## 1. Introduction

Surface oceanic submesoscale currents consist of coherent fronts, filaments, and eddies with spatial scales on the order of 0.1–10 km, and time scales on the order of hours to days, that are dynamically characterized by *O*(1) Rossby and Richardson numbers (Thomas et al. 2008; McWilliams 2016). Due to their significant ageostrophic vertical velocities, they are thought to be instrumental in transferring properties from the surface of the ocean to the interior, thereby affecting a wide range of physical and biogeochemical processes (Thomas et al. 2008; McWilliams 2016; Mahadevan 2016). In addition, they are believed to influence the dispersion of contaminants (Poje et al. 2014) and to provide an important route for kinetic energy dissipation (Capet et al. 2008c; Molemaker et al. 2010; Barkan et al. 2015), thus impacting the general circulation of the oceans.

Submesoscale fronts and filaments are identified as anisotropic flow features with large magnitudes of vorticity, horizontal divergence, buoyancy gradients, and velocity gradients. Their generation is explained by a variety of processes, including mixed layer instabilities (Boccaletti et al. 2007; Fox-Kemper et al. 2008), deformation frontogenesis (Hoskins and Bretherton 1972), and boundary layer turbulence (Gula et al. 2014; McWilliams et al. 2015; Sullivan and McWilliams 2018) all of which rely on the conversion of available potential energy (APE) to kinetic energy (McWilliams 2016). Because of the importance of local advection to their dynamical evolution; their plausible interaction with boundary layer turbulent processes (Bachman and Taylor 2016; McWilliams 2017), surface waves (McWilliams and Fox-Kemper 2013; Hamlington et al. 2014; Suzuki et al. 2016), and internal waves (Whitt and Thomas 2015; Barkan et al. 2017c; Thomas 2017); and their small temporal and spatial scales, they are challenging to study theoretically, numerically, and observationally.

*f*is the Coriolis frequency;

In the current paper we propose an asymptotic model that highlights the role of the horizontal convergence induced by the secondary circulation in governing submesoscale frontogenesis (section 2). Under the proposed model, analytical solutions for the inviscid evolution of all gradient fields in a Lagrangian reference frame can be obtained. The validity of the asymptotic model is verified with respect to submesoscale-resolving numerical simulations in the northern Gulf of Mexico (section 4) and drifter observations (section 5) collected as part of the Lagrangian Submesoscale Experiment (LASER) during January–February 2016 (details of the numerical setup and data analysis are provided in section 3). The asymptotic model is then applied to estimate vertical buoyancy fluxes during frontogenesis (section 6), and a summary and discussion of the results is given in section 7.

## 2. The asymptotic model

^{1}An important spectral identity relates the PSD of the velocity gradient with the PSDs of the vorticity and the divergence fields, namely,

^{2}Indeed, the velocity PSD and the corresponding PSDs of the velocity gradient, vorticity, and divergence in our submesoscale-resolving numerical simulations (Fig. 2) illustrate the importance of the divergence field at submesoscales (i.e., horizontal scales smaller than 10 km), even though the velocity PSD is somewhat steeper than

*k*

_{h}= 10

^{−4}m

^{−1}and gradually decreases at higher wavenumbers, suggesting that the magnitudes of

*ζ*and

*δ*are asymptotically similar at these (submeso) scales.

*f*,

*u*and

*υ*to be the cross-front and alongfront velocities, which correspond to the alongfront length scale

*L*in the

*y*direction and the cross-front length scale

*l*in the

*x*direction. Variable

*T*is the time scale, and

*ε*is the anisotropy ratio, and

*λ*is the aspect ratio. We further choose the distinguished limit

^{3}

*u*is purely ageostrophic (appendix A). Furthermore,

*h*subscripts denote horizontal vector components, and the equations are written in dimensional form. The Laplacian of the pressure

*g*and ag subscripts denoting the geostrophic and ageostrophic vorticity components, respectively. Note that

*ζ*× Eq. (12c) +

*δ*× Eq. (12d) = Eq. (12b), in agreement with Eq. (10).

