Dimensionally AGM,Redi=L2/T, where L is a length scale and T a timescale. Most modern climate models use some version of the closure first proposed by where 1/T is given by the growth rate of baroclinically unstable waves assuming the growth rate scales as that described by . In this case, assuming thermal wind balance, one obtains 1/T=f/Ri1/2=f|dU/dz|/N≈|Sρ|N, where Sρ is the slope of density surfaces . In climate models, the slope is usually taken as some average slope over a range of depths (for example 100–2000 m in ). All else being equal, these approximations predict that growth rates and diffusion coefficients should be very small in the center of the subtropical gyres where isopycnals are relatively flat. An example is shown in Fig. a for the GFDL CM2Mc model of . Because of its role in moving light tropical waters into polar regions, particularly within the Southern Ocean, AGM has a first-order impact on the large-scale circulation of the ocean . Values much larger than 1000 m2s-1 in the main pycnocline tend to produce unrealistic smoothing of the pycnocline and suppression of the large-scale overturning circulation unless unrealistically high values of vertical mixing are applied in the tropics. Thus, in models where ARedi=AGM, values applied in the center of subtropical gyres are often very low, on the order of a few hundred square meters per second.

Measuring AGM directly from observations in nearly impossible, but several observational estimates of ARedi do exist; these estimates seem incompatible with the values of AGM described above. Modern estimates of ARedi using a variety of methods find relatively large values in the ocean interior. For example , using SF6 released into the thermocline, estimated an east–west diffusion of 1500 m2s-1 and a north–south diffusion of 600 m2s-1. , looking at the Guinea Dome, found a value of 1000–1200 m2s-1 in the east–west direction and 500 m2s-1 in the north–south direction. Estimates with subsurface floats yield even larger values, with estimating values of 2000–9000 m2s-1 in the equatorial Pacific and finding values of 1500–3000 m2s-1 in the North Atlantic. A recently developed “pseudo-observational” approach uses satellite-derived surface geostrophic velocities to drive tracer or particle transport simulations . Lateral diffusion coefficients can then be calculated directly from such simulations. applied this method globally, producing the map of near-surface ARedi shown in Fig. c. Finally, eddy diffusion coefficients can be estimated from eddy-resolving numerical models . All these diverse estimates suggest that ARedi can be several times larger than the values commonly used for AGM. This is the isopycnal mixing paradox.

One question is whether this paradox persists into the deep ocean. There are only a few direct estimates of ARedi below the main thermocline. , in the Santa Barbara Basin, found a diffusion coefficient of 10–20 m2s-1 on spatial scales of 10 km. in the Brazil Basin found a value of around 100 m2s-1. Both of these, however, are at depths where lateral motion is likely to be strongly physically constrained by topography. There are no observational studies of which we are aware that directly estimate lateral diffusion between the base of the thermocline and depths associated with the tops of the mid-ocean ridges. In the few models where it is allowed to vary with depth, AGM is assumed to drop in the deep ocean, as eddy kinetic energy is lower there. A zonal average of AGM illustrating this is shown for the NCAR CESM model in Fig. b. However, if the surface values of ARedi are too low, forcing it to decrease with depth may actually give less realistic deep values than the vast majority of parameterizations which simply prescribe values of 500–1000 m2s-1 uniformly throughout the water column. Evaluating the extent to which this is the case is a major thrust of this paper.

One possible resolution of the paradox is the highly anisotropic nature of eddy transport. and show that eddy diffusion can be up to 10 times stronger in one direction (generally the zonal direction) than in the perpendicular direction. Yet all the climate models used in the CMIP5 (Coupled Model Intercomparison Project Phase 5) exercise employ isotropic (scalar) diffusivities for both AGM and ARedi. Experiments are ongoing regarding the consequences of anisotropic eddy diffusion for ocean models (Reckinger and Fox-Kemper, 2015). It is not clear, however, how well the observational estimates capture the details of anisotropy, given that a particular challenge in making such measurements is removing imperfectly known background shear flows (see for example LaCasce et al., 2014, who resolved this issue by looking at the cross-stream diffusivity only). Such shear flows may also cause tilts in the major and minor axes of the dispersion tensor that reflect resolved flows rather than details of the turbulence. Employing higher subgrid-scale diffusivities in the along-stream direction in models where one already resolves the large-scale flow may thus count the impact of the shear dispersion twice.

