Simple Coupled Midlatitude Climate Models

Transcription

1 016 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 Simple Cupled Midlatitude Climate Mdels LYNNE D. TALLEY Scripps Institutin f Oceangraphy, University f Califrnia, San Dieg, La Jlla, Califrnia (Manuscript received 3 June 1997, in final frm 10 September 1998) ABSTRACT A set f simple analytical mdels is presented and evaluated fr interannual t decadal cupled cean atmsphere mdes at midlatitudes. The atmsphere and cean are each in Sverdrup balance at these lng timescales. The atmsphere s temperature respnse t heating determines the spatial phase relatin between SST and sea level pressure (SLP) anmalies. Vertical advectin balancing heating prduces high (lw) SLP lying east f warm (cld) SST anmalies, as bserved in the Antarctic circumplar wave (ACW), the decadal Nrth Pacific mde, and the interannual Nrth Atlantic mde. Znal advectin in an atmsphere with a rigid lid prduces lw SLP east f warm SST. Hwever, if an ad hc equivalent bartrpic atmspheric respnse is assumed, high SLP lies east f warm SST. Relaxatin t heating prduces behavir like the bserved Nrth Atlantic decadal pattern, with lw SLP ver warm SST. Meridinal advectin in the atmsphere cannt prduce the bserved SST/SLP patterns. The dminant balance in the cean s temperature equatin determines the phase speed f the mdes. The cupled mde is nndispersive in all mdels examined here, indicating the need fr additinal prcesses. Fr mdes with an SST SLP ffset as bserved in the ACW and Nrth Pacific, Ekman cnvergence acting as a heat surce causes eastward prpagatin relative t the mean cean flw. Sverdrup respnse t Ekman cnvergence, acting n the mean meridinal temperature gradient, causes westward prpagatin relative t the mean cean flw. When the cean temperature adjusts thrugh surface heat flux alne, the mde is advected by the mean cean flw and is damped. Relaxatin t heating in the atmsphere, when perating with Sverdrup respnse in the cean, prduces the nly cmplete slutin presented here that exhibits grwth, with an e-flding timescale f rder (100 days). This slutin appears apprpriate fr the Nrth Atlantic decadal mde. In Nrthern Hemisphere basins, with meridinal bundaries, the same sets f dynamics create the bserved SST SLP phase relatin. An additinal factr is the creatin f SST anmalies thrugh variatins in the western bundary current strengths, which are related t the znally integrated wind stress curl ver the whle basin. If bartrpic and hence fast adjustment is assumed, the resulting psitive feedback can maintain r strengthen the cupled anmalies in the Nrth Pacific and interannual Nrth Atlantic mdes. 1. Intrductin Interannual and decadal variability in the midlatitude upper cean and atmsphere has received increased attentin in recent years as the time series available fr describing such mdes have lengthened. The dminant spatial mdes f variability in the Nrth Pacific, Pacific Nrth American (PNA) pattern, and Nrth Atlantic, Nrth Atlantic scillatin (NAO) pattern, with frequency spectra much brader than El Niñ were described sme time ag, with refinements cntinuing. An eastward prpagating cupled mde in the Suthern Ocean has recently been described (White and Petersn 1996), and has a phase relatinship between sea surface temperature (SST) and sea level pressure (SLP) anmalies Crrespnding authr address: Dr. Lynne D. Talley, Scripps Institutin f Oceangraphy 030, Physical Oceangraphy Research Divisin, La Jlla, CA ltalley@ucsd.edu similar t that in the Nrth Pacific decadal mde and the Nrth Atlantic interannual mde (Kushnir and Held 1996) (cartns in Figs. 1a and 1b). The Nrth Atlantic s decadal mde has a different phase pattern, with lw SLP slightly east f warm SST (Kushnir and Held 1996) (cartn in Fig. 1c). The general questins fr interannual t decadal timescales are whether midlatitude SST anmalies can frce the atmsphere directly and lcally, as well as being created by anmalus atmspheric frcing (which culd well cme frm the Trpics), and which f several pssible mechanisms dminate in creating anmalies in the atmspheric circulatin and in the cean s SST. The well-dcumented spatial relatinships between SST and SLP anmaly centers (reviewed in sectin ) are a diagnstic fr the validity f varius cupled mechanisms. The eastward prpagatin f the Suthern Ocean mde shuld als be reprduced. In sectin 3 varius sets f simple, midlatitude cupled cean and atmsphere mechanisms are investigated 1999 American Meterlgical Sciety

2 AUGUST 1999 TALLEY 017 FIG. 1. Schematic f SST and SLP anmalies in the (a) the Suthern Ocean, and (b) a Nrthern Hemisphere (NH) basin, based n White and Petersn (1996) fr the Suthern Hemisphere (SH) and many surces fr the NH (e.g., Tanimt et al. 1993; Palmer and Sun 1985; Pitcher et al. 1988; Kushnir 1994). The schematic mean cean circulatin is shwn in wide black cnturs (Antarctic Circumplar Current in the Suthern Ocean and subtrpical/subplar gyres in the NH). SST anmalies are shwn in red (warm) and blue (cld). SLP anmalies are shwn in green (high) and range (lw). The example selected fr (b) is high SST in the western bundary regin ringed by cld SST, which induces high SLP centered in the basin, and is applicable t the Nrth Pacific decadal and the Nrth Atlantic interannual mdes.

3 018 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 t see what kinds f balances can and cannt prduce the bserved spatial phasing f SST and SLP anmalies. The Antarctic Circumplar Wave (ACW) and Nrth Pacific decadal and Nrth Atlantic interannual mdes appear clsest t the slutins f sectins 3b(3) and 3c(3). The Nrth Atlantic decadal mde appears t be best described by the slutin f sectin 3e(). In sectin 4 the midbasin slutin f sectin 3b(3), which matches the Nrth Pacific decadal SST SLP pattern, is extended partially t a basin with western and eastern bundaries, and a tendency fr psitive feedback thrugh bartrpic spinup f the circulatin is described. These varius mdels shuld be useful fr evaluatin f mre cmplex mdels and bservatins. They are similar in cncept t the equatrial SST mde intrduced by Neelin (1991) t describe aspects f El Niñ.. Backgrund Gyre-scale sea surface temperature and sea level atmspheric pressure anmaly patterns at shrt t lng timescales have been dcumented in the midlatitude Nrth Pacific (e.g., Rden and Reid 1961; Davis 1976; Namias et al. 1988; Cayan 199; Trenberth and Hurrell 1994; Deser and Blackmn 1995; Tanimt et al. 1993). Mst shw a high SLP anmaly dwnstream (east) f a warm SST anmaly, and lw SLP east f cld SST, at many timescales. Fr example, Tanimt et al. (1993) shw that n bth shrt and lng timescales (perids shrter than 4 mnths up t perids lnger than 5 years), a warm SST anmaly centered bradly at 40N, verlying the separated Kurshi and Oyashi currents (abut 35N and 40N, respectively) is assciated bth with cld SST in the eastern and nrtheastern Pacific and with high SLP shifted eastward and slightly nrthward relative t the warm SST (Fig. 1b). The recently described mde f interannual variability fr the Suthern Ocean, the circumplar wave (White and Petersn 1996), shws this same phase relatin between SST and SLP anmalies in the 3 t 7 yr band: high SLP centers east f high SST centers (Fig. 1a). The gestrphic wind anmaly is thus pleward ver the warm anmalies and equatrward ver the cld, which is the same as fr the Nrthern Hemisphere SST SLP mdes described abve. The anmalies prpagate eastward at a phase speed f abut 6 8 cm s 1 and als include anmalies f the Antarctic sea ice edge. At a speed f 6 cm s 1 at 56S, they take 1 years t circle the glbe and at 8 cm s 1, 9 years. With the bserved znal wavenumber f (see als Qiu and Jin 1997), a warm anmaly appears in a given lcatin every 4 t 6 years. White et al. (1998) shw that the SST and SLP anmalies als spiral meridinally, an aspect that is nt treated herein. In the Nrth Atlantic, Palmer and Sun (1985) shw warm SST anmalies in the Gulf Stream Nrth Atlantic Current separatin regins assciated with high SLP centered t the east and slightly t the nrth. Kushnir and Held s (1996) cmpsite fr interannual variability is similar t this picture, and als includes an SST anmaly f ppsite sign (cld) farther t the nrtheast and an SLP anmaly f ppsite sign (lw) nrth f the cld SST anmaly the pattern which this suggests fllws the mean cean circulatin (Nrth Atlantic Current). Hwever, Kushnir and Held s (1996) cmpsite fr decadal variability differs frm this pattern, with lw SLP verlying and slightly east f a warm SST anmaly (Fig. 1c). In the Nrthern Hemisphere, the znal length scale f the cupled mde appears set by the width f the cean basin. In the Suthern Ocean, the mde is znal wavenumber, hence with a wavelength f abut km (White and Petersn 1996; Qiu and Jin 1997). This may be related t the lcatin f Tasmania/ New Zealand and Suth America/Antarctic Peninsula, which create tw subtrpical gyres at the latitude f the Antarctic Circumplar Current (the Pacific gyre and the Atlantic/Indian gyre). Alternatively, internal dynamics related t Rssby waves r instabilities may set the scale. An intrinsic mde f the Suthern Hemisphere, the Pacific Suth American telecnnectin, may als set the spatial scale (e.g., M and White 1985; Karly 1989; Christph et al. 1998), althugh this scale may still be related t the dispsitin f land and cean. Observed interannual t decadal atmspheric anmalies appear t be equivalent bartrpic (Palmer and Sun 1985; Kushnir and Held 1996). The previusly mdeled atmspheric anmalies that mst resemble the bservatins are equivalent bartrpic except fr a significant shift f the surface anmalies t the east (Palmer and Sun 1985; Pitcher et al. 1988). a. Sea surface temperature anmalies The maximum SST anmalies assciated with the bserved climate mdes are n the rder f 1C in the Nrthern Hemisphere (e.g., Miller et al. 1994; Kushnir 1994) and n the rder f 0.5C in the Suthern Ocean (e.g., White and Petersn 1996). What sets the lcatin f the maximum anmalies and what are likely maintenance mechanisms? 1) High meridinal SST gradients (frnts) are cllcated with the SST anmaly centers. Shifting the frntal lcatins even a small amunt results in a strnger SST anmaly than in ther latitude bands the bserved temperature change acrss the Kurshi is 3 4C/1 lat, while acrss the Antarctic Circumplar Current frnts the change can be 1C/1 lat. Hwever, the frntal regins are much narrwer meridinally than the anmalus SST regins. ) Strm tracks verlay the maximum SST gradients. Changes in the strength f the westerlies and vigr f the strms and/ r a meridinal shift in these patterns culd create lcally anmalus surface heat flux and mixed layer entrainment and hence SST anmalies. Shifts in the winds als affect the lcatin f maximum Ekman advectin, which can thus als result in SST anmalies. 3) In the

