakdown from *z*_{c} > 50 km. The vortex terminus is very sharp throughout the stratosphere.

*z*>

*z*

_{b}. In reality, we expect some wave reflection to occur at

*z*=

*z*

_{b}. For example, Killworth and McIntyre (1985) show that a Rossby wave critical layer changes from an absorber to a perfect reflector of waves as it matures. We can crudely model the effects of wave reflection by making the wave forcing shorter, since the reflected (downward) wave packet partially cancels the upward wave activity flux and reduces the persistence of the net wave forcing (Harnik 2009; Kodera et al. 2016). In Figs. 6e and 6f, we apply wave forcing only for 5 days. We accomplish this by adding

*z*= 22 km, below which

*A*and

*z*

_{b}, therefore

*z*

_{b}remains steady. Greater and longer forcings cause the vortex terminus to reach lower altitudes (not shown).

To highlight the role of wave–mean flow interaction in the threshold behavior, in Figs. 6g and 6h we repeat the same experiment with *α* = 0. Without wave–mean flow interaction, the pulse of wave activity propagates upward without modifying

## 4. Effects of damping

*S*is nonconservative sources-sinks of

*A*, whereas

*c*

_{gz}is given by Eq. (18).

^{9}Note that

*u*

_{REF}now varies with time due to nonconservative processes. Note also that Eqs. (35)–(37) ensure

### a. Radiative damping

*τ*(

*z*):

*τ*

^{−1}increases linearly from (30 days)

^{−1}at

*z*= 10 km to (5 days)

^{−1}at

*z*= 50 km and remains (5 days)

^{−1}for

*z*> 50 km. Similar values are used by Holton and Mass (1976) and Plumb and Semeniuk (2003). With (39) and the same diffusion coefficient as before, we solve Eqs. (35)–(37) numerically.

*t*

_{on}= 2 days,

*t*

_{0}= 6 h). The subsequent evolution is qualitatively similar to the undamped case, except the descent of the vortex terminus is much slower: 38 days after the forcing is switched on, it is still 4–5 km above the lower boundary, whereas it reached the boundary in less than 20 days in the adiabatic case (Figs. 6a,b). Also, after the vortex breakdown,

*τ*

^{−1}varies in

*z*much more slowly than exp(−

*z*/

*H*), the approximate solution to Eq. (40) is

*F*

_{1}is independent of height and

*F*

_{1}= 0 for

*z*>

*z*

_{b}since the rhs of Eq. (41) vanishes as

*z*→ ∞. This leads to

*u*

_{REF}with

*u*

_{REF}−

*H*/(

*Cτ*),

*F*

_{0}with

*F*

_{1}, and

*F*

_{z}with

*u*

_{REF}−

*H*/(

*Cτ*) in Eq. (33), that is, radiative damping slows down the descent.

*t*

_{on}= 2,

*t*

_{off}= 7 days in Eqs. (30) and (34)], not only is the vortex breakdown interrupted halfway, but the vortex recovery proceeds in a discontinuous fashion (Figs. 7c,d). In a stark contrast with Figs. 6c and 6d,

*A*and

*z*=

*z*

_{b}) that moves up at a near constant rate. The upward movement of this “recovery line” is due to a gap in the wave activity flux across it. After the wave forcing is shut off,

*z*

_{b}since

*A*≈ 0. However,

*F*

_{z}is positive for

*z*>

*z*

_{b}, since

*A*≈ [

*u*

_{REF}−

*H*/(

*Cτ*)]/

*α*> 0. Due to this gap in the flux, the recovery line

*z*=

*z*

_{b}must move up, and the rate of the movement is given by

*H*= 7 km and

*τ*≈ 5 days, the ascent rate is 1.47 km day

^{−1}, roughly consistent with Fig. 7c (≈1.4 km day

^{−1}).

### b. Mixing

*A*in Eq. (39) with a flow-dependent damping through

*S*:

*τ*(

*z*) is the radiative damping time introduced earlier. When

*A*= 0),

*τ*

_{mix}recovers the radiative damping time

*τ*. When

*τ*

_{mix}minimizes to 0.2

*τ*. The above formula is ad hoc and meant as a crude parameterization of mixing inhibition by the mean flow (e.g., Ferrari and Nikurashin 2010).