### a. Frontogenetic tendencies and rates

^{4}

### b. Analytical solutions in a Lagrangian reference frame

*τ*notation is to remind the reader that the solutions are in a Lagrangian reference frame. The 0 subscript denotes the initial values of the different fields at time

*a*is assumed to be constant. It can be shown that any cyclonic ageostrophic vorticity will prevent a finite time singularity from occurring,

^{5}and we therefore pick

The analytical solutions for the divergence field [Eqs. (16d) and (18d) and appendix C] are shown in Fig. 3 (similar Lagrangian evolutions are found for the other gradient fields; not shown). These solutions illustrate how sensitive submesoscale frontogenesis is to even small values of ageostrophic vorticity (black and reds curve compared with the blue curve), which can substantially speed up the gradient sharpening. In all cases, the Lagrangian evolution and inviscid blowup times are independent of the physical mechanisms that initiate submesoscale frontogenesis. This implies that these analytical solutions [Eqs. (16d) and (18d) and appendix C] are generic with respect to the gradient evolution during submesoscale frontogenesis, as long as the scaling assumptions [Eq. (7)] are valid. In the following sections, we test and validate the asymptotic model [Eqs. (12a)–(12d)] against submesoscale resolving numerical simulations and drifter measurements in the northern Gulf of Mexico.

## 3. Models and measurements

In this section we describe the numerical setup and drifter measurements we used to validate the asymptotic model. The results follow in sections 4 and 5.

### a. Model setup

The numerical simulations used in this study are carried out with the Regional Oceanic Modeling System (ROMS; Shchepetkin and McWilliams 2005) using a one-way nesting procedure (Mason et al. 2010) and focusing on solutions in the northern Gulf of Mexico (GoM) region with an approximately 500-m, nearly isotropic, horizontal resolution. These 500-m solutions are gradually nested down from a 7-km ROMS solution of the entire Atlantic basin (Barkan et al. 2017a). The atmospheric forcing is climatological with a QuikSCAT-based daily product of scatterometer wind stresses (Risien and Chelton 2008), CORE (Large and Yeager 2009) monthly heat-flux atmospheric forcing, and HOAPS (Andersson et al. 2010) monthly freshwater atmospheric forcing. No tidal forcing is used, and a daily river forcing is applied based on daily river volume flux data from the USGS (http://waterdata.usgs.gov/nwis/rt) for the year 2010. This forcing methodology largely eliminates the generation of internal gravity waves and allows us to focus on the dynamics of submesoscale currents. The vertical mixing of momenta and tracers are parameterized using the *K*-profile parameterization (KPP; Large et al. 1994), and the horizontal diffusion of tracer and momenta are associated with the third-order upstream-biased advection scheme (appendix D). The analysis in Figs. 5 and 6 is carried out based on two years of equilibrated solutions for winter (January–March) and summer (July–September) months, using a twice-per-day output frequency. Figures 7–9, 11, D1, E1, and E2 are based on offline particle advection, using a 15-min output frequency. Additional information about the numerical setup is provided in Barkan et al. (2017a).

### b. Drifter measurements

LASER took place in northern GoM near the site of the Deepwater Horizon oil spill during the winter of 2016, as part of the Consortium for Advanced Research on the Transport of Hydrocarbons in the Environment (CARTHE). Over a thousand surface drifters were deployed during LASER, along with aerial and shipboard observations to help define the dynamics (D’Asaro et al. 2018). The analysis shown in section 5 is based on drifter observations collected between 20 January and 20 February 2016.

The mostly biodegradable drifters consisted of a floater extending 5 cm below the surface, which contained a GPS tracking and satellite communication system, and a drogue extending 60 cm below the surface, which was hanging beneath the floater on a flexible tether (Novelli et al. 2017). On some floats, the drogue separated from the floater during storms. The detection of drogue loss for the drifters is described in Haza et al. (2018), where a combination of algorithms based on position transmission data and comparison of nearby drifter tracks were used. Here, we apply a more restrictive approach to exclude undrogued drifters by exclusively examining changes in data transmission intervals. The motivation for this is that surface velocity gradient computations based on drifter tracks are very sensitive to velocity differences between nearby drifters. Using both drogued and undrogued drifters leads to anomalously large vorticity and divergence values. When the drifters are without a drogue, they have a tendency to flip over in the water, temporarily placing the antenna of the satellite communication system underneath other components. This leads to detectable increases in transmission intervals (Haza et al. 2018). The default position transmission interval is 5 min for the LASER drifters. Therefore, if the transmission interval averaged over 60 position updates exceeds 6 min the drifter is rejected from further analysis. While this approach may remove some drifters that were still drogued, it works well to avoid noise and anomalous values in the computed surface velocity gradient values. After the undrogued drifter data are removed we apply local polynomial regression [locally weighted scatterplot smoothing (LOWESS)] curve fitting (Cleveland and Devlin 1988) to the raw GPS position drifter tracks to reduce GPS position noise. The span used for the curve fitting is set to be 10 points. The curve fitting also removes the occasional *O*(100) m errors in GPS fixes that occurred a limited number of times due to unknown reasons [see Fig. S2 in the supplemental material of D’Asaro et al. (2018)]. Drifter positions are subsequently interpolated to integer 15-min intervals.

