A Study of the Stratospheric Major Warming and Subsequent Flow

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JOURNAL OF THE ATMOSPHERIC SCIENCES

VOLUME 58

A Study of the Stratospheric Major Warming and Subsequent Flow Recovery during the Winter of 1979 with an Isentropic Vertical Coordinate Model JOON-HEE JUNG, CELAL S. KONOR, CARLOS R. MECHOSO,

AND

AKIO ARAKAWA

Department of Atmospheric Sciences, University of California, Los Angeles, Los Angeles, California (Manuscript received 15 May 2000, in final form 28 February 2001) ABSTRACT The principal goal of this paper is to gain further insight into the dynamical processes during the stratospheric major warming of February and early March 1979, with a special emphasis on the recovery stage. To achieve this goal, first the entire evolution of the warming event is numerically simulated using an isentropic vertical coordinate model. Then the results from the following complementary points of view are quantitatively analyzed: wave effects on the mean flow, potential enstrophy conversion and transport, and potential vorticity redistribution on a synoptic chart. There is an indication that wavenumber 1 during the recovery stage amplifies through a mechanism within the stratosphere and propagates downward. The simulated Eliassen–Palm flux field shows that the amplified wave 1 is responsible for the mean flow acceleration in the recovery stage. It is therefore concluded that the in situ amplification mechanism for wave 1 plays a crucial role in the dynamics of the flow recovery. In order to examine the mechanism, detailed analyses of the simulated eddy potential enstrophy budget are performed. It is found that there is an ‘‘anticascade’’ of potential enstrophy in the recovery stage through which wave 1 amplifies in the midstratosphere. This result is consistent with a synoptic description of the event in terms of potential vorticity distribution on isentropic surfaces.

1. Introduction The major warming of the northern stratosphere in February and early March 1979 occurred during the period of the First Global Atmospheric Research Programme (GARP) Global Experiment (FGGE), which included intensified efforts on global observations following the launch of the TIROS-N and Nimbus-7 satellites. A number of studies on the warming event have been performed using observational data (Quiroz 1979; Labitzke 1981; Palmer 1981; McIntyre and Palmer 1983; Gille and Lyjak 1984; Smith et al. 1984; Dunkerton and Delisi 1986; Baldwin and Holton 1988), model simulations (Butchart et al. 1982; Palmer and Hsu 1983; Smith and Avery 1987; Fairlie et al. 1990; Manney et al. 1994), and numerical forecasts (Simmons and Stru¨fing 1983; Mechoso et al. 1985, 1986). Most studies listed above focused on the prewarming, onset and maturity stages of the warming event and relatively little attention was paid to the later stage. Yet Manney et al. (1994) successfully simulated the entire evolution of the event using an isentropic vertical coordinate model. They showed that, in both observation and simulation, the event evolves through the following Corresponding author address: Dr. Joon-Hee Jung, Dept. of Atmospheric Sciences, University of California, Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095-1565. E-mail: [email protected]

q 2001 American Meteorological Society

three stages: onset, maturity, and recovery stages. These stages are, respectively, characterized by the enhanced upward propagation of wave activity and the breakdown of the polar vortex into two vortices, the separation of these two vortices by a narrow tongue of air with low potential vorticity intruding from the Tropics into the polar region, and the recombination of the two vortices to form a single vortex. Manney et al. (1994), however, performed no quantitative analysis of the recovery stage. The motivation of this paper is to better understand the key dynamical processes during the entire evolution of the event, from the onset through recovery stages. There is no question that connections with the troposphere are crucial to the onset of the event, whether they depend on transient forcing (Matsuno 1971; Holton 1976) or steady forcing (Holton and Mass 1976; Yoden 1987; Scaife and James 2000) of planetary waves at the tropopause level. Mechoso et al. (1985, 1986) further demonstrated that sudden warmings can be better predicted with a higher-resolution model primarily due to a better forecast of the flow in the upper troposphere. The situation is, however, less clear as the warming event develops since a link with the troposphere beyond the linear theory would have to be established across the critical level. Here, we question the extent to which the evolution of the event is driven by the troposphere. In particular, we are interested in identifying processes responsible for the recovery of the flow.

