# A conservation theorem for the $f$ plane

###### Abstract

Ertel’s potential vorticity theorem is essentially a clever combination of two
conservation principles. The result is a conserved scalar $q$ that accurately
reflects possible vorticity values that fluid parcels can possess and acts as
a tracer for fluid flow. While true at large scales in the ocean and
atmosphere, at increasingly smaller scales and in sharply curved fronts, its
accuracy breaks down. This is because Earth’s rotation imparts angular
momentum to fluid parcels and the conservation of absolute angular momentum
$L$ restricts the range of centripetal accelerations possible in balanced
flow; this correspondingly restricts vorticity. To address this discrepancy,
we revisit Ertel’s original derivation and obtain a new conserved scalar $Lq$
that more properly reflects the behavior of fluid parcels at these small
horizontal scales. Application of the theorem is briefly discussed, with an
emphasis on better understanding oceanic submesoscale and polar mesoscale
flows.

##  1 Introduction

Ocean dynamics at small-scales fronts have garnered considerable attention in
recent years. This attention has been evident in both observational and
modelling portions of the community. New advancements in observing
systems–including those from floats (D’Asaro et al., 2011), gliders (Thompson
et al., 2016; du Plessis et al., 2019), and long-range surface vehicles such
as SailDrones (Gentemann et al., 2020)–have increased our capability to
resolve small-scale phenomena. The result is that velocity and density
gradients at horizontal scales between $1$ and $10$ km–previously only
inferred from spacecraft (e.g. Flament et al., 1985; Scully-Power, 1986; Munk
et al., 2000) and long-term moored measurements (e.g. Bane et al., 1989; Lilly
and Rhines, 2002; Buckingham et al., 2016)–are now becoming resolved in
targeted studies (e.g. Thomas and Lee, 2005; D’Asaro et al., 2011; Thomas et
al., 2013; Adams et al., 2017; Naveira Garabato et al., 2019). At the same
time, computational resources have increased at an exponential rate,
permitting scientists the ability to realistically simulate dynamics at these
fine scales. At present, numerical models are capable of providing realistic
ocean simulations for the globe at horizontal resolutions approaching $1$ km
(https://data.nas.nasa.gov/ecco/data.php). Within nested regional
configurations, horizontal grid resolutions of $100$ m are possible (Onken et
al., 2020), with the result that oceanic phenomena with $e$-folding scales of
several hundred meters can be resolved.

Oceanic flows at these small spatial scales are commonly referred to as
submesoscale processes (Thomas et al., 2008; McWilliams, 2016) in order to
distinguish them from larger-scale counter-parts, referred to as mesoscale
processes. At mid-latitudes, these terms correspond to horizontal scales
smaller than $10$ km (submesoscale) and larger than $30$ km (mesoscale), where
the transition between these scales is roughly defined by the first-mode,
baroclinic deformation radius $R_{d}$ (Chelton et al., 1998; Smith, 2007).
However, at high latitudes, $R_{d}$ approaches $1$-$10$ km (Timmermans et al.,
2008; Nurser and Bacon, 2014) such that assigning absolute scales to these
phenomena is problematic. In the ocean community, this has motivated a
dynamical definition for this class of fluid motion.

###  1.1 Dynamical definition of the oceanic submesoscale

Processes within the oceanic submesoscale regime are dynamically characterized
by relative vorticity ${\zeta=(\nabla\times\mathbf{u})\cdot\mathbf{\hat{k}}}$
values that rival the vertical component of Earth’s vorticity
$f=2\mathbf{\Omega}\cdot\mathbf{\hat{k}}$ (Thomas et al., 2008; McWilliams,
2016). Here, $\mathbf{u}$ is velocity, $2\mathbf{\Omega}$ is planetary
vorticity, and $\mathbf{\hat{k}}$ is the vertical unit vector. This contrasts
with that found for mesoscale processes, where $f$ is typically an order of
magnitude greater than relative vorticity, corresponding to quasi-geostrophic
(QG) flow.

Additionally, the submesoscale regime is typically characterized by enhanced
vertical shears or, equivalently, as a consequence of thermal wind balance
(TWB)
${\partial_{z}\mathbf{u}_{h}=\frac{1}{f}\mathbf{\hat{k}}\times\nabla_{h}b}$,
enhanced horizontal buoyancy gradients. In Cartesian coordinates oriented
relative to the front, we write this balance as
$f\partial_{z}\overline{v}=\partial_{x}\overline{b}$, where
${M^{2}=\partial_{x}\overline{b}}$ denotes the mean cross-frontal buoyancy
gradient. In these expressions, $\mathbf{u_{h}}$ is the horizontal velocity,
$\overline{v}$ denotes the mean velocity in the along-front direction,
$\partial_{z}\overline{v}$ is the mean vertical shear, $b=-g\rho/\rho_{o}$
denotes buoyancy ($g$ is gravity, $\rho$ is density, and $\rho_{o}$ is a
reference density), and $x$ and $y$ are cross-front and along-front
coordinates, respectively.

Finally, within boundary layers such as the ocean surface or bottom boundary
layers, vertical stratification $N^{2}=\partial_{z}\overline{b}$ is often
reduced to an extent that fluid parcels within the submesoscale dynamical
regime have a greater propensity for vertical motion. These two
characteristics are succinctly quantified by the gradient Rossby number
(${\mathrm{Ro}=\overline{\zeta}/f}$) and gradient Richardson number
(${\mathrm{Ri}=N^{2}/|\partial_{z}\overline{v}|^{2}}$), both of which have
values approaching $1.0$ within the oceanic submesoscale regime (Thomas et
al., 2008; McWilliams, 2016). In contrast, processes within the quasi-
geostrophic regime are characterized by reduced relative vorticity
($\mathrm{Ro}\ll 1$) and elevated stratification and/or reduced vertical shear
($\mathrm{Ri}\gg 1$).

###  1.2 Accounting for centripetal accelerations

It is common to assume that the mean flow within fronts is in approximate
geostrophic and hydrostatic balance–i.e. TWB. Additionally, the effect of
viscous forces have been considered (McWilliams, 2016). These are reasonable
approximations for density fronts with horizontal scales larger than $R_{d}$
(Pedlosky, 1987). However, at increasingly smaller scales, the momentum
balance can easily shift from geostrophic to cyclogeostrophic balance,
reflecting the growing importance of centripetal accelerations. Indeed, vortex
generation by baroclinic or barotropic instabilities leads to such a balance.
Together with a hydrostatic assumption, this implies a gradient wind balance
(GWB): $(f+2\overline{v}/r)\partial_{z}\overline{v}=\partial_{r}\overline{b}$.
[For a vector representation of this balance, see McWilliams (1985b) or Grooms
(2015).] Factoring out the Coriolis parameter $f$ from the quantity in
parentheses immediately leads to a nondimensional parameter which quantifies
the impact of centripetal accelerations on the vertical shear:
${\mathrm{Cu}=2\overline{v}/(fr)}$. This “curvature” number also scales with
the ratio of centripetal to Coriolis accelerations (Shakespeare, 2016). In the
expression above, $r$ is the cross-front coordinate, such that
$M^{2}=\partial_{r}\overline{b}$ is the radial gradient buoyancy gradient and
implicitly contains information regarding frontal curvature. For clarity, we
note that ${\mathrm{Cu}>0}$ for cyclonic curved fronts and ${\mathrm{Cu}<0}$
negative for anticyclonic curved fronts111In vortices, $r$ is the distance
from the vortex center and is everywhere positive, while $\overline{v}>0$ for
cyclones and $\overline{v}<0$ for anticyclones. In meandering baroclinic
frontal flows, we can replace $r$ with a signed radius of curvature $R$ so
long as the along-front flow $\overline{v}>0$.. Moreover, in the limit
$\mathrm{Cu}\rightarrow 0$ one recovers TWB. GWB is therefore descriptive of
curved fronts and vortices, and yet includes TWB as a limiting case. This
motivates a slightly refined definition of the oceanic submesoscale,
consistent with the former but where curvature effects are explicitly
accounted for. In this study, we define the oceanic submesoscale as being a
dynamical regime in which the mean flow is in approximate hydrostatic and
cyclogeostrophic balance (i.e. GWB), permitting gradient Rossby, Richardson,
and curvature numbers of order-one: $(\mathrm{Ro},\mathrm{Ri},\mathrm{Cu})\sim
1$.

###  1.3 Motivation

In a previous study (Buckingham et al., 2020a, b), it was suggested that a
unique conservation principle may be present within highly curved fronts and
vortices (i.e. “vortex flow”) on the $f$-plane. Moreover, this principle was
invoked when proposing a mechanism for the evolution of small-scale (i.e.
submesoscale and polar mesoscale) vortices in the ocean. The implication was
that fluid parcels within curved baroclinic fronts and vortices do not simply
conserve the Ertel potential vorticity (PV) (Ertel, 1942), and therefore
undergo vortex stretching and tilting to conserve this quantity. Rather, fluid
parcels adjust barotropic and baroclinic components of another scalar
quantity,222Buckingham et al. (2020a, b) suggested that the generalized
Rayleigh discriminant $\Phi=2Lq/r^{2}$ (Kloosterziel et al., 2007) was
conserved following fluid parcels in highly curved fronts and vortices.
However, as demonstrated below, this statement is incorrect: it is $Lq$ or
${r^{2}\Phi}$ that is conserved following fluid parcels. This difference is
critical because it implies cross-frontal motion will modify the stability
“seen” by fluid parcels. which is proportional to the product of the Ertel PV
($q$) and the vertical component of absolute angular momentum ($L$). If true,
the fact that this additional term $L$ enters the conserved scalar provides an
added constraint to the problem, thereby making parcel motion within highly
curved baroclinic flows differ from those in which PV alone is the conserved
scalar. As will be demonstrated below, this places constraints on vorticity.

The purpose of this manuscript is therefore to provide a rational argument for
the statement that “the product of the absolute angular momentum and Ertel PV
is conserved following fluid parcels.” Moreover, we wish to assess under which
conditions such a statement is true. In doing so, we lay a more formal
foundation for the analysis of submesoscale baroclinic flows in which
centripetal accelerations cannot be neglected.

###  1.4 Outline

The outline of this study is as follows. We first derive a conservation
theorem for the new scalar quantity ${Lq}$ (section 2). This derivation
closely follows that of Ertel (1942) but it includes a brief presentation of
absolute angular momentum conservation–a topic typically neglected in
oceanographic studies. Second, the application of the resulting theorem is
briefly discussed in section 3 and its limitations mentioned in section 4. We
conclude the study in section 5.

