Lagrangian pair dispersion in upper-ocean turbulence in the presence of mixed-layer instabilities
LLagrangian pair dispersion in upper-ocean turbulencein the presence of mixed-layer instabilities
Stefano Berti a) and Guillaume Lapeyre Univ. Lille, ULR 7512 - Unité de Mécanique de Lille Joseph Boussinesq (UML), F-59000 Lille,France LMD/IPSL, CNRS, École Normale Supérieure, PSL Research University, 75005 Paris,France
Turbulence in the upper ocean in the submesoscale range (scales smaller than the deformation radius) plays an importantrole for the heat exchange with the atmosphere and for oceanic biogeochemistry. Its dynamics should strongly dependon the seasonal cycle and the associated mixed-layer instabilities. The latter are particularly relevant in winter andare responsible for the formation of energetic small scales that extend over the whole depth of the mixed layer. Theknowledge of the transport properties of oceanic flows at depth, which is essential to understand the coupling betweensurface and interior dynamics, however, is still limited. By means of numerical simulations, we explore the Lagrangiandispersion properties of turbulent flows in a quasi-geostrophic model system allowing for both thermocline and mixed-layer instabilities. The results indicate that, when mixed-layer instabilities are present, the dispersion regime is localfrom the surface down to depths comparable with that of the interface with the thermocline, while in their absencedispersion quickly becomes nonlocal versus depth. We then identify the origin of such behavior in the existence offine-scale energetic structures due to mixed-layer instabilities. We further discuss the effect of vertical shear on theLagrangian particle spreading and address the correlation between the dispersion properties at the surface and at depth,which is relevant to assess the possibility of inferring the dynamical features of deeper flows from the more accessiblesurface ones.
I. INTRODUCTION
Oceanic motions at scales larger than few tens of km arequasi-horizontal due to the pronounced stratification of sea-water and Earth’s rotation and are characterized by quasi-two-dimensional turbulence. At scales around 300 km (inthe mesoscale range), coherent structures (almost circularvortices) with depths reaching 1000 m contain most of thekinetic energy in the ocean. At scales around 10 km ( i.e. inthe submesoscale range) and over the water column the flowis populated by smaller eddies and filamentary structures as-sociated with strong gradients of physical properties (suchas temperature), which play an important role in both phys-ical and biogeochemical budgets . Such small scales arefound mainly in the mixed layer (the first ≈
100 m belowthe surface), a weakly stratified layer lying on top of a morestratified one known as the thermocline.Two mechanisms leading to the generation of these finescales have been proposed. On one side, they can be pro-duced by the stirring due to larger scale eddies . In sucha case, however, they are confined close to the surface. Onthe other side, they can result from mixed-layer instabilities,in which case they extend all over the mixed layer . Whenthe latter is sufficiently deep, as is the case in winter, thepotential energy contained in surface buoyancy gradients atmesoscales, can give rise to baroclinically unstable modeswith horizontal scale of O ( − ) km that grow over timescales of O ( ) day. It has to be noted that the first mechanismdoes not depend on the mixed-layer depth, which stronglydiffers from one season to another. As such, it cannot ac-count for seasonal variations of the intensity of turbulence a) Email: [email protected] (at small scales), which has been observed to be a distinctivefeature of submesoscale flows .In order to explore the impact of mixed-layer instabilitieson submesoscale turbulence, an attractive quasi-geostrophic(QG) model was recently proposed . It describes the dy-namics of two coupled fluid layers having different stratifi-cation properties giving rise to both mixed-layer and thermo-cline instabilities, thus permitting a comparison of the twomechanisms mentioned above. In the absence of a mixedlayer, at sufficiently small scales the model essentially givessurface quasi-geostrophic (SQG) dynamics , which areconsidered a paradigm of mesoscale-driven submesoscalegeneration. It should be noted, however that in this case sub-mesoscales are trapped at the surface. As shown in Ref. 15,by including baroclinic mixed-layer instabilities, the modelgives rise to turbulent flows characterized by energetic sub-mesoscales down to the thermocline, which positively com-pare with those observed in the field in winter.In this work we adopt the above QG model to carry outnumerical simulations resolving both the mesoscale and sub-mesoscale ranges, in realistic conditions for the winter mid-latitude ocean. Our main goal is to investigate the role ofmixed-layer instabilities on the spreading process of La-grangian tracer particles. Here, we ignore non-geostrophicmotions such as inertia-gravity waves that act at small scalesand at high frequencies, having also some effect on La-grangian dispersion .Lagrangian statistics allow access to the stirring operatedby turbulent flows, which plays an essential role in trans-port processes ( e.g. of biogeochemical tracers), as well as forsurface energy and heat exchanges, at different scales .Several previous studies have addressed the relative disper-sion of pairs of surface drifters from experimental data, andits relation with the statistical properties of the underly-ing turbulent flows (see, e.g. , Refs. 19, 21–25), but the re- a r X i v : . [ phy s i c s . f l u - dyn ] F e b sults vary from region to region and are not always conclu-sive about dispersion regimes. Interestingly, however, sev-eral evidences of enhanced dispersion at submesoscales havebeen recently provided , which ask for a more de-tailed understanding of the physical processes acting at thesescales.Below the surface, the knowledge of flow properties is stilllimited, due to the complexity of performing measurementsat depth. In this respect, Lagrangian approaches can reveala useful tool to understand the coupling between the surfaceand interior dynamics. Not many studies of relative disper-sion at depth, from float trajectories, at small temporal andspatial scales are available. There is, however, some evi-dence, at rather large depths in the western Atlantic, of dis-persion being local , meaning governed by eddies of thesame size as the pair separation distance, at scales betweensome tens and some hundreds of km, or controlled by meanshear (up to 100 km). Nonlocal dispersion, i.e. mainly dueto the largest eddies, was detected in the same area at scalessmaller than 40 km and, more recently, in the Antarctic Cir-cumpolar Current at depths between 500 and 2000 m, in astudy resolving the 1 −
100 km scale range .Learning, from observations or numerical simulations,how submesoscale turbulence affects the spreading of La-grangian particles at different depths seems appealing alsoin view of the high-resolution velocity data expected fromfuture satellite altimetry, as the SWOT mission . As La-grangian statistics reflect Eulerian ones, such as energy spec-tra (see, e.g. , Refs. 32–35), they may serve to assess the rangeof validity, in terms of spatial scales, of the satellite-derivedflow. Furthermore, the characterization of dispersion proper-ties below the surface can be informative about the possibil-ity to extrapolate information from the surface to depth.In our numerical study, we examine particle-pair separa-tion statistics at different depths, relying on both fixed-timeand fixed-scale indicators. From a methodological pointof view, our approach shares some similarity with that ofRef. 36, where however the focus was not on mixed-layerinstabilities, and with that of Ref. 37, where the effect ofthe latter was mainly considered for its signature on dis-persion close to the surface. By contrasting the results ob-tained with different turbulent flow dynamics, i.e. gener-ated by thermocline-only or mixed-layer only instabilities, orboth, we aim at identifying the resulting dispersion regimes,with an emphasis on their general features. Furthermore, bystudying the correlation of dispersion properties at the sur-face and in deeper layers, we address the question of infer-ring the dynamical features of deeper flows from the moreaccessible surface ones.This article is organized as follows. The model adoptedfor the turbulent dynamics is presented in Sec. II; the mainstatistical properties of the turbulent flows are illustrated inSec. III. The results of the analysis of Lagrangian pair sepa-ration are reported in Sec. IV, where we separately focus onhorizontal dispersion at different depths (Sec. IV A) and onthe correlation of its properties along the vertical (Sec. IV B).Finally, discussions and conclusions are presented in Sec. V. II. MODEL
We consider a QG model (Fig. 1) consisting in two cou-pled fluid layers (aimed to represent the mixed layer andthe thermocline) with different stratification. Such a modelcan give rise to both meso and submesoscale instabilitiesand subsequent non-linear turbulent dynamics that comparewell with observations of wintertime submesoscale flows .While the former instabilities are due to classical baroclinicinstability, the latter are associated with mixed-layer insta-bilities. FIG. 1. Schematics of the 2-layer model.
