Upper-Tropospheric Potential Vorticity Fluctuations and the Dynamical Relevance of the Time Mean

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1 1JULY 2001 SWANSON 1815 Upper-Tropospheric Potential Vorticity Fluctuations and the Dynamical Relevance of the Time Mean K. L. SWANSON Department of Mathematical Sciences, University of Wisconsin Milwaukee, Milwaukee, Wisconsin (Manuscript received 15 March 2000, in final form 18 November 2000) ABSTRACT Analysis of frequency distribution functions of potential vorticity (PV) on middleworld isentropic surfaces that intersect the tropopause reveals that dynamics on these surfaces are dominated by the tropopause separating subtropical and polar PV pools with definitive characteristic values. The dominance of the tropopause on these surfaces suggests that the subseasonal transient variance should be a function solely of the time mean PV. Exploiting this relation, it is shown that knowledge of the annual mean PV distribution allows quite skillful deduction of interannual anomalies in transient variance. This analysis is extended to the dynamic tropopause, where it is shown that polar and subtropical jets clearly emerge in fields of the most frequent tropopause potential temperature. Implications of these results on the current understanding of how patterns of interannual variance arise are discussed. 1. Introduction The time mean upper-tropospheric vorticity budget has long been the natural starting point for the study of extratropical interannual variability. The natural partition of this budget into terms necessary to support Rossby wave propagation (Hoskins and Karoly 1981), a Rossby wave source term containing the forcing associated with divergent flow (Sardeshmukh and Hoskins 1988), and an additional forcing term due to transients provides a clean diagnostic framework within which to examine such variability, and progress toward understanding the relative importance of each of the terms in this budget to the overall response represents one of the hallmark achievements of the Tropical Ocean Global Atmosphere (TOGA) decade (Trenberth et al. 1998). The explanatory power of dynamics in the vicinity of time mean upper-tropospheric vorticity fields belies what at first sight appear to be reasonable prior expectations that such dynamics should not be applicable. Philosophically, this possibility has received comment numerous times; Pedlosky (1987, chapter 7.1) notes that the stability of the time mean state cannot in general be used as an argument for or against the proposal that observed fluctuations in that state are the result of an instability process, while Andrews (1984) points out the intrinsic ambiguity of examining linearized dynamics about time mean states. More recently, Morgan (1994) Corresponding author address: Kyle Swanson, Department of Mathematical Sciences, University of Wisconsin Milwaukee, Milwaukee, WI kswanson@csd.uwm.edu has addressed whether it is possible to define an observationally based dynamical state that is more relevant to the study of midlatitude cyclogenesis than the time mean. Empirically, on interannual timescales it is well appreciated that the upper troposphere exhibits substantial variability, primarily due to nonlinear internal processes (e.g., Kumar and Hoerling 1995). This variability opens the possibility that the time mean may not be representative of the underlying atmospheric state, or alternatively, that other fields may exist that provide an equally good representation as the time mean, depending on one s particular perspective. This certainly would be the case, for example, if the underlying frequency distribution functions (FDFs) of various dynamical quantities were either strongly skewed or bimodal. Several analyses focusing on the structure of FDFs do exist in the literature; White (1980) and Nakamura and Wallace (1991) show that FDFs of upper-tropospheric geopotential height in general are positively skewed poleward of the midlatitude jets and negatively skewed equatorward. In contrast, Swanson and Pierrehumbert (1997) find that FDFs of lower-tropospheric temperature are Gaussian, which they attribute to damping of temperature anomalies by the underlying surface. In this work, we extend these prior studies by examining the FDFs of isentropic potential vorticity (PV) on so-called middleworld isentropic surfaces that intersect the tropopause (Hoskins et al. 1985, hereafter HMR; Hoskins 1991). Compared to vorticity, PV has the advantage that it is materially conserved by two-dimensional motions on isentropic surfaces under adiabatic conditions. In 2001 American Meteorological Society

