Understanding mixing efficiency in the oceans: Do the nonlinearities of the equation of state matter?
aa r X i v : . [ phy s i c s . f l u - dyn ] J u l Manuscript prepared for Ocean Sci.with version 2.3 of the L A TEX class copernicus.cls.Date: 30 August 2018
Understanding mixing efficiency in the oceans:Do the nonlinearities of the equation of state for seawater matter?
R. Tailleux Department of Meteorology, University of Reading, United Kingdom
Abstract.
There exist two central measures of turbulentmixing in turbulent stratified fluids that are both caused bymolecular diffusion: 1) the dissipation rate D ( AP E ) ofavailable potential energy AP E ; 2) the turbulent rate ofchange W r,turbulent of background gravitational potentialenergy GP E r . So far, these two quantities have often beenregarded as the same energy conversion, namely the irre-versible conversion of AP E into
GP E r , owing to the wellknown exact equality D ( AP E ) = W r,turbulent for a Boussi-nesq fluid with a linear equation of state. Recently, how-ever, Tailleux (2009) pointed out that the above equality nolonger holds for a thermally-stratified compressible, with theratio ξ = W r,turbulent /D ( AP E ) being generally lower thanunity and sometimes even negative for water or seawater,and argued that D ( AP E ) and W r,turbulent actually repre-sent two distinct types of energy conversion, respectively thedissipation of AP E into one particular subcomponent of in-ternal energy called the ‘dead’ internal energy IE , and theconversion between GP E r and a different subcomponentof internal energy called ’exergy’ IE exergy . In this paper,the behaviour of the ratio ξ is examined for different strat-ifications having all the same buoyancy frequency N ver-tical profile, but different vertical profiles of the parameter Υ = αP/ ( ρC p ) , where α is the thermal expansion coeffi-cient, P the hydrostatic pressure, ρ the density, and C p thespecific heat capacity at constant pressure, the equation ofstate being that for seawater for different particular constantvalues of salinity. It is found that ξ and W r,turbulent dependcritically on the sign and magnitude of d Υ /dz , in contrastwith D ( AP E ) , which appears largely unaffected by the lat-ter. These results have important consequences for how themixing efficiency should be defined and measured in prac-tice, which are discussed. Correspondence to:
R. Tailleux([email protected])
As is well known, turbulent diffusive mixing is a physicalprocess that it is crucially important to parameterise well innumerical ocean models in order to achieve realistic simula-tions of the water mass properties and of the so-called merid-ional overturning circulation (Gregg , 1987), which are twoessential components of the large-scale ocean circulation thatmay interact with Earth climate. For this reason, much efforthas been devoted over the past decades toward understandingthe physics of turbulent mixing in stratified fluids, one impor-tant goal being the design of physically-based parameterisa-tions of irreversible mixing processes for use in numericalocean climate models.At a fundamental level, turbulent molecular diffusion instratified fluids is important for at least two distinct — al-though inter-related — reasons: 1) for transporting heat dif-fusively across isopycnal surfaces – a process often referredto as ‘diapycnal mixing’; 2) for dissipating available poten-tial energy, which contributes for a significant fraction — of-ten called the mixing efficiency — of the total dissipationof available mechanical energy
M E , i.e., the sum of total ki-netic energy KE and available potential energy AP E , whichare defined by: KE = Z V ρ v dV, (1) AP E = Z V ρ ( gz + I ) dV | {z } P E − Z V ρ ( gz r + I r ) dV | {z } P E r , (2)where ρ is the density, v = ( u, v, w ) is the three-dimensionalvelocity vector, g is the acceleration of gravity, z is the verti-cal coordinate increasing upward, and I the specific internalenergy. The APE is defined as in Lorenz (1955) as the differ-ence between the potential energy P E of the fluid (i.e., thesum of the gravitational potential energy
GP E plus internal R. Tailleux: Is mixing efficiency affected by nonlinear equation of state?energy IE ) minus the potential energy P E r of a referencestate that is the state of minimum potential energy achiev-able in an adiabatic re-arrangement of the fluid parcels. Asshown by Winters & al. (1995), the AP E and
P E r play afundamental in the modern theory of turbulent mixing ow-ing to the fact that by construction P E r is only affected byirreversible processes; as a result, measuring the time evolu-tion of the reference state provides a direct and objective wayto quantify the amount of irreversible mixing taking placeduring turbulent mixing events, which is now commonly ex-ploited to diagnose mixing in numerical experiments, e.g.,Peltier & Caulfield (2003).In the oceans, turbulent diapycnal mixing is required totransfer heat downward from the surface at a sufficientlyrapid rate to balance the cooling of the deep ocean by high-latitudes deep water formation. In the oceanographic litera-ture, the most widely used approach to parameterise the ver-tical (diapycnal) eddy diffusivity K ρ is based on the Osborn-Cox model (Osborn and Cox (1972)): K ρ = ε P N = γ mixing ε K N , (3)which expresses K ρ in terms of either the turbulent viscouskinetic energy dissipation ε K or turbulent diffusive dissi-pation of available potential energy ε P , where N is thesquared buoyancy frequency, and γ mixing = ε P /ε K is theratio of the APE to KE dissipation, which is often called the‘mixing efficiency’, e.g. Lindborg & Brethouwer (2008).Expressing K ρ in terms of ε K appears to have been first pro-posed by Lilly & al. (1974) and Weinstock (1978) in thecontext of stratospheric turbulent mixing, and adapted to theoceanographic case by Osborn (1980). The definition ofmixing efficiency as a dissipations ratio adopted in this pa-per appears to have been first proposed by Oakey (1982).Since both ǫ P and ε K are linked to the dissipation of me-chanical energy of which KE and AP E represent the twomain dynamically important forms, Eq. (3) makes it clearthat turbulent diapycnal mixing is directly related to the me-chanical energy input in the oceans, but this link has been sofar very rarely exploited in numerical ocean models. Rather, K ρ is often regarded as a tunable parameter whose valueis adjusted to reproduce the main observed features of theoceanic stratification. Such an approach was used by Munk(1966), who assumed the stratification to obey the verticaladvective/diffusive balance: w ∂θ∂z = ∂∂z (cid:18) K ρ ∂θ∂z (cid:19) , (4)where θ is the potential temperature, and w the vertical ve-locity. Physically, Eq. (4) states that the upward advec-tion of cold water is balanced by the downward turbulentdiffusion of heat, the rate of upwelling being set up bythe rate of deep water formation. By using Eq. (4) as amodel for stratification profiles in the Pacific, Munk (1966)concluded that the canonical value K ρ = 10 − m / s was apparently needed to explain the observed structure of theoceanic thermocline. Subsequently, however, the validity ofMunk (1966)’s approach was questioned, as several observa-tional studies found K ρ in the ocean interior to be typicallysmaller by an order of magnitude than Munk’s value, e.g.,see Ledwell & al (1998) and the review by Gregg (1987).However, it seems widely recognised today that K ρ is highlyvariable spatially, prompting Munk & Wunsch (1998) to re-interpret the value K ρ = 10 − m . s − as resulting from theoverall effect of weak interior values combined with intenseturbulent mixing in coastal areas or over rough topography.While the above approach is useful, it does not exploitthe link between K ρ and the mechanical sources of stir-ring suggested by Eq. (3). Clarifying this link was pio-neered by Munk & Wunsch (1998), who translated the ad-vection/diffusion balance into one for the gravitational po-tential energy budget, which they argue must be a balancebetween the rate of GP E loss due to cooling and the rate of
