Arham Amin Khan Department of Mechanical Engineering, University of Delaware,
Newark, Delaware, USA Joseph J. KuehlJoseph Kuehl, Department of Mechanical
Engineering, University of Delaware, 130 Academy Street, Newark, DE 19716,
USA. Department of Mechanical Engineering, University of Delaware, Newark,
Delaware, USA

# Multiple Equilibrium States of the Loop Current in the Gulf of Mexico

###### Abstract

It is known that western boundary currents, which encounter a gap in their
supporting boundary, assume two dominant steady states: a loop current state
and a gap leaping state, and that transitions between these states display
hysteresis. However, a question of whether the idealized geometries considered
to date apply to the Gulf of Mexico Loop Current (LC) remained. Here, the
nonlinear potential vorticity advection-diffusions equations are solved, for
Gulf of Mexico topography, using Newton’s Method. We demonstrate that, in
application to the LC in the Gulf of Mexico, the original conclusions do hold
and additionally describe peculiarities of the more realistic steady states
that have implications for the LC modeling and forecasting.

††journal: jpo

##  1 Introduction

Responding to the external forcing and intrinsic dynamics, the Loop Current
(LC) in the Gulf of Mexico exhibits dramatic variations. In a retracted state,
the LC leaps directly from the Straits of Yucatan near Cuba to the Florida
Straits, and in an extended state, the LC penetrates far into the Gulf of
Mexico and forms a large eddy which then may pinch off and drift westward
(Donohue et al. 2016b, a; Lipphardt et al. 2008) (Figure 1 upper right and
left panels, respectively). This process exerts a major control over the
strength of hurricanes (Kafatos et al. 2006), dispersion of pollutants
(Weisberg et al. 2017), offshore energy operations (Koch et al. 1991), and
even coastal ecosystem health (Hetland et al. 1999; Weisberg et al. 2014).
Despite decades of scientific inquiry and major field program initiatives, the
fundamental problem of LC predictability remains (Committee on Advancing
Understanding of Gulf of Mexico Loop Current Dynamics et al. 2018). Sheremet
(2001) originally discovered that an idealized western boundary current with
Munk profile running along a straight wall with a gap, for the same governing
parameters, may be in two steady states: penetrating into the western basin
due to the $\beta$-effect and leaping across the gap due to flow inertia.
Furthermore, the time variability of the current will involve a hysteresis, a
dependence on prior evolution. These results were later confirmed in rotating
table laboratory experiments for more general barotropic boundary currents
(Sheremet and Kuehl 2007; Kuehl and Sheremet 2009) as well as for baroclinic
boundary currents (Kuehl and Sheremet 2014; McMahon et al. 2020). In addition,
more advanced modeling studies have confirmed the existence of multiple steady
states with hysteresis for idealized “real ocean parameters” (Wang et al.
2010; Yuan and Wang 2011; Song et al. 2019; Mei et al. 2019; Yuan et al.
2019). A question of whether this idealized geometry applies to realistic
western boundary currents remained. In particular, whether the orientation of
the gap and the upstream jet pointing into the gap may completely change the
current system behavior. In this paper we demonstrate, that in application to
the LC in the Gulf of Mexico, the original conclusions do hold and
additionally describe peculiarities of the more realistic steady states that
have implications for LC modeling and forecasting.

##  2 Model

In order to understand the role of the lateral boundary constraints on LC
pathways the potential vorticity advection-diffusion and Poisson equations

$J(\psi,q)=\nu\nabla^{2}\omega,\qquad\nabla^{2}\psi=\omega$ (1)

are solved on a $\beta$-plane for realistic Gulf of Mexico geometry, where
$q=\beta y+\omega$ is the potential vorticity, $\beta$ is the beta effect, $y$
is the northward distance, $\omega$ is the relative vorticity, $\psi$ is the
flow stream function. The lateral boundary is taken to be the 250m isobath
where the bottom rapidly drops from the shelf to the deep continental slope
(Figure 1). We assume only the planetary beta effect by considering the swift
flow, roughly in the top 500m layer, of a fixed thickness, the portion of the
flow which is largely shielded from the deep bathymetry by the main
thermocline. The wind forcing enters the problem via the inflow boundary
condition as an integral effect over the subtropical gyre.

