The effect of mechanical stirring on buoyancy-driven circulations
aa r X i v : . [ phy s i c s . f l u - dyn ] J un Tellus , 000–000 (0000) Printed 29 October 2018 (Tellus L A TEX style file v2.2)
The effect of mechanical stirring on horizontal convection
By R TAILLEUX ⋆ and L ROULEAU Department of Meteorology, University of Reading, Earley Gate, PO Box 243, Reading, United Kingdom (Manuscript received 24 November 2003; in final form 25 November 2003)
ABSTRACT
The theoretical analysis of the energetics of mechanically-stirred horizontal con-vection for a Boussinesq fluid yields the formula: G ( AP E ) = γ mixing G ( KE ) + (1 + γ mixing ) W r,laminar where G ( AP E ) and G ( KE ) are the work rate done by the buoyancy and mechani-cal forcing respectively, γ mixing is the mixing efficiency, and W r,laminar is the back-ground 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 ( AP E )) even for a relatively low mixing ef-ficiency, whereas this is only possible in absence of mechanical stirring if γ mixing ≫ γ mixing G ( KE ) ≫ (1 + γ mixing ) W r,laminar . This result explains why the buoyancy-driven overturning cell in the laboratory experiments by Whitehead et al. (2008) isamplified by the lateral motions of a stirring rod. The formula implies that the ther-modynamic efficiency of the ocean heat engine, far from being negligibly small as iscommonly claimed, might in fact be as large as can be thanks to the stirring done bythe wind and tides. These ideas are further illustrated by means of idealised numericalexperiments. A non-Boussinesq extension of the above formula is also given. A fundamental objective of ocean circulation theory isto characterise and quantify the relative importance of themechanical and thermodynamical forcing in driving and stir-ring the oceans. Although the oceans are highly nonlineara system, ocean circulation theory nevertheless historicallyevolved under the assumption that separate theories for thewind-driven and buoyancy-driven circulation could be de-veloped, as it the two kind of circulation could be regardedas somehow decoupled. From a thermodynamic viewpoint,this is somehow justified from the fact that mechanical forc-ing, unlike the buoyancy forcing, does not modify the fluidparcels’ buoyancy, suggesting that distinct effects shouldbe observable. In the thermodynamic engineering literature,mechanical forcing is commonly regarded as “shaft” work,which alters the energy of the fluid without altering its totalentropy. In oceanographic textbooks, wind-driven circula-tion theories usually occupy a more prominent place thatbuoyancy-driven circulation theories. Indeed, it turns out tobe relatively easy to link the near surface transport (Ekmantheory) or the vertically-integrated transport (Sverdrup the-ory) to the wind stress or its spatial derivatives in a quan-titative relatively accurate way. Theories for the buoyancy-driven circulation, on the other hand, are often more quali-tative, and at best offer only tentative scaling arguments to ⋆ Corresponding author.e-mail: [email protected] relate the strength of the large-scale overturning cell takingplace in the meridional/vertical plane, e.g. Colin de Verdi`ere(1993). From a physical viewpoint, the two main salient in-gredients of the buoyancy-driven circulation have been tra-ditionally associated with high-latitude cooling, envisionedas the destabilising mechanism setting up the meridionaloverturning circulation into motion, and turbulent diapyc-nal mixing, envisioned as the process required to carry heatfrom the surface at the depths cooled by deep water forma-tion, thus precluding the oceans to fill up with dense andsalty waters.Physically, however, the buoyancy-driven circulationcan only be regarded as independent of the mechanical forc-ing if one can establish that the stirring required to maintainturbulent diapycnal mixing is predominantly sustained bythe work rate done by the surface buoyancy fluxes G ( AP E )(i.e., the rate of available potential energy production, seeLorenz (1955)). A crucial question, therefore, is how large is G ( AP E )? In their observational analysis of ocean energetics,Oort et al. (1994) found G ( AP E ) = 1 . ± . G ( AP E ) to be significant, and hence thatturbulent diapycnal mixing must be primarily be sustainedby the work rate done by the mechanical forcing due to thewind