GP E increase due to turbulent diffusive mixing, i.e., (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E r ) dt (cid:12)(cid:12)(cid:12)(cid:12) cooling ≈ (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E r ) dt (cid:12)(cid:12)(cid:12)(cid:12) mixing , (5)this result being obtained by multiplying Eq. (4) by α θ ρ gz ,after some manipulation involving integration by parts andthe neglect of surface heating, where α θ is the thermal expan-sion, g the acceleration of gravity, ρ a reference density, and z the vertical coordinate pointing upward. The subscript r is added here because it can be shown that Munk & Wunsch(1998) must actually pertain to the background GP E r bud-get, rather than the AGP E budget, as shown by Tailleux(2009). This follows from the fact that cooling and turbulentmolecular diffusion act as a
GP E sink and source only forthe background
GP E r , as it is the opposite that holds for AGP E . If one assumes that density is primarily controlledby temperature for simplicity, the effect of mixing on
GP E r is thus given by: (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E r ) dt (cid:12)(cid:12)(cid:12)(cid:12) mixing = − Z V ρ α θ gz ∂∂z (cid:18) K ρ ∂θ∂z (cid:19) dV = Z V ρ K ρ N (cid:18) zα θ ∂α θ ∂z (cid:19) dV, (6)by using the result that N = α θ g∂θ/∂z in absence ofsalinity effects, and by assuming z = 0 at the ocean sur-face, and no flux through the ocean bottom. In their paper,Munk & Wunsch (1998) neglected the nonlinearities of theequation of state, which amounts to regard α θ as constant, inwhich case the above expression becomes: (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E r ) dt (cid:12)(cid:12)(cid:12)(cid:12) mixing ≈ Z V ρ K ρ N dV. (7). Tailleux: Is mixing efficiency affected by nonlinear equation of state? 3By using Eq. (3), assuming γ mixing constant, this formulacan be rewritten as follows: (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E r ) dt (cid:12)(cid:12)(cid:12)(cid:12) mixing = Z V ρ ε P dV | {z } D ( AP E ) = γ mixing Z V ρ ε K dV | {z } D ( KE ) (8)where D ( AP E ) and D ( KE ) are the total volume-integrateddiffusive dissipation of available potential energy and vis-cous dissipation of kinetic energy respectively. To conclude,Munk & Wunsch (1998) linked the dissipation to productionterms by assuming the balance D ( KE ) = G ( KE ) , where G ( KE ) is the work rate done by the mechanical forcing dueto the winds and tides. As a result, the above formula yields: G ( KE ) = 1 γ mixing (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E ) dt (cid:12)(cid:12)(cid:12)(cid:12) cooling . (9)By estimating the rate of GP E loss due to cooling tobe . , and by using the canonical value γ mixing =0 . , Munk & Wunsch (1998) concluded that G ( KE ) = O (2 TW) of mechanical energy input was required to sus-tain turbulent diapycnal mixing in the oceans. Since the workof the wind stress against the surface geostrophic velocityis widely agreed to be O (1 TW) , Munk & Wunsch (1998)suggested that the shortfall should be explained by the workrate done by the tides. The issue remains controversial, how-ever, because the role of the surface buoyancy forcing is notsufficiently well understood, as discussed in Tailleux (2009).Another important issue in assessing the uncertaintiesassociated with Eq. (9) concerns the importance ofthe nonlinearities of the equation of state, neglected byMunk & Wunsch (1998). As is well known, the nonlinear-ities of the equation of state are mostly responsible for thefluid “contracting upon mixing”. This contraction is respon-sible for the actual increase in GP E r due to mixing to beless than for a linear equation of state. In Eq. (6), this canbe seen from the fact that ∂α θ /∂z is usually positive for astably stratified fluid. Since z is negative by assumption, itfollows that a correction factor is required that modifies Eq.(8) as follows: (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E ) dt (cid:12)(cid:12)(cid:12)(cid:12) mixing = (1 − C m ) D ( AP E )= (1 − C m ) γ mixing D ( KE ) . (10)where C m > , which in turn modifies Munk & Wunsch(1998)’s constraint (Eq. (9)) as follows: G ( KE ) = 1 ξγ mixing (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E ) dt (cid:12)(cid:12)(cid:12)(cid:12) cooling (11)where ξ = 1 − C m < . Based on the above arguments,Munk & Wunsch (1998)’s results are expected to underesti-mate the constraint on G ( KE ) , to the extent that γ mixing andthe rate of GP E r loss due to cooling can be kept fixed. This point was first pointed out by Gnanadesikan & al. (2005)who emphasised the importance of cabelling. Discussing thevalue of C m or ξ to be used in Eq. (9) is beyond the scopeof this paper. Note, however, that it is possible to constructstratifications with ξ not only smaller than one but also pos-sibly even negative, as discussed by Fofonoff (1998, 2001).The latter cases are interesting, because they are such that GP E r decreases upon mixing, not increases , in contrast towhat is usually assumed.The main reason that GP E r is often assumed to increaseas a result of turbulent mixing stems from that for an incom-pressible fluid with a linear equation of state, Eq. (8) statesthat: (cid:12)(cid:12)(cid:12)(cid:12) d ( GP E ) dt (cid:12)(cid:12)(cid:12)(cid:12) mixing = D ( AP E ) , (12)i.e., that GP E r increases at the same rate that AP E de-creases, which is classically interpreted as implying that thediffusively dissipated
AP E must be irreversibly convertedinto
GP E r , e.g., Winters & al. (1995). Tailleux (2009)pointed out, however, that Eq. (12) is at best only a good ap-proximation, not a true equality, since in reality the rates of GP E r increase and AP E decrease are never exactly equal,and sometimes even widely different, because of the nonlin-ear character of the equation of state.In order to better understand how the net change in
