# Near-inertial echoes of ageostrophic instability in submesoscale filaments

Erin Atkinson1<EMAIL_ADDRESS>James C. McWilliams2 and
Nicolas Grisouard1  1Department of Physics, University of Toronto, 60 St.
George Street, Toronto, Ontario, Canada M5S 1A72Department of Atmospheric and
Oceanic Sciences, University of California, Los Angeles, Los Angeles, CA, USA

###### Abstract

Ocean submesoscales, flows with characteristic size around 10m–10km, are
transitional between the larger, rotationally-constrained mesoscale and three-
dimensional turbulence. In this paper we present simulations of a submesoscale
ocean filament. The strong vertical transport associated with submesoscale
fronts and filaments ventilates the deeper water and provides a route for the
observed forward cascade of energy in the ocean. In our case, the filament is
strongly sheared in both vertical and cross-filament directions and is
unstable. Instability indeed dominates the early behaviour with a fast
extraction of kinetic energy from the vertically sheared thermal wind. The
filament is expected to be subject to instabilities, including symmetric and
Kelvin-Helmholtz instabilities. An instability emerges with a circulation that
does not match either, though symmetric modes become apparent later. The
action of the instability is sufficiently rapid that the filament does not
respond in a geostrophically balanced sense. Instead, it later exhibits
vertically sheared near-inertial oscillations with higher amplitude as the
initial minimum Richardson number decreases. Horizontal gradients strengthen
only briefly as the fronts restratify. These unstable filaments can be
generated by strong mixing events at pre-existing stable structures. The
oscillations must also be considered when designing initial conditions for
numerical experiments of submesoscale flows, where the initial state is
intended to be balanced.

##  1 Introduction

The submesoscale ocean is dominated by highly horizontally anisotropic
features. Fronts — confined lateral changes in density — remain balanced by
the Coriolis acceleration of an along-front jet. In the open ocean, these
features are created primarily at the edges of mesoscale vortices such as
those created by western boundary currents. As they are drawn out by the
strain flow in these regions, horizontal gradients in velocities and tracers
become larger, eventually leading to the onset of a new dynamical regime,
where geostrophic and hydrostatic balances no longer describe the flow well
(Taylor & Thompson, 2022; Barkan et al., 2019).

The strain flow and turbulence drive an ageostrophic secondary circulation
around the front (McWilliams, 2021; Sullivan & McWilliams, 2018). The
resulting vertical velocities provide a route for transport of tracers,
momentum and energy between the surface layer and deep water (Klein et al.,
2019; Verma et al., 2019). Particularly, the submesoscale is a link between
the larger scale, horizontally non-divergent flows and smaller scale
turbulence that facilitates the observed forward cascade of energy in the
ocean (Srinivasan et al., 2023).

In the absence of turbulence, horizontal gradients will approach infinity in
finite time when forced by a strain flow (Hoskins & Bretherton, 1972). For a
realistic front, some effective horizontal diffusion due to turbulence will
arrest the frontogenesis before this singularity is reached, though this
process is not well-understood. The turbulence, namely the vertical momentum
mixing induced by surface cooling or wind stress, acts on the vertically
sheared thermal wind velocity. If only these processes act in the submesoscale
mixed layer, the expected momentum balance is the turbulent thermal wind
balance between the Coriolis acceleration of an along-front jet, the
horizontal pressure gradient and vertical momentum mixing (Gula et al., 2014;
Bodner et al., 2023).

This study investigates the evolution of a dense filament resembling the
configuration of Sullivan & McWilliams (2018), in the initially low-turbulence
regime. A filament is a pair of opposing, interacting fronts. The vertical
vorticity of a dense filament in thermal wind balance is large and positive in
the centre between two smaller magnitude negative vorticity regions.

Of particular interest in this study is symmetric instability (SI). A
symmetrically unstable flow is unstable to perturbations that exchange fluid
parcels along isopycnals, with no variation (hence symmetric; Stone, 1966)
along the direction of mean flow. The instability manifests as highly
anisotropic rolls, along the direction of the frontal jet, with major axis
aligned with isopycnals. SI extracts kinetic energy from the thermal wind
velocity shear (Taylor & Thompson, 2022; Haine & Marshall, 1998).

