arXiv subjects

Remi Tailleux

Publications and source records attributed to Remi Tailleux.

18 recordsLinked to original sources

arXiv2025

Energetically consistent localised APE budgets for local and regional studies of stratified flow energetics

Because it allows a rigorous separation between reversible and irreversible processes, the concept of available potential energy (APE) has become central to the study of turbulent stratified fluids. In ocean modelling, it is fundamental to the parameterisation of meso-scale ocean eddies and of the turbulent mixing of heat and salt. However, how to apply APE theory consistently to local or regional subdomains has been a longstanding source of confusion due to the globally defined Lorenz reference state entering the definition of APE and of buoyancy forces being generally thought to be meaningless in those cases. In practice, this is often remedied by introducing heuristic `localised' forms of APE density depending uniquely on region-specific reference states, possibly diverging significantly from the global Lorenz reference state. In this paper, we argue that across-scale energy transfers can only be consistently described if localised forms of APE density are defined as the eddy APE component of an exact mean/eddy decomposition of the APE density, for which a new physically more intuitive and mathematically simpler framework is proposed. The eddy APE density thus defined exhibits a much weaker dependency on the global Lorenz reference state than the mean APE, in agreement with physical intuition, but with a different structure than that of existing heuristic localised APE forms. Our framework establishes a rigorous physical basis for linking parameterised energy transfers to molecular viscous and diffusive dissipation rates. We illustrate its potential usefulness by discussing the energetics implications of standard advective and diffusive parameterisations of the turbulent density flux, which reveals potential new sources of numerical instability in ocean models.

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arXiv2024

Negative APE dissipation as the fundamental criterion for double diffusive instabilities

The background potential energy (BPE) is the only reservoir that double diffusive instabilities can tap their energy from when developing from an unforced motionless state with no available potential energy (APE). Recently, Middleton and Taylor linked the extraction of BPE into APE to the sign of the diapycnal component of the buoyancy flux, but their criterion can only predict diffusive convection instability, not salt finger instability. Here, we show that the problem can be corrected if the sign of the APE dissipation rate is used instead, making it emerge as the most fundamental criterion for double diffusive instabilities. A theory for the APE dissipation rate for a two-component fluid relative to its single-component counterpart is developed as a function of three parameters: the diffusivity ratio, the density ratio, and a spiciness parameter. The theory correctly predicts the occurrence of both salt finger and diffusive convection instabilities in the laminar unforced regime, while more generally predicting that the APE dissipation rate for a two-component fluid can be enhanced, suppressed, or even have the opposite sign compared to that for a single-component fluid, with important implications for the study of ocean mixing. Because negative APE dissipation can also occur in stably stratified single-component and doubly stable two-component stratified fluids, we speculate that only the thermodynamic theory of exergy can explain its physics; however, this necessitates accepting that APE dissipation is a conversion between APE and the internal energy component of BPE, in contrast to prevailing assumptions.

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arXiv2024

A Simple and Transparent Method for Improving the Energetics and Thermodynamics of Seawater Approximations: Static Energy Asymptotics (SEA)

The static energy encodes all possible information about the thermodynamics and potential energy (and all related forces) of stratified geophysical fluids. In this paper, we develop a systematic methodology, called static energy asymptotics, that exploits this property for constructing energetically and thermodynamically consistent sound-proof approximations of the equations of motion. By approximating the static energy to various orders of accuracy, two main families of approximations are (re-)derived and discussed: the pseudo-incompressible (PI) approximation and the anelastic (AN) approximation. For all approximations, the background and available potential energies (in Lorenz sense) can be constructed to match their exact counterparts as closely as feasible and to be expressible in terms of the exact (as opposed to ad-hoc) thermodynamic potentials. For hydrostatic motions, the AN approximation (of which the Boussinesq approximation is a special case) has the same structure as that of legacy Seawater Boussinesq primitive equations. The energetics of such models could therefore be made transparently traceable to that of the full Navier-Stokes equations at little to no additional cost, thus allowing them to take full advantage of the Gibbs Sea Water (GSW) library developed as part of the new thermodynamic standard for seawater TEOS-10

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arXiv2023

On the determination of the 3D velocity field in terms of conserved variables in a compressible ocean

