Water mass ventilation provides an important link between the atmosphere and the global ocean circulation. In this study, we present a newly developed, probabilistic tool for offline water mass tracking. In particular, NEMOTAM, the tangent-linear and adjoint counterpart to the NEMO ocean general circulation model, is modified to allow passive-tracer transport. By terminating dynamic feedbacks in NEMOTAM, tagged water can be tracked forward and backward in time as a passive dye, producing a probability distribution of pathways and origins, respectively. To represent surface (re-)ventilation, we optionally decrease the tracer concentration in the surface layer and track this concentration removal to produce a ventilation record.

Two test cases are detailed, examining the creation and fate of North Atlantic Subtropical Mode Water (NASMW) and North Atlantic Deep Water (NADW) in a

The intricate process by which the atmosphere and ocean exchange properties has decisive effects on oceanic and atmospheric circulation, biochemistry, and climate. Tracing the pathways of newly ventilated ocean water masses is a multidisciplinary pursuit, for which an entire toolbox of methods has been developed.

Thermocline ventilation and water mass formation is an area with a long history of inquiry. From charts of hydrographic sections,

Observational studies have been able to capture snapshots of this turbulent behaviour locally in passive-tracer dye-release experiments

Simulations of larger numbers of Lagrangian particles in sophisticated eddy-resolving ocean general circulation models

This study presents a method for tracking water masses by means of passive-tracer deployment in the tangent-linear and adjoint model (TAM) developed for the NEMO OGCM

The model's adjoint and tangent-linear components can respectively track the origins and fate of passive-tracer-tagged water in this manner.

TAM construction is typically a laborious process

Several studies have bypassed such complications by computing approximations to the true adjoint of tracer transport in an OGCM. An early application of an adjoint approximation to the tracking of water masses was developed in

Another pseudo-adjoint passive tracer approach is presented by

The work presented herein is of a similar nature to the above studies but repurposes an existing TAM for passive-tracer-tracking. In this sense its adjoint is the true adjoint of the nonlinear model, rather than a bespoke approximation.

We demonstrate the efficacy of the development through case studies of two climatically important water masses of the North Atlantic, whose formation regions are closely aligned with major components of the Atlantic Meridional Overturning Circulation (AMOC). These are North Atlantic Subtropical Mode Water (NASMW) or Eighteen Degree Water, formed in the vicinity of the Gulf Stream and North Atlantic Deep Water (NADW), formed in the subpolar North Atlantic.

NASMW is the name given to a homogeneous body of water found in the upper region of the western subtropical gyre (STG). It was first identified as early as the Challenger expedition, where repeat soundings revealed its unusual uniformity

NASMW formation has historically been attributed to surface heat loss during winter

Newly ventilated NASMW follows the subtropical gyre circulation, travelling north with the Gulf Stream

Multiple studies have sought to quantify or describe the fate of NASMW.

A quite different water mass in nature, NADW is one of the two primary high-density water masses formed in polar waters which act as preconditioners of the thermohaline component of the AMOC

Both source waters are exported southward following formation. The classical view from in situ current measurements

Beyond the Equator, NADW mixes laterally and vertically with ambient water masses, eventually entering the Indian and Pacific basins via the Antarctic Circumpolar Current

Here, we present a new tool which allows us to track water masses to and from their point of formation and apply it to NASMW and NADW. We demonstrate the use of this development by considering the life cycles of the above water masses in a new framework. The paper is set out as follows. In Sect.

The tangent-linear method describes the evolution of perturbations to the ocean state in a linear framework. Perturbations are described by vectors following the structure of the ocean state vector, containing values for all prognostic variables at all locations. The linear evolution of an initial perturbation

The adjoint approach considers the sensitivity of properties of interest to earlier perturbations. For a “cost function”

Perturbations

In the case of propagating a passive tracer,

Hence, in summary, the tangent-linear model

The model used herein is the NEMO 3.4 OGCM

Outcrop locations of NASMW (red) and NADW (blue) from a 60-year climatology of the nonlinear model, following their respective definitions in Sects.

Zonally averaged distribution of NASMW (red) and NADW (blue) in a 60-year climatology of the nonlinear model, following their respective definitions in Sects.

To isolate the purely passive response detailed in Eq. (

Adaptation of NEMOTAM time-stepping procedure for the tangent-linear (blue) and adjoint (red) cases. Steps outlined in green are new additions to the scheme.

