Coupled Ocean–Atmosphere Dynamics in a Simple Midlatitude

3704 JOURNAL OF CLIMATE VOLUME 14

Coupled Ocean±Atmosphere Dynamics in a Simple Midlatitude Climate Model

DAVID FERREIRA AND CLAUDE FRANKIGNOUL Laboratoire d'OceÂanographie Dynamique et de Climatologie, UniteÂ mixte de recherche CNRS-ORSTOM-UPMC, UniversiteÂ Pierre et Marie Curie, Paris, France

JOHN MARSHALL Program in Atmospheres, Oceans, and Climate, Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 22 February 2000, in ®nal form 16 January 2001)

ABSTRACT Midlatitude air±sea interactions are investigated by coupling a stochastically forced two-layer quasigeostrophic channel atmosphere to a simple ocean model. The stochastic forcing has a large-scale standing pattern to simulate the main modes of low-frequency atmospheric variability. When the atmosphere interacts with an oceanic mixed layer via surface heat exchanges, the white noise forcing generates an approximately red noise sea surface temperature (SST) response. As the SST adjusts to the air temperature changes at low frequency, thus decreasing the heat ¯ux damping, the atmospheric spectra are slightly reddened, the power enhancement increasing with the zonal scale because of atmospheric dynamics. Decadal variability is enhanced by considering a ®rst baroclinic oceanic mode that is forced by Ekman pumping and modulates the SST by entrainment and horizontal advection. The ocean interior is bounded at its eastern edge, and a radiation condition is used in the west. Primarily in wintertime conditions, a positive feedback takes place between the atmosphere and the ocean when the atmospheric response to the SST is equivalent barotropic. Then, the ocean interior modulates the SST in a way that leads to a reinforcement of its forcing by the wind stress, although the heat ¯ux feedback is negative. The coupled mode propagates slowly westward with exponentially increasing amplitude, and it is fetch limited. The atmospheric and SST spectral power increase at all periods longer than 10 yr when the coupling with the ocean interior occurs by entrainment. When it occurs by advection, the power increase is primarily found at near- decadal periods, resulting in a slightly oscillatory behavior of the coupled system. Ocean dynamics thus leads to a small, but signi®cant, long-term climate predictability, up to about 6 yr in advance in the entrainment case.

1. Introduction and Deser 1995). As discussed by Deser and Blackmon There has been growing interest in understanding the (1993), the decadal SST ¯uctuations in the North At- nature of the air±sea interactions controlling extratrop- lantic are related to the wintertime atmospheric anom- ical sea surface temperature (SST) variability on cli- alies in broadly the same way as on shorter timescales, matic timescales. As reviewed by Frankignoul (1985), with stronger winds overlying cooler SSTs, while dif- the interannual SST anomalies mainly re¯ect the re- ferent relations are seen on interdecadal and longer time- sponse of the oceanic surface mixed layer to the day- scales. Halliwell and Mayer (1996) showed that in the to-day changes in the local air±sea ¯uxes, which act as westerlies local anomalous turbulent heat ¯ux is effec- a stochastic forcing. Away from frontal zones, the main tive in forcing the wintertime SST anomalies down to SST anomaly forcing is by surface heat exchanges and the decadal period, although in the western North At- vertical entrainment, and turbulent heat ¯uxes also con- lantic, the geostrophic variability strongly modulates the tribute to SST anomaly damping (Frankignoul et al. SST ¯uctuations (Halliwell 1998). The decadal signal 1998). Although the decay time of the SST anomalies may have an enhanced variance around periods of about is typically 3 months, that of the heat content in the 12±14 yr (Deser and Blackmon 1993; Sutton and Allen mixed layer is longer and wintertime SST anomalies 1997), but a dominant SST timescale does not require generally recur during the following autumn (Alexander an active ocean±atmospheric coupling; it could arise through the interplay between stochastic atmospheric forcing with a ®xed spatial pattern and oceanic advec- Corresponding author address: Claude Frankignoul, Laboratoire tion (Saravanan and McWilliams 1998), or re¯ect SST d'OceÂanographie Dynamique et de Climatologie, UniteÂ mixte de recherche CNRS-ORSTOM-UPMC, Case 100, UniversiteÂ Pierre et Ma- modulation by stochastically forced geostrophic ¯uc- rie Curie, 4 place Jussieu, 75252 Paris, France. tuations (Frankignoul et al. 1997; Jin 1997). E-mail: [email protected] General circulation model (GCM) experiments with

᭧ 2001 American Meteorological Society 1SEPTEMBER 2001 FERREIRA ET AL. 3705

Decadal oscillations sustained by a positive midlatitude ocean±atmosphere feedback were invoked by Latif and Barnett (1994) to explain decadal variability in the North Paci®c in the ECHO coupled general circulation model. In their scenario, in an anomalously strong subtropical gyre, the Kuroshio transports more warm water northward and increases the SST in the western and central Paci®c. The atmospheric response to the SST anomaly is such that it sustains the latter via a positive heat ¯ux feedback while decreasing the wind stress curl, which after some delay spins down the subtropical gyre and eventually yields a SST anomaly of the reversed sign. GroÈtzner et al. (1998) have suggested that a similar coupling is at play in the North Atlantic. However, the midlatitude ocean is less active in ECHAM1/LSG and the heat ¯ux feedback negative (Zorita and Frankignoul 1997; Frankignoul et al. 2000), and the ocean is primarily passive in the Geophysical Fluid Dynamics Lab- oratory model (Delworth 1996). Since cause and effect are dif®cult to distinguish in coupled simulations, more studies are needed to understand the nature of the air± sea coupling and establish statistical signatures that could be tested with observed or coupled model data. Early basic studies of the midlatitude coupling have been reviewed by Frankignoul (1985) and Barsugli and FIG. 1. (top) NAO pattern obtained as the leading mode (47% of Battisti (1998). The latter coupled a stochastically the variance) of wintertime sea level pressure over the North Atlantic forced slab (or energy balance) atmosphere to a slab from an EOF analysis. Monthly anomalies were computed from the ocean via surface heat exchanges. The coupling allowed NCEP reanalysis for the period 1948±99 and then averaged to form for an adjustment between air temperature and SST that wintertime (Dec±Mar) anomalies. (bottom) The power spectrum (ar- bitrary units) of the ®rst principal component was computed using reduced the heat ¯ux feedback at low frequencies, en- the multitaper method. The power law in dashed line was obtained hancing the oceanic and atmospheric variability (see by a least squares ®t for periods between 2 and 20 yr. also Saravanan and McWilliams 1998). Longer timescales are introduced in the coupled system by the ocean dynamics. Frankignoul et al. (1997, hereafter FMZ) prescribed SST anomalies generally suggest that there showed that the baroclinic response of the ocean to sto- is some atmospheric response in fall and winter, but it chastic wind stress forcing was red with a decadal dom- is model dependent. For example, in Kushnir and Held inant timescale, and Jin (1997) that the geostrophic cur- (1996), the response to a North Atlantic SST anomaly rents distorted the climatological SST gradient, leading is very weak and baroclinic, while in Palmer and Sun (for realistic SST damping time) to a very weak decadal (1985), Ferranti et al. (1994), and Peng et al. (1995), it peak under a spatially organized wind stress forcing, is stronger and equivalent barotropic, the SST anomaly and a more pronounced one with active atmospheric primarily displacing the storm track and altering the coupling. However, the coupling was based on ad hoc upper-tropospheric eddy vorticity ¯ux. Rodwell et al. assumptions for the zonally averaged quantities. A bal- (1999) showed that much of the observed low-frequency ance between meridional wind anomalies and low-level variability of the North Atlantic oscillation (NAO) can convergence, plus the assumption that the latter is pro- be reproduced by an atmospheric GCM forced by the portional to the SST perturbation, was used instead by observed changes in SST and sea ice. In addition, Czaja White et al. (1998) to study the wind stress feedback and Frankignoul (1999) showed that North Atlantic SST upon free oceanic Rossby waves. Empirical coupling anomalies have an impact on the observed atmospheric approaches have also been used, as in Weng and Neelin variability in late spring and early winter. These studies (1998) and Neelin and Weng (1999), who used the re- suggest that the ocean could lead to a small predict- sponse of atmospheric GCMs to prescribed SST to spec- ability of the wintertime NAO, consistent with the slight ify the wind stress and heat ¯ux feedback acting on SST redness of its spectrum discussed in Hurrel and van anomaly, or in Xu et al. (1998), who coupled an oceanic Loon (1997) and Wunsch (1999). The NAO redness is GCM to a statistical atmosphere derived from a coupled illustrated in Fig. 1, where the spectral slope is slightly GCM, although this procedure biases air±sea ¯uxes to- steeper than in Wunsch (1999) because the NAO is de- ward positive feedback (Frankignoul 1999). A dynam- ®ned by a large-scale pattern rather than by the tradi- ically more consistent approach, albeit simple enough tional NAO index. to be tractable analytically, was used by Goodman and 3706 JOURNAL OF CLIMATE VOLUME 14

TABLE 1. Standard value of the parameters.