and tides, leading to the idea that the buoyancy-drivencirculation is actually mechanically-driven. MW98 further- c (cid:13) AUTHOR RUNNING HEAD more sought to quantify the magnitude of the mechanicalsources of stirring required to sustain diapycnal mixing inthe oceans. Their main result is the following formula: G ( KE ) = W r,forcing γ mixing , (1)where G ( KE ) is the work rate done by the mechanicalsources of stirring, W r,forcing is the rate at which the back-ground gravitational potential energy GP E r decreases asthe result of high-latitude cooling, and γ mixing is the so-called mixing efficiency, e.g., Osborn (1980). MW98 esti-mated W r,forcing ≈ . γ mixing = 0 . G ( KE ) = O (2 TW) was required to sustain the observedrate of turbulent diapycnal mixing and its associated pole-ward heat transport of about 2 PW. Since the wind forcingprovides no more than about 1 TW, this result suggests anapparent shortfall of about 1 TW of mechanical stirring thatMW98 argued must only come from the work rate done bythe tides, spawning much research over the past decade onthe issue of tidal mixing.MW98’s study prompted much debate over the pastdecade about the relative importance of the surface buoy-ancy forcing, and about whether Sandstrom’s theorem re-ally implied for G ( AP E ) to be negligible. Recently, Tailleux(2009) revisited a number of misconceptions about thenature of energy conversions in turbulent stratified fluidsby extending the available potential energy framework ofWinters et al. (1995) to the fully compressible Navier-Stokesequations (CNSE thereafter). One of the main result was todemonstrate from first principles that: G ( AP E ) ≈ W r,forcing , (2)which physically states that the production rate of AP E isapproximately equal to the rate at which
GP E r decreases asthe result of high-latitude cooling. This result is extremelyimportant, because it rigourously proves that it is inconsis-tent to assume W r,forcing to be large while simultaneouslyassuming G ( AP E ) to be negligible, as done in MW98. Asa result, rather than establishing that the work rate doneby surface buoyancy forcing is small, MW98’s study actu-ally establishes precisely the contrary, since their estimate W r,forcing ≈ . G ( AP E ) ≈ . G ( AP E ). Moreover, Tailleux (2009) was able to showthat a generalisation of MW98’s Eq. (1) for a non-Boussinesqfluid can be rigourously derived from first principles by com-bining the mechanical energy budget for
AP E and
GP E r separately, the final result taking the form: G ( KE ) = 1 + (1 − ξ ) γ mixing ξγ mixing G ( AP E )= 1 − ξR f ξR f G ( AP E ) , (3)where ξ is a non-Boussinesq nonlinearity parameter which issuch that −∞ < ξ γ mixing is themixing efficiency, and R f = γ mixing / (1 + γ mixing ) is the so-called flux Richardson number, e.g., Osborn (1980). MW98’sEq. (1) is recovered in the limiting case ξ = 1 describing aBoussinesq fluid with a linear equation of state, by using theabove result that G ( AP E ) ≈ W r,forcing . The fact that the generalisation of MW98’s result canbe written as a formula linking G ( KE ) and G ( AP E ) ratherthan G ( KE ) and W r,forcing raises the question of whether itis really appropriate to interpret Eq. (3) (and hence Eq. (1))as a constraint on the amount of G ( KE ) required to sustaindiapycnal mixing in the oceans, as suggested by MW98. In-deed, it seems that such an interpretation implicitly assumesthat both G ( AP E ) and γ mixing can be regarded as fixed insome sense, but whether this is the correct way to interpretEq. (3) is unclear, given that the latter can be rewritten inthe two following ways: G ( AP E ) = ξγ mixing − ξ ) γ mixing G ( KE ) , (4)or as: ξR f = G ( AP E ) G ( KE ) + G ( AP E ) , (5)i.e., under the form of a constraint either on G ( AP E ) oron ξR f . Note that from observations, plausible numbers forthe wind and buoyancy forcing are G ( KE ) ≈ G ( AP E ) ≈ . ξR f ≈ . . . . Another way to pose the problem initially formulated byMW98 would be to ask the question of whether the abovevalue for ξR f can be ruled out from what we know aboutmixing efficiency (and the role played by the nonlinearity ofthe equation of state, but we know less about the latter).Although the value γ mixing ≈ R f ≈ . R f = 0 . γ mixing = 1) have beenreported by Dalziel et al. (2008) for buoyancy-driven mixingassociated with Rayleigh-Taylor instability. Since G ( AP E )is comparable to G ( KE ), values of γ mixing intermediate be-tween those for mechanically- and buoyancy-driven mixingshould be expected.In trying to figure out the exact physical meaning of Eq.