The Basic Effects of Atmosphere Ocean Thermal Coupling on Midlatitude Variability*

Transcription

1 VOL. 55, NO. 4 JORNAL OF THE ATMOSPHERI SIENES 15 FEBRARY 1998 The Basic Effects f Atmsphere Ocean Thermal upling n Midlatitude Variability* JOSEPH J. BARSGLI IRES, niversity f lrad, Bulder, lrad DAVID S. BATTISTI Department f Atmspheric Sciences, niversity f Washingtn, Seattle, Washingtn (Manuscript received 30 Octber 1996, in final frm 7 May 1997) ABSTRAT Starting frm the assumptin that the atmsphere is the primary surce f variability internal t the midlatitude atmsphere cean system n intraseasnal t interannual timescales, the authrs cnstruct a simple stchastically frced, ne-dimensinal, linear, cupled energy balance mdel. The cupled system is then dissected int partially cupled and uncupled systems in rder t quantify the effects f cupling. The simplicity f the mdel allws fr analytic evaluatin f many quantities f interest, including pwer spectra, ttal variance, lag cvariance between atmsphere and cean, and surface flux spectra. The mdel predicts that cupling between the atmsphere and cean in the midlatitudes will enhance the variance in bth media and will decrease the energy flux between the atmsphere and the cean. The mdel als demnstrates that specificatin f histrical midlatitude sea surface temperature anmalies as a bundary cnditin fr an atmspheric mdel will nt generally lead t a crrect simulatin f lw-frequency atmspheric thermal variance. This mdel prvides a simple cnceptual framewrk fr understanding the basic aspects f midlatitude cupled variability. Given the simplicity f the mdel, it agrees well with numerical simulatins using a tw-level atmspheric general circulatin mdel cupled t a slab mixed layer cean. The simple mdel results are als qualitatively cnsistent with the results btained in several ther studies in which investigatrs cupled realistic atmspheric general circulatin mdels t cean mdels f varying cmplexity. This suggests that the experimental design f an atmspheric mdel cupled t a mixed layer cean mdel wuld prvide a reasnable null hypthesis against which t test fr the presence f distinctive decadal variability. 1. Intrductin There is n dubt that the midlatitude atmsphere and ceans frm a cupled system, but hw imprtant is this cupling n intraseasnal and lnger timescales? It is very difficult, if nt impssible, t quantify frm bservatins alne the relative imprtance f the atmsphere and cean in determining midlatitude lw-frequency variability. Any deductin f the relative imprtance f the atmsphere and cean will require reference (implicitly r explicitly) t an underlying mathematical r statistical mdel. At the cmplex end f the mdeling spectrum, studies using realistic atmspheric general circulatin mdels (AGMs) have been used t examine the impacts f cupling n the natural climate variability in the midlatitudes (e.g., * Jint Institute fr the Study f the Atmsphere and Ocean ntributin Number 38. rrespnding authr address: Dr. Jseph Barsugli, IRES, niversity f lrad, ampus Bx 449, Bulder, O jjb@cdc.naa.gv; david@atms.washingtn.edu Schneider and Kinter 1994; Manabe and Stuffer 1996; Lau and Nath 1996; Bhatt et al. 1998; Bladé 1997; Nitsche 1996). The methdlgy used in each f these studies invlves the cmparisn f tw integratins f the AGM. First, the AGM is integrated using a prescribed sea surface temperature (SST) field (typically, the bserved annual cycle). The secnd integratin allws fr interactins between the AGM and either an cean general circulatin mdel r a surface cean mixed layer mdel. Figure 1, reprduced frm Manabe and Stuffer (1996), illustrates a large, rbust effect seen in each f these studies: cmpared t the uncupled integratin, the cupled mdel shws significant enhancement in the variance f the surface air temperature (typically a dubling f the variance due t cupling), with the increased variance arising mainly frm changes at lw frequencies. The riginal mtivatin fr this paper came frm the results in Barsugli (1995, hencefrth B95), which nted the abve effect n surface temperature variance in a numerical mdel cnsisting f a tw-level atmspheric GM with znally symmetric bundary cnditins run with perpetual annual-mean inslatin, and cupled t a 50-m deep, glbal mixed layer cean mdel. The md American Meterlgical Sciety 477

2 478 JORNAL OF THE ATMOSPHERI SIENES VOLME 55 FIG. 1. The rati f the standard deviatin f the 5-yr-mean surface air temperatures frm tw runs f the GFDL R15 AGM: (fixed climatlgical SST)/(cupled t OGM). [Reprduced frm Manabe and Stuffer (1996).] el was designed t minimize variability in the Trpics s that the variability due t intrinsic midlatitude atmsphere cean interactins culd be islated. B95 shwed that the strng enhancement f thermal variance (defined as variance in the temperature and assciated thermal wind fields) due t cupling ccurs mainly fr timescales lnger than the e-flding decay time fr a mixed layer temperature anmaly, ML, which is apprximately 4 mnths fr that numerical mdel. Here ML can be estimated directly frm the autcrrelatin functin fr SST anmalies, r by dividing the effective heat capacity f the mixed layer by an empirically derived cefficient relating net surface flux anmalies t SST anmalies. Barsugli shwed that cupling reduces the verall variance f surface fluxes seen by the atmsphere, especially fr timescales greater than ML. Finally, B95 demnstrated that the structures f the atmspheric circulatin and surface flux anmalies in the midlatitudes were relatively insensitive t the atmsphere cean feedback and argued that the primary effect f cupling was t selectively enhance the natural lw-frequency variability in the midlatitude atmsphere thrugh reduced thermal damping. The atmspheric structures mst affected by cupling were the quasi-statinary, equivalent bartrpic Rssby waves that dminate the lw-frequency variability in this mdel. Lau and Nath (1996) came t a similar cnclusin regarding selective enhancement using the GFDL AGM [als cnfirmed by Bladé (1997) and Nitsche (1996)], as d Bhatt et al. (1998) in their study using the M1. The strength and rbustness f the abve results frm varius mdels lead us t ask: an we explain the basic effects due t cupling using a much simpler mdel? This study is directed at answering that questin, emplying a simple, stchastically frced, cupled atmsphere cean mdel. We start by reviewing the wrk f Hasselmann (1976) and f Frankignul and Hasselmann (1977), wh cnsidered the simplest stchastic mdel f SST variability, dt/dt F T. Here F represents mixed layer frcing anmalies (with any cntributin by feedback remved), T the sea surface temperature anmalies, and a feedback parameter. They assumed the pwer spectrum t F t be white (i.e., flat) fr