Research Article

Atmospheric blocking as a traffic jam in the jet stream

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Science  06 Jul 2018:
Vol. 361, Issue 6397, pp. 42-47
DOI: 10.1126/science.aat0721

A traffic jam of air

Persistent meandering of the jet stream can cause atmospheric blocking of prevailing eastward winds and result in weather extremes such as heat waves in the midlatitudes. Nakamura and Huang interpret the poorly understood origins of these systems as the meteorological equivalents of traffic congestion on a highway and show how they can be described by analogous mathematical theory. Climate change may affect the frequency of blocking as well as its geographic distribution, reflecting a simultaneous shift in the structure of the stationary atmospheric waves and the regional capacity of the jet stream.

Science, this issue p. 42

Abstract

Atmospheric blocking due to anomalous, persistent meandering of the jet stream often causes weather extremes in the mid-latitudes. Despite the ubiquity of blocking, the onset mechanism is not well understood. Here we demonstrate a close analogy between blocking and traffic congestion on a highway by using meteorological data and show that blocking and traffic congestion can be described by a common mathematical theory. The theory predicts that the jet stream has a capacity for the flux of wave activity (a measure of meandering), just as the highway has traffic capacity, and when the capacity is exceeded, blocking manifests as congestion. Stationary waves modulate the jet stream’s capacity for transient waves and localize block formation. Climate change likely affects blocking frequency by modifying the jet stream’s proximity to capacity.

Winds in Earth’s mid-latitudes blow predominantly eastward, and their speeds increase with altitude to form a jet stream in the middle to upper troposphere. Winds steer cyclones and anticyclones, and these weather systems in turn cause the jet stream to meander over thousands of kilometers. This undular pattern also migrates eastward, forming the transient Rossby waves (1, 2). However, occasionally the jet stream develops persistent meandering in a certain region, disrupting the passage of the transient waves—a condition known as blocking (35). A block can last for a few days to more than a week and often brings about anomalous, sometimes extreme, weather in the mid-latitudes. For example, the unprecedented heat wave that claimed tens of thousands of lives in Europe in the summer of 2003 was due at least partially to a strong anticyclonic blocking (6). In late October 2012, a highly meandering jet stream due to North Atlantic blocking steered Superstorm Sandy into a surprising westward path to make landfall on the New Jersey coast of the eastern United States (7).

Figure 1, A and B, illustrates observed blocking over the eastern Pacific and Euro-Atlantic sectors of the Northern Hemisphere [data are from (8)]. The crowded contours of the 500-hPa geopotential height (Z500, roughly a midtropospheric stream function) and high wind speeds define the jet stream, and its pronounced poleward excursion in the respective regions marks blocking. In the boreal winter, blocking often occurs when a preexisting quasistationary ridge amplifies and obstructs transient waves. For example, in the longitude-time diagram of Z500 (Fig. 1C), the diagonal streaks of transient waves are suppressed around 20 to 40°W in late January and mid-February. Though this suggests a role of stationary waves in block formation (9), the onset of blocking is still poorly understood, and it remains a challenging problem in numerical weather prediction (10, 11). Previously proposed mechanisms of block formation and maintenance include forcing by transient eddies and intrinsic instability of low-frequency dynamics (1219), yet no definitive theory exists for the onset criterion. Because of incomplete mechanistic understanding, even the definition of blocking remains somewhat subjective, and various blocking indices do not agree on the effects of climate change on them (20, 21).

Fig. 1 A blocked jet stream disrupts wave propagation.

(A) Z500 (500-hPa geopotential height) (contours every 50 m) and horizontal wind speed (color) in the Northern Hemisphere extratropics for 7 January 2005 (18 UTC). The star identifies the center of a block. (B) As in (A) but for 26 January 2005 (06 UTC). (C) Z500 at 49.5°N as a function of longitude and time (1 December 2004 to 31 March 2005). The contour interval is 50 m. (D) As in (C) but for column LWA. In all panels, the zero longitude corresponds to the International Reference Meridian. Data were obtained by using the ERA-Interim reanalysis product (8).

