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# A survey of submesoscale currents

- James C. McWilliams
^{1}Email authorView ORCID ID profile

**6**:3

https://doi.org/10.1186/s40562-019-0133-3

© The Author(s) 2019

**Received:**30 August 2018**Accepted:**14 March 2019**Published:**25 March 2019

## Abstract

Submesoscale currents are pervasive throughout the ocean. They have intermediate space and time scales—neither mesoscale nor microscale—that have made them elusive for measurements and modeling until recently. In this brief article, a survey is presented of their primary characteristics and interpretive explanations, intended for a broad audience of physical and biogeochemical oceanographers. Besides their identifying scales, submesoscale currents are distinctive in their flow patterns, their essential dynamical processes, and their consequences for transport, mixing, and dissipation in the general circulation. There are two primary submesoscale populations, a frontal one in the near-surface layer with its typically reduced stratification, and another vortical one, generated in topographic wakes, that (sparsely) fills the oceanic interior.

## Keywords

- Submesoscale currents
- Density fronts and filaments
- Coherent vortices
- Topographic waves

## Introduction

^{1}The name is also apt in the sense that the primary source for SMC energy comes from mesoscale eddies by a downscale transfer. The principal SMC generation mechanisms are (1) extraction of available potential energy (due to horizontal buoyancy gradients) in the weakly stratified surface layer either through baroclinic instability or frontogenesis, and (2) topographic-drag vorticity generation in flows along a sloping bottom, followed by boundary current separation and wake instability. These phenomena partly manifest a loss of hydrostatic, geostrophic momentum balance, and they exhibit a turbulent energy cascade forward toward even smaller scales. Thus, SMC dynamics typically go beyond quasigeostrophy, which is the generally successful theoretical framework for mesoscale eddies, while still being strongly influenced by Earth’s rotation and the generally stable density stratification in the ocean. That is, the Rossby and Froude numbers:

*V*is a characteristic horizontal velocity scale, \(\ell\) and

*h*horizontal and vertical length scales,

*f*the Coriolis frequency, and

*N*the Brunt–Vaisala or stratification frequency.) Thus, a Rossby number

*Ro*measures the relative magnitude of momentum advection relative to the Coriolis force, and a Froude number

*Fr*measures the ratio of an advecting velocity to the phase speed of an internal gravity wave. Small values for these numbers indicate the dynamical importance of rotation and stratification. Nevertheless, the evolution of SMCs is primarily through advection, which distinguishes it from the inter-gravity waves that can occupy similar scale ranges in the ocean; their mutual influence is a topic of current research, but, in some first approximation, their interactions are weak.

The science of SMCs has blossomed in recent years. The delay, compared to other more familiar types of oceanic currents, is mainly due to technical barriers. The SMC space and time scales are awkwardly in between the finer scale sampling from ships, sparse buoys, and floats, and the larger scale sampling from most satellite sensors; their simulation requires large computations that encompass both the mesoscale and submesoscale ranges; and the relevant theories involve difficult nonlinear dynamics. The more recent empowering technologies are high-resolution surface images, multiply-nested computational simulation methods, and, in a few instances, massive swarms of surface drifters (as in the CARTHE experiments in the Gulf of Mexico). As yet there is no widely deployable SMC measurement technique for the subsurface ocean, so simulations are leading the way. Autonomous gliders and ship-towed instruments can provide submesoscale spatial sampling along their tracks, but they are limited to two dimensions and thus often have difficulty in distinguishing SMCs and inertia-gravity waves.

^{2}Their originating source is forcing by surface winds and air–sea buoyancy fluxes at the energetic scales of the atmosphere, i.e., mostly on the planetary scale comparable to the size of oceanic basins. The direct oceanic response is oceanic currents on the basin and inter-basin scales, including their narrower transport closures as boundary and equatorial currents. On the other hand, the sink is energy dissipation and information loss that can only be completed at the microscale due to molecular viscosity and conductivity. Currents of different types must connect the source and sink across the intervening scales and dynamical regimes. The first step is geostrophic instabilities of the forced circulation, yielding mesoscale eddies. Through the force-balanced constraints of geostrophic and hydrostatic balance, they are inhibited from further transfers to smaller scales; i.e., they have an inverse cascade of energy (Charney 1971). Leakages out of the mesoscale eddies by partial violation of these force balances continue downscale. Three middle-scale “routes” are depicted in Fig. 1: (1) spontaneous emission of inertia-gravity waves from currents, either in the interior or as bottom lee waves, followed ultimately by energy transfer to smaller scales; (2) partly ageostrophic instabilities and forward energy cascade of non-wave (partly balanced) currents; and (3) turbulent bottom drag on currents that generates both bottom boundary-layer turbulence and topographic vortical wakes. Below this middle-scale range, three-dimensional turbulence completes the connection to the microscale. While no accurate global accounting of these three routes is yet available, the SMC role in the latter two routes is almost certainly a dominant one. Along the pathway toward the microscale, the character of the currents changes from being highly anisotropic with \(h/\ell \ll 1\) and relatively small vertical velocity to approaching isotropy with \(h \sim \ell\), as in Kolmogorov’s paradigm for universal turbulent behavior at high Reynolds number. In addition, the local