Explicit expressions of the 3D velocity field in terms of the conserved quantities of ideal fluid thermocline theory, namely Bernoulli function, density, and potential vorticity, are generalised here to a compressible ocean with a realistic nonlinear equation of state. The most general such expression is the `inactive wind' solution, an exact nonlinear solution of the inviscid compressible Navier-Stokes that satisfies the continuity equation as a consequence of Ertel's potential vorticity theorem. Such expressions are shown to be non-unique due to the non-uniqueness of the choice of Bernoulli function and in general approximately differ by the magnitude of their vertical velocity component. Due to the thermobaric nonlinearity of the equation of state, the expression of the 3D velocity field for a compressible ocean is found to resemble its ideal fluid counterpart only if constructed in terms of the available form of Bernoulli function as per Lorenz theory of available potential energy (APE). APE theory also naturally defines a quasi-material approximately neutral density variable called Lorenz reference density, which in turn defines a potential vorticity variable minimally affected by thermobaric production, thus providing all necessary tools for extending most results of ideal fluid thermocline theory to a compressible ocean.

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arXiv2023

On the Links Between Thermobaricity, Available Potential Energy, Neutral Directions, Buoyancy Forces, and Lateral Stirring in the Ocean

The various ingredients of Lorenz theory of available potential energy (APE) are shown to hold the key for understanding how to develop a first-principles theory of lateral stirring and lateral stirring surfaces in the oceans embedded in the study of the full Navier-Stokes equations for compressible seawater. This theory establishes that it is the existence of thermobaric forces acting along isopycnal surfaces that makes stirring in seawater fundamentally different from that in a simple fluid and is the ultimate cause for the non-existence of neutral surfaces. It also establishes that the `true' neutral directions are those perpendicular to an APE-based form of the P vector previously identified by Nycander, contrary to what has been assumed so far. Where thermobaric forces are small enough to be neglected, our theory establishes that the Lorenz reference density (LRD) surfaces entering APE theory are very accurately neutral and represent the relevant definition of lateral stirring surfaces. Where thermobaric forces are large, however, lateral stirring becomes strongly coupled with vertical stirring and complicates the identification of the `right' lateral mixing surfaces. Importantly, rewriting the momentum balance equations in their thermodynamic form using Crocco-Vazsonyi theorem and removing the dynamically inert part of the Bernoulli function proves decisive for obtaining most results. The new results have important implications for the theory of isopycnal analysis and ocean mixing parameterisations.

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arXiv2020

Spiciness theory revisited, with new views on neutral density, orthogonality and passiveness

This paper clarifies the theoretical basis for constructing spiciness variables optimal for characterising ocean water masses. Three essential ingredients are identified: 1) a material density variable $γ$ that is as neutral as feasible; 2) a material state function $ξ$ independent of $γ$, but otherwise arbitrary; 3) an empirically determined reference function $ξ_r(γ)$ of $γ$ representing the imagined behaviour of $ξ$ in a notional spiceless ocean. Ingredient 1) is required because contrary to what is often assumed, it is not the properties imposed on $ξ$ (such as orthogonality) that determine its dynamical inertness but the degree of neutrality of $γ$. The first key result is that it is the anomaly $ξ' = ξ- ξ_r(γ)$, rather than $ξ$, that is the variable the most suited for characterising ocean water masses, as originally proposed by McDougall and Giles (1987). The second key result is that oceanic sections of normalised $ξ'$ appear to be relatively insensitive to the choice of $ξ$, as first suggested by Jackett and McDougall (1985). It is also argued that orthogonality of $\nabla ξ'$ to $\nabla γ$ in physical space is more germane to spiciness theory than orthogonality in thermohaline space, although how to use it to constrain the choices of $ξ$ and $ξ_r(γ)$ remains to be fully elucidated. The results are important for they unify the various ways in which spiciness has been defined and used in the literature. They also provide a rigorous theoretical basis justifying the pursuit of a globally defined material density variable maximising neutrality. To illustrate the latter point, this paper proposes a new implementation of the author's recently developed thermodynamic neutral density and explains how to adapt existing definitions of spiciness/spicity to work with it.

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arXiv2020

The generalised buoyancy/inertial forces and available energy of axisymmetric compressible stratified vortex motions

Adiabatic and inviscid axisymmetric perturbations to a stable reference vortex in gradient wind balance are known to experience two kinds of restoring forces: one that is proportional to both the perturbation density and the reference pressure gradient, and one that is purely radial and proportional to the squared angular momentum perturbation. We show that the work required to move a fluid parcel against such forces from its equilibrium to actual position is path-independent and formally equivalent to the available energy accounting for momentum constraints previously constructed by Andrews (2006) and Codoban and Shepherd (2006). Physically, this work represents the energy of the unbalanced part of the vortex and hence a form of eddy energy. It can be partitioned into available acoustic energy, slantwise available potential energy and centrifugal potential energy. We show that the conditions required for these energies to be positive definite correspond to the classical conditions for symmetric stability, valid for both small and finite-amplitude perturbations. The new available energy framework possesses various advantageous features over previous approaches that shed new light on how the thermodynamic and mechanical sinks/sources of energy control the intensification of the balanced and unbalanced parts of a warm core cyclonic vortex. These features should prove useful in the future to re-examine and clarify the links between the different existing paradigms of tropical cyclone (TC) intensification within a single unifying theoretical framework.