Our water mass tracking procedure is as follows. We begin by defining the water which we intend to track, for example using a temperature–salinity (TS) range. To track the evolution of newly formed water of this type, we identify all surface locations where T and S properties meet our definition in the nonlinear model trajectory. We then inject into the tangent-linear model a dye concentration of 1 in these locations and run the model forward. To identify the origins of existing water using the adjoint model, the procedure is similar. We begin by identifying model grid cells in the nonlinear trajectory (at all depths) which satisfy our TS definition. The budget vector is populated using the volume of these grid cells, with zero values elsewhere. This budget vector is then propagated back in time using the adjoint model to produce a sensitivity of the budget to earlier concentrations.
In both cases, we remove any tracer which reaches the surface using the surface temperature-restoring scheme of the model. This scheme restores concentrations towards zero with a timescale of 60 days for a 50 m mixed layer

The default advection scheme of the nonlinear model, used to compute the background state around which the model is linearised, is a total variation diminishing (TVD) scheme. This is a flux-corrected transport scheme

An alternate scheme which is first-order linear (thus compatible with tangent-linear models and added to NEMOTAM for this study) is the trajectory-upstream scheme. This is an approximation to the classical upstream scheme, whereby the upstream direction is inferred from the trajectory velocity field (avoiding nonlinear sign functions). As the velocity field cannot be modified in the passive tracer case, however, the method is everywhere equivalent to the classical scheme. The scheme is forward in both time and space and offers the advantage of not producing negative tracer values. Nevertheless, this scheme also possesses an artificial (positive-valued) diffusion term, leading to a disproportionate spread of tracer, particularly in the vertical direction.

To strike a balance between the relative merits and caveats of the linear schemes described above, an approximation to the weighted-mean scheme of

Comparison of advection schemes in NEMOTAM. To assess the schemes, one of the runs of this study (the tangent-linear run of Sect.

In order to test performance, the model was run for 100 d at 2

Results of performance tests for two model configurations (ORCA2, upper section, and ORCA025, lower section). Top rows: trajectory storage requirements per output. Lower rows: runtime per model day for four model modes, including nonlinear with trajectory output (NLT), nonlinear without trajectory output (NL), tangent linear (TL), and adjoint (AD). The standard deviation of 10 runs is given as a ± uncertainty. Dashes indicate tests which failed due to insufficient memory.

The general order of time efficiency is consistent across all tests; the linear forward model is between 2.5 (16 cores) and 3.1 (64 cores) times as fast as the nonlinear model without trajectory output in ORCA2 and from 1.9 (64 cores) to 2.0 (512 cores) times as fast in ORCA025. The adjoint is slightly less efficient. The nonlinear runs which produce trajectory output are the slowest across the board, due to the high level of output. Memory use appears higher in the linear model than the nonlinear, such that the linear model could not be run on two 64GiB compute nodes alone in ORCA025 (Table 1, column two).

The model shows good scaling in the ORCA025 configuration at all CPU arrangements tested here but begins to worsen in ORCA2 beyond 128 CPU cores. Further, the added time required for trajectory output in nonlinear ORCA2 runs on 128 CPU cores can be considerable. This is possibly due to the generation of a very large number of files during the model run, as the number of files generated is proportional to the number of CPU cores, the trajectory write frequency, and the run length. The required storage size remains relatively constant for runs with a larger number of CPU cores, despite the greater number of files (Table

An additional limitation which we discovered during our longest experiments comes from system file-number limits, which can readily be reached for typical systems for very long runs using many cores.

We now apply the developments of Sect.

Model NASMW has a consistent outcrop in a single area (Fig.

Mean-year time series of NADW (blue) and NASMW (red)

As outlined in Sect.

Evolution of NASMW-tagged tracer in the tangent-linear model. Black and white shading indicates depth-integrated probability density (corresponding to the likelihood that NASMW is found in that region) at 1, 2, 5, 10, and 50 years. Inset percentages show the global integral of this field, i.e. the total proportion of tracer not yet re-ventilated at the surface. The coloured line tracks the centre of mass of the tracer from initialisation to its current position, with colours indicating the mean depth of the tracer. The purple contour shows the distribution of the original tracer injection.

Our NASMW is short lived. Persistent proximity to the surface leads to the re-ventilation of 95 % of the initialised tracer over the 60-year run. Of that which remains in the ocean, 90 % is transformed and no longer fits our definition of NASMW (Fig.

Evolution of model NASMW in TS space. Shading indicates likelihood that the water mass found in a particular TS class after 1, 2, 5, 10, and 50 years. The red box marks the TS range used to define the water mass in this study. Contours show the density at the average depth level of the tracer.

