Effects of the Mixed Layer Time Variability on ... - AMS Journals

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VALDIVIESO DA COSTA ET AL.

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Effects of the Mixed Layer Time Variability on Kinematic Subduction Rate Diagnostics MARIA VALDIVIESO DA COSTA Department of Oceanography, The Florida State University, Tallahassee, Florida

HERLÉ MERCIER

AND

ANNE MARIE TREGUIER

Laboratoire de Physique des Océans, CNRS-IFREMER-UBO, Plouzané, France (Manuscript received 2 September 2003, in final form 27 August 2004) ABSTRACT An eddy-resolving primitive equation general circulation model is used to estimate water-mass subduction rates in the North Atlantic Ocean subtropical gyre. The diagnostics are based on the instantaneous kinematic approach, which allows the calculation of the annual rate of water-mass subduction at a given density range, following isopycnal outcrop positions over the annual cycle. It is shown that water-mass subduction is effected rapidly (⬃1–2 months) as the mixed layer depth decreases in spring, consistent with Stommel’s hypothesis, and occurs mostly over the area of deep late-winter mixed layers (ⱖ150 m) across the central North Atlantic in the density range 26 ⱕ ␴ ⱕ 27.2. Annual subduction rates O(100–200 m yr–1) are found south and east of the Gulf Stream extension in the density range of subtropical mode waters from roughly 26.2 to 26.6. In the northeastern part of the subtropical gyre, annual subduction rates are somewhat larger, O(250 m yr–1), from a density of about 26.9 east of the North Atlantic Current to 27.4 (upper cutoff in this study). The overall basin-integrated subduction rate for subtropical mode waters (26.2 ⱕ ␴ ⱕ 26.6) is about 12.2 Sv (Sv ⬅ 106 m3 s⫺1), comparable to the total formation rate inferred from the surface density forcing applied in the model of roughly 11 Sv in this density range. In contrast, basin-integrated rates for denser central water (26.8 ⱕ ␴ ⱕ 27.2) provide a vanishingly small net subduction. In this range, eddy correlations (⬍30 days) between the surface outcrop area and the local subduction rate counteract the net subduction by the mean flow (deduced from monthly averaged model fields). Comparison with estimates of the annual subduction rate based on the annual mean velocity and late-winter mixed layer properties alone, as is usual in climatological and coarse-resolution model analyses, indicates a mismatch of at least 8 Sv in the density range where the model forms subtropical mode water. This mismatch is primarily due to time-varying mixed layer processes rather than small-scale mixing not resolved explicitly by the model. Our diagnostics based on the instantaneous kinematic approach provide a more complete picture of the watermass formation process than diagnostics based only on air–sea flux or late-winter mixed layer model data. They reveal the crucial importance of both the seasonal mixed layer cycle and mesoscale eddies to the overall formation rate and provide thus a valuable tool for the analysis of water-mass formation rates in eddy-resolving numerical simulations at basin scale.

1. Introduction Kinematic subduction rate diagnostics are a widely used tool for the description of processes that couple the dynamics of the mixed layer and the ocean interior. This paper is concerned with some uncertainties that influence such calculations. The problem is to predict the annual rate of water-mass subduction at which fluid escapes irreversibly into the main thermocline from the kinematic relationship between the velocity field and the mixed layer depth. Stommel (1979) was the first to

Corresponding author address: Maria Valdivieso Da Costa, GIP MERCATOR-OCEAN, 8/10 rue Hermes, Parc Technologique du Canal, 31520 Ramonville St. Agne, France. E-mail: [email protected]

© 2005 American Meteorological Society

JPO2693

suggest that the annual subduction rate is virtually independent of the detailed seasonal mixed layer cycle and can be diagnosed from the late-winter mixed layer properties alone. This hypothesis appears to be supported by kinematic subduction rate diagnostics based on climatological data. For instance, Marshall et al. (1993) showed that the effective subduction period diagnosed from climatological data in the North Atlantic Ocean is rather short, about 1–2 months in late-winter to early spring, over most of the subtropical gyre. Since then, most subduction rate diagnostics from both observations (e.g., Qiu and Huang 1995; Karstensen and Quadfasel 2002) and model simulations (e.g., New et al. 1995; Williams et al. 1995; Spall et al. 2000) have been based on the steady limit, in which the velocity field is given by the time-averaged velocity over a mixed layer

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cycle and the mixed layer depth is a snapshot of its late-winter pattern. A relevant issue is the reliability of such diagnostics for the time-dependent and eddying upper ocean. A potentially important source of errors is the limited spatial and temporal resolutions of the available data for subsequent subduction rate estimations. One might expect these errors to be relatively large if in reality the mixed layer is, during some part of its seasonal cycle, determined by small-scale oceanic motions that are either poorly resolved or not resolved at all by the data: for example, the motions that occurs in and near a mixed layer front (see, e.g., Pollard and Regier 1992; Follows and Marshall 1994). For the study of the subduction process in the real ocean it is, therefore, relevant to know whether effects of time-dependent behavior of the mixed layer are indeed relatively large or not. This is an issue that has not been investigated yet in detail, though its conclusions could question the reliability of steady diagnostics based only on late-winter mixed layer properties. The possible importance of mesoscale eddies has been discussed by Marshall (1997) and an attempt at quantification has been made by Hazeleger and Drijfhout (2000), but both studies dealt with idealized flow configurations. One method of estimating the error in the annual subduction rate due to limited data resolution is to investigate the sensitivity of the computed diagnostics to the resolution of the observations. Unfortunately, the physical processes involved in the seasonal cycle of the mixed layer are rather difficult to observe directly in the ocean as their origin are intimately connected with both mechanical and thermodynamical atmospheric forcing (e.g., Nurser and Marshall 1991; Marshall and Marshall 1995). As an alternative, the systematic study of this resolution sensitivity should be attempted within ocean general circulation models (OGCMs) wherein simulations for many years at basinscale eddy resolution have been performed notably for the North Atlantic (e.g., Bryan and Holland 1989; Bleck et al. 1989; Marshall et al. 1997; Treguier et al. 2001). In the present study, the output from an eddyresolving OGCM is used to quantify the effects of timedependent behavior of the mixed layer on subduction rate diagnostics in the North Atlantic subtropical basin. Our diagnostics are based on the kinematical approach, which allows the calculation of the net annual subduction rate at a given density range by integrating the local subduction rates over the outcrop areas following instantaneous isopycnal positions over the annual cycle. The purposes of this study are 1) to quantify and discuss the errors caused by our data for subduction diagnostics being steady at the end of winter and annually averaged, rather than instantaneous, as required by the method we use, 2) to estimate the contribution of mesoscale eddy pro-

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cesses to net annual subduction rates in the context of this eddy-resolving model, and 3) to establish the relation between water-mass formation rates due to subduction and water-mass formation rates as deduced from the surface density flux of the model. To perform these tasks, the results from a 20-yr run of the OGCM forced with analyzed air–sea fluxes are assumed to represent the climatological seasonal circulation of the North Atlantic Ocean at the spatial and temporal resolutions of the model (1⁄6° horizontal resolution, 5-day temporal resolution, 46 vertical levels). Section 2 introduces the primitive equation numerical model used to generate the velocity fields and mixed layer cycle, and presents the method to evaluate the net-annual subduction rate in a particular density range. Some characteristics of the model mixed layers at the end of winter and estimates of the annual subduction rate across the base of the wintertime (deepest) mixed layer are presented and compared with climatologies in section 3. In section 4, we quantify the effects of mixed layer time-dependence on our calculations. First, we test Stommel’s (1979) hypothesis at the spatial and temporal resolutions of the model. Then, we quantify the error in our estimates caused by limited data resolution in time, giving preliminary estimates of the importance of the eddies to subduction on the subtropical gyre. A comparison with water-mass formation rates as deduced from surface density fluxes alone is made in section 5. An integrated view of the results is given in the outcrop density space by the diagnostic of the time- and basin-integrated water-mass formation rates in the different density classes of the North Atlantic subtropical basin. A conclusion of this diagnostic study is given in section 6.