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arXiv2018

APE dissipation is a form of Joule heating. It is irreversible, not reversible

Available Potential Energy (APE) dissipation plays a central role in the description of mixing in turbulent stratified fluids. The dominant paradigm is that it converts APE into background gravitational potential energy ${\rm GPE}_r$, and that the APE thus converted can be infinitely recycled back into APE by external buoyancy fluxes such as high-latitude cooling in the oceans. In this paper, we argue that such a paradigm is unphysical, because its corollary is that APE dissipation is neither truly dissipative nor irreversible, while also violating energy conservation in more subtle ways. In this paper, we prove from first principles that in reality, APE dissipation is a form of Joule heating, which --- like viscous dissipation --- can only increase ${\rm GPE}_r$ via locally expanding the fluid parcels, a tiny effect. ${\rm GPE}_r$ thus primarily increases at the expense of the exergy of the stratification, a subcomponent of the background internal energy, regardless of whether the flow is laminar or turbulent. The results greatly clarify the energetics of mechanically- and buoyancy-driven circulations. As a side benefit, our results yield a new physical principle justifying why turbulent mixing tends to homogenise the fluid's materially conserved variables rather than relax the fluid towards thermodynamic equilibrium.

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arXiv2018

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 primitive 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.

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arXiv2017

Assessment of algorithms for computing moist available potential energy

Atmospheric moist available potential energy (MAPE) has been traditionally defined as the potential energy of a moist atmosphere relative to that of the adiabatically sorted reference state defining a global potential energy minimum. Finding such a reference state was recently shown to be a linear assignment problem, and therefore exactly solvable. However, this is computationally extremely expensive, so there has been much interest in developing heuristic methods for computing MAPE in practice. Comparisons of the accuracy of such approximate algorithms have so far been limited to a small number of test cases; this work provides an assessment of the algorithms' performance across a wide range of atmospheric soundings, in two different locations. We determine that the divide-and-conquer algorithm is the best suited to practical application, but suffers from the previously overlooked shortcoming that it can produce a reference state with higher potential energy than the actual state, resulting in a negative value of MAPE. Additionally, we show that it is possible to construct an algorithm exploiting a theoretical expression linking MAPE to Convective Available Potential Energy (CAPE) previously derived by Kerry Emanuel. This approach has a similar accuracy to existing approximate sorting algorithms, whilst providing greater insight into the physical source of MAPE. In light of these results, we discuss how to make progress towards constructing a satisfactory moist APE theory for the atmosphere. We also outline a method for vectorising the adiabatic lifting of moist air parcels, which increases the computational efficiency of algorithms for calculating MAPE, and could be used for other applications such as convection schemes.

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arXiv2017

On the local view of atmospheric available potential energy

The possibility of constructing Lorenz's concept of available potential energy (APE) from a local principle has been known for some time, but has received very little attention so far. Yet, the local APE framework offers the advantage of providing a positive definite local form of potential energy, which like kinetic energy can be transported, converted, and created/dissipated locally. In contrast to Lorenz's definition, which relies on the exact from of potential energy, the local APE theory uses the particular form of potential energy appropriate to the approximations considered. In this paper, this idea is illustrated for the dry hydrostatic primitive equations, whose relevant form of potential energy is the specific enthalpy. The local APE density is non-quadratic in general, but can nevertheless be partitioned exactly into mean and eddy components regardless of the Reynolds averaging operator used. This paper introduces a new form of the local APE that is easily computable from atmospheric datasets. The advantages of using the local APE over the classical Lorenz APE are highlighted. The paper also presents the first calculation of the three-dimensional local APE in observation-based atmospheric data. Finally, it illustrates how the eddy and mean components of the local APE can be used to study regional and temporal variability in the large-scale circulation. It is revealed that advection from high latitudes is necessary to supply APE into the storm track regions, and that Greenland and Ross Sea, which have suffered from rapid land ice and sea ice loss in recent decades, are particularly susceptible to APE variability.