4 AUGUST 1999 TALLEY 019 Nrthern Hemisphere, the maximum SST gradient ccurs in the general regin f western bundary current separatin (which is related t the ffshre frntal lcatins). Change in strength f the subtrpical and subplar gyres affects western bundary current transprts. A strng subtrpical western bundary current creates a warm SST anmaly and a strng subplar western bundary current creates a cld SST anmaly. Increased strength in bth gyres culd enhance the meridinal SST gradient itself, which culd then bth be mre sensitive t meridinal advectin and culd als enhance barclinic frcing f the atmsphere. SST anmalies are driven by the atmsphere, thrugh changes in cean circulatin (e.g., Rden and Reid 1961; Latif and Barnett 1994, 1996) and surface layer prperties and flw. Circulatin changes include western bundary current strength and separatin lcatin, lcatin f the subtrpical/subplar bifurcatin in the east, and subductin and advectin f anmalies. Subductin and subsurface advectin f temperature anmalies arund the subtrpical gyre as a feedback has been suggested by Latif and Barnett (1994, 1996) and thers; such subductin has been dcumented fr the Nrth Pacific by Deser et al. (1996) and Schneider et al. (1998). In the surface layer, strnger westerlies accmpanied by enhanced strminess and penetratin f strm tracks farther t the east increase heat lss frm the cean mixed layer t the atmsphere and cause deeper cean mixing, which entrains clder water int the mixed layer (e.g., Cayan 199; Miller et al. 1994). Ekman transprt als changes (e.g., Palmer and Sun 1985). The relative imprtance f varius mechanisms depends n lcatin (Miller et al. 1994). Subplar Nrth Pacific patterns f SST and SLP shw that cld SST in the Gulf f Alaska regin is assciated with high SLP (strnger anticyclnic frcing) (Davis 1976; Cayan 199; Miller et al. 1994). Lw SLP in the central Pacific (deep Aleutian Lw) spins up the subplar circulatin, advecting mre cld water int the separated Oyashi regin (Sekine 1988; Hanawa 1995), and warm water nrthward int the Gulf f Alaska. Cheltn and Davis (198) shw increased suthward flw in the Califrnia Current during high SLP perids. These bservatins suggest that cld SST in the Gulf f Alaska is due t reduced nrthward flw f warm water frm the subtrpics, as als substantiated by changes in bmb-prduced tritium inventries in the nrthern Califrnia Current (VanScy and Druffel 1993). b. Sea level pressure anmalies Midlatitude atmspheric SLP anmalies can be created by remte frcing frm the Trpics (e.g., Lau and Nath 1996; Graham et al. 1994), and vertical r hrizntal advective respnses t lcal heat surces (Smagrinsky 1953), which can be either deep r shallw (e.g., Hskins and Karly 1981). The treatment in the fllwing sectins includes nly lcal frcing. At lng time and space scales in the atmsphere the Sverdrup balance between vertical stretching and meridinal wind hlds (Hskins and Karly 1981; Palmer and Sun 1985). On the ther hand, relative vrticity may becme imprtant in the upper trpsphere where the mean wind is strnger (W. White and S.-C. Chen 1997, persnal cmmunicatin). The main questin here fr the atmsphere is the dminant balance in the temperature equatin fr a given latitude and SST anmaly size, and the assciated vertical distributin f heating. Large SST anmalies at midlatitude can frce a lcal respnse (Pitcher et al. 1988). Midlatitude trpspheric heating at interannual timescales appears t be deep, peaked in midtrpsphere (W. White and S.-C. Chen 1997, persnal cmmunicatin). The resulting lw-level cnvergence and vrtex stretching create pleward wind ver warm SST anmalies thrugh Sverdrup respnse (Gill 1980), which results in high SLP t the east f high SST (Palmer and Sun 1985) as bserved in the ACW and Nrth Pacific (Fig. 1 schematic). Hrizntal advectin f atmspheric temperature anmalies als creates a phase shift between SST and SLP anmalies. Whether this creates high r lw SLP east f high SST depends n the vertical bundary cnditins assumed fr the atmsphere (sectin 3). If, as in Qiu and Jin (1997), it is assumed that the atmsphere is equivalent bartrpic (pressure perturbatin f the same sign at all heights, and maximum perturbatin velcities at the tp f the trpsphere), then high SLP ccurs east f warm SST as bserved. If instead the maximum perturbatin velcities are at the grund r midtrpsphere, then this temperature balance causes lw SLP t lie east f high SST (Hskins and Karly 1981); in the upper trpsphere the pressure anmaly wuld be f ppsite sign. If atmspheric heating is balanced simply by a thermal damping term with n advectin, then the atmsphere temperature anmalies are in phase with SST; thermal wind places maximum wind anmalies ver maximum SST gradients, and hence high SLP directly ver high SST. This culd be the dminant mechanism fr the Nrth Atlantic decadal anmalies described by Kushnir and Held (1996). c. Cupled prcesses The number f pssible cupled mechanisms is large. The rbust phase relatin between the midlatitude SST and SLP anmalies suggests that midlatitude mdels be cnsidered, but external frcing might be necessary t maintain their strength. Petersn and White (1998) shw that external frcing fr the Suthern Ocean anmalies can be prvided thrugh telecnnectin with the western trpical Pacific. Telecnnectin frm the western trpical Pacific t the Nrth Pacific thrugh the Hadley circulatin has been demnstrated fr El Niñ timescales

