1. Introduction
Mode waters are identified as a local maximum in a volumetric census in a temperature–salinity (T–S) diagram tied to a water mass with weak stratification and low potential vorticity (Hanawa and Talley 2001). The mechanism that selects the temperature, salinity, and density class where a particular mode water accumulates is not well understood. The formation of mode water has long been attributed to wintertime air–sea buoyancy loss and convection (e.g., Worthington 1959). However, because of the large thermal inertia of mode waters, their temperature is insensitive to wintertime heat loss, implying that their temperature–salinity relation is not set by air–sea fluxes (Warren 1972). Instead, processes in the ocean interior likely play a key role in the selection of their water mass properties. Mode waters are found on the equatorward side of major ocean fronts, suggesting a role for frontal processes in their generation and maintenance (Marshall et al. 2009). Thomas and Shakespeare (2015) described a mode water formation mechanism involving the mixing of cross-front temperature–salinity contrasts, cabbeling, and frontogenesis that has the potential to select the density, temperature, and salinity of a particular mode water.
The fronts that border mode waters are characterized by density-compensated temperature and salinity contrasts that decrease in magnitude with depth. Along-isopycnal mixing of the disparate water masses across the fronts leads to an increase in density of the water through cabbeling (McDougall 1987). Cabbeling has been invoked to explain the formation of water masses such as North Pacific Intermediate Water (Talley and Yun 2001) and intermediate, deep, and bottom waters in the Southern Ocean (Foster 1972; Klocker and McDougall 2010; Urakawa and Hasumi 2012; Groeskamp et al. 2016). Thomas and Shakespeare (2015) posited that cabbeling could also generate mode waters at fronts because the vertical structure of the cross-front T–S contrast makes cabbeling-driven diapycnal mass fluxes convergent and prone to filling isopycnal layers over time. Their theory explored the steady-state behavior of this process in which mixing is balanced by frontogenetic strain (e.g., caused by confluent flow on the gyre- or mesoscale) that squeezes together isotherms and isohaline surfaces, thereby sustaining the cross-front T–S gradient. In this limit, the diapycnal mass flux scales with the strain, the equilibrated width of the front, and the square of the isopycnal cross-front temperature contrast. The theory predicts that mode water should form in the isopycnal layer where the magnitude of the diapycnal gradient of the cross-front temperature contrast squared is largest. In the theoretical model, however, the volume of isopycnal layers does not change in time because the frontogenetic strain is spatially uniform. While the theory illustrates how cabbeling selects the mode water isopycnal layer, since the layer never actually fills, it cannot be used to predict the temperature and salinity of the mode water that results. Moreover, the theory parameterizes cross-front mixing with a constant, lateral turbulent diffusivity and therefore greatly simplifies the submesoscale mixing processes that allow cabbeling to occur. In this article, we extend the theory of Thomas and Shakespeare (2015) using numerical simulations configured with a more realistic flow with spatially variable strain and capable of resolving submesoscale instabilities, with the goal to elucidate the underlying physics that determines the T–S relation of mode waters when they are formed through cabbeling and frontogenesis.
2. Model configuration
The suite of simulations. The theoretical steady-state half-width of the front is
3. Results
Figure 2 shows snapshots of the surface temperature and density in the reference simulation, as described in the previous section. At time zero there is a 50-km-wide temperature (and salinity) front that is invariant in the x direction, and the density does not vary in the horizontal. The background flow field (superimposed on Fig. 2a) acts on the front, driving cross-frontal confluence between 50 < x < 150 km and diffluence outside this region, as seen in the 25-day temperature (Fig. 2c). A filament of dense water forms in the confluent region as a result of cabbeling, as anticipated from theory. The filament continues to sharpen and densify until it becomes unstable and rolls up into eddies, as seen in the 50-day density snapshot (Fig. 2f). The eddies grow in size and are advected along the front toward the diffluent region, leading to an accumulation of mode water in this region by 100 days (Fig. 2h). A vertical transect of the vorticity, temperature, and vertical velocity through such an eddy is shown in Figs. 3a, 3c, and 3e. The cyclonic eddy (vorticity ~0.1f) is composed of a largely uniform mode water mass (T ~ 18°C and density anomaly 0.13 < Δρ = ρ − ρ0 < 0.15 kg m−3) between the surface and 100-m depth and has only weak vertical velocity on the periphery, suggesting no significant active frontogenesis.
