Ocean Dynamics 42001) 52: 36±49 Ó Springer-Verlag 2001

Till Kuhlbrodt Á Sven Titz Á Ulrike Feudel Stefan Rahmstorf A simple model of seasonal open ocean convection

Part II: Labrador Sea stability and stochastic forcing

Received: 25 April 2001 / Accepted: 10 July 2001

Abstract Aspects of open ocean deep convection vari- Keywords North Atlantic deep convection Á ability are explored with a two-box model.In order to Labrador Sea Á Bistability Á Stochastic climate place the model in a region of parameter space relevant model Á Decadal variability to the real ocean, it is ®tted to observational data from the Labrador Sea.A systematic ®t to OWS Bravo data allows us to determine the model parameters and to 1 Introduction locate the position of the Labrador Sea on a stability diagram.The model suggests that the Labrador Sea is in Deep convection in the North Atlantic is a sensitive part a bistable regime where winter convection can be either of the world ocean's thermohaline circulation 4THC).Its ``on'' or ``o€ '', with both these possibilities being stable intensity, location, and variability in¯uence the north- climate states.When shifting the surface buoyancy ward heat transport of the THC 4Rahmstorf 1995a), the forcing slightly to warmer or fresher conditions, the only pathway of North Atlantic Deep Water 4NADW) steady solution is one without winter convection. 4Wood et al.1999), and the total meridional overturning We then introduce short-term variability by adding a 4Delworth et al.1993; Rahmstorf 1995b) in models. noise term to the surface temperature forcing, turning Observational data from the past decades 4Lazier 1980; the box model into a stochastic climate model.The Dickson et al.1988; Belkin et al.1998) display great surface forcing anomalies generated in this way induce interannual to decadal variability in the occurrence and jumps between the two model states.These state tran- depth of deep convection events.The role of deep con- sitions occur on the interannual to decadal time scale. vection in the THC is the removal of heat from the deep Changing the average surface forcing towards more ocean, thus balancing the downward penetration of heat buoyant conditions lowers the frequency of convection. due to various diapycnal mixing processes 4Munk and However, convection becomes more frequent with Wunsch 1998). stronger variability in the surface forcing.As part of the While freshwater ¯uxes are important for the back- natural variability, there is a non-negligible probability ground strati®cation, convection events are usually for decadal interruptions of convection.The results triggered by strong heat loss through the ocean surface, highlight the role of surface forcing variability for the reducing the vertical density gradient until the water persistence of convection in the ocean. column becomes statically unstable.Vigorous vertical mixing down to 2000 m and more results 4Send and Marshall 1995).Convection events have an extent of only a couple of days in time and some 10 km in space Responsible Editor: JoÈ rg-Olaf Wol€ 4Marshall and Schott 1999).As this is subgrid scale in T.Kuhlbrodt Á S.Titz Á S.Rahmstorf most ocean general circulation models 4OGCMs), con- Institute of Physics, University of Potsdam, vection is parameterized by convective adjustment Postfach 60 15 53, 14415 Potsdam, Germany schemes 4Rahmstorf 1993; Klinger et al.1996).These U.Feudel schemes remove static instability in the water column at Institute for Chemistry and Biology of the Marine Environment, each time step by mixing vertically adjacent grid cells. University of Oldenburg, Postfach 2503, 26111 Oldenburg, Germany Although this may lead to a too intense mixing 4Lilly et al.1999) and grid-scale instabilities may occur 4Cessi T.Kuhlbrodt 4 &) Á S.Rahmstorf 1994; Molemaker and Dijkstra 2000), convective ad- Potsdam-Institut fuÈ r Klimafolgenforschung, Postfach 60 12 03, 14412 Potsdam, Germany justment has proven to be a satisfactory parameteriza- e-mail: [email protected] tion for many applications. 37 The most simple conceptual model of deep convection observed time series 4Frankignoul and Hasselmann consists of two boxes, one for the permanently mixed 1977).Hasselmann's mechanism was found to trigger surface layer and one for the deep ocean 4Welander decadal variability of the THC in an OGCM 4Weisse 1982).It includes di€erent boundary conditions for the et al.1994), where the water column in the Labrador Sea surface ¯uxes of heat and salt, as well as a strongly integrated noisy freshwater forcing.In other models, nonlinear dependence of vertical mixing between the two decadal variability is generated by stochastic excitation boxes on the vertical density gradient.Under certain of internal ocean modes 4Delworth et al.1993) or by a boundary conditions, this leads to a bistable regime of local convective oscillator 4Pierce et al.1995; Rahmstorf the water column where convection can be either per- 1999). manently ``on'' or permanently ``o€ '', depending on the Here, we e€ectively combine the basic ideas of initial condition.This is due to a positive salinity feed- Hasselmann's stochastic climate model and Welander's back in the presence of surface freshwater input: once two-box model to a stochastic climate model of deep convection is interrupted 4e.g., by a freshwater anomaly), convection.Adding stochastic perturbations to the the surface water will become less and less saline, making surface forcing of the extended two-box model 4devel- a restart of convection increasingly harder to achieve. oped in Part I) mimics the essential role of buoyancy Using Welander's box model, Lenderink and Haarsma forcing variability in triggering deep convection 41994) showed that large regions of a model North 4Marshall and Schott 1999; Sathiyamoorthy and Atlantic were bistable due to this feedback. Moore 2001).The frequency of jumps and the resi- In Part I of this paper 4Rahmstorf 2001), Welander's dence times in both model states 4i.e., with and without model is extended in two ways.First, temperature and winter convection) are studied, together with their de- salinity of the deep box are introduced as variables pendence on model parameters.The interplay of the rather than as prescribed ®xed values.Heat and salt can di€erent timescales involved is shown to result in then accumulate in the deep box during nonconvecting decadal variability. phases and are released by convective mixing.In this The OWS Bravo data are analyzed and interpreted in way, the heat ¯ow from the deep ocean through the the following section.The third section gives a short convecting water column to the atmosphere is modeled. summarizing description of the box model and explains Second, a seasonal cycle is introduced in the boundary how it was ®tted to the OWS Bravo time series.The conditions for the upper box.This reduces convection to stability diagram in the non-stochastic case is discussed a short period in winter, rather than occurring perma- in section 4.The ®fth section is devoted to the in¯uence nently.The model demonstrates why the convection of stochastic forcing on the model dynamics, and the events are so short: slow processes like advection and paper ends with some conclusions. diapycnal mixing replenish the heat store of the deep layer throughout the year, while a few days' time is en- ough to release the accumulated heat to the atmosphere 2 Observational data via the much faster processes of convective mixing and surface exchange. Long time series of hydrographic data that show clear Welander's model has been used by some authors signs of deep convection events are rare.The data from 4Lenderink and Haarsma 1994; Pierce et al.1995; Ocean Weather Ship 4OWS) Bravo are exceptional due Lenderink and Haarsma 1996; Hirschi et al.1999) to to their location and their sampling rate.OWS Bravo analyze output from OGCMs.Here, we use the modi®ed was located in the central Labrador Sea close to the area version to analyze observational data from the Labrador of the deepest convection events.In the time from Sea.The data from Ocean Weather Ship 4OWS) Bravo January 1964 through September 1974 the sampling rate 4Lazier 1980) show how convection was switched o€ and of the data varied between 6 h and 2 months.This on again in the course of the Great Salinity Anomaly enables the derivation of a time series of monthly means 4GSA) of the years 1968±1972 4Dickson et al.1988).By that clearly re¯ects the winter open-ocean deep convec- ®tting the model to the OWS Bravo data we can locate tion events. the Labrador Sea in a stability diagram.The results The original data 4Lazier 1980) were interpolated to suggest that the convecting state is only marginally sta- standard depth levels.Potential temperature 4 h) and ble; anomalies in the surface forcing can trigger state potential density 4r0) were computed with the standard transitions very easily. formulas 4Fofono€ and Millard 1984).To obtain The e€ect of short-term variability on the ocean monthly mean values, the data of each month were surface layer can be represented by adding a stochastic binned and averaged at each depth level.Missing term to the surface forcing.The most simple form of monthly means were interpolated linearly.Subsequently, oceanic response to stochastic forcing was analyzed by the data were averaged for an upper layer 40±50 m) and Hasselmann 41976).In his stochastic climate model, an a deep layer 4200±2000 m).The intermediate level ocean surface layer of ®xed thickness acts as a reservoir 450±200 m) was left out because, on the one hand, this and integrates white noise temperature forcing from the layer still shows substantial seasonal variations, but on atmosphere, which leads to a red noise variance spec- the other hand, it is not part of the surface mixed layer trum of sea surface temperature similar to that seen in throughout the year. 38 The resulting time series of monthly means are given compensate for the deep ocean being saltier than the in Fig.1.The winters 1969±1971 show the impact of the surface waters, the upper layer must become colder than Great Salinity Anomaly 4GSA, described by Dickson the deep ocean before the vertical density gradient van- et al.1988) that suppressed deep convection in the ishes and deep convection starts 4see Part I, section 4). Labrador Sea by the advection of a large freshwater In order to consider seasonal and interannual vari- anomaly.In consequence, temperatures and salinities ability separately, we ®rst computed the mean seasonal followed di€erent trends in both layers: cooling and cycle of the time series 4Fig.2).In the upper layer, the freshening in the upper layer leading to less dense temperature cycle has its minimum in February and its waters, while the deep ocean is becoming slightly warmer maximum in September, with an amplitude of 2:2 C. and saltier.The upper layer values show a strong sea- The salinity cycle lags by about 1 month: the cycle with sonal cycle.When the potential density di€erence be- an amplitude of 0.13 psu peaks in March and reaches its tween the two layers is small, this indicates deep minimum in October.The r0 cycle lies in between with convection.For a number of reasons, this di€erence is an amplitude of 0.29 kg mÀ3.The deep layer seasonal not exactly zero: the mixing of the layers occurs only cycles 4not shown) are almost 2 orders of magnitude during a few days, but the data are averaged monthly; smaller.They show an annual warming of 0 :12 C and a the mixing does not occur necessarily exactly at the ship density decrease of 0.015 kg mÀ3, both of which start in site and throughout the whole water column.In some April after the convection season, and reach their winters the upper layer overshoots in temperature extremum in November/December.Deep-layer salinity 4though not very visible in the monthly means): to variations are very small.The temperature cycle is mostly forced by ¯uxes of latent and sensible heat in winter and short-wave radiation in summer 4Smith and Dobson 1984).Freshwater sources of poorly known strength 4Canadian runo€, meltwater, local precipita- tion, and low-salinity in¯ow from the Arctic Sea) ac- count for the salinity cycle 4Lilly et al.1999). In a second step, we subtracted the seasonal cycle from each time series in Fig.1.The resulting time series 4Fig.3) show the variability excluding the seasonal cy- cle.Three distinct phases stand out, marked by either the occurrence or the absence of convection.In the ®rst convective phase 4phase 1) from January 1964 to March 1968 the values in the upper box ¯uctuate with hardly any interannual trend.However, there are trends in the deep layer.Time series of single depth levels reveal that these trends are more pronounced in deeper layers.With maximum convection depth varying from year to year, the deeper layers sometimes remain untouched and accumulate heat over more than 1 year.

