Topographic Stabilization of Ocean Rings

Abstract

Plain Language Summary
Swirls of circular currents tens to hundreds of kilometers in diameter known as ocean rings are critically important for transporting heat and nutrients throughout the ocean. Such structures usually reside in the upper ocean (top 1,000 m) and can last from months to several years. Ocean rings are often emitted by strong currents, such as the Gulf Stream, Agulhas, and Kuroshio. However, scientists are not entirely sure what allows rings to maintain their strength and persist for long periods. In particular, early theoretical models suggest that such large‐scale vortices are unstable and therefore should quickly disintegrate. In this study, we suggest that the resolution of the vortex longevity conundrum could lie in an unexpected direction—topography. The seafloor contains numerous underwater mountains, ridges, and valleys which, surprisingly, can dramatically affect rings that spin several kilometers above the bottom. Using numerical simulations with flat and realistically varying bottom, we demonstrate that eddies above rough topography persist much longer than their counterparts with identical parameters above a flat seafloor. We also show that there is a critical height of the bottom roughness which allows the surface‐intensified rings to remain stable and maintain their structure for years.

Full text

1. Introduction
Coherent vortices are among the most intriguing but still poorly understood features of the World Ocean. Many
are formed by instabilities of intense, large-scale flows, which lead to meanders that break away forming a ring of
current (Robinson,2012). Others spontaneously emerge from irregular patterns of active mesoscale variability,
are generated by topographic features, or through the interactions between other vortices (Jia etal.,2011). There
is general consensus regarding their importance in controlling the distribution of heat, salt, nutrients, pollutants,
and other tracers on planetary scales (Kamenkovich etal., 1986; Robinson,2012; Vallis,2019). Rings in the
North Atlantic, for instance, represent only 10%–15% of the ocean surface but are largely responsible for the
transport of life-sustaining zooplankton and nutrients to the Sargasso Sea (Ring Group,1981). A particularly
challenging and long-standing dynamical problem in the theory of coherent vortices concerns their longevity and
stability. Large-scale vortices with radii of 50–150km in the ocean are known to last for months and even years
(Chen & Han,2019; Chelton etal.,2011; Fu etal.,2010). However, stability analyses and numerical experi-
ments performed with seemingly realistic vortex patterns (Benilov & Flanagan,2008; Dewar & Killworth,1995;
Killworth etal.,1997; Mahdinia etal.,2017; Sokolovskiy & Verron,2014; Yim etal.,2016) reveal their strong
baroclinic instability and the tendency to disintegrate on the time-scale of several weeks. For a comprehensive
discussion of the dynamics and properties of baroclinic instability, the readers are referred to the monographs by
Vallis(2017) and Smyth and Carpenter(2019).
There have been several attempts to address the stability conundrum. Stable solutions can be constructed using the
sign-definite potential vorticity (PV) gradient criterion for vortex stability (Dritschel,1988). Specific examples of
Abstract Coherent large-scale vortices in the open ocean can retain their structure and properties for
periods as long as several years. However, the patterns of potential vorticity in such vortices suggest that they
are baroclinically unstable and therefore should rapidly disintegrate. This study proposes a plausible explanation
of the longevity of large-scale ocean rings based on bottom roughness, which restricts flow in the lower layer
and thereby stabilizes the eddy. We perform a series of simulations in which topography is represented by the
observationally derived Goff-Jordan spectrum. We demonstrate that topography stabilizes coherent vortices
and dramatically prolongs their lifespan. In contrast, the same vortices in the f lat-bottom model exhibit strong