GP E r correlates with the total amount of AP E diffusively dissi-pated during an irreversible turbulent mixing event, it is use-ful to examine the process of turbulent mixing in the light ofclassical thermodynamic transformations. To make progress,the conditions under which the diffusive exchange of heat be-tween fluid parcels takes place need to be known, but in prac-tice this is problematic, for it would require solving the fullcompressible non-hydrostatic Navier-Stokes equations downto the diffusive scales. Fortunately, it is often the case thatstratified fluids at low Mach numbers are close to hydrostaticequilibrium, suggesting that the diffusive heat exchange be-tween parcels may reasonably be assumed to occur at ap-proximately constant pressure. If so, irreversible diffusivemixing must then be close to be a process conserving the to-tal potential energy
P E = AP E + P E r of the system, whichimplies that any amount ∆ AP E diff < of diffusively dissi-pated AP E must be irreversibly converted into background
P E r , viz., ∆ P E r = − ∆ AP E diff > . (13)The implications for the net change in GP E r can be deter-mined from the definitions P E r = GP E r + IE r , AP E = AGP E + AIE , and IE = AIE + IE r , which imply: ∆ GP E r = − ∆ AP E diff − ∆ IE r (14) R. Tailleux: Is mixing efficiency affected by nonlinear equation of state?where ∆ IE r is the net change in background internal energytaking place during the irreversible mixing event. As a result,the quantities ξ and C m previously defined become: ξ = ∆ GP E r | ∆ AP E diff | = 1 − ∆ IE r | ∆ AP E diff | , (15) C m = ∆ IE r | ∆ AP E diff | . (16)Eqs. (15) and (16) are important, because they establish thatthe nonlinearities of the equation of state — which are re-sponsible for the temperature and pressure dependence of α — can give rise to internal energy changes ∆ IE r com-parable in magnitude with ∆ AP E diff and ∆ GP E r dur-ing a turbulent mixing event. Such large IE r changes mustin turn be associated with potentially large compressibilityeffects whose work against the pressure field may also ex-pected to be large, as first demonstrated by Tailleux (2009).In other words, the above formula suggest that the nonlin-earities of the equation of state may give rise to significantnon-Boussinesq effects. So far, however, most numericalocean models still make the incompressible and Boussinesqapproximations, while at the same time using some versionof the nonlinear equation of state for seawater. Such an ap-proach yields values of ξ and C m that are predicted by Eq.(6), but since those values ultimately derive from initiallymaking the Boussinesq approximation, it is unclear whetherthey can take into account the nonlinear character of theequation of state in a fully consistent manner.In fact, even when the net change ∆ IE r appears to besmall or negligible, seemingly justifying the incompressibleassumption, Tailleux (2009) argues that compressible effectsmay still be large, because one may show that ∆ IE r can bedecomposed as follows: ∆ IE r = ∆ IE exergy + ∆ IE , (17)where IE and IE exergy = IE r − IE are two subcom-ponents of IE r called the ‘dead’ and ‘exergy’ components.Physically, IE represents the internal energy of a notionalthermodynamic equilibrium state of uniform temperature T ,whereas IE exergy represents the internal energy associatedwith the vertical stratification of the reference state. An im-portant result of Tailleux (2009) is that the net changes in IE exergy and IE are related at leading order to ∆ GP E r and ∆ AP E diff as follows: ∆ GP E r ≈ − ∆ IE exergy , (18) ∆ IE ≈ − ∆ AP E diff > , (19)to a very good approximation in a nearly incompressible fluidsuch as water or seawater. These relations state that turbulentmolecular diffusion primarily dissipates AP E into ‘dead’ in-ternal energy IE , while simultaneously causing a transferbetween GP E r and IE exergy . Physically, the former effect results in an increase of the equivalent thermodynamic tem-perature T , whereas the latter effect results in the smoothingout of dT r /dz . This contrasts with the standard interpretationthat turbulent molecular diffusion irreversibly converts AP E into
GP E r , as proposed by Winters & al. (1995). The dif-ferences between the two interpretations are schematicallyillustrated in Fig. 1. The main reason why compressibilityeffects may be important even if ∆ IE r ≈ is because vol-ume changes are primarily determined by ∆ IE exergy , not by ∆ IE or ∆ IE r .As regards to the empirical determination of the mixingefficiency γ mixing = ε P /ε K , the above remarks are impor-tant because ∆ GP E r and | ∆ AP E diff | are currently widelythought to physically represent the same quantity, promptingmany studies to actually estimate ε P from measuring the netchanges in GP E r , e.g., McEwan (1983a,b); Barry (2001).For the reasons discussed above, however, this makes senseonly if ξ can be ascertained to be close to unity, as if not,the relevant value of ξ is then required. One of the main ob-jective of this paper is to establish that the behaviour of ξ isclosely connected to the sign and amplitude of the followingparameter: ddz (cid:18) αPρC p (cid:19) (20)where α is the thermal expansion coefficient, P is the pres-sure, ρ is the density, and C p is the specific heat capac-ity at constant pressure, while salinity is assumed to beuniform throughout the domain. Physically, the parameter Υ = αP/ ( ρC p ) = P Γ /T , where Γ is the adiabatic lapserate, represents the fraction of the amount of heat δQ re-ceived by a parcel in an isobaric process that can be convertedinto work. As a result, Υ is expected to be the main param-eter controlling the net change in GP E r due to the turbulentdiffusive heat exchange between fluid parcels.From the viewpoint of turbulent mixing, the main diffi-culty posed by a nonlinear equation of state is to make itpossible for different vertical stratification to share the sameprofile N ( z ) without necessarily having the same Υ( z ) verti-cal profile. From a dynamical viewpoint, this is not expectedto be a problem as long as the dynamical evolution of KE and AP E , as well as D ( KE ) and D ( AP E ) , remain mostlycontrolled by N ( z ) at leading order, as is usually assumed.If so, the dissipations ratio D ( AP E ) /D ( KE ) , and hencethe bulk mixing efficiency γ mixing , can then be assumed tobe unaffected by the nonlinearities of the equation of state atleading order. The main objective of this paper is to verifythat D ( AP E ) appears indeed to be largely insensitive to the Υ( z ) vertical profile, and hence mostly controlled by N ( z ) .If so, we can safely conclude that it must also be the case for D ( KE ) , since there is even less reasons to believe that thelatter could be affected by Υ( z ) . This could be directly ver-ified through direct numerical simulations of turbulent strat-ified mixing using a fully compressible Navier-Stokes equa-tions solver, which we hope to report on in the future. On. Tailleux: Is mixing efficiency affected by nonlinear equation of state? 