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I greo y welcome this unique Encyclopedio of six volumes conloining 340confributions, eoch of opproximolely 4000 words, on oll moior ospecls ofolmospheric science ond cognofe subjecfs such os oceonogrophy ond hydrology,

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An imprint of Elsevier Science

STATIONARY WAVES (OROGRAPHIC ANDTHERMALLY FORCED)

S Nigam, University of Maryland, College Park, MD,USA

E DeWeaver, University of Wisconsin, Madison, WI,USA

Copyright 2003 Elsevier Science Ltd. All Rights Reserved.

Introduction

The term stationary waves refers to the zonallyasymmetric features of the time-averaged atmosphericcirculation. They are also referred to as standingeddies, where standing refers to the time averagingover a month to season, and eddy is a generic term forzonally asymmetric patterns. The zonal asymmetriesof the seasonal circulation are particularly interestingbecause they occur despite the longitudinally uniformincidence of solar radiation on our planet. Stationarywavesmust arise, ultimately, due to asymmetries at theEarth’s surface – mountains, continent–ocean con-trasts, and sea surface temperature asymmetries.Understanding precisely how the stationary wavesare generated and maintained is a fundamental prob-lem in climate dynamics.

Stationary waves have a strong effect on the climatethrough their persistent northerly and southerly sur-face winds, which blow cold and warm air. Advectionof moisture by the stationary wave flow contributes tohydroclimate variations over the continents. Beyondtheir direct advective impact, stationarywaves controlthe location of stormtracks – the preferred paths ofsynoptic weather systems in the midlatitudes, and thezone of tropical–extratropical interaction in thesubtropics. Stationary waves are important also onlonger time scales, since interannual climate variabil-ity projects substantially on the zonally asymmetriccomponent of the flow. Finally, stationary wavescontribute significantly to the maintenance of thecomplementary zonally symmetric circulation, inboth climatological and anomalous states; the contri-bution is through quadratic fluxes of meridionalmomentum and heat. Stationary waves are thus afundamental feature of the general circulation of thetroposphere.

Observed Structure

Stationary waves are stronger in the Northern Hem-isphere because of greater orography and continenta-lity.Wave amplitudes in the NorthernHemisphere are

largest during winter, modest during the transitionseasons of spring and autumn, and weakest duringsummer. The Southern Hemisphere stationary wavesand their seasonal variation are substantially smallerin comparison.

Northern Hemisphere Winter Structure

Because of the geostrophic balance condition, station-ary waves in the upper-level flow can be convenientlydisplayed using the height of the 300 hPa pressuresurface. The geostrophic wind blows along the heightcontours,with lower heights to the left in theNorthernHemisphere, and with a speed proportional to thegradient of the height field. The height of the 300 hPasurface varies considerably with latitude and longi-tude (Figure 1A), with the mean height being close to9 km. The polar vortex is clearly recognizable in thisprojection. The vortex is due to insolation andplanetary rotation, both zonally symmetric inputs,but the vortex has notable departures from symmetry:troughs over northern Canada and western Siberia,and ridges over the eastern Atlantic and Pacific. (Thezonally asymmetric component of the field whichhighlights the troughs and ridges is shown later, inFigure 2A.) The regions where the height contours areclose together correspond to strong westerly (comingfrom the west) jets: the Asian–Pacific and NorthAmerican jets.

Stationary waves at the Earth’s surface can beidentified using the sea-level pressure field, which inelevated areas is the surface pressure reduced to sealevel. The lightly shaded regions in Figure 1B aresurface lows, and the dark regions are highs. Lows arefound over both ocean basins, the Aleutian Low in thePacific and the Icelandic Low in the Atlantic. TheAleutian Low is centered off the tip of the AleutianIslands chain, and the counterclockwise flow aroundthe low brings southerly marine air to coastal CanadaandAlaska, lessening the severity of thewinter season.To the south of the Icelandic Low is a high-pressurecenter known as the Azores High. Strong onshoresurface flowoccurs between the Icelandic Low and theAzores High, again lessening the severity of coastalwinters in Europe. Much higher surface pressure canbe found over central Asia in a center called theSiberian High. Between the Siberian High and theAleutian Low is a region of strong northerly flow,which brings down colder air and lowers near-surfacetemperature along the east coast of Asia. The winter

STATIONARY WAVES (OROGRAPHIC AND THERMALLY FORCED) 2121

sea-level pressure field can be broadly characterized asbeing high over the continents and low over thecomparatively warmer northern oceans. Since sea-

level pressure is related to column temperature,vertical coherence of the continent–ocean temperaturecontrast in the lower troposphere is key, as discussedlater.

The stationary wave pattern changes considerablybetween the surface and 300 hPa, and these changesare highlighted in Figure 2. The top panels show eddyheights at the 300 and 850 hPa levels, revealing thetroughs and ridges. These features are displacedwestwardwith increasing height, i.e., westward tilted,assuming that the same features are being tracked atthe two levels. The low-level trough over the Pacific ispositioned 15–201westward of the Aleutian Low, andgives way to a trough centered on the east Asian coastat 300 hPa, which is associated with the Asian–Pacificjet. The 850 hPa trough over the North Atlantic islikewise shifted relative to the Icelandic Low, andmigrates further westward towards Hudson Bay atupper levels; it brings cold Arctic air into the centraland eastern United States and Canada. The low-levelfeature over Eurasia (Figure 2B) is more definitelylinked to the surface Siberian High, but there is nocorresponding feature of significance present at theupper level – in contrast with the vertically coherentstructure of the Azores High.

The vertical structure of stationary waves is plottedin Figures 2C and D, which are cross-sections of theeddy height field at 401 N and 601 N. The shading inthese panels depicts the eddy temperature field. (Inhydrostatic balance, this is the vertical derivative ofthe height field in log(p) coordinates.) These plotsallow for the tracking of features.The northern sectionshows the pronounced westward tilt of the east Asiantrough, the Rocky Mountain ridge, and the AzoresHigh. The tilt is a consequence of meridional temper-ature advection by the associated geostrophic wind,which induces cooling to the west (east) of the low(high). Interestingly, connection with the prominentsurface features is not strong, except in case of thehighs. The Aleutian and Icelandic Lows, in particular,are quite shallow (p0 800 hPa), exhibiting littleconnectivity to the westward displaced upper-leveltroughs. The southern section (401 N) nicelyreveals the limited vertical extent of the SiberianHigh, in contrast with the deep structure of the AzoresHigh.