Here, instead, we investigate another possible resolution: the non-equivalence between AGM and ARedi. Despite the common assumption that they are equal, there is ample evidence that the two coefficients can differ significantly. Linear quasigeostrophic theory can be used to show that the diffusion coefficient governing lateral buoyancy transport (corresponding to AGM) is very different from the coefficient governing along-isopycnal PV (potential vorticity) transport, which is asymptotically equal to ARedi . This theory has been tested in fully nonlinear primitive equation simulations . These studies all show ARedi to be larger than AGM. Our experiments are designed to test whether elevated values of ARedi throughout the water column give unrealistic results for helium isotopes and radiocarbon.

Given a decay of 238U of 4.26 MeV =6.02×10-13 J and a ratio of R=3He/4He of 1.6×10-5 as in , we expect one molyear-1 of 3He to be associated with a global heat flux of about 800 MW. It is estimated that U decay contributes about 10–20 TW of heating, implying a flux of around 12 500–25 000 mol 3He year-1. However, , using a box model calibrated with observations of 3He, estimated a much lower flux of 1070 molyear-1. This has led to suggestions from geochemists that the mantle as a whole does not convect, so that helium is trapped in the deep mantle while heat escapes . While alternative explanations for this difference have been proposed see for example the exact value of the mantle helium flux remains an important calibration point for geochemistry.

Box models are, however, a crude representation of the true rate at which the ocean is ventilated. As each box is rapidly homogenized, the effective transport of tracer from the deep ocean to the surface may be much larger than in the real world. This means that box models with the right mean overturning fluxes could overestimate the speed at which tracers make it out of the deep ocean, and thus would require too large a source of 3He in the deep ocean. On the other hand, the neglect of mixing fluxes may mean that box models can underestimate the true flux.

simulated helium isotopes in a suite of ocean general circulation models, which (presumably) have a more realistic representation of the processes that overturn the ocean. However, this paper found large ranges in the inventory of mantle helium. Given the differences in model construction it was far from clear why this was the case. revisited this problem using a suite of ocean-only models developed to look at the differing roles of vertical and lateral diffusion in setting global circulation. In these models AGM and ARedi were varied together, with the result that increasing the lateral mixing generally reduced vertical exchange. Examination of the relationship between δ3He and Δ14C in these models showed that those models with the most realistic radiocarbon, and thus presumably the most realistic ventilation of the deep ocean, tended to retain too much mantle 3He. The solution proposed was to reduce the flux of mantle 3He yet further to 527 molyear-1.

However, if ARedi changes are not directly tied to changes in AGM, as may be the case in the real ocean, the picture is potentially different. As shown in , increasing ARedi tends to destratify the high latitudes, allowing more exchange with the deep ocean and causing sea ice to retreat. Thus, increasing ARedi would be expected to reduce mantle 3He. The impact on radiocarbon is, however, less clear, given the long timescales involved in equilibration of carbon isotopes. An illustrative model of this is schematized in Fig. , in which a source of tracer is introduced into the deep box, circulation transports tracer between the interior box and the surface box producing a flux Mo×(Cd-Cs) and gas exchange restores the surface box to an equilibrium concentration with a flux Ma×(Cs-Ceq). At equilibrium, both fluxes are equal to S and the deep concentration is just Cd=Ceq+S×(1/Mo+1/Ma) and similarly Cs=Cd×Mo+Ceq×MaMo+Ma.

Suppose (as we do) that the deep Pacific is primarily ventilated from the Southern Ocean. For radiocarbon, the surface concentrations in the Southern Ocean are closer to the deep ocean mean of -170 ‰  than to zero, implying that Mo is actually a little bigger than Ma (consistent with the idea that it takes about 10 years to equilibrate the surface mixed layer). Thus, about half of the buildup of radiocarbon in the deep Pacific is due to slow gas exchange and will not be changed by increasing the ocean circulation. By contrast, helium isotopes at the surface of the Southern Ocean are in equilibrium with the atmosphere, implying a much larger Ma. This implies that helium is more sensitive to changes in mixing than is radiocarbon. Could resolving the isopycnal mixing paradox by breaking ARedi and AGM apart result in a less stringent helium–heat paradox as well? Or are changes in the effective mixing rate over the range of potential ARedi too small to make a difference?

