# Part I: One possible answer to the "missing mixing" problem

The observed exponential stratification profile in the mid-depth interior of the global ocean was first explained by the advective-diffusive balance proposed by Munk 1966. However, tracer release experiments have shown that the vertical mixing in the mid-depth open ocean is much weaker than that required by the advective-diffusive theory. An alternative explanation involves the Southern Ocean (SO) wind which steepens the isopycnal surfaces, and the SO eddies which slump them. The competition between the wind and eddies leads to an isopycnal slope that will map the SO surface density distribution to a vertical profile in the interio. However, we have found that the resultant stratification is not exponential as observed, when vertical mixing is negligible.

Observed mid-depth interior profiles of potential density (upper), and vertical derivative of potential density (lower), and corresponding exponential fits.

Idealized GCM simulations are designed which uses a re-entrant channel to include SO dynamics. Different configurations of vertical mixing are used: uniform weak mixing (MinMix); uniform strong mixing (MunkMix); boundary mixing (MargMix). We found that the MargMix case simulates an exponential interior stratification profile, equivalent to that simulated in the MunkMix case, while the MinMix case failed to simulate an exponential profile and therefore inconsistent with observations. Munk-like temperature budget is observed in the marginal regions where mixing is strong, and it is transported into the interior, giving rise to the exponential profile.

Horizontal maps of the vertical mixing (colors) distribution. Dashed: SO channel.

Simulated temperature profile, averaged in the interior (black dots). Exponential fit (black line). Vertical integral of the advective-diffusive balance in the interior (chased blue line). In the last panel: vertical temperature profile averaged in regions of enhanced mixing (red dots), and exponential fit to this profile (red line).

# Part II: Mid-depth ocean interior stratification: Southern Ocean dynamics vs interior vertical diffusivity

In the theory of how SO dynamics determine the interior mid-depth stratification, the role played by interior vertical diffusivity is assumed to be neligible. However, we have shown that the resultant stratification profile is not exponential as observed. In the second part, we used eddy-permitting simulations where vertical mixing is only changes north of the channel to study how the SO and interior vertical mixing interact to determine the interior stratification.

(a,b,c) Surface forcing terms; (d) gray-color contours: bathymetry; colors: a snapshot of relative vorticity.

The SO dynamics can make the mid-depth interior ocean stratified even though interior vertical mixing is weak. However, the stratification profile only becomes exponential when interior vertical mixing is strong enough. The larger the interior vertical mixing, the more exponential the interior stratification profile becomes.

The interior-averaged profiles of vertical derivative of potential density, simulated (blue line), and exponential fit (red dashed line).

As interior vertical diffusivity is increased, the isopycnal slope in the SO becomes steeper, and the SO eddies become more energetic. This implies that (1) the SO isopycnal slopes are not only determined by the near-vanishing MOC condition (a balance between SO wind and eddies), but also responding to interior changes; (2) the SO eddies are not sufficiently efficient to draw APE from the mean-state slopes, and maintain a state of "marginal criticality".

(left) Averaged isopycnal slope that connects to interior between 1-3 km in the Southern Ocean; (right) Averaged eddy kinetic energy in different parts of the model domain.