*Ro*and

*Fr*systematically increase as

*h*and \(\ell\) decrease.

The paper is organized by considering the two SMC surface-layer and topographic populations separately in “Lines on the surface” and “Topographic wakes” sections, and it ends with a summary in “Final remarks” section.

This article is adapted from a lecture given at the 2018 annual meeting of the Asia Oceania Geosciences Society (AOGS) in Honolulu, HI. It is intended for a broad audience, and it is more about my experiences and opinions, with some illustrations, rather than about the full evidence and literature behind them. A more extensive and scholarly review article is McWilliams (2016), while this paper is intended to complement it for a more general audience.

## Lines on the surface

SMCs can be difficult to discern in single vertical profiles or time series because of potential confusions with other types of flows. Among the most useful observations are surface images, especially those with high horizontal resolution (subsurface images are rare, but see Fig. 14). By the addition of an extradimension and with dense sampling, patterns emerge, and with experience, they can be interpreted for their underlying phenomena. Images in a horizontal plane (i.e., at the oceanic surface) allow a recognition of sharp convergence lines and small, horizontally recirculating vortices. Several examples are presented in this section.

Besides the recognition of lines of large horizontal density gradients (i.e., fronts), many of these lines exhibit meanders, which is suggestive of frontal instability, likely due to the associated vertical or horizontal shear in the mixed layer. The former is a type of baroclinic instability but with preferred length scales that are in the submesoscale range due to the small surface-layer baroclinic deformation radius (Boccaletti et al. 2007; Fox-Kemper et al. 2008). The latter is a form of classical shear instability that is engendered by the sharpening horizontal gradients caused by active frontogenesis. Both types can be partly ageostrophic because of the large value of the Rossby number.

Such images and photographs are highly informative, but their information is intrinsically subjective. It is quite difficult to get in situ measurements that cover the indicated patterns, though there have been some successes. More generally, however, the quantitative science of SMCs has been advanced by computational simulations. An example for the offshore Gulf Stream is Fig. 6. The experience has been that, in realistic simulations with active mesoscale eddies and associated horizontal density gradients, a sufficiently fine-grid resolution will lead to the spontaneous emergence of SMCs that first arise in the weakly stratified surface layer. The necessary resolution varies with conditions (e.g., region or season), but it is around \(dx \approx 1\) km; such simulations can be referred to as “submesoscale-permitting”, because the full range of submesoscale variability extends down to \(\approx\) 10–100 m, and the latter would have to be reached to be fully “submesoscale-resolving”. Nevertheless, simulations show approximately self-similar scaling behavior when *dx* is varied within this submesoscale range. The associated kinetic-energy horizontal-wavenumber spectra are relatively shallow, \(E(k) \sim k^{-\,\gamma }\), with \(\gamma \approx 2\), where *k* is the horizontal wavenumber. This differs from simulations and altimetric sea-level measurements that show generally steeper spectra (larger \(\gamma\) values) in the mesoscale range. The value of the simulation results is mainly in the phenomenological discoveries they have enabled. This is illustrated in Fig. 6 by the variety of different SMC patterns associated with different mesoscale environments. Once the phenomenology is known, then detailed diagnostic analyses and theoretical explanations can be adduced.

One dynamical frontier for submesoscale simulations is the onset of essentially non-hydrostatic behavior. Most SMC simulations to date are made with hydrostatic models. Some estimates for this lower size limit for SMCs are where frontogenesis is arrested by frontal instability and/or where the currents in the forward energy cascade reach scales where rotation and stratification influences cease to be significant (i.e., relevant *Ro* and *Fr* values are large; Sullivan and McWilliams 2018). Both estimates would yield a horizontal scale in the 10–100 m range. Whether, in fact, there are important non-hydrostatic effects on SMCs at larger scales remains to be further tested.