Due to rapid re-exposure, the e-folding time of our NASMW is just 60 days. This is shorter than the estimations of

To track existing model NASMW back to its source locations, we again follow the procedure outlined in Sect.

Most NASMW propagated with the adjoint model reaches the surface quickly; 70 % of the tracer-tagged water mass is under 5 years old. During this early stage, ventilation occurs predominantly within the subtropical gyre recirculation, in the neighbourhood of the initial NASMW outcrop (Fig.

Surface origins of model NASMW as determined by the backtracking budget analysis (adjoint model simulation). Shading indicates probability density. This corresponds to the likelihood that model NASMW is formed in a given region during the time periods [0,5], (5,10], (10,30], and (30,50] years. Inset percentages show the global integral (that is, the total proportion of the budget which is formed during each time period). Note that, contrary to Fig.

For water over 5 years old, current patterns begin to have a distinct influence on formation. There is a clear signature of the Gulf Stream on the youngest model NASMW. This evolves backward as the adjoint propagates, eventually leaving an imprint of the entire subtropical gyre (Fig.

Surface origins of model NASMW as determined by the backtracking budget analysis (adjoint model simulation) in TS space. Shading indicates the proportion of the budget originating from a particular TS class at the surface within 50 years. The red box marks the TS definition of NASMW used in this study. Contours show surface density.

We also consider the timescales involved with NASMW formation in the model. By recording the time at which tracer is removed from the budget by the surface restoring scheme, we may construct a probability distribution function (PDF) of water mass age (Fig.

Probability distribution of model NASMW age (hatched bars) and age of NASMW restricted to be found below the mixed layer depth (blue bars). The percentage of the NASMW budget formed in a given 0.5-year bin is indicated by its associated bar. The expected value of the distribution is marked by a solid line.

We finally address the asymptotic tail of the PDF, which represents the oldest waters found within NASMW. Our findings suggest a high-latitude source makes up a large fraction of this water, having followed a pathway from outside of our defined thermohaline range from the surface to eventually contribute to the makeup of NASMW (Fig.

As in Sect.

Model NADW volume is almost constant year round at

As with NASMW, the dye injection

SP-NADW rapidly sinks, with tracer reaching an average depth of 1235 m after 0.2 years. It initially moves quickly westward, departing the surface region around Cape Farewell. It then spreads throughout the Labrador basin at depth and extends into the Irminger Sea (Fig.

As in Fig.

As in Fig.

The tracer patch then moves southward, steadily deepening. Its deepest average point, 2466 m, is reached after some 22 years. Its mean position is at first closely tied to the DWBC. However, beyond the Flemish Cap, it takes a more central course through the basin interior. While interior southward routes generated by deep eddies have been found parallel to the DWBC in recent profiling float data

The tracer initialised in the model is quickly sequestered and is thus not vulnerable to re-exposure. Indeed, while 27 % of the initial volume of SP-NADW is re-ventilated within the first decade, only a further 24 % is removed during the rest of the century (Fig.

As in Fig.

As in Fig.

The forward evolution of A-NADW (Fig.

As in Fig.

Of all of the initialised A-NADW, that which ultimately reaches the Atlantic basin represents just 3.8 %. We may use the velocity fields of the nonlinear model to estimate transport pathways of this passive tracer into the basin. Consider an idealised case with two openings into the basin (here taken to represent the Denmark Strait and west of the Reykjanes Ridge), at points

Passive tracer injected at the A-NADW outcrop rapidly moves through TS space (Fig.

As in Fig.

Our simulation shows that surface-borne A-NADW does not proceed to form a substantial part of subsurface NADW in the simulation. Within 0.6 years, 78 % of the tracer has been re-exposed to the atmosphere. Of that which remains, some 98 % has left the NADW TS class. The little tracer which stays in our defined NADW class is persistent, eventually following a similar trajectory to SP-NADW through TS space.

The two tangent-linear experiments (following water from each of the two distinct NADW outcrop regions) suggest that the more northeasterly outcrop contributes quantitatively little to the NADW bulk. Hence, any surface origins of NADW in the Arctic of the model are likely found outside of the NADW TS class.

As before, we take a water mass budget at the end of the nonlinear model simulation (in this case for NADW after 400 model years) and provide it to the adjoint model.
Using this approach, 86 % of the NADW budget in model can be traced to its creation within 400 years (Fig.

As in Fig.

During the first year, only 0.13 % of the total volume of model NADW is tracked back to the surface. The strong presence of shallower NADW during this period leads to a clean signature of the two outcrops (Fig.

