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Title Coupling between Wind-Driven Currents and Midlatitude Storm Tracks
Permalink https://escholarship.org/uc/item/644887v9
Journal Journal of Climate, 14(6)
Authors Primeau, F. Cessi, P.
Publication Date 2001-03-01
DOI 10.1175/1520-0442(2001)014<1243:CBWDCA>2.0.CO;2
License https://creativecommons.org/licenses/by/4.0/ 4.0
Peer reviewed
eScholarship.org Powered by the California Digital Library University of California 15 MARCH 2001 PRIMEAU AND CESSI 1243
Coupling between Wind-Driven Currents and Midlatitude Storm Tracks
FRANCËOIS PRIMEAU AND PAOLA CESSI Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California
(Manuscript received 15 December 1999, in ®nal form 30 May 2000)
ABSTRACT A model for the interaction between the midlatitude ocean gyres and the wind stress is formulated for a shallow-water, spherical hemisphere with ®nite thermocline displacement and the latitudinal dependence of the long Rossby wave speed. The oceanic currents create a temperature front at the midlatitude intergyre boundary that is strongest near the western part of the basin. The intergyre temperature front affects the atmospheric temperature gradient in the storm track region, increasing the eddy transport of heat and the surface westerlies. The delayed adjustment of the gyres to the wind stress causes the westerly maximum to migrate periodically in time with a decadal period. The behavior of the model in a spherical geometry is qualitatively similar to that in a quasigeostrophic setting except that here the coupled oscillation involves oceanic temperature anomalies that circulate around the subpolar gyre, whereas the quasigeostrophic calculations favor the subtropical gyre. Another difference is that here there is a linear relationship between the period of the coupled oscillation and the delay time for the adjustment of ocean gyres to changes in the wind stress. This result departs from the quasigeostrophic result, in which the advection timescale also in¯uences the period of the decadal oscillation.
1. Introduction vertical shear. Depending on the particular con®guration The possibility of midlatitude ocean±atmosphere in- of the climatological storm track with respect to the teractions is exciting, because it brings into play the SST-induced heating anomaly, the anomalous eddy ¯ux- ocean in a role beyond that of a passive integrator of es can reinforce or reduce the perturbation in the mean noisy atmospheric forcing. As such, it allows for the ¯ow. Because the eddy ¯uxes are mostly due to transient possibility of enhanced predictability in midlatitude eddies, this scenario emphasizes the necessity of ap- weather patterns on decadal timescales. propriately resolving or parameterizing the baroclinic Although there is some observational evidence of a processes that maintain the storm track in midlatitudes. correlation between decadal ¯uctuations in atmospheric In a previous study, Cessi (2000) has formulated a sea level pressure (SLP) and in sea surface temperatures model that captures a feedback loop between storm (SST) anomalies (e.g., Nakamura et al. 1997; Trenberth tracks and the oceanic currents in which the SST, and Hurrell 1994), the dominant mechanism for the cou- through their coupling to the atmospheric heat budget, pling has not been determined. One of the obstacles in in¯uences the baroclinic eddy activity in the atmosphere analyzing this process, besides the obvious inadequacy and consequently the surface wind stress that drives the of the observational database on decadal timescales, is wind-driven ¯ows advecting the SSTs. The model cou- that different atmospheric general circulation models ples two simple modules for the ocean and atmosphere; (AGCM) respond very differently to similarly pre- namely, Stommel's model for the ocean gyres and scribed SST anomalies [cf. Peng and Whitaker (1999) Green's (1970) parameterization for baroclinic eddies and the references therein]. The discrepancies in the for the midlatitude transport of heat and momentum in response are largely due to differences in the models' the atmosphere. The only prescribed forcing in the mod- climatologies and eddy statistics. This is a crucial prob- el is the net absorbed shortwave heat ¯ux at the top of lem if, as suggested by Peng and Whitaker (1999), the the atmosphere, and although the basic strati®cations in main effect of anomalous SST is to alter the eddy ¯uxes the atmosphere and ocean are prescribed, the momentum of heat and vorticity via small changes in the mean and heat budgets, based on conservation laws, produce a remarkably realistic climatologyÐsurface westerlies at midlatitudes, with a well-de®ned storm track forced by surface heat ¯uxes on the ¯anks of the intergyre Corresponding author address: Francois Primeau, Canadian Centre thermal front. Furthermore, the delayed adjustment of for Climate Modelling and Analysis, Meteorological Service of Can- ada, University of Victoria, P.O. Box 1700, Victoria, BC V8W 2Y2, the gyres by slowly propagating baroclinic Rossby Canada. waves produced a self-sustained oscillation of the at- E-mail: [email protected] mospheric storm track.
᭧ 2001 American Meteorological Society 1244 JOURNAL OF CLIMATE VOLUME 14
Recently, Miller et al. (1998) have shown that the In the atmosphere, we consider the zonally averaged, oceanic component of the feedback loop is in place. An vertically integrated heat and momentum balances. Thus increase in the intensity of the westerly winds across the redistribution of heat and momentum by baroclinic the midlatitudes in the late 1970s to early 1980s pro- eddies must be parameterized in terms of the zonally vided the opportunity for a case study of the oceanic averaged quantities. We adopt the parameterizations of response to a persistent change in the wind stress. Spe- Green (1970) and Stone (1972) as detailed in the fol- ci®cally, there is evidence of a spinup of the gyres with lowing. a western-intensi®ed thermocline response, together with an SST signal in the Kuroshio±Oyashio extension. a. Vertically integrated zonally averaged heat budget Encouraged by the recent observations, we revisit Cessi's model and inquire into the effects of the ide- The atmosphere is assumed to adjust instantaneously alized model con®guration. One unsatisfactory aspect to the ocean, so that the zonally and vertically integrated that can be abandoned without sacri®cing the simplicity heat budget is given by of the model, is the idealized -plane geometry. The (cos͗͘)ץ planetary scale of the wind-driven ocean circulation ϱ 1 C dz ץ suggests that a more realistic spherical geometry be pa ͵ a cos 0 [] used. On a sphere, the westward phase speed of Rossby waves becomes a function of latitude, an effect that ϭ͗Qio͘Ϫ͗Q ͘Ϫs͗Fao͘, (1) might be important given the quasigeostrophic result where the angle brackets indicate a zonal average, and that the timescale for Rossby waves to cross the basin is the latitude. Here, Q is the net absorbed short- sets the period of the oscillation in conjunction with the i wave heat ¯ux at the top of the atmosphere. The Q gyre advection time. In the present study, we have re- o is the outgoing longwave radiation that is parameter- formulated Cessi's (2000) model by recasting the ocean ized by linearizing the ``gray Stefan±Boltzmann law,'' module in terms of the planetary geostrophic equations. Q ϭ G() 4 , about a mean value, ⌰ (in kelvins), The planetary geostrophic equations, unlike the quasi- o so that geostrophic equations can retain the full variation of the
