## 1. Introduction

In several ocean basins, long-lived and semistationary mesoscale anticyclones (ACs) appear above topographic bowls.^{1} Examples are the Mann Eddy (Mann 1967), Lofoten Basin Eddy (e.g., Ivanov and Korablev 1995; Köhl 2007; Søiland et al. 2016), and the Rockall Trough Eddy (Le Corre et al. 2019). The eddies have a clear climatological signature in sea surface height, as seen from satellite altimetry (Fig. 1). The first two have been repeatedly sampled in hydrographic surveys since their first discoveries. In the elongated bowl-like Iceland Basin long-lived ACs also appear to be common (Martin et al. 1998; Wade and Heywood 2001; Read and Pollard 2001; Zhao et al. 2018).

The aforementioned ACs all occur in mid- to high-latitude North Atlantic^{2} seas or basins, where stratification is relatively weak, and the Coriolis parameter *f* is relatively high. This results in a larger tendency for currents to respond to seabed depth *H* variations (Salmon 1998) and follow ambient potential vorticity *f*/*H* contours (Isachsen et al. 2003). The latter are often well approximated by *H* contours due to the limited relative *f* variation in higher latitudes compared with relative *H* variation across basins (Nøst and Isachsen 2003). Thus the effect of topographic depressions may be enhanced in these areas and play a role in the formation of the observed ACs.

Significant thermohaline fluxes and transformations occur in these basins, processes of climatic significance as part of the Atlantic meridional overturning circulation (AMOC). It is still unclear whether the semistationary ACs impact these regional processes. Tulloch and Marshall (2012) have shown that AMOC variability is associated with a spatial pattern of density variability that peaks in the region of the Mann Eddy, in two ocean general circulation models. The Lofoten Basin Eddy attains anomalously large wintertime mixed layer depths (Yu et al. 2017). It encompasses a large reservoir of heat, absorbing warm eddies shed from poleward flowing boundary currents (Raj et al. 2015; Søiland et al. 2016). Richards and Straneo (2015) presented evidence for water mass transformation within a Lofoten Basin anticyclonic eddy.

Several different mechanisms have previously been suggested to sustain these eddies. Ivanov and Korablev (1995) suggested that wintertime intermediate-depth convection can regenerate the Lofoten Basin Eddy following a slow annual erosion. Köhl (2007) presented evidence that the Lofoten Basin Eddy is maintained by repeated mergers with ACs, which are shed from the Norwegian Atlantic Current and descend into the Lofoten Basin due to planetary and topographic beta drift (see Carnevale et al. 1991). The importance of eddy mergers versus convection in Lofoten Basin Eddy maintenance has also been discussed by (e.g., Søiland and Rossby 2013; Raj et al. 2015; Volkov et al. 2015; Bashmachnikov et al. 2017; Bosse et al. 2019; Trodahl et al. 2020). Le Corre et al. (2019) showed that the Rockall Trough Eddy is similarly replenished by ACs, resulting from frictional vorticity generation at the adjacent topographic slope. Zhao et al. (2018) showed that Iceland Basin ACs are likely formed locally through mixed baroclinic–barotropic instability of the adjacent slope current. Rossby (1996) suggested that the Mann Eddy is spawned from ACs released from the anticyclonic side of the adjacent North Atlantic Current.

In contrast, several theoretical and numerical studies have shown that in the absence of large-scale external flow, mesoscale variability tends to produce rectified prograde^{3} mean flows. That is, cyclonic (anticyclonic) circulation develops over topographic depressions (bumps), consistently with vorticity stretching (compression). This is predicted by “enstrophy-minimization” theory (Bretherton and Haidvogel 1976),^{4} as well as statistical mechanics theories of topographic turbulence (starting with Salmon et al. 1976). This tendency is commonly known as the “Neptune” effect (e.g., Merryfield et al. 2001). With an imposed large-scale impinging flow, dynamical circulation theories also predict development of a prograde mean circulation over seamounts, i.e., a Taylor Cap (Hogg 1973; Huppert and Bryan 1976), as indeed had been observed in the ocean^{5} (Hogg 1973; Brechner Owens and Hogg 1980; Richardson 1981; Freeland 1994; White and Mohn 2004). Alternative mechanisms must therefore explain the presence of long-lived ACs in topographic “bowls.”

