1. Introduction

Stratospheric sudden warmings (SSWs) are arguably the most dramatic manifestation of wave–mean flow interaction in the atmosphere, characterized by a precipitous drop in the westerly wind speed (major SSWs cause a wind reversal) and a sharp rise in the polar stratospheric temperatures (Scherhag 1952; Limpasuvan et al. 2004; Charlton and Polvani 2007). They occur usually every other winter in the Northern Hemisphere and have substantial effects on surface weather and climate (Limpasuvan et al. 2004; Charlton and Polvani 2007; Lubis et al. 2018a). Theoretical and numerical studies have long recognized the role of the planetary waves of tropospheric origin in modifying the zonal-mean state of the stratosphere during SSWs (Matsuno 1971; Holton and Mass 1976; Holton 1976; Davies 1981). Using mechanistic models, they showed that wave transience (i.e., a change of wave activity with time) is the primary mechanism for a reversal of the mean flow and the potential vorticity (PV) gradient during the major warming events. Dynamically, this corresponds to the violation of the nonacceleration condition (Charney and Drazin 1961), wherein wave activity (negative angular pseudomomentum) of planetary waves is converted to the angular momentum of the mean flow (McIntyre and Palmer 1983).

As far as the authors are aware, a full budget of wave activity during SSWs has not been considered previously, likely because the traditional diagnostic theory is tailored to small-amplitude waves and is unsuitable for large-wave events such as SSWs (e.g., Davies 1981; Smith 1983, 1985). For example, wave activity based on the linear quasigeostrophic dynamics diverges when the gradient of zonal-mean PV vanishes (Schoeberl and Smith 1986; Andrews et al. 1987). There is also a conceptual difficulty in separating large-amplitude eddies and the mean state through the Reynolds decomposition when the mean state is already modified by the eddies (Solomon and Nakamura 2012). This poses a challenge to closing the wave activity budget using data.

The other hitherto underexplored aspect of SSWs is the effect of eddy-induced mixing on the mean flow during SSWs. The disruption of the polar vortex during SSWs exerts significant influence on the distribution of tracer constituents because of changes in the transport and mixing properties of the atmospheric flow (Plumb 2002). The enhanced isentropic mixing after SSWs is associated with strong large-scale stirring due to Rossby wave breaking in the stratosphere and is often characterized by the stretching and folding of material lines, creating fine filaments in the PV fields (e.g., Nakamura 1995; Allen and Nakamura 2001; de la Cámara et al. 2018). Under adiabatic conditions, enhanced isentropic mixing is directly associated with an increase in the equivalent length of tracer contours (Nakamura 1996). The increasing equivalent length is indicative of the cascade of tracer variance at fine scales. At these fine scales, increasing equivalent length makes the small-scale diffusion become more intense, and therefore, the diffusivity that is apparent in the large-scale structure increases. The increased mixing after the SSW events can persist for more than 2 months in the lower stratosphere (de la Cámara et al. 2018; also see below in Fig. 4). Despite this evidence, it remains unclear how such mixing can affect wave activity evolution and circulation around the polar vortex.

The goal of this paper is to circumvent these difficulties and to elucidate the relative importance of wave transience at finite amplitude and mixing in the transformation of the polar vortex during SSWs using the budget of finite-amplitude Rossby wave activity (FAWA; Nakamura and Zhu 2010; Nakamura and Solomon 2010; Lubis et al. 2018a). FAWA is a generalization of the linear pseudomomentum/wave activity beyond small-amplitude wave limit. Its budget quantifies the contributions of advective transport and irreversible mixing to its evolution and can be computed readily from data. The quantitative difference between FAWA and its small-amplitude approximation is generally much larger than the errors associated with other approximations such as quasigeostrophy (QG) in the midlatitudes; therefore, FAWA greatly improves the budget calculation. As we will show below, the FAWA budget allows one to attribute the forcing of the zonal-mean flow to wave transience, mixing, and diabatic effects. This helps us to better understand the driving mechanisms for the formation, breakdown, and recovery phases of the polar vortex during SSWs.

The paper is organized as follows. Section 2 reviews the basic theory of wave activity and data used in this study. Then a decomposition of the eddy forcing of the mean flow into different contributions, as well as its variation for different types of SSWs, is presented in sections 3–5. Conclusions and discussion are provided in section 6.

2. Theory and data

a. Finite-amplitude wave activity

FAWA (denoted by ) is defined as the areal displacement of QGPV (q) from zonal symmetry (Nakamura and Zhu 2010):

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where is equivalent latitude, defined in such a way that the area covered by the two integrals is identical (see Fig. 1 for an illustration):

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where is pressure pseudoheight, is scale height, t is time, is the radius of Earth, q is QGPV defined as , is the Coriolis parameter, is the rotation rate of Earth, ϕ is latitude, ζ is relative vorticity, is the background density, θ is potential temperature, and is the average potential temperature over the sphere. The is the Lagrangian-mean QGPV with respect to equivalent latitude (blue dashed line in Fig. 1) on each z surface, and . Equation (1) applies to eddies with arbitrary amplitude and in the small-amplitude limit recovers the familiar expression for linear waves (Nakamura and Solomon 2010). Using this diagnostic, the difficulty associated with overturning PV contours is eliminated, so the budget of wave activity can be evaluated throughout the duration of SSW events.