In this article, we propose a mechanism of block formation on the basis of observation and simple mathematical theory. The main tool of investigation is a recently developed metric for the jet stream’s meandering, finite-amplitude local wave activity (LWA, herein denoted by the symbol A) (22, 23). Based on the displacement of quasigeostrophic potential vorticity (24), LWA is a dynamic field that quantifies Rossby wave packets and their interaction with eastward wind on regional scales [see materials and methods (MM) in supplementary materials for theoretical background and the definitions of mathematical symbols]. Given that blocking signal is largely tropospheric and vertically coherent (5, 22), we make extensive use of density-weighted, vertically averaged (“column”) LWA, Embedded Image (the angle bracket denotes the column average). The column budget of LWA reveals marked similarity between atmospheric blocking and traffic congestion on a highway. The analogy enables us to understand block formation with the well-known traffic theory, which not only predicts a threshold of onset but also aids in the formation of testable hypotheses for the influence of climate change on blocking.

Jet stream behaviors analyzed with column LWA

To unravel the dynamics of block formation, we rely on the following properties of column LWA: (i) It qualitatively distinguishes blocking from wave propagation; (ii) it quantifies compensating tendencies between the jet stream’s meander and the speed of eastward wind; and (iii) it obeys a simple budget evaluable with data. Figure 1D shows column LWA in the same format as Fig. 1C. Diagonal streaks and clusters of large LWA events are readily recognized. The former delineate eastward propagation of transient Rossby waves, whereas the latter identify blocking. The distinction is more pronounced than that in Fig. 1C, and it is easy to isolate major blocking events such as those in Fig. 1, A and B.

Figure 2A shows in color the temporal covariance between column LWA Embedded Image and column eastward wind Embedded Image for the boreal winter [December-January-February (DJF)], averaged for 1979 to 2016. Here and throughout the article, both quantities are multiplied by the cosine of latitude Embedded Image to make an explicit connection to angular momentum. Covariance is almost everywhere negative (red indicates zero): That is, when the jet stream meanders more, the eastward wind slows down (and is sometimes reversed to westward). The most strongly negative covariance is found in the downstream of the climatological jets (contours), where blocks form frequently (25, 26). At the locations of maximum covariance, A (9°W, 45°N) and B (147°W, 42°N), a mutually compensating tendency of Embedded Image and Embedded Image is evident in the scatter diagrams (Fig. 2, B and C), which suggests an approximate relation Embedded Image(1)where u0 and α are empirically determined positive constants. This relation is associated with the conservation of the longitudinal average of Embedded Image through Kelvin’s circulation theorem, although locally the conservation is not perfect, so there is significant scatter and α deviates from 1 (22, 27, 28). The minimum value of column LWA is finite at both locations (Fig. 2, B and C), which we interpret as the stationary wave component of LWA (23).

Fig. 2 LWA and eastward wind covary negatively.

(A) Color indicates the covariance between Embedded Image and Embedded Image in the boreal winter, and contours indicate DJF climatology of Embedded Image expressed in meters per second. The locations of peak covariance are labeled A (9°W, 45°N) and B (147°W, 42°N). Data were obtained by using the ERA-Interim reanalysis product for 1979 to 2016 (8). (B) Scatter diagram of 4-day average Embedded Image versus 4-day average Embedded Image for the same period at location B. The cyan box represents the DJF climatological mean. The red diamonds represent periods in which Embedded Image and Embedded Image are simultaneously in the top and bottom 5 percentiles, respectively, of all 858 sampled values (listed in tables S1 and S2). The orange line is a least-squares fit whose slope is indicated. (C) Same as (B) but for location A.