5 00 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 by Tyrell and Karly (1996), with the dwnward limb f the Hadley cell frcing atmspheric Rssby waves in the regin f the Kurshi; this culd pssibly als be a surce f SST frcing fr this regin, r culd mre generally be the surce f changed wind stress curl, which spins up the cean gyre circulatin, thus creating SST anmalies. Lau and Nath (1996) cnclude that midlatitude Nrth Pacific anmalies are frced thrugh telecnnectins with the trpical Pacific and als shw that midlatitude cupled prcesses can prvide psitive feedback t maintain the anmalies. Christph et al. (1998), in their cupled GCM fcused n the Antarctic circumplar wave, cnclude that the Suthern Hemisphere Pacific Suth American pattern is the mst imprtant factr in frcing the ACW, rather than telecnnectins tied t El Niñ. They suggest that the ACW may nt be a true cupled mde, but rather the respnse f SST in the advective Antarctic Circumplar Current t a statinary atmspheric pattern. Simple analytical mdels incrprating just a few f the pssible mechanisms help t islate pssible cupled mdes. Predminantly analytical studies f the midlatitude cupled cean atmsphere include thse f Pedlsky (1975), Palmer and Sun (1985), Liu (1993), Qiu and Jin (1997), White et al. (1998), and Saravanan and McWilliams (1998). The treatment here is mst similar t Liu s with respect t varius cean mechanisms. The principal difference here is treatment f the atmsphere s respnse, which is assumed here t be either in Sverdrup balance (Palmer and Sun 1985) with strngest meridinal winds ver the strngest SST anmalies r including znal atmspheric advectin f the znal atmspheric temperature gradient. Pedlsky s (1975) scaled analysis f a similar cupled system emphasizes the finite amplitude feedback instability f a barclinic wave in the atmsphere, which draws n the heat surce f the SST anmalies. SST develpment is based n cean Sverdrup transprt acting n the meridinal temperature gradient fr simplicity. A simpler atmsphere is assumed here, and the direct effect f Ekman flw and the weaker effect f znal temperature gradients are included fr the cean. Saravanan and McWilliams (1998) use stchastic atmspheric frcing with a preferred length scale and find that cmbinatin with advectin in a slab cean sets the timescale, as is the general finding here. Their results depend n the relative strength f znal advectin and heating and n the strength f atmspheric damping cmpared with the flux between the cean and atmsphere (as in the cmparisn f results frm sectins 3b and 3e belw). Their meridinal advectin mechanism is stchastic and they d nt include vertical velcity in the atmsphere in a fundamental way. Qiu and Jin (1997) used a tw-layer quasigestrphic cean mdel with advectin f a backgrund meridinal temperature gradient by meridinal gestrphic flw and znal advectin f temperature anmalies. Because it has tw layers, their cean mdel admits barclinically unstable slutins, which dictate the time and hence space scale f the mde. A critical assumptin is that the atmsphere is equivalent bartrpic, the ramificatins f which are explred belw in sectin 3c. Their grwing cupled mde has phase-shifted SLP and SST anmalies resulting mainly frm eastward advectin f the SLP anmaly by the znal wind. Fr Nrthern Hemisphere basins with gyres f restricted znal extent, the cupled mdel f Latif and Barnett (1994, 1996), described abve, and the analytical/numerical results f Jin (1997) are relevant. Jin assumed that the atmsphere damps t match the cean thermal frcing (as in sectin 3e belw); the meridinal SST anmalies create znal wind anmalies, which cuple t the cean thrugh its Sverdrup transprt. The cean includes barclinic Rssby waves, which prvide the imprtant decadal feedback fr the clsed Nrthern Hemisphere basin that he explred. When frced by stchastic winds, the cean respnse has a brad peak at the basin mde. When cupled t the atmsphere, the decadal respnse peak is greatly heightened, thrugh psitive feedback. The fllwing sectin wrks thrugh varius simplified cupled mechanisms fr the znally peridic Suthern Ocean. It is pssible in this way t eliminate sme cupled prcesses since they d nt reprduce the bserved phase relatins between SST and SLP. The remaining chices can be useful fr analyses f cupled numerical mdels. 3. Simple cupled mdels fr the prpagating Antarctic Circumplar Wave In bth the ACW and the Nrth Pacific decadal mde, a large-scale warm SST anmaly is flanked t the east by a high atmspheric SLP anmaly, and cld SST is flanked t the east by lw SLP (schematics in Fig. 1 and references abve). Because f the peridicity f these anmalies in the circumplar wave (znal wavenumber ), it is pssible t say that these patterns are nearly in quadrature, nt just slightly displaced (White and Petersn 1996). These characteristics are mdeled in a simple way here using a quasi-ne-dimensinal (active znal variatin nly) mdel in bth the atmsphere and cean, with meridinal variatin cnfined t a linear backgrund cean temperature gradient. The cean mdel includes SST change thrugh znal advectin by the mean flw, Ekman pumping acting n the upper-layer depth as a prxy fr heating, meridinal advectin due t Sverdrup respnse t Ekman pumping, and surface heat flux, which in reality depends n wind directin and speed amng ther factrs (e.g., Cayan 199). Only a very simple apprach t cean heating/cling is included. Mst f these atmsphere and cean mechanisms are included variusly, but nt tgether, in Palmer and Sun (1985), Pedlsky (1975), Liu (1993), and Qiu and Jin (1997).

6 AUGUST 1999 TALLEY 01 TABLE 1. Parameters. Parameter Functin Magnitude Ekman pumping effect n cean temperature 10 Km 1 h E Ekman layer thickness 100 m h S Surface velcity relative t Sverdrup flw 500 m H A Trpsphere height 10 km Wind speed relative t wind stress 10 m skg 1 SST effect n atmspheric heating s 1 1 Surface wind relative t SST amplitude m s 3 K 1 r Atmsphere thermal damping rate s 1 Ocean thermal damping rate s 1 Ocean mean density 100 kg m 3 dt/dy Ocean mean SST gradient 10 K/000 km Atmsphere mean temperature 50 K /z Atmsphere mean temperature gradient 70 K/10 km u A Atmsphere mean znal wind 10 m s 1 The atmsphere here is heated and cled directly abve the temperature anmalies with maximum heating at midtrpsphere. Clearly the actual heating distributin has a mre cmplicated vertical structure, but as lng as the maximum heating is nt at the grund, this chice prvides the simplest framewrk fr the basic physics in the lwer trpsphere. Fur atmspheric respnses t midlevel heating anmalies are cnsidered: 1) vertical advectin giving rise t a Sverdrup respnse similar t the trpical mdel f Gill (1980), ) znal advectin f the anmalus temperature, 3) meridinal advectin f the anmalus temperature, and 4) relaxatin t the heating surce. Vertical advectin (adiabatic heating) leads easily t the bserved ffset between the SST and SLP anmalies in the ACW, while neither atmspheric relaxatin nr meridinal advectin can. Atmspheric relaxatin is the mst likely respnse fr the Nrth Atlantic decadal mde based n its different SST SLP phase structure. Fr simplicity, it is assumed that respnse t warm and cld anmalies is f similar magnitude but ppsite effect althugh an asymmetric respnse is seen in Pitcher et al. (1988). Further elabratins f this mdel, nt cnsidered here, rapidly becme cmplex, invlving, fr instance, prpagatin f Rssby waves in bth the cean and atmsphere at differing phase speeds, the majr variatins in mixed layer depth acrss the Antarctic Circumplar Current r Kurshi/Gulf Stream, variatins in the meridinal cean temperature gradient, subductin and subsurface westward advectin f temperature anmalies in the cnfined subtrpical gyres nrth f the ACC alng with sea surface prpagatin thrugh subplar gyres suth f the ACC, etc. Eddy fluxes and strm tracks in the atmsphere are nt cnsidered explicitly here the vrticity balance is shwn t be dminated by the Sverdrup balance fr these lng time and space scales, and the effect f eddy fluxes in the temperature equatin is parameterized as Newtnian cling. a. Ocean and atmsphere mdel frmulatins The mdeled cean cnsists f a single layer f unifrm depth, representing the cean s mixed layer. Simple vertical structure is permitted in the trpsphere in rder t calculate the vertical stretching and t check the appearance f barclinicity, which can be cmpared with bservatins. The simplest mdel gemetry is peridic in the znal (x) directin and assumes n meridinal (y) dependence hence is ne-dimensinal in the znal directin. The next simplest gemetry, with a degree f tw-dimensinality, includes a linear backgrund meridinal temperature gradient in the cean. The large cllectin f parameters endemic t even an idealized cupled mdel is listed in Table 1 with units and a brief descriptin f their functin. Many different balances in the atmsphere and cean are explred with results summarized in Table. 1) OCEAN MODEL Assume that the sea surface temperature and znal and meridinal velcities are cmpsed f a backgrund state and an anmaly: T(x, y, t) T(y) T(x, y, t) u(x, y, t) u u(x, y, t) (x, y, t) (x, y, t), (1a c) where T and u are the backgrund SST and znal velcity and T, u, and are the mdeled anmalies. Temperature evlves accrding t T T T (u u) we T, () t x y where T is the anmalus surface heat flux, which damps the anmaly t zer. The Ekman pumping w E arises frm the curl f the wind stress, ; is a psitive, empirical cnstant (with units C m 1 ) and is like the upper layer s mean vertical temperature derivative T/z. This Ekman pumping frcing f SST increases SST in the near-surface thermcline in regins f cnvergence and dwnwelling, and decreases SST in regins f divergence/upwelling. Such a respnse t vertical advectin, but als inversely prprtinal t the changing surface layer depth (which here is cnstant), is ften used