The model dynamics are specified by the following evolu-tion equations (more details in Ref. 15): ∂ t θ i + J ( ψ i , θ i ) + U i ∂ x θ i + Γ i ∂ x ψ i = r ∇ − θ i + D s ( θ i ) , (1)where J is the Jacobian operator. The fields θ i (with i = , ,
3) are three δ − PV (potential vorticity) sheets at theocean surface ( z = z = − h )and at the bottom of the thermocline ( z = − H ), respec-tively. The variables ψ i stand for the streamfunctions at eachdepth, through which the horizontal flows can be expressedas u i = ( u i , v i ) = ( − ∂ y ψ i , ∂ x ψ i ) .The δ − PV sheets are related to the buoyancy field b = b ( x , y , z , t ) by: θ = − f b ( z = ) N m , (2) θ = f (cid:20) b ( z = − h + ) N m − b ( z = − h − ) N t (cid:21) , (3) θ = f b ( z = − H ) N t , (4)where f is the Coriolis frequency, and N m and N t are theBrunt-Väisälä frequencies for the mixed layer and the ther-mocline, respectively. Due to the QG assumption, the buoy-ancy field is also related to the streamfunction by b = f ∂ z ψ .The coupling between θ i and ψ i can be expressed, in Fourierspace, by ˆ θ i = L i j ˆ ψ j (5)with the hat denoting the horizontal Fourier transform and L i j the elements of the matrix: L = f k − coth µ m N m csch µ m N m csch µ m N m − coth µ m N m − coth µ t N t csch µ t N t csch µ t N t − coth µ t N t (6)where k is the modulus of the horizontal wavenumber, and µ m = N m kh / f and µ t = N t k ( H − h ) / f are non-dimensionalwavenumbers.In view of the ensuing discussions, an interesting featureof this model is that it allows the computation of the hori-zontal velocity field at any depth, once the streamfunction atthe discrete levels 1 , , U = U = − Λ m h , U = − Λ m h − Λ t ( H − h ) and mean merid-ional PV gradients with Γ = f Λ m / N m , Γ = − f Λ m / N m + f Λ t / N t , Γ = − f Λ t / N t . Here Λ m and Λ t account for con-stant vertical shear in the mixed layer and in the thermocline,respectively. III. TURBULENT FLOW PROPERTIES
The evolution equations (1), with (5) and (6), are nu-merically integrated by means of a pseudospectral methodon a doubly periodic square domain of side L at resolution512 , starting from an initial condition corresponding to astreamfunction whose Fourier modes have random phasesand small amplitudes such that the resulting kinetic energyspectrum is constant in the range of wavenumbers consid-ered. The code was adapted from an original one developedby Ref. 38 and previously used in Refs. 33 and 39. In themodel, to remove energy from the largest scales we use hy-pofriction with coefficient r , while small-scale dissipation(and numerical stability) is assured through an exponentialfilter D s ( θ i ) acting beyond a cut-off wavenumber k c .In our numerical simulations, we adopt realistic parame-ter values for the (wintertime) midlatitude ocean (similarlyto Ref. 15), as listed in table I. We further choose a domainlinear size L =
500 km and a grid spacing ∆ x (cid:46) r (cid:39) . · − m − s − and the non-dimensional cut-off wavenumber for the expo-nential filter is k c =
50 (corresponding to an inverse wave-length of 0 . − ). Let us mention that, even if the resultspresented here are in dimensional units, the numerical in-tegration is carried out using non-dimensional variables, inwhich times are made non-dimensional using the advectivetime-scale L / ( π u ) , where u = .
12 m s − is taken as thetypical velocity.We consider three cases: the thermocline only case (TC),the mixed-layer only case (ML), the full model (F), whichare specified by the values of the depths h and H in table II.For the thermocline only case, the parameter h does not havea physical meaning and its value is set to H /
2; moreover wetake N m = N t = · − s − . In the following we will focuson dynamics down to depth z = −
500 m, for all the threeconsidered cases.