Although separate formation mechanisms were previously suggested for each of the abovementioned semistationary oceanic ACs, there is not presently a unified dynamical understanding of AC formation and longevity above bowl-like topographic depressions. With the aim of achieving such an understanding, in this study we pose an idealized topographic-turbulence problem. Specifically, we conduct numerical simulations of the free evolution of mesoscale eddies randomly initialized over topographic depressions. We restrict our attention to the layered primitive equations, in one or two density layers, allowing wide sweeps of parameter space.

A description of the numerical experiments configurations is given in section 2. In section 3 we show that ACs form consistently over isolated topographic bowls in a very wide set of circumstances in single-layer experiments, and characterize the range of dynamical regimes that emerge in our simulations. In section 4 we investigate the degree of cross-topography exchange involved in AC formation, and its dynamical mechanisms. In section 5 we show that bowl ACs form from barotropic as well as baroclinic initial conditions in two-layer experiments, and investigate the dependence of its final vertical structure on the initial conditions. A discussion and comparison with previous results is presented in section 6. A summary and conclusions are given in section 7.

## 2. Methods

### a. Layered primitive equations model

We conduct our simulations using the layered primitive equations model AWSIM (Stewart and Dellar 2016). A layered primitive equations model (McWilliams 2006) represents the simplest setting for studying topographic turbulence with finite amplitude topography and finite amplitude circulation effects. A primitive equation model is preferred over a quasigeostrophic model because some of the observed ACs (section 1) have a high Rossby number [*ζ* = *O*(*f*)] and reside in areas with *O*(1) changes in *H* (Søiland et al. 2016; Yu et al. 2017; Le Corre et al. 2019).

The main model equations and its numerical scheme are described more fully in Stewart and Dellar (2016). Here we summarize salient aspects of the model. The dynamical equations are discretized via finite differences on an Arakawa C grid. The spatial discretization of the momentum and thickness equations (Stewart and Dellar 2016) is essentially identical to Arakawa and Lamb (1981) for the present experiments. Time stepping scheme follows the third-order Adams–Bashforth scheme (Durran 1991). The model conserves total energy, layer-wise potential enstrophy, and mass to machine precision in the absence of explicit dissipation (Stewart and Dellar 2016). A rigid upper lid condition is applied for computational efficiency, and surface pressure is diagnosed at each time step by numerical solution of the associated elliptic equation using a multigrid method. Gridscale accumulation of energy and enstrophy is controlled using a hyperviscous operator in the momentum equation (Griffies and Hallberg 2000).

The model can also evolve a passive tracer using an advection–diffusion equation, which was implemented in several of the experiments we conducted. We used the flux-limited tracer advection scheme of Kurganov and Tadmor (2000), which allows us to integrate the tracer equation without any explicit diffusion operator in the tracer advection equation.

### b. Main experiments

*H*) structure:

*H*

_{0}is depth far outside of the bowl,

*H*

_{b}is bowl depth relative to

*H*

_{0},

*R*

_{b}is bowl radius, and

*r*is distance from bowl center. We call

*W*

_{b}the slope half-width, since most of the topographic variation (76%) occurs over a distance ±

*W*

_{b}from slope center, and since the slope decreases appreciably at larger distances. We are interested mostly in small

*W*

_{b}/

*R*

_{b}values, which prescribe relatively flat bowl interiors (section 1). For example, with

*W*

_{b}/

*R*

_{b}= 1/6, the slope magnitude at

*r*=

*R*

_{b}−

*W*

_{b}is already an order of magnitude smaller compared with at

*r*=

*R*

_{b}. Examples of several

*H*profiles with different parameter choices used in our experiments are shown in Fig. 2. Experiments with more complex topography are discussed in section 6.