7 0 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 TABLE. Cupled mde behavirs. Atmsphere Znal advectin, assuming equivalent bartrpic Temperature relaxatin u A (/x) Q A 0 Q A r Vertical advectin Znal advectin w (/z) Q A u A (/x) Q A Ocean SLP lw ver SST high Barclinic atmsphere Advected with the mean cean flw Damped due t atmsphere relaxatin SLP high dwnstream f SST high Equivalent BT atmsphere Eastward prpagatin relative t mean cean flw SLP lw dwnstream f SST high Barclinic atmsphere Westward prpagatin relative t mean cean flw Ekman pumping SLP high dwnstream f SST high T/t u(t/x) w E Barclinic atmsphere Eastward prpagatin relative t mean cean flw SLP lw ver SST high Barclinic atmsphere Advected with the mean cean flw Damped due t cean SLP high dwnstream f SST high Equivalent BT atmsphere Advected with the mean cean flw Damped due t cean SLP lw dwnstream f SST high Barclinic atmsphere Advected with the mean cean flw Damped due t cean SLP high dwnstream f SST high Barclinic atmsphere Advected with the mean cean flw Damped due t cean Surface heat flux T/t u(t/x) T SLP lw ver SST high Barclinic atmsphere Advected with the mean cean flw Damped due t cean Grwth due t atmsphere relaxatin SLP high dwnstream f SST high Equivalent BT atmsphere Westward prpagatin relative t mean cean flw Damped due t cean SLP lw dwnstream f SST high Barclinic atmsphere Eastward prpagatin relative t mean cean flw Damped due t cean SLP high dwnstream f SST high Barclinic atmsphere Westward prpagatin relative t mean cean flw Damped due t cean Sverdrup respnse and surface heat flux T/t u(t/x) v (dt/dy) T S in trpical cean mdels (e.g., Zebiak and Cane 1987; Clement et al. 1996). Hwever, this mechanism des nt appear t be dminant in upper-cean respnse in midlatitudes (Miller and Schneider 1998). Instead, a Sverdrup respnse, included belw in Eqs. (6) (8) is likely the mst imprtant means fr changing SST in respse t the wind. The unifrm backgrund surface velcity u is taken t be the average gestrphic eastward speed f the Antarctic Circumplar Current. The surface velcity anmaly u has a thermal wind and an Ekman part: u(x, y, t) u(x, g y, t) u E(x, y, t) h (y) g T dz, (3) f y f h 0 E where is the thermal expansin cefficient and h is the cnstant thickness f the layer cntaining the temperature anmaly; is the cnstant backgrund density, f is the Crilis parameter, u E is the Ekman flw as- sciated with the anmalus meridinal wind stress, and h E is the Ekman layer thickness, which in the simplest mdel culd be chsen equal t h. The thermal wind prtin is zer thrughut since the backgrund meridinal temperature gradient is assumed t be either zer (cncentratin n Ekman pumping respnse) r cnstant (cncentratin n Sverdrup advectin respnse), s n mean gradient in T develps. Hence nly the Ekman prtin f u cntributes t the anmaly develpment in this simple mdel. The Ekman pumping w E fr () is (y) (x) (x) (x) 1 ( ) we k, f f x y f (4) where is the meridinal derivative f f. The wind stress is assumed t have a steady, unifrm znal mean cmpnent and bth meridinal and znal anmalies: (x, t) i[ (x) (x) (x, y, t)] j (y) (x, y, t). (5) The cean s anmalus meridinal velcity is imprtant when there is a backgrund temperature gradient. The surface velcity has gestrphic and Ekman cmpnents: h (x) g T g E f x fh 0 E (x, y, t) dz. (6) The planetary gestrphic balance is assumed fr the cean: w g f. (7) z When this is integrated vertically, the gestrphic prtin f (6) is cnsidered t arise frm Sverdrup balance:

8 AUGUST 1999 TALLEY 03 FIG.. Ocean respnse t meridinal wind anmaly pattern, with a mean meridinal temperature gradient. The diagram is apprpriate fr the NH; the warming and cling tendencies are crrect fr the SH as well. The wind causes znal Ekman transprt u E cnvergence, which causes Ekman pumping w E and hence meridinal Sverdrup transprt S. Ekman pumping in sme mdels is a prxy fr heat cnvergence (dwnwelling warming) and divergence (upwelling cling). Sverdrup transprt acts n the mean temperature gradient. These tw effects ppse each ther. The Sverdrup transprt mechanism is the mre likely physically. 1 f g w E. (8) h S Here h S has units f meters and relates the surface velcity t the Sverdrup transprt and thus depends n hw the latter is distributed vertically; it is nt the depth f an actual layer. All advective changes in SST in () ccur thrugh Ekman pumping, illustrated in Fig.. In Ekman cnvergence regins, where w E 0, the pumping term n the right side f () yields warming (Fig. vertical velcity). On the ther hand, Sverdrup respnse t the same dwnward Ekman pumping is equatrward flw, which, when it acts n the mean meridinal temperature gradient, yields cling (Fig. meridinal velcity). ) ATMOSPHERE MODEL The atmsphere acts n the cean thrugh anmalus heat flux and thrugh wind anmalies, assumed t be gestrphic and in thermal wind balance. The wind is assumed t have a znal mean, u A, and znal and meridinal anmalies, u A and A. The meridinal wind anmaly A is gestrphic: 1 pa A g f A and, (9a,b) A x z f x where p A is the pressure, A is the mean density, and and are the ptential temperature anmaly [see (14) belw] and mean, making the Bussinesq apprximatin. The wind velcity is assumed t be prprtinal t the wind stress: A (y), (10) where has units f m skg 1 and is cnstant. The linear, quasigestrphic vrticity equatin fr the atmsphere is assumed: A A wa ua A f, (11) t x z where the relative vrticity is A ua A. x y Fr quasigestrphy, the Rssby number, U/fL, and the aspect rati, H/L, are small, where U, H, and L are characteristic scales fr the circumplar wave. These assumptins are clearly satisfied fr any reasnable wind speed with length scale f 6000 km, yielding a Rssby number f rder The timescale f the cupled mde is several years. This is much lnger than the timescale f atmspheric Rssby waves, fr which the first tw terms in (11) are imprtant. The znal length scale is large. Thus the nndimensinal parameters L /U and LT are large, where T is the characteristic timescale. [Fr length scales f rder 6000 km (half wavelength f the circumplar wave), wind speeds f 10 m s 1, and timescales f several years, L /U O(50); LT O(5000).] The atmsphere s vrticity balance thus reduces t the steady, linear Sverdrup balance: w f A A, (1) z which is the analg t (7) fr the cean. Balance (1) was used by Gill (1980) fr the Trpics, and by Hskins and Karly (1981) and Palmer and Sun (1985).

