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  • View in gallery

    A schematic of the model setup with a basin north of 42°S and a zonal channel to the south with a 1908-m-deep bottom ridge across the channel. The forcing fields are a surface relaxation of temperature and a zonal wind.

  • View in gallery

    (a) Zonal wind stress. (b) Relaxation rate for surface temperature. (c) Time- and zonal-mean sea surface temperature (solid line) and the target temperature θ* for surface relaxation (dashed line). (d) Time-mean surface heat fluxes, with contour interval 20 W m−2. Shading denotes surface heat loss.

  • View in gallery

    The 6-hourly-mean fields. (a) Sea surface temperature, with contour interval 4°C for θ ≥ 4° and 2°C for θ ≤ 4°C. (b) Mixed layer depth Hm, with shading interval 100 m for Hm ≥ 120 m and 30 m for Hm ≤ 120 m. (c),(d) The region marked by the dashed-line boxes in (a),(b). (c) Sea surface temperature, with shading interval 0.6°C and contour interval 0.3°C. (d) Mixed layer depth, with same shading interval as in (b). (e),(f) The region south of 48°S. (e) Sea surface temperature, with the same contour interval as in (c). (f) Surface heat fluxes, with shading divides at −5, −15, and −30 W m−2 and a contour interval of 5 W m−2.

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    The 4-yr time-mean fields. (a) Sea surface height, with contour interval of 0.2 m. (b) Sea surface temperature, with contour interval of 2°C. (c) Zonal-mean temperature (solid lines, with contour intervals ranging from 0.5° to 2°C) and zonal-mean mixed layer depth (dashed line). (d) Mixed layer depth, with contour interval of 50 m.

  • View in gallery

    The diagnosed terms corresponding to (1) integrated from 48° to 54°S, with H = 300 m. (a) The advective term on the lhs, (thick solid line); the combined two eddy terms on the rhs, (dashed line); the combined noneddy terms on the rhs, (dotted line); and the remaining term (thin solid line). (b) The mean vertical shear term, (thick solid line); the Ekman contribution, (thin dashed line); the surface fluxes term, (dotted line); and the vertical advection term, (thin solid line).

  • View in gallery

    Here, 6-hourly-mean data over 1 yr are used unless otherwise specified. (a) Subduction streamfunctions: net subduction net (thick solid line), Eulerian-mean subduction eu (dashed line) and eddy subduction eddy (dotted line), and Ekman transport (thin solid line). (b) The mass budget within the mixed layer: the net subduction (thick solid line), diapycnal flow from surface fluxes (thick dashed line), diapycnal flow from vertical diffusion (dotted line), the volume changes over the 1-yr time period (thin dashed line), and the residual of the mass budget (thin solid line). The plus symbols are the difference between the diapycnal transport derived from 6-hourly surface fluxes and from 1-yr-mean surface fluxes. (c) Net subduction using 6-hourly-mean data [solid line, same as the thick solid line in (a)] and using daily-mean data (dashed line). (d) Vertical diffusion from 1 yr of 6-hourly-mean data [thick solid line, same as the dotted line in (b)], from 1 yr of daily-mean data (dashed line), and from 4 months of 2-hourly-mean data (thin solid line).

  • View in gallery

    Schematics illustrate the relationship between subduction and isothermal transport (note that dx is omitted in the schematics). (a) Subduction Sdxdy between isotherms θ1 and θ2 across the base of mixed layer, the meridional isothermal transport Vdxdz below the mixed layer, and the vertical isothermal transport Wdxdy across the constant z intersecting mixed layer base. (b) If θ1 moves to at a different time, then the difference between Sdxdy and Vdxdz includes the diapycnal flow in the mixed layer, which could be large. (c) For the same motion of θ1, the difference between Sdxdy and Wdxdy is the diapycnal flow below the mixed layer, which could remain small.

  • View in gallery

    Isothermal transport streamfunctions (contour interval of 1 Sv). (a) Ψeu(θ, z), the vertical transport streamfunction from the Eulerian time-mean flows. (b) Ψeddy(θ, z), the eddy vertical transport streamfunction. (c) Ψnet(θ, z), the net vertical transport streamfunction. (d) Γnet(y, θ), the net meridional transport streamfunction. (e) , the net vertical transport streamfunction in (c) remapped onto (y, z) space using equivalent latitude. (f) , the net meridional transport streamfunction in (d) remapped onto (y, z) space using equivalent depth. The zonal-mean mixed layer (the x’s) is superimposed in all plots.

  • View in gallery

    Comparisons between subduction streamfunction and the value of isothermal transport streamfunction at the base of zonal-mean mixed layer. (a) The net subduction streamfunction net remapped to the equivalent latitude (solid line). The value of remapped net vertical isothermal transport streamfunction at the zonal-mean mixed layer base (dashed line). The value of remapped net meridional isothermal transport streamfunction at the zonal-mean mixed layer base (dotted line). (b) The remapped eddy subduction streamfunction eddy (thick solid line), the remapped eddy vertical isothermal transport streamfunction at the zonal-mean mixed layer base (thin solid line), the approximate eddy vertical transport streamfunction (dashed line), and the approximate eddy meridional transport streamfunction (dotted line).

  • View in gallery

    (a),(b) The zonal accumulative local subduction from loc (solid line), (dashed line,) and (dotted line). (c),(d) The zonal accumulation of local mass budget (20): the local net subduction loc [solid lines are the same as in (a) and (b)], the zonal transport term U (dotted line), and the surface fluxes–driven diapycnal flow term from (dashed line) and the sum of the latter two terms (thin solid line). (e),(f) Meridionally averaged mixed layer depth within the specified temperature ranges. The temperature ranges are (a),(c),(e) θ ≤ 1.6°C and (b),(d),(f) 1.6°C ≤ θ ≤ 5.0°C.

  • View in gallery

    (a) The zonal accumulative local eddy subduction streamfunction for θ ≤ 1.6°C (solid line, the same as the dotted line in Fig. 10a), the approximation from the VER term (dashed line) and from the LAT + VER term (dotted line). (b) As in (a), but for temperature range 1.6°C ≤ θ ≤ 5.0°C.

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Eddy Subduction and the Vertical Transport Streamfunction

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  • 1 National Oceanography Centre, Southampton, United Kingdom
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Abstract

Subduction—the transport of fluid across the base of mixed layer—exchanges water masses and tracers between the ocean surface and interior. Eddies can affect subduction in a variety of ways. First, eddies shoal the mixed layer by restratifying water columns through baroclinic instabilities. Second, eddies induce an isopycnic transport that leads to the entrainment of warm waters and subduction of cold waters, which effectively counters the wind-driven overturning circulation. In this study, the authors use an idealized model to examine these two mechanisms by which eddies influence subduction and to discuss how eddy subduction may be better approximated using the concept of vertical transport streamfunction than the conventional meridional transport streamfunction.

Corresponding author address: Dr. A. J. George Nurser, Marine Systems Modelling Group, National Oceanography Centre, Southampton SO14 3ZH, United Kingdom. E-mail: agn@noc.ac.uk

Abstract

Subduction—the transport of fluid across the base of mixed layer—exchanges water masses and tracers between the ocean surface and interior. Eddies can affect subduction in a variety of ways. First, eddies shoal the mixed layer by restratifying water columns through baroclinic instabilities. Second, eddies induce an isopycnic transport that leads to the entrainment of warm waters and subduction of cold waters, which effectively counters the wind-driven overturning circulation. In this study, the authors use an idealized model to examine these two mechanisms by which eddies influence subduction and to discuss how eddy subduction may be better approximated using the concept of vertical transport streamfunction than the conventional meridional transport streamfunction.