For a Boussinesq fluid with equal momentum and buoyancy diffusivities, the
stability of a flow to symmetric perturbations can be diagnosed by the sign of
Ertel potential vorticity $q$. Fluid unstable to SI has
$fq=f(f\hat{\boldsymbol{z}}+\nabla\times\boldsymbol{u})\cdot\nabla b<0,$ where
$\boldsymbol{u}=u\hat{\boldsymbol{x}}+v\hat{\boldsymbol{y}}+w\hat{\boldsymbol{z}}$
and $b$ are velocity and buoyancy fields respectively and $f$ is the Coriolis
frequency (Emanuel, 1979). SI competes with gravitational, inertial,
baroclinic and Kelvin-Helmholtz instabilities, and dominates in regions where
the Richardson number of the flow lies in the range $0.25<Ri<0.95$ (Stone,
1966; Thomas et al., 2013). The small size and strong baroclinicity of
submesoscale fronts often satisfies the condition for SI (Thomas et al., 2013;
Gula et al., 2014).

In this paper, we present a set of simulations, unstable to various
instabilities owing to the low Richardson number ($Ri\leq 0.2$). Our results
focus on the spin-down of the fronts and their subsequent instability. We show
that instabilities act quickly to remove kinetic energy and excite near-
inertial oscillations within the fronts that briefly strengthen horizontal
buoyancy gradients at the surface. Finally, we argue that numerical studies of
similar features must be designed with this instability and subsequent
oscillations in mind, or risk drawing conclusions that may not be
generalisable to balanced flows in the real ocean.

##  2 Model

The simulations presented in this paper are of a set of buoyancy filaments in
a rotating, Boussinesq fluid, namely,

$\frac{\text{D}\boldsymbol{u}}{\text{D}t}+f\boldsymbol{\hat{z}}\times\boldsymbol{u}=-\nabla\phi+b\boldsymbol{\hat{z}}+\nabla\boldsymbol{\cdot}(\nu\nabla\boldsymbol{u})+\boldsymbol{F}_{\boldsymbol{u}},$
$None$ $\frac{\text{D}b}{\text{D}t}=\nabla\boldsymbol{\cdot}(\kappa\nabla
b)+F_{b},\quad\text{and}\quad\nabla\boldsymbol{\cdot}\boldsymbol{u}=0,$ $None$

with geopotential $\phi$, kinematic eddy viscosity $\nu$, and buoyancy
diffusivity $\kappa$. The terms $\boldsymbol{F_{u}}$ and $F_{b}$ are velocity
and buoyancy forcing terms. We elaborate on these quantities below. The
equation set ($None$) is integrated using Oceananigans, a Julia package for
simulation of incompressible fluid flows in an oceanic context (Ramadhan et
al., 2020).

Figure 1: a) Schematic of the simulation setup and an example initial
condition (not to scale). Black contours are isopycnals and the heatmap shows
the form of the balanced thermal wind velocity. The vertical grid size
smoothly transitions between two uniform values, such that about 96 grid cells
are in the mixed layer, and 32 grid cells comprise the remainder of the
vertical domain (not shown). b) The local Richardson number for the reference
state with $Ri_{\text{min}}=0.1$. Fluid inside the red, dashed contour has
$fq<0$.

A schematic of the simulation setup is shown in figure 1, and details of the
initial condition are in Appendix A. The grid size is
$(N_{x},N_{y},N_{z})=(1024,1024,128)$ and the simulation domain is
$(x,y)\in[-5L,5L)^{2}$, where $L$ is a measure of the initial surface
separation between the centres of the two counter-flowing jets, and
$z\in[-2.5H,0]$, where $H$ is the initial depth of the mixed layer. The grid
spacing is uniform in the horizontal directions. The vertical grid spacing
smoothly transitions between two uniform values at the pycnocline, such that
approximately $3/4$ of the cells are in the surface mixed layer ($z>-H$),
while the remainder are in the deep water. Quantities are non-dimensionalised
such that $f=L=1$, although we sometimes keep the notations $f$ and $L$ for
physical clarity.

The domain is periodic in both horizontal directions. In the vertical,
boundary conditions are no-flux — the filament is not forced by wind stress or
surface cooling, namely,

$u_{z}=v_{z}=w=b_{z}=0\quad\text{at}\quad z=\\{-2.5H,0\\},$

where $\phi_{i}$ denotes differentiation of $\phi$ with respect to coordinate
$i$.