frequencies lwer than sme high-frequency limit, yielding a red spectrum fr T. The SST spectra frm their stchastic mdel was shwn t be a reasnable fit t bserved SST spectra in the nrthern Pacific Ocean away frm dynamically active regins. The term feedback as used in the abve papers refers primarily t a strng damping effect that surface fluxes are presumed t have at lw frequencies n an SST anmaly. nambiguus separatin f the frcing frm the feedback is a difficult matter. It is suggested in Frankignul and Reynlds (1983) that it is best t determine the spectrum f F and the value f indirectly by tuning these parameters t fit the pwer spectrum f bserved SST. Nevertheless, they attempt t calculate sme knwn terms in F directly as well. Fr illustrative purpses, they then split F and int knwn and unknwn parts and calculate the lag-crrelatins between SST and the unknwn frcing, frm which the feedback has nt been entirely remved. Frankignul (1985) reviews this and ther wrk n mdeling atmsphere cean interactin in midlatitudes. We apprach the prblem differently, creating a stchastically frced cupled mdel, shwn schematically in Fig.. Our apprach has several advantages. First, as this is a cupled mdel, the feedback due t surface heat fluxes will be built int the mdel. Secnd, we will be able t cnsider feedback due t the atmspheric dynamical respnse t SST anmalies, albeit in a very apprximate manner. Third, the cupled stchastic mdel can be easily cnfigured t represent three cmmnly

3 15 FEBRARY 1998 BARSGLI AND BATTISTI 479 FIG.. Diagram f simple energy balance mdel n which Eqs. (1) and () are based. See appendix A fr definitin f symbls. used GM experimental designs: cupled (slab mixed layer), uncupled (fixed climatlgical SST), and prescribed time-dependent SST (frm bservatins r a cupled mdel). Finally, we have identified a plausible candidate fr a truly white frcing f the cupled system in the nnlinear dynamical frcing f the free atmsphere temperature equatin. This frcing can be seen in its purest frm in an AGM run with fixed SSTs. 1 We want t emphasize that the physics f the surface heat fluxes in Frankignul and Hasselmann (1977) is the same as in ur mdel, fllwing naturally frm a linearizatin f the bulk frmulas fr surface fluxes. Hwever, ur frmulatin emphasizes the rle f surface heat fluxes in reducing the internal damping in the cupled cean atmsphere system at lw frequencies, allwing fr greater thermal variance in bth the cean and atmsphere. We will discuss the rle f surface fluxes in cupled and uncupled mdels in mre detail in sectins 3 and 4. Simple cupled stchastic mdels are nt new. Fr example, Kim and Nrth (199) applied a cupled stchastic energy balance mdel t study varius aspects f the cupled atmsphere mixed layer deep cean system, building n earlier wrk f Nrth et al. (1983). Nrth and ahalan (1981) briefly cmpare the expected timescales fr scenaris where an AGM is run with slab mixed layer, fixed SST, r surface energy balance lwer bundary cnditins. Zubarev and Demchenk (199) use a cupled stchastic mdel similar t urs t investigate the relative rles f atmspheric versus ceanic stchastic frcing. Their frmalism encmpasses ur mdel; hwever, the parameter values assumed by these authrs leads t excessive sensitivity t cupling. Finally, highly idealized, deterministic uncupled mdels were used by Schpf (1985), Frankignul (1985, 4), and Martzke and Pierce (1997) t 1 Nte that in ne example Frankignul and Hasselmann (1977) use the bartrpic vrticity equatin t mdel uncupled atmspheric wind variability and use meridinal velcity anmalies as a prxy fr anmalies in air sea temperature difference. Hwever, they d nt cnsider the rle f the atmspheric dynamical respnse t SST anmalies. investigate the rle f the atmspheric respnse t a SST anmaly. These studies cncentrate n the dependence f the decay time f an SST anmaly n the spatial scale f the anmaly. Frankignul (1985) als cnsiders the quasi-steady cupled case. The paper is structured as fllws. In sectin we describe and analyze a stchastically frced cupled energy balance mdel that captures the basic effects f cupling n lw-frequency variability. In sectin 3 we shw pwer spectra f SST, atmspheric temperature and surface fluxes, as well as lagged linear regressins between atmspheric temperature and SST. In sectin 4 we present a discussin f the salient results frm this study alng with their implicatins. In sectin 4c we summarize results frm several recent studies in which cupled, uncupled, and prescribed SST experiments were perfrmed using full atmspheric GMs and discuss the results frm these studies in light f the expectatins frm the simple mdel presented here. nclusins are presented in sectin 5.. A simple stchastically frced energy balance mdel f cupled variability a. Mdel develpment The mdel we prpse t accunt fr the basic effects f cupling between the atmsphere and the cean in the midlatitudes is displayed schematically in Fig.. It is a ne-dimensinal thermdynamic mdel fr the upper (slab) cean cupled t a graybdy atmsphere. The single spatial dimensin represents a typical pint in the midlatitudes, althugh we als interpret the mdel variables in terms f individual hrizntal mdes f the atmsphere cean system. There is an assumed randm frcing assciated with the ubiquitus dynamical mtins in the midlatitude atmsphere. The equatins fr this mdel, linearized abut the climatlgical mean state, are as fllws [see appendix A; als cf. Schpf (1985)]: dt a (T T ) T F a sa s a a (1) dt dt (T T ) T s s. () dt Subscripts a and refer t atmsphere and cean respectively; T is the anmalus temperature; the heat capacity; s the linearized cefficient f cmbined latent, sensible, and lngwave heat flux (values f sa and s differ nly slightly); and a, the radiative damping f each cmpnent t space. Surface heat fluxes are calculated using the surface air temperature T s. The term F represents the dynamical cmpnent f the frcing, which we take t be stchastic. We assume that the surface air temperature anmaly is linearly related t the free atmsphere temperature