Schematic for visualizing the finite-amplitude wave activity. Wave activity is defined as the areal displacement of QGPV (q) from zonal symmetry (i.e., the differences between the green and the yellow regions). The dashed contour is the latitude line , and the solid contour is the contour line of . The white arrows indicate the eddy fluxes that instigate the meridional areal displacement in the Q contour.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Schematic for visualizing the finite-amplitude wave activity. Wave activity is defined as the areal displacement of QGPV (q) from zonal symmetry (i.e., the differences between the green and the yellow regions). The dashed contour is the latitude line , and the solid contour is the contour line of . The white arrows indicate the eddy fluxes that instigate the meridional areal displacement in the Q contour.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Schematic for visualizing the finite-amplitude wave activity. Wave activity is defined as the areal displacement of QGPV (q) from zonal symmetry (i.e., the differences between the green and the yellow regions). The dashed contour is the latitude line , and the solid contour is the contour line of . The white arrows indicate the eddy fluxes that instigate the meridional areal displacement in the Q contour.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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b. Generalized Eliassen–Palm relation for finite-amplitude eddies

The generalized Eliassen–Palm (E–P) relation as introduced by Andrews and McIntyre (1976) describes the wave activity budget for small-amplitude waves in the following form:

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where is wave activity for small-amplitude waves; the bar and prime denote the Eulerian zonal average and the departure from it, respectively; F is the generalized E–P flux, which represents radiation stress of the wave and equals the group velocity times the wave activity density for a slowly modulated, small-amplitude wave:

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; D denotes nonconservative effects on wave activity; u and υ are the zonal and meridional wind components, respectively; and represents terms of third (and higher) order of wave amplitude. The terms also include advection of wave activity by eddies and are ignored in the small-amplitude limit. However, these ignored terms become quickly comparable to, sometimes greater than, the other terms when wave amplitude is large. Therefore, the budget is hard to close for large-scale wave events in the atmosphere without the higher-order terms, which complicates the interpretation.

In the finite-amplitude limit, the E–P relation that is applicable for both small- and large-amplitude eddies can be expressed based on FAWA theory (Nakamura and Zhu 2010) as

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where the zonal mean is evaluated at . On the right-hand side of Eq. (5), is FAWA (Eq. 1), is effective diffusivity (Nakamura 1996; Nakamura and Ma 1997; Haynes and Shuckburgh 2000) and is diabatic source/sink of wave activity defined as:

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where , and is the diabatic heating term [e.g., Eq. (3.2.14) in Andrews et al. (1987)]. Unlike the E–P relation for the small-amplitude wave activity, Eq. (5) does not involve a cubic term in eddy amplitude.

Equation (5) is written as an expression for the E–P flux divergence, since that term represents the eddy forcing of the zonal-mean angular momentum in the stratosphere:

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where is the meridional component of the residual circulation and is nonresolved zonal momentum forcing (e.g., frictional torque, small-scale gravity wave drag). Now, let be the fraction of the right-hand-side terms of Eq. (7) that drives the Coriolis torque of the residual circulation:

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In the adiabatic limit, γ is largely determined by the aspect ratio of eddy forcing (E–P flux divergence) and becomes smaller as the forcing becomes taller (Pfeffer 1987; Nakamura and Solomon 2010, 2011). In the steady state, γ approaches unity. As we will see later (in Figs. 7c and 7d), and γ is nearly independent of time in the stratosphere [consistent with Eq. (7.2.14) of Andrews et al. (1987)]. By eliminating the E–P flux divergence from Eqs. (7) and (8),

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Thus, the acceleration of the zonal-mean angular momentum is governed by the sum of wave activity tendency (wave transience), diffusive flux of PV (mixing), diabatic wave source/sink, and nonresolved zonal momentum forcing. This is a generalized form of nonacceleration relation, and we will evaluate mainly the three terms in the square brackets throughout the life cycle of SSWs. We will see that, of the three, the first two play predominant roles around the stratospheric polar vortex. The last term in Eq. (9) is found to be much smaller in the lower and middle stratosphere based on the residual of the budget (see later in the discussion of Fig. 8 and Figs. S6 and S7 in the online supplemental material).

Mixing is driven by effective diffusivity ; combines the effects of subgrid-scale mixing and large-scale stirring. If the former is given by regular diffusion, takes the form of (Nakamura 1996; Nakamura and Zhu 2010)

View Expanded

where κ is the diffusion coefficient, is horizontal gradient operator, and denotes the area-weighted average around the wavy PV contour with the value Q. The diffusivity ratio may be interpreted as the square of the normalized equivalent length of the PV contours (Nakamura 1996). The ratio is therefore a measure of the elongation of PV contours (stirring) so that high indicates regions of strong stirring/mixing under near-adiabatic conditions with negligible sources and sinks. Equivalent length has been widely used for diagnosing irreversible mixing and diffusive flux in the stratosphere (e.g., Nakamura 1996; Shuckburgh and Haynes 2003).

Modern GCMs rarely use regular diffusion for subgrid-scale mixing, and the diffusive flux is often implicit. In this case, the form of κ is not known and may vary with location and time. Therefore, we evaluate the second term on the right-hand side of Eq. (5) as a residual of the budget, assuming that all the terms in Eq. (5) are well balanced (e.g., Yu et al. 2014). We then estimate by dividing the term by the PV gradient. As an a posteriori validation, we will compare the result of with the direct calculation of normalized equivalent length . We will show later that and normalized equivalent length behave similarly, suggesting that the estimates of effective diffusiviity based on the wave activity budget are reliable and that the details of κ do not affect the variability of significantly.

d. Classification of event type

SSW events are identified following Charlton and Polvani (2007), as the day when the zonal-mean zonal wind at 60°N and 10 hPa is reversed to easterly during the boreal winter (November–March). Each SSW event in the same winter must be separated by consecutive westerlies of at least 20 days. If the zonal-mean zonal wind does not return to westerly for at least 10 consecutive days before the end of April, then the event is not considered an SSW but a final warming instead and is excluded in the analysis. In the 38-yr period of MERRA-2, we identify 25 SSWs (0.66 yr⁻¹), while in the 100-yr simulation with WACCM, we have 63 (0.63 yr⁻¹) events.