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To achieve the goal of this study, we first simulate the entire evolution of the major warming of February and early March 1979. As Manney et al. (1994) did, we use an isentropic vertical coordinate model. A major advantage of isentropic vertical coordinates is that coordinate surfaces coincide with material surfaces under adiabatic conditions. Moreover, Ertel’s potential vorticity for the quasi-static system can be simply expressed as g(2V sinw 1 k · = u 3 v)(2]p/]u) 21 , where V is the earth’s angular velocity, w is latitude, k is vertical unit vector, and v is horizontal velocity. Since the conservation of Ertel’s potential vorticity plays a central role in the large-scale dynamics of the stratosphere, we expect that isentropic vertical coordinate models like Hsu and Arakawa’s (1990), in which the advantages described above are maintained even after vertical discretization, can simulate the entire evolution of the sudden warming reasonably well. The simulated Eliassen–Palm (EP) flux and potential vorticity fields, as well as the simulated potential enstrophy budget, are then analyzed. The EP flux has been widely used as an indicator of the propagation of wave activity (Dunkerton et al. 1981; Palmer 1981; Butchart et al. 1982; Palmer and Hsu 1983; Mechoso et al. 1985, 1986). O’Neill and Pope (1988), however, discussed potential limitations of the EP flux approach when westerlies breakdown and flows become highly nonlinear. Potential vorticity field, on the other hand, has been used mainly to describe nonlinear processes without following the traditional practice of separating flow quantities into zonal means and sinusoidal wave components (McIntyre and Palmer 1983; Butchart and Remsberg 1986; Dunkerton and Delisi 1986; Baldwin and Holton 1988; O’Neill and Pope 1988; Manney et al. 1994). Also, potential enstrophy budget has been analyzed for investigating the nonlinear interactions between waves (Smith 1983; Smith et al. 1984; Tao 1994). In this study, we take the advantage of each approach mentioned above and combine the results of the analyses to understand the key dynamical processes during the entire evolution of the event. We try to minimize the calculation errors in the potential enstrophy budget analysis by using formulations derived from the model’s discrete governing equations. In this way, detailed decomposition of the budget is accomplished without being contaminated by computational errors and inconsistencies between prediction and diagnosis. The paper is organized as follows. Section 2 reviews the observed features of the February and early March 1979 stratospheric warming. Section 3 describes the isentropic vertical coordinate model used in the study. Sections 4 and 5 present simulations of the warming event with observed and truncated lower boundary conditions, respectively. In these sections, we discuss the model’s performance in simulating the event and the extent to which the evolution of the event is driven by the troposphere. In section 6, the simulated EP flux and eddy potential enstrophy budget are analyzed. Based on

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these analyses, we discuss the roles of waves 1 and 2 in each stage and the nonlinear amplification of wave 1 in the recovery stage. Finally, section 7 presents summary and conclusions. 2. Observed features of the February 1979 stratospheric major warming A detailed description of the major warming of February and early March 1979 is given by Andrews et al. (1987 and references therein). Here, we review only basic features in the flow and temperature evolution during the event. The observational dataset is obtained from two sources. Below 10 hPa, we use temperature, geopotential height and wind from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996) with a horizontal resolution of 2.58 longitude 3 2.58 latitude on the 100-, 70-, 50-, 30-, 20-, and 10hPa isobaric surfaces. Above 10 hPa, we use temperature and geopotential height fields compiled by the National Meteorological Center (now NCEP) with a horizontal resolution of 58 longitude 3 2.58 latitude on the 5-, 2-, 1-, and 0.4-hPa isobaric surfaces. Since winds are not available above 10 hPa, we produce estimates by using the geostrophic relationship except at low latitudes between 208N and 208S, where a linear interpolation in the meridional direction is used. Figure 1 shows 10-hPa geopotential height and temperature fields during the warming event. On 17 February, there is a large and elongated cyclone centered near 608E over the polar region and an anticyclone over the Aleutian islands. The lowest temperature is approximately 210 K near the center of the cyclone and the warmest is about 240 K over North America. From 17 to 21 February, the cyclone splits into two, the stronger of which develops over Eurasia and the weaker over North America. An anticyclone forms near 108W over the Atlantic Ocean. From 21 to 25 February, the cyclones separate further and the anticyclones merge across the Pole. A pool of air with temperatures up to about 255 K develops east of the cyclone over Eurasia. From 25 February to 1 March, the cyclone over North America starts weakening whereas the anticyclone in the polar region remains strong. The temperature increases in the entire polar region. After 1 March, the Eurasian cyclone moves toward the Pole where it is joined by the remainder of the North American cyclone. The temperature over the polar region starts decreasing. In brief, along with a general warming of the polar region, the zonal wave 1 structure of the geopotential height associated with one cyclonic vortex displaced from the Pole evolves into a wave 2 structure with two distinct cyclonic vortices. Later, the wave 2 structure returns to a wave 1 structure as the temperature over the polar region decreases. In view of the flow evolution outlined in this section, we will consider the warming

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FIG. 1. Observed (NCEP–NCAR reanalysis) geopotential height (contours) and temperature (grayscale) at 10 hPa, on selected days between 17 Feb and 5 Mar 1979. Projection is polar stereographic with the outer edge at 208N. The contour interval is 0.2 km and the grayscale is designated on the figure. The thick line represents 30.6-km geopotential height.

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and recovery stages of the event separately. These stages are represented by the periods 17–27 February and 28 February–5 March, respectively.

m [ 2]p/]u.

(7)

(u*, y *) [ (mu, my );

(8)

In addition,

1 K [ (u 2 1 y 2 ); 2

3. Model description The model we use is a global version of the isentropic vertical coordinate model developed by Hsu and Arakawa (1990). This model predicts the horizontal velocity on isentropic surfaces and the mass between isentropic surfaces. The Montgomery potential is diagnosed from the vertical distribution of mass. The space finite-difference schemes of the model are especially designed to maintain the advantages of isentropic vertical coordinates in modeling, such as 1) the discretization problem for the vertical advection terms is virtually eliminated, and 2) the simple form of potential vorticity makes it easier to maintain its conservation in a vertically discrete system. In particular, potential enstrophy is conserved under adiabatic frictionless processes. The model includes diffusion schemes for layers with very small mass.

