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 |u0〉 to the ocean state vector is described by the equation 1|u(t)〉=Ψ(t,t0)|u0〉, where |u(t)〉 provides the condition of the perturbation at time t; Ψ(t,t0) is known as the propagator matrix, and we have used the “bra–ket” notation of in which row and column vectors are respectively written as bras (〈a|) and kets (|b〉) such that closed bra–ket terms become scalar (through the scalar product 〈a|b〉=c).

The adjoint approach considers the sensitivity of properties of interest to earlier perturbations. For a “cost function” J (mapping the state vector to such a property), which is scalar valued and linear, that is J(u)=F|u (for some F), the sensitivity is given by 2∂J∂|u0〉=Ψ†(t0,t)|F〉, where Ψ†(t0,t) is the adjoint of the propagator matrix, correspondent to its transpose in a Euclidean inner product space. These relationships are derived in full in , where a thorough treatment of the linear method and its limitations may be found, beyond the simplified, inherently linear use case of the present study. Finally, we have 3〈F|Ψ(t,t0)|u0〉=〈u0|Ψ†(t0,t)|F〉 following the definition of the adjoint as a linear operator.

Perturbations |u0〉 such as those in Eq. () typically induce an active or dynamic response . For temperature or salinity perturbations, this corresponds to a modified density field evoking changes in streamlines by modifying pressure gradients. This occurs in tandem with the passive advection and diffusion of the perturbed tracer. We may represent this by splitting the propagator into active (A) and purely passive (P) components Ψ(t,t0)=ΨA(t,t0)+ΨP(t,t0). Note that we use the term “purely passive” for ΨP because the active component ΨA encompasses all dynamic interaction between the unperturbed ocean state and the perturbed field. This itself includes a partially passive element (for example passive transport of dynamically modified tracers). The purely passive component ΨP(t,t0), on the other hand, involves no such interactions.

In the case of propagating a passive tracer, ΨA(t,t0) vanishes. Our perturbation is now merely an initial injection of dye of a specified concentration, |c0〉, and so Eq. () collapses to 4|c(t)〉=ΨP(t,t0)|c0〉. This dye is unable to feed back on the ocean state and so as a perturbation is trivial. We will hence refer to initial perturbations as “injections” and evolved perturbations as dye concentrations, as is more appropriate for the passive tracer case. The common term “cost function”, which relates to optimisation problems, is also a misnomer in the more simple context of propagating passive tracers. In our analysis, 〈F| acts on the concentration vector |c(t)〉 to produce a tracer budget. As such we will more suitably refer to cost functions as budget co-vectors, 〈B|.

Hence, in summary, the tangent-linear model ΨP describes the time evolution of dye concentrations |c(t)〉 in response to initial dye injections |c0〉. Accordingly, the adjoint model ΨP† describes the sensitivity of tracer budget co-vectors 〈B| to earlier dye injections |c0〉.

The model used herein is the NEMO 3.4 OGCM with the tangent-linear and adjoint package NEMOTAM . In particular, the former (nonlinear model) provides the unperturbed, nonlinear model trajectory. Meanwhile the latter (linear model) provides a model implementation of the propagator and its adjoint. Both models are used in the ORCA2-LIM configuration. This consists of a 2∘ grid, with refinement of the equatorial and Mediterranean regions . There are 31 vertical levels, whose height follows a hyperbolic tangent function of depth, ranging from 10 m at the surface to 1000 m. There are two ice layers in the background state. The ice model is not linearised or coupled to NEMOTAM, although in the context of this study, ice dynamics are inherently unaffected (as a tracer which is passive cannot affect ice). The model is forced using the single, repeating normal year of the CORE forcing package . We therefore consider our results in a climatological, as opposed to historical, context. In this configuration, model NASMW (see Sect. for more details on the definition and properties of NASMW in the nonlinear model) most persistently outcrops in the neighbourhood of the time-mean North Atlantic barotropic stream function maximum, which is 38.8 Sv (1 Sv =1×106 m3 s-1) (Fig. , red shading and grey contours). Below the surface, it occupies a narrow latitudinal band at depths of up to 310 m (Fig. , red shading). Conversely, model NADW (as defined and described in Sect. ) most often surfaces in the region close to the barotropic stream function minimum of -25.3 Sv (Fig. , blue shading and grey contours). At most latitudes, NADW occupies depths below the local time-mean meridional stream function maximum (Fig. , blue shading and grey contours). The overall time-mean North Atlantic meridional stream function maximum is 16.1 Sv, occurring at 42∘ N at a depth of 870 m.