5the other hand, the net change in GP E r is expected to beextremely sensitive to Υ( z ) . Most of the paper is devotedto verify that this is indeed the case, and to find ways to re-late the net change in GP E r to the sign and magnitude of d Υ /dz . Section 2 provides a theoretical formulation of theissue discussed. Section 3 discusses the methodology, whilethe results are presented in Section 4. Finally, section 5 sum-marises and discusses the results. ( KE ) ,available potential energy ( AP E ) , and background gravita-tional potential energy ( GP E r ) : d ( KE ) dt = − C ( KE, AP E ) − D ( KE ) , (21) d ( AP E ) dt = C ( KE, AP E ) − D ( AP E ) , (22) d ( GP E r ) dt = W r,mixing = W r,laminar + W r,turbulent , (23)where C ( KE, AP E ) is the so-called buoyancy flux, whichphysically represents the reversible conversion between KE and AP E , while all other terms represent irreversible pro-cesses, with D ( KE ) denoting the viscous dissipation of KE , D ( AP E ) the diffusive dissipation of AP E , and W r,mixing the rate of change of GP E r due to moleculardiffusion, which is customarily decomposed into a turbulent and laminar contribution. Note that the above equations aredomain-averaged, not local formulations, which are expectedto be well suited for understanding laboratory experiments ofturbulent mixing for which lateral fluxes of AP E and KE can be ignored.As discussed by Tailleux (2009), Eqs. (21- 23) providea unifying way to describe the energetics of both the incom-pressible Boussinesq and compressible Navier-Stokes equa-tions, by adapting the definitions of the energy reservoirsand energy conversion terms to the particular set of equa-tions considered. Explicit expressions for D ( AP E ) and W r,mixing are given by Tailleux (2009) in the particularcases of: 1) a Boussinesq fluid with a linear and nonlin-ear equation of state in temperature; 2) for a compressiblethermally-stratified fluid obeying the Navier-Stokes equa-tions of state with a general equation of state depending ontemperature and pressure. These expressions are recalledfurther below for case 2). While W r,laminar is well under-stood to be a conversion between IE and GP E r , the na-ture of the energy conversions associated with D ( AP E ) and W r,turbulent is still a matter of debate. Currently, it is widelyassumed that D ( AP E ) and W r,turbulent represent the samekind of energy conversion, namely the irreversible conver-sion of AP E into
GP E r owing to the fact that for a Boussi-nesq fluid with a linear equation of state (referred to as theL-Boussinesq model hereafter), one has the exact equality D ( AP E ) = W r,turbulent . It was pointed out by Tailleux(2009) that this equality is a serendipitous artifact of the L-Boussinesq model, which does not hold for more accurateforms of the equations of motion. More generally, Tailleux(2009) found that the ratio ξ = W r,turbulent /D ( AP E ) isnot only systematically lower than unity for water or sea-water, but can in fact also takes on negative values, as pre-viously discussed by Fofonoff (1962, 1998, 2001) in a se-ries of little known papers. In other words, the equality D ( AP E ) = W r,turbulent is only a mathematical equality,not a physical equality, by defining a physical equality as amathematical equality between two quantities that persistsfor the most accurate forms of the governing equations ofmotion. To clarify the issue, Tailleux (2009) sought to un-derstand the links between D ( AP E ) , W r,mixing and internalenergy, by establishing the following equations: d ( IE ) dt ≈ D ( KE ) + D ( AP E ) , (24) d ( IE exergy ) dt ≈ = − [ W r,laminar + W r,turbulent ] | {z } W r,mixing , (25)which demonstrate that the viscously dissipated KE and dif-fusively dissipated AP E both end up into the dead part ofinternal energy IE , whereas W r,mixing represent the con-version rate between GP E r and the ’exergy’ component ofinternal energy IE exergy . A schematic energy flowchart il-lustrating the above points is provided in Fig. 1. R. Tailleux: Is mixing efficiency affected by nonlinear equation of state?2.2 Efficiency of mixing and mixing efficiencyThe APE framework introduced by Winters & al. (1995) fora Boussinesq fluid with a linear equation of state, and ex-tended by Tailleux (2009) to a fully compressible thermally-stratified fluid, greatly simplifies the theoretical discussion ofthe concept of mixing efficiency. To that end, it is useful tostart with the evolution equation for the total “available” me-chanical energy M E = KE + AP E , obtained by summingthe evolution equations for KE and AP E , leading to: d ( M E ) dt = − [ D ( KE ) + D ( AP E )] . (26)Eq. (26), along with Eq. (24), are very important, forthey show that both viscous and diffusive processes con-tribute to the dissipation of M E into deal internal energy IE . From this viewpoint, understanding turbulent diapyc-nal mixing amounts to understanding what controls the ratio γ mixing = D ( AP E ) /D ( KE ) , that is, the fraction of the to-tal available mechanical energy dissipated by molecular dif-fusion rather than by molecular viscosity. The amount of MEdissipated by molecular diffusion, i.e., D ( AP E ) , is impor-tant, because it is directly related to the definition of turbulentdiapycnal diffusivity, as said above in relation with Eq. (3).The link between the dissipation mixing efficiency andmore traditional definitions of mixing efficiency can be clari-fied in the light of the above energy equations, by investigat-ing the energy budget of a notional “turbulent mixing event”,defined here as an episode of intense mixing followed andpreceded by laminar conditions (i.e., characterised by veryweak mixing), during which KE and AP E undergo a netchange change ∆ KE < and ∆ AP E < . As far as weunderstand the problem, most familiar definitions of mixingefficiency appear to implicitly assume ∆ AP E ≈ , as is thecase for a turbulent mixing event developing from a unsta-ble stratified shear flow for instance, e.g., Peltier & Caulfield(2003). This point can be further clarified by comparing theenergetics of turbulent mixing events developing from theshear flow instability with that developing from the Rayleigh-Taylor instability, treated next, which by contrast can be re-garded as having the idealised signature ∆ KE ≈ and ∆ AP E < .In the case of the stratified shear flow instability, assumedto be such that ∆ KE < and ∆ AP E ≈ , integrating theabove energy equations over the time interval over which theturbulent mixing event takes place yields: ∆ KE = − C ( KE, AP E ) − D ( KE ) , (27) C ( KE, AP E ) − D ( AP E ) , (28) It is usually assumed that the time average should be shortenough that the viscous dissipation of the mean flow can be ne-glected. Alternatively, one should try to separate the laminar fromthe turbulent viscous dissipation rate. The following derivations as-sume that the viscous dissipation is dominated by the dissipation ofthe turbulent kinetic energy rather than that of the mean flow. ∆ GP E r = W r,mixing = W r,turbulent + W r,laminar , (29)where the overbar denotes the time integral over the mix-ing event. For a Boussinesq fluid with a linear equationof state, Winters & al. (1995) showed that D ( AP E ) = W