4 480 JORNAL OF THE ATMOSPHERI SIENES VOLME 55 TABLE 1. The standard parameter values as defined in the text and in appendix A. Parameter Value Parameter Value a Jm K 1 a Jm K 1 b 0.5 sa 3.9 W m K 1 c 1 s 3.4 W m K 1 d 1.08 a.8wm K Wm K 1 N 1 ad ad 0Wm K 1 z.4 bc anmaly, T s ct a. We take the Furier transfrm (t ) f Eqs. (1) and () and divide thrugh by sa t yield ita ata T F() (3) it cta dt. (4) We have made the fllwing substitutins: a / sa, a a / sa c, d / s 1, ( / a )( sa / s ), and F F/ sa. A tilde dentes a time-dmain variable, and an unadrned variable the crrespnding Furier transfrm variable. Explicit reference t the independent variables t and will be used nly fr emphasis r t avid cnfusin. The derivatin f Eqs. (1) and () frm a mre detailed energy balance mdel is presented in appendix A, where reasnable values f the mdel parameters are als justified. These parameter values, shwn in Table 1, will be referred t as the standard parameters and are used in the examples t fllw. Equatins (3) and (4) as they stand are nt suitable fr cmparing cupled and uncupled systems because the dynamical frcing term F includes the effects f cupling and will differ between cupled and uncupled runs. T illustrate this pint, we calculate the pwer spectrum f T a. Frm Eq. (3) we have (i a) T a F T FT* T F*, (5) where * dentes cmplex cnjugatin. The term T a is the pwer spectral density fr atmspheric temperature, and FT* is the Furier transfrm f the lag-cvariance functin between F and T. The lag-cvariance terms in Eq. (5) indicate that we must accunt fr the dependence f F n T in ur thery f the effects f cupling. In the analysis that fllws we will split the dynamical frcing int an SST-frced deterministic part and a purely randm part as fllws: F (b 1)T N, where b is a real cnstant. We have assumed that the dynamical respnse is prprtinal t the SST anmaly at the lw frequencies f interest. We assume that the pwer spectrum f N is independent f the cupling t the cean, hence inherent t the atmsphere. When substituted int Eq. (3), the ttal thermal and dynamical respnse becmes bt, and we will refer t b as the atmspheric respnse parameter. In actuality the temperature respnse f the free atmsphere t diabatic heating is accmplished largely by dynamical adjustment; therefre we will fcus nly n the ttal respnse in the rest f this paper. With the abve assumptins abut the atmspheric respnse, Eqs. (3) and (4) are in the standard frm f a tw-variable linear system (with stchastic frcing nly in the atmsphere equatin) : ita ata bt N (6) it ct dt. (7) a The cefficients a and d represent damping f the atmsphere and cean respectively, and the cefficients b and c represent the cupling between atmsphere and cean. At this pint it is useful t define a cupling cefficient bc, which represents the feedback due t atmsphere cean cupling, and a stability parameter z ad/, which results frm the cmpetitin between this feedback and damping. Nte that the atmspheric damping parameter, a, cntains a dependence n the parameter c. b. Methdlgy The design f the numerical experiments in B95 will be repeated using the simple mdel presented in this paper. This design cnsisted f three mdel runs as fllws. 1) upled: The cupled mdel was run first. ) ncupled: (a) The atmsphere mdel was run with SST fixed t be the znal mean f the climatlgy f the cupled run. (b) The slab mixed layer mdel was integrated in diagnstic mde, frced with the time histry f winds and temperatures frm run a. 3) MOGA [Midlatitude Ocean, Glbal Atmsphere, after Lau and Nath (1994)]: (a) The atmsphere mdel was run with SST prescribed t be the time histry f SSTs frm the cupled run. (b) The slab mixed layer mdel was integrated in diagnstic mde, frced with the time histry f winds and temperatures frm run 3a. Equatins (6) and (7) can be used t mdel the cupled, uncupled, and MOGA experiments as fllws. The cupled mdel (dented by superscript ) slves Eqs. (6) and (7) as a cupled set: at a bt N (8) T ct. (9) a Fr illustrative purpses, an even simpler system may be cnstructed by replacing Eq. (6) with T a M bt, where M is a stchastic prcess with a specified pwer spectrum, perhaps derived frm the utput f an uncupled GM run.

5 15 FEBRARY 1998 BARSGLI AND BATTISTI 481 Given N, ne can slve fr Ta and T. We have intrduced the abbreviated ntatin a (i a) and (i d). Fr the uncupled system, dented by superscript, we get the fllwing set f equatins: at a N (10) T ct. (11) a Equatin (10) was btained by setting T 0inEq. (6), since SST is fixed at climatlgical values. Given N, ne can slve fr T a. Equatin (11) is the diagnstic equatin fr a slave cean driven by the uncupled atmsphere, which we slve t get T. Finally, t mdel the prescribed SST experiment (MOGA, dented by superscript M), we set T T in the atmsphere equatin and use the MOGA atmsphere t frce a slave mixed layer cean, yielding the fllwing equatins: M M at a N bt (1) M M T ct. (13) a In additin, we assume that N M and T are uncrrelated. Recall that the direct effect f the prescribed SST n the dynamical frcing, including any rerganizatin f the atmspheric eddy fluxes, is accunted fr by the dynamical respnse term (b 1) T. This latter assumptin will simplify the calculatin f pwer spectra fr the case f prescribed SST and is, in fact, central t the success f this simple mdel. We have assumed that the natural variability f the atmsphere is unaffected by cupling t the surface. Mathematically, N M, N, and N shuld be cnsidered as separate realizatins f the same randm prcess, with the same pwer spectra. The superscripts n N included in Eqs. (8) (13) nly clarify the abve assumptin and will be drpped fr all the fllwing calculatins. The mdels in Eqs. (8) (13) are frmally equivalent t linear Markv prcesses. The uncupled system can be cast as ne-dimensinal Markv prcesses: T a as a first-rder prcess, and T as a secnd- rder prcess. The cupled system can be cast as a tw-dimensinal, first-rder Markv prcess, and the MOGA system as a fur-dimensinal, first-rder Markv prcess, subsuming the cupled system. Hwever, we have chsen t keep the mdels in a frm that highlights the physical parallelism between the systems. The methdlgy f the cupled, uncupled, and MOGA experiments as expressed in Eqs. (8) (13) illuminates the effect f cupling the atmsphere and cean in the midlatitudes. The results f the uncupled experiment, alng with the diagnstic SST field, serve as the uncupled null hypthesis against which we test fr the basic effects f thermal cupling. Our analysis f the MOGA experiment has bearing n understanding the rle f midlatitude SST anmalies in seasnal climate predictin. In additin, the MOGA experiment aids in the interpretatin f the rle f the midlatitudes in the simulatins dne FIG. 3. Pwer spectrum f 5-day-mean vertical-mean atmspheric ptential temperature frm a lng run f the tw-level mdel f B95 with fixed SSTs (thin line). Spectral estimates are the average estimates frm 16 recrds f 500 days each, using a Hanning windw. N significance is claimed fr any individual spectral peak. The thick line is the best-fit stchastic mdel spectrum frm Eq. (16), with damping timescale f days. as part f the Atmspheric Mdel Intercmparisn Prject (Gates 199), in which the glbal histrical SST field was specified as a bundary cnditin fr realistic AGMs. These diagnstic mixed layer mdel experiments prvide a cmmn currency fr cmparing the atmspheric variability in the