␤ 1.8 ϫ 10Ϫ11 mϪ1 sϪ1 f 8 ϫ 10Ϫ5 sϪ1 Ϫ1 co Ϫ2cms

La 700 km Ϫ3 ␳a 1.3 kg m 3 Ϫ3 ␳o 10 kg m Ϫ6 Ϫ3 Ϫ1 Cpo 4 ϫ 10 kg m K

Ha 10 km

Ho 4500 m Ϫ3 CD 1.5 ϫ 10 Ϫ6 Ϫ1 ␥r 1.5 ϫ 10 s Ϫ6 Ϫ1 ␥a 4 ϫ 10 s Ϫ7 Ϫ1 ␥s 1 ϫ 10 s Ϫ8 Ϫ1 ␥e 3 ϫ 10 s Ϫ1 U1 18ms Ϫ1 U2 7ms k 4 ϫ 10Ϫ7 mϪ1

FIG. 2. Schematic longitude±height cross section of the coupled ocean±atmosphere system. Marshall (1999, hereinafter GM), who coupled a two- layer quasigeostrophic channel atmosphere to a baroclinic oceanic Rossby wave via an oceanic mixed layer, and showed that the wind stress curl feedback could anomaly (these are derived from ®rst principles in ap- lead to growing coupled modes. However, the calcu- pendix A): lation was done for an unbounded ocean. On the other atmosphere: hand, the western boundary current plays a key role in (␪ ϭ⌫ aT ϩ F (1( ץthe simple models of Cessi (2000) and Marshall et al. (⌫ϩTxV (2001), with linear dynamics and simple parameteri- sea surface temperature: zations relating its ¯uctuations to heat transport.

(teoT ϭ ␥ (T Ϫ Tץ The goal of the present paper is to re®ne GM's model by taking into account some of the effects of the ®nite width of the ocean, which limits the growth of the cou- ϩ ␥so(␪ Ϫ T) ϩ Q (2) pled perturbations, and use FMZ's statistical approach 1 (xo␺ ϭϪ␣␺ s, (3 ץto␺ ϩץ :to quantitatively estimate the decadal variability that ocean results from the natural variability of the weather forc- co ing. For algebraic simplicity, we only consider large where To is the ocean interior (thermocline) temperature atmospheric zonal scales and linear perturbations to a anomaly; ␺s is the atmospheric streamfunction at the basic oceanic state at rest, which is a reasonable ap- surface, to which the stress blowing over the ocean is proximation for the ®rst baroclinic mode (Sirven and proportional; co is the speed of ®rst baroclinic oceanic Frankignoul 2000), and we do not represent the western Rossby waves; and the ⌫s and ␥s are inverse thermal boundary current, using instead a radiation condition. timescales. Other parameters are de®ned below. A sche- The focus is thus on the air±sea coupling in the central matical representation of the model is given in Fig. 2. and eastern parts of an ocean basin, admittedly a severe The atmospheric model from which (1) is derivedÐ limitation. The coupled model is formulated in section see appendix AÐis a steady two-level quasigeostrophic 2. The response of the atmospheric model to prescribed ␤-plane atmosphere linearized about a mean zonal wind SST is discussed in section 3 and the coupling with the with vertical shear. The transient response of the at- oceanic mixed layer in section 4. The SST modulation mosphere is neglected assuming equilibrium on the by interior motions is taken into account in section 5. timescales we are interested in. The atmosphere loses The in¯uence of the wind stress feedback onto the oce- heat to the ocean mixed layer at the rate ⌫a and includes anic and atmospheric spectra is discussed in section 6 a radiative damping at the rate ⌫r (⌫T ϭ⌫a ϩ⌫r). and some statistical signature of the air±sea interaction Atmospheric dynamics is encapsulated in V, the Dopp- documented. Conclusions are presented in section 7. ler-shifted phase speed of a free Rossby wave in the atmosphere, a rather complicated function of the mean 2. A simple coupled model winds, ␤, the meridional wavelength, and strati®cation [see (A14)]. Resonance occurs for V ϭ 0 when hori- Our simple coupled model can be succinctly stated zontal advection of heat exactly balances adiabatic heat- by three equations for our three main variables, ␪ the ing, so that only dissipation and surface heat exchanges atmospheric temperature anomaly, T the SST anomaly, limit the amplitude. The stochastic forcing F represents and ␺o the ocean interior geostrophic streamfunction the (white noise) forcing of temperature by baroclinic 1SEPTEMBER 2001 FERREIRA ET AL. 3707

␺ץ Q ϭ ar o . (6) xץoo Since the dominant modes of natural variability of the atmosphere at low frequency have a geographically ®xed spatial pattern that is not sensitive to SST forcing, we represent the stochastic forcing as a standing pattern:

Ä Ϫi␻t F ϭ Ck(w)e sin(kx ϩ ␸) sin(ly), (7) where ␻ is frequency, k is zonal wavenumber, l is meridional wavenumber, ␸ is the phase at the eastern

boundary, and Ck is the amplitude. To mimic the NAO (Fig. 1), we choose k ϭ 4 ϫ 10Ϫ7 mϪ1 (zonal wavenumber 2) as standard parameter, and ␸ is determined for each l to obtain a well-located peak at x ϭϪ2000 km for␺˜ , as sketched in Fig. 2 (see also Fig. 11, bottom). FIG. 3. Amplitude (thick solid) and phase (dashed±dotted) of the The observations suggest that the meridional wave- air temperature response to ®xed SST anomalies as a function of the length 2 /l should be on the order of 6000 km, meridional wavelength l (for zonal wavenumber 2). The amplitude ␭y ϭ ␲ is normalized so that it is 1 at resonance. A negative phase corre- satisfying condition (A8). The forcing is a stationary sponds to V Ͼ 0 and a downstream response. The vertical solid and random process with zero mean and we assume that the dashed lines denote barotropicity and resonance, respectively. Shad- frequency spectrum of F is white at low frequency. ing corresponds to equivalent barotropic conditions. Note that

R setting To ϭ Qo ϭ 0 in (2), the oceanic interior motions eddies. Therefore, in this model, the atmosphere has a have no in¯uence on the atmosphere, as in FMZ; white spectrum in the absence of air±sea interaction, R the interplay between the free atmosphere and the and redness only appears through thermal damping (sec- oceanic surface layer is a generalization of Barsugli tion 4) or active air±sea coupling (section 5). Consistent and Battisti (1998) or Saravanan and McWilliams -␺˜ ϭ 0 (no atmospheric dy ץwith observations of low-frequency large-scale atmo- (1998), who assume V spheric modes such as NAO (see Fig. 1), we have as- x namics) and ␥ e ϭ 0 (no entrainment); and sumed in (1) that the zonal scale is much larger than R our equations for the ocean, atmosphere, and coupling the meridional oneÐsee (A8). physics are those of GM, except for the inclusion of The SST anomaly can be changed by air±sea ¯uxes a stochastic forcing and a radiative damping in the Ϫ1 at the rate␥ s , by entrainment of thermocline water of atmosphere. Ϫ1 temperature To at the rate␥ e and by horizontal advection of mean SST by geostrophic meridional current anomalies, Qo. 3. The atmospheric response to a ®xed SST The ocean is assumed to exist in a semi-in®nite do- anomaly main bounded to the east by land (in x ϭ 0) where there To understand the role of atmospheric dynamics, it is a no-normal ¯ow condition (see Fig. 2). There are no is interesting to ®rst study the response of the atmo- western boundary currents and we impose a radiation sphere to a ®xed SST anomaly. Assuming an eikx form condition in the west. The ocean interior variability is for the latter and neglecting the stochastic forcing, (1) governed by ®rst baroclinic Rossby waves excited by can be solved for ␪: winds ␺s. According to our simple atmospheric modelÐ see (A15)Ðthe surface winds are related to the atmo- T spheric temperature ␪ through the relation: ␪ ϭ⌫a , (8) ⌫ϩT iVk ␺ ϭϪm␪, (4) s and the surface wind response is given by (4). The trans- where m is a function of the mean zonal winds and fer function between T and ␪ has an imaginary part, controls the relative strength of the barotropic and bar- which results in a zonal phase shift. Figure 3 illustrates oclinic modes. The thermocline temperature To is as- the amplitude and phase of the response as a function sumed to result from the adiabatic undulation of the of l for zonal wavenumber 2; shading denotes meridi- isopycnal surfaces underneath the mixed layer and thus onal wavelength for which the atmosphere is equivalent can be related to the baroclinic streamfunction ␺o: barotropic (atmospheric ¯uctuations have the same sign throughout the atmosphere and increasing amplitude T ϭ r ␺ , (5) ooo with height). For V Ͼ 0, the phase is negative and the where ro is an appropriate scaling factorÐsee (A17). atmospheric response is shifted downstream of the SST The source Qo due to advection can also be related to while it shifted upstream for V Ͻ 0. ␺oÐsee (A18): The amplitude of the air temperature response de- 3708 JOURNAL OF CLIMATE VOLUME 14 pends on the relative size of the thermal equilibration Ϫ1 timescale⌫ T and the advective-propagative timescale (Vk)Ϫ1 (the time for a free Rossby wave to travel a distance kϪ1). At resonance (V ϭ 0 and l ϭ 2␲/5590 km for our standard parameters in Table 1), the forced response is maximum and in phase with SST (vertical dashed±dotted line in Fig. 3). Near resonance, one has

Vk K ⌫T, that is the thermal equilibration time is much shorter than the advective-propagative time, and the air temperature adjusts closely to the SST. This corresponds to the thermal equilibration discussed by Shutts (1987), Marshall and So (1990), and GM. In this limit, the atmospheric response is equivalent barotropic, a warm high (cold low) settles over a warm (cold) SST anomaly.