Comparison of the two cross-sections indicates astriking difference in vertical variation of the eddyheights, particularly in the Eastern Hemisphere. In thenorthern section, the wave amplitude keeps growingwith height up until the tropopause, and even beyond.The structure is indicative of upward propagation ofstationary wave energy into the polar lower strato-sphere. (Note that thewave’s phase is stationary, so thephase velocity is zero, but its group velocity – the

(A)

(B)

9

8.5

9.5

9

H

L

H

L

Figure 1 (A) Average height of the 300hPa pressure surface in

northern winter months (December, January, February, and

March: DJFM). The average is over 20 winter seasons (December

1979 through March 1999), and is computed from the reanalysis

fields produced by the US National Center for Environmental

Prediction (NCEP). The contour interval is 100m. (B) Average sea-

level pressure (SLP) for the samemonths and years,with a contour

interval of 5.0 hPa. Sea-level pressure data come fromTrenberth’s

analysis, which is archived at NCAR. Dark (light) shading repre-

sents values above 1015hPa (below 1010hPa). The letters ‘L’ and

‘H’ designate the prominent centers of action: the Aleutian Low,

Siberian High, Icelandic Low, and the Azores High. Map domain

begins at 201 N.

2122 STATIONARY WAVES (OROGRAPHIC AND THERMALLY FORCED)

velocity of energy propagation – is not zero.)In contrast, the 401 N structure is indicativeof trapping of wave energy within the troposphere.The eddy height at the 10 hPa level, displayedusing shaded contours in Figure 3, reveals the presenceof a large-amplitude stationary wave at an altitudeof nearly 30 km. The zonal wavelength of thispattern is evidently close to the circumference ofthe latitude circle, i.e., the largest possible. Bothobservations and theory (see Rossby Waves) suggestthat disturbances of such large wavelengths canpropagate into the stratosphere. The wave pattern inFigure 3 moves the center of the polar vortex awayfrom the geographical pole and reduces the strength ofthe vortex.

Equatorial westerly duct An important circulationfeature in the deep tropics during northern winter isthe presence of strong upper-level westerlies(B10m s�1) over the Pacific and Atlantic longitudes.This is notable because the equatorial belt isotherwise occupied by easterly winds. Zonal windsat 200 hPa are shown in Figure 4A, with theeasterly region shaded. A vertical section at theequator (Figure 4B) shows westerly zones to beconfined to the near-tropopause region (100–300 hPa), with maximum values (B15m s� 1) at200 hPa. The origin of equatorial westerly zones isnot well understood, but their absence in northernsummers and El Nino winters suggests thattheir occurrence is linked to the absence of

(A) (B)

(C)

(D)

200

400

600

8001000

200

400

600

8001000

0° 60° E 60° W120° E 120° W180° 0°AH

40° N

60° N

ILSH AL

00

0

0

0

0

0

Figure2 (A)Eddyheight at 300hPaduringnorthernwinter (DJFM), or height of the300 hPapressure surfaceafter subtracting the zonal

average. The contour interval is 50m, with dark (light) shading for positive (negative) values in excess of 50m. (B) Eddy height at 850 hPa

for the same period, with a contour interval of 25m and dark (light) shading for positive (negative) values in excess of 25m. The thick

contours inBenclose regionswhere the surfacepressure is less than850hPa. In these regions, thepressure surface is interpolatedbelow

ground. (CandD) 1000–100 hPazonal–vertical cross-sectionsof eddyheight and temperatureat (C) 601Nand (D) 401N.Contour intervalfor eddy height is 50m, and dashed contours represent negative values. Eddy temperature is plotted in 3K contours with dark (light)

shading for positive (negative) values in excess of 3K, and zero contours suppressed. ‘SH’, ‘AL’, ‘IL’, and ‘AH’ are the surface lows and

highs of Figure 1B.


strong convection in the central equatorial Pacific andAtlantic longitudes.

Rossby wave propagation theory (see RossbyWaves) suggests that tropical easterlies are an effective

dynamical barrier, shielding the equatorial zonefrom the influence of midlatitude perturbations.Openings in this barrier, or westerly ducts, thusprovide a conduit for equatorward penetration ofmidlatitude waves during northern winter – a timelyopening, since the midlatitude stationary andtransient wave activity is most vigorous in winter.Interaction between midlatitudes and the equatorialzone can impact convection and water vapor distri-bution in the tropics and subtropics. Lateral mixingfrom extratropical intrusions can also influence tracertransports.

Northern Hemisphere Summer Structure

The northern polar vortex is much weaker in summerthan in winter. The summer vortex is shown at the150 hPa level in Figure 5A, and is evidently quitesymmetric. It also lacks the tight meridional gradientsthat characterized the winter vortex. A somewhathigher level was chosen for displaying the summerpattern in order to capture fully the divergent mon-soonal flow and accompanying rotational circulationsover the warmer landmasses. The upper-level asym-metries include the very prominent anticyclone overTibet, and troughs over the subtropical ocean basinswhich are easier to appreciate in the eddy height plots,shown later.

31

30

Figure 3 Average height of the 10hPa pressure surface in

northern winter (DJFM). Thick contours show the height field in

500m increments. The eddy height is plotted at 100m intervals,

with dark (light) shading for positive (negative) values in excess of

100m. The zero contour for eddy height is suppressed.

(A)

(B)

45° N

45° S

0°

200

400

600800

10000° 90° E 90° W180° 0°

−5

0

5

−10

10

10

0

Figure 4 (A) The 200hPa zonal wind in northern winter (DJFM), contoured in 10ms� 1 intervals. Regions where the zonal wind blows

from the east are shaded. (B) Zonal–vertical cross-section of zonal wind at theEquator, contoured in 5ms� 1 increments, with shading for

easterly regions. Top level in (B) is 50 hPA.


The summertime sea-level pressure (Figure 5B) hasalmost a reversed winter structure. Two subtropicalanticyclones of comparable strength are present in theocean basins, underneath the upper-level troughs.They are referred to as the Pacific High and theBermuda High. The Bermuda High is the summerequivalent of the Azores High, which expands whilethe Icelandic Low retreats northward during thetransition from winter to summer. The Pacificsector undergoes a similar winter to summer transi-tion. The subtropical anticyclones constitutethe descending branch of the regional Hadleycells which are driven by deep convection in thetropics. Descending motions induced to the north-west of subtropical monsoonal heating may alsocontribute to anticyclone development.

Over the continents, sea-level pressure is low duringsummer. A large region of low sea-level pressure ispresent over Asia beneath the Tibetan anticyclone(which is actually centered over northern India). Thecontinental-scale anticyclone is an integral element ofthe Asian monsoon circulation, being the rotationalresponse to deep heating.

The cross-section of eddy height at 301N (Figure5C) shows the internal baroclinic structure that istypically produced by deep heating in the tropics. The

Tibetan anticyclone reaches maximum amplitude at150 hPa, the level displayed in Figure 5A. Over theoceans, the structure is also baroclinic, but the Pacificand Bermuda Highs are evidently shallow features –although not as shallow as theirwinter counterparts inFigure 2C. The strong positive temperature centeredover the Tibetan plateau is caused by latent heatrelease in the monsoon rains. On the other hand,negative temperatures over the Pacific and BermudaHighs are produced, in part, from the long-waveradiative cooling to space.

