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Tellus (1977), 29,289-305

Stochastic climate models, Part I1Application to sea-surface temperature anomalies andthermocline variability

By CLAU DE FRA NKIG NOUL and KLAUS HASSELMANN, Max-Planck-lnstitutfur Meteorologie.Bundesstr. 55,2000 Hamburg 13, FGR(Manuscript received December 10, 1976; in final form February 11, 1977)

ABSTRACTThe concept of stochastic climate models developed in P art I of this series (Hasselmann , 1976 ) isapplied to the investigation of the low frequency variability of the upper ocean. It is shown thatlarge-scale, long-time sea surface temperature (SST) anomalies may be explained naturally asthe response of the oceanic surface layers to short-time-scale atmospheric forcing. The w hite-noise spectrum of the atmospheric input produces a red response spectrum, with most of thevariance concentrated in very long periods. Without stabilizing negative feedback, the oceanicresponse would be nonstationary, the total SST variance growing indefinitely with time. Withnegative feedback, the response is asympto tically stationary. These effects are illustrated throughnumerical experiments with a very simple ocean-atmosphere model. The model reproduces theprincipal features and orders of magnitude of the observed SST anomalies in mid-latitudes.Independent support of the stochastic forcing model is provided by direct comparisons ofobserved sensible and latent heat flux spectra with SST anomaly spectra, and also by thestructure of the cross correlation functions of atmospheric surface pressure and SST anomalypatterns. The numerical model is further used to simulate anomalies in the near-surfacethermocline through Ekman pumping driven by the curl of the wind stress. The results suggestthat short-time-scale atmospheric forcing shou ld be regarded as a possible candidate for theorigin of large-scale, low-period variability in the seasonal therm ocline.

1. IntroductionIn Part I of this series (Hasselmann, 1976,referred to in the following as I) a stochastic modelof climate Variability was considered in which slowchanges of climate were interpreted as the integralresponse to continuous random excitation by short-time-scale weather disturbances. To illustrate the

basic concepts of the model we consider here avery simple climatic system consisting only ofthe upper layer of the oce an, which is driven by therandom fluxes of heat and momentum across theair-sea interfac e. The model reproduce s the prin-cipal features and orders of magnitude of observedmid-latitude sea surface temperature (SST) vari-ability in the period range of approximately onePresent address: Department of Meteorology,Massachusetts Institute of Technology, Cambridge,MA02139, U.S.A.

month to two years, and is not inconsisten t with theobserved therm ocline variability in the period rangeof a few weeks to half a year. In view of its strongsimplifications the model should nevertheless beviewed only as an example of the basic statisticalstructure which may be expected generally for atwo-time-scale climatic system, rather than anattempt to model quantitatively the upper laye rs ofthe ocean. A number of models of the internaldynamics of the upper layers a s h e ocean havebeen proposed, most of them more sophisticatedthan the one considered here (cf. Nder, 1975 orThompson, 1976 for reviews), but the principalfeatures of the input-response relations with whichwe sha ll be concerned are independent of the detailsof the model. Another very simple climatic modelapplicable for longer time scales in the range 1 to105 years is discussed in Part I11 of this series(Lemke, 1977).

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290 C. FRANKIGNOUL AND K. HASSELMANNBecause of their long persistence, large-scaleSS T anomalies have attracted considerable atten-tion in the context of long-range weather fore-

casting. Strong indications of SST-weatherrelationships have been revealed through the statis-tical analyses of extended time series (cf. Bjerknes,1966; Namias, 1969). Numerical experiments havealso clearly demonstrated statistically significantmodifications of weather patterns by criticallyplaced SST anomalies, particularly in the tropics(e.g. Shukla, 1975; Rowntree, 1976), although theeffects in mid-latitudes are often more difficult todetect in the presence of the higher natural vari-ability of the atmosphere (Salmon 8c Hendershott,1976; Chervin et al. 1976).Since anomalous thermal or circulation patternsin either the ocean or the atmosphere may signifi-cantly modify the conditions in the other medium,it has often been suggested that SST anomaliesarise through ocean-atmosphere feedback pro-cesses, in which the presence of an SST anomalymodifies the weather patterns in su ch a man ner tha tthe anomaly is strengthened (e.g. Namias, 1959,1969)-although in som e situations the atm os-phere may also tend to destroy an existing SSTanomaly (Namias, 1975). W e suggest here th at thedevelopment of SST anomalies can be understoodmore simply, without invoking feedback mechan-isms, as the integral response of the ocean torandom short-time-scale forcing by the atmos-phere-in acco rdan ce with an earlier prop osal byMitchell (1 966).Following the notation of I, the heat balanceequation governing the evolution of an SSTanomaly y ( t ) a t a given position in the ocean (ordefined as an areal average) is assumed to have thesimple form

where u is a forcing function determined by thefluxes of latent and sensible heat, short- and long-wave radiation and momentum across the air-seainterface. The forcing fun ction is regarded as para-meterized in terms of various atmosphericboundary-layer variables x ( t ) and mixed-layerparameters. For simplicity, we shall retain only theSST anomaly as mixed-layer parameter in u (cf.Section 3).The basic assumption of the stochastic forcingmodel in I is that the characteristic correlation time

scale r, of the atmospheric variables x ( f ) is smallcompared with the time scale ry of the responsey ( t ) . For integration times t < r,,, the variable y canthen be regarded as con stant in the right-hand sideof ( l . l ) , and the evolution of th e SST change Sy(f)= y ( t )- o relative to an initial temperature y = y oat time t =0 is given by the equa tion

(1.2)where u ( t ) may be regarded as a given, randomfunction o f t . Dividing the variables into mean an dfluctuating com ponen ts, Sy = (Sy) + y, u = ( u ) +u where the means (. ..) are defined as ensembleaverages over the atmo spheric states x for given yo ,eq. (1.2) may then be divided into a mean part anda fluctuating part, given bydY- u ( t )dt (1.3)Equation (1.3) corresponds to the well-knowncontinuous random-walk equation first consideredby Taylor (1921). For a statistically stationaryinput u ( t ) the response is nonstationary, thevariance of y increasing linearly with time fort 9 r,Q f 2 ) = 2 Dt fo r r, < t < rr (1.4)

mwhere the diffusion coefficient D = .jJ-m ( r ) , andR ( r ) = ( v ( t + r ) u ( t ) ) denotes the input co-variance function.The variance spectra F(w) of the input u andG ( w ) of the response y are related in the corres-ponding frequency intervals through