Away from resonance, for Vk/⌫T large, the atmospheric response is smaller and zonally shifted from the SST anomaly (the forced response limit of Shutts 1987). Near barotropicity (V ϭϱand l ϭ 2␲/5236 kmϪ1), the air FIG. 4. Power spectra of SST (solid lines) and air temperature temperature anomaly vanishes and the forced response (dashed lines) at x ϭϪ4000 km when there is no mixed layer mod- is minimum. ulation by the geostrophic ¯uctuations. Forcing is at zonal wavenumber 2 with ␭y ϭ 5400 km (thin lines) and ␭y ϭ 5600 km (thick The phase of the forced surface pressure is easily lines). Plots have been made by (arbitrarily) choosing that the low- obtained from Fig. 3 noting that air temperature (or wind frequency spectral level of SST for ␭y ϭ 5400 km is one. shear) and surface pressure anomalies have the same sign only for equivalent barotropic conditions, other- wise they are in quadrature. Above land (x Ͼ 0), the solution is obtained taking

⌫a ϭ 0 in (11) and there is no propagation. To match the solutions at x ϭ 0 and remain bounded, the free 4. Thermal coupling between the atmosphere and mode must be added above either land or ocean, de- the mixed layer pending on the sign of V. However, the free solution We ®rst neglect the SST modulation by wind-driven has a zonal decay scale that is too short for the ap- geostrophic currents in the ocean. Above the ocean, (1) proximation (A8) to remain valid. Rather than using the full set of equations, which would require tedious al- holds while the SST Eq. (2) reduces to (setting To ϭ Q ϭ 0): gebra involving a scale of the order of the Rossby radius, o we simply neglect the free solutions and use (11) above

␥s the ocean but recognize that the solution breaks down T ϭ ␪ ϭ b␪, (9) close to the eastern boundary. ␥To Ϫ i␻ If the frequency spectrum of the air temperature ␪ where is the (inverse) damping timescale ␥ To ϭ ␥ s ϩ ␥ e were white, then (9) would reduce to the ®rst-order of SST anomalies, and b(␻) is a transfer function be- Markov process considered by Frankignoul and Has- tween ␪ and T that describes the frequency dependence selmann (1977). The system (9)±(10) is similar the one of the coupling between SST and the atmosphere. It is discussed by Barsugli and Battisti (1998), except for the understood that ␪ ϭ ␪(x, ␻) and T ϭ T(x, ␻) are Fourier inclusion of explicit atmospheric dynamics. As in their components. study, the atmospheric spectrum S␪(␻), which is easily Replacing in (1) yields: computed from (11), is not white because of the air± sea temperature adjustment. As illustrated in Fig. 4 -dashed lines), S (␻) is white for ␥ K ␻, then it in) ץ ⌫ϩV ␪ ϭ C sin(kx ϩ ␸), (10) ␪ To x k creases as frequency decreases because the SST startsץ΂΃ adjusting to the air temperature anomaly and reduces where ⌫ϭ⌫ Ϫ b⌫ ϭ (1 Ϫ b)⌫ ϩ⌫. Since the T a a r the air±sea contrast and thus the heat ¯ux. For ␻ K steady-state assumption prevents baroclinic instability, ␥ To, the spectrum becomes white again, but at a higher the solution consists of a free mode plus a forced so- level since the SST is fully adjusted. Correspondingly, lution. The free mode varies like exp(Ϫ⌫x/V) and prop- the surface heat ¯ux spectrum (not shown) decreases at agates with a varying amplitude since ⌫ is complex. low frequency toward a lower white noise level. The The forced solution is given by SST anomaly spectrum (Fig. 4, solid lines), computed from (9), broadly resembles that of a ®rst-order Markov Ck ␪ ϭ [⌫ sin(kx ϩ ␸) Ϫ Vk cos(kx ϩ ␸)] process, although the air±sea adjustment, and thus the (Vk)22ϩ⌫ reduced thermal damping, has resulted in an increased for x Ͻ 0. (11) persistence of both air temperature and SST anomalies. 1SEPTEMBER 2001 FERREIRA ET AL. 3709

Note that the spectra have been estimated at x ϭϪ4000 namely changes in the relative importance of thermal km for comparison with the coupled case, but the spec- equilibration and heat advection. tral shape depends little on x. However, because of (7) the spectral level varies strongly in the zonal direction (see Fig. 11, section 6a). 5. Fully coupled case

The power enhancement E ϭ S␪(0)/S␪(␥ To K ␻)of ␪ at very low frequency is identical to the ratio of cou- When SST modulation by the geostrophic motions is pled-to-uncoupled low-frequency atmospheric variance. included, we eliminate ␪ and T from (1)±(6) and obtain The latter is easily obtained by comparing (11) with the AB C solution for ®xed SST [i.e., with T ϭ 0 in (1)]. The ͦͦ ͦ parameter E is a measure of the air±sea temperature |||||| V ␣m Ä ,␺ ϭ F( ץ␺ Ϫ c(␥ ϩ a(ץxoxoexo(Ϫik ϩץadjustment at low frequency. It depends on the degree 1 ϩ to which thermal equilibration is achieved, hence on ΂΃⌫⌫ both k and l (see section 3). For small | Vk/⌫ | , thermal (12) equilibration occurs and results in a strong air±sea tem- where k ϭ ␻/c is the wavenumber of a free oceanic perature adjustment, hence E is large (the atmospheric o o spectrum is reddened). Correspondingly, the persistence Rossby wave of frequency ␻ and the combination of of SST anomaly is long and the spectral level is high coupling constant c is (Fig. 4, thick lines). For Vk ϭ 0 (either because k ϭ 0 ⌫ rb or there is resonance), E and the persistence of SST c ϭ ␣m ao. (13) anomaly are maximum, the latter reaching about 7 ⌫ ␥s months for our standard values. For large | Vk/⌫ | , the Term A represents atmospheric dynamics, term B rep- advective-propagative timescale becomes shorter, the resents Rossby wave propagation, and term C is the air±sea temperature adjustment is small, and E decreas- coupling between the thermocline and the atmosphere es. Then, the persistence of SST anomaly is short, and via the oceanic mixed layer. the spectral level is lower (thin lines). For Vk ϭϱ The free modes of the coupled system are solutions (barotropic case, no air temperature anomaly), the at- of the homogeneous part of (12). They vary like e␦x mospheric spectrum is white and the persistence of SST where ␦ must satisfy the quadratic equation: anomaly minimum, given, as expected from (9), by ␥ Ϫ1 (3 months for our standard values). To V The averaged power enhancement in coupled models 1 ϩ ␦ (␦ Ϫ ikoe) ϭ c(␥ Ϫ a␦). (14) is about 2.5 in Manabe and Stouffer (1996); it is 1.7 in ΂΃⌫ BladeÂ (1997). Also, the latter noticed that it was scale As expected, the two solutions ␦1 and ␦ 2 are an oceanic dependent, the lowest wavenumbers being most strongly Rossby wave and the free solution of section 4, both ikx enhanced. If we assume for simplicity an e dependence modi®ed by the coupling. For the same reason as above, for all variables (which avoids the zonal nonhomoge- we neglect the second solution and only keep the cou- neity caused by our choice of forcing pattern), choose pled Rossby wave, even though the atmospheric solu- 5400 km (an intermediate value between the ex- ␭y ϭ tion is then discontinuous at x ϭ 0 and the solution is treme cases V 0 and V ), and neglect (which ϭ ϭϱ ␥e not valid near the eastern ocean boundary. The retained corresponds to the above GCM experiments), we ®nd solution is obtained by adding the forced part of the that E decreases with increasing zonal wavenumber: E solution to the coupled mode: ϭ 13.4, 4, 1.9, and 1.4 at zonal wavenumber 0, 1, 2, and 3, respectively (E ϭ 5.2, 2.9, 1.7, and 1.3 when ␺ ϭ ␣ e␦1x ϩ ␣ sin(kx ϩ ␸) ϩ ␣ cos(kx ϩ ␸), (15) considering entrainment and decreases further if surface o 12 3 friction is added but increases if l is closer to resonance). where ␣ 2 and ␣3 are determined by injecting the solution Thus, the predictions of our simple model are broadly in (12), and ␣1 is then readily obtained from (A21). The consistent with the GCMs. Contrary to Barsugli and geostrophic response to the stochastic forcing at each Battisti (1998), our model is able to reproduce the wave- ␻ is thus given by a forced standing response plus a number dependence of the power enhancement dis- free westward-propagating one that is generated at the cussed in BladeÂ (1997). In their study, they expected eastern boundary to satisfy the no-normal ¯ow condi- ``the dynamical part of the atmospheric response to act tion. The latter resembles an oceanic Rossby wave but, generically as a negative feedback, with atmospheric because of the air±sea coupling, its amplitude increases heat ¯uxes partially offsetting the diabatic effects of an or decreases as it propagates westward, depending on SST anomaly,'' which is consistent with our results. the parameters. Note that an increase in amplitude cor- However, they parameterized it as a linear scale-inde- responds to the temporal instability investigated by GM. pendent negative feedback. Our model suggests that the If the coupling is weak enough, one can derive simple wavenumber dependence of the power enhancement expressions of the roots ␦1 and ␦ 2 and thus of the air± seen in GCMs could be due to atmospheric dynamics, sea feedback. Details are given in appendix B. 3710 JOURNAL OF CLIMATE VOLUME 14

