Differential Diffusion in the Ocean

1. Differential Diffusion in the Ocean Tian Tian

2. Contents • What is differential diffusion? • Why differential diffusion is important? • The existing evidence of differential diffusion • Discussions and Conclusions

3. It is generally assumed that the diffusivity representing the mixing process is the same for salt and heat. • In present study,all numerical models parameterize unresolved interior diapycnal fluxes as an eddy diffusivity K times a mean property gradient normal to the isopycnals. • Most models also use the same eddy diffusivities for heat and salt. However, the transport of heat and salt in the ocean by small scale processes are different. • The best known of these processes are double-diffusive processes (salt fingering and double-diffusive layering), which transport heat and salt unequally because of much larger molecular diffusivity for heat (10-3m2s-1) than for salt (10-5m2s-1).

4. Gargett (1998) raises the possibility that heat and salt may mix at different rates in stratified turbulence. • The conceptual picture of differential mixing is that the scalar fields are stirred by the large eddies of the turbulent flow and strained to the scalar dissipation scales in which molecular diffusion removes the variance. • If the turbulence diffusion is weak, the scalar variance would not be removed before the fluid stirred by the eddies restratifies. Therefore, more of the scalar with the large molecular diffusivity would be transferred vertically by the mixing. • Since the molecular diffusivity for temperature T is around 100 times larger than the salinity S, the net vertical diffusivity for heat is larger than for salt.

5. Why differential diffusion is important? • The differential diffusion of heat and salt occurs at small spatial scales, however, it has potentially huge effect on large-scale ocean processes, such as thermohaline circulation and the global resupply of nutrients. • Gargett and Holloway (1992) set different vertical diffusivities for T and S in the sense of expecting differential diffusion in a model for ‘North Atlantic’ ocean basin, an increase of 50% in the meridional overturning circulation. • The magnitude of such changes are less in the sense of global range (Merryfield, Holloway & Gargett, 1999), differential diffusion may important in diurnal and seasonal pycnoclines where vertical diffusion is associated with intermittent turbulence occurring in the presence of strongly stable stratification. • Differential mixing by turbulence can drive lateral thermohaline intrusions (Hebert, 1998; Merryfield, 2002) and then affect horizontal mixing rates as well.

6. Lab evidence: Fig. 1. Comparison between directly measured entrainment (mixing) velocity associated with grid-stirring on one side of a density interface produced respectively by (filled circles) temperature and (open circles) salt. The dimensionless mixing rates obtained by Turner (1968) are plotted here against a Richardson number Ri0 calculated with turbulent length l1 and velocity u1 scales characteristic of the grid turbulence. (From Turner, 1973)

7. Lab evidence: Fig. 2. Entrainment rate as a function of Richardson number Ri0, for grid-stirring on both sides of a density interface produced by temperature alone (open circles), salt alone (crosses), and both temperature and salt (triangles) present simultaneously. Rundown experiments proceed from right to left. In the simultaneously T- and S-stratified run, density entrainment first follows the T-entrainment curve, but finally tends towards the S-entrainment curve as the initial contrast in T is mixed away more rapidly than the initial S contrast (From Altman & Gargett, 1990).

8. Lab evidence: Fig. 3. Ratio of the effective diffusivity of fluroscein-dextran to that of rhodamine as a function of the effective diffusivity of rhodamine. The lower dashed line shows the molecular diffusivity ratio. The squares are the experiments with salt-only stratification. The circles are the experiments with the stratification due to salt and dextran in equal fractions. For each data point, the solid lines are the standard deviation in estimates of the diffusivity and diffusivity ratio. (From Hebert et al, 2003)

9. Model evidence: Fig. 4. Time dependent fluxes for run I (low initial energy, red spectrum) Top: advective fluxes FT (dotted line), FS (solid line), and flux differential FT-FS (dashed line). Bottom: cumulative fluxes for FT (dotted line), FS (solid line), and FT-FS (dashed line). All fluxes are in units of the thermal conductive flux.