TABLE I. Main physical parameters of the model; in the presentstudy Λ m = Λ t = Λ .Vertical shear Λ − s − Mixed-layer buoyancy frequency N m · − s − Thermocline buoyancy frequency N t · − s − Coriolis frequency f − s − TABLE II. Depth values used for the three different models (seetext). Model TC ML FMixed-layer depth h
250 m 100 m 100 mTotal depth H
500 m 1000 m 500 m
A. Spatial structure and kinetic energy spectra
The spatial organization of the horizontal flow can beinspected by plotting the vorticity field ζ = ∂ x v − ∂ y u =( ∂ x + ∂ y ) ψ . Some snapshots of ζ at a given time afterthe system reached the statistically steady state are shownin Fig. 2, after normalization by the root-mean-square (rms)value ζ rms = (cid:104) ζ (cid:105) / (with brackets indicating a spatial av-erage). In the figure, each column corresponds to a differentmodel (TC, ML, F, from left to right). In the top, middle andbottom row ζ / ζ rms is shown at the surface, at z = −
100 mand at z = −
500 m, respectively.In all the examined cases, the surface fields are charac-terized by a whole range of active scales. In the TC case,vorticity is prominently organized in a tangle of long fila-ments (Fig. 2a). Eddies of different sizes are also present, sothat the flow field is characterized by both features simulate-nously. As argued in Ref. 15, in this case the dynamics ofsurface buoyancy anomalies decouple from those at the bot-tom at sufficiently small scales, essentially giving rise to tur-bulent flows of surface quasi-geostrophic (SQG) type. Then,as only the mesoscale instability is here present, small-scaleeddies are generated by a roll-up instability of larger flowfeatures . Small-scale eddies rapidly decay with depth andat z = −
100 m only the largest structures are still present(Fig. 2d). At z = −
500 m the TC vorticity field is statisti-cally equivalent to its surface counterpart, due to SQG-likedynamics at level z = − H and symmetry of the dynamicswith respect to the half total depth.The situation is quite different for the ML case (centralcolumn of Fig. 2). In the presence of mixed-layer instabil-ities, eddies, initially of the same size as the scale of thesubmesoscale instability, grow until being balanced, in a sta-tistical sense, by hypofriction. Several coherent vortices arevisible in the surface vorticity field. One can also remark thatfilaments are now shorter and less intense. In sharp contrastwith the TC case, vorticity varies very little, in a statisticalsense, over the first 100 meters below the surface, pointingto energetic submesoscales in this whole depth range. Belowthe mixed layer, small scales decay in a way similar to whatobserved for TC, until at z = −
500 m only very large vor-
Thermocline only (TC) Mixed-layer only (ML) Full model (F) x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 (a) Surface (b) Surface (c) Surface x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 (d) z = −
100 m (e) z = −
100 m (f) z = −
100 m x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 x (km) y ( k m )
250 300 350 400 450050100150200250 −2.5−2−1.5−1−0.500.511.522.5 (g) z = −
500 m (h) z = −
500 m (i) z = −
500 m
FIG. 2. Vorticity field, normalized by its rms value, for the TC, ML and F cases (columns from left to right) at the surface (a, b, c), at z = −
100 m (d, e, f) and at z = −
500 m (g, h, i). In all panels, a zoom on the subregion [ , ] × [ , ] km of the total domain is shown. ticity patches are found. Notice that here the thermocline isvirtually absent, as H = h , which accounts for the differ-ences observed at z = −
500 m with respect to other cases.The picture in the full model (right column of Fig. 2) issimilar to that of the ML case in the mixed layer and theupper thermocline. However, at larger depths, the flow re-covers energy at small scales, due to the effect of the finite-depth thermocline and SQG-like dynamics at its bottom. At z = −
500 m, vorticity has a rather filamentary structure inwhich several small eddies are immersed.A statistical characterization of these turbulent flows canbe provided by their kinetic energy spectrum E ( k ) , whichis shown in Fig. 3 for each model. The top row shows the spectra (as a function of the horizontal wavenumber) at someselected depths, while in the bottom row a more completedescription for varying depth is reported. Here the dashedlines correspond to the reference depths considered for thehorizontal Lagrangian dispersion (Sec. IV A).In the TC case the spectrum at the surface (and at thebottom) displays a scaling behavior that is rather close to E ( k ) ∼ k − / , as expected in SQG turbulence. In the inte-rior, the energetic content of the largest scales is comparableto the corresponding value at the surface, but the spectrumrapidly falls off, due to the decay of small eddies with depth.Indeed, already at z = −
100 m, E ( k ) is found to be definitelysteeper than k − . -3 -2 -1 -3 -2 -1 (a) E ( k ) ( m s - ) k (km -1 )surfacez=-100 mz=-150 mz=-500 mk -5/3 k -3 -3 -2 -1 -3 -2 -1 (b) E ( k ) ( m s - ) k (km -1 )surfacez=-100 mz=-150 mz=-500 mk -5/3 k -3 -3 -2 -1 -3 -2 -1 (c) E ( k ) ( m s - ) k (km -1 )surfacez=-100 mz=-150 mz=-500 mk -5/3 k -3 (d) 0.01 0.1 1inverse wavelength (km -1 )-500-400-300-200-100 0 dep t h ( m ) -3 -2 -1 s pe c t r a l den s i t y ( m s - ) (e) 0.01 0.1 1inverse wavelength (km -1 )-500-400-300-200-100 0 dep t h ( m ) -3 -2 -1 s pe c t r a l den s i t y ( m s - ) (f) 0.01 0.1 1inverse wavelength (km -1 )-500-400-300-200-100 0 dep t h ( m ) -3 -2 -1 s pe c t r a l den s i t y ( m s - ) FIG. 3. Kinetic energy spectra, temporally averaged over several flow realizations in the statistically steady state, at some selected depths forthe TC (a), ML (b) and F (c) cases. The variation of spectra with depth is shown in (d), (e), (f) for the TC, ML and F cases, respectively. Herethe dashed lines indicate the reference depths considered for the horizontal Lagrangian dispersion (Sec. IV A).
In the ML case, kinetic energy spectra are similar in abroad range of wavenumbers over the mixed layer, whichis then fully energized. At the surface and at the base ofthe mixed layer their scaling is not far from k − / , thoughslightly steeper at large scale, as also observed in Ref. 15.Below z = −
100 m, the spectrum shows a fast decrease withthe wavenumber and becomes steeper than k − , due to lessand less intense small scales at larger and larger depths. Weremark that we verified that the value of the total depth H does not considerably affect the spectral properties of the tur-bulent flows down to z = −
500 m.When both the mixed layer and the thermocline arepresent (case F), E ( k ) is very similar to the spectrum foundin the previous case, both in the mixed layer and in the upperthermocline. Nevertheless, close to z = −
500 m it displaysenergetic small scales again, due to the dynamics at the bot-tom. At this depth, similarly to what occurs at the surface, ascaling range with spectrum not far from k − / is observedat relatively small scales, while at large scales the spectrumis steeper and tends to approach k − . B. Turbulence intensity at varying depth
Here we consider the variation with depth of the typicalintensities of the turbulent flow velocities and of their gradi-ents, both of which are expected to be relevant for the trans-port of Lagrangian particles.We first examine the rms turbulent velocity u rms = (cid:104)| u | (cid:105) / , and compare it to the intensity of the zonal meanflow U = U ( z ) ˆ x = Λ z ˆ x (see Sec. II and table I). The behav-ior of both quantities as a function of the depth | z | is reportedin Fig. 4, where the inset shows the (inverse) turbulence in- tensity, U / u rms , versus depth. Generally speaking, one cansee from these plots that turbulence becomes weaker, whilethe mean flow gains importance, with depth. The way thisoccurs, however, depends on the model dynamics. While forthe TC and ML cases the mean flow can become comparableto typical turbulent velocity fluctuations, in the full model U / u rms never exceeds 0 .