a. Entrainment case We ®rst consider the SST modulation by entrainment and neglect the in¯uence of geostrophic advection [taking a ϭ 0 in (14)]. In the weak coupling case, the spatial growth or decay of the Rossby wave as it propagates westward depends on the frequency and is given by the real part of (B3). We further simplify the solution and assume that ␻ is suf®ciently low for the SST tendency

to be neglected (␻ K ␥ To), yielding

␮ 1 ϩ ␣r ⌫ ␥ ΂΃2 Re(␦ ) ϭ oa e . (16) 1 2 raTo⌫ ␥ Vk 1 ϩ o ΂΃⌫ The condition for westward growth is then: ␮ 1 ϩϽ0or␮ ϽϪ2, (17) 2 which from (A10) solely depends on l and the mean zonal wind. For a given zonal wind, only a limited range FIG. 5. Sketch of the growing coupled modes in a horizontal plane of meridional wavelength with equivalent barotropic in (top) the entrainment case and (bottom) the advection case. The dashed and continuous lines represent oceanic baroclinic pressure ¯uctuations is unstable. Note that the zonal wavenumber and sea level pressure, respectively. Shading denotes a positive SST and thus the phase speed are also modi®ed by the cou- anomaly. Only a half-wavelength is represented. pling [see (B4) and Fig. 6 later]. The mode grows because a positive subsurface temperature anomaly [depressed isopycnal, see (A17)] and that near-barotropic conditions are poorly represented its associated positive SST anomaly generate a positive in our model.

air temperature anomaly. In equivalent barotropic con- Figure 6 shows the growth rate Re(␦1) (top) and the ditions, the latter is associated to an anticyclonic surface zonal wavenumber Im(␦1) (bottom) of the free modes circulation that causes downward Ekman pumping and from the exact solution of (14) as a function of ␻ and deepens the isopycnal further thus acting as a positive l for the standard values (Table 1). Relation (17) still feedback (Fig. 5, top). On the other hand, for non- determines the domain of signi®cant westward growth equivalent barotropic conditions, a positive air temper- when ␻ increases, so that instability primarily occurs ature results in a cyclonic surface circulation leading to for meridional wavelengths ranging between 5200 and negative feedback and a spatial decay of the coupled 5800 km. It is thus relevant to the NAO problem. In mode. the most unstable conditions, the instability length scale The condition (17) is similar to the instability con- reduces to 5000 km, resulting in a large amplitude in- dition of GM, except for the lack of dependence on crease across an ocean basin. However, as shown below, zonal wavenumber that results from our approximation this growth rate only pertains to winter conditions. The (A8). In the unbounded domain considered by GM, zonal wavenumber broadly matches that of an oceanic these were the modes that coupled with the ocean and Rossby wave in the long-wave approximation, although led to temporal growth. In the present case, the modes it depends somewhat on l at the largest periods. In the grow from the eastern boundary, but their amplitude region of strong growth rate, the zonal wavelength is remains bounded because the fetch is limited. indeed slightly larger resulting in an increased phase

The growth rate depends on Vko/⌫, hence on the de- speed. Note that, because of (A8), only the low-fre- gree of thermal equilibration (see section 3). Here, the quency part of the ®gure is consistent with our simpli-

advective±propagative timescale Vko is set by the scale fying assumptions. For periods shorter than about 10 ko of the oceanic Rossby wave and hence depends lin- years, the coupled mode should have been calculated early on ␻. Thus, the largest growth rates are found at with full atmospheric dynamics. the lowest frequencies that favor thermal equilibration. Note that when approaching barotropic conditions ( ␮ b. Advection case → Ϫϱ), the ratio of surface wind to wind shear anomaly becomes very large, widely increasing the ef®ciency of A similar analysis can be done for the SST modu- the wind stress feedback. However, taking into account lation by geostrophic advection, neglecting the entrain- surface friction would strongly damp the response so ment term in (14). The free mode is also an oceanic 1SEPTEMBER 2001 FERREIRA ET AL. 3711

Ϫ8 Ϫ1 FIG. 6. (top) Contours of the growth rate (10 m ) and (bottom) FIG. 7. As in Fig. 6 but for advection. zonal wavenumber (10Ϫ7 mϪ1) of the coupled mode as a function of frequency and meridional wavelength for the entrainment case. West- ward decay is shaded. (bottom) Dashed lines correspond to zonal wavenumber of a free Rossby wave in the long-wave approximation. ing a positive feedback. If V Ͻ 0, the atmospheric perturbation is upstream of the SST anomaly, creating a Rossby wave that grows or decays as it propagates west- southward SST advection and acting as a negative feed- ward. In the low-frequency limit, one ®nds from (B5): back. Hence, a downstream atmospheric response is V ␮ necessary for growth. This is in contrast with the en- 2 ako 1 ϩ trainment case where the instability could occur when ␣r ⌫⌫ 2 Re(␦ ) ϭ oa . (18) the atmospheric response was shifted both east or west 1 2 raTo⌫ ␥ Vk of the SST anomaly. 1 ϩ o ΂΃⌫ As before, a thermally equilibrated atmospheric response (Vko/⌫ K 1) favors the unstable air±sea inter- The condition for westward growth is thus action. However, the strength of the coupling primarily ␺ and hence on k , so that the growth ץdepends on ␷ ϭ ␮ o x o o 1 ϩ V Ͻ 0, (19) rate increases with frequency. Figure 7 shows the growth ΂΃2 rate and the zonal wavenumber computed from (33) as and the only solution is 1 ϩ ␮/2 Ͻ 0 and V Ͼ 0. Because a function of l and ␻. Even without approximation, con- of the latter condition, the instability is more scale se- dition (19) remains approximately valid over all fre- lective than in the entrainment case and westward quencies. The strongest growth rates are reached at high growth only occurs for meridional wavelengths ranging frequency and are found closer to resonance (V ϭ 0) as between about 5600 and 5800 km. A northward ¯ow frequency increases. Indeed, the phase shift between in the ocean advects warm water from the south, creating SST and meridional advection increases and thus pro- a positive SST anomaly that, in turn, generates a positive vides the required phase quadrature with the mechanical air temperature anomaly (Fig. 5, bottom). For an equiv- forcing in conditions that become nearer to thermal alent barotropic atmosphere, the latter causes an anti- equilibration. However, the zonal scale is then too short cyclonic surface circulation, hence a transfer of negative for (A8) to be acceptable (and applicable to NAO stud- vorticity into the ocean interior. If V Ͼ 0, the atmo- ies), so that only low frequencies (zonal wavenumber) spheric perturbation is downstream of the SST anomaly are considered. Note that, as in the entrainment case, and it reinforces the northward SST advection, provid- the zonal wavelength is larger at low frequency com- 3712 JOURNAL OF CLIMATE VOLUME 14

FIG. 8. (top) As in Fig. 7 but for summertime conditions. The area of westward growth rate for wintertime conditions is indicated by the two thick solid lines. pared to the uncoupled case, hence resulting in an increased phase speed.

