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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.
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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.
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,
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,
Gulliver, L. T., & Radko, T. (2022).
Topographic stabilization of ocean
rings. Geophysical Research Letters,
49, e2021GL097686. https://doi.
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)
where ψi is the streamfunction in layer i, which determines the lateral velocity
𝜕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)
where g′=gΔρ/ρ0 is the local reduced gravity. The formulation is simplified by assuming identical density differ-
ences between adjacent layers (Δρ=ρiρi−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 gtot = (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
=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:
0(− 2)
typical values of parameters in the doubly periodic domain Equation3 suggested by Nikurashin etal.(2014) and
used in the present study are
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
≈ 31.6
the simulations are initiated by using a Gaussian streamfunction pattern for the upper layer:
̄ 1=−exp(−2),
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
𝜕 ̄𝜓 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
−1 and α=9.5×10
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
𝜕𝑡 +𝐽
̄𝜓 1,𝑞
1, ̄𝑞 1
where primes denote perturbations:
, ̄
The linear simulations were initiated by the random distribution of ψi 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
 𝑑
The energy is non-dimensionalized by its initial value E′(0), and the growth rate (λ) of the vortex instability is
evaluated accordingly:
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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 
(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
−1. The corresponding instability timescale is
−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 ηHn 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
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(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.
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... In some cases, the topography may produce a secondary circulation that reinforces mean flow (Holloway, 1987;1992) or create energy cascades that ultimately produce relatively narrow zonal currents (Vallis and Maltrud, 1993). Rough bathymetry can also dramatically alter the stability characteristics of large-scale currents, which was demonstrated for parallel flows (LaCasce et al., 2019;Radko, 2020) and axisymmetric oceanic vortices known as rings (Gulliver and Radko, 2022a;2022b). The latter study argued that the suppression of baroclinic instability by rough seafloor could add years to ocean rings' lifespan. ...
... Following Gulliver and Radko (2022a;2022b), we construct bottom topography as a sum of Fourier modes with random phases and spectral amplitudes conforming to (2). However, in the present simulations, the nominal RMS height of topography ðg 0 Þ and its nominal wavenumbers ðk 0 ; l 0 Þ are scaled down to match the dimensions of the rotating tank. ...
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This study presents numerical analogs of laboratory experiments designed to explore the interaction of broad geophysical flows with irregular small-scale bathymetry. The previously reported "sandpaper" theory offered a succinct description of the cumulative effect of small-scale topographic features on large-scale flow patterns. However, initial investigations have been conducted using numerical models with simplified quasi-geostrophic equations that may inadequately represent the dynamics realized in the world's oceans. This investigation advances previous efforts by using a fully nonlinear Navier-Stokes model configured for rotating tank experiments to (i) validate theory and (ii) offer guidance for future physical experiments that will ground-truth theoretical ideas.
... In contrast, the vortex above irregular topography remains coherent and nearly steady throughout the entire simulation. This topographic stabilization is consistent with our earlier findings (Gulliver & Radko 2022) and calls for a more systematic analysis. ...
... Another robust tendency revealed by this analysis is the topographic stabilization of axisymmetric vortices. While this effect has already been observed in topography-resolving simulations of Gulliver and Radko (2022), the more efficient parametric model permits its systematic exploration over a wide range of governing parameters. We argue that the conditions for topographic stabilization are unrestrictive and commonly met in nature, which could explain the abundance of long-lived mesoscale eddies in all ocean basins (e.g. ...
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This study explores the impact of small-scale variability in the bottom relief on the dynamics and evolution of broad baroclinic flows in the ocean. The analytical model presented here generalizes the previously reported barotropic 'sandpaper' theory of flow-topography interaction to density-stratified systems. The multiscale asymptotic analysis leads to an explicit representation of the large-scale effects of irregular bottom roughness. The utility of the multiscale model is demonstrated by applying it to the problem of topography-induced spin-down of an axisymmetric vortex. We find that bathymetry affects vortices by suppressing circulation in their deep regions. As a result, vortices located above rough topography tend to be more stable than their flat-bottom counterparts. The multiscale theory is validated by comparing corresponding topography-resolving and parametric simulations.