New eddies are continually generated in the confluent region, sustaining the water mass transformation at the front. A y transect across the confluent front at 100 days is shown in Figs. 3b, 3d, and 3e. The temperature field shows the sharp front with O(f) vorticity between the warm southern and cold northern water masses. The vertical velocity field possesses the structure predicted by Thomas and Shakespeare (2015) (cf. Fig. 1) with water drawn up on the sides of the density front and sinking strongly in the center as it is densified via cabbeling. This circulation is associated with a lifting of isopycnals at the front and a filling of the mode water layer (0.13 < Δρ < 0.15 kg m−3), which adopts a lenticular shape and strong anticyclonic flow. The mode water layer continues to deepen as cabbeling continues and ultimately the front rolls up into an eddy similar to that already described (Figs. 3a,c,e).
Prior to 50 days, the system APE increases as cabbeling creates the dense filament along the strain axis. There is negligible KE at this time as seen in Fig. 4a. However, the kinetic energy increases rapidly as the instability initiates at 50 days, implying that the eddies are drawing their kinetic energy from the available potential of the density filament, consistent with the mechanism of baroclinic instability. The eddies enhance mixing at the front, therefore amplifying cabbeling and creating more APE. The strengthening baroclinic eddies continue to draw down this APE. Eventually, cabbeling weakens as the northern and southern water masses are consumed and the APE (and total mechanical energy, E = APE + KE) peaks at 97 days. There is negligible increase in the total kinetic energy between 100 and 160 days, indicating a balance between kinetic energy production via instability at the front and kinetic energy destruction via viscous dissipation. The total mechanical energy decreases consistently over this period, predominantly due to viscous dissipation (see Fig. 4b; with a contribution from irreversible mixing of density, not shown). The system energy is essentially steady by 200 days with equal APE and KE.
a. Mode water properties
The depth–density and density–temperature water mass distribution for the reference simulation is shown in Fig. 5 at 0, 100, and 200 days. At time zero, the domain is linearly stratified throughout (initial stratification is indicated by the dashed line in Fig. 5a), and the water mass is bisected into the warmer southern water and northern cooler water (dashed lines in Fig. 5d), with water in the initial front distributed between these two extremes. By 100 days (Figs. 5b,e) a maximum is visible in the water mass distribution at Δρ = 0.14 ± 0.01 kg m−3, corresponding to the newly formed mode water. The maximum extends from the surface to about 125-m depth and over temperatures of 16.5° to 19.5°C. As will be shown in section 3c, this maximum in the water mass distribution is associated with the eddies seen previously in Fig. 2h. By 200 days, the initial northern and southern water masses have been completely consumed. Figure 5c shows an almost uniform density distribution over the domain, with a weakly stratified (compared to the initial state) mode water layer concentrated between 50- and 125-m depth. At this time, Fig. 5f shows that the temperature and density range of the mode water has been extended, with two distinct maxima, at the same density as previously, but with two different temperatures. The higher temperature maximum results from the mode water being fluxed around the background eddies and back into the confluent zone, where it mixes with the warmer southern water mass. Similarly, the lower temperature maximum results from the mode water being fluxed around the background eddies and back into the confluent zone, where it mixes with the cooler, northern water mass.
b. Mass transformation rate
c. Mode water sources
Here, we use Lagrangian floats to investigate the sources and spatial distribution of the mode water in our simulations. As described in section 3a, the mode water is identified at 100 days as the maximum in the ρ–T–S volume distribution shown in Fig. 5e, around which we have drawn a pink-dashed triangle. Model grid points (x, y, z) with ρ–T–S values within this triangle at 100 days are identified as locations where mode water exists. In these locations, we initialize a Lagrangian parcel at every tenth such grid point in the horizontal (which equates to one parcel in every 1.25 km × 1.25 km square within regions of mode water) and in each vertical level, leading to a total of just over 25 000 parcels. The spatial location of each parcel at the 100-day tagging time is shown in Fig. 7c, with colors indicating the density anomaly (Δρ = ρ − ρ0) of the parcel. The mode water parcels are located over depths from 0 to 150 m and are located within eddies, predominately in the diffluent region of the domain. As seen in Fig. 7c, these eddies are weakly stratified with a density difference of approximately 0.02 kg m−3 over their 150 depth, which equates to a stratification of 1.3 × 10−6 s−2—an order of magnitude smaller than the background stratification (N2 = 10−5 s−2).