2. Diagnosis of the annual rate of water-mass subduction in a numerical model a. Ocean model The model used here is the ATL6 Atlantic configuration developed in the framework of the “CLIPPER” project, a French contribution to the World Ocean Circulation Experiment (WOCE). The model is based on the primitive equation code OPA8.1 (Oce´an Paralle´lise´) developed at the Laboratoire d’Oce´anographie Dynamique et de Climatologie (Madec et al. 1998). It is a second-order finite-difference model with a rigid lid in z coordinates on the vertical. The horizontal grid is a Mercator isotropic grid with resolution 1⁄6° at the equator, covering the Atlantic ocean from Drake passage to 30°E and from Antarctica (75°S) to 70°N. The vertical grid has 42 geopotential levels with a grid spacing of 12 m at the surface and 200 m below 1500 m. A horizontal biharmonic operator is used for lateral mixing of tracers and momentum, with a coefficient varying as the third power of the grid spacing. The vertical mixing of mo-

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mentum and tracers is calculated using a second-order closure model (Madec et al. 1998). In case of static instability the vertical mixing coefficients are set to a very large value (⬃1 m2 s⫺1). The bathymetry is calculated from Smith and Sandwell (1997). Buffer zones are defined in the Norwegian Sea, Baffin Bay, and Weddell Sea. There are four open boundaries at Drake Passage, 30°E between Africa and Antarctica, in the Gulf of Cadiz, and at 70°N in the Nordic seas. The treatment of these open boundaries is similar to the 1⁄3° model version described in detail by Treguier et al. (2001). The model is initialized using the seasonal climatology of Reynaud et al. (1998). The model integration starts in the Northern Hemisphere winter season (15 February). Seasonal values of temperature and salinity are interpolated linearly in time to serve as relaxation fields in the buffer zones. The surface forcing fields are derived from the European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis ERA-15. The heat flux is formulated as suggested by Barnier et al. (1995), using their feedback coefficient relaxation to the Reynolds SST field. The evaporation minus precipitation (E ⫺ P) fluxes are formulated as a pseudosalt flux, including the river runoff (Treguier et al. 2001). The model is run during 28 years using a repeated seasonal cycle (average of the years 1979–93 of the ECMWF reanalysis), and 5-day averages of the model variables are stored for diagnostic purposes. The subduction rate diagnostics in the present work are based on these 5-day averaged model fields of the final 20 years. The model resolution (1⁄6°) is not sufficient to fully resolve eddies and boundary currents: for example, the model Gulf Stream separates too far north from the coast, as usual for z-coordinate models at that resolution. Despite such local discrepancies, the eddy kinetic energy (EKE) level of the model is of the same order of magnitude as the one measured by satellite altimetry. The Agulhas eddies in the Cape Basin are well represented at that resolution (Treguier et al. 2003) and the model forced by interannual wind and fluxes reproduces the interannual variability of surface EKE in the North Atlantic (Penduff et al. 2004). The ATL6 model thus appears suitable for a first quantification of the effect of eddies on subduction. Higher-resolution models exist but they do not provide long enough time series for our purpose.

b. Method The idea behind the method presented here is that, by calculating the annual subduction rate as a function of the outcropping density (rather than as a function of the geographically fixed coordinate) following instantaneous isopycnal positions over the annual cycle, it is possible to take all contributing factors into account that transfer water into the interior in any given density range. This method is based on the kinematical approach by integrating the local subduction rates over

the outcrop areas and in density classes. It has also been applied by De Miranda et al. (1999) in their model data analysis of subduction for the South Atlantic Ocean. Hazeleger and Drijfhout (2000) discussed the method but could not apply it because of limitations in their model archive. The instantaneous subduction rate S, that is, the volume flux per unit horizontal area through the moving mixed layer base, is kinematically defined from the volume balance of a fluid column in the mixed layer neglecting the contributions from evaporation and precipitation as (e.g., Cushman-Roisin 1987):

冋

S共x, t兲 ⫽ ⫺

册

⭸h ⫹ ⵱ · 共uh兲 , ⭸t

共1兲

where h is the instantaneous thickness of the mixed layer, u ⬅ (u, ␷) is the horizontal component of the velocity vector in the mixed layer, x ⬅ (x,y) are the horizontal coordinates, ⵱ ⬅ (⳵/⳵x, ⳵/⳵y) is the horizontal gradient operator, and t is time. Subduction is positive downward, hence the negative sign that precedes the summation (1). Physically, the two terms of (1) can be interpreted as follows: the first term represents the subduction caused by mixed layer retreat and the second, subduction resulting from convergence of horizontal transport in the mixed layer. Integrating the local subduction rates from (1) for a control volume between neighboring isopycnals with density ␴ and ␴ ⫹ ␦␴, which outcrop at the surface and intersect the mixed layer base z ⫽ ⫺h(x, t), we obtain the instantaneous volume flux entering (leaving) the seasonal thermocline in the density band between these two isopycnals: S共␴, t兲 ⫽ ⫺

冕冋 AS共␴,t兲

册

⭸h ⫹ ⵱ · 共uh兲 dA, ⭸t

共2兲

where dA is the element of the surface outcrop area defined as AS(␴, t) ⫽ {x : ␴ ⱕ ␴h(x, t) ⬍ ␴ ⫹ ␦␴}, and ␴h is the instantaneous density in the mixed layer assumed as vertically homogeneous. Averaging now over a mixed layer cycle, the net annual subduction rate in the density band between ␴ and ␴ ⫹ ␦␴ is given by S共␴兲 ⫽ ⫺

1 ℑ

冕再冕冋 ℑ

0

AS共␴,t兲

册冎

⭸h ⫹ ⵱ · 共uh兲 dA ⭸t

dt, 共3兲

where ℑ ⫽ 1 yr. Evaluating the right-hand side of (3) precisely requires integrating the terms over the outcrop areas following instantaneous isopycnal positions during the course of the year. Although this calculation is straightforward, it requires detailed information about the seasonal cycle of the mixed layer. Errors in the calculation of the annual-average subduction rate based on (3) may thus result from the lack of data of the mixed layer properties as continuous functions of time. To examine the effects of these errors