In addition, we also analyze characteristic of different type of SSW events based on 1) the geometry of the vortex breakdown, including splitting and displacement types (Charlton and Polvani 2007), and 2) the speed of vortex recovery, including reflective and absorbing types (Kodera et al. 2016; Lubis et al. 2018b). The algorithm used for classification of vortex-displacement or vortex-splitting events is similar to the method outlined by Charlton and Polvani (2007). The method involves identifying the number and relative sizes of cyclonic vortices during the evolution of the warming. Furthermore, the SSW events are also classified into reflective (rapid recovery) or absorptive (slow recovery) vortex events. An individual SSW event is defined to be reflective when the total eddy heat flux (averaged over 45°–75°N at 100 hPa) remains negative for more than two out of seven days, on and after the maximum temperature during an SSW event, while the remaining events are classified as absorptive types. The observed SSW dates used for this analysis are listed in Table 1.

Table 1.

Dates and types of SSW events identified in MERRA-2; “D” indicates a vortex displacement, “S” indicates a vortex split, and “A” and “R” denote absorptive and reflective SSW events, respectively. The bottom row shows the total cases and the count for each event type.

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3. Wave activity (angular pseudomomentum) budget

Figure 2 shows the time-lagged composites of the anomalies of the terms in Eq. (5) during SSW events, averaged between 50° and 70°N in MERRA-2 (hereafter simply observation) and in the complex CESM1(WACCM) simulation. We first focus on the observation (left column in Fig. 2). Prior to the peak of the warming events, there is a significant negative eddy-forcing anomaly (anomalous E–P flux convergence) in the observation (Fig. 2a). This anomalous convergence exhibits similar pattern and strength to the wave activity tendency anomaly (Fig. 2b) for the same period, indicating that the increasing FAWA largely accounts for the anomalous E–P flux convergence prior to the warming events. This is consistent with the growth of the planetary-wave disturbances that propagate from the troposphere into the stratosphere and interact strongly with the mean flow there (Matsuno 1971; Limpasuvan et al. 2004). There is also a significant effect of mixing (Fig. 2c) and a smaller effect of diabatic heating (Fig. 2d) in the upper stratosphere, all contributing constructively to the E–P flux convergence anomaly prior to the warming events. As a result, the zonal-mean zonal wind is decelerated rapidly (Fig. 2a, contours).

Time–height composites of wave activity budget terms [Eq. (5)] during SSWs averaged between 50° and 70°N in (left) 38-yr MERRA-2 and (right) 100-yr CESM1(WACCM) simulation. (a),(e) Eddy forcing , (b),(f) tendency of FAWA density , (c),(g) irreversible wave dissipation due to diffusive flux of PV or mixing , and (d),(h) diabatic source/sink of FAWA anomalies. The contour intervals are in logarithmic powers of 2: ±[0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, …] m s⁻¹ day⁻¹. The black dashed (solid) contour line denotes negative (positive) zonal-mean wind anomaly with contour interval of 10 m s⁻¹, and the thick line represents zero. The number of events is given above (a) and (e). The green solid (dashed) contour lines indicate 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Time–height composites of wave activity budget terms [Eq. (5)] during SSWs averaged between 50° and 70°N in (left) 38-yr MERRA-2 and (right) 100-yr CESM1(WACCM) simulation. (a),(e) Eddy forcing , (b),(f) tendency of FAWA density , (c),(g) irreversible wave dissipation due to diffusive flux of PV or mixing , and (d),(h) diabatic source/sink of FAWA anomalies. The contour intervals are in logarithmic powers of 2: ±[0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, …] m s⁻¹ day⁻¹. The black dashed (solid) contour line denotes negative (positive) zonal-mean wind anomaly with contour interval of 10 m s⁻¹, and the thick line represents zero. The number of events is given above (a) and (e). The green solid (dashed) contour lines indicate 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Time–height composites of wave activity budget terms [Eq. (5)] during SSWs averaged between 50° and 70°N in (left) 38-yr MERRA-2 and (right) 100-yr CESM1(WACCM) simulation. (a),(e) Eddy forcing , (b),(f) tendency of FAWA density , (c),(g) irreversible wave dissipation due to diffusive flux of PV or mixing , and (d),(h) diabatic source/sink of FAWA anomalies. The contour intervals are in logarithmic powers of 2: ±[0.25, 0.5, 1, 2, 4, 8, 16, 32, 64, …] m s⁻¹ day⁻¹. The black dashed (solid) contour line denotes negative (positive) zonal-mean wind anomaly with contour interval of 10 m s⁻¹, and the thick line represents zero. The number of events is given above (a) and (e). The green solid (dashed) contour lines indicate 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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After the peak of the warming events, there is a strong compensating effect between the wave activity tendency anomaly and the anomalies in the other two right-hand-side terms of Eq. (5) (mixing and diabatic heating; Figs. 2b–d), resulting in a much reduced eddy-forcing anomaly (Fig. 2a). Of the two, mixing plays a dominant role (Fig. 1c), and the diabatic heating accounts for a less significant fraction of wave activity tendency (Fig. 2d). The predominant balance between the anomalies in wave activity tendency and mixing (plus diabatic heating) in Eq. (5) keeps the anomaly in the E–P flux divergence small, so the tendency of zonal wind in Fig. 2a (indicated by the contours) is much weaker after the events. This demonstrates that after the peak of the warming events, mixing and radiative heating largely act to offset wave activity tendency, weakening the acceleration in the zonal-mean flow, particularly in the lower stratosphere. Thus, there is a significant asymmetry in the tendencies of the zonal-mean zonal wind before and after the SSW events, and the role of mixing and other diabatic effects are to slow down the recovery of the polar vortex.