Coriolis parameter and are not restricted to small hor- Qo ϭ A ϩ Bs. (2) izontal variations in the thickness of the wind-driven layer. The constants A and B are prescribed, and if the at- The plan for the paper is as follows. In section 2, we mosphere were in radiative equilibrium they would de- describe the atmospheric module, comprising vertically termine the mean surface temperature, as well as the integrated, zonally averaged heat (section 2a) and mo- difference in temperature between the pole and the equa- mentum (section 2b) budgets. In section 3, we present tor. The term F is the ¯ux of heat from the atmosphere the oceanic module based on the planetary geostrophic ao equations for the momentum budget (sections 3a, 3b) into the ocean. The zonally averaged air±sea heat ¯ux and the thermodynamic budget, based on an advection± must be weighed by the fraction of a latitude circle, s, diffusion equation for the SST (section 3c). In section occupied by the ocean. Following Haney (1971), the 4, we present the climatology of the model, and in sec- ¯ux of heat through the ocean's surface is given by the tion 5, we describe the variability produced by the cou- approximation pled model that we contrast with the results obtained Fao ϭ (s Ϫ Ts), (3) from a quasigeostrophic -plane formulation by Cessi (2000). In section 6, a box model is analyzed to clarify where Ts is the SST, and is the bulk heat transfer the dependence of the oscillation's period on some of coef®cient. The Cpa is the speci®c heat of the atmosphere the parameters. Finally in section 7, we present a dis- at constant pressure, and is the density of the atmo- cussion and summarize the results. sphere, assumed to be a function of height only. The zonally averaged heat ¯ux ͗͘, is parameterized to be down the mean gradient, that is, 2. The model atmosphere ͗͘ץ The strategy of the model is similar to that used in ͗͘ϭϪ . (4) ץ a Cessi (2000) except that spherical polar coordinates are used. We make the simplifying assumption that on the The form (4) has been adopted by Green (1970) as a timescales of interest, that is, much longer than a month, plausible representation of the heat ¯ux by midlatitude the atmosphere is in equilibrium with the ocean. Thus, baroclinic eddies, and has also been derived by Cessi there are two atmospheric diagnostic variables, the sur- (1998) for the mean meridional circulation in the Trop- face potential temperature, s, and the surface wind ics. Pavan and Held (1996) summarize the dependence stress, s, that are in equilibrium with the two oceanic of the eddy diffusivity on the mean heat gradients. prognostic variables, the upper-ocean temperature, T, Here, for simplicity, we take it to be a function of height and the thermocline thickness, h. only 15 MARCH 2001 PRIMEAU AND CESSI 1245
ϭ s exp(Ϫz/d), (5) accelerate the westerlies in the midlatitudes. However, a local relationship between the potential vorticity and where and d are constants. s heat eddy ¯uxes and their respective mean gradients is The potential temperature pro®le in the atmosphere found for baroclinic eddies growing on large-scale jets is taken to be (Pavan and Held 1996). Thus, the diffusive closure
͗(,z)͘ϭSz ϩ s(), (6) schemes that apply for conserved quantities when there is a scale separation between the mean and eddy ®elds, where the strati®cation S is a prescribed constant, and are appropriate for heat and potential vorticity. Green the dynamical part of the potential temperature is in- (1970) shows that in the quasigeostrophic approxima- dependent of height. Thus the vertical resolution of tion, it is possible to relate the ¯ux of momentum to the atmosphere is equivalent to that of a two-level the ¯uxes of potential vorticity and potential tempera- model. Last, the vertical structure of the density is ture (i.e., heat). taken to be For our purpose, it is necessary to apply the ideas of
ϭ s exp(Ϫz/D). (7) Green (1970), to a spherical geometry, and here we are guided by White (1977) who has generalized Green's With the above speci®cations, the heat budget for the model to spherical polar coordinates. An appropriate atmosphere leads to the following equation for the zon- de®nition of the quasigeostrophic potential vorticity on ally averaged surface temperature , s the sphere is given by White (1977):
sץ Dd 1 ץ U cos) fo)ץ Vץ ϪC cos 1 , q ϭ 2⍀ sin ϪϪ ϩץ pa ssa2 cos D ϩ d zSץץ ץ[] a cos[] ϭ͗Qio͘Ϫ͗Q ͘Ϫs͗Fao͘. (8) (10)
The boundary conditions are chosen to ensure that there where f o ϭ 2⍀ sin o is a typical midlatitude value of is conservation of heat in the hemisphere, that is, the Coriolis parameter, and a is the radius of the earth. In this de®nition, the planetary vorticity is exactly rep- ץ s ϭ 0at ϭ 0, . resented, but the stretching term is approximated by 2 replacing f with f o. This approximation, supplementedץ by the use of f o in the thermal wind relation for the b. Vertically integrated zonally averaged zonal velocity, momentum budget g (١, (11 fozv ϭ k ϫ The longitudinally and vertically integrated zonal mo- ⌰ mentum balance for the atmosphere in statistical steady guarantees that the total energy and vorticity are ap- state is given by propriately conserved (Mak 1991). With this de®nition, -͗u͘ cos2) the following relationship between eddy ¯uxes of mo)ץ ϱ 1 Ϫ dz ϭ . (9) mentum, potential vorticity, and potential temperature sץ a cos2 ͵ 0 is obtained: ץ In the above, s is the zonally averaged zonal component 1 (͗uЈЈ͘ cos2) of the surface stress, and the left-hand side is the ver- 2 ץ tically integrated convergence of momentum ¯ux. a cos Therefore, at least when mountain torque can be ne- ͗ЈЈ͘ ץ glected compared to the viscous surface drag, the sur- ϭϪ͗ЈqЈ͘ ϩ fo . (12) zSץ face stress can be estimated through knowledge of the lateral momentum ¯ux. Green (1970), showed that an Following Green (1970), the eddy ¯uxes of potential estimate of the momentum ¯ux can be obtained assum- vorticity and of potential temperature are parameterized ing that it is dominated by baroclinic eddies, ͗u͘ ഠ down the zonal mean gradient, that is, ͗uЈЈ͘, where primes indicate departures from the zonal q͗͘ץ average, and making the quasigeostrophic approxima- ͗ЈqЈ͘ϭϪ , (13) ץ tion. The momentum transport by the mean overturning a ͗͘ץ circulation, ͗u͗͘͘, is neglected, in accord with the qua- sigeostrophic approximation. Fortunately, this is not a ͗ЈЈ͘ϭϪ . (14) ץ bad approximation, even in the Tropics, for the verti- a cally averaged momentum ¯ux. To make further pro- Here, is the same eddy diffusivity used in the at- gress, the eddy ¯uxes must be related to zonal mean mospheric heat budget, and its expression is given in quantities. Simply parameterizing the momentum ¯ux (5). Substituting (12), (13), and (14) into (9) and using as downgradient diffusion of the zonally averaged wind (10), we obtain an equation for the surface stress that is not appropriate because the eddies are observed to contains only zonal mean quantities: 1246 JOURNAL OF CLIMATE VOLUME 14