r,turbulent . If we combine the latter result with the AP E budget (i.e., Eq. (28)), one sees that one has the triple equal-ity: C ( KE, AP E ) = D ( AP E ) = W r,turbulent . (30)The triple equality Eq. (30) suggests that any of the threequantities C ( KE, AP E ) , D ( AP E ) , or W r,turbulent can apriori serve to measure “the fraction of the kinetic energy thatappears as the potential energy of the stratification”, whichis the traditional definition of the flux Richardson numberproposed by Linden (1979). Historically, the buoyancy flux C ( KE, AP E ) is the one that was initially regarded as thenatural quantity to use for that purpose in an overwhelmingmajority of past studies of turbulent mixing. As a result, mostexisting studies of turbulent mixing define the turbulent di-apycnal diffusivity, mixing efficiency, and flux Richardsonnumber in terms of the buoyancy flux as follows: K fluxρ = C ( KE, AP E ) N , (31) γ fluxmixing = C ( KE, AP E ) D ( KE ) , (32) R fluxf = C ( KE, AP E ) C ( KE, AP E ) + D ( KE ) . (33)It is easily verified that the above equations are consistentwith those considered by Osborn (1980) for instance. Phys-ically, however, there are fundamental problems in using thebuoyancy flux to quantify irreversible diffusive mixing, be-cause as pointed out by Caulfield & Peltier (2000), Staquet(2000) and Peltier & Caulfield (2003), C ( KE, AP E ) rep-resents a reversible energy conversion, which usually takeson both large positive and negative values before settling onits long term average D ( AP E ) . Moreover, as pointed out be-low, the buoyancy flux is only related to irreversible diffusivemixing only if ∆ AP E ≈ holds to a good approximation,for otherwise, it becomes also related to the irreversible vis-cous dissipation rate as shown by the KE budget (Eq. (27)).Eq. (30) makes it possible, however, to use either D ( AP E ) or W r,turbulent instead of C ( KE, AP E ) in the definitions(31) and (32). For this reason, both Caulfield & Peltier(2000) and Staquet (2000) proposed to measure the effi-ciency of mixing based on W r,turbulent , i.e., K GP Erρ = W r,turbulent N , (34) γ GP Ermixing = W r,turbulent D ( KE ) , (35). Tailleux: Is mixing efficiency affected by nonlinear equation of state? 7 R GP E r f = W r,turbulent W r,turbulent + D ( KE ) , (36)such a definition being motivated by Winters & al. (1995)’sinterpretation that D ( AP E ) and W r,turbulent represent thesame energy conversion whereby the diffusively dissipated AP E is irreversibly converted into
GP E r . The parame-ter R GP E r f was called the “cumulative mixing efficiency”by Peltier & Caulfield (2003) and modified flux Richardsonnumber by Staquet (2000). As argued in Tailleux (2009), itis D ( AP E ) , rather than W r,turbulent , that directly measuresthe amount of KE eventually dissipated by molecular dif-fusion via its conversion into AP E , suggesting that the fluxRichardson number should actually be defined as: R DAP Ef = D ( AP E ) D ( KE ) + D ( AP E ) . (37)While the above formula makes it clear that all above defini-tions of R f are equivalent in the particular case considered,it is easily realized that they will in general yield differentnumbers if one relaxes the assumption ∆ AP E ≈ in Eq.(28), as well as the assumption of a linear equation of state,yielding a ratio ξ = W r,turbulent /D ( AP E ) that is gener-ally lower than unity and sometimes even negative for wateror seawater. For this reason, it is crucial to understand thephysics of mixing efficiency at the most fundamental level.From the literature, it seems clear that most investigators’sidea about the flux Richardson number is as a quantity com-prised between and . From that viewpoint, the dissipationflux Richardson number R DAP Ef is the only quantity that sat-isfies this property under the most general circumstances, ascases can easily be constructed for which both W r,turbulent and C ( KE, AP E ) are negative. Indeed, cases for which ξ < are described in this paper, whereas C ( KE, AP E ) iseasily shown to be negative in the case of a turbulent mixingevent for which all mechanical energy is initially providedentirely in AP E form. In that case, assuming ∆ AP E < and ∆ KE ≈ in the above energy budget equations yields: C ( KE, AP E ) = D ( AP E ) + ∆
AP E = − D ( KE ) , (38)which shows that this time, C ( KE, AP E ) directly measuresthe amount of viscously dissipated kinetic energy, ratherthan diapycnal mixing. The latter case is relevant to under-stand the energy budget of the Rayleigh-Taylor instability,see Dalziel & al (2008) for a recent discussion of the latter.2.3 Link between D ( AP E ) and W r,mixing In order to help the reader understand or appreciate why theratio ξ = W r,turbulent /D ( AP E ) is generally lower thanunity for water or seawater, and hence potentially signif-icantly different from the predictions of the L-Boussinesqmodel, it is useful to examine the structure of W r,mixing and D ( AP E ) in more details. As shown by Tailleux (2009), the analytical formula for the latter quantities in a fully com-pressible thermally-stratified fluid are given by: W r,mixing = Z V α r P r ρ r C pr ∇ · ( κρC p ∇ T ) dV, (39) D ( AP E ) = − Z V T − T r T ∇ · ( κρC p ∇ T ) dV, (40)where as before α is the thermal expansion coefficient, P isthe pressure, C p is the specific heat capacity at constant pres-sure, ρ is density, with the subscript r indicating that valueshave to be estimated in their reference state. The parame-ter Υ = αP/ ( ρC p ) plays an important role in the problem.Physically, it can be shown that in an isobaric process duringwhich the enthalpy of the fluid parcel increases by dH , theparameter Υ represents the fraction of dH that is not con-verted into internal energy, i.e., the fraction going into work(and hence contributing ultimately to the overall net changein GP E r ). As a result, Υ plays the role of a Carnot-like ther-modynamic efficiency. In Eq. (39), Υ r denotes the value that Υ would have if the corresponding fluid parcel was displacedadiabatically to its reference position.In order to compare these two quantities, we expand T asa Taylor series around P = P r , viz., T = T r + Γ r ( P − P r ) + . . . (41)where Γ r = α r T r / ( ρ r C pr ) is the adiabatic lapse rate. Atleading order, therefore, one may rewrite D ( AP E ) as fol-lows: D ( AP E ) = Z V α r ( P r − P ) ρ r C pr T r T ∇ · ( κρC p ∇ T ) dV + . . . = W r,mixing + Z V ( T r − T ) T α r P r ρ r C pr ∇ · ( κρC p ∇ T ) dV − Z V α r T r Pρ r C pr T ∇ · ( κρC p ∇ T ) dV + · · · (42)These formula shows that D ( AP E ) can be written as thesum of W r,mixing plus some corrective terms. One sees thatthe L-Boussinesq model’s results derived by Winters & al.(1995) can be recovered in the limit T ≈ T r , P ≈ − ρ gz , α r / ( ρ r C pr ) ≈ α / ( ρ C p ) , ρC p ≈ ρ C p , where thesubscript refers to a constant reference Boussinesq value,yielding: D ( AP E ) ≈ W r,mixing − W r,laminar = W r,turbulent . (43)These results, therefore, demonstrate that the strong corre-lation between D ( AP E ) and W r,mixing originates in bothterms depending on molecular diffusion in a related, but nev-ertheless distinct, way, the differences between the two quan-tities being minimal for a linear equation of state. The factthat the two terms are never exactly equal in a real fluidclearly refutes Winters & al. (1995)’s widespread interpre-tation that D ( AP E ) and W r,turbulent physically represents R. Tailleux: Is mixing efficiency affected by nonlinear equation of state?the same energy conversion whereby the diffusively dissi-pated AP E is irreversibly converted into