When Eq. (43) is used instead of *S*_{rad} in Eq. (39), the numerical results change substantially. (In this set of experiments, we also triple the diffusion coefficient to 30 m^{2} s^{−1}). Figures 8a and 8d correspond to Figs. 7a and 7d, the only difference being *S* used in Eq. (35). In both the switch-on and the short pulse forcings, FAWA is heavily damped by mixing after it reaches maximum in the upper stratosphere (Figs. 8a,d). In fact, in both cases *A* survives only for a short duration in a very similar way, suggesting an important role of mixing in determining the duration of a wave event. The main difference between the two forcing types is the response of *promotes* vortex recovery when the wave forcing is short. White contours in Fig. 8b identify regions in which

Figure 9 shows relationships between *A* and *A* and *u*_{REF} at *z* = 32 km during the two experiments shown in Fig. 8. In both cases *A* and Δ*u* form a linear relation as expected (Figs. 9a,c), whereas *u*_{REF} monotonically decreases as *A* grows and decays. Only in the recovery stage of Fig. 9d, *u*_{REF} is radiatively restored and the hysteresis loop is complete.^{10} The qualitative resemblance between Figs. 9 and 1 (except that Fig. 1d does not form a closed loop partly due to the seasonal demise of the polar vortex) is another evidence that the model is capturing the rudimentary dynamics of the observed SSWs.

## 5. Observational evidence for threshold behavior

*F*

_{z}is a nonlinear (unimodal) function of FAWA due to wave–mean flow interaction and (ii) the mode of

*F*

_{z}defines the threshold for a spontaneous growth of FAWA. The WKB theory based on a vertically propagating stationary Rossby wave predicts

*F*

_{z}to be a quadratic function of

*A*[Eq. (20)], but the foregoing conclusions hold for a more generalized form of Eqs. (19) and (21):

*M*(

*ξ*) is any smooth, nonnegative, unimodal function that does not depend on

*z*, and Γ(

*z*) is defined in Eq. (21). For the WKB theory

*M*(

*ξ*) = (1 −

*ξ*)

*ξ*. In the steady state, the lhs of Eq. (44) vanishes and Eq. (23) is replaced by

*M*(

*ξ*

_{c}) =

*M*

_{max}. The threshold behavior occurs at

*ξ*=

*ξ*

_{c}. If the observed flux fits the above form and if we can demonstrate the modality of

*M*with respect to

*ξ*and exceedance of

*ξ*

_{c}during SSWs, it will be a strong evidence for a threshold behavior, irrespective of the WKB theory or how the wave activity flux is generated. Since [e.g., Andrews et al. (1987), their Eq. (3.5.6); NS10, their Eq. (7)]

*M*at multiple altitudes using reanalysis data as

*ξ*. If the data matches the WKB theory, we should observe

*M*(

*ξ*) = (1 −

*ξ*)

*ξ*at all altitudes.

Figure 10 shows *M*(*ξ*) during the growing stage of SSW (60-day lead plus 2-day lag) for 5, 7, 10, and 20 hPa at 60°N (daily composites of 26 events in ERA-Interim and 25 events in MERRA-2). In Eq. (47), we have assumed *n* = 2 for all events to evaluate *C* [Eq. (17)] but used instantaneous *u*_{REF} since it varies significantly from one season to another. In doing so, we are effectively treating Γ(*z*) as a function of time also. This brings about some scatter in our analysis, but for both reanalysis products, data points for 5-, 7-, and 10-hPa collapse on average on a single curve (dashed red) with a clearly defined peak around *ξ* ≈ 0.7(=*ξ*_{c}). The 20-hPa data trace a similar curve (dashed green) but its peak is lower and shifted to the left of the other curve. On both curves, the last few days around the event are on the right-hand side of the peak (*ξ* > *ξ*_{c}). This is highly suggestive of a threshold behavior, and at least for 5–10 hPa, the height-independent *M*(*ξ*) is consistent with Eq. (44). On the other hand, although the diagnosed *M*_{max} is close to theory (0.25), its location *ξ*_{c} is shifted from the theoretical prediction (0.7 as opposed to 0.5). It may be that the weak dependence of *u*_{REF} on *A* due to mixing toward the event (Figs. 1c and 1d) modifies the form of *M*(*ξ*) from theory. Some discrepancies are undoubtedly related to the limitation of the WKB approximation employed in section 4. For example, *M*(*ξ*) remains positive even after the zonal-mean zonal wind is reversed; this is likely due to a memory effect of rapidly evolving Rossby waves – an aspect that cannot be handled by the quasi-steady assumption in the WKB theory.