cupled, uncupled, and MOGA runs in terms f the effect f the atmsphere n SST variability. c. mments n the stchastic frcing and the cupling parameters The nise frcing N() and the cupling parameters b and c represent the distillatin f cmplicated atmspheric dynamics and deserve further cmment. Figure 3, adapted frm B95, shws that the lw-frequency prtin f the spectrum f atmspheric temperature frm a lng integratin f the tw-level mdel is well apprximated by Eq. (16), assuming white nise frcing and a ttal damping timescale f abut days. The salient features reprduced are the flat spectrum at lw frequencies and the drp-ff tward higher frequencies. The damping timescale is abut half that f the standard parameters, with the discrepancy prbably due t neglecting the dynamical damping term discussed in appendix A, althugh the chice f an effective heat capacity f the atmsphere is als an issue. Equatin (10) is nt meant t be a sphisticated mdel f atmspheric lw-frequency variability but nly an apprximate representatin that we use as a starting pint t illustrate the effects f cupling. We reiterate that nnlinear ptential vrticity dynamics is driving the lw-frequency variability f the atmsphere. The nise therefre results frm the nnlinearities in the equatins f mtin, as well as frm the representatin f a multidimensinal system

6 48 JORNAL OF THE ATMOSPHERI SIENES VOLME 55 in terms f ne variable. Fr this simple mdel, we fcus slely n the thermal cmpnent f the ptential vrticity frcing because numerical experiments lead us t believe that the greatest effect f cupling will be n temperature (and assciated thermal wind) variance, which we refer t as thermal variance. Fr simplicity, in the analytic calculatins presented in sectins 3b d we will assume N t be white nise f unit amplitude. We have parameterized the atmspheric respnse t an SST anmaly as bt, which includes the linear dynamic and diabatic respnse as well as the prtin f the nnlinear respnse that can be linearly parameterized in terms f SST. Based n B95 we expect the dynamical prtin f the respnse t act generically as a negative feedback, with atmspheric heat fluxes partially ffsetting the diabatic effects f an SST anmaly. That is, we expect that 0 b 1. (Other pssibilities are discussed in sectin 4a.) The simplest measure f b is btained by slving the time mean f Eq. (1) fr the case f a GM with steady SST anmaly frcing (superscript S ): S S at a bt [cf. Zubarev and Demchenk (199), with their Xˆ Y equivalent t ur b, and Ŷ X c]. In a similar vein, ne can estimate the atmspheric respnse frm the linear regressin f the atmspheric respnse t timevarying SST anmalies, chsing an apprpriate midlatitude SST anmaly index as the basis fr the regressin. Barsugli des just that fr a tw-level idealized atmspheric GM, and shws that the lcal free-atmsphere temperature respnse is rughly 0.5 Kelvin per degree f SST anmaly. Because a 1, it is reasnable t chse b 0.5 as a bulk measure f the free atmsphere temperature respnse. A gd indicatin f the effects f the nnlinear dynamical feedback in the tw-level mdel f B95 can be btained by cmparing the linear and nnlinear respnses t typical SST anmalies. The linear respnse t the same SST anmaly is rughly twice that f the nnlinear respnse and has similar hrizntal structure in the regin f the largest SST frcing, indicating that the nnlinearity is acting primarily as a damping n the linear respnse. The prprtinality between T a and T s can be estimated frm the lw-frequency variance f the uncupled diagnstic cean, Eq. (11), which is frced by the uncupled atmsphere: d c T T. mparing the lw-frequency prtin f the spectra f SST and f the vertically integrated ptential temperature frm the tw-level mdel f B95 indicate an apprximate value f c 0.8 fr that measure f the free atmsphere temperature. Fr simplicity f interpretatin, we will assume c 1 fr the standard parameters. a In general, the cupling cefficient will depend n the vertical structures f the uncupled atmspheric variability, f the stchastic frcing, and f the diabatic effects assciated with cupling, and n hw well these prject n ne anther. These dependencies in turn are a functin f the hrizntal structure f lw-frequency variability. In light f this cmplexity we view as a characteristic cupling cefficient fr the entire cupled system. We have chsen the vertical mean temperature as the apprpriate free atmsphere temperature variable because f the dminant rle played by equivalent bartrpic, quasi-statinary structures in lw-frequency variability. Later we shall see hw deep and shallw atmspheric mdes can be interpreted in terms f varying strengths f this cupling cefficient. 3. Slutins a. Pwer spectra in the cupled, uncupled, and MOGA systems As nted abve, the stchastic mdel can be used t interpret cupled, uncupled, and MOGA experiments. The stchastic mdel allws us t predict hw cupling affects variance (pwer) in the three experiments. In the cupled case (superscript ) we slve the cupled set f Eqs. (8) and (9) fr the pwer spectral densities P a () Ta and P() T as fllws: N Pa (14) a c N P. (15) a Fr the uncupled (superscript ) case we slve Eq. (10) fr the atmspheric pwer spectrum. The pwer spectrum fr T, the slave cean temperature, fllws frm the diagnstic cean equatin (11). In that case we get fr the pwer spectra: N Pa (16) a c N P. (17) Fr the MOGA case (superscript M), we slve Eq. (1), assuming that N and T are uncrrelated. Thus, when M we calculate the pwer spectrum f T a the crss-terms between N and T are zer. As in the uncupled case, the MOGA atmsphere is used t frce a slave mixed layer cean. The pwer spectra fr the MOGA case are as fllws: a N M Pa 1 (18) a a c N M P 1. (19) a a

7 15 FEBRARY 1998 BARSGLI AND BATTISTI 483 FIG. 4. Pwer spectra f atmsphere and cean temperature fr the cupled, MOGA, and uncupled cases. The standard parameters (see Table 1) are used. Plts f these quantities are shwn in Fig. 4 fr the standard parameters, with N assumed t be white nise with unit amplitude. Nte that the uncupled atmsphere pwer is pure red nise the result f a firstrder Markv prcess with autcrrelatin time f a 1, r in dimensinal units, a a ( sa a ) 1. Fr the standard parameters a 4 days. We are nw in a psitin t draw sme general cnclusins abut variance in cupled and uncupled runs. The MOGA pwer spectrum fr either atmsphere r cean temperature can be expressed in terms f the respective cupled and uncupled pwer spectra as fllws: M Pa, Pa, 1 a (0) a M Pa, P a,. (1) At this stage we have nt assumed anything abut the shape f the spectrum f the frcing term N. We can see that at all frequencies, bth the cupled and MOGA runs have mre variance than the uncupled run, at least fr reasnable values f. The cmparisn f the MOGA and