Fig. 1a±c Time series of monthly means obtained from the OWS Bravo data set: potential temperature 4a), salinity 4b), and potential density 4c) of the upper layer 4dashed) and the deep layer 4dots). Interpolated values are indicated by circles.The large minimum density di€erence in the winters from 1969 to 1971 is an indication for Fig. 2 Mean seasonal cycle of the upper layer 40±50 m) from OWS the absence of deep convection, which led to the cooling and Bravo data for potential temperature 4solid), salinity 4dashed), and freshening of the upper layer potential density 4dash±dotted) 39

Table 1 Trends during the GSA.Trends of potential temperature, salinity and potential density in the upper layer 4index 1) and the deep layer 4index 2) during the GSA 404/68 to 09/71) from the time series without seasonal cycle depicted in Fig.3

Quantity h1 S1 r0;1 Trend 4yr)1) )0.28 °C )0.11 psu )0.060 kg m)3

Quantity h2 S2 r0;2 Trend 4yr)1) 0.070 °C 0.0072 psu )0.0013 kg m)3

year until the end of the time series.The upper layer returns to more saline and dense conditions but remains cool, in a state clearly di€erent from phase 1.Possibly this is a consequence of the deep convection chimney being farther away from the ship now; the larger winter gap between upper and lower layer salinity 4compared to phase 1, see Fig.1b) suggests this.The deep layer jumps back to a colder and less saline state, and a further cooling and freshening trend sets in, albeit with little e€ect on density. In summary, the OWS Bravo data show a transition from a state of annual convection to stable strati®cation and back to convection.We will now discuss to what extent the simple box model can help to understand these transitions.