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on our results, we will estimate independently the annual subduction rate based on the steady kinematic approach similar to the one used by Marshall et al. (1993) in their climatological data analysis of subduction for the North Atlantic. Marshall et al. have defined the annual subduction rate, Sann, as the total volume per unit horizontal area per year of water crossing the base of the winter mixed layer, z ⫽ ⫺H, and entering the permanent thermocline. Under the assumption that there is no interannual variability and considering z ⫽ ⫺H as a fixed surface given by the maximum depth of the wintertime mixed layer at any position, they show that the annual subduction rate is given by Sann共x兲 ⫽ ⫺共uH · ⵱H ⫹ wH兲,

共4兲

where uH and wH are the annual-mean horizontal and vertical velocity components evaluated at z ⫽ ⫺H, respectively. The first term in (4) represents the horizontal advection of fluid across the sloping mixed layer base, the so-called lateral induction term, and the second, the volume flux due to the vertical motion. To evaluate the annual volume flux across the base of the deepest winter mixed layer in a particular density range, we integrate (4) between two isopycnals ␴ and ␴ ⫹ ␦␴ intersecting H: Sann共␴兲 ⫽ ⫺

冕

AW共␴兲

共uH · ⵱H ⫹ wH兲 dA,

共5兲

where dA represents the element of the surface outcrop area defined as AW(␴) ⫽ [x : ␴ ⱕ ␴H(x) ⬍ ␴ ⫹ ␦␴} and ␴H is the mixed layer density at the time when the mixed layer depth is the largest, that is, H. This approach does not take into account the time dependence in the surface mixed layer (either interannual, seasonal, or eddy time scales). A comparison between the rates from both approaches [(3) and (5)] may allow one to judge whether these temporal changes contribute to the net annual water-mass subduction or not in the context of our model. Note that the annual average of (1) is not equal to (4); this is discussed in detail by Hazeleger and Drijfhout (2000). It is worth noting that the process by which fluid passes from the mixed layer into the main thermocline is not a pure kinematic process but is intimately connected with the thermodynamical forcing at the ocean’s surface (e.g., Nurser and Marshall 1991; Marshall and Marshall 1995). Another method to evaluate the rate of water-mass formation in a particular density range is to evaluate the water-mass transformation rate due to air– sea fluxes alone, making use of the buoyancy equation; a buoyancy loss at the surface leads to a volume flux of water from lower density to higher density. This method, suggested by Walin (1982), was applied by Speer and Tziperman (1992) to estimate the formation rate of water masses from climatological data of the North Atlantic Ocean. Following Speer and Tziperman, the annual mean water-mass transformation rate, F(␴),

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through an isopycnal surface ␴ driven by air–sea fluxes of heat and freshwater is given by F共␴兲 ⫽

1 ℑ

冕冋冕冉 ℑ

0

1 ␦␴

AS共␴,t兲

⫺

␣Q ⫹ ␳␤WS Cp

冊册

dA dt, 共6兲

where ␳ is the sea surface density, ␣ is the thermal expansion coefficient, Q is the heat flux, W is the freshwater flux (evaporation minus precipitation), S is the sea surface salinity, Cp is specific heat, and dA is the element of the surface outcrop area defined as in (2). All of these terms are functions of location and time and the time integral represents the average over ℑ ⫽ 1 yr. Positive (negative) values of F(␴) imply a transformation to greater (lower) densities and the convergence (divergence) of F(␴) over a density range between ␴ and ␴ ⫹ ␦␴ accounts for the overall water formation (destruction) over that range during one year. The main difference between this calculation and our approach is that (6) is based completely on air–sea fluxes without any additional information about the circulation or dynamics within the mixed layer. A comparison between the rates from both approaches may allow one to know which part of the water mass formed at the surface can penetrate into the permanent thermocline. This portion corresponds to the net annual subduction rate calculated from (3).

c. Calculation In the following sections, water-mass formation rates due to subduction (1)–(5) and air–sea density fluxes (6) are estimated using 5-day averaged data of the last 20 years of the 1⁄6°-resolution integration over the North Atlantic basin (extending from 10° to 70°N). The mixed layer thickness h is diagnosed from the density field, which is calculated from the temperature and salinity with the equation of state used in the model integration. To define h we adopt a uniform definition related to a maximum density difference of 0.01 kg m⫺3 between the surface and the last model level within the mixed layer. At every grid point in the basin, the mixed layer tendency, ⳵h/⳵t, is evaluated using a centered differencing scheme. The divergence of the horizontal transport, ⵱ · (uh), is evaluated as in the model integration—that is, from volume fluxes through each face of a grid box surrounding a mass point using centered differencing. The air–sea density flux is evaluated from heat and freshwater fluxes and the relaxation. Annual subduction rates (4) and (5) are evaluated from the annual mean velocity and late-winter mixed layer depth, H, which is diagnosed over the climatological year as the annual maximum depth of the mixed layer at any grid point. The area integrals in (3), (5), and (6) are replaced by a sum of all grid-point terms, multiplied by a boxcar function that is equal to 1 when the outcropping density is in the interval [␴, ␴ ⫹ ⌬␴], and zero elsewhere (see

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FIG. 1. (a) Depth in meters and (b) density in kilograms per cubic meter of the deepest late-winter mixed layer in the North Atlantic model domain.

Speer and Tziperman 1992 for details). A sampling density interval ⌬␴ of 0.1 kg m⫺3 is generally used corresponding to dividing the total outcropping density range from roughly 22 to 28 into 60 equal density classes. The sensitivity of the calculations of S(␴) to the mixed layer depth criteria and density sampling interval is briefly presented in the appendix.

3. End-of-winter mixed layer characteristics In this section the regional distribution of the modeled mixed layer depth and density at the end of the winter, that is, when the mixed layer depth is at its deepest, and estimates of the annual subduction rate using (4) are presented. These results are used here both to test the realism of our numerical simulation by comparison with previous estimates based on climatologies and to compare with instantaneous subduc-

tion rate diagnostics presented in the subsequent sections. Figure 1a shows the depth of the maximum winter mixing over the North Atlantic model domain, which corresponds approximately to the March mixed layer depth at every location. The general pattern of H is not unrealistic when compared with the March mixed layer depth field deduced from the Levitus (1982) climatology (see Fig. 4a of Marshall et al. 1993). Over most of the model subtropical gyre the maximum mixed layer depth is on the order of 100 m and becomes increasingly deeper toward higher latitudes. North of roughly 30°N a tongue of relatively deeper mixed layers extends northeastward across the central North Atlantic, reaching about 300-m depth in the northern Sargasso Sea and 450-m depth south of the British Isles. In the northern part of the model, the mixed layer reaches 500–700-m depth between Scotland and Iceland, and it is particu-

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FIG. 2. Annual subduction rate per unit horizontal area, Sann(x) (m yr–1), across the base of the deepest late-winter mixed layer. Positive values imply a volume flux into the thermocline. A six-point boxcar window smoother has been applied along each axis. Solid lines represent the zero contours of Sann(x).