instability and fragmentation on the timescale of weeks. A critical depth variance exists that allows vortices to
remain stable and circularly symmetric indefinitely. Our investigation underscores the essential role played by
topography in the dynamics of large- and meso-scale flows.
Plain Language Summary Swirls of circular currents tens to hundreds of kilometers in diameter
known as ocean rings are critically important for transporting heat and nutrients throughout the ocean. Such
structures usually reside in the upper ocean (top 1,000m) and can last from months to several years. Ocean
rings are often emitted by strong currents, such as the Gulf Stream, Agulhas, and Kuroshio. However, scientists
are not entirely sure what allows rings to maintain their strength and persist for long periods. In particular, early
theoretical models suggest that such large-scale vortices are unstable and therefore should quickly disintegrate.
In this study, we suggest that the resolution of the vortex longevity conundrum could lie in an unexpected
direction—topography. The seafloor contains numerous underwater mountains, ridges, and valleys which,
surprisingly, can dramatically affect rings that spin several kilometers above the bottom. Using numerical
simulations with flat and realistically varying bottom, we demonstrate that eddies above rough topography
persist much longer than their counterparts with identical parameters above a flat seafloor. We also show that
there is a critical height of the bottom roughness which allows the surface-intensified rings to remain stable and
maintain their structure for years.
GULLIVER AND RADKO
Published 2022. This article is a U.S.
Government work and is in the public
domain in the USA.
Topographic Stabilization of Ocean Rings
L. T. Gulliver1 and T. Radko1
1Department of Oceanography, Graduate School of Engineering and Applied Sciences, Naval Postgraduate School, Monterey,
CA, USA
Key Points:
• Coherent surface-intensified vortices
with scales greater than the radius of
deformation are stabilized by rough
bottom topography
• The lifespan of large vortex rings
increases with the increasing
amplitude of bottom topography
• A critical threshold of the depth
variance exists, above which the
vortices are linearly stable
Correspondence to:
L. T. Gulliver,
ltgulliv@nps.edu
Citation:
Gulliver, L. T., & Radko, T. (2022).
Topographic stabilization of ocean
rings. Geophysical Research Letters,
49, e2021GL097686. https://doi.
org/10.1029/2021GL097686
Received 6 JAN 2022
Accepted 2 MAR 2022
Author Contributions:
Conceptualization: T. Radko
Data curation: L. T. Gulliver
Formal analysis: L. T. Gulliver
Investigation: L. T. Gulliver
Methodology: T. Radko
Software: L. T. Gulliver
Supervision: T. Radko
Validation: L. T. Gulliver
Visualization: L. T. Gulliver
Writing – original draft: L. T. Gulliver
Writing – review & editing: T. Radko
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such solutions include the models of Meddies—North Atlantic salt lenses containing water masses of Mediter-
ranean origin (Sutyrin & Radko,2016; Radko & Sisti,2017). The numerical simulations performed by Sutyrin
& Radko(2016) and Radko and Sisti(2017) confirmed that the sign-definite PV gradient models produce stable
and robust vortices, maintaining their initial structure for several years or more. However, the technique used in
these studies suffers from a significant limitation. It can be applied only to relatively small-scale vortices with
radii comparable to, or less than, the first radius of deformation. Attempts to derive analogous solutions for larger
vortices result in nearly barotropic structures. In this sense, the sign-definite models are unrepresentative of most
observed baroclinic, surface-intensified ocean rings with radii in the range of 50–150km, substantially exceeding
the radius of deformation (Chelton etal.,2011; Olson,1991).
To develop fully baroclinic solutions for large-scale vortices, Benilov(2018) proposed an asymptotic model in