cupled runs is mre cmplicated. Fr lw frequencies, ad, the cupled variance exceeds the MOGA variance. Fr higher frequencies, the direct frcing by the SST anmalies exceeds the internal variance. Hwever, due t the lng timescales assciated with the cean there is little pwer at these higher frequencies. It is instructive t cnsider the ratis f variance between different systems in the limit as 0 (r equivalently, 0) as fllws: a z, () (z 1) 1 M, 1 (3) (z 1) z,m, (4) (z 1) 1 where z ad/ as befre. We have defined the rati, P (0)/P (0), with the ther ratis defined analgusly, and have drpped the subscripts because these ratis are the same fr the atmsphere and cean variables. (This equality f variance ratis des nt necessarily hld in mre realistic mdels, as discussed in sectin 4a.) The rati f pwer at lw frequencies depends nly n the stability parameter z. A plt f the abve pwer ratis as a functin f z is shwn in Fig. 5. The standard parameters crrespnd t z.4, and this value is indicated by a vertical line in the plt. As previusly mentined, z is a stability parameter fr the cupled system. Fr z 1, the cupled system is stable, and the pwer in the system is maintained by the stchastic frcing. Large values f the stability parameter, z k 1, crrespnd t either large damping, large negative atmspheric feedback, r inefficient cupling between the free atmsphere and the surface. In this case the MOGA variance appraches the uncupled variance, while the cupled variance appraches the uncupled variance, thugh much mre slwly. In the limit as z 1, which is ff the scale f Fig. 5, the cupled and MOGA variances becme equal, and bth apprach infinity. This limit crrespnds t the unrealistic case where psitive atmspheric feedback cmpletely cunteracts damping. Even in the case where we allw n

8 484 JORNAL OF THE ATMOSPHERI SIENES VOLME 55 FIG. 5. Rati f pwer in SST between the cupled and uncupled (slid line), cupled and MOGA (dashed line), and MOGA and uncupled (dash dt line) cases in the limit as the frequency appraches zer. The thick vertical line at z.4 indicates the standard parameters. atmspheric dynamical feedback (i.e., we allw nly the thermal respnse f the atmsphere s that b 1, resulting in z 1.5 fr the standard parameters), we get excessively large ratis between cupled and uncupled runs. Fr z 1, we have linearly unstable atmsphere cean interactin fr which the stchastic mdel prpsed here is inapprpriate. b. Integrated variance The differing effect that cupling has n the atmsphere versus the cean can be seen if we take the integral ver all frequencies in Eqs. (14) (19) t get the ttal variance. We will assume white nise frcing f unit amplitude (i.e., N 1) t simplify the integratin. The results fr the uncupled case are the simplest (see e.g., R Math Tables): Pa d (5) a 0 0 P d, (6) ad(a d) where d d/. Fr the cupled system we will shw nly the rati f the ttal cupled pwer t the ttal uncupled pwer, where we define, a a a 0 0 P d P d (the crrespnding ratis fr ceanic variance and fr the MOGA system are defined analgusly). The integrals fr the cupled and MOGA systems are mre cmplicated than fr the uncupled system. The exact FIG. 6. Rati f ttal variance between cupled and uncupled runs fr SST (slid line) and atmspheric temperature (dashed line) as a functin f the stability parameter z. results are shwn in appendix B. If we expand the exact results in the small parameter d/(a), keeping nly the leading rder in, we get, a 1 O( ), (7) z 1 z, (exact), (8) z 1, where z is as befre; is pltted as a functin f z, in Fig. 6. Fr the standard parameters, a 1.03 and, The parameter is prprtinal t a, the rati f the effective heat capacity f the atmsphere t that f the cean mixed layer. Therefre, is inversely prprtinal t the mixed layer depth h. The crrespnding ratis fr the MOGA case are as fllws: M, a 1 O( ) z(z 1) (9) 1 M, 1 O(). (z 1)(z 1) (30) Because the bulk f atmspheric variability lies in higher frequencies where cupling has little effect, the rati f cupled t uncupled ttal atmspheric variance is near unity. On the ther hand, the bulk f mdel ceanic variance is at lw frequencies where the effect f cupling is strng, s the cean rati is substantially greater than 1. This reflects what happens in the twlevel mdel f B95, as the ttal atmspheric variance is nly slightly affected by cupling, but the ceanic variance is substantially increased by cupling (see sectin 4c). The effect f cupling n the ttal ceanic pwer is strictly independent f in the mdel presented here, and thus is independent f the mixed layer depth.

9 15 FEBRARY 1998 BARSGLI AND BATTISTI 485 FIG. 7. Pwer spectra f surface flux, defined as Q T s T, fr cupled, MOGA, and uncupled cases using the standard parameters. The fluxes int the atmsphere ( ), the fluxes int the diagnstic cean mdel ( ), and the cupled fluxes (]) are shwn. The symbls are defined in sectin 3c. c. Surface flux spectra We can cmpute the pwer spectra f surface fluxes in a straightfrward manner frm the mdel equatins (8) (13), s the frmulas are nt shwn here. Fr the cupled system, we define Q] Ts T cta T t be scaled symmetric surface flux. Fr cnvenience, we are neglecting the terms in the surface fluxes, that are nt prprtinal t the air sea temperature difference. If these terms were included, the fluxes that frce the cean wuld strictly g t zer fr 0, but therwise the discrepancy is small. Fr the MOGA system, there are tw sets f fluxes, thse that frce the atmsphere (with SSTs specified frm the cupled mdel), Q Ts T, and thse that frce the diagnstic M M M M M cean, Q Ts T. Likewise, fr the uncupled system we have Q Ts and Q Ts T. These pwer spectra are pltted in Fig. 7. At lw frequencies M the fluxes that frce the cean satisfy Q] Q Q, as in B95. Given the simplicity f ur cean mdel, it is therefre nt surprising that the resulting SST variance fllws the same rdering. Als, at lw frequencies Q Q ]. This relatin is a manifestatin f the reduced thermal damping in the cupled system, and it agrees with the numerical mdel results f B95 and Bhatt et M al. (1998). Finally, Q Q at lw frequencies. That is, there is a large excess f pwer in the lw-frequency fluxes that the MOGA atmsphere sees, cmpared t thse that the uncupled and cupled atmspheres see. This result was smewhat unexpected in the numerical experiments f B95, particularly in light f the fact that bth the atmspheric and ceanic thermal variance f the MOGA run were intermediate between the uncupled and cupled runs. In the present mdel, this excess is a direct result f assuming that a large fractin f the natural variability is independent f the prescribed SST anmalies. The implicatins fr ne-way frced experiments using numerical mdels are discussed in sectin 4b. d. Lag cvariance FIG. 8. Autcrrelatin functin fr atmspheric temperature (thick lines) and SST (thin lines) fr the cupled (dash dt), MOGA (dashed), and uncupled (slid) cases. Figure 8 shws the autcrrelatin functins fr T and T a fr the three systems with the standard parameters. The autcrrelatin functins are simply the nrmalized Furier transfrms f the pwer spectra shwn