2 1816 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 principle, the dominance of advection makes the distribution of PV fluctuations easier to understand. Additionally, in contrast to geopotential height, PV is a fundamentally local quantity. Yet, when coupled with the invertibility principle, its distribution effectively determines other dynamical fields, given the distribution of potential temperature at the surface (HMR). Below, PV frequency distributions are shown to provide a window into the underlying dynamical processes on these middleworld isentropic surfaces. The character of middleworld PV that emerges from this analysis suggests a much richer definition of the relevance of the time mean, namely, that it is a mirror of the fluctuations themselves. The outline of the paper is as follows: in section 2, the time mean fields and fields of most frequent PV on middleworld isentropic fields are shown to contrast sharply, with the latter revealing the dominance of the tropopause on these middleworld surfaces. Section 3 focuses on the implicit connection between transients and the time mean that arises from the dominance of the tropopause, while section 4 explores the nature of the dynamic tropopause itself. Results are discussed and conclusions drawn in section Fluctuations in upper-tropospheric PV In a frictionless, adiabatic atmosphere in hydrostatic balance, the conservation of circulation gives the material conservation of the PV 1 P a, (2.1) where g 1 p/ plays the role of density in isentropic coordinates, and a is the component of the vorticity perpendicular to the isentropic surface (HMR). The PV is conserved by two-dimensional motions on an isentropic surface; similarly, the potential temperature is conserved by two-dimensional motions on an iso-pv surface. Further, if the interior PV and boundary distributions are known, then the balanced motion may in principle be determined by inversion of a quasi-elliptic operator. Following HMR, PV magnitudes in the troposphere typically increase poleward and upward to values greater than 1 PVU (1 PVU 10 6 Ks 1 kg 1 ). Above the extratropical tropopause, the magnitude is about 4 PVU and increases rapidly poleward and upward. While the precise value of PV used to designate the tropopause is not critical for the present purposes, consistent with prior studies we equate the tropopause with the 2-PVU surface (HMR; Morgan and Nielson-Gammon 1998). The PV fields used herein were constructed from the National Centers for Environmental Prediction reanalysis 4 day 1 spectral fields archived at the National Oceanic and Atmospheric Administration s (NOAA) Climate Diagnostics Center for the period and were interpolated onto isentropic surfaces following the methodology of HMR. We focus on the winter (January February) season. The most striking characteristic of middleworld PV frequency distributions is the sharp contrast between the mean and mode (most frequent) PV values virtually everywhere, 1 as apparent from even cursory examination of Fig. 1. While the mean field is a smooth function of latitude on each surface, with obvious qualitative resemblance to the streamfunction, the mode field is characterized by a sharp PV jump dividing nearly homogeneous subtropical and polar PV pools. In only a few regions, most notably on the 330- and 345-K surfaces at the termination of the Pacific and Atlantic storm track, does the mode PV field show a smooth latitudinal gradient. The significance of this structure is readily confirmed using a jackknife procedure of progressively neglecting one winter from the analysis; the resultant changes in the mode fields are qualitatively negligible even for samples as short as one winter. The binary nature of the mode PV fields on these middleworld isentropic surfaces suggests that the underlying dynamical events generating the fluctuations draw upon distinct, relatively homogeneous subtropical and stratospheric PV pools whose characteristic values are roughly invariant over the decadal timescales of the time series here. This behavior contrasts with surface temperature distributions over the continents, which are approximately Gaussian in spite of instantaneous sharp temperature gradients in frontal zones because of strong damping to the underlying surface (Swanson and Pierrehumbert 1997). Figure 2 demonstrates the changes in the PV distributions that occur as one moves poleward along 180 on the 315-K surface; for subtropical latitudes (25 N), the frequency distribution function (FDF) is dominated by fluctuations tightly constrained to the characteristic subtropical PV value of 0.5 PVU. However, as one moves poleward (40 N in the figure), a secondary maxima in the FDF emerges with PV values greater than 2 PVU, indicating air of stratospheric origin. Sufficiently close to the pole, this secondary maxima dominates the subtropical maxima (i.e., incursions of stratospheric air at that location become sufficiently frequent), and the mode shifts to this stratospheric maxima, apparent in the 55 N slice in the figure. To explore the dynamical consequences of this behavior further, we focus on the 315-K surface and consider the following thought experiment: for each time slice, let us assign the observed polar pool value of 5.5 PVU to every location where the instantaneous 315-K PV is greater than 2 PVU, and the subtropical value of 0.5 PVU to those locations where the 315-K PV is less than 2 PVU. This decomposition yields a hypothetical time mean field wherein the PV value at each location 1 Calculation of the mode requires explicit calculation of frequency distribution function (FDF) of PV values at that point. For the case here, the FDFs are constructed by sorting the individual PV values at a given point and for a given isentropic surface into bins of width PVU. The results below are not sensitive to the choice of the bin width, as bins of width 0.25 PVU give virtually identical results.

