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

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

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

Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state

In this paper, the concept of available potential energy (APE) density is extended to a multicomponent Boussinesq fluid with a nonlinear equation of state. As shown by previous studies, the APE density is naturally interpreted as the work against buoyancy forces that a parcel needs to perform to move from a notional reference position at which its buoyancy vanishes to its actual position; because buoyancy can be defined relative to an arbitrary reference state, so can APE density. The concept of APE density is therefore best viewed as defining a class of locally defined energy quantities, each tied to a different reference state, rather than as a single energy variable. An important result, for which a new proof is given, is that the volume integrated APE density always exceeds Lorenz’s globally defined APE, except when the reference state coincides with Lorenz’s adiabatically re-arranged reference state of minimum potential energy. A parcel reference position is systematically defined as a level of neutral buoyancy (LNB): depending on the nature of the fluid and on how the reference state is defined, a parcel may have one, none, or multiple LNB within the fluid. Multiple LNB are only possible for a multicomponent fluid whose density depends on pressure. When no LNB exists within the fluid, a parcel reference position is assigned at the minimum or maximum geopotential height. The class of APE densities thus defined admits local and global balance equations, which all exhibit a conversion with kinetic energy, a production term by boundary buoyancy fluxes, and a dissipation term by internal diffusive effects. Different reference states alter the partition between APE production and dissipation, but neither affect the net conversion between kinetic energy and APE, nor the difference between APE production and dissipation. We argue that the possibility of constructing APE-like budgets based on reference states other than Lorenz’s reference state is more important than has been previously assumed, and we illustrate the feasibility of doing so in the context of an idealised and realistic oceanic example, using as reference states one with constant density and another one defined as the horizontal mean density field; in the latter case, the resulting APE density is found to be a reasonable approximation of the APE density constructed from Lorenz’s reference state, while being computationally cheaper

Negative APE dissipation as the fundamental criterion for double diffusive instabilities

Understanding the energetics and mechanisms of double diffusive instabilities in stratified fluids remains challenging. These instabilities can arise from a mechanically unforced stably stratified state with no available potential energy (APE), thus requiring the background potential energy (BPE) to be dynamically active rather than passive. Middleton and Taylor proposed a criterion linking BPE extraction to APE with the diapycnal component of the diffusive buoyancy flux, but it only predicts diffusive convection instability, not salt finger instability. We argue that the extraction of BPE into APE is better determined by the sign of the APE dissipation rate, which relates to turbulent diapycnal mixing diffusivity and mixing efficiency. Our theory predicts that in doubly-diffusive fluids, the APE dissipation rate can be enhanced, suppressed, or even have the opposite sign compared to a simple fluid, depending on the diffusivity ratio, density ratio, and a spiciness parameter measuring density-compensated thermohaline variations. In the laminar or weakly-turbulent regime, negative APE dissipation predicts the occurrence of both salt finger and convective diffusive instability. However, only convective diffusive instability persists in the turbulent regime. To sustain salt finger instability in turbulent conditions, density-compensated thermohaline (spiciness) variations are essential. We derive new criteria for the range of density ratios where double diffusive instabilities are active, resembling those in laminar regimes where the Prandtl number is replaced by the inverse of the mixing efficiency. This work significantly advances our understanding of double diffusive instabilities by providing a unifying framework that elucidates their mechanisms even in general turbulent regimes.Comment: 10 pages 1 figur