The coarse-resolution Earth system models do a reasonable job at reproducing the large-scale hydrography of the ocean. Figure a shows the horizontally averaged temperature in the models. The horizontal axis is cut off at 12∘C (corresponding to a depth of around 200 m) in order to highlight the differences at depth. The relative lack of sensitivity of the mean thermocline to changes in ARedi stands in strong contrast to AGM, which plays a major role in setting temperatures at these depths . In the deep ocean, there is a general tendency for the models with higher levels of mixing to produce more realistic cold temperatures.

Salinity (Fig. b) shows a quite different behavior. The observed mean salinity profile is low at the ocean surface as a result of fresh mixed layers at high latitudes, it exhibits a subsurface maximum associated with the subtropical gyres and a subsurface minimum associated with the formation of mode and intermediate waters. The lowest mixing case (AREDI400, black line) tends to overestimate the strength of polar haloclines, while the higher mixing cases (AREDI1200, AREDI2400 and ABER2D) all capture the near-surface salinity maximum while erasing vertical gradients in salinity below about 1000 m. The AREDI800 simulation does a good job of capturing the deeper salinity structure but less of a good job near the surface. Together the temperature and salinity plots illustrate the difficulties in “tuning” Earth system models – parameterizations that improve one field may not improve another.

Radiocarbon (Fig. c) is generally too high in the models, both at the surface and at depth. This suggests that either our rates of air–sea exchange, vertical exchange (due to advection and diffusion), or both are too vigorous. In general, increasing mixing tends to increase radiocarbon in the deep ocean, though the intramodel differences are generally smaller than the model–observation differences. To first-order, the average depletion in the deep ocean is within about 15 ‰  of observations, implying an error in radiocarbon age of around 120 years.

Helium isotope anomalies (Fig. d) are also too large in the deep ocean (implying too little ventilation in contrast to radiocarbon). As expected, increasing ARedi tends to reduce the deep isotope anomaly. Also as expected, the relative change appears to be much larger for helium than it is for radiocarbon. While the deep overprediction of mantle helium is striking, the total mantle helium inventory is less sensitive, with the average value for AREDI400 of 13.2 % and AREDI2400 of 9.3 % bracketing the observed value of 11.1 %. The models capture most of the observed horizontally averaged δ3He signal.

As discussed in , increasing ARedi results in destabilizing the high-latitude Southern Ocean and North Pacific. Models with more intense eddy mixing stir more salt into the high-latitude surface layer. This reduces the salinity contrast across the winter halocline, which is the dominant factor in determining vertical exchange in the subpolar North Pacific and Southern Ocean. The results of this can be seen in Fig. . The AREDI400 simulation with the lowest mixing captures the value of the minimum in radiocarbon in the North Pacific, though the depth at which this minimum is seen is too great. As ARedi increases the result is to bring more young water down in the North Pacific, reducing the correlation between the observed and modeled zonal mean from 0.88 (AREDI400) to 0.71 (AREDI2400), and substantially increasing the RMSE (root mean squared error) in radiocarbon concentration from 22 (AREDI400) to 42 ‰  (AREDI2400). The spatially varying diffusion tends to act like low diffusion in the south, but higher diffusion in the north, and so also produces enhanced, unrealistic northern sinking, though not to the same extent as the AREDI2400 case. The differences between the AREDI400 and AREDI800 runs are largely due to an increase in the gradient between the Southern Ocean and the North Pacific, with the Southern Ocean values of radiocarbon remaining relatively unchanged, suggesting that lateral transport is important here. However, the differences between AREDI1200, AREDI2400 and AREDI800 show the impact of decreases in high-latitude stratification (particularly in the North Pacific) as well. In recent work we show that the destratification of the North Pacific also results in unrealistic uptake of anthropogenic carbon in this region.

Mantle helium shows a behavior somewhat similar to radiocarbon, but with the difference that the highest values are found in the tropical South Pacific rather than in the far north (Fig. a). The AREDI400 case largely captures the location of the maximum zonal mean δ3He anomaly but, as with radiocarbon, increasing ARedi results in ventilation that removes mantle helium from the far northern Pacific, decreasing the correlation with observations and raising the mean error. There is, however, some sense that the lower values of ARedi result in an excessively high peak of mantle helium in the tropics, so that the RMSE in zonally averaged δ3He is actually lowest (4.5 %) for the AREDI800 simulation rather than the AREDI400 simulation (5.5 %), despite having a lower correlation coefficient (0.74 vs. 0.78). The ABER2D simulation does less well than the AREDI800 in this case, though the peak value in the South Pacific is somewhat better captured.