The eddy height fields during summer are displayedin Figure 6. The Tibetan anticyclone is the prominentfeature at upper levels. Baroclinic structure is evidentin the NorthernHemisphere, with upper-level troughspositioned over the subtropical highs. Also evident atthe upper level is a weak ridge over North Americathat is associated with the local monsoon system,which includes the Mexican monsoon. The westernedge of the Bermuda High produces low-level south-erly flow, which brings in significant amounts ofmoisture from the Gulf of Mexico into the US GreatPlains. A notable low-level feature in the SouthernHemisphere is the Mascarene High centered south ofMadagascar, which generates strong easterlies alongits northern flank (recall that the flow around a

200

400

600800

10000° 60° E 60° W120° E

30° N

120° W180° 0°BHPH(C)

TA

(B)(A)

14.3

14

13.7

14

H H

L

0

0

0

00

Figure5 (A) Average height of the 150hPa pressure surface in northern summermonths (June, July, and August: JJA). The average is

over 20 summer seasons (June 1980 through August 1999), and is computed from NCEP reanalysis. The contour interval is 100m. (B)

Average sea-level pressure for the samemonths, plotted as in Figure 1. The letters ‘L’ and ‘H’ designate the prominent centers of action:

theBermudaHigh, thePacificHigh, and thebroad regionof lowpressureassociatedwith theAsianmonsoon.Mapdomainbeginsat 151N.(C) The 1000–100hPa zonal–vertical cross-section of eddy height and temperature at 301N, with contours and shading as in Figure 2D.

‘TA’ gives the location of the Tibetan anticyclone, which is enclosed by the 14.3 km contour near the top of panel (A).


Southern Hemisphere High is counterclockwise).After turning northward along the African coast andcrossing the Equator, this flow evolves into the south-westerly monsoon flow over the Arabian Sea.

In the Asian monsoon circulation, equatorial andcross-equatorial flows play an important role, andthese cannot be appreciated in the height field, sincethe geostrophic relationship breaks down at theEquator. The summer circulation figures are thuscomplemented with a vector-wind plot at 850 hPa(Figure 6C); only the zonally asymmetric componentsof winds are plotted. Strong cross-equatorial flowoccurs along the east coast of Africa, bringing mois-ture to the Asian continent. Easterly flow is found allalong the Equator, particularly along the southernflank of the Pacific and Bermuda Highs.

Southern Hemisphere Stationary Waves

The Southern Hemisphere has much less land thanthe Northern Hemisphere, resulting in weaker as-ymmetries at its lower boundary. A more zonallysymmetric circulation, with smaller-amplitude sta-tionary waves, is thus expected. Due to the largerfraction of ocean, the seasonal cycle will also bemuted. The seasonal change in surface temperature,for example, will be smaller than in the NorthernHemisphere.

The southern vortex is shownduring theDecember–March (southern summer) and June–August (southernwinter) periods in Figure 7. As before, the wintervortex is shown at 300 hPa and the summer one at thehigher 150 hPa level. Thick lines mark the height

(A)

90° N

45° N

45° S

0°

(B)

90° N

45° N

45° S

0°

(C)

90° N

45° N

45° S90° E 90° W 0°180°

0°

0°

Figure 6 Eddy height at (A) 150 hPa and (B) 850hPa during northern summer (JJA), with a contour interval of 25m and dark (light)

shading for positive (negative) values in excess of 25m. (C) Eddy wind vectors at 850hPa. Regions where the eddy wind speed is in

excess of 5ms�1 are shaded, and the longest arrow represents a wind speed of 18ms� 1. Eddy winds with speeds below 2ms� 1 are

suppressed. As in (B), the thick closed contours in (C) surround mountainous regions where the surface pressure is less than 850hPa.


contours while the shaded region shows the corre-sponding eddy height patterns. Note that in theSouthern Hemisphere the flow around a low isclockwise rather than counterclockwise. The southernvortex is considerably more symmetric than thenorthern one. Eddy heights are thus smaller, andcontoured at 25m in both summer and winter (Figure7). The summer and winter patterns are both domi-nated by the wave number 1 component in the highlatitudes so that opposite points along a latitude circlehave opposite polarities. The wave component exhib-its similar phase and amplitude structure in the twoseasons, indicating a significant role of Antarcticorography in its forcing. The subtropics shows greaterseasonality, with a ridge over northern Australia insummer; this upper-level feature is linked to theAustralian monsoon outflow.

The extent of zonal asymmetries at the surface isexamined using sea-level pressure which is contouredwith a 2.5 hPa interval as opposed to 5.0 hPa in theNorthern Hemisphere. The summer distribution(Figure 7C) is much like the one in the NorthernHemisphere (Figure 5B), with high-pressure cells

occupying the midlatitude ocean basins. In summer,the subtropical highs are interrupted by continentalheat lows, caused by the warmer land temperatures.The winter sea-level pressure (Figure 7D) is morezonally symmetric, unlike the Northern Hemispherewhere asymmetries are most pronounced duringwinter (cf. Figures 1 and 2). A prominent feature ofthe southern winter pattern is the Mascarene Highextending from Africa to Australia, which generatesstrong south-easterly flow along its northern flank. Itslinkage with south-westerly flow over the northernIndianOcean andAsian summermonsoon can be seenin Figure 6C.

The vertical structure of Southern Hemispherestationary waves is shown in Figure 8 at 301 S insummer and 601 S in winter – the latitude of thesubtropical highs and the polar wavenumber 1 pat-tern, respectively. Contour intervals in Figure 8A are10m for height and 1.5K for temperature, as opposedto 50m and 3.0K in northern summer (Figure 5C). Asin the Northern Hemisphere, the subtropical highshave a baroclinic structure with upper-level troughssuperimposed on surface highs. The heat lows over

(A) (B)

(C) (D)

14

13

9

8.2

Figure7 (A)Height of the150 hPapressuresurface in theSouthernHemisphereduringDJFMmonths (southernsummer). (B)Height of

the 300hPa surface during JJA months (southern winter). Thick solid contours show the total height field in 200m increments, while thin

contours represent the eddy height. The contour interval for eddy height is 25m, with dark (light) shading for positive (negative) values in

excess of 25m. The zero contour for eddy height is suppressed. (C, D) Sea-level pressure for (C) DJFM and (D) JJA months, in 2.5 hPa

increments, with dark (light) shading for values above 1015hPa (below 1012.5 hPa). Map domain is from the Equator to the South Pole.

Sea-level pressure values over Antarctica are unreliable and hence suppressed.