We employ 3-hourly outputs of the three-dimensional model velocity fields to advect the Lagrangian parcels both forward and backward in time.4 The locations of the parcels at time zero and 50 and 200 days are shown in Figs. 7a, 7b, and 7d, respectively. The parcels are initially evenly distributed between the northern and southern sides of the front and over the full 0–150-m depth (0–0.16 kg m−3 density) range. As time proceeds the parcels are drawn in horizontally along the line of maximum confluence (x = 100 km) by the barotropic background strain flow and by 50 days have accumulated in the confluent regions on either side of the front. As the instability grows (50 to 100 days) these two stratified water masses are mixed at the front, forming the weakly stratified eddies. By 200 days the mode water has spread over most of the domain to both the north and south of the original front.
Figure 8a displays the paths of 50 parcels in density–depth space between 0 and 100 days. The 50 parcels start at different initial depths (densities) and random horizontal positions. Initially, parcels remain fixed in density–depth space as they are advected horizontally by the strain flow. As they approach the front, the parcels begin to rise (consistent with the vertical velocity field at the front; Fig. 3f) and marginally densify [density increases by O(0.01–0.03) kg m−3]. However, the majority of the densification occurs as parcels reach their maximum height, with most parcels densifying to the mode water density class (0.13–0.14 kg m−3) between 0- and 25-m depth. The newly densified mode water parcels then sink at the front in a vertical jet (as seen in Fig. 3f). These parcel paths are at odds with the linear model of Thomas and Shakespeare (2015), which proposed dense water formation by essentially separate horizontal mixing of water masses at each depth, after which the density of parcels reaches ρ100. Instead, here we have upward advection of water to near the surface, where the vast majority of cabbeling occurs, thus producing well-mixed mode water with a density different than ρ100, as argued previously. To further emphasize this behavior, Fig. 8b displays the minimum, maximum, and final (100 day) depths averaged over all parcels starting at a given initial depth (density). The mean 100-day depth of any parcel initially above 100 m is close to 75 m, regardless of its initial depth. On average, all such parcels rise to a minimum depth of 0–25 m prior to densifying and sink to a maximum depth of 90–100 m after densifying.
d. Diffusive water mass transformation
Heretofore, we have focused on the water mass transformation due to cabbeling only. However, the use of a horizontal diffusivity in the simulations implies that there is significant mass transformation purely due to diffusion [e.g., (17)]. This transformation occurs wherever there are horizontal gradients in density (i.e., density fronts) such as on the edge of the eddies containing the mode water. The horizontal diffusion acts to lighten the relatively higher-density, near-surface mode water and densify the relatively low-density, stratified water surrounding the eddies. This effect can be seen in the lightening of mode water parcels (to 0.1–0.12 kg m−3) near the surface in Fig. 7 between 100 and 200 days. To quantify the net effect of diffusive transformation, Fig. 9 displays the water mass transformation from both cabbeling (as shown previously; Fig. 6) and diffusion and their sum (the total transformation). Figure 9b implies that diffusion is removing volume from around the 0.134 kg m−3 mode water created via cabbeling (maximum volume flux divergence) and fluxing it into the 0.1–0.12 kg m−3 layer (maximum volume flux convergence), exactly as observed in Fig. 7. However, this “leakage” to lighter density classes does not affect the essential mechanism of mode water formation via cabbeling, as described in previous sections. Instead, the diffusion tends to modify the density of the mode water after it has been created by cabbeling.
e. Sensitivity
Here, we investigate the sensitivity of the mode water properties to the initial water mass distribution (section 1) and rate of mass transformation to various model parameters such as stratification, strain, viscosity, and diffusivity (section 2). In addition to the reference case discussed previously, there are a further 13 simulations described in this section, which are summarized in Table 1.