GP E r . In real-ity, D ( AP E ) and W r,turbulent represent two distinct typesof energy conversions that happen to be both controlled bystirring and molecular diffusion in related ways, which ex-plains why they appear to be always strongly correlated, andeven exactly equal in the idealised limit of the L-Boussinesqmodel. If one accepts the above point, then it should be clearthat what is now required to make progress is the understand-ing of what controls the behaviour of the parameter ξ , sincethe knowledge of the latter is obviously crucial to make infer-ences about turbulent diapycnal mixing from measuring thenet changes of GP E r for instance. The purpose of the nu-merical simulations described next is to help gaining insightsinto what controls ξ . To get insights into how the equation of state of seawa-ter affects turbulent mixing, we compared D ( AP E ) and W r,turbulent for a number of different stratifications havingthe same buoyancy frequency vertical profile N , but differentvertical profiles with regard to the parameter αP/ ( ρC p ) , asillustrated in Fig. 2. The quantities D ( AP E ) and W r,mixing were estimated from Eqs. (39) and (40), while W r,turbulent was estimated from W r,turbulent = W r,mixing − W r,laminar , (44)where W r,laminar was obtained by taking T = T r inthe expression for W r,mixing . The quantities D ( AP E ) and W r,turbulent were estimated numerically for a two-dimensional square domain discretised equally in the hori-zontal and vertical direction. In total, 27 different stratifica-tions were considered, all possessing the same squared buoy-ancy frequency N illustrated in the left panel of Figure 2,but different mean temperature, salinity, and pressure result-ing in different profiles for the αP/ ( ρC p ) parameter illus-trated in the right panel of Figure 2. In all cases considered,the pressure varied from P min to P max = P min + 10 dbar ,with P min taking the three values (0 dbar, 1000 dbar, 2000dbar). In all cases, the salinity was assumed to be con-stant, and taking one of the three possible values S = (30Psu, 35 psu, 40 psu). With regard to the temperature pro-file, it was determined by imposing the particular value T max = T ( P min ) at the top of the fluid, with all remain-ing values determined by inversion of the buoyancy fre-quency N common to all profiles by an iterative method.The imposition of a fixed buoyancy profile N , salinity S ,pressure range, and minimum temperature T min was foundto yield widely different top-bottom temperature differences T ( P min ) − T ( P max ) , ranging from a few tenths of degrees toabout degrees C depending on the case considered, as seenin Fig. 3. In each case, the thermodynamic properties ofthe fluid were estimated from the Gibbs function of Feistel (2003). Specific details for the temperature, pressure, andsalinity in each of the 27 experiments can be found in Table1 along with other key quantities discussed below.Numerically, the two-dimensional domain used to quan-tify D ( AP E ) and W r,turbulent was discretised into N pi × N pj points in the horizontal and vertical, with N pi = N pj =100 . Mass conserving coordinates were chosen in the verti-cal, and regular spatial Cartesian coordinate in the horizontal.For practical purposes, the vertical mass conserving coordi-nate can be regarded as standard height z , as the differencesbetween the two types of coordinates were found to be in-significant in the present context, and thus chose ∆ x = ∆ z .In order to compute D ( AP E ) and W r,turbulent for turbulentconditions, we modelled the stirring process by randomlyshuffling the fluid parcels adiabatically from resting initialconditions. Shuffling the parcels in such a way requires acertain amount of stirring energy, which is equal to the avail-able potential energy AP E of the randomly shuffled state.
For each of the 27 particular reference stratifications consid-ered, synthetic turbulent states were constructed by gener-ating hundreds of random permutations of the fluid parcels,thus simulating the effect of adiabatic shuffling by the stirringprocess, in each case yielding a particular value of D ( AP E ) , W r,mixing , W r,turbulent and AP E . One way to illustratethat W r,turbulent depends more sensitively on the equationof state than D ( AP E ) is by plotting each quantity as a func-tion of AP E , as illustrated in Fig. 4. Interestingly, the figureshows that all values of D ( AP E ) appear to be close to a lin-ear straight line, with no obvious sensitivity to the particularvalue of Υ . In contrast, the right panel of Fig. 4 demonstratesthe sensitivity of W r,turbulent to Υ , as a separate curve is ob-tained for each different stratification. Note that one shouldnot construe from Fig. 4 that D ( AP E ) is a linear functionof AP E . Physically, D ( AP E ) depends both on the AP E ,as well as on the spectrum of the temperature field. It sohappens that the method used to randomly shuffle the parcelstends to artificially concentrate all the power spectrum at thehighest wavenumbers, the effect of which being to suppressone degree of freedom to the problem, which is responsiblefor the appearance of a linear relationship between D ( AP E ) and AP E in Fig. 4. It is easy to convince oneself, however,that stratifications can be constructed which have the samevalue of
AP E , but widely different values of D ( AP E ) .In order to understand how the equation of state affects W r,turbulent , it is useful to rewrite W r,mixing as given byEq. (39) as follows: W r,mixing = − Z V κρC p ∇ T · ∇ (cid:18) α r P r ρ r C pr (cid:19) dV ≈ − Z V ρκC p ∂∂z r (cid:18) α r P r ρ r C pr (cid:19) ∂T r ∂z r k∇ z r k dV + · · · (45). Tailleux: Is mixing efficiency affected by nonlinear equation of state? 