To prevent internal wave reflections from the bottom boundary, a sponge layer
forcing $\boldsymbol{F}_{\boldsymbol{u}},F_{b}$ is included for all fields.
The fields are relaxed towards the reference state
($\boldsymbol{u}_{0},b_{0}$) at a rate that decays quadratically away from the
bottom boundary, namely,

$\boldsymbol{F}_{\boldsymbol{u}}=-\sigma(z)(\boldsymbol{u}-\boldsymbol{u}_{0})\quad\text{and}\quad
F_{b}=-\sigma(z)(b-b_{0})$ $None$
$\text{with}\quad\sigma(z)=\frac{1}{2}\frac{N_{0}}{2\pi}\times\left\\{\begin{array}[]{ll}(2z/H+4)^{2},&z\leq-2H,\\\
0,&z>-2H.\end{array}\right.$ (3)

The isotropic eddy viscosity $\nu$ is parameterised as a Smagorinsky-Lilly
turbulence closure with a Smagorinsky constant $C=0.16$ (Smagorinsky, 1963;
Lilly, 1967), and we set $\kappa=\nu$. The adaptive time-stepping routine aims
to achieve a maximum Courant-Friedrichs-Levy number of 0.15.

##  3 Experiments

The reference state of the simulation is a buoyancy profile $b_{0}(x,z)$, and
a jet $\boldsymbol{u}_{0}=v_{0}(x,z)\boldsymbol{\hat{y}}$ in thermal wind
balance, namely,

$v_{0}=\frac{1}{f}\int_{-\infty}^{z}\frac{\partial b_{0}}{\partial
x}\;\text{d}z.$

The reference buoyancy contains a weakly-stratified surface layer above a
strongly-stratified fluid. The height of the transition between these two
stratifications varies, providing the horizontal buoyancy gradient that makes
up the filament. The exact definition of the functional forms of $v_{0}$ and
$b_{0}$ are given in appendix A.

The reference state is invariant in the down-front direction $y$ and the
initial evolution introduces little variation along this direction for the
large-scale velocity and buoyancy fields (i.e. the filament does not meander).
As such, we define the mean state to simply be the down-front average of the
full 3D state, with averaging operator

$\overline{\phi}(x,z,t):=\frac{1}{10L}\int_{-5L}^{5L}\phi(x,y,z,t)\;\text{d}y.$
(4)

The primary non-dimensional numbers that determine the dynamical regime of a
submesoscale mixed layer flow are the Rossby, Richardson and (turbulent) Ekman
numbers (Bodner et al., 2023). For the base state, the local values of each
are, respectively,

$\displaystyle Ro=\frac{1}{f}\frac{\partial v_{0}}{\partial x},\quad
Ri=\frac{\partial b_{0}/\partial z}{(\partial v_{0}/\partial
z)^{2}}\quad\text{and}\quad
Ek=\frac{\partial\overline{w^{\prime}v^{\prime}}/\partial z}{v_{0}f}.$ (5)

With no surface forcing, the filament is initially in a low-turbulence
($Ek\sim 0$) regime. We parametrise our initial conditions by extremal values
of the local Rossby and Richardson numbers as follows.

This study is concerned with the submesoscale ocean, where advection and the
Coriolis acceleration are comparable in magnitude: $Ro\sim 1$. To limit the
number of possible instabilities, the simulations have minimum vertical
vorticity above $-f$, which excludes inertial instability. We use
$Ro_{\text{min}}=-0.8$ here. Given the prescribed filament shape, this results
in a maximum $Ro_{\text{max}}=1.9$. For typical horizontal velocities in the
ocean this corresponds to a jet width of order $1-10~{}\text{km}$ at mid-
latitudes.

The submesoscale mixed layer often has $Ri\lesssim 1$. For the filament shape
used here, the Richardson number is approximately constant with depth in the
mixed layer, and attains a minimum value in the centre of each front. We
present the results of a set of low-Ri filaments, with a range of minimum
Richardson numbers: $Ri_{\text{min}}\in\\{0.0,0.1,0.2\\}$.

The remainder of the non-dimensional parameters, as well as the relationship
between $Ro_{\text{min}}$, $Ri_{\text{min}}$ and dimensional parameters, is
given in appendix A.

