Piecewise Tendency Diagnosis of Weather Regime...

15 AUGUST 2003 1941E V A N S A N D B L A C K

q 2003 American Meteorological Society

Piecewise Tendency Diagnosis of Weather Regime Transitions

KATHERINE J. EVANS AND ROBERT X. BLACK

School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia

(Manuscript received 19 July 2002, in final form 10 March 2003)

ABSTRACT

Piecewise tendency diagnosis (PTD) is extended and employed to study the dynamics of weather regimetransitions. Originally developed for adiabatic and inviscid quasigeostrophic flow on a beta plane, PTD partitionslocal geopotential tendencies into a linear combination of dynamically meaningful source terms within a potentialvorticity (PV) framework. Here PTD is amended to account for spherical geometry, diabatic heating, andageostrophic processes, and is then used to identify the primary mechanisms responsible for Northern Hemisphereweather regime transitions.

Height tendency patterns obtained by summing the contributions of individual PTD forcing terms correspondvery well to actual height tendencies. Composite PTD analyses reveal that linear PV advections provide thelargest dynamical forcing for the weather regime development over the North Pacific. Specifically, linear bar-oclinic growth provides the primary forcing while barotropic deformation of PV anomalies provides a secondarycontribution. North Atlantic anticyclonic anomalies develop from the combined effects of barotropic deformation,baroclinic growth, and nonlinear eddy feedback. The Atlantic cyclonic cases develop primarily from barotropicdeformation and nonlinear eddy feedback. All four weather regime types decay primarily due to enhanced waveenergy propagation away from the primary circulation anomaly. In some cases, regime decay is aided bydecreasing positive contributions from barotropic deformation as the circulation anomaly attains a deformedhorizontal shape. The current results 1) provide quantitative measures of the primary mechanisms responsiblefor weather regime transition and 2) demonstrate the utility of the extended PTD as a concise diagnostic paradigmfor studying large-scale dynamical processes in the midlatitude troposphere.

1. Introduction

The ability to predict weather events beyond typicalsynoptic timescales is often influenced by the occur-rence of persistent flow anomalies (PFA; Branstator1990), which act to modulate regional patterns of pre-cipitation, synoptic storm activity, and surface temper-atures (Dole 1986; Edmon 1980; Namais 1978; Lau1988). In order to optimize long-term meteorologicalforecasts, it is necessary to obtain a more complete un-derstanding of the time evolution of these large-scaleweather regimes.

Large-scale circulation anomalies in the NorthernHemisphere wintertime circulation have been charac-terized in a number of independent investigations (Wal-lace and Gutzler 1981; Dole and Gordon 1983; Black-mon et al. 1984; Cheng and Wallace 1993). They aredefined as a marked deviation of the local circulationfield from climatological values, reach their maximumamplitude near 300 hPa, and tend to form particularlyover the northern Pacific and Atlantic Oceans (Dole andGordon 1983; Black and Dole 1993). In this study, the

Corresponding author address: Dr. Robert X. Black, School ofEarth and Atmospheric Sciences, Georgia Institute of Technology,Atlanta, GA 30332-0340.E-mail: [email protected]

onset of persistent cyclonic flow anomalies over theNorth Pacific (PCO) is first studied in detail because theresults can be readily contrasted with earlier more syn-optically oriented analyses of Dole and Black (1990)and Black and Dole (1993). Next, results from the lessfrequently studied Pacific anticyclonic, Atlantic cyclon-ic, and Atlantic anticyclonic cases are presented. Forillustrative purposes, the mean upper tropospheric po-tential vorticity (PV) ( ) field and mean geostrophicqwind vectors ( g) are displayed in Fig. 1a, and the totalv(mean plus anomaly) PV field (q) and the total geo-strophic wind vectors (vg) associated with mature Pa-cific cyclonic events are displayed in Fig. 1b. In theupper troposphere, Pacific cyclonic anomalies are as-sociated with intensified westerly flow across the centralNorth Pacific, an amplified ridge along the western U.S.coast, and an enhanced trough over eastern North Amer-ica.

The growth mechanisms for PFAs are not fully un-derstood. Several major theories have been proposed,including instability of the time-mean flow (Simmonset al. 1983; Frederiksen 1983; Swanson 2001), large-scale transient development (Farrell 1989; Cash and Lee2001), nonlinear feedback by synoptic-scale eddies(Shutts 1986), and local and remote forcing via anom-alous topographic and diabatic sources (Hoskins and

1942 VOLUME 60J O U R N A L O F T H E A T M O S P H E R I C S C I E N C E S

FIG. 1. Wind vectors overlaying the 300-hPa quasigeostrophic PV field for the (a) climatologicalmean state and (b) the total fields for period 1318 (3 days, 18 h after onset) during compositePCO. Potential vorticity contours every 0.2 3 1024 s21 for the PV fields.

Karoly 1981; Navarra 1990). Black (1997) diagnosedthe wave activity source locations during PFA life cyclesand found results suggesting local wave sources duringonset and decay. Potential enstrophy analyses and PVinversions attribute the onset of the event to baroclinicand, to a lesser extent, barotropic processes (Black andDole 1993). These analyses did not directly assesschanges in the circulation anomalies themselves, how-ever, and thereby were not able to assess circulationgrowth due to superposition, deformation and wavepacket propagation mechanisms. In particular, when aPV anomaly becomes more circular there are associatedincreases in the perturbation circulation and kinetic en-ergy, even though the areally averaged perturbation PVmay not change (Nielsen-Gammon and Lefevre 1996,hereafter NGL). A simulated anticyclonic North PacificPFA exhibited initial growth through barotropic energy

conversions, with mean flow interactions and synoptic-scale eddy feedback contributing comparably in a ki-netic energy budget (Higgins and Schubert 1994). Con-sistent with these results, linear dynamic terms domi-nated an empirical orthogonal function (EOF) stream-function analysis of the growth phase of a simulatedPFA (Feldstein 1998). Although many of these resultsprovide important initial insight into key PFA formationprocesses such as heat and momentum fluxes, there hasbeen little quantitative analysis of the specific physicalgrowth mechanisms, specifically linear baroclinic andbarotropic growth and nonlinear interactions. Note thatthe direct role of diabatic and ageostrophic processes isalso unknown. To this end this study represents a sub-stantial new contribution toward understanding the keymechanisms responsible for regime transition.

To determine the physical mechanisms driving PCO,


it is necessary to diagnose a flow measure such as theheight tendency, phrased in a meaningful mechanisticdiagnostic framework. Height anomalies are more close-ly linked to circulation anomalies than potential enstro-phy or wave activity. Here we retain the powerful andsuccinct PV formalism while linking it to height ten-dency, which is more directly related to changes in geo-strophic circulation. This is accomplished using a qua-sigeostrophic (QG) potential vorticity (QGPV) frame-work, which not only has a conservation relation butalso a linear balance condition permitting unambiguousflow partitioning. With this ‘‘PV thinking,’’ NGL de-veloped the piecewise tendency diagnosis (PTD) toquantitatively assess development mechanisms for anupper-level trough. The PTD partitions the tendency ofQGPV into dynamically meaningful components andthen inverts each component to deduce its associatedheight tendency. In particular, the PTD method allowsthe study of perturbation growth associated with su-perposition and/or deformation mechanisms.