The non-integrable singularity of the responsespectrum (1.5) a t w = 0 is consistent with the non-stationarity of y . For large t & r, the variance ofyis contained almost entirely in a linearly growing 6function at zero frequency. In accordance with thespectral interpretation of (1.4), the dnusion coeffi-cient can be expressed rathe r simply in terms of thespectral density of the input a t zero frequency,D = nF(0) (1.6)Th e input spectrum is white in the range 5 < w 6

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STOCHASTIC CLIMATE MODELS 29 1GI, and the spectrum F ( 0 ) at zero frequency ineqs. (1.5) and (1.6) should in fact be interpretedmore rigorously as referring to this ban d, since theside condition t Q ry precludes a meaningfuldefinition of the spectrum for frequencies of orderr, r less.For t = O(ry),y can no longer be regarded asconstant in the right-hand side of (1.2) and (1.3),and feedback effects become importa nt. A net posi-tive feedback converts the linear growth in therange r, Q t Q ry into an exponential instability,and can therefore be ruled out as a realistic pos-sibility. Negative feedback, however, leads to anasymptotic balance between the random forcingand the feedback damping, yielding a statisticallystationary response. For the simplest case of alinear feedback, eq. (1.1) takes the form

-Y = u ( t )- 1y 1 > 0, const.dtan d the structure function

of fixed depth (copper plate model). The conceptis tested both numerically, in a Monte-Carlo simu-lation using a simple two-dimensional turbulentmodel atmosphere (Sections 2 and 3), and againstreal input and response data (Section 4). In generalit is found that the atmospheric input anomalyspectra are white in the relevant period range froma few weeks to a few years, as required, and thatthe SST spectra conform with the predicted form(1.9). Despite the crude copper-plate model of themixed layer, the orders of magnitude of the inputand response spectra are also in reasonable agree-ment with theory.In Section 5 the stochastic forcing model isapplied to variability in the seasonal thermocline.In this case the driving term is the Ekman suctionproduced by the curl of the wind stress. For periodsshorter than about 6 months, the /?-effect does notplay a role, so that the linear response of the mixedlayer can be characterized by an equation of theform ( l . l ) , and a time-scale separation betweeninput and response is again possible. Comparison

(1.7)

[6y2(t)l= W o + t )of the response is given by

2 01[dy2]=- 1 - -A? fo r t > r,

of the model simulations with observations suggesttha t the short-term random wind forcing may yielda non-negligible contribution to the variability ofthe seasonal thermocline. However, quantitativecomp arisons are more difficult in this case becauseof inadequate observational data on the wave-number-frequency spectra of the wind-stress curl.(1.8)

Ensemble values over a l ly an d x (corresponding totime averages over periods r % r,) have beendenoted here by wuare brackets [...I to dis- 2* The mode- -tinguish them from th e ensemble values (. . .) over(which correspond to time averages Ran dom atmos pheric forcing is simulated in ourOver intermediate periods satisfying rx T r y ~ . numerical stocha stic air-sea interaction model

bulence, in a form developed by Kruse (1975) fornumerical investigations of the nonlinear transfer

for fixedThe equivalent spectral relations to (1.8) are given using a barotropic model of qU~i-geos t roPhic ur-by

within large-scale atmospheric motions in an equili-brium state. T he model is of the type discussed, forexample, by Lilly (1972) and Rhines (1975). I t(1.9) provides a statistically stationary wind field, fromwhich heat and momentum anomaly fluxes areinferred using bulk transfer formulae. The atmos-pheric system is not affected by the underlyingoce ans, i.e. the re is no direct feedback fro m theocean to the atmosphere.Kruses model represents a non-divergent, one-layer atmosphere in the B-plane approximation,bounded by two latitudes and zonally cyclic. Thedependent variable is the stream function which

An important feature of eqs. (1.8) and (1.9) is that,although the atmospheric forcing can be regardedas essentially uncorrelated white noise, the SSTcorrelation time scale is very much larger, beingdetermined solely by the relaxation time arisingfrom th e linear feedback.W e apply these sto chastic forcing concepts nowto a very simple SST model in which the upperlayer of the ocean is represented by a mixed layerTellus 29 (1977), 4

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292 C. FRANKIGNOUL AND K. HASSELMANNobeys the equation

-rV2 v /whered B a- - u,-dt at ax

(2.1)

Here x and y are eastward and northwardCartesian coordinates, U, constant velocity takento be 5 m sec-I, and J denotes the Jacobianoperator. The dissipation terms simulate a combi-nation of internal viscous friction an d surfa ce drag.The boundary conditions are

andW(X,y , t ) = W (X kPL, Y , (2.3)Jt is assumed in the following that L, =L , = 8800km (hemispheric model). Equations (2.1)-(2.3)are solved spectrally by expanding Y n terms ofharmonic functions. In our numerical experimentsthe series is square truncated after wavenumber 8.Wavenumber k in the model corresponds tolongitudinal wavenumber n 2 4k in the atmos-phere. The flow is assumed to be symmetric about4 5 O latitude 0,= LJ 2) to reflect the mid-latitudemaximum of eddy kinetic energy (this also has theadvantage of reducing the number of Fourier co-efficients). A constant forcing is imposed at wave-numbers (2,3) and (3,2) to simulate baroclinicinstability. The prognostic equations for theFourier amplitudes were solved by alternating aBaer-Platzman an d an Adam s-Bashfort time-stepping sch em e with a time step of 1 h.After the wind field had reached an a pproxim atesteady state (characterized by constant energy andenstrophy spectra), the equations were stepped fora further period of 512 simulated days. Fourieramplitudes were saved at 12-h intervals only, sincethere was little energy at frequencies higher than acycle per day. T he wind field w as construc ted on a16 x 16 grid by inverse Fou rier transforma tion.Fig. 1 shows the power spectrum of thesimulated wind data. The general features of

P = 1,2,. .