(3), it is important to discuss the nature and structure of G ( KE ) and G ( AP E ) in more details. Thus, the work ratedone by the wind stress is given by: G ( KE ) = Z S τ · u s dS (6)where τ is the wind stress at the ocean surface, and u s is theocean surface velocity. The important point about G ( KE ) isthat it is a correlation between the external forcing (the windstress) and a parameter depending upon the particular stateof the system (the surface velocity). In other words, it is es-sential to realize that G ( KE ) cannot be determined from theknowledge of the external forcing alone. This is important,because Eq. (6) expresses the possibility for the wind-drivencirculation to be controlled by the buoyancy forcing, to theextent that the latter is able to influence the surface veloc-ity. In order to gain insight into the potential importanceof the buoyancy control of the wind-driven circulation, itwould be interesting to compute G ( KE ) for a purely windforced homogeneous ocean model. Doing so, however, is be-yond the scope of this paper. For the present purposes, weshall assume that G ( KE ) is primarily determined by the c (cid:13) , 000–000ECHANICALLY-STIRRED BUOYANCY-DRIVEN CIRCULATIONS wind forcing, and that buoyancy forcing only alters it as asecond order effect, which seems plausible on the basis thatthe near surface circulation in the oceans is usually thoughtto be the signature of the wind forcing rather than the buoy-ancy forcing.With regard to the work rate done by surface buoyancyfluxes, it was shown by Tailleux (2009) to be given by thefollowing expression: G ( AP E ) = Z S T − T r T Q surf dS, (7)in the case of a compressible thermally-stratified fluid forcedby surface heat fluxes, where T is the surface temperature ofthe fluid parcels, and T r the temperature the surface parcelswould have if displaced adiabatically to their level in Lorenz(1955)’s reference state, while Q surf is the diabatic rate ofheating cooling/heating due to the surface heat fluxes. Use-ful approximations for G ( AP E ) can be obtained by expand-ing T as a Taylor series expansion around the surface pres-sure P a , i.e., T ≈ T r + Γ r ( P a − P r ) + O (( P a − P r ) ), whereΓ r = α r T r / ( ρ r C pr ) is the adiabatic lapse rate, leading to: G ( AP E ) ≈ − Z S α r ( P r − P a ) ρ r C pr Q surf dS ≈ Z S α r gz r C pr Q surf dS, (8)by using the approximation P r − P a ≈ − ρ gz r , the lat-ter expression being actually the Boussinesq approxima-tion of G ( AP E ) derived by Winters et al. (1995) providedthat one regards the thermal expansion α r and specificheat capacity C pr as constant (the suffix r means that thevariables have to be estimated in Lorenz (1955)’s refer-ence state). As for G ( KE ), G ( AP E ) also takes the formof a correlation between the external forcing (the heat-ing/cooling rate at the surface) and a parameter dependingon the particular state of the system, namely ( T − T r ) /T ≈ α r ( P a − P r ) / ( ρ r C pr ) ≈ α r gz r /C pr . Eqs. (7) and (8) makeit possible, therefore, for the buoyancy-driven circulation tobe mechanically-controlled. Physically, this is expected, be-cause mechanical forcing is widely agreed to increase diapy-cnal mixing, which should produce a deeper thermocline,thereby allowing dense plumes to penetrate deeper, thusincreasing G ( AP E ) and hence the buoyancy-driven circu-lation. Interestingly, this appears consistent with the labo-ratory experiments by Whitehead et al. (2008) which pro-vide evidence that the lateral motion of a stirring rod cangreatly enhance the strength of horizontal convection (seeHughes et al. (2008) for a review on this topic). This paperexamines the possibility of interpreting this result as theconsequence that G ( AP E ), and hence the buoyancy-drivencirculation, is increased by the work rate done by the stir-ring rod. To that end, Section 2 seeks to extend Eq. (3) toalso describe the purely buoyancy-driven case for which it isnot valid. The physical meaning of the extended formula isthen explored by means of idealised numerical experimentsin Section 3. Finally, section 4 presents a discussion of theresults.