which the strongly stratified active surface layer was placed above a deep, nearly motionless, and more homo-
geneous abyssal layer. This configuration allowed for vortices with radii exceeding the radius of deformation
to remain stable and thereby helped to resolve the stability conundrum. The question that arises regarding such
models is whether the asymptotic limit of the deep abyssal layer adequately represents typical oceanic conditions.
Numerical simulations presented here using surface-intensified vortices in the flat-bottom ocean model with the
realistic mean ocean depth and stratification pattern indicate that large vortices can still be unstable and rapidly
disintegrate. This finding implies the presence of some other stabilizing mechanisms controlling the evolution
of ocean rings.
This paper explores the possibility that the stability and dynamics of large-scale rings could be affected by
the non-uniform bottom topography. The significance of flow-topography interactions for large-scale ocean
processes is widely recognized (Alvarez etal., 1994; Holloway, 1992). Okane and Frederiksen (2004), for
instance, showed that topography acts to localize energy and enstrophy transfers, which impacts the lifetimes of
larger scale coherent structures. In fact, some of the longest-lived coherent structures are situated over prominent
topographic features (Dewar,1998; de Miranda et al.,1999), which can be attributed to the mechanisms origi-
nally advocated by Bretherton and Haidvogel(1976) and Verron and Le Provost(1985). Topographic influences
on baroclinic instability have been explored by Hart(1975), Benilov(2005), Rabinovich etal.(2018), and Brown
etal.(2019). Promising attempts have also been made to parameterize the effects of topography using statistical
circulation models (Alvarez etal.,1994; Chavanis & Sommeria,2002; Frederiksen & O’Kane,2005; Merryfield
& Holloway,2002; Okane & Frederiksen,2004; Polyakov,2001). However, most studies consider relatively large
topographic scales (Chen & Kamenkovich,2013; Radko & Kamenkovich,2017) or isolated topographic features
(Benilov,2005; Verron & Le Provost,1985; Zavala Sansón,2019).
In contrast, our analysis is focused on irregular bathymetric patterns with dominant scales that are less than the
size of the phenomena of interest, a configuration hereafter referred to as the “sandpaper” model. We demonstrate
that eddy stresses generated by the rough bottom topography adversely affect the circulation in the abyssal zone.
This effect draws associations with the sandpaper glued face-up to a smooth surface, like non-skid on the deck of
a vessel, preventing the slippage of large objects in stormy conditions. Baroclinic instability, on the other hand,
is principally driven by the interaction of flow patterns located at different levels (e.g., Phillips,1951). Thus, it is
likely that the sandpaper effect, which constrains the flow in the deep ocean,can suppress the baroclinic instabil-
ity of large-scale vortices, ultimately extending their lifespan. In this regard, it should be noted that vortices in the
reduced gravity models, where the abyssal layer is a priori assumed to be quiescent, tend to be relatively stable
and persistent (Radko,2021; Radko & Stern,1999,2000). The tendency of sub-mesoscale bottom roughness to
stabilize large- and mesoscale parallel flows has already been demonstrated (LaCasce etal.,2019; Radko,2020).
The present investigation provides evidence that this effect is also at work in ocean rings and could account for
their remarkable longevity.
This study is organized as follows. In Section2 we present the model configuration and governing multilayer
quasi-geostrophic equations. Topography is represented by the observationally derived spectrum of Goff and
Jordan(1988). Section3 describes the numerical experiments and linear stability analyses conducted using the
minimal two-layer model. To ensure that our results are not biased by the crude representation of stratification,
selected simulations are reproduced using the ten-layer model (Section4). Conclusions are drawn in Section5.