3 1JULY 2001 SWANSON 1817 FIG. 1. Time mean (top row) and mode (bottom row) PV fields on the 315-, 330-, and 345-K isentropic surfaces. Contour interval is 1 PVU. is proportional to the fraction of time that point spends within the stratosphere. This field, shown in Fig. 3, can be directly compared to the time mean 315-K PV field of Fig. 1; the resemblance is apparent. If, as this experiment suggests, the time mean PV distribution on a given middleworld isentropic surface merely reflects the relative frequency of stratospheric air at a given location, the link between subseasonal transient variability of the tropopause on these middleworld surfaces and the time mean PV field is then straightforward; high transient activity coincides with weak latitudinal PV gradients, and low transient variability coincides with strong PV gradients. This explicit FIG. 2. Potential vorticity frequency distribution functions (FDFs) for 25, 40, and 55 N along 180 on the 315-K surface. The 25 N FDF has been scaled in half to facilitate comparison with the other FDFs. FIG. 3. Cumulative distribution function of the stratosphere on the 315-K surface, as described in the text. Contour interval is 1 PVU.

4 1818 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 FIG. 4. Along-gradient component of the 315-K PV flux. Contour interval is 0.5 PVU m s 1, with negative contours dashed. connection contrasts with traditional linear approaches that treat the time mean as a quantity that can be imposed apart from the inherent underlying transient variability. Moreover, the prevalence of the tropopause suggests that at leading order, middleworld dynamics are not dominated by the downscale cascade of enstrophy that would rapidly act to dissipate such a sharp gradient, but rather by the quasi-reversible fluctuations of the tropopause itself. This can be verified explicitly by examining the along-gradient component of PV fluxes on the 315-K surface u P n, where n P/ P is the along-pv gradient unit normal. Figure 4 shows that areas of significant countergradient fluxes of PV occur over North America and Europe. Local magnitudes of these countergradient fluxes are sufficiently large that they nearly cancel the downgradient fluxes upon zonal averaging. This contrasts with the quasigeostrophic along-gradient PV fluxes examined by Plumb (1986), which were shown to be downgradient almost everywhere. Presumably, this difference lies in the nature of quasigeostrophic PV and its inability to properly treat the large horizontal contrast in static stability across the tropopause on middleworld isentropic surfaces. 3. Transients and the time mean a. Variability structure and interannual variability of transients The connection between subseasonal transient fluctuations and the time mean implicit in the dominance of instantaneous middleworld PV distributions by the tropopause has a number of important (and useful) ramifications. Provided the time mean PV distribution results from a wandering tropopause separating homogeneous subtropical and polar PV pools with respective PV values P sub and P pol, it is straightforward to show that the transient PV variance e P 2 is related to the time mean PV P by the expression FIG. 5. Scatterplot of PV variance as a function of time mean PV at all Northern Hemisphere grid points. The curve representing the theoretical variance for a purely bimodal PV distribution is shown for comparison. (P P sub)(ppol P ) P [P sub, P pol] e(p ) (3.1) 0 otherwise. Figure 5 shows a scatterplot of the observed subseasonal transient PV variance as a function of the time mean PV at all Northern Hemisphere locations on the 315-K surface. Also shown is the theoretical expression (3.1), where (P pol, P sub ) (5.5, 0.5) PVU are chosen consistent with the 315-K mode field of Fig. 1. Not surprisingly, the theoretical expression generally exceeds the actual variance by roughly 50%, which reflects the fact that (3.1) is an upper bound on the variance independent of the time mean PV distribution, provided instantaneous PV values remain in the range [P sub, P pol ]. However, the observed variance clearly exhibits the inverted parabolic character of the theoretical expression (3.1). An empirical e( P) relation may be constructed that represents the scatterplot in Fig. 5 better than the theoretical curve (3.1). In particular, the function where 2 4 8s e(p ) 100Ppols e, (3.2) s P/P pol provides a curve that lies within the densest concentration of points in Fig. 5. Figure 6 compares the spatial structure of observed variance pattern and the variance pattern estimated using (3.2) with the 315-K time mean PV distribution as the input. The observed and estimated patterns differ by less than 0.5 PVU 2 everywhere. It is vital to note that the observed PV variance does not exhibit the longitudinal variation that other quadratic transient quantities exhibit, for example, perturbationsquared streamfunction. This property of the PV variance indicates that the width of the variance region, not