Corresponding author address: Dr. A. J. George Nurser, Marine Systems Modelling Group, National Oceanography Centre, Southampton SO14 3ZH, United Kingdom. E-mail: agn@noc.ac.uk

1. Introduction

Subduction (the transport of fluid across the base of mixed layer) of mode waters—a thick layer of fluids formed during the winter mixed layer deepening—is a key mechanism for the sequestration of anthropogenic CO2 into the ocean and so has an important role in the global carbon cycle (Sabine et al. 2004). Over the Southern Ocean, the “hot spots” for mode water formation and subduction include the southeastern Indian Ocean and southeastern Pacific Ocean where winter mixed layers are particularly deep (McCarthy and Talley 1999). Following the current, the fluid is entrained/subducted into the mixed/thermocline layer as it enters the region of deep/shallow mixed layer (Sallée et al. 2010). Because the mixed layer depth in these regions has large spatial variations, so does the subduction. Local subduction is much larger than integrated over the large-scale as entrainment and subduction partly cancel each other. Such cancellation does not necessarily apply to the transport of biogeochemical tracers because there might be local sources/sinks for these tracers.

If there are no eddies in the ocean, then local subduction is linked to local surface buoyancy flux and local Ekman transport (Nurser and Marshall 1991). The presence of eddies means that accurate estimates of subduction hinge on first understanding the roles of eddies in subduction and second representing them correctly in the estimate of subduction.

The effect of eddies on the kinematic subduction (Cushman-Roisin 1987) is manifested in a variety of ways. First, by extracting available potential energy stored in the steeply inclined isopycnals within the mixed layer, baroclinic eddies bring warm water on top of cold water and so the mixed layer shoals as a consequence of the restratification (Nurser and Zhang 2000). Thus, eddies affect subduction through the modification of mixed layer depth. Second, time-averaged total subduction of water with a given temperature is not simply an Eulerian time average. The eddy stirring of density means that there may be cancellations between subduction and entrainment of water with the same temperature but at different locations or times. In other words, there are eddy correlations between subduction rate and the area with water of a given temperature (Marshall 1997).

The effect of eddies on subduction may also be interpreted from the thermodynamic perspective. First, integrating over the large scale, water masses formed at the surface must be eventually transported into the thermocline through subduction and so water mass transformation and subduction are intimately related (Marshall 1997; Marshall et al. 1999). For this reason, the subduction rate may be inferred from the surface fluxes, similar to inferring water mass transformation rate from surface fluxes following Walin (1982). Eddies play a role here because of the imprint of eddy stirring on surface fluxes: warm eddies lead to heat loss to the atmosphere and cold eddies to heat gain. Second, it has been speculated that eddy diffusion might be important for driving diapycnal flows (Marshall 1997; Marshall and Radko 2003). However, it is unclear whether the appropriate eddy diffusion should be the time-mean eddy buoyancy flux across the time-mean isopycnals or the time mean of the diffusive flux across instantaneous isopycnals, which are distorted by eddy stirring.

Part of the challenge in quantifying regional subduction in the real ocean is because estimates of eddy subduction rely on some kind of approximation. Because subduction is linked to lateral isopycnic transport below the mixed layer (Marshall 1997), the approximation for lateral eddy isopycnic transport has been applied to eddy subduction (Sallée et al. 2010). However, it is known that the approximation of lateral isopycnic eddy transport does not perform well in the upper ocean (McDougall and McIntosh 2001). Instead, the notion of vertical eddy transport was proposed for the lower atmosphere (Held and Schneider 1999) and was linked to the vertical isopycnic transport in the upper ocean (Nurser and Lee 2004a). Thus, the vertical eddy transport may provide an alternative estimate of eddy subduction.

The aim of this study is to use an idealized Southern Ocean model to examine the effect of eddies on the subduction with the emphasis on the local distribution of subduction. In particular, we will

  1. demonstrate the impact of eddies in shaping mixed layer depth,
  2. construct a locally varying net subduction that includes eddies, and
  3. suggest Eulerian approximations to both net subduction and eddy subduction.

2. Model

The model is the Nucleus for European Modeling of the Ocean (NEMO) version v3.3.1 (Madec 2008). The horizontal resolution is 0.1° with 346 × 346 grid points in a Mercator projection. The southern and northern most latitudes are 56° and 31.65°S, respectively. The depth is 3000 m in 41 vertical levels with variable spacing (5 m in the top grid, increasing to ~30 m at a depth of 200 m and over 100 m below 800 m). The configuration consists of a basin north of 42°S with a lateral boundary of one grid point wide at 0° longitude and a zonal reentrant channel south of 42°S (Fig. 1). In the channel, there is a bottom ridge below 1908 m across the channel, one grid point wide and located at 0° longitude.

Fig. 1.
Fig. 1.

A schematic of the model setup with a basin north of 42°S and a zonal channel to the south with a 1908-m-deep bottom ridge across the channel. The forcing fields are a surface relaxation of temperature and a zonal wind.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

Tracers are advected using the total volume dissipation (TVD) scheme (Zalesak 1979; Levy et al. 2001; Madec 2008), which uses a combination of simple upstream and central schemes to prevent the development of local extrema. The horizontal explicit diffusion is set to be zero. The model mixes vertically with a vertical diffusivity that is the sum of a constant (1.2 × 10−5 m2 s−1) and that given from a simple turbulent kinetic energy (TKE) scheme. The TKE scheme, described in Madec et al. (1998), is based on that described by Gaspar et al. (1990) and Blanke and Delecluse (1993). It involves a prognostic equation for TKE, and a length scale set by the vertical scale of the buoyancy oscillations of a particle with vertical velocity . No additional explicit convective scheme is used.

The model is forced with zonal wind, constant in time and longitude. The maximum wind stress is 0.2 N m−2 at 55°S and the minimum is −0.08 N m−2 at the northern boundary (Fig. 2a). Temperature is the only active tracer, with a linear equation of state depending only on temperature. The initial surface temperatures are 18.7° and 1°C at the northern and southern boundaries, respectively. The initial subsurface temperature at the northern and southern boundaries decreases exponentially in the vertical to 0°C at the bottom with a depth scale of 500 m. For the rest of the domain, the initial temperature is linearly interpolated between the north and south boundaries. The surface heat flux, −λ(θθ*), is a relaxation of surface temperature θ to a target temperature θ*. The target temperature is constant in longitude with a steep meridional gradient in the middle of the domain (Fig. 2c, dashed line). Within about 1° latitude band from the boundaries, the relaxation rate λ is 40 W m−2 K−1. Outside the band, the relaxation rate is 20 W m−2 K−1 over 48°–55°S, decreasing northward to 2 W m−2 K−1 at 42°S and to 0.2 W m−2 K−1 at 40°S, and continues to decrease to the northern boundary (Fig. 2b).

Fig. 2.
Fig. 2.

(a) Zonal wind stress. (b) Relaxation rate for surface temperature. (c) Time- and zonal-mean sea surface temperature (solid line) and the target temperature θ* for surface relaxation (dashed line). (d) Time-mean surface heat fluxes, with contour interval 20 W m−2. Shading denotes surface heat loss.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

The model is integrated for 17 yr with a time step of 480 s. Water masses in the upper ocean reach a statistically steady state, although they are still drifting in the deeper ocean. The main dataset for diagnoses is the 6-hourly running-mean data over year 14 (only one year because of the large data volume). Two additional datasets are also used for comparisons: daily running mean data over years 14–17 and 2-hourly running-mean data for the first 4 months of year 14. For the calculation of subduction, we will use the years 14–17 time-mean mixed layer depth as the control surface.