... LaCasce et al. 2019; Radko 2020) of the significance of bathymetry is the dramatic impact of sea-floor roughness on the intensity of mesoscale variability, traditionally defined as flow components with a lateral extent of 10-100 km. Particularly relevant to the present investigation are the findings of Gulliver & Radko (2022), who analysed the effects of irregular topography on the stability and longevity of ocean rings. This study explored the parameter regime in which the lateral extent of primary flows greatly exceeded that of individual topographic features -the configuration aptly dubbed the 'sandpaper model'. ...
... The name was chosen to invoke the associations with fine abrasive particles of sandpaper that may be individually insignificant but have a tangible cumulative effect in grinding down much larger objects. A series of simulations in Gulliver & Radko (2022) revealed dramatic dissimilarities in the evolution of coherent vortices in flat-bottom basins and in the presence of realistic topographic patterns. The set-up of these experiments is illustrated by the schematic diagram in figure 1. ...
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This study examines the impact of small-scale irregular topographic features on the dynamics and evolution of large-scale barotropic flows in the ocean. A multiscale theory is developed, which makes it possible to represent large-scale effects of the bottom roughness without explicitly resolving small-scale variability. The analytical model reveals that the key mechanism of topographic control involves the generation of a small-scale eddy field associated with considerable Reynolds stresses. These eddy stresses are inversely proportional to the large-scale velocity and adversely affect mean circulation patterns. The multiscale model is applied to the problem of topography-induced spin-down of a large circularly symmetric vortex and is validated by corresponding topography-resolving simulations. The small-scale bathymetry chosen for this configuration conforms to the Goff-Jordan statistical spectrum. While the multiscale model formally assumes a substantial separation between the scales of interacting flow components, it is remarkably accurate even when scale separation is virtually non-existent.
... The present study can be advanced in numerous directions by systematically increasing the complexity and realism of the model configuration, including, for instance, the atmospheric forcing, continental boundaries, bottom topography, or higher vertical modes. Furthermore, the bottom drag in the present version is described by linear friction, which may not represent the essential effects of rough topography (Gulliver and Radko, 2022b). Nevertheless, even the analysis of the most basic system suggests that the vortex dynamics and eddy-induced transport could be profoundly different in EB and WB flows and should be explored further. ...
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We identify and explore the fundamental differences in the dynamics of mesoscale vortices in eastward background (EB) parts in mid-latitude ocean gyres and in westward background (WB) return flows. In contrast to eddy behavior in EB flow, a systematic meridional drift of eddies in WB flow results in poleward expulsion of cold-core cyclones and equatorward expulsion of warm-core anticyclones from the unstable zone with a negative potential vorticity gradient (PVG). Consequently, heat can be transferred further by upper ocean vortices intrinsically coupled with deep opposite sign partners. Such structures can drift through the stable zone with positive PVG in both layers. This mechanism of lateral transfer is not captured by local models of homogeneous turbulence. The crossflow drift is related to the coupling of the upper vortices with opposite sign deep eddies shifted eastward. The abyssal vortices can be viewed as lee Rossby waves induced by their upper-layer partners and described analytically in the vicinity of the latitude of marginal stability. Here, we show how such self-amplifying hetons, emerging in homogeneous turbulence, saturate when they approach locally stable regions of inhomogeneous currents. The presented results indicate that subtropical regions with return WB flows in the upper layer favor long-distance heat transport by spatially coherent eddies in accordance with observations and motivate the development of non-local parameterizations of eddy fluxes.