##  2 Derivation

Ertel’s (1942) PV theorem is essentially an intersection of two conservation
principles: vorticity and density. It is logical to presume that the inclusion
of a third conservation principle together with its corresponding conditions
could permit a new vorticity theorem subject to these additional limitations.
This is the central concept behind the present study, where the third
conservation law is provided by the evolution equation for absolute angular
momentum (Figure 1).

Figure 1: Venn diagram conceptually depicting the intersection of three
conservation principles: absolute vorticity $\mathbf{\omega_{a}}$, density
$\rho$, and the vertical component of absolute angular momentum ${L}$. Here,
the sources and sinks in each equation are implied. While Ertel (1942) focused
on the intersection of density and vorticity conservation (i.e. $q$
conserved), this study examines the intersection of density, vorticity, and
absolute angular momentum conservation (i.e. $q$ & $L$ conserved). We
emphasize that conservation equations for $\rho$, $\mathbf{\omega_{a}}$, and
$L$ are not independent, thereby making this intersection possible.

###  2.1 Governing equations

The equations of motion describing the balance of forces per unit mass of a
fluid parcel within a rotating reference frame are (Batchelor, 1967; Pedlosky,
1987; Cushman-Roisin, 1994)

$\frac{D\mathbf{u}}{Dt}+2\mathbf{\Omega}\times\mathbf{u}=-\frac{1}{\rho}\nabla
p+\underbrace{\mathbf{g^{*}}+\mathbf{a_{c}}}_{\mathrm{}~{}\mathbf{g}}+\frac{\mathscr{F}}{\rho},$
(1)

where it is understood that all terms are evaluated within the rotating
reference frame. Here, ${D/Dt=\partial_{t}+\mathbf{u}\cdot\nabla\mathbf{u}}$
denotes the material or substantial derivative, $\mathbf{r}$ is the position
vector, $\mathbf{\Omega}$ is the angular rotation rate
($|\mathbf{\Omega}|~{}=~{}2\pi/$day $\approx~{}7.22~{}\times 10^{-5}$ s-1 for
Earth) and assumed to be constant, $2\mathbf{\Omega}\times\mathbf{u}$ is the
Coriolis acceleration,
${\mathbf{a_{c}}=-\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})=|\mathbf{\Omega}|^{2}\mathbf{r_{\perp}}}$
is the centrifugal acceleration due to the rotation of the reference frame,
$\rho$ is density, $p$ is pressure, $\mathbf{g^{*}}$ is the acceleration due
to gravity, and $\mathscr{F}$ denotes the frictional force.

It is customary to combine centrifugal and gravitational accelerations into a
resultant acceleration ${\mathbf{g}=\mathbf{g^{*}}+\mathbf{a_{c}}}$, or
effective gravity. The resultant is then approximately perpendicular to
geopotential surfaces and, hence, oriented vertically333Local changes to the
gravitational potential, for example, due to irregular topography or
seamounts, will perturb $\mathbf{g^{*}}$ from its mean direction. (Cushman-
Roisin, 1994). For clarity, we illustrate planetary vorticity, gravitational
acceleration, gravity, and centrifugal acceleration vectors (Figure 2). Mass
conservation is given by the continuity equation

$\frac{\partial\rho}{\partial t}+\mathbf{\nabla}\cdot(\rho\mathbf{u})=0.$ (2)

An equation of state is necessary to relate $\rho$ to known or measured
variables. In the ocean, this is a complex function of temperature, salinity,
and pressure. For simplicity, we will assume the density is known perfectly.

Figure 2: Illustration of vectors present within the equations of motion on
the sphere (cf. Equation 1) and $f$ plane approximation (cf. Equation 3). In
(a), we depict planetary vorticity $2\mathbf{\Omega}$ (orange), the position
vector $\mathbf{r}$ (heavy black), components of the position vector
$\mathbf{r_{\perp}}$ and $\mathbf{r_{||}}$ (gray), and vertical unit vector
$\mathbf{\hat{k}}$ (green). In (b), we depict the gravitational vector
$\mathbf{g^{*}}$ (black), the centrifugal acceleration vector
${\mathbf{a_{c}}=-\mathbf{\Omega}\times(\mathbf{\Omega}\times\mathbf{r})=|\mathbf{\Omega}|^{2}\mathbf{r_{\perp}}}$
(red), and the vector resultant, or effective gravity
${\mathbf{g}=\mathbf{g^{*}}+\mathbf{a_{c}}}$ (dashed black). We also
illustrate the surface of Earth as represented by a sphere (solid blue) and
oblate sphere, or spheroid (dashed blue). The unit vector $\mathbf{\hat{k}}$
is anti-parallel to $\mathbf{g}$ and, therefore, approximately perpendicular
to the spheroid’s surface. Figure 3: A cylindrical coordinate system on an
$f$-plane at latitude $\theta=\theta_{o}$: (a) perspective view and (b) plan
view, illustrating the orthogonal unit basis
$(\mathbf{\hat{r}},\mathbf{\hat{\phi}},\mathbf{\hat{k}})$, position vector
${\mathbf{r_{c}}=(r,\phi,z)}$ (red), where the angle $\phi$ is defined
relative to an eastward direction, and a vector $\mathbf{r_{o}}$ (yellow)
which helps define the origin of the cylindrical coordinate system. Although
not shown, the velocity is ${\mathbf{u}=(u,v,w)}$ and its components point in
$\mathbf{\hat{r}}$, $\mathbf{\hat{\phi}}$, and $\mathbf{\hat{k}}$ directions,
respectively.

#### Restriction to small horizontal scales

The corresponding equations of motion valid under an approximation of a
constant rotation rate of the reference frame (i.e. $f$-plane) are formally
obtained by expressing Equation 1 in spherical coordinates, scaling the
equations of motion, and discarding terms multiplied by $|ds|/R_{e}\ll 1$ or
smaller, where $|ds|=R_{e}d\theta$ denotes a meridional arc length and $R_{e}$
is the mean radius of Earth (Grimshaw, 1975). The result is a vectorized set
of equations comparable to Equation 1 except where
$2\mathbf{\Omega}\times\mathbf{u}$ is now evaluated at a specific latitude
$\theta_{o}$:

$\frac{D\mathbf{u}}{Dt}+2\mathbf{\Omega_{o}}\times\mathbf{u}=-\frac{1}{\rho}\nabla
p+\mathbf{g}+\frac{\mathscr{F}}{\rho}.$ (3)

Note that making the $f$ plane approximation does not alter the continuity
equation (cf. Equation 2).

For later reference, we detail the above terms in cylindrical polar
coordinates. In cylindrical coordinates, where the triad of orthogonal unit
vectors $(\mathbf{\hat{r}},\mathbf{\hat{\phi}},\mathbf{\hat{k}})$ point in
radial, azimuthal, and vertical (upward) directions, respectively, we denote
the position vector by ${\mathbf{r_{c}}=(r,\phi,z)}$ and velocity by
${\mathbf{u}=(u,v,w)}$ (Figure 3). The material derivative is then
${D/Dt=\partial_{t}+\mathbf{u}\cdot\nabla=\partial_{t}+u\partial_{r}+(v/r)\partial_{\phi}+w\partial_{z}}$.
Finally, the frictional force is $\mathscr{F}=(F_{r},F_{\phi},F_{z})$ and
effective gravity is $\mathbf{g}=(0,0,-g)$. Note that the choice of a
cylindrical coordinate system on the $f$ plane complicates expression of
$\mathbf{\Omega_{o}}$ owing to its variation with azimuth angle $\phi$. To
retain generality in our derivation below, we use the vector form of Equation
3 together with the full Coriolis vector $\mathbf{\Omega_{o}}$.

###  2.2 Absolute vorticity conservation

The following two sections can be found elsewhere (e.g. Pedlosky, 1987;
Müller, 1995; Vallis, 2017), but are repeated here for completeness. Equation
3 can be recast in terms of the absolute vorticity or “Stokes” form
(Batchelor, 1967):

$\frac{\partial{\mathbf{u}}}{\partial{t}}+\mathbf{\omega_{a}}\times\mathbf{u}=-\frac{1}{\rho}\nabla
p+\nabla\left[\mathbf{g}\cdot\mathbf{r_{c}}-(\mathbf{u}\cdot\mathbf{u})/2\right]+\frac{\mathscr{F}}{\rho},$
(4)

where
${\omega_{a}=\nabla\times\mathbf{u_{a}}=2\mathbf{\Omega_{o}}+\nabla\times\mathbf{u}=2\mathbf{\Omega_{o}}+\mathbf{\omega}}$
is absolute vorticity (i.e. the sum of relative and planetary vorticity) and
${\mathbf{u_{a}}=\mathbf{u}+\mathbf{\Omega}\times\mathbf{r_{c}}}$ is absolute
velocity. Taking the curl of Equation 4 gives

$\frac{\partial{\mathbf{\omega}}}{\partial{t}}+\nabla\times\left(\mathbf{\omega_{a}}\times\mathbf{u}\right)=\frac{\nabla\rho\times\nabla
p}{\rho^{2}}+\nabla\times\left(\frac{\mathscr{F}}{\rho}\right).$ (5)

Using the identity (e.g. Riley et al., 2006)

${\nabla\times\left(\mathbf{A}\times\mathbf{B}\right)}=\mathbf{A}\nabla\cdot\mathbf{B}+(\mathbf{B}\cdot\nabla)\mathbf{A}-\mathbf{B}\nabla\cdot\mathbf{A}-(\mathbf{A}\cdot\nabla)\mathbf{B},$
(6)

and noting that the planetary vorticity is constant,444This is true regardless
of the chosen coordinate system since the vector $\mathbf{2\Omega}$ remains
unchanged. one can re-write the vorticity equation as

$\frac{D\mathbf{\omega_{a}}}{Dt}=\mathbf{\omega_{a}}\cdot{\nabla\mathbf{u}}-\omega_{a}\nabla\cdot\mathbf{u}+\frac{\nabla\rho\times\nabla
p}{\rho^{2}}+\nabla\times\left(\frac{\mathscr{F}}{\rho}\right).$ (7)

This equation states that the absolute vorticity of a fluid element is
modified by (i) shearing motion that tilts or re-orients the vorticity vector
or by vortex stretching, (ii) compressibility, (iii) baroclinicity, which
alters the center of mass relative to that found when density contours and
pressure contours are parallel, and (iv) frictional forces. This is the basis
for vortex stretching and tilting interpretations of PV.

