Comments on eddies, mixing and the large-scale ocean circulation

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1 Comments on eddies, mixing and the large-scale ocean circulation John Marshall Massachusetts Institute o Technology Abstract. Some recent laboratory and numerical studies o idealized ocean gyres and circumpolar currents are reviewed in the context o what they might tell us about the relative role o eddy transer and mixing in setting the structure o the large-scale ocean circulation.. Introduction We briely review here some recent studies that address the relative role o geostrophic eddies and smallscale mixing in the large-scale ocean circulation and the maintenance o the thermocline. The starting point or our discussion is the time mean buoyancy equation, written here in a amiliar orm: v b = µ k s z z b v 0 b 0 () where v is the Eulerian mean velocity and, as usual, variables have been separated in to mean quantities (represented by an overbar) and perturbations rom this mean (represented by a prime) due to transient eddies. In Eq. () the eect o small-scale mixing processes has been represented by k s acting on the vertical (in z) stratiication. The eect o buoyancy transer due to geostrophic eddies is the last term. In ideal luid thermocline theory e.g. Luyten, Pedlosky and Stommel (983) both terms on the rhs o Eq. () are neglected and a surace buoyancy distribution is mapped adiabatically down in to the interior. In the diusive thermocline e.g. Samelson and Vallis (997), Vallis 2003 (and reerences therein) small scale mixing is retained and the eddy lux divergence is (conceptually at least) set to zero. Mixing plays a role at the base o the thermocline, joining the ideal luid above to the abyss below, as in the model o Salmon (990). It is commonly argued that the neglect o v 0 b 0 in Eq.() is justiied because:. geostrophic eddy luxes are adiabatic anyway, aren t they? In this case v 0 b 0 can be written entirely as an advective lux (as is made explicit by residual mean theory - see section 2) and so eddies cannot play any role in diabatic processes. 2. eddy luxes are only large in special regions such as boundary currents and jets and are probably negligible elsewhere. So Eq.() with v 0 b 0 0 will suice over most o the ocean most o the time. Using evidence rom idealized numerical experiments o eddying gyres and circumpolar currents, we will argue here that both o these statements are misleading: our conceptual models need to be modiied to take in to account the role o eddies in diabatic processes and the global implications o local eddy eects. Point. isdiscussedinsection2andpoint2insection3. In section 4 we summarize and conclude. 2. Geostrophic eddies and diabatic processes The last term on the r.h.s. o () represents the eddy lux divergence. Some valuable lessons about its character can be learned rom the case o small amplitude, non-breaking eddies on a zonal low, or which it can be shown (e.g., Plumb [979], McDougall and McIntosh [996]) that the i th component o the lux can be written v 0 i b0 = (K ij + L ij ) j b where j / x j and where K and L are symmetric and antisymmetric tensors. The symmetric part is diusive and can be related to processes (growth o parcel displacements and non-conservation o the tracer) known to lead to meaningul transport. The lux associated with L, however, is a skew lux which is equivalent to advection, rather than diusion, and is requently nonzero even when no real transport is occurring. The advective component can be subsumed in to a retransormation o the mean velocity using the residual mean theory o Andrews and McIntyre (976). To illustrate how to proceed in a general ramework in which the small amplitude assumption is not made,

2 2 MARSHALL we specialize Eq.() to two dimensions thus : v b y + w b z = µ k s z z b v0 b y w0 b z (2) Our goal now is to express Eq.() in terms o the residual circulation, Ψ res : Ψ res = Ψ + Ψ (3) where Ψ is the streamunction or the Eulerian mean low and Ψ is the streamunction or the advective component associated with eddies. The key step is to note that i the eddy lux v 0 b 0 lies in the b surace, then v 0 b 0 can be written entirely as an advective transport, v b, where, ollowing Held and Schneider (999), v, the eddy-induced velocity, is deined in terms o a streamunction Ψ given by: Ψ = w0 b 0 b y. (4) Here w 0 b 0 is the vertical eddy buoyancy lux and b y is the mean meridional buoyancy gradient. More precisely, to express Eq.() in terms o Ψ res, we eliminate (v, w) using (3) and (4), to obtain 2 : b v res y + w b res z = µ k s z z b ( µ) v0 b y 0, (5) where µ is given by: µ µ w0 b µ = 0. (6) v 0 b 0 s ρ and s ρ = b y (7) is the slope o mean buoyancy suraces. Note that we have made no approximations here Eq.