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Recently, Jánosi et al. (2019) introduced the concept of a “vortex proxy” based on an observation of strong correlations between integrated kinetic energy and integrated enstrophy over a large enough surface area. When mesoscale vortices are assumed to exhibit a Gaussian shape, the two spatial integrals have particularly simple functional forms, and a ratio of them defines an effective radius of a “proxy vortex”. In the original work, the idea was tested over a restricted area in the Californian Current System. Here we extend the analysis to global scale by means of 25 years of AVISO altimetry data covering the (ice-free) global ocean. The results are compared with a global vortex database containing over 64 million mesoscale eddies. We demonstrate that the proxy vortex representation of surface flow fields also works globally and provides a quick and reliable way to obtain coarse-grained vortex statistics. Estimated mean eddy sizes (effective radii) are extracted in very good agreement with the data from the vortex census. Recorded eddy amplitudes are directly used to infer the kinetic energy transported by the mesoscale vortices. The ratio of total and eddy kinetic energies is somewhat higher than found in previous studies. The characteristic westward drift velocities are evaluated by a time-lagged cross-correlation analysis of the kinetic energy fields. While zonal mean drift speeds are in good agreement with vortex trajectory evaluation in the latitude bands 30–5∘ S and 5–30∘ N, discrepancies are exhibited mostly at higher latitudes on both hemispheres. A plausible reason for somewhat different drift velocities obtained by eddy tracking and cross-correlation analysis is the fact that the drift of mesoscale eddies is only one component of the surface flow fields. Rossby wave activities, coherent currents, and other propagating features on the ocean surface apparently contribute to the zonal transport of kinetic energy.
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This study explores the dynamics of intense coherent vortices in large-scale vertically sheared flows. We develop an analytical theory for vortex propagation and validate it by a series of numerical simulations. Simulations are conducted using both stable and baroclinically unstable zonal background flows. We find that vortices in stable westward currents tend to adjust to an equilibrium state characterized by quasi-uniform zonal propagation. These vortices persist for long periods, during which they propagate thousands of kilometers from their points of origin. The adjustment tendency is realized to a much lesser extent in eastward background flows. These findings may help to explain the longevity of the observed oceanic vortices embedded in predominantly westward flows. Finally, we examine the influence of background mesoscale variability induced by baroclinic instability of large-scale flows on the propagation and persistence of isolated vortices.
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A new, energetically and dynamically consistent closure for the lee wave drag on the large scale circulation is developed and tested in idealized and realistic ocean model simulations. The closure is based on the radiative transfer equation for internal gravity waves, integrated over wavenumber space, and consists of two lee wave energy compartments for up-and downward propagating waves, which can be co-integrated in an ocean model. Mean parameters for vertical propagation, mean-ow interaction, and the vertical wave momentum flux are calculated assuming that the lee waves stay close to the spectral shape given by linear theory of their generation. Idealized model simulations demonstrate how lee waves are generated and interact with the mean flow and contribute to mixing, and document parameter sensitivities. A realistic eddy-permitting global model at 1/10 ° resolution coupled to the new closure yields a globally integrated energy flux of 0.27 TW into the lee wave field. The bottom lee wave stress on the mean flow can be locally as large as the surface wind stress and can reach into the surface layer. The interior energy transfers by the stress are directed from the mean flow to the waves, but this often reverses, for example in the Southern Ocean in case of shear reversal close to the bottom. The global integral of the interior energy transfers from mean ow to waves is 0.14 TW, while 0.04 TW is driving the mean ow, but this share depends on parameter choices for non-linear effects.
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This study explores the long-term impact of a weak initial departure from circular symmetry in coherent equivalent-barotropic vortices on their dynamics and evolution. An algorithm is developed which makes it possible to construct models of vortices that initially propagate with prescribed velocity. These solutions are used as the initial conditions for a series of numerical simulations. Simulations indicate that seemingly minor perturbations of dipolar form can control the propagation of vortices for extended periods, during which they translate over distances greatly exceeding their size. The numerical results are contrasted with the linear model, which assumes that the non-axisymmetric component of circulation in the vortex interior is relatively weak. The linear solutions reflect the self-propagation tendencies of coherent vortices to a much lesser degree, which underscores the role of fundamentally nonlinear mechanisms at play. The remarkable ability of quasi-monopolar vortices to retain the memory of weak initial perturbations helps to rationalize the wide range of the observed propagation velocities of coherent long-lived vortices in the ocean.