(9)

M is the Montgomery potential defined by M [ c p T 1 F;

(10)

P is the Exner function, P [ c p (p/p o ) k ;

(11)

and Q is the diabatic heating rate. b. Upper and lower boundary conditions

]u 1 ] ]u 2 qy * 1 (K 1 M ) 1 u˙ 5 0, ]t a cosw ]l ]u

(1)

The vertical model domain covers the stratosphere above the 400-K surface. For the simulation, which will be referred to as ‘‘control,’’ the Montgomery potential is prescribed on the lower boundary. In the hypothesistesting experiments (see section 5), we use a special lower boundary condition to minimize spurious reflections of downward propagating waves. Further details on this lower boundary condition are given in section 5. At the upper and lower boundaries, we assume that there is no diabatic heating (Q 5 0). To reduce spurious reflections of upward propagating waves at the upper boundary, we use a Rayleigh-type friction in the top four model layers, with coefficients increasing upward. For the top layer, the coefficient is 2.8 3 10 26 s 21 .

]y 1 ] ]y 1 qu* 1 (K 1 M ) 1 u˙ 5 0, ]t a ]w ]u

(2)

c. Spatial and time discretization

a. Governing equations The governing equations written in spherical coordinates are

]m 1 ]u* 1 ]t a cosw ]l 1

1 ] ] (y * cosw) 1 (mu˙ ) 5 0, a cosw ]w ]u

(3)

]M 5 P, ]u

(4)

Q u˙ 5 , P

(5)

d. Additional model features

where l is longitude, w is latitude, and a is earth’s radius. Further, q is potential vorticity defined by f 1 q[ where

[

]

1 ] ] y2 (u cosw) a cosw ]l ]w m

,

The vertical discretization of the model follows Hsu and Arakawa (1990). The horizontal velocity and mass are predicted for the layers and the vertical mass flux representing diabatic heating is calculated at the interfaces. The horizontal discretization is based on Arakawa and Hsu (1990) generalized to the spherical coordinate system following Arakawa and Lamb (1981). The time discretization follows Konor and Arakawa (1997).

(6)

To avoid the generation of artificially strong winds in layers with small mass, we include the vertical momentum diffusion scheme of Konor and Arakawa (1997). This diffusion acts only when the thickness of the layer becomes very small. At this stage of development, the model does not include comprehensive physical processes. Instead, the thermal forcing is given in the form of a Newtoniantype heating in which pressure is relaxed to its zonally averaged initial value with a timescale of about 20 days in the lower stratosphere and about 5 days in the upper

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stratosphere. (Recall that, in the u coordinate, the space distribution of pressure determines the thermal structure of the model atmosphere.) This formulation is a crude representation of radiative processes; yet it is useful for the objective of this paper. e. Model resolution Preliminary experiments suggested that model results are less sensitive to horizontal than to vertical resolution. For the simulation presented here, the horizontal resolution is 58 longitude 3 48 latitude. We use a 30-s time step, which is short enough to maintain linear computational stability without any zonal filtering near the Poles. There are 23 model layers, spaced nearly evenly in height, resulting in a vertical resolution of about 1.7 km except for the highest layer whose depth is approximately 0.7 hPa. The potential temperatures of the isentropic layers are 414, 444, 479, 518, 561, 607, 654, 710, 770, 831, 901, 974, 1058, 1160, 1263, 1358, 1470, 1600, 1724, 1858, 2005, 2145, and 2688 K. 4. Simulation of the February 1979 stratospheric major warming The model integration starts on 17 February 1979 and is 20 days long to cover the entire evolution of the warming event. We obtain initial pressures on the model’s isentropic surfaces by linear interpolation using observed temperatures on pressure levels. The initial horizontal velocities are also obtained by linear interpolation from those observed. [Even though the initial fields are obtained from two data sources (see section 2), we have not found any sign of inconsistency in the simulation results.] During the integration, the Montgomery potential is prescribed at the lower boundary using a linear time interpolation from the daily NCEP– NCAR reanalysis. Figure 2 shows a selection of simulated 10-hPa geopotential height and temperature fields. A comparison of Fig. 2 with Fig. 1 indicates that the evolution of geopotential height is realistically simulated. The complete splitting of the cyclone over the polar region and its later recovery are both captured. The locations of cyclones and anticyclones are also well simulated, although simulated cyclones are slightly weak. The simulated temperature field shows a realistic evolution of the warming, especially east of the cyclone over Eurasia and in the polar region. However, the temperature gradient near the center of the Eurasian cyclone is somewhat weak. Additionally, the simulated rate of temperature decrease over the polar region after 1 March is smaller than observed. Yet the model’s success in simulating the flow recovery after the warming with such a relatively coarse vertical resolution is encouraging. Manney et al. (1994) attributed their similar success to the use of an isentropic vertical coordinate. Figure 3 shows meridional cross sections of observed