(5) is simply Eq.(2) rewritten. ut note also that The ull 3-d treatment can be ollowed in, or example, Ferrari, 2003, in this volume. 2 To arrive at (5) rom (), decompose the eddy luxes (v 0 b 0, w 0 b 0 ) into an along b component (w 0 b 0 /s ρ, w 0 b 0 ) and the remaining horizontal component (v 0 b 0 w 0 b 0 /s ρ, 0) seemarshall and Radko, The divergence o the along b component is then written as an advective transport (w 0 b 0 /s ρ, w 0 b 0 )=v b y + w where Ψ is given by Eq. (4). This is combined with mean low advection in () to yield the lhs o (5). The divergence o the diapycnal (horizontal) eddy lux, leads to the last term on the right hand side o (5). 0 where w b v b s Figure. Eddy luxes, v 0 b 0 = v 0 b 0, w 0 b 0, and (time and azimuthally-averaged) mean buoyancy suraces, b, with slope s ρ, diagnosed rom the numerical simulations o circumpolar low described in Karsten et al (2002). The parameter µ measures the degree to which v 0 b 0 crosses b suraces. In the interior µ the eddies are almost entirely adiabatic. ut as the surace is approached µ v 0 b 0 becomes horizontal crossing the outcropping b suraces with strong diapycnal component. we have not been entirely successul in absorbing the eddy lux in to a retransormation o the mean. The last term on the rhs o Eq.(5) remains i the eddies are adiabatic. The parameter µ controls the magnitude o the diapycnal eddy lux: i µ =, then the eddy lux is entirely along b suraces, the horizontal diapycnal component vanishes and the advective transport captures theentireeddylux; i µ =0, horizontal diapycnal eddy transport makes a contribution to the buoyancy budget. I eddies were adiabatic, then µ =and Eq.(5) reduces to Eq.(2) with the eddy terms set to zero. Then we would simply have to reinterpret our coarse-grained velocity ield as a residual low. ut the ollowing simple kinematic argument tells us that eddies cannot be adiabatic everywhere. In the interior o the ocean eddy luxes are indeed likely to be aligned in mean buoyancy suraces because mixing in the thermocline is observed to be weak see Ledwell et al (993). ut, as the surace is approached, vertical motion is inhibited and eddy luxes must become directed horizontally, crossing outcropping buoyancy suraces, as observed in the numerical experiment shown in Fig.. O course there are myriad mixing processes at work in the upper boundary layer o the

3 Eddies, mixing and the large-scale ocean circulation 3 ocean to provide the small-scale mixing processes necessary to allow irreversible lateral eddy transer. ut what are the rate controlling processes? I it is the rate at which eddies strain tracer contours driving gradients on the small scale, then the small scale may be slaved to the large-scale eddy ield stirring the surace. Diagnosis o the eddy resolving polar cap calculations presented in Karsten et al (2002), show that the interior eddy lux is indeed closely adiabatic see Fig.. ut, as the surace is approached, w 0 b 0 tends to zero, leaving a horizontal eddy lux directed across b suraces. Having argued in this section that eddy luxes have an important diapycnal component, in the next section we will show that, although oten conined to local regions, they can have global implications. 3. Maintenance o the thermocline in a turbulent ocean What is the probable (possible) role o eddies in the integral buoyancy budgets o ocean gyres? Consider Fig.2, showing a hydrographic section running N S through the Atlantic ocean. We clearly see the bowl o the thermocline dipping down in the subtropics o each hemisphere. This is created by the pumping down o warm water rom the surace in the Ekman layers as in the sketch and the sucking up o cold water around the poles and at the equator. ut what happens to the warm water pumped down rom the surace. How is equilibrium established? As sketched in the schema o the warm water lens at the bottom o Fig.2, there are two likely possibilities. The irst is that the heat is diused away in to the interior by small scale mixing processes, presumably associated with the breaking o internal waves. Indeed Samelson and Vallis (997) describe such a scenario in their numerical solutions o the planetary geostrophic equations, obtaining an interior, diusively controlled boundary layer Salmon (990) that joins an upper ideal regime to the abyss. ut there is another possibility. Large-scale eddies associated with the instability o the gyre (localized to the western boundary currents) could break o blobs o warm water and lux them laterally, thereby equilibrating the system, as sketched