9by using an integration by parts, assuming insulated bound-aries, and using the approximation ∇ T ≈ ∇ T r + O ( T − T r ) ,by noting that the reference quantities depend only upon z r .Eq. (45) suggests that W r,mixing and W r,turbulent are pri-marily controlled by the vertical gradient of Υ = αP/ ( ρC p ) ,and that both W r,mxing and W r,turbulent are likely to be pos-itive only when d Υ /dz is negative. This is obviously the casewhen the vertical variations of α/ ( ρC p ) can be neglected, asin this case d Υ /dz ≈ α/ ( ρC p ) dP/dz ≈ − αg/C p < , as-suming the pressure to be hydrostatic. The case when thevertical gradient of αP/ ( ρC p ) is positive was extensivelydiscussed by Fofonoff (1962, 1998, 2001), and can be easilyencountered in the oceans.In all experiments considered, we found the ratio ξ = W r,turbulent /D ( AP E ) to be systematically lower thanunity, as already pointed out in Tailleux (2009). In orderto better understand how d Υ /dz controls the behaviour of W r,turbulent , the ratio ξ = D ( AP E ) /W r,turbulent was av-eraged over all randomly shuffled states separately for eachstratification, the results being summarised in Fig. 5 and Ta-ble 1, along with the minimum value of d Υ /dz , as well aswith the top-bottom difference ∆Υ = Υ( P min ) − Υ( P max ) .Panels (a) and (c) show that as long that d Υ < , the equality W r,turbulent ≈ D ( AP E ) holds to a rather good approxima-tion, up to a factor of 2, the approximation being degradedat the lowest temperature and salinity. Note, however, thatin the cases considered, ξ > only at atmospheric pressure,with ξ being systematically negative at P min = 1000 dbars and P min = 2000 dbar respectively. Both Table 1 andFig. 5 (a) and (c) show that ξ becomes increasingly neg-ative as [ d Υ /dz ] min becomes increasingly large and posi-tive, the worst case being achieved for the lowest T , lowestsalinity, and highest pressure. As a further attempt to un-derstand this behaviour, we also computed the average ratio AGP E/AP E for each particular reference stratification. In-terestingly, we find that the classical case ξ ≈ coincide with AP E ≈ AGP E , as expected in the Boussinesq approxima-tion. We find, however, that the decrease in ξ coincides with AGP E being an increasingly bad approximation of
AP E .As the latter implies that
AIE becomes increasingly impor-tant, it also implies that compressible effects become increas-ingly important. This suggests, therefore, that the effects of anonlinear equation of state are apparently strongly connectedto non-Boussinesq effects, a topic for future exploration.The key point of the present results is that while thereexist stratifications such that W r,turbulent ≈ D ( AP E ) toa good approximation, and hence that conform to classicalideas about turbulent mixing in a Boussinesq fluid with a lin-ear equation of state, there also exist stratification for which W r,turbulent and D ( AP E ) differ radically from each other.The main reason why this is not more widely appreciatedis suggested by the results summarised in Table 1, whichshows that W r,turbulent ≈ D ( AP E ) appears to hold wellunder normal temperature and pressure conditions, which areusually those encountered in most laboratory experiments of turbulent mixing. In that case, the classical results ofBoussinesq theory are applicable, and there is no problems inmeasuring the mixing efficiency of turbulent mixing eventsfrom measuring the net change in GP E r , as often done, e.g.,Barry (2001), in accordance with the definition of mixing ef-ficiency proposed by Caulfield & Peltier (2000) and Staquet(2000), since ξ ≈ to a good approximation. Temperature,salinity, and pressure conditions in the real oceans can bevery different than in the laboratory, however, especially inthe abyss. In the latter case, the present results suggest notonly that ξ can potentially become very large and negative,but that the discrepancy between AGP E and
AP E can be-come significant to the point of making the Boussinesq ap-proximation and the neglect of compressible effects very in-accurate. This point seems important in view of the currentintense research effort devoted to understanding tidal mixingin the abyssal oceans that was prompted a decade ago by theinfluential study by Munk & Wunsch (1998). The point isalso important because values of mixing efficiency publishedin the literature have been traditionally been reported withoutmentioning the associated value of ξ , which may explain partof the spread in the published values, and adds to the uncer-tainty surrounding this crucial parameter. The present resultssuggest that an important project would be to seek to recon-struct the missing values of ξ , which is in principle possible ifsufficient information about the ambient conditions are avail-able. The nonlinearities of the equation of state for water or sea-water make it possible for a stratification with given meanvertical buoyancy profile N to have widely different ver-tical profiles of the parameter Υ = αP/ ( ρC p ) , depend-ing on particular oceanic circumstances. The main resultof this paper is that the sign and magnitude of d Υ /dz greatly affect W r,turbulent — the turbulent rate of change of GP E r — while they correspondingly little affect D ( AP E ) ,the dissipation rate of AP E . As a result, the ratio ξ = W r,turbulent /D ( AP E ) is in general lower than unity, andsometimes even negative, for water or seawater. For this rea-son, the fact that D ( AP E ) and W r,turbulent happen to beidentical for a Boussinesq fluid with a linear equation of stateappears to be a very special case, which is rather misleadingin that it fails to correctly address the wide range of valuesassumed by the parameter ξ in the actual oceans, while alsoleading to the widespread erroneous idea that the diffusivelydissipated AP E is irreversibly converted into
GP E r , andhence that turbulent mixing always increase GP E . As far aswe understand the problem, based on the analysis of Tailleux(2009), D ( AP E ) and W r,turbulent represent two physicallydistinct kinds of energy conversion, the former associatedwith the dissipation of AP E into ‘dead’ internal energy, andthe latter associated with the conversion between
GP E r and0 R. Tailleux: Is mixing efficiency affected by nonlinear equation of state?the ’exergy’ part of internal energy. The former is alwayspositive, while the latter can take on both signs, dependingon the particular stratification.From the viewpoint of turbulence theory, the present re-sults indicate that the equality D ( AP E ) = W r,turbulent obtained in the context of the L-Boussinesq model byWinters & al. (1995) should only be construed as implyinga strong correlation between D ( AP E ) and W r,turbulent , notas an indication that the diffusively dissipated AP E is con-verted into