To isolate the purely passive response detailed in Eq. (), we set velocity and sea surface height (SSH) modifications to zero in the NEMOTAM time stepping procedure (Fig. ). Further feedbacks involving parameterisations such as vertical and eddy-induced mixing are already absent in NEMOTAM due to approximations in linearisation and so did not require further modification. Additionally, eddy-induced advective velocities are computed online by the nonlinear model and not stored explicitly in the trajectory. They may be recomputed from other trajectory variables, but this is not the default behaviour of NEMOTAM. As such, they were not included for the passive-tracer experiments in this study. They have been since been enabled in the passive-tracer source code as an optional online calculation. A preliminary comparative test suggests that errors are minor for our case studies (positive and negative concentration differences tend to cancel out, but even summing absolute differences, we do not exceed ∼ 6 % relative to the injection volume). We expect the difference to be greater in steep isopycnal regions such as the Southern Ocean.

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 . In the adjoint case, a record of this restoring leads to a spatio-temporal probability distribution of the surface origins of the tracer.

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 FCT;, which balances an upstream scheme with a centred scheme using a nonlinear weighting parameter described by . This scheme is preferable to others available in the model for non-eddying configurations , but its nonlinearity means that it is not suitable for a TAM. As such, the standard advection scheme of NEMOTAM is a linearised counterpart in the form of a second-order forward-time–central-space (FTCS) finite difference scheme. A caveat of such schemes is the presence of an negative artificial diffusion coefficient, which can lead to negative quantities of positive-definite tracers . This encourages down-gradient diffusion of otherwise zero concentrations, acting as a source of further negative tracer when combined with the surface restoring scheme. This is particularly problematic in the case of passive-tracer advection as, unlike in the fully active TAM, negative quantities are unphysical.

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 was coded into NEMOTAM. Like the TVD scheme, the true weighted-mean scheme is a linear combination of the classical upstream and FTCS methods. As before, we approximate the classical upstream method by using the trajectory-upstream scheme. The weighting parameter is determined dynamically from the trajectory. Strongly advective regions lean towards upstream advection, while strongly diffusive regions favour the centred formulation. Despite inheriting issues from both FTCS and the trajectory-upstream schemes, it is preferable to either of these individually. Its negative tracer concentrations are less persistent than those of the TVD and FTCS schemes, and its anomalous vertical spread was found to be less than half that of the upstream scheme (Fig. ). Some issues have been raised with multivariate and time-dependent extrapolations of the weighted-mean scheme e.g.. However, nonlinear flux-corrected transport schemes are required in order to improve on the weighted-mean scheme , and these are not compatible with the linear model. We thus proceed using our approximation to the weighted-mean scheme (in both the horizontal and vertical directions) for the water mass case studies detailed in this paper. We remark that it is entirely possible to incorporate a nonlinear advection scheme into NEMOTAM but that the resulting quasi-linear model would not have a true adjoint, which we require for our study.