We first extend the PTD to account for spherical ge-ometry, diabatic processes, and ageostrophic effects.The present goal is to 1) demonstrate that the diagnosticsucceeds in capturing the actual height tendencies; 2)perform a simple breakdown into contributions due tolinear geostrophic and nonlinear geostrophic advections,ageostrophic processes, and diabatic heating forcings;and 3) provide an initial indication of the primarygrowth mechanisms. The data and methodology are in-troduced in section 2, the results from this diagnosticare presented in section 3, and a discussion of theseresults follows in section 4.

2. Data and PTD methodology

The observational data used in this study is the Na-tional Centers for Environmental Prediction–NationalCenter for Atmospheric Research (NCEP–NCAR) re-analysis dataset. The assimilation provides dynamicallyconsistent gridpoint estimates of difficult-to-measurequantities, such as diabatic and radiative heating rates.Further details pertaining to the data sources and assim-ilation methods are documented elsewhere (Kalnay etal. 1996).

To generate the mean state and identify cases withinthe time period of interest, we have extracted 4 timesdaily (0000, 0600, 1200, and 1800 UTC) instantaneousrecords of the Northern Hemisphere geopotential height;temperature; and zonal, meridional, and vertical windsat 12 pressure levels. Six-hourly averaged vertical dif-fusion terms and diabatic heating rates were extractedat 28 model levels. By adapting selection and unpackingsubroutines provided by NCAR, the data were translatedfrom a standard gridded binary (GRIB) output onto ver-tical pressure coordinates and 2.58 latitude by 2.58 lon-gitude horizontal grids. For consistency with the dy-namic records these values were combined so that theyrepresent an average spanning 6 h prior until 6 h after

a given time period. The final data grids were collectedfor 15 consecutive winters inclusive from 1979–80 to1993–94. One winter season is defined as the 120-1 dayperiod spanning 16 November to 15 March. The com-posite cases were selected as in Dole and Gordon (1983)except using 500-mb-height anomaly thresholds of6100 m for periods greater than 7.5 days (e.g., Blackand Evans 1998). Once individual PFAs of each type(e.g., PCO) were identified, we created a compositeanomaly by designating the first time when each eventcrosses the magnitude threshold as day 0, or onset timeand the last time as day 0 of the decay time. Then weensemble-averaged the set of anomalies at this andneighboring time records (as in Black and Dole 1993;Higgins and Schubert 1994; Black 1997). Earlier anal-yses of temporal behavior show that PFA magnitudescan be interrupted by transient (;1 day) mobile dis-turbances (Dole and Gordon 1983); however, the presentgoal was to pinpoint mechanisms governing low-fre-quency behavior. Therefore a nine-point ‘‘light’’ low-pass time filter (Blackmon et al. 1986) was applied toeach term in Eq. (9) after being inverted.

Our approach was to employ an extension of NGL’sPTD to isolate the relative importance of the hypothe-sized physical mechanisms described in section 1 uponthe atmospheric height tendency. Developed by NGL,PTD adapts the piecewise potential vorticity inversiondiagnostic (Davis and Emanuel 1991; Shapiro andFranklin 1999) to partition the local time tendency ofPV into an array of advective and nonconservative forc-ing terms. These forcing terms are then inverted sepa-rately using relaxation methods to obtain the associatedthree-dimensional height tendency induced by the forc-ing. This provides a quantitative methodology for as-sessing the contributions of individual dynamic andphysical sources.

Quasigeostrophic PV is conserved in adiabatic andinviscid geostrophic flow, so the local time rate ofchange is dependent only on advection by the geo-strophic wind (vg) and ageostrophic sources/sinks dueto diabatic heating (SH), friction (SF), and non-QG ef-fects (SAG). As derived in the appendix, the sphericalform of the QGPV equation is given by

]q5 2v · =q 1 S 1 S 1 S , (1)g H F AG]t

where

21 ] F 1 ] cosf ]Fq 5 f 1 1

2 2 2 1 2f (a cosf) ]l a cosf ]f f ]f

] 1 ]F1 f (2)1 2[ ]]p s ]pP

is QGPV in spherical coordinates and


] 2QRS 5 f ,H [ ]]p s pP

]F1 ](F cosf)f lS 5 2 , (3)F [ ]a cosf ]l ]f

where Ff and Fl are the meridional and zonal com-ponents of the frictional force per unit mass, respec-tively, and the diabatic heating Q has units of degreesKelvin per second. Other symbols are defined in theappendix. Friction SF was found to be negligible in allcases and is subsequently disregarded. However, SAG

provides modest contributions so it is included in theanalysis (see appendix for SAG component). The systemgiven by Eqs. (1)–(3) is derived from the basic equationsof motion (in spherical geometry) after expanding themin terms of the Rossby number and retaining the first-and second-order terms. We note that the scale analysisprovides upper bounds on the potential role of variousforcing terms and does not ensure their importance inregime transition.

Inverting (1) to solve for the geopotential F, whereq 5 f 1 L (F) gives the height tendency equation:

]F ]q215 L 1 2]t ]t21 21 215 L (2v · =q) 1 L (S ) 1 L (S ). (4)g H AG

Because L is independent of time, we can exchange Land the local time derivative operators.

For this study, the only lateral boundary condition isF 5 0 at the southern boundary (taken to be 108N)because continuity is applied across the North Pole andalong longitudinal circles (the results are insensitive tothe placement of the southern boundary). To make theupper (pt 5 100 hPa) and lower (p0 5 1000 hPa) bound-ary conditions for the elliptic inversion homogeneous,surface temperature anomalies are included as effectivePV anomalies, as in Bretherton (1966):

u9 u9 q 2 f d(p 2 p ) 1 f d(p 2 p ) 5 L (F). o tdu du ref ref

dp dp (5)

In our numerical solution (which is based on successiverelaxation methods), the surface temperature contribu-tions are included by applying Neumann conditions (viathe hydrostatic equation) at the upper and lower bound-aries.

The extended PTD is the piecewise form of Eq. (4).Terms on the right-hand side can be partitioned intocomponents associated with specific dynamical andphysical mechanisms. Because the total height tendencyis expressed as a linear combination of the forcing terms,the tendency associated with one or a selected group ofterms can be evaluated separately and then assessedrelative to the total height tendency.