I Ilb61 Hz 1(llmonlh) ( I l w e e k )

- 5

Fig.1. Simulated spec trum of the wind velocity at x = 0,y =L,/4.observed wind spectra are reproduced: a constantenergy level at low frequencies, a peak at periods ofseveral days, and a strong drop off at higherfrequencies. Nevertheless, compared with observedwind spectra above the ocean (Byshev & Ivanov,1969; Wunsch, 1972; Frost, 1975), the simulatedspectrum is nearly one order of magnitude toosmall at low frequencies and the energy peak isnearly one order of magnitude too high. However,this deficiency can be readily corrected for whencomparing our computed oceanic responsespectra against observations, since the input-response relations are linear. The wavenumberspectrum of the wind (Fig. 2) reproduces reason-ably well wavenumber spectra observed in thetroposphere (for example by Kao & Wendell,1970), although observations suggest a somewhatflatter spectrum at low wavenumbers and a lesssteep decay proportional to k- at high wave-numbers. This latter discrepancy arises from therather strong dissipation and limited wavenumberrange of the model, which prevents the formationof an enstrophy inertial range. It should also bepointed out that the model wavenumber spectrummay show additional deficiencies when applied tothe marine boundary layer.Despite its oversimplifications, the atmosphericmodel provides a simulation of a statistically

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STOCHASTIC CLIMATE MODELS 29 3

1 2 3 4 5 7 10k

iyn.2/m-

lo-*'

Fig. 2. Simulated wavenumber spectrum of windvelocity (continuous lines), total heat flux (dotted h e )and wind-stress curl (dashed lines). The values areaverages over eight independentestimates.stationary wind field which is not too unrealistic.From the wind field, the heat and momentumanomaly fluxes which drive the ocean can then becalculated using bulk transfer formulae (Section 3).Th e main advan tage of using a numerical model tosimulate the forcing rather than observed data isthat one obtains in this way a complete three-dimensional spectral description of the forcingfunctions with respect to frequency and horizontalwavenumber, and this is difficult to extract fromreal data.

3. Simulated sea surface temperatureanomalies(a ) The mixed-layer model

The simplest models of the uppermost layer ofthe o cean assum e a completely mixed, horizontallyhomogeneous layer of depth h and temperature Toverlying a seasonal thermocline with prescribedtemperature structure (e.g. Kraus & Turner, 1967;Denman, 1973; Niiler, 1975; Thompson, 1976).Ignoring the effects of salinity and horizontal ad-vection, th e evolution of the layer paramete rs h, Tar e then governed by t wo prognostic equ ations of

the form

dt

dhdt- f,(U)(3.1)

(3.2)

Equation (3.1) s a heat balance equation in whichf, enotes the sum of (a) the total flux of latent heatHL and sensible heat H , through the air-sea inter-face (dependent on the air temperature To,humidity q and wind speed U through the standardbulk transfer relations), (b) the net short- an d long-wave radiation R through the sea surface, and (c)the heat exchanged with the layer below. Thesecond equation (3.2) s an em pirical relation basedon an energy argument in which it is postulatedtha t a fixed (small) fraction of the turbulent energyproduced in the mixed layer is used to thicken thelayer by entrainment of fluid from below. It isnormally assumed that the kinetic energy con-verted into potential energy by the entrainmentprocess is supplied to the mixed layer by the wind,so that the source function f,depends only on thewind speed (apart from internal parameters of thesystem such as the stratification of the thermo-cline).

We shall simplify this model further by retainingonly eq. (3.1) and assuming that the mixed-layerdepth remains constant. In addition, only thecontribution (a) to f, will be considered. Thevariations in radiation flux (b) are ignored, sincethey cannot be incorporated in our simple atmos-pheric model, a nd th e neglect of the heat exchange(c) with the lower layer is consistent with theneglect of entrainment. If we apply the usual bulktransfer relations, we may then parameterizef, nthe form

where

and C,, is the bulk transfer coefficient for thesensible heat flux, B the (Bowen) ratio of the latentheat flux to the sensible heat flux, pa and pw th edensities of air and water, and C; and Cr th especific heats of air an d water.

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29 4 C . FRANKIGNOUL AN D K. HASSELMANN0.04

l y l s0

10.04 I1.o

O C

0.5

0

-0.5 J 15 00100 200 300 d a y sFig. 3 . Simulated SST variations (without and with feedback effect) and total heat flux at x = 0, y = L J 4 . The heatflux time series is subsampled at 5-day intervals and the SST ime series are low-passed, using a quadratic Lanczosfilter (cut-off frequency 8 - lo-' Hz).(OC)* /Hz

1o5

10'

1o3

lo i.-

sensible\ 'heat f l u x \r . \

y l r ) ~ H z10 2

cools, and when it blows from the south, the waterwarms. With this crude parameterization anequation formally analogo us to eq. (1.1) is obtainedfor the rate of change of SST,d T pa C;: KvlUl- = C , , ( l + B )

10' dt pw c; h (3.5)In our simulations, we have set K = 0.25 (" C/m/s)and taken C,, = gm~ m - ~ ,w= 1 gm ~ m - ~ ,;= 0.24 cal gm-' (" C)-I,C; = 0.96 cal gm-' ("C )-', and h = 25 m. Amixed-layer depth of 25 m has been suggested byThompson (1976) for low mid-latitudes (30" N) onthe basis of a com parison of the observed seasonalSST cycle with predictions by a copper-platemodel. The effective mixed-layer depths at higherlatitudes are considerably greater ( a value of 1 0 0 mis taken in the following section for station India at

B = 3, p" = 1.2510 0

lo- '

(10-7 I l b - 6 Hz( 1 rnonth)Fig. 4 . Simulated spectrum of sensible heat flux (dashedlines) and SST anomaly (continuous lines) at x = 0, y =L J 4 . The ar rows indicate the 95% confidence interval.

To incorporate heat transfer into our at-mospheric model we assume that the air-seatempe rature difference is proportional to the north-south velocity V , (To- T )= K V , K = const. Thuswhen the wind blows from the north, the water

59"N). With our choice of K , the r.m.s. air-seatemperature difference generated by the model is1.25 "C .(b ) Simulated SSTanomalies neglecting eedback

Time series of the stoch astic atmosph eric forcingaccording to eq. (3.5)were constructed a t each gridpoint, a nd the SST hanges were then calculated bystraightforward integration. Fig. 3 illustrates theintegral response of the SS T to the rapidly varyingfluxes. To draw attention to the evolution of the lowfrequencies in the SS T fluctuations, the SS T timeTellus 29 (1977 ), 4

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STOCHASTIC CLIMATE MODELS 295series have been low-pass filtered. The longer theintegration time in (3.5), the larger the amplitude ofthe SST oscillations, as expected from the non-stationarity state of the climatic response for thecase w ithout feedback.