Australia and southern Africa are quite shallow andintense: this is typical of arid regions where rainfalland mid-tropospheric latent heating do not occur inresponse to the surface heat low.Thewinter height andtemperature structures (Figure 8B) are plotted using25mand 1.5K intervals, as opposed to 50mand3.0Kinnorthernwinter (Figures 2C,D), due to their relativeweakness. The southern winter pattern evidentlychanges littlewith height. There ismuch lesswestwardtilt in comparison with the northern winter structure(Figure 2C), indicating less upward propagation ofwave energy. Although westerlies are necessary forupward propagation, theoretical considerations sug-gest that propagation is hindered by the presence ofexcessive westerlies (westerlies exceeding the Rossbycritical velocity), the southern winter vortex issubstantially stronger than its northern counterpart(cf. Figures 1A and 7B; note the larger contouringinterval in the latter figure).

Transience in the Atmosphere

The above review of stationary wave structure doesnot convey the extent to which these waves arerepresentative of the instantaneous circulation. Forexample, how stationary (or transient) is the upper-level circulation during northern winter? Can thestationarywaves be ‘seen’ on synopticweather charts?The degree to which these charts depart from theclimatological pattern is a measure of the strength of

the transient flow. Transient activity is estimated innorthernwinter in Figure 9 because it is expected to bestrongest in this season. The greater vertical shear ofthe thermally balanced Asian–Pacific and Atlantic jetsin winter makes them prone to hydrodynamic insta-bility, which in the context of geostrophic flows iscalled baroclinic instability. Baroclinic instabilityproduces transient disturbances on subweekly timescales.

The extent to which the stationary wave structure isrepresentative of instantaneous flow is depicted inFigure 9 by projecting the daily, instantaneous(00UTC), 300 hPa circulation on the climatologicalwave pattern (Figure 2A) during the winters of 1980/81 and 1989/90. Correlation – a measure of thestructural similarity of the two maps (without regardto amplitude) – is plotted on the y-axis. The correla-tion ranges from 0.2 to 0.8 in these winters, indicatingthat the climatological pattern accounts for up to 65%of the spatial variance. High correlation is howeverachieved only on a few days in each winter. Moretypically, the correlation is between 0.5 and 0.6.Interestingly, the correlation drops and recovers over a2–3 week period, 1–2 times each winter, revealing theestablishment time scale of the climatological pattern.Dynamical analysis of such episodes, especially of therecovery phase, can shed light on the establishmentmechanisms of stationary waves.

The question of whether the climatological wavepattern can be ‘seen’ on synoptic charts is addressed inFigures 9B and C, which show the instantaneous

0°60° W

60° SJJA

30° SDJFM

120° W180°120° E60° E0°

(A)

200

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1000

200

400

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1000(B)

0

0

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00

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0Figure 8 Zonal–vertical cross-sections of eddy height and temperature in the Southern Hemisphere at (A) 301S in DJFM and (B) 601Sin JJAmonths. In (A), the contour interval for eddy height is 10m, with dashed contours for negative values. The contour interval for eddy

temperature in (A) is 1.5K, with dark (light) shading for positive (negative) values in excess of 1.5K, and zero contours suppressed.

In (B), contour intervals for eddy height and temperature are 25m and 1.5K, respectively, with plotting conventions as in panel (A).


(00UTC) wave pattern on two days: 1 February 1981,when the spatial correlation is high (0.74; Figure 9B),and 16 February 1990, when the correlation is low(0.29; Figure 9C). The climatological pattern (Figure2A) can be clearly recognized in the former plot, butnot in the latter. Even when structurally similar, thepatterns can evidently have very different waveamplitudes; the contour interval is 50m in Figure 2Abut 100m in Figure 9B.

Forcing of Stationary Waves

Stationary waves are generated, ultimately, by thezonal asymmetries at the Earth’s surface: orography,continent–ocean contrasts, and sea surface tempera-ture gradients. Through dynamic and thermodynamicinteractionswith the zonal-mean flow, and subsequentmutual interactions, surface inhomogeneities producezonally asymmetric circulation and precipitation fea-tures at upper levels. Comprehensive numerical mod-els of the atmosphere,which include coupling betweenphysical and dynamical processes, are able to realis-ticallymodel the observed stationarywaves. In a sense,the often posed question – on relative contribution oforography and other processes in forcing of stationarywaves – has been addressed by such prognostic generalcirculation models (GCMs). Comparison of GCM

simulations obtained with and without orographyprovide insight. In these assessments, the change in theheating distribution is attributed to orographic forc-ing, whose circulation impact is found to be compa-rable to that of all other processes put together.

Historically, answers to the above question weresought in a frameworkwhere ‘orographic forcing’ wasused more restrictively – to refer to the dynamicalforcing of flow from mechanical diversion. In suchanalysis, the entire heating distribution, regardless ofits origin (e.g., from condensation in adiabaticallycooled upslope flow), was regarded as an independentforcing. This framework was adopted, perhaps, be-cause mechanical diversion of flow by an orographicbarrier is conceptually easier to model. Such studieslead to the rapid advancement of stationary wavetheory, including construction of potential vorticityconserving models for the responses to orography,meridional and vertical wave propagation analy-sis, and understanding of troposphere–stratosphereinteraction.

Theoretical Considerations

Large-scale atmospheric motions in the extratropicsare approximately hydrostatic and quasi-geostrophic(QG) in character. The hydrostatic approxima-tion recognizes the operative balance between the

0.8

0.6

0.4

0.2

1 DDec 16 Dec 1 Jan 16 Jan 1 Feb 16 Feb 1 Mar 16 Mar(A)

1980/811980/81 1989/901989/90

((B) (C)

Figure 9 (A) Spatial correlation between the instantaneous (00UTC) and climatological 300hPa eddy heights during 1980/81 (solid

curve) and 1989/90 (dashed curve) winters. Correlations are area weighted and include height data from 201 N to the north pole.

(B) 300 hPa eddy height on 1 February 1981, when the spatial correlation was 0.74. (C) Eddy height on 16 February 1990, when the

correlation was 0.29. In (B) and (C), the contour interval is 100m (twice the interval in Figure 2A), with dark (light) shading for positive

(negative) values in excess of 100m.


horizontally varying pressure and density perturbat-ions, while the QG approximation acknowledges thenear-balance between the Coriolis force and thehorizontal pressure gradient. QG flow is thus domi-nated by the rotational component. Its evolution,however, is determined, in part, by the comparativelyweaker divergent flow component, as described by thevorticity equation

qzqt

þ Vh � =zþ bv ¼ �ð f þ zÞð= � VhÞ ½1�

Here, z is the QG relative vorticity ð¼ kk�ð=�VhÞ ¼=2cÞ, c is the geostrophic streamfunction, f is theCoriolis parameter with qf=qy ¼ b, and Vh is thehorizontal QG flow. The right-hand term is theproduct of absolute vorticity ðf þ zÞ and horizontalconvergence, and is often called the ‘stretching term’because convergent flow leads to stretching of vortextubes.