The changes in deep tracers are reflected in changes in the scatterplot of deep δ3He vs. Δ14C (Fig. ). As the mixing increases, the peak seen in the southeastern Pacific becomes smoothed out and the maximum depletion decreases. However, all five runs lie close to the observed relationship. Since the δ3He values are linear in the source term, it is easy to see that doubling this term to the value of will lead to a strong offset. The results of appear to be robust against changes in ARedi that are not directly associated with changes in AGM. Resolving the isopycnal mixing paradox thus does not affect the helium–heat paradox.

This result is contrary to what might be expected from the changes in the tracers seen in Figs.  and , which show clear changes in the extreme values of these tracers in the deep ocean and substantially greater penetration of young waters as ARedi increases. However, these changes in the endpoints do not substantially change the effective rate at which tracer signals make their way from the deep ocean to the surface. Our results thus suggest that the key barrier to transfer between the deep ocean and the surface is not mixing across the mixed layer base but transport from the ocean interior to the high-latitude regions. Such transport is accomplished both by advection and turbulent diffusion. If one considers a Péclet number for the deep South Pacific UL/ARedi, deep velocities U are on the order of 1 mms-1 but the length scale L between where He is injected and where isopycnals reach the surface is on the order of 4000 km. Thus, even for the highest value of ARedi, the Péclet number is greater than 1 and advection dominates mixing.

Examination of changes in overturning supports this idea. We can define the overturning as Mover=∫z1z2∫xwxev×dx×dz, where v is the northward velocity, and xw,xe,z1 and z2 are the limits of integration in the west, east, and vertical directions. The integral is taken at 30∘S in the Pacific. If the integral is performed relative to the surface (z1=z,z2=0, Fig. a) the similarity in the northward transport of surface water is emphasized, but differences appear in the deep ocean. Note that the value does not go to zero at the bottom because water that flows into the southern Pacific leaves through the Indonesian throughflow and Bering straits. This net flux does change as ARedi changes. Summing the northward transport from the bottom of the ocean upward (z1=H,z2=z where H is the ocean depth, Fig. b), we see that the models with constant ARedi all have similar inflows of deep water of around 10 Sv (only the ABER2D simulation is noticeably lower). The main thing that changes between the simulations is whether this water is largely returned below the thermocline (as happens for the higher mixing cases) or whether some small fraction of it upwells into the tropical thermocline and is returned through the Indonesian throughflow (as appears to be the case for AREDI400). From the point of view of flushing the deep tropical Pacific, however, it is largely irrelevant which pathway is taken. Instead, what matters is the throughflow, which is relatively constant across the different models.

Both the models and observations show high values in the southeastern Pacific, where a rapidly spreading rift releases large amounts of mantle helium. The sharpness of this plume, however, varies substantially between the models. In Fig. , the distributions at 2500 m are overlaid with the actual observations between 2250 and 2750 m (filled squares). Differences in color thus highlight locations where the model is mismatched with the in situ observations. In Fig. a we see that there are some differences between the gridded and in situ observations. This reflects the fact that in order to extrapolate over the relatively coarse observations in the deep ocean to a global data set, had to define a radius of influence that effectively smooths the data and may be comparable to the size of some of the features seen in the model output. For this reason, we have not chosen to present RMSEs, as these would imply a greater degree of precision than is justified given the coarseness of the sampling and the model.

However, even a qualitative comparison makes it clear that the mixing exerts a strong control over how well the models match the observations. In the AREDI400 run (Fig. b), the symbols over the East Pacific Rise show up as lower than the model, which produces a peak value exceeding 70 %, while the highest values in the observations are around 45 %. By contrast, the AREDI2400 model produces no values higher than 36 %. The AREDI800, AREDI1200 and ABER2D models all produce peaks that are qualitatively similar to observations.

This is not merely the impact of the depth chosen. Figure  shows a depth–longitude section of δ3He cutting through the center of the plume between latitudes of 20 and 30∘ S. Again, we see that the observations over the ridge show up as too low in the AREDI400 run, too high in the AREDI2400 run and close to the simulated values for the AREDI800, AREDI1200 and ABER2D simulations (though note that the highest values in the models are displaced downwards relative to observations). This suggests that very low diffusion coefficients in the deep southeastern Pacific are not consistent with the release of helium there, in contrast to current theory. The values that are most consistent with the observations are comparable to what is found nearer to the surface. This suggests that the isopycnal mixing paradox is not simply a matter of ARedi being much larger than AGM in the surface layer but also in the deep ocean.