Understanding mixing efficiency in the oceans: Do the nonlinearities of the equation of state matter?

There exist two central measures of turbulent mixing in turbulent stratified fluids, both caused by molecular diffusion: 1) the dissipation rate D(APE) of available potential energy (APE); 2) the turbulent rate of change Wr,turbulent of background potential energy GPEr. So far, these two quantities have often been regarded as the same energy conversion, namely the irreversible conversion of APE into GPEr, owing to D(APE)=Wr,turbulent holding exactly for a Boussinesq fluid with a linear equation of state. It was recently pointed out, however, that this equality no longer holds for a thermally-stratified compressible fluid, the ratio \xi=Wr,turbulent/D(APE) being then lower than unity and sometimes even negative for water/seawater. In this paper, the behavior of the ratio \xi is examined for different stratifications having the same buoyancy frequency N(z), but different vertical profiles of the parameter \Upsilon = \alpha P/(\rho C_p), where \alpha is the thermal expansion, P the hydrostatic pressure, \rho the density, and C_p the isobaric specific heat capacity, the equation of state considered being that for seawater for different particular constant values of salinity. It is found that \xi and Wr,turbulent depend critically on the sign and magnitude of d\Upsilon/dz, in contrast with D(APE), which appears largely unaffected by the latter. These results have important consequences for how the mixing efficiency should be defined and measured.Comment: 17 pages, 5 figures, 1 Table, accepted in Ocean Science (special issue on seawater) on July 10th 200

On the validity of single-parcel energetics to assess the importance of internal energy and compressibility effects in stratified fluids

It is often assumed on the basis of single-parcel energetics that compressible effects and conversions with internal energy are negligible whenever typical displacements of fluid parcels are small relative to the scale height of the fluid (defined as the ratio of the squared speed of sound over gravitational acceleration). This paper shows that the above approach is flawed, however, and that a correct assessment of compressible effects and internal energy conversions requires considering the energetics of at least two parcels, or more generally, of mass conserving parcel re-arrangements. As a consequence, it is shown that it is the adiabatic lapse rate and its derivative with respect to pressure, rather than the scale height, which controls the relative importance of compressible effects and internal energy conversions when considering the global energy budget of a stratied fluid. Only when mass conservation is properly accounted for is it possible to explain why available internal energy can account for up to 40 percent of the total available potential energy in the oceans. This is considerably larger than the prediction of single-parcel energetics, according to which this number should be no more than about 2 percent

Irreversible compressible work and APE dissipation in turbulent stratified fluid

Although it plays a key role in the theory of stratified turbulence, the concept of available potential energy (APE) dissipation has remained until now a rather mysterious quantity, owing to the lack of rigorous result about its irreversible character or energy conversion type. Here, we show by using rigorous energetics considerations rooted in the analysis of the Navier-Stokes for a fully compressible fluid with a nonlinear equation of state that the APE dissipation is an irreversible energy conversion that dissipates kinetic energy into internal energy, exactly as viscous dissipation. These results are established by showing that APE dissipation contributes to the irreversible production of entropy, and by showing that it is a part of the work of expansion/contraction. Our results provide a new interpretation of the entropy budget, that leads to a new exact definition of turbulent effective diffusivity, which generalizes the Osborn-Cox model, as well as a rigorous decomposition of the work of expansion/contraction into reversible and irreversible components. In the context of turbulent mixing associated with parallel shear flow instability, our results suggests that there is no irreversible transfer of horizontal momentum into vertical momentum, as seems to be required when compressible effects are neglected, with potential consequences for the parameterisations of momentum dissipation in the coarse-grained Navier-Stokes equations

Isoneutral control of effective diapycnal mixing in numerical ocean models with neutral rotated diffusion tensors