1) Temperature–depth structure
In section 3a, we proposed a method to determine a priori the properties of the mode water generated in our simulations. Here, we show that this method is robust for various initial water mass distributions.
The 100% cabbeling temperature profile T100 [(13)] is shown in blue on each subplot in Fig. 10 plotted against the initial density of the water mass [dashed; equivalent to depth since
2) Model parameters
The values’ horizontal diffusivity (κh = 3 m2 s−1) and viscosity (νh = 10 m2 s−1) in the reference simulation are “subgrid turbulent” parameters whose values are chosen to prevent the collapse of gradients below the grid scale. As such, it is of interest to determine the sensitivity of our results and in particular the rate of water mass transformation via cabbeling to these parameters. Figure 12 displays the maximum water mass transformation with time (Fig. 12a) and cumulative water mass transformation across all density classes (in units of domain volume; Fig. 12b) for five simulations with different viscosity and/or diffusivity. The reference simulation is shown in black. Simulations where only the viscosity is varied are shown as dashed curves. The water mass transformation tends to decrease with increased viscosity, presumably due to the instabilities being partially suppressed. The same effect is seen for increased diffusivity; that is, increased explicit mixing (diffusivity) has the effect of reducing the net mixing by preventing sharp gradients and the generation of the instabilities responsible for this mixing. For κh = 5 m2 s−1 (blue), the growth of the instability is delayed by about 25 days, and for κh = 10 m2 s−1 (red), the instability is entirely suppressed. These results suggest that reducing the viscosity and/or diffusivity toward more realistic values (e.g., by using a higher-resolution model) will either not change, or possibly increase, the rate of water mass transformation.
Figures 12c and 12d display analogous plots of water mass transformation for simulations with 0.5, 2, and 4 times the background stratification used in the reference case. The half-stratification case also has an increased horizontal diffusivity (κh = 5 m2 s−1; see Table 1), which is necessary to keep the Richardson number [(9)] above 0.25 and thereby prevent poorly resolved frontal shear instabilities. Thomas and Shakespeare (2015) predict that the water mass transformation scales inversely with the stratification, which is largely consistent with these simulations. In particular, halving the stratification (red) results in a more-than-doubled peak water mass transformation of 1.3 Sv. The case with doubled stratification (green) has a Richardson number of Ri = 1.0 (see Table 1) and so is only weakly unstable, leading to a peak water mass transformation of less than 0.2 Sv. The case with 4 times stratification has Ri = 2 and does not manifest an instability, leading to negligible transformation.
Last, in Figs. 12e and 12f, we show the water mass transformation for simulations with 1/3 and 2 times the reference background strain. In each simulation the reference diffusivity is modified by the same factor (i.e., 1/3, 2) to keep the frontal width and Richardson number unchanged. Both the larger and smaller strains result in less water mass transformation than the reference simulation. This behavior appears to be the result of two competing effects: (i) increasing the strain increases the supply of water to the front, which should increase water mass transformation, but (ii) increasing the strain rate also tends to suppress instabilities at the front (Bishop 1993; Spall 1997), thus reducing the amount of mixing.
4. Application to observed mode waters
We now analyze the water mass properties of observed mode waters near the Kuroshio, Gulf Stream, and Subantarctic Front to evaluate the potential relevance of cabbeling, mixing, and frontogenesis to their formation. The observations are from WOCE line P10 in the North Pacific (cruise expedition designation 49NZ20050525), WOCE line A22 in the North Atlantic (cruise expedition designation 316N200310), and the OISO-01 cruise at the Subantarctic Front to the northeast of Kerguelen Island (Lo Monaco and Metzl 2007). At each front, the cross-front temperature contrast ΔTf decreases with density near the isopycnal layers of each mode water (Fig. 13). The theory predicts that cabbeling and frontogenesis should therefore drive a water mass transformation [(18)] that is convergent in these density classes. For constant strain and mixing, the maximum mode water formation occurs in the isopycnal layer where the diapycnal gradient of (ΔTf)2 is largest in magnitude. This isopycnal layer (indicated by the dashed lines in Fig. 13) was calculated using the observed cross-front temperature contrasts at each front and compares favorably with the actual density class of the mode waters.