GP E r . As the present results show, the correla-tion between the two rates strongly depends on the nonlin-earities of the equation of state. Fundamentally, D ( AP E ) and W r,turbulent appear to be correlated because they bothdepend on molecular diffusion, and on the gradient of theadiabatic displacement ζ = z − z r of the isothermal surfacesfrom their reference positions. Based on the present results,the ratio ξ = W r,turbulent /D ( AP E ) appears to be deter-mined at leading order mostly by the sign and magnitude of d Υ /dz = d/dz [ αP/ ( ρC p ) . Further work is required, how-ever, to clarify the precise link between ξ and d Υ /dz underthe most general circumstances, which will be reported in asubsequent paper.The present results are important, because they show thatthe two following ways of defining a flux Richardson number R f and mixing efficiency γ mixing , viz., γ DAP Emixing = D ( AP E ) D ( KE ) , (46) R DAP Ef = D ( AP E ) D ( AP E ) + D ( KE ) (47)called the dissipation mixing efficiency and flux Richardsonnumber by Tailleux (2009), and γ GP Ermixing = W r,turbulent D ( KE ) , (48) R GP Erf = W r,turbulent W r,turbulent + D ( KE ) , (49)as proposed by Caulfield & Peltier (2000) and Staquet(2000), which are equivalent in the context of the L-Boussinesq model, happen to be different in the context ofa real compressible fluid, as the conversion rules γ GP Ermixing = ξγ DAP Emixing , (50) R GP Erf = ξR DAP Ef − (1 − ξ ) R DAP Ef . (51)now involve the parameter ξ . Note that historically the fluxRichardson number was defined by Linden (1979) as “Thefraction of the kinetic energy which appears as the poten-tial energy of the stratification.” Physically, the kinetic en-ergy that appears as the potential energy of the stratificationis the fraction of kinetic energy being converted into AP E and ultimately dissipated by molecular diffusion. This frac-tion is therefore measured by D ( AP E ) , not by W r,turbulent ,since the latter technically represents the “mechanically-controlled” fraction of internal energy converted into GP E r ,if one accepts Tailleux (2009)’s conclusions. From thisviewpoint, it is R DAP Ef rather than R GP E r f that appears tobe consistent with Linden (1979)’s definition of the fluxRichardson number, and hence γ DAP Emixing rather than γ GP E r mixing that is consistent with Osborn (1980)’s definition of mixingefficiency.From a practical viewpoint, however, the above conceptualobjections against γ GP E r mixing and R GP E r f do not mean that it isequally physically objectionable to seek estimating the effi-ciency of mixing from measuring the net changes in GP E r taking place during a turbulent mixing event, as is commonlydone, e.g., Barry (2001). Such a method is perfectly valid,owing to the correlation between D ( AP E ) and W r,turbulent .The present results show, however, that such an approach re-quires the knowledge of the parameter ξ , which is usuallynot supplied. For most laboratory experiments performed atatmospheric pressure, the issue is probably unimportant, as ξ appears to be generally close to unity in that case. The issuebecomes more problematic, however, for measurements car-ried out in the ocean interior, as there is less reason to assumethat ξ ≈ will be necessarily verified. A critical review ofpublished values of γ mixing would be of interest, in order toidentify the cases potentially affected by a value of ξ signifi-cantly different from unity.So far, we have only considered the case of an equation ofstate depending on temperature and pressure only, by holdingsalinity constant. In practice, however, many studies of tur-bulent mixing are based on the use of compositionally strat-ified fluids. Understanding whether ξ can be significantlydifferent from unity in that case remains a topic for futurestudy. Acknowledgements.
The author acknowledges funding from theRAPID programme. Rainer Feistel, Michael McIntyre and ananonymous referee are also thanked for provided comments that ledto significant improvements in presentation and clarity.
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Ann. Rev. of Fluid Mech. , ,281–314.... fff t2 R. Tailleux: Is mixing efficiency affected by nonlinear equation of state? KE IEexergyGPErAPE GPErAPE IEoKE IED(KE)C(KE,APE) Wr,laminarWr,laminarWr,turbulentC(KE,APE) D(KE)= D(APE)D(APE)A) New view of energetics of turbulent mixing (Tailleux, 2009)B) Classical view of energetics of turbulent mixing (Winters et al, 1995)+ D(APE) ξ Fig. 1.
A) New view of the energetics of freely decaying turbulent stratified mixing as proposed by Tailleux (2009) versus B) the earlierinterpretation proposed by Winters et al. (1995). In the new view, internal energy IE is subdivided into a dead part IE and exergy part IE exergy . The double arrow linking IE exergy and GP E r means that both W r,laminar and W r,turbulent can be either positive or negativein general. . Tailleux: Is mixing efficiency affected by nonlinear equation of state? 13 −4 p−p min (dbar) N ( s − ) −6 −5 −4 −3 −2 p−p min (dbar) α P / ( ρ C p ) Fig. 2. (Top panel) The squared buoyancy frequency N common to all stratifications considered. (Bottom panel) The thermodynamicefficiency-like quantity αP/ ( ρC p ) corresponding to the 27 different cases considered. Note that the Fofonoff regime, i.e., the case forwhich GP E decreases as the result of mixing, is expected whenever the latter quantity decreases for increasing pressure. The classical caseconsidered by the literature, i.e., the case for which
GP E increases as the result of mixing corresponds to the case where the latter quantityincreases with increasing pressure on average (see Table 1 for more details). T ( p m i n ) − T ( P m a x ) Fig. 3.
Distribution of the top-bottom temperature difference T ( P min ) − T ( P max ) as a function of the experiment number. . Tailleux: Is mixing efficiency affected by nonlinear equation of state? 15 APE D ( APE ) APE W r ,t u r bu l en t Fig. 4. (Left panel) The dissipation rate of
AP E as a function of
AP E , each point corresponding to one particular experiment. Note thatthere is no obvious dependence on the stratification. (Bottom panel) The rate of change W r,turbulent as a function of AP E . This time, eachstratification is associated with a different curve. R a t i o ξ = W r ,t u r bu l en t / D ( APE ) (a) 0 10 20 3011.11.21.31.41.5 Experiment number A G PE / APE (b)0 10 20 30−3−2−101 x 10 −5 Experiment number M i n d / d z ( α P / ( ρ C p )) (c) 0 10 20 30−10−505 x 10 −5 Experiment number ∆ α P / ( ρ C p ) (d) Fig. 5. (a) The averaged ratio ξ = W r,turbulent /D ( AP E ) as a function of the experiment number; (b) The averaged ratio AGP E/AP E as a function of the experiment number; (c) The minimum value of d/dz [ αP/ ( ρC p ) as a function of the experiment number; (d) Thetop-bottom difference of αP/ ( ρC p ) as a function of the experiment number. . Tailleux: Is mixing efficiency affected by nonlinear equation of state? 17 Table 1.
Averaged values of the two ratios ξ = W r,turbulent /D ( AP E ) and AGP E/AP E for the 27 different types of stratificationsconsidered in this paper. The quantities [ d Υ /dz ] min and ∆Υ refer to the minimum value of the vertical derivative of Υ = αP/ ( ρC p ) andtop-bottom difference of Υ respectively. S is the salinity used in the equation of state for seawater, T is the mean temperature of the profileconsidered, and P min denotes the minimum value of the vertical pressure profile. The top-bottom temperature differences are displayed inFig. 3, while the pressure interval is
10 dbar in all cases. The tabulated values demonstrate that increasingly negative values of ξ coincidewith increasingly large positive values of d Υ /dz , as well as with with the increasing importance of non-Boussinesq compressible effectsassociated with an increasing discrepancy between AGP E and
AP E . The standard case for which ξ ≈ is achieved close to atmosphericpressure. The maximum negative value of ξ occurs for the lowest S , lowest T , and largest P min values considered.Expt ξ AGP E/AP E [ d Υ /dz ] min × ∆Υ × S(psu) T ( ◦ C ) P min ( dbar ))