Our starting point is the theoretical description of theenergetics of mechanically and thermodynamically forcedturbulent stratified fluids recently derived by Tailleux (2009)pertaining to a thermally-stratified fluid governed by thefully compressible Navier-Stokes equations (NCSE there-after), which builds upon the available potential energyframework previously introduced by Winters et al. (1995)for a Boussinesq fluid with a linear equation of state. InTailleux (2009)’s description, the energetics is described atleading order by means of the following five evolution equa-tions: d ( KE ) dt = − C ( KE, AP E ) + G ( KE ) − D ( KE ) , (9) d ( AP E ) dt = C ( KE, AP E ) + G ( AP E ) − D ( AP E ) , (10) d ( GP E r ) dt = W r,turbulent + W r,laminar − W r,forcing , (11) d ( IE ) dt = (1 − Υ ) ˙ Q net + D ( AP E )+ D ( KE ) − G ( AP E ) , (12) d ( IE exergy ) dt = Υ ˙ Q net − W r,turbulent − W laminar , (13)where KE is the volume-integrated kinetic energy, AP E is Lorenz (1955)’s volume-integrated available potential en-ergy,
GP E r is the volume-integrated gravitational poten-tial energy of Lorenz (1955)’s reference-state, IE is thevolume-integrated of a subcomponent of internal energy( IE ) that we call the dead part of IE , IE exergy is thevolume-integrated of another subcomponent of IE called theexergy. Physically, variations in IE are associated with vari-ations in the equivalent thermodynamic equilibrium temper-ature T ( t ) of the system, whereas variations in IE exergy re-flect variations in the reference temperature profile T r ( z, t ).The other important conversion terms are C ( KE, AP E ),the so-called buoyancy flux that represents the reversibleconversion between KE and AP E ; G ( AP E ), the productionrate of
AP E that physically represents a conversion between IE and AP E ; W r,laminar and W r,turbulent represent thelaminar and turbulent rate of exchange between IE exergy and GP E r due to molecular diffusion; D ( KE ) is the dis-sipation rate of KE into internal energy IE by molecularviscous processes; D ( AP E ) is the dissipation rate of
AP E into internal energy IE by molecular diffusive processes.The parameter Υ ≪ Q net splits between IE exergy and IE .As previously shown by Winters et al. (1995), a numberof conversion terms appear to be strongly correlated to eachother. This is the case for D ( AP E ) and W r,turbulent , owingto both terms: 1) being controlled by molecular diffusion;2) being controlled by the spectral distribution of the AP E density, see Holliday et al. (1981) and Roullet et al. (2009)for a discussion of the latter concept. For a compressiblethermally-stratified fluid, Tailleux (2009) showed that this c (cid:13) , 000–000 AUTHOR RUNNING HEAD correlation between D ( AP E ) and W r,turbulent can be writ-ten under the general form: W r,turbulent = ξD ( AP E ) (14)where ξ is a parameter that measures the importance ofthe nonlinearity of the equation of state, such that ξ = 1for a Boussinesq fluid with a linear equation of state, but −∞ < ξ G ( AP E )and W r,forcing owing to both terms being controlled by thesurface buoyancy forcing. Tailleux (2009) shows that to agood approximation, one usually has: G ( AP E ) ≈ W r,forcing . (15)For the sake of brevity, the reader is referred to Tailleux(2009) for the details about the explicit forms of all theterms entering the above energy equations, as they are unim-portant for the arguments developed in this paper. Fig. (1)schematically illustrates the energetics of a mechanically andthermodynamically forced stratified fluids associated withthe above equations. For all practical purposes, G ( AP E )and G ( KE ) represent the work rate done by the buoyancyand mechanical forcing respectively, whereas D ( AP E ) and D ( KE ) represent the two terms dissipating the “available”mechanical energy ME = AP E + KE . G ( AP E )A fundamental question posed by this paper is whatcontrols the magnitude of the buoyancy-driven circulation,that is, the circulation drawing its energy from G ( AP E ),with and without mechanical stirring acting on the fluid. Inorder to answer this question, we first need to introduce acouple of parameters that are traditionally used to measurethe efficiency of turbulent mixing, the so-called mixing effi-ciency γ mixing and the flux Richardson number R f , whichare defined here as follows: γ mixing = D ( AP E ) D ( KE ) , (16) R f = γ mixing γ mixing = D ( AP E ) D ( AP E ) + D ( KE ) . (17)The physical rationale for such definitions, as well as theirconnection to other existing definitions, was discussed in de-tails in Tailleux (2009) to which the reader is referred to fordetails. From a practical viewpoint, the differences in ex-isting definitions is unimportant, as least in the context ofa Boussinesq fluid, as then all definitions are then numeri-cally equivalent, even if based on different physical assump-tions. In particular, the present definitions allow to recoverMW98’s Eq. (1), as discussed below. To distinguish themfrom existing definitions, Tailleux (2009) refer to the above γ mixing and R f as the ”dissipative” mixing efficiency andflux Richardson number respectively.In a second step, the mechanical energy balance is con-structed by summing the steady-state version of the KE and AP E equations, which yields: G ( AP E ) + G ( KE ) = D ( AP E ) + D ( KE ) . (18)Now, by combining this equation with the definition of theflux Richardson number (i.e., Eq. (17), one may write: D ( AP E ) = R f [ G ( AP E ) + G ( KE )] . (19)Next, we turn to the steady-state GP E r balance, viz., W r,turbulent + W r,laminar = W r,forcing . (20)As mentioned above, one has to a very good approximationthe following equality W r,forcing ≈ G ( AP E ). For mathe-matical rigour, we introduce a parameter ξ ≈ W r,forcing = ξ G ( AP E ). Using the defi-nition of ξ above, the GP E r budget provides the followingrelationship: ξD ( AP E ) + W r,laminar = ξ G ( AP E ) , (21)which after some algebra eventually yields: G ( AP E ) = ξR f ξ − ξR f G ( KE ) + W r,laminar ξ − ξR f . (22)Eq. (22) is one of the most important result of this paper,as it represents one way to demonstrate the interconnectionof the work rate done by the wind and buoyancy forcing ingeneral. A useful limit is the case of the widely used Boussi-nesq model for which ξ = ξ = 1, in which case Eq. (22)simplifies to: G ( AP E ) = R f − R f G ( KE ) + W r,laminar − R f = γ mixing G ( KE ) + (1 + γ mixing ) W r,laminar . (23)Eq. (22) shows that in absence of mechanical forcing, thework rate done by the surface buoyancy fluxes is given by: G ( AP E ) = W r,laminar ξ − ξR f (24)or, for a Boussinesq fluid: G ( AP E ) = W r,laminar − R f = (1 + γ mixing ) W r,laminar . (25)The latter two results are interesting, because they showthat although W r,laminar is small by construction, it doesnot forbid in principle G ( AP E ) to be large provided that ξ − ξR f or 1 − R f can become small enough. The problem,however, is that the latter conditions require R f to be closeto unity for a Boussinesq fluid, or equivalently γ mixing ≫ γ mixing ≈ .