FIG. 9. Power spectra of (top) ␺o, (middle) SST, and (bottom) air c. Summer conditions temperature for the uncoupled case (dashed line) and for the entrainment case (solid line) at x ϭϪ4000 km with ␭y ϭ 5400 km. In summer conditions, the zonal ¯ow is about one- half its wintertime counterpart, and hence we take U1 Ϫ1 Ϫ1 ϭ 12 m s and U 2 ϭ 4ms . Because of the weaker stead, we limit ourself to zonal wavenumber 2 and pri- winds and the presence of a well-strati®ed seasonal ther- marily consider two reference meridional wavelengths mocline, entrainment plays little role in the SST dy- ␭y ϭ 5600 km and ␭y ϭ 5400 km that favor the ad- namics (Alexander and Deser 1995; Alexander and Pen- vection and entrainment mechanisms, respectively. land 1997) and the bulk of the geostrophic modulation Spectra and correlations are presented for x ϭϪ4000 Ϫ6 yT ϭϪ3 ϫ 10 ЊC km, which is far enough from the eastern boundary toץ takes place via advection. Using Ϫ1 Ϫ2 Ϫ1 m , hmix ϭ 50 m, and ␭ ϭ 20Wm K as repre- allow for a signi®cant development of the unstable sentative summer values, we ®nd that the magnitude of modes while remaining within the limits of a schemat- the westward growth due to SST advection is similar ical Atlantic basin, away from the western boundary to the wintertime case but is restricted to meridional current region where higher-order ocean dynamics dom- wavelengths that are 20% shorter than in winter (Fig. inate and need to be represented. To clarify the statistical 8). The instability thus takes place on scales that are signature of the coupling, the spectra are shown for too short to be of interest in the present context. periods between 100 and 2.5 yr, even though (A8) is Note that the modes that were growing during winter only valid for periods down to about 10 yr. are damped during summer, but the damping is very weak, nearly 1 order of magnitude smaller than the win- a. Power spectra ter growth, and it can be neglected. Nonetheless, the growth rates based on winter conditions should be The passive response of ␺o in the absence of SST scaled down on a yearly basis, because of the inter- modulation, obtained from (3), (4), and (11), provides mittency of the instability. the reference spectrum discussed in FMZ, except for the slight redness of the atmospheric forcing. The oceanic response has a red spectrum that decays as Ϫ2 at high 6. Statistical signatures of the interaction ␻ frequency, since the ocean then responds as an integrator Realistic statistical signatures of the two-way cou- to the stochastic Ekman pumping, and ¯attens at low pling should be established by considering that the sto- frequency where the ¯uctuations reach Sverdrup equi- chastic forcing has a ®nite wavenumber bandwidth. librium (Fig. 9, top, dotted line). The dominant time- However, this would requires tedious integrations. In- scale (which clearly appears in the variance conserving 1SEPTEMBER 2001 FERREIRA ET AL. 3713

FIG. 10. (left) Variance spectrum and (right) autocorrelation of 1- yr averaged ␺o at x ϭϪ4000 km with ␭y ϭ 5400 km for a passive ocean (dashed line) and for the entrainment case (solid line). Variance has been normalized so that maximum uncoupled variance is 1. form in Fig. 10) is decadal and determined by the time it takes a long Rossby wave to propagate from the eastern boundary. The peaks (troughs) seen at high frequency correspond to an in-phase (out of phase) relation between the forced response and the Rossby wave generated at the eastern boundary. They are linked to our choice of a single wavenumber for the forcing and would be smoothed for a more realistic forcing pattern FIG. 11. Variance of (top) ␺o, (middle) SST, and (bottom)␺˜ as a having a ®nite wavenumber bandwidth. The variance of function of x for a passive ocean (dashed line) and for the entrainment

␺o strongly increases westward (Fig. 11), but less so case (solid line) with ␭y ϭ 5400 km. Thin (thick) line is for yearly than for zonally independent forcing (FMZ) since the (5 yr) averages. Normalization was made independently for each forcing variance peaks at 2000 km from the eastern variable to have a maximum yearly averaged uncoupled variance of 1. boundary. When the SST is modulated by entrainment, the power spectrum of ␺o shows for our reference value ␭y ϭ would be 80 for ␭y ϭ 5300 km and 3.6 for ␭y ϭ 5600 5400 km that the positive windstress feedback enhances km). On the other hand, the atmospheric power en- the power by 40% at periods larger than about 10 yr, hancement is weaker and shows little sensitivity to l, although the dominant period remains unchanged (Figs. ranging between 2.3 and 3.5; since the atmosphere re- 9 and 10, continuous line). The enhancement depends sponds less as it approaches the barotropic state, the somewhat on l (70% for ␭y ϭ 5300 km and 40% for larger SST enhancement has little effect. The atmo- ␭y ϭ 5600 km), but not the range of enhanced periods. spheric spectrum is thus only slightly red behaving as Figure 9 compares the SST (middle) and the air tem- ␻Ϫ0.5 at periods of several decades or less. This is con- perature spectra in the absence of geostrophic ¯uctua- sistent with the slight redness of the NAO pattern in tions in the ocean interior (dashed line) and in their Fig. 1, in contrast to the uncoupled case where the at- presence when the SST modulation occurs by entrain- mospheric spectrum became white for periods longer ment (solid line). At low frequency, the geostrophic than 5 yr. variability strongly enhances the power, but this pri- When the geostrophic modulation occurs via SST ad- marily results from the SST modulation by the interior vection, the coupled modes for our reference value ␭y motions, not from the positive air±sea feedback whose ϭ 5600 km (close to resonance), are most unstable at relative importance is given by the ␺o spectra. Note that high frequency so that a power enhancement is found the SST and␺˜ spectra are little in¯uenced by the in- at periods shorter than about 15 yr in the ␺o-spectrum phase and out-of-phase relations between free and (Fig. 12, top), with a corresponding shift of the dominant forced ␺o modes because ␺o has limited power at periods timescale toward a shorter period (Fig. 13). As com- shorter than decadal. Thus, our estimates remain ap- pared to the case with no geostrophic variability, cou- proximately valid at high frequency even though (A8) pling by advection substantially enhances the SST and is no longer satis®ed. The in¯uence of the geostrophic ␺˜ spectra at periods between 10 and 20 yr (Fig. 12, modulation on the SST spectrum is very large and sen- middle and bottom). For the three variables, a power sitive to l (the enhancement factor is 13 in Fig. 9, but decrease is seen at lower frequency because the forced 3714 JOURNAL OF CLIMATE VOLUME 14

FIG. 13. As in Fig. 10 but for the SST advection case with ␭y ϭ 5600 km.

sin(T␻/2) h(␻) ϭ , (20) T␻/2 which is the equivalent to using a running ®lter of width T, with T ϭ 1 or 5 yr. Lagged correlation is then computed from the analytical solutions by an inverse Fourier transform, assuming in addition that the stochastic forc-

FIG. 12. As in Fig. 9, but for the SST advection case with ␭y ϭ ing F is a ®rst-order autoregressive process with a decay 5600 km. time ␯Ϫ1 of 10 days (a ®nite energy signal). This has no effect on the spectra at the low frequencies of interest, and free modes are then slightly in quadrature, inter- but note that the correlations used in this section are acting negatively. However, farther away from the east- only approximate as they are based on an integration ern boundary, that is, for a much larger basin than the over all frequencies while our solution for the free cou- Atlantic, a small power increase would be found instead. pled modes is only valid for frequency shorter that de- At high frequency, the succession of peaks and throughs cadal. This should have little impact in the entrainment seen in the SST and␺˜ spectra is much more visible case as the coupling primarily enhances the low fre- than in entrainment case as the meridional geostrophic quencies. For reference we ®rst discuss the passive case of sec- velocity varies as ␻␺o and thus has most of its variance at high frequency. As pointed out before, the high fre- tion 4 where there is no SST modulation by the geo- quencies must be considered with much caution since strophic variability, only using ␭y ϭ 5400 km as l then the coupled mode then has too short a scale for (A8) has very little in¯uence on the correlations. The yearly to be valid. For wavelengths differing by more than a averaged SST anomalies are somewhat correlated from one year to the next, but become uncorrelated at 2-yr few tenth of kilometer from resonance (␭y ϭ 5590 km), coupling by advection would hardly affect the spectra, interval (Fig. 14, top left, dashed line). Because of the re¯ecting the narrowness of the high growth rate region strong SST imprint that is predicted by our simple at- in Fig. 7. Advection effects are thus only substantial mospheric model, yearly air temperature anomalies have near resonance. However, the amplitude of the latter in a corresponding, if slightly weaker, year-to-year persis- the two-layer channel model is unrealistically large (Eg- tence (Fig. 14, top right). The cross correlation between ger 1977; Held 1983), and a more realistic atmospheric SST and surface heat ¯ux (not shown) displays the model should be used. change of sign between lead and lag conditions that is characteristic of a negative heat ¯ux feedback (Fran- kignoul 1985; Frankignoul et al. 1998), whereas that b. Cross correlation and atmospheric predictability between air temperature and SST (Fig. 14, bottom) Cross-correlation functions generally provide a more peaks when SST lags by a few months, which corre- stringent test of cause to effect relationships than power sponds to the 1-month lag that would be found with spectra, and they are directly linked to the predictability monthly data (Frankignoul and Hasselmann 1977). The issue. To emphasize the long timescales, we discuss cor- correlation is negligible when SST leads␺˜ by more than relation based on averaged data. To do so, Fourier com- about a year, so that there is no predictive skill beyond ponents have been multiplied by 1 yr for the yearly averaged␺˜ . This corresponds to the 1SEPTEMBER 2001 FERREIRA ET AL. 3715