Figure 3 compares the DJF climatology of column LWA (color) and Z500 (contours) with the composites of the large LWA events marked by the red diamonds in Fig. 2, B and C. These diamonds identify 4-day periods in which Embedded Image and Embedded Image are simultaneously in the top and bottom 5 percentiles, respectively, of all 858 sampled periods (24 for location A and 22 for location B, listed in tables S1 and S2). The climatology in Fig. 3A shows large LWA events on the poleward and downstream sides of the jet maxima (packed contours of Z500), as well as stationary ridges over the west coasts of North America and Europe, in agreement with Fig. 1C. The composite of the Atlantic events (Fig. 3B) shows a protruding ridge over the eastern North Atlantic, with a peak LWA in the close vicinity of the height anomaly. The Pacific events display a similar structure, except the ridge tilts in the opposite way and the LWA maximum is weaker (Fig. 3C). Both composites exhibit characteristics of a block: enhanced regional meandering of the jet stream that persists at least for a few days. Figures S1 and S2 repeat the analysis using eddy kinetic energy and eddy potential enstrophy instead of LWA. These more traditional Eulerian metrics are less successful at identifying blocks—to a varying degree they capture the properties of the displaced jets rather than those of the blocks themselves.

Fig. 3 LWA in climatology and blocked states.

(A) DJF climatology of Embedded Image (color) and Z500 (contours, in meters) in the Northern Hemisphere extratropics. (B) Same as (A) but a composite of 24 anomalous Atlantic events. (C) Same as (A) but a composite of 22 anomalous Pacific events. The analysis locations (A and B) are the same as those in Fig. 2.

The column budget of LWA reads (22, 23)Embedded Image(2)where the right-hand side (RHS) terms are (I) convergence of the zonal LWA flux (the form of Embedded Image will be discussed shortly), (II) divergence of the meridional eddy momentum flux, (III) meridional eddy heat flux at the lower boundary, and (IV) nonadiabatic sources and sinks of LWA (see MM for the definitions of variables). Equation 2 is similar to the linearized (small-amplitude) wave activity budget (29) yet applies to waves of arbitrary amplitude. Term IV does not subsume any cubic or higher-order term commonly neglected in the small-amplitude limit. The relative importance of the terms in Eq. 2 depends on the time scale (23). On synoptic time scales, term I dominates the RHS; when evaluated with data for locations A and B in Fig. 2A, this term explains about 70 percent of the total variance of the left-hand side, whereas the other terms play much smaller roles (table S3). (Term IV is evaluated as the residual of the budget, so it inevitably contains some analysis error.) The finding suggests that, to the lowest order, the jet stream may be viewed as a waveguide in longitude. This renders the column budget of LWA a one-dimensional (1D) transport problem—perhaps an oversimplification for the mature stage of blocking but adequate for periods leading up to the block formation. We therefore take a closer look at the zonal LWA flux Embedded Image in Eq. 2 and its relation to column LWA.

Zonal flux of LWA and analogy to traffic flow

The explicit form of Embedded Image is (22, 23)Embedded Image(3)Two of the RHS terms (F1 and F3) are associated with group propagation of the Rossby waves in the reference state (including advection by uREF, the zonal wind speed in a wave-free reference state) (MM). The sum F1 + F3 varies approximately linearly with column LWA, where the proportionality constant is the mean zonal group velocity of the waves (see the orange diamonds in Fig. 4, A and C). This linear relation corresponds to the diagonal streaks in Fig. 1D with modest magnitudes of LWA. The remaining term (F2) represents nonlinear modification of the flux by large-amplitude waves. With this term included, the total flux Embedded Image (blue diamonds) exhibits considerable scatter against column LWA [because of the irregular covariance of ue and qe (the deviation in potential vorticity) in F2]. However, the majority of the blue diamonds reside below the orange cluster, and their separation becomes more pronounced at greater LWAs. The curve fits and quartile plots in Fig. 4, B and D, further summarize this. Despite the considerable range of bars in the quartile plots, it is fair to say that Embedded Image maximizes at an intermediate value of column LWA (50 to 60 ms−1), beyond which it decreases with increasing LWA. The deviation of Embedded Image from F1 + F3 means that F2 is predominantly negative and more so at higher LWAs. This is because at large wave amplitudes, the zonal wind is decelerated relative to the reference state, so Embedded Image (Eq. 1). From Eq. 3 and eq. S2a, we then expect Embedded Image, which prevails over Embedded Image at large values of Embedded Image and suppresses Embedded Image. For this reason, we use a parabolic curve fit for Embedded Image in Fig. 4, B and D. The fitted blue curves trace the median values reasonably well at high LWAs and come close to the orange lines at lower values (20 to 40 ms−1).