in the schematic diagram at the bottom o Fig.2. In terms o eddy luxes this appears as a diapycnal lux directed laterally across outcropping buoyancy suraces parallel to the sea surace, as shown in Fig.. Figure 2. A hydrographic section (WOCE A6) through the Atlantic along the path marked, cutting across the subtropical gyres o both hemispheres. The schematic diagram shows the pattern o Ekman pumping which results in warm surace waters being pumped down in to the interior o the ocean. We discuss here the relative importance o small scale mixing acting in the interior, k v, and lateral geostrophic eddy luxes, v 0 h 0,whereh is the depth o the warm lens, in allowing a steady state to be achieved. 3.. A laboratory analogue: equilibration o a warm,pumpedlensonan plane This possibility has been recently studied in both a laboratory setting and by numerical experiment. Fig.3

4 4 MARSHALL shows a laboratory lens, ormed by the pumping o warm water down rom the base o a rotating heated disc. Experimental details can be ound in Marshall et al (2002). As can be seen rom the photograph looking down on the apparatus, geostrophic eddies orm and sweep water away rom under the disk allowing an equilibrium to be established in which the rate o creation o available potential energy by the action o pumping and dierential heating, is exactly balanced by its rate o conversion in to eddy energy by baroclinic instability. This is just the balance hypothesized in Gill, Green and Simmons (974). Simple theory discussed in Marshall et al (2002) predicts that, as supported by the graph reproduced in Fig.3, the depth o the lens, h ³ w Ek r,where is the rotation rate o the tank, w Ek is the Ekman pumping rate, r is the radius o the heating disc and = w Ek g 0, is the imposed buoyancy lux associated with the pumping down o warm water o reduced gravity g 0. In this experiment, then, the stratiication o the lens at equilibrium is controlled by eddy transer rather than small scale mixing. Diapycnal eddy luxes play a key role in the integral buoyancy budget o the lens. ut what about more realistic scenarios with a β eect and western boundary currents that are closer analogues o ocean gyres? Equilibration o numerical ocean gyres on a β plane In an attempt to address the role o β and the presence o boundaries, Radko and Marshall (2003) carried out high resolution numerical experiments o miniature ocean basins both single gyre and double gyre as can be seen in Fig.4. The depth o these gyres, in Sverdrup balance, scales as in classical theory ³ wek 2 2 h βg r, rather than the Marshall et al (2002) 0 plane result given in section 3.. However, eddies play a crucial role in integral balances o these numerical gyres. To diagnose the importance o eddies in the maintenance o the gyres, Radko and Marshall (2003) compared the diapycnal volume lux due to eddies with that due to small-scale mixing. oth eddy and small scale mixing terms in Eq.() result in a inite cross isopycnal volume lux per unit area denoted below by w 3 dia. In a statistically steady state, this lux can be conveniently expressed in terms o v 0 b 0 and z ks z b by rewriting our equations using buoyancy (rather than z) as a vertical coordinate: w dia = w(x, y, b) D z(b) =w u Dt x z(b)+v y z(b). (8) where D/Dt is the total derivative. Eq.(8) is urther simpliied using the expression or the isopycnal slope in z-coordinates: x z(b) = b x y z(b) = b y,. (9) When (9) is substituted in (8), the result is h 2 w Ek r 2 w Ek r Figure 3. On the top let we show a schematic diagram o the warm, pumped lens studied in the laboratory by Marshall et al (2002). A photograph rom the experiment itsel is shown on the bottom let we see eddies orming around the periphery o the warm lens which lux warm luid away rom under the rotating heated disc, arresting the deepening o the lens. The observed depth o penetration at equilibrium is plotted against the theoretical prediction on the top right. In this experiment, diapycnal eddy luxes are central to achieving an equilibrium. v b = w dia, Comparing this with the buoyancy equation (), we arrive at the expression or the cross-isopycnal lux z ks z w dia = b v 0 b 0, (0) The eddy-driven component is thus readily isolated as: w dia_eddy = v 0 b 0, () Clearly the relative importance o eddies and smallscale mixing in setting w dia is an important measure 3 Note that the diapycnal velocity w dia, is oten denoted by w in thermocline theory see, or example, Pedlosky, 996. Unortunately the notation is also commonly used in oceanography to represent eddy-induced velocities, a dierent quantity. Here, to avoid conusion we use w dia to denote a diapycnal velocity.