3 The simple boxmodel

3.1 Model equations

Fig. 3a, b Time series of monthly means with subtracted seasonal Before discussing the ®tting of the convection box model cycle for the upper layer 4a) and the deep layer 4b) of potential to the observational data, we recall the equations of the temperature 4solid), salinity 4dashed), and potential density 4dash± model and extend them slightly.The reader is referred to dotted) Part I of this paper for a detailed description of the model and the derivation of its equations.The model The phase from April 1968 to September 1971 4phase consists of two boxes, a shallow upper box 4index 1) 2) is characterized by the GSA passing the Labrador Sea representing the annual mixed layer and a large deep and suppressing convection there.Phase 2 begins after box 4index 2) representing the waters below the seasonal the last convection event and ends with the upper-layer thermocline.The ratio of the box depths is termed hÃ. salinity starting to rise again.Annual trends of all For the sake of simplicity, the box depths are ®xed, so quantities during phase 2 are given in Table 1.The e€ects like variable convection depth or mixed layer decoupling of the two layers induced by advective deepening are not included.The model variables are the freshening 4Dickson et al.1988) in the upper layer leads temperatures T1, T2 and the salinities S1, S2 in both there to cooling, further freshening, and a density de- boxes.These four variables are relaxed towards pre- à à à crease.While the cooling comes to a halt already in early scribed relaxation temperatures and salinities T1 , S1 , T2 , à 1970, the strong freshening continues until late 1971.In S2 .Through the use of three di€erent time scales s1T , s1S, the deep layer, the waters are becoming slightly warmer and s2, the di€erent coupling strengths of the heat and and more saline, mostly by lateral mixing from adjacent salt ¯uxes into box 1 and 2 are accounted for.The two water masses.Potential density in the deep layer shows a time scales s1T and s1S of the upper box forcing represent weak decreasing trend that is clearly smaller than in the di€erent feedbacks a€ecting temperature and salinity phase 1 as the warming and salini®cation partly com- forcing 4similar to mixed boundary conditions); included pensate in their e€ect on density. in this forcing are surface ¯uxes as well as lateral ¯uxes Phase 3 is again characterized by annual convection due to advection and mixing.The use of a single time events.Starting in October 1971, strong wind mixing of scale s2 for the deep box is motivated by the eddy the surface mixed layer caused it to deepen and entrain transfer ¯uxes at depth.In Part I, the basic equation salt from below 4Dickson et al.1996).Additional strong system 4Eq.6) was re®ned by adding a seasonal cycle cooling then achieved a vigorous deep convection event with amplitude AT to the upper box relaxation temper- à in early 1972.Afterwards, convection occurs again every ature T1 , in order to include seasonality and achieve a 40 short winter convection period instead of year-round of sc is much smaller than the other time scales involved, convection, in accordance with the observations.To we assume sc ! 0 and use the common parameteriza- facilitate comparison with observations and given the tion for deep convection, known as convective adjust- pronounced seasonal cycle in upper layer salinity in the ment 4Rahmstorf 1993; Klinger et al.1996).The water data, we add a seasonal cycle 4with amplitude AS and column is checked at each time step for hydrostatic zero mean) to the upper box salinity forcing as well.A stability 4we used dt ˆ 2 days).Nothing is done in the phase shift w between temperature and salinity cycles is case of stable strati®cation, but any occurring instability introduced as an adjustable parameter.Finally, keeping is instantaneously removed by complete mixing.Thus, the dimensions of the variables clari®es the physical the numerical integration scheme has two parts. meaning of the parameters.Time is in units of 1 year. The full equation set now reads: 1.Integrate forward Eq.41) to Eq.44) one time step without the vertical mixing terms.If we start at time i, Âà dT1 1 1 à ^i‡1 ^i‡1 ^i‡1 ^i‡1 ˆ †‡T À T T À A cos †À 2pt T this gives preliminary values T1 ; S1 ; T2 ; S2 à 2 1 1 T 1 dt h sc †Dq s1T for the variables at time i ‡ 1. 1† 2.Apply the convective adjustment scheme to obtain i‡1 i‡1 i‡1 i‡1 the ®nal values T1 ; S1 ; T2 ; S2 of the vari- dS 1 1 ÂÃables.If Dq  0, the ®nal values are identical to the 1 ˆ †‡S À S Sà À A cos †À 2pt ‡ w S à 2 1 1 S 1 preliminary ones; if Dq > 0, the two columns are dt h sc †Dq s1S mixed: 2† i‡1 i‡1 à ^i‡1 à ^i‡1 ÀÁ T1 ˆ T2 ˆ h T1 ‡ †1 À h T2 and dT2 1 1 à ˆ †‡T1 À T2 T2 À T2 3† i‡1 i‡1 à ^i‡1 à ^i‡1 dt sc †Dq s2 S1 ˆ S2 ˆ h S1 ‡ †1 À h S2 : 6† dS 1 1 ÀÁ 2 ˆ †‡S À S Sà À S : 4† dt s †Dq 1 2 s 2 2 c 2 3.2 Fitting the box model to the OWS Bravo data In each of the equations, the ®rst term on the right-hand side represents convective vertical mixing, and the sec- We adjust the model parameters 4see Table 2 for a ond term the horizontal and surface heat/salt ¯ux.We complete list) to ®nd the best ®t of the model to the recall that the four relaxation temperatures and salini- OWS Bravo data, and use a least-squares ®t procedure ties, the three relaxation timescales, the two amplitudes, for this purpose.We de®ne a cost function K as the sum and the phase shift are model parameters 4see Table 2 of the quadratic distances of each monthly averaged for a detailed list), but that the time scale of vertical model variable time series T1, S1, T2, S2 4weighted by the convective mixing sc is a strongly nonlinear function of thermal and haline expansion coecients to have a the model variables through the density di€erence common density unit) to the observed values.The op- timal parameter set minimizes K.As discussed above the Dq ˆ q À q ˆÀa †‡T À T b †S À S ; 5† 1 2 1 2 1 2 OWS Bravo data show two di€erent states with con- where a and b are the thermal and haline expansion vection 4phases 1 and 3, possibly due to di€erent surface coecients of the linearized equation of state of sea- forcing and/or convection locations), but the model can water, see Eq.44) in Part I. For stable strati®cation have only one convecting solution with the same forcing. 4Dq  0) the vertical mixing is very weak, so sc has a Hence we restrict our analysis to phases 1 and 2 of the large value.In the case of unstable strati®cation OWS Bravo data.4An analysis based on phases 2 and 3 4Dq > 0) convection starts, i.e., vigorous vertical mixing gives similar results).The 10-dimensional parameter with a timescale sc of the order of days.Since this value space is spanned by a 10-dimensional matrix.For every