In order to test performance, the model was run for 100 d at 2∘ resolution (the ORCA2 configuration used in our case studies) and 5 d at 0.25∘ resolution the ORCA025 configuration,. Each configuration was run in four different modes (nonlinear, nonlinear with TAM trajectory production, tangent linear with passive tracer, and adjoint with passive tracer). Trajectory files were written once per day, and the linear advection scheme was the weighted-mean scheme described in Sect. . Each test was conducted with a range of parallelisation arrangements (16, 32, 64, 128, 256, and 512 CPU cores) and repeated 10 times to account for system variability (Table ). The tests were conducted on a local HPC system, with 72 compute nodes, each offering two eight-core 2.6 GHz processors (Intel Xeon E5-2650 v2) and 64 GiB, connected by an InfiniBand QDR network.

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 , row one).

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.  to the problem of tracking the origins and fate of NASMW in the ORCA2-LIM simulation. There is no universally accepted definition of NASMW, with differing definitions leading to conflicting results between studies . While the method allows us to define water masses in terms of any model variable, we choose for simplicity to utilise the common approach of a temperature–salinity range. In particular, we define model NASMW as water lying in the temperature band [17,19] ∘C, as in the original definition of , with salinity in the range of [36.4,36.6] psu. As in some other studies e.g., we impose a third criterion, in our case that NASMW has a thickness of at least 125 m. This condition ensures homogeneity and isolates the water mass from the adjacent Madeira Mode Water (MMW) to its east .

Model NASMW has a consistent outcrop in a single area (Fig. ) and does not extend far south of its outcrop region below the surface (Fig. ). The outcrop area of NASMW in the model peaks at the beginning of April at 1.3×106 km2 (Fig. ), coinciding with peaks in volume (4.2×105 km3) and thickness (310 m) of the water mass. These maxima occur a month before the annual maximum local mixed layer depth (MLD) of 430 m. Following this, rapid stratification due to summer warming shoals the mixed layer. The outcrop area concurrently diminishes until the water mass is completely sheltered from air–sea exchange. This period of submergence extends from early June to early December, as the upper surface of the water mass progressively deepens to a maximum of 101 m. The volume also decreases, towards 1.3×105 km3 at the annual minimum.

As outlined in Sect. , we begin by identifying all water matching our NASMW definition in the surface layer of the nonlinear model at a given time. We choose this to be the time when the outcrop is maximal (Fig. ). We then inject a concentration of 1 into the tangent-linear model at these locations and propagate it for 60 years. Newly ventilated model NASMW initially resides close to the surface (Fig. ). Here, its behaviour is governed by surface currents, which fractionate the water mass. At the point of application, 37 % of the tracer lies in the Gulf Stream. Most of this water follows the recirculation of the subtropical gyre, while a minority is exported to the subpolar region (2.5 % over 4 years). Our quantification of subpolar export is on the same order of that found under similar conditions by , where inter-gyre exchange was studied in a purpose-built Lagrangian study.

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. ). As time progresses, because of vertical mixing together with near-surface tracer removal by the restoring scheme, remaining tracer tends to reach deeper, colder, and fresher waters. Using the passive tracer approach, it is not possible to establish a full, qualitative description of the fate of NASMW. However, it can be observed that only around 5 % of all model NASMW re-ventilation occurs within its surface outcrop. The rest is mixed out of the TS class, either remaining in the system for the remainder of the 60-year simulation or (more commonly) resurfacing elsewhere.

Due to rapid re-exposure, the e-folding time of our NASMW is just 60 days. This is shorter than the estimations of (1 year) and (3 years), but this is likely due to inherent differences in formulation. In the above studies, Lagrangian particles released in NASMW are not removed upon contact with the surface, so as to account for re-ventilation. Accounting for re-exposure, suggest around 75 days, more closely aligned with our own findings.