The power spectrum of the simulated sensibleheat flux anomaly at a fixed location is shown inFig. 4. It has the same features as the simulatedwind spectrum (Fig. I) . In particular, it is essentiallywhite at low frequencies, in agreement withobservation (and as required by theory) althoughthe energy level is about one order of magnitudelower than observed flux da ta (Section'4). Thesimulated flux spectrum implies a diffusion co-efficient according to relation (3.5) of D z 0.25(C)2 year-', i.e. the ran do m atmos phe ric forcingproduces a stand ard deviation in the SST of about0.7 OC in one year. As predicted by relation (1.5),the simulated SST anomaly spectrum is propor-tional to th e inverse frequency squared at low fre-quencies (Fig. 4). This is again in agreement withthe obse rvations, and the energy level is also abo utone order of magnitude lower than observed mid-and high-latitude levels, in accordance with theorder of magnitude underestimate of the simulatedinput spectrum. A more detailed c omp arison withthe ob serv ation s is given in Section 4.The wavenumber spectrum of the SST anomaliesis proportional to the wavenumber sp ectrum of theatmosp heric input (Fig. 2, dotted lines). In con trastto the wind spectrum, which is highest at wave-number 1 , the max imum variance of the simulatedSST o ccurs at wavenumbers 2 and 3. This corres-ponds roughly to the observed scale of thedominant SST anomaly patterns, which aretypically several thousand kilometers in diameter,and is also consistent with the observation th at thedom inant scales of the SS T anomalies app ear to besomewhat smaller than the scales of air tem-perature or sea-level pressure anomalies (e.g.Kraus & Morrison, 1966; Davis, 1976).Although our ocean-a tmosph ere model is admit-tedly highly simplified, the m ain features of the S STspectral response to short time scale weatherforcing appear to be reproduced reasonab ly well inthe numerical experiments. A s discussed in Section4, further processes (e.g. radiation fluxes of Ekmantransport) will need to be considered in morequantitative models. However, a s long as these canbe represented by short-time-scale "weather vari-ables", they will yield only an additional whitenoise input. Depending on their correlation with theTellus 29 (1977), 4

sensible a nd la tent heat fluxes considered here, theywill produce lower or higher energy levels of theSST anomalies, but no changes in the basicstructure of the spectrum. Other effects whichshould be incorporated in more detailed modelsinclude slow changes in the coupled system (e.g.seasonal variations of the mixed-layer depth),which will mod ulate the oceanic response.

U p to this point we have also omitted feedbackeffects. The observations (Section 4) suggest thatthe ch aracteristic feedback time of SST anomaliesis of the order of 6 months, so that the results ofthis section can be applied only for periods shorterth an this time scale.(c ) Simulated SST anomalies includingfeedback

F or small temperature anomalies, the functionf ,in eq. (3.1), dT/d t = f,/h (h = const), can beexpanded with respect to T. Writing f , = ull+ i,an d defining T= 0 to co rrespond t o an equilibriumtempe rature for which ull= 0, eq. (3.1) then takesthe form of a first-order autoregressive (Markov)process

dt hwhere 1 = ( a [ , l / a T ) , , , is a constant feedbackfactor. F or a stable system w ith negative feedback,1 s positive (cf. eq.) (1.7)).Since our atmospheric model contains no ther-modynamics, we cannot simulate the feedbackexplicitly in the coupled system. However, we canestimate the feedback factor by expanding the bulkformula (3.4) with respect to T. Assuming the airtempe rature to remain constant, this yields

where (lUl) s the mean wind speed. Taking (IUl)= 8 m sec-' an d C,,, B and h as given in Section3a, one obtains 1 = (1.7 month)-'. Thi s feedbackfactor is larger than inferred from observations,presumably because of the unrealistic assumptionof a constant air temperature (cf. Section 4). Thevalue was nevertheless used in our model experi-ments to illustrate the stabilizing influence of anegative feedback in our rather short (512) day)simulation runs (Fig. 3). Th e decrease in amplitudeof the lowest frequency SST oscillation as com-pared with the case without feedback is clearly dis-

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296 C. FRANKIGNOUL AN D K. HAS SEL MA cernible. Th e computed SST spec tra also showed acorresponding flattening at low frequencies, asexpected.Our simulation experiments may be comparedwith the work of Salmon Br Hendershott (1976),wh o simulated one yea r of air-sea interactionprocesses by coupling a two-layer, thermallyvariable atmospheric model to a copper plateoceanic mixed layer. With this more realisticmodel, which simulated the observed northernhemisphere seasona l cycle in fair detail, they founda frequency spectrum of SST anom aly very similarto the spectrum represented in Fig. 4. This supportsthe premise that the principal features of the SSTvariability are independent of the details of thecoupling and can be inferred from the generalconcept of a white forcing spectrum and a linearauto-regressive integrator response, with a largetime-constant separation between input andresponse.For comparison, Fig. 3 also shows the charac-teristic explosive b ehaviour of the processes withlinear positive feedback, 1 < 0. Because of the sh orttime sca les involved, this cas e can be ruled o ut forSST anomalies, but it has been suggested that itmay be relevant for more inert components of theclimatic system, such as the ice sheets. It should benoted that a positive linear feedback does notnecessarily imply overall climate instability, as non-linear terms can stabilize the response at largeramplitudes.

4. Cornparkon with observationsThe principal fe atures of the stochastic two-scalemodel of SST variability are that the frequencyspectra of SST anomalies obey an inverse squarelaw for frequencies in the range r, < o < r i 1 ,flattening to a constant level at lower frequencies,and that the atmospheric input spectra are whitethrougho ut this frequency range. Examples of dataexhibiting these features are given in Figs. 5 and 6.Fig. 5 shows the spectra of SST anomaly a t the

Atlantic Oc ean weather sh ip India (59ON, 19O W),and Fig. 6 sh ows the spectra of latent a nd sensibleheat flux at the same location (after Frost, 1975).The seasonal signal has been subtracted from theSST time series, but not from the latent andsensible heat flux series. Approximately 16 year s of

( Ilmonth)Fig. 5 . Spectrum of SST anomaly at Ocean WeatherShip India for the period 1949-1964 (after Frost, 1975).The arrows indicate the 95% confidence interval. Thesmooth curve was calculated from relation (4 .1) with h =100m, L = (4.5 month)-.

LATENTH E A TFLUX

SENSIBLEHEATFLUX

Fig. 6 . Spectra of latent and sensible heat flu at OceanWeather Ship India for the period 1949-1964 (afterFrost, 1975).data were available, and the SST anomalyspectrum was com puted from 14-day average data,yielding a Nyqvist frequency of about 1 cycle permonth. Daily average s of the fluxes were compu tedfrom the flux time series obtained from the threehourly wind and temperature data using the bulkformulae, thereby avoiding the usual dficulties ofestimating fluxes from climatologically averagedwind and tem perature fields.