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and simulated zonal-mean zonal wind on 17 (initial state) and 25 February (warming stage), and on 5 March (recovery stage). A comparison with the observations indicates that the breakdown of westerlies and the acceleration of easterlies in high latitudes are well simulated. On 5 March, the simulated westerlies also resemble those observed in the lower stratosphere, but in the upper stratosphere simulated winds are still easterly. The existence of an effective upper boundary due to the use of a thick top layer is a likely contributor to this error in the upper stratosphere. The evolution of Rossby–Ertel potential vorticity (PV 5 gq; g 5 9.8 m s 22 ) on the 831-K isentropic surface from the observation and the simulation is shown in Fig. 4. On 25 February, we can see clearly an elongated tongue of high-PV air that appears to be pulled out of the vortex over North America. This high-PV tongue and associated small-scale features seem to show the main features of planetary-wave breaking (McIntyre and Palmer 1983). Meanwhile, low-PV air is advected from low latitudes into the polar region east of the Eurasian vortex. The low-PV tongue approximately coincides with the region of temperature increase shown in Fig. 1c, indicating that the increase is associated with air descend. By 5 March, the Eurasian vortex with a reduced intensity is relocated across the Pole. The close agreement with the observation shows that the model captures major features in the PV evolution through the warming and recovery stages, such as the complete breakdown of the initial vortex over the polar region into two vortices, the advection of low-PV air from low to high latitudes, and the recovery of the high-PV vortex over the polar region. In view of these results, we conclude that the model’s simulation of the warming and subsequent flow recovery can be used as the basis for hypothesis-testing experiments. In this regard, the results obtained so far will be labeled as ‘‘control’’ in the remainder of the paper. 5. Hypothesis-testing experiments In order to better understand the extent to which the wave forcing at the lower boundary is important for the warming and subsequent flow recovery, we perform two hypothesis-testing experiments using truncated initial and lower boundary conditions (see Table 1). The initial conditions for both experiments consist of the zonal mean and the wave 1 and 2 components of the velocity and pressure fields. The Montgomery potential at the lower boundary in experiment I consists of the zonal mean and waves 1 and 2, while that in experiment II consists only of the zonal mean and wave 2. As in control, the zonal-mean component at the lower boundary corresponds to that observed. The wave components at the lower boundary, on the other hand, are not always taken from observations as a revised lower boundary condition is used to minimize spurious reflections of downward propagating waves. The revised condition is

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FIG. 2. As Fig. 1 but for simulated geopotential height and temperature.

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FIG. 3. Meridional cross sections of observed (a), (b), and (d) and simulated (c) and (e) zonal-mean zonal wind, on selected days. Westerlies and easterlies are shown as thick solid lines and dotted lines, respectively. The contour interval is 10 m s 21 . Thin horizontal solid lines show isentropic surfaces that correspond to the vertical levels of the model for the simulations.

based on the direction of vertical propagation of wave activity, which is inferred by inspection of the phase tilt of the wave in the Montgomery potential field between the lowest two model layers at each latitude. If the di-

rection is upward—that is, if the tilt is westward—the Montgomery potential for the wave component at the lower boundary exactly corresponds to that observed. Otherwise, the component is obtained through extrap-

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FIG. 4. Evolution of the Rossby–Ertel potential vorticity on the 831-K isentropic surface obtained from observations (a) and (b) and the simulation (c). Projection is polar stereographic with an inner circle of 608N and outer edge at 208N. The contour interval is 50 potential vorticity units (PVU 5 1026 m 2 K kg 21 s 21 ). The regions of low potential vorticity (less than 200 PVU) are shaded.

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TABLE 1. Description of the initial and lower boundary conditions for experiments I (EXP I) and II (EXP II).

Experiments EXP I EXP II

Initial condition Horizontal winds and pressure (17 Feb 1979) Zonal mean 1 wave 1 1 wave 2 Zonal mean 1 wave 1 1 wave 2

Lower boundary condition Montgomery potential (prescribed* throughout the integration) Zonal mean 1 wave 1 1 wave 2 Zonal mean 1 wave 2

* The lower boundary condition is modified to avoid internal reflections when the wave activity is propagating downward near the lower boundary.

olation from the interior. A meridional smoothing of the extrapolated values is performed to avoid excessively strong latitudinal gradients. The integrations also start on 17 February and are 20 days long. By comparing experiments I and II with control, we attempt to gain insight into the roles played by the different wave components during the warming and subsequent flow recovery. Figures 5a,b show the 10-hPa geopotential height and temperature fields on 21 February obtained in experiments I and II. In the warming stage both experiments closely resemble control (see Fig. 2) although the cyclone over Eurasia is slightly weaker and the cyclone over North America is slightly stronger in experiment II than in experiment I. The temperature fields in both experiments also show quite realistic warming in the polar region. The same fields on 5 March are shown in Figs. 5c,d. Experiment I successfully captures the recovery of the cyclone over the polar region. The temperature fields also resemble those of control. Experiment II, however, fails to reproduce control’s success in recovering the cyclone over the polar region. Based on these experiments, we conclude that while wave 2 at the lower boundary plays a key role during the warming stage, the existence of wave 1 at the lower boundary is crucial for the later vortex recovery. This statement, however, should be taken cautiously as we discuss toward the end of this section. To take a closer look at the behavior of waves 1 and 2, we plot time–height cross sections of the wave amplitudes of the Montgomery potential at 708N for experiments I (Figs. 6a,c) and II (Figs. 6b,d). In experiment I, wave 1 goes through two notable amplification periods, one between day 0 and day 4, and the other between day 10 and day 17. On the contrary, wave 2 goes through an early amplification; after which it decays almost monotonically at all levels. These features exist in both control and observation (not shown). In experiment II, the amplification of wave 1 in the initial condition takes place during the first few days but the later amplification hardly exists. We also note that the decay of wave 2 is slow after day 9, especially in the midstratosphere. Judging from the failure of experiment II in simulating the recovery stage, the amplification of