2. On the other hand, we are not aware of anypublished value for γ mixing pertaining to horizontal convec-tion, so that sufficient empirical evidence to be really conclu-sive. The idealised numerical experiments presented belowseeks to get insight into this issue.Perhaps the most important feature of Eq. (22) is to re-veal that even a small amount of mechanical forcing is suffi-cient to radically alter the nature of horizontal convection, asit suggests that the latter becomes mechanically controlledwhen G ( KE ) is such that ξR f G ( KE ) ≫ W r,laminar , whichis possible even for relatively low values of γ mixing — a veryimportant point. This theoretical result suggests thereforethat G ( AP E ) can be dramatically enhanced by the presenceof mechanical stirring, provided that the latter provides pos-itive work to the system (i.e., such that G ( KE ) > c (cid:13) , 000–000ECHANICALLY-STIRRED BUOYANCY-DRIVEN CIRCULATIONS IEIEoKE APE GPEr exergy
G(KE) D(KE) G(APE)(1−Yo)Qnet Yo QnetD(APE)C(KE,APE) W r,mixing W r,forcing Figure 1.
Schematic diagram of the energy conversions taking place in a mechanically and thermodynamically forced turbulent thermally-stratified fluid corresponding to Eqs. (9-13) previously derived by Tailleux (2009). −8−7.3−6.6−5.9−5.2−4.5−3.8−3.1−2.4−1.7−1−0.3(a)x z
10 20 30 40 501020304050 −16−14.5−13−11.5−10−8.5−7−5.5−4−2.5−1(b)x z
10 20 30 40 501020304050−7−6.4−5.8−5.2−4.6−4−3.4−2.8−2.2−1.6−1−0.4(c)x z
10 20 30 40 501020304050 −10−8.1−6.2−4.3−2.4−0.51.43.35.27.19(d)x z
10 20 30 40 501020304050
Figure 2.
Streamfunction for the different experiments: (a) No mechanical forcing; (b) Clockwise mechanical forcing; (c) Weak anti-clockwise mechanical forcing; (d) Strong anti-clockwise mechanical forcing.
The physical implications of the formula given by Eqs.(22) and (23) are explored in the following by means of nu-merical experiments aiming at illustrating the main effectsmechanical stirring can have on buoyancy-driven circula-tions.
To that end, the idealised problem of horizontal con-vection previously considered by Paparella et al. ( 2002) isinvestigated here, the novelty being the addition of mechan- ical stirring to the problem modelled here as a forcing termin the vorticity equation. The numerical implementation ofsuch a model is that previously described by Marchal (2007).The equations solved by the numerical model are the Boussi-nesq equations for a fluid with a linear equation of state: ∂ω∂t = J (Ψ , ω ) − gα ∂T∂x + ν ∇ ω + F ( x, z, t ) (26) ∂T∂t = J (Ψ , T ) + κ ∇ T, (27)where ω = ∇ Ψ is the vorticity, T is the temperature, J ( a, b ) = ∂a/∂x∂b/∂z − ∂a/∂z∂b/∂x is the Jacobian op- c (cid:13) , 000–000 AUTHOR RUNNING HEAD z z z z Figure 3.