larly deep off the southeast coast of Greenland and in the Labrador Sea, where it can exceed 2000 m. The regional distribution of the density at the base of the deepest mixed layer (Fig. 1b) show that there is a broad range of outcrops densities in March, from about 23 on the southern flank of the subtropical gyre to 27.8 in the northwestern part of the subpolar gyre, generally oriented from southeast to northwest. The region of very deep mixed layers in the Irminger and Labrador Seas closely matches the area in which the mixed layer density exceeds 27.8 in March. The excessive depths and densities of these model mixed layers may result from the forcing (a repeated seasonal cycle without interannual variability) or from the lack of small scale eddies (not resolved by the model grid). Those eddies have been shown to contribute significantly to the restratification between two winters, thus avoiding excessive deepening (Chanut 2003). The region of unrealistic mixed layers is confined and, as we will see below, does not significantly affect the main subduction zones in the model subtropical gyre of interest here. We now evaluate the annual subduction rate, Sann, as the annual average volume flux per unit horizontal area across the base of the deepest winter mixed layer using (4). The field of Sann is shown in Fig. 2. The subduction rates are positive between 50 and 100 m yr–1 in the central subtropical gyre and near zero or slightly negative along the eastern boundary. A zone of relatively higher subduction rates extends across the basin from approximately 30°N , 70°W to 45°N, 15°W. In this band the subduction rates are typically on the order of 100–

200 m yr–1 but reach 250–300 m yr–1 in localized areas. This band corresponds to the main area of formation of mode waters in the North Atlantic subtropical gyre, the density of which ranges roughly from 26 to 27.2. The region of winter deep mixed layer (⬃300 m thick) and weak meridional density gradient, from roughly 26.2 to 26.4 located south and east of the Gulf Stream, corresponds to the site of formation of the subtropical mode water, which is known for its large volume and uniform properties near 18°C and 36.5 psu (Worthington 1959; McCartney 1982). A second region of formation, associated with the southward turn of a part of the North Atlantic Current, lies in the northeastern part of the gyre between 40° and 45°N off the coast of Europe. Outcrop densities here are between 26.9 and 27.1 and are thus in the range of subpolar mode waters (McCartney and Talley 1982). In comparison, the climatological estimates of Marshall et al. (1993; see their Fig. 5) found subduction rates of the order of 50–100 m yr–1 over most of the subtropical gyre. The subduction band found here has rates somewhat higher, and possibly more realistically, reaches farther to the northeast, stretching across the whole of the subtropical gyre. To the north of this band, modeled subduction rates have smaller-scale features and very large positive and negative values exceeding ⫾1000 m yr–1 in many places. This pattern is not seen in the climatological analysis of Marshall et al. that showed widespread entrainment, typically –100 and –300 m yr–1, over the southern part of the subpolar gyre. The rather noisy pattern found here results primarily from the high resolution in space of the

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FIG. 3. Instantaneous subduction rates, S(␴, t), in the density range 24.1–27.2 using a density bin of ⌬␴ ⫽ 0.1. In each panel, the horizontal axis corresponds to time in months. In the vertical axis, shown in units of Sverdrup, S(␴, t) ⬎ 0 indicates volume flux out of the mixed layer and S(␴, t) ⬍ 0 into the mixed layer.

OGCM simulation; however, an independent analysis of the first 10 years and the last 10 years of the simulation resulted in virtually the same patterns and magnitudes for Sann.

4. The role of the mixed layer time dependence We now examine how the time-dependent behavior of the mixed layer influences subduction rate diagnostics over the modeled North Atlantic. We are interested in the annual rate of water-mass subduction evaluated by “online” integration over the year using (3), which we called S(␴). To improve our understanding of S(␴), we first consider examples of its seasonal variation.

a. The seasonal cycle The instantaneous detrainment/entrainment rate across the base of the mixed layer, S(␴, t) [see (2)], is shown in Fig. 3 as a function of time at selected density classes (⌬␴ ⫽ 0.1) in the range 24.1–27.2. As the outcrop densities in the subtropical basin are oriented mainly from southeast to northwest, these examples are

thought to mimic the southern (Figs. 3a–c), central (Figs. 3d–f), and northeastern (Figs. 3g–i) parts of the subtropical gyre. In the subtropical basin, the entrainment [S(␴, t) ⬍ 0] and detrainment [S(␴, t) ⬎ 0] of the mixed layer are primarily controlled by the annual cycle of the mixed layer depth (Fig. 4) because of the seasonal surface buoyancy forcing. These processes take place alternately. During the winter season, the cooling of surface water results in an entrainment of the mixed layer. Near the end of winter, entrainment slows down and is overtaken by subduction when the surface buoyancy flux changes sign at the beginning of spring. The distribution of depths at which water of a given density is being subducted from (or entrained in to) the mixed layer varies as the isopycnal locations migrate over the year. For isopycnals in the density range 23– 25.6, outcropping from roughly 10° to 25°N in March (Fig. 1b), the wintertime mixed layers are shallow (⬃50–75 m) of comparable depth as the summer mixed layers (see solid lines in Fig. 4). In this range, subduction O(2–10 Sv) (Sv ⬅ 106 m3 s⫺1) occurs typically from March to June; however, its efficiency is low because

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b. Net annual subduction rates 1) REGIONAL

FIG. 4. Mixed layer thickness (m) as a function of time at selected density classes (⌬␴ ⫽ 0.1) in the range 24.1–27.1.

entrainment takes place during the remaining twothirds of the year and compensates the seasonal detrainment as the mixed layer retreats in the spring and early summer (Figs. 3a–c). For denser isopycnals (␴ ⬎ 26) outcropping north of roughly 30°N in March, the late-winter mixed layers are substantially deeper (⬎150 m) than the summer mixed layers (⬍20 m: see dashed lines in Fig. 4) because of strong atmospheric cooling in the northern part of the model. In this case, water detrained as the mixed layer retreats rapidly in spring can penetrate sufficiently deeply by the time the mixed layer deepens again from late autumn to midwinter and, as a result, enters the permanent thermocline. This is illustrated clearly in Fig. 3 by the maximum instantaneous detrainment rates in April O(100 Sv) in the class range 26.3–27.1, whereas entrainment rates are smaller, O(20–30 Sv) from December to March (Figs. 3e–h). In this range detrainment of the mixed layer starts generally in March after the mixed layer has reached its annual deepest value and is maximal around the end of April. From these examples, the period of subduction lasts over 2 months of the year, typically from March to May, in the central subtropical gyre (Figs. 3e,f) and it becomes somewhat shorter toward the northeastern part of the gyre (Figs. 3g,h) where the seasonal mixed layer cycle is more pronounced. This is generally consistent with the effective subduction period diagnosed from climatological data by Marshall et al. (1993), about 1–2 months, over most of the central subtropical gyre. These results show that the seasonal subduction in the modeled subtropical gyre is relatively important for isopycnals in the density range 26–27.2 outcropping in the central and northeastern subtropical gyre where the winter mixed layer is thick (ⱖ150 m), whereas it is relatively less important than the seasonal entrainment for lighter isopycnals (␴ ⬍ 26), outcropping in the southern part of the gyre where the winter mixed layer is shallow (⬍100 m). In the class range 26–27.2, the subduction is effected rapidly on the same time scales as in the observations, and the characteristics of the modeled subducted waters are biased to late-winter and early spring properties, reflecting Stommel’s (1979) so-called mixed layer demon.