Using the cleaned and interpolated drifter tracks described above, at each time, all possible combinations of three drifters are considered. A triplet of drifters is the smallest cluster from which velocity gradients in two directions can be determined, thus maximizing the number of velocity gradient values that can be applied for statistical analysis.^{6} Every drifter triplet forms a triangle from which we can obtain scale and aspect ratio information. The scale is defined as the root-mean-square (RMS) of the distance between the triangle drifters and the aspect ratio is defined as the ratio of the triangle height to triangle base (an equidistant triangle has an aspect ratio of ^{−5} s^{−1} (^{−9} s^{−2} for velocity gradient variances. These estimates are based on an assumed velocity RMS error of 4 cm s^{−1} at 2-km scales (Ohlmann et al. 2017) and are consistent with the RMS errors found in the model data (appendix E).

## 4. Analysis of simulated fronts and filaments

In this section we test the asymptotic model developed in section 2 against submesoscale-resolving numerical simulations in the northern GoM (section 3a). These numerical simulations are carried out in the same region where the LASER field campaign took place (section 3b) and provide dynamical context for the analysis of drifter measurements that follows (section 5). The general circulation of the GoM consists of a meandering Loop Current to the south, which frequently sheds mesoscale warm-core Loop Current eddies, and active river inflow associated with the Mississippi–Atchafalaya River system (Oey et al. 2005). The corresponding submesoscale currents in the northern part of the basin are affected by the mesoscale strain associated with the Loop Current eddies, the Mississippi–Atchafalaya River system, and are seasonally and spatially variable (Barkan et al. 2017a,b; Choi et al. 2017). This intrinsic dynamical variability makes this region suitable to test our theoretical predictions in a rich mesoscale- and submesoscale-turbulent environment.

### a. Eulerian analysis of model solutions

The asymptotic model [Eqs. (12a)–(12d)] and the inviscid solutions [Eqs. (16) and (18) and appendix C] derived in section 2 are most naturally evaluated in a Lagrangian reference frame. Nevertheless, before we carry out a thorough comparison using Lagrangian particles (section 4b), it is instructive to examine a few Eulerian computed statistical measures. We focus here on the advective frontogenetic tendencies and rates, which are most commonly analyzed in the context of frontogenesis (e.g., Hoskins 1982; Capet et al. 2008b). The analysis of the remaining terms is delayed to the next subsection.

First, we would like to demonstrate the advantage of the frontogenetic rates [*T*; Eqs. (13a)–(13d)] over the frontogenetic tendencies [

The probability density functions (PDFs) of *unbalanced* turbulent processes and secondary instabilities, we do not expect the asymptotic model, which is *balanced* and focuses on inviscid dynamics, to adequately describe frontolysis.

To further assess the validity of Eq. (14) we choose a more stringent measure and examine the equality using scatterplots of

### b. Lagrangian analysis of model solutions

Figures 5 and 6 suggest that our asymptotic theory should be most applicable to strong frontogenetic regions, which in our numerical simulations are most often found during wintertime. This is expected because submesoscale currents are most energetic during winter, when the mixed layer is deepest (Callies et al. 2015). We therefore focus our Lagrangian analysis on the month of February, during which the LASER experiment took place (section 5). Specifically, we aim to carefully quantify how the various terms in the asymptotic model [Eqs. (12a)–(12d)] agree with the corresponding terms in the full equation set [Eqs. (1)–(4)]. To this end 5625 equally spaced Lagrangian particles are seeded at the surface within the region shown in Fig. 1 (rectangular box in Fig. 4, top). The different terms in Eqs. (1)–(4) are interpolated onto the particle locations as they evolve with the horizontal surface velocities, and tracked for 48 h beginning on 3 February. The same seeding and tracking procedure is repeated six more times until 17 February. The time period and seeding location are chosen to match the time and place of the LASER field experiment (section 3b). To quantify the dynamics during submesoscale frontogenesis an algorithm is developed to identify events during which all gradient fields

The different terms in Eqs. (1)–(4) averaged over all of the frontogenetic events are shown in Figs. 7 and 8, where the breakup to two figures is done to facilitate the presentation. The agreement with our asymptotic model for the *O*(10) events (appendix D). In addition, Fig. 7 shows that the increase in all gradient fields (

The

## 5. Measured fronts and filaments

In the previous section we demonstrated a good agreement between the asymptotic model [section 2; Eqs. (12a)–(12d)] and the numerical solutions, during winter. We further identified TTW as the main mechanism responsible for the generation of the surface convergent motions that lead to submesoscale frontogenesis. In this section, we test the applicability of the asymptotic regime to the surface drifter measurements collected during LASER.