The physical problem has two governing parameters: the western boundary
current inertial length scale $L_{I}=\sqrt{U/\beta}$ and the Munk viscous
layer width $L_{M}=(\nu/\beta)^{1/3}$ dependent on the coefficient of lateral
turbulent diffusion $\nu$, where $U=Q/(HL)$ is the zonal velocity scale in the
Sverdrup interior, expressed in terms of the total subtropical gyre transport
$Q$, thermocline depth $H$ and the subtropical gyre meridional scale $L$. The
corresponding nondimensional parameters are $\lambda_{I}=L_{I}/L$ and
$\lambda_{M}=L_{M}/L$. However, the structure of the western boundary current
depends on the boundary layer Reynolds number
$R=UL_{I}/\nu=(L_{I}/L_{M})^{3}$.

Figure 1: Gulf of Mexico topography with light grey indicating isobaths
5:100:3500m and dark grey indicating isobaths 250, 500, 1000, 2000m. Grey mask
indicates land. Superimposed are sea surface height contours for: Upper Left)
on and around the 193rd day of 2015. Upper Right) on and around the 57th day
of 2017. Lower) on and around the 154th day of 2017. Colorbar units are in
$cm$.

The numerical model utilized in this study was developed to support a series
of rotating table laboratory experiments. Both the experimental setup and
validation of the numerical model have been well-established in the literature
(Sheremet and Kuehl 2007; Kuehl and Sheremet 2009, 2014; McMahon et al. 2020),
so only a brief summary is provided here. In nondimensional form, (1) has
$\lambda_{M}$ instead of $\nu$ and the potential vorticity becomes
$q=y+\delta_{I}^{2}\omega$. The domain, as partially seen in figures 1 and 2,
spans the Gulf of Mexico and part of the western Atlantic Ocean (77 - 98 west
longitude and 13 - 30 north latitude). The kinematic conditions for solving
the elliptic equation are $\psi=0$ along all points on the North American
continent and the southern boundary. The inflow is prescribed as
$\psi=\Psi_{B}(y)$ along the eastern boundary, varying from 0 at the
southeastern corner to 1 at Cuba (corresponding to the dimensional value of
the upstream western boundary current transport $Q$), and remaining 1 to the
northeastern corner. The outflow through the northern boundary east of Florida
is specified with the Neumann condition. The dynamical conditions are no-slip,
zero velocity, at all land boundaries. The values of vorticity at the solid
boundaries were calculated assuming the antisymmetry of the tangential
velocity component as it was extended outside the fluid domain, which reduces
to the formula by Thom (1933) for a straight wall.

The numerical problem is solved using standard finite differences on a
rectangular grid dividing the domain into $N_{x}\times N_{y}$ cells. For small
boundary layer Reynolds numbers $R=(\lambda_{I}/\lambda_{M})^{3}$ simple
explicit iterations with treating the nonlinear terms as perturbations work
well, but for the moderate $R$ the iterations fail to converge. In this case
Newton’s method has be to employed for finding steady solutions. We consider a
state vector $X=(\omega,\psi)$ consisting of values at all grid nodes
including the boundaries, the size of this vector is
$M=(N_{x}+1)*(N_{y}+1)*2$. Substituting an initial guess $X_{0}$ into (1)
results in the vector of residuals $F(X_{0})$ at each grid node of the same
size $M$. In order to find the next iteration $X_{1}$ that brings residual
closer to vanishing $F(X)=0$, we need to calculate the Jacobian matrix
$J_{F}[X_{0}]$ (of size $M\times M$ which depends on $X_{0}$) of all first-
order partial derivatives of $F$ with respect to $X$ and then solve the linear
system

$J_{F}[X_{0}](X_{1}-X_{0})=-F(X_{0})$ (2)

The iterations $X_{i}$ then continue until the residual vanishes. The
iterations were stopped when the residual function (at each node and the
overall standard deviation) reduced to and plateaued at $O(-12)$. It usually
sufficed to take 5-7 steps for that. The elements of the Jacobian matrix can
be calculated analytically by considering the variational problem
(nondimensional) corresponding to 1.

$J(\delta\psi,q)+J(\psi,\delta
q)-\lambda_{M}^{3}\nabla^{2}\delta\omega=0,\qquad-\nabla^{2}\delta\psi-\delta\omega=0,$
(3)

The variations of the boundary conditions are trivial. Finite difference
approximations result in a sparse banded type of $J_{F}$, and the grids of
size upto $1000\times 1000$ can be solved on a computer with 24 GiB of
operational memory. For the present calculation we used the horizontal spacing
of $1/20$ of a degree of arc in both directions or $5.55km$.