Another useful form of the vorticity equation is obtained as follows. One can
replace the divergence term in the vorticity equation using Equation 2, and
rewrite the material derivative of absolute vorticity per unit mass as

$\frac{D}{Dt}\left(\frac{\mathbf{\omega_{a}}}{\rho}\right)=\frac{1}{\rho}\left[\frac{D\mathbf{\omega_{a}}}{Dt}-\frac{\mathbf{\omega_{a}}}{\rho}\frac{D\rho}{Dt}\right],$
(8)

allowing one to express Equation 7 as the conservation of absolute vorticity
per unit mass:

$\frac{D}{Dt}\left(\frac{\mathbf{\omega_{a}}}{\rho}\right)=\left(\frac{\mathbf{\omega_{a}}}{\rho}\cdot\nabla\right)\mathbf{u}+\frac{\nabla\rho\times\nabla
p}{\rho^{3}}+\left(\nabla\times\frac{\mathscr{F}}{\rho}\right)\frac{1}{\rho}.$
(9)

###  2.3 Conservation of density

The next step displays the creativity of Ertel. Following Pedlosky (1987)
(Ertel assumes $D\rho/Dt=0$), we write the conservation of a scalar quantity
$\lambda$ as

$D\lambda/Dt=\frac{\partial{\lambda}}{\partial{t}}+\mathbf{u}\cdot\nabla\lambda=\Psi.$
(10)

Taking the inner product of $\nabla\lambda$ and Equation 9, one obtains

$\nabla\lambda\cdot\frac{D}{Dt}\left(\frac{\mathbf{\omega_{a}}}{\rho}\right)=\nabla\lambda\cdot\left[\left(\frac{\mathbf{\omega_{a}}}{\rho}\cdot\nabla\right)\mathbf{u}\right]+\nabla\lambda\cdot\frac{\nabla\rho\times\nabla
p}{\rho^{3}}+\frac{\nabla\lambda}{\rho}\cdot\left(\nabla\times\frac{\mathscr{F}}{\rho}\right).$
(11)

Incorporating $\nabla\lambda$ into the material derivative on the left-hand-
side (LHS),555This follows from
${\mathbf{A}\cdot\frac{D(\nabla\lambda)}{Dt}=\mathbf{A}\cdot\nabla\frac{D\lambda}{Dt}-\nabla\lambda\cdot\left(\mathbf{A}\cdot\nabla\mathbf{u}\right)}$,
where we have used ${\mathbf{A}={\mathbf{\omega_{a}}}/{\rho}}$. we obtain

$\frac{Dq}{Dt}=\frac{D}{Dt}\left(\frac{\mathbf{\omega_{a}}}{\rho}\cdot\nabla\lambda\right)=\frac{\mathbf{\omega_{a}}}{\rho}\cdot\nabla\Psi+\nabla\lambda\cdot\frac{\nabla\rho\times\nabla
p}{\rho^{3}}+\frac{\nabla\lambda}{\rho}\cdot\left(\nabla\times\frac{\mathscr{F}}{\rho}\right).$
(12)

This states that the quantity,
${q=(\mathbf{\omega_{a}}}/\rho)\cdot\nabla\lambda$, is conserved following
fluid parcels if the RHS is zero. Choosing, for example, density as our scalar
quantity, $\lambda=\rho$, while requiring frictional and diabatic processes to
be zero so that the flow is inviscid and density is conserved, we see that all
three terms on the RHS vanish. This is Ertel’s (1942) vorticity theorem. More
generally, $\lambda$ can be any variable so long as it is a function of
density and pressure–i.e. a “thermodynamic function” (Pedlosky, 1987).

###  2.4 Absolute angular momentum conservation

One of the contributions of Rayleigh (1917) was to demonstrate that, if a
vortex is axisymmetric, then the azimuthal momentum equation can be multiplied
by $r$ and re-expressed as a conservation equation for the angular momentum
per unit mass: ${D/Dt(r\overline{v})=0}$, where $\overline{v}$ denotes the
azimuthal velocity. Application of this approach to a fluid parcel in a
rotating reference frame with constant rotation rate also permits such a
rearrangement: ${DL/Dt=0}$, where $L=r\overline{v}+fr^{2}/2$ is now the
_absolute angular momentum_ , and is the sum of relative angular momentum
($r\overline{v}$) and planetary angular momentum imparted by the rotating
reference frame. Importantly, the absolute angular momentum of a fluid parcel
in a vortex on the $f$ plane is exactly the same as if the vortex were located
at the center of the rotating reference frame, where $r$ is the magnitude of
the position vector (Kloosterziel and van Heijst, 1991). This motivates the
following vector representation.

We orient our coordinate system so that its origin is at the center of a
curved front or vortex (cf. Figure 3). Taking the cross product of the
position vector $\mathbf{r_{c}}$ and each of the terms in Equation 3, one
obtains after some effort

$\frac{D\mathbf{m_{a}}}{Dt}=-\mathbf{\Omega_{o}}\times\mathbf{m}-\frac{\mathbf{r_{c}}\times\nabla
p}{\rho}+\mathbf{r_{c}}\times\mathbf{g}+\frac{\mathbf{r_{c}}\times\mathscr{F}}{\rho},$
(13)

where ${\mathbf{m_{a}}=\mathbf{r_{c}}\times\mathbf{u_{a}}}$ and
$\mathbf{m}=\mathbf{r_{c}}\times\mathbf{u}$ are absolute and relative angular
momentum, respectively. Using the definition of absolute velocity,
${\mathbf{u_{a}}=\mathbf{u}+\mathbf{\Omega_{o}}\times\mathbf{r_{c}}}$, we
observe that ${\mathbf{m_{a}}=\mathbf{m}+\mathbf{m_{\Omega}}}$ is the sum of
relative angular momentum ${\mathbf{m}=\mathbf{r_{c}}\times\mathbf{u}}$ and
planetary angular momentum
${\mathbf{m_{\Omega}}=\mathbf{r_{c}}\times(\mathbf{\Omega_{o}}\times\mathbf{r_{c}})}$
in a manner analogous to absolute vorticity $\mathbf{\omega_{a}}$.

For our purposes, we wish to isolate the vertical component of absolute
angular momentum. We take the inner product of Equation 13 and the vertical
unit vector $\mathbf{\hat{k}}$ to obtain

$\frac{DL}{Dt}=-\left(\mathbf{\Omega_{o}}\times\mathbf{m}\right)\cdot\mathbf{\hat{k}}-\frac{\mathbf{r_{c}}\times\nabla
p}{\rho}\cdot\mathbf{\hat{k}}+(\mathbf{r_{c}}\times\mathbf{g})\cdot\mathbf{\hat{k}}+\frac{\mathbf{r_{c}}\times\mathscr{F}}{\rho}\cdot\mathbf{\hat{k}},$
(14)

where we have introduced the notation
${L=\mathbf{m_{a}}\cdot\mathbf{\hat{k}}}$ to denote the vertical component of
absolute angular momentum, consistent with the literature (Holton, 1992;
Shakespeare, 2016). Thus, the vertical component of absolute angular momentum
$L$ of a fluid parcel is modified by torques due to pressure, gravitation, and
friction, as well as a torque produced by Earth’s rotation acting on the
relative angular momentum $\mathbf{m}$. For cases when $\mathbf{m}$ is not
vertical, the latter reduces $L$, tilting the absolute angular momentum vector
away from the vertical.

#### Angular momentum conservation on the spheroid

While our interest is in small-scale fronts and vortices, it is nonetheless
helpful to compare the conservation equation above (cf. Equation 13) with that
obtained for the oblate sphere (e.g Barnes et al., 1983; Peixoto and Oort,
1992; Bell, 1994). This was considered, for example, by Egger (2001). In this
case, the position vector $\mathbf{r}$ extends from Earth’s center to the
fluid parcel (Figure 2). Computing the cross product of $\mathbf{r}$ and the
more general equations of motion (cf. Equation 1), one obtains

$\frac{D\mathbf{m_{a}}}{Dt}=-\mathbf{\Omega}\times\mathbf{m_{a}}-\frac{\mathbf{r}\times\nabla
p}{\rho}+\mathbf{r}\times\underbrace{(\mathbf{g}-\mathbf{a_{c}})}_{\mathbf{g^{*}}}+\frac{\mathbf{r}\times\mathscr{F}}{\rho},$
(15)

where now the absolute, relative, and planetary angular momentum are given,
respectively, by ${\mathbf{m_{a}}=\mathbf{m}+\mathbf{m_{\Omega}}}$,
${\mathbf{m}=\mathbf{r}\times\mathbf{u}}$, and
${\mathbf{m_{\Omega}}=\mathbf{r}\times(\mathbf{\Omega}\times\mathbf{r})}$.
Expanding the first term on the RHS and examining only the planetary portion
of this term, we see that Earth’s rotation induces a torque with magnitude
${|-\mathbf{\Omega}\times\mathbf{m_{\Omega}}|=|\mathbf{r_{||}}||\mathbf{\Omega}||\mathbf{\Omega}\times\mathbf{r_{\perp}}|=|\mathbf{\Omega}|^{2}|\mathbf{r_{\perp}}||\mathbf{r_{||}}|}$
that is directed eastward.666Note:
${\mathbf{\Omega}\times\mathbf{r}=\mathbf{\Omega}\times\mathbf{r_{\perp}}}$
and
${\mathbf{A}\times(\mathbf{B}\times\mathbf{C})=(\mathbf{A}\cdot\mathbf{C})\mathbf{B}-(\mathbf{A}\cdot\mathbf{B})\mathbf{C}}$
allow us to simplify the planetary angular momentum:
${\mathbf{m_{\Omega}}=|\mathbf{r_{\perp}}|^{2}\mathbf{\Omega}-|\mathbf{r_{||}}||\mathbf{\Omega}|\mathbf{r_{\perp}}}$.
Similarly, the torque induced by the centrifugal acceleration has magnitude
${|-\mathbf{r}\times\mathbf{a_{c}}|=|\mathbf{\Omega}|^{2}|\mathbf{r_{\perp}}||\mathbf{r_{||}}|}$
but is directed westward. That is, the two terms cancel and Equation 15 can be
written simply as

$\frac{D\mathbf{m_{a}}}{Dt}=-\mathbf{\Omega}\times\mathbf{m}-\frac{\mathbf{r}\times\nabla
p}{\rho}+\mathbf{r}\times\mathbf{g}+\frac{\mathbf{r}\times\mathscr{F}}{\rho}.$
(16)

We now note that Equation 16 is identical to Equation 13 except where
$\mathbf{r_{c}}$ is replaced by $\mathbf{r}$. Thus, while a formal proof
remains, we argue that absolute angular momentum is conserved on the $f$ plane
similarly as how it is conserved on the sphere.777Egger (2001) did not
demonstrate this vector cancellation and led him to conclude that angular
momentum conservation was different in the $f$ plane approximation than on the
spheroid (in the limit of small $|ds|/R_{e}$). We disagree with this statement
for the reasons stated above. This may be why, for sufficiently small
horizontal scales and for balanced (i.e. hydrostatic) flows in which the
meridional component of Coriolis is neglected, the volume-integrated, vertical
component of absolute angular momentum is approximately conserved (Egger,
2001, Fig. 2e).