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This paper examines factors contributing to the remarkable longevity of coherent vortices in the subtropical westward flows. Baroclinic vortices embedded in large-scale vertical shears generate Rossby waves which form an opposite sign eddy associated with inertial Taylor columns on the beta-plane. The combination of the vortex and lee Rossby wave can be viewed as a hetonic dipole that induces meridional drift and heat flux leading to self-amplification of vortices in baroclinically unstable flows. The analytical tractability is achieved by considering the marginally stable flow, where the beta-effect is nearly compensated by the potential vorticity gradient (PVG) associated with the meridional slope of the density interface. In the two-layer model such compensation can occur in the upper layer with the westward flow or in the lower layer with the eastward flow on top. In baroclinically unstable mean flows, vortices are shown to intensify due to the Lagrangian conservation of potential vorticity inside their cores, drifting meridionally in the layer with negative PVG and supporting baroclinic turbulence. The theory is confirmed by numerical simulations indicating that for westward flows in subtropical oceans, reduced PVG in the upper layer provides favorable conditions for eddy longevity and pathways.
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The dependence of abyssal hill roughness on mid‐ocean ridge (MOR) spreading rate is an important indicator for faulting and volcanism. I reanalyze this relationship using a global gravity‐based prediction of root‐mean‐square (RMS) heights, enabling dense sampling of RMS/spreading rate space and thus a far more detailed examination than possible with bathymetric data. RMS histograms are multimodal, indicating previously unrecognized complexity in roughness versus spreading rate that cannot be characterized by a single trend. Modal peaks are used to define four different types of abyssal hills, three of which can be associated with axial valley, transitional, and axial high MOR morphology. The most abundant type at any one spreading rate bin is used to define a “characteristic” trend that transitions abruptly from axial valley to axial high types across half rates of ~20 to 30 mm/yr. Abyssal hills outside this trend are associated geographically with anomalously “hot” and “cold” mantle regions.
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This study explores the control of mesoscale variability by topographic features with lateral scales that are less than the scale of the eddies generated by baroclinic instability. These dynamics are described using a combination of numerical simulations and an asymptotic multiscale model. The multiscale method makes it possible to express the system dynamics by a closed set of equations written entirely in terms of mesoscale variables, thereby providing a physical basis for the development of submesoscale parameterization schemes. The submesoscale topography is shown to influence such fundamental properties of mesoscale variability as the meridional eddy-induced transport and eddy kinetic energy. It is argued that the adverse influence of submesoscale topography on baroclinic instability is ultimately caused by the homogenization tendency of potential vorticity in the bottom density layer. The multiscale model formally assumes a substantial separation between the scales of interacting flow components. However, the comparison of asymptotic solutions with their submesoscale-resolving numerical counterparts indicates that the multiscale method is remarkably accurate even when scale separation is virtually non-existent.
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This study attempts to identify the equilibration mechanisms of baroclinic instability and investigate the effects of the orientation of the background flow and topography on eddy‐induced transport. The analysis is based on growth rate balance theory, which assumes that nonlinear equilibration occurs when the growth rate of the primary baroclinic instability becomes comparable to that of the secondary instabilities of amplifying modes. Two‐layer quasi‐geostrophic numerical simulations are performed and compared to growth rate balance theory in order to analytically predict the cross‐stream fluxes of potential vorticity. The model performs remarkably well in predicting the effects of variation in zonal topographic slope and background flow orientation. We find that meridional topographic slopes affect the baroclinic instability in an inherently nonlinear way. A predictive model based on conservation of potential vorticity is developed for the optimal slope that maximizes the transport characteristics of the baroclinic instability.
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The increase of temporal resolution from weekly to daily and spatial resolution from ~1° to ~(1/4)° in merged altimeter sea surface height data along with their over two decades of time series opens an unprecedented opportunity to reveal the “spectrum” of eddy variabilities from days to years in time and from mesoscale to semimesoscale in space. Eddies with a lifetime shorter than 1 month or longer than 1 year are classified here as short lived and long lived, respectively. Contrasting the variabilities of short- and long-lived eddies could be an effective way to explore their geographic origins and dynamic formations. In the time domain, the population fluctuations of short- and long-lived eddies are basically dominated by annual and interannual component, respectively. The magnitude of kinematic and dynamic properties of oceanic eddies is positively correlated with their lifetime in general. Statistically, the properties of short-lived (long-lived) eddies experience a symmetric (an asymmetric) growth and decay with a single flat peak during their life cycles. In the space domain, eddy occurrence is observed as a high probability event, which can reach every corner of the ocean. However, homes of short- and long-lived eddies are highly regionalized and are geographically separated. A prominent “young eddy belt” is observed in the tropical oceans for the first time. These findings suggest two fundamental characteristics with regard to short- and long-lived mesoscale eddies: residing in largely separated geographic zones under different mechanisms while following a similar pattern of intrinsic life cycle in terms of property evolution.