Table 2 Model parameters. The model parameters with the Parameter Value Uncertainty Description values determined from ®tting à the model to OWS Bravo data. T1 4.4 °C 4.0±4.6 °C Upper box restoring temperature à The set of these parameter va- S1 33.5 psu # Upper box restoring salinity lues is called the `optimal para- à T2 4.1 °C 3.9±4.3 °C Deep box restoring temperature meter set'.The uncertainty Sà 34.97 psu # Deep box restoring salinity range is spanned by all para- 2 meter sets whose cost function s1T 5 months 3±9 months Restoring timescale of upper box temperature value exceeds the minimum by s1S 8 years 6±11 years Restoring timescale of upper box salinity less than 10%.The parameters s2 20 years 14±28 years Restoring timescale of deep box marked with # were not de- à AT 6.4 °C 5.0±7.8 °C Amplitude of seasonal cycle added to T1 termined through the cost à function, but directly from ob- AS 4.5 psu 3±6 psu Amplitude of seasonal cycle added to S1 servational data 4see section 2 w 0.6 months )0.5±1.5 months Phase shift of the seasonal cycles and 3) hà 1/36 # Ratio of box depths 41 possible parameter combination out of this matrix the the latitude band of the Labrador Sea 4but excluding the cost function K is computed from a model run with Labrador Sea itself) and between 200 and 2000 m depth à  à convecting state initial conditions.The onset of the GSA are T2 ˆ 4:3 C and S2 = 34.97 psu. These values are in the model is achieved by adding an anomalous salt almost equivalent to assuming Irminger Sea conditions ¯ux of À0:8 psu aÀ1 to the upper box for a period of 3 for the nonconvecting Labrador Sea.They yield a deep months in spring 1968 to mimic the arrival of an box time scale of s2 ˆ 20 years.In comparison with advected freshwater anomaly.The crucial idea here is other studies on the exchange rates in the deep Labrador that we aim to ®nd one single parameter set that yields a Sea 4Khatiwala and Visbeck 2000), this value is rather realistic model behavior in both states 4convecting and large; yet it is in agreement with the small trends in the nonconvecting) with the same forcing; the prescribed deep layer during the GSA 4see Table 1).Finally, we salt ¯ux anomaly provides a brief ``kick'' which induces applied one further assumption, namely to restrict the a state transition in the model. length of the winter convection event twce to less than 20 It turns out that K and its derivatives with respect to days, as low cost function values were in some cases the parameters are smooth functions and behave in a reached also with excessively long convection periods. physically understandable manner.However, both All the major conclusions of this paper are insensitive to physical intuition and the objective analysis show that the somewhat arbitrary assumptions described in this à à à the ten free model parameters are underdetermined by paragraph and hold for a wide range of S1 , T2 , S2 , and the ®t, i.e., the problem is ill-posed. This is because the maximum twce.The constraints from the OWS Bravo OWS Bravo data contain a steady convecting state data are sucient to determine the stability properties 4phase 1), but not a steady nonconvecting state.Phase 2 discussed below. of the OWS Bravo data displays the initial trends after The ®tting procedure now arrives at a global mini- cessation of convection, but does not reveal which mum of the cost function K and is repeated with equilibrium values the three variables S1, T2,andS2 parameter matrices of higher resolution 4in parameter would eventually reach in the nonconvecting state.Only space) to localize the global minimum more exactly.The à the trend in T1 stops in 1970, so that T1 can be deter- optimal parameter set thus determined is shown in mined.The impact of the missing nonconvecting state Table 2.The value for hà is a result of our analysis in the on the parameter determination can be clari®ed through previous section and not of the ®tting procedure. the model equations.Take Eq.42) for the upper box Changes in hà 4by assuming an upper box depth of salinity.We neglect the seasonal cycle here because it has 100 m, say) lead to quantitatively slightly di€erent re- no impact on the long-term trend.If no convection sults.To measure the parameter uncertainty we deter- occurs, Eq.42) then reduces to: mined all parameter combinations for which the cost function remains within 10% of its minimum value.This ÀÁ dS1 1 à de®nes an uncertainty range for each parameter 4except ˆ S1 À S1 : 7† dt s1S the ®xed ones). A comparison of a model run using the optimal When dS1=dt and an initial S1 are known from the data, parameter set with the OWS Bravo data 4Fig.4) shows then on the right-hand side of Eq.47) for any arbitrary that many relevant features of the data are captured by à choice of S1 a corresponding value of s1S can be found to the model.This includes the two upper box seasonal ful®l the equation.Thus, one of the two parameters in cycles during phase 1, leading to a winter convection à à Eq.47) is free.The situation is similar for T2 and S2 ; and event each year.The minimum di€erence between S1 since Eqs.43) and 44) are coupled through the common and S2 is very small in the model time series because the time scale s2, one second degree of freedom arises here. model mixes the two boxes completely, whereas for a In short, the least-squares ®t procedure constrains the number of reasons the complete mixing is not visible in ten-dimensional parameters space to a two-dimensional the observational data.In this cyclostationary state of subspace.This null space is clearly seen when attempting the model, without stochastic forcing, there is no inter- to minimize the cost function. annual variability in any variable.Phase 2 starts with a Two further constraints are thus needed to close the negative salt ¯ux anomaly in the model run.Convection problem, i.e., to obtain a global minimum in K.One ceases, and the model reproduces the observed trends of could assume arbitrary values of s1S and s2; we opt for all four variables: the upper box cools and freshens à à à making assumptions about S1 and either T2 or S2 .This strongly, while the deep box warms and becomes more option is equivalent to assuming the equilibrium mean saline.4The trends in the deep box are hardly visible on values of the three variables S1, T2, S2 in the non- the scale of Fig.4).In accordance with the observational convecting state.The upper box salinity is expected to lie data, the model deep box is not an in®nite reservoir of between the 34.7 psu of the convecting Labrador Sea heat and salt 4as in former versions of the box model, and the approximately 32 psu of the North Paci®c at the e.g., Lenderink and Haarsma 1994), but receives di€u- same latitude.From the mean salinity distribution in the sive ¯uxes from the neighbouring waters.The end of à Labrador Sea 4Levitus 1982) a value of S1 ˆ 33:5 psu phase 2, marked by the beginning salini®cation of the à à seems plausible.For the deep box parameters T2 and S2 , upper layer in the OWS Bravo data, is achieved in the the average values for the waters in the North Atlantic in model by a cold and saline anomaly in the upper box 42

à à forcing.Only a strong anomaly in surface forcing is the upper box buoyancy forcing, that is in T1 and S1 . capable of turning convection on again.It is clearly seen Changing other parameters will lead to similar pictures. from the trends in Fig.4 that the longer convection is 4However, reducing the model time scales by 1 order of o€, the lighter the surface layer becomes and the magnitude leads to decadal oscillations which will be stronger an anomaly must be to restart convection.This described elsewhere.) The parameter space section along point is studied in greater detail in the next section. these two axes 4Fig.5) shows the stable model states: the After convection is started again, the model returns to convecting state, the nonconvecting state, and a bistable its previous convecting state.The di€erent phase 3 state domain.The states are cyclostationary due to the pres- of the OWS Bravo data cannot be captured by the ence of the seasonal cycles.The domain boundaries are model. given in a good approximation by the analytical expressions for the necessary conditions for the non- convecting state: ÀÁÀÁ 4 Stability of the Labrador Sea à à à à a T1 À AT À T2 > b S1 À AS À S2 ; 8† The stable states of the model under varying parameters and for the convecting state: are now explored.Of particular interest are changes in ÀÁÀÁs ‡ s hà a T à À A À T à < b Sà À A À Sà 1T 2 ; 9† 1 T 2 1 S 2 à s1S ‡ s2h compare Eqs.49) and 412) in Part I.Since we introduced seasonal cycles such that convection now occurs around the temperature minimum and the salinity maximum in the upper box, the respective amplitudes are subtracted and added in Eqs.49) and 412) in Part I.The shape of Fig.5 is in agreement with earlier studies 4Lenderink and Haarsma 1994).We can thus conclude that the