Mixed layer depth

A rich structure of fronts and eddies permeates the entire model domain as seen from the fields of surface temperature and mixed layer depth (Figs. 3a,b). The influence of small-scale features on the mixed layer depth can be seen from a closer examination of a subdomain surrounding an eddy at the south western corner (marked in dashed line in Figs. 3a,b). Within the subdomain, the warm eddies and filaments (Fig. 3c) are associated with a shallower mixed layer (Fig. 3d). The imprint of eddies on the surface forcing is also evident with stronger heat loss associated with warm eddies and filaments (Figs. 3e,f show south of 48°S).

Fig. 3.
Fig. 3.

The 6-hourly-mean fields. (a) Sea surface temperature, with contour interval 4°C for θ ≥ 4° and 2°C for θ ≤ 4°C. (b) Mixed layer depth Hm, with shading interval 100 m for Hm ≥ 120 m and 30 m for Hm ≤ 120 m. (c),(d) The region marked by the dashed-line boxes in (a),(b). (c) Sea surface temperature, with shading interval 0.6°C and contour interval 0.3°C. (d) Mixed layer depth, with same shading interval as in (b). (e),(f) The region south of 48°S. (e) Sea surface temperature, with the same contour interval as in (c). (f) Surface heat fluxes, with shading divides at −5, −15, and −30 W m−2 and a contour interval of 5 W m−2.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

The following describes the years 14–17 time-mean fields. In the basin part of the domain, surface heat flux almost vanishes where the relaxation of sea surface temperature is very weak, but there is a strong heat flux into the ocean within the 1° latitude band of the northern boundary (Fig. 2d). This ensures that warm water near the northern boundary does not cool down too much as it flows southward along the western boundary of the basin. Indeed, the sea surface height and sea surface temperature (Figs. 4a,b) show that the western boundary current carries warm water (18°C) south along the east coast, turning eastward at 42°S before returning northward. Immediately south of the basin, the frontal region between 42° and 46°S is maintained by the surface temperature relaxation (Fig. 2c, solid line). The gyre circulation of warm surface water from the north keeps the upper ocean stratified and so the (time mean) mixed layer is no more than 30 m deep in the basin (Figs. 4c,d). In the channel part of the domain, in response to the southward extension of the warm boundary current, the surface heat flux is mostly heat loss (except very near the southern boundary) and stronger to the west (Fig. 2d). The mixed layer deepens eastward, reaching over 300 m deep toward the ridge and then shoals immediately east of the ridge. Notice that the deepest mixed layer is located downstream of, rather than at the same location as, the region with strongest heat loss.

Fig. 4.
Fig. 4.

The 4-yr time-mean fields. (a) Sea surface height, with contour interval of 0.2 m. (b) Sea surface temperature, with contour interval of 2°C. (c) Zonal-mean temperature (solid lines, with contour intervals ranging from 0.5° to 2°C) and zonal-mean mixed layer depth (dashed line). (d) Mixed layer depth, with contour interval of 50 m.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

The zonal variations of the time-mean mixed layer depth in the channel region can be explained by diagnosing the evolution of surface stratification (Lee et al. 2011). As in their Eq. (6), at a given (x, y), the time mean of the evolution of the vertical integral of the temperature difference between the surface and a given depth z = −H is
e1
where θ is temperature, u is the horizontal velocity, and the subscript 0 indicates variables evaluated at z = −H. Here, is the contribution from the vertical velocity. The remaining terms are the surface forcing and diffusion . The overbar indicates the time mean and the prime indicates the the deviation from the time mean. The sign is positive for increasing stratification and negative for decreasing stratification. The lhs is the change of time-mean stratification following the time-mean flow at z = −H and is made up of a tendency term and an advective term. On the rhs, the first term is the advection of time-mean temperature due to the time-mean vertical shear, and the second and third terms are the correlations between eddy vertical shear and temperature gradient and between eddy flows at z = −H and time-varying stratification, respectively. For our diagnosis, we set H = 300 m and integrate (1) meridionally between 48° and 54°S where mixed layer depth varies most. The diagnoses shown in Fig. 5 are based on daily-mean fields over years 14–17; similar results hold when using 6-hourly-mean fields over year 14 (not shown).
Fig. 5.
Fig. 5.

The diagnosed terms corresponding to (1) integrated from 48° to 54°S, with H = 300 m. (a) The advective term on the lhs, (thick solid line); the combined two eddy terms on the rhs, (dashed line); the combined noneddy terms on the rhs, (dotted line); and the remaining term (thin solid line). (b) The mean vertical shear term, (thick solid line); the Ekman contribution, (thin dashed line); the surface fluxes term, (dotted line); and the vertical advection term, (thin solid line).

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

Over the time-averaging period T = 4 yr, the tendency term is insignificant. The advective term on the lhs (Fig. 5a, thick solid line) shows that following the flow the time-mean stratification increases rapidly over 0°–2°E, followed by a large destratification over 2°–4.5°E and a slower but consistent destratification east of 8°E. The zonal variation of stratification is a manifestation of mixed layer depth variation. Of the terms on the rhs of (1), the mean vertical shear flow (Fig. 5b, thick solid line) is dominated by the Ekman flow (Fig. 5b, thin dashed line), which destratifies water columns by transporting cold water northward. Surface cooling and vertical advection both destroy stratification (thin solid line and dotted line in Fig. 5b). The three noneddy terms, , , and , combined together contributes to destratification (dotted line in Fig. 5a). The diffusion is small (thin line in Fig. 5a).

The two eddy terms combined together (dashed line in Fig. 5a) increases the stratification. This is largely the result of baroclinic instability leading to warm water overriding cold water. It is known that eddy restratification leads to a shallower mixed layer (Nurser and Zhang 2000; Oschlies 2002). In our case, the eddy restratification is everywhere of the same order as the destratification resulting from all noneddy terms combined. In particular, immediately east of the ridge, the eddy restratification dominates, which is why the mixed layer there shoals even though Ekman transport and surface cooling are strong. Thus, eddies are crucial for shaping the mixed layer depth.

Over 0°–10°E, all terms are large compared to the rest of the region because of warmer water and steeper temperature gradients in the vicinity of the western boundary current. This is also where eddy restratification intensifies. This highlights the role of the land barrier and bottom ridge, which allows the relatively warmer western boundary current to continue into the channel region. Once temperature gradients vary zonally, the mixed layer depth is determined by the delicate balance between dynamics (eddies and vertical advection) and forcing (mechanical and thermal). In a pure zonal channel without bottom ridge or land barrier, the time-mean mixed layer would be zonally symmetric. The location of the deep mixed layer depth relative to the western boundary current in the model bears some resemblance to the location of the mode water formation region over the southeastern Indian Ocean, where deep mixed layer lies downstream of the Agulhas Return Current. It would be interesting to see whether eddy restratification plays a part in shaping the mixed layer depth there.

3. Large-scale integrated subduction

In the real ocean, the winter mixed layer base is often used as the control surface for permanent subduction. Because there is no seasonal cycle in the model, we use the time-mean mixed layer base for the control surface throughout the study.

a. The kinematic perspective

Subduction is usually defined as the transport of fluid across the base of mixed layer, which is equivalent to the divergence of lateral volume flux in the mixed layer,
e2
where u = (u, υ) is the horizontal velocity and Hm is the depth of the time-mean mixed layer. The sign convention is such that s is positive for the upward entrainment into the mixed layer and negative for downward subduction out of the mixed layer.
We define the net subduction streamfunction to be the time-mean subduction of water with temperature colder than θ,
e3
where the overbar indicates the time average and θb(x, y, t) is the temperature at the base of the mixed layer. The phrase “net” is to indicate that the time averaging is taken after the areal integral and “streamfunction” is used because the net subduction of fluid with temperature between θ and θ + Δθ is given by the divergence of subduction streamfunction, (∂net/∂θθ.
Similarly, the Eulerian-mean subduction streamfunction is
e4
where and are the Eulerian time-mean quantities. Eulerian-mean subduction differs from the net subduction in that in the former the time averaging is taken before the areal integral while in the latter it is after. The eddy subduction streamfunction is the difference between the two,
e5
Thus, eddy subduction involves the deviations from the time-mean quantities of both the subduction s and the temperature θ.