Slowly-evolving stratified flow over rough topography is subject to substantial drag due to internal motions, but often numerical simulations are carried out at resolutions where this “wave” drag must be parameterized. Here we highlight the importance of internal drag from topography with scales that cannot radiate internal waves, but may be highly non-linear, and we propose a simple parameterization of this drag that has a minimum of fit parameters compared to existing schemes. The parameterization smoothly transitions from a quadratic drag law ( ) for low- (linear wave dynamics) to a linear drag law ( ) for high- flows (non-linear blocking and hydraulic dynamics), where N is the stratification, h is the height of the topography, and u 0 is the near-bottom velocity; the parameterization does not have a dependence on Coriolis frequency. Simulations carried out in a channel with synthetic bathymetry and steady body forcing indicate that this parameterization accurately predicts drag across a broad range of forcing parameters when the effect of reduced near-bottom mixing is taken into account by reducing the effective height of the topography. The parameterization is also tested in simulations of wind-driven channel flows that generate mesoscale eddy fields, a setup where the downstream transport is sensitive to the bottom drag parameterization and its effect on the eddies. In these simulations, the parameterization replicates the effect of rough bathymetry on the eddies. If extrapolated globally, the sub-inertial topographic scales can account for 2.7 TW of work done on the low-frequency circulation, an important sink that is redistributed to mixing in the open ocean.
Some oceanic and atmospheric flows may be modelled as equivalent-barotropic systems, in which the horizontal fluid velocity varies in magnitude at different vertical levels while keeping the same direction. The governing equations at a specific level are identical to those of a homogeneous flow over an equivalent depth, determined by a pre-defined vertical structure. The idea was proposed by Charney (J. Met., vol. 6 (6), 1949, pp. 371-385) for modelling a barotropic atmosphere. More recently, steady, linear formulations have been used to study oceanic flows. In this paper, the nonlinear, time-dependent model with variable topography is examined. To include nonlinear terms, we assume suitable approximations and evaluate the associated error in the dynamical vorticity equation. The model is solved numerically to investigate the equivalent-barotropic dynamics in comparison with a purely barotropic flow. We consider three problems in which the behaviour of homogeneous flows has been well established either experimentally, analytically or observationally in past studies. First, the nonlinear evolution of cyclonic vortices around a topographic seamount is examined. It is found that the vortex drift induced by the mountain is modified according to the vertical structure of the flow. When the vertical structure is abrupt, the model effectively isolates the surface flow from both inviscid and viscous topographic effects (due to the shape of the bottom and Ekman friction, respectively). Second, the wind-driven flow in a closed basin with variable topography is studied (for a flat bottom this is the well-known Stommel problem). For a zonally uniform, negative wind-stress curl in the homogeneous case, a large-scale, anticyclonic gyre is formed and displaced southward due to topographic effects at the western slope of the basin. The flow reaches a steady state due to the balance between topographic, β, wind-stress and bottom friction effects. However, in the equivalent-barotropic simulations with abrupt vertical structure, such an equilibrium cannot be reached because the forcing effects at the surface are enhanced, while bottom friction effects are reduced. As a result, the unsteady flow is decomposed as a set of planetary waves. A third problem consists of performing simulations of the wind-driven flow over realistic bottom topography in the Gulf of Mexico. The formation of the so-called Campeche gyre is explored. It is found that such circulation may be consistent with the equivalent-barotropic dynamics.
Cambridge Core - Oceanography and Marine Science - Instability in Geophysical Flows - by William D. Smyth