ϱ (͗U͘ cos) f ͗͘ Ϫ ϭ 2⍀ sin ϪϩϪo f ͗͘ dz. (15) s ͵ aacosSSao 0 Ά []zz ·
Applying the thermal wind relation (11) to the potential small horizontal variations in layer thickness, as is ap- temperature pro®le (6), the zonally averaged zonal wind propriate for the wind-driven ocean circulation on the can be related to the surface temperature and to the planetary scale. The planetary geostrophic equations surface wind Us through also neglect inertia, and this is a useful approximation for our purposes, since we wish to isolate coupled sץ g ͗U͘ϭ͗Us͘Ϫ z. (16) ocean±atmosphere interactions. Ocean models that re- tain the nonlinear effects of relative vorticity advectionץ a⌰2⍀ sino Last, the surface wind can be related to the surface stress can exhibit intrinsic variability even with prescribed, through the linear drag law, steady wind stress. The permanent thermocline is modeled by a single s ϭ s␥͗Us͘. (17) interface separating two ¯uid layers of uniform but dif- Substituting the vertical pro®les, for ͗U͘, ͗͘, , and ferent density. Only the top layer, of thickness h, is set into (15) and performing the vertical integrations, we into motion by the wind stress. Below the interface the obtain a ®nal expression relating the surface stress to ¯uid is assumed to be at rest and isolated from the wind- the surface temperature pro®le, driven circulation. The density jump across z ϭϪh is constant, and gives rise to a reduced gravity gЈ. Al- s d* (s cos) though the momentum and mass balance for the ocean Ϫ s ␥ a2 cos model will ultimately be expressed with only one prog- [] nostic equation for the thickness of the wind-driven lay- 2⍀ sin d*g(cots) f er, h, the dynamics are more transparent by considering ϭϪd* ϩϩos , (18) ss a 2⍀⌰a3 cos Sda the momentum and mass balances for the Ekman and [] interior layers separately. where
Dd a. The Ekman layer d* ϭ . (19) D ϩ d The layer of thickness h is partitioned into two layers: The boundary conditions are a top Ekman layer of uniform thickness he, overlying an interior layer of thickness h Ϫ he. The dynamics of the Ekman layer is governed by ϭ 0at ϭ 0, , s 2 fue ϭϪgЈh /a Ϫ ␣r e, (21) supplemented by the constraint of no net surface torque: Ϫf e ϭϪgЈh /a cos Ϫ rueseϩ /(h ), (22) /2 (u ϭ w . (23 ´ ١ cos2 d ϭ 0. (20) Ϫh ͵ s eh e e 0 The equations are expressed in spherical polar coordi- The latter constraint can be enforced because in (18) nates, with the longitude, and the latitude. r is the the scale height of the eddy diffusivity, d, is considered Rayleigh drag coef®cient, and the parameter ␣ Ͼ 1, to be unknown. which multiplies the drag coef®cient in the meridional In summary, (8) and (18) are the diagnostic equations momentum equation increases the dissipation in the zon- for the model atmosphere that determine ͗͘ and s once al direction. This anisotropy in the frictional parame- the net absorbed shortwave heat ¯ux, ͗Qi͘, is speci®ed terization allows us to adequately resolve the western and once the oceanic surface temperature Ts is known. boundary layer without excessively damping interior The former is speci®ed in the present model, while the perturbations. The choice of a stronger drag coef®cient latter is determined by examining the oceanic heat and in the zonal direction is also consistent with the nu- momentum dynamics. merical simulations of Haidvogel and Keffer (1984), which showed that the variations of the Coriolis param- eter with latitude makes eddy diffusivities more effec- 3. Model ocean tive in the zonal direction. The other parameters are gЈ, The ocean model is based on the planetary geostroph- the reduced gravity, and a, the radius of the earth. Sub- ic equations in spherical coordinates. This model, unlike stituting the horizontal velocity from (21) and (22) into the quasigeostrophic equations, can retain the full var- the continuity equation (23) we obtain the expression iation of the Coriolis parameter and is not restricted to for the Ekman pumping we, in terms of the oceanic 15 MARCH 2001 PRIMEAU AND CESSI 1247
(h١T)] ϭ Fao. (32)´ UhT) Ϫ Ah١)´ ١ prognostic variable h and the atmospheric diagnostic Cw[(hT)t ϩ variable . s On the left-hand side, the second term is the divergence of the heat ¯ux carried by the vertically averaged geo- b. Below the Ekman layer strophic plus Ekman currents U, and the third term pa- rameterizes the transport by mesoscale eddies as a dif- ١T. The ,The dynamics of the thermocline layer, of depth h Ϫ fusive ¯ux down the temperature gradient h , is governed by e right-hand side of (32) is the ¯ux of heat through the top of the ocean, given by (3), and C is the heat fu ϭϪgЈh /a Ϫ ␣r, (24) w capacity of water. We assume that there is no exchange Ϫf ϭϪgЈh /(a cos) Ϫ ru, (25) of heat through the bottom of the layer, at z ϭϪh. The oceanic heat budget (32) is guaranteed to conserve heat (h Ϫ h )u ϩ w ϭ 0. (26)´ ١ h ϩ th e e when supplemented by the insulating boundary condi-
.We have assumed that the bottom of the thermocline, z tions Ah١hT ´ nà ϭ 0 on the solid walls ϭϪh, is a material surface and thus that there is no With this formulation, the oceanic temperature T is exchange of density with the water below the thermo- not completely passive, because it affects the atmo- cline. spheric potential temperature through the air±sea heat A single prognostic equation in h is obtained com- ¯uxes. Thus, through the atmospheric (8) and oceanic bining the continuity equations for the Ekman and in- (32) heat budgets and the wind stress balance (18), T terior layers to eliminate the Ekman pumping velocity indirectly in¯uences the velocity ®eld (U, V) by which we. This equation is most simply expressed in terms of it is advected. The oceanic velocity is given by (31), the vertically averaged velocities, (U, V), de®ned as and with the speci®cation of the incoming shortwave radiation the model is closed. U ϵ [hueeϩ (h Ϫ h e)u]/h, (27)
V ϵ [hee ϩ (h Ϫ h e)]/h. (28) 4. Climatology The vertically averaged velocity is entirely determined In this section we describe the climatology of a typical by h and s and is given by solution, obtained for the set of parameters listed in (␣r22ϩ f )U ϭϪgЈ[ fh /a ϩ ␣rh /(a cos)] Table 1. The only forcing for the coupled system is the in-
ϩ rs /(h) (29) coming net shortwave radiation, ͗Qi͘ that is constructed by ®tting a simple polynomial to the data of Stephens 22 (␣r ϩ f )V ϭ gЈ[ fh/(a cos) Ϫ rh /a] et al. (1981), that is,
Ϫ f /(h). (30) 2 s ͗Qi͘ϭϪQ1 sin ϩ Q 2 sin ϩ Q3, (33) Last, the evolution equation for h is and is plotted in Fig. 1 (solid curve). Also shown in Fig. 1 is the outgoing longwave radiation, Q , (dashed Uh) ϭ 0. (31) ͗ o͘)´ ١ h ϩ t curve) that requires knowledge of the atmospheric tem-