Due to compressibility of air, evolution of thethermodynamic state is conveniently described usingpotential temperature, y ¼ Tðp0=pÞR=Cp , which isconserved in adiabatic motion; p0 is the referencepressure (1000 hPa). Potential temperature thuschanges only in response to diabatic heating

.Q, as

follows:

qyqt

þ Vh � =yþwqyqz

¼.Q

Cpðy0=T0Þ ½2�

where y0 ¼ T0ðp0=pÞR=Cp , with T0 ¼ Tðp0Þ.Since stationary waves refer to the zonally varying

component of the flow (here onward denoted byprime), their dynamics can be described, to first order,by linearizing eqns [1] and [2] about the zonal-meancirculation, �UUðy; zÞ and �yyðy; zÞ. The linearized equa-tions are valid for small-amplitude perturbations:

z0t þ �UUz

0x þ v0ðb� �UUyyÞ � �f ð= � V 0

hÞ ½3�

y0t þ �UUy

0x þ v0�yyy þw0�yyz � ð .Q0y0=T0cpÞ ½4�

For convenience, subscripts are used to denote thepartial derivatives.

Orographic Forcing and Response

The forcing and propagation of stationary waves canbe discussed using eqns [3] and [4]. In contrast withdiabatic heating, which is explicitly present as right-hand forcing in the thermodynamic equation, themechanical forcing by orography ðh0Þ is implicitlypresent through its kinematic impact on verticalvelocity at the lower boundary ðwsÞ. In the presenceof the zonal-mean circulation, the linearized verticalvelocity, w

0s, equals

�UUh0x.

In simplified treatments of orographic intera-ction, the geophysical fluid is additionally consideredto be homogeneous and incompressible ð= � Vh ¼�qw=qzÞ, so that response is determinable using thevorticity equation alone – this ‘shallow water’ ap-proximation is indeed reasonable for the interaction ofoceanic flows with underwater topography, but some-what limited in capturing aspects of the atmosphericinteraction. In shallow water theory, density (ortemperature) is constant, and the horizontal flow,including horizontal divergence, is height independ-ent. Assuming that a rigid lid is placed at the top of thefluid ðz ¼ HÞ, so that vertical velocity vanishes there,one obtains qw0=qz � �ð �UUh

0xÞ=H. The forced waves

are then modeled by eqn [5], where �UU has beenadditionally assumed to be latitude independent, andperturbation vorticity is dissipated (e.g., by Ekmanspin-down) on an e�1 time scale:

½q=qt þ �UU q=qxþ e�ðc 0xx þ c

0yyÞ

þ bc0x � �ð f=HÞ �UUh

0x ½5�

To understand the forced response, consider anarbitrary Fourier component of the geostrophicstreamfunction: c0ðx; y; tÞ ¼ Realfcck; le

iðkxþly�orÞg,where the hat denotes the complex amplitude corre-sponding to zonal and meridional wavenumbers, kand l, and associated frequency o. For such a pertur-bation, eqn [5] yields the solution

cc ¼ f hh

H½ðk2 þ l2Þð �UU � cÞ= �UU � b= �UU � ieðk2 þ l2Þ=ð�UUkÞ�½6�

For stationary waves ðo ¼ 0Þ, the zonal phase speed,cð¼ o=kÞ, vanishes. This simplifies the first term in thedenominator to ðk2 þ l2Þ. In the presence of dissipa-tion, the orography and streamfunction are not inphase, since the denominator in eqn [6] is complex.The possibility of resonance is also indicated in theinviscid case ðe ¼ 0Þ, when ðk2 þ l2Þ ¼ b=�UU. Dissipa-tion however limits the wave amplitude at resonance,with cc / ihh. The streamfunction and orography are901 phase-shifted (or in quadrature) in this case, withthe trough in the flow being a quarter wavelengthdownstreamof themountain ridge.When forcing is onlarger scales, ðk2 þ l2Þob= �UU, planetary vorticity ad-vection dominates zonal advection of relative vorticityin balancing the orographically induced vorticity onupslopes and downslopes. Both the real and imaginaryparts of the denominator in eqn [6] are negative in thiscase, which puts the trough within a quarter wave-length downstream of the ridge.


It is interesting that troughs in the observed 300 hPastationary wave pattern (cf. Figure 2A) are alsodownstream of the orographic features, but the extentto which these are forced by local orographyremains somewhat uncertain, as discussed later.Also, the assumed Fourier representation of thestreamfunction implies the presence of meridionalboundaries which confine wave energy to a midlati-tude channel – a set-up conducive for resonance. Thesinusoidal zonal structure is also unrealistic, sinceorographic features are generally localized. In nature,the wave energy propagates away zonally, meridion-ally, and vertically from the localized forcingregion, thus calling into question the validity of thesolution [6].

The pedagogically useful shallow-water modelof orographic interaction is limited for other reasonsas well. The tropospheric flow cannot be assumed tobe homogeneous and incompressible, since cooling(heating) from adiabatic expansion (compression)during ascent (descent) is important in the thermody-namic budget. Moreover, a flow configuration whichsatisfies the vorticity eqn [3] can be unbalanced fromthe viewpoint of this budget. For example, whenplanetary vorticity advection dominates the left-hand side in balancing the upslope divergent flow ineqn [3], the thermodynamic budget [4] is unbalancedin the absence of condensation heating (precipitation),since both adiabatic ascent and equatorwardflow lead to colder temperatures. The generation oforographic response must thus be understood byconsidering the vorticity and thermodynamic equa-tions together, so that any implications that one mayhave for the other are fully accounted for. Suchconsiderations lead to the development of the QGpotential vorticity equation.

QG Potential Vorticity Equation

The prediction equation for QG flow that doesnot explicitly reference the divergent flowcomponent is called the QG potential vorticityequation. Although it can be derived quitegenerally, the focus here is on its simplified linearizedversion, when the zonal-mean flow �UU isindependent of latitude. The equation is derivedby eliminating the divergent flow from eqns [3]and [4]. In the zn ¼ ðRT0=gÞ lnðp0=pÞ co-ordinate, which reduces to geometric height in anisothermal atmosphere, the QG potential vorticityequation is

q0t þ �UUq0

x þ v0�qqy ¼ R

Hcpr0

qqzn

r0.Q0

N2

� �½7�

where

q0ðx; y; zÞ ¼ c0xx þ c0

yy þf 2

N2r0

qqzn

r0qc0

qzn

� �

and

�qqyðzÞ ¼ b� f 2

N2r0

qqzn

r0q�UUqzn

� �½8�

Since divergent flow is not referenced by this equation,it is of some interest to examine the manifestation oforographic forcing in this analysis framework. Notsurprisingly, this forcing enters as a lower boundarycondition, but in the thermodynamic equation [4].This is because of the direct reference to verticalvelocity in eqn [4], in contrast with the vorticityequation [3] which refers only to its vertical gradient.With w

0s ¼ �UUh

0x, the boundary condition conveying

orographic forcing is

y0t þ �UUy

0x þ v0�yyy

¼ � �UUh0x�yyz þ ð .Q0y0=T0cpÞ at the surface ½9�

Assume for purposes of this discussion that diabaticheating vanishes at the surface, so that only adiabaticcooling (warming) is occurring on the upslope (down-slope). In steady flows, the heating can be balanced byzonal eddy advection and/or meridional advection ofmean temperature. If upslope cooling is compensatedby the latter, the upslope flow will be poleward, and ahigh-pressure center will be positioned over themountain ridge near the surface. The response atupper levels depends upon the zonal scale of moun-tains: large wavelengths will propagate into thelower stratosphere, and phase lines will tilt westwardwith increasing height, all as depicted in theFigure 10A schematic. It is interesting that althoughthermal advection and vertical wave propagationare absent in the shallow water model, thehorizontal structure of the long-wavelength solution(in the presence of damping, eqn [6]) is nottoo different from that indicated at upper levels inFigure 10A.