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arXiv2017

A new process-based vertical advection/diffusion theoretical model of ocean heat uptake

The vertical upwelling/diffusion model (VUDM) has historically played a key role in shaping our ideas about how the heat balance is achieved in the ocean. Its has been and is still widely used in many applications ranging from the estimation of transfer coefficients to the parameterisation of ocean heat uptake in Simple Climate Models (SCMs). Its conceptual value as a realistic theoretical model of the ocean heat balance has become increasingly unclear over the years however, because: 1) the different ways in which upwelling has been linked to high-latitude deep water formation and downgradient diffusion linked to vertical/diapycnal mixing have remained imprecise and somewhat ad-hoc so far; 2) other effects such as isopycnal mixing, density-compensated temperature anomalies, meso-scale eddy-induced advection and the depth-varying ocean area have all be demonstrated to affect actual ocean heat uptake as well, but their incorporation into existing VUDM frameworks has been problematic. In this paper, a new process-based vertical advection/diffusion theoretical model of ocean heat uptake is constructed that resolve all above difficulties. This new model is obtained by coarse-graining the full three-dimensional advection/diffusion for potential temperature carried by ocean climate models, by using the same isopycnal analysis as in the theory of ocean water masses. The resulting model describes the temporal evolution of the isopycnally-averaged thickness-weighted potential temperature in terms of an effective velocity that depends uniquely on the surface heating conditionally integrated in density classes, an effective diapycnal diffusivity controlled by isoneutral and dianeutral mixing, and an additional term linked to the meridional transport of density-compensated temperature anomalies by the diabatic residual overturning circulation.

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arXiv2017

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

The current view about the mixing of heat and salt in the ocean is that it should be parameterised by means of a rotated diffusion tensor based on mixing directions parallel and perpendicular to the local neutral vector. However, the impossibility to construct a density variable in the ocean that is exactly neutral because of the coupling between thermobaricity and density-compensated temperature/salinity anomalies implies that the effective diapycnal diffusivity experienced by any possible density variable is partly controlled by isoneutral diffusion when using neutral rotated diffusion. Here, this effect is quantified by evaluating the effective diapycnal diffusion coefficient for five widely used density variables: Jackett and McDougall (1997) $γ^n$, Lorenz reference state density $ρ_{ref}$ of Winters et al. (1996), Saenz et al. (2015), and three potential density variables $σ_0$, $σ_2$ and $σ_4$.Computations use the World Ocean Circulation Experiment climatology, assuming either a uniform value for isoneutral mixing or spatially varying values inferred from an inverse calculation. Isopycnal mixing contributions to the effective diapycnal mixing yields values systematically larger than $10^{-3}$ $\text{m}^2/\text{s}$ in the deep ocean for all density variables, with $γ^n$ suffering the least from the isoneutral control of effective diapycnal mixing, and $σ_0$ 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 $10^{-4}$ $\text{m}^2/\text{s}$. This work highlights the potential pitfalls of estimating diapycnal diffusivities by means of Walin-like water masses analysis or in using Lorenz reference state for diagnosing spurious numerical diapycnal mixing.

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arXiv2011

Thermodynamics/dynamics coupling and thermodynamic consistency of Boussinesq and anelastic binary fluids with an arbitrary nonlinear equation of state

This paper shows that the energetics of Boussinesq and anelastic fluids possesses a term that can be identified as the approximation $δW_{ba}$ to the compressible work of expansion/contraction $δW =-P {\rm d}\upsilon$, where $P$ is the pressure and $\upsilon$ is the specific volume. It follows that Boussinesq and anelastic fluids admit explicit compressible effects and conversions between internal energy and mechanical energy, under the form of apparent changes in gravitational potential energy resulting from changes in density by diabatic and adiabatic effects. From the knowledge of $δW_{ba}$, the corresponding approximation to the "heat" $δQ_{ba}$ can be constructed in a consistent way by requiring that the Maxwell relationships be satisfied, ultimately leading to the construction of a well defined approximation to the internal energy and ultimately of the full range of known thermodynamic potentials. These properties make it possible to endow common forms of the Boussinesq and anelastic approximations with fully consistent energetics and thermodynamics, even when diabatic effects and an arbitrary nonlinear equation of state for a binary fluid are retained, without loss of accuracy. In that case, it can be shown that the sum of kinetic energy and enthalpy is a conservative quantity, which plays the role of the total energy in the Boussinesq and anelastic approximations for both diabatic and adiabatic motions. This implies that gravitational potential energy can be regarded as the difference between enthalpy and internal energy, and hence as a pure thermodynamic property of the fluid. The results have implications for our understanding of turbulent mixing in stratified fluids, as well as for correcting the energetics of current numerical ocean general circulation models, which are discussed.