It only requires knowledge of the wind stress s and it perature, and here the time-averaged solution is used. is independent of the Ekman layer thickness he.We At low latitudes the incoming radiation exceeds the out- impose the mass-conserving boundary condition that going radiation, while at high latitudes, the opposite is U ´ nà vanishes at the solid walls. true. To maintain this equilibrium, the wind-driven gyres in the ocean and the parameterized baroclinic eddies in the atmosphere carry poleward the heat absorbed at low c. Thermodynamics latitudes. This increases the surface temperature at high Although uniform density layers are a useful ideali- latitudes where the excess heat is radiated back to space. zation for the dynamics of the wind-driven ocean, ther- The time- and zonally averaged surface temperature modynamics are cumbersome to represent. We thus al- pro®le is plotted in Fig. 2 (solid curve), along with the low the temperature T to vary horizontally within the temperature pro®le that would be obtained for a radi- layer, without accounting for the horizontal pressure gra- ative equilibrium solution, (dashed±dotted curve), dients and associated velocity vertical shears that should RE ϭ (͗Q ͘ϪA)/B. (34) accompany lateral temperature gradients that are not s i compensated by salinity. The temperature at the equator is reduced from the ra- Thus our oceanic heat balance treats T as a passive diative equilibrium temperature by about 10ЊC and scalar, vertically uniform throughout the layer, which is warmed at the pole by about 30ЊC, as a consequence of advected by the wind-driven velocity ®eld, (30). Fol- the northward transport of heat by the ocean and the lowing the treatment in Young (1994) the vertically in- atmosphere. tegrated upper-ocean heat content hT is governed by The time-averaged surface wind stress is plotted in 1248 JOURNAL OF CLIMATE VOLUME 14
TABLE 1. List of parameters. Radiation parameters Ϫ2 Q1 322.72 W m Incoming radiation polynomial coef®cient Ϫ2 Q2 75.26 W m Incoming radiation polynomial coef®cient Ϫ2 Q3 309.4 W m Incoming radiation polynomial coef®cient A 200WmϪ2 Outgoing longwave parameterization coef®cient B 2.4 ϫ 10Ϫ2 WmϪ2 ЊC Ϫ1 Outgoing longwave parameterization coef®cient Atmospheric parameters Ϫ3 s 1.25 kg m Surface density of air D 8000 m Scale height for reference atmospheric density S 5.0 ϫ 10Ϫ3 ЊCmϪ1 Atmospheric basic strati®cation Ϫ1 Ϫ1 Cpa 100Jkg ЊC Speci®c heat of air 6 2 Ϫ1 s 2.7 ϫ 10 m s Eddy diffusivity coef®cient ⌰ 273.0ЊK Reference temperature for Boussinesq approximation Ϫ4 Ϫ1 fo 1.0284 ϫ 10 s Reference Coriolis parameter Coupling parameters 23WmϪ2 ЊC Ϫ1 Bulk heat transfer coef®cient ␥ 2.4 ϫ 10Ϫ2 msϪ1 Drag coef®cient s 110/360 Fraction of latitude circle occupied by the ocean Ocean parameters W 110Њ of longitude Width of ocean basin H 1000 m Initial thermocline depth gЈ 0.022 m sϪ2 Reduced gravity 1000 kg mϪ3 Density of water Ϫ1 Ϫ1 Cw 4000 J kg ЊC Speci®c heat of water 2 Ϫ1 Ah 150 m s Horizontal eddy diffusivity coef®cient for temperature r 6.0 ϫ 10Ϫ8 sϪ1 Ocean Rayleigh drag coef®cient ␣ 10 Anisotropy coef®cient for Rayleigh drag General parameters a 6.37 ϫ 106 m Radius of the earth ⍀ 7.2722 ϫ 105 sϪ1 Earth rotation rate g 9.8msϪ2 Acceleration due to gravity
FIG. 1. Zonally averaged incoming shortwave radiation ͗Qi͘ and time- and zonally averaged Ϫ2 outgoing longwave radiation ͗Qo͘ at the top of the atmosphere (W m ), as a function of the sine of the latitude for run 5. 15 MARCH 2001 PRIMEAU AND CESSI 1249
FIG. 2. Zonally and time-averaged temperature pro®le (ЊC) as a function of sin for run 5. RE The dashed±dotted curve gives the atmospheric surface temperature,s , for the radiative equi-
librium. The dashed curve shows s for the case with baroclinic eddies in the atmosphere, but
with no ocean. The solid curve shows s , for the case with both the atmosphere and the ocean transporting heat meridionally. The dotted curve gives the time- and zonally averaged ocean temperature.
Fig. 3, (solid curve). North of 29ЊN there are westerlies, In Fig. 4, the transport stream function for the ocean with a maximum6msϪ1 surface wind speed at 58ЊN is contoured. The ocean circulation consists of three corresponding to a surface wind stress of 0.18 N mϪ2. counter-rotating western-intensi®ed gyres (there is a At low latitudes there are easterlies with a maximum weak cyclonic gyre in the Tropics). The maximum trans- surface wind speed of 4 m sϪ1 corresponding to a surface port is 54 Sv (1 Sv ϭ 106 m3 sϪ1) in the subpolar gyre, stress of 1.1 ϫ 10Ϫ1 NmϪ2. For reference, the surface and 41 Sv in the subtropical gyre. The meridional trans- stress that would be obtained if there were no ocean is port, driven by the curl of the wind stress in the interior, plotted with the dashed curve. The difference between is returned in swift western boundary currents that are the two pro®les is due to the thermal feedback of the con¯uent between the subtropical and subpolar gyres. wind-driven ocean heat transport on the driving winds. This results in a stretching deformation ®eld that con- Clearly the largest effect of the ocean on the atmosphere centrates the isotherms along the intergyre boundary. In is in the region of the westerlies' maximum. As detailed Fig. 5, the ocean temperature is contoured. There is a in the following, the heat transport by the ocean is north- strong temperature front near 55ЊN in the region be- ward at all latitudes, and tends on average to reduce tween the subpolar and the subtropical gyres. In Fig. 6, the pole-to-equator temperature gradient. However, the the heat ¯ux through the ocean surface is contoured, reduction in the gradient in the surface temperature ®eld with shading indicating regions of heat ¯ux into the does not happen at all latitudes. The con¯uent currents atmosphere. The heat ¯ux ®eld has a dipole structure in the ocean produce a strong thermal front at the bound- that straddles the temperature front, with a maximum ary between the subtropical and subpolar gyres that lo- heat loss to the atmosphere of approximately 450 W cally intensi®es the atmospheric temperature gradient mϪ2 south of the front and a maximum heat ¯ux into and strengthens the surface westerlies over the corre- the ocean of 200 W mϪ2 north of the front. This dis- sponding latitude band (cf. the dotted curve in Fig. 2, tribution of surface heat ¯ux tends to erode the oceanic which is the zonally averaged ocean temperature). Be- temperature front. The front is nevertheless maintained cause the surface westerlies are driven by eddy ¯uxes by the vigorous western boundary currents that advect of momentum, the acceleration of surface ¯ow implies cold water southward in the subpolar gyre and warm a local intensi®cation of the atmospheric eddy activity water northward in the subtropical gyre. In the atmo- by the oceanic ¯ow. sphere, the converse is trueÐthe surface heat ¯ux pat- 1250 JOURNAL OF CLIMATE VOLUME 14
FIG. 3. The time-averaged surface stress for run 5, (solid curve) and for reference the surface stress that would be obtained if there were no ocean (dashed curve). Although the northward heat transport by the ocean tends on average to reduce the pole-to-equator temperature gradient and the zonally averaged surface winds, the ocean currents produce a strong thermal front at the intergyre boundary that increases the atmospheric temperature gradient and strengthens the atmospheric jet over the corresponding latitude band. tern tends to strengthen the baroclinicity, which is then The impact of the air±sea heat ¯ux is best understood balanced by the baroclinic eddies that erode the tem- by considering the heat budget for the whole system. perature gradient. Thus, the heat ¯uxes through the air± The time- and zonally averaged heat budget for the sea interface are responsible for maintaining the in- ocean is given by creased eddy activity in the atmospheric storm track.