Fig. 4 LWA and its zonal flux at the jet exit.

Relationships between Embedded Image (abscissa) and the flux terms on the RHS of Eq. 3 (ordinate). (A and B) 9°W, 45°N (location A). (C and D) 147°W, 42°N (location B). [(A) and (C)] Four-day averages of F1 + F3 (orange) and Embedded Image (blue) against the 4-day average of column LWA, sampled during DJF of 1979 to 2016. The red diamonds correspond to those in Fig. 2, B and C. [(B) and (D)] Diagrams simplifying (A) and (C) with a least-squares fit (orange), a chi-square fit with a parabolic curve (blue), and quartile plots. The quartile plots show maximum, minimum, and median values, together with the upper and lower quartiles of Embedded Image, for 10-ms−1 intervals of Embedded Image. Boxes and bars were not drawn if the interval contains fewer than five data points. The cyan squares in (A) and (C) and blue squares in (B) and (D) indicate the DJF climatological mean values.

There is a close parallelism between Fig. 4 and the so-called fundamental diagram of traffic flow in transportation engineering (30, 31). The latter describes traffic flow (or flux, the number of cars passing through a point of highway per unit of time) as a function of traffic density (the number of cars per unit of length of highway). For a major highway and a network of highways, traffic flow maximizes at an intermediate value of traffic density, which defines highway capacity (30, 32). When the traffic is light, most cars travel at or near the speed limit, so the traffic flow is proportional to traffic density, just like the orange lines in Fig. 4, B and D, if we substitute traffic density for column LWA. However, as the traffic becomes heavier, the increasing density slows down the traffic because of changes in the drivers’ braking practices and other driving behaviors, which will eventually suppress the flow despite the increasing density. This is analogous to the blue curves in Fig. 4: Waves slow the winds and limit the growth of the LWA flux (i.e., the effect of F2 on Embedded Image). When this happens, a traffic bottleneck (shock) forms quickly because a decreasing flow in the direction of traffic causes convergence and hence rapid accumulation of density (33, 34). Traffic congestion therefore occurs at the high-density–low-flow end of the fundamental diagram after the highway capacity is reached. Similarly, the red diamonds in Fig. 4, A and C, corresponding to the block-producing red diamonds in Fig. 2, B and C, occupy high LWA and low zonal LWA flux. The analogy suggests that atmospheric blocking occurs much the same way as traffic congestion: The jet stream possesses a capacity for the zonal LWA flux, which when exceeded triggers block formation.

To test this hypothesis with a simple model, we first documented the observed life cycle of blocking for a reference. Figure 5 shows a composite of the 24 Atlantic events listed in table S1, in longitude-time diagrams of column LWA, column zonal wind, and Embedded Image along 45°N. The composite shows gradual buildup and decay of the column LWA anomaly and the simultaneous deceleration (reversal) and acceleration of the zonal wind. The life span of the event is approximately a week, and the signal is longitudinally isolated (Fig. 5, A and B). The zonal LWA flux turns slightly negative within the block, but in the immediate upstream (40° to 10°W) there is a prolonged period of enhanced flux, with particularly large values appearing from 13 to 7 days before the event (Fig. 5C). This supports the notion that an elevated wave activity flux in the upstream (due, for example, to anomalous storm track activity) triggers a block formation downstream (35, 36).

Fig. 5 Composite life cycle of Atlantic blocks.

(A) Composite longitude-time diagram of Embedded Image along 45°N for the 24 Atlantic events listed in table S1 in supplementary materials. The black contour indicates 60 ms−1. Day 0 corresponds to the peak of blocking. (B) Same as (A) but for Embedded Image. The white contour indicates 0. (C) Same as (A) but for Embedded Image. The gray and black contours indicate 0 and 600 m2 s−2, respectively.