Since *ξ*_{c} ≈ 0.7, a threshold behavior is expected to occur when *r* at 60°N and *z* = 32 km are compared for 18 winters with SSWs and non-SSW winters using ERA-Interim. While *r* stays above 0.3 for the majority of the non-SSW winters, the winters with SSW experience rapid decrease in *r* after it reaches 0.3, which marks the onset of the event. There is a wide spread after the event, as some events recover quickly (“reflecting” type in Table 2) while others recover more slowly (“absorbing” type), but the threshold behavior during the onset period is shared by all events analyzed. There were a few winters in which *r* dropped below 0.3 temporarily but a full SSW did not follow, presumably because the wave forcing from below was terminated prematurely.

Given the ability of *r* to identify an SSW threshold, we propose it as a useful instantaneous diagnostic for the proximity of the polar vortex to breakdown and a tool to understand the intermittency of SSWs. Figure 12 shows height–time cross sections of *u*_{REF} (color shading) and *r* (contours, plotted for 0.4 or less) at 60°N for strong-vortex winters and weak-vortex winters. To differentiate vortex strengths, we have computed the January–March NAM index based on the first EOF of the 50-hPa geopotential height anomaly north of 30°N. We have chosen 15 years in which the index exceeded 0.5 standard deviation as strong-vortex winters, and 11 years in which it exceeded −0.5 standard deviation as weak-vortex winters (Perlwitz and Graf 2001; Baldwin and Dunkerton 1999). (See caption for the list of winters in each category.) Composite for the strong-vortex winters shows a strong and unperturbed *u*_{REF} in the stratosphere through January–February, and *r* drops below threshold only at the end of March (final warming, Fig. 12a). SSWs are characteristically absent, as exemplified by the streak of SSW-free winters during the mid-1990s. Whereas during the weak-vortex winters, both *u*_{REF} and *r* are significantly smaller already at the beginning of January (Fig. 12b). Then *r* drops below 0.3 from late January through February, starting at the top of the stratosphere and subsequently at lower altitudes, as SSWs form around this time of winter. Mixing associated with SSWs sharply decreases *u*_{REF} from the mid- to lower stratosphere, and it remains weak through the rest of the winter. However, persistence of *u*_{REF} in the upper stratosphere increases after SSW, and this significantly delays the timing of final warming and easterly transition compared to the strong-vortex winters (Fig. 12b), consistent with the findings of previous studies (Hu et al. 2014; Lubis et al. 2017).

A weaker *u*_{REF} is more conducive to SSWs because it lowers Γ [Eq. (21)], allowing the threshold to be reached in the mid- to lower stratosphere with less wave forcing [Eq. (45)]. On the other hand, if *u*_{REF} is too weak, it reduces the growth rate, and hence suddenness, of SSWs [Eq. (26)]. This may be the reason why final warming is more pronounced during the strong-vortex winters (Fig. 12a). Since *u*_{REF} depends on mixing and radiative damping associated with preceding wave events, early season troposphere–stratosphere coupling likely plays a key role in the preconditioning of *u*_{REF} for SSWs. Interestingly, during the weak-vortex winters there are pockets of below-threshold *r*s in the lower troposphere (Fig. 12b). Their connection to SSWs is unclear from this figure alone, but it strongly suggests that wave coupling between the troposphere and the stratosphere is more active during the weak-vortex winters.

## 6. Summary and discussion

In the spirit of Jucker (2016), we have sought canonical properties of SSWs that transcend type variations. In particular, we have highlighted the role of wave–mean flow interaction [Eq. (9)] giving rise to suddenness and intermittency of SSWs through a threshold behavior. The threshold arises from a competition between an increasing wave activity *A* and a decreasing zonal-mean zonal wind *F*_{z}. This renders *F*_{z} a quadratic function of *A* [Eq. (20)] and the mode of *F*_{z} defines the threshold [Eq. (24)]. Once *A* exceeds the threshold value, vortex breakdown proceeds spontaneously through a positive feedback between *A* and *F*_{z}, until *F*_{z} between the two, the boundary migrates downward (Fig. 6). Mathematically this process is analogous to a traffic jam on a highway, where the jam expands backward by absorbing the incoming traffic. The resultant 1D model is akin to the traffic jam model of atmospheric blocking considered by Nakamura and Huang (2017, 2018) and Paradise et al. (2019). Further simplified from previous theoretical models (e.g., Holton and Mass 1976; Plumb and Semeniuk 2003), this model permits certain analytical predictions besides the threshold, including the growth rate of SSW [Eq. (26)], migration speeds of the vortex terminus [Eq. (33)] and the recovery line [Eq. (42)]. The model’s response has been tested for a variety of wave forcing and damping and shown to capture the salient features of observed SSWs (Fig. 8) despite the obvious limitations of the WKB theory it is built on.