##  3 Results

Steady states of the nonlinear system are found with Newton’s method applied
to the discrete finite-difference approximation of the equations. This is
particularly important, as one can then obtain flow patterns that are not
obscured by the external variability (i.e. we are able to identify the
underlying dynamics governing the LC). Furthermore, we can determine if
distinctly different equilibrium flow patterns are possible for the same
values of governing parameters such as the western boundary current transport,
as was originally predicted by Sheremet (2001). Indeed, our investigation
shows that for the LC restricted by the geometry of Yucatan and Florida, there
exist 5 distinct equilibria (3 stable and 2 unstable) relevant for LC
predictability and forecasting. The behavior of the system is summarized in
Figure 2. We fixed the turbulent viscosity to a typical value $\nu=450$
$m^{2}s^{-1}$ used in other numerical models to produce realistic western
boundary currents: $300$ $m^{2}s^{-1}$ in tropics and $600m^{2}s^{-1}$ in
northern parts of the Gulf Stream (Large et al. 2001). The Reynolds number,
$R$, then was varied, thus varying the boundary current transport since $R\sim
Q^{3/2}$. For realistic western boundary currents the inertial effects must
dominate over friction, $R>1$. In our calculations $R=10$ corresponds to a
typical western boundary current inertial width of $L_{I}\approx 60km$. That
value comes from the Gulf Stream transport of $30$ Sverdrups or $30\times
10^{6}m^{3}s^{-1}$ spread meridionally over $L=1112km$ and vertically over the
$e$-folding depth scale of $360m$. With $\beta=2.09\cdot 10^{-11}(m\cdot
s)^{-1}$ for latitude of $24$N, the Sverdrup interior velocity scale is
$U=7.5cms^{-1}$.

The LC can be conveniently characterized by its extent $X_{p}$, which we
define as the farthest penetration distance of the streamline $\psi=1/2$ into
the Gulf of Mexico from Cuba (shown by red line in panels A and C). The
maximum value of stream function $\psi_{max}$ is also helpful for
characterizing different states. The stream function is scaled by the upstream
western boundary current transport $Q$. All the flow that enters the
computational domain in the Carribean Sea (the unit transport) must exit as
the Gulf Stream northward along Florida. However, the $\psi_{max}$ can exceed
unity as the LC drives the recirculation eddy inside the Gulf of Mexico
(Panels B-D). The tracing of $X_{p}$ and $\psi_{max}$ for the steady solutions
are shown in the last panels at lower right as a function of $R$. The letters
A through G along the $X_{p}$ curve (red) correspond to the flow patterns
shown in the corresponding Panels.

If we start from the linear solution, $R=0$, no-advection, (Panel A) the
streamlines are blocked by Yucatan and do not penetrate westward. As the
advection increases, the LC rapidly penetrates into the Gulf of Mexico
reaching the northern coast at $R=2.5$. After that the flow pattern (like the
one shown in Panel B for $R=10$) changes little until $R\simeq 30$. Then the
intensity of the LC eddy increases substantially. Near $R\simeq 40$ the LC
eddy occupies nearly the whole area between Yucatan and Florida and its
transport increases by the factor 1.4-2. There are actually six steady
solutions possible for the same $R$ in that region. They have similar patterns
(as shown in Panel C) but differing by $\psi_{max}$ and by the points at which
the gyre touches the coast.

Further tracing the solution branches, takes us back to smaller $R$ and a
smaller LC eddy (Panel D). There are three folds near $R=10$ and
correspondingly four more steady solutions marked by D,E,F,G each representing
a different extent of the LC. Finally, in the state G the LC assumes its
retracted state and remains such as $R$ increases beyond.

##  4 Discussion

Figure 2: The streamfunction patterns, the maximum streamfunction, and the LC
extent illustrating tracing of steady solutions as the boundary Reynold
number, $R$, is varied. The panels B,E,G (stable sates) and panels D,F
(unstable states) represent five solutions for exactly same parameter $R=10$.
The panel/state letters are marked along the LC Extent vs $R$ (red) plot in
the lower right. The definition of LC extent is marked by red line and red
circles in panels A and C. The streamfunction contour interval is 0.1 of the
total transport. The land mask was generated based on the 250m isobath, except
Cayman, Jamaica and small islands in the Carribean were ignored in order to
specify the zonal inflow.