A simple model of blocking life cycle

To conceptualize what we have observed so far, we propose the nonlinear partial differential equationEmbedded Image(4)where x is longitude and t is time. Equation 4 governs the evolution of the transient-wave component of column LWA, Â. It is derived from Eqs. 2 and 3 with Eq. 1 and other simplifying assumptions, and by partitioning LWA into stationary- and transient-wave components: A(x,t) = A0(x) + Â(x,t) (see MM for derivation). [C(x) – αÂ]ÂF is the nonlinear flux of Â, where C(x) ≡ uREF + cg − 2α A0(x) denotes the background group velocity modulated by the stationary wave A0 (uREF + cg is a constant zonal group velocity in the reference state). The dependence of C on A0 introduces interaction between stationary and transient waves. The parameter α is an empirical constant introduced in Eq. 1, which is assumed to be independent of space and time. The last three terms in Eq. 4 replace terms II to IV in Eq. 2 with a simple source and sink of  plus numerical diffusion that keeps  smooth. By analogy, Eq. 4 also describes the traffic flow problem (33, 34). In that context, C(x) plays the role of speed limit. Table S4 summarizes parallel interpretations of Eq. 4 for traffic and blocking problems.

We integrate Eq. 4 from  = 0 with a constant forcing Ŝ > 0 (representing a homogeneous transient-wave source associated with storm activities and nonadiabatic processes), a prescribed stationary wave A0(x) with zonal wave number 2, and periodic boundary conditions in x (see MM for the details of the experiment). Eventually the total LWA, A = A0 + Â, settles into a steady state with two peaks, notably higher than A0 because of the forcing. This corresponds to Fig. 3A. Subsequent evolution of A is depicted in Fig. 6A and movie S1. To mimic anomalous weather activity in the upstream (35), we increase the forcing Ŝ temporarily with a peak value around 44°W and day −4. (A highway analogy would be a sudden merging of traffic that increases traffic density.) In a few days, A begins to grow downstream, most markedly at a point slightly upstream of the preexisting ridge. The growth culminates in a shock formation around 15°W (movie S1), a well-known behavior in the traffic flow problem and an indication of wave breaking in our context. See our previous work (37) as to how shock formation in one dimension relates to wave breaking and block formation in two dimensions. After a maximum value is reached, A slowly recedes to the steady state through damping, while the shock migrates downstream.

Fig. 6 Blocking life cycle simulated by Eq. 4.

(A) Longitude-time diagram of A = A0 + Â for a 1D experiment. The black contour indicates 52 ms−1. Day 0 corresponds to the peak of the wave event. (B) Same as (A) but for u. The white contour indicates 11 ms−1. (C) Same as (A) but for F. The contours indicate 608, 656, and 704 m2 s−2. In all panels, only a part of the computational domain is shown. See movie S1 for more details.

Figure 6 also shows u ≡ 40 − αA (eastward wind) (Fig. 6B) and F = [C(x) − αÂ] (zonal LWA flux) (Fig. 6C) for this experiment. Broad agreements with Fig. 5 attest to the ability of Eq. 4 to capture the rudimentary characteristics of blocking. There are also notable differences: For example, in Fig. 6, u and F do not turn negative. (Traffic may come to a halt, but it never reverses on its own.) This is one of the limitations of the 1D model. Another discrepancy is that the growth phase of the wave event is shorter in Fig. 6, with the peak flux concentrated within 5 to 2 days prior to blocking (Fig. 6C), whereas in the observation the peak flux occurs farther in advance (Fig. 5C). This is probably because of the specific form of forcing used in the experiment. The decay of the block in the simulation is dictated by a constant damping rate, but in reality processes such as the meridional redistribution of momentum by waves (term II in Eq. 2) may alter this rate once the block matures. Generally, we regard Eq. 4 as a minimal model of block formation; to capture the fuller details of life cycle, one needs a 2D (or 3D) model.