Temperature field for the different experiments: (a) No mechanical forcing; (b) Clockwise mechanical forcing; (c) Weakanti-clockwise mechanical forcing; (d) Strong anti-clockwise mechanical forcing. erator, g is the acceleration due to gravity, α is the thermalexpansion coefficient assumed to be constant, ν is the kine-matic viscosity, κ is the thermal diffusivity, and F is a termaimed at modelling the effect of mechanical stirring.Following Marchal (2007), the above system is madedimensionless as follows: t = ( L /κ ) t ∗ , ( x, z ) = L ( x ∗ , z ∗ ), ω = ( κ/L ) ω ∗ , T = ∆ T T ∗ , and Ψ = κ Ψ ∗ , where the starredquantities are the dimensionless ones. The dimensionlessforms of the above equations become, after dropping thestart for clarity: ∂ω∂t = J (Ψ , ω ) + P r (cid:16) − R a ∂T∂x + ∇ ω (cid:17) + F ∗ , (28) ∂T∂t = J (Ψ , T ) + ∇ T, (29)where R a = gα ∆ T / ( νκ ) is the Rayleigh number, and P r = ν/κ is the Prandtl number. In the numerical experimentsdescribed here, we used P r = 10 and R a = 10 . For com-parison, note that typical oceanic values are R a = O (10 ),e.g., Paparella et al. ( 2002).As in Paparella et al. ( 2002), the fluid is forced by asurface temperature boundary condition varying linearly in x . The numerical resolution used for the experiments is 51 ×
51 in a 2-D square geometry. At equilibrium, the forcingterm in the vorticity equation is associated with the workrate: G ( KE ) = Z Z V F Ψ dxdz (30)which can be in principle either positive or negative. Four idealised experiments were considered:(i) Purely buoyancy-driven (i.e., no mechanical forcing);(ii) Clockwise Mechanical forcing;(iii) Weak anti-clockwise mechanical forcing;(iv) Strong anti-clockwise mechanical forcing.The purely buoyancy-driven case is associated with aclock-wise thermally-direct circulation, and represents the“control” reference simulation to be compared with themechanically-stirred ones. The purpose of Experiment (ii)is to gain insight into the case for which the mechanical stir-ring tends to re-enforce the existing thermally-direct clock-wise circulation, implying G ( KE ) >
0. From Eq. (23), theexpectation is that the overturning circulation should in-crease both as the result of increased G ( AP E ), as well asfrom the direct effect of the mechanical forcing acting as asource of clockwise vorticity. In such a case, therefore, KE , AP E , Ψ, D ( AP E ) and G ( AP E ) all should increase.In experiments (iii) and (iv), however, the anti-clockwise forcing works against the existing buoyancy-drivencirculation. If too weak, as designed to be the case for (iii),the mechanical forcing is unable to reverse the sense of theexisting thermally-direct circulation anywhere, resulting in G ( KE ) <
0. According to Eq. (23), a decrease in all quan-tities KE , AP E , Ψ, D ( AP E ), and G ( AP E ) is expected inthat case. If the clockwise forcing is strong enough, however,it can locally generate a anti-clockwise circulation associ-ated with G ( KE ) >
0, making it possible in that case forall above quantities to be increased. c (cid:13) , 000–000ECHANICALLY-STIRRED BUOYANCY-DRIVEN CIRCULATIONS G(KE) G(APE) D(APE) γ mixing (a) 0 5.98 4.12 1.78(b) 7.54 10.1 9.52 1.04(c) -0.77 4.72 2.58 1.49(d) 9.83 8.50 7.18 0.60 Table 1.
Values of the work rate done by the mechanical forcing G ( KE ), rate of AP E production G ( AP E ), diffusive
AP E dissi-pation rate D ( AP E ), and mixing efficiency γ mixing for (a) Nomechanical forcing; (b) Clockwise mechanical forcing; (c) Weakanti-clockwise mechanical forcing; (d) Strong anti-clockwise me-chanical forcing.KE APE Ψ max H max (a) 0.13 4.04 7.2 3.15(b) 0.64 6.25 15.5 5.09(c) 0.09 5.49 6.08 2.19(d) 0.51 11.1 9.67 2.75 Table 2.
Values of the volume-integrated kinetic energy KE ,volume-integral available potential energy AP E , maximum valueof the thermally-direct overturning streamfunction, and maxi-mum value of the heat transport for: (a) no mechanical forcing;(b) Clockwise mechanical forcing; (c) Weak anti-clockwise me-chanical forcing; (d) Strong anti-clockwise mechanical forcing.