5 1JULY 2001 SWANSON 1819 TABLE 1. Anomaly pattern correlation coefficients (ACs) between modeled and observed interannual anomalies in subseasonal transient PV variance. JF of El Niños 7LaNiñas 7 NAO 7 NAO Total AC FIG. 6. (top) Observed and (bottom) empirically modeled PV variance on the 315-K surface. Contour interval is 0.5 PVU 2, with values greater than 3 PVU 2 shaded. the absolute magnitude of the variance itself, is the relevant measure of transient activity on these middleworld isentropic surfaces. The relationship between the subseasonal transient PV variance and the time mean provides significant deductive skill of interannual anomalies in PV variance in the sense described by Whitaker and Sardeshmukh (1998), namely: given a particular time mean PV distribution, to what extent is it possible to deduce the average transient statistics? Table 1 shows the anomaly pattern correlation coefficients (ACs) between the observed and empirically deduced interannual anomalies in variance on the 315-K surface for 13 individual winters, as well as the ACs between the actual and empirically deduced anomalies for an average over seven El Niño events (1958, 66, 73, 83, 87, 92, 98), seven La Niña events (1968, 71, 74, 76, 85, 89, 99), seven winters with strongly positive North Atlantic oscillation (NAO) indices (1972, 73, 75, 83, 89, 92, 93), and seven winters with strongly negative NAO indices (1963, 69, 70, 77, 79, 80, 85). The expression (3.2) was used to generate the empirical anomalies for the subseasonal transient variance; however, the quoted anomaly correlations are quite insensitive to the details of these empirical curves, as use of the theoretical expression (3.1) yields essentially identical ACs, albeit overestimating the relative magnitudes of the anomalies themselves. The empirically modeled anomalies in variance encompass a substantial fraction of the interannual variability of the observed anomalies, as the AC averaged over the 13 selected years is 0.6. As an example of this agreement, consider the observed and modeled anomalies in variance for the seven El Niño and seven La Niña winters, respectively, shown in Fig. 7. The anomalous variance maxima (minima) in the eastern Pacific may be identified with the downstream (upstream) extension (retraction) of the storm track during El Niño (La Niña) winters noted by Lau (1988). Of course, the fact that transient variance patterns respond to time mean anomalies is well appreciated (e.g., Lau 1988; Robinson 1991; Branstator 1995). However, the analysis here suggests a nearly trivial link between the total and subseasonal transient variability and the time mean, arising solely from the dominance of the tropopause in instantaneous middleworld PV distributions. The markedly smaller AC for La Niña compared to El Niño winters results from the failure of the e( P) relation to capture the increase in transient variance over the Aleution low during La Niña. To understand why this occurs, we explicitly consider the FDFs for the La Niña and El Niño winters in this region. The FDFs for the point (60 N, 150 W) for El Niño and La Niña winters, shown in Fig. 8, provide an adequate characterization of behavior in this region. Consistent with the e( P) deduction of insignificant anomalies in variance, the mean of these distributions is roughly unchanged between El Niño and La Niña. However, a marked increase in the frequency of subtropical PV incursions during La Niña winters is apparent, which may be identified with the marked increase in blocking events in

6 1820 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 FIG. 7. (top) Observed and (bottom) empirically modeled 315-K PV variance anomalies during (left) El Niño and (right) La Niña winters. Contour interval is 0.25 PVU 2, with negative contours dashed. FIG. 8. Frequency distribution functions on the 315-K surface for the point 60 N, 150 W for El Niño (solid) and La Niña (dashed) winters. For reference, the climatological FDF, not shown for clarity, is virtually indistinguishable from the El Niño FDF. this region during La Niña events (Chen and van den Dool 1997). These increased incursions lead to the positive anomalies in transient variance. In principle, the e( P) relation could capture this increase in subtropical PV incursions and accompanying increase in subseasonal transient variance. However, a coincident offsetting shift in the stratospheric most frequent PV value in this area also occurs. Given these offsetting structural changes, it is far from apparent that any local theory based upon the time mean could capture the observed anomalies in variance. An even more marked change in FDF structure is found when comparing anomalies in variance for positive and negative NAO winters. Figure 9 shows that the e( P) modeled anomalies are negative (positive) over the European North Atlantic sector during positive (negative) NAO winters. While the model s deduction of reduced (enhanced) variance over Greenland and the North Atlantic mimic respective observed anomalies in variance, there is also enhanced (reduced) variance at 60 N over Europe during positive (negative) NAO winters in the observations not captured by the model. The FDFs at the point (60 N, 0 ), shown in Fig. 10, indicate why this discrepancy occurs; the FDF structure shifts