*ψ*, i.e.,

*υ*= ∂

_{x}

*ψ*,

*u*= −∂

_{y}

*ψ*. The streamfunction is defined by its discrete Fourier transform,

*x*and

*y*directions, respectively. The dependence of the eddy energy on wavelength is set by

*K*

_{0}, the wavenumber at which the spectral power peaks, and

*dK*=

*K*

_{0}/8, the exponential width of the spectral power maximum. The phase

*θ*

_{k}of each Fourier component (

*k*,

*l*) is randomly generated from a uniform [0, 2

*π*] distribution. The factor

*N*is a normalization constant that is selected to make domain-averaged kinetic energy density equal to a prescribed value

*E*. An example of the resulting initial conditions is shown in Fig. 2a.

Table 1 lists our main single-layer experiments. We divide the experiments into batches according to the values of the geometric parameters *H*_{0}, *H*_{b}, *R*_{b}, *W*_{b}, and *λ*_{0} = 2*π*/*K*_{0}. In each batch we fix these parameters and vary the initial kinetic energy density *E*. Each experiment is then referred to via the naming convention B*n*E*m*, where *n* is the batch number (left column of the corresponding row in Table 1), and *m* is the experiment number within that batch. We set *m* equal to 1, 2, 3, … to denote the experiments with 1st, 2nd, 3rd, … lowest initial kinetic energies within each batch.

Summary of our main single-layer experiments. Each experiment is later referred to by a name of form: B*n*E*m*, where *n* is the batch number (left column of appropriate row in the table), and *m* is the experiment number within the same batch. Parameters pertaining to the topography and initial conditions (IC) are given in each column. An experiment number *m* of 1, 2, 3, … corresponds to the 1st, 2nd, 3rd, … lowest initial kinetic energy experiment within each batch (rightmost column). For example, experiment B1E5 (for which diagnostics are shown in Figs. 3 and 4) refers to the fifth lowest initial energy (0.05 m^{2} s^{−2}) among the experiments in the first row (*n* = 1).

The values of *H*_{0} used are typical of deep seas (~2 km) or abyssal ocean depths (~4 km). The values *H*_{b} = 0.1–0.5, *R*_{b} = 150–300, and *W*_{b} *=* 50–100 km, are loosely based on values relevant for the Mann Eddy basin, although the latter is significantly more complicated than a symmetric bowl shape. The selected kinetic energy density values span (and exceed by a factor of ~2) the range of typical ocean velocities, i.e., velocity scales from centimeters per second to several meters per second. We set the Coriolis parameter to a value of *f* = 10^{−4} s^{−1} in all cases.

Unless otherwise specified, we conduct our experiments in square domain with a side length of 1000 km. We use periodic boundary conditions to facilitate comparison with topographic turbulence theories. We conducted several experiments with wall boundary conditions or with periodic domains multiple times larger, but we found no appreciable differences in the resulting bowl circulation. The horizontal resolution *dx* used is uniform and is either ≈2 or 1 km, corresponding to 512^{2} or 1024^{2} grid cells, respectively. The latter was used only in batch 1. In all cases *dx* ≤ *λ*_{0}/46. We reran various experiments from several different batches with the grid spacing decreased by a factor of 2–4 to check the resolution sensitivity, but these experiments exhibited little quantitative and qualitative differences. Biharmonic momentum dissipation with constant coefficient *A*_{4} = 0.01*dx*^{3}*U*_{0} was used, where *U*_{0} = 1.5max|**u**_{0}(**x**)|, and **u**_{0}(**x**) is the initial fluid velocity at position **x**. Rerunning several experiments with Smagorinsky-like (Griffies and Hallberg 2000) biharmonic dissipation operator instead resulted in negligible quantitative differences. We also verified that kinetic energy does not accumulate at the grid scale over time—a sign of insufficient gridscale dissipation. Minimal experiment duration was 500 days. Model output was saved in 5-day (1-day) averages in batches 1–5 (6–9), except where noted otherwise.

*c*(

*x*,

*y*,

*t*= 0) =

*c*

_{0}(

*r*), where

*r*is distance from bowl center, as follows:

*W*

_{c}is set to a value of 10 km to minimize the width of the transition in the tracer concentration without introducing numerical artifacts in the calculation.