*g*is gravity, \(\rho\) is density, and \(\rho _0\) a mean value) gradient will initially sharpen at an exponential rate \(\sim \exp [\alpha t]\) as a function of time

*t*, if the gradient is favorably aligned in relation to a barotropic deformation flow with a uniform strain rate, no horizontal divergence, and no vorticity; i.e., \(\mathbf{u}= (u_d,v_d,0)\) with \(u_d = - \, \alpha x / 2\) and \(v_d = \, \alpha y / 2\), where \(\alpha\) is the horizontal strain rate and (

*x*,

*y*,

*z*) and \((u,\ v,\ w)\) are the coordinates and velocity components in the (east, north, and upward) directions (Fig. 7a). Because of this density structure, there is an associated circulation, both along the axis and mostly geostrophic in

*v*, and in the cross-axis plane with ageostrophic

*u*and

*w*. If the front is uniform along the axis or if an average is taken in this direction (denoted by angle brackets), then the cross-axis velocity is 2D non-divergent, and it can be represented with a secondary-circulation streamfunction \(\Phi\) defined by the following:

*b*and \(\Phi\) fields for a surface front are shown in Fig. 8. Interestingly, the monopole \(\Phi\) pattern is qualitatively the same as for a front in a deformation flow (Fig. 7), and the same similarity occurs for dense filaments. Thus, the TTW circulations are also frontogenetic due to the surface convergence on the dense side or center. The last panel in Fig. 8 shows the Lagrangian tendency for the SMC horizontal shear variance, \(T^\mathbf{u}\ = \ D|\nabla \mathbf {u}|^2/Dt\): it is strongly positive on the upper dense side of the front. Thus, differential advection by the secondary circulation is the cause of frontogenesis in both the buoyancy gradient and velocity shear.

*E*(

*k*) (Fig. 12) shows a broad range of variability from the submesoscale peak with its characteristic spectrum slope exponent of \(\gamma \approx 2\) into a more fully 3D range with a smaller value of \(\gamma \approx 5/3\), as expected for boundary-layer turbulence (Thorpe 2005; McWilliams 2016).

Thus, there is an intimate relation between submesoscale currents and boundary-layer turbulence near the surface, with the turbulence providing important mixing effects (e.g., in a TTW evolution) and the former providing important additional TKE excitation and modifying the mixing behavior. This is a research frontier that is almost completely wide open.

## Topographic wakes

Wakes are a familiar fluid dynamical phenomenon: flow past an obstacle generates vorticity in a boundary layer and then separates in the lee; if the Reynolds number^{3} is not small, then the velocity shears within both the boundary layer and the wake are unstable and generate turbulence. The question is how to translate this for the ocean, which involves stratification, rotation, and bottom slopes not side walls. In some instances, the currents are diverted horizontally to approximately follow bathymetric contours without much vorticity generation, and in others, the flow is diverted vertically and generates internal gravity lee waves propagating vertically into the interior. However, here, my focus is on instances of significant vorticity generation that lead to unstable wakes, locally enhanced diapycnal mixing and energy dissipation, and formation of submesoscale coherent vortices (SCVs) that are advected into and widely populate the interior ocean (McWilliams 1985).

*q*by the geometric argument depicted in Fig. 13: because the bottom boundary layer decreases an interior mean flow to zero at the sloping bottom, there must be an associated horizontal shear (i.e., vertical vorticity, \(\zeta ^z\)) along a horizontal line extending out into the interior. This is a flow-structure argument, and it needs to be extended to encompass the actual rate of \(\zeta ^z\) generation by the along-slope gradients in bottom stress. Nevertheless, the sketch indicates why along-slope near-bottom flows are necessarily a source of vertical vorticity of the flow, and vertical vorticity is a common ingredient in lateral (barotropic) shear instability with small

*Ro*and

*Fr*values. Furthermore, there is little impetus for currents to separate from a flat-bottom boundary against the gravitational barrier of a stable vertical stratification, whereas it is much more common for currents along a slope to separate while on an intersecting isopycnal surface, whether aided by boundary curvature or even spontaneously.