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2. Formulation
2.1. Governing Equations
In order to represent the evolution of a large-scale surface-intensified vortex, we consider a quasi-geostrophic
n-layer, rigid lid model (Charney,1948; Pedlosky,1983).
⎧
⎪
⎨
⎪
⎩

 +(,
)=∇4,=1, ..., (-1)


+(,
)+

(,
)=∇4
,
(1)
where ψi is the streamfunction in layer i, which determines the lateral velocity
(
𝑢𝑖,𝑣
𝑖)=
(
−𝜕𝜓𝑖
𝜕𝑦
,𝜕𝜓𝑖
𝜕𝑥 ),
Hi is the
layer thickness, and n represents the bottom layer. Here, η is the variation in the bottom depth, ν is the viscosity,
and J is the Jacobian:

(𝑎 )≡




−




.
The Coriolis parameter f in Equation1 is assumed to be constant, which makes it possible to unambiguously
analyze the stability characteristics of circularly symmetric vortices. The PV (qi) is expressed in terms of stream-
function as follows:
⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
1=∇
21+2
′1
(2−1)
=∇
2+2
′
(−1 ++1 −2)=2, ..., (
− 1)
=∇
2+2
′
(−1 −),
(2)
where g′=gΔρ/ρ0 is the local reduced gravity. The formulation is simplified by assuming identical density differ-
ences between adjacent layers (Δρ=ρi − ρi−1). In the following simulations, we use the mid-latitude Coriolis
parameter of f=10
−4 s
−1 and the minimal value of viscosity needed to maintain the numerical stability of simu-
lations ν=30 m 2s
−1. The net reduced gravity of g′tot = (n − 1)g′=0.01 m s −2 is based on the density variation
over the entire water column with a representative ocean depth of

=
∑
=1
=4,000 m
.
2.2. Bottom Roughness Model
To determine the effects of bottom roughness on the stability of coherent vortices, it is imperative to introduce a
realistic model of the ocean depth variability. Our work is based on the empirical spectrum of topography derived
by Goff and Jordan(1988) using the ocean depth sampling provided by actual echo-sounding systems:

ℎ=2
0(− 2)
(2)300

1+


20

2
+


20

2
−∕2
(3)
typical values of parameters in the doubly periodic domain Equation3 suggested by Nikurashin etal.(2014) and
used in the present study are

=3
.
5
,
0
=1
.
8×10
−4
m
−1,
0
=1
.
8×10
−4
m
−1
(4)
where k and l are zonal and meridional wavenumbers respectively. In the following calculations, we construct
bottom topography as a sum of Fourier modes with random phases and spectral amplitudes conforming to the
Goff-Jordan spectrum Equation3. The coefficient η0 controls the height of topographic features. It will be system-
atically varied to quantify the link between the seafloor roughness and the vortex stability. The representative
seafloor variability in the ocean (Goff & Jordan, 1988) corresponds to the RMS bathymetric height of Hrms ∼
305m. A recent analysis of satellite marine gravity data reveals that Hrms varies between 10 and 400m in differ-
ent regions of the World Ocean (Goff,2020). These estimates will guide our exploration of the parameter space.
It should be noted that the quasi-geostrophic model is designed to represent relatively large-scale (low Rossby
number) flow patterns. Therefore, we exclude from the topographic spectrum all Fourier modes with wavelengths

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of Lmin=10km or less. This restriction makes it possible to keep the effective Rossby number below 0.075 in
all simulations, thereby satisfying assumptions of the quasi-geostrophic model Equation1. To be specific, the
Rossby number is defined here as Ro=|∇
2ψ1|/f and averaged over the vortex area. The typical pattern of seafloor
generated in this manner is shown in Figure1.
3. Two-Layer Model
3.1. Model Configuration
To assess the effects of bottom roughness on vortex stability, a series of simulations are conducted. The govern-
ing quasi-geostrophic Equations1 and2 are solved using the pseudospectral dealiased Fourier-based algorithm
employed and described in our previous works (Radko & Kamenkovich,2017; Sutyrin & Radko, 2021). The
temporal integrations use the fourth-order Runge-Kutta scheme with the adjustable time-step based on the CFL
(Courant-Fredrichs-Lewy) condition. The computational domain of lateral extent Lx × Ly=1,500km×1,500km
is resolved by Nx × Ny=1,024 × 1024 grid points.
The experiments in this section are performed with the two-layer (n=2) version of the model, with layer heights
of H1=1,000m and H2=3,000m. The upper layer represents the main mid-latitude thermocline, and the lower
layer—the abyssal ocean. Thus, the baroclinic radius of deformation based on the thermocline depth in this
configuration is

≡
√
′1

≈ 31.6
km
(5)
the simulations are initiated by using a Gaussian streamfunction pattern for the upper layer:
̄ 1=−exp(−2),
(6)
where r is the distance from the ring center. The lower layer, on the other hand, is assumed to be initially quiescent
(
 𝐴 2=0
) where the overbar denotes the basic state. This pattern represents an isolated surface-intensified vortex.
Figure 1. The model configuration. The upper plane shows the pattern of the sub-surface potential vorticity in the cyclonic
vortex. The vortex is located above the irregular seaf loor (brown surface) with the depth variance of Hrms=305m. Only a
fraction (400×400km in x and y) of the computational domain is shown.