Fig. 5 Stable model states depending on the upper box buoyancy forcing parameters.Convection is either ``on'' or ``o€ ''.These two stable states overlap in a bistable domain.The asterisk denotes the position of the optimal parameter set in the parameter space. à à Fig. 4a±c Comparison of model output 4solid upper box; dash±dotted The stable states were determined for varying only T1 and/or S1 , deep box) and observed time series 4dashed upper layer; dotted deep keeping the other parameters from the optimal set constant.For à layer) for temperatures 4a), salinities 4b), and densities 4c).The model S1 > 34:97 psu the upper box would be forced to a higher salinity was run with the optimal parameter set.All graphs are in monthly than the deep box; this case does not occur in high-latitude deep means.The GSA, also called phase 2 in the text, was started by adding convection as in the North Atlantic.For other parameter sets within à à a negative salt ¯ux anomaly in the upper box during April to June the uncertainty range or with di€erent values of S1 or S2 , the slopes of 1968.The GSA was stopped by a positive salt ¯ux anomaly during the domain boundaries change, but the marginal position of the October to December 1971 accompanied by a cold anomaly in the parameter set itself is a robust feature.The inclusion of sea-ice e€ects à upper box temperature forcing.See text for further explanation would a€ect the diagram for very low T1 values 43 presence of the seasonal cycles does not change the basic introduce variability that generates these anomalies in stability properties of Welander's 41982) box model. the context of the simple box model used here. The position of the model Labrador Sea ± repre- Observations 4Lilly et al.1999; Marshall and Schott sented by the optimal parameter set ± in the parameter 1999) suggest that heat ¯ux anomalies 4due to weather space section is marked by the asterisk in Fig.5.It is in activity) and freshwater ¯ux anomalies 4by advection) the bistable domain, i.e., both states, convecting and act together to trigger or suppress convection.Yet the nonconvecting, are steady states of the model under the heat ¯uxes clearly prevail in their contribution to the given conditions.A suciently large anomaly can overall buoyancy forcing 4Sathiyamoorthy and Moore switch convection on or o€.Moreover, the model is 2001).Hence, we focus on heat ¯ux variability as the located very close to the domain where only the non- primary variability component. convecting state is stable.Changing the buoyancy The characteristic time scale of synoptic cyclones is forcing by a few tenths of a degree or a few tenths of a few days.Following the concept of Hasselmann's psu will lead to the convecting state becoming uncon- 41976) stochastic climate model, we parameterize syn- ditionally unstable.In other words, there are two optic-scale variability by a noise term added to the possible ways for suppressing convection.In the ®rst surface heat ¯ux forcing.To estimate this noise term way, convection is temporarily switched o€ by an quantitatively, a 52-year-long time series of daily net anomaly, but can be restarted later by an opposite surface heat ¯ux from NCEP reanalysis data 4Kalnay anomaly, while the average properties of the buoyancy et al.1996) was analyzed.The analysis ± given in detail forcing 4i.e., the model parameters) do not change. This in the Appendix ± motivates a noise term consisting of is the GSA case depicted in Fig.4, and this may be the red noise nt with a decorrelation time of about 6 days case with a 50 year-long spell of convection in the times a standard deviation r, so that Eq.41) is Labrador Sea being switched ``on'' among centuries extended to: without convection that Tett et al.41997) found in a dT 1 coupled GCM.In the second way, under a slowly 1 ˆ †T À T à 2 1 changing buoyancy forcing 4e.g., T à or SÃ) the con- dt h sc Dq† 1 1 Âà vecting state eventually becomes unstable and convec- 1 à ‡ T1 À AT cos 2pt†‡rnt À T1 : 10† tion stops.This scenario could apply to the global s1T warming GCM run of Wood et al.41999), in which With this step, two classical conceptual models ± for Labrador Sea convection stops early in the 21st cen- deep convection 4Welander 1982) and for high-fre- tury.The role of the anomalies triggering state transi- quency atmospheric forcing of the ocean 4Hasselmann tions in both cases is examined in detail in the next 1976) ± are combined to give a simple model that could section. be called a stochastic climate model of deep convection. We checked systematically how the stability diagram Extended by the seasonal cycles and ®tted to the OWS changes for di€erent choices of SÃ, Sà 4or T Ã) and 1 2 2 Bravo data, the box model is now suitable to study the maximum twce, as well as for parameter sets in the un- variability of deep convection in the Labrador Sea. certainty range de®ned by a 10% change in the cost function.Equations 48) and 49) show that those pa- rameter changes a€ect the width of the bistable domain: à à for instance, for a larger di€erence 4S1 À S2 ) the bistable 5.2 Stochastic forcing and state transitions à domain 4in terms of T1 ) is wider.However, the distance of the model solution from the border of the bistability Thirty years from a model run with stochastic forcing à domain varies only by some tenths of a degree on the T1 are displayed in Fig.6.Several times, convection is in- à axis and similarly small amounts on the S1 axis.This is a terrupted for a few years, indicated by the small mini- consequence from the OWS Bravo data which explicitly mum density di€erence Dq between the upper and the show that a freshwater anomaly equivalent to 0.2 psu deep box 4Fig.6c). The upper box temperature T1 induced a transition from the convecting to the non- 4Fig.6a) is the only variable directly in¯uenced by the convecting state.The precarious position of the Labra- noise, so it shows the strongest variability.Apart from dor Sea, in the bistable domain but close to the the convective mixing induced by that variability, the nonconvecting domain, is therefore a robust feature of other three variables evolve in an unperturbed way. the ®tted model. Similarly to the GSA in the OWS Bravo data 4Fig.1), in the non-convecting years the upper box tends to freshen 4Fig.6b) and cool 4Fig.6a), until a cold anomaly re- 5 A stochastic climate model of deep convection starts convection again.In contrast to the observed GSA, the nonconvecting phase in this model run is 5.1 Motivation kicked o€ by a warm anomaly in the upper box, not a freshwater anomaly.This is due to the fact that the In the previous section, the essential role of surface stochastic variability appears in T1 only, not in S1,soby buoyancy forcing anomalies switching convection on construction only temperature anomalies can appear. and o€ became clear.We look for a meaningful way to However, through the Hasselmann mechamism the 44 than freshwater) that prevents convection in 1968. Convection then cannot recover by itself but requires another substantial trigger event.The longer convection has been o€, the larger the trigger needs to be.Had the winter of 1972 not been such a harsh one, subsequent winters would have needed to be even colder to restart Labrador Sea convection. This hypothesis is consistent with the conclusion of Dickson et al.41996), who analyzed the 1972 convection onset in the Labrador Sea in detail in the observed data. They conclude that the jump-like rise of the upper layer salinity is explicable only by anomalously strong wind forcing that mixes saline intermediate waters into the mixed layer; advective processes cannot lead to such strong changes.In other words, observational data show that the termination of the GSA in the Labrador Sea was achieved by anomalous weather conditions at the ocean surface, not by its internal dynamics. The presence of noise leads to a qualitative change in the model's stability behavior: the sharp domain boundaries depicted in Fig.5 are replaced by more gradual changes in the frequency of the occurrence of convection.As a measure we use nc, the fraction of years with convection out of all years in a model run. Figure 7 shows how nc depends on the noise strength r.Using the optimal parameter set with the convecting state as initial condition and increasing r 4Fig.7a), nc drops close to zero for weak noise.This re¯ects the marginal position of the optimal parameter set in pa- rameter space 4Fig.5). Any small perturbation shifts the model into the nonconvecting state, but the small perturbations are not able to induce a jump back to Fig. 6a±c Time series of monthly means from the stochastically convection.For r > 12 C, the convecting state is forced model: temperature 4a), salinity 4b), and density 4c) of the upper reached in some cases.The fraction of convective years box 4solid) and the deep box 4dash±dotted).The model was run with the optimal parameter set and a standard deviation r ˆ 18 Cofthe rises quickly and asymptotically reaches a value of stochastic forcing.The 30 model years shown include several nc  0:75.In this regime of strong noise, the noise interruptions of the convecting state.The di€erence to the observed tends to override the deterministic stability properties. GSA 4cf.Fig.1) is that the positive buoyancy anomaly needed to stop This feature becomes clearer when considering changes convection is achieved here by local heat ¯uxes rather than advective à freshwater ¯uxes.See text for details in the surface buoyancy forcing T1 in addition.The contour plot in Fig.7b displays how nc depends on r à and T1 .For r ˆ 0, Fig.7b corresponds to the deter- upper box integrates the weather noise to 4intra-)sea- ministic 4not stochastically forced) parameter space sonal temperature anomalies 4because s1T ˆ 5 months). section 4Fig.5) with only the nonconvecting state being à  à Thus, integrated synoptic heat ¯ux anomalies and ad- stable for T1 > 4:5 C.For low noise and large T1 there vected intraseasonal freshwater anomalies have the same is a large, wedge-shaped domain where almost no impact on Dq. convection events occur.The shape of this domain can The key point here is that a comparatively short be understood when thinking of the convecting state à anomaly 4lasting for a couple of months) triggers the becoming less and less stable for larger T1 .Then, for à state transition, and that the long-term trends evolving larger T1 a smaller amount of noise is needed to trigger afterwards are due to the internal dynamics until the a jump into the nonconvecting state.Since the non- à next anomaly triggers the next state transition.We hy- convecting state becomes more stable for larger T1 ,a pothesize that this picture of a bistable water column larger amount of noise is necessary to trigger jumps holds for the Labrador Sea.This implies that the falling back into the convecting state.Figure 7b also shows surface salinity from 1968 to late 1971 does not result that beyond this wedge-shaped domain the noise is from anomalous freshwater input during this whole capable of keeping the model in the convecting state time, nor does the return to convecting conditions in for more than half of the time even when this state is 1972 result from an end to the anomalous forcing. unstable in the deterministic case.The exact extent of Rather, the falling salinity requires only a brief trigger this domain depends on the respective parameter set: à anomaly 4which could even have been thermal rather for instance, a lower S1 leads to a less dense upper box 45 5.3 Stability and residence times