###  2.5 A vorticity theorem for the $f$ plane

We are now in a position to combine conservation laws (cf. Equations 12 and
14). It is simple to show that if $\frac{DA}{Dt}=0$ and if $\frac{DB}{Dt}=0$,
then $\frac{D}{Dt}(AB)=0$. Although elementary, this is the logic behind the
following step. We multiply Equation 12 by
${L=\mathbf{m_{a}}\cdot\mathbf{\hat{k}}=(\mathbf{m}+\mathbf{m_{\Omega}})\cdot\mathbf{\hat{k}}}$
and add this to $q=({\mathbf{\omega_{a}}}/{\rho})\cdot\nabla\lambda$
multiplied by Equation 14. This gives

$\displaystyle\frac{D}{Dt}(Lq)$ $\displaystyle=$ $\displaystyle
L\left[\frac{\mathbf{\omega_{a}}}{\rho}\cdot\nabla\Psi+\nabla\lambda\cdot\frac{\nabla\rho\times\nabla
p}{\rho^{3}}+\frac{\nabla\lambda}{\rho}\cdot\left(\nabla\times\frac{\mathscr{F}}{\rho}\right)\right]$
(17) $\displaystyle+$ $\displaystyle
q\big{[}-{\left(\mathbf{\Omega_{o}}\times\mathbf{m}\right)}\cdot\mathbf{\hat{k}}-\frac{\mathbf{r_{c}}\times\nabla
p}{\rho}\cdot\mathbf{\hat{k}}$ $\displaystyle+$
$\displaystyle\left(\mathbf{r_{c}}\times\mathbf{g}\right)\cdot\mathbf{\hat{k}}+\frac{\mathbf{r_{c}}\times\mathscr{F}}{\rho}\cdot\mathbf{\hat{k}}\big{]}.$

where it is emphasized that $\mathbf{r_{c}}$ is the position vector in the
cylindrical coordinate system and $\mathbf{g}$ is directed anti-parallel to
the unit vector $\mathbf{\hat{k}}$ (Figure 3). This equation states that the
scalar quantity $Lq$ is conserved following fluid parcels on the $f$ plane if,
for non-zero $L$ and $q$, all of the following conditions are met:

  1. 1.

density is conserved ($\Psi=0$)

  2. 2.

the fluid is inviscid ($\mathscr{F}=0$)

  3. 3.

    1. (a)

the fluid is barotropic ($\nabla\rho\times\nabla p=0$) or

    2. (b)

the fluid is baroclinic ($\nabla\rho\times\nabla p\neq 0$) and $\lambda$ is a
“thermodynamic function”

  4. 4.

relative angular momentum $\mathbf{m}$ is directed vertically so that
${(\mathbf{\Omega_{o}}\times\mathbf{m})\cdot\mathbf{\hat{k}}=0}$

  5. 5.

pressure torques are zero or orthogonal to the vertical so that
${(\mathbf{r_{c}}\times\nabla p)\cdot\mathbf{\hat{k}}=0}$

  6. 6.

perturbations in Earth’s gravitational field are zero so that
${(\mathbf{r_{c}}\times\mathbf{g})\cdot\mathbf{\hat{k}}=0}$, and

While the above conservation theorem may find reduced application when
compared to Ertel’s PV theorem owing to the number of aforementioned
conditions, several simplifications are possible. For adiabatic and inviscid
baroclinic flows, selecting $\lambda$ as a thermodynamic variable (e.g.
$\lambda=-\rho g$) satisfies conditions (1)-(3). For geophysical flows of the
type considered here, the flow is nearly two-dimensional such that
$\mathbf{m}$ points approximately vertically. In this case,
$\mathbf{\Omega_{o}}\times\mathbf{m}$ will be tangent to Earth’s surface and
condition (4) can be met. In the absence of geopotential perturbations,
condition (6) is met. Finally, assuming symmetry in the direction of the flow,
pressure gradient torques are zero. (This may not be true if perturbations due
to baroclinic instability break such symmetry.) However, undulating topography
introduces pressure torques. This includes ocean bottom topography or rough
ice (i.e. for vortex flows found beneath sea ice or ice shelves). Thus,
condition (5) is met for azimuthally symmetric flow away from boundaries.
Conditions (1)-(3) correspond exactly to Ertel’s PV theorem, while conditions
(4)-(6) ensure conservation of $L$. In conclusion, we have a conservation
theorem valid on the $f$ plane that is different than Ertel’s PV theorem and
yet, at least in curved balanced flows away from boundaries, has the potential
to satisfy all of the aforementioned conditions. If these conditions are met,
the product of the vertical component of absolute angular momentum and Ertel
PV ($Lq$) is conserved following fluid parcels.888It is not clear how best to
refer to the quantity $Lq$. We were at first tempted to refer to $Lq$ as a
generalized form of PV since fluid parcels have possible vorticity values that
are set by the sign of $Lq$ through the stability discriminant
$\Phi=2Lq/r^{2}$ (Buckingham et al., 2020a, b). However, the theorem’s
validity is confined to small horizontal scales such that $Lq$ is not
universally conserved. For this reason, the term submesoscale potential
vorticity might be a suitable alternative. Note: the scalar
$\Phi/f=2Lq/(fr^{2})=(1+\mathrm{Cu})q$ is perhaps a better variable to be
named submesoscale potential vorticity since it shares the same units as $q$
and applies to straight and curved fronts. One recovers the Ertel PV in the
limit $\mathrm{Cu}\rightarrow 0$. In any case, to avoid conflict with the
Ertel PV and given its established relationship to angular momentum (Rayleigh,
1917; Solberg, 1936; Fjortoft, 1950), we adopt the term potential momentum
below in reference to $Lq$, reflecting that changes in angular momentum can
occur as a result of alterations in the baroclinic nature of the fluid.

##  3 Discussion

It is helpful to consider how such a conservation principle might find
practical application. Given the restriction to the $f$ plane, we propose that
the conservation theorem will find greatest application in understanding
vortex flows away from the tropics in the oceanic submesoscale regime (cf.
section 11.2). These will include (1) polar mesoscale eddies found under ice
in the Arctic (D’Asaro, 1988; Timmermans et al., 2008; Zhao et al., 2014), (2)
simulated vortices in the laboratory (Stegner et al., 2004; Kloosterziel et
al., 2007; Lazar et al., 2013), (3) hydrothermal vents and convective plumes
(Helfrich and Battisti, 1991; D’Asaro et al., 1994; Legg and McWilliams, 2001;
Deremble, 2016), and (4) meddies and other mid-latitude coherent vortices
formed through interactions of currents with topography, cyclogeostrophic
adjustment of buoyant waters, and vertical convection through buoyancy fluxes
in boundary layers (McDowell and Rossby, 1978; McWilliams, 1985a; Riser et
al., 1986; Bane et al., 1989; Konstianoy and Belkin, 1989; Lilly and Rhines,
2002; Bosse et al., 2016; Meunier et al., 2018). We note that it might also
find relevance in understanding (5) parcel motion within highly curved fronts
(MacKinnon et al., 2021, M. Freilich 2020, personal communication). Owing to
its generic nature, the concept may also find relevance for other geophysical
flows such as in the atmosphere or on other planets.

While presenting a framework for understanding sources and sinks of $Lq$ or
“potential momentum” (e.g. Haynes and McIntyre, 1987, 1990; Marshall and
Nurser, 1992) is beyond the scope of this study, one can nevertheless
conceptually consider the theorem’s application to the aforementioned flows by
expressing Equation 17 for an axisymmetric vortex. This is done below for a
vortex set at high latitudes and is followed by a brief discussion of the
theorem’s imprint on relative vorticity.

###  3.1 Application to the axisymmetric vortex

We now consider the evolution of $Lq$ within an axisymmetric vortex in
cyclogeostrophic and hydrostatic balance (i.e. GWB). We require frictional and
diabatic effects to be weak enough such that this balance holds. This will
consequently result in weak cross-frontal or radial fluid motion (Eliassen,
1951). Formally, we state that perturbations from the balanced state are small
such that
${\mathbf{u}=\mathbf{\overline{u}}+\mathbf{u^{*}}\approx(0,\overline{v},0)}$
and ${b=\overline{b}+b^{*}\approx\overline{b}}$, where the overbar denotes
mean quantities and asterisks (∗) denote deviations from this state. We
neglect compressibility effects, make the Boussinesq approximation, and define
$\lambda=-g\rho$, allowing the Ertel PV to be written as
${q=\mathbf{\omega_{a}}\cdot\nabla b}$, where $b=-g\rho/\rho_{o}$ is buoyancy
and $\rho_{o}$ is a constant reference density. Finally, we set the meridional
component of Coriolis to zero on the basis that the flow is sufficiently
distant from the Equator.