It is desirable to analyze PFA as perturbations froma climatological-mean framework and consider three-dimensional spatial variability. Therefore we split eachvariable into its 15-winter time mean (overbar) and per-turbations from this mean (prime). We restrict our at-tention to the advective and nonadvective PV sources/sinks only at levels within a 500–250-hPa layer. It isthe PV at these levels that is most important in deter-mining the upper-level circulation anomalies. Potentialvorticity anomalies at lower levels contribute a rela-tively small amount (less than 10%) to the height ten-dency observed at upper levels. Furthermore, the ad-vection of upper-level PV by the winds associated withlower-level PV anomalies (including surface tempera-ture contributions) is accounted for [see term B in Eq.(9)] so errors do not accumulate. Applying this restric-tion (q ù qU), substituting the expanded variables (ov-erbar and prime) into Eq. (4), and removing the timederivatives of mean variables gives

]F9 ]q9U U215 L 1 2]t ]t21 215 L (2v · =q ) 1 L (2v9 · =q )g U g U

21 211 L (2v · =q9 ) 1 L (2v9 · =q9 )g U g U

21 211 L (S 1 S9 ) 1 L (S 1 S9 ) (6)H H AG AG

(NGL). When the time-mean balance of (6) is subtract-ed, it becomes

]F9U 21 215 L (2v9 · =q ) 1 L (2v · =q9 )g U g U]t211 L (v9 · =q9 2 v9 · =q9 )g U g U

21 211 L (S9 ) 1 L (S9 ). (7)H AG

The key to PTD is to partition the first term on therhs in (7) further. Based upon earlier studies (Black andDole 1993; NGL), we chose to partition the QGPVanomaly field into groups that best encompass upper-(U 5 500 to 100 hPa) and lower-level (L 5 1000 to600 hPa) PV anomalies, respectively. Dividing the geo-strophic velocity into components associated with theupper and lower PV anomaly fields gives

21L (2v9 · =q )g U

21 215 L (2v9 · =q ) 1 L (2v9 · =q ). (8)gU U gL U

Thus, Eq. (7) becomes

2v9 · =q 2 v9 · =q 2 v · =q9 gU U gL U g U]F9U | | | | | |21 5 L | | |]t A B C

1 (2v9 · =q9 1 v9 · =q9 ) 1 S9 1 S9 g U g U H AG| | | | | | .| | |

D E F2N(9)

Partitioning Eq. (4) in this way isolates local height


changes due to linear geostrophic advections (terms A,B, and C), nonlinear geostrophic advections (term D),diabatic heating anomalies (term E), and several ageos-trophic forcings.1 Fundamental physical interpretationsof terms A, B, and C are provided in NGL. FollowingNGL, term A represents upstream wave propagation as-sociated with upper-level Rossby wave advecting themean PV gradient. This embodies both (a) simple west-ward Rossby-wave retrogression (e.g., a monochromaticwave pattern embedded in a zonally uniform basic state)and (b) local wave growth/decay associated with down-stream energy propagation. Term B represents thechanges in an upper-level PV anomaly due to the influ-ence from low-level PV anomalies. Baroclinic growthis included in this term. Term C is the advection of thePV anomaly wave pattern by the geostrophic mean flow,which includes horizontal translation and the effect ofhorizontal and vertical deformation of PV anomalies dueto mean flow asymmetries, including barotropic growth.The most significant ageostrophic terms in regime tran-sition are ageostrophic advection of potential vorticity(term F), relative vorticity stretching (term H), and theageostrophic generation of thermal vorticity (term K;see appendix).

3. Results

a. PCO: A prototype

The PTD is first applied to the onset of a compositecyclonic persistent flow anomaly over the North Pacific(PCO). We focus on height tendencies at 300 hPa, thevertical level of maximum perturbation amplitude. Sec-tion 2 describes how the case was extracted. The de-velopment period for PCO is defined as day 23 to day14 from onset, which encompasses the full develop-ment period. Fourteen 12-h development ‘‘periods’’ aredesignated so that information from the reanalysis datafields is available at the beginning, end, and midpointof each period. They are denoted as the number of daysand hours between the midpoint of the period and theonset day. For example, ‘‘2218’’ refers to minus 2 daysand 18 h, which is the period covering 23 days to 22days, 12 h from onset.

The synoptic characteristics of PCO are summarizedfor background in Figs. 2–5 which indicate three distinctdevelopment stages. The first stage, which lasts fromperiods 2218 to 2106, begins with a cold air masssituated over East Asia (Fig. 2a), anomalously high PVair over Japan (Fig. 4a) and a weakening upper-levelridge over the central North Pacific (Fig. 3a). Synoptic-scale cyclogenesis then ensues as a westward-tilting lowpropagates east from Japan (Fig. 5a). During the middle

1 In the current calculations, v0 (see appendix) has been approxi-mated by assimilated values of v in the second-order forcing terms( ). This was done for practical reasons and results in errors atS9AG

third order.

stage of development (2018 to 1 106) the developingcyclone flow anomaly slows down, becomes zonallyelongated and equivalent barotropic, and begins large-scale growth (Figs. 3b,c; 4b,c; and 5b). The third stageof development is marked by surface warm advectionnortheast of the low (Fig. 2e) and the reformation of awestward tilting structure (Fig. 5c). This is accompaniedby continued large-scale circulation growth as the upper-level perturbation increases in magnitude and attains amore round horizontal shape (Figs. 3e,f). The suggestionis that development prior to the onset day is baroclinic,whereas both barotropic and baroclinic growth are im-portant thereafter. A two-stage growth process has alsobeen suggested previously in both observational studies(Dole and Black 1990) and theoretical predictions (Fred-eriksen 1982; Frederiksen 1983).

The actual 300-hPa height tendency associated withchanges in upper-level PV [i.e., the lhs of Eq. (9)] iscontrasted in Fig. 6 with the net height tendency inducedby the sum of all the forcing terms on the rhs of (9). Ifthe PTD method was perfect, Fig. 6a would exactlymatch Fig. 6b. Note that all the nonlinear terms arecalculated individually for each case and then compos-ited so that individual case behavior such as nonlineareddy feedback is retained. For PCO the sum of forcingsunderestimates the actual magnitude change by only 3%over the development period. The spatial pattern cor-relation between the actual and calculated tendency at300 hPa over the region of interest (108 to 708N,1122.58E to 1208W) is 0.87. During peak development,it reaches 0.94 for individual periods.

The regional contributions of the terms in Eq. (9) overthe 7-day development period are displayed in Fig. 7.Term B, the baroclinic growth term, contributes moststrongly to anomaly development. Strong cancellationexists between terms A and C due to the competingeffects of downstream advection by the mean flow andupstream Rossby wave propagation. It appears that termC, which includes barotropic growth, may provide apositive contribution over the eastern North Pacific.Terms D (nonlinear) and E–N (non-QG, including dia-batic heating) are generally weaker and out of phasewith the height tendency. The non-QG terms generallyhelp improve the spatial correspondence between theactual tendencies and the net forcing.