Heating Response

The stationary wave response to heating can be qual-itatively understood from the thermodynamic equation[4]. In the deep tropics, horizontal variations of geopo-tential (and temperature) are much smaller since it isdifficult to maintain them in the presence of the weakCoriolis force. Consequently, horizontal temperatureadvection is ineffective in balancing diabatic heating ineqn [4].Away fromthe surface, heating is thusbalanced,almost entirely, by adiabatic cooling, with the vertical


profile of w0 closely following that of heating. Asubstantial portion of heating in the tropics resultsfrom deep convection, which produces strongest heat-ing in the mid-to-upper troposphere, as shown later inFigure 11C. Such heating distribution leads to conver-gence (divergence) in the lower (upper) troposphere,which results in vortex stretching (squashing). Therotational response to the induced vorticity depends onthe horizontal forcing scale: if the scale is large, thestretching is offset by poleward advection of planetaryvorticity, which is tantamount to the surface low beingpositionedwestwardof theheat source,as schematicallyillustrated in Figure 10B.

In the midlatitudes, heating does not extend asdeeply into the troposphere as in the tropics. Heatingin the Pacific and Atlantic stormtracks, for example, isconfined mostly to the lower troposphere, as shownlater in Figure 11B. Midlatitude heating is offset to alarge extent by horizontal temperature advection;larger temperature gradients are sustainable in mid-latitudes due to the greater Coriolis force. Large-scaleheating in midlatitudes is balanced, mostly, by coldadvection from the north; the near-surface low isthus positioned eastward of the heating. Interestingly,vertical motion in the vicinity of midlatitude heatingis determined by vorticity balance considerations –a complete reversal of the tropical situation:cold advection from the north brings with it highervorticity air as well, and this induced vorticityadvection must be offset if a steady state is to bemaintained. The compensation is accomplished byvortex compression, which has implications for thetemperature field.

Wave Propagation

The qualitative arguments discussed above arehelpful in understanding the nature of response inthe forcing region. The stationary wave responseis however not confined to the forcing region alone,since Rossby waves propagate zonally, meridionally,and vertically, carrying the disturbance (energy)into the far field (unforced region). The energypropagation, or group velocity, characteristics dependboth on the perturbation scale and structure of thebasic state. Some zonal-mean zonal wind configura-tions encourage Rossby wave propagation, whileothers impede it. Basic state flow can thus profoundlyimpact wave propagation into the tropics and thestratosphere.

Theoretical analysis helps to focus on the basic stateattributes that are influential, e.g., the direction andcurvature of the zonal-mean zonal wind. A usefulquantity in wave propagation analysis is the refractiveindex which seizes on these and other relevantattributes. A display of refractive index variations isoften helpful, since it conveys, to first order, the wavepropagation pathways, as waves are generally refract-ed towards higher refractive index regions. Suchanalysis suggests that midlatitude stationary wavesare refracted towards the Equator, drawn there by thelarge index values resulting from diminishing westerlywinds. The tropical easterlies, in contrast, present aneffective dynamical barrier to equatorward propaga-tion of midlatitude stationary waves. In the vertical,waves with large horizontal scales alone can propa-gate upward, but only when the upper-level westerliesare not too strong.

Mid-lat

L

Tropics

L

L

H C

W

(A)

(B)

(C)

Figure10 Schematicdepictionof the longitude–height response

forced by (A) westerly flow over midlatitude orography, (B) tropical

heating, and (C) midlatitude heating, all taken from Hoskins and

Karoly (1981). The orographic response is shown for the long-

wavelength case, and is determined from both dynamic and

thermodynamic (i.e., quasi-geostrophic potential vorticity) consid-

erations. The arrows depict vertical motion, and circled crosses

and dots denote poleward and equatorward flow, respectively. H

and L denote the pressure ridge and trough, with the lines showing

the vertical tilt of the pressure wave. W and C are the warmest and

coldest air, respectively.


Diabatic Heating in Northern Winter

Diabatic heating plays a prominent role in the forcingof stationary waves. In stationary wave theory, it is anexplicit forcing in the QG potential vorticity equation[7], and even orographic forcing in this theoreticalframework manifests as surface heating [9].

In nature, heating resulting from the change ofphase of water substance, turbulent eddy diffusion,and short-wave and long-wave radiative fluxes isreferred to as diabatic heating. (Note that the temper-ature of air parcels can change even without anydiabatic heating, from adiabatic compression orexpansion.) In contrast with Earth’s orography, whosehighly accurate measurements are widely known, thethree-dimensional structure of diabatic heating is onlybeginning to be described. The main reason why theheating distribution has remained uncertain is that,unlike other quantities, heating is not directly meas-ured. It is, instead, estimated, usually as a residual inthe thermodynamic budget. Since the heating estimate

is only as good as the quality of atmospheric data fromwhich it is diagnosed, the quality of atmosphericanalysis is critical for the diagnosis. Fortunately, datacoverage and quality and analysis methods have allimproved in the last two decades, and are reflected inthe modern reanalysis data sets. Heating diagnosisfrom one such data set, the US National Centers forEnvironmental Prediction (NCEP) reanalysis, isshown in Figure 11.

The mass-weighted vertical average of diabaticheating,

1

ðps � 100ÞZ ps

100

.Qc�1

p dp

is shown during northern winter in units of K day�1;here, ps is the surface pressure, cp is the specific heat ofair at constant pressure, and the integration is from thesurface to 100 hPa. Key features in Figure 10A includethe heating centers in the extratropical Pacific andAtlantic basins, which effectively define the two

60° N

30° N

60° S

30° S

0°

180°60° E 120° E0°

200

400

600

800

1000

200

400

600

800

1000120° W 60° W

5° N

0°

37.5° N

b

c

(C)

(B)

(A)

Figure 11 (A) Mass-weighted vertical average of diabatic heating, calculated as a residual from the thermodynamic equation. The

winter season (DJFM)diagnosis is obtained fromNCEP reanalysis fields for 20winter seasons (1979/80–1998/99). The contour interval is

0.5Kday� 1, with dark (light) shading for positive (negative) values in excess of 0.5Kday� 1, and zero contours suppressed. (B,C) Zonal–

vertical (1000–100hPa) cross-section of diabatic heating at (B) 37.51 N and (C) 51 N, with contours and shading as in panel (A). The

latitudes of the cross-sections in (B) and (C) are marked with thick lines at the edges of panel (A). A 9-point smoother is applied to the

heating field before plotting.


midlatitude stormtracks. The northern continents, incontrast, constitute the cooling regions. In the tropics,heating is strong over the South Pacific convergencezone and the Amazon basin. A narrow zone of heatingis also present in the Pacific just northward of theEquator; this intertropical convergence zone (ITCZ) ismuch stronger during northern summer when it ispositioned a few degrees farther northward and fullyextended across the Pacific basin.