The results are illustrated by the figs. (2) and (3), whichshow the streamfunction and temperature distributions forthe 4 different experiments considered. In addition, Tables 1and 2 summarise how several important quantities vary as afunction of the mechanical forcing imposed. As expected, allthe plots for Ψ show evidence of a well-marked thermally-direct cell in all cases. Fig. (2) (b) shows that the strengthof the overturning cell is greatly enhanced by the addition ofclockwise mechanical forcing. This can be explained by theobserved increase in G ( AP E ) (Table 1), as well as by the di-rect effect of the mechanical forcing as a source of clockwisevorticity. Fig. 3 (b) shows that the increase in G ( AP E ) isassociated with the dense plumes penetrating deeper in thatcase. Fig. (2) (c) shows that a weak anti-clockwise mechani-cal forcing decreases the buoyancy-driven circulation, whichis expected from the theory when G ( KE ) < G ( AP E ), despite an apparent increase in thethermocline depth (Fig. 3) which one would normally asso-ciate with an increase in G ( AP E ). As shown in Table 2, thisoccurs because of a reduction in the strength of the surfaceheat flux, which results in a decrease in the maximum heattransport H max . Fig. (2) (d) shows that the anti-clockwisemechanical forcing is strong enough to generate a thermally-indirect circulation, resulting in G ( KE ) >
0, which increasesthe overturning strength. In that case, there is no ambiguitythat the increase in the latter is entirely explained by theincrease in G ( AP E ), given that the mechanical forcing actsas a source of anti-clockwise vorticity, which can only tendto weaken the buoyancy-driven thermally-direct cell.In the literature, there is a tendency to regard the mix-ing efficiency γ mixing as some kind of universal parameterwhose value is relatively constant and uniform throughoutthe oceans. It is interesting, therefore, to examine whether this is the case in the present numerical experiments. Tothat end, the viscous dissipation D ( KE ) and diffusive dis-sipation D ( AP E ) were estimated for all experiments, thevalues being reported in Table 1. As it turns out, γ mixing is found to vary widely across the different experiments. Itsmaximum value is achieved in the purely buoyancy-drivencase, for which γ mixing ≈ .
78. Interestingly, this value issignificantly larger than the value γ mixing ≈ AP E driving the instability is not entirely available forturbulent mixing. For this reason, Tailleux (2009) suggestedthat higher values of mixing efficiency could in principle beachieved for buoyancy-driven turbulent mixing events if theabove constraint could be relaxed. As it turns out, hori-zontal convection does not suffer from the limitations at-tached to the Rayleigh-Taylor instability, making it possibleto achieve values of γ mixing significantly large than unity.The addition of mechanical stirring, however, is seen in Ta-ble 1 to systematically reduce γ mixing , the largest reductionoccurring for the strong anti-clockwise mechanical forcing.In that case, a value of γ mixing = 0 . γ mixing = 0 . γ mixing = 0 . G ( AP E ) and G ( KE ) are compa-rable in magnitude in reality.Another interesting question is what is the effect of themechanical stirring on the heat transport, which was alsoraised in MW98’s study. The last column of Table 2 providesinsight into this issue by providing the maximum value of theheat transport for each experiment. The most striking resultis that the heat transport appears to be drastically increasedwhen the mechanical forcing supports the thermally-directcirculation, but decreased when the mechanical forcing op-poses the latter. In the actual oceans, the wind forcing cre-ates surface Ekman cells that are alternatively helping oropposing the large-scale thermally direct cell in the Atlanticocean responsible for the Atlantic heat transport. It seemstherefore difficult to conclude at this stage whether the over-all net effect of the mechanical forcing in the oceans is todirectly contribute to the strength of the overturning circu-lation by being a net source of vorticity of the right sign, or ifits contribution is only indirect and limited to its increase of G ( AP E ) and hence of the strength of the buoyancy-drivencomponent of the overturning.