Around $R=10$ (which correspond to typical oceanographic conditions) there are
5 steady solutions (B,D,E,F,G) possible in the system. According to the
alternating stability of branches in bifurcation theory, the equilibria B,E,G
(increasing $R$) are stable, while D and F (decreasing $R$) are unstable; see
Ierley and Sheremet (1995); Sheremet (2002) for further details. We emphasize
that we mean here the stability with respect to only one, but very
significant, eigenvalue whose imaginary part is zero (a non-oscillatory eigen
mode). At the folds of the solution branches, the real part of this eigenvalue
(representing the exponential decay or amplification of the mode) also
vanishes, the eigenvalue becomes zero, which gives rise to the non-uniqueness
of the steady solutions and multiple steady states. We note that there may be
other modes affecting the linear stability, but those are oscillatory and may
lead to Hopf bifurcations. Conducting a full stability analysis is beyond the
scope of this paper, which will be done in a subsequent work.

The stability type of the states is extremely important to Loop Current
predictability/forecasting, as the unstable states D and F act as a separatrix
between the stable basins of attraction corresponding to solutions B, E, G.
This can be understood by considering the simpler case of vibrating system
analysis. In vibrating systems analysis, the state and evolution of the system
are often cast into a state space (or phase space) diagram. This diagram is
constructed by first identifying steady state solutions of the system and then
identifying their stability type. The system is repelled from unstable
solutions and attracted to stable solutions. Thus, the system evolution in the
state space can be identified with unstable solutions delineating the basins
of attraction of the stable solutions. Here, we view the LC as a dynamical
system, similar to Lugo-Fernández (2007), and we can interpret the unstable
states as the higher-dimensional separatrices of our system. That is, flow in
the state D, if perturbed, will evolve toward B or E depending on the sign and
structure of perturbation. Similarly, the flow in state F will tend to evolve
toward E or G. Thus, knowing the structure of unstable LC states, allows for
insight into the transitional dynamics between the stable LC states.

To validate our approach, we compared our results to the satellite altimeter
Sea Surface Height (SSH) dataset for the Gulf of Mexico. The dataset is
available from GCOOS (URL https://geo.gcoos.org/ssh/data1/) and now is
spanning the years 2004-2019 (we excluded the year 2020 since it currently
only spans January and February). Details of the data and its processing can
be found on the GCOOS website and in Leben et al. (2002). We measured the
extent of the LC as a distance from the point (-84.250E, 22.666N) at the
northwestern coast of Cuba to the point along the 240km wide swath directed
along the azimuth of -35 degrees from true north where the isoline of half LC
transport is located. In the SSH field that corresponds to the isoline of
35cm, half the difference between the values at Cuba and Yucatan or between
Cuba and Florida, which on average is about 70cm. This SSH difference is a
proxy for the total geostrophic transport of the current. The LC extent is
expressed in terms of the latitude of that point (Figure 3 lower left Panel,
green dots; the blue line is a linear fit). The same algorithm for LC extent
is applied to our steady numerical solutions and the data are superimposed as
red dots; we matched the mean value of $\Delta$ SSH=70cm to correspond to the
realistic numerical case with R=10 discussed above.

Most of SSH derived data points fall in the range of LC latitudes between our
states D and G and correspond to similar transports. There is a small but
noticeable decrease of LC Extent with increased transport of the current (note
the linear fit, blue line). Next we performed statistical averaging of SSH
fields by selecting only the fields with the LC extent within a small range
(plus or minus 0.1 degrees latitude) around the intersection points of blue
and red curves, and thus obtained averages corresponding to states D, E and G
illustrated in the other panels of Figure 3. They indeed have remarkable
resemblance to the numerically calculated patterns of Figure 2 (note the same
arrangement for ease of comparison). The state G is a stable fully retracted
LC, E is also a stable state with a medium extent. The state D is unstable,
and, as we mentioned earlier, it acts as a boundary of the stable basin. If LC
surpasses that extent, it will tend to evolve towards state B, continue to
grow and in real ocean will pinch off a westward drifting LC eddy. That is why
the state D is not realized in the SSH observations.

Figure 3: The LC Extent (lower left panel): from the satellite altimetry SSH
fields (green dots) and a linear fit (blue line); the red dots are LC Extent
from steady numerical solutions. The statistically averaged SSH fields
corresponding to the LC near the states D,E,G are shown the surrounding
panels, the SSH units are cm. The red rectangle in the upper left panel shows
the swath where the 35cm SSH contour was searched.