10 486 JORNAL OF THE ATMOSPHERI SIENES VOLME 55 in Fig. 4. learly, cupling significantly increases persistence f bth atmspheric and ceanic temperature anmalies. Likewise, the lag cvariance between T and T a is the Furier transfrm f T a T*, where the asterisk dentes the cmplex cnjugate. The derivatin f frmulas fr lag-cvariance frm Eqs. (8) (13) is straightfrward and is nt shwn. The lag-cvariance functin can then be used t derive the lag-linear regressin functin, which is shwn in Fig. 9. Atmspheric variance, which is strngly dependent n averaging perid, des nt appear in the denminatr f the linear regressin, reducing the sensitivity f this measure t the averaging perid used. Fur curves are shwn in Fig. 9. The uncupled case (curve a) shws strng asymmetry in lag. This asymmetry is characteristic f a system with large thermal inertia (the cean), frced by a stchastic system with a fast decrrelatin time (the atmsphere), as seen in Hasselmann (1976). The MOGA and cupled systems (curves b and c) shw a prgressively larger cmpnent that is symmetric in lag. The regressin between the prescribed SST and the MOGA atmsphere (curve d) is almst entirely symmetric in lag. This symmetry is the result f the assumptin that the effects f SST anmalies n the atmsphere are instantaneus. The analgus lag crrelatin functins (fr pentad means) are similar in shape t thse in Fig. 9, but with maximum crrelatins f arund 0.4 fr curves a c and 0.1 fr curve d. These curves are cnsistent with the lagged linear regressin maps fr these fur cases shwn in B95. Fr the standard parameters, the lag-crrelatin peaks fr atmsphere leading cean by abut 10 days, a little mre than twice the decrrelatin time f the atmsphere. 4. Discussin a. General cmments abut cupling t slab mixed layers T put the effects f cupling in a larger framewrk, cnsider an atmspheric GM cupled t a slab mixed layer cean f cnstant depth h. The value f h determines the nature f the lwer bundary cnditin n the atmspheric thermdynamic equatin. In the extreme case f h 0, the lwer bundary cnditin becmes an instantaneus surface energy balance (SEB). The ther extreme, h, crrespnds t fixed SST, where atmspheric temperature anmalies f all frequencies are damped equally by surface fluxes. The fixed SST case shuld exhibit the mst damping due t surface fluxes, and the SEB case the least. Fr intermediate values f h the cupling acts t damp nly highfrequency atmspheric temperature anmalies. Fr timescales lnger than ML, the SST can adjust t atmspheric temperature anmalies and the surface fluxes are reduced t near zer. As h 0, the timescale f the mixed layer becmes cmparable t r shrter than FIG. 9. Lagged linear regressin between T a and T. urve a: T a, M M M T ; curve b: T, T ; curve c: T, T ; curve d: T, T. a a a the synptic timescale in the atmsphere. Because barclinic cnversin is the ultimate surce fr much f midlatitude variability, the entire spectrum f variability may be affected. Therefre, we have avided cnsideratin f the SEB case. Instead we chse the ther extreme, h (the uncupled system, with fixed SST) as the basis fr cmparisn with the ther systems. In the cupled and uncupled cases, Eqs. (8) (11), the stchastic dynamics f the atmsphere is the nly surce fr lw-frequency variability in the cupled mdel. In the MOGA framewrk hwever, the prescribed SST anmalies act as an additinal external frcing at lw frequencies [see Eqs. (1) and (13)], s we expect that the MOGA run will have mre variance than the uncupled run with its fixed, znally symmetric SST. The questin is Hw much mre? It seems reasnable t assume, as we have dne, that the bulk f the midlatitude nnlinear atmspheric variability is uncrrelated with the prescribed SST anmalies, at least fr SST anmalies f mdest amplitude. Evidence f this in a mre realistic mdel is seen, fr example, in the predictability study f Miller and Rads (1990), where inclusin f actual SST anmalies in midlatitudes did little t enhance predictability. Hence, it is nt surprising that the MOGA atmsphere temperature variance is less than the cupled mdel variance. Fr example, Eq. (4) with the standard values fr the parameters indicates that the lw-frequency variance shuld be 1.93 times greater in the cupled framewrk than in the MOGA framewrk. Barsugli (1995) argues that large respnses t SST anmalies are nt generic t atmsphere cean cupling, but rather depend n the special circumstances invlving such factrs as the climatlgical mean statinary waves, the psitin f the SST anmalies, r land cean temperature cntrasts. Tw cmmnly seen respnses f realistic AGMs t prescribed SST anmalies are a surface-trapped lw ver a warm SST anm-

11 15 FEBRARY 1998 BARSGLI AND BATTISTI 487 aly and an equivalent bartrpic high dwnstream frm a warm SST anmaly (Palmer and Sun 1985; Peng et al. 1995). Bth f these respnses are cnsistent with the simple mdel presented in this paper, prvided that the interactin is stable. A large bartrpic respnse wuld indicate that SST anmalies cuple effectively t natural bartrpic variability f the atmsphere, prbably thrugh precipitatin r eddy fluxes, resulting in a large effective value f b. Sme mdels (e.g., Latif and Barnett 1994; Palmer and Sun 1985) shw evidence f a large psitive feedback; that is, ad. Such a linearly unstable system is nt amenable t treatment by the present stchastic mdel and, like any linearly unstable system, requires a mechanism fr (statistical) equilibratin in rder t be a cmplete thery. The instability in the abve experiments appears t arise largely because f a strng atmspheric respnse t SST anmalies. One wuld expect t see tw phenmena in the presence f such strng upward cupling, which, as far as we knw, are nt present in the bservatins: First, lag lead crrelatins between SST and the free atmsphere wuld be largely symmetric abut zer lag. Secnd, prescriptin f midlatitude SST anmalies wuld significantly enhance predictability f the atmsphere in hindcast studies. b. One-way frced experiments and spurius surface fluxes In Fig. 7 we shw that atmspheric mdels with prescribed SST can have large spurius surface fluxes at lw frequencies. The largest spurius fluxes, defined as M Q / Q ], ccur fr small values f b and fr values f near the stability bundary (i.e., ad). These spurius fluxes d nt present a serius prblem fr making shrt-term frecasts, as these frecast mdels are initialized with the crrect