Figure 1b shows the local $Ri$ of the reference state, and the region with
$fq<0$ at the outer edges of the filament.

Before we impose the reference state, there is a brief initialisation process.
We pre-initialise the simulation at $t=-1$ with a state identical to the
reference state at $|x|\rightarrow\infty$. We then add noise to all velocities
and run the simulation until $t=0$. This produces fields
$\boldsymbol{u}_{\text{init}}=\overline{\boldsymbol{u}}_{\text{init}}+\boldsymbol{u}_{\text{init}}^{\prime}$
and $b_{\text{init}}=\overline{b}_{\text{init}}+b_{\text{init}}^{\prime}$. The
initial state of the filament phase of the simulation is taken to be the
reference state plus the fluctuation terms from the initialisation phase,
namely,

$\boldsymbol{u}(t=0)=v_{0}\hat{\boldsymbol{y}}+\boldsymbol{u}_{\text{init}}^{\prime}\quad\text{and}\quad
b(t=0)=b_{0}+b_{\text{init}}^{\prime}.$

The mean, balanced state provides the energy source for instabilities with the
nature of the instability determining the dominant energy source. Energy
transfer from fluctuations to the mean kinetic energy of a laterally or
vertically sheared velocity profile is given by the lateral or vertical shear
production respectively

$\text{LSP}=\overline{\boldsymbol{u}}_{x}\boldsymbol{\cdot}\overline{u^{\prime}\boldsymbol{u}^{\prime}},\quad\text{VSP}=\overline{\boldsymbol{u}}_{z}\boldsymbol{\cdot}\overline{w^{\prime}\boldsymbol{u}^{\prime}}.$
(6)

Mean potential energy is transferred via the buoyancy flux,

$\text{BFLUX}=-\overline{w^{\prime}b^{\prime}}.$ (7)

##  4 Results

Figure 2: Evolution of the mean state for the simulation with
$Ri_{\text{min}}=0.1$. Three snapshots are shown at
$ft/2\pi=\\{0.8,1.5,2.2\\}$. The heatmap shows $\overline{v}$. Black contours
are mean isopyncals $\overline{b}$. Orange (purple) contours are
(anti-)clockwise streamlines.

###  4.1 General evolution

The reference filament state does not satisfy the boundary conditions and
there is an initial adjustment phase that creates an Ekman layer at the
surface. For the first 2–3 inertial periods, the filament remains
approximately uniform in the down-front direction. Towards the end of our
simulations, and especially visible at the surface, the isopycnals begin to
meander due to baroclinic instability. The domain is not large enough to
support many baroclinic modes; following Stone (1966), the fastest-growing
baroclinic mode wavelength $\lambda$ can be estimated for an Eady model with
parameters comparable to the centre of a jet

$\lambda=\frac{2\pi v_{0}}{f}\sqrt{\frac{1+Ri_{\min{}}}{5/2}}\approx 5,$ (8)

which is half the downfront length of our domain. We ran a verification
simulation with a longer down-front domain that resolved many wavelengths of
this instability (not shown), and found that the baroclinic instability
develops too slowly to be significant in the time period of interest in this
paper. The rest of this discussion is concerned with the evolution of the
filament before the meandering and shortly after the adjustment at the
surface, which can be understood from the perspective of the down-front mean
state.

Figure 2 outlines the evolution of the mean state during the first inertial
periods. Shortly after the filament is allowed to evolve freely, a strong
secondary circulation on the scale of the fronts emerges (figure 2a). The
secondary circulation acts to restratify the fronts. At the surface, the
circulation increases front strength: horizontal convergence pushes isopycnals
together immediately under the surface (figure 2b), before the trend reverses
somewhat and fronts weaken (figure 2c). The maximum horizontal buoyancy
gradient does not occur in the centre of the filament, but at approximately
$x=\pm 0.4$. The evolution is contained in the region with negative potential
vorticity and the fronts behave somewhat independently, rather than as a
filament.

This happens within less than an inertial period and therefore, the frontal
strengthening and arrest mechanisms do not appear to be balanced at some
maximum front strength. This is very suggestive of a damped oscillation within
the filament.

###  4.2 Initial evolution is a fast instability

In this section, we show that the dynamics in the first inertial period
contains a low-$q$ instability of the vertical shear. The form of the
instability consists of thin rolls aligned with isopyncals in vertical
($x$-$z$) slices that extract energy from the mean vertical shear via a
negative VSP. However, there is significant down-front variation of the flow.