The diagnosis below uses years 14–17 time-mean mixed layer for Hm and year 14 of 6-hourly-mean velocities for θ and s. To calculate quantities in θ space, we use the binning method as described in Lee et al. (2007), which ensures heat content and heat transport are conserved while converting data from z coordinate to temperature bins. There are 52 temperature bins with the bin size about 0.4°C. We tested with doubling the number of temperature bins and results are not sensitive to the bin number. The same binning procedure is used throughout the rest of the paper.

Figure 6a shows the Eulerian-mean, eddy, and net subduction streamfunctions. The Eulerian-mean flows give 3.6 Sv (1 Sv ≡ 106 m3 s−1) of entrainment of cold waters (less than 1.6°C) and 2.6 Sv of subduction of warm waters (between 1.6° and 16°C) (dashed line). The Eulerian-mean subduction is close to the Ekman transport (thin solid line). The eddy subduction, on the other hand, has a maximum of 5.3 Sv at 1.6°C, substantially larger than the Eulerian-mean subduction. Formally, the eddy subduction is the correlation between the rate of subduction and the area between two isotherms, but it is closely linked to the eddy transport between the two isotherms vertically spaced below the base of the mixed layer (Marshall 1997; also the schematic in Fig. 7a). The baroclinic eddies induce a poleward surface flow as a result of their flattening of the isotherms just below the base of mixed layer. Consequently, warm waters are entrained and colder waters are subducted, which is opposite to the Eulerian-mean subduction.

Fig. 6.
Fig. 6.

Here, 6-hourly-mean data over 1 yr are used unless otherwise specified. (a) Subduction streamfunctions: net subduction net (thick solid line), Eulerian-mean subduction eu (dashed line) and eddy subduction eddy (dotted line), and Ekman transport (thin solid line). (b) The mass budget within the mixed layer: the net subduction (thick solid line), diapycnal flow from surface fluxes (thick dashed line), diapycnal flow from vertical diffusion (dotted line), the volume changes over the 1-yr time period (thin dashed line), and the residual of the mass budget (thin solid line). The plus symbols are the difference between the diapycnal transport derived from 6-hourly surface fluxes and from 1-yr-mean surface fluxes. (c) Net subduction using 6-hourly-mean data [solid line, same as the thick solid line in (a)] and using daily-mean data (dashed line). (d) Vertical diffusion from 1 yr of 6-hourly-mean data [thick solid line, same as the dotted line in (b)], from 1 yr of daily-mean data (dashed line), and from 4 months of 2-hourly-mean data (thin solid line).

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

Fig. 7.
Fig. 7.

Schematics illustrate the relationship between subduction and isothermal transport (note that dx is omitted in the schematics). (a) Subduction Sdxdy between isotherms θ1 and θ2 across the base of mixed layer, the meridional isothermal transport Vdxdz below the mixed layer, and the vertical isothermal transport Wdxdy across the constant z intersecting mixed layer base. (b) If θ1 moves to at a different time, then the difference between Sdxdy and Vdxdz includes the diapycnal flow in the mixed layer, which could be large. (c) For the same motion of θ1, the difference between Sdxdy and Wdxdy is the diapycnal flow below the mixed layer, which could remain small.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

Thus, there is a cancellation between eddies and wind for the subduction. However, it is not a complete cancellation. The net subduction (Fig. 6a, thick solid line) gives 0.8 Sv of entrainment of cold waters (less than 0.4°C), 2.5-Sv subduction of slightly warm waters (between 0.4° and 1.6°C), and 1.6-Sv entrainment of warm water (between 1.6° and 9°C). There is almost no entrainment or subduction for waters between 9° and 16°C. Waters warmer than 16°C will not be discussed because these waters are in the basin part of the domain.

We compare the net subduction streamfunctions described above with that using daily-mean data over year 14 (Fig. 6c). The daily-mean data result in a maximum net subduction 0.3 Sv smaller, which is about 16% of the maximum subduction from 6-hourly-mean data. Because the Eulerian-mean subduction is the same as before, the 0.3-Sv difference in subduction is entirely the eddy part. Thus, higher-frequency data give slightly stronger eddy subduction, but the difference from using daily-mean data is small, at least for the kinematic subduction.

b. The thermodynamic perspective

The mass balance discussed in Marshall et al. (1999) gives the thermodynamic aspect of subduction. It is basically the water mass transformation of Walin (1982) applied to the mixed layer. It states that, in a steady state, the net subduction streamfunction (equivalent to the diapycnal volume transport, from cold to warm, across the θ isotherm in the mixed layer) is maintained by the divergence of ml(θ), the total nonadvective heat input into the mixed layer where its temperature is colder than θ,
e6
The sources of nonadvective (diabatic) heat input ml = Q + ver + res include surface forcing Q, vertical diffusion in the mixed layer ver, and other diffusion in the mixed layer res.
Figure 6b shows the contributions toward the mixed layer diapycnal flow (and so subduction streamfunction) from these heat inputs. First, the component from surface forcing Q is calculated from
e7
where θs is the temperature at the surface. The surface heat flux consistently cools waters with temperature 0.8°–15°C (thick dashed line). This is because the southward extension of the western boundary current brings warmer water into the channel region and so surface temperature relaxation has to cool to keep waters at the target temperature. The surface cooling gives 2.4 Sv of subduction for water colder than 1.6°C and entrainment for water between 1.6° and 9°C. For water between 9° and 16°C, there is almost no diapycnal flow from the surface forcing. The maximum diapycnal transport of 2.4 Sv is at the temperature 1.6°C because of the large area of water (and so large total heat loss) at that temperature. The simple areal average gives the largest heat loss at 4.5°C.
The second constituent of diapycnal flow is derived from the explicit vertical diffusion ver as
e8
where κ is the model’s explicit vertical diffusivity, which is time dependent and is derived from the mixed layer scheme. In (8), it is crucial that the divergence of vertical diffusive flux is binned into the temperature bins at (x, y, z, t) within the time-mean mixed layer. At a given time, the temperature within the time-mean mixed layer is not necessarily vertically uniform and so vertical diffusion occurs not only at the base of the mixed layer but also inside the mixed layer. The diagnosis shows that (solid line in Fig. 6d; also the dotted line in Fig. 6b) vertical diffusion warms water colder than 8°C. This is caused by the surface heat loss over these waters together with our choice of not allowing convection in the model, so the water column is not always stably stratified in the mixed layer. When cold water is above warm water, the vertical diffusion gives rise to the heat input.

Adding together the diapycnal flows driven by surface forcing and by vertical diffusion does not completely balance the net subduction. So, there are other terms to be considered. First of all, changes of the volume of fluids colder than θ in the mixed layer are small (thin dashed line in Fig. 6b), so we can assume a steady state. Because the model is run without explicit lateral diffusion, the remaining diapycnal source res is calculated from the residual. The residual diapycnal flow is small (thin solid line in Fig. 6b). This term is attributed to implicit numerical diffusion associated with the advection scheme and to any inaccuracy in the calculations of ver.

To address the latter, we repeat the calculations of ver with two other datasets: 1 yr of daily-mean data and 4 months of 2-hourly-mean data (Fig. 6d). Clearly, there is a large difference in the vertical diffusion when the running-mean data are downgraded from every 6 h to daily. However, increasing the time resolution from 6-hourly to every 2 h makes little difference to the vertical diffusion. Thus, we are confident with our calculation of vertical diffusion based on the 6-hourly-mean data. It should be said that the ultimate assurance would be to check against online calculation, but this is technically difficult for our model. Nevertheless, our findings raise an important issue regarding data sampling in eddy-resolving models. The common practice is to archive running-mean data at an interval of an order 3–5 days for global eddy-resolving models and daily for smaller regional models. Our results suggest that for the upper ocean even 1-day-mean data may give inaccurate vertical diffusion of heat, which is crucial for the heat budget in the mixed layer. It is worth comparing with the sensitivity of advective transport to the time resolution. Recall that in section 4a and Fig. 6c we found that kinematic subduction from daily-mean data is only slightly weaker compared to that from 6-hourly-mean data.