Figure 8 is a time series of the calculated forcingsover the fourteen 12-h time periods of the developmentsequence following the developing height minimum.The time periods are plotted relative to their distancefrom onset day, so period 2218 (which covers 23 daysto 22 days, 12 h from onset) is plotted as 22.75 daysrelative to onset day. All the forcing terms from Eq. (9)are incorporated in this analysis and sum to the ‘‘net’’value. The intensity of the event is defined as the localminimum of the perturbation height field. So the rateof intensification is the height tendency at the localheight minimum. Thus, for each time period, we identifythe contribution of each forcing term to the height ten-


FIG. 2. Sea level pressure contours (solid, every 4 hPa) overlaying 1000- to 500-hPa thickness contours (dashed,every 100 m) for the composite PCO (a) 2218, (b) 2106, (c) 2006, (d) 1106, (e) 1206, and (f ) 1318 time periods;‘‘L’’ and ‘‘H’’ designate the local surface pressure low and high centers.


FIG. 3. 300-hPa anomalous geopotential height (negative contours are dashed, every 20 m) for the composite PCO,(a) 2218, (b) 2106, (c) 2006, (d) 1106, (e) 1206, and (f ) 1318 time periods.

dency that occurs at the location of the height minimumfor the midpoint of the period. In doing so, we clearlyidentify local perturbation growth and remove tenden-cies associated with monochromatic wave translation

and propagation (see NGL, p. 3123 for further discus-sion of this issue).

With the information from Fig. 8, a quantitative mea-sure of the contribution of terms in Eq. (9) toward


FIG. 4. 300-hPa anomalous QGPV 3 104 (negative contours are dashed, every 0.1 3 1024 s21) for the compositePCO, (a) 2218, (b) 2106, (c) 2006, (d) 1106, (e) 1206, and (f ) 1318 time periods.

growth can be assessed. Note the excellent correspon-dence between the actual height tendencies over each12-h period and the net forcing [rhs of (9)]. Also, thetransition from synoptic to large-scale development is

clearly marked by the jump in the height tendency atday 21. As we deduced from Fig. 7b, term B (the bar-oclinic growth term) is the primary contributor to thedevelopment sequence. Term C (the mean flow advec-


FIG. 5. Anomalous height field at 408N for the composite PCOduring periods (a) 2218, (b) 1006, and (c) 1318. Contours every10 m.

FIG. 6. (a) 300-hPa height change due to upper-level PV changes[lhs of (9)], (b) sum of forcing terms in rhs of (9) at 300 hPa over7-day development period of composite PCO over the North Pacific(contours every 50 m).

tion term) is also a positive contributor after day 21when the anomaly intensifies more quickly. Integratingthe area under the curves for terms B and C from day21 forward allows a quantitative comparison; term Bmakes up 85% of the sum of the two terms leaving acontribution of 15% for term C. So baroclinic growthdominates barotropic growth during large-scale devel-

opment. The Rossby wave propagation term (A) is al-most always opposing growth, which implies that waveenergy is propagating away from the anomaly of inter-est. The nonlinear (D) and non-QG (E–N) terms (in-cluding diabatic heating) are small throughout, bothweakly contributing to growth just prior to onset. Atearly stages the dynamics is consistent with synoptic-scale cyclogenesis.

The linear PV advection forcings are clarified byoverlaying the corresponding wind vectors and PV fieldsfor each term. These are displayed in Fig. 9 for a rep-resentative time during large-scale development. Figure9a is the breakdown of term A, the Rossby wave ret-rogression term, with 300-hPa geostrophic winds as-sociated with upper-level PV anomalies overlaying themean PV field. The cyclonic winds linked with the up-per-level PFA are centered at about 167.58W and induceheight rises downstream and height falls upstream byadvection of the mean PV field. Figure 9b is a similarplot, but with vectors representing 300-hPa geostrophicwinds due to lower-level PV anomalies overlaying themean PV field. Here, upstream surface cold and down-stream surface warm anomalies jointly act to producea northwesterly wind flow that extends upward to the300-hPa level, advecting high PV southeastward towardthe growing perturbation. Term C is broken down inFig. 9c with mean wind vectors overlaying the anom-alous PV field at period 1318. The anomaly is locatedin a jet exit region, with winds decreasing at the westernedge of the anomaly and slight diffluence around 1708–1608W near the anomaly center. The net effect of this


FIG. 7. Height change due to inverted forcing terms in Eq. (9): (a) term A; (b) term B; (c) term C; (d) sum of terms A, B, and C; (e)term D; and (f ) sum of terms E–N. Contour interval is 50 m for (a)–(c) and 20 m for (d)–(f ). Regions greater than 300 m are shaded lightly;regions less than 2300 m are shaded darkly.

scenario is that the mean wind field is tending to makethe anomaly more round, which implies that horizontaldeformation is a potential growth mechanism during thisperiod.

To pinpoint the precise nature of the contribution byterm C, the barotropic growth term, the PTD is parti-tioned further (e.g., NGL) into forcings due to horizontaltranslation of PV anomalies, deformation of the PVanomaly field due to horizontal mean flow asymmetries,and vertical superposition of PV anomalies:

21 21L (2v · =q9 ) 5 L (2v · =q9 )g U ADM U

211 L (2v9 · =q9 )HW U

211 L (2v9 · =q9 ), (10)V U

where the time-mean total wind field has been averagedin all three spatial dimensions in addition to time v4DM,and the remaining three-dimensional variability of thetime-mean field (determined by isolating fromv4DM

) has been divided into vertically averaged, , andv v9g HW

horizontally averaged, , components that are embed-v9Vded in a frame of reference moving with the anomaly.

Applying this to the existing term C shows that virtuallyall the contribution is due to the horizontal deformationof the PV anomaly (not shown).

To test the robustness of the composite results, we alsoapplied the PTD to single case of persistent flow anomalydevelopment within the PCO composite (not shown). Itoccurred during January 1985, within a winter closelyresembling long-term climatology. As might be expected,the individual forcing terms are larger in magnitude thanin the composite due to the implicit spatial and temporalsmoothing of the compositing process. However, the spa-tial pattern correlations between the actual and net heighttendency fields are comparable to the composite case.Although there are quantitative differences with the com-posite analyses, there is a robust qualitative similarity,with baroclinic and barotropic growth mechanisms offsetby Rossby wave propagation acting to radiate energyaway from the growing anomaly.

b. Other types of weather regime development

Once the PTD was successfully applied to the onsetof Pacific cyclonic anomalies, the same technique was


TABLE 1. Magnitude estimation and spatial correlation for the onset of each of the four PFAs investigated, namely the Pacific and Atlanticcyclonic and anticyclonic events. Magnitude estimation [1(2) is over (under)] gives the percent difference between the height tendency ofthe observed and PTD calculated at the height minimum/maximum. The spatial correlation is a regional three-dimensional correlation ofgrid points of each 12-h height tendency. The main number is the average 12-h correlation over the development period and the number inparentheses is the correlation for the 12-h height tendency during maximum development.