4. In the absence of the mixed layer, U > u rms / U / u rms reaches its peak value ( ≈
1) close to the bottom, namely for300 m < z <
400 m, then slightly decreasing to reach 0 . z = −
500 m, due to more intense turbulence at the bottom. Inthe presence of the mixed layer, u rms remains essentially un-changed in the first 100 meters from the surface. Below themixed layer it decreases, but the mean flow becomes com-paratively relevant only at quite large depths (in the ML case U > u rms / | z | >
400 m). Moreover, while in the ML case U / u rms (cid:39) z = −
500 m, in the full model the more en-ergetic turbulent dynamics at the bottom of the thermoclinepartially compensate the importance of the mean flow at thelargest depths, where U / u rms ≈ .
35 at most.If the rms velocity gives information about the intensity ofthe turbulent flow, what matters in the separation process ofadvected Lagrangian particles are the velocity gradients. Thelatter can be quantified by the rms vorticity ζ rms ( z ) , whichis shown in Fig. 5 for the three models. Its decrease withdepth is evident in all cases. In the TC case, as for u rms ,the symmetric behavior with respect to | z | =
250 m resultsfrom the dynamics at the bottom. The trace of the latter isalso visible in the full model, where it causes an increase of ζ rms at the largest depths. This feature is absent in the MLcase, where ζ rms monotonously decreases below the mixedlayer. In the presence of the latter (ML and F cases), the u r m s ( m s - ) |z| (m)TCMLFU(z) U / u r m s |z| (m) FIG. 4. Typical turbulent velocity fluctuations u rms and mean flowintensity U ( z ) as a function of depth for the TC, ML and F cases.Inset: ratio U ( z ) / u rms versus depth. The rms velocities shown hereare temporally averaged over several flow realizations in the statis-tically steady state. ζ r m s ( da y - ) |z| (m)TCMLF 10 -2 -1
0 100 200 300 400 500 ζ r m s /f |z| (m) FIG. 5. Typical turbulent vorticity fluctuations ζ rms as a function ofdepth for the TC, ML and F cases. In the inset ζ rms is normalizedby the Coriolis frequency f and plotted in semilogarithmic scale.The rms vorticities shown here are temporally averaged over severalflow realizations in the statistically steady state. rms vorticity is always larger above z = −
100 m than deeperbelow, and typically larger than in the TC case.
IV. LAGRANGIAN PAIR DISPERSION
In the following we will consider the horizontal disper-sion properties of an ensemble of Lagrangian tracer particlesmoving at fixed depth z ∗ in the turbulent flows produced bythe TC, ML and F models. The equation of motion of theseparticles is d x i dt = v ( x i ( t ) , z ∗ , t ) , i = , ..., N , (7)where x i = ( x i , y i ) denotes the horizontal position of parti-cle i and v ( x , y , z ∗ , t ) = u ( x , y , z ∗ , t ) + U ( z ∗ ) the total veloc-ity field at the particle position (at depth z ∗ ) resulting from the sum of the turbulent component u , computed from thestreamfunction in Eqs. (A.1-A.2), and the mean flow Λ z ∗ ˆ x .In our numerical experiments, Eq. (7) is integrated us-ing a fourth-order Runge-Kutta scheme and bicubic inter-polation in space of the velocity field at particle positions .We assume that the particle motion occurs in an infinite do-main and use the spatial periodicity of the Eulerian flow tocompute the Lagrangian velocities outside the computationalbox.The particles are seeded in the turbulent flows once thelatter have reached statistically steady conditions. At eachconsidered depth particles are initially placed in triplets, uni-formly spread (on the horizontal) over the spatial domain.The number of triplets is M = × = x and onealong y , both of which are characterized by an initial sepa-ration R = ∆ x / (cid:39)
500 m (with ∆ x (cid:39) R = (cid:113) R x + R y (where R x and R y are the separations along x and y , respec-tively), for pairs initially along x . We verified that therewas no major difference in the statistics when consider-ing dispersion in the x or y direction, despite the presenceof the mean zonal shear at depth. In this study we onlyconsider original pairs, and we choose as reference depths z = , − , − , − , −
500 m, except where explicitlymentioned.In Sec. IV A we examine horizontal dispersion at differentdepths using both fixed-time indicators, as relative disper-sion (as a function of time) and fixed-scale ones, asthe finite-size Lyapunov exponent (FSLE, or FSLE-I) .In Sec. IV B, we address the properties of the relative mo-tion of subsurface particles with respect to surface ones, byanalyzing the so-called FSLE of the 2 nd kind (FSLE-II) . A. Horizontal dispersion
Here we are interested in assessing how the horizontal dis-persion process varies in the vertical. In particular we aimat identifying different dynamical regimes and at higlightingpossible transitions among them as a function of depth.The first diagnostic we consider is relative dispersion,which is defined as (cid:104) R ( t ) (cid:105) = (cid:104)| x i ( t ) − x j ( t ) | (cid:105) , (8)where the average is over all pairs ( i , j ) such that at t = | x i ( ) − x j ( ) | = R .Assuming that relative velocity is independent of the par-ticle pair separation, at sufficiently short times a ballistic be-havior (cid:104) R (cid:105) − R ∼ t is expected . At intermedi-ate times, for which dispersion scales are within the iner-tial range of the turbulent cascade, the expected behavior de-pends on the form of the kinetic energy spectrum (see Ref. 51for a compact review). Assuming a power-law spectrum, E ( k ) ∼ k − β , the value of the exponent β then determinesthe expected dispersion regime. For a rough flow, for which -4 -2 ζ rms t) ( ζ rms t) ζ rms t(a) ( < R > - R ) / R t ζ rms (z)surfacez=-100 mz=-250 mz=-350 mz=-500 m -4 -2 ζ rms t) ( ζ rms t) ζ rms t(b) ( < R > - R ) / R t ζ rms (z)surfacez=-100 mz=-250 mz=-350 mz=-500 m -4 -2 ζ rms t) ( ζ rms t) ζ rms t(c) ( < R > - R ) / R t ζ rms (z)surfacez=-100 mz=-250 mz=-350 mz=-500 m FIG. 6. Relative dispersion (after subtraction of the initial value R and normalization by it) as a function of the time rescaled by the rmsvorticity ζ rms (Fig. 5) at the reference depths for the TC (a), ML (b) and F (c) cases. β <
3, relative dispersion should scale as (cid:104) R (cid:105) ∼ t / ( − β ) (see Refs.19, 52, and 53), which includes Richardson su-perdiffusive behavior (cid:104) R ( t ) (cid:105) ∼ t for β = /