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STOCHASTIC CLIMATE MODELS 297Both the S ST spectra a nd the flux spe ctra agreewell with the form predicted by theory. Theflattening of the SST spectra at high frequenciesmay be attributed to aliasing. The flattening at lowfrequencies corresponds to a linear feedback factor

1 = (4.5 months)-'. Kraus & Morrison (1966,Table 7) have calculated the correlation functionrT1(r) f monthly average SST nomalies at stationIndia from 12 corresponding years of data. Fittinga function e-"' (predicted from eq. (1.7), seeappendix) to the given values of rrT(r)also givesgood agreement for 1 = (4.5 months)-'. This feed-back factor is smaller than the value R = (1.7months)-' which was com puted in Section 3 fromthe bulk formula under the assumption thatchanges in SST were not accom panied by changesin air temperature. In fac t it is known that the heatexchange a cross the s ea surface rapidly leads to anadjustment of the surface air temperature to a valuefairly close to the local SST, so that SST anomaliesalways produce air temperature anomalies of thesame sign and only slightly smaller amplitude (cf.Thom pson, 1976). Thus the net feedback factor, asdetermined by the true air-sea tem peratu re dif-ference in the bulk form ula, may be expected to beconsiderably smaller than 1= (1.7 months)-'.The ratio of the latent and sensible heat fluxspectra t o the SS T spectrum is also consistent withthe simple copper-plate model described in theprevious section. From (3.1) and (3.3) we find

(4.1)

where G (w ) is the SST spectrum, and FJO) andF,(O) are the (constant) low frequency levels of thesensible and latent heat flux spectra, respectively.At the high latitud e of station India, these fluxes arelikely to dominate the local heat balance, althoughradiation effects may be important in summer.Dimensional arguments (see also Gill & Niiler,1973) suggest furthermore that heat advection byanomalous wind driven currents play a minor rolein mid- and high latitudes . It has also been assumedhere for simplicity that the sensible and latent heatflux variations are uncorrelated, which is probablyincorrect, but the degree of correlation is un-important, since F,(O) % FJO). Equation (4.1) is inagreement with the data if a value h = 100 m ischosen, which is consistent with typical depths ofthe seasonal thermocline at station India (Frost,

st ructurccorrclat ion r6 Y 2 1 function

-0.5 I I 1 I I c0 1 2 3 4 5 6y e a r sFig. 7. -* . structure function [Sy(t)l f the NorthPacific SST anomaly data of Namias (1969). -:predicted structure function for D = 0.04(C)fy-', =(0.75 yr)-I. ----* . correlation function of the SSTanomaly data. ... .. input correlation function infer-red from differenced temperature time series. The arrowsindicate the variance of the correlation estimates(approximatelyf the 95 %confidence intervals).1975). The energy levels correspond to a diffusioncoefficient (eqs. (1.4), (1.6)) of D = 1.5(C)2/yr,which m eans that without feedback the short-time-scale atmospheric flux variations would producer.m.s. SST anomalies of order (2 - (1.5))1'2= 1.7 OCin a year .Another example of a local temperature variancespectrum, in this case for 13 years of data takenat 10-m depth at the Panulirus station nearBermuda (32O N, 65O W ) can be found in Fig. 16of Wunsch (1972). the seasonal signal was notremoved, but the w-* behaviour of the spectrumfor frequencies above 1 cpy is neverthelessapparent. The energy level is here also of thesame order as at station India, but a quantitativecom parison of the input-response energy levels isnot possible in this case because the relevant inputdata was not given.Figs. 7 and 8 show correlation (structure)functions and power spectra of spatially average dSST anomalies and atmospheric pressureanomalies for the North Pacific. The structurefunction of Nam ias' (1969) 25-year time series ofSST an omaly da ta, averaged over the entire Pacificnorth of 20N, Fig. 7, shows initially a linearincrease with lag, as predicted by the stochasticforcing model without feedback, followed by aflattening to a constant value, as expected when thestabilising feedback comes into play at large timelags (eqs. (1.4) and (1.8)). Continuous time seriesdata for the integrated air-sea fluxes over theNorth Pacific are not available, but the equivalent

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298 C . PRANKIGNOUL AND K. HASSELMANN100

Luga 10u0._L-!!

1 0 i 2 3 4f r e q u e n c y ( c y c le s / y e a r )

Fig. 8. Spectrum of the dominant EOF of the NorthPacific SST anomaly and sea level pressure (after Davis,1976). The smooth curve represents the predicted SS Tspectrum for 1= 6 month)- with app ropria te choice ofthe level of the white input spectrum.forcing function u ( t ) in the lag range 7x < t < 7,(see relation (1.7)) has been evaluated here simplyby differentiating the integrated SST time series.The input and response correlation functions ( r , ,and rTI .n Fig. 7) exhibit the structure and timescale separation assumed in the stochastic model.In this exam ple the essentially zero-lag w idth of theauto-correlation function of the atmosphericforcing follows directly from the initially linearform of the autocorrelation function of theanomaly, and is shown simply for illustration,rather than as independent corroboration of themodel. The diffusion coefficient, Le. the rate ofgeneration of SST anomalies, can be estimatedfrom th e slope of [Sy2(t)lat small t : D z 0.04(OC)*yr-I. Thi s value is smaller than at statio n India, butis not unreasonable, when considering that the vari-ability of the fluxes at a fixed position should beconsiderably higher than the variability of thefluxes averaged over the North Pacific, to whichonly the longest wavelengths contribute. The feed-back relaxation time for the North Pacific is of theorder I- = 0.75 year, somewhat larger than thestation India value, but the 3-month time incre-ment of Namias time series is too coarse for anaccurate calculation. Namias & Born (1974) givethe correlation function calculated from monthlyaverages of this SST data, at lags up to 24 months.An exponential decay with 1 = (1 year)- providesa good fit at lags larger than 6 months, but theinitial decay is faster, suggesting that the feedbackprocess may be more complex than the single-variable equa tion (1.7).This could be d ue to other