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wave 1 and the rapid decay of wave 2 between day 10 and day 17 seem to be closely related to the dynamics of the flow recovery. We now deduce the direction of vertical propagation of wave activity from Fig. 7, which shows time–height cross sections at 708N of the phases of waves 1 and 2 for experiment I. (Here ‘‘phase’’ means the longitude of maximum.) Wave 1 tilts westward with height until day 10 (27 Feb) and eastward afterward. On the other hand, wave 2 tilts westward with height during the entire simulation period. These features appear throughout high latitudes although we show only the cross sections at 708N. As we discuss in section 6a, a westward (eastward) tilt with height normally implies upward (downward) propagation of wave activity. Thus, it seems that wave 2 always propagates upward; on the contrary, wave 1 propagates upward before day 10 and downward afterward. These results suggest that the early amplification of waves 1 and 2 in the stratosphere is induced by the upward propagation of their counterparts in the troposphere. The later amplification of wave 1, however, must be due to a mechanism within the stratosphere since the wave is propagating downward. This means that the existence of wave 1 at the lower boundary during that stage does not matter for the evolution of wave 1 above. Then, the failure of experiment II in simulating the recovery stage suggests that the amplification of wave 1 during the recovery stage needs the existence of wave 1 at the lower boundary during the preceding stage. In the following section, we further examine the behavior of waves 1 and 2 during the warming and recovery stages. 6. Analyses of experiment I In this section we analyze the results obtained in experiment I in more detail since it produces a realistic simulation in a framework simpler than control. Our main goal is to understand the roles of waves 1 and 2 in the evolution of the event and the mechanism for the later amplification of wave 1. To facilitate the analysis, we separate the entire simulation period into three stages based on the evolution of wave 1 (Fig. 6a). They will be referred to as stage I (days 0–4), stage II (days 5– 9), and stage III (days 10–17). Roughly, stage I and stage II correspond to the period of warming and stage III corresponds to the recovery. We first examine the EP flux for waves 1 and 2 during the entire evolution. Then, we focus on the eddy potential enstrophy budget to obtain insights into the amplification of wave 1 during stage III. a. Eliassen–Palm flux The EP flux based on linear wave theory is a useful tool for diagnosing the forcing of the mean flow by

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FIG. 5. Simulated geopotential height (contours) and temperature (grayscale) at 10 hPa, from EXP I (left) and EXP II (right). Upper panels are for the warming stage, on 21 Feb, and lower panels are for the recovery stage, on 5 Mar. Figure arrangement is as in Fig. 1.

waves. As Edmon et al. (1980) pointed out, the EP flux can be also used as an indicator of the propagation of wave activity. When the meridional gradient of the zonal-mean (quasigeostrophic) potential vorticity is positive, as is normally the case, the direction of the EP flux is the same as that of wave activity propagation. Dunkerton et al. (1981), Palmer (1981), and Butchart et al. (1982) are good examples of studies on stratospheric sudden warmings that use the EP flux as a diagnostic. Here, we use the EP flux formalism generalized to

the primitive equation system with the u coordinate, which is given by Andrews (1983) and Andrews et al. (1987). Nevertheless, we display the EP flux, F [ (F(w) , F(u) ), and its divergence, = · F, on (w, z) plots, in which z [ 2H ln(p/p s ), where H is a scale height (7 km) and p s is a standard reference pressure (1000 hPa), following two steps. First, we interpolate (F(w) , F(u) ) to a grid with uniform spacing in both w and z to obtain (F9(w) , F9(z)). Second, we scale the interpolated values in the following way:

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FIG. 6. Time–height cross sections of the amplitude of the Montgomery potential for zonal waves 1 (upper) and 2 (lower) obtained from EXP I (left) and EXP II (right). The ordinate represents the layers of the model (the corresponding heights are designated on the right side of the frames). The contour interval is 5 3 10 2 m 2 s 22 . The regions where amplitudes are larger than 55 3 10 2 m 2 s 22 are shaded.

5

Fˆ (w) [ Cw 3 F9(w) Fˆ (z) [ Cz 3 F9(z) .

(12)

Here, Cw and C z are used to eliminate the geometric distortion inherent in the use of different vertical and latitudinal scales. In the following diagrams, Cw 5 0.009 and C z is the area mean of ]z/]u, which takes into account the variations of u with height. For waves 1 and 2, Fig. 8 shows averages over each stage of Fˆ 5 (Fˆ(w) , Fˆ(z) ) and DF given by DF [

g 1 = · F, m a cosw

(13)

which is the mean zonal force per unit mass by waves. In stage I, the EP fluxes of both waves 1 and 2 are upward. Since the meridional gradient of the zonal-mean potential vorticity is positive everywhere in the domain, this indicates that wave activity propagates upward for both waves 1 and 2. The contours of DF show a con-

vergence of the EP flux in high latitudes also for both waves 1 and 2. In particular, there is polar focused convergence of the wave 1 flux. This convergence is consistent with the deceleration of westerlies and the development of easterlies during the warming stage (Fig. 3c). During stage II, the EP fluxes of both waves 1 and 2 are still upward but are focused equatorward. The contours of DF for wave 2 show almost the same convergence pattern as in stage I. For wave 1, however, the contours of DF show a more expanded divergence pattern between 708 and 608N than in stage I. During stage III, on the other hand, waves 1 and 2 behave in a completely different way. The wave 1 EP fluxes are downward in high latitudes, while those of wave 2 are still upward. Since the meridional gradient of the potential vorticity is positive in the lower stratosphere, this indicates that wave 1 propagates downward from the midstratosphere, while wave 2 still propagates upward from the lower boundary. This result implies that the stage

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use an isentropic vertical coordinate model that maintains conservation of total potential enstrophy in a discrete system under adiabatic and frictionless processes, we have a computational advantage in this analysis. The zonal-mean eddy potential enstrophy equation with an isentropic coordinate is given by m