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Given the invariance of quasi-geostrophic equations with respect to the transformation (ψ, y) → (−ψ, −y), we consider
only cyclonic eddies (A > 0), without loss of generality. The coefficients A and α control the vortex intensity and
size. Here they are determined by insisting that the maximal azimuthal velocity is
𝑉
max
≡max
𝑟
(||||
𝜕 ̄𝜓 1
𝜕𝑟
||||)
=1
ms−1
and the radius of maximal velocity is rmax=70 km—values that are representative of the observed ocean rings
(Olson,1991). These parameters correspond to A=6×10
6 m
2s
−1 and α=9.5×10
−11m
−2.
3.2. Nonlinear Simulations
Seventeen simulations were conducted in which the amplitude of topographic roughness was systematically
increased from the flat-bottom limit (Hrms=0) to the maximal RMS depth variance of Hrms=400m (Goff,2020).
Each experiment was extended for two years of model time. Snapshots of the upper layer PV for three selected
simulations are shown in Figure2 at t=0, 70days, 100days, and 2years. The eddy in the flat seafloor experi-
ment (Figures2a–2d) becomes unstable in just over two months (68days). The primary vortex loses form, breaks
apart, and re-forms into much smaller eddies that irregularly transit the computational domain throughout the
rest of the simulation.
In all simulations with the depth variance of Hrms < 150 m, the vortex lifespan is extremely limited, ranging
from 62 to 75 days. However, as Hrms is elevated above 150 m, the eddy lifespan begins increasing rapidly.
Figures2e–2h present the PV patterns for the simulation with Hrms=175m. The primary vortex in this experi-
ment persists for 105days, outlasting its flat-bottom counterpart by over a month. For Hrms=200m and above,
the primary vortices do not break apart for the full two years—the entire duration of simulations. A representative
example of such stable systems (Hrms=305m) is shown in Figures2i–2l.
3.3. Linear Stability Analysis
To further explore the bathymetric stabilization, we perform the linear stability analysis of the primary vortex
Equation6. The purpose of this inquiry is two-fold. The first objective is the confirmation of the link between
vortex longevity, stability, and the bottom roughness. We wish to ascertain that the fragmentation of vortices
for low Hrms is caused by linear instability, rather than by some unidentified nonlinear processes. Likewise, we
would like to demonstrate that the vortex persistence for large Hrms can be attributed to their enhanced stability.
The second benefit of the linear stability analysis is the opportunity to unambiguously determine the instability
growth rate (λ) and its dependence on Hrms. We begin by linearizing the governing Equation1 about the basic
state Equation6. For a two-layer system, this linearization yields
⎧
⎪
⎨
⎪
⎩
𝜕𝑞′
1
𝜕𝑡 +𝐽
(
̄𝜓 1,𝑞
′
1
)
+𝐽
(
𝜓′
1, ̄𝑞 1
)
=𝜈∇4𝜓
′
1
𝜕𝑞′
2
𝜕𝑡
+𝑓
𝐻2
𝐽
(
𝜓′
2,𝜂
)
=𝜈∇4𝜓′
2,
(7)
where primes denote perturbations:
(

′

,
′
)=(


,
)−(
̄

, ̄
)
.
The linear simulations were initiated by the random distribution of ψ′i and system Equation7 was integrated in
time using the pseudo-spectral algorithm, analogous to the one used for the nonlinear simulations (Section3.2).
The evolution of the linear system is eventually dominated by the fastest exponentially growing mode with a
well-defined growth rate. To quantify the magnitude of the perturbation and its evolutionary pattern, we use the
spatially integrated specific energy