It is quite interesting that our analysis identifies 3 stable LC states, when
the traditional view has always been that of a 2 state system: extended and
retracted. Though this should not be too surprising given the difficulty in
identifying ‘true Loop Current states’ from time-dependent observational or
numerical datasets (Kuehl et al. 2014; Weisberg and Liu 2017). However, the
stable flow patterns identified do have precedent in the observational data
records. Figure 1 identifies sea surface height fields from the historical
satellite altimetry data that roughly correspond to the stable LC states
identified in Figure 2: Figure 1 upper left panel is a looping state that was
present on and around the 193rd day of 2015 and corresponds to state B. Figure
1 upper right panel is a leaping state that was present on and around the 57th
day of 2017 and corresponds to state G. Figure 1 lower panel is an
intermediate state that was present on and around the 154th day of 2017 and
corresponds to state E. Note, while we can not guarantee that these
observational flow patterns are ‘true Loop Current states’, all flow fields
identified persisted for approximately 2 months.

To further investigate the SSH observational data, a Smoothed Orthogonal
Decomposition (SOD) (Chelidze and Zhou 2006; Khan et al. 2020) investigation
was conducted. SOD considers the constrained maximization problem

$\underset{\phi}{max}\|X\phi\|\hskip 28.45274pt\textrm{subject to}\hskip
28.45274pt\underset{\phi}{min}\|V\phi\|,$ (4)

where $V$ is the temporal derivative of $X$. The corrsponding Rayleigh’s
quotient becomes

$\underset{\phi}{max}\left\\{\lambda(\phi)=\frac{\|X\phi\|}{\|V\phi\|}\right\\}\rightarrow\underset{\phi}{max}\left\\{\lambda(\phi)=\frac{\phi^{T}\Sigma_{x}\phi}{\phi^{T}\Sigma_{v}\phi}\right\\},$
(5)

where $\Sigma_{v}$ is the covariance matrix of V. Variational techniques may
be applied and result in a generalized singular value decomposition (GSVD)
problem for $X$ and $V$ or correspondingly the generalized eigenvalue problem
$\Sigma_{x}\phi_{i}=\lambda_{i}\Sigma_{v}\phi_{i}$. It follows that SOD
considers those modes which maximize amplitude variance projection, while at
the same time being as smooth in time a possible. As nature tends to behave in
a “smooth” fashion, this “smoothness” property has been shown to be more
physically relevant than the amplitude based Proper Orthogonal Decomposition
(POD, or EOF, or PCA) analysis (Chelidze and Liu 2008). In the case of the
Gulf of Mexico, we might expect the dominant LC states to be separated by
time-scale which is indeed what is observed (Figure 4). The top row of the
figure identifies (from left to right) the dominant signal of an extended LC
state, an intermediate LC state, and a retracted LC state. The middle row
panels identify a periodic eddy-shedding LC state, and the bottom row panels
illustrate oscillatory behavior of the retracted state. These results, in
combination with numerical work presented above, strongly suggest the presence
of at least three dominant LC states in the Gulf of Mexico, in addition to a
periodic eddy-shedding states similar to that identified by Kuehl and Sheremet
(2014).

Figure 4: SOD modes shapes calculated from SSH observational data. Mode are
identified by mode number, the most dominant being 1 and reducing as mode
number increases. Grey identifies dry land.

The identified states are also consistent with our current understanding of
the governing vorticity dynamics of LC systems and their structure highlights
the importance of current-boundary interaction. The LC system essentially
involves a combination of: 1) a boundary current separation problem. 2) a
current reattachment problem. 3) a vorticity budget closure problem.

The question of flow separation from a curved boundary (Yucatan Peninsula) is
among the most difficult in fluid mechanics. A criterion based on the
cancellation of the inviscid flow singularity at the point of separation is
known as Brillouin-Villat condition after Brillouin (1911); Villat (1914) who
proposed it. The analogy of the gap leaping problem and the teapot effect was
discussed in Sheremet (2001). The separation of pouring flow from the teapot
spout was investigated by Vanden-Broeck and Keller (1989) who constructed
ideal fluid solutions with separation from arbitrary points downstream of the
lip and concluded that it is the conditions in the far field downstream that
are crucial. Thus, in the gap leaping problem it follows that the position of
flow reattachment (to the west Florida coast) exerts a more significance
control over the LC state than the separation point (i.e. the reattachment
point controls the separation point, not vice-versa). The present
calculations, with viscosity, suggest that the range of movement of the
reattachment point is much greater in comparison.