3. In such acase ( β <
3) the dispersion process is referred to as a localone, meaning that the growth of the separation distance be-tween two particles in a pair is governed by eddies of thesame size as the separation itself . When β >
3, instead,the flow is smooth and the expectation for relative disper-sion is (cid:104) R ( t ) (cid:105) ∼ exp ( λ L t ) (see Refs. 19 and 54), where λ L is the Lagrangian maximum Lyapunov exponent. Such anexponential growth of (cid:104) R (cid:105) is typically referred to as a non-local dispersion regime, meaning governed by the largest ed-dies . Finally, for separations much larger than the largestcharacteristic flow scale, a diffusive behavior (cid:104) R (cid:105) ∼ t is ex-pected, due to essentially uncorrelated particle velocities.To identify different dispersion regimes, it is useful toperform a rescaling of the considered variables. A relevantquantity is, in this respect, the rms vorticity ζ rms , which ac-counts for the intensity of typical velocity gradients. The be-havior of (cid:104) R ( t ) (cid:105) is reported in Fig. 6. Here time is rescaledwith 1 / ζ rms , which provides an estimate of the typical timeover which trajectory pairs loose memory of their initial con-dition; relative dispersion is plotted after subtraction of itsinitial value R and normalization by the latter. Through thisrepresentation we are able to detect several distinct behav-iors, which correspond to the different dispersion regimesthat are realized in the course of time. The quite nice col-lapse of the data further indicates the generality of the ob-served spreading mechanisms. As it can be seen, indepen-dently of the model and of the depths, when t is smaller thana time of order 1 / ζ rms , a clear ballistic behavior ( ∼ t ) isfound. In the opposite limit of very large times, all curvesindicate diffusive behavior ( ∼ t ), as expected. At interme-diate times, relative dispersion approaches a t scaling, sug-gesting Richardson local dispersion, at the surface (for allmodels), at the base of the mixed layer (for the ML and Fcases) and at the bottom of the thermocline (for the TC andF models). These results are in fair agreement with the ex-pectation based on the shape of the kinetic energy spectrum,which in these cases is close to E ( k ) ∼ k − / (Fig. 3). Forthe remaining cases ( i.e. in the interior of the TC system,below the mixed layer in the ML one, and in the upper ther-mocline for the F case), the collapse of the curves (for fixed model and different depths), points to a common dispersionregime characterized by fast growth in time (meaning fasterthan t ) of (cid:104) R (cid:105) . Here, based on E ( k ) being steeper than k − ,we should expect exponential growth of the squared separa-tion distance (nonlocal dispersion). Even when examined ona lin/log scale, the data, however, do not quantitatively sup-port this picture and do not allow to measure λ L (not shown).A possible reason for such a difficulty is that relative disper-sion is constructed as an average at fixed time . Indeed, inthe presence of large variability as a function of the initialpair location and/or time, as it is found to be the case here(not shown), (cid:104) R ( t ) (cid:105) does not allow the detection of the cor-rect scaling behavior. An illustration of this effect showing aspurious anomalous regime for a system of point vortices isdocumented in Ref. 55, while Ref. 53 reports the difficultyto detect Richardson’s scaling from (cid:104) R ( t ) (cid:105) , but not from theFSLE, in direct numerical simulations of three-dimensionalhomogeneous isotropic turbulence.Let us now consider dispersion indicators at fixed lengthscale, which are less affected by the superposition of differ-ent regimes associated with particle pairs having differentseparation distances at the same time. We first consider rela-tive diffusivity, defined as κ =
12 d (cid:104) R ( t ) (cid:105) d t . (9)This is presented in Fig. 7, as a function of the separationdistance δ = (cid:104) R ( t ) (cid:105) / , after rescaling time by 1 / ζ rms anddistance by R . This diagnostic returns a picture more in ad-equacy with the theoretical expectations based on the shapeof the kinetic energy spectrum. Even if in some cases thecurves show some wiggles, on average we find that the scale-by-scale relative diffusivity reasonably scales according tothe dimensionally expected behaviors, δ (corresponding toa spectral exponent β > δ / (corresponding to β = / -2 -1
1 10 100 1000( δ /R ) ( δ /R ) (a) κ / ( ζ r m s R ) δ /R surfacez=-100 mz=-250 mz=-350 mz=-500 m -2 -1
1 10 100 1000( δ /R ) ( δ /R ) (b) κ / ( ζ r m s R ) δ /R surfacez=-100 mz=-250 mz=-350 mz=-500 m -2 -1
1 10 100 1000( δ /R ) ( δ /R ) (c) κ / ( ζ r m s R ) δ /R surfacez=-100 mz=-250 mz=-350 mz=-500 m FIG. 7. Relative diffusivity as a function of the separation distance δ = (cid:104) R ( t ) (cid:105) / , after rescaling time with ζ − and distance with R , at thereference depths for the TC (a), ML (b) and F (c) cases. The dashed and dash-dotted lines respectively correspond to δ / (Richardson localregime) and δ (nonlocal regime) for reference. -3 -2 -1
1 10 100 1000( δ /R ) -2/3 ( δ /R ) -2 (a) λ ( δ ) / ζ r m s ( z ) δ /R surfacez=-100 mz=-250 mz=-350 mz=-500 m -3 -2 -1
1 10 100 1000( δ /R ) -2/3 ( δ /R ) -2 (b) λ ( δ ) / ζ r m s ( z ) δ /R surfacez=-100 mz=-250 mz=-350 mz=-500 m -3 -2 -1
1 10 100 1000( δ /R ) -2/3 ( δ /R ) -2 (c) λ ( δ ) / ζ r m s ( z ) δ /R surfacez=-100 mz=-250 mz=-350 mz=-500 m FIG. 8. FSLE-I, normalized by the rms vorticity ζ rms as a function of the separation distance normalized by its initial value, δ / R , at thereference depths for the TC (a), ML (b) and F (c) cases. scale ranges as those from relative dispersion) onto generalbehaviors determined by the kinetic energy spectrum, pro-vides a first clear evidence of universal dispersion regimescontrolled by the dynamical properties of the turbulent flows.The dispersion rate at fixed length scale is quantified bythe FSLE. Since here we consider the separation process oftwo particles advected by the same flow starting from dif-ferent positions, we refer to the FSLE-I, which is computedas λ ( δ ) = log r (cid:104) τ ( δ ) (cid:105) , (10)where the average is over all pairs and τ ( δ ) is the timeneeded to observe the growth of separation from a scale δ to a scale r δ (with r > . The amplificationfactor was set to r = √