9 04 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 The ptential temperature equatin fr the atmsphere with a heating surce and linear damping is u A A wa Q A(x, y, z, t) r, t x y z (13) where the ptential temperature cnsists f a mean and an anmaly: (x, y, z, t) (y, z) (x, y, z, t). (14) Equatin (13) is linearized abut the znal mean wind u A (z); Q A is a surce f ptential temperature (heating). It is chsen t be prprtinal t the SST anmaly and maximum well abve the grund: Q A Q (x, y, t) sin(z/h A ) T(x, y, t) sin(z/h A ), (15) where H A is chsen here t be 10 km, the height f the trppause, and is a psitive cnstant (with units f s 1 ). Smagrinsky s (1953) heating functin chice was similar t (15) and als included expnential dependence, which allwed the maximum heating height t vary. A very simple frm is used here since the emphasis is n the surface wind and the prcess that creates its directin. This is insensitive t the height f maximum heating. Scaling f the atmsphere s temperature equatin (13) was discussed by Hskins and Karly (1981). Assuming a timescale f 3 yr, temperature anmaly f 1 K, mean znal wind f 10 m s 1, anmalus meridinal wind f 1ms 1, vertical velcity f 10 4 ms 1 based n (1), meridinal temperature gradient f 10 K/5000 km, and vertical temperature gradient f 10 K/5 km, the timedependent term in (13) is rder 0 times smaller than vertical advectin, and the tw hrizntal advectin terms are rder 7 10 times larger than vertical advectin. There is likely partial cancellatin f the hrizntal advectin terms, s the seemingly larger magnitude f these individual terms cmpared with vertical advectin might nt hld. Hence vertical advectin can als be cnsidered in dminant balances. Bundary cnditins fr the atmsphere must be impsed t btain slutins. The mst straightfrward assume rigid bundaries at the grund and trppause: w A 0 at z 0, H A. (16) In the fllwing subsectins varius simplificatins f the cean () and atmsphere (13) temperature equatins islate different prcesses in each that lead t the bserved spatial phasing f SST and SLP anmalies. Results are summarized in Table. It will be seen that the chice f atmspheric prcess determines the SST SLP relatin. Eastward and westward phase prpagatin directins are als determined fr each cmbinatin f prcesses, and are fund t depend largely n the chice f cean prcess since time dependence resides in the cean temperature equatin. Clearly the actual slutins include all f the terms t sme degree. The subsectins are rganized accrding t which atmspheric prcess is assumed dminant (clumns f Table ); several cean prcesses are cnsidered within each subsectin (rws f Table ). Based n the SST SLP pattern in the many slutins presented belw, the mst relevant slutins are sectin 3b(3) and pssibly 3c(3) fr the ACW and Nrth Pacific decadal and Nrth Atlantic interannual mdes, and sectin 3e() fr the Nrth Atlantic decadal mde. Ocean damping can be included in any f these slutins and just causes decay. b. Atmsphere: Vertical advectin balancing heating The first chice fr a dminant balance in the atmsphere s temperature equatin (13) is between vertical advectin and heating: w A(x, y, z) wa N Q A(x, y, z, t) z g T(x, y, t) sin(z/h A). (17) The vertical velcity anmaly is upward in heating and dwnward in cling regins. It satisfies the bundary cnditins (16). (Nte that nly anmalies are cnsidered, s symmetry f heating and cling really means symmetry in the degree f lcal atmspheric heating abut the mean.) The atmsphere s vrticity equatin (1) yields f wa f A G1T(x, y) cs(z/h A) z g G1, (18a,b) N HA where N is assumed cnstant. The meridinal wind at the grund is thus pleward ver warm SST and equatrward ver cld SST (Fig. 3a) as was fund by Gill (1980) fr his trpical slutins, and by White et al. (1998) fr slutins pertaining t the circumplar wave. Abve z H A /, the meridinal wind and SLP anmaly signs reverse. The vertical structure is barclinic rather than equivalent bartrpic, but des have the bserved spatial SLP SST phasing in the lwer trpsphere. The meridinal wind anmalies cause Ekman pumping (4), which then acts n the SST, thrugh either upper-layer heat cnvergence (ne-dimensinal slutin) r Sverdrup advectin f the mean temperature gradient (quasi-tw-dimensinal slutin), as described next. Damping f the cean s SST anmaly thrugh the heating term can be included thrughut withut mdifying the effect f these tw basic mechanisms. 1) OCEAN UPPER-LAYER HEAT CONVERGENCE (TABLE, ROW 1, COLUMN 1) Assume fr simplicity that the mean cean temperature is unifrm (dt/dy 0). Then SST evlutin () becmes

10 AUGUST 1999 TALLEY 05 FIG. 3. Meridinal wind anmaly (y) (gray arrws), Ekman transprt u E (thin znal arrws), Ekman pumping w E, and Sverdrup transprt S superimpsed n schematic warm (W) and cld (C) SST anmalies and a mean meridinal cean temperature gradient. (a) Assuming vertical advectin balancing heating in the atmsphere (Table, clumn 1) and als znal advectin balancing heating in the atmsphere with an ad hc equivalent bartrpic assumptin (Table, clumn 3). This might be the best balance fr the ACW and the Nrth Pacific decadal and Nrth Atlantic interannual mdes. (b) Assuming znal advectin balancing heating in the atmsphere with rigid-lid bundary cnditins (Table, clumn ). (c) Assuming relaxatin f atmspheric temperature t lcal heating (Table, clumn 4).

11 06 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 shcks is interesting but nt imprtant t this simple mdel and is hence ignred. A znally peridic slutin t (19) relevant t the circumplar wave, ignring u E (x, t), is T T cs(kx t). (1) The frequency and znal wavelength are related nndispersively thrugh k(u u e ). () The assciated anmalies f gestrphic meridinal wind velcity (18), sea level atmspheric pressure (9a), and cean Ekman pumping (4) are FIG. 4. Slutin fr vertical advectin balancing heating in the atmsphere. Of the slutins presented here, this is the mst likely fr the ACW and the Nrth Pacific decadal and the Nrth Atlantic interannual mdes. (See Fig. 9 fr further elabratin n the Nrth Pacific mde.) Units are arbitrary. (a) Heating Q A, vertical velcity w A, and meridinal wind A with signs apprpriate directly abve a maximum psitive SST anmaly. (b) Anmalies f SST T, SLP p A, meridinal surface wind A, and Ekman pumping w E. Psitive w E (upwelling) cls SST directly, while pleward Sverdrup flw in the same lcatin warms SST. where T T [u ue u E(x, t)] 0, (19) t x (y) e 1 E 1 f he h E u G and u G T. (0a,b) Advectin by the unifrm backgrund flw u is augmented by the psitive definite u e, which arises frm the Ekman pumping term in (). As a reminder, the psitive parameter is the prprtinality factr between Ekman pumping and SST anmaly (), relates wind speed t wind stress (10), and relates vertical velcity in the atmsphere t the SST anmaly T (15). Znal Ekman advectin u E (x, t) arising frm the meridinal wind stress anmalies creates a nnlinearity and hence shcks n the east side f the warm anmalies. In the presence f diffusin (nt included), this prduces a frm f Burger s equatin, where the shck width depends n diffusivity and prpagatin speed. Since the znal temperature gradient des nt feed back int the meridinal wind [e.g., Eq. (18)], the presence f the f A(z 0) G1 T cs(kx t) (3a) f A T A 1 p (z 0) G sin(kx t) (3b) k we G1 kt sin(kx t). (3c) These are illustrated in Figs. 3a and 4 fr the Nrthern Hemisphere ( f 0). Nte that the surface wind stress is pleward ver warm anmalies, atmspheric pressure is high east f a warm anmaly, and Ekman pumping is dwnward (negative) under the high pressure. The time- and space scales f the cupled mde are related by () and must be set externally. The scales and speeds are discussed in sectin 3f. In this slutin, regardless f the cean mdel, the relative phase f the temperature, pressure, and wind stress anmalies at the grund in the Suthern Hemisphere matches the east west phasing f White and Petersn s (1996) circumplar wave and the dminant interannual surface patterns in the Nrthern Hemisphere and decadal pattern in the Nrth Pacific (e.g., Tanimt et al. 1993; Palmer and Sun 1985; Kushnir 1994), althugh the cntinental bundaries cmplicate the Nrthern Hemisphere znal prpagatin, as discussed belw in sectin 4. Hwever, because this mdel includes a reversal f meridinal winds and change in sign f the pressure anmaly abve the midtrpsphere heating surce (15), it des nt match the bservatins, which suggest equivalent bartrpic behavir. [It has been suggested (W. White and S. -C. Chen 1997, persnal cmmunicatin) that relative vrticity in the upper trpsphere, which may becme imprtant because the mean wind is s strng there, can alter these vertical advective/heating slutins s that they appear t be equivalent bartrpic.] With the SST anmaly set by surface layer heat cnvergence and divergence, the cupled mde (1) (3) prpagates eastward relative t the mean cean flw. Using the parameters f Table 3, the phase speed is abut 4cms 1 (see sectin 3f).