We did investigate the sensitivity of results to boundary conditions by
varying the coefficient of proportionality $k$ in the formula connecting the
boundary vorticity $\zeta_{0}$ to the nearby tangential velocity
$(\psi_{1}-\psi_{0})/h$: $\zeta_{0}=k*(\psi_{1}-\psi_{0})/h^{2}$, where the
values $\zeta_{0}$ and $\psi_{0}$ are at the boundary and $\psi_{1}$ is the
value of the stream function one grid step $h$ away from the boundary. The
standard case (no-slip) reported here (Thom (1933), $k=2$) and the partial
slip ($k=1$) give very close flow patterns since they produce realistic flow
separation from Yucatan and reattachment at Florida. In fact, the pattern
corresponding to state E can be maintained for $k$ as small as 0.43. In
contrast, if the slip condition (no stress, $k=0$) were specified, it would
prevent the current from separating and it would follow the coastline around
Yucatan into the western Gulf of Mexico.

The vorticity dynamics of LC systems have been thoroughly addressed by the
series of works by Sheremet and Kuehl (referenced above), and recalling
Pichevin and Nof (1997), can be applied to understand dynamical balances of
each state.

  * •

The low momentum branch B, has insufficient inertia to separate from the
Yucatan topography before 24.5 north latitude. However, unlike state A, it
does have enough momentum to overshot the latitude of southern Florida and
must eventually make a right turn as describe originally by Reid (1972). This
state would normally result in a periodic eddy shedding state if not for the
northern Gulf of Mexico Slope. The vorticity advection into the LC system can
be balanced by a periodic eddy shedding state (Pichevin and Nof 1996; Kuehl
and Sheremet 2014) or in the steady state through interactions with northern
Gulf of Mexico topography.

  * •

The high momentum branch G, has sufficient inertia to separate early from the
Yucatan topography. The effective radius of curvature of the current flowing
directly from the Yucatan to the Florida Straits is larger than the distance
between the two coastlines and thus the current traverses this gap fairly
undisturbed.

  * •

The middle branch E is a balance between the the low and high inertial
branches. The current has sufficient inertial to separate from the Yucatan
topography and leap to the west Florida coast. However, increased
contact/interactions with the west Florida coast are required to dissipate
excess vorticity advected into the Gulf of Mexico, very similar to that
documented in Kuehl and Sheremet (2014).

Note, it seems counterintuitive that the flow, as the intertia is increaded,
will leap from Yucatan to Florida, despite the fact the the jet initially
points to the interior of the Gulf. The inertial flow due to the
$\beta$-effect has a characteristic width $L_{I}$. If this scale becomes
comparable with or larger than the gap half width, then the current cannot
squeeze into the Gulf of Mexico, in other words, the incoming and outgoing
boundary current would merely not fit through the gap (see the curvature based
transition in Kuehl and Sheremet (2014)).

Conclusions: We identified multiple equilibrium states of the Gulf of Mexico
Loop Current by solving with Newton’s method the nonlinear steady potential
vorticity advection-diffusion equation restricted by the realistic lateral
boundaries. This approach allows us to find both the stable and unstable
states (flow patterns) of the system that are dynamically balanced. In
contrast, other approaches such as modal decomposition (Empirical Orthogonal
Functions, Singular Value Decomposition, Dynamic Mode Decomposition, etc.),
machine learning, self-organizing maps, etc. are based on observational (or
numerical) data correlations. We intentionally restricted the physics to the
quasigeostophic dynamics to capture the most important mechanism of the LC
formation: the balance between the planetary $\beta$-effect that promotes the
LC penetration into the Gulf of Mexico and the inertia that promotes the
current to leap directly from Yucatan to the Florida Straits. We are currently
working on generalizing this approach to include additional physical effects:
topographic $\beta$-effect, stratification, turbulent viscosity model in the
framework of realistic numerical models with a goal of using the obtained
states as a basis for analysis of the system predictability.

###### Acknowledgements.

The authors are thankful to the National Science Foundation, USA for funding
this research via grant number 1823452, and to the National Academies of
Sciences, Engineering and Medicine (NASEM) UGOS-1 via grant number 2000009918
and UGOS-2 via grant number 200011071. The authors are also grateful to two
anonymous reviewers for their valuable comments and suggestions for improving
the manuscript. The numerical code used in this work can be found at
http://sites.udel.edu/kuehl-group/software/ .

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