The hypothesis is that SCVs are generated primarily in topographic wakes where the drag-induced \(\zeta\) and *q* are large enough to make the ensuing vortices strong enough to resist the early disruption by encounters with other, weaker interior currents. Alternatively, a localized vertical mixing event in a stratified region, followed by geostrophic adjustment, can do so, as well (McWilliams 1985; Bosse et al. 2017). Because bottom currents, stable stratification, and topographic slopes are ubiquitous in the ocean at all depths, so SCVs are common, albeit occupying only a small volumetric fraction of the oceanic interior (one estimate for Meddies is \(< 10\%\) of the middle depths of the Eastern Subtropical Atlantic).

*Ro*is small (Fig. 16). Another example is a realistic simulation of the Subtropical Eastern-Boundary Current System west of North America. There, the poleward California undercurrent flows along the continental slope. It manifests the drag-induced vorticity generation scenario described above. Where it separates, a strong centrifugal instability

^{4}arises in the wake (with \(fq < 0\)), and submesoscale vortices emerge and mutually interact to form a California Undercurrent SCV (a Cuddy; Fig. 17). Many different Cuddies have been detected off the U.S. West Coast by acoustically tracked recirculating, trapped subsurface float trajectories.

These unstable wakes exhibit strong submesoscale turbulence with a forward energy cascade to dissipation and mixing of material concentrations both along and across density surfaces. Thus, this class of topographic SMCs can have widespread influences on the oceanic interior. In my view, it is important to explore these phenomena further, both computationally and observationally.

## Final remarks

*w*in the surface layer. More generally, SMCs enhance material exchanges between the turbulent boundary layers and the interior.

In summary, SMCs are active over much of the ocean with large seasonal and geographical variability. They have a distinctive dynamics by being advective, partly ageostrophic, and frontogenetic. There are at least two distinct populations: one associated with surface-layer frontogenesis and the other associated with topographic wakes. Both populations are tightly coupled with the local microscale turbulence, and thus, they are a significant cause of intermittency, heterogeneity, and non-stationary behavior in the surface and bottom boundary layers.

An important open question is how active SMCs are in the ocean interior. Idealized simulations of rotating, stratified turbulence indicate that they should be so, at least in some places where *Ro* and *Fr* are not too small (Molemaker et al. 2010; Kafiabad and Bartello 2016), e.g., within strong currents and eddies. As yet no realistic oceanic simulations or measurements unambiguously show this to be true, but neither has this issue yet been pushed very hard. Were it to be true, then one would expect a relatively shallow kinetic-energy spectrum, forward energy cascade, and elevated dissipation and diapycnal mixing rates. Of course, SCVs are abundant in the interior, but, by the hypothesis stated in “Topographic wakes” section, these are most likely to have been generated near the topography or in local mixing zones and then moved into the wider ocean.

However, Submesoscale Coherent Vortices (SCVs), once formed and freely moving within the interior ocean, can have survival lifetimes of years (McWilliams 1985).

The energy and information cycles for the tides and surface gravity waves are largely separate from the general circulation cycle.

\({Re} = VL/\nu\), where \(\nu\) is the molecular viscosity. It is a common parameter indicating how strong momentum advection is compared to momentum diffusion.

Centrifugal (or symmetric or inertial) instability occurs when the Ertel potential vorticity *q* changes sign within the local domain. Thus, it can only occur when *Ro* or *Fr* is large. In both the surface and topographic SMC populations, it often is triggered by potential vorticity fluxes through the top and bottom boundaries, respectively.

## Declarations

### Authors’ contributions

JM wrote the manuscript. The author read and approved the final manuscript.

### Acknowledgements

This paper was written during a visit to the Kavli Institute for Theoretical Physics, supported in part by the National Science Foundation under Grant No. NSF PHY-1748958. I appreciate continuing research support from the National Science Foundation and the Office of Naval Research. I also appreciate the invitation from AOGS to present a Distinguished Lecture for the Ocean Sciences Section.

### Competing interests

The author declares no competing interests.