anomaly modulating-processes such as an ad-vection or diffusion.An analysis of 28 years of monthly averagedNorth Pacific SST da ta (north of 20N ) has beencarried out by Davis (1976). The SST anomalyfields were correlated with fields of atmosphericsurface pressure anomalies. Both fields weredecomposed first into their empirical orthogonalfunctions (EOFs). Fig. 8 shows the variancespectra of the amplitudes of the d om inan t first EOFof each field. An exponential decay with A = (6months)-* provides an excellent fit to the cor-relation function of the dominant SST modereported by D avis (his Fig. 6) .This feedback factorhas been used to construct the predicted SSTspectra behaviour in Fig. 8. The agreement with theobserved spectrum is good throughout the wholefrequency range (again, aliasing may explain thesomewhat higher energy at high frequencies), andthe pressure spectrum is flat, as expected for anatmosp heric variable.A direct verification of the model by comparisonof the inpu t a nd response e nergy levels is again notpossible for Davis data because of the lack ofappropriate time series measurements of thespatially integrated surface heat fluxes. However,Davis calculated the cross correlation function forthe amplitudes of the first EOFs of the SST andsurface pressure anomaly, and this provides arather critical test of the basic premise of themodel, namely that SST anomalies represent theintegral response to short-time-scale atmosphericforcing.We assume that the amplitude of a surfacepressure anomaly may be chosen in place of theanomalous surface heat flux as the atmosphericvariable determining the rate of change of an SSTanomaly. This is permissible if the anomalouscirculation patterns associated with the pressureanomalies are correlated with anomalous heatfluxes (as was the case, for example, for the heatfluxes computed from the wind field in the simplesimulation model in Section 3). The variables y andv in eq. (1.7)ma y then be interpreted as the ampli-tudes of the first EOFs of the SST and surfacepressure anomaly fields, respectively (except for aproportionality factor which ca n be normalized tounity). Multiplication of (1.7) by v ( t - 7) andaveragin g yields th e relationd- R,, ( r )=R , , (7)- 1R,, (7)dt (4.2)

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STOCHASTIC CLIMATE MODELS 299for the covariance functions

andR J r ) = [ d t + r)u(t)lAssuming white-noise atm ospheric forcing R r ) =AS($ and R y y ( r )= 0 for r < 0 (causality), thesolution of equation (4.2) is given byR y v ( r ) A e-AT for r > 0

= O for r

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300 C. FRANKIGNOUL AND K . HASSELMANNThe response of a stratified ocean subject todifferent forms of forcing has been studied invarious contexts by many authors. A general de-compo sition of the forcing functions and responseinto inertial-gravity and geostrophic motions usingprojection operators is given, for example, inHasselmann (1970). In the linear,f-plane approxi-mation, the rate of change of the quasi-geostrophiccomponent of the displacement (relative to thefree surface) of an isotherm immediately below theEkm an layer satisfies the simple Ek man pumpingequation

(5.1)

where S s the stress applied at the surface andf theCoriolis parameter. (Equation (5.1) is normallyderived from the continuity equation aylat = V Mand the geostrophic relation f A M = S for thevertically integrated mass transport M of a quasi-stationary Ekman layer, but is in fact validrigorously for the quasi-geostrophic compon ent ofthe re sponse in the f-plane , independent of theassumption of quasi-stationarity.) If the barotropicmode, which yields a negligible contribution to thenear-surface isotherm al displacement, is excluded,the pp la ne restoring forces can be neglected to firstorder for time scales up to s everal months, and theappropriate f-plane equation (5.1) hen .reduces t othe basic form (1.3) of a stochastically drivensystem without feedback.The generation of thermocline displacementsaccording to (5.1)was investigated in an experi-ment, using the atmospheric model of Section 2 tosimulate the wind forcing through the bulk relationS = C,P,IUIU, with C,= 1.5. lOW. The windstress was first constructed a t each grid point, andits curl was then taken in the Fourier domain usingthe fast Fourier Transform method. The powerspectrum of s imulated wind-stress curl is approxi-mately white at low frequencies, as other atmo-spheric spectra, w ith a spec tral level of the order of10-9-10-10yn2 cm-6/Hz (10-7-10-6 ewton2m-6/Hz). The spatially averaged r.m.s. Ekmansuction velocity is approximatelyFig. 10 shows the computed power spectrum ofthermocline vertical displacement at a typicallocation (x = 0, y = 1543). he associated white-noise level of the input spectrum in this case is4 . dyn2 cm-6/Hz, which correspo nds to a

m s-*.

m21Hz

107.

lo6,

lo5

T H E R M O C L IN E

10-7 I HZ( 1 I mont h)

Fig. 10. Spectrum of simulated thermocline displace-ments at x = 0, = LJ3. The arrows indicate the 80%confidence interval.diffusion coefficient D = 63 m z year-. Thus in theabsence of feedba ck, the random short-time forcingby Ekm an pumping would produce a r.m.s. vertica ldisplacement of 1 1 m in a year.However, it is possible that our atmosphericmodel strongly underestimates the variancespectrum of the wind-stress curl. Observations atBermuda (Wunsch, 1972) suggest that the white-noise level of the wind-stress spectrum may benearly one order of magnitude larger than in themodel. Furthermore, the wavenumber spectrum ofthe simulated wind-stress curl shows a maximumcontribution to the variance near wavenumber 5 ,which corresponds to a dominant wave length of2500 km (Fig. 2). The strong decrease of thespectrum at higher wavenumbers is probably un-realistic and may be attributed to the large coeffi-cient of dissipation and the truncation of the modelat wavenumber 8.Dimensional arguments suggest in fact that ifthe wind kinetic energy decays as k- (or less) inthe enstrophy inertial range, the wind-stress curlspectrum should decay as IC (or less), so thatscales not represented in our atmospheric modelare likely to contribute significantly to the wind-stress curl variance. This is in agreement with the

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STOCHASTIC CLIMATE MODELS 301P E R I O D

FREQUENCY ( CP H 1Fig. 11. Spectraof temperature variations in the seasonal thermocline (from Bernstein& White, 1974). Data are fromvarious stations in the m id-Pacific between 30N and 43 N at depths of 150 m (1 -4) and 200 m (5). Frequency slopesof -2 are indicated.recent work of Saunders (1976), who showed thatthe cu rl of the wind s tress estimated from long-termaveraged wind data was underestimated by a factorone half when computed from a 5 O square grid ascompared with a 1O square grid.Fig. 10 may be compared with the spectra oftemperature variations measured at variousstations in the North Pacific at 150 and 200 mdepth within the the rmocline by Bernstein & White(1974), Fig. 1 1 . Although the exact form of thespectrum is uncertain due to the composite natureof the data, the general trend is consistent with apower law. Taking a typical thermal stratifi-cation of 5 10-ZoC/m (cf. Bernstein & White,