### c. Coherent monopole and dipole experiments

*H*

_{0}= 4,

*R*

_{b}= 300,

*W*

_{b}= 50, and

*H*

_{b}= 0.5 km. A domain length of 1500 or 2000 km was used, with little difference between the two, and with 1024

^{2}grid cells. All other parameters were identical to those described above. The prescription of a single vortex (centered at initial location

*r*

_{0}(m

^{−1}):

### d. Eddy detection

To track the central AC we use an algorithm based on an Okubo–Weiss parameter (OWP) threshold. The OWP is defined by OWP = *s*^{2} − *ζ*^{2}, where *s* is horizontal strain [*s*^{2} = (*u*_{x} − *υ*_{y})^{2} + (*u*_{x} + *υ*_{y})^{2}], and *ζ* = *υ*_{x} − *u*_{y} is the vorticity within an isopycnal layer. Eddies are characterized by negative OWP values, i.e., vorticity dominating over strain. Within a single time sample and isopycnal layer, a grid cell (with index *i*) is identified as a vortex-core candidate if the OWP in that cell satisfies OWP_{i} < −*a*STD_{OWP}, where STD_{OWP} is the spatial standard deviation of OWP for the same time sample and isopycnal layer, and *a* is a constant factor. Drawing on previous studies (Pasquero et al. 2001; Isern-Fontanet et al. 2006; Volkov et al. 2015), we set *a* = 0.2.

A connected region of model grid cells is identified as containing a vortex “core” if each cell was identified as a candidate, if *ζ* is singly signed in the connected region, and if the total area *A* is larger than an imposed minimum *r*_{min} = 10 km, equivalent to 5 grid cells for the lowest-resolution experiments reported here. The detected eddy radius is defined as *A* is the detected (connected) area size. The detected eddy vorticity is defined as the spatially averaged vorticity within the detected area. In simulations that develop a bowl-trapped AC, we found that the AC could be tracked accurately by searching for the strongest AC within the bowl at any given time.

## 3. Emergence of barotropic bowl-trapped anticyclones

We begin by describing the results of a representative single-layer experiment in some detail, in section 3a. In section 3b we consider the results of all standard single-layer experiments (section 2b and Table 1) and identify nondimensional parameters that approximately constrain the properties of the bowl-trapped AC. In section 3c we investigate the long-time fate and stability of the emergent bowl AC. In section 3d, we present a vorticity budget for the evolution of the trapped AC and of the bowl slope current.

### a. Emergent circulation over a bowl: Case study

In this subsection we describe the results of experiment B1E5 (Table 1), as an illustrative and typical example of bowl AC formation and of its properties in our experiments. The results of free evolution from the random initial conditions (section 2) are graphically summarized in Figs. 3 and 4, and a movie (Animation S1 in the online supplemental material).

Two main circulation patterns emerge within the bowl. One is a cyclonic slope current, i.e., propagating with shallower water to its right (cyclonic in the present case). It is associated with positive vorticity in the inner slope region and somewhat interior to it, and azimuthal velocity peak at midslope. The second emergent circulation pattern is a central (bowl-trapped) AC. The AC is apparent by its negative vorticity around the bowl center (Fig. 3), and by its negative (retrograde) azimuthal velocity, peaking at a radius of about 60 km (Fig. 4a). Multiple (mainly cyclonic) eddies survive outside of the bowl (Fig. 3d), although in time they tend to merge into a smaller number (see Animation S1).

The emergence and intensification of the trapped AC is related to repeated merging of ACs (Fig. 3). Downslope migration of ACs contributes to these mergers. In contrast, cyclones are cleared from the center of the bowl, leaving a diffuse cyclonic layer over most of the bowl interior, up to the center of the slope (Fig. 3d). A consequence of this redistribution of cyclonic and AC vorticity is that potential vorticity (PV) becomes segregated, with low PV material forming the center AC (Fig. 4b). Material transport and vortex cross-slope propagation are further investigated in section 4.

The emergence of a prograde slope current is to be expected based on topographic turbulence theories posed by previous studies (section 1), whereas the emergence of a central AC is not. This is underlined by the late-time streamfunction–PV relation (Fig. 4); this relation is multivalued, contrary to the predictions of topographic turbulence theories (e.g., Bretherton and Haidvogel 1976). This point and further comparison with topographic turbulence theories are discussed in section 6. However, the evolution time scales of the slope current and of the AC are similar (