1 2

] q92 ]t 2

5 2y *

1 2 2 q9y *9 a ]w 1 mD

1 ] q92 a ]w 2

1 ]q

1 m q9NA ,

(14)

where

1

m q9NA [ 2q9 u*9

FIG. 7. Time–height cross sections of the phase of the Montgomery potential for zonal waves 1 (a) and 2 (b) for EXP I. The ordinate represents the layers of the model (the corresponding heights are designated on the right side of the frames). The contour interval is 108.

b. Eddy potential enstrophy budget The eddy potential enstrophy budget provides a theoretical framework for investigating the potential enstrophy sources and sinks associated with waves. Smith (1983) and Smith et al. (1984) discussed the relative importance of each process contributing to the local change of potential enstrophy of planetary waves. Tao (1994) performed similar analysis with an isentropic coordinate model. Both noted, however, that the budget analysis involving high-order derivatives was very sensitive to data accuracy and calculation errors. Since we

(15)

See the appendix for the derivation of (14) and the expression for the diabatic effects, m D. In this equation, the local time change of zonal-mean eddy potential enstrophy, which essentially represents the wave activity, is due to advection by the mean meridional flow, linear conversion (wave–mean flow interaction), diabatic effects, and nonlinear transport and conversion (wave– wave interaction). Since we focus on the time evolutions of waves 1 and 2, we further derive the eddy potential enstrophy equation for each wave component. Multiplying the eddy potential vorticity equation (A6) by q9 for waves 1, 2, and the remainder of the waves and zonally averaging the result, we obtain the corresponding zonalmean eddy potential enstrophy equations,

m III amplification of wave 1 is due to a mechanism within the stratosphere. In the same stage, the contours of DF show a divergence pattern for wave 1 and a convergence pattern for wave 2. Since westerlies are accelerated in high latitudes in stage III (Fig. 3e), it appears that wave 1 plays the main role in the flow recovery, competing with the deceleration effect on westerlies by wave 2.

2

1 ]q9 1 ]q9 1 y *9 . a cosw ]l a ]w

] 1 ]q9 1 ]q (q9(n)2 /2) 5 2y * q9(n) 2 q9(n)y *9 ]t a ]w a ]w 1 mD(n) 1 m q9(n) NA ,

(16)

where the subscript n represents 1, 2, and 31. The subscripts (1), (2), and (31) denote zonal waves 1, 2, and wavenumbers 3 and higher, respectively. The nonlinear terms of (16) are identified as m q9(1) NA 52 1

1 ] 1 [( y *9q9(1)2 ) 1 (q9(1)y *9q9(2) ) 1 (q9(1)y *9q9(31) )] a ]w 2 C12 C131 1 , 2 2

(17)

m q9(2) NA 52 2

1 ] 1 [( y *9q9(2)2 ) 1 (q9(1)y *9q9(2) ) 1 (q9(2)y *9q9(31) )] a ]w 2 C12 C 231 1 , 2 2

(18)

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FIG. 8. Meridional cross sections of the EP flux Fˆ and the eddy forcing DF for waves 1 (left) and 2 (right) averaged over each stage. Upper, middle, and lower panels represent stages I, II, and III, respectively. Arrows are EP fluxes and their scales are specified on each frame in units of kg m s 22 K 21 . Thin and thick solid contours in the figure show the convergence and the divergence of EP fluxes, respectively; the contour interval is 2 m s 21 day 21 . Thick dash–dot lines show q y 5 0, thus the region inside of the line (which is shaded gray) has negative mean potential vorticity gradient.

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m q9(31) NA 52 2

1 ] 1 2 [( y *9q9(31) ) 2 (q9(1)y *9q9(31) ) 2 (q9(2)y *9q9(31) )] a ]w 2 C131 C 231 2 , 2 2

(19)

where we assume that the mass flux is nondivergent. The quantities C12 , C131 , and C 231 are defined by

[

1

Cnm [ 2 q9(n) u*9

1

2

1 ]q9(m) 1 ]q9(m) 1 y *9 a cosw ]l a ]w

2 q9(m) u*9

2

]

1 ]q9(n) 1 ]q9(n) 1 y *9 , a cosw ]l a ]w

(20)

where n 5 1 or 2 and m 5 2 or 31. In (17)–(19), we regard the terms associated with y *9 as the nonlinear transport and C12 , C131 , and C 231 as the nonlinear conversion terms between waves specified with subscripts (1), (2), and (31). We first calculated each term in (14) using the finitedifference expressions derived from the model’s discrete governing equations. Then, we examined the residual of (14) to see how well the balance is maintained in the discrete system. In this test, we obtained the local time change of zonal-mean eddy potential enstrophy by running the model for several time steps. The result was then compared with the change determined from the right-hand side of the equation. We found this residual to be negligible almost everywhere in the domain throughout the entire simulation period. Based on this calculation accuracy, we perform detailed decomposition of the potential enstrophy budget as in (16). In the following we focus on stage III and only show the effects of the linear conversion and nonlinear terms of the budget since those of mean meridional advection and diabatic source/sink turn out to be relatively small. Figure 9 shows latitude–height sections of the dominant terms in (16) averaged over stage III. During this stage, the processes associated with wave 1 are dominant. Through linear conversion, wave 1 loses potential enstrophy to the mean flow. This loss is consistent with the acceleration of mean westerlies and the result of the EP flux analysis for wave 1 (Fig. 8e). Through nonlinear processes, however, the potential enstrophy of wave 1 significantly increases in high latitudes. Such an increase more than compensates the loss of potential enstrophy to the mean flow, resulting in the intensification of wave 1 activity near the polar region. For wave 2, on the other hand, the gain from the mean flow is comparable to the loss through nonlinear processes, and hence the wave 2 activity hardly changes. The strong net tendency to increase the potential enstrophy of wave 1 near the polar region between layers 5 and 15 (23–41 km) (see Fig. 9e) corresponds well to the amplification of this wave component during the same