DISTRIBUTION

We now investigate how the annually averaged subduction evaluated from the instantaneous kinematic approach (3) is spatially distributed in our model. Figure 5 shows the net annual subduction rate, S(␴), per unit horizontal area in the density range 26–27.2 using an outcrop density window, ⌬␴ ⫽ 0.2. This figure gives an integrated view on the regional distribution of the net annual effective subduction/entrainment process as the isopycnal outcrops in the mixed layer move over the year. Note that the outcrops in late winter are deflected to the southwest in the subtropical gyre (Fig. 1b) but in spring they move rapidly to the northeast responding to the seasonal cycle in the forcing. In the density classes between 26 and 27.2, the subduction rates are typically of the order of 100–250 m yr–1 and occur mostly across the central North Atlantic roughly from 30°N, 70°W in the Sargasso Sea to 52°N, 10°W in the northeastern part of the subtropical gyre. As expected, in all classes net subduction takes place exclusively from the late-winter outcrop windows, when the mixed layer rapidly retreats from its deepest value in spring (H ⬎ 200 m; refer to thin solid line in Fig. 5) and the outcrops reach their southern most limit (denoted by heavy solid lines in Fig. 5). The reentrainment process (upwelling through the base of the mixed layer) occurs mostly just to the north of the subduction zone when the mixed layer steadily deepens in late autumn and early winter as the outcrops return to the winter positions. The broad pattern and magnitudes of S(␴) resemble those of Sann obtained using the simple definition of (4) and shown in Fig. 2, but the differences are important and we consider two examples to illustrate why. Typical values of S(␴) are O(100–200 m yr–1) in the region of the Gulf Stream extension within the class range 26– 26.6 (Figs. 5a–c), consistent to those of Sann in a similar area. However, as the distribution of winter mixed layer depths varies as the subduction locations migrate during the spring, S(␴) exists over a larger area, while Sann is confined along the southern flank of the area of deep winter mixing (see Fig. 2). In the northeastern part of the subtropical gyre, S(␴) is also widespreaded and reaches a maximum, about 250 m yr–1, just south and east of the North Atlantic current (from 40° to 52°N, 10° to 20°W) within the class range 27–27.2 (Fig. 5f). For these geographical locations, the patterns of Sann are not single signed and exist over a smaller area. Our estimates from the instantaneous kinematic approach (3) show thus that the band of net subduction rates O(100–250 m yr–1) stretching across the subtropical gyre exists over a larger area and so, in an integrated sense, is very important. They also show, possibly more realistically, maximum subduction rates O(250 m yr–1) in the northeast corner of the subtropical gyre near the

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FIG. 5. Net annual subduction rate per unit horizontal area, S(␴) (m yr⫺1), in the density range 26–27.2 using a density bin of ⌬␴ ⫽ 0.2. In each panel, blank values denote where the annual-averaged volume fluxes do not exist in that density class. Heavy solid lines indicate the positions of the late-winter outcrops. Thin solid line indicates the position at which the mixed layer is 200 m deep in late winter.

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region where 11°–12°C subpolar mode waters are subducted by the southward recirculation of a part of the North Atlantic Current (McCartney and Talley 1982).

2) BASIN-INTEGRATED

SUBDUCTION

One way of quantitatively estimating the difference between instantaneous and steady diagnostics is to compare basin-integrated subduction rates from both approaches with respect to the outcrop densities, that is, S(␴) versus Sann(␴)[(3) and (5)]. Figure 6 shows the net annual subduction rate S(␴) over the density range between 22 and 27.4, using a density bin of ⌬␴ ⫽ 0.1. It shows how much water leaves the moving mixed layer and enters the interior (or vice versa) as a function of density during one year. Of the two components that make up S( ␴ ), the contribution from the timedependent term (⳵h/⳵t) is clearly dominant. This may be attributed to a correlation between the local subduction rates caused by mixed layer retreat and the seasonal varying outcrop areas over which the subduction takes place. In the density range of 22–25.9, S(␴) shows sligthly negative values with a minimum of –1 Sv at ␴ ⫽ 25.3– 25.4. This implies a net flux of mass from the interior to the mixed layer in these density ranges, integrated across the basin, despite the fact that in the interior of the subtropical gyre the divergence of the horizontal transport in the mixed layer (solid line in Fig. 6) is directed downward over this latitude range (at roughly from 10° to 25°N in March, Fig. 1b). This because in most of the subtropical basin the water entering the mixed layer during the entrainment phase (in autumn and winter) is the water that was detrained to the seasonal thermocline at locations where the mixed layer is shallow (⬍100 m), not from the permanent thermocline (see Figs. 3a–c). For densities greater than 26, S(␴) is positive with two maxima associated with the production of the two main mode waters in the North Atlantic. The first peak around 26.2–26.6 corresponds to the formation of the subtropical mode water (STMW) of the Sargasso Sea. The total formation rate for STMW if defined between 26.2 and 26.6 is 12.2 Sv and is dictated by the local change in the mixed layer depth weighted by the surface outcrop area (dashed line in Fig. 6), with a very small positive contribution from the divergence of the horizontal transport in the mixed layer. This rate agrees well with the value of 14.1 Sv by Woods and Barkmann (1986), who estimated the annual rate of STMW formation using a Lagrangian mixed layer model forced by surface heat flux and Ekman pumping. Woods and Barkmann emphasized the importance of the dynamical role of the mixed layer, especially the horizontal gradient of mixed layer depth and its seasonal cycle. The fact that the time-dependent term is here clearly dominant confirms their conclusion and may be attributed to the Lagrangian correlation of seasonally vary-

FIG. 6. Annual mean subduction rate, S(␴) (Sv), over the density range 22.1–27.4 using a density bin of ⌬␴ ⫽ 0.1. Gray line with circles: net annual subduction rate; solid line: convergence of the horizontal transport in the mixed layer; dashed line: mixed layer depth tendency. A positive value of S(␴) in a particular density class implies a volume flux out the moving mixed layer base in that class.

ing mixed layer depth and density. The second subduction peak in Fig. 6 of about 2 Sv at ␴ ⫽ 27.3 corresponds to the formation of subpolar mode water (SPMW), which is found around the subpolar gyre, from a density of about 27 east of the North Atlantic Current up to the density of the Labrador seawater near 27.8 (McCartney and Talley 1982). The formation rate for SPMW, about 3.5 Sv, is underestimated in this study because the upper cutoff on the S(␴) curve was chosen to ␴ ⫽ 27.4, in order to avoid regions where the winter mixed layer depth is particularly deep in the northern part of the model (see Fig. 1a). It is interesting to note that although annual subduction rates O(150–250 m yr⫺1) are found locally within the class range 26.8–27.2 (Figs. 5e– f), the overall subduction rate integrated across the basin is close to zero in this range. This is due to the fact that very high entrainment rates exceeding –300 m yr⫺1 are found north of the model subduction band in the northeast Atlantic (Fig. 5f). This suggests that all of the water subducted in the northeastern part of the gyre between roughly 40° and 50°N from the outcrop window 26.8–27.2 returns to the surface layer as the North Atlantic Current flows northward. We consider now the amount of water subducted from the mixed layer into the permanent thermocline with late-winter water properties, which we called Sann(␴) (Fig. 7). This is obtained by summing the annual volume flux across the base of the deepest mixed layer over the end-of-winter outcrop areas using (5). In the density range of 22–25.8, Sann(␴) is near zero and has a pronounced minimum of –2.4 Sv at ␴ ⫽ 25.8. This minimum corresponds to very high entrainment rates with values greater than –500 m yr⫺1 found within the Gulf Stream (see Fig. 2) and is primarily dictated by the lateral induction term (solid line in Fig. 7). Apparently, this large contribution of the lateral induction flux is due to the strong meridional gradient of the late-winter mixed layer base and strong northward flow near the western boundary current. In the density range of 26 to 26.8, Sann(␴) is positive with two peaks: one of about 1 Sv at ␴ ⫽ 26.2 and the other of about 1.9 Sv at ␴ ⫽ 26.4. The formation rate for subtropical mode waters (class