5 Eddies, mixing and the large-scale ocean circulation 5 w dia_eddy Evaluate w dia_eddy w Ek -plane gyres Single Double Figure 5. A schematic o a warm water lens. The continuous line is a temperature surace outcropping at the sea surace. Thevolumeoluidwithinthelensisincreasedbythe pumping o warm water down rom the surace, w Ek ;itcan be reduced by the action o diabatic eddy luxes, w dia_eddy. Their ratio, α, is a measure o the importance o diapycnal eddy luxes in the integral balance o the lens. Figure 4. Near surace temperature ields rom high resolution numerical simulations o ocean gyres on a β plane single gyre on the let, double gyre on the right. From Radko and Marshall (2003). Instantaneous ields are shown in the top panels, time-mean ields below. In each case the initial conditions were one o uniorm temperature interior stratiication is set up through the collusion o patterns o surace warming/cooling, Ekman pumping, (resolved) eddy processes and small scale mixing. o the relative role o eddies and small scale mixing in the buoyancy budget. This, as sketched in Fig.5, can be ascertained by diagnosing the quantity wdia_eddy α = hw Ek i rom our simulation, where hi are appropriate area integrals. Fig.6 (and legend) presents results o diagnostics o α rom the double-gyre experiment shown on the r.h.s o Fig.4. We see that in the upper ocean (corresponding to the warmest luid) α exceeds In other words, o the volume o warm water pumped down rom the surace Ekman layer, more than 85% o it is luxed away by eddies. From the horizontal maps o w dia_eddy,we see that this occurs almost entirely in the region o the model s separated Gul Stream and is associated with diapycnal eddy luxes. Deeper down eddies play a lesser role, but are more important than small scale mixing in the volume budget within all temperature suraces. I these numerical simulations are any guide, the role o small scale mixing in the thermocline o gyres may be supplanted by eddy transer in global budgets o buoyancy and potential vorticity. Moreover eddies may have an increasingly dominant role to play as the surace is approached. Similar conclusions have recently been drawn by Henning and Vallis (2003), who ind that small scale mixing remains a player at the base o the thermocline but is dominated by eddies on shallower layers. This is consistent with the results presented in Radko and Marshall (2003) and summarized in Fig.6. Motivated by the similarity in the pattern o w dia_eddy and 2 h (where h is layer depth) shown in Fig.6, Radko and Marshall (2003) modiy LPS theory to take account

6 6 MARSHALL w dia_eddy w Ek the study o the lens o Marshall et al (2002) outlined in section 3.. Again, these eddy luxes are diapycnal in nature. = 0.87 T w dia_eddy Figure 6. α(t ) diagnostics rom the double gyre experiment shown in Fig.4. The top right panel shows the geography o an outcropping temperature surace. The bottom panel shows the pattern o w dia_eddy and 2 h,whereh is the depth o the temperature surace shown in the top panel. Diabatic eddy luxes are concentrated in the region o the model s separated Gul Stream. eddy current ront o w dia_eddy. They are able to ind solutions in closed basins in which eddy processes are responsible or the equilibration o the gyre. Moreover western boundary layers set the character o the interior potential vorticity distribution the boundary layer is active The role o eddies in Circumpolar Currents The laboratory experiment described in Marshall et al (2002) see Fig.3 and numerical studies o circumpolar low by Karsten et al (2002), suggest that eddies may play a central role in setting the depth and stratiication o the Antarctic Circumpolar Current. The ideas are urther illustrated in a recent laboratory study by Cenedese et al (2003). As shown in Fig.7, the pumping down o light water in the subtropics, and the sucking up o dense luid around Antarctica, is represented in the laboratory by (one needs to lip things over!) pumping o salty (and hence dense) luid up rom below over the periphery and extracting resher (and hence lighter) luid rom the centre. In this way one creates a donut o salty luid which thickens through the action o pumping rom below. ut the thickening is arrested by baroclinic instability which sweeps blobs o dense eddying luidtowardthecentrewhereit is sucked out o the system. The height o the equilibrated donut scales like h ³ 2 w Ek r,justasin depth time Figure 7. The laboratory experiment described in Cenedese et al (2003), enquiring in to the processes that control the depth and stratiication o the Antarctic Circumpolar Current. On the top let we see the experiment rom above. Salty (dense) luid is pumped up through holes in the base o a circular, rotating tank. An equal mount o luid is extracted over the centre o the tank. The deepening o the salty donut o luid (dark in the picture) a vertical salinity section through which can be seen on the top right panel is arrested by the baroclinic instability o the azimuthal current associated with it. Eddies sweep the salty luidintowardthecentrewhereitissuckedoutothe tank. A time series o the depth o the salty donut measured close to the wall, is shown in the bottom right panel. We see the initial rise as luid is pumped in rom the base, and the equilibrium level as lateral luxes begin to balance the vertical inlux.

