Diapycnal motion , diffusion , and stretching of tracers in the ocean

Small-scale mixing drives the diabatic upwelling that closes the abyssal ocean overturning circulation. Measurements of in-situ turbulence reveal that mixing is bottom-enhanced over rough topography, implying downwelling in the interior and stronger upwelling in a sloping bottom boundary layer. However, in-situ mixing estimates are indirect and the inferred vertical velocities have not yet been confirmed. Purposeful releases of inert tracers, and their subsequent spreading, have been used to independently infer turbulent diffusivities; however, these Tracer Release Experiments (TREs) provide estimates in excess of in-situ ones. In an attempt to reconcile these differences, Ruan and Ferrari (2021) derived exact buoyancy moment diagnostics, which we here apply to quasi-realistic simulations. We show in a numerical simulation that traceraveraged diapycnalmotion is directly driven by the tracer-averaged buoyancy velocity, a convolution of the asymmetric upwelling/downwelling dipole. Diapycnal spreading, however, involves both the expected contribution from the tracer-averaged in-situ diffusion and an additional non-linear diapycnal stretching term. These diapycnal stretching effects, caused by correlations between buoyancy and the buoyancy velocity, can either enhance or reduce tracer spreading. Diapycnal stretching in the stratified interior is compensated by diapycnal contraction near the bottom; for simulations of the Brazil Basin Tracer Release Experiment these nearly cancel by coincidence. By contrast, a numerical tracer released near the bottom experiences leading-order stretching that varies in time. These results suggest mixing estimates from TREs are not unambiguous, especially near topography, and that more attention should be paid towards the evolution of tracers’ first moments. 10

(1) where u is the velocity vector, ∇ = , , is the gradient operator, and is an isotropic 134 turbulent diffusivity (assumed to be the same for all tracers). Buoyancy, tracer concentrations, 135 and velocity have been filtered on spatial and temporal scales larger than those associated with 136 small-scale turbulence, and the filtered scalar fluxes are parameterized as an enhanced diffusive flux 137 F = − ∇ , where the effective turbulent diffusivity is much larger than the molecular diffusivity.

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For simplicity of exposition, we here approximate density as a linear function of temperature; thus, 139 density , buoyancy , and temperature are all proportional and will be used interchangeably 140 throughout: ≡ − 0 ≈ , where 0 is a reference density and is the thermal contraction 141 coefficient. (Salinity can be easily included as long as it is assumed to also be linearly proportional 142 to density.) 143 a. Exact tracer-weighted buoyancy-moment models 144 In his classic paper, Taylor (1922) demonstrates that the growth rate of half the second moment 145 of a 1D tracer distribution in physical space is exactly equal to its diffusivity. Ruan  which is the magnitude of the diapycnal velocity through buoyancy space (e.g. Marshall et al. 155 1999). We have taken an additional step of converting to physical velocity and diffusivity units by 156 normalizing by the appropriate tracer-weighted powers of the buoyancy gradient.
so c 1,3 stretches so c 1,2 contracts (T 1 − T 2 )(ω 1 − ω 2 ) < 0  where (x) is the Delta function. The evolution of the first moment (3) is simply given by the twice 168 the average buoyancy velocity of the two patches, where we use the shorthand ≡ (x ). The evolution of the centered second moment (4), is given where Δ ≡ − and Δ ≡ − are buoyancy velocity and temperature differences between 172 the two patches, respectively. While the first moment tendency is simply given by the average of the 173 two patches' tendencies, the centered second moment tendency includes an additional non-linear 174 interaction term. If the warmer patch upwells faster than the colder patch (Δ Δ > 0), this term 175 drives diapycnal stretching (e.g. 1,3 and 3,2 in Figure 1); conversely, Δ Δ < 0 corresponds to diapycnal contraction (e.g. 1,2 in Figure 1).

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A corollary of (7)  and is thought to provide much of the available potential energy that drives sub-inertial abyssal    Within the first few eddy turnover timescales, the released tracer blobs are stirred into a web 222 of filaments along isopycnals by submesoscale eddies (e.g. Figure 3). While the BBTRE and

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At the other extreme, the Bottom tracer is released entirely in the BBL and thus upwells vigorously 286 upon release, with ≈ >0 (Figure 6g). As some of the tracer eventually spreads into the SML 287 above and the strictly negative contribution <0 grows, the net upwelling of the tracer weakens 288 over time .

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The Bottom tracer is released near the bottom of a weakly stratified depression along the canyon thalweg, and its average stratification increases dramatically over the first 200 days (Figure 6a,b). Thus, the early diapycnal upwelling and spreading is enhanced when converting to physical space because the buoyancy surfaces are on average much further apart than they are later on (see normalization in equations 3 and 4).  The Crest release is perhaps the most interesting: at first, the Crest tracer is in the SML far above

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In practice, however, instantaneous measurement of the tracer-weighted temperature (or tem-309 perature variance) tendency is infeasible. Instead, practical methods are akin to estimating the 310 time-average of the right hand-side of eq. 3 (or eq. 4) from finite differencing of the tracer-weighted 311 volume-averaged temperature (or temperature variance) between observational surveys at 0 and 1 , typically representing two separate cruises separated  The key to reconciling the two diagnostics is that the tracer distributions, while initially compact, Tracer kernels with widths less than the thickness of the BBL accurately reproduce its -structure,   The plotted quantity is the summand in eq. 14, which are integrated such that the contributions from each bin where e ≡ u · n − |∇ | n is the diapycnal velocity and V (˜ < ) is the volume enclosing any water defined (see Figure B1).

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In practice, meaningful evaluation of this integral in the slope-native configuration requires