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FIG. 7. Annual mean subduction rate, Sann(␴) (Sv), over the density range 22.1–27.4 using a density bin of ⌬␴ ⫽ 0.1. Gray solid line with circles: net annual subduction rate; solid line: lateral induction term; dashed line: vertical pumping at the base of the deepest mixed layer. A positive value of Sann(␴) in a particular density class implies a volume flux across the base of the deepest mixed layer in that class.

26.2–26.6) estimated from Sann(␴) is about 4.2 Sv with 1.2 Sv due to the Ekman pumping at the base of the mixed layer (dashed line in Fig. 7) and 3 Sv due to the lateral induction flux. This rate agrees well with the value of 3.2 Sv obtained by Qiu and Huang (1995; see their Table 3), who estimated the annual rate of subtropical mode water formation by Lagrangian integrations using a wintertime mixed layer density and depth and an annual flow field derived from Levitus climatology. For densities greater than 26.8, Sann(␴) shows a minimum of about –1.7 Sv at ␴ ⫽ 27.1 and a maximum of 2.3 Sv at ␴ ⫽ 27.3. Notice that, although the Ekman pumping is directed downward and thus makes a positive contribution to subduction, the negative values of Sann(␴) found in the class 26.9–27.2 imply a net transfer of fluid from the interior to the mixed layer in this range. This result is consistent with the obduction rate peaks between 26.8 and 27.2 found by Qiu and Huang (1995) using the Levitus climatology.

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Last, it is important to note that in the density range where subduction occurs mostly in the model subtropical gyre, roughly from 26 to 27.2, S(␴) and Sann(␴) show generally the same sign with a maxima at roughly the same density ␴ ⫽ 26.4. However, the overall subduction implied by the instantaneous method is much higher in this density range: about 12 Sv are estimated from S(␴) as compared with only 4 Sv from Sann(␴). We will return to this point later in section 5.

c. The effect of transient eddies The regional distribution of S(␴) from Fig. 5 shows that the main subduction zones in the model subtropical gyre are closely correlated with frontal regions where the existence of mesoscale eddies is likely. The eddy effect is difficult to isolate because eddies have time scales close to seasonal. Our model is forced with a repeated seasonal cycle without interannual variability, which allows us to define a mean seasonal cycle by averaging the model results monthly and over 15 years. The contribution of eddies to subduction is then defined as the difference between S(␴) calculated from the 5-day averages and the rate Sm(␴) deduced from the climatological monthly averages. The contribution of the eddies to the seasonal subduction when globally averaged over the model domain is illustrated for the different density classes in Fig. 8. As expected, the largest impact of eddies can be found in the density range between 26 and 27.2, where very large instantaneous subduction rates O(100 Sv) were estimated over a period between March and May (Fig. 3). In this density range the eddies can increase the local subduction rates by about 20–30 Sv in April, but they do not alter the effective period of subduction,

FIG. 8. Monthly differences (Sv) between total S(␴) and monthly mean Sm(␴) subduction rates in the density range 22.1–27.4 using a density bin of ⌬␴ ⫽ 0.1.

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which remains about 1–2 months, after the end of winter. We now investigate the contribution of the eddies to the annual mean subduction rate. Figure 9 shows the horizontal distribution of the total subduction S(␴), the annual averaged monthly subduction Sm(␴), and their difference (eddy subduction), per unit horizontal area integrated over the outcrop windows 26.2–26.6 (Figs. 9a–c) and 26.8–27.2 (Figs. 9d–f). In the class 26.2–26.6, which spans the densities of subtropical mode waters, eddies tend to enhance subduction within the region of winter deep mixed layer southeast of the Gulf Stream. Local eddy rates O(100–200 m yr⫺1) are found along 35°N (Fig. 9c), and thus the total subduction is dominated here by the eddy contribution. Furthermore, in the region where most reentrainment takes place on the northeast side of the wintertime outcrop area, the eddy contribution counteracts the subduction associated with the mean flow (Fig. 9b). In the class 26.8–27.2, which spans the densities of subpolar mode waters in the northeastern part of the subtropical gyre, eddies tend to enhance subduction along the southern edge of the area of deep winter mixed layer (⬃400 m thick) east of the North Atlantic Current (see Fig. 1a). Here eddy rates up to 300 m yr⫺1 are found between 40° and 45°N, while to the north of this band subduction rates are reduced by the eddies (Fig. 9f). So the eddy contribution counteracts the subduction/entrainment by the mean flow (Fig. 9e) at most places of the wintertime outcrop window in this class range. Such cancellation between the eddy effect and the mean flow on subduction has been suggested by Marshall (1997) in the case of the Antarctic Circumpolar Current. The overall contribution of the eddies to S(␴) can be made clear by comparing basin-integrated subduction rates (Fig. 10). The difference between annual mean subduction rates estimated from the 5-day and monthly averages is O(2 Sv) or smaller in the class range 26.2– 27.2 and negligible for all other density ranges. This suggests that, although the subduction process can be modified locally by the presence of fronts and eddies, the overall eddy contribution to S(␴), when averaged over many seasonal cycles and integrated across the basin, is much smaller than the contribution by the mean flow in all density classes ␴ ⬍ 26.8. Our number of 2 Sv is close to the one estimated by Hazeleger and Drijfhout (2000), using an Eulerian kinematic approach and an idealized model of the Gulf Stream region. For denser isopycnals in the class range 26.8–27.2, the eddy contribution counteracts the subduction by the mean flow, and thus provides zero subduction in that range. Notice that the amplitude of the eddy contributions O(100–300 m yr⫺1), as obtained from the spatial patterns in Figs. 9c and 9f, is comparable to the estimates by Follows and Marshall (1994) and Spall (1995), who found eddy subduction rates O(150 m yr⫺1) using scaling arguments and a channel model of an idealized jet, respectively.