atmspheric state frm the cupled system (i.e., frm bservatins). Hwever, a prblem may arise fr seasnal predictins where SST is specified, whether specified as strict persistence f bserved SST anmalies r whether these anmalies are damped back t climatlgy ver sme empirical mixed layer timescale. Specifying midlatitude SST anmalies cnstrains the amplitude f atmspheric temperature anmalies thrugh the spurius (damping) surface fluxes. It wuld be better t make seasnal frecasts using an ensemble f runs f AGMs cupled t mixed layer mdels than t use fixed bundary cnditins. The same prblem arises in climate simulatin studies in which the histry f glbal SSTs is specified, r in establishing cntrl run statistics fr studies f decadal variability. The true variability in the cupled system is nt necessarily well apprximated by frcing the atmsphere with realistic SSTs fr mdels with small r mderate values f the atmspheric respnse parameter b. Numerical experiments in which atmspheric quantities are prescribed as the frcing fr an cean mdel suffer frm a different prblem. The merits f specifying surface air temperatures and winds versus specifying surface heat fluxes has been debated in the literature. Bth methds suffer frm prblems, as bth surface fluxes and air temperature are strngly cupled t the cean temperature. If the flux frcing cmes frm an uncupled r MOGA experiment, then extreme SST variance at lw frequencies will result because f spurius surface fluxes. The excessive lw-frequency flux frcing is ften cmpensated by applying an arbitrary feedback term prprtinal t the SST anmaly. Neglecting wind-frced surface flux variance, this methd is linearly equivalent t specifying surface air temperatures and winds, as lng as the crrect value f the feedback is chsen. This is seen in the fllwing extensin f Eq. (): dt (T T ) T Q ( )T s s s. dt Hwever, GM bundary layer frmulatins usually cntain a strng nnlinear dependence n the static stability f the bundary layer. mputing the frcing frm atmspheric surface air temperature and wind, using a sphisticated nnlinear surface flux parametrizatin avids the assumptin f linearity, but the cmplexity f sme GM bundary layer frmulatins may make this an undesirable chice in practice. The pwer spectrum f surface fluxes used t frce a slab mixed layer mdel must tend tward zer at lw frequencies, r else the temperature variance wuld be unacceptably large. If a mre cmplicated cean mdel is frced with spurius lw-frequency fluxes, this frcing will be cmpensated by ther ceanic prcesses entrainment, advectin, cnvectin, diffusin at these timescales. In fact, substantial lw-frequency surface flux variance in a cupled mdel r in bservatins is a signature f such prcesses. A ptentially serius prblem arises if ne uses surface fluxes diagnsed frm bservatins t frce cean mdels. Observatinal errrs, which prject nt all frequencies, can lead t ptentially large errrs in lwfrequency temperature variance (e.g., Rnca and Battisti 1997). Adding a simple linear relaxatin term, as discussed abve, is nt entirely satisfactry, as there exists sme frequency belw which the temperature variance is determined primarily by the errr variance in the surface fluxes. In effect, randm (uncrrelated in time) bservatinal errr in surface fluxes results in a lw-frequency limit belw which ne is unable t mdel temperature variance reliably. c. mparisn with mre cmplete atmsphere and cean mdels Barsugli (1995) perfrmed uncupled, MOGA, and cupled integratins in which the atmsphere was mdeled by the glbal, tw-level primitive equatins with parameterized cnvectin and radiatin [similar t the

12 488 JORNAL OF THE ATMOSPHERI SIENES VOLME 55 TABLE. Enhancement f pwer in the simple stchastic mdel with the standard parameter values and in the tw-level mdel f B95. Quantities in italics are determined using the average pwer fr perids lnger than 65 days. dentes the vertical mean ptential temperature in the tw-level atmsphere. Rati (Eq. in text), () M, (3),M (4), a (7), (8) M, (9) a M, (30),M a,m Stchastic mdel Tw-level GM SST ().69 (1.80).00 (1.38) 1.34 (1.30) mdel f Held and Suarez (1978)]. SST was either prescribed r simulated using a 50-m slab cean mdel. The mdel gemetry was idealized t be a glbal cean. The cupled stchastic mdel qualitatively reprduces the rati f ttal variance between cupled, MOGA, and uncupled runs in B95, as shwn in Table. The spectra f cean and atmsphere temperature fields and surface flux fields frm B95 are reprduced in Fig. 10. These are qualitatively cnsistent with the results frm the stchastic energy balance mdel, as seen in a cmparisn f Fig. 10a t Fig. 4, and Fig. 10b t Fig 7. The results frm the three experiments in B95 supprt the hypthesis fr the twfld effects f cupling: the reductin f thermal damping at lw frequencies and the relatively weak direct effect f frcing by SST anmalies. There are, hwever, tw majr discrepancies between the variance ratis predicted by Eqs. () (4) frm the stchastic cupled mdel and thse calculated frm the tw-level GM f B95. First, the lw-frequency variance enhancement in the atmsphere is generally less than that f the cean (see Table ), and the enhancement decreases as ne passes frm the surface t 50 mb (nt shwn). learly, in the tw-level mdel there can exist variability with a vertical structure that has n signature in surface temperature, and thus des nt participate in cupling. This additinal atmspheric variability dilutes the enhancement f variance ne wuld expect frm a univariate mdel such as we have presented. The secnd majr discepancy is seen in the wavenumber-frequency spectra f B95 enhancement f variance is a strng functin f znal wavenumber. The explanatin lies in the mdal structure f variability in the tw-level mdel. There are tw hrizntal mdes that participate strngly in the cupling, a deep, equivalent bartrpic mde with znal wavenumber k 4, and a shallw k 1 mde. The frmer is the dminant mde f lw-frequency variability in the uncupled case and is merely enhanced by the cupling t the mixed layer. The latter appears strngly in the cupled run but nly very weakly in the uncupled run, and can be explained as a surface-trapped advective, cupled mde (Frankignul 1985; B95). The k 1 mde als appears FIG. 10. (a) As in Fig. 4 but fr pwer spectra f SST and atmspheric vertical mean ptential temperature frm integratins f the tw-level mdel f B95. (b) As in Fig. 7 but fr pwer spectra f ttal surface fluxes frm the tw-level mdel. strngly in the MOGA run spectrum, suggesting a large value f assciated with this mde. These discrepancies, and the assciated dynamical reasns fr them, are indicative f what t expect frm the analysis f mre realistic mdels. Manabe and Stuffer (1996) present results frm the GFDL R15 AGM in