Diabatic heating has a complicated vertical struc-ture which changes with latitude and season. Thechanges with latitude are shown in Figures 10B andC,which depict height–longitude cross-sections throughthe midlatitude stormtracks (37.51 N) and the ITCZ(51 N). The stormtrack heating is evidently strongestnear the surface, with peak values close to 6Kday� 1,and diminishes rapidlywith height. Latent heating dueto precipitation in baroclinically unstable synoptic-scale disturbances is the primary contributor tostormtrack heating. Diabatic cooling, on the otherhand, is comparatively weaker, and focusedmore nearthe tropopause in this estimation, for reasons that arenot clear.

The vertical structure of ITCZ heating is strikinglydifferent. Although the entire column is being heated,heating is generally strongest in the mid-to-uppertroposphere. For example, over the tropical Pacificwarm pool – the site of persistent deep convection –heating is strongest (B3Kday� 1) at 400 hPa. Incontrast, heating over land (e.g., equatorial Africa) isstrongest near the surface due to sensible heating. Theheating structure over Central America is also similar,except that elevated surface heating there has pro-duced some deep convection as well.

Interaction with Transients

The climatological stationary waves coexist withvigorous atmospheric motions occurring on a varietyof time scales (Figure 9), and there are strong interac-tions between these transient motions and the station-ary waves. Transient motions are the instantaneousdepartures of the flow from its climatological state,and the time mean of transient motion thus vanishes,by definition. However, fluxes of heat and vorticity bytransients do not vanish in general. For example, thecontribution of transients to the advection terms ineqn [2] can be written as

V0 0h � =y0 0 þw0 0 qy0 0=qz

� = � ðV 0 0hy

0 0Þ þ qðo0 0y0 0Þ=qp ½10�

where the double prime denotes the transient compo-nent ando is the vertical velocity inp-coordinates. The

right-hand side of eqn [10] is the heat-flux divergencefrom transient motions. In synoptic systems, north-ward (southward) transient motions are typicallyaccompanied by positive (negative) temperature fluc-tuations, so that heat flux diverges to the south of astormtrack and converges to the north. The heat-fluxdivergence acts as heat sources and sinks for the time-mean flow, and the stationary waves respond to thisthermal forcing just as they respond to diabaticheating. Likewise, the convergence of transient vor-ticity flux ð�= � ðV 0 0

h z0 0Þ � qðo0 0z0 0Þ=qpÞ provides

sources and sinks of vorticity for the stationary waves.The net effect of transient thermal and vorticity

fluxes on stationary waves is not easy to characterize.However, it is clear that transient forcing is strongenough in northern winter to exert a powerfulinfluence on stationary waves. The 700 hPa heat-fluxconvergence by perturbations lasting less than 1month is superimposed in Figure 12A on the localwinter eddy temperature pattern. The two fieldsevidently oppose each other. For example, transientheat fluxes diverge from the warmer regions over theAtlantic and the west coast of North America andconverge in the colder regions above north-easternCanada. Thus, throughout most of the northernmidlatitudes, transient thermal fluxes have a dampingeffect on the lower tropospheric stationary eddytemperature pattern. Furthermore, the forcing bytransient thermal fluxes is on the order of 0.5Kday�1,, while the eddy temperatures are about 4K. In theabsence of other processes, it would take little morethan a week for the thermal fluxes to reduce dramat-ically the 700 hPa eddy temperature field. Such areduction also implies a substantial weakening of theupper-level geopotential pattern, since temperature isthe vertical derivative of geopotential in hydrostaticbalance.

While transients are an important influence onstationary waves, the stationary waves can be equallyinfluential for the transients. One way in whichstationary waves organize transients is by creatinglocalized regions of strong cyclogenesis. Synopticsystems tend to develop in regions of strong lowertropospheric temperature gradients, and such regionsare present off the coasts ofAsia andNorthAmerica inFigure 12A. The ability of the local temperaturegradient to enhance the growth of synoptic systemscan be measured by the Eady growth parameter ðeÞ:

e ¼ 0:31j=Tj

jTðT0 q ln y= g qzÞj1=2

Its plot in Figure 12B shows large values in the sameregions where stormtrack heating occurs in Figure11A. Comparison of these panels suggests that


stationary waves play an important role in determi-ning the locations of stormtracks. In addition to thiseffect on growth rate, stationarywaves can also have amechanical effect on stormtracks, by changing the

steering winds that determine storm paths and byexchanging mechanical energy with the storms.

The Eady growth rate is applicable to synoptictransients, which grow by extracting energy from thethermal gradients (or vertical shear) of the climato-logical state. Synoptic systems, in fact, account for lessthan half of the transient forcing in the vorticity andthermodynamic equations for the climatological sta-tionary waves. Furthermore, the slower transients(those with time scales between, say, 10 days and 1month) are quite distinct from synoptic transients.They do not generally travel along concentratedstormtracks, nor do they typically grow by extractingenergy from the climatological temperature gradients.Although these transients are certainly influenced bythe climatological stationary waves, the nature of thisinfluence is rather complex and cannot be easilysummarized.

Modeling of Northern WinterStationary Waves

Modeling of orographically forced stationary wavesdates back to the seminal paper of Charney andEliassen in 1949, in which linear shallowwater theorywas applied to the longitudinal distribution of oro-graphy at 451N.The earlier discussion here, includingthe development of eqn [5] and its solution [6], largelyfollows the analysis reported in that paper. Charneyand Eliassen found the midlatitude mountains to berather influential, accounting for almost all of theobserved signal in their analysis.

Since that time, diabatic heating due to continent–ocean contrasts, and transient fluxes of heat andmomentum have also been advocated as importantmechanisms for the generation of stationary waves. Inthe intervening period, the atmosphere has been moreclosely observed, both spatially and temporally, andthere has been a tremendous increase in computationalpower for modeling studies. A reassessment of therelative roles of various forcing mechanisms is thus inorder. In effect, more complete versions of thedynamical and thermodynamical equations can nowbe solved globally at high resolution, and verifiedagainst the extensive record of upper-air observationsthat have been compiled since.