We now express Equation 17 in cylindrical coordinates. The vertical component
of absolute angular momentum is
${L=\mathbf{m_{a}}\cdot\mathbf{\hat{k}}=(\mathbf{m}+\mathbf{m_{\Omega}})\cdot\mathbf{\hat{k}}}$,
where ${\mathbf{m}=\mathbf{r_{c}}\times\mathbf{u}}$ and
${\mathbf{m_{\Omega}}=\mathbf{r_{c}}\times(\mathbf{\Omega}\times\mathbf{r_{c}})}$
are the relative and planetary angular momentum. Together with
${\mathbf{r_{c}}=(r,\phi,z)}$, ${\mathbf{u}=(u,v,w)}$, and
${\mathbf{\Omega}=(0,0,f)}$, we find $L=r\overline{v}+fr^{2}/2$. The Ertel PV
for this vortex is ${q=(2\mathbf{\Omega}+\nabla\times\mathbf{u})\cdot\nabla
b}$. Together with GWB
${(f+2\overline{v}/r)\partial_{z}\overline{v}=\partial_{r}\overline{b}=M^{2}}$,
we have
${q=(f+\overline{\zeta})N^{2}-(f+2\overline{v}/r)|\partial_{z}\overline{v}|^{2}}$,
where we have neglected the horizontal vorticity owing to its smallness
relative to other terms. Thus, the relative vorticity associated with the
balanced state is
${\overline{\zeta}=(1/r)\partial_{r}(r\overline{v})=\overline{v}/r+\partial_{r}\overline{v}}$
and vertical stratification is ${N^{2}=\partial_{z}\overline{b}}$. With these
definitions in hand, Equation 17 becomes

$\displaystyle\frac{D\Pi}{Dt}=S_{\mathscr{F}+\mathscr{D}+\mathscr{P}+\mathscr{g}+\Omega},$
(18)

where the potential momentum is

$\Pi=Lq=\left(r\overline{v}+\frac{fr^{2}}{2}\right)\left[\left(f+\overline{\zeta}\right)N^{2}-\left(f+\frac{2\overline{v}}{r}\right)|\partial_{z}\overline{v}|^{2}\right]=\frac{r^{2}\Phi}{2}$
(19)

and $S_{\mathscr{F}+\mathscr{D}+\mathscr{P}+\mathscr{g}+\Omega}$ represents
sources and sinks of momentum due to frictional, diabatic, pressure, and
gravitational sources, as well as Earth’s rotation acting on the relative
momentum vector (i.e. RHS of Equation 17). $\Phi$ is the generalized Rayleigh
discriminant and consists of barotropic and baroclinic components (Buckingham
et al., 2020a):

$\Phi=\underbrace{(f+2\overline{v}/r)(f+\overline{\zeta})N^{2}}_{barotropic}-\underbrace{(f+2\overline{v}/r)^{2}|\partial_{z}{\overline{v}}|^{2}}_{baroclinic}=\chi^{2}N^{2}-M^{4}$
(20)

where ${\chi^{2}=(f+2\overline{v}/r)(f+\overline{\zeta})}$ is the generalized
Rayleigh discriminant for barotropic vortices (Kloosterziel and van Heijst,
1991; Mutabazi et al., 1992). As in the introduction, we define gradient
Rossby, gradient Richardson, and curvature numbers as
${\mathrm{Ro}={\overline{\zeta}}/{f}}$,
${\mathrm{Ri}={N^{2}}/{|\partial_{z}{\overline{v}}|^{2}}}$, and
${\mathrm{Cu}={2\overline{v}}/{fr}}$, allowing us to also write the potential
momentum as

$\Pi=Lq=\left(\frac{f^{2}N^{2}r^{2}}{2}\right)\Phi^{\prime},$ (21)

where

$\Phi^{\prime}=L^{\prime}q^{\prime}=\left(1+\mathrm{Cu}\right)\left(1+\mathrm{Ro}\right)-\left(1+\mathrm{Cu}\right)^{2}\cdot\mathrm{Ri}^{-1}.$
(22)

is a nondimensional form of the generalized Rayleigh discriminant,
$L^{\prime}=1+\mathrm{Cu}$ is a nondimensional form of absolute angular
momentum, and $q^{\prime}$ is the nondimensional Ertel PV. Expanding Equation
18, we find

${\frac{D\Phi}{Dt}=2{S_{\mathscr{F}+\mathscr{D}+\mathscr{P}+\mathscr{g}+\Omega}}-\frac{2u}{r}\Phi}.$
(23)

Thus, in the absence of sources and sinks of potential momentum
(${S_{\mathscr{F}+\mathscr{D}+\mathscr{P}+\mathscr{g}+\Omega}=0}$) and
assuming no cross-frontal motion (${u=0}$), the stability of the vortex is
constant: $D\Phi/Dt=0$. However, if a fluid parcel moves radially ($u\neq
0$)–even in the absence of sources and sinks of potential momentum–there must
be a corresponding change in the stability of the flow. This point above
contrasts with statements made by Buckingham et al. (2020a). In particular, in
their discussion, Buckingham et al. (2020a) speculated that $\Phi$ and
$\Phi^{\prime}$ might be conserved quantities following fluid parcels within
meandering fronts and vortices. As evidenced by Equation 23, this statement is
not entirely true. We can expect cross-frontal motion as a fluid parcel is
advected along its path in a meandering flow (Bower, 1989; Samelson, 1992)
such that $\Phi$ is not conserved. Additionally, $D\Phi/Dt=0$ does not imply
$D\Phi^{\prime}/Dt=0$, except if $N^{2}$ and $r^{2}$ do not change following a
fluid parcel (cf. Equation 21). These conditions can be met within the
axisymmetric vortex ($\partial_{\phi}\overline{b}\approx 0$ and $u\approx 0$)
but will not generally be met following fluid parcels within a meandering
front. These points therefore correct and/or qualify statements made by
Buckingham et al. (2020a, b). It also demonstrates that, to the extent that
$\Phi$ can be used as a predictor for along-isopycnal motion, the theorem may
find use in understanding parcel motion in curved baroclinic fronts.

In general, application of the vorticity theorem requires detailed knowledge
of each of the terms represented by
${S_{\mathscr{F}+\mathscr{D}+\mathscr{P}+\mathscr{g}+\Omega}}$. For example,
for a polar vortex located under ice but away from seamounts,
$\mathbf{\Omega_{o}}\times\mathbf{m}\approx 0$, the gravitational torque is
zero, frictional and diabatic terms are likely nonzero, and depending upon the
topography of the ice, gradients in pressure and the corresponding pressure
torque could also be nonzero (K. Nicholls, 2020; personal communication). In
such a situation, specification of these source/sink terms can be non-trivial.
However, a unique situation exists in which all of these terms are
approximately zero. This case is considered below.

#### Submesoscale and polar mesoscale vortices

Coherent baroclinic vortices with radii of $1$-$10$ km are believed to be
ubiquitous in the world’s oceans and are generated in close proximity to
boundaries (Figure 4). Despite that they include polar mesoscale vortices,
this class of vortices has been termed “submesoscale coherent vortices” (SCVs)
owing to their comparable size, intensity and stratification as those found at
mid-latitudes (McWilliams, 1985a). In particular, they are marked by
significantly reduced vertical stratification and unique watermass properties.
They appear to have been first documented as a distinct water-mass in
hydrographic observations near the Bahamas in the Atlantic (McDowell and
Rossby, 1978). Since, however, they are now regarded as ubiquitous in the
global oceans (e.g. McCoy et al., 2020).

The prevailing understanding is that SCV formation can happen via a horizontal
shear instability, for example, past topography or within an island wake
(Barkley, 1972; Stegner, 2014; Gula et al., 2016; MacKinnon et al., 2019;
Srinivasan et al., 2019), vertical shear or mixed layer baroclinic instability
(Haine and Marshall, 1998; Eldevik and Dysthe, 2002; Boccaletti et al., 2007),
as well as convection of boundary layer fluid (Legg and McWilliams, 2001;
Bosse et al., 2016). This set of boundary layers include the ocean surface
boundary layer (OSBL), bottom boundary layer (BBL), and ice-ocean boundary
layer (IOBL). At formation, vortices will typically spin up in such a way that
an anticyclone is preferred. This occurs principally due to conservation of
$q$ (with $fq>0$ initially) (Spall, 1995; Thomas, 2008) but conservation of
$L$ nevertheless plays an important role owing to centripetal accelerations
experienced by fluid parcels. It follows from the preceding derivation that
Equation 17 applies, so long as the fluid is continuous in the azimuth
direction such that $L$ can be appropriately defined.

One of the consequences of Equation 17 is that geophysical vortices which
reside away from boundaries and perturbations in Earth’s geopotential
approximately conserve the product of $L$ and $q$. In the absence of sources
and sinks of potential momentum
(${S_{\mathscr{F}+\mathscr{D}+\mathscr{P}+\mathscr{g}+\Omega}=0}$), we can
safely approximate cross-frontal motion within a vortex as zero ($u\approx
0$). By virtue of Equation 23, this also implies that the Rayleigh
discriminant $\Phi$ is conserved: ${D\Phi/Dt\approx 0}$. To understand the
consequences of these statements, we revisit the generation and evolution of
SCVs.

While the details of this evolution remain unclear without corroborating
laboratory or model support, the following arguments are reasonable. Now, the
low-stratified boundary layer fluid will be characterized by $\mathrm{Ri}\sim
1$. If this fluid is trapped within the vortex core,999One measure of this
trapping is the ratio of the azimuthal speed $|\overline{v}|$ to the wave
propagation speed $c$ (Samelson, 1992; Chelton et al., 2011). Given an
internal wave speed $c=NH$ and maximum velocity scale $v_{m}$, this condition
is equivalent to ${v_{m}/(NH)>1}$. then the fluid with low $\mathrm{Ri}$ will
persist even as the vortex subducts or is advected away from the boundary
(Figure 4). It immediately follows that the results from Buckingham et al.
(2020b) apply. That is, symmetric instability will be active within cyclonic
vortices, while anticyclonic vortices will remain marginally stable, decaying
over long time scales due to weak inertial-symmetric instabilities. A
timescale for the decay of the cyclonic vortex can be estimated from the
growth rate of symmetric disturbances: ${T=2\pi/\sigma}$, where
${\sigma=f(-\Phi^{\prime})^{1/2}}$ is an approximate growth rate of symmetric
disturbances under a simplified axisymmetric vortex model (Buckingham et al.,
2020a, Appendix A). An analytical time scale for the anticyclone is unknown to
the author, although anecdotally it has been estimated at months to years
based on watermass properties (McDowell and Rossby, 1978; McWilliams,
1985a).101010Mahdinia et al. (2017) observe a timescale greater than $50$ eddy
“turnaround times” within stability analysis of Gaussian vortices.

One point not considered above is the advection of the vortex into a different
ocean environment. In this case, the vortex must alter barotropic and
baroclinic components of $\Phi$ so as to keep $\Phi$ constant. Thus, vertical
stratification, vorticity, vertical shear, and curvature (cf. Equation 20)
must change, where vorticity is related to curvature through
${\mathrm{Ro}=\partial_{r}\overline{v}/f+\mathrm{Cu}/2}$. If, however, the
vortex is unable to alter these parameters quickly enough, energy will be
dissipated until an equilibrium is reached. In non-dimensional form (cf.
Equations 21 and 22), we find that $\mathrm{Ro}$, $\mathrm{Ri}$, and
$\mathrm{Cu}$ must change in some way so as to conserve the non-dimensional
form of $Lq$ (i.e. $\Phi^{\prime}$). This can be seen as a form of cyclo-
geostrophic adjustment (Stegner et al., 2004). Note: Equation 21 applies but
${D/Dt(N^{2})\neq 0}$.