A summary of the mass budget in the mixed layer is as follows. To the first order, surface cooling drives 2.2 Sv of entrainment of fluid with temperature 2°–8°C. This is countered by a 0.8-Sv warming as a result of combined explicit vertical diffusion (0.5 Sv) and implicit numerical diffusion (0.3 Sv). Thus, nearly 36% of the diapycnal flow comes from the diffusion (23% from explicit vertical diffusion and 13% from implicit diffusion). Broadly speaking, our results agree with Cerovečki and Marshall (2008), who found that explicit vertical and explicit lateral diffusion make relatively small contributions to the water mass transformation. The diapycnal flows from our model explicit diffusion have a similar order of magnitude to theirs, although ours are slightly stronger, which might be due to a deeper mixed layer.

From the kinematic perspective, the effect of eddies are embedded in the net transport across the base of the mixed layer. From the thermodynamic perspective, the role of eddies is manifested in the surface forcing term. Surface heat fluxes respond to the sea surface temperature variabilities generated by eddies: warm eddies lead to surface heat loss and cold eddies to surface heat gain. This eddy effect may be quantified by comparing the diapycnal flow diagnosed using 6-hourly-mean surface heat fluxes with that using annual-mean data. Figure 6b (plus symbols) shows that the difference between the two calculations is about 20% of the diapycnal flow from 6-hourly data. This is small compared to the kinematic eddy subduction. The small difference between 6-hourly and annual-mean data for surface fluxes may seem puzzling and also contradicts Cerovečki and Marshall (2008) where they found the water mass transformation rate using daily-mean surface forcing differs substantially from that using the annual-mean data. The explanation is given by the following thought experiment.

Consider an eddy with temperature θw located at the latitude y = yc. Suppose the heat flux at that location is , where is the target temperature at y = yc. For the purpose of calculating water mass transformation, c is binned into the temperature bin Bw determined by the temperature θw. In the annual-mean data, there are no eddies and the fluid parcel with the temperature θw is now located at a different latitude y = yw. The target temperature at y = yw is and the surface heat flux for the fluid parcel is . Now, the crucial step is that w is also binned into the same temperature bin Bw because the temperature of the fluid parcel is θw. Therefore, the difference for the heat flux in the bin Bw between 6-hourly and annual-mean data is , proportional to the latitudinal variation of the target temperature. Thus, the larger meridional gradients in the target temperature, the larger the diapycnal flow driven by eddy-induced surface fluxes will be.

In our model setup, the target temperature (Fig. 2c) remains 1°C south of 50°S, increasing to 2°C at 47°S and to 4°C at 45°S. In other words, south of 45°S, the target temperature varies little and so the diapycnal flow driven by surface heat flux would not be so sensitive to using annual or 6-hourly-mean data. In the frontal region between 42° and 45°S, the meridional gradient of the target temperature is large, and so the effect of eddies on the surface heat flux–driven diapycnal flow would be greater. Indeed, Fig. 6b shows that over the temperature 6°–11°C, the diapycnal flow driven by the time-varying part of the surface flux becomes relatively larger compared to that over the temperature 2°–5°C. The reason why our result is different from Cerovečki and Marshall’s (2008) is that our main area for subduction is south of the front, whereas theirs is at the frontal region. We would also argue that our model setup is not completely unrealistic. In the southeast Indian Ocean, the sites of mode water formation and subduction are to the north of (rather than at) the Subantarctic Front. Thus, the meridional gradients of surface temperature in the formation region are not as steep as in the frontal region.

c. Relating subduction to the isothermal transport streamfunction

The temperature of the fluid changes little as it crosses the base of mixed layer unless vertical diffusion is very strong there. Away from the mixed layer base, the transport will be largely isothermal (at the advective time scale). Thus, subduction is related to isothermal transport beneath the mixed layer.

The isothermal transport streamfunctions are traditionally constructed from the meridional transport of fluid colder than a given temperature across a constant latitude (McIntosh and McDougall 1996). It gives a meridional overturning circulation. However, this construction becomes problematic in the upper ocean, where isotherms are more vertically inclined so meridional flow no longer follows isotherms. In addition, occasional outcropping of isotherms means that the meridional isothermal transport streamfunction does not separate the transport in the mixed layer from that below the mixed layer.

An alternative to the meridional isothermal transport streamfunction is the vertical isothermal transport streamfunction, suggested by Nurser and Lee (2004a). In their framework, the vertical transport streamfunction is given by the upward transport of fluid colder than a given temperature across a constant height surface,
e9
where w is the vertical velocity. The net vertical transport streamfunction is the time average of ψ,
e10
This gives a vertical overturning circulation, in contrast to the meridional overturning circulation. This isothermal transport, integrated at a constant z, does not mix properties at different levels. It, however, will mix properties at different y. Similarly, the Eulerian-mean vertical transport streamfunction is
e11
and the eddy vertical isothermal transport streamfunction is Ψeddy = Ψnet − Ψeu.
The mass balance for fluid colder than θ at constant z is [Nurser and Lee 2004a, Eq. (5.17)]
e12
where is the horizontal area of fluid colder than θ at constant z and is the total heat input into the area per unit depth. In a steady state, the vertical divergence of the vertical transport of fluid colder than θ at constant z, , is balanced by the diapycnal flow across the isotherm θ which is driven by the heat input at z. Integrating (12) vertically between the surface and z and averaging in time, we have
e13
The net vertical transport of fluid colder than θ at constant z is balanced by the time mean of diapycnal flow across θ above z. It is worth noting that (13) is similar to the mass balance in the mixed layer (6) where the reference surface is the base of the time-mean mixed layer rather than constant z.

All diagnoses for the rest of the paper are performed using 6-hourly-mean data. The vertical transport streamfunctions above 280 m are shown in Fig. 8. The Eulerian-mean vertical streamfunction (Fig. 8a) Ψeu(θ, z) shows the typical wind-driven overturning circulation with the upwelling of cold water (less than 0.6°C) and downwelling of warmer water (between 0.6° and 6°C). The Ekman transport is spread over the top 30 m rather than over the mixed layer because near surface the vertical viscosity is about 4 × 10−2 m2 s−1, giving an Ekman layer about 20 m thick.

Fig. 8.
Fig. 8.

Isothermal transport streamfunctions (contour interval of 1 Sv). (a) Ψeu(θ, z), the vertical transport streamfunction from the Eulerian time-mean flows. (b) Ψeddy(θ, z), the eddy vertical transport streamfunction. (c) Ψnet(θ, z), the net vertical transport streamfunction. (d) Γnet(y, θ), the net meridional transport streamfunction. (e) , the net vertical transport streamfunction in (c) remapped onto (y, z) space using equivalent latitude. (f) , the net meridional transport streamfunction in (d) remapped onto (y, z) space using equivalent depth. The zonal-mean mixed layer (the x’s) is superimposed in all plots.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

The net vertical transport streamfunction (Fig. 8c) Ψnet(θ, z) has two cells. The mass balance (13) implies that the two cells are driven by diabatic sources. The shallow cell centered at 0.6°C and z = 60 m is driven by surface warming from the surface heat fluxes and subsurface cooling from explicit vertical diffusion, which occurs mostly near the southern boundary. The deeper cell centered at 2°C involves the downwelling of colder waters and upwelling of warm waters. This cell is consistent with the cooling of water warmer than 1.6°C driven by surface fluxes (Fig. 6b).