To track existing model NASMW back to its source locations, we again follow the procedure outlined in Sect. . This consists of taking a budget of NASMW in the final year of a nonlinear simulation and propagating this budget backward using the adjoint model. We begin the adjoint run at the same point in the annual cycle as the tangent-linear run, when model NASMW volume is at its maximum (Fig. ). We bisect the mode water budget co-vector 〈BMW| into tracer above the mixed layer depth (〈BMWU|) and tracer below (〈BMWL|) and propagate each part separately. This allows us to explore the potential for a resilient layer of older NASMW below the MLD . By linearity, the union of the propagated budget vectors is equivalent to the propagated budget vector of NASMW as a whole, as follows: Ψ†(t0,t)(|BMWU|〉+|BMWL〉)=Ψ†(t0,t)|BMW〉

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. ). The 60-year mean formation location is also within this region, at 32∘ N, 58∘ W. This is almost coincident with the core of NASMW formation determined by and agrees with the air–sea exchange-based estimate of .

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. ). Simultaneously, newly formed mode water in the eastern North Atlantic (MMW) begins to cross into NASMW within 10 years. The signature of the Mediterranean outflow is particularly strong on the very oldest model NASMW, which also contains contributions from the subpolar region. The culmination of all of these water types throughout the 60-year evolution is evident when the sources of NASMW in the model are viewed in TS space (Fig. ). Also apparent is that the primary source of NASMW has a warmer signature, which reflects the advection of warmer waters from the south, which cool and subduct to form the water mass. While the surface distribution suggests that a relatively small neighbourhood dominates the formation of model NASMW (Fig. ), it should be borne in mind that the seasonal variability in surface properties (particularly the outcrop area) reflects strongly on the thermohaline properties of NASMW at formation (Fig. ). This suggests that the NASMW surface outcrop is not, in fact, the dominant origin of NASMW in the model, contrary to the intuition provided by laminar models of the ventilated thermocline e.g.. The implication is that mode water formation is not a passive process and that ocean dynamics must play a fundamental role.

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. ). There is a visibly lower skewness for NASMW lying below the MLD (Fig. , blue bars). Also clear is the seasonal cycle of shielding brought on by summer stratification, with less NASMW reaching the surface during these periods. Due to the simulation beginning at the annual maximum in NASMW production, peak NASMW formation occurs almost instantly. For tracer below the mixed layer (3.78 % of the total) the peak does not occur until the outcrop maximum of the third year. From the PDFs, the expected age of the model NASMW constituents above and below the MLD are, respectively, 4.70 and 8.79 years. It is evident that NASMW located below the MLD is shielded from renewal, with nearly double the expected age of model NASMW as a whole.

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. ). Indeed, we find that for the period from 30 to 50 years, 68.3 % of NASMW originates at latitudes of 45∘ or higher. The idea of a distant source of the oldest NASMW is concurrent with the few studies which have considered it. used an ideal-age tracer to construct a histogram similar to our own, acknowledging a remote source of the very oldest NASMW. found that at least 20 % of NASMW stems from other regions and highlighted the potential importance of subpolar latitudes.

As in Sect. , we now consider the properties and behaviour of NADW in the ORCA2-LIM simulation. We define NADW to fall within a temperature–salinity band of [2,4] ∘C and [34.9,35.0] psu, in close alignment with the definition of . Analysis of the nonlinear model trajectory (about which the model is linearised) shows that there are two distinct latitudes at which water of this TS signature persistently outcrops in the simulation (Fig. , blue shading). These correspond to the (subpolar) Labrador–Irminger seas region, southwest of the Greenland–Scotland Ridge, and the (Arctic) Greenland Sea region, northeast of the ridge (Fig. , blue shading). The outcrop oscillates between the two regions with the seasonal cycle. While observed NADW is known to form only in extreme winters and has a strong interannual signature , our use of repeated forcing implies that formation has little year-to-year variation. The Labrador–Irminger outcrop peaks at the end of March, around a month after the annual local mixed layer depth maximum of ∼ 1 km. At this peak, water in our NADW TS class occupies 5.3×105 km2 at the surface (Fig. ), before the area diminishes with the shoaling mixed layer. The Greenland Sea NADW outcrop in the model peaks in November, with a maximal extent of 2.3×105 km2. At this time, surface NADW is exclusively found northeast of the Greenland–Scotland Ridge, and the water mass is shielded from ventilation elsewhere. On average, 66 % of the total annual outcrop area is southwest of the ridge, and 34 % is to its northeast.