2, but we checked the robustness ofthe results with respect to this choice.Let us recall that, dimensionally, one expects the FSLEto scale as λ ( δ ) ∼ δ ( β − ) / for a kinetic energy spectrum E ( k ) ∼ k − β (see Refs. 19 and 33). If β < β = / λ ( δ ) ∼ δ − / , a behavior that is directlyrelated to Richardson superdiffusive regime, (cid:104) R ( t ) (cid:105) ∼ t .When β >
3, instead, i.e. when the advecting flow is smooth, the FSLE is expected to be constant, which indicates a nonlo-cal dispersion regime. Finally, at scales much larger than thelargest eddies, the FSLE has a diffusive scaling λ ( δ ) ∼ δ − .Figure 8 reports the FSLE-I, rescaled by ζ rms , as a functionof the separation distance rescaled by its initial value, δ / R .The results quite clearly indicate that dispersion is nonlo-cal (constant FSLE-I, with λ ( δ ) ≈ . ζ rms ) over a broadrange of scales up to O ( ) R , in the interior of the TCsystem, as well as below the mixed layer in the ML case.It is also the case for the F model, provided | z | is not toolarge. Close to the vertical boundaries and in the mixed layer,when present, dispersion is instead always local. The valueof λ ( δ ) at the smallest separations, which should providean estimate of λ L , is found to be quite close to 0 . ζ rms . Inthe range δ < O ( ) R , extending up to the largest activeflow scales, the FSLE-I displays a power-law dependence onthe separation distance compatible with δ − / (the theoret-ical expectation for Richardson superdiffusion), at least onaverage. Finally at scales larger than O ( ) R , λ ( δ ) ∼ δ − in all cases, indicating a diffusive behavior in this range.Summarizing, once properly rescaled with the rms vor-ticity, relative dispersion, relative diffusivity and the FSLE-I return a coherent picture that allows to identify differentdispersion regimes and to relate them with the statistical fea-tures of the turbulent flows. In particular, the analysis revealsa transition of behavior with depth. The dispersion processis found to be local at the surface, while it becomes nonlocalat depth, due to the decay of small eddies. However, while FIG. 9. Schematic illustration of typical trajectory pairs used forthe computation of the FSLE-II. Here δ ( t ) is the horizontal sepa-ration distance between trajectories x i ( t ) and x j ( t ) , evolving ondifferent depth levels ( z i and z j , respectively). The gray curve is theprojection of the trajectory at depth on the upper level. The initialposition of particles i and j is the same on the horizontal and is hereindicated by a dot. this occurs rapidly with increasing depth in the absence ofmixed layer instabilities, when the latter are present the tran-sition is moved to larger depth, below the mixed layer, due toenergetic submesoscale dynamics in the whole mixed layer. B. Vertical correlation of horizontal dispersion properties
In the previous section, horizontal dispersion propertieswere discussed, depth by depth, in particular using the 1 st -kind FSLE. We are now interested in the relative motionbetween particles seeded at different depths. This amountsto considering the evolution of pairs of trajectories startingfrom the same initial position, on the horizontal, but withdynamics governed by different flows (see Fig. 9). Theirspreading process can be examined using a modified typeof FSLE, as proposed in the context of predictability stud-ies .To take this approach, we consider the positions at time t , x i ( t ) and x j ( t ) , of particles initialized on different levels ( z i and z j ) and advected by the horizontal flow at their depth: d x i dt = v ( x i ( t ) , z i , t ) , d x j dt = v ( x j ( t ) , z j , t ) , with x i ( ) = x j ( ) . We can still define as δ = | x i ( t ) − x j ( t ) | the horizontal separation distance for pair ( i , j ) , i.e. as if thetwo particles were at the same level (or, in other terms, byprojecting x i ( t ) on the plane z = z j , as in Fig. 9). We thenintroduce the FSLE-II with a definition analogous to that ofthe FSLE-I, Eq. (10), and denote it λ v ( δ ) . In the following,we will always consider that one particle is at the surface(hence, e.g. , z j = u rms with | z | in Fig. 4), andvertical shear can play a relevant role. It is not difficult toobtain from dimensional arguments that, due to the meanshear, λ v ( δ ) (cid:39) log ( r ) Λ | z | δ − (recall that the mean shear is Λ = Λ m = Λ t in the present simulations, see table I). Thistype of contribution to the FSLE-II can be expected to belarge where U ( z ) / u rms is large (see inset in Fig. 4). A simi-lar scaling of λ v ( δ ) can also arise, more generally, from theshear due to the typical difference (versus depth) of the totalvelocity (cid:104)| v ( z ) − v ( ) | (cid:105) / , where v ( z ) = U ( z )+ u ( x , y , z , t ) (see also Eq. (7)), from which one would expect λ v ( δ ) (cid:39) log ( r ) (cid:104)| v ( z ) − v ( ) | (cid:105) / δ − .Another point to bear in mind is that, in all (TC, ML, F)cases, also the small-scale energetic content of our flows isreduced in the upper thermocline, or below the mixed layer.This situation is close to the one discussed in Ref. 47, whichconsiders the separation of two particles advected by twoflow fields v and v (cid:48) that have identical energy spectra at largescales but one of which has no scales smaller than a cut-offlength (cid:96) ∗ . In such a case, the distance between the two corre-sponding trajectories should be | x ( t ) − x (cid:48) ( t ) | ∼ | v − v (cid:48) | t ∼ v ∗ t , where v ∗ is the typical velocity difference, as long as | x ( t ) − x (cid:48) ( t ) | < (cid:96) ∗ . Thus, dimensionally, one has that theFSLE-II should scale as λ v ( δ ) ∼ δ − for δ small enough.At larger scales, the difference between the two flows has nomore influence and the FSLE-II typically recovers the behav-ior of the FSLE-I λ ( δ ) . In our case, an analogous reasoning(with v and v (cid:48) the velocities at the surface and at depth, re-spectively) would imply that a critical length scale (cid:96) ∗ shouldmark the transition between the behaviors λ v ( δ ) ∼ δ − and λ v ( δ ) ∼ λ ( δ ) , provided the previously discussed shear con-tribution is weak enough.Let us now illustrate the results, which are obtained us-ing 4096 original pairs, as for the computation of the FSLE-I(but now selecting one particle at the surface and anotherone at depth). Our interest is mainly focused on the scalerange between O ( ) km and O ( ) km. The FSLE-II isshown in Fig. 10 for the reference depths and two additionalones, z = −