′()= 1
2∬
1
|
∇′
1
|
2+2
|
∇′
2
|
2+
2

′
(
′
1−′
2
)
2
 𝑑
(8)
The energy is non-dimensionalized by its initial value E′(0), and the growth rate (λ) of the vortex instability is
evaluated accordingly:

=1
2

ln
(

′
()
′(0)
).
(9)

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Figure 2. Snapshots of the upper-layer potential vorticity q1 (x,y) in three representative simulations at t=0days, 70days, 100days, and 2years. Panels (a–d)
represent the flat-bottom simulation (Hrms=0), (e–h) —Hrms=175m, and (i–l) show the experiment performed with the nominal observed value of Hrms=305m. As
Hrms increases, the lifespan of vortices increases as well.

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Figure3a presents 
1
2
ln
(

′
()
′
(0) )
as a function of time for the topographic variance ranging from Hrms=0–400m. In
each case, after a brief initial period of adjustment (∼3months), the perturbation starts growing exponentially, as
evidenced by the straight lines representing the energy records in logarithmic coordinates (Figure3a). The slopes
of these lines represent the growth rates (λ), which are evaluated as a best linear fit to the data in Figure3a over
the interval t>3months. The resulting growth rates are plotted as a function of Hrms in Figure3b.
The linear analysis in Figure3b leads to two key conclusions regarding bathymetric stabilization: (a) the vortex
instability monotonically weakens with the increasing Hrms, and (b) there exists a critical value of the depth vari-
ance Hcr ∼ 250m above which vortex becomes linearly stable (λ<0). It should be emphasized that the nominal
value of the observed depth variance Hrms=305m suggested by Goff and Jordan(1988) exceeds Hcr. Thus, the
stability condition Hrms>Hcr is commonly met in the World Ocean. The apparent longevity of the observed ocean
rings could therefore be attributed to the sandpaper effect—the adverse action of the small-scale topographic
variability on the abyssal circulation. The foregoing stability analysis is also fully consistent with the nonlinear
simulations (Section3.2) revealing rapid vortex fragmentation for Hrms<Hcr and their long-term persistence for
Hrms>Hcr. For low values of Hrms, the growth rates are λ ∼ 10
−6s
−1. The corresponding instability timescale is
λ
−1 ∼ 10days, which also reflects the vortex evolution in the nonlinear experiments.
4. Ten-Layer Model
The two-layer model of bathymetric stabilization (Section3) offered a basic proof of concept and permitted the
efficient exploration of the parameter space. However, it is prudent to ensure, at this point, that our inferences
are not biased by the model's minimalistic representation of stratification. To this end, we present our second
set of experiments in which the number of layers (n) is increased to 10. The height of each layer (Hi) increases
from 100m on the surface, to 2,000 m at i=10. The lower layer is chosen to be relatively deep to ensure that the
quasi-geostrophic requirement η ≪ Hn is met. To represent the surface-intensified ocean rings, we consider the
circulation pattern that exponentially decreases with depth
Figure 3. Linear analysis of the two-layer quasi-geostrophic model. Panel (a) shows the natural log of the net perturbation energy for values of Hrms from 0 to 400m.
Panel (b) presents the growth rate (λ) as a function of roughness (Hrms), illustrating the systematic weakening of vortex instability with increasing variance of sea-floor
topography.