Model NADW volume is almost constant year round at 4.9×107 km3 (18.7 % of the model North Atlantic), deviating by less than 0.5 % annually.

As with NASMW, the dye injection |c0〉 coincides with the time step corresponding to the annual maximum outcrop extent, which here falls in April. It should be noted that all of the locations corresponding to this maximum are southwest of the Greenland–Scotland Ridge, in the subpolar Labrador–Irminger region. To investigate the nature of the Arctic outcrop, we follow a second injection of dye, in November, when its own outcrop extent is maximal. At this point, the water mass exclusively surfaces within the Arctic circle. We thus refer to these distinct surface waters as SP-NADW (for the southwestern, subpolar outcrop) and A-NADW (for the northeastern, Arctic outcrop). The tangent-linear model was run for 400 years with the SP-NADW dye injection and 60 years with the A-NADW dye injection. The run lengths were determined by the rate of surface tracer re-ventilation in each case.

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. ), with the rest remaining in the ocean interior.

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. a). On multidecadal timescales, tracer from NADW can be traced back to locations spanning most of the northern North Atlantic (Fig. b), dominated still by the region surrounding its subpolar outcrop. Of particular interest on these timescales is the signature of the Mediterranean Outflow. It has been proposed from hydrographic data that the northward penetration of Mediterranean Water can intermittently reach the subpolar gyre, influencing LSW . However, we may only speculate as to whether this mechanism is present in our simulation.

Modelled NADW spanning the entire 400-year run (Fig. c) has sources throughout the Atlantic basin, with a notable contribution from the eastern boundary of the South Atlantic, local to the Benguela Current. The Labrador–Irminger sector remains the primary source of NADW at all considered timescales, however.

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. a). This may further be partitioned into waters originating from the two climatic regions (i.e. subpolar and Arctic) associated with the distinct outcrops of NADW (Fig. b and c, respectively). Overall, 45 % of the total NADW formed during the 400-year run may be attributed to the subpolar region. The dominant TS class associated with NADW formation in the model matches that of our definition ([2,4] ∘C and [34.9,35.0] psu), though there are contributions from a cluster of subpolar water types in the range [2,10] ∘C and [34.5,35.5] psu. Despite the substantial seasonal surface exposure of NADW in the Arctic (34 % of the annual outcrop area; see Sect. ), this region ultimately accounts for only 17 % of modelled NADW formation in 400 years. This agrees with our findings in the tangent-linear model (Sect. ) that surface-borne NADW in the Arctic is subject to rapid re-ventilation and does not contribute to the NADW bulk. It further suggests that there is no other narrowly defined TS band which outcrops in the Arctic from which modelled Arctic NADW originates. As such, NADW from this region generally derives from a broad range of waters colder and fresher than we define NADW to be.

NADW that does not originate from either of these two regions of the North Atlantic makes up a substantial proportion (24 %; Fig. d) of the budget. Of this, 4 % is formed at high latitudes in the Southern hemisphere. Overall, 5 % of the total originates from outside the Atlantic.

As in Sect. , we may also consider the temporal distribution of simulated NADW formation (Fig. ). Age peaks in the 13th year, during which 0.6 % of the total budget can be traced back to the surface. The expected age is 112 years. It should be noted that the mean age of model NADW is likely slightly higher. As with any water mass studied in this manner, a proportion will always remain in the ocean (closing the overall budget). Due to this, surface ventilation does not quite account for 100 % of the total budget during our 400-year run. We limit our runs to this length due to technical constraints. In Sect. , we remarked on the unusual formation regions associated with the very oldest North Atlantic Subtropical Mode Water. This was associated with its much shorter life cycle. As such, the corresponding tail of the PDF of NADW age holds fewer surprises. Much like the youngest NADW considered here, the oldest NADW originates predominantly from winter convection in the Labrador and Irminger seas of the model.