The eddy vertical transport streamfunction (Fig. 8b) Ψeddy shows the downwelling of cold water and upwelling of warm water, consistent with the baroclinic instability flattening isotherms. Eddy transport is strongest in the upper 50 m. It is not clear how to explain this from baroclinic instability, but it may be explained in terms of the mass balance in (13). The sum of Eulerian-mean transport and eddy transport must be such that the net transport corresponds to the diapycnal flow. Below the surface, the net transport can only be driven by the diffusion of heat. Thus, eddy transport has to cancel the Eulerian-mean transport at each depth where diffusion is small, implying that eddy transport is strong in the Ekman layer where the Eulerian-mean flow is strong.

The mass balance in (13) is a useful guide for interpreting the roles of eddies, winds, and diabatic sources in setting the overturning circulation. That is, the net transport is determined by the diapycnal flow, which is largely driven by the surface fluxes (diffusion is relatively small). The Eulerian-mean transport is mainly determined by the wind and so eddy transport, for both the strength and the distribution, is determined by wind and diabatic sources. In our experiment, the eddy transport is stronger than the Ekman transport because surface flux cools waters warmer than 0.8°C, directing flows from warm to cold and resulting in downwelling of colder waters and upwelling of warm waters: the same sense as eddy transport. Because surface fluxes determine the net transport, the eddy transport must be stronger than the Ekman transport in order for this to happen.

We are interested in the relationship between the vertical transport streamfunction and the subduction streamfunction defined in section 3a. For simplicity, consider the two-dimensional case in (y, z). Define to be the value of the vertical transport streamfunction at the base of the mixed layer: that is, the total upward transport of fluid colder than θ across the horizontal plane that intersects the base of mixed layer (see the schematic in Fig. 7a). Where there are no spatial or temporal variations in either the mixed layer depth or temperature, equals the subduction streamfunction. In general, does not correspond exactly to the subduction streamfunction. As isotherms vary in space and time, the vertical transport across constant z differs from subduction by the diapycnal flows across isotherms. If the intersection takes place below the mixed layer, then the discrepancy between subduction and vertical transport is small because diapycnal flow is small below the mixed layer (see the schematic in Fig. 7c). On the other hand, if the intersection takes place inside the mixed layer, although diapycnal flow may be large the distance between the constant z and the base of the mixed layer is small, so the diapycnal transport (velocity × vertical distance) is not too large. Therefore, we expect the vertical transport of fluid with temperature colder than θ is reasonably close to the subduction of fluid with temperature colder than θ.

To compare the vertical transport streamfunction with the subduction streamfunction, we need to put them in some sort of common coordinates. For this purpose, we use the equivalent latitude coordinates. For each θ, the equivalent latitude is the latitude such that south of has the same area as the time-mean area of fluid with temperature colder than θ. For the net subduction streamfunction net(θ), the temperature at the mixed layer base is used to find the equivalent latitude , which then defines . Figure 9a (solid line) shows the remapped subduction streamfunction . For example, the subduction of fluid colder than 1.6°C (Fig. 6a, solid line) is remapped to equivalent latitude 49°S. Similarly, the vertical transport streamfunction Ψnet(θ, z) is remapped onto (y, z) space using the equivalent latitude at each z. Figure 8e shows the remapped vertical transport streamfunction . From this, define to be the value of at the zonal-mean mixed layer depth. Figure 9a shows that (dashed line) indeed agrees well with the true subduction streamfunction (solid line).

Fig. 9.
Fig. 9.

Comparisons between subduction streamfunction and the value of isothermal transport streamfunction at the base of zonal-mean mixed layer. (a) The net subduction streamfunction net remapped to the equivalent latitude (solid line). The value of remapped net vertical isothermal transport streamfunction at the zonal-mean mixed layer base (dashed line). The value of remapped net meridional isothermal transport streamfunction at the zonal-mean mixed layer base (dotted line). (b) The remapped eddy subduction streamfunction eddy (thick solid line), the remapped eddy vertical isothermal transport streamfunction at the zonal-mean mixed layer base (thin solid line), the approximate eddy vertical transport streamfunction (dashed line), and the approximate eddy meridional transport streamfunction (dotted line).

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

For completeness, we also calculate the conventional meridional isothermal transport streamfunction Γnet(y, θ), which is defined as the meridional transport of fluid warmer than θ across a latitude y,
e14
where υ is the meridional velocity. The eddy meridional transport streamfunction Γeddy(y, θ) is also defined accordingly. Note that we use warmer rather than colder so the resultant streamfunction has the same sign convention as the vertical transport streamfunction. The meridional isothermal streamfunction (Fig. 8d) Γnet(y, θ) also gives two cells: the clockwise cold water cell (less than 1.6°C) spreads over 55°–45°S and counterclockwise warm water cell (greater than 1.6°C) north of 52°S.

For comparison, the meridional isothermal transport streamfunction Γnet(y, θ) is remapped from (y, θ) space into (y, z) space using the equivalent depth of θ at each y, where is similar to equivalent latitude but using vertical area at constant y instead of horizontal area at constant z. The remapped meridional isothermal transport streamfunction (Fig. 8f) is broadly similar to the remapped vertical transport streamfunction (Fig. 8e) . The largest difference is to the south of 52°S where the cell near the southern boundary in the vertical transport streamfunction is absent in the meridional isothermal transport streamfunction. This is because in these latitudes the averaging in (y, θ) space results in vertical averaging, leading to cancellations between vertical diffusion and surface forcing.

We now look at how well the meridional isothermal transport streamfunction compares to the subduction streamfunction. Like before, define to be the value of remapped meridional isothermal transport streamfunction at the base of zonal-mean mixed layer. It shows that (Fig. 9a, dotted line) differs from the subduction streamfunction net much more than does. The reason for this is that, for a given θ, its y position varies with time and so the meridional isothermal transport at constant y might either miss some of the transport in the mixed layer or include the transport below the mixed layer (see the schematic in Fig. 7b). In either case, the error is likely to be large. Given that isotherms are more vertically inclined in the mixed layer, it is not surprising that the vertical transport streamfunction performs better than the meridional transport streamfunction in recovering the subduction.

d. Approximating eddy subduction

A similar comparison is carried out for the eddy component. Using the same procedure as before, both eddy subduction eddy and eddy vertical transport streamfunction Ψeddy are remapped to the equivalent latitude coordinate and from the latter its value at the base of zonal-mean mixed layer gives . Figure 9b shows that the vertical eddy transport streamfunction (thin solid line) is very close to the eddy subduction streamfunction eddy (thick solid line). Thus, to approximate eddy subduction, as it is often required in estimating subduction in the real ocean, it is sensible to start with the approximation of eddy vertical isothermal transport streamfunction.

Held and Schneider (1999) introduced a vertical eddy streamfunction Φw(y, z) suitable for the atmospheric boundary layer. Nurser and Lee (2004b) show that Φw(y, z) is also an approximation to the eddy vertical isothermal transport streamfunction,
e15
where the overbar indicates the time mean, prime indicates the deviation from the time mean, [ ] is the zonal average, and L(y, z) is the along channel length. The subscript y is the meridional derivative at constant z. Note that the Ψeddy on the lhs is calculated in (θ, z) and remapped onto (y, z) whereas the Φw(y, z) on the rhs is calculated directly in (y, z) coordinates.

From (15), we define to be the value of Φw(y, z) at the base of the zonal-mean mixed layer. Figure 9b shows that (dashed line) compares well with the true eddy subduction eddy (thick solid line). This is better than we expected since (15) is derived from the Taylor expansion in the limit of small-amplitude perturbations. That is, time-varying temperature is assumed to be small departure from the time-mean temperature. Thus, (15) does not always hold whenever there are large-scale meanders or detached eddies. However, it seems that may still be used to approximate eddy subduction.