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5. Water-mass transformations To improve our understanding of the subduction process in our model, it is instructive to consider the net annual rate of formation of surface water induced by air–sea fluxes. A complete description of this process is not given here, but a few remarks are useful at this point to complement the previous sections. The spatial distribution of the annual averaged surface density flux applied in the model is shown in Fig. 11. This surface flux is the sum of the climatological fluxes and the relaxation (Barnier et al. 1995). Annual mean water-mass formation sites in the model subtropical gyre are seen along the western boundary current and in the Gulf Stream, where density fluxes of more than 14 ⫻ 10⫺6 kg m⫺2 s⫺1 are found near 38°N, 70°W. The annual mean density loss is typically ⫺2 to ⫺4 (⫻ 10⫺6) kg m⫺2 s⫺1 and is confined to near-coast regions along the eastern boundary. We now apply Speer and Tziperman’s (1992) technique of diagnosing annual mean water-mass transformation rates directly from the surface density forcing. Figure 12a shows the net annual transformation rate F(␴) over the density range between 22 and 28, calculated using (6) with a surface density bin of ⌬␴ ⫽ 0.1. The general structure of F(␴), as inferred from Fig. 12a, is qualitatively consistent to that derived by Speer and Tziperman (1992) using the climatology of Isemer and Hasse (1987) and a density bin of 0.1. Except at higher density ranges, where the peak found here around 27.3 is not seen in climatologies because of the limitation of the Isemer and Hasse dataset. In the density range 22– 24, F(␴) shows negative values with a minimum of about –5 Sv at ␴ ⫽ 23.3. The surface forcing tends to create lighter water in this range, which is consistent with climatologies. For densities greater than 24, F(␴) is positive with two maxima: one of about 19 Sv at ␴ ⫽ 26.1 and the other of roughly 21 Sv around ␴ ⫽ 27.3. The amount of water mass formed (destroyed) by air– sea interaction within a particular density interval is given by the convergence (divergence) of F(␴) over that interval. In the density range between 26.1 and 26.8, where the model forms subtropical mode water, the surface forcing leads to an overall formation rate of about 11 Sv, which is very close to our estimate of about 12.2 Sv from interior data using (3) and also agrees well with the value of 14 Sv obtained by Speer and Tziperman (1992) in this range. Seasonal transformation curves depicted in Fig. 12b reveal that almost all the annual production in the density range 26.1–26.8 occurs during the month of January, while most of subtropical mode water formation occurs one month later. This is shown by the maximum transformation rate at ␴ ⫽ 26.4 in mid-February of about 120 Sv, while the transformation rates at ␴ ⫽ 26.1 and ␴ ⫽ 26.8 are both smaller and almost equal at this time. This suggests that the water-mass subduction in the range 26.1–26.8 takes some time to respond to the

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FIG. 9. (a), (d) Annual subduction rate S(␴) from 5-day means; (b), (e) annual subduction rate Sm(␴) from monthly means; and (c), (f) eddy subduction rate in the class ranges 26.2–26.6 and 26.8–27.2. All values are expressed in meters per year. Blank values denote where the annual-averaged volume fluxes do not exist in a particular density class. Heavy solid lines indicate the positions of the late-winter outcrops. Thin solid line indicates the position at which the mixed layer is 200 m deep in late winter.

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FIG. 10. Annual mean subduction rates (Sv) over the density range 22.1–27.4 using a density bin of ⌬␴ ⫽ 0.1. Total S(␴) (gray line with circles) is estimated using 5-day averaged fields. Mean Sm(␴) (black dashed line with diamonds) is estimated using monthly averaged fields.

surface forcing since detrainment of the mixed layer starts in March and therefore lags the transformation process by about 1 month as shown in Fig. 3 in this density range. It is now instructive to compare these results with those obtained by Marshall et al. (1999) using a noneddy-resolving model of the North Atlantic with parameterized bolus velocity and seasonally varying surface forcing. Marshall et al. using the Speer and Tziperman’s (1992) approach with a density bin of 0.2 derived a water-mass formation rate of 12 Sv over the density range between 26 and 27. In addition, they gave a water-mass formation rate due to subduction of about 2 Sv over this range, using a control volume bounded by monthly mean isopycnals that intersect the base of the winter mixed layer from below. According to Marshall et al., mixing processes at the base of the winter mixed layer and in the seasonal thermocline were responsible for the difference between both estimates in their model. In contrast, from the results presented here, the gradient of F(␴) seems a good estimator of S(␴) in the class range 26–27, which closely matches the area of

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deep winter mixed layers in the central North Atlantic (Fig. 1a). This result suggests that diffusive processes make a minor contribution in the surface mixed layer in our numerical model. It is interesting to note at this point that, from the orders of magnitude provided in section 4d(2), Sann(␴) is a factor 3 lower than S(␴) in density range between 26 and 27, but is close to the estimate from Marshall et al. (1999) based on a low-resolution model. This difference between S(␴) and Sann(␴) is not negligible and may be due to the sampling of the local mixed layer densities where the late-winter density is equal to ␴ rather than where the instantaneous density is equal to ␴. To understand this effect, Fig. 13 shows the change in the mixed layer density during the period when subduction occurs most—that is, between March and April when the mixed layer rapidly retreats. The main feature of interest here is the region of increased density (by typically 0.05–0.1) that extends across the basin and overlies the subduction zone (refer to the position of the 200-m late-winter mixed layer depth contour, which lies close to the subduction zone in Figs. 2 and 5). The region of increased mixed layer density in the Gulf Stream extension (between 30° and 40°N) is very closely correlated with the region where the atmospheric cooling by the model air–sea fluxes is strongest (Fig. 11). It is possible, therefore, that the total subduction estimated from Sann(␴) using (5), roughly 4 Sv in the class range 26–27, is an underestimate of the amount of water subducted in this range, owing to the use of the end-of-winter data that have lower surface density than the actual density during water-mass formation events. On the contrary, the result of about 12 Sv estimated from S(␴) using the instantaneous method (3) reflects the overall water-mass formation driven directly by the air–sea fluxes in that density range. This

FIG. 11. Annual mean surface density flux (10 –6 kg m⫺2 s⫺1) estimated from the air–sea fluxes of heat and freshwater surface applied in the model.

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FIG. 12. (a) Annual mean transformation rate, F(␴) (Sv), over the density range 22–28 using a density bin of ⌬␴ ⫽ 0.1. Positive (negative) values of F(␴) imply a transformation to greater (lower) densities. (b) Instantaneous transformation rate, F(␴), at selected densities: ␴ ⫽ 26.1 (black solid line), ␴ ⫽ 26.4 (gray solid line), and ␴ ⫽ 26.8 (dashed line).

suggests that the major limitation of the definition based on (5) is the assumption of time independence. This because it neglects the seasonal cycle of the mixed layer and thus water-mass conversion of the seasonal thermocline, which is available for further conversion across the base of the mixed layer into the modeled subtropical mode water at later times. Subduction induced by mesoscale eddies cannot be invoked to explain the differences between S(␴) and Sann(␴) in our model because, when the effect of eddies is filtered out by using climatological monthly mean fields [Sm(␴)], the subduction rate in the range 26–27 remains close to the total S(␴) (see Fig. 10).