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̄
=−exp(∕0)exp
(
−
2),
(10)
where
 𝐴 
(i=1,...,10) is the basic streamfunction in layer i, H0=1,000m is the effective vertical extent of the
ring, and Zi is the location of the i-th layer center.
As in the two-layer case, A and α are chosen to yield the ring radius of rmax =70km and the maximal surface
velocity of Vmax=1m s −1. The resulting speed pattern is shown in Figure4a, where the length of each bar repre-
sents the maximal azimuthal velocity in each layer and its vertical extent—the layer height. The chosen velocity
pattern is consistent with observations of ocean rings. For instance, Figure4b shows four profiles of geostrophic
velocity in the interior of a South-Atlantic Agulhas ring (from Casanova-Masjoan etal.,2017). These measure-
ments reveal that the velocity decreases with depth in a nearly exponential manner with the e-folding scale of
approximately 1,000m, which is reflected in the employed vortex model Equation10.
The snapshots of the upper layer PV in Figure5 reveal the evolution of vortices in experiments with Hrms values
of 0m, 175m, and 305m. As with the two-layer experiment, the vortex lifespan monotonically and dramatically
increases with the increasing depth variance. In the flat bottom simulation (Figures 5a–5e), the vortex loses
coherence at t ∼ 120days, later than in the corresponding two-layer case, splitting into multiple smaller eddies.
The vortex in Figures5f–5j (Hrms=175m) does not exhibit signs of instability for six months before breaking
apart, as seen in Figure 5j. Importantly, the vortex simulated using the nominal observed depth variance of
Hrms=305m remains coherent throughout the entire 2-year long experiment.
5. Conclusions
The key outcome of this study is the discovery and validation of the “sandpaper” effect. We demonstrate that
irregular topography of realistic magnitude and spatial pattern can stabilize large surface-intensified ocean rings
by suppressing the circulation in the abyssal layer. Our numerical simulations indicate that vortices above a rough
seafloor are much more robust and have larger lifespans than those over a flat bottom. The larger the magnitude
of the depth variability, the more stable and long-lived are the modeled rings. Furthermore, we show that there is
Figure 4. (a) Plot of the maximal azimuthal velocity (Vmax) in each isopycnal layer coded by color. (b) Four profiles of geostrophic velocity at CTD stations monitoring
an Agulhas ring during the period 8 Feb – 10 March 2010 (modified from Casanova-Masjoan etal.,2017).

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Figure 5. The same as Figure2 for the ten-layer simulations at t=0, 4months, 6months, and 2years. Panels (a–d) (e–h), and (i–l) represent the experiments
performed for Hrms=0, 175m, and 305m, respectively. As in the two-layer experiments (Figure2), the lifespan of vortices increases with increasing Hrms.

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a critical value (Hcr∼250m) of the depth variance above which the large-scale vortices are linearly stable. This
condition (Hrms>Hcr) is unrestrictive and is met in many ocean regions.
Our results offer a plausible resolution of the long-standing stability conundrum—why large-scale ocean rings
are stable and long-lived even when they satisfy the formal instability condition (Dritschel,1988). Concurrently,
this work draws attention to the limitations of commonly used flat-bottom circulation theories and motivates
efforts to parameterize the effects of bottom roughness on larger scales of motion.
The present study can be advanced in numerous directions by systematically increasing the complexity and
realism of the model configuration. The minimal model used in our investigation can be extended to more
general frameworks, such as the shallow-water and Navier-Stokes systems. In this regard, we note that the present
quasi-geostrophic configuration a priori excludes several potentially significant topographic effects, including
the lee-wave bottom drag (e.g., Eden et al., 2021; Klymak etal., 2021). Thus, the calculations in this study
could underestimate the effects of variable topography on the dynamics of ocean rings. Furthermore, the present
version of the sandpaper model does not represent topographic features with lateral extents less than 10km —the
limitation posed by the quasi-geostrophic approximation. However, such sub-mesoscale topographic features are
widespread in the ocean (e.g., Goff,2020) and can also affect the dynamics and stability of large-scale flows
(Radko,2020). Nevertheless, even the analysis of the most basic system suggests that the topographic influences
on ocean rings could be profound and should be explored further.
Data Availability Statement
Data created during this study are made openly available at Figshare repository, discoverable at https://doi.
org/10.6084/m9.figshare.17696003.v1.
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Acknowledgments
Support of the National Science Founda-
tion (grant OCE 1828843) is gratefully
acknowledged. The authors thank Drs.
Terrence O’Kane, Caitlin Whalen, and
the anonymous reviewer for helpful
comments.

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