7 1JULY 2001 SWANSON 1821 FIG. 9. As in Fig. 7 but for (left) positive and (right) negative NAO index winters. FIG. 10. Frequency distribution functions on the 315-K surface for the point 60 N, 0 for positive (solid) and negative (dashed) NAO index winters. markedly between positive and negative NAO winters, with the former marked by a strongly bimodal FDF while the latter FDF is unimodal. As in the La Niña case outlined above, enhanced (reduced) bimodality leads to positive (negative) anomalies in variance, suggesting that the qualitative change in the FDF structure is responsible for the anomalies in variance over Europe unresolved by the e( P) relation. Again, it is far from obvious whether any local theory based upon the time mean could reproduce the anomalies induced by this change in FDF structure. Even given the above exceptions, the deductive skill implicit in the e( P) relationship provides a benchmark against which various theories of anomalies in subseasonal transient variance as a function of an imposed time mean flow can be compared. While the author is unaware of dynamical theories for transient variability encompassing the entire subseasonal frequency band, a number of authors, for example, Branstator (1995), Zhang and Held (1999), and Whitaker and Sardeshmukh (1998), have recently discussed theories for storm track high-frequency transient statistics based upon stochastic forcing of dynamical models linearized about the time mean flow. In particular, Whitaker and Sardeshmukh examine the ability of their statistically stationary stochastic model to simulate interannual shifts in storm track structure, given the appropriate interannual anomalies in the time mean flow as the input for their model, for the same 13-yr sample as in Table 1. The average anomaly correlation coefficient for their stochastic mod-

8 1822 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 el was 0.23, 2 compared to an average anomaly correlation of 0.60 for the subseasonal anomalies in variance here. It must be emphasized that an e( P ) relation is not a theory for transient statistics in the sense of these stochastic models, as it only deduces the transient PV variance. However, the larger fraction of variance encompassed by the e( P) relation suggests the time mean contains more information regarding subseasonal transients than suggested by naive extrapolation of the stochastic storm track model ACs. b. Transient forcing of interannual anomalies The approximate e( P) nature of the total transient variance provides insight into the manner by which transients force interannual anomalies and specifically highlights the apparent dominance of that forcing by effectively inviscid processes. The budget for the time mean PV distribution on an isentropic surface has the form u P (u P ) S, (3.3) where S represents the nonconservative redistribution of PV, the bars (primes) denote time mean (transient) quantities as usual, and advection by divergent motions is neglected, following Brunet et al. (1995). Linearizing dynamics about this time mean flow yields the PV variance equation 1 u e (u P ) P D, (3.4) 2 where D is the variance dissipation. If e e( P), it is straightforward to manipulate this equation into the form 1 e u P (u P ) P D. (3.5) 2 P Neglecting the dissipation term in the variance budget, this expression provides an explicit relationship between the along-gradient component of the PV flux and the basic-state wind across contours of constant time mean PV. The along-gradient derivative of this flux is one component forcing the time mean flow, that is, u P (u P ) S [(u P n)n (u P s)s] S (3.6) [ ] 1 e u n n (u P s)s S, (3.7) 2 P where n P/ P is the unit vector normal along the time mean PV gradient, s is the unit vector orthogonal to n, and again, where variance dissipation is neglected. Figure 11 shows the anomalous 315-K PV tendency during El Niño winters based upon the observed 2 There is a discrepancy between the value of 0.3 quoted in by Whitaker and Sardeshmukh in their text and the average of the values quoted in their Table 1 of 0.23; we use the latter for comparison here. FIG. 11. (top) Observed and (bottom) modeled anomalous PV tendencies during El Niño winters. Contour interval is 0.01 PVU day 1, with negative contours dashed. along-gradient PV flux convergence [( u P n)n], along with the modeled tendency anomaly {1/2[( e/ P) u n]n}, constructed using the observed El Niño and climatological winter rotational wind fields on the 315- K surface from the reanalysis data, along with the respective PV fields and the expression (3.2). Both fields in the figure have been truncated to T12 spectral resolution for clarity. The two tendencies share the same dominant feature, namely a positive PV tendency anomaly centered near (40 N, 130 W); this anomalous tendency may be identified with the anomalous negative streamfunction tendencies in this region that are the primary signature of anomalous extratropical transient forcing during El Niño (Hoerling and Ting 1994). The primary discrepancy between the modeled and observed tendencies is the underestimation of the magnitude of the negative tendency anomaly centered over the Aleutions. This agreement suggests that these anomalous transient tendencies are simply a by-product of changes in the advection of transient PV variance, quantifying an idea first espoused by Branstator (1995) using arguments based upon barotropic wave ray tracing. The climatological winter PV tendencies (not shown)