FIG. 4. Contour plot of the time-averaged transport streamfunction FIG. 5. Contour plot of the time-averaged ocean surface temperature for run 5. The contour interval is 10 in units of Sverdrups (10 6 m3 ®eld for run 5. The contour interval is 4ЊC, and the dashed contours sϪ1), and the dashed contours are negative. are negative. 15 MARCH 2001 PRIMEAU AND CESSI 1251
FIG. 7. The time- and zonally averaged northward heat transports FIG. 6. Contour plot of the time-averaged surface heat ¯ux between (run 5) in watts, for the ocean (ONHT dotted), the atmosphere (ANHT the ocean and atmosphere for run 5. The shaded regions indicate a dashed), and for the atmosphere plus ocean (ONHT ϩ ANHT solid). heat ¯ux from the ocean to the atmosphere. The contour interval is Also plotted for reference is the transport of heat by the atmosphere Ϫ2 50Wm . if there were no ocean (dashed±dotted).
Tץ A ץ 1 h zonally averaged heat transport are accounted for. In Cww cos ͗VhT͘Ϫ h any event, as Thompson and Wallace (2000) show, theץ[] a Όץ a cos ϭ͗F ͘, (35) dominant mode of atmospheric variability has an es- ao sentially zonal structure. where the left-hand side is the divergence of the zonally The ®rst term on the right-hand side is the atmo- averaged ocean north-ward heat transport. The global spheric northward heat transport, ANHT, and the second heat budget can be obtained by eliminating the surface term is the ocean northward heat transport, ONHT. The ¯ux in the atmospheric heat budget (8), multiplying by left-hand side is the integrated residual from the equator the area element for a latitude band, 2a 2 cos d, and to between the incoming shortwave radiation and the integrating in latitude: outgoing longwave radiation. Figure 7 shows the north- Radiative Imbalance ward heat transports as a function of the sine of the ͦ latitude. The oceanic heat transport, (dotted curve) is || northward everywhere, with two prominent local max- (͗Q ͘ϪB Ϫ A)2a2 cosЈ dЈ ima at the latitudes of largest mass transport in the sub- ͵ is tropical and subpolar gyres. In the subtropical gyre, 0 ONHT peaks at 1.5 ϫ 1015 W and in the subpolar gyre 15ץ 1 ϭϪ2a cosC d* s it peaks at 0.5 ϫ 10 W. The atmospheric northward heat transport (dashed curve) peaks at 3.2 ϫ 1015 Wץ pa ss a ||near the latitude of the strong thermal front over the ͦ ANHT ocean. This maximum in ANHT is reduced and shifted northward from its position for the no-ocean case T (dashed±dotted curve). This shift toward the latitudeץ h ϩ s2a cosC ͗VhT͘ϪA . (36) where the ONHT is minimum is due to the reductionץ ww hΌa ||in ANHT over the latitude band in the subtropical gyre ͦ where ONHT is largest: the atmosphere must transport ONHT more where the ocean transports less. Note that although the factor s appears explicitly only Although many features of the observed climatology in the ONHT term, its presence is implicit in the ANHT are reproduced in our idealized model, others are poorly term since s appropriately weighs the air±sea heat ¯ux represented. The ocean temperature in the subpolar gyre that determines the zonally averaged surface tempera- is in general too cold. This is probably due to the ab- ture and therefore atmospheric ¯ow that transports the sence of convection, of sea ice, and of a thermohaline heat (see also section 6). The zonally averaged nature component to the ocean circulation in our model. As a of the atmospheric module cannot capture the zonal var- result, the temperature gradient at the intergyre bound- iations of the storm tracks, but their in¯uence on the ary is overestimated, and the resulting air±sea heat ¯ux- 1252 JOURNAL OF CLIMATE VOLUME 14
TABLE 2. List of runs.
Run No. gЈ (m2 sϪ1) Basin width (Њ) 1 0.120 110 2 0.064 110 3 0.032 110 4 0.022 140 5 0.022 110 6 0.022 105 7 0.022 100 8 0.022 95 9 0.022 90 10 0.022 85 11 0.022 75 12 0.022 70 13 0.022 65 14 0.021 110 15 0.018 110 16 0.016 110
es are closer to winter time values than to the annual FIG. 8. Time±latitude plot of the anomalous zonally averaged sur- mean. face wind stress for run 5. Dashed contours are negative and the contour interval is 5 ϫ 10Ϫ4 NmϪ2. 5. Variability The climate variability captured by the model consists cally migrates northward before weakening and rein- of a periodic oscillation involving both the atmosphere tensifying to the south. and the ocean. In the atmosphere, the divergence of the The zonally averaged ocean temperature as a function meridional transport of heat and momentum by baro- of time is contoured in Fig. 9. The shaded contours are clinic eddies oscillates in response to the slowly evolv- proportional to the north±south temperature gradient. ing sea surface temperature ®eld. The surface wind During one period, the temperature front ®rst intensi®es stress, balanced by the divergence of the momentum near 52ЊN, drifts northward approximately 3Њ of latitude, transport, also oscillates, and provides a time-dependent and then weakens before reintensifying to the south. The forcing to the ocean circulation. perturbation of the intergyre temperature front produces In the ocean, long baroclinic Rossby waves are con- temperature anomalies that are then advected around tinually generated near the eastern wall, and propagate both gyres. The evolution of the temperature anomalies westward into the interior, in a continuous attempt to can be seen in Fig. 10, which shows the temperature bring the ¯ow into Sverdrup balance with the changing wind stress. The meridional currents induced by the waves perturb the intergyre temperature front and pro- duce SST anomalies that then alter the atmospheric tem- perature. Therefore the intergyre boundary is the lati- tudinal band where ¯uctuations in the oceanic currents can most affect the atmosphere. Moreover, because the SST gradients are strongest in the western part of the basin where the western boundary currents are con¯u- ent, the feedback to the atmosphere is delayed by the time it takes for the waves to cross the basin. This de- layed feedback prevents dissipative processes in the ocean and in the atmosphere from completely damping out the oscillation. The typical behavior is illustrated with run 5 from Table 2. The time-averaged ®elds and heat balances for this run were presented in section 4. In Fig. 8, the anom- alous surface wind stress in the latitude band of the intergyre boundary is contoured as a function of time through two cycles each lasting 18.4 yr. Positive and FIG. 9. Time±latitude plot of zonally averaged ocean temperature negative wind stress anomalies form near 55ЊN and drift ®eld (lines) and ocean temperature gradient (shading) showing the poleward, under the in¯uence of the evolving SST ®eld. steepening and northward migration of the intergyre temperature front The result is an atmospheric storm track that periodi- near 55ЊN east of the western wall for run 5. 15 MARCH 2001 PRIMEAU AND CESSI 1253