Modelled NADW spanning the entire 400-year run (Fig.

The propagated budget vector can be separated into different source regions and signatures. For instance, 31 % of the ventilated NADW can be traced back to the Irminger Sea, versus just 14 % to the Labrador Sea in the model. We may also construct a volumetric census of model NADW source waters in thermohaline space (Fig.

As in Fig.

NADW that does not originate from either of these two regions of the North Atlantic makes up a substantial proportion (24 %; Fig.

As in Sect.

As in blue shading of Fig.

We have presented a newly developed addition to the NEMOTAM tangent-linear and adjoint modelling framework for the NEMO OGCM. This package allows tangent-linear and adjoint tracking of passive-tracer transport. Our framework is rooted in concentrations and probabilities, comparable to the statistical properties of a high-resolution Lagrangian approach with a large (theoretically infinite) number of particles.
The development was achieved by deactivating dynamic feedbacks in the time-stepping routine of NEMOTAM and embedding an alternative advection scheme suitable for passive-tracer transport. This advection scheme, proposed by

We have exhibited the use of this tool in two case studies concerning the tracking of North Atlantic-borne water masses, North Atlantic Subtropical Mode Water, and North Atlantic Deep Water. The versatility of the method has been demonstrated through the calculation of several quantities pertaining to these water masses. If a sufficiently long trajectory is used, a near-complete statistical distribution of surface formation can be constructed. This allows diagnosis of expected age and expected origin location, as well as the rate of eradication and re-ventilation of newly ventilated waters.

The linearity of the method ensures that the water being tracked can be partitioned into several components. When these components are propagated separately, the tracking of the whole is equivalent to the tracking of their union.

The results of our case studies show good agreement between the model configuration and common aspects of prior observational and computational studies. We have shown, for example, that, in our simulation, on average, the expected surface origin of tracer initialised within NASMW is 32

For simulated NADW, we have shown (in the tangent-linear model) that an Arctic outcrop of water with its signature to the northeast of the Greenland–Scotland Ridge contributes little to its final form. However, we have found (by backtracking) that Arctic surface waters still make a contribution to simulated NADW formation (17 % here) but from a broad range in thermohaline space. It is understood that OW is produced by the cooling and freshening of North Atlantic inflow to the Greenland Sea

Nevertheless, there are several more intricate details of North Atlantic water mass formation and transport which are the subject of ongoing investigation and as such deserve a dedicated study at higher resolution. For example, the importance of fine-scale bathymetry for an accurate description of on-shelf NADW formation is well described

Despite good broad agreement between the passive-tracer pathways and those noted in previous studies of these water masses, there is an interesting disparity between the forward and backward modes within the TAM itself. This originates from using a TS-based description to inform the initial tracer distribution. For example, the backtracking method suggests that NASMW predominantly originates in slightly warmer surface waters than those of the outcrop used to inform the forward model. Meanwhile, A-NADW, while occupying over a third of the simulated NADW annual outcrop, ultimately contributes almost nothing to the subsurface water mass. These deviations highlight the approximation used by many water-mass-tracking model studies – thermohaline characteristics are not a purely passive tracer. Water parcels experience changes in their thermohaline properties, and so water in a particular TS class at depth is not exclusively related to the same TS class at the surface through a passive advection pathway.

TAM use at high resolution is typically limited, due to baroclinic instability. However, this is not detrimental for passive tracer tracking, due to lack of dynamic feedbacks. As such, our tool may be used in conjunction with higher-resolution configurations of NEMO

Although we have presented the development in the context of water mass tracking, there are many potential further applications. Ocean heat uptake pathways and carbon sequestration have been studied by means of modelled passive tracers

Despite TAM use for sensitivity analysis being traceable to the 1940s

NEMO v3.4 is available from

DS co-developed the source code, ran simulations, and wrote this paper. The development of the source code was overseen by SM, who provided further suggestions and additional technical assistance. FS proposed and supervised the project and assisted with experiment design. All authors discussed and contributed to the final paper.

The authors declare that they have no conflict of interest.

This research was supported by the Natural and Environmental Research Council UK (SMURPHS, grant no. NE/N005767/1, and MESO-CLIP, grant no. NE/K005928/1, and the SPITFIRE DTP) and the DECLIC and Meso-Var-Clim projects funded through the French CNRS–INSU LEFE programme.

This paper was edited by Qiang Wang and reviewed by Erik van Sebille and one anonymous referee.