It is straightforward to estimate how the stability of the states 4with respect to perturbations) changes quantita- tively with varying model parameters.Consider a par- ticle in an ideal double-well potential.If the particle is initially in one well, then added noise will rattle it.To leave the initial potential well and jump to the other one, the particle has to overcome a potential di€erence DU.If the noise is Gaussian distributed, eventually one per- turbation, occurring after time tr, is large enough for the particle to hop into the other well.In this ideal case there is a simple relation between the particle's mean residence time htri in one well, the potential di€erence DU and the standard deviation r of the added noise: 2 DU ˆ r loghitr ‡ C : 11† This equation directly follows from Arrhenius' formula 4Gardiner 1994).The constant C is a function of the potential's curvature at the well bottom and at the po- tential hill that separates the two wells. The two stable states of the box model 4with four variables) cannot be expressed as minima of an 4one- dimensional) potential.However, there is a way to use Eq.411) for our purposes. The potential U can be interpreted as a quasipotential 4see Freidlin and Wentzell, 1998, for a rigorous de®nition).Then, the potential di€erence DU is the necessary perturbation strength for a state transition.The larger DU is, the more stable is the state.The mean residence times htci and htni in the convecting and the nonconvecting state can be estimated from long model runs.We do this for leaving r constant Fig. 7 a Dependence of the fraction n of convection years in a long à c and varying T1 only.With the help of Eq.411) we are model run 4106 years) on the noise level r.The model was run with the then able to give a quantitative estimate of the relative unchanged optimal parameter set and convecting state initial conditions.The deterministic case, without stochastic forcing, is at stability of the two states as a function of the surface r ˆ 0. b Contour plot of nc as a function of r and additionally of the forcing.Figure 8 shows how the logarithms of the resi- upper box temperature forcing T Ã.The dashed line denotes the à 1 dence times change with varying T1 .For low values of position of the graph shown in a.With a nonconvecting initial T Ã, the convecting state is clearly the more stable one. à  1 condition, nc ˆ 0forr ˆ 0 for all values of T1 > À4 C 4i.e. in the bistable regime, see Fig.5). The lower part of b would change Conversely, for warm surface forcing the stability of the accordingly nonconvecting state increases strongly.In the case of the optimal parameter set 4dashed line in Fig.8), the non- convecting state is about twice as stable as the convec- in the nonconvecting state, so convection is harder to ting one.This gives a quantitative understanding of trigger, and the domain becomes wider. Fig.7a: weak noise can provide the anomalies to jump In the Appendix it is shown that for the Labrador Sea into the nonconvecting state, but anomalies twice as conditions in the box model r is likely to be near or large, necessary for the jump back, occur only extremely larger than 15 C.This means that the model is located rarely.Note the contrast between the sharp stability in a domain where nc is sensitive to changes in the sur- domain borders of the deterministic model 4Fig.5) and face forcing.There are two ways of making convection the smooth shape of the stability curves for the sto- occur less often: either by decreasing the variability r or chastic case. à by increasing the surface temperature T1 4or equivalently From Fig.8 we see that the mean residence times for à decreasing S1 ); but convection can still occur even when the optimal parameter set and a standard deviation of  the convecting state is unstable in the deterministic case. r ˆ 18 C in the stochastic forcing are htciˆ3:5 years à If an increase in T1 is taken as a crude representation of and htniˆ11 years.Thus, the average time for the a global warming scenario, then these results suggest model to jump from one state to the other and back is that the frequency of Labrador Sea convection could about 15 years.In other words, the typical time scale for decrease substantially due to a future warming 4and/or the variability is in the decadal range.This is clearly freshening) unless variability increases strongly at the di€erent from the synoptic time scale of the stochastic same time. surface forcing.The e€ect of the weather noise is here to 46