Figure 4: Concept of the generation and evolution of small-scale, coherent
vortices in the ocean interior, adapted from Buckingham et al. (2020b).
Although other formation mechanisms are possible, here vortices are depicted
as forming from a front in the ocean surface boundary layer (OSBL) and flow-
topography interaction in the bottom boundary layer (BBL), with each color
denoting a different boundary layer fluid (blue=OSBL, yellow=BBL). In general,
in regions of low stratification, conservation of $Lq$ tends to generate flow
with anticyclonic vorticity. However, instances exist when cyclones are
produced, such as by abrupt topography. Immediately following formation, the
cyclone has large relative vorticity (at time $t=0$). However, due to reduced
core stratification and corresponding low Richardson numbers, the cyclone must
modify shears, stratification, and centripetal accelerations to ensure
${Lq>0}$. We propose that this occurs via energy loss resulting from symmetric
instability. Eventually, the cyclone becomes stable ($t=t_{2}$) but its final
vorticity is reduced. In contrast, the anticyclone, being weakly or marginally
stable, is not as significantly affected and maintains its energy for a
greater number of inertial periods (${t=0,t=t_{1},}$ … ${,t=t_{2}}$). A time
scale for the decay of the cyclone is given in the main text and is inversely
proportional to the square root of $-\Phi^{\prime}$ (cf. Equation 22).

###  3.2 Comment on the distribution of vorticity in the oceans

We close this study with one final comment. It is generally understood that
the distribution of vorticity as measured at small horizontal scales in the
oceans has two characteristics. Expressed in non-dimensional form, the
distribution of vorticity $\zeta/f$ is positively skewed (mostly cyclonic) in
straight fronts and negatively skewed (mostly anticyclonic) in eddying or
vortex flows. This has been noted, for example, in upper ocean observations
and model simulations when examined at submesoscale resolution (Rudnick, 2001;
Shcherbina et al., 2013; Buckingham et al., 2016). For reference, we refer the
reader to Shcherbina et al. (2013, Figure 5b). While the former can be
rationalized in terms of PV conservation (Hoskins and Bretherton, 1972), the
latter has not fully been explained. Here, we offer a simple explanation for
this observation.

Note that requiring positive potential momentum $Lq>0$ for all time is
equivalent to requiring balanced flow ($\Phi>0$) for all time. In this case,
$D/Dt(Lq)=0$ together with an initial positive state $Lq>0$ places constraints
on the sign of $\Phi$ and determines the distribution of relative vorticity in
the oceans. This is analogous to how $Dq/Dt=0$ together with an initial
positive state ${fq>0}$ determines the distribution of relative vorticity at
straight fronts (Buckingham et al., 2016). Another way to state this is that
the statistics of relative vorticity are determined by the possible set of
Rossby numbers which ensure the stability discriminant is positive: $\Phi>0$
or $\Phi^{\prime}>0$. If one requires that Equation 20 or 22 be positive and
solves for the set of Rossby numbers which ensure this is true, the negative
skewness discussed above will emerge at low $\mathrm{Ri}$ (e.g. Figure 13 of
Buckingham et al., 2020b). While this skewness is revealed due to the effect
of curvature on the Ertel PV $q$, helping to stabilize anticyclonic flow while
de-stabilizing cyclonic flow through a tilting of the vorticity vector, $L$
plays an important role in bounding anticyclonic vorticity.

##  4 Limitations

In this study, we have stressed the vertical component of absolute angular
momentum. As a result, Equation 17 cannot be used predict $Lq$ since $L$ is
coupled to the other components of $\mathbf{m_{a}}$ through
${\mathbf{\Omega}\times\mathbf{m}}$. As stated above, this term arises from
Earth’s rotation and tilts the absolute angular momentum vector away from the
vertical, reducing the magnitude of $L$. This torque is greatest at the
Equator and zero at the poles. Such a case might arise when a tropical vortex
wobbles about its vertical axis of rotation–i.e. when the relative angular
momentum $\mathbf{m}$ is not entirely vertical. This does not make Equation 17
incorrect but simply limits its utility in certain cases.

The author has not attempted to reframe the theorem for the $\beta$-plane,
although this would indeed be a useful exercise since intense vortices and
tropical instability waves are found in the vicinity of the Equator
(Marchesiello et al., 2011; Holmes et al., 2013; Simoes-Sousa et al., 2021).
Progress in this area might be made by examining the work of Grimshaw (1975)
and Kloosterziel et al. (2017). In conclusion, these restrictions–i.e. the (1)
restriction to the $f$ plane and (2) growing importance of
${\mathbf{\Omega}\times\mathbf{m}}$–are the two main limitations to the
theorem. A third limitation might also be that the fluid must be continuous in
the azimuth direction in order to properly define $L$, but at small horizontal
scales this is easily achieved.

It may be worth noting that the $f$ plane approximation together with angular
momentum conservation principles have previously been successful for
investigating tropical cyclone dynamics (Houze, 1993), thus indicating that
these limitations–which grow for vortices closer to the Equator–may not be so
severe. It is probable that a more elegant intersection of these three
principles (i.e. Figure 1 but where $L$ is replaced by $\mathbf{m_{a}}$), will
be presented in the future.

##  5 Conclusions

In this study, we have presented a conservation equation valid on the $f$
plane. It is an extension of Ertel’s PV theorem to flow at small horizontal
scales such that absolute angular momentum is appropriately conserved. The
combination of absolute angular momentum conservation together with Ertel’s
theorem has implications for the motion of fluid parcels. In particular, we
discovered two important consequences of the theorem: (1) shear,
stratification, and centripetal accelerations are modified in concert in an
effort to conserve $Lq$ and (2) asymmetry or skewness in the distribution of
relative vorticity results directly from this conservation principle,
permitting the occurrence of stable anticyclonic curved flow, while limiting
the occurrences of cyclonic curved flow at low Richardson numbers (Buckingham
et al., 2020a, b). That is, if $Lq>0$ initially, then $D/Dt(Lq)=0$ has
important consequences for the range of vorticity values seen at small
horizontal scales in the ocean. While this may find clearest application in
explaining why SCVs are overwhelmingly anticyclonic (McDowell and Rossby,
1978; Riser et al., 1986; D’Asaro, 1988; Bane et al., 1989; Timmermans et al.,
2008; Zhao et al., 2014), the theorem will also find use in understanding
Lagrangian parcel motion within highly curved baroclinic fronts.

The topic of absolute angular momentum conservation has received little
attention in oceanography texts, while this same topic has received
considerable attention in the atmospheric literature (Holton, 1992; Peixoto
and Oort, 1992; Barnes et al., 1983; Bell, 1994). This appears to be due to
the presence of continental boundaries in the ocean but which are absent in
the atmosphere (Griffies, 2004), causing PV rather than absolute angular
momentum to be a more universally conserved quantity at large horizontal
scales (Pedlosky, 1987). An exception may be in the Southern Ocean, where
obstacles to zonal flow are absent (Straub, 1993). However, for small-scale
geophysical flows in which centripetal accelerations are present and Earth’s
rotation plays a dynamically important role,111111A helpful point here is the
following: Earth’s rotation imparts angular momentum to fluid parcels and can
limit centripetal accelerations present in anticyclonic flows through the
constraint, $L^{\prime}=1+\mathrm{Cu}>0$, assuming $\mathrm{Cu}>-1$ initially.
This is analogous to how Earth’s rotation imparts vorticity to fluid parcels
and limits horizontal shear present in anticyclonic flows through the
constraint, $1+\mathrm{Ro}>0$. the conservation of absolute angular momentum
finds its place. Submesoscale and polar mesoscale flows are ideal examples in
which such a conservation principle may apply. It is in the context of these
phenomena that the combined theorem should find greater use.

###### Acknowledgements.

This study was made possible by an Individual Fellowship from the European
Commission on centrifugal instability (2019-2020). The derivation was made in
isolation during a two-month period (December 2020-January 2021) following the
fellowship. Although numerous individuals might be mentioned, the author
wishes to acknowledge the following individuals: George Nurser, Guillaume
Roullet, Stephen Griffies, Keith Nicholls, Greace Crystle, Jonathan Gula,
Xavier Carton, Charly DeMarez, and Thomas Meunier. Of these, George Nurser
made an important comment to the author regarding PV conservation (May 2015);
Charly DeMarez served as an important sounding block as these ideas were
forming (January-February 2020) and should be acknowledged; and Greace Crystle
patiently and kindly listened to the author’s musings and frustrations,
playing a quiet but significant role in this study. Comments by several
anonymous reviewers are also acknowledged. Subsequent conversations with other
scientists (e.g. Amit Tandon, Glenn Flierl, and Bruno Deremble) have been very
helpful. This article is dedicated to Alex Stegner, who mentored numerous
individuals in this field, and we would like to highlight the recent work of
Suraj Singh and Manikandan Mathur which is closely related to this problem in
the viscous context.