1974, Fig. 8) and the Coriolis parameter for thelatitude 40 , the average spectrum F,(u) = lo-'a-2 (C)2/cph (a = frequency in cph) of thethermocline temperature variation T shown in Fig.14 corresponds to a spectrum of thermoclinedisplacement F, , z 10-5v-2 m2 S - ~ / H Z v inHertz). This is about two orders of magnitudelarger than the model data.Another example of observed thermocline vari-ability can be found in Fig. 16 of Wunsch (1972),which shows the variance spectra of temperaturefluctuations at 100 meters depth at the Panulirusstation in Bermuda. The spectrum decreasedapproximately as w 2 or periods ranging between

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302 C . FRANKIGNOUL A N D K. HASSELMANN1 month and 1 year and begins to flatten forperiods longer than about 1 year. The energy levelof the tempe rature spectrum is similar to that in theNorth Pacific, but the corresponding displacementspectrum is difficult to estimate, since seasonalvariations of the vertical temperature gradient arevery pronounced at this depth (cf. Schroder &Stommel, 1969).However, the equivalent displacement spectra ofboth Bernstein-White and Wu nsch a re clearly con-siderably higher than the spectra simulated by themodel. Taking a wind stress white-noise level of5 . lo5 dyne2 ~ m - ~ / H zestimated from Wunsch,1972), he North Pacific data require that most ofthe wind-stress variance would need to be con-centrated around wave lengths as short as 220 kmin order to produce the observed thermoclinedisplacements by Ekman pumping. Bernstein &White (1974) uggest, in fac t, that the North Pacificthermocline variability is ca used by internal baro-clinic eddies, rather than wind forcing. It should benoted, however, that the examples of thermoclinedisplacement spectra discussed here may not bevery representative for other oceanic regions.Dantzler (1976) showed, for example, that thereare large areas in the North Atlantic where ther.m.s. depth displacement of the 15 OC isotherm (atdepths ranging from 100 to 500 m) is at least afactor 4 maller than near Berm uda.In summary, it remains an open questionwhether variability in the sea sonal thermocline canbe explained by atmospheric forcing, despite theagreement of the shape of the temperature fre-quency spectrum of a white-noise atmosphericforcing model. This question can be resolved onlyby a more detailed analysis of the wavenumberspectrum of the wind-stress curl, particularly in theregion of high wavenumbers. But it can be con-cluded that at all events local atmosph eric forcingshould be regarded as a serious candidate for theorigin of variability in the seasonal thermocline(and possibly below) in certain regions of theOcean.

6. ConclusionWith regard to SST va riability it has been shownthat in mid-ocean regions away from intensecurrents or thermal fronts the principal statisticalproperties of SST an omalies can be explained by asimple model in which the atmosphere acts as a

white-noise generator and the ocean as a first-orderMa rkov integrator of the atmosph eric input.The mechanism was demonstrated numericallyby a M onte-Carlo simulation, using an idealizedtwo-dimensional model of atmospheric turbulence,and was verified independently by m easurem ents.According to this picture, the evolution of SSTanomalies is unpredictable, except that, oncegenerated, they tend to decay w ith an e-folding timeof the order of f year. The evolution of theanomalies is independent of any feedback with theatmosphere.It should be stressed, however, that we havepresented only a first-order picture and haveignored important effects such as the seasonalmodulation, interactions within the ocean andhigher-order anomaly-pattern interactions. Ouranalysis was based either on single-station data,spatially averaged data or on the interactionbetween the dominant first EOF's of SST andatmospheric pressure anomaly. A more detailedanalysis using a more sophisticated model and amore extensive data base may well reveal higherorder feedback processes of practical significancefor long-range weather forecasting.The analysis of variability in the seasonalthermocline provided some evidence, from thegeneral structure of the frequency spectra, of w hite-noise atmospheric forcing, but a quantitativecomparison of the input and response spectraremained inconclusive because of inadequateinformation on the w avenum ber-frequency spectraof the wind-stress curl. Since the atmosphericforcing may be expected to lie in the same range ofthe wavenumber spectrum as the oceanic meso-scale eddies, which contain the m ajor portion of thekinetic energy of the ocean, it is important toclarify whether a significant part of the eddy energynear the surface (and possibly deeper) could begenerated by local atmospheric forcing. We con-clude that, on the basis of present evidence, thismechanism must be considered as a possibleimportant source of mesoscale eddy energy incertain regions of the ocean and should be investi-gated further.

7. AcknowledgementsThe authors wish to thank Burkhard Kruse forproviding the atmospheric model and Tim Barnettfor helpful discussions.

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STOCHASTIC CLIMATE MODELS 303~ ~ i 1 - 2 e ( n + l ) v T-. Appendix r Y Y ( n ) (-(vT- n = -1 , -2, . . .Effect of smoothing on the correlation betweenclimate and weather variables

Here we generalize the result (4.3) to include afinite corre lation of the atmospheric forcing and theeffect of data smoothing. Following Munk (1960)an d k i t h (1973) we represent the weather variablev ( t ) in eq. (1.7) by a first-order Markov process,whose autoco rrelation is given by= ( (v;:1))/2 n = O

21vT2I I - n = l , 2 , ...( v T - 1 ) e-dT

(A7)vv(r)= (A 1)The precise form of rvv is not critical. Equation(4.2) and the corresponding equation for the co-variance function R,(r) = b ( t + r)y(t)l then yieldthe correlation functions

fo r r G O (A2)

where we have assumed 1 e v, consistent withou r two-time scale approximation.If the anomaly data is time averaged over aninterval T, the covariance function R f P (5) be-tween the smoothed variables a nd B is gwen by

dt R, (t + 7- )(A41

For an averaging time T i n the interval v- @ T eA-I, one finds, after some algebra

r, ( n T )= 1 n = Oe y T e-Dl!lTl

5- n = f l , f 2 , . ..~ ( v T - )(A5)

-ryy (n )N n = 0 , + 1 , + 2 , ...(A6)

Relation (A5) is an approximate form of therelation ( 5 ) given by Munk (1960) in his dis-cussion of the increased persistence of atmo-spheric variables by smoothing. Relation (A6)shows that the correlation function of the climatevariable is not affected by smoothing, as expected.However, the cross correlation between the clim ateand weather variable is increased considerablythrough smoothing. Also, the shape of the curve ismodified.The increase of the correlation between climateand weather variables by sm oothing can be readilyunderstood in terms of the variance spectra.Equations (1.5), (1.6) and (1.9) show that theclimate response is driven by the low-frequencypart of the inpu t spectrum. By low-pa ss filtering theatmospheric input the main part of the spectrumwhich does not contribute to the climatic responseis removed, whereas the essential low-frequencyregion of the input spectrum is left unchanged.Thus the residual input signal will exhibit a muchhigher correlation with the climatic response thanthe unfiltered input.For v = (4 day)- and 1 = (6 month)-], repre-sentative of the weather and SST anomalies re-spectively, the zero-lag unsmoothed correlationrTv(0)is 0.15 (similar to the value reported bySalmon & Hendershott (1976)), and the cor-relation function has a maximum of 0.27 when theweather variable leads the SST by 12 days. Ifmonthly averaged data are used, the predictedzero-lag correlation is iTv(0)= 0.31, and amaximum correlation of 0.52 occurs when v leadsT by one month. If seasonal averages are used,even higher corre lations result, 0.5 at zero lag and amaximum of 0.6 at lag one (one season). This isconsistent with the high correlation factors repor-