12 AUGUST 1999 TALLEY 07 Vertical advectin; Ekman respnse TABLE 3. Estimated znal phase prpagatin speeds relative t mean cean speed. Mdel Phase speed Phase speed (cm s 1 ) u e G 1 4 Vertical advectin; Sverdrup respnse Znal advectin, rigid-lid; Ekman respnse Znal advectin, rigid-lid; Sverdrup respnse u u u f dt G h dy Sv 1 G f u ez sz 1 dt G fua h S dy S A Znal advectin, equivalent bartrpic; Ekman respnse Znal advectin, equivalent bartrpic; Sverdrup respnse u u ezbt szbt 1 f ua 1 dt fuahs dy 10 Decay rate Decay rate (s 1 ) Atmsphere relaxatin; Ekman respnse; damping kg 3 Atmsphere relaxatin; Sverdrup respnse; damping kg ( km wavelength) ( km wavelength) ) OCEAN SURFACE HEAT FLUX (TABLE, COLUMN 1, ROW ) If the cean changes nly thrugh heating and cling rather than Ekman pumping and vertical advectin is still the dminant prcess balancing atmspheric heating, the cean temperature balance () is T T u T, (4) t x where warm anmalies cl (and the atmsphere abve heats) and the ppsite. This simply damps the anmalies: t T Te cs[k(x ut)] (5a) f t A(z 0) G1 Te cs[k(x ut)] (5b) f A t p A(z 0) G1 Te sin[k(x ut)]. (5c) k The signal is advected with the mean cean velcity. The meridinal wind is still prprtinal t T and hence high pressure lies east f warm SST, which matches bservatins. This behavir will be included in all f the remaining slutins withut further cmment. 3) OCEAN SVERDRUP ADVECTIVE RESPONSE (TABLE, COLUMN 1, ROW 3) If the mean cean temperature gradient, which is equatrward almst everywhere, is included, then the equatrward (pleward) Sverdrup flw (8) caused by Ekman dwnwelling (upwelling) can als create SST anmalies. The effect f this Sverdrup advectin mechanism n the cupled mde prpagatin is f ppsite sign t the mixed layer heat cnvergence mechanism, which was just explred in (19) (3) (Fig. ). Assume that the backgrund meridinal temperature gradient dt/dy is cnstant. The prblem is then pseudne-dimensinal, with T and independent f y and u 0. Using (8) with (4) and (10) fr the meridinal advectin term in (), and nw als including the cean surface flux term prprtinal t temperature, which damps the SST anmalies, we have T T f dt T u G1 T, (6) t x h dy x where G 1, defined in (18b), is psitive. Assuming again a znally peridic slutin, the anmalies are t T Te cs[k(x (u u )t)] (7a) f t A G1 Te cs[k(x (u u Sv)t)] (7b) f A t pa G1 Te sin[k(x (u u Sv)t)] k (7c) f t g G1 kt e sin[k(x (u u )t)] Sv h (7d) s f dt usv G 1. (7e) hs dy The slutin is identical t (3) except fr the phase speed and inclusin f damping and is illustrated in Figs. Sv s

13 08 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 3a and 4, using the Sverdrup respnse arrws. The wind remains pleward ver warm and equatrward ver cld SST anmalies. This creates Ekman pumping (suctin) and equatrward (pleward) cean Sverdrup transprt east f warm (cld) SST. This advects cld (warm) water int the cean east f warm (cld) SST anmalies. Therefre the mde prpagates westward relative t the mean cean flw u since u sv is negative fr bth the Suthern and Nrthern Hemispheres. The phase speed is 18 cm s 1 fr the parameters chsen in Table 3 (sectin 3f). If the initial cnditins were truly tw-dimensinal, with temperature anmalies limited in the meridinal directin, then there wuld als be anmalus znal winds giving rise t meridinal Ekman flw, E. If the temperature anmalies were symmetric in the nrth suth directin and the backgrund meridinal temperature gradient unifrm (bth f which are fairly reasnable first-rder assumptins), the meridinal Ekman advectin wuld accmplish tw things: pumping under high pressure and suctin under lw, as already mdeled abve, and strengthening the meridinal gradient f T under high pressure regins and weakening it in the lw pressure regins. That is, it wuld be frntgenetic under highs and frntlytic under lws. This might have sme effect n the atmspheric circulatin, but it likely wuld nt be as imprtant as a simple advectin f warm r cld water. c. Atmsphere: Znal advectin balancing heating Next is cnsidered separately the result f balancing heating in the atmsphere by znal advectin in (13): f A u A u A QA T sin(z/h A) (8) x g z using thermal wind (9). If the znal mean wind u A is unifrm and heating is chsen as in (15), (8) yields A V (x, y, t) G T(x, y, t) cs(z/h A) fu A HA wa W Vz G T(x, y, t) sin(z/h A) fu A gh G A. (9) The signs f the meridinal wind anmaly at the grund and hence the SLP anmaly depend n the unknwn functin V. If the rigid bundary cnditins (16) are applied t the vertical velcity, then V W 0. The vertical velcity structure is smewhat cunterintuitive, with sinking in heating regins and rising in cling regins. This is required by the vrticity balance, the relatin between vertical shear f A and heating, and the rigid bundary cnditins. The surface wind is equatrward ver warm SST and pleward ver cld (Figs. FIG. 5. Slutin fr znal advectin balancing heating in the atmsphere, with rigid upper and lwer bundary cnditins (16). This slutin des nt match any bserved SST SLP phasing. Units are arbitrary. (a) Heating Q A, vertical velcity w A, and meridinal wind A with signs apprpriate directly abve a maximum psitive SST anmaly. (b) Anmalies f SST T, SLP p A, meridinal surface wind A, and Ekman pumping w E. Psitive w E (upwelling) cls SST directly, while pleward Sverdrup flw in the same lcatin warms SST. 3b and 5). A lw pressure anmaly lies dwnstream f warm SST, which is ppsite t bservatins f all decadal mdes. 1) OCEAN HEAT CONVERGENCE RESPONSE (TABLE, COLUMN, ROW 1) If the cean respnse t the meridinal wind anmaly is thrugh Ekman heat cnvergence nly, then frm (4) and (10) w T G f u x E T T (u u ez) T t x uez G, (30) f u where u ez is negative definite. A znally peridic slutin is A A

14 AUGUST 1999 TALLEY 09 t TTe cs{k[x (u u ez)t]} (31a) t A(z 0) G Te cs{k[x (u u ez)t]} (31b) fu A A t p A(z 0) G Te sin{k[x (u u ez)t]}. fu A k (31c) The atmsphere pressure anmaly is lw east f warm SST. Under the lw SLP, Ekman upwelling causes cling, which therefre results in westward prpagatin at speed u ez relative t the mean cean flw. The meridinal winds are strngest directly ver the anmalies because the znal atmspheric temperature gradient must be largest there. ) OCEAN SVERDRUP RESPONSE (TABLE, COLUMN, ROW 3) If the cean s Sverdrup respnse t Ekman pumping advects the backgrund meridinal temperature gradient dt/dy, then T T (u u sz) T t x 1 dt usz G. (3) fua h s dy The slutin fr T, A, and p A is identical t (31) with u ez replaced by u sz. East f a warm SST anmaly, SLP is lw, there is Ekman upwelling, and pleward Sverdrup advectin f warm water. Therefre the warm anmaly prpagates eastward. FIG. 6. Slutin fr znal advectin balancing heating in the atmsphere, with an ad hc equivalent bartrpic assumptin with velcity f the same sign increasing upward. This prduces the crrect spatial phase fr the ACW, Nrth Pacific decadal, and Nrth Atlantic interannual mdes but requires a nnrigid vertical velcity bundary cnditin at the trppause. Units are arbitrary. (a) Heating Q A, vertical velcity w A, and meridinal wind A with signs apprpriate directly abve a maximum psitive SST anmaly. (b) Anmalies f SST T, SLP p A, meridinal surface wind A, and Ekman pumping w E. Psitive w E (upwelling) cls SST directly, while pleward Sverdrup flw in the same lcatin warms SST. 3) EQUIVALENT BAROTROPIC ATMOSPHERE (TABLE, COLUMN 3) Can znal advectin in the atmsphere lead t the crrect spatial relatin between SST and SLP anmalies? Qiu and Jin (1997) assumed an equivalent bartrpic respnse in the atmsphere, which they assumed t be dminated by znal advectin. They did nt apply the rigid bundary cnditins. If a rigid lid is nt impsed at z H A and we make an ad hc chice f V 0 in (9) such that the meridinal wind is in the same directin at all heights and increases upward, fr example, 1 V (x, y, t) (1 G) T(x, y, t), (33) fu A where 1 is a psitive cnstant with units m s 3 K 1, then the surface wind is pleward ver warm SST and high SLP lies t the east f warm SST (Figs. 3a and 6), as bserved. The ppsite is true if the znal wind anmaly is chsen t be minimum at the tp and maximum at the grund. Assuming Ekman heat cnvergence (Table, clumn 3, rw 1) rather than Sverdrup respnse in the cean, the surface wind anmaly, the cean Ekman pumping, and the SST anmaly equatin are 1 A (z 0) T (34a) fu A 1 T we (34b) fu x T T (u u ezbt) T, (34c) t x where the prpagatin speed relative t the mean cean flw u 1 f u ezbt is psitive definite. A wave slutin is A A