Event onset Pacific cyclonic Pacific anticyclonic Atlantic cyclonic Atlantic anticyclonic

Magnitude estimationSpatial correlation

23%0.87 (0.94)

,1%0.89 (0.96)

266%0.72 (0.90)

13%0.98 (0.98)

TABLE 2. Percent contribution by each term in Eq. (9) to regimeonset during maximum development for the four anomalies analyzed.The percent contribution for each term (those that do not contributeto growth are not included and are disignated as a starred entry) iscalculated by summing its value at each 12-h time period duringmaximum growth (data used for the time series plots) and calculatinga percent compared to the other terms. The periods of maximumdevelopment for each case are days 21 to 1 4 (Pacific cyclonic),11 to 14 (Pacific anticyclonic), 23 to 14 (Atlantic cyclonic), and23 to 12 (Atlantic anticyclonic).

Event onsetPacific

cyclonicPacific

anticyclonicAtlanticcyclonic

Atlanticanticyclonic

Term ATerm BTerm CTerm DTerms E–N

*8515**

*9109**

**

5842*

*293140*

applied to 1) the onset of other persistent flow anomalies(Pacific anticyclonic, Atlantic cyclonic and anticyclon-ic) and 2) the decay phases of the same four PFA types.

When the PTD is applied to the other onset cases,overall we find a very good correspondence betweenthe actual and calculated height tendencies (see Table1). The main exception is that the PTD underestimatesmagnitudes during Atlantic cyclonic onset. All four PFAtypes exhibit multiple development stages. During earlydevelopment the primary circulation anomaly is gen-erally smaller scale and less well organized as it prop-agates eastward to the key region. Around onset day,the anomaly patterns become quasi stationary and zon-ally elongated and experience large-scale growth. At-lantic cyclonic onset is an exception in that the mainanomaly growth occurs during the early phase when itis less mobile and more zonally elongated than the othercases.

The Pacific anticyclonic events form similarly to PCOwith analogous development stages. Both baroclinic andbarotropic processes contribute, although barotropicgrowth is even less important than during PCO (referto Table 2). Analyzing vector overlay plots like Fig. 9for the Pacific anticyclonic onset illustrate that an up-stream surface warm anomaly has associated cyclonicflow at upper levels advecting PV northward, causinga PV decrease within the upper-level anticyclonic flowanomaly. Further, as for PCO the primary circulationanomaly is located in the diffluent jet exit region, con-tributing to term C.

Over the Atlantic, the onset of the cyclonic cases

results from barotropic and secondary nonlinear pro-cesses while the anticyclonic cases developed from acombination of baroclinic, barotropic, and nonlinearprocesses (again refer to Table 2 for quantitative values).Further analysis of Atlantic cyclonic onset evolution(not shown) indicates that there are smaller contribu-tions from wave energy propagation (at initial times)and baroclinic growth (later times). Also, this evolutionoccurs more slowly and evenly than the other eventsand, as mentioned above, exhibits a distinct structuralevolution. The inability of the PTD to more accuratelyrepresent the dynamics is assumed to be associated withthese differences. During Atlantic cyclonic onset theheight tendency magnitude is smaller at each time periodand is thus subject to more cumulative error. Atlanticanticyclonic onset closely follows Pacific anticycloniconset, except that baroclinic growth contributesthroughout development rather than only after day 0,and nonlinear effects are significant (Table 2).

Contrasting the developments of the four PFA types,the largest differences are between the Pacific and At-lantic cases rather than between cyclonic and anticy-clonic events. Over the Pacific, baroclinic growth is theprimary forcing whereas over the Atlantic, barotropicgrowth is primary with comparable contributions fromnonlinear eddy feedback and baroclinic growth (Atlanticanticyclonic). Note that term A opposes developmentin all four cases.

c. Weather regime decay

Weather regime decay is even less well understoodthan regime development. Consequently, PTD is alsoapplied here to the decay cycle of composite North Pa-cific and North Atlantic cyclonic and anticyclonic re-gimes. The same procedures used to study onset are alsoapplied for decay (see section 2). The decay cycle forall four composites takes place between day 24 andday 13 inclusive, where day 0 is the time period atwhich each case falls below the height anomaly thresh-old of 100 m.

Like onset, regime decay exhibits distinct evolution-ary stages. However, the decay cycles generally proceedmore slowly and evenly (except Atlantic anticyclonic)and occur as follows. First, the event diminishes in mag-nitude slightly while retaining its basic structure andposition. Second, the event rapidly dissolves while un-dergoing significant horizontal and vertical structural


FIG. 8. Contribution from calculated forcings to the height tendency at the height minimum forthe 7-day development period of the composite PCO. Equation (a) terms A, B, and C are linear;D is nonlinear; E–N (labeled Non-QG) are the diabatic heating, perturbation ageostrophic ad-vection, and other non-QG forcings. ‘‘Net’’ is the sum of all the terms.

changes. PTD-forcing time series for Pacific cyclonicdecay are shown here to illustrate these characteristics.

Figure 10 is set up in a similar manner as Fig. 8 (PCO)with the height tendency rate (in meters per 12 hours)plotted for each time period. Pacific cyclonic decay oc-curs primarily through a net horizontal export of Rossbywave energy downstream from the primary circulationanomaly (term A). The nonlinear and non-QG terms pro-vide small contributions, which appear to help tip thebalance toward perturbation decay. Nonetheless, we notethat at the beginning, term A is 35 m21 12 h larger inmagnitude than at the end of PCO (Fig. 8), whereas thenon-QG terms change less. Interestingly, the primaryterms contributing to PCO (B and C) actually increasesomewhat during the early stages of decay. However,there is also a gradual decrease in terms B and C afterday 21, which helps to facilitate the later stages of decay.So decay occurs primarily from an increase in term Awith very minor contributions from non-QG forcings.

Thus, it is the enhancement in wave energy propa-gation away from the anomalies between onset to decaythat drives the eventual decay. Synoptic analyses (notshown) show an increase in the strength of downstreamheight anomalies during decay. The decay of the Pacificanticyclonic and Atlantic PFAs (not shown) occurs ina generally similar way. Pacific anticyclonic decay re-sults from a strengthening of term A with minor con-tributions from the non-QG terms in tandem with a de-cay of one of the key formation mechanisms, namelybarotropic growth (term C). Atlantic anticyclonic decayoccurs very much the same way (increase in A anddecrease in C), but with more equal contributions from

the non-QG terms. Atlantic anticyclonic decay occursa little differently. Terms A, B, and C are all initiallystronger during early decay than the end of onset, butthen B and C, the growth terms during onset, drop offsignificantly, and decay ensues. The non-QG forcing isstill very small. Thus, in all four cases, term A is thelargest contribution toward decay with slight variationin the details.