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304 C. FRANKIGNOUL AND K. HASSELMANNted for time-averaged data, for example by Nam ias(1969). Equation (A7)was used to construct thepredicted curve in the low er panel of Fig. 9 (Section

4), taking Y = (8.5 day)-' in acco rdanc e with thevalue quoted by Davis (1976) or the 1st EOF ofthe North Pacific surface pressure anom aly.

REFERENCESBernstein, R. L. & White, W. B. 1974. Time and lengthscales of baroclinic eddies in the central North Pacificocean. J. Phys. Oceanogr. 4,613-624.Bjerknes, J. 1966. A possible response of the atmo-spheric Hadley circulation to equatorial anom alies ofocean temp erature. Tellus 28,820-828.Byshev, V. . & Ivanov, Y. A. 1969. The time spectra ofsome characteristics of the atmosphere above theocean. Izuestiya Akademii nauk SS S R 5,17-28.Chervin, R. M., Washington, W. . & Schneider, S. H.1976. Testing the statistical significance of theresponse of the NCAR General Circulation Model to

North Pacific Ocean surface temperature anomalies. J.Atm os. Sciences 33 .4 13-423.Clark, N. E. 1972. Specification of sea surface tempera-ture anomaly patterns in the eastern North Pacific. J.Phys. Oceanogr. 2 , 391-404 .Davis, R. E. 1976. Predictability of sea surfacetemperature and sea level pressure anomalies overthe North Pacific ocean. J . Phys. Oceanogr. 6, 249-266.Dantzler, H. L. Jr. 1977. Potential energy maxima in thetropical and subtropical North Atlantic. J. Phys.Oceanogr. (submitted to).Denman, K. L. 1973. A time-dependent model of theupper ocean. J . Phys. Oceanogr. 3, 173-184.

Emery, W. J . 1976. The role of vertical motion in theheat bu dget of the upper N ortheastern Pacific Ocean.J . Phys. Oceanogr. 6,299-305.Frost, M. R. 1975. Stress, evaporative heat fluxand sensible heat flux distributions of the NorthAtlantic (mid-latitude) and their contribution tothe production of large scale sea surface tem-perature anomalies, Master thesis, SouthamptonUniversity.Gill, A. E. 1975. Evidence for mid-ocean eddies inweather ship records. Deep-sea Res. 22, 647-652.Gill, A. E. & Niiler, P. P. 1973. The theory of the

seasonal variability in the ocean. Deep-sea Res. 20 ,141-177.Hasselmann, K. 1970. Wave-driven inertial oscillations.Geophys. Fluid Dyn. 1,463-502.Hasselmann, K. 1976. Stochastic climate models. Part I,theory. Tellus 28 , 473-485 .Kao, S.R. & Wendell, L. L. 1970. Th e kinetic energy oflarge-scale atmospheric motion in wavenumber-frequency space: I. Northern Hemisphere. J. Atmos.Sci. 27, 359-375.Kraus, E. B. & Morrison, R . E. 1966. Local interactionsbetween the sea and the air at monthly and annualtime scales. Quart. J . Royal Meteor. Soc. 92, 114-127.

Kraus, E. B. & Turner, J. S. 1967. A one-dimensionalmodel of the seasonal thermocline: 11. The generaltheory and its consequences, Tellus 19,98-106.Kruse, B. 1975. Untersuchung groBskaliger nicht-linearer Wellenwechselwirkungen mit einem n umeri-schen spektralen Einschichtenmodell, Diplomarbeit,Universitat Hamburg.Leith, C. E. 1973. The standard error of time-averageestimates of climatic means. J. Applied MeteorologyLemke, P. 1977. Stochastic climate models, Part 3.Application to zonally averaged energy models. Toappear in Tellus.Lilly, D. K. 1972. Numerical simulation studies of two-dimensional turbulence: I. Geophys. Fluid Dyn. 3,Mitchell, J. M. Jr. 1966. Stochastic models of air-seainteraction and climatic fluctuation. (Symp. on theArctic Heat Budget and Atmospheric Circulation,Lake A rrowhead, Calif., 1966). Mem. RM-5233-NSF,The Rand Corp., Santa Mon icaMunk, W. H. 1960. Smoothing and persistence. J.Atmosph. Sci. 17,92-93.Namias, J . 1959. Recent seasonal interactions betweenNorth Pacific waters and the overlying atmosphericcirculation. J . Geophys. Res . 6 4 , 6 31-646 .Namias, J . 1969. Seasonal interactions between theNorth Pacific ocean and the atmosphere during the1960's.Month. Weath. Rev. 97, 173-192.Namias, J. 1975. Stabilization of atmospheric circu-lation patterns by sea surface temperature. J . Mar.Res. 33, Suppl., 53-60.Namias, J. & Born, R. M. 1974. Further studies oftemporal coherence in the North Pacific sea surfacetemperatures. J.Geophys. Res . 79, 893-894.Niiler, P. P. 1975. Deepening of the w ind-mixed layer.J.Marine Res. 33,405-422.Rhines, P. B. 1975. Waves and turbulence on a beta-plane. J.Fluid Mech . 69 , 417-443 .Rowntree, P. R. 1976. Response of the atmosphere to atropical Atlantic ocean temperature anomaly. Quart.

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STOCHASTIC CLIMATE MODELS 30 5numerical experiment with the GFDL model. J. models of the surface mixed layer of the ocean. J .Atmos. Sci. 32,503-5 11. Phys. Oceanogr.6 , 4 9 6 5 0 3 .Taylor, G. I. 1921. Diffusion by continuous movements. Wunsch , C. 1972. Bermuda sea level in relation to tides,weather, and baroclinic fluctuations. Rev. Geophys.Thompson, R. 0 .R. Y. 1976. Climatological numerical SpucePhys. 1 0 , 1 4 9 .ROC.ond. Math. SOC. 0, 196.

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