FIG. 14. (top) Autocorrelation of yearly averaged SST and air temperature and (bottom) cross

correlation between the two variables at x ϭϪ4000 km for the entrainment case with ␭y ϭ 5400

km (solid line) and ␭y ϭ 5600 km (dashed±dotted line). Reference for the uncoupled case is

given for ␭y ϭ 5400 km (dashed line). limited predictability of the one-dimensional model of persistence increases with the distance from the eastern Bretherton and Battisti (2000). boundary, because of the zonal dependence of the geostrophic ¯uctuations. As seen in the cross correlation between␺˜ and SST 1) COUPLING BY ENTRAINMENT (Fig. 14, bottom), the long geostrophic timescale en- Figure 10 (right) compares the autocorrelation func- hance the atmospheric predictability since there is a small but persistence correlation when SST preceeds the tion of ␺o at x ϭϪ4000 km in the passive and coupled cases. In both cases, the persistence is large and the atmosphere by as much as 5±8 yr, depending on l. Using autocorrelations much smoother than the power spectra. the square of the lagged correlation as a coarse measure This occurs because the phase relations between the free of predictive skill (the fraction of variance that can be and forced oceanic response that caused its peaks and predicted), about 2% (depending on ␭y) of the yearly throughs corresponded to speci®c frequencies whereas climatic variability could thus be predicted from yearly the autocorrelation is an integral over all frequencies. SST data several years before. Such predictive skill is Since coupling by entrainment primarily increases the very small, but it increases for averages over a longer power of ␺o at periods longer than a decade but does duration (averaging reduces the unpredictable short not signi®cantly alter the spectral shape, it changes little timescale variability). For instance, the predictive skill the geostrophic persistence. However, the geostrophic at 5-yr lead reaches 4% using 3-yr averages and ranges

¯uctuations strongly alter the autocorrelation function from 1% to 9% using 5-yr averages for ␭y between 5600 of SST. Figure 14 (solid and dashed±dotted lines) clearly and 5400 km. In most cases,␺˜ can be equally well shows the two SST timescales in the coupled case; a predicted from earlier values of␺˜ or SST, while pre- short timescale that corresponds to that of the oceanic dictions based on ␺o are inferior (see below). However, mixed layer, and a longer one that re¯ects the geo- in practice,␺˜ is likely to be obscured by other sources strophic modulation. Like the power spectrum, the au- of variability. Since improved predictive skill could be tocorrelation function is sensitive to l and the long-term obtained from more sophisticated forecast models, they persistence is maximum in near-barotropic conditions. may be of some practical value. This demonstrates that Although the correlation levels are lower and the sen- Bretherton and Battisti's (2000) conclusion that true sitivity to l smaller, the two timescales also appear in NAO predictability is limited to 6 months or so is a the autocorrelation of␺˜ . In both cases, the long-term direct consequence of the simplicity of their ocean mod- 3716 JOURNAL OF CLIMATE VOLUME 14

FIG. 15. Cross correlation between yearly averaged air temperature and ␺o at x ϭϪ4000 km

for the entrainment case for ␭y ϭ 5400 km (solid line) and ␭y ϭ 5600 km (dashed±dotted line).

Reference for the uncoupled case is given for ␭y ϭ 5400 km (dashed line). el, which neglects the geostrophic variability of the to those described in Fig. 5. When the atmosphere leads, ocean interior. the thermocline motions re¯ect the oceanic response to

The cross correlation between ␺o and␺˜ is given in the wind stress forcing. When the ocean leads, a de- Fig. 15. In the passive case (dashed line), it has a broad pressed thermocline is associated with a warm SST peak when ␺o lags␺˜ by a few years and is negligible anomaly and a downstream sea level pressure high (see when it leads. The adjustment time at x ϭϪ4000 km also their Fig. 8). As in the present model, there is little is about 3.1 yr, which is approximately the Rossby wave change of patterns between lead and lag conditions and transit time from the center of action of the wind stress the antisymetric distribution with lag of the covariance

(see Jin 1997). In the coupled case, the cross correlation is broadly consistent with that seen between␺˜ and ␺o still primarily re¯ects the wind stress forcing of the in Fig. 15. A positive ocean±atmosphere feedback sim- ocean interior, but there now is a small predictability ilar to that investigated here may thus be at play in the when the ocean leads. Also, the delay between the wind coupled model. Note that is the GCM case, the maxi- stress forcing and ␺o is shorter (2.2 yr for ␭y ϭ 5400 mum correlation between ␺o and the atmosphere is km and 2.7 yr for ␭y ϭ 5600 km), consistent with the found when the latter leads by 1 yr, which is shorter increase in phase speed (Fig. 6). This can be compared than in Fig. 15. However, the 1-yr delay pertains to to the short delay found between the NAO and the basinwide patterns and is indeed expected to be much changes in the Gulf Stream path, if the latter re¯ect an shorter than at 4000 km from the eastern boundary. adjustment to the changes in Sverdrup transport. Using The zonal distribution of the variance in the coupled observations, Taylor and Stephens (1998) found a 2-yr case is shown in Fig. 11 (solid line) for yearly (thin) lag while Frankignoul et al. (2001) found a 1±1.5-yr and 5-yr (thick) averages. Coupling with the ocean in- lag, and Halliwell (1998) obtained a 2-yr lag in a sim- terior results in a strong westward intensi®cation of the ulation with an oceanic GCM. SST variance but, as mentioned above, this is mostly The entrainment case bears resemblance with the cou- due is due to the geostrophic modulation, not to the pled ocean±atmosphere mode seen in the ECHAM1/ active coupling whose in¯uence is that seen for ␺o. The LSG coupled GCM (Frankignoul et al. 2000). As shown atmospheric variance increases only slightly in the cen- in Fig. 16, the mostly second dominant mode of a max- tral and western part of the basin. It is thus primarily imum covariance analysis based on a singular value determined by the uncoupled dynamics of the atmo- decomposition (SVD) between the baroclinic pressure sphere, in agreement with GCM experiments (e.g., Sar- at 250 m and the sea level pressure has similar patterns avanan 1998). Note that the 5-yr averaging has little 1SEPTEMBER 2001 FERREIRA ET AL. 3717

FIG. 16. Maps for (left) the baroclinic pressure at 250 m in the ocean (Pa) and (right) the sea level pressure (Pa) for the (mostly) second dominant SVD mode at signi®cant lags between Ϫ4 and 1 (positive when the atmosphere leads) when their (upper-left side) signi®cance level is below 5%. The SVD times series are normalized so that the maps indicate typical amplitudes. Positive (negative) isolines are in black (white). Lag, mode number, and correlation are also indicated (after Frankignoul et al. 2000).

in¯uence on ␺o, which is dominated by long timescales, variability seen in Fig. 12. Note that we have only con- but signi®cantly reduces the SST and␺˜ variance. sidered 5-yr averages in Fig. 17, to remove some of the in¯uence of the periods which are ill represented by our model. Several observational studies have reported an 2) COUPLING BY SST ADVECTION oscillatory behavior at decadal periods for SST and sea As discussed above, coupling by advection only sig- level pressure in the North Atlantic (e.g., Deser and ni®cantly enhances the geostrophic response in near- Blackmon 1993) and subsurface temperature (Molinari resonant conditions, and the enhancement increases with et al. 1997) in the North Atlantic. The present study frequency. Since the power density of ␺o strongly de- suggests that it might be linked to ocean±atmosphere creases at periods shorter than 10 yr, the impact of the coupling and SST advection by the wind-driven geo- positive wind stress feedback is strongest at a period of strophic motions. about 12 yr, resulting in a slightly oscillatory behavior. Thus, the autocorrelation function of ␺ changes sign o 7. Conclusions and has a small minimum at a lag of half this period (Fig. 13). Because of the strong imprint of geostrophic Using a two-layer quasigeostrophic channel atmo- ¯uctuations on SST, the oscillatory behavior also ap- sphere, we have studied the coupling between geo- pears in the autocorrelation and in the cross correlation strophic motions in the ocean and the natural variability of SST and␺˜ (Fig. 17), re¯ecting the enhanced decadal of the atmosphere at the decadal timescale, and we have 3718 JOURNAL OF CLIMATE VOLUME 14

FIG. 17. (top) Autocorrelation of 5-yr-averaged SST and air temperature and (bottom) cross

correlation between the two variables at x ϭϪ4000 km for the advection case with ␭y ϭ 5600 km (solid line). Reference for the uncoupled case is given in dashed line. established the statistical signatures of the interaction. at low frequency, it is well represented in our simpli®ed