Fig. 8 Mean residence times in the convecting state 4htci, solid)andin the nonconvecting state 4htni, dash±dotted) in dependence on the mean à surface temperature forcing T1 for constant standard deviation 4r ˆ 18 C) of the stochastic surface temperature forcing.The dashed line indicates the position of the optimal parameter set: the convecting state lasts 3.5 years and the nonconvecting state 11.2 years, on average excite intraseasonal variability in the mixed layer, which Fig. 9a, b Distribution of residence times in the convecting state 4a) and the nonconvecting state 4b).The frequency of every single in turn triggers interannual to decadal variability.The residence time is given as a fraction of the total number of residences deep ocean, being isolated from the atmosphere nearly during a 105 year run.Hence, the distributions are approximate at all times, ``sees'' the synoptic variability through the probability density functions.The fraction axis is logarithmically ``window'' of deep convection events ± but responds to scaled; the small bar in panel a at tc ˆ 25 years corresponds to one this forcing with its own typical decadal time scale.In single occurrence.The model was run with the optimal parameter set and a standard deviation of r ˆ 18 C, yielding a fraction of this way, deep convection is a prominent example of convective years nc ˆ 0:26.The tail of the distribution in 4 b)was time-scale interactions in the climate system. cut arbitrarily; the maximum tn is 526 years The analysis can be carried one step further by ex- tending our view from the mean residence times htci and htni to their distributions ps tc† and ps tn† that are and Rhein 2000; J.Holfort personal communication), is equivalent to the stationary probability density functions not necessarily a sign of a global climatic trend, but 4pdfs).This draws a more complete and accurate picture could be within the natural variability properties of a of the variability.The di€erent shapes of the two pdfs in convective water column. Fig.9 stand out.The residence times of the convecting The di€erence between the two distributions in Fig.9 state 4Fig.9a) are distributed following a straight can be understood qualitatively in the framework of line, with tc > 20 years occurring only rarely during a ``runs'' introduced by von Storch and Zwiers 41999).A 100 000-year model run.In comparison, ps tn† has a bent ``run'' is de®ned as the time that a stochastic process shape, with high probability density for very short resi- spends uninterruptedly on one side of its mean value. dence times and some occurrences of tn exceeding 20 Von Storch and Zwiers analyzed AR41) processes with years.These features are obscured when considering the varying autocorrelation coecient a1, and found that mean value only: while the mean residence time is for a1 ˆ 0 4white noise) the run length pdf decreases htniˆ11 years, we learn from ps tn† that the time series exponentially.In a logarithmic plot this pdf of the run will contain many cases of only a few years without lengths, or residence times, appears as a straight line, as convection, but also some occasions where convection is in Fig.9a.For red noise 4 a1 > 0), long residence times interrupted for more than 100 years.With the are more probable at the expense of intermediate times, probability distributions of the residence times 4Fig.10) which gives the bent shape of the graph in Fig.9b.In this can be quanti®ed.For instance, there is a 10% other words, the linear shape of p tc† in Fig.9a means probability for the nonconvecting state to last longer that the probability for a convection stop is equal in all than 13 years, but the convecting state will do so only years, whereas the exponential shape of p tn† in Fig.9b with a probability of 1.5%. Hence, the observation of stems from the diminishing probability of leaving the two decades without deep convection, as in the years nonconvecting state with increasing residence time tn. 1982±2001 in the Greenland Sea 4Rhein 1996; Visbeck This feature again re¯ects the positive salinity feedback. 47 to interrupt it, suggesting that convection might have ceased for much longer if 1972 had not been an anomalously harsh winter over the Labrador Sea.Note the contrast with the hypothesis of Dickson et al. 41996), and Lilly et al.41999), who conclude that anomalous surface conditions 4be it in freshwater ad- vection or local heat ¯uxes) lasted throughout the 4 years of the GSA and were needed to suppress deep convection.Our results suggest that a di€erent mecha- nism might have been working here: a short-term per- turbation switched convection o€, and the subsequent evolution was governed by internal, local dynamics until another perturbation switched convection on again. We found that the position of the model Labrador Sea in the bistable domain is very close to the border to the monostable domain without stable convecting state, irrespective of the parameter assumptions about the stable nonconvecting state.This position is precarious: changing the ocean's surface forcing by about 1 °C to- wards warmer conditions leads to the convecting state becoming unconditionally unstable.Such a shutdown of Fig. 10 Probability distribution of residence times in the convecting Labrador Sea convection occurred in a global warming state 4tc, solid) and the nonconvecting state 4tn, dash±dotted)fromthe scenario computed with a coupled AOGCM 4Wood et al. same model run as in Fig.9. For any time t, the probability 1999).It is not clear yet whether the additional feed- distribution gives the probability of the residence time being smaller than or equal to t backs in complex models lead to a less precarious stability of deep convection. With the model being in the bistable domain, anomalies in the forcing are essential to excite state 6 Conclusions transitions.Therefore, we included stochastic variability in the model as in the stochastic climate model of The aim of this paper is to understand some basic sta- Hasselmann 41976), to parameterize weather variability bility and variability properties of open-ocean deep over the Labrador Sea.In this way, two seminal con- convection.The simplest possible model for this purpose ceptual models, for deep convection 4Welander 1982) is a two-box model of a water column in a potentially and for high-frequency atmospheric forcing of the ocean convective part of the ocean.The simplicity of the model 4Hasselmann 1976), are combined to give a ``stochastic allows the exploration of large parts of the parameter climate model of deep convection''.The parameters of space.Observational data from the Labrador Sea show the noise term are estimated from a daily time series of phases with and without convection and are used for the surface heat ¯ux. parameter determination of the model.Not all of the We conclude that realistic noise amplitudes are large model parameters are well constrained by the dataset. enough to blur the clear picture of the stability diagram. Some properties of a steady nonconvecting state in the Even when the convecting state is unstable in the de- Labrador Sea have to be assumed in order to close the terministic case, the variability excites frequent convec- problem of parameter determination. tion events.Conversely, the observation of intermittent With the parameter set obtained, the position of the convection in the real ocean or in a model does not allow Labrador Sea in a stability diagram can be determined. direct inferences about the stability of the underlying For a certain region in the parameter space the model deterministic state. has two stable states, with convection being either ``on'' When stochastic variability is taken into account, a or ``o€'' each winter.With the Labrador Sea parameters warming 4and/or freshening) at the surface will not lead the model is located in this bistable domain, so that to a complete stop of convection at a certain threshold, anomalies in the forcing are capable of triggering jumps but rather to a decline in the frequency of convection from the convecting to the nonconvecting state and events.The frequency of state transitions depends on the back.The model shows that lasting anomalies similar to noise level in a highly nonlinear way.There are two the Great Salinity Anomaly 41968±1972) can be trig- plateaus for the frequency value: weak noise triggers gered by short-term anomalies in the surface conditions hardly any jumps, and with strong noise the jumps be- suppressing convection in one winter.This mechanism come very frequent and occur every few years.In be- of transitions between two stable states can explain the tween is a small range of noise levels where the jump basic properties of the time series from OWS Bravo. frequency steeply rises.For the plausible parameter The longer a nonconvecting phase lasts, the harder it is range, the position of the Labrador Sea is in the region 48 of this steep rise, which re¯ects its sensitivity to changes Apart from the decorrelation length, the second parameter in surface forcing. we need to estimate for the noise term in Eq.410) is the standard deviation r.Using the heat ¯ux time series in Eq.4A1) to force the The interannual to decadal jumps are triggered by the temperature of an ocean surface layer we write: intraseasonal density anomalies in the upper box, which in turn are excited by weather noise in the surface heat dT cpq0h1 ˆ Q0 ‡ AQ cos †‡ 2pt rQnt À kT ; A4† ¯ux.Advective freshwater anomalies have not been dt À1 À1 modeled here, but would have the same impact on the with the speci®c heat capacity cp ˆ 3990 J kg K , a reference À3 density q0 ˆ 1028 kg m , the surface layer depth h1 ˆ 50 m, and a model dynamics.The probability distribution of the restoring constant k.Setting c ˆ c q h , this reads: residence times in the nonconvecting state shows that 0 p 0 1 dT Q0 AQ rQ k there is a small, but not negligible probability for the ˆ ‡ cos †‡ 2pt nt À T : A5† nonconvecting state lasting a decade or longer.In con- dt c0 c0 c0 c0 trast to a deterministic understanding of the system, this Comparison with Eq.410) shows how the standard deviation r of means that convection may start again after a long break the stochastic forcing in Eq.410) is related with the standard devi- ation r of the heat ¯ux time series: due to natural variability.The recent 19-year-long Q s1T cessation of deep convection in the Greenland Sea is r ˆ rQ : A6† thus not necessarily due to a long-term climatic trend c0 but could be part of the normal stochastic variability With a dependence on the averaging interval again, we ®nd À2  properties of convection.On the other hand, a surface rQ ˆ 120±140 W m , which translates into r ˆ 8±9 C.This is a rather high value, larger than the average ¯ux Q0 ± and still the warming or freshening trend as may result from an- seasonal cycle of the standard deviation itself, reaching its maxi- thropogenic global warming can be expected to sub- mum in winter, has not been accounted for here.Sathiyamoorthy stantially reduce the frequency of convection in the and Moore 42001), in their analysis of the buoyancy ¯ux at OWS Labrador Sea and possibly elsewhere. Bravo derived from weather ship data, obtain a similar result.The dominant role of the synoptic-scale variability of the heat ¯ux in the Labrador Sea is highlighted here again.For simplicity we Acknowledgements The work on this paper was supported by the have added noise only to the surface temperature forcing.To Deutsche Forschungsgemeinschaft 4Sfb 555 and Heisenberg- obtain a realistic variability in the whole surface buoyancy ¯ux, Programm).The paper is a contribution to a cooperation of the pronouncedly higher values of r have to be assumed since the universities of Potsdam, Germany, and Lisbon, Portugal, variability of surface freshwater ¯uxes and of advective transports promoted by the Deutscher Akademischer Auslandsdienst.We are of heat and salt have not been considered here.A range of grateful to Eva Bauer, Miguel Morales Maqueda, Adam Monahan, r ˆ 15±20 C seems therefore plausible.For the numerical inte- Axel Timmermann, and Udo Schwarz for inspiring discussions. gration we applied a semi-implicit Milstein scheme following Comments of two anonymous reviewers helped to clarify the paper. Kloeden and Platen 41999, Chap.10). Its theoretical convergence is twice as good as with an ordinary Euler scheme, which is rel- evant regarding the discontinuities arising from the convective Appendix adjustment.