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## Authors’ Affiliations

## References

- Adams K, Hosegood P, Taylor J, Sallee JB, Bachman S, Torres R, Tamper M (2017) Frontal circulation and submesoscale variability during the formation of a southern ocean mesoscale eddy. J Phys Ocean 47:1737–1753View ArticleGoogle Scholar
- Boccaletti G, Ferrari R, Fox-Kemper B (2007) Mixed layer instabilities and restratification. J Phys Ocean 37:2228–2250View ArticleGoogle Scholar
- Bosse A, Testor P, Mayot N, Prieur L, D’Ortenzio F, Mortier L, Goff HL, Gourcuff C, Coppola L, Lavigne H, Raimbault P (2017) A submesoscale coherent vortex in the Ligurian Sea: from dynamical barriers to biological implications. J Geophys Res Oceans 122:1–22View ArticleGoogle Scholar
- Charney JG (1971) Geostrophic turbulence. J Atmos Sci 28:1087–1095View ArticleGoogle Scholar
- D’Asaro E, Shcherbina A, Klymak JM, Molemaker J, Novelli G, Gigand C, Haza A, Haus B, Ryan E, Jacobs GA, Huntley HS, Laxagne HJM, Chen S, Judt F, McWilliams JC, Barkan R, Krwan AD, Poje AC, Ozgokmen TM (2018) Ocean convergence and dispersion of flotsam. PNAS 115:1162–1167View ArticleGoogle Scholar
- Fox-Kemper B, Ferrari R, Hallberg RW (2008) Parameterization of mixed layer eddies. Part I: theory and diagnosis. J Phys Ocean 38:1145–1165View ArticleGoogle Scholar
- Gower J, Hu C, Borstad G, King S (2006) Ocean color satellites show extensive lines of floating sargassum in the Gulf of Mexico. IEEE Trans Geosci 44:3619–3625View ArticleGoogle Scholar
- Gula J, Molemaker MJ, McWilliams JC (2015) Gulf Stream dynamics and frontal eddies along the southeastern U.S. Seaboard. J Phys Ocean 45:690–715View ArticleGoogle Scholar
- Kafiabad KA, Bartello P (2016) Balance dynamics in rotating stratified turbulence. J Fluid Mech 796:914–949View ArticleGoogle Scholar
- Kessouri F, McWilliams JC, Bianchi D, Renault L, Deutsch C, Frenzel H (2019) Effects of submesoscale circulation on the nitrogen cycle in the California Current System. Global Biogeochem Cycles (submitted) Google Scholar
- Mahadavan A (2016) The impact of submesoscale physics on primary productivity of plankton. Ann Rev Mar Sci 8:161–184View ArticleGoogle Scholar
- McWilliams JC (1985) Submesoscale, coherent vortices in the ocean. Rev Geophys 23:165–182View ArticleGoogle Scholar
- McWilliams JC (2016) Submesoscale currents in the ocean. Proc R Soc A 472:20160117–132View ArticleGoogle Scholar
- McWilliams JC (2017) Submesoscale surface fronts and filaments: secondary circulation, buoyancy flux, and frontogenesis. J Fluid Mech 823:391–432View ArticleGoogle Scholar
- McWilliams JC, Colas F, Molemaker MJ (2009) Cold filamentary intensification and oceanic surface convergence lines. Geophys Res Lett 36:18602View ArticleGoogle Scholar
- Molemaker MJ, McWilliams JC, Capet X (2010) Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J Fluid Mech 654:35–63View ArticleGoogle Scholar
- Molemaker MJ, McWilliams JC, Dewar WK (2015) Submesoscale instability and generation of mesoscale anticyclones near a separation of the California Undercurrent. J Phys Ocean 45:613–629View ArticleGoogle Scholar
- Munk W, Armi L, Fischer K, Zachariasen F (2000) Spirals on the sea. Proc R Soc A 456:1217–1280. https://doi.org/10.1098/rspa.2000.0560 View ArticleGoogle Scholar
- Papenberg C, Kaleschen D, Krahmann G, Hobbs RW (2010) Ocean temperature and salinity inverted from combined hydrographic and seismic data. Geophys Res Lett 37:04601View ArticleGoogle Scholar
- Riser SC, Owens WB, Rossby HT, Ebbesmeyer CC (1986) The structure, dynamics, and origin of a small scale lens of water in the Western North Atlantic thermocline. J Phys Ocean 16:572–590View ArticleGoogle Scholar
- Scully-Power P (1986) Navy oceanographer shuttle observations, STS 41-G Mission Report. NUSC. Technical Document 7611Google Scholar
- Srinivasan K, McWilliams JC, Molemaker MJ, Barkan R (2019) Submesoscale vortical wakes in the lee of a seamount. J Phys Ocean (in press) Google Scholar
- Sullivan PP, McWilliams JC (2018) Frontogenesis and frontal arrest for a dense filament in the oceanic surface boundary layer. J Fluid Mech 837:341–380View ArticleGoogle Scholar
- Thorpe SA (2005) The Turbulent Ocean. Cambridge University Press, Cambridge, p 439View ArticleGoogle Scholar