15 030 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 t TTe cs{k[x (u u ezbt)t]} (35a) 1 t A ezbt fu A 1 t p A(z 0) A Te sin{k[x (u u ezbt)t]}. u A k (z 0) Te cs{k[x (u u )t]} (35b) (35c) The anmaly prpagates eastward faster than the mean cean flw. This enhancement results frm the Ekman pumping respnse t meridinal wind anmalies (and nt frm the mean znal wind). The meridinal winds are strngest directly ver the SST anmalies because f the mean znal wind, which mves the atmsphere s temperature anmaly dwnstream f the SST anmaly. With this ad hc equivalent bartrpic chice fr A, high surface pressure is fund east f warm SST, as bserved; this is als true at all heights in the trpsphere, with wind anmalies increasing with increasing height. If instead the surface wind had been chsen t be westward, reducing t zer wind at the trppause, then lw surface pressure wuld be fund east f warm SST. If an cean Sverdrup advective respnse is assumed in this ad hc equivalent bartrpic mdel and cean surface fluxes are included (Table, clumn 3, rw 3), the slutin is identical t (35), with u ezbt replaced by u szbt 1 dt fuahs dy 1 t g szbt f uahs kt e sin{k[x (u u )t]}, (36a b) where g is the meridinal Sverdrup surface flw. Since dt/dy and f are f ppsite sign in bth hemispheres, u szbt is negative. Fr this slutin a high SLP east f warm SST causes Ekman pumping and equatrward Sverdrup flw f cld water. This results in westward prpagatin f the mde. In this subsectin we have seen that the SLP SST phase relatin depends again n the dminant atmspheric respnse, whereas the directin f prpagatin depends n the cean respnse (since the time dependence is in the cean temperature evlutin). With rigid bundaries at the grund and trppause, a lw SLP anmaly is fund east f high SST. Smagrinsky (1953) used the same rigid bundary cnditins and his slutins, which include all f the terms tgether in the temperature equatins, shw such a respnse, suggesting that znal advectin in the atmsphere is an imprtant part f his verall slutin. The meridinal wind anmaly reverses at midtrpsphere and the slutin is barclinic. Hwever, the bserved interannual/decadal mdes have high SLP east f warm SST and an equivalent bartrpic atmsphere. If the latter assumptin is made and the bundary cnditins n vertical velcity are disregarded, then it is pssible t find a znal advective slutin that matches the bservatins. d. Atmsphere: Meridinal advectin balancing heating A third tw-term chice frm the atmsphere s temperature equatin (13) is between meridinal advectin f the mean meridinal temperature gradient and heating: d A Q A(x, y, z, t). (37) dy Using (15) fr Q A and the vrticity equatin (1), we btain A T(x, y, t) sin(z/h A) d/dy H w W A A T cs(z/h A ). (38) fd/dy This tw-term balance is nt well psed since it is nt pssible t apply bth bundary cnditins (16) fr w A with nly ne unknwn. Because the mean temperature decreases tward the ples, this balance prduces equatrward winds, bringing cld air int heating regins and pleward winds ver cling regins, which is the ppsite frm the bservatins. Als nte that fr this frm (15) f Q A, the wind is zer at the grund, althugh it might be reasnable t assume that fr a less idealized chice, the wind culd be in the same directin at the grund as at all ther heights. Finally, much f this term in a full mdel culd be canceled by a prtin f the znal advectin term. Thus fr several reasns it appears unlikely that a dminant balance between the meridinal advective atmsphere term and heating hlds. The remainder f this slutin is nt presented here, and it is nt included in Table. e. Atmsphere: Relaxatin t heating A final tw-term chice fr the atmsphere s temperature equatin (13) is between heating and damping: 0 Q A r. (39) Of all slutins presented herein, this ne appears t be the mst apprpriate fr the Nrth Atlantic decadal mde. The meridinal wind anmaly frm the thermal wind relatin (9b) is then A g 1 QA. (40) z f r x Using the sinusidal frm (15) fr Q A, (40) yields T A V G cs(z/h A) fr x HA T wa W Vz G sin(z/h A), (41) f f r x

16 AUGUST 1999 TALLEY 031 T Te t 1 cs[k(x ut)] 1t A G kte sin[k(x ut)] fr 1t pa GA Te cs[k(x ut)] r 1 Gk. 3 (43) This mde is advected at the mean cean flw speed. The cean surface flux term () damps the mde, as befre. The atmsphere s relaxatin respnse als causes decay due t the displacement f the meridinal wind anmaly with respect t the SST anmaly, which means that the Ekman pumping anmaly is in phase with and reducing the SST anmaly. In sectin 3f it is seen that the additinal decay term can be smewhat larger than the basic decay. FIG. 7. Slutin fr relaxatin t heating in the atmsphere. This slutin prduces the crrect SST SLP phasing fr the Nrth Atlantic decadal mde. Units are arbitrary. (a) Heating Q A, vertical velcity w A, and meridinal wind A. The sign fr Q A is apprpriate directly abve the maximum psitive SST anmaly, and signs fr w A and A are apprpriate half a wavelength t the east [see (b) and Fig. 3c]. (b) Anmalies f SST T, SLP p A, meridinal surface wind A, and Ekman pumping w E. Psitive w E (upwelling) cls SST directly, while pleward Sverdrup flw in the same lcatin warms SST. where T is the surface temperature and the psitive cnstant G is defined in (9). The bundary cnditins (16) yield V 0 W 0 0. Here we find pleward wind east f warm SST and equatrward wind east f cld SST, hence lw SLP ver warm SST and high SLP ver cld SST (Figs. 3c and 7). This is similar t Kushnir and Held s (1996) bservatin fr a decadal Nrth Atlantic mde. 1) OCEAN EKMAN HEAT CONVERGENCE (TABLE, COLUMN 4, ROW 1) The equatin fr SST evlutin () retaining the heat cnvergence term and including surface heat flux as well is T T T u G3 T t x x G G3, (4) f r f where G 3 is psitive. A znally peridic slutin t (41) and (4) is ) OCEAN SVERDRUP RESPONSE (TABLE, COLUMN 4, ROW 3) If the cean has a Sverdrup advectin respnse t Ekman pumping, rather than mixed layer heat cnvergence, then (41) and (8) yield T T T u G4 T t x x G 1 dt G4, (44) f r hs dy where G 4 is psitive definite since the signs f the meridinal temperature gradient and Crilis parameter are ppsite in each hemisphere. The slutin is identical t (43) but with G 3 replaced by G 4 : T Te t cs[k(x ut)] Gk. 4 (45) Because the Sverdrup flw advects cld (warm) water int cld (warm) SST anmalies, this mde can grw if the advectin effect is larger than the cean heat flux damping that is, if is negative. This is the nly cmplete slutin presented in this paper that can grw withut external frcing. In sectin 3f, the grwth rate is evaluated and can be n the rder f 100 days. Incmplete slutins fr basins with western and eastern bundaries, presented in sectin 4, als have psitive feedback and grwth. If western and eastern bundaries were included with this slutin, the western bundary current anmalies wuld create negative feedback, hence decay which culd be ffset by grwth due t midbasin cean Sverdrup respnse (45). 3) OCEAN SURFACE HEAT FLUX (DAMPING) (TABLE, COLUMN 4, ROW ) If the cean temperature balance is (4), the slutin is identical t (43) but with damping due t the cean s