The analysis of a 52-year-long time series of daily net surface heat ¯ux at the OWS Bravo site is presented here.This time series was References extracted from the NCEP database 4Kalnay et al.1996).The aim is an estimate for the standard deviation and the decorrelation length of the noise term in Eq.410). Assuming that the heat ¯ux time Belkin IM, Levitus S, Antonov J, Malmberg S-A 41998) Great series Q can be decomposed into an average ¯ux Q , a seasonal Salinity Anomalies in the North Atlantic.Prog Oceanogr 41: 0 1±68 cycle with amplitude AQ, and a noise term nt with standard devi- ation r , Cessi P 41994) A simple box model of stochastically forced ther- Q mohaline ¯ow.J Phys Oceanogr 24: 1911±1920 Q ˆ Q0 ‡ AQ cos 2pt†‡rQnt ; A1† Delworth T, Manabe S, Stou€er RJ 41993) Interdecadal variations of the thermohaline circulation in a coupled ocean±atmosphere it turns out that, for averaging intervals of 1 to a few days, the model.J Clim 6: 1993±2011 autocorrelation function of the noise process nt falls o€ to zero only Dickson RR, Lazier J, Meincke J, Rhines P, Swift J 41996) Long- after the ®rst few lags.In other words, nt can be modeled by an term coordinated changes in the convective activity of the AR41) process North Atlantic.Prog Oceanog 38: 241±295 nt ˆ a1ntÀ1 ‡ ft ; A2† Dickson RR, Meincke J, Malmberg S-A, Lee AJ 41988) The ``Great Salinity Anomaly'' in the northern North Atlantic 1968±1972. where the value nt at time t is determined by the value ntÀ1 at the Prog Oceanog 20: 103±151 previous time step times the autocorrelation at lag 1, a1, plus a Fofono€ NP, Millard RC Jr 41984) Algorithms for computation of random value ft from a Gaussian white noise process.With values fundamental properties of sea water.Unesco Tech Pap Mar Sci for a1 estimated from the NCEP time series, the decorrelation time 44, UNESCO, Paris as de®ned in von Storch and Zwiers 41999): Frankignoul C, Hasselmann K 41977) Stochastic climate models, Part II: Application to sea surface temperature anomalies and 1 ‡ a1 s ˆ ; A3† thermocline variability.Tellus: 29: 289±305 D 1 À a 1 Freidlin MI, Wentzell AD 41998) Random perturbations of dy- lies between 5 and 7 days, depending on the averaging interval of namical systems.2nd edn.Springer, Berlin, Heidelberg, New the time series.This decorrelation time is just the typical time scale York for synpotic activity.Using this red noise forcing in the model Gardiner CW 41994) Handbook of stochastic methods for physics, 4instead of pure white noise) is a more realistic parameterization of chemistry and the natural sciences.Springer, Berlin, Heidelberg, the high-frequency heat ¯ux variability and renders the model New York results more robust to changes in the time step of the numerical Hasselmann K 41976) Stochastic climate models, Part I: Theory. integration scheme. Tellus 28: 473±485 49

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