The above results are found to be robust, in a qualitative sense, with the results from the 100-yr complex GCM simulation with CESM1(WACCM) (Figs. 2e–h). It is shown that the enhanced E–P flux convergence prior to the peak of the warming events is mainly attributed to a stronger wave-transience effect (Figs. 2e,f). Only after the peak of the warming events, mixing and radiative heating strongly compensate the wave-transience effects (Figs. 2g,h), resulting in a smaller eddy forcing and thus slower recovery of the zonal-mean flow in the lower stratosphere. In the models, however, the time scale of the anomalous E–P flux divergence in the upper stratosphere after the peak of the warming events is longer compared to that in the observation (Fig. 2e). This pattern is consistent with a faster wave decay and reduced wave dissipation anomalies in the model (Figs. 2f–h). Despite such discrepancies, the general conclusion obtained from the CESM1(WACCM) is similar to the observation.

The results so far demonstrate that the irreversible wave dissipation due to mixing plays an important role in eddy forcing after the peak of the warming events. Since the dissipation of wave activity is proportional to the product of and [Eq. (5)], changes in either quantity affect this term. To evaluate their relative importance in the enhancement of wave dissipation, we further decompose the irreversible wave dissipation at ~800 K (about 10 hPa) into contribution from and (Figs. 3a–c). The results show that the main source in the irreversible wave dissipation arises from changes in instead of changes in (the increase in the former increases the product despite the decrease in the latter). It is consistent with the view that large-scale stirring during the SSW event leads to an enhanced dissipation of potential enstrophy, which erodes the PV gradient (with respect to equivalent latitude). Qualitatively similar results are found in the CESM1(WACCM) simulation, although the peaks of each quantity are relatively stronger compared to the observations (see red and blue curves in Figs. 3a–c). Furthermore, given the assumptions made to calculate (section 2b), one may question whether the changes in are robust. As an a posteriori validation, we computed the diffusivity ratio using the right-hand-side expression of Eq. (10) (Fig. 3d). If Eq. (10) holds, changes in can only arise from the changes in equivalent length (Nakamura 1996; Allen and Nakamura 2001). A cursory inspection of Fig. 3d reveals a strong resemblance between the anomalies of and (Figs. 3b,d). This suggests that the estimates of based on the wave activity budget are reliable and that the details of κ do not affect the variability of significantly. This result, therefore, confirms the insights gained from the budget analysis above, that the increased irreversible wave dissipation after the vortex breakdowns is due mainly to increased .

Time-lagged composites of anomalous (a) diffusive flux of PV , (b) effective diffusivity , (c) meridional gradient of PV , and (d) diffusivity ratio , averaged between 50° and 70°N at 800 K (about 10 hPa) in MERRA-2 (blue lines) and 100-yr CESM1(WACCM) simulation (red lines). Note the units in (a)–(c) are in natural log of dissipation. The linear decomposition of (a) theoretically gives (b) and (c). Red and blue shading indicate plus and minus one standard deviation from the mean of each term.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Time-lagged composites of anomalous (a) diffusive flux of PV , (b) effective diffusivity , (c) meridional gradient of PV , and (d) diffusivity ratio , averaged between 50° and 70°N at 800 K (about 10 hPa) in MERRA-2 (blue lines) and 100-yr CESM1(WACCM) simulation (red lines). Note the units in (a)–(c) are in natural log of dissipation. The linear decomposition of (a) theoretically gives (b) and (c). Red and blue shading indicate plus and minus one standard deviation from the mean of each term.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Time-lagged composites of anomalous (a) diffusive flux of PV , (b) effective diffusivity , (c) meridional gradient of PV , and (d) diffusivity ratio , averaged between 50° and 70°N at 800 K (about 10 hPa) in MERRA-2 (blue lines) and 100-yr CESM1(WACCM) simulation (red lines). Note the units in (a)–(c) are in natural log of dissipation. The linear decomposition of (a) theoretically gives (b) and (c). Red and blue shading indicate plus and minus one standard deviation from the mean of each term.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Figure 4 shows composites of anomalies as a function of time lag and equivalent latitude at isentropic surfaces; can be considered as a measure of the elongation of PV contours (stirring) so that high indicates regions of strong mixing. In the observation, enhanced mixing properties first appear at the middle latitudes around lag 0 and migrate poleward (Fig. 4a) and downward (Figs. 4c–f) as the PV gradients at the vortex edge start weakening. This positive anomaly persists for several weeks, consistent with the result of de la Cámara et al. (2018). This behavior of the anomalies is consistent with the evolution of the irreversible wave dissipation after the warming events that propagate downward in time (Fig. 2c). The CESM1(WACCM) run captures quite reliably the changes in isentropic mixing associated with SSWs as compared to the observation. Yet there is a slight difference in the magnitude and propagating time scale in anomalies (Fig. 4b and Fig. S1). A stronger magnitude in in the model is consistent with stronger diffusive flux of PV anomalies (Figs. 3a,d) as compared to the observation. Overall, the results indicate that enhanced mixing after the vortex breakdown (as shown by and anomalies) increases irreversible wave dissipation, which in turn dampens wave activity tendency and slows down vortex recovery. This highlights the importance of mixing in the asymmetry between breakdown and recovery of the polar vortex during SSWs.