In this paper, the effect of mechanical stirring onbuoyancy-driven flows was investigated both theoreticallyand numerically. The most important theoretical result arethe formula Eqs. (22) and (23) linking the work done bythe mechanical stirring to the work rate done by the surfacebuoyancy fluxes via the bulk mixing efficiency of the system,valid for the compressible Navier-Stokes equations and for c (cid:13) , 000–000 AUTHOR RUNNING HEAD the Boussinesq model respectively. These formula are a fur-ther generalisation of that derived by Tailleux (2009), whichextends Munk et al. (1998)’s previous result on the energyrequirement on the mechanical sources of stirring to sustaindiapycnal mixing in the oceans. The actual physical mean-ing of the formula Eqs. (22) and (23) is not entirely clear,however. Physically, the formula only seem to express theexistence of a mechanical control on buoyancy-driven flowsand vice-versa, rather than a true constraint on G ( KE ), incontrast with the interpretation put forward by Munk et al.(1998). This idea is further re-enforced by the structure of G ( KE ) and G ( AP E ), which demonstrate that the work ratedone by the mechanical and buoyancy forcing does not sim-ply depend on the external forcing (i.e., the wind and surfacebuoyancy fluxes), but also on the actual state of the system.In other words, the work rate done by either the wind andbuoyancy forcing depends sensitively on the work rate doneby the other kind of forcing. In this paper, this idea is exam-ined from the viewpoint of the buoyancy-driven circulation,by demonstrating that the presence of mechanical stirringcan have dramatic effects on the value of G ( AP E ) owingto its direct effect on diapycnal mixing and hence on thestructure of the thermocline. The other way that mechanicalforcing can affect G ( AP E ) is by modifying the near surfacetemperature, thereby modifying the net flux into the sys-tem, since the flux is also to be determined as part of thesolution when a surface temperature boundary condition isimposed.In the purely buoyancy-driven case, Eq. (22) demon-strates that large values of G ( AP E ) are in principleachievable provided that the “dissipative” mixing efficiency γ mixing = D ( AP E ) /D ( KE ) defined in Tailleux (2009) islarge enough. The particular numerical experiment consid-ered in this paper suggests that this is possible, since thevalue γ mixing ≈ γ mixing = 0 . D ( AP E )and hence G ( AP E ) also increase with the Rayleigh num-ber. On the other hand, Paparella et al. ( 2002)’s “anti-turbulence theorem” states that the viscous dissipation ofkinetic energy is bounded from above by a bound that isindependent of the Rayleigh number. If D ( AP E ) increaseswith R a with no corresponding increase in D ( KE ), thenone can infer that the dissipative mixing efficiency γ mixing must also increase with the Rayleigh number. Testing thishypothesis further will be an important future research goal.As shown by Paparella et al. ( 2002), the “anti-turbulence theorem” places a stringent constraint on themagnitude of the viscous dissipation rate of kinetic en-ergy that can be sustained by buoyancy forcing alone. Asa consequence, mechanical forcing appears critical to pro-duce significant amount of viscous dissipation of kinetic en-ergy D ( KE ). However, because of the relatively low mix-ing efficiency of mechanically-driven mixing, it is usuallythe case that the addition of mechanical forcing to hori- zontal convection results in relatively modest increase in D ( AP E ) compared to that in D ( KE ). The overall conse-quence is to decrease the mixing efficiency γ mixing of thesystem as compared to the purely buoyancy-driven case, asverified in the numerical experiments considered here forwhich G ( KE ) >
0. The decrease in mixing efficiency result-ing from mechanically stirring horizontal convection doesnot preclude an increase in G ( AP E ) however, as expectedfrom the theoretical formula, and verified in the numeri-cal experiments. In our opinion, this is the fundamentalreason why the strength of the overturning circulation inWhitehead et al. (2008)’s laboratory experiments appears tobe greatly enhanced by the action of the stirring rod. De-pending on the particular circumstances, it appears possiblefor the mechanical forcing to contribute directly to the ob-served overturning increase, in addition to the indirect effectit has on the increase of G ( AP E ), as in the particular caseof the clockwise mechanical forcing experiment. In the caseswhere the mechanical forcing acts anti-clockwise, however,only the increase in G ( AP E ) appears to be directly respon-sible for the increase in the overturning. The latter case islikely to be the one pertaining to Whitehead et al. (2008)’slaboratory experiments, as it seems difficult to see how thelateral motions of the stirring rod could contribute to thevorticity source required to drive a thermally direct cell.The present results make it hard to regard Eq. (22) asa constraint on the mechanical sources of stirring, as pro-posed by Munk et al. (1998), given that G ( AP E ) is likelymore sensitive to a change in the mechanical forcing than G ( KE ) to a change in the buoyancy forcing, although thishas not been directly verified. The other main difficulty ininterpreting Eq. (22) as a constraint on G ( KE ) comes fromthe fact that the value of γ mixing , far from having the valueof 0 . γ mixing = 1 in Munk et al. (1998)’s paper, the requirementon the mechanical sources of stirring reduces drastically to G ( KE ) = 0 . γ mixing might differfor different equilibria. c (cid:13) , 000–000ECHANICALLY-STIRRED BUOYANCY-DRIVEN CIRCULATIONS This work was created using the Tellus L A TEX 2 ε classfile. This study was supported by the NERC funded RAPIDprogramme. The author acknowledges comments by J.Whitehead, J. Nycander, and an anonymous referee on anearlier draft which were helpful in improving the presentmanuscript. REFERENCES
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