It is worthwhile seeing whether the meridional counterpart can provide a similar approximation to eddy subduction. The approximation to the eddy meridional isothermal transport streamfunction Γeddy(y, θ) is usually expressed as (e.g., McIntosh and McDougall 1996)
e16
Once more, the remapping procedure is carried out for on the lhs. This approximation works well in the interior of the ocean but is likely to fail near surface where isotherms outcrop and the stratification is weak. Define to be the value of Φυ at the base of zonal-mean mixed layer. We found that (dotted line in Fig. 9b) is not as close to the eddy subduction as is . Our conclusion is that the vertical eddy transport streamfunction does a better job than the meridional eddy transport streamfunction as an approximate eddy subduction: that is, but .

4. Local subduction

Spatial variations in the mixed layer depth (Fig. 3b) imply that both entrainment and subduction must occur for the fluid with the same temperature as it enters and leaves the region of deep mixed layer. This information is lost when integrating subduction over a large region inside which entrainment and subduction partly cancel. Local subduction and entrainment are particularly important for tracer transport between surface layer and ocean interior because of the different locations of tracer sources and sinks.

The net subduction at a location is not just time-averaged subduction at that location because this would miss the eddy subduction. In the Southern Ocean, the deepening and shallowing of mixed layer depth along the major current gives rise to local entrainment and local subduction. Thus, the simplest way to reconstruct the local subduction is to assume the local variation occurs in the zonal direction. In addition, the time-mean isotherms in the Southern Ocean are roughly monotonic in the meridional and so they may be used as pseudo-meridional coordinate. Based on this assumption, we extend the net subduction streamfunction from a function of θ to a function of (x, θ). In the following, we use the approach of Lee et al. (2011) to reconstruct such net local subduction, which includes eddy subduction.

a. Subduction and mass budget at (x, θ)

Recall that the net subduction streamfunction in (3) is the areal integral of s over the whole region. To preserve the zonal dimension, the areal integral is rearranged to define the net local subduction streamfunction (per unit x) as a function of (x, θ),
e17
That is, at a given x, subduction of fluid with temperature colder than θ0 is integrated only in the meridional direction. The net local subduction streamfunction loc includes eddies’ contribution because the temporal variations of the area (i.e., dx × the temporal variations of meridional distance) between two isotherms are included in the time averaging. Clearly, integrating loc along the zonal direction gives exactly the same net subduction streamfunction defined in (3),
e18
Similarly, the Eulerian-mean local subduction streamfunction is
e19
and the local eddy subduction streamfunction is .

Figure 10a (solid line) shows the net local subduction streamfunction loc for fluid colder than 1.6°C. This is shown as an eastward accumulation, : for example, the value at 10°E in the figure is the net subduction between 0° and 10°E of fluid colder than 1.6°C. It shows a net subduction of ~7 Sv between 0° and 5°E, followed by an alternating entrainment and subduction of ~1 Sv between 5° and 20°E. East of 20°E, the entrainment is consistent but weaker. Although zonally integrated net subduction for this temperature range is only about 2 Sv, the local variation is up to 7 Sv. Fluid is subducted where the mixed layer shallows and entrained where it deepens (see the mixed layer depth in Fig. 10e). The Eulerian-mean subduction (dashed line in Fig. 10a) has a similar zonal variation, but the eddy subduction (dotted line in Fig. 10a) behaves differently. Despite the large variation up to 100 m in mixed layer depth eddies consistently give subduction.

Fig. 10.
Fig. 10.

(a),(b) The zonal accumulative local subduction from loc (solid line), (dashed line,) and (dotted line). (c),(d) The zonal accumulation of local mass budget (20): the local net subduction loc [solid lines are the same as in (a) and (b)], the zonal transport term U (dotted line), and the surface fluxes–driven diapycnal flow term from (dashed line) and the sum of the latter two terms (thin solid line). (e),(f) Meridionally averaged mixed layer depth within the specified temperature ranges. The temperature ranges are (a),(c),(e) θ ≤ 1.6°C and (b),(d),(f) 1.6°C ≤ θ ≤ 5.0°C.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

For temperature between 1.6° and 5°C, the net local subduction streamfunction loc(x, θ) varies only about 1 Sv (Fig. 10b, solid line), in accord with smaller mixed layer depth variation in this temperature range (Fig. 10f). Eddy subduction gives consistent entrainment (Fig. 10b, dotted line) and, as before, is not so sensitive to the local variations of the mixed layer depth.

The mass budget in the mixed layer can also be extended to the local basis. Assuming a steady state, the local mass budget in the mixed layer at (x, θ) is similar to (6) but with an extra term taking into account the zonal flows,
e20
where is the time mean of the vertically integrated zonal transport in the mixed layer of fluid colder than θ0 at the base of mixed layer and is the total heat input per unit x in the mixed layer for fluid with temperature colder than θ. The diabatic sources for include the surface fluxes and diffusion. The mass balance in (20) states that local net subduction is not simply controlled by the diapycnal flows like the large-scale integrated net subduction. Both the kinematic and thermodynamic aspect plays a role in balancing the local subduction.

Figure 10c shows the terms from (20) as eastward accumulations for temperature colder than 1.6°C. The zonal transport term (dotted line) U(x, θ) is remarkably close to the true subduction (thick solid line). This implies that the diapycnal flow term is small. However, because the zonal integral of zonal divergence is zero, net subduction has to come from the diapycnal flow. This means that, although diapycnal flow is locally small relative to the zonal transport, it still cannot be ignored. One component of diapycnal flow is from the surface heat flux , which is calculated according to the local version of (7). Figure 10c (dashed line) shows that does not have large zonal variation, but its persistent sign gives a net diapycnal flow of −2.4 Sv. The sum of the surface forcing–driven diapycnal flow and the zonal transport (Fig. 10c, thin solid line) is close to the true net local subduction streamfunction.

For temperature between 1.6° and 5°C, the surface flux term (dashed line, Fig. 10d) is of the same order as the zonal transport term (dotted line, Fig. 10d), although the latter follows the local variation of true subduction (thick solid line, Fig. 10d) more closely. As for the other temperature range, the sum of zonal transport and surface flux–driven diapycnal flow is close to the true net local subduction (cf. thin solid line and thick solid line in Fig. 10d). The two examples suggest that the local subduction may be reasonably recovered from combining just two contributions: the zonal transport in the mixed layer U and the diapycnal flow driven by surface forcing . This result may be useful for estimating local subduction in the real ocean. However, in our diagnosis both terms are calculated from the 6-hourly-mean data. Before the method can be applied to the real ocean, the sensitivity to the time resolution needs to be addressed, which could be an interesting topic for future study.

b. The Eulerian approximation of local eddy subduction

We have shown in section 3c that the large-scale integrated eddy subduction can be reasonably approximated by the eddy vertical transport streamfunction at the base of mixed layer, . Here, we investigate whether a similar approximation is feasible for the local eddy subduction.