4. Discussion

The results in section 3 provide a quantitative indi-cation of the forcing mechanisms during PFA life cycles.The most important contributors in all the onset anddecay sequences are the linear quasigeostrophic advec-tion terms. Nonlinear eddy feedback and several non-QGPV terms are less important, depending on the PFAtype and portion of the life cycle. The piecewise ten-dency diagnostic does a remarkable job accounting forthe observed height tendency patterns in most cases.The average regional spatial correlations are as high as0.94, which is an improvement upon other recent heighttendency analyses. In a study of anticyclone develop-ment Tan and Curry (1993) obtain similar correlations(0.80–0.85) with rather large magnitude errors ($50%).In a separate study of individual cyclone developmentusing an extended version of the QG height tendencyequation (Tsou et al. 1987), the 300-hPa correlation ofthe observed and calculated tendencies obtained is about0.65 for a 48-h time period. The magnitude discrep-ancies in the present analysis range from 1% in thePacific anticyclonic onset sequence to 66% over the


FIG. 9. Wind vectors overlaying the upper-level PV field for terms (a) A, (b) B, and (c) C, inEq. (9) at 300 hPa for period 1318 (3 days, 18 h after onset) for the composite PCO. Potentialvorticity contours every 0.2 3 1024 s21 for the PV fields.

Atlantic during cyclonic onset. In most cases the errorsare less than 20%.

The primary quantitative improvements of our ap-proach appear related to (a) incorporating spherical ge-ometry (all forcing terms), (b) including ageostrophicadvections (term F), and (c) accounting for diabatic

heating (term E). Although the magnitude discrepancyis smaller using the extended PTD compared to earlierstudies, several factors limit the quantitative ability ofour analysis. These include errors due to time samplingand spatial discretization. For example, the reanalysisdataset allows for smaller differencing increments than


FIG. 10. Contribution from calculated forcings to the height tendency at the height minimumfor the 7-day decay period of the composite Pacific cyclonic decay. Equation (9) terms A, B, andC are linear; D is nonlinear; E–N (labeled Non-QG) are the diabatic heating, perturbation ageos-trophic advection, and other non-QG forcings. ‘‘Net’’ is the sum of all the terms.

most observed datasets have provided in the past, butthe PTD still requires finite differencing calculationsover each development period. Thus, errors may ariseif the tendencies are nonlinear over the 12-h period.Nothing that Atlantic cyclonic onset is the only case inwhich baroclinic growth is not important, we speculatethat sampling and discretization errors most stronglyimpact forcing terms C and D. Supplementary analyses(not shown) indicate that the impact of analysis incre-ments and lateral boundary conditions is insignificant.

Our analyses using the extended PTD provides aquantitative measure of the important source terms driv-ing both onset and decay of observed PFA events, someof which can be closely linked to physical mechanisms.Term C captures local wave growth or decay associatedwith horizontal deformation of PV anomalies by themean flow. By applying PTD following the height ex-trema, information on wave packet propagation is lim-ited to the growth and decay due to wave interaction.Term B, which measures advection of mean PV by up-per-level winds due to lower-level PV features, isolatesbaroclinic growth as it measures growth dependent onthe relative location of lower-level features. If the large-scale anomaly were purely barotropic, the winds at up-per levels due to a lower-level anomaly would be zeroat the height minimum. The secondary contributions ofdiabatic heating (term E) and ageostrophic advection ofQGPV (term F) provides evidence that, even at largescales, non-QG processes can be important.

The current study complements previous PFA studiessince many of the past studies address different researchquestions with quite different methods and datasets.

Nonetheless, in regions of overlap we find very goodconsistency between the current results and other PFAstudies. The likely significance of linear dynamics ispredicted by theoretical (Frederiksen 1982, 1983; Swan-son 2001) and modeling studies (Cash and Lee 2001;Winkler et al. 2001). For PCO, Black and Dole (1993)performed potential enstrophy analyses and deducedminimal net contributions from nonconservative andnonlinear processes; instead, linear baroclinic processesare the primary contributor with barotropic processescontributing to a lesser degree. However, their analysiswas limited by the fact that it was unable to accountfor the role of barotropic deformation of PV anomaliesupon circulation growth.

The Pacific anticyclonic onset has been investigatedusing observed and model events. Feldstein (1998)found that growth of a simulated anticyclonic PFA wasdue to linear dynamic processes, but did not identifythe precise forcing mechanism. Black’s (1997) analysisof wave activity shows a net export of wave activity toremote regions from the key region during Pacific an-ticyclonic onset. In our case PTD attributes growth tobaroclinic growth and secondarily horizontal deforma-tion and that non-QG terms weakly oppose onset. Theseresults differ from those of Higgins and Schubert (1994),who identified barotropic conversions as the maingrowth mechanism for an anticyclonic event. However,their simulated analysis using a limited-area kinetic en-ergy budget was restricted to a single vertical level.

Over the Atlantic region, only the formation of blockshas received much attention in past research. A recentinvestigation of composite block formation over Europe


provided results suggesting that incoming wave activityflux was a primary contributor to growth and that tran-sient eddy feedback was less than 45% responsible(Nakamura et al. 1997). Higgins and Schubert (1994)discuss results (not shown in their study) for the de-velopment of a simulated Atlantic positive PFA in whichbarotropic energy conversions are the most importantforcing, with equal contributions from the mean flowand synoptic-scale eddies. Black (1997) found that, sim-ilar to other PFA cases investigated, there is a net waveexport up and away from the lower troposphere nearthe key region. Here, we find that the Atlantic anticy-clonic event formed due to baroclinic horizontal defor-mation and nonlinear eddy feedback. These results areconsistent with Nakamura et al. (1997) as they focus onthe development 4 days before the peak height of acomposite where the anomaly maximum center hasshifted, which could easily coincide with days 12 to14 in our time frame where wave propagation is im-portant. Also, it is not clear that identical phenomenaare identified for study (blocks versus PFA).

Very little research to date has focused on the decayof PFA. One study using the streamfunction tendencyequation performed calculations for simulated Pacificevents and found that both the positive and negativephases decay linearly with equal contributions fromlow-frequency eddies and the self-interaction of high-frequency eddies during cyclonic anomaly decay andcontributions from high-frequency eddies during anti-cyclonic anomaly decay (Feldstein 1998). In our anal-ysis, linear propagation of wave energy away from theanomaly is primarily responsible with secondary assis-tance from non-QG terms. The primary role of wavepropagation during decay is consistent with the waveactivity flux analysis of Black (1997).

Cross comparison of the four PFA types providesadditional insight as general trends may have implica-tions for the role of local geography and the existenceof preconditioning. The development process for thePacific cyclonic and anticyclonic onset periods are verysimilar, with baroclinic and barotropic processes con-tributing to growth. In both cases, wave energy is prop-agating away from the anomaly, especially during laterdevelopment. Their similarities are also manifested intheir vertical and horizontal structure during this time,for both exhibit 1) a mobile slow growth stage (days23 to 21) while the anomaly moves toward the keyregion and 2) a fast-growing stage (days 21 to 14)when the anomaly is mostly stationary. Clearly the pres-ence of a strong low-level anomaly coupled with theexistence of a zonally elongated event in the jet exitregion are involved in the development of both PacificPFA.