9 1JULY 2001 SWANSON 1823 show similar good agreement at the ends of both the storm tracks, with the agreement somewhat less satisfactory at the upstream end of the Pacific track and over Asia, where the subseasonal variance is not as well described as a simple function of P. In this sense, the e( P) relation may be much more successful as a perturbation model than as a model for the total eddy fluxes, which is not surprising, given the intrinsic inviscid assumption underlying this relation. As discussed by Held et al. (1989), interannual anomalies in transient forcing, at least during El Niño, generally involve the displacement of the downstream end of the storm track; given the skill of the e( P) relation at reproducing such displacements, its ability to model associated transient forcing is not surprising. 4. Tropopause potential temperature fluctuations The natural emergence of the dynamic tropopause in the isentropic fields of most frequent PV shown in Fig. 1 suggests further examination of the intrinsic two-dimensional dynamics of the tropopause itself (e.g., Juckes 1994). While the importance of the tropopause to the development of extratropical cyclones has long been recognized [see Davis and Emmanuel (1991) for a review], its role in variability on longer timescales remains relatively unexplored. In part, this is due to the tendency to focus on relatively smooth time mean vorticity fields, quite simply because it is the change in those fields that is the central concern. However, by smoothing out the tropopause in the construction of the time mean, and further, by considering the time mean tropopause as somehow distinct from the underlying transient variability, the possibility exists that certain dynamical links may be distorted. To this end, in this section we concentrate on the potential temperature on the dynamic tropopause, which we approximate as the 2-PVU surface. Our method for calculating tropopause follows Morgan and Nielson-Gammon (1998) and consists of interpolating onto the 2-PVU surface from above for each individual time slice. In places where there are tropopause folds, this process is necessarily ambiguous, as no functional relationship exists between the dynamic tropopause and a latitude longitude coordinate system. However, for the long time series considered here, the number of tropopause folds at any given geographic location is small enough that it does not cloud the analysis. Following the analysis procedure of section 2, Fig. 12 compares the time mean and mode tropopause fields. Similar to the time mean upper-tropospheric PV fields, the time mean tropopause field resembles the streamfunction for the flow, increasing smoothly from values of 300 K over the pole to 370 K in the subtropics, with enhanced gradients marking the midlatitude jets. The mode tropopause field is marked by two regions of enhanced latitudinal gradients marking the subtropical and polar jets. The former divides a pool of FIG. 12. (top) Time mean and (bottom) mode potential temperature on the dynamic tropopause. Contour interval is 10 K. homogeneous 370 K in the subtropics, presumably associated with angular momentum conserving outflow in the Hadley circulation (Held and Hou 1980), from a weak tropopause gradient region in the midlatitudes. Enhanced mode gradients downstream of both the Pacific and Atlantic storm tracks indicate the emergence of the polar jet; such a feature cannot be distinguished in the time mean field. If one approximates the instantaneous tropopause fields in terms of three regions of constant in the manner of the thought experiment of section 2, that is, assigning each location where 310 K the value 300 K, each location where 320 K 350 K the value 335 K, and each location where 350 K the value 370 K, it is possible to construct a hypothetical time mean tropopause field analogous to the hypothetical time mean PV field of Fig. 3. Doing so yields a field that is essentially indistinguishable from the time mean field of Fig. 12; as such, from a certain perspective the time mean tropopause distribution may be considered to result from the wandering of the polar and subtropical tropopause breaks, just as the time mean 315-K PV field may be considered to result from the wandering of the tropopause itself.