50 m and z = −
150 m, respectively above andbelow the mixed layer, when present. At small depths theslope of the FSLE-II is close to that of the FSLE-I (recall that λ ( δ ) ∼ δ − / at the surface, before the onset of the diffusiveregime δ − at the largest values of δ ). This is particularlyevident for the ML and F cases, where the behavior ∼ δ − / is observed over a broader range of separations, namely fromfew to slightly less than 100 km. For the TC model, alreadyquite close to the surface ( z = −
50 m in Fig. 10a), at sep-arations δ < O ( ) km, however, the slope of the FSLE-IIgets definitely larger in absolute value ( λ v ( δ ) ∼ δ − ). Asthe vertical shear is still weak at such depths, it is possibleto associate this change of scaling with the missing smallscales in the deeper flow. It also appears reasonable, here,that the crossover scale is (cid:96) ∗ ≈
10 km, as this value is alsoclose to the length scale of the mixed-layer instability (capa-0 -2 -1
1 10 100 1000 δ -2/3 δ -1 δ -2 (a) λ v ( δ ) ( / da y ) δ (km)z=-50 mz=-100 mz=-150 mz=-250 mz=-350 mz=-500 m -2 -1
1 10 100 1000 δ -2/3 δ -1 δ -2 (b) λ v ( δ ) ( / da y ) δ (km)z=-50 mz=-100 mz=-150 mz=-250 mz=-350 mz=-500 m -2 -1
1 10 100 1000 δ -2/3 δ -1 δ -2 (c) λ v ( δ ) ( / da y ) δ (km)z=-50 mz=-100 mz=-150 mz=-250 mz=-350 mz=-500 m FIG. 10. FSLE-II for the TC (a), ML (b) and F (c) cases. The settings are as in Fig. 8 but particles in a pair are now selected such that, initially,the first one is at the surface and the second one is at a depth z , with no horizontal separation between them. Two additional depths, withrespect to the computation of the FSLE-I, are here shown: z = −
50 m and z = −
150 m. Darker symbols correspond to larger depths. -2 -1
1 10 100 1000(a) λ v ( δ ) /[ l og (r) < | v ( z )- v ( ) | > / δ - ] δ (km)z=-50 mz=-100 m z=-150 mz=-250 m z=-350 mz=-500 m -1
1 10 100 1000 λ v ( δ ) δ / δ (km) -2 -1
1 10 100 1000(b) λ v ( δ ) /[ l og (r) < | v ( z )- v ( ) | > / δ - ] δ (km)z=-50 mz=-100 m z=-150 mz=-250 m z=-350 mz=-500 m -1
1 10 100 1000 λ v ( δ ) δ / δ (km) -2 -1
1 10 100 1000(c) λ v ( δ ) /[ l og (r) < | v ( z )- v ( ) | > / δ - ] δ (km)z=-50 mz=-100 m z=-150 mz=-250 m z=-350 mz=-500 m -1
1 10 100 1000 λ v ( δ ) δ / δ (km) FIG. 11. FSLE-II for the TC (a), ML (b) and F (c) cases; darker symbols correspond to larger depths. In the main panels λ v ( δ ) is compensated by the expectation in the presence of the typical vertical shear due to both the mean flow and turbulent fluctuations,log ( r ) (cid:104)| v ( z ) − v ( ) | (cid:105) / δ − (where v ( x , y , z , t ) is the total velocity at depth z ). The insets show λ v ( δ ) compensated by δ − / ; the dashedlines are guides to the eye for constant values. ble of energizing the submesoscale in the first 100 m belowthe surface), which is absent in this model. A similar transi-tion to λ v ( δ ) ∼ δ − becomes evident only below the mixedlayer ( z = −
150 m in Figs. 10b,c) in the ML and F cases,for which small scales are energetic down to z = −
100 m.Further below the surface, the FSLE-II is in all cases close to δ − (though slightly steeper for the ML and F models, where λ v ( δ ) ∼ δ − α , with α (cid:39) .
25 and 1 .
15, respectively) overa more extended range of separations, due to the missingsmall scales, but now also due to the vertical shear becomingmore important with depth and eventually dominating. In-terestingly, some indications about the relevant role of verti-cal shear were recently documented also in a more realistic,albeit more specific, numerical study addressing pair disper-sion at submesoscales in the Bay of Bengal . At the largestdepths, a flattening of the FSLE-II at the smallest separationsis seen, particularly for the ML and F cases. It should benoted, however, that in this range of scales and depths, dueto the large velocity differences involved, the results may beaffected by the finite temporal resolution of the data.To better appreciate the contribution from the verti-cal shear, in Fig. 11 we report λ v ( δ ) compensated bythe expectation in the presence of the vertical shear aris-ing from both the mean flow and the turbulent velocity,log ( r ) (cid:104)| v ( z ) − v ( ) | (cid:105) / δ − (main panels). The compen-sation by the contribution from the mean vertical shear only, i.e. log ( r ) Λ | z | δ − , was not found to be sufficient to accountfor the behavior of the FSLE-II (not shown). On the contrary,the FSLE-II compensated by the total-shear prediction ap-proaches the constant value 1 (Fig. 11), particularly at largerdepths. Note that the collapse of the different curves corre-sponding to different depths is much better for the TC casethan for the ML and F cases. For the TC case, the devi-ations at the smallest depths (e.g. z = −
50 m) for separa-tions δ >
10 km are related to the scaling λ v ( δ ) ∼ δ − / ,as shown by the inset of Fig. 11a. Indeed, due to still en-ergetic eddies in both flows at these scales, and weak verti-cal shear at these depths, as already observed from Fig. 10a,in this range the FSLE-II is close to the FSLE-I. In theML and F cases, when depths larger than the mixed-layerdepth ( h =
100 m) are considered, the compensated (by δ − )FSLE-II is fairly close to 1 in a broad range of horizontalseparations (2 km ≤ δ ≤
100 km). More important devia-tions are observed at smaller depths (inside the mixed layer)and can be attributed to the scaling λ v ( δ ) ≈ δ − / (insets ofFig.11b and c), related to the similar small-scale energeticcontent of the flows at the surface and below it, in this rangeof depths.1 V. CONCLUSIONS