Interestingly, the temperatures of the observed mode waters all exceed the temperature T100 that would result if the water masses that bound the fronts were mixed isopycnally in equal proportions (e.g., dotted lines in Fig. 13). They are instead very similar to the temperatures of the water masses on the warm side of the fronts. As described in section 3e, this does not, however, rule out cabbeling as a potential mechanism for their formation. The temperature profiles at the three fronts are all biased to the warm water mass; therefore, a combination of cross-front and vertical mixing could allow cabbeling to generate mode waters with similar properties to the warm water mass, in an analogous fashion to what is seen in our simulations. Having said this, there are processes present in the ocean that are absent in our simulations that could shape the formation of mode waters at observed fronts. We will discuss these potential limitations of our model and opportunities for future research in the next section.
5. Discussion
Here, we have investigated the formation of mode water via cabbeling in an idealized simulation of a strained thermohaline front. We initialized the simulation in a uniformly stratified state and imposed a balanced background flow field consisting of four mesoscale eddies driving a confluent strain rate of O(10−6) s−1 across the front. The background flow draws cold northern and warm southern water into the front, where horizontal mixing results in the formation of a dense filament via cabbeling, as predicted by Thomas and Shakespeare (2015). Here, we have selected parameters such that this filament is baroclinically unstable, as is the case for thermohaline fronts such as the Gulf Stream north wall. The submesoscale baroclinic instabilities at the front drive enhanced horizontal mixing with effective diffusivities of O(100) m2 s−1, exceeding the explicit diffusivity in the simulations by two orders of magnitude. This turbulent mixing causes significant water mass transformation at the front with peak values of O(0.5–1) Sv for the 200-km-long front, consistent with the theoretical prediction [(18)] of Thomas and Shakespeare (2015). The resolved turbulent mixing is strengthened, and the water mass transformation increases as the explicit viscosity or diffusivity in the model is reduced, suggesting our results are robust and are not a consequence of artificially high turbulent parameters.
Unlike classical mechanisms of mode water formation, which rely on air–sea fluxes, the cabbeling mechanism investigated here prescribes the temperature and salinity of the mode water, given knowledge of the temperature–salinity profiles with the depth of the source water masses on either side of the front. Key to this mechanism is that mode water formation by the submesoscale instabilities is a fundamentally three-dimensional process. Essentially all water mass transformation in the simulations occurs in the surface 25 m, with water from as deep as 125 m being drawn up to the surface prior to transformation. The mode water forms at the front above 25 m and then sinks to fill the mode water layer at depth. This layer gradually expands up to the surface and ultimately detaches from the front as an eddy. The occurrence of essentially all cabbeling mass transformation near the surface is in contrast to the linear theory of Thomas and Shakespeare (2015) where mass transformation occurred at each vertical level via direct horizontal mixing. This difference means that mode water forming in the simulations is well mixed since the source water is a combination of water mass from multiple depths.
The linear theory of Thomas and Shakespeare (2015) predicts that the depth of maximum mode water formation via cabbeling is where the stratification associated with the cabbeled density profile [(14)] is minimized. Owing to the vertical exchange of water described above, here we argue that the density of the mode water is the average of the cabbeled density profile between this depth and the surface. Similarly, the temperature of the mode water is the average of the northern and southern profiles between this depth and the surface. We showed that this methodology for calculating the mode water properties is robust for a variety of northern and southern temperature–depth profiles, at least for a fully compensated front.
We have made a number of simplifications and assumptions in configuring the present model. The imposed background strain field is highly idealized compared to the strain fields associated with gyres or mesoscale eddies in the ocean. However, these real phenomena would have the same effect on the thermohaline front—that is, sharpening the front, thereby promoting instability and continually fluxing water mass in toward the front—albeit in a far more temporally and spatially variable fashion.