The triangle method described in section 3b allows us to compute velocity gradients from drifter observations and to validate the asymptotic model with respect to the

A good agreement is found between the frontogenetic tendencies of divergence and velocity gradient variance

## 6. Vertical buoyancy fluxes during submesoscale frontogenesis

*l*, with a constant lateral buoyancy gradient

*l*and

To quantify the relationship between buoyancy gradient sharpening and APE and to test the scaling [Eq. (22)], we repeat the seven particle seeding experiments described in section 4a at a depth of 10 m, where the vertical velocity does not vanish. Applying the same methodology as in the preceding sections, frontogenetic events are identified and the Eulerian-computed terms in Eq. (1) are interpolated onto the particle positions (appendix D). In addition to the horizontal and vertical advective tendencies

The Lagrangian evolution of

## 7. Summary and discussion

Submesoscale-current turbulence consists of flow structures with spatial scales of *O*(0.1–10) km, temporal scales on the order of hours to days, in a dynamical regime characterized by *O*(1) Rossby and Richardson numbers (Thomas et al. 2008; McWilliams 2016). Phenomenologically, the flow exhibits anisotropic patterns of lines or streaks with large magnitudes of buoyancy and velocity gradients, cyclonic vorticity, and convergence (Fig. 1). The associated velocity power spectral density (PSD) is shallow (

These dynamical, phenomenological, and spectral characteristics motivate us to develop an asymptotic theory that focuses on the formation of the fronts and filaments, that is, the Lagrangian evolution of

Submesoscale resolving numerical simulations and drifter measurements collected during the LASER field campaign confirm the applicability of the asymptotic regime to strong frontogenesis in the northern GoM, during winter (Figs. 7, 10a–c). Furthermore, through careful investigation of all viscid and inviscid terms in the evolution equations of the modeled gradient fields [Eqs. (1)–(4)], we demonstrate that the dominant mechanism responsible for the generation of the ageostrophic convergent motions is boundary layer turbulence (TTW; Gula et al. 2014; McWilliams et al. 2015; Sullivan and McWilliams 2018). In the TTW model the vertical mixing term associated with boundary layer turbulence, which can be substantial, is largely balanced by the pressure and Coriolis terms (Fig. 8), and generates ageostrophic motions that drive frontogenesis. As a result the corresponding Lagrangian evolution of the frontogenetic fields is dominated by the frontogenetic tendencies [

Our asymptotic model and solutions make no assumptions about the underlying physical processes that produce the ageostrophic convergent motions that lead to gradient sharpening (e.g., deformation, mixed layer instabilities, boundary layer turbulence), and it can therefore be regarded as a generic model for submesoscale frontogenesis. If TTW is an important frontogenetic mechanism globally, as it is in the Gulf of Mexico, then it implies that boundary layer turbulence plays a significant role in determining the transport of materials from the mixed layer to the ocean interior.

The clearest limitation of our model is ignoring the onset of frontal instabilities that are most likely the cause of frontal arrest (McWilliams and Molemaker 2011; Sullivan and McWilliams 2018). We also note that our theory ignores the effects of near-inertial motions and internal gravity waves, which might overlap in scales (Callies and Ferrari 2013) and can alter the dynamics at fronts (Whitt and Thomas 2015; Barkan et al. 2017c; Thomas 2017). In addition, the theory does not consider the dynamical effects of surface gravity waves, which has been previously shown to affect frontal evolution (McWilliams and Fox-Kemper 2013; Hamlington et al. 2014; Suzuki et al. 2016; McWilliams 2018). Nevertheless, we argue that it is instructive to first develop a theory for the underlying “balanced,” submesoscale-current dynamics, and leave the incorporation of frontal instabilities and wave interactions to future work.

## Acknowledgments

We thank three anonymous referees for comments that greatly improved the presentation of this work. This work was made possible by a grant from The Gulf of Mexico Research Initiative through the CARTHE consortium. Data are publicly available through the Gulf of Mexico Research Initiative Information and Data Cooperative (GRIIDC) at https://data.gulfresearchinitiative.org (doi:

## APPENDIX A

### Derivation of the Scaled Frontogenetic Tendency Equations

*b*is the buoyancy;

*λ*are described in Eq. (8).

*h*denotes horizontal vector components. The index notation is chosen to concisely write the terms on the right hand side. Because the velocity gradient tensor

## APPENDIX B

### The Asymptotic Model in Coordinate Invariant Form

*ϕ*is a nondimensional velocity potential;

*ψ*is a nondimensional streamfunction; and

*δ*(the coordinate invariant strain rate is

*δ*].