FIG. 10. Sequence of temperature anomalies, (shaded contours) for run 5, being advected around the subtropical gyre (unshaded contours). Negative anomalies have dashed contours, and positive anomalies have solid contours. The contour interval is 0.4ЊC, and anomalies greater than 0.2ЊCor less then Ϫ0.2ЊC are shaded. anomalies greater than 0.2ЊC (shaded) superimposed on strength contribute to a different degree to the zonally the streamfunction ®eld. Alternating cold (dashed con- averaged temperature in the subtropical and subpolar tours) and warm (solid contours) temperature anomalies gyres, with the anomalies in the subtropical gyre hav- are formed at the intergyre boundary and are then ad- ing the smaller relative contribution. The relaxation to vected by the currents. The strongest anomalies are a zonally averaged atmosphere tends to homogenize found in the subpolar gyre, where they propagate east- oceanic temperature zonally resulting in more dilution ward across the basin, and are then advected poleward of an anomaly of given area in the subtropical gyre toward the basin boundary, where they are stretched out than in subpolar gyre. This geometrical discrimination by the northern boundary current. is most effective for anomalies generated very close In the subtropical gyre, the anomalies are much to the intergyre boundary, and cannot be captured by weaker and dissipate quickly, contrary to the result the quasigeostrophic approximation. obtained in the quasigeostrophic calculations (Cessi Remarkably, Fig. 11b shows that the ratio of the area- 2000) where the stronger anomalies circulate in the integrated temperature variance in the subtropical versus subtropical gyre. In Fig. 11a we plot the area-weighed subpolar gyres drops as gЈ is decreased. Typically, in- variance of the zonally averaged temperature anoma- creasing the delay in a delayed differential equations lies for the subtropical and subpolar gyres as a function tends to destabilize solutions that are close to neutral of the control parameter gЈ, and in Fig. 11b we plot the ratio of the two. The ratio of the area-integrated (see section 6). Based on this result, one would expect anomalies is close to unity, although the area occupied that oscillators in the subtropical gyre that have shorter by the subtropical gyre is over 4 times larger than that delays than those in the subpolar gyre might be desta- of the subpolar gyre. This implies that for all values bilized by increasing the delay time (decreasing gЈ). of gЈ, the local temperature anomalies are stronger in Instead, the result that the variance of the zonally av- the subpolar gyre. For the particular case of gЈϭ0.022 eraged temperature rise and fall in unison in both gyres m 2 sϪ1 , the stronger subpolar anomalies can readily be as the control parameter gЈ is varied, indicates that a seen from Figs. 9 and 10. Thus, because of the dif- single unstable delayed oscillator acting at the latitude ference in the basin width between high and low lat- of the intergyre front is responsible for producing the itudes on a sphere, anomalies of comparable area and variance in the temperature ®eld. 1254 JOURNAL OF CLIMATE VOLUME 14
FIG. 11. (a) Area-weighed variance of the zonally averaged temperature variance as a function of the reduced gravity gЈ for the subtropical (dashed) and subpolar (solid) gyres. (b) Ratio of the subtropical to subpolar area-weighed variance of the zonally averaged temperature.
a. Heat balance tudes in the subpolar gyre, the phase lag for the peak in ONHT is between /2 and . This phase lag is con- In the time-dependent zonally averaged heat balance sistent with that obtained with a linear-delayed ordinary an additional heat storage rate term for the ocean must differential equation with two feedback terms, one with be added to (36), and one without delay. In such a model, unstable os- Storage rate cillatory solutions must have the delayed feedback term ͦ ||(ONHT) lag the nondelayed feedback term (ANHT) by more than a quarter period, but less than a half period hT͗͘ץ sC 2a2 cosЈ dЈ (see section 6). tץ ww ͵ 0 The ocean heat storage rate can be decomposed into a part due to isopycnal heaving and a part due to iso- ϭϪONHT Ϫ ANHT pycnal temperature changes, that is, ϩ [͗Q ͘(Ј) Ϫ B Ϫ A]2a2 cosЈ dЈ. (37) Storage rate ͵ is 0 Tץ hץ ഠ sC T ϩ h 2a2 cosЈ dЈ. (38) tץ tץ Figure 12 shows a time series of each term in (37) ͵ ww 0 evaluated at 56ЊN, the latitude where the anomalous Ό ocean northward heat transport is greatest. Two cycles We have approximated each term on the right-hand side of the oscillation are shown. The maximum anomalous by taking the time average, denoted by an overline, of ONHT is more than 3 times larger than the maximum the undifferentiated ®eld. In this way the roles of the ANHT, but a large part of this transport is stored in the layer thickness changes versus the temperature changes ocean and returned during the next phase of the oscil- on an isopycnal surface can be isolated. Figure 13 shows lation. The residual between the transport and storage a time series of the heat storage term decomposed into of heat by the ocean is mostly balanced by the atmo- the above two terms. As would be expected from a spheric heat transport. The radiative imbalance is small geostrophic scaling the heaving term is signi®cantly throughout the oscillation. The peak in the oceanic heat smaller than the term due to an isopycnal change in transport lags behind the peak atmospheric transport by temperature. Hence the ¯uctuations in the thermocline slightly more than a quarter period. In fact, at all lati- displacement do not contribute signi®cantly to the time- 15 MARCH 2001 PRIMEAU AND CESSI 1255
FIG. 12. Time series of anomalies in the ocean northward heat transport (ONHT dash±dotted), the atmospheric northward heat transport (ANHT dashed), the ocean heat storage rate (solid), and the radiative imbalance (dotted), at 56ЊN for run 5. dependent thermodynamic balance. Since the model for- 2⍀ cos gЈh c ϭϪ . (39) mulation does not allow a heat ¯ux through the bottom x a (2⍀ sin)2 of the dynamically active layer, the large spatial vari- ations in the layer thickness provide a geographically For the basin geometry used in our study, the crossing varying oceanic heat capacity. time, Tϫ, as a function of the latitude is given by 4as222⍀ sin T ϭ , (40) b. Timescales ϫ gЈh The oceanic response to a change in the wind stress, where, s ϵ W/360, is the fraction of the latitude circle and thus the thermal feedback on the atmosphere, is occupied by the ocean basin. delayed by the adjustment time for the wind-driven cir- We have conducted two series of numerical exper- culation. This delay is equal to the time for a baroclinic iment in order to explore how the period of the oscil-
Rossby wave to propagate across the basin. In Fig. 14, lation varies with a change in the timescale Tϫ . In one the position of a wave front originating at the eastern series of experiments we varied the reduced gravity gЈ, wall is plotted at successive times. The basin crossing and in the other we varied the width, W, of the ocean time as a function of latitude can be inferred by the basin. Table 2 summarizes the numerical experiments. intersection of the wave front with the western wall. In Fig. 15, the period of the oscillation is plotted as a
Variations of the Coriolis parameter and in the mean function of the crossing time, Tϫ , evaluated at the lat- thermocline depth make the speed of the Rossby waves itude of the intergyre temperature front. The circles a function of position. For long Rossby waves, the var- indicate the runs in which the width of the basin was iation in the thickness h, enters the phase speed only in changed and the squares indicate the runs in which gЈ the Rossby radius of deformation. The contribution from was varied. For the series of runs in which gЈ was the dependence of the potential vorticity gradient on the varied, there exists a linear relationship between the layer thickness is exactly canceled by the Doppler shift period of the oscillation and the crossing time. For the caused by the ¯uid velocity that is proportional to the series of runs in which the basin width was varied the gradient in layer thickness. The phase speed of the long relationship is also nearly linear except perhaps for a Rossby waves is then simply hint of curvature in the runs with the narrowest basins. 1256 JOURNAL OF CLIMATE VOLUME 14
FIG. 13. Time series of the heat storage rate shown in Fig. 12, decomposed into a term due to isopycnal heaving (dashed±dotted line) and one due to isopycnal changes of temperature (dashed line).