Over the Atlantic, anticyclonic onset has similaritiesto its Pacific PFA counterparts while cyclonic onset israther different. The anticyclonic anomaly developsvery quickly due to the combined effect of baroclinicand barotropic processes, and anomalous nonlinear ef-

fects. The cyclonic anomaly develops with primary con-tributions from barotropic deformation and nonlinearfeedback. Unlike the Pacific events, nonlinear eddyfeedback is significant over the Atlantic during bothgrowth phases. During early development, the Atlanticevents are less mobile and grow from anomalies situatedclose to the key region.

The differences between the Atlantic and Pacific casesmay be associated with relative geography due to 1)Atlantic PFA located closer to land and 2) smaller ex-tension of the jet stream over the Atlantic. Their dif-ferences also have important implications for the ulti-mate predictability of PFA growth. The dominant roleof the linear dynamics during Pacific PFA developmentmeans that these systems may, in practice, be identifiedprior to large-scale growth and persistence. However,the presence of the smaller-scale variable interactionsduring Atlantic PFA growth prevents the practical taskof predicting Atlantic onset much beyond the currentforecasting limits.

Baroclinic amplification has been identified as an im-portant growth mechanism, especially for the PacificPFA, so an appropriate question at this point is whetherthe nature of the growth is due to transient nonmodaldevelopment or three-dimensional instability of thetime-mean flow. If development is mostly governed bytransient nonmodal development, then, synoptically, aninitial disturbance would be present before the onset oflarge-scale development (Borges and Hartmann 1992).Further, the presence of stable modes from the contin-uous spectrum allows for time-varying horizontal andvertical structure (Farrell 1989). However, if growth oc-curs from the fastest growing unstable mode, then wav-etrains of fixed spatial structures would be expected toemerge during large-scale development. Looking at thevertical structure of anomalous height at day 22.75 dur-ing Pacific cyclonic development (Fig. 5a), it is clearthat a smaller-scale upper-level disturbance is presentand seems to initiate the large-scale development (alsoBlack and Dole 1993). So the event appears to growfrom an existing anomaly. Further, during the initialdevelopment phase the vertical structure evolves froma westward tilt toward a more equivalent barotropicstructure (Fig. 5b), which is consistent with nonmodalbaroclinic growth.

5. Conclusions

The piecewise tendency diagnosis (PTD) of NGL isextended to formally account for spherical geometry,diabatic heating, and ageostrophic dynamic processes.The extended PTD method is then employed to identifythe primary source mechanisms resulting in the for-mation and decay of persistent anomalous weather re-gimes over the North Pacific and Atlantic Oceans. Indoing so we demonstrate the utility of PTD in studyinglarge-scale dynamic processes in the midlatitude tro-posphere. To first order, it is found that weather regime


transitions over the Pacific are governed by linear dy-namic mechanisms. Specifically, for both cyclonic andanticyclonic cases, weather regime onset is primarilydue to baroclinic growth with secondary contributionsfrom the barotropic deformation of PV anomalies. At-lantic anticyclonic cases develop as a result of baro-tropic deformation, which plays a relatively larger rolethan for the Pacific cases, baroclinic growth, and anom-alous nonlinear eddy feedback. Atlantic cyclonic regimedevelopment is due primarily to barotropic deformationand nonlinear eddy feedback. Diabatic and ageostrophicprocesses provide only small contributions during re-gime development, generally opposing growth. Weatherregime decay begins gradually mainly via downstreamwave energy propagation with very small contributionsfrom diabatic heating (Atlantic cases) and ageostrophicPV advections (all cases except Atlantic cyclonic).Large-scale weather regime dissipation then occurs pri-marily because of 1) an increase in Rossby wave energypropagation away from the primary circulation anomalyand, in some cases, 2) a decrease in the contributionfrom barotropic deformation as a local wave source.

Acknowledgments. We thank the Scientific Comput-ing Division (SCD) at the National Center for Atmo-spheric Research (NCAR) for providing data unpackingand interpolation routines and for computing time. Thiswork is supported by NSF under Grants ATM-9634667and ATM-0001346 and by the NASA Global Modelingand Analysis Program under Grant NAG5-10374. Wethank the anonymous reviewers for their helpful com-ments and guidance.

APPENDIX

QG Potential Vorticity

To optimize the diagnostic accuracy of our calcula-tions, the QGPV equation is derived in spherical co-ordinates by formally including terms that are secondorder in terms of the Rossby number. These terms rep-resent local PV sources and provide small correctionsto the first-order QGPV equation.

The spherical QGPV equation is derived from theinviscid equations of motion in spherical and isobariccoordinates assuming hydrostatic balance and a thin at-mosphere (Haltiner and Williams 1980). We begin withthe vorticity equation in spherical coordinates:

D(j 1 f ) ]v 1 ]v ]y 1 ]v ]u5 (j 1 f ) 2 1

Dt ]p a cosf ]l ]p a ]f ]p

]F1 ](F cosf)f l1 2 , (A1)[ ]a cosf ]l ]f

where j is relative vorticity in spherical coordinates,D/Dt the three-dimensional total derivative (also ex-pressed in spherical coordinates), Fl and Fw are thezonal and meridional components, respectively, of the

frictional force per unit mass, and the other variableshave their conventional meteorological interpretations.In obtaining (A1), no further approximations are madeto the system of equations; the curvature terms foundin the spherical momentum equations are not neglectedbut actually drop out. Equation (A1) is identical to thevorticity equation (4.21) presented in Holton (1992),except that the vorticity and total derivative are ex-pressed in spherical coordinates.

Characteristic variable scales are chosen in order todevelop successive approximations to (A1). Typicalscales for midlatitude weather regimes include those forthe horizontal velocity U ; 10 m s21, horizontal lengthL ; 106 m, and Coriolis parameter f 0 ; 1024 s21, leadingto a Rossby number Ro of 0.1. In choosing a verticalvelocity scale, we adopt the common assumption that thehorizontal wind divergence scales as (URo)/L.

Noting that, under our assumptions L/a ; Ro, weobtain the following expected magnitudes for the termsin (A1):

2] y df ]v U1 v · = j ; ; f ; ,

21 2[ ]]t a df ]p L

2]v ]j 1 ]v ]y 1 ]u ]v Uj ; v ; 2 1 ; Ro .