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arXiv2009

On the energetics of stratified turbulent mixing, irreversible thermodynamics, Boussinesq models, and the ocean heat engine controversy

A key issue in stratified turbulence theory concerns the nature of the link between D(APE), the dissipation rate of available potential energy APE, and W_{r,turbulent}, the turbulent rate of change of background gravitational potential energy GPE_r, which are both controlled by molecular diffusion. For Boussinesq fluids with a linear equation of state, this link is simply W_{r,turbulent}=D(APE), widely interpreted as implying that GPE_r increases at the expense of APE, in contrast with the laminar case where GPE_r increases at the expense of internal energy (IE). This idea is revisited here by regarding IE as the sum of three distinct subcomponents: available internal energy (AIE), exergy (IE_{exergy}), and dead internal energy (IE_0). In this new view, D(APE) is the dissipation rate of APE into IE_0, while both W_{r,laminar} and W_{r,turbulent} convert IE_{exergy} into GPE_r. The equality W_{r,turbulent}=D(APE) thus states that IE_{exergy} is converted into GPE_r at the same rate as APE is dissipated into IE_0. For non-Boussinesq fluids, the equality D(APE)=W_{r,turbulent} is at best a good approximation, for W_{r,turbulent} is generally smaller than D(APE), and sometimes even negative for a strongly nonlinear equation of state. In a second step, the link between stirring and mixing is examined for a wind-and buoyancy-driven thermally stratified ocean to determine whether these constrain the mechanical sources of stirring, as recently advocated. It is established that the coupling between stirring and mixing cannot refute the traditional buoyancy-driven view of the so-called meridional overturning circulation, in contrast to recent claims. In fact, the buoyancy forcing appears to be as important as the mechanical forcing in stirring and driving the large-scale ocean circulation.

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arXiv2009

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 ξ=Wr,turbulent/D(APE) being then lower than unity and sometimes even negative for water/seawater. In this paper, the behavior of the ratio ξis examined for different stratifications having the same buoyancy frequency N(z), but different vertical profiles of the parameter Υ= αP/(ρC_p), where αis the thermal expansion, P the hydrostatic pressure, ρthe 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 ξand Wr,turbulent depend critically on the sign and magnitude of dΥ/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.

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arXiv2009

The effect of mechanical stirring on buoyancy-driven circulations

The theoretical analysis of the energetics of mechanically-stirred horizontal convection for a Boussinesq fluid yields the formula: G(APE) = γ_{mixing} G(KE) + (1+γ_{mixing}) W_{r,laminar} where G(APE) and G(KE) are the work rate done by the buoyancy and mechanical forcing respectively, γ_{mixing} is the mixing efficiency, and W_{r,laminar} is the background rate of increase in gravitational potential energy due to molecular diffusion. The formula shows that mechanical stirring can easily induce a very strong buoyancy-driven overturning cell (meaning a large G(APE)) even for a relatively low mixing efficiency, whereas this is only possible in absence of mechanical stirring if γ_{mixing} >> 1. Moreover, the buoyancy-driven overturning becomes mechanically controlled when $γ_{mixing} G(KE) >> (1+γ_{mixing}) W_{r,laminar}$. This result explains why the buoyancy-driven overturning cell in the laboratory experiments by \cite{Whitehead2008} is amplified by the lateral motions of a stirring rod. The formula implies that the thermodynamic efficiency of the ocean heat engine, far from being negligibly small as is commonly claimed, might in fact be as large as can be thanks to the stirring done by the wind and tides. These ideas are further illustrated by means of idealised numerical experiments. A non-Boussinesq extension of the above formula is also given.

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arXiv2007

Thermodynamic inadmissibility of the incompressible hydrodynamics description of turbulent stratified fluid flows at low Mach numbers

The incompressible Navier-Stokes equations currently represent the primary model for describing stratified turbulent fluid flows at low Mach number. The validity of the incompressible assumption, however, has so far only been rigorously established for adiabatic motions. Here, we show from first principl es that the use of available energetics and thermodynamics considerations applied to a turbulent mixing event associated with stratified shear flow instability r efutes the widespread idea that the incompressible assumption is also valid when diabatic irreversible effects are important. The main consequence is that dynamics and thermodynamics are strongly coupled in stratified turbulence. This departs strongly from the currently accepted wisdom, and calls for a complete revisiting of the physical processes governing stratified turbulence at low Mach numbers.

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