(top) composites of anomalies as a function of time lag and equivalent latitude for (a) MERRA-2 and (b) CESM1(WACCM) at 800-K isentropic surface (about 10 hPa). (bottom) As in the top panel, but as a function of latitude and isentropic surfaces for (c) lag −5, (d) lag 0, (e) lag +5, and (f) lag +10 days in MERRA-2. The black contour lines indicate the zonal-mean zonal wind with contour interval of 5 m s⁻¹. The white solid (dashed) contour lines indicate the 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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(top) composites of anomalies as a function of time lag and equivalent latitude for (a) MERRA-2 and (b) CESM1(WACCM) at 800-K isentropic surface (about 10 hPa). (bottom) As in the top panel, but as a function of latitude and isentropic surfaces for (c) lag −5, (d) lag 0, (e) lag +5, and (f) lag +10 days in MERRA-2. The black contour lines indicate the zonal-mean zonal wind with contour interval of 5 m s⁻¹. The white solid (dashed) contour lines indicate the 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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(top) composites of anomalies as a function of time lag and equivalent latitude for (a) MERRA-2 and (b) CESM1(WACCM) at 800-K isentropic surface (about 10 hPa). (bottom) As in the top panel, but as a function of latitude and isentropic surfaces for (c) lag −5, (d) lag 0, (e) lag +5, and (f) lag +10 days in MERRA-2. The black contour lines indicate the zonal-mean zonal wind with contour interval of 5 m s⁻¹. The white solid (dashed) contour lines indicate the 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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4. Dependence on the types of SSWs

Having illustrated the importance of finite-amplitude wave transience, mixing, and diabatic effects during SSWs, we now proceed to apply the same analysis on different types of SSW events (Figs. 5 and 6). As we discussed in the method description, we classify SSWs based on 1) the geometry of the vortex breakdown, including splitting and displacement types, and 2) the speed of vortex recovery, including reflective and absorbing types.

As in Figs. 2a–d, but for split and displacement types of SSW events in MERRA-2. The number of events is given above (a) and (e). The black solid (dashed) contour lines indicate 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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As in Figs. 2a–d, but for split and displacement types of SSW events in MERRA-2. The number of events is given above (a) and (e). The black solid (dashed) contour lines indicate 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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As in Figs. 2a–d, but for split and displacement types of SSW events in MERRA-2. The number of events is given above (a) and (e). The black solid (dashed) contour lines indicate 95% (99%) significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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As in Figs. 2a–d, but for reflective and absorptive types of SSW events in MERRA-2. The number of events is given above (a) and (e). The black solid (dashed) contour lines indicate 95% (99%) significant level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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As in Figs. 2a–d, but for reflective and absorptive types of SSW events in MERRA-2. The number of events is given above (a) and (e). The black solid (dashed) contour lines indicate 95% (99%) significant level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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As in Figs. 2a–d, but for reflective and absorptive types of SSW events in MERRA-2. The number of events is given above (a) and (e). The black solid (dashed) contour lines indicate 95% (99%) significant level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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In general, both splitting and displacement types of SSWs have similar eddy-forcing characteristics prior to the warming events. In particular, the anomalous E–P flux convergence is balanced by the wave-transience effects, with the irreversible dissipation and diabatic term only playing a minor role (Figs. 5a–h). The magnitude of the anomalous E–P flux convergence in the split events is slightly larger compared to those in the displacement events (Figs. 5a,e).

After the peak of the warming events, both wave activity tendency and irreversible wave activity dissipation last longer and play more significant roles in the wave activity budget of the splitting type than the displacement type (Figs. 5b,f). The greater dissipation reflects an enhanced mixing (equivalent length) in the splitting events (Fig. S2). In either type, irreversible wave dissipation is significantly compensated for by the wave activity tendency. However, the compensation is more complete and persistent for the splitting type, causing weak eddy forcing after day 5 and thus delayed recovery of the polar vortex (Fig. 5a). On the other hand, the compensation is incomplete and short lived for the displacement type, which sustains a modest positive eddy forcing between days 4 and 10, contributing to a partial restoration of the polar vortex (Fig. 5e). The results indicate that the splitting type of SSWs produce greater mixing and hence significantly delay the vortex recovery.

Comparing the reflective (rapid recovery) and absorptive (slow recovery) SSW events (Fig. 6), we see that the anomalous E–P flux convergence in the absorptive events is stronger and persists longer in duration compared to the reflective SSW events (Figs. 6a,b). The anomalous E–P flux convergence is balanced by the wave-transience effects, with the irreversible dissipation and diabatic effects playing a minor role. After the vortex breakdowns, a significant positive E–P flux divergence anomaly is observed during the reflective SSW events (from days 1 to 10; Fig. 6a). This anomalous divergence is mainly attributed to the wave-transience anomaly (decrease in wave activity) and a smaller wave dissipation. The decreased wave activity during this period could be associated with a destructive interference between upward wave activity and downward wave reflection due to a formation of vertical reflecting surface (critical layers) after the warming events (Tomikawa 2010; Kodera et al. 2016; Lubis et al. 2016, 2018b). In contrast, an anomalous E–P flux convergence persists during the absorptive SSW events, resulting from a smaller wave-transience anomaly and a greater irreversible wave dissipation after the vortex breakdown. The stronger wave dissipation during these events reflects stronger eddy-induced mixing compared to the reflective events (Fig. S3). Beyond days 10, the compensation between wave-transience effect and mixing keeps the divergence of the E–P flux small, which explains why the polar vortex recovery is slow in the absorptive events. These results are robust and confirmed by the long-term simulation with the CESM1(WACCM) (see Figs. S4 and S5).