21 2[ ]]p ]p a cosf ]l ]p a ]p ]f L

Defining SF as the combined effect of the frictionalterms in (A1), in the scale analysis we adopt the con-servative choice of bounding SF by the leading termsin (A1). The dependent variables are then expanded interms of Ro where the geostrophic winds are first-ordersolutions to the momentum equations (i.e., v 5 vg 1va ; j 5 jg 1 ja). To suitably expand the vertical ve-locity field, we first make use of the thermodynamicequation

] du ]u9ref1 v · = u9 1 v 1 v 5 H, (A2)1 2]t dp ]p

where u is potential temperature [decomposed into areference state uref(p) and perturbation u9 values], andH represents changes in u due to diabatic heating. Asfor the vorticity equation, we assess expected magni-tudes for the various terms in the equation (e.g., Haltinerand Williams 1980)

2] f U01 v · = u9 ; ,1 2]t R2du f Uref 0 2v ; (Ro) Ri,

dp R2 2]u9 f U f U0 0v ; Ro; H & ,

]p R R

where R is the gas constant and Ri is the Richardsonnumber (taken to be ;100); H is taken to be boundedby the leading terms [first two terms on lhs of (A2)].Equation (A2) is then solved for v:


H 2 D u9 1 (H 2 D u9)H Hv 5 5 ,(u ) 1 (u9) [1 1 (u9) /(u ) ] (u )ref p p p ref p ref p

(A3)

where DH is the total derivative for isobaric motion andthe subscript p denotes ]/]p.

Accounting for our assumed Ri scale, the ratio(u9)p/(uref)p ; Ro, and we perform a Taylor series ex-pansion to second order:

(u9)1 pø 1 2 . (A4)[1 1 (u9) /(u ) ] (u )p ref p ref p

Upon substitution into (A3), we obtain (accurate to sec-ond order)

(u9)(H 2 D u9) (H 2 D u9)pH Hv ø 2 . (A5)(u ) (u ) (u )ref p ref p ref p

The operator DH is expanded in terms of the Rossbynumber as DH 5 Dg 1 Da with the definitions

]D 5 1 v · =; D 5 v · =,g g a a]t

where vg and va are the geostrophic and ageostrophiccomponents, respectively, of the horizontal velocityfield. Expanding (A5) in this way leads to

(H 2 D u9) (u9) (H 2 D u9) (u9)D u9 D u9g p g pa av ø 1 2 1 2 1 .5 6 5 6(u ) (u ) (u ) (u ) (u ) (u )ref p ref p ref p ref p ref p ref p (A6)| | | | | | | || | | |

v v v v0 a b c

Scale analysis reveals that v0 is first order (in terms ofRo), va and vb are second order, and vc is third order.We can use a simple substitution to rewrite (A6) as

(u9) (u9)p pv ø v 1 v 2 v 2 v . (A7)0 a 0 a(u ) (u )ref p ref p

The first- and second-order approximations to v can bewritten, respectively, as

(H 2 D u9)gv ø v 5 ,0 (u )ref p

(u9)D u9 pav ø v 2 2 v . (A8)0 0(u ) (u )ref p ref p

Returning to the problem of formally approximating (A1),we expand u, y, v, and j in terms of the Rossby numbermaking use of (A8). The first-order (terms ;U2/L2) ap-proximation to (A1) is

y H 2 D u9df ]g gD j 1 5 f 1 S . (A9)g g F[ ]a df ]p (u )ref p

At second order (terms ;RoU 2/L2 and larger), we ob-tain the following approximation:

(y 1 y ) ]jdfg a gD (j 1 j ) 1 D j 1 1 vg g a a g 0a df ]p

H 2 D u9 (u9)] ] D u9g pa5 f 2 f 1 v0[ ] [ ]]p (u ) ]p (u ) (u )ref p ref p ref p

]y ]u]v 1 ]v 1 ]vg g0 0 01 j 2 1 1 S ,g F]p a cosf ]l ]p a ]p ]f(A10)

where we have substituted for v0 in the first-order terms[as in (A9)]. Collecting terms on the left-hand side andexpanding terms on the right-hand side (using the ther-mal wind relation) leads to

]jgD (j 1 f ) 1 D j 1 D (j 1 f ) 1 vg g g a a g 0 ]p

y] H ] u9 df ] u9 ] u9 y df ] u9g a5 f 2 D f 1 2 D f 1g a5 6 5 6[ ] [ ][ ] [ ] [ ]]p (u ) ]p (u ) a df ]p (u ) ]p (u ) a df ]p (u )ref p ref p ref p ref p ref p

(u9)f 1 ]u9 ]u 1 ]u9 ]y ] ] u9 ] ] 1 ]v pa a 02 + 2 v f 1 f v u9 2 f0 01 2 5 6 5 6[ ] [ ](u ) a cosf ]l ]p a ]f ]p ]p ]p (u ) ]p ]p (u ) ]p (u )ref p ref p ref p ref p

]y ]u]v 1 ]v 1 ]vg g0 0 01 j 2 1 1 S . (A11)g F]p a cosf ]l ]p a ]p ]f


To help restructure (A11) into the desired potentialvorticity form, we define the ‘‘thermal’’ vorticity qt 5f (]/]p)[u9/(uref )p] and the diabatic heating term SH 5f (]/]p)[H/(uref )p]. Upon substitution and rearrange-ment of terms,

]v yq df0 tD q 5 S 2 D q 2 D j 1 j 1g H a g a g ]p fa df| | | | | | | | | |

| | | | |E F G H J

f 1 ]u9 ]u 1 ]u9 ]y ]qa a2 1 2 v01 2(u ) a cosf ]l ]p a ]f ]p ]pref p| | | |

| |K L

(u9)]v ] ] 1p02 f 1 f v u90 5 6[ ]]p (u ) ]p ]p (u )ref p ref p| |

|M

]y ]u1 ]v 1 ]vg g0 02 1 1 S ,Fa cosf ]l ]p a ]p ]f(A12)| | | |

| |N P

where

] u9q 5 j 1 f 1 f (A13)g [ ]]p (u )ref p

is the quasigeostrophic potential vorticity, which is con-served for adiabatic and inviscid geostrophic motion[i.e., terms on the right-hand side of (A12) are negli-gible]. Using geostrophic wind balance,

1 ]F 1 ]Fu 5 2 ; y 5 , (A14)g gaf ]f af cosf ]l

and the hydrostatic equation leads to the quasigeo-strophic balance condition in which q is expressed solelyin terms of the geopotential F:

21 ] F 1 ] cosf ]Fq 5 f 1 1

2 2 2 1 2f (a cosf) ]l a cosf ]f f ]f

] 1 ]F1 f . (A15)1 2]p s ]pp

This is the spherical geometry representation ofquasigeostrophic potential vorticity. The term s p 52(R/p)( p/ps )k (duref /dp) is an alternate form of staticstability (Holton 1992).

Terms E and P in (A12) represent diabatic and fric-tional sources/sinks of q, respectively, whereas termsF–N make up the group of ageostrophic terms denotedSAG in the main text. The latter terms act as second-order corrections to the usual quasigeostrophic equa-tions providing local sources and sinks of q, analogousto terms E and P.

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