Figure 3: Terms extracting energy from the mean state for each of the
simulations integrated over the mixed layer. Figure 4: At $t/2\pi=0.25$,
comparison of an $x$-$z$ slice at $y=0$ of a) vertical velocity $w$ and b)
vertical shear production in the left side of the filament for the simulation
with $Ri_{\text{min}}=0.1$. The red, dashed line in the top left panel locates
$z=-0.05$, namely, the depth of the $x$-$y$ slice of vertical velocity
displayed in c). The horizontal slice of $w$ is coarse-grained with a Gaussian
filter with a half-width of one grid cell for clarity. A video timeseries of
this plot appears in the supplementary material as Movie 1. Figure 5: As in
figure 4 but at $t/2\pi=0.5$. Note the difference in vertical velocity and
shear production scales compared to figure 4.

Figure 3 shows an initial sharp peak in the VSP, greater for the simulations
with smaller minimum Ri, before $t/2\pi\lessapprox 0.8$. Other sources of
turbulent kinetic energy are negligible in comparison.

The instantaneous vertical velocity fields show the form of the instability.
Figures 4 and 5 show the VSP and horizontal ($x$-$y$) and vertical ($x$-$z$)
slices of the vertical velocity $w$ at $t/2\pi=0.25$ and $0.5$. In the
vertical slices, the velocity field consists of thin cells, aligned with
isopycnals. A horizontal slice through the emerging cells (figure 4c) reveals
that they are not aligned along the down-front direction, as would be expected
of SI. The cells appear rotated away from the symmetric axis. Later (figure 5)
cells emerge that appear to align with the filament and resemble an expected
SI circulation. Videos of the vertical velocity and VSP as shown in figures 4
and 5 are presented in the supplementary material.

Once the instability saturates and gives way to turbulence, there is a large
production of turbulent kinetic energy via negative VSP. The saturated regions
in figure 4a,b above $z=-0.05$ are associated with a larger VSP signature. The
maximum in VSP moves down over time, and figure 5c shows the horizontal state
as the coherent cells are destroyed by secondary instabilities and the
turbulence begins to isotropise in the $x$-$y$ plane. The extraction of energy
is fast, and can be seen in figure 3 to grow, peak and subside within the
first inertial period. The consequence is that the filament is now out of
geostrophic balance, as shown in the following section.

###  4.3 The initial instability echoes throughout the front as an inertial
oscillation

Following the onset of instability, the evolution of the filament is not
balanced. Indeed, the process in the previous section happens within less than
an inertial period, and in the resulting evolution the front is not stabilised
at some minimum width. We highlight the consequences of the short-lived
turbulent energy production production using the linearised Sawyer-Eliassen
equation for the mean secondary circulation streamfunction $\psi$ forced by
by-products of turbulent vertical fluxes as we measure them in our
simulations. That is, we use

$\nabla^{2}\psi_{tt}+\underbrace{f^{2}\psi_{zz}-J(fv_{0},\psi_{z})+J(b_{0},\psi_{x})}_{L\psi}=\underbrace{f\overline{w^{\prime}v^{\prime}}_{zz}-\overline{w^{\prime}b^{\prime}}_{xz}}_{G},$
(9)

in which we have neglected $O(\psi^{2}$) terms, and where

$(\overline{u},0,\overline{v})=(-\psi_{z},0,\psi_{x}),\quad\text{and}\quad
J(a,b)=a_{z}b_{x}-b_{z}a_{x}.$ $None$

In doing so, we neglect terms due to the sub-grid momentum and buoyancy
fluxes, as well as the remainder of the turbulent flux terms. The turbulent
velocity flux $\overline{w^{\prime}v^{\prime}}$ induces a forcing on $\psi$.
Integrating this equation over the left half of the interior mixed layer gives
a leading-order evolution equation for the circulation around the
corresponding front $C_{\text{IML}}=-\iint\nabla^{2}\psi\text{d}x\text{d}y$,
namely,

$\frac{\text{d}^{2}C_{\text{IML}}}{\text{d}t^{2}}=\int_{-0.9H}^{-0.1H}\text{d}z\int^{0}_{-L_{x}/2}\text{d}x\Big{[}f^{2}\psi_{zz}-J(fv_{0},\psi_{z})+J(b_{0},\psi_{x})-f\overline{w^{\prime}v^{\prime}}_{zz}+\overline{w^{\prime}b^{\prime}}_{xz}\Big{]}.$
(11)