which the AGM was (i) integrated using the prescribed annual cycle f SST and (ii) cupled t a 50-m slab mixed layer, and (iii) cupled t an cean GM. They fund that cupling acted t enhance the variance f SST and surface air temperature in the midlatitudes, with the greatest increase ccurring at lwest frequencies (see their Figs. 1 5). In Fig. 1 we have reprduced their Fig. 4b, shwing the rati f standard deviatins between their cupled GM and fixed SST runs. The carse cntur interval in their figures makes it difficult t determine quantitatively the extent f the agreement between their GM and the stchastic mdel. Hwever, their figures indicate that the rati f cupled-t-uncupled variance (derived by squaring their standard deviatins) in surface air temperature in midlatitudes is rughly.5 fr 5-yr averaged fields, whereas the stchastic mdel yields a rati between.4 (perid 5 yr) and.9 (perid infinity). Significantly, the ratis f variance between the mdels with the cean GM and the slab mixed layer are near unity ver much f the pen ceans (their Fig. 4a). Furthermre, the GFDL cupled AGM/OGM yields midlatitude spectra f surface air temperature and SST (their Fig. 1) that are remarkably similar in structure t that frm the

13 15 FEBRARY 1998 BARSGLI AND BATTISTI 489 simple stchastic mdel. Hwever, the GFDL mdel yields slightly mre variance in SST than in surface air temperature (unlike the stchastic mdel) fr perids greater than abut 0 years, presumably due t variability assciated with changes in the ceanic thermhaline circulatin. Bladé (1997) used the same atmsphere GM as in Manabe and Stuffer (1996) t examine the effect f midlatitude cupling n the variability in the midlatitude atmsphere. In this study, the cntrl integratin was cmpared t an integratin in which the atmsphere was cupled t a 50-m slab cean in the midlatitudes f bth hemispheres; bth integratins featured perpetual January slar frcing. Bladé reprted that the influence f midlatitude cupling significantly enhanced the midlatitude lwer trpspheric temperature and circulatin variance at the lwest frequencies. Fr example, the rati f cupled-t-uncupled pwer in the 850-mb air temperature ver the midlatitude ceans at lwest frequencies is apprximately 1.6. Bladé (1997) furthermre nted that the effect f midlatitude cupling in the full GFDL GM was quantitatively similar t that fund in the idealized tw-level GM studies f B95. Bhatt et al. (1998) examined the differences in the atmspheric circulatin anmalies that result frm cupling the NAR M1 R15 AGM t the variabledepth mixed layer mdel f the Nrth Atlantic Ocean. Bth cupled and cntrl integratins were 35 yr lng and include a seasnal cycle. mpared t the cntrl integratin, the variance f the wintertime averaged surface air temperature in the cupled integratin increased by a factr f abut.3. The net surface heat fluxes decreased by a factr f abut 0.6. upling als enhanced the persistence f the principal mde f variability in the atmsphere. These results are in agreement with predictins frm the stchastic mdel. While the simple mdel we present was develped t explain enhancement f internally generated midlatitude lw-frequency variability, it is als applicable t externally frced variability. Lau and Nath (1996) reprted the results frm a series f experiments with the GFDL R15 AGM n the effect f midlatitude cupling n trpically frced midlatitude variability. In these experiments they examined the prtin f the midlatitude variability in the cean and atmsphere that is assciated with ENSO via atmspheric telecnnectins ( the atmspheric bridge ) by prescribing bserved trpical Pacific SST anmalies under the AGM. The effect f the cupling in the midlatitudes n the ENSOfrced midlatitude variability was determined by examining differences between tw ensembles f integratins: the first ensemble held midlatitude SST fixed at climatlgical values (TOGA), while the secnd ensemble allwed the midlatitude atmsphere t interact with a 50-m slab cean (TOGA-ML). The principal results f Lau and Nath s study are qualitatively cnsistent with the stchastic mdel results. Specifically, the ENSOrelated variance in the surface air temperature ver the Nrth Pacific Ocean, which is predminantly at lw frequencies, increased abut furfld due t cupling. 3 The variance and persistence f the 500-mb and surface atmspheric circulatin anmalies assciated with the ENSO were als enhanced by the midlatitude atmsphere cean cupling, cnsistent with the enhancement in the temperature variance via the thermal-wind relatin. The structure f the midlatitude ENSO-related anmalies was fund t be insensitive t midlatitude cupling. In a similar investigatin t Lau and Nath (1996), Alexander (199) cncluded that air sea interactin primarily acts t damp cean anmalies, whereas we cme t the ppsite cnclusin. The reslutin f this seeming cntradictin is that he cmpared his cupled SSTs t SSTs frm an cean mdel frced by surface fluxes frm his uncupled atmsphere integratins (dt / dt Q in ur ntatin). As discussed in sectin 3c, this latter methdlgy prduces spuriusly large SST variability, and des nt serve t illuminate the basic rle f cupling. In fact, what Alexander (199) calls partially cupled SST is equivalent t ur uncupled SST, and indeed shws abut half the temperature signal cmpared t his cupled SST away frm the cntinents in winter, thus cnfirming ur viewpint. d. aveat cncerning the use f this mdel t interpret GM results The main virtues f the mdel in Eqs. (8) (13) are its simplicity and flexibility. One is tempted t use it t quantitatively diagnse utput frm atmspheric GMs cupled t slab mixed layer ceans. aveat emptr: the mdel as it stands is versimplified in several imprtant aspects, making it unsuitable t use fr quantitative diagnsis. We believe that the majr versimplificatins are as fllws: the assumptin f a first-rder Markv prcess fr an uncupled atmsphere mdel, the assumptin f purely thermal cupling, and the neglect f mdal structure in the atmsphere. We will discuss these belw. The uncupled atmsphere mdel f Eq. (10) des nt generally capture the spectrum f lw-frequency variability in mre realistic GMs. Hwever, the exact frm taken by the uncupled atmsphere equatin is nt critical t understanding the basic effects f atmsphere cean cupling. Fr example, the 850-mb temperature pwer spectrum frm and uncupled run f the GFDL R15 AGM (Bladé 1997), taken at grid pints ver the Nrth Pacific Ocean, cannt be fit by Eq. (10). Hwever, an ad hc fit t the GM spectrum may be btained using Eq. (10) with an 8-day damping time- 3 This enhancement was estimated by cmparing the squares f the linear regressin cefficients at the bull s-eye in the Nrth Pacific (Figs. 3c and 1c f Lau and Nath) fr the TOGA and TOGA-ML runs.