To provide an intuitive, qualitative understanding as to how mixing leads to a prolonged weak vortex from a synoptic perspective, we analyzed daily PV and streamfunction maps at 530 K for two major SSW events: one an event in February 2009 that represents a split-absorptive SSW event (Fig. 7a) and the other an SSW event in December 2001 that represents a displacement-reflective-type event (Fig. 7b). The contorted dark green curve is the contour of , and the red curve is the latitude line . The dark green contour encloses approximately the same area as the polar cap north of 60°N. Based on maps of PV during such SSWs, we can clearly see that the PV contour in the split event is more disturbed and more elongated compared to the displacement event. In particular, during the split event (Fig. 7a), the PV contour is relatively close to zonal symmetry at lag −10 days, and the associated equivalent lengths are small at this period (Fig. 7c). At lag +10 days and onward, however, there is an extreme elongation of the PV contour that leads to enhanced equivalent length (Fig. 7c). The enhanced equivalent length is indicative of a cascade of tracer variance to fine scales at which diffusion is more efficient, meaning that the effective diffusivity increases.¹ The increased diffusivity that is apparent in the large-scale structure causes a stronger diffusive flux of PV (Fig. 5c) and thus a prolonged weak vortex, as indicated by a persistent weak streamfunction gradient after the vortex breaks up (Fig. 7a). The opposite synoptic conditions can be seen in the displacement-type SSW (Fig. 7b). A much lesser elongation of the PV contour after breakup of the vortex translates into both a smaller equivalent length (Fig. 7c) and the diffusive flux of PV (Fig. 5g). This condition leads to fast recovery of the polar vortex, as indicated by a stronger gradient of the streamfunction after the vortex breaks up.

(a),(b) Snapshots of PV fields at 530 K (about 50 hPa) during (a) February 2009 split-absorptive SSW event and (b) December 2001 displacement-reflective SSW event. The contorted dark green curve is the contour of , and the red curve is the latitude line . The dark green contour encloses approximately the same area as the polar cap north of 60°N. The cyan contour shows the streamfunction with interval 1 × 10⁷ m² s⁻¹. (c) Evolution of diffusivity ratio during each SSW event at 60°N and 530 K.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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(a),(b) Snapshots of PV fields at 530 K (about 50 hPa) during (a) February 2009 split-absorptive SSW event and (b) December 2001 displacement-reflective SSW event. The contorted dark green curve is the contour of , and the red curve is the latitude line . The dark green contour encloses approximately the same area as the polar cap north of 60°N. The cyan contour shows the streamfunction with interval 1 × 10⁷ m² s⁻¹. (c) Evolution of diffusivity ratio during each SSW event at 60°N and 530 K.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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(a),(b) Snapshots of PV fields at 530 K (about 50 hPa) during (a) February 2009 split-absorptive SSW event and (b) December 2001 displacement-reflective SSW event. The contorted dark green curve is the contour of , and the red curve is the latitude line . The dark green contour encloses approximately the same area as the polar cap north of 60°N. The cyan contour shows the streamfunction with interval 1 × 10⁷ m² s⁻¹. (c) Evolution of diffusivity ratio during each SSW event at 60°N and 530 K.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Evidence of this behavior was anticipated on theoretical grounds by McIntyre (1982) and in the mechanistic model of Matsuno (1971), in which the irreversible mixing during major warmings took place by stretching out of PV contour to such an extent that dissipation could take over and suppress the eddy forcing.

5. Factors controlling vortex recoveries

The previous results suggest that the finite-amplitude wave transience and eddy-induced mixing play leading roles in eddy-forcing evolution during SSWs. Therefore, it is worth investigating further how important these two factors are in determining the duration of the vortex recovery after SSW events.

Figure 8 shows the time evolution of wave activity as a function of time lag and height, as well as the anomaly composites of the leading terms in Eq. (9) at 10 hPa averaged over 50°–70°N. A cursory inspection shows that rapid vortex recovery involves weak wave transience (Fig. 8a), while a much stronger and more persistent is observed during a slow vortex recovery (Fig. 8b). Furthermore, the budget analysis shows that overall deceleration is driven by the E–P flux divergence (eddy forcing), which is partially offset by the Coriolis torque of the residual circulation , as expected from theory (Andrews et al. 1987; Figs. 8c,d). The role of unresolved processes term² , on the other hand, is negligible in the lower and middle stratosphere (see Figs. S6 and S7). In rapid recovery events, the deceleration is slightly weaker and more short lived compared to slow recovery events. Much of the change in eddy forcing during these events can be accounted for by changes in wave transience, while other terms play a minor role (Fig. 8c). This suggests that the rapid recovery events involve a weak wave transience that follows a nonacceleration relation reversibly. In contrast, a stronger and more persistent deceleration (longer-lived forcing) is observed during slow recovery events. The changes in the diffusive flux of PV (irreversible wave dissipation) mostly dominates the eddy forcing after the peak of warming event, while other terms are of secondary importance (Fig. 8d). This indicates that the slow recovery events are characterized by stronger wave dissipation due to mixing that deviates from a classical nonacceleration relation.