In addition, we have assumed that cabbeling is the dominant nonlinearity in the equation of state and that thermobaricity is by comparison negligible. We can test this assumption a posteriori using the model output. Thermobaricity results in a modification to the density −ρ0(∂PαT)δTδP given a change in pressure δP when combined with a change in temperature δT. For the temperatures and salinity values used in our simulations, the variation in thermal expansion coefficient with pressure ∂PαT is ~2 × 10−8 K−1 dbar−1. In the simulations, fluid parcels experience vertical displacements and hence pressure changes of order 100 dbar (e.g., Fig. 8), which if combined with ~5° C temperature change for a fluid parcel mixed in the front, would yield a thermobaric density change of 0.01 kg m−3. This value is an order of magnitude weaker than the 0.1 kg m−3 density increase caused by cabbeling, indicating that neglecting thermobaric effects in our simulations was justified.
A further major assumption in our model is that of a fully compensated thermohaline front and the corresponding use of an artificially large cross-frontal temperature difference (10°C) and weak stratification to ensure the front becomes unstable. More realistic partially compensated fronts are unstable and have large turbulent mixing, which could enhance mode water formation. However, the fronts have much smaller cross-frontal temperature difference and stronger stratification, which could substantially reduce the resultant water mass transformation as per the scaling [(18)]. Further, it is unclear how the frontogenetic circulation at a noncompensated density front would modify the generation rate or T–S properties of the mode water. The thermally direct ageostrophic circulation associated with frontogenesis (e.g., Hoskins et al. 1978) would tend to preferentially bring water to the surface on the lower-density—typically the warmer, equatorward—side of the front, and this could significantly bias mode water properties toward the T–S properties of the warmer water. The thermally direct circulation could also lead to a preferential accumulation of the mode water at depth on the lighter side of the front, since it is associated with a flow at the base of the front from the denser to the lighter side.
In contrast to the front used in this study, the fronts where mode waters are observed have bounding water masses with properties that are maintained by sources and sinks of heat and salt on the large scale. Therefore, while mode water formation in our simulations is a transient process where formation terminates after the two distinct northern and southern water masses are mixed across the whole domain, at observed fronts it could be sustained. In addition, our model does not include any surface buoyancy fluxes, which are known to contribute significantly to mode water formation (Speer and Forget 2013). We anticipate that surface buoyancy loss will enhance the mass transformation via cabbeling by promoting the vertical exchange of fluid and bringing more water mass to the surface, but the water mass properties may be different to those predicted from cabbeling alone, particularly if the seasonality of the fluxes is considered. Conversely, the properties of the mode water formed would be different to those predicted by buoyancy fluxes alone since lateral mixing and cabbeling would be involved. Evidence to this effect is found in wintertime observations of Eighteen Degree Water formed near the Gulf Stream. Here, Joyce et al. (2013) determined that the temperature and salinity of the mode water could not be explained solely by air–sea fluxes but required lateral mixing of cool freshwater from the north across the front.
Collectively, these processes found at observed fronts will give rise to a complicated steady-state mode water formation, which is likely to be different from the transient version seen in the present simulations. We will investigate these processes in a future work. Nevertheless, it is likely the cabbeling mechanism described here will still play an important role in determining the properties of the mode water.
Acknowledgments
L. N. T. was supported by NSF Grant OCE-1459677. C. J. S. acknowledges funding from the ARC Centre of Excellence for Climate System Science Grant Number CE1101028.
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While cabbeling can create density inversions in general, in our simulations the vertical exchange or mixing of cabbeled water described in section 3c happens sufficiently quickly that inversions are avoided.
Parameters are chosen to approximately match the Gulf Stream region based on a reference salinity of S0= 36.05 and reference temperature of T0 = 18°C. The parameter values for ρ0 = ρ(T0, S0) = 1026.5 kg m−3, αT = ∂Tρ(T0, S0)/ρ0, βS = ∂Sρ(T0, S0)/ρ0, and c are obtained from the full nonlinear equation of state.
Uppercase P denotes the net field,
A second-order Runge–Kutta method is used to advect parcels. To obtain the correct parcel paths, the total flow field (model perturbation plus imposed background eddy) must be used for the advection.