Despite its crudeness, Eq. 4 provides a useful threshold for wave breaking and subsequent block formation. Without the last three terms, Eq. 4 develops a shock after F saturates; that is, after it reaches the capacity C2/4α (or equivalently, Â exceeds C/2α) where C' < 0 (i.e., the background group velocity decreases downstream) (38). Figure S3 describes the evolution of Γ, the saturation level of LWA flux defined asEmbedded Image(5)during the experiment. Γ is already as high as 0.9 initially (day −17) at 9°E, where C is at minimum because of large stationary wave A0. Γ first reaches 1 around 0° on day −3.5, where C′ < 0, after which the shock forms in the upstream. Therefore, even with additional terms in Eq. 4, the flux saturation (Γ = 1) is an important threshold for the onset of blocking. The precise timing and location of saturation in the real atmosphere may be hard to predict because of large uncertainty in Embedded Image (Fig. 4), but regions with higher values of Γ (closer to saturation) are more conducive to block formation. The exit regions of the storm tracks are prime examples for this, where diffluence of the jet stream associated with stationary waves keeps the flux capacity small (analogous to a reduced speed limit on a highway) and a large LWA flux is frequently seeded by cyclogenesis in the upstream. The observed climatological mean states indicated by the blue boxes in Fig. 4, B and D, are close to the vertices of the blue curves, analogous to Γ ~ 1 in the model.

Discussion

The present study provides a theoretical framework to quantify processes that cause block formation. Previously, Charney and DeVore (39) characterized blocking as a metastable state of the atmosphere arising from the resonance between topographic forcing and the Rossby waves, but they were unable to identify the mechanism of transition into this state. In contrast to their theory, Eq. 4 has only one stable equilibrium. Such equilibrium could reach capacity and form a perpetual block (shock) if the forcing Ŝ were excessive. However, in the experiment described above (and in the real atmosphere), steady state is subcritical (nonblocking) (Fig. 3A). When a localized, transient forcing (e.g., explosive cyclogenesis) is applied, it causes the downstream flow to reach capacity for a limited time, giving rise to a blocking episode.

Climate change potentially affects blocking frequency by altering the jet stream’s proximity to capacity (Γ). Though much of the ongoing debate concerns the role of reduced meridional temperature gradients associated with polar amplification (21), Eq. 5 suggests that many factors affect Γ: a shift in the stationary waves, reference-state jet speed, transient-wave LWA flux, or any combination of these. As the response of Γ to climate change is likely nonuniform, it can affect blocking frequency heterogeneously. For example, the vast majority of the 24 Atlantic events in table S1 occurred after 1997, whereas the opposite is true for the 22 Pacific events in table S2. Linear trends in the DJF seasonal mean of column LWA north of 40°N show that over the years LWA has decreased in the Pacific, with generally opposite trends in the zonal wind (fig. S4). At 147°E, 42°N (location B), we find a statistically significant shift in the distributions of LWA and zonal wind to corroborate this trend (fig. S5). The increased eastward wind may have rendered the jet stream less saturated and contributed to the decreased blocking frequency in table S2.

Supplementary Materials

www.sciencemag.org/content/361/6397/42/suppl/DC1

Materials and Methods

Figs. S1 to S5

Tables S1 to S4

References (4043)

Movie S1

References and Notes

Acknowledgments: Funding: This work has been supported by NSF grant AGS1563307. Author contributions: N.N. laid out the conceptual framework for the finite-amplitude wave activity diagnostic and identified the connection between atmospheric blocking and the traffic flow problem. He formulated and conducted the 1D experiment and also produced all but two figures. C.S.Y.H. worked out the LWA budget as part of her Ph.D. dissertation study and performed most of the data analysis described in this article. She also produced figs. S4 and S5. Competing interests: The authors do not have any competing interests. Data and materials availability: To evaluate the terms in Eqs. 1 to 3 and to create Figs. 1 to 5; figs. S1, S2, S4, and S5; and tables S1 to S3, we used the European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-Interim reanalysis product (8). Six-hourly global pressure–level analysis for u, v (meridional wind speed), temperature, and geopotential was obtained from http://apps.ecmwf.int/datasets/data/interim-full-daily/levtype=pl/ with 1.5° by 1.5° horizontal grid resolution. We have also used the daily North Atlantic oscillation index downloaded from NOAA’s Climate Prediction Center at http://www.cpc.ncep.noaa.gov/products/precip/CWlink/pna/nao_index.html to compute data in the last column of table S1. The python package that contains the modules to compute LWA and uREF from ERA-Interim reanalysis data can be found at https://github.com/csyhuang/hn2016_falwa. The scripts to produce the analyses in this paper are contained in the folder https://github.com/csyhuang/hn2016_falwa/tree/master/examples/nh2018_science.
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