6. Conclusions The main purpose of the present work has been to clarify and quantify the role of the mixed layer time variability on kinematic subduction rate diagnostics from an eddy-resolving-resolution OGCM of the North

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Atlantic. Diagnostics of Sann (4) based on data of the last 20 years of the 1⁄6° numerical integration show that many important large-scale features of the subduction process across the base of the winter (deepest) mixed layer in the North Atlantic have been successfully simulated (Fig. 2). For instance, Sann O(50–100 m yr⫺1) over most of the central subtropical gyre, a narrow band of Sann O(150–250 m yr⫺1) stretching from southwest to northeast along the southern flank of the area of deep winter mixing, and strongly negative values of Sann ⬎ ⫺500 m yr⫺1 within the Gulf Stream are found. This suggests that this dataset can be used with confidence and the results of our diagnostics can be applied to the real ocean. Furthermore, diagnostics using the velocity and mixed layer cycle, as revealed by 5-day temporal resolution model fields, indicate a subduction period of about 1–2 months between March and May over the model subtropical gyre, suggesting that fluid in the class range (26–27.2) is only transferred from the mixed layer into the main thermocline after the end of the winter (Fig. 3). This is consistent with both climatological estimates of Marshall et al. (1993) and model data estimates of Williams et al. (1995) that found a period of about 1–2 months for most of the North Atlantic subtropical basin. Although these results generally support Stommel’s (1979) hypothesis, our diagnostics based on the instantaneous kinematic approach lead to a correction to steady diagnostics deduced from the annual mean velocity and late-winter mixed layer properties alone, and hence affects the estimate, from that diagnostics, of the annual rate at which waters are subducted from the mixed layer into the permanent thermocline. The regional distribution of S(␴) (Fig. 5) indicates that the band of net subduction rates O(150–250 m yr ⫺1 ) stretching across the whole subtropical gyre has a larger extent and so in an integrated sense is very important. Inspection of the difference between basin-integrated subduction rates, that is, S(␴) versus Sann(␴) (Figs. 6 and 7), indicates a mismatch of at least 10 Sv in the range 26–27.2, which spans the densities of mode waters in the North Atlantic subtropical gyre. For densities between 26.2 and 26.6, where subduction occurs south

FIG. 13. Changes in the mixed layer density deduced by subtracting values for Mar from those of Apr. The solid line is the position at which the mixed layer is 200 m deep in late winter.

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and east of the Gulf Stream, S(␴) is about 12 Sv, consistent with Woods and Barkmann (1986), while Sann(␴) is only about 4 Sv, which is consistent with previous estimates of the large-scale subduction rate based on climatologies (e.g., Marshall et al. 1993) and coarseresolution models (e.g., Marshall et al. 1999). For densities between 26.8 and 27.2, where subduction occurs in the northeastern part of the gyre east of the North Atlantic Current, S(␴) is close to zero while Sann(␴) is negative (upwelling through the base of the winter mixed layer) of about –2 Sv. If we assume that these differences between S(␴) and Sann(␴) are real and not a consequence of uncertainties in our calculations (which include the criterion for the mixed layer depth and the choice of the sampling density interval; see Figs. A1 and A2), the main possibility to explain this discrepancy seems to be that the temporal changes of the mixed layer make important contributions to subduction. Implicitly contributing factors for S(␴), which are not considered in Sann(␴), arise from the seasonally varying surface buoyancy forcing and from mesoscale transient eddies. We provide here the first direct estimate of eddy subduction at the basin scale. In the class range 26.2–26.6, the overall eddy contribution to S(␴) is O(2 Sv) (Fig. 10). This is small in comparison with the total formation rate inferred by the air–sea transformation rate F(␴) (Fig. 12a), which is O(11 Sv) in this density range, consistent with S(␴). Downstream in the class range 26.8–27.2, the total amplitude of the eddies is also O(2 Sv); however, the most important result is that here the eddy contribution counteracts the subduction by the mean flow and thus provides zero subduction in this range. These results demonstrate that the annual rate of water-mass subduction in the modeled subtropical gyre depends on both the mixed layer cycle due to the seasonal air–sea forcing and mesoscale eddies, with the seasonal cycle dominating subtropical mode water, and eddies counteracting subduction from the mean flow for the subpolar mode water. The fact that in this study, the air–sea transformation F(␴)—that converge into the range where the model forms subtropical mode water—is a good estimator of S(␴), suggests that diffusion processes make a relatively minor contribution to watermass subduction in the present model. This investigation shows clearly that the process of subduction deserves timely consideration as the OGCMs become increasingly refined. In particular, these results demonstrate that subduction rate diagnostics based on the instantaneous kinematic method provide a much more complete picture of the subduction process than steady diagnostics based on late-winter mixed layer properties. This is because they implicitly account for water-mass conversion of the seasonal thermocline and mesoscale eddies. Obviously, the success of this approach depends on the accuracy of the velocity fields and mixed layer cycle predicted by the OGCMs. The results reported here using an eddy-

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resolving OGCM indicate that it is essential to describe the upper boundary layer of the numerical ocean models adequately in both time and space. On the contrary, increasing the spatial resolution alone when the temporal resolution is low will not improve the diagnostics substantially. In the future, this diagnostic study needs to be repeated at different temporal and grid resolutions of the model fields to refine these conclusions. Acknowledgments. We are grateful to the two anonymous reviewers for comments toward clarifying the text. MVDC acknowledges EPSHOM (Etablissement Principal du Service Hydrographique et Océanographique de la Marine) and IFREMER for support, and LPO (Laboratoire de Physique des Océans) for hosting during this work. The computations were performed at the IDRIS (Institut du Développement et des Ressources en Informatique Scientifique). Discussions with Bill Dewar as MVDC was at the Florida State University were appreciated.

APPENDIX Sensitivity of S(␴) to the Mixed Layer Depth Criterion and the Density Sampling Interval Our calculations of S(␴), Sann(␴), and F(␴) are based on a number of assumptions that include the criteria for the mixed layer thickness, h, and the choice of the density sampling interval, ⌬␴. The results discussed in sections 3, 4, and 5 are obtained for h defined by the depth at which the density increases by 0.01 kg m⫺3 over the surface value and a density bin ⌬␴ of 0.1. Here we calculate the annual mean subduction rate S(␴) for year 1990 using a different mixed layer base density and three choices of ⌬␴. The results are presented in Figs. A1 and A2. A density increase within the range 0.01–0.05 kg m⫺3 over the surface value is typically used as criterion when calculating the mixed layer thickness from model data (e.g., Williams et al. 1995; De Miranda et al. 1999). The close agreement between the subduction curves using 0.01 and 0.05 kg m⫺3 (Fig. A1) indicates that the

FIG. A1. Annual mean subduction rate, S(␴) (Sv), over the density range 22.1–27.4 (⌬␴ ⫽ 0.1), using two different definitions for the mixed layer depth: Circles are estimates for a density increase of 0.01 kg m⫺3 over the surface value (as, e.g., De Miranda et al. 1999). Asterisks are estimates for a density increase of 0.05 kg m⫺3 (as, e.g., Williams et al. 1995).

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FIG. A2. Annual mean subduction rate, S(␴) (Sv), over the density range 22.1–27.4 using three different sampling intervals for the mixed layer density: ⌬␴ ⫽ 0.05 (small circles), ⌬␴ ⫽ 0.1 (asterisks), and ⌬␴ ⫽ 0.2 (diamonds).

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