(top) composites of pseudomomentum/wave activity (color shading) and zonal wind (black contour lines) averaged over 50°–70°N for (a) rapid and (b) slow recovery of the polar vortex during SSWs. The contour interval of the zonal-mean wind is 5 m s⁻¹, and the thick line represents zero. The dots indicate 95% significance level. (bottom) Typical balance of terms in Eq. (8) at 10 hPa for (c) rapid and (d) long recovery of the polar vortex. The color shading indicates 95% significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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(top) composites of pseudomomentum/wave activity (color shading) and zonal wind (black contour lines) averaged over 50°–70°N for (a) rapid and (b) slow recovery of the polar vortex during SSWs. The contour interval of the zonal-mean wind is 5 m s⁻¹, and the thick line represents zero. The dots indicate 95% significance level. (bottom) Typical balance of terms in Eq. (8) at 10 hPa for (c) rapid and (d) long recovery of the polar vortex. The color shading indicates 95% significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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(top) composites of pseudomomentum/wave activity (color shading) and zonal wind (black contour lines) averaged over 50°–70°N for (a) rapid and (b) slow recovery of the polar vortex during SSWs. The contour interval of the zonal-mean wind is 5 m s⁻¹, and the thick line represents zero. The dots indicate 95% significance level. (bottom) Typical balance of terms in Eq. (8) at 10 hPa for (c) rapid and (d) long recovery of the polar vortex. The color shading indicates 95% significance level based on a 1000-trial Monte Carlo test.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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Another way to evaluate the extent to which the dynamics of rapid recovery events are conservative is to integrate Eq. (9) and assume that the mixing and other nonadiabatic processes are of secondary importance. Equation (9) then reduces to

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where based on Figs. 8c and 8d (the black and blue curves). This relation is approximately observed in the middle to upper stratosphere around the time of reflective SSW events (Fig. 8a), as indicated by the alignment of the color (A*) and contours . In this case, the quick termination of the easterly wind is directly related to an adiabatic decay of wave activity, which may be linked to other internal dynamics of winter stratosphere (Tomikawa 2010; Kodera et al. 2016; Lubis et al. 2016). In contrast, a much stronger and a more persistent is observed during absorptive events. Here, although the color and contours are nearly parallel shortly prior to the event (Fig. 8b), the behavior of and deviates appreciably from Eq. (9) after these events. While the peak of descends relatively quickly from the upper stratosphere to the upper troposphere, the descent of is more gradual, and the easterly zone persists much longer than the rapid recovery cases. The deviations from this congruence between the changes in and underscore the importance of mixing and other nonconservative processes during the slow recovery events. Thus, it satisfies the following relationship:

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Finally, the relative importance of eddies and nonconservative processes in maintaining the zonal-mean zonal wind can be also directly examined by inverting the zonal angular momentum equation [Eq. (7)] with the adiabatic forcing [first term on the right-hand side of Eq. (5)] and nonadiabatic forcing [last two terms on the right-hand side of Eq. (5)] separately, following the method outlined in Nakamura and Solomon (2010) and Lubis et al. (2018a). Essentially, the product of the inversion gives the net contribution of eddies and nonconservative processes to the total zonal-mean zonal wind . Figure 9 shows the anomaly composites of and for the rapid and slow recoveries of the polar vortex. It can clearly be seen that most of anomalies in the rapid case are attributed to the anomalies, with a small contribution from (Figs. 8a,b). On the other hand, the long-lived deceleration in slow recovery events is largely attributable to , while the explains most changes taking place before and during the onset of the events (Figs. 8c,d). This emphasizes our previous results that the apparent deceleration of the polar circulation in the rapid recovery events is primarily an adiabatic, reversible redistribution of momentum between the zonal flow and the pseudomomentum associated with the waves. Only during the slow recovery events, mixing and diabatic forcing of the flow cause irreversible modification to the mean flow and slow down vortex recovery.

The net contribution of eddies (color shading) and nonconservative processes ; (color shading) to the zonal wind anomalies (black contour lines) averaged over 50°–70°N for (a),(b) rapid and (c),(d) slow recovery of the polar vortex during SSWs. The color shadings are only drawn for anomalies that are statistically significant at the 95% confidence level. The contour line interval is 5 m s⁻¹, and the zero wind lines is omitted. Note that the red (blue) color shading indicates negative (positive) values.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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The net contribution of eddies (color shading) and nonconservative processes ; (color shading) to the zonal wind anomalies (black contour lines) averaged over 50°–70°N for (a),(b) rapid and (c),(d) slow recovery of the polar vortex during SSWs. The color shadings are only drawn for anomalies that are statistically significant at the 95% confidence level. The contour line interval is 5 m s⁻¹, and the zero wind lines is omitted. Note that the red (blue) color shading indicates negative (positive) values.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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The net contribution of eddies (color shading) and nonconservative processes ; (color shading) to the zonal wind anomalies (black contour lines) averaged over 50°–70°N for (a),(b) rapid and (c),(d) slow recovery of the polar vortex during SSWs. The color shadings are only drawn for anomalies that are statistically significant at the 95% confidence level. The contour line interval is 5 m s⁻¹, and the zero wind lines is omitted. Note that the red (blue) color shading indicates negative (positive) values.

Citation: Journal of the Atmospheric Sciences 75, 11; 10.1175/JAS-D-18-0138.1

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6. Conclusions and discussion

There have been considerable advances in theories and data availability since the first observation of SSWs by Scherhag (1952). Still, some dynamical aspects of SSWs lack quantitative assessments, such as the roles of finite-amplitude eddies and irreversible wave dissipation due to mixing in the transformation of the polar vortex during SSWs. This is likely because of the fact that the linearized form of pseudomomentum is restricted to small-amplitude disturbances and is not applicable to large-wave events such as SSWs, particularly when the gradient of zonal-mean PV reverses. There is also a conceptual difficulty in defining large-amplitude eddies through the Reynolds decomposition, when the zonal-mean state is already modified by the eddies. By employing the FAWA diagnostic (Nakamura and Zhu 2010), we have been able to elucidate the processes that make up eddy forcing , including the finite-amplitude and irreversible aspects during SSWs. In this framework, the difficulty associated with overturning PV contours during large-scale wave-breaking events is eliminated, and thus, the significant departure of wave activity from the exact nonacceleration relation can be directly attributed to nonconservative terms of wave activity, including irreversible mixing and diabatic effects.