The frontal circulation appears dominated by decaying near-inertial
oscillations. The peak in VSP and thus $\overline{w^{\prime}v^{\prime}}$ is
large, and occurs in less than one inertial period, before significant change
in the mean state; this suggests that the subsequent behaviour will be
unbalanced and lead to oscillations. A comparison of the terms in equation
(11) is shown for each of the simulations in figure 6. The circulation shows
clear decaying oscillatory behaviour with period $2\pi$, with linear terms
dominating for small $Ri_{\min{}}$. Frontal arrest is not necessarily due to
horizontal diffusion, but rather the restoring phase of the oscillation. The
strength of these oscillations is dependent on the Richardson number of the
flow: a smaller $Ri_{\min{}}$ produces a faster and stronger extraction of
kinetic energy and in turn leads to higher amplitude oscillations.

Figure 6: The circulation around the left half of the interior mixed layer as
in equation (11). The black line is the actual evolution of the circulation.
The blue line is the contribution of the linear terms, the orange line is the
contribution of the by-products of turbulent flux terms, and the green line is
their sum. $L\psi$ and $G$ are defined as in equation (9)

##  5 Discussion and conclusions

We investigated the evolution of low-Richardson number mixed layer filaments
with numerical simulations. A low-$q$ instability quickly extracts energy from
the vertical shear. This puts the filament in an unbalanced state, and results
in vertically sheared inertial oscillations. The oscillations are well-
described by the linear terms in the Sawyer-Eliassen equation for the down-
filament streamfunction, following a forcing due to the vertical turbulent
flux of down-front momentum.

Vertical mixing caused by fast-acting instability in a front is capable of
producing secondary circulations superficially similar to those produced when
a front is forced by a strain flow or turbulence. Rather than solely
restratifying the mixed layer, the instability at least initially strengthens
surface buoyancy gradients.

We believe that this initial circulation is at least partially responsible for
the filamentogenesis observed by Sullivan & McWilliams (2018). The authors
integrate up to slightly past maximum front strength, so would not see
inertial oscillations resulting from an instability-induced imbalance. Here we
run smaller, yet still substantial, simulations of a similar setup and observe
a similar behaviour up to that point in time, without any forcing by surface
cooling or winds.

Wienkers et al. (2021) observe a similar instability-to-inertial-oscillations
pathway that begins with a strong symmetric instability in 2.5D direct
numerical simulations of finite-width, zero Richardson number frontal zones
(i.e., constant lateral buoyancy gradients in domains that are periodic in the
cross-front direction). The behaviour of our filament with $Ri_{\min{}}=0$ is
remarkably similar. For instance, the leftmost plot of figure 3 shows
reoccuring creation of turbulent kinetic energy via negative VSP associated
with the steepest isopycnals at each phase of the inertial oscillation, much
like figure 4 in the above reference. Our simulations are three-dimensional
and reveal that, while it shares features of SI, the instabilities that
eventually lead to inertial oscillations have significant down-front
variation, at least initially.

Submesoscale structures in a realistic, time-varying ocean can show inertial
oscillations associated with the diurnal variation of vertical diffusivity as
shown by Dauhajre & McWilliams (2018). We demonstrate here that the changes in
effective diffusivity need not come from changes in surface forcing, but can
arise without forcing, from turbulent kinetic energy production due to
submesoscale instabilities acting on a front.

Frontal features created in the prototypical sense, via a confluent strain
flow, cross a range of scales before they reach the regime of the filament
presented here. We believe that this work indicates there is understanding to
be gained by spinning up fronts in-situ for idealised numerical studies of
turbulence, rather than beginning with what we think a submesoscale front
ought to look like. In particular, the transition from $Ro\ll 1$ to $Ro\sim 1$
frontogenesis as these scales remain poorly resolved by global ocean models.