10 1824 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 The idea that we originally sought to test based upon the above realization was that forced interannual modes of variability could be characterized by shifts in the most frequent tropopause position, with the tropopause break variance about that position remaining unchanged, while free interannual modes of variability arise primarily from changes in the variance of the tropopause breaks, leaving the most frequent break positions unchanged. However, changes in the structure of the mode field coincident with interannual patterns of variability suggest that this picture of the tropopause is oversimplified. Figure 13 shows the mode tropopause composited for the El Niño, La Niña, positive NAO index, and negative NAO index winters. During El Niño (Fig. 13), the subtropical tropopause break moves 5 poleward over the eastern Pacific, while the polar tropopause break loses much of its coherence over the northern Pacific. In contrast, during La Niña, the subtropical break migrates deep into the Tropics and loses much of its coherence, while the polar jet slides slightly poleward and is much more coherent. These changes are consistent with the aforementioned downstream extension (retraction) of the Pacific storm track during El Niño (La Niña), along with the accompanying transfer of landfalling eddy activity equatorward (poleward) on the Pacific coast of North America (Trenberth and Hurrel 1994). Insofar as the relative tightness of most frequent tropopause gradient for the two tropopause breaks adequately represents the quality of each break as a waveguide, the redistribution of transient eddy activity described above is natural. Positive NAO index winters are marked by a much sharper Atlantic sector polar jet than the time mean, while during negative NAO index winters, the polar jet in the Atlantic sector cannot be distinguished. This behavior is reminiscent of the model studies of Williams (1988) and Lee (1997) regarding the emergence and maintenance of multiple jets in geophysical flows, suggesting the Atlantic sector may be marginal in the sense that it makes the transition from a single to two jets due solely to internal dynamics. These structural changes to the mode tropopause field, particularly for different phases of the NAO, suggest that viewing the dynamical tropopause as the composite of two wandering tropopause breaks is an oversimplification. 5. Discussion and conclusions Analysis of PV distributions on middleworld isentropic surfaces reveals that these distributions are bimodal over a substantial fraction of the globe. This bimodality is most apparent in the mode (most frequent) fields, which are characterized by a single jump in the PV marking the tropopause, in contrast with the time mean fields, which exhibit smooth latitudinal PV gradients (Fig. 1). While it has been recognized for some time that instantaneous upper-tropospheric PV fields are characterized by a jump at the tropopause, this analysis shows that the magnitude of the jump is not random, but rather has a definitive value characteristic to a particular isentropic surface. Apparently, the assumption of homogeneous PV in the troposphere may be even more valid than the time mean itself suggests (Sun and Lindzen 1994). A direct implication of the dominance of the tropopause in middleworld PV dynamics is that the transient variance of the tropopause and the structure of the middleworld time mean PV fields are inextricably linked. The simple assumption that the variance is a function solely of the time mean isentropic PV allows for quite skillful deduction of interannual anomalies in the isentropic PV variance given corresponding anomalies in the time mean PV fields. In addition, anomalous mean flow forcing associated with extratropical interannual variability appears to result simply from changes in the advection of this variance. Confining inquiry to the dynamic tropopause (identified with the 2-PVU surface), the field of the most frequent tropopause also differs from the respective time mean (Fig. 12) and is characterized by well-defined tropopause breaks, one in the subtropics extending from roughly 360 to 330 K and another marking the polar jet from 320 to 300 K. These breaks separate three regions of roughly homogeneous : a tropical region presumably associated with angular momentum conserving motions in the outflow branch of the Hadley circulation, a midlatitude region that emerges most clearly downstream of the midlatitude storm tracks, and a polar region. Northern Hemispheric interannual variability is marked by significant changes to the mode tropopause field; most notably, during La Niña the subtropical tropopause break is subducted deep into the Tropics, while during the positive phase of the North Atlantic oscillation (NAO), the polar jet in the Atlantic sector is more defined in comparison to climatology. The clear emergence of subtropical and polar jet structures in the mode tropopause fields is a surprising result; from the perspective of the most frequent tropopause fields, the positive phase of the NAO marks the existence of two well-defined jets in the North Atlantic sector, while the negative phase of the NAO is a state marked by a single jet. There are a number of philosophical issues raised in this study that certainly warrant further study. First, understanding how the behavior of linear and nonlinear models based upon mode versus mean basic states differ is an obvious extension, as there is no guarantee that the meridional Rossby wave propagation that underlies the current understanding of the extratropical response to El Niño events will be similar for these different basic states. More deeply, the connection between qualitative changes in the FDF structure in physical space in the North Pacific during the El Niño cycle and in the North Atlantic during the NAO cycle, and multimodality in phase space (e.g., Palmer 1993; Smyth et al. 1999) is