A detailed analysis of the wave activity budget reveals the following key results:

1. An increase in pseudomomentum/wave activity prior to a warming event largely compensates for the anomalous E–P flux convergence, while the total wave dissipation due to mixing (enstrophy dissipation) and diabatic heating rates only plays a secondary role. This indicates that deceleration of polar circulation prior to a warming event is primarily a conservative, adiabatic wave–mean flow interaction.

2. After the vortex breaks down, irreversible wave dissipation due to mixing and diabatic heating rates compensates for wave transience (decay of tendency), resulting in a weak divergence of the E–P flux and, hence, delaying the recovery of the polar vortex. Enhanced irreversible wave dissipation is largely attributable to eddy-induced mixing and is consistent with increased effective diffusivity after the onset of such events.

3. A rapid recovery of the polar vortex is characterized by a weaker wave-transience effect that follows a nonacceleration relation reversibly [symmetric breakdown and recovery; Eq. (11)]. In this case, the wave activity is directly related to the E–P flux convergence through neglect of its dissipation mechanisms.

4. A slow recovery involves a stronger and more persistent effect of irreversible wave dissipation due to mixing, a deviation from the classical nonacceleration relation [Eq. (12)]. This indicates that eddy-induced mixing and diabatic forcing of the flow cause irreversible modification to the state of the polar vortex after the vortex breaks down.

The first point is consistent with the prevailing view of SSWs and confirms the results of the earliest modeling studies (e.g., Matsuno 1971; Holton 1976) yet with the full budget of observed wave activity. Our result helps gain insights into the relevance of PV dynamics in understanding the dramatic change in stratospheric flow associated with SSWs. In particular, it is suggested that the change in the large-amplitude wave saturation on wave breaking and irreversible wave dissipation (due to associated wave breaking that elongates and stirs the PV contours) should be taken into account in order to fully understand the eddy-forcing evolution during SSW events. The former largely accounts for a stronger E–P flux convergence prior to a warming event, while the latter dampens the wave-transience effects, resulting in irreversible modification to the state of the polar vortex.

The results also demonstrate that enhanced effective diffusivity after the vortex breakdown aids in increasing wave dissipation and dampens wave-transience effects, resulting in a slow recovery of the polar vortex. The importance of mixing in delaying the vortex recovery is consistent with the role of nonconservative processes in maintaining easterly wind anomalies in the aftermath of stratospheric weakening events (Lubis et al. 2018a). Our current results clearly demonstrate that in periods when the vortex undergoes dramatic changes (such as those during slow recovery or splitting types of SSW events), the irreversible wave dissipation induced by the diffusive flux of PV offsets the tendency of wave activity and thereby dampens the E–P flux divergence. This prolongs the recovery of the westerlies. Since the effect of mixing on eddy forcing is pronounced only after the breakdown of the polar vortex, the role of mixing is to introduce asymmetry between the breakdown and recovery phases of SSWs.

Previous studies have argued that radiative processes are important for the time scale over which the vortex recovers in the lower stratosphere (e.g., Kohma et al. 2010; Hitchcock and Shepherd 2013). In particular, Hitchcock and Shepherd (2013) showed that SSW events that occur during the period of polar jet oscillation (PJO-SSWs) of the stratospheric flow in the lower stratosphere because of mainly weak radiative cooling. Through the thermal wind balance, the weak radiative cooling causes less acceleration of the zonal-mean wind, leading to slow recovery of the vortex after such SSWs. This mechanism can be seen to be the direct effect of radiative damping on the “mean flow.” Here, we argue that an extended recovery of the stratospheric flow is also attributed to the direct effect of radiative damping and diffusive flux of PV on the “wave amplitude,” which in turn affects the mean flow through the E–P relation. For example, there is clear evidence that PJO-SSWs exhibit significantly stronger and persistent mixing compared to non-PJO-SSWs (de la Cámara et al. 2018). Based on our results, such an increase in eddy-induced mixing leads to stronger irreversible wave dissipation, thus acting to suppress eddy forcing. This results in an extended, slower recovery of the vortex in PJO-SSW years, similar to the absorptive SSWs. Our results, therefore, suggest that an understanding of the time scale of rapid and slow recoveries of the polar vortex would require an understanding of the behavior of the wave transience and diffusive flux of PV (mixing) in addition to the vertical structure of radiative time scales and climatological diabatic heating (Hitchcock and Shepherd 2013).

The results reported herein are robust, in a qualitative sense, with the results from the CESM1(WACCM) simulation. To our knowledge, this study presents the first analysis that elucidates the relative importance of wave transience at finite amplitude and irreversible mixing on the mean flow at various stages of SSWs using reanalysis and a long model run. This study moves SSW research a step forward by considering mixing not only as an agent of tracer transport (e.g., Allen and Nakamura 2001; Manney and Lawrence 2016; de la Cámara et al. 2018) but also as the source of wave dissipation that affects eddy forcing during SSW events. Quantifying the role of wave activity and mixing through this framework enhances our understanding of the processes that determine the state of the polar vortex; hence, this may potentially improves the predictability of the extratropical stratosphere.