7 Eddies, mixing and the large-scale ocean circulation 7 4. Conclusions To conclude I want to make three points inspired by the oregoing examples and discussion:. Geostrophic eddy transer is oten erroneously neglected in avor o small scale mixing in discussions o large scale buoyancy balances. 2. Numerical and laboratory experiments suggest that eddy transer, rather than small-scale mixing, may play a central role in maintaining the structure o the thermocline, particularly in the southern ocean, but also in subtropical ocean gyres. 3. Geostrophic eddies are likely to play a key role in diabatic processes, particularly near the upper surace, where eddy luxes inevitably have a diapycnal component. 5. Acknowledgements John Marshall would like to thank Chris Garrett and Peter Müller or organizing and inviting him to the Aha Huliko a Hawaiian Winter Workshop. He also thanks Timour Radko or their collaboration in the exploration o ideas outlined here. Many enjoyable conversations are acknowledged with Alan Plumb and Ra Ferrari o MIT. 6. Reerences Andrews and McIntyre (976) Planetary waves in horizontal and vertical shear: the generalized Eliassen- Palm lux and the mean zonal acceleration. JAtmos. Sci., 33, Cenedese, C., J. Marshall and J.A. Whitehead (2003). A laboratory model o thermocline depth and exchange across circumpolar ronts. submitted to JPO. Ferrari, R. (2003): Residual circulation in the ocean. Aha Huliko a Hawaiian Winter Workshop: this volume. Gent, P.R. and J.C. McWilliams (990) Isopycnal mixing in ocean circulation models, J. Phys. Oceanogr, 20, Gill, A.E, J.S.A Green and A. J. Simmons (974) Energy partition in the large-scale ocean circulation and the production o mid-ocean eddies. Deep Sea Research., 2, Held, I., and T. Schneider, 999: The surace branch o the zonally averaged mass transport circulation o the troposphere. J. Atmos. Sci., 56, Henning and Vallis (2003) The eect o mesoscale eddies on ocean stratiication: Parts I and II - in preparation. Karsten, R., Jones, H and J. Marshall (2002) The role o eddy transer in setting the stratiication and transport o a Circumpolar Current. J. Phys. Oceaongr. vol.32, No, Ledwell, J.R., Watson, A.J. and C.S. Law (993) Evidence or slow mixing across the pycnocline rom an open-ocean tracer-release experiment. Nature, 364 (6439): Luyten, J., J. Pedlosky and H. Stommel, (983) The ventilated thermocline.j. Phys. Oceanogr.3, Marshall, J, Jones, H, Karsten, R and R. Wardle (2002) Can eddies set ocean stratiication? J. Phys. Oceaongr. vol.32., No, Marshall, J. and T. Radko (2003) Residual mean solutions or the Antarctic Circumpolar Current and its associated overturning circulation. to appear JPO. McDougall, T. J., and P. C. McIntosh: The temporalresidual mean velocity. Part I: Derivation and the scalar conservation equation. J. Phys. Oceanogr., 26, (996). Pedlosky, J. (996) Ocean circulation theory. Plumb, R.A. (979) Eddy luxes o conserved quantities by small-amplitude waves. J. Atmos. Sci., 36, Radko, T. and J. Marshall (2003) Eddy-induced diapycnal luxes and their role in the maintenance o the thermocline. submitted to JPO. Radko, T and J. Marshall (2003) The leaky thermocline: submitted to J. Phys. Oceaongr. Salmon, R (990): The thermocline as an internal boundary layer J. Mar. Res.,48, Samelson, R. G. and G.K. Vallis (997): Large-scale circulation with small diapycnal diusion: the twothermocline limit. J. Mar. Res. Vallis, G. K Mean and eddy dynamics o the main thermocline. In Nonlinear Processes in Geophysical Fluid Dynamics, O. U. Velasco Fuentes, J. Sheinbaum and J. Ochoa (editors), Kluwer Academic Publishers. Dordrecht, The Netherlands, pp4-72.