## References

  * Adams et al. (2017) Adams, K. A., P. Hosegood, J. R. Taylor, J.-B. Sallée, S. Bachman, R. Torres, and M. Stamper, 2017: Frontal circulation and submesoscale variability during the formation of a Southern Ocean mesoscale eddy. Journal of Physical Oceanography, 47 (7), 1737–1753, 10.1175/JPO-D-16-0266.1.
  * Bane et al. (1989) Bane, J. M., L. M. O’Keefe, D. R. Watts, J. C. J. Nihoul, and B. M. Jamart, 1989: Mesoscale Eddies and Submesoscale, Coherent Vortices: Their Existence Near and Interactions with the Gulf Stream, Elsevier Oceanography Series, Vol. 50, 501–518. Elsevier, 10.1016/S0422-9894(08)70204-6.
  * Barkley (1972) Barkley, R. A., 1972: Johnston Atoll’s wake. Journal of Marine Research, 30 (2), 201–216.
  * Barnes et al. (1983) Barnes, R. T. H., R. Hide, A. A. White, and C. A. Wilson, 1983: Atmospheric angular momentum fluctuations, length-of-day changes and polar motion. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 387 (1792), 31–73, 10.1098/rspa.1983.0050.
  * Batchelor (1967) Batchelor, G. K., 1967: An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, 1–658 pp., 10.1017/CBO9780511800955.
  * Bell (1994) Bell, M. J., 1994: Oscillations in the equatorial components of the atmosphere’s angular momentum and torques on the Earth’s bulge. Quarterly Journal of the Royal Meteorological Society, 120 (515), 195–213, 10.1002/qj.49712051512.
  * Boccaletti et al. (2007) Boccaletti, G., R. Ferrari, and B. Fox-Kemper, 2007: Mixed layer instabilities and restratification. Journal of Physical Oceanography, 37 (9), 2228–2250.
  * Bosse et al. (2016) Bosse, A., and Coauthors, 2016: Scales and dynamics of submesoscale coherent vortices formed by deep convection in the northwestern Mediterranean Sea. Journal of Geophysical Research: Oceans, 121 (10), 7716–7742, 10.1002/2016JC012144.
  * Bower (1989) Bower, A. S., 1989: Potential vorticity balances and horizontal divergence along particle trajectories in Gulf Stream meanders east of Cape Hatteras. Journal of Physical Oceanography, 19 (11), 1669–1681, 10.1175/1520-0485(1989)019¡1669:PVBAHD¿2.0.CO;2.
  * Buckingham et al. (2020a) Buckingham, C. E., J. Gula, and X. Carton, 2020a: The role of curvature in modifying frontal instabilities, part 1. Journal of Physical Oceanography.
  * Buckingham et al. (2020b) Buckingham, C. E., J. Gula, and X. Carton, 2020b: The role of curvature in modifying frontal instabilities, part 2. Journal of Physical Oceanography.
  * Buckingham et al. (2016) Buckingham, C. E., and Coauthors, 2016: Seasonality of submesoscale flows in the ocean surface boundary layer. Geophys. Res. Lett., 43, 2118–2126, 10.1002/2016GL068009, 2016GL068009.
  * Chelton et al. (1998) Chelton, D. B., R. A. deSzoeke, M. G. Schlax, K. El Naggar, and N. Siwertz, 1998: Geographical variability of the first baroclinic rossby radius of deformation. Journal of Physical Oceanography, 28 (3), 433–460, 10.1175/1520-0485(1998)028¡0433:GVOTFB¿2.0.CO;2.
  * Chelton et al. (2011) Chelton, D. B., M. G. Schlax, and R. M. Samelson, 2011: Global observations of nonlinear mesoscale eddies. Progress in Oceanography, 91 (2), 167–216, 10.1016/j.pocean.2011.01.002.
  * Cushman-Roisin (1994) Cushman-Roisin, B., 1994: Introduction to Geophysical Fluid Dynamics. Prentice Hall, Englewood Cliffs, N.J.
  * D’Asaro et al. (2011) D’Asaro, E., C. Lee, L. Rainville, R. Harcourt, and L. Thomas, 2011: Enhanced turbulence and energy dissipation at ocean fronts. Science, 332 (6027), 318–322.
  * D’Asaro et al. (1994) D’Asaro, E., S. Walker, and E. Baker, 1994: Structure of two hydrothermal megaplumes. Journal of Geophysical Research: Oceans, 99 (C10), 20 361–20 373, 10.1029/94JC01846.
  * D’Asaro (1988) D’Asaro, E. A., 1988: Observations of small eddies in the Beaufort Sea. Journal of Geophysical Research: Oceans, 93 (C6), 6669–6684, 10.1029/JC093iC06p06669.
  * Deremble (2016) Deremble, B., 2016: Convective plumes in rotating systems. Journal of Fluid Mechanics, 799, 27–55, 10.1017/jfm.2016.348.
  * du Plessis et al. (2019) du Plessis, M., S. Swart, I. J. Ansorge, A. Mahadevan, and A. F. Thompson, 2019: Southern Ocean seasonal restratification delayed by submesoscale wind–front interactions. Journal of Physical Oceanography, 49 (4), 1035–1053, 10.1175/JPO-D-18-0136.1.
  * Egger (2001) Egger, J., 2001: Angular momentum of $\beta$-plane flows. Journal of the Atmospheric Sciences, 58 (17), 2502–2508, 10.1175/1520-0469(2001)058¡2502:AMOPF¿2.0.CO;2.
  * Eldevik and Dysthe (2002) Eldevik, T., and K. B. Dysthe, 2002: Spiral eddies. Journal of Physical Oceanography, 32 (3), 851–869, 10.1175/1520-0485(2002)032¡0851:SE¿2.0.CO;2.
  * Eliassen (1951) Eliassen, A., 1951: Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophysica Norvegica, 5 (2), 1–42.
  * Ertel (1942) Ertel, H., 1942: Ein neuer hydrodynamischer wirbelsatz. Meteorol. Z., 59, 271–281.
  * Fjortoft (1950) Fjortoft, R., 1950: Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geophys. Publ., 17 (6), 1–52.
  * Flament et al. (1985) Flament, P., L. Armi, and L. Washburn, 1985: The evolving structure of an upwelling filament. Journal of Geophysical Research: Oceans, 90 (C6), 11 765–11 778, 10.1029/JC090iC06p11765.
  * Gentemann et al. (2020) Gentemann, C. L., and Coauthors, 2020: Saildrone: Adaptively sampling the marine environment. Bulletin of the American Meteorological Society, 101 (6), E744–E762, 10.1175/BAMS-D-19-0015.1.
  * Griffies (2004) Griffies, S. M., 2004: Fundamentals of Ocean Climate Models. Princeton University Press, Princeton, 1–528 pp., 10.2307/j.ctv301gzg.
  * Grimshaw (1975) Grimshaw, R. H. J., 1975: A note on the $\beta$-plane approximation. Tellus, 27 (4), 351–357, 10.3402/tellusa.v27i4.9982, URL https://doi.org/10.3402/tellusa.v27i4.9982.
  * Grooms (2015) Grooms, I., 2015: Submesoscale baroclinic instability in the balance equations. Journal of Fluid Mechanics, 762, 256–272.
  * Gula et al. (2016) Gula, J., M. J. Molemaker, and J. C. McWilliams, 2016: Topographic generation of submesoscale centrifugal instability and energy dissipation. Nature Communications, 7, 10.1038/ncomms12811.
  * Haine and Marshall (1998) Haine, T. W., and J. Marshall, 1998: Gravitational, symmetric, and baroclinic instability of the ocean mixed layer. Journal of physical oceanography, 28 (4), 634–658.
  * Haynes and McIntyre (1987) Haynes, P. H., and M. E. McIntyre, 1987: On the evolution of vorticity and potential vorticity in the presence of diabatic heating and frictional or other forces. Journal of the Atmospheric Sciences, 44 (5), 828–841, 10.1175/1520-0469(1987)044¡0828:OTEOVA¿2.0.CO;2, URL https://doi.org/10.1175/1520-0469(1987)044¡0828:OTEOVA¿2.0.CO;2.
  * Haynes and McIntyre (1990) Haynes, P. H., and M. E. McIntyre, 1990: On the conservation and impermeability theorems for potential vorticity. Journal of the Atmospheric Sciences, 47 (16), 2021–2031, 10.1175/1520-0469(1990)047¡2021:OTCAIT¿2.0.CO;2, URL https://doi.org/10.1175/1520-0469(1990)047¡2021:OTCAIT¿2.0.CO;2.
  * Helfrich and Battisti (1991) Helfrich, K. R., and T. M. Battisti, 1991: Experiments on baroclinic vortex shedding from hydrothermal plumes. Journal of Geophysical Research: Oceans, 96 (C7), 12 511–12 518, 10.1029/90JC02643.
  * Holmes et al. (2013) Holmes, R. M., L. N. Thomas, L. Thompson, and D. Darr, 2013: Potential vorticity dynamics of tropical instability vortices. Journal of Physical Oceanography, 44 (3), 995–1011, 10.1175/JPO-D-13-0157.1.
  * Holton (1992) Holton, J. R., 1992: An Introduction to Dynamic Meteorology. 3rd ed., Academic Press, 1–507 pp.
  * Hoskins and Bretherton (1972) Hoskins, B. J., and F. P. Bretherton, 1972: Atmospheric frontogenesis models: mathematical formulation and solution. Journal of the Atmospheric Sciences, 29, 11–37.
  * Houze (1993) Houze, R., 1993: Cloud dynamics, International Geophysics Series, Vol. 53. Academic Press, 573 pp.
  * Kloosterziel et al. (2007) Kloosterziel, R. C., G. F. Carnevale, and P. Orlandi, 2007: Inertial instability in rotating and stratified fluids: barotropic vortices. Journal of Fluid Mechanics, 583, 379–412, 10.1017/S0022112007006325.
  * Kloosterziel et al. (2017) Kloosterziel, R. C., G. F. Carnevale, and P. Orlandi, 2017: Equatorial inertial instability with full Coriolis force. Journal of Fluid Mechanics, 825, 69–108, 10.1017/jfm.2017.377.
  * Kloosterziel and van Heijst (1991) Kloosterziel, R. C., and G. J. F. van Heijst, 1991: An experimental study of unstable barotropic vortices in a rotating fluid. Journal of Fluid Mechanics, 223, 1–24, 10.1017/S0022112091001301.
  * Konstianoy and Belkin (1989) Konstianoy, A. G., and I. M. Belkin, 1989: A survey of observations on intrathermocline eddies in the world ocean, 821–841. Elsevier Oceanography Series, Elsevier Academic Press, 10.1016/S0422-9894(08)70223-X.
  * Lazar et al. (2013) Lazar, A., A. Stegner, R. Caldeira, C. Dong, H. Didelle, and S. Viboud, 2013: Inertial instability of intense stratified anticyclones. part 2. laboratory experiments. Journal of Fluid Mechanics, 732, 485–509, 10.1017/jfm.2013.413.
  * Legg and McWilliams (2001) Legg, S., and J. C. McWilliams, 2001: Convective modifications of a geostrophic eddy field. Journal of Physical Oceanography, 31 (4), 874–891, 10.1175/1520-0485(2001)031¡0874:CMOAGE¿2.0.CO;2.
