In traditional and geophysical fluid dynamics, it is common to describe stratified
turbulent fluid flows with low Mach number and small relative density variations by means of the incompressible
Boussinesq approximation. Although such an approximation is often interpreted as decoupling the
thermodynamics from the dynamics, this paper reviews recent results and derive new ones
that show that the reality is actually more subtle and complex when diabatic
effects and a nonlinear equation of state are retained. Such an analysis reveals indeed:
(1) that the compressible work of expansion/contraction remains of comparable importance as the
mechanical energy conversions in contrast to what is usually assumed; (2) in a Boussinesq fluid,
compressible effects occur in the guise of changes in gravitational potential energy due to
density changes. This makes it possible to construct a fully consistent description of the
thermodynamics of incompressible fluids for an arbitrary nonlinear equation of state;
(3) rigorous methods based on using the available potential energy and potential enthalpy budgets
can be used to quantify the work of expansion/contraction
A large class of fluid flows of interest in traditional and geophysical fluid dynamics are characterised by fluid velocities much smaller than the speed of sound (low Mach number), strong density stratification yet with small relative density variations, and high Reynolds number (i.e., they are turbulent). It has become common practice to regard such flows as incompressible or nearly incompressible, and to describe them by means of particular approximations to the fully compressible Navier-Stokes equations known as the Boussinesq and anelastic approximations. Traditionally, such approximations are derived in the idealised context of purely adiabatic motions, hence excluding irreversible processes such as molecular viscous and diffusive processes. The equation of state is also usually linearised, so that density anomalies become proportional to temperature anomalies. In that case, it is easily shown that the resulting approximations filter out acoustic waves and thus fully decouple the dynamics from the thermodynamics, in such a way that the mechanical energy (the sum of kinetic energy (KE) and gravitational potential energy (GPE)) and the internal energy (IE) obey independent conservation laws.
The coupling between dynamics and thermodynamics is reinstated, however, when the Boussinesq and anelastic approximations are appended with a representation of irreversible processes due to molecular viscous and diffusive processes, and/or when the nonlinearities of the equation of state are no longer neglected. Such an approach is used for instance in the field of numerical ocean modelling. The way the coupling between the dynamics and thermodynamics manifests itself in that case is through the appearance of nonconservative terms in the local balance equation for the mechanical energy, which indicates the need for conversions with internal energy if total energy is to be conserved. Whereas the approximate forms of KE and GPE are close to their nonapproximated forms, the approximate form of IE has remained so far very mysterious, given that only the conversion terms with IE are explicitly represented in the Boussinesq and anelastic approximations, not IE itself. Since the approximate form of IE is in general unknown and left implicit, it is in general not straightforward to determine whether the approximation obtained by adding viscous and diffusive terms has a well-defined energy budget. Moreover, understanding the role of IE in such approximations may give rise to misinterpretations as in the case discussed by Tailleux [
The lack of explicit knowledge about the form assumed by the internal energy in the Boussinesq and anelastic approximations makes it difficult to understand the precise role played by internal energy (and hence the dynamics/thermodynamics coupling) in incompressible or nearly incompressible turbulent stratified fluid flows. Until now, this difficulty has been systematically avoided by assuming the conversions to and from internal energy, and hence the dynamics/thermodynamics coupling, to be dynamically unimportant at leading order. Recently, however, Tailleux [
The main objective of this paper will be to provide insights into the dynamics/thermodynamics coupling in weakly compressible turbulent stratified fluid flows. In particular, it will serve as follows. It will provide a general overview of the fundamental physical and technical issues involved in understanding the coupling between dynamics and thermodynamics in turbulent stratified fluids. It will explain the importance of isolating the part of the total potential energy (gravitational + potential) that is available for reversible conversions into kinetic energy, as well as the part of the internal energy available for It will show how to construct explicitly the full range of known thermodynamic potentials, including entropy, internal energy, enthalpy, that have previously remained implicit in the Boussinesq and anelastic approximations. It will show how it is possible to establish a formal correspondence between the work of expansion/contraction in compressible fluids, and the changes in gravitational potential energy in the Boussinesq and anelastic approximations. It will show how the above results can help clarify and provide new insights into the energetics of turbulent mixing in turbulent stratified fluid flows, as discussed in Tailleux [ It will discuss the general issue of how to estimate the overall work of expansion/contraction in steady-state buoyancy driven circulations, which has received much attention over the past decade in relation with understanding the relative importance of the surface buoyancy fluxes in driving and stirring the ocean circulation.