10. Model evidence: Fig. 5. Time dependent fluxes for run II (high initial energy, blue spectrum) Top: advective fluxes FT (dotted line), FS (solid line), and flux differential FT-FS (dashed line). Bottom: cumulative fluxes for FT (dotted line), FS (solid line), and FT-FS (dashed line). All fluxes are in units of the thermal conductive flux.

11. Model evidence: Fig. 6. Results of direct numerical simulation of T and S mixing by two-dimensional turbulence, displayed in the space of {Re, Fr} calculated from initial turbulence properties. (a) Ratio of cumulative T-flux to cumulative S-flux. The dashed line schematically divides the region of unequal cumulative fluxes (ratio>1, below the line) from that with ratio~1. (b) Maximum instantaneous flux difference FT – FS, in units of thermal conductive flux. Above the dashed line, FT-FS exceeds roughly twice the thermal conductive flux. These results suggest that differential diffusion is important (fluxes greater than twice molecular, with cumulative flux ratios >1) in the parameter range roughly between the two dashed lines. (From Merryfield et al., 1998).

12. Observational evidence: Fig. 7. The upper panels show free-fall CTD profiles of T (light lines) and S (heavy lines) taken from an anchored ship at (A) high water slack tide and (B, C) as tidal flow increased over the subsequent three hours. Ordinary turbulent mixing of the initial water column would retain the original linear relation shown in (A), merely decreasing the range of T and S. Instead, the turbulence (revealed variously by large density overturning scales, critical mean shears and large vertical velocities, not shown here) results in a strongly nonlinear deformation of the T/S relation. The observed change in the T/S relation (lower panel) is in the sense that would be associated with differential diffusion, i.e. KT> KS. (From Gargett, 2003).

13. Observational evidence: Fig. 8. Potential temperature and salinity sampled at 2-db intervals at Oden 91 station 17 (88°00.3'N, 85°03.3'E) in Amundsen Basin,after Anderson et al. (1994). Approximate depth ranges having background gradients in the diffusive convective, salt fingering, and double-diffusively stable regimes are indicated, based on a subjective removal of the effects of intrusions from the potential T and S profiles. Dotted lines bracket strong intrusive features in the stable Upper Polar Deep Water.

14. Observational evidence: Fig. 9. Statistics of the diffusivity ratio based on large-eddy transports d0 (top) and based on the scalar dissipation rates dx(bottom). (From Nash & Moum, 2002)

15. Double diffusion and differential diffusion: Double diffusion is different from differential diffusion. The former case is driven by the release of potential energy from the gravitationally unstable in salinity or temperature. For the salt fingering regime, and flux ratio between T and S is less than 1, Turner angles between 45° and 90°; for the diffusive convective regime, and flux ratio between T and S is greater than 1, Turner angles between -45° and -90°. Differential diffusion case is not restricted by gravitationally unstable of scalars in mean field. Because it is caused by incomplete turbulent mixing which vertically transported heat is more than that of salt, we expected flux ratio between T and S is greater than 1. Differential diffusion could be happened in Turner angles between -45°and 45° where the stratification is stable both in temperature and salinity.

16. Differential diffusion is stirred by large eddies of turbulent flow is strained to scalar dissipation scales within a time depends on Schmidt number and Reynolds number. For very diffusive scalars (small Sc), scalar variance can be entirely erased within the decay time scale of the velocity field; for nondiffusive scalars (very high Sc) will have no transfer of variance to their much smaller diffusive scales within the same period. Rehamann (1995) estimated total time required to strain and mix scalars (TM) and turbulence decay time (TD). When TM>>TD, weak turbulence occurs, parcel displaced by turbulence will return to its original place by decay of the eddy before there is time for significant transfer scalar variance to its dissipation scale, this corresponding to differential diffusion; when TM<<TD, there is enough time for scalar variance to be transferred to small scales, mixing completely finished before the turbulence itself decayed away, this has no differential diffusion.