The two points at the ends of the dashed line in Fig. dinary differential equation (see section 6) shows that 15 produced only damped oscillations, they correspond for a ®xed delay, an increase in the strength of the to Runs 4 and 13 in Table 2. delayed feedback produces a longer period, consistently The linear relationship between the period and the with Fig. 15. delay time differs from that obtained by Cessi (2000) with a quasigeostrophic formulation for the ocean. There, the period of the oscillation was found to scale as the geometric mean of the advective timescale and of the delay. Here, the temperature anomalies advected by the subpolar gyre are diffused along the northern boundary before any fraction can be reinjected at the intergyre boundary. Thus, the advective timescale does not enter in the ocean±atmosphere feedback, which only operates at the intergyre boundary. In Fig. 15, the slope of the linear relationship between the delay and the period differs depending on whether the delay is varied by changing the width of the basin or by changing the reduced gravity gЈ. We ®nd that increasing the delay by increasing the speed of the Ross- by waves through gЈ produces a larger increase in the period than a comparable increase in the delay obtained by making the basin wider. The difference is due to the fact that increasing the width of the basin not only in- creases the delay, but it also increases the strength of FIG. 14. Plot of a Rossby wave front at successive times. The front the thermal feedback on the wind stress by increasing is initially aligned with the eastern wall. The travel time for a Rossby the coupling between the ocean and the atmosphere wave to cross the basin at various latitudes can be deduced from the through the parameter s in (8). A simple delayed or- intersection of the front with the western wall. 15 MARCH 2001 PRIMEAU AND CESSI 1257
FIG. 15. Period of the oscillation as a function of the time for a Rossby wave in the ocean to cross the basin at the latitude of the intergyre temperature front. The circles indicate runs in which the the width of the basin was varied with gЈϭ0.022 ®xed, and the squares indicate runs where gЈ was varied, but with the width of the basin ®xed at 110Њ of longitude. The solid and dashed lines are least squares linear ®ts to the data.
6. A delayed oscillator box model We illustrate some properties of delayed oscillators by considering a simple delayed ordinary differential equation
xt ϩ ␣x(t) ϩ x(t Ϫ t 0) ϭ 0. (41) We motivate the equation with a two-box model for the ocean atmosphere heat budget. Our purpose is not to obtain a quantitative model for the coupled system but simply to illustrate some of the qualitative properties of delayed oscillators. The following heuristic derivation provides a guidance to interpret the different terms in the equation, and to suggest how the parameters vary as the fraction of a latitude circle occupied by the ocean, s, is changed. In particular, we will discuss how the stability, the period of the oscillation, and the phase lag between the oceanic and atmospheric heat transport de- pend on the parameters. We divide the hemisphere into two regions: one is to the south of the latitude separating the subtropical gyre from the subpolar gyre and the other is to the north of it (Fig. 16). We assume that the changes in the radiative FIG. 16. Delayed oscillator box model con®guration. Two atmo- balance can be neglected compared to the changes in spheric boxes exchange heat with two ocean boxes representing the subpolar and subtropical gyres. The north±south dimension of each the heat transports by the ocean and atmosphere. We then consider the heat budget integrated over each box, box is L. The temperaturesTЈa1 ,TЈa2 ,TЈo1 , andTЈo2 represent box-av- eraged deviations from climatology. and linearized around the steady balance, 1258 JOURNAL OF CLIMATE VOLUME 14
dTaЈ1 by the anomalous wind stress and are therefore pro- Ca1ao1ϭϪANHTЈϩsLFЈ , (42) dt portional to the wind stress at a time t Ϫ t 0, where t 0 is the the averaged spinup time for the ocean gyre, and dTaЈ2 is a fraction of the Rossby wave crossing time, T , C ϭ ANHTЈϩsLFЈ , (43) ϫ a2ao2dt de®ned in (40). Since both the atmospheric transport anomalies and the wind stress anomalies are due to the dTЈ o1 same physical process, namely baroclinic eddies in the sCo1ao1ϭϪONHTЈϪsLFЈ , (44) dt atmosphere acting on the atmospheric temperature gra- dient, the wind stress is also proportional to the anom- dToЈ2 sCo2ao2ϭ ONHTЈϪsLFЈ . (45) alous atmospheric temperature difference. However, the dt gyre anomalies lag the wind stress anomalies by the
We assume that the air±sea heat ¯uxes are proportional gyre spinup time, t 0. Using (53), we can relate the oce- to the temperature difference between the ocean and the anic heat transport to ⌬TЈ time lagged by the delay t 0. atmosphere, Finally, the contribution of the oceanic heat transport on the integrated heat balance must be weighed by the Fao1Ј ϭ (TЈo1 Ϫ TЈa1ao2), and FЈ ϭ (TЈo2 Ϫ TЈa2 ). (46) fraction of a latitude circle occupied by the ocean basin. As in the full coupled model, we assume that the This introduces an extra s dependence in ONHTЈ atmospheric heat storage is negligible for the decadal sL˜ timescales of interest, so that we have the following two ONHTЈϭs ⌬TЈ(t Ϫ t0). (55) slave relations for the atmospheric heat budgets: (2␣Ä ϩ sL) This expression should be compared with (36), where 0 ϭϪANHTЈϩsLFao1Ј (47) only the s that weighs the contribution of the oceanic
0 ϭ ANHTЈϩsLFao2Ј . (48) heat ¯ux on the zonally integrated heat budget appears Taking the difference between (44) and (45) and using explicitly, the other s dependence is implicit in the mass the slave relations (47) and (48) to eliminate the air± transport term Vh. sea heat ¯uxes F and F ,weget With these assumptions, we obtain a single delayed ao1 ao2 differential equation for ⌬TЈ, the temperature difference d⌬TЈ between the ocean boxes, sM ϩ ONHTЈϩANHTЈϭ0, (49) dt d⌬TЈ sL␣Ä sM ϩ⌬TЈ where dt (2␣Ä ϩ sL) || ⌬TЈϵToЈϪ1 ToЈ2, and (50) ͦ ANHTЈ CC M ϵ o1 o2 . (51) s˜ 2(C ϩ C ) ϩ sL ⌬TЈ(t Ϫ t0) ϭ 0. (56) o1 o2 (2␣Ä ϩ sL) To close the system, we need to express ANHTЈ and || ͦ ONHTЈ in terms of the ocean temperature difference ONHTЈ ⌬TЈ. To obtain ANHTЈ, we assume that the atmospheric Last, we introduce the the following quantities to sim- heat transport due to baroclinic eddies is proportional plify the notation to the atmospheric temperature difference ␣Ä L sL˜ ␣ ϵ , and  ϵ . (57) ANHTЈϭ␣Ä (TЈa1 Ϫ TЈa2). (52) M(2␣Ä ϩ sL) M(2␣Ä ϩ sL) By combining Eqs. (46), (47), (48) and (52), we can With the simpli®ed notation, the box model is expressed express the atmospheric temperature difference in terms as a single delayed ordinary differential equation of the oceanic temperature difference d⌬TЈ sL ϩ ␣⌬TЈ(t) ϩ ⌬TЈ(t Ϫ t0) ϭ 0. (58) TЈϪTЈϭ ⌬TЈ, (53) dt a1 a2 (2␣Ä ϩ sL) In Eq. (58), the parameter ␣ Ͼ 0 controls the ef®ciency and thus obtain with which the atmospheric eddies can remove oceanic sL␣Ä temperature anomalies, the parameter  controls the ANHTЈϭ ⌬TЈ. (54) strength of the thermal feedback through the atmo- (2␣Ä ϩ sL) spheric wind stress on the ocean gyre heat transport, The oceanic heat ¯ux anomalies are due to wind- and the parameter M is the thermal inertia of the system. driven gyre anomalies acting on the mean ocean tem- The delayed feedback parameter  is a monotonically perature gradient. The gyre anomalies in turn are forced increasing function of s. The delay t0 is proportional to 15 MARCH 2001 PRIMEAU AND CESSI 1259