11 1JULY 2001 SWANSON 1825 FIG. 13. Mode tropopause potential temperature fields for (left) positive and negative El Niño winters, and (right) positive and negative NAO index winters. Contour interval is 10 K. a topic of intrinsic importance to our ultimate understanding of the nature of extratropical interannual variability. Establishing a connection between these two would provide explicit physical evidence that multimodality in fact exists and has a readily discernible signature in observed meteorological fields. Finally, the apparent functional dependence of the transient variance upon the time mean PV shown in section 3 in principle provides a parameterization of certain aspects of transient feedback on remotely forced quasi-stationary wave trains. Since this parameterization has a trivial computational cost, it can be easily implemented in numerical models and the resultant modifying effects upon such wave trains readily explored. Acknowledgments. The comments of three anonymous reviewers helped focus and clarify the manuscript substantially. This research was supported by the NOAA Office of Global Programs under Grant NA86GP0211. REFERENCES Andrews, D. G., 1984: On the stability of forced non-zonal flows. Quart. J. Roy. Meteor. Soc., 110, Branstator, G. W., 1995: Organization of storm track anomalies by recurring low-frequency circulation anomalies. J. Atmos. Sci., 52, Brunet, G., R. Vautard, B. Legras, and S. Edouard, 1995: Potential vorticity on isentropic surfaces: Climatology and diagnostics. Mon. Wea. Rev., 123, Chen, W. Y., and H. M. van den Dool, 1997: Asymmetric impact of tropical SST anomalies on atmospheric internal variability over the North Pacific. J. Atmos. Sci., 54, Davis, C. A., and K. A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev., 119, Held, I. M., and A. Y. Hou, 1980: Nonlinear axially symmetric circulations in a nearly inviscid atmosphere. J. Atmos. Sci., 37, , S. W. Lyons, and S. Nigam, 1989: Transients and the extratropical response to El Niño. J. Atmos. Sci., 46, Hoerling, M. P., and M. Ting, 1994: Organization of extratropical transients during El Niño. J. Climate, 7, Hoskins, B. J., 1991: Towards a PV- view of the general circulation. Tellus, 43AB, , and D. J. Karoly, 1981: The steady linear response of a spherical atmosphere to thermal and orographic forcing. J. Atmos. Sci., 38, , M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, Juckes, M., 1994: Quasigeostrophic dynamics of the tropopause. J. Atmos. Sci., 51, Kumar, A., and M. P. Hoerling, 1995: Prospects and limitations of

12 1826 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 58 seasonal atmospheric GCM predictions. Bull. Amer. Meteor. Soc., 76, Lau, N.-C., 1988: Variability of the observed midlatitude storm tracks in relation to low frequency changes in the circulation pattern. J. Atmos. Sci., 45, Lee, S., 1997: Maintenance of multiple jets in a baroclinic flow. J. Atmos. Sci., 54, Morgan, M. C., 1994: An observationally and dynamically determined basic state for the study of synoptic scale waves. Ph.D. thesis, Massachusetts Institute of Technology, 123 pp., and J. W. Nielsen-Gammon, 1998: Using tropopause maps to diagnose midlatitude weather systems. Mon. Wea. Rev., 126, Nakamura, H., and J. M. Wallace, 1991: Skewness of low frequency fluctuations in the tropospheric circulation during the Northern Hemisphere winter. J. Atmos. Sci., 48, Palmer, T. N., 1993: A nonlinear dynamical perspective on climate change. Weather, 48, Pedlosky, J. P., 1987: Geophysical Fluid Dynamics. 2d ed. Springer- Verlag, 710 pp. Plumb, R. A., 1986: Three-dimensional propagation of transient quasi-geostrophic eddies and its relationship with the eddy forcing of the time mean flow. J. Atmos. Sci., 43, Robinson, W. A., 1991: The dynamics of low-frequency variability in a simple model of the global atmosphere. J. Atmos. Sci., 48, Sardeshmukh, P. D., and B. J. Hoskins, 1988: The generation of global rotational flow by steady idealized tropical divergence. J. Atmos. Sci., 45, Smyth, P., K. Ide, and M. Ghil, 1999: Multiple regimes in Northern Hemisphere height fields via mixture model clustering. J. Atmos. Sci., 56, Sun, D.-Z., and R. S. Lindzen, 1994: A PV view of the zonal mean distribution of temperature and wind in the extratropical troposphere. J. Atmos. Sci., 51, Swanson, K. L., and R. T. Pierrehumbert, 1997: Lower-tropospheric heat transport in the Pacific storm track. J. Atmos. Sci., 54, Trenberth, K. E., and J. W. Hurrel, 1994: Decadal atmospheric ocean variations in the Pacific. Climate Dyn., 9, , G. W. Branstator, D. Karoly, A. Kumar, N.-C. Lau, and C. Ropelewski, 1998: Progress during TOGA in understanding and modeling global teleconnections associated with tropical sea surface temperatures. J. Geophys. Res., 103 (C7), Whitaker, J. S., and P. D. Sardeshmukh, 1998: A linear theory of extratropical synoptic eddy statistics. J. Atmos. Sci., 55, White, G. H., 1980: Skewness, kurtosis, and extreme values of the Northern Hemisphere geopotential heights. Mon. Wea. Rev., 108, Williams, G. P., 1988: The dynamical range of global circulations, I. Climate Dyn., 2, Zhang, Y., and I. M. Held, 1999: A linear stochastic model of a GCM s midlatitude storm tracks. J. Atmos. Sci., 56,

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