It is still advantageous of linearize the system ofequations, at least initially, since this allows theinfluence of individual forcing terms to be examinedseparately. Of course, the forcing terms can havestrong mutual interactions. For example, heating cancause eddy flow which then impinges on mountains,and the subsequent orographically induced uplift cangenerate convection and lead to further heating.

(A)

(B)

Figure 12 (A) The 700hPa eddy temperature (thick contours)

andheat-fluxconvergenceby transientmotions (shadingandwhite

contours) in northern winter months (DJFM). The contour interval

for temperature is 2K, and dashed lines represent negative values.

The contour interval for heat-flux convergence is 0.5Kday� 1, with

dark (light) shading for positive (negative) values in excess of

0.5Kday� 1. Zero contours for temperature and heat-flux conver-

gence are suppressed, and regions where the surface pressure is

less than 700hPa are masked out. (B) Eady growth parameter in

northern winter (DJFM), calculated from the 700hPa temperature

gradients. The contour interval is 0.1 day�1, with dark shading for

values in excess of 0.6 day� 1.


However, linear diagnostic models serve a valuablepurpose in assessing the relative importance of thedifferent forcing terms in different regions. They alsoindicate the degree towhich linear perturbation theoryapplies to stationary waves.

A linear simulation of northern winter stationarywaves is presented in Figure 13. The linear model usess ¼ p=ps as the vertical coordinate (here ps is thesurface pressure) so that mountains do not intersectthe model levels. There are 15 levels in the vertical,ranging from 1000 to 25 hPa, and the horizontalresolution is 7.51 in the zonal direction and about 2.51in the meridional direction. As in eqns [3] and [4], themodel is linearized about the zonal-mean climatolog-ical state. The momentum and thermal dissipation inthe model is roughly equivalent to a 5-day dampingtime scale in eqns [3] and [4]. The forcing consists ofthree-dimensional diabatic heating (Figure 11), oro-graphic height, and transient fluxes of heat andmomentum.

The 300 hPa response obtained with all forcings isshown inFigure 13A, and plotted using the conventionused in Figure 2A, with which it should be compared.

The linear model can simulate the ridge over theAtlantic, the low off the east Asian coast, and thetrough over Canada. A notable flaw in the simulationis the weakness of the ridge over the Rockies. Thesolution shows that linear perturbation theory canexplain many, though not all, aspects of the observedstationary wave pattern.

The model response when forced separately bydiabatic heating, mountains, and transients is shownin panels (B)–(D), respectively; note that contourinterval in these three panels is half that in panel (A).All three forcings contribute significantly to the totalpattern. The heating response includes jets in theAsian–Pacific and Atlantic sectors. Heating is evident-ly important in establishing the ridge over the Atlanticand northern Europe, and contributes significantly tothe trough over Canada as well.

The response to mountains in panel (C) showstroughs downstream of the Himalayan–Tibetan com-plex and the Rockies, as suggested by the shallowwater solution [6] and also QG potential vorticityconsiderations (cf. Figure 10A), in each case for longwavelengths. Thus, orography contributes to the

(A) (B)

(C) (D)

Figure13 (A) The300-hPaheight responseof a linear stationarywavemodel forcedby heating,mountains, and transient fluxes of heat

andmomentum. The contour interval is 50m, with dark (light) shading for positive (negative) values in excess of 50m. (B–D)Response of

the model when forced separately by (B) heating, (C) mountains, and (D) transient fluxes. The contour interval in (B–D) is 25m.


forcing of the jets as well. The high amplitudes directlyover Greenland and Tibet are a consequence of thelinearization of the hydrostatic equation in s-coordi-nates. Examination of geopotential heights gives asomewhat misleading impression that waves generat-ed by mountains propagate primarily in the zonaldirection. Examination of the modeled streamfunc-tion (a more suitable variable for describing therotational response in the tropics; not shown), how-ever, reveals considerable equatorward propagationofthe forced waves.

The forcing by submonthly transients (panel D)produces a somewhat intricate pattern with no clearrelationship to the synoptic stormtracks. Transientsare apparently responsible for a large part of theresponse over eastern Atlantic and northern Europe.

Studies of stationary wave dynamics have tradi-tionally focused on the question of the relativeimportance of the various forcing terms in generatingthe observed pattern. Yet recent simulations such asthe one in Figure 13 show clearly that the northernwinter stationary waves do not constitute a simplelinear response to a single form of forcing. Further-more, linearized equations, such as eqns [3] and [4],neglect the advection of eddy heat and vorticity by thestationary waves themselves, and also the effect ofeddy winds impinging on the mountains. These termsplay an important role in generating some features ofthe stationary wave pattern, such as the ridge over theRockies. Future examinations of stationary wave

dynamics will have to assess not only the relativeimportance of various forcing terms but their mutualinteractions, and the nonlinear interactions of thestationary waves themselves.

See also

Climate Variability: Seasonal to Interannual Variability.Coriolis Force. Cyclogenesis. Dynamic Meteorol-ogy: Overview; Waves. Stratosphere–TroposphereExchange: Global Aspects.

Further Reading

Gill AE (1982) Atmosphere–Ocean Dynamics. Orlando:Academic Press.

Grotjahn R (1993) Global Atmospheric Circulations: Ob-servations and Theories. New York: Oxford UniversityPress.

Holton JR (1992) An Introduction to Dynamic Meteorol-ogy. New York: Academic Press.

Hoskins BJ andKaroly DJ (1981) The steady linear responseof a spherical atmosphere to thermal and orographicforcing. Journal of the Atmospheric Sciences 38:1179–1196.

Hoskins BJ and Pearce R (1983) Large-Scale DynamicalProcesses in the Atmosphere. London: Academic Press.

James IN (1994) Introduction to Circulating Atmospheres.Cambridge: Cambridge University Press.

Saltzman B and Manabe S (eds) (1985) Advances inGeophysics, vol. 28, Issues in Atmospheric and OceanicModeling. Orlando: Academic Press.

STRATOSPHERE–TROPOSPHERE EXCHANGE

Contents

Global Aspects

Local Processes

Global Aspects

JRHolton, University ofWashington, Seattle,WA,USA

Copyright 2003 Elsevier Science Ltd. All Rights Reserved.

Introduction

The troposphere and the stratosphere are separated bya boundary called the tropopause, whose altitudevaries from about 16 km in the tropics to about 8 km

near the poles. The troposphere is characterized byrapid vertical transport andmixing caused by weatherdisturbances; the stratosphere is characterized by veryweak vertical transport and mixing. The tropopausethus represents a boundary between the troposphere,where chemical constituents tend to be well mixed,and the stratosphere, where chemical constituentstend to have strong vertical gradients. The two-wayexchange of material that occurs across the tropo-pause is important for determining the climate andchemical composition of the upper troposphereand the lower stratosphere. This cross-tropopause

STRATOSPHERE–TROPOSPHERE EXCHANGE / Global Aspects 2137

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