We explored Lagrangian pair dispersion in stratifiedupper-ocean turbulence. We focused on the identification ofdifferent dispersion regimes and on the possibility to relatethe characteristics of the spreading process at the surface andat depth. The latter question is particularly relevant to assessthe possibility of inferring the dynamical features of deeperflows from the experimentally more accessible ( e.g. by satel-lite altimetry) surface ones. In this sense, Lagrangian dis-persion statistics can provide useful information to under-stand the coupling between the surface and interior dynam-ics. Some perspectives on the use of further particle-basedapproaches to this subject, which is key to understand howsubmesoscale flows participate in biogeochemical and heatbudgets, are discussed in Ref. 57.Tracer particles were advected by turbulent flows charac-terized by energetic submesoscales close to the surface, bothin the presence (ML and F cases) and in the absence (TCcase) of mixed-layer instabilities. The numerical simulationsof the model dynamics were carried out using realistic pa-rameter values for the midlatitude ocean. Even if the pres-ence of a mixed layer has a signature at the surface in termsof a less filamentary flow field, its main effect is to ener-gize the full upper part of the water column and, hence, tostrongly impact the vertical variation of the statistical fea-tures of turbulence. Kinetic energy spectra close to k − / arefound at the surface in all models. They are instead steeperthan k − at depth, due to the decay of small eddies. How-ever, while this change of behavior occurs already close tothe surface in the TC case, in the ML and F cases, it onlymanifests below the depth of the mixed layer ( z = −
100 m).The different statistical indicators examined, once prop-erly rescaled to take into account the typical intensity of ve-locity gradients, allowed to group the data at different depthsand from different models into only two universal behaviors,corresponding to nonlocal and local dispersion, which are inagreement with the dimensional expectations based on ki-netic energy spectra. Therefore, our results indicate a cleartransition of dispersion regime with depth, which is quitegeneric. The spreading process is local at the surface. In theabsence of a mixed layer it very soon changes to nonlocal atsmall depths, while in the opposite case this only occurs atlarger depths, below the mixed layer.It is here worth commenting on our results from a dimen-sional point of view. Horizontal dispersion is always foundto be diffusive-like at spatial scales larger than O ( ) kmand at times t >
30 day from the release. The intensity ofthe dispersion process, as quantified, e.g. , by the FSLE-I,decreases with depth in all models, except close to the bot-tom boundary in the TC and F cases (due the small-scaleflows gaining energy again there). In the nonlocal-dispersioncases, the flat behavior of the FSLE-I in a broad range ofscales ( O ( ) km < δ < O ( ) km) allows to estimate theLyapunov exponent λ L . The latter is found to be of or-der 0 .
01 day − in the TC case and 0 . − in the MLand F ones. Under local-dispersion, the behaviors foundare compatible with Richardson superdiffusion from few km to about 100 km. The scale-by-scale dispersion rate, λ ( δ ) ,is considerably enhanced at submesoscales, reaching values ≈ ( . − . ) day − in the TC case and ≈ − , compat-ible with surface-drifter observations in different regions ,in the ML, F cases.We further investigated the transition from local to nonlo-cal dispersion, with increasing depth, by means of the FSLE-II. Our results indicate that, in the absence of a mixed layer,dispersion properties rapidly decorrelate from those at thesurface. In the ML and F cases, instead, a similar phe-nomenon occurs only below the mixed layer. The transitionis sharper in the TC model. However, the relation betweendispersion at depth and at the surface appears in this case tobe largely controlled by the vertical shear (due to the totalvelocity), as revealed by the very good collapse of the datarescaled by the prediction based on it. This suggests that,even in this case, it should be in principle possible to inferhow a tracer at depth separates from one at the surface, if theshear is known, or to parameterize it using information at thesurface only. In the presence of mixed-layer instabilities, theresults indicate that the statistical properties of the spreadingprocess at the surface can be considered as a good proxy ofthose in the whole mixed layer. The vertical-shear predictionfor the FSLE-II still appears reasonable, particularly belowthe mixed layer and for separations ranging from few km to ≈
100 km, but now the agreement is essentially limited tothe order of magnitude, which makes it more difficult to es-tablish a link between the interior and surface dispersion.Finally, based on the above considerations, in our opinion,this study provides evidence of the interest for future satel-lite altimetry, as the SWOT mission , that should providesurface velocity fields at unprecedented high resolution, alsoin the light of understanding subsurface ocean dynamics. ACKNOWLEDGMENTS
This work is a contribution to the joint CNES-NASASWOT projects “New dynamical tools” and DIEGO and issupported by the French CNES TOSCA program.
Appendix: Streamfunction at arbitrary depth
The streamfunction ψ at a generic depth z can be ex-pressed, in Fourier space (with k the horizontal wavenum-ber) in terms of ψ , ψ , ψ asˆ ψ ( k , z ) = µ m (cid:20) ˆ ψ ( k ) sinh (cid:18) µ m z + hh (cid:19) −− ˆ ψ ( k ) sinh (cid:16) µ m zh (cid:17)(cid:105) (A.1)in layer 1 ( − h < z ≤
0) and asˆ ψ ( k , z ) = µ t (cid:20) ˆ ψ ( k ) sinh (cid:18) N t N m µ m z + Hh (cid:19) −− ˆ ψ ( k ) sinh (cid:18) N t N m µ m z + hh (cid:19)(cid:21) (A.2)in layer 2 ( − H ≤ z ≤ − h ).2 REFERENCES J. C. McWilliams, “Submesoscale currents in the ocean,” Proc. R. Soc. A , 20160117 (2016). M. Lévy, “The modulation of biological production by oceanic mesoscaleturbulence,” Lect. Notes Phys. , 219––261 (2008). R. Ferrari, “A frontal challenge for climate models,” Science ,316–317 (2011). Z. Su, J. Wang, P. Klein, A. F. Thompson, and D. Menemenlis, “Oceansubmesoscales as a key component of the global heat budget,” Nat. Com-mun. , 775 (2018). A. Bracco, G. Liu, and D. Sun, “Mesoscale-submesoscale interactions inthe Gulf of Mexico: from oil dispersion to climate,” Chaos, Solitons andFractals , 63–72 (2019). L. Siegelman, “Energetic submesoscale dynamics in the ocean interior,”J. Phys. Oceanogr. , 727–749 (2020). G. Lapeyre and P. 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