To set the stage for the analysis, we ®rst considered the model, yielding a small power enhancement of the ␺o coupling between the atmosphere and the oceanic mixed spectrum for decadal and longer periods. The coupling layer, thus extending the work of Barsugli and Battisti also decreases the apparent delay between the interior (1998) and Saravanan and McWilliams (1998) to the response and the wind stress forcing. In our standard case of a dynamical atmosphere. This has allowed us wintertime conditions, the range of unstable meridional to investigate the wavenumber dependence of the SST wavelength is between 5200 and 5800 km, but larger adjustment to air temperature change that is found at meridional scales could have been obtained for stronger Ϫ1 low frequency showing that there is an increase in power mean winds. For instance, taking U1 ϭ 21ms and Ϫ1 enhancement with increasing zonal scale, as seen in U2 ϭ 10ms as representative of extreme wintertime GCM experiments (BladeÂ 1997; Manabe and Stouffer conditions, the range of unstability is between 5800 and 1996). 6400 km. These scales are roughly consistent with that The coupling with the ocean interior occurs via en- of NAO, suggesting that the latter may well be sustained trainment and meridional advection. In both cases, a at low frequency by the coupling with the geostrophic positive feedback may occur between the ocean interior variability in the ocean interior. However, as entrainment and the atmosphere, leading to Rossby-like coupled is not effective in summer, the instability is intermittent, modes that grow as they propagate westward. In the and the predicted growth rates somewhat overestimated. entrainment case, the vertical displacement of isopyc- Note that, although the spatial structure of the stochastic nals generate subsurface temperature anomalies that re- forcing would need to be adapted, the model should sult in SST anomalies and, through surface heat ex- also be applicable to the North Paci®c, yielding larger changes, air temperature anomalies of the same sign. amplitudes and longer dominant periods as the basin is Positive feedback only occurs for equivalent barotropic larger. conditions because they are needed for the wind stress The geostrophic modulation by entrainment signi®- to reinforce the original geostrophic ¯uctuations. Then, cantly affects the SST and air temperature spectra at the atmospheric pressure is nearly in phase with the SST periods longer than about 10 yr, slightly reddening the perturbation that is itself only slightly downstream of predicted atmospheric spectrum down to periods of 20 the thermocline depth perturbation, and the positive or 30 yr. Such reddening is found in the observed NAO wind stress feedback can overcome the negative heat spectrum (Hurrel and van Loon 1997; Wunsch 1999). ¯ux one. Since coupling by entrainment is most ef®cient Coupling by entrainment does not result in a quasi- 1SEPTEMBER 2001 FERREIRA ET AL. 3719 oscillatory behavior but it yields a signi®cant correlation Acknowledgments. This research was supported in between SST and the atmosphere when the former leads part by a grant from the PNEDC and Grant ENV4- the latter by a few years. Using 5-yr average, the pre- CT98-0714 of the European Community (SINTEX). dictive skill in our model is signi®cant, reaching up to David Ferreira was supported by a bursary from the 9% 5 yr in advance. The overall in¯uence of the cou- DGA-CNRS and John Marshall received support from pling on the atmosphere is nonetheless weak and the NOAA's Of®ce of Global Programs. spatial pattern remains determined by uncoupled atmospheric dynamics. The statistical signatures of the APPENDIX A interaction suggest that this mechanism may be at play in the ECHAM1/LSG coupled model where a weak pos- Derivation of the Coupled Model itive ocean±atmosphere feedback was detected (Fran- Here we derive the coupled model whose properties kignoul et al. 2000). Indeed, the oceanic and atmo- are studied in the body of the paper. spheric perturbations of the coupled model were approximately in phase, and there was no change of sign between lead and lag conditions in the cross-correlation a. Atmosphere function between ␺o (or SST) and␺˜ . Whether coupling The atmospheric ¯uctuations are assumed to be in by entrainment is also prevalent with a less diffusive equilibrium with the ocean and governed by the linear ocean model remains to be established. quasigeostrophic vorticity and thermodynamic equa- In the SST advection case, geostrophic advection of tions in a ␤ plane: warm water from the south, say, generates a positive ץץץ SST anomaly and then a positive air temperature anom- 2 (Uaaٌ ␺ ϩ ␤␺ aϭ fw a (A1 zץ xץ xץ aly. Provided the atmospheric response to the SST is ␪ qץ ␪ץץ -equivalent barotropic and shifted downstream, the sur face wind stress reinforces the northward meridional U ␪ ϩ ␷ ϩ w ϭϪ␥␪, (A2) zC rץ yץ xץaaa velocity, hence acting as a positive feedback. The ef- pa

®ciency of the mechanism increases with frequency and where Ua(z) is the mean zonal wind (only function of is maximum on timescales of a few years. Unfortu- the vertical coordinate), wa and ␷ a are the vertical and nately, the latter are not well represented in our model meridional velocity, ␪ is the potential temperature, f is where the zonal scale was assumed to be much larger the Coriolis parameter, ␤ is its meridional gradient, ␥ r than the meridional one, and further investigations are represents radiative and other damping, q is the diabatic needed. In any case, coupling by SST advection intro- heating per unit mass, and Cpa is the air heat capacity. duces a small oscillatory behavior at the decadal time- The vorticity equation (A1) is evaluated at the upper scale, as observed in the North Atlantic. This is also and lower levels of a two-level atmosphere, and the predicted by the scenario of Latif and Barnett (1994). thermodynamic Eq. (A2) evaluated at midlevel (see Fig. However, the present model differs from the latter since 2). De®ning barotropic and baroclinic components: the heat ¯ux feedback is negative, not positive, and the () ϭ (Ç) ϩ (Ç) , () ϭ (Ç) Ϫ (Ç) , (A3) air±sea coupling takes place over the whole basin, albeit 12 12 with stronger effects as one moves westward, and not equations for the evolution of barotropic and baroclinic via the western boundary current ¯uctuations. potential vorticity can be writtenÐsee Eqs. (24) and The simplicity of our coupled model should be em- (25) of GM: phasized. Its main weakness for the atmosphere is a Ã 22Ä (␺˜ ϭ 0 (A4ٌץ␺ˆ ϩ U ץ ˆ␤ xxx␺ˆ ϩٌץU crude vertical resolution, the lack of mean horizontal shear, and the neglect of the synoptic variability. As 2 Ä 22Ã ˜x␺ ץ ˆ␤ ␺˜ Ϫ ␺˜ ϩٌץ␺ˆ ϩ U ץ ˜␤ xxx␺ˆ ϩٌץshown by Palmer and Sun (1985) and others, transients U L2 and the changes of the storm track play an active role ΂΃a in setting the low-frequency response of the atmosphere 4 ␥ to SST anomalies, hence they may strongly in¯uence ϭ ␥␺˜ Ϫ a T ϩ F (A5) 2 Ta s the nature of the ocean±atmosphere coupling. The ocean Lraa΂΃ is also oversimpli®ed, in particular as we have neglected Ä 2 with ␤ˆ ϭ 2␤, ␤Ä ϭ 2U/,LaT␥ a ϭ ␭/␳aCpaHa, and ␥ ϭ western boundary dynamics, although the western a ␥ a ϩ ␥ r. In (A5), a white noise stochastic forcing Fs boundary currents and their associated meridional heat has been included to represent stirring by baroclinic transport play a key role in several mechanisms of de- instability and other nonrepresented dynamics. Also, ␪ cadal variability (Latif and Barnett 1994; Cessi 2000; has been converted to␺˜ using the thermal wind relation Marshall et al. 2001). Although the robustness of our thus analysis needs to be addressed in a more realistic model, 2f␪ the present results seem nonetheless relevant to the air± a ␪ ϭ ␺˜ ϭ ra␺˜ , (A6) sea coupling over most of the midlatitude ocean basins. gHa 3720 JOURNAL OF CLIMATE VOLUME 14