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stage. Primarily, the net tendency is due to the nonlinear processes as described above. To further investigate this tendency, we separate the nonlinear terms in (16) to transport and conversion processes following (17)–(20). Figure 10 shows those effects for waves 1 and 2 (the sum of the three effects represents the nonlinear effect shown in Figs. 9c,d). It appears that through nonlinear conversion wave 1 gains potential enstrophy from wave 2 and loses a part of it to waves with higher wavenumbers. The large loss of potential enstrophy of wave 2 into wave 1 is almost compensated by the gain from waves with higher wavenumbers and that by transport. From these, we infer that the potential enstrophy transition from waves with higher wavenumbers into wave 1 via wave 2 (‘‘anticascade’’) should be attributed to the significant tendency to increase of wave 1 shown in Fig. 9. This anticascade of potential enstrophy is associated with the poleward extension of small-scale features in the potential vorticity field accompanied by the Eurasian vortex. To further elaborate this point, we show the potential vorticity fields on the 1160 K isentropic surface in Fig. 11 on selected days. The PV structure of wave 2 on day 4 (stage I) suggests that the Eurasian vortex captures the low-PV air from low latitudes and elongates it toward and across the polar region (day 8). During stage III, as the low-PV tongue further extends poleward, its leading part is cut off (days 12 and 14). This cutoff part of the low-PV tongue and the high-PV Eurasian vortex inside the 608N latitudinal circle form a wave 1 structure (day 14). In longitude, the low-PV tongue is a high zonal wavenumber feature in low latitudes while it is a low zonal wavenumber feature in high latitudes. The poleward extension of low-PV tongue associated with the Eurasian vortex appears as an anticascade of potential enstrophy in the budget analysis. 7. Summary and conclusions In February and early March 1979 the northern stratosphere experienced a major warming associated with a breakdown and subsequent recovery of the polar night vortex. A number of studies on the prewarming, onset and maturity stages of the warming event have been performed using observational data, model simulations and numerical forecasts. The principal goal of this paper is to gain further insight into the dynamical processes during the warming event, with a special emphasis on the recovery stage. To achieve this goal, we numerically simulated the entire evolution of the warming event and then quantitatively analyzed the results. For the numerical simulation, we used an isentropic vertical coordinate model that is a generalization of Hsu and Arakawa (1990) to the global domain. The discretization used in this model maintains most of the advantages of isentropic vertical coordinates. In particular, the model conserves total potential enstrophy under adiabatic frictionless conditions. The vertical domain of the model covers the stratosphere above the 400-K isentropic surface.

2644


FIG. 9. Meridional cross sections of major terms in the eddy potential enstrophy budget for waves 1 (left) and 2 (right) averaged over stage III: linear conversion between waves and mean flow (a) and (b), nonlinear transport and local conversion between waves (c) and (d), and sum of both (e) and (f ). The ordinate represents the model layers, whose representative heights are given on the right side of the frames. Thin and thick solid contours in the figure show negative and positive values, respectively; the contour interval is 4 3 10 216 s 23 .

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FIG. 10. Meridional cross sections of terms representing the nonlinear transport and local conversion for waves 1 (left) and 2 (right): nonlinear transport (a) and (b), conversion between waves 1 and 2 (c) and (d), conversion between waves 1 and 31 (e), and conversion between waves 2 and 31 (f ). The contour interval is 4 3 10 216 s 23 .

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FIG. 11. Normalized potential vorticity on the 1160-K isentropic surface on selected days. Projection is polar stereographic with an inner circle of 608N and outer edge at 208N. Contour interval is 300 3 10 27 s 21 . Low potential vorticity (less than 900 3 10 27 s 21 ) regions are shaded.

Physical processes are represented by a simple Newtoniantype heating or cooling. NCEP–NCAR datasets were used for initial and boundary conditions. Integrations started on 17 February 1979 and were 20 days long to cover the entire evolution of the warming event. When the initial and lower boundary conditions were taken from observations, the model produced a realistic evolution of the major warming. We believe that the successful simulation of the event, especially that of the vortex recovery after the warming period, can be attributed to use of the isentropic vertical coordinate model in which potential vorticity dynamics can be quite accurately represented. Analyses of the simulated evolution of waves 1 and 2 showed that wave 1 went through two amplification periods, one during the warming stage and the other during the recovery stage. Wave 2, on the other hand, was amplifying during the warming stage and decaying afterward. Based on two hypothesis-testing experiments using truncated initial and lower boundary conditions, we conclude that the early amplification of waves 1 and