In Nurser and Lee (2004b), the eddy flux of temperature is separated into two parts: the advective part gives an eddy-induced velocity and the diffusive part gives an across isothermal eddy flux. The eddy diffusive flux in this context is diffusion across the time-mean isotherms, which is not the same as the diapycnal flow across instantaneous isotherms. There is a freedom of choice for the direction of the eddy diffusive flux m. This freedom stems from the fact that the eddy diffusive flux needs not be directed normal to isotherms; it only needs to have the same normal projection as the full eddy flux. As shown in Nurser and Lee (2004b) for the two-dimensional, zonally integrated flux in the zonal channel, choosing m to be the unit meridional vector, m = j, gives the vertical eddy transport streamfunction,
eq1
and choosing m as the unit vertical vector, m = k, gives the meridional eddy transport streamfunction,
eq2
These are the two zonally integrated streamfunctions already discussed in section 3.
Such decomposition is also possible for the three-dimensional case. In this case, the natural choice of the direction for eddy diffusive flux is the one perpendicular to isotherms . In the mixed layer, and so the natural choice would be . However, a simpler choice is to let m = j but keep the fully three-dimensional expression for the vector streamfunction. Following Nurser and Lee (2004b), the three-dimensional vector streamfunction for m = j is
e21
The eddy-induced velocity, defined as the curl of the vector streamfunction, is thus
e22
where
e23
Given the eddy-induced velocity as above, the eddy subduction is therefore
e24
This can be rewritten to isolate the contribution from each streamfunction,
e25
where LAT represents the first two terms combined and VER is the last term. The term LAT may be interpreted as lateral induction. It does not vanish in general because the derivatives do not commute when Hm varies spatially. The term VER is not exactly the vertical velocity because it also includes the meridional derivative of mixed layer depth. If the mixed layer has no spatial variation, then LAT vanishes and eddy subduction is just the vertical velocity wj.

The following compares j with the true eddy subduction for water colder than 1.6°C. Although the true eddy local subduction is already a function of (x, θ), the approximation j is still a function of (x, y). To compare like with like, j at each x is integrated meridionally over the region where the time-mean temperature at the base of the time-mean mixed layer depth is colder than 1.6°C. The comparison is shown as eastward accumulation (Fig. 11a). It shows that the approximation from the VER alone (dashed line) is fairly close to the true eddy subduction (solid line), except over 0°–10°E. In this region, the LAT term makes up the differences and so LAT + VER (dotted line) improves the approximation to the true eddy subduction although it also becomes noisy.

Fig. 11.
Fig. 11.

(a) The zonal accumulative local eddy subduction streamfunction for θ ≤ 1.6°C (solid line, the same as the dotted line in Fig. 10a), the approximation from the VER term (dashed line) and from the LAT + VER term (dotted line). (b) As in (a), but for temperature range 1.6°C ≤ θ ≤ 5.0°C.

Citation: Journal of Physical Oceanography 42, 11; 10.1175/JPO-D-11-0219.1

A similar test is carried out for temperature between 1.6° and 5.0°C (Fig. 11b). The true eddy local subduction gives consistent entrainment and is larger over 0°–10°E (solid line). For this region, the VER term alone (dashed line) fails to approximate the true eddy entrainment, but LAT + VER does improve the approximation considerably (dotted line). These two examples suggest that near the western boundary current region where eddy activity is strong the LAT term could be important. Away from this boundary current region, the term VER gives a reasonably good approximation of local eddy subduction.

5. Discussion

Given the challenges of estimating net subduction in the ocean, this modeling study sets out to assess the possibilities of recovering net subduction including both local variation and eddy effects. The main conclusions are as follows:

  1. First and foremost, eddies influence subduction by modifying mixed layer depth. Eddy restratification is known to shallow the mixed layer. In our model experiment, the eddy restratification reaches the same order as the destratification from the combined cooling because of surface fluxes, wind, and vertical advection. Near the western boundary current, the eddy restratification dominates, leading to a shallower mixed layer there and so breaking up the zonal symmetry of the mixed layer depth along the channel. The location of a deep mixed layer downstream of the boundary current is not unlike that in the southeast Indian Ocean where the deep winter mixed layer lies downstream of the Agulhas Current. It would be interesting to see whether eddy restratification plays a similar role in modifying mixed layer depth there as it does in the model.
  2. Integrated over the large scale, what really matters is the net subduction (the sum of Eulerian-mean and eddy subduction), which is controlled by diabatic heat sources. This is the thermodynamic perspective and is a powerful one because it provides a tool for inferring subduction. However, care must be taken in order to account for all the diabatic sources. For example, the surface fluxes need to have sufficient spatiotemporal resolution to include the imprint of eddies. In the model, the diapycnal flow driven by surface fluxes using annual-mean data gives an error about 20% of that using 6-hourly-mean data. Another diabatic source is the vertical diffusion. This term is of similar order to the eddies’ influence on the surface fluxes. In addition, the calculation of vertical diffusion is found to be sensitive to the time resolution of the data. In our case, vertical diffusion using daily-mean data differs considerably from that using 6-hourly-mean data. Although thermodynamic inference of subduction is an attractive option, the calculation is challenging in practice, especially for the diffusive sources. In the model, using surface fluxes alone without diffusion yields diapycnal flows about 20% more than what is needed to balance the net subduction.
  3. The net subduction is closely linked to the isothermal vertical transport streamfunction. This link lies in the fact that isotherms near the surface are inclined more vertically and so the transport along isotherms across the base of mixed layer are also more vertical. In fact, if the mixed layer is flat then subduction is the vertical transport. The mass balance for vertical transport across a constant z horizontal plane is similar to the mass balance for subduction by replacing the reference surface from the horizontal plane to the surface forming the base of mixed layer. Taking all these together, it is perhaps not too surprising that (large-scale integrated) net subduction is better approximated by the vertical isothermal transport streamfunction than by the traditional meridional isothermal transport streamfunction. The approximation works well not only for net subduction but also for eddy subduction and so one might wish to revisit the parameterization of eddy subduction from the vertical transport streamfunction point of view.
  4. The overarching goal of this study is to construct net subduction at a local basis. The principal difficulty is that by definition net subduction involves spatial averaging with respect to isotherms and by doing so its (x, y) positions are lost. In the region where time-mean isotherms are monotonic in the meridional direction, such as the zonal channel in our model or the Antarctic Circumpolar Current in the Southern Ocean, the contours of constant isotherms can be used as a pseudo-meridional coordinate. In this sense, local subduction may be constructed at given (x, θ). Such local net subduction has the true x position, whereas its y position can only be the time-mean y position of θ at x. This construction reveals that for the same isotherm there are regions of entrainment where the mixed layer deepens and subduction where the mixed layer shallows. The local entrainment and subduction can be much larger than zonally averaged subduction when the mixed layer depth variation is large.The mass balance for the local net subduction in (20) involves both a kinematic part (the zonal transport) and the thermodynamic part (the diabatic sources). The former gives the local variations of subduction whereas the latter gives the net subduction. Where the mixed layer depth varies strongly, the kinematic part dominates over the thermodynamic part. However, because the kinematic part gives zero net subduction, it is still necessary to include the thermodynamic part, however small locally. This suggests that local net subduction could be approximated from combining zonal transport and diabatic sources. It would be interesting to see whether this approximation can be applied to the real ocean using, for example, satellite altimetry–derived velocities and surface flux data.The local eddy subduction does not have similar regional variations to the local net subduction. By and large, eddies give upward entrainment of warm water and downward subduction of cold water, regardless of mixed depth deepening or shallowing. This suggests that the zonally varying pattern of net subduction is mainly the result of the time-mean geostrophic flow at the base of mixed layer. However, eddies do modify the zonal variation of subduction as we have seen that the large-scale integrated net subduction is the end result of the compensation between eddy and mean parts. To approximate local eddy subduction, we use the kinematic approach. To first order, the local vertical eddy transport streamfunction is reasonably successful over the most of the domain. Near the western boundary current region, the zonal eddy transport streamfunction is also required to give the lateral induction. This result seems robust as it is tested in region where mixed layer depth varies either 100 or 40 m over 30° longitude. Thus, our findings suggest that the eddy vertical transport streamfunction could be used to estimate net subduction not only integrated over the large-scale but also locally.

Acknowledgments

This work is funded by the U.K. NERC standard Grant NE/F020252/1. The authors thank David Smeed and Alberto Naveira-Garabato for useful communication. We also thank two reviewers for giving constructive comments on the manuscript.

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