where ␪a is a typical atmospheric temperature and Ha the Doppler-shifted phase speed of a free Rossby wave the depth of the troposphere. The surface heat ¯ux Q in the atmosphere. has been set proportional to the air±sea temperature dif- Over land (x Ͼ 0), we set ␥ a ϭ 0 in (A5): land is ference, so that assumed to have a negligible heat capacity so that its temperature is always adjusted to that of the atmosphere Q ␭ q ϭϭ(T Ϫ ␪), (A7) and there are no anomalies in diabatic heating. ␳apaaCH ␳ apaaCH The surface streamfunction is linearly extrapolated where T is the SST anomaly, ␭ is the heat exchange from uppers levels: coef®cient, and ␳a is the air density. We are mostly interested in large-scale, low-frequen- 1 ␮␪ ␺s ϭ ␺ˆ Ϫ ␺˜ ϭϪ 1 ϩϭϪm␪. (A15) cy midlatitudes ¯uctuations. Since such ¯uctuations in 22΂΃ra the atmosphere have patterns with a zonal scale much For simplicity, we neglect the effect of surface friction larger than the meridional one, we suppose that on atmospheric perturbations. 22 2 (ϭϪy l , (A8ץϳٌ where l is meridional wavenumber. Although this ap- b. Sea surface temperature proximation is somewhat limiting, as discussed below, it simpli®es considerably the algebra and allows (A4) The ocean is topped by a slab mixed layer of constant to be written thus depth hmix. Assuming a mean meridional SST gradient y, we represent the evolution of the mixed layerץ/Tץ ␺ˆ ϭϪ␮␺˜ (A9) temperature anomalies T by TQץ Tץ with ϩ ␷ ϭ ␥ (T Ϫ T) Ϫ , (A16) y ␳ Chץt oeoץ UÄ ␮ ϭ . (A10) opomix ␤ˆ UÃ Ϫ where ␷ o is the meridional surface geostrophic velocity, 2 l To is the temperature anomaly below the mixed layer, Q is the surface heat ¯ux as de®ned in (A7), C is the The relative strength of the barotropic and baroclinic po water heat capacity, ␳ is the water density, and ␥ pa- modes is controlled entirely by ␮. On scales close to o e rameterizes entrainment and vertical mixing. For sim- that of stationary barotropic Rossby waves l ϭ plicity, we have neglected effects such as Ekman ad- Ã Ϫ1 ͙␤ˆ /U (ϭ2␲/5236 km for our standard values), | ␮ | vection, horizontal mixing, and mixed layer depth var- is very large and atmospheric perturbations are nearly iation, and so the above leads to an overestimate of the barotropic. When ␮ is large and negative (␮ ϽϪ2 for role of the surface heat exchanges in generating the SST our choice below of extrapolation to the surface), the anomalies. Following GM, we assume that the thermal ¯uctuations are equivalent barotropic. For ␮ large and anomalies being entrained re¯ect the adiabatic undu- positive, they also keep the same sign, but the amplitude lation of the isopycnal surfaces underlying the mixed decreases with height. When | ␮ | is small, the pertur- layer, and so can be related to vertical displacements ␨ bations change sign in the vertical: in the long-wave ϭ ␺ (x, y, t)Z(z), where Z is the vertical pro®le of the limit (l 2 K ˆ /UÃ ), they are highly baroclinic ( ) o ␤ ␺1 ϳϪ␺ 2 ®rst mode displacement. Then, the subsurface temper- 2 Ã while in the short-wave limit (l k ␤ˆ /U) the perturbation ature anomaly can be expressed in term of the baroclinic is larger in the lower layer, the ratio ␺1/␺ 2 being scaled Ä Ã streamfunction by by ϪU 2/U1 (or␺˜ /␺ˆ by ϪU/U). Tץ Tץ :Rearranging (A5) it can be written T ϭϪ oo␨ ϭϪ Z(h )␺ ϭ r ␺ , (A17) z mix oooץ zץ o (␪ ϭ⌫ aT ϩ F (A11( ץϩTxV⌫) z is the temperature gradient below theץ/T oץ with where mixed layer. Similarly, the geostrophic advection term 4␥a,r in (A16) can be expressed in term of by ⌫ϭ , (A12) ␺o a,r 22 lLa ␺ץ␺ooץ Tץ Tץ ␷ ooϭ ␾(0) ϭϪar , (A18) xץ xץ yץ yץ (ϭ⌫ϩ⌫Tar, (A13⌫ where ⌫ and ⌫ are the (inverse) thermal and radiative .yץ/Tץ(a r where aro ϭϪ␾(0 timescale of the potential vorticity anomaly and Then, (A16) becomes ␺ץ Tץ ˆ␤ UÃ 2 ˜␤ V ϭϪUÄ ␮ ϩ ␮ ϩ l2 ϩϪ(A14) ϭ ␥ (r ␺ Ϫ T) ϩ ␥ (␪ Ϫ T) ϩ ar o (A19) 22 22 eoo s o xץ tץ ll΂΃ Lla represents atmospheric dynamics and is a measure of with ␥ s ϭ ␭/␳oCpohmix. 1SEPTEMBER 2001 FERREIRA ET AL. 3721 c. Ocean interior anomaly damping timescale due to heat ¯ux is thus about four months. Choosing a typical temperature of We consider a semi-in®nite ocean bounded to the east 290 K for a 10-km height troposphere gives r ϭ 4.5 by land for x Ͼ 0 (Fig. 2). Oceanic western boundary a ϫ 10Ϫ7 KsmϪ2. To estimate r and a, we use ␾(0) ϭ dynamics are not represented and our calculation is lim- o 3 and Z(100) ϭϪ1.5 ϫ 10Ϫ2 smϪ1, from Richman et ited to x ϾϪL. The ocean is linearized about a state T ϭ 1.2 ϫ 10Ϫ2ЊCmϪ1 and ץ al. (1977), so that for of rest and obeys quasigeostrophic ␤-plane dynamics. zo T ϭϪ5 ϫ 10Ϫ6ЊCmϪ1 we ®nd r ϭ 2 ϫ 10Ϫ4 Ks ץ As shown in FMZ the barotropic mode can be neglected y o mϪ2 and a ϭ 7.5 ϫ 10Ϫ2 msϪ1. Last, as in GM, we at low frequencies, and so our ocean interior is repre- relate the wind stress anomaly to the surface wind speed sented by a ®rst baroclinic mode forced by stochastic anomaly by linearizing the bulk drag law about the mean Ekman pumping. The geostrophic streamfunction is giv- surface value us: en by ␺o(x, y, t)␾(z) where the ®rst mode vertical pro®le 0 2 ␶ ϭ 2␳ Cuu. (A25) ␾(z) is normalized by #ϪHo ␾(z) dz ϭ Ho. In the long- aDss wave approximation, it evolves according to Comparing (A22) and (A25), we obtain ␺␾o (0) ␾(0) (ϫ ␶, (A20 ١ txo␺ ϭץϩץ ␣Јϭ2␳aDsCu . (A26) cooo␳ H ␤ ␳ooH where co is the speed of the ®rst baroclinic Rossby wave, Ϫ3 Ϫ1 Using CD ϭ 1.5 ϫ 10 and u s ϭ 5ms , we ®nd ␣Ј ␶ is the surface wind stress, and Ho is the constant depth ϭ 1.3 ϫ 10Ϫ8 sϪ1. of the ocean. We impose a radiation condition in the west and a no-normal ¯ow condition at the eastern APPENDIX B boundary: The Weak Coupling Case ␺o ϭ 0atx ϭ 0. (A21) Over the ocean we assume that the surface wind stress If the coupling is weak enough, that is, the SST mod- perturbation is proportional to the surface wind: ulation by the ocean interior entails a small perturbation of the free modes, one can derive simple expressions ␾(0) 2 for the roots of (14): (ϫ ␶ ϭ ␣Јٌ ␺s, (A22 ١ ␳ooH c⌫ M ␦ Ӎ ik ϩ a ϩ  (B1) so that, (A20) becomes 1 o 2V ⌫  ik ϩ  o V ␺o (txos␺ ϭϪ␣␺ (A23ץϩץ co ⌫ c⌫ M ␦ Ӎ ϪϪ a Ϫ  (B2) 2 V 2V ⌫ with   iko ϩ ␣Јl2 V ␣ ϭ . (A24) ␤ with ⌫ ca2 ⌫ M ϭ 2␥ ϩ aikϪϩ . d. Numerical parameters eo΂΃V 2 V Standard values for the parameters are chosen to be When the SST modulation occurs by entrainment and representative of winter parameters (Table 1), but note the in¯uence of geostrophic advection is neglected, tak- that some of them differ from those of GM. Our choice ing a ϭ 0, (B1) simpli®es to for the atmosphere is traditional, using 35Њ±40ЊNas c␥e reference latitudes and a rather strong radiative damping ␦1 ϭ iko ϩ Ϫ1 Vko ␥ r ϳ (8 days) , to make up for the neglect of surface 1 ϩ i friction. The heat exchange coef®cient ␥ a is derived ⌫ using ␭ ϭ 40 W mϪ2 KϪ1. An atmospheric heat capacity 7 Ϫ2 Ϫ1 Ϫ1 ␮ of 10 Jm K implies ␥ a ϳ 3 days. The oceanic 1 ϩ Ϫ1 ␣roa⌫ ␥ e 2 ``entrainment'' feedback ␥ e ϳ (1 yr) may seem weak ϭ iko ϩ . (B3) but is broadly consistent with the observed persistence raTo⌫ (␥ Ϫ i␻) Vko of midlatitude sea surface salinity anomalies (Hall and 1 ϩ i ⌫ Manabe 1997), which are not directly affected by the ΂΃ heat ¯ux feedback and thus provide a direct estimate of The mode is a westward-propagating oceanic Rossby its averaged strength. The heat ¯ux damping factor ␥ s wave whose wavelength has been modi®ed by the cou- was derived using a mixed layer depth hmix ϭ 100 m. pling (imaginary part of the second term). It grows (de- In the absence of air temperature adjustment, the SST cays) as it propagates when the real part of ␦1 is negative 3722 JOURNAL OF CLIMATE VOLUME 14

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