FIG. 17. (left panel) Growth rate scaled by the delay time t0 as a function of ␣t0 and t0. Solid contours indicate positive growth rates, and dashed contours indicate negative growth rates. The thick solid contour indicates the neutral curve. The contour interval is 0.5. (right panel) Period of the oscillation scaled by the
delay time t0 as a function of ␣t0. The neutral curve is plotted with the thick solid line. The contours can also be interpreted as the phase lag between ONHTЈ and ANHTЈ with the phase lag given by 2 divided by the contour value. the time for long Rossby waves to cross the basin. It is Hence, the contours in the right panel of Fig. 17 give inversely proportional to the basin width. the phase lag, with the lag given by 2 divided by the To explore the behavior of Eq. (58), we seek solutions contour value. The neutral stability curve (thick solid in the forme(iϩ)t . The characteristic equation for the line in panel b) intersects the /2 phase lag contour frequency and the growth rate , are when ␣ ϭ 0. Hence, the phase lag for unstable param- 2 Ϫ2t 22 eter values with ␣ Ͼ 0 is always greater than /2 and (t ) e 0 ϭ (t ϩ ␣t ) ϩ (t ) , (59) 0000 tends toward, but never exceeds as ␣ and  tend to
t0000ϭϪ(t ϩ ␣t ) tan(t ). (60) in®nity. A graphical solution of t and t as functions of ␣t In summary, we have shown how the stability, the 0 0 0 period of the oscillation and the phase lag between the and t0 is given in Fig. 17. The contours in the right panel show the growth rate. The thick solid line is the delayed and the nondelayed feedback term depends on neutral curve separating stable (dashed lines) from un- some parameters. Delayed oscillators that are close to the neutral stability curve will in general become un- stable solutions (solid lines). For a ®xed delay, t 0, de- creasing ␣ or increasing  tends to destabilize the mode. stable when the delay time is increased. We have shown Also, the neutral curve lies above the line ␣ ϭ , and that the phase lag between the delayed feedback term asymptotes to this line as ␣ → ϱ. Hence, stable solutions (ONHTЈ) and the nondelayed feedback term (ANHTЈ) that are close to the neutral curve will tend to become must be between a quarter period and a half period for destabilized if the delay is increased while keeping ␣ growing oscillations. We have also shown that the pe- and  ®xed. riod of the oscillation decreases when the delayed feed- The period in multiples of the delay t is contoured back coef®cient is increased, and that this coef®cient is 0 a monotonically increasing function of the coupling pa- on the right panel of Fig. 17. For a ®xed t 0 , increasing the strength of the delayed feedback coef®cient,  rameter s. leads to oscillations with shorter periods. Since  is an increasing function of s, the simple box model sug- 7. Summary and discussion gests that increasing s with a ®xed delay time should lead to shorter periods. This is consistent with the full In this study we have extended the model introduced model where increasing the width of the basin (i.e., by Cessi (2000) for the interaction between the wind increasing s) for a ®xed delay time leads to a shorter and the wind-driven component of the oceanic heat period. transport. As in the previous formulation, highly sim- One can also deduce the phase lag of the ocean north- pli®ed and parameterized modules for the ocean and ward heat transport with respect to the atmospheric atmosphere are coupled through the global thermal bal- northward heat transport from the right panel in Fig. ance. The present model replaces the quasigostrophic 17. From Eq. (56) the phase lag between ONHTЈ and formulation with planetary geostrophy in a spherical ANHTЈ is given by hemisphere. In the latter formulation the delay time for the adjustment of the oceanic heat transport to a change t phase lag ϭ 0 2. (61) in the wind stress is a function of latitude. Despite this period difference, the delayed negative feedback mechanism 1260 JOURNAL OF CLIMATE VOLUME 14 identi®ed by Cessi (2000) is still effective at producing switching mechanism they propose. Here we show that a self-sustained oscillation. oscillatory behavior can be obtained solely because of In both models, the intergyre boundary is the site at delayed negative feedback due to Rossby wave propa- which the ocean±atmosphere coupling is most effective gation. and is where the time-dependent ¯uctuations in both In this study we have increased the realism of the oceanic and atmospheric variables are localized. This is formulation of the coupled model introduced by Cessi the region where the time-mean temperature gradients (2000) while retaining much of the simplicity of the are maximum in both the atmosphere and the ocean, original formulation. As such it is an additional contri- and where the westerlies peak. In this sense, the modeled bution to a hierarchy of models of increasing realism feedback is between the separated western boundary necessary for understanding midlatitude interaction be- currents and the midlatitude storm track, through ex- tween the wind and the heat transport of wind-driven changes of anomalous heat ¯uxes at the air±sea inter- currents. The fact that other sources of variability are face. absent in the present model made it possible to clearly Although the mechanism for generating the oscilla- identify the mechanism responsible for producing the tion is the same in the quasi- and planetary geostrophic variability. Future work should study how additional formulations, two important qualitative differences have sources of variability affect the characteristics of the been identi®ed. One difference is that in the spherical oscillation and guide us in attempting to detect it in geometry the temperature anomalies advected into the natural data or GCM simulations. subpolar gyre are stronger than those circulating in the subtropical gyre, which is the opposite of the quasi- Acknowledgments. Our research is supported by the geostrophic calculations. The preferred migration of National Science Foundation and by the Department of temperature anomalies into the subpolar gyre is due to Energy. a geometrical effect on the sphere and it resembles the observations by Sutton and Allen (1997) of the slow northeastward progression of North Atlantic SST. Our REFERENCES calculations indicate that the ultimate fate of the SST Cessi, P., 1998: Angular momentum and temperature homogenization generated at the western end of the intergyre boundary in the symmetric circulation of the atmosphere. J. Atmos. 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