## APPENDIX C

### Solution to Eqs. (12a)–(12d) for Time-Varying Ageostrophic Vorticity

*δ*based on the recursion relation [Eq. (C5)] is plotted in Fig. 3 for different

*a*values (red curves). The solution for the other fields can then be determined by separation of variables because these power series are integrable.

## APPENDIX D

### Closing the Balance and the Selection Algorithm for Frontogenetic Events

#### a. Closing the balance

To close the balances in Eqs. (1)–(4) we carry out the following procedure:

- The different terms in the momentum and tracer equations are computed online in a way that is exactly consistent with ROMS’s discretization, and are saved every 15 min.
- The required derivative operations are applied to the balances computed in step 1 to arrive at Eqs. (1)–(4). We carefully compute the various advection terms because ROMS is written in volume flux form. The
${V}_{\text{mix}}$ terms are computed based on the KPP (Large et al. 1994) mixing profiles and values. The${H}_{\text{diff}}$ terms in ROMS are fourth-order hyperviscous/diffusive and are associated with the advection schemes for tracers and momenta. They are computed as the difference between a third-order upstream-biased and a fourth-order central advection schemes. - Two-dimensional Lagrangian trajectories are computed offline using the surface velocities output at the same fifteen minute intervals. The different terms computed in steps 1 and 2 are then interpolated onto the particle positions. Note that the different Lagrangian tendency terms consist of the interpolated three-dimensional material derivatives. The Lagrangian errors shown in Fig. 7 are computed as the difference between the Lagrangian and Eulerian
${\mathcal{F}}_{\text{hor}}$ terms (${\mathcal{F}}_{\text{hor}}^{\text{Lag}}-{\mathcal{F}}_{\text{hor}}^{\text{Eul}}$ ), where the${\mathcal{F}}_{\text{hor}}^{\text{Eul}}$ terms are computed directly from the model and are interpolated onto the particle locations. The${\mathcal{F}}_{\text{hor}}^{\text{Lag}}$ terms are computed as$\begin{array}{l}{\mathcal{F}}_{\text{hor}}^{{\text{Lag}}_{\delta}}={\displaystyle \frac{\partial}{\partial \tau}}\hspace{0.17em}\delta -{\nabla}_{h}\cdot {\displaystyle \frac{{D}_{{h}_{\text{adv}}}}{Dt}}{\mathbf{u}}_{h},\\ {\mathcal{F}}_{\text{hor}}^{{\text{Lag}}_{\zeta}}={\displaystyle \frac{\partial}{\partial \tau}}\hspace{0.17em}\zeta -{\nabla}_{h}\times {\displaystyle \frac{{D}_{{h}_{\text{adv}}}}{Dt}}{\mathbf{u}}_{h},\end{array}$

*τ*-differentiated following the Lagrangian trajectories. The second term on the right-hand-side corresponds to the divergence/curl of the Eulerian-computed horizontal material derivative (subscript

*h*) in advective form (subscript adv) that is interpolated onto the particle locations. Because any particle advection code is subject to the discretization errors of the modeled Eulerian velocities, a Lagrangian advection error is always expected. As shown in Fig. 7, these errors are largest at the final stage of frontogenesis when the gradient fields are largest and, hence, have spatial scales closest to the grid resolution.

#### b. Selection algorithm

The algorithm used to select for frontogenetic events is based on the gradient fields

Figure D1 illustrates the events selected in the model solutions using the above algorithm for seven particles (gray curves) that collected onto the same submesoscale feature. Although the variability among the particles is considerable, their mean evolution agrees well with the theoretical prediction [Eq. (16d)]. A similar agreement was found for other submesoscale structures as long as we averaged over *O*(10) particles (not shown).

## APPENDIX E

### Error Biases in the Triangle-Based Gradient Computation Method

To test the ability of the triangle method (section 3b) to quantify dominant balances during frontogenetic events we apply it to one of our particle release experiments in the model. Because the modeled particles have no position error and the subgrid-scale energy is small, it allows us to extend the triangle aspect ratios to

Finally, we compute the undersampling errors of the triangle method in the model based on the RMS differences between the triangle-based gradient computation and the “true” Eulerian values. These come out to be

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^{1}

Note that the Garrett–Munk (Garrett and Munk 1972) internal wave continuum spectral slope is also close to

^{2}

For quasigeostrophic flow, which is horizontally nondivergent,

^{3}

Choosing another limit would mean that the

^{4}

In this scaling regime

^{5}

When

^{6}

Increasing the number of drifters in a cluster will reduce the error in the computed gradients, but only at a rate proportional to the square root of the drifter number (Ohlmann et al. 2017).