17 03 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 FIG. 8. Eastward gestrphic velcity at the sea surface relative t the cean bttm, acrss the Antarctic Circumplar Current (ACC) alng 150W (central Pacific), 88W (eastern Pacific), and in Drake Passage, based n Wrld Ocean Circulatin Experiment sectins P16, P19, and A1. The ACC is lcated between the heavy dts, based n water prperty changes (nt shwn). The mean eastward speed fr the ACC fr each sectin is given. alne. Thus lw SLP lies ver high SST, with bth the atmsphere and cean temperatures respnding passively t the heat flux frm ne t the ther. f. Phase speeds and timescales fr znally peridic slutins Each f the slutins given abve yields a phase speed fr the cupled mde relative t the mean cean flw. Since the purpse f these mdels is t understand a set f pssible mechanisms thrugh the simplest balances (in the usual sense f linear stability analyses), gd crrespndence between the predicted and bserved prpagatin speeds shuld nt be expected. The predicted prpagatin directins (west r east) are rbust. The rder f magnitude shuld be abut right. It is nt clear what mean cean znal velcity is relevant fr these relatively large-scale anmalies. The full latitude range f the eastward Antarctic Circumplar Current includes tw r three narrw bands f cm s 1 embedded in much weaker eastward flw, as illustrated in Fig. 8 with several Wrld Ocean Circulatin Experiment hydrgraphic sectins crssing the ACC in the Pacific frm 150W t Drake Passage. Peak speeds are 40 cm s 1 and mean speeds are 5 10 cm s 1. The bserved Antarctic circumplar wave anmaly speed f 6 8 cm s 1 is f the same rder as the average eastward flw speed and much lwer than the peak frntal speeds, arguing fr either statinarity r westward prpagatin f the mde relative t the mean cean flw. Eastward flw in the Nrth Atlantic and Nrth Pacific currents is similarly banded. Table 3 summarizes the phase speeds f the varius znally peridic mdes based n parameter chices listed in Table 1. A latitude f 45 was chsen. All phase speeds are less than 0 cm s 1 and mst are smaller than 5 10 cm s 1, which is f the same rder as the mean flw speed f the Antarctic Circumplar Current (see sectin 3g belw). The bserved ACW and Nrth Pacific SST SLP spatial ffset is prduced by the mdels with vertical advectin balancing heating in the atmsphere and with znal advectin balancing heating with an ad hc equivalent bartrpic assumptin. With an cean Sverdrup respnse, the ACW prpagates westward fr bth f these mdels. Therefre the bserved ACW prpagatin speed cannt be used t differentiate between these tw mdels. Since the Sverdrup respnse prduces westward prpagatin, it is the mre likely ceanic mechanism than mixed layer heat cnvergence

18 AUGUST 1999 TALLEY 033 respnse since the latter prduces eastward phase speeds. The balance that appears t dminate the Nrth Atlantic decadal mde is lcal adjustment f the atmsphere s temperature t heating (relaxatin). This mde prpagates with the mean flw speed in these znally peridic slutins and grws if a Sverdrup respnse is assumed in the cean. The e-flding time fr the mde (Table 3) is abut 100 days fr a km wavelength, which is apprpriate fr the Nrth Atlantic where the half-wavelength is abut 5000 km. g. Summary f znally peridic slutins Tw scenaris fr interannual t decadal midlatitude mdes have been bserved: 1) warm SST anmaly with high SLP anmaly lying t the east (ACW, Nrth Pacific decadal mde, Nrth Atlantic interannual mde), and ) lw SLP lying abve warm SST (Nrth Atlantic decadal mde) (references cited in the intrductin). These behavirs can be prduced thrugh simple balances withut resrting t the cmplicatins f instabilities r remte frcing. The spatial relatin f SST and SLP anmalies is given in the mdels by the dminant balance chsen fr the atmsphere s temperature balance. Vertical advectin (adiabatic heating) balancing diabatic heating and znal advectin f temperature anmalies balancing diabatic heating (with an ad hc and hence trubling assumptin f an equivalent bartrpic atmsphere) bth prduce the bserved SST SLP ffset fr the first scenari. If the Nrth Atlantic decadal mde were znally peridic, then the mst apprpriate dminant balance wuld be relaxatin f the atmsphere s temperature t lcal heating, as nted by Kushnir and Held (1996). This slutin grws and s culd maintain itself against damping. The eastward phase speed f the mdeled mdes depends n chice f the cean mechanism. Predicted phase speeds are within the range f the bserved mde prpagatin speeds, which d nt differ much frm the mean flw speed. The ne grwing slutin has an e- flding timescale f abut 100 days, which is encuraging. The vertical structure f the mdes in the atmsphere is a ptential diagnstic. Hwever, the mdels here are s idealized that matching vertical structure is very tentative. Mst are barclinic in the sense f winds and SLP anmaly sign reversing in the upper trpsphere relative t the lwer. In the znal advectin/heating mdel, an equivalent bartrpic structure (winds f same sign and increasing upward) had t be impsed t prduce the bserved SST SLP relatin, but it is nt clear that the required disregard f bundary cnditins is physically crrect. W. White and S. -C. Chen (1997, persnal cmmunicatin) shwed frm bservatins that relative vrticity shuld nt be neglected in the upper trpsphere. When they included upper-trpsphere relative vrticity in a mdel with a vertical advective/heating balance in the atmsphere, the bserved cupled mde behavir at the grund was retained and, mrever, an equivalent bartrpic atmsphere was prduced. Little is gained in this simple znally peridic mdel by adding znal channel bundaries r cnfining the anmalies t a limited meridinal regin. As mentined abve, the main additin t the slutin wuld be meridinal SST frntgenesis (frmatin f a znal frnt) under high SLP and SST frntlysis under lw SLP. These d nt change the grss sense f lcatin f the SST anmalies and SLP anmalies relative t each ther. The wavelength f the cupled mde is nt determined by any f these slutins. It must be impsed externally r thrugh the true, mre cmplicated nnlinear dynamics including instability (e.g., Qiu and Jin 1997). One pssibility is that the timescale is set by external frcing, such as prpagatin f the ENSO signal suthward in the Pacific (Petersn and White 1998) r that the space scale is set externally [as invked by Saravanan and McWilliams (1998)] perhaps thrugh an intrinsic Pacific Suth American mde (Christph et al. 1998). The time and space scales are related thrugh whatever dispersin relatin is mst applicable r the space scale is set by the basin scales in the Nrthern Hemisphere and by the tw (Pacific and Atlantic/Indian) gyres f the Suthern Hemisphere; then advectin and Rssby wave prpagatin set the timescale (Latif and Barnett 1994; Cessi 1999). 4. Tendencies in bunded basins f the Nrthern Hemisphere In the Nrthern Hemisphere the cntinental bundaries preclude the znally peridic, prpagating cupled mde described in sectin 3. Nevertheless, the decadal Nrth Pacific and interannual Nrth Atlantic mdes shw high (lw) SLP centered east f warm (cld) SST, as in the circumplar wave. Thus ne might lk t the vertical advective atmsphere and either Sverdrup r Ekman respnse in the cean fr basic balances in these mdes. The Nrth Atlantic decadal mde appears mst like the atmsphere relaxatin mde f sectin 3e, which als includes grwth if an ceanic Sverdrup respnse is assumed. What are sme pssible effects f bundaries n these mdes? Negative feedback arises frm increased transprt f cld waters int the eastern subtrpical gyre in the case f a warm SST anmaly in the west (Cheltn and Davis 198; Miller et al. 1994). These are subducted and advected arund the gyre (Zhang and Levitus 1997; Deser et al. 1996). Westward barclinic Rssby wave prpagatin als carries the uplifted thermcline signal frm the eastern bundary (Latif and Barnett 1994; Jin 1997; Cessi 1999, amng thers). Bth f these have timescales f mre than a decade. Shrt-term ceanic adjustment t the winds created

19 034 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 9 FIG. 9. Idealized bunded basin cupled mde assuming vertical advectin balancing heating in the atmsphere. This slutin is likely relevant fr the Nrth Pacific decadal mde. (a) Mean western bundary current transprt (blue slid) based n mean Sverdrup transprt. Anmalus western bundary current transprt (blue dashed), assuming the SST