Examination of reanalysis data lends strong support for the threshold behavior during the onset of SSWs. In the mid- to upper stratosphere, the observed wave activity follows a canonical unimodal function of FAWA and the onset of SSW lies on the right-hand side of its mode (Fig. 10). Consistent with this, the time series of *r* suggests *r* ≈ 0.3 as a “tipping point” for SSW’s onset (Fig. 11). The utility of *r* as an instantaneous diagnostic for the proximity to vortex breakdown has been illustrated (Fig. 12).

Given that the threshold is reached on average 4–5 days prior to an SSW event (Fig. 11), *r* should provide a forecast skill of SSW with a few days of lead time without a need for running a numerical weather prediction model. More importantly, the threshold may be used to better constrain the effects of climate change on the frequency of SSWs. Currently climate models’ confidence in the future projection of SSW frequency is low (e.g., Ayarzagüena et al. 2018). The lack of convergence among the model projections is likely due to model-to-model variations in (i) the threshold itself and (ii) the fashion in which the threshold is reached. On the first point, analyses like Figs. 10 and 11 should reveal model biases. Models with *ξ*_{c} > 0.7 likely underestimate SSW frequencies for the same wave forcing, and vice versa. On the second point, one must ultimately understand how *u*_{REF} and *A* are determined and how they respond to climate perturbations. Particularly important is the early season behavior (preconditioning) of *u*_{REF} as it determines the subsequent wave forcing required to reach the threshold [Eq. (47)]. Sensitivity of the model prediction of *u*_{REF} and its impact on the SSW frequency to the early season troposphere–stratosphere wave coupling is a worthy topic for further investigation.

## Acknowledgments

This work has been supported by NSF Grants AGS1563307 and AGS1909522. The authors thank two anonymous reviewers for constructive critiques on the early version of the manuscript.

## APPENDIX

### Dispersion Relation in the Mercator Coordinate

^{2}

*ϕ*, reads

^{A1}

*ψ*′ is the perturbation geostrophic streamfunction and the following coefficients are considered “slowly varying”:

*k*=

*n*/

*a*(

*n*is the integer zonal wavenumber) is the equatorial wavenumber and (

*Y*,

*Z*,

*T*) ≡ (

*γy*,

*γz*,

*γt*), (

*γ*≪ 1) is the slow coordinate in the meridional plane and time. Substitution in Eq. (A3) yields the following dispersion relation:

*ω*= 0) Eq. (A6) gives

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^{1}

The rhs of Eq. (6) is the nonconservative sources and sinks of the rhs of Eq. (7). The slight difference in the first term of Eqs. (6) and (7) arises from the approximation made in deriving Eq. (3)—latitudinal dependence of *f* does not cause divergence in the horizontal wind—a benign but necessary assumption to avoid mathematical complication. This discrepancy does not arise in the *β*-plane approximation.

^{2}

We ignore the slight offset in *y* and the variation of *α* in data. These do not affect the subsequent results significantly.

^{4}

Slow variation in the wave packet relative to the wave phase is admittedly not always a best approximation in the real atmosphere. Here we use the WKB theory as a guide and verify its outcome with observed data in section 5.

^{5}

Equation (19) of NZ10 relates the PV gradient of the reference state with that of the instantaneous zonal-mean state through the curvature of FAWA.

^{6}

In the small-amplitude limit, *A* is a quadratic function of *q*′ so the meridional wavenumber of *q*′ should be half of that of *A*. Furthermore, from Eq. (A2), wavenumber in the Mercator coordinate is cos*ϕ* times the wavenumber in the local Cartesian coordinate.

^{7}

∂/∂*T* and ∂/∂*Z* have been rescaled to ∂/∂*t* and ∂/∂*z* using (*Z*, *T*) = (*γz*, *γt*), *γ* ≪ 1, and the cos*ϕ* factor is dropped.

^{8}

This is in essence a critical line where the WKB approximation breaks down formally.

^{9}

We ignore the effect of damping on the group velocity in this study.

^{10}

The area enclosed by the loop is proportional to the pseudoenergy lost to mixing.