Issues pertaining to the coupling between the dynamics and thermodynamics in turbulent-stratified fluids are fundamentally rooted in the description of the energetics of a fully compressible fluid. For most fluids of interest, this can be achieved in the context of the compressible Navier-Stokes equations. To fix ideas, these equations are given here for a binary fluid with a nonlinear equation of state such as seawater, namely,
In order to discuss the nature of the coupling between dynamics and thermodynamics for a compressible fluid, we first form a local evolution equation for the mechanical energy
In order to make this apparent, we now turn to the derivation of a local evolution equation for the internal energy. From the well-known expression of its total differential form:
As further motivation and justification for investigating the coupling between the dynamics and thermodynamics in turbulent-stratified fluids, let us simply point out that the work of expansion/contraction plays a central role in the energetics of purely buoyancy-driven circulations, as in horizontal convection, for example, Hughes and Griffiths [
Two main classes of “sound-proof” approximations are commonly used for the study of weakly compressible fluid flows at low Mach numbers, such as those typical of oceanic and atmospheric flows. These are the Boussinesq approximation, for example, Boussinesq [
Prior to the studies by Ingersoll [
The previous approaches to establishing the energetic consistency of the BA system by Ingersoll [
Tailleux [
The possibility to regard the term
In the previous approaches by Ingersoll [
Assuming that
The above derivations therefore make it clear that it is possible to use any existing knowledge of the thermodynamic properties of the fluid considered to reconstruct the relevant approximations for a BA fluid, and also that the approximations underlying the construction of the BA system of equations only slightly alter the form of the thermodynamic properties, which remain close to that of a fully compressible fluid.
In this section, we discuss the consequences of the above results for our understanding of the energetics of turbulent mixing in stratified fluids, which were originally pointed out by Tailleux [
The classical view on the energetics of turbulent mixing has usually revolved on the idea that during a turbulent mixing event, although most of the initial kinetic energy of the parallel shear flow is lost to viscous dissipation as expected, a significant fraction (often reported to be around 20 percent) appears to be lost to gravitational potential energy. This can be mathematically formulated by considering the evolution equation for the kinetic energy, namely,
As first recognised by Lorenz [
Let us now return to the problem of understanding the energetics of a turbulent mixing event, again defined as an episode of intense turbulent mixing characterised by potentially high values of AGPE preceded and followed by laminar conditions for which
Evolution of ideas about the energetics of turbulent stratified mixing. (a) Interpretation seemingly following the result of classical energetics arguments based on the three reservoirs
Classical Energetics
Winters et al. [
Tailleux [
It is important to remark that the validity of Winters et al. [
The above difficulty is easily resolved, however, if one accepts to regard the term
The latter issue was addressed by Tailleux [
Experimental setup of an experiment showed to the author by Peter Rhines during a visit to his GFD Laboratory in Seattle in June 2008, suggesting that compressibility effects increase with the degree of turbulent in stably stratified fluids. An Erlenmeyer flask is filled up with water at room temperature. A vertical stratification is then created by bringing the upper part of the water near boiling point, and a magnetic stirrer is dropped in the fluid. The top of the flask is then sealed up with a cork through which a thin glass tube is inserted in order to magnify the variations of the air-water interface. In laminar conditions (a), the fluid is near resting conditions, and the air-water interface moves downward very slowly as the result of molecular diffusion and/or radiative cooling. Activating the magnetic stirrer generates stratified turbulence (b) that results in the spectacular drop of the air-water interface, most likely as the result of the well-known contraction upon mixing effect that is due to the temperature dependence of the thermal expansion coefficient.
Having determined that the classical Boussinesq and anelastic approximations formally possess a representation of the coupling between the dynamics and thermodynamics that is comparable to that of the fully compressible Navier-Stokes equations, we now seek to illustrate some aspects of such coupling in the particular case of a purely buoyancy-driven circulation driven by differential heating/cooling imposed at the top boundary of the domain, as in the oceanic case, a configuration generally referred to as “horizontal convection,” see Hughes and Griffiths [
The above issue was first addressed for a Boussinesq fluid with a linear equation of state governed by (
The equation of state of seawater is strongly nonlinear, however, so that it is unclear whether (
The most natural approach to estimating the volume-integrated work of expansion/contraction for a compressible ocean with a nonlinear equation of state is perhaps to regard the specific volume as a function of entropy and pressure (we discard salinity temporarily for simplicity), which allows one to expand the rate of expansion/contraction as follows:
A significant breakthrough in understanding how to get around the difficulty posed by the term
In this section, we aim to further clarify the above issues, by showing how the above results can be extended to a fully compressible ocean by considering the budget of potential enthalpy. To simplify the discussion, the effects of salinity are discarded. The potential enthalpy, introduced in the oceanographic context by McDougall [
Reflecting in hindsight on the different ideas that have been developed over the past decade to generalise Paparella and Young [
For over a century, the coupling between the dynamics and thermodynamics has been overwhelmingly regarded as being of little or no relevance to the understanding of stratified turbulence in weakly compressible fluids such as water or seawater. As a result, the study of such fluids has been nearly systematically been carried out in the context of dynamical approximations, such as the Boussinesq approximation or anelastic approximation, that decouples the dynamics and thermodynamics at leading order. It is not necessarily fully realized, however, that such a decoupling is achieved only in absence of diabatic effects and for a linear equation of state. Indeed, coupling is unavoidably reintroduced whenever any one of the latter effects is retained. If so, a key issue is what can we say about the nature of such a coupling in that case? How does it compare with that of the fully compressible Navier-Stokes equations?
In order to address these questions, the present paper first showed that the particular anelastic approximation derived by Pauluis [
An important feature of the energetics of the BA system is that there is