  * Lilly and Rhines (2002) Lilly, J. M., and P. B. Rhines, 2002: Coherent eddies in the Labrador Sea observed from a mooring. Journal of Physical Oceanography, 32, 585–598.
  * MacKinnon et al. (2019) MacKinnon, J. A., M. H. Alford, G. Voet, K. L. Zeiden, T. M. Shaun Johnston, M. Siegelman, S. Merrifield, and M. Merrifield, 2019: Eddy wake generation from broadband currents near palau. Journal of Geophysical Research: Oceans, 124 (7), 4891–4903, 10.1029/2019JC014945.
  * MacKinnon et al. (2021) MacKinnon, J. A., and Coauthors, 2021: A warm jet in a cold ocean. Nature Communications, 12 (1), 2418, 10.1038/s41467-021-22505-5.
  * Mahdinia et al. (2017) Mahdinia, M., P. Hassanzadeh, P. S. Marcus, and C.-H. Jiang, 2017: Stability of three-dimensional Gaussian vortices in an unbounded, rotating, vertically stratified, Boussinesq flow: linear analysis. Journal of Fluid Mechanics, 824, 97–134, 10.1017/jfm.2017.303.
  * Marchesiello et al. (2011) Marchesiello, P., X. Capet, C. Menkes, and S. C. Kennan, 2011: Submesoscale dynamics in tropical instability waves. Ocean Modelling, 39 (1), 31–46, 10.1016/j.ocemod.2011.04.011, URL http://www.sciencedirect.com/science/article/pii/S1463500311000849.
  * Marshall and Nurser (1992) Marshall, J. C., and A. J. G. Nurser, 1992: Fluid dynamics of oceanic thermocline ventilation. Journal of Physical Oceanography, 22 (6), 583–595, 10.1175/1520-0485(1992)022¡0583:FDOOTV¿2.0.CO;2.
  * McCoy et al. (2020) McCoy, D., D. Bianchi, and A. L. Stewart, 2020: Global observations of submesoscale coherent vortices in the ocean. Progress in Oceanography, 189, 102 452, 10.1016/j.pocean.2020.102452.
  * McDowell and Rossby (1978) McDowell, S. E., and H. T. Rossby, 1978: Mediterranean water: an intense mesoscale eddy off the bahamas. Science, 202, 1085–1087.
  * McWilliams (1985a) McWilliams, J. C., 1985a: Submesoscale, coherent vortices in the ocean. Reviews of Geophysics, 23 (2), 165–182, 10.1029/RG023i002p00165.
  * McWilliams (1985b) McWilliams, J. C., 1985b: A uniformly valid model spanning the regimes of geostrophic and isotropic, stratified turbulence: Balanced turbulence. Journal of Atmospheric Sciences, 42 (16), 1773–1774, 10.1175/1520-0469(1985)042¡1773:AUVMST¿2.0.CO;2.
  * McWilliams (2016) McWilliams, J. C., 2016: Submesoscale currents in the ocean. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 472 (2189), 1–32, 10.1098/rspa.2016.0117.
  * Meunier et al. (2018) Meunier, T., and Coauthors, 2018: Intrathermocline eddies embedded within an anticyclonic vortex ring. Geophysical Research Letters, 45 (15), 7624–7633, 10.1029/2018GL077527.
  * Müller (1995) Müller, P., 1995: Ertel’s potential vorticity theorem in physical oceanography. Reviews of Geophysics, 33 (1), 67–97.
  * Munk et al. (2000) Munk, W., L. Armi, K. Fischer, and F. Zachariasen, 2000: Spirals on the sea. Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, The Royal Society, Vol. 456, 1217–1280.
  * Mutabazi et al. (1992) Mutabazi, I., C. Normand, and J. E. Wesfreid, 1992: Gap size effects on centrifugally and rotationally driven instabilities. Physics of Fluids A: Fluid Dynamics, 4 (6), 1199–1205, 10.1063/1.858238.
  * Naveira Garabato et al. (2019) Naveira Garabato, A. C., and Coauthors, 2019: Rapid mixing and exchange of deep-ocean waters in an abyssal boundary current. Proceedings of the National Academy of Sciences, 116 (27), 13 233, 10.1073/pnas.1904087116.
  * Nurser and Bacon (2014) Nurser, A. J. G., and S. Bacon, 2014: The Rossby radius in the Arctic Ocean. Ocean Sci., 10, 967–975, 10.5194/os-10-967-2014.
  * Onken et al. (2020) Onken, R., B. Baschek, and I. M. Angel-Benavides, 2020: Very high-resolution modelling of submesoscale turbulent patterns and processes in the Baltic Sea. Ocean Science, 16 (3), 657–684, 10.5194/os-16-657-2020.
  * Pedlosky (1987) Pedlosky, J., 1987: Geophysical Fluid Dynamics. 2nd ed., Springer-Verlag, Springer New York, New York, 1–710 pp.
  * Peixoto and Oort (1992) Peixoto, J. P., and A. H. Oort, 1992: Physics of Climate. 1st ed., American Institute of Physics Press, 1–520 pp.
  * Rayleigh (1917) Rayleigh, L., 1917: On the dynamics of revolving fluids. Proc. Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 93, 148–154.
  * Riley et al. (2006) Riley, K. F., M. P. Hobson, and S. J. Bence, 2006: Mathematical Methods for Physics and Engineering: A Comprehensive Guide. 3rd ed., Cambridge University Press, Cambridge, 10.1017/CBO9780511810763.
  * Riser et al. (1986) Riser, S. C., W. B. Owens, H. T. Rossby, and C. C. Ebbesmeyer, 1986: The structure, dynamics, and origin of a small-scale lens of water in the western North Atlantic thermocline. Journal of Physical Oceanography, 16 (3), 572–590, 10.1175/1520-0485(1986)016¡0572:TSDAOO¿2.0.CO;2.
  * Rudnick (2001) Rudnick, D. L., 2001: On the skewness of vorticity in the upper ocean. Geophysical Research Letters, 28 (10), 2045–2048, 10.1029/2000GL012265.
  * Samelson (1992) Samelson, R. M., 1992: Fluid Exchange across a Meandering Jet. Journal of Physical Oceanography, 22 (4), 431–444, 10.1175/1520-0485(1992)022¡0431:FEAAMJ¿2.0.CO;2.
  * Scully-Power (1986) Scully-Power, P., 1986: Navy Oceanographer Shuttle observations, STS 41-G Mission Report. Tech. rep., DTIC Document, Newport, Rhode Island.
  * Shakespeare (2016) Shakespeare, C. J., 2016: Curved density fronts: Cyclogeostrophic adjustment and frontogenesis. Journal of Physical Oceanography, 46, 3193–3207.
  * Shcherbina et al. (2013) Shcherbina, A. Y., E. A. D’Asaro, C. M. Lee, J. M. Klymak, M. J. Molemaker, and J. C. McWilliams, 2013: Statistics of vertical vorticity, divergence, and strain in a developed submesoscale turbulence field. Geophysical Research Letters, 40 (17), 4706–4711, 10.1002/grl.50919.
  * Simoes-Sousa et al. (2021) Simoes-Sousa, I. T., I. C. A. Silveira, A. Tandon, G. R. Flierl, C. H. A. Ribeiro, and R. P. Martins, 2021: The barreirinhas eddies: Stable energetic anticyclones in the near-equatorial south atlantic. Frontiers in Marine Science, 8, 28, 10.3389/fmars.2021.617011.
  * Smith (2007) Smith, K. S., 2007: The geography of linear baroclinic instability in Earth’s oceans. Journal of Marine Research, 65 (5), 655–683, 10.1357/002224007783649484.
  * Solberg (1936) Solberg, H., 1936: Le mouvement d’inertie de l’atmosphere stable et son role dans la theorie des cyclones. Sixth Assembly, Paul Dupont, Edinburgh, Union Geodesique et Geophysique Internationale, 66–82.
  * Spall (1995) Spall, M. A., 1995: Frontogenesis, subduction, and cross-front exchange at upper ocean fronts. Journal of Geophysical Research: Oceans, 100 (C2), 2543–2557, 10.1029/94JC02860, URL https://doi.org/10.1029/94JC02860.
  * Srinivasan et al. (2019) Srinivasan, K., J. C. McWilliams, M. J. Molemaker, and R. Barkan, 2019: Submesoscale vortical wakes in the lee of topography. Journal of Physical Oceanography, 49 (7), 1949–1971, 10.1175/JPO-D-18-0042.1.
  * Stegner (2014) Stegner, A., 2014: Oceanic Island Wake Flows in the Laboratory, chap. 14, 265–276. American Geophysical Union (AGU), 10.1002/9781118856024.ch14.
  * Stegner et al. (2004) Stegner, A., P. Bouret-Aubertot, and T. Pichon, 2004: Nonlinear adjustment of density fronts. Part 1. the Rossby scenario and the experimental reality. Journal of Fluid Mechanics, 502, 335–360, 10.1017/S002211200300781X.
  * Straub (1993) Straub, D. N., 1993: On the transport and angular momentum balance of channel models of the Antarctic Circumpolar Current. Journal of Physical Oceanography, 23 (4), 776–782, 10.1175/1520-0485(1993)023¡0776:OTTAAM¿2.0.CO;2.
  * Thomas (2008) Thomas, L. N., 2008: Formation of intrathermocline eddies at ocean fronts by wind-driven destruction of potential vorticity. Dynamics of Atmospheres and Oceans, 45 (3), 252–273, 10.1016/j.dynatmoce.2008.02.002.
  * Thomas and Lee (2005) Thomas, L. N., and C. M. Lee, 2005: Intensification of ocean fronts by down-front winds. J. Phys. Oceanogr., 35, 1086–1102.
  * Thomas et al. (2008) Thomas, L. N., A. Tandon, and A. Mahadevan, 2008: Submesoscale processes and dynamics. Ocean Modeling in an Eddying Regime, Geophys. Monogr. Ser, 177, 17–38.
  * Thomas et al. (2013) Thomas, L. N., J. R. Taylor, R. Ferrari, and T. M. Joyce, 2013: Symmetric instability in the Gulf Stream. Deep Sea Research Part 2, 91, 91–110.
  * Thompson et al. (2016) Thompson, A. F., A. Lazar, C. E. Buckingham, A. C. Naveira Garabato, G. M. Damerell, and K. J. Heywood, 2016: Open-ocean submesoscale motions: A full seasonal cycle of mixed layer instabilities from gliders. J. Phys. Oceanogr., 10.1175/JPO-D-15-0170.1.
  * Timmermans et al. (2008) Timmermans, M.-L., J. Toole, A. Proshutinsky, R. Krishfield, and A. Plueddemann, 2008: Eddies in the Canada Basin, Arctic Ocean, observed from ice-tethered profilers. Journal of Physical Oceanography, 38 (1), 133–145, 10.1175/2007JPO3782.1.
  * Vallis (2017) Vallis, G. K., 2017: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. 2nd ed., Cambridge University Press, Cambridge, 10.1017/9781107588417.
  * Zhao et al. (2014) Zhao, M., M.-L. Timmermans, S. Cole, R. Krishfield, A. Proshutinsky, and J. Toole, 2014: Characterizing the eddy field in the Arctic Ocean halocline. Journal of Geophysical Research: Oceans, 119 (12), 8800–8817, 10.1002/2014JC010488.