It is well known that there are infinite number of ways of constructing a globally-defined density variable for the ocean, with each possible density variable having a priori its own distinct diapycnal diffusivity. Because no globally-defined density variable can be exactly neutral, numerical ocean models tend to use rotated diffusion tensors mixing separately in the directions parallel and perpendicular to the local neutral vector at rates defined by the isoneutral and dianeutral mixing coefficients respectively. To constrain these mixing coefficients from observations, one widely used tool are inverse methods based on Walin-type water masses analyses. Such methods, however, can only constrain the diapycnal diffusivity of the globally defined density variable $\gamma$ —such as $\sigma_2$ —that underlies the inverse method. To use such a method to constrain the dianeutral mixing coefficient therefore requires understanding the relations between the different diapycnal diffusivities. However, this is complicated by the fact that the effective diapycnal diffusivity experienced by is necessarily partly controlled by isoneutral diffusion owing to the unavoidable misalignment between iso- surfaces and the neutral directions. Here, this effect is quantified by evaluating the effective diapycnal diffusion coefficient pertaining to five widely used density variables: Jackett and McDougall (1997) $\gamma^n$, Lorenz reference state density $\rho_{ref}$ of Saenz et al. (2015), and three potential density variables $\sigma_0$, $\sigma_2$ and $\sigma_4$. Computations are based on the World Ocean Circulation Experiment climatology, assuming either a uniform value for the isoneutral mixing coefficient or spatially varying values inferred from an inverse calculation. Isopycnal mixing 15 contributions to the effective diapycnal mixing yield values consistently larger than 10^(-3) m^2/s in the deep ocean for all density variables, with $\gamma^n$ suffering the least from the isoneutral control of effective diapycnal mixing, and $\sigma_0$ the most. These high values are due to spatially localised large values of non-neutrality, mostly in the deep Southern Ocean. Removing only 5% of these high values on each density surface reduces the effective diapycnal diffusivities to less than 10^(-4) m^2/s. The main implication of this work is to highlight the conceptual and practical difficulties of relating the diapycnal mixing diffusivities inferred from global budgets or inverse methods relying on Walin-like water mass analyses to locally defined dianeutral diffusivities. Doing so requires the ability to separate the relative contribution of isoneutral mixing from the effective diapycnal mixing. Because it corresponds to a special case of Walin-type water mass analysis, the determination of spurious diapycnal mixing based on monitoring the evolution of the Lorenz reference state may also be affected by the above issues when using a realistic nonlinear equation of state. The present results thus suggest that part of previously published spurious diapycnal mixing estimates could be due to isoneutral mixing contamination

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter

Generalised patched potential density and thermodynamic neutral density: two new physically-based quasi-neutral density variables for ocean water masses analyses and circulation studies

In this paper, two new quasi-neutral density variables — generalised patched potential density (GPPD) and thermodynamic neutral density γT — are introduced, which are showed to approximate Jackett and McDougall (1997) empirical neutral density γn significantly better than the quasi-material rational polynomial approximation γa previously introduced by McDougall and Jackett (2005b). In contrast to γn, γT is easily and efficiently computed for arbitrary climatologies of temperature and salinity, both realistic and idealised, has a clear physical basis rooted in the theory of available potential energy, and does not suffer from non-material effects that makes γn so difficult to use in water masses analysis. In addition, γT is also significantly more neutral than all known quasi-material density variables, such as σ2, while remaining less neutral than γn. Because unlike γn, γT is mathematically explicit, it can be used for theoretical as well as observational studies, as well as a generalised vertical coordinate in isopycnal models of the ocean circulation. On the downside, γT exhibits inversions and degraded neutrality in the polar regions, where Lorenz reference state is the furthest away from the actual state. Therefore, while γT represents progress over previous approaches, further work is still needed to determine whether its polar deficiencies can be corrected, an essential requirement for γT to be useful in Southern Ocean studies for instance

Local available energetics of multicomponent compressible stratified fluids

We extend the local theory of available potential energy (APE) to a general multicomponent compressible stratified fluid, accounting for the effects of diabatic sinks and sources. As for simple compressible fluids, the total potential energy density of a fluid parcel is the sum of its available elastic energy (AEE) and APE density. These respectively represent the adiabatic compression/expansion work needed to bring it from its reference pressure to its actual pressure and the work against buoyancy forces required to move it from its reference state position to its actual position. Our expression for the APE density is new and derived using only elementary manipulations of the equations of motion; it is significantly simpler than existing published expressions, while also being more transparently linked to the relevant form of APE density for the Boussinesq and hydrostatic primitivve equations. Our new framework is used to clarify the links between some aspects of the energetics of Boussinesq and real fluids, as well as to shed light on the physical basis underlying the choice of reference state(s) in local APE theory