2 is induced by upward propagation from the troposphere, but the later amplification of wave 1 takes place through a mechanism within the stratosphere. During the recovery stage, therefore, wave 1 at the lower boundary hardly affects the stratospheric flow above. (Wave 1 forcing at the lower boundary during the preceding stage, however, plays an important role in the later amplification of wave 1 and flow recovery through producing the proper evolution of the stratospheric flow.) This suggests that the dynamics of the in situ amplification of wave 1 is closely related to that of the flow recovery. The importance of wave 1 in the flow recovery can be seen in the simulated EP flux field, which shows that wave 1 plays a primary role in the mean-flow transition from easterlies to westerlies. In order to investigate the in situ amplification mechanism of wave 1 during the recovery stage, we analyzed the eddy potential enstrophy budget. This analysis requires elaborate computational procedures since it involves highly derived quantities so that the result can be very sensitive to calculation errors. We tried to minimize

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JUNG ET AL.

these errors by using a formulation derived from the model’s discrete governing equations, instead of using one directly descretized from the continuous form of the eddy potential enstrophy equation. The results show that, during the recovery stage, wave 1 amplifies by gaining potential enstrophy from waves with higher zonal wavenumbers through nonlinear conversions. We also point out that such an ‘‘anticascade’’ of potential enstrophy from waves with higher zonal wavenumbers into wave 1 is associated with the poleward intrusion of the Eurasian vortex and an accompanying tongue of low PV. One of the main points of the paper is to relate different ways of viewing the same phenomena: from the points of view of wave effects on the mean flow, potential enstrophy conversion and transport, and potential vorticity redistribution on a synoptic chart. The vortex recovery stage of the major warming event, which is characterized by a wave 1 structure, is described from these complementary viewpoints.

m

]q 1 ]q 1 ]q 1 u* 1 y* ]t a cosw ]l a ]w 5q

5

] [ ]6

[

] 1 ] ] ] ]y (mu˙ ) 1 u˙ (u cosw) 2 u˙ , ]u a cosw ]w ]u ]l ]u (A1)

m

12

12

12

] q2 1 ] q2 1 ] q2 1 u* 1 y* ]t 2 a cosw ]l 2 a ]w 2 ] (mu˙ ) ]u

5 q2

1q

5

] [ ]6

[

1 ] ] ] ]y u˙ (u cosw) 2 u˙ , (A2) a cosw ]w ]u ]l ]u

where l is longitude, w is latitude, a is earth’s radius, q is potential vorticity, m [ 2]p/]u, and (u*, y *) [ (mu, mv). To derive the eddy potential vorticity and eddy potential enstrophy equations, we divide the variables into their zonal means and eddy components as

Acknowledgments. We wish to thank to John Farrara for useful comments on the text. Extensive comments of three reviewers (William Lahoz, Gloria Manney, and an anonymous reviewer) helped to improve the manuscript. This research was funded under NASA NAG5-4420, NAG5-6386, and NCCS-5149; CalSpace CS-22-98; NSF ATM-9613979; and DOE DE-DG03-98ER625. This was also in part funded through a G7 project, ‘‘Technology for Climate Change Prediction,’’ supported by the Ministry of Environment, Korea, and conducted by the Global Environment Laboratory, Yonsei University.

u* y* m u˙ q

5 5 5 5 5

u* 1 u*9  y * 1 y *9 m 1 m9  , u˙ 1 u˙ 9 q 1 q9 

 

(A3)

where APPENDIX A [

Derivation of the Zonal-Mean Eddy Potential Enstrophy Equation With the spherical coordinates, the potential vorticity and potential enstrophy equations can be written as

m 1 m9 ø m u* ø m u; mu˙ ø m u˙ ;

1 2p

E

2p

A dl

and A9 [ A 2 A.

(A4)

0

Substituting (A3) into (A1) and using the approximations



y* ø m y ; (mu˙ )9 ø m u˙ 9

u*9 ø mu9;



y *9 ø m y 9 ,

(A5)

 

we can obtain the eddy potential vorticity equation given by

m

]q9 1 ]q9 1 ]q9 1 ]q ] ] 1 u* 1 y* 1 y *9 5 q (m u˙ 9) 1 q9 (m u˙ ) 1 m(N 2 N ) ]t a cosw ]l a ]w a ]w ]u ]u 1 2

where N 5 N A 1 N D ,

5 [ y u 5 [ w l u

]6

1 ] ] ] u˙ (u9 cosw) 1 u˙ 9 (u cosw) a cosw ]w ]u ]u 1 a cos

]6

] ] 9 ]y ˙ 1 u˙ 9 , ] ] ]u

(A6)

2648


1

2



. ] 1 ] ] [ q9 (m u˙ 9) 1 u˙ 9 (u9 cosw) 2 ] ]]l [u˙ 9 ]]yu9]6 ]u a cosw 5]w [ ]u

mNA [ 2 u*9

mND

1 ]q9 1 ]q9 1 y *9 a cosw ]l a ]w

VOLUME 58



(A7)



Multiplying (A6) by q9 and zonal averaging the result, we obtain the zonal-mean eddy potential enstrophy equation m

1 2

] q92 ]t 2

5 2y *

1 2 2 q9y *9 a ]w 1 mD

1 ] q92 a ]w 2

1 ]q

1 m q9NA ,

(A8)

where

[

mD [ q q9

]

] ] (m u˙ 9) 1 q92 (m u˙ ) ]u ]u

5

]6

[

1 q9

1 ] ] ] u˙ (u9 cosw) 1 u˙ 9 (u cosw) a cosw ]w ]u ]u

2 q9

1 ] ]y 9 ]y u˙ 1 u˙ 9 1 m q9ND . a cosw ]l ]u ]u

1

2

(A9)

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