Submesoscale fronts and filaments are the sites of various instabilities, the
consequences on the global ocean and best parametrisation of which remains an
open question. Here we have shown that instability alone is capable of
strengthening surface buoyancy gradients in a mixed layer filament via a
multi-stage process in which the initial instability acts quickly, puts the
fronts out of balance and the resulting adjustment begins with a convergent
secondary circulation on the scale of the fronts that strengthen them near the
surface. For our experiments, frontal arrest is not necessarily due to
turbulence and can be described simply as the restoring phase of an inertial
oscillation. The instability in question shares characteristics with SI, in
agreement with previous studies, but the initial emergence has significant
down-front variation, before symmetric modes appear later. We also comment on
the consequences for studies of submesoscale flows, and suggest that the most
fruitful way to investigate how these structures evolve in the real ocean,
from a computational perspective, may be to grow, rather than impose,
analogues.

[Acknowledgements]The authors ackowledge fruitful discussions with William R.
Young, Leif N. Thomas, John Taylor and Matthew N. Crowe. Computations were
performed on the Mist supercomputer at the SciNet HPC Consortium (Loken et
al., 2010; Ponce et al., 2019). SciNet is funded by Innovation, Science and
Economic Development Canada; the Digital Research Alliance of Canada; the
Ontario Research Fund: Research Excellence; and the University of Toronto.
[Funding]We acknowledge the support of the Natural Sciences and Engineering
Research Council of Canada (NSERC), [funding reference number
RGPIN-2022-04560]. [Declaration of interests]The authors report no conflict of
interest [Data availability statement]Replication code for this manuscript can
be found at https://doi.org/10.5281/zenodo.11997619 (Atkinson, 2024) [Author
ORCIDs]E. Atkinson, https://orcid.org/0000-0002-5913-3137; J.C. McWilliams,
https://orcid.org/0000-0002-1237-5008; N. Grisouard,
https://orcid.org/0000-0003-4045-2143

##  Appendix A Parametrisation of the reference state

The reference state of the filament is assumed to be independent of the down-
filament direction and described by a buoyancy profile

$b_{0}(x,z)=N_{0}^{2}z+(N^{2}-N_{0}^{2})\lambda
H\;g\\!\left(\frac{z-h(x)}{\lambda H}\right),$ (12)

with a jet $v_{0}(x,z)$ in thermal wind balance ($f{\partial v_{0}}/{\partial
z}={\partial b_{0}}/{\partial x}$), with $v_{0}\rightarrow 0$ as
$z\rightarrow-\infty$. The above equations are defined in terms of a filament
shape function

$h(x)=H\gamma\left(\frac{x}{\ell}\right)\quad\text{with}\quad\gamma(s)=-1+\frac{\delta}{2}\left[\text{erf}\left(s+\frac{1}{2\alpha}\right)-\text{erf}\left(s-\frac{1}{2\alpha}\right)\right].$
$None$

and mixed layer transition function $g(\xi)=\ln(1+e^{\xi})$. $N_{0}$ and $N$
are deep-water and mixed layer buoyancy frequencies respectively, $f$ is the
Coriolis frequency, $L$ is the distance between front centres, $H$ is the
depth of the mixed layer and $\ell$ is the frontal half-width. We parametrise
the state by varying $N$ and $N_{0}$ to achieve prescribed minimum local
Rossby and Richardson numbers, given respectively by

$Ro_{\text{min}}=\frac{(N^{2}-N_{0}^{2})\beta^{2}}{f^{2}\alpha^{2}}\max_{s}\Big{(}\gamma\gamma_{s}\Big{)}\quad\text{and}\quad
Ri_{\text{min}}=\frac{f^{2}\alpha^{2}N^{2}}{\beta^{2}(N^{2}-N_{0}^{2})^{2}}\min_{s}\Big{(}\gamma_{s}^{-2}\Big{)}.$
$None$

The full set of input parameters is shown in table 1.

Front width-separation ratio | $\alpha=\ell/L$ | 1.5
---|---|---
Mixed layer aspect ratio | $\beta=H/L$ | 0.1
Mixed layer depth change | $\delta$ | -0.25
Mixed layer transition width | $\lambda$ | 0.025
Minimum Rossby number | $Ro_{\text{min}}$ | -0.8
Minimum Richardson number | $Ri_{\text{min}}$ | $\in\\{0,0.1,0.2\\}$
Table 1: Filament parameters

## References

  * Atkinson (2024) Atkinson 2024 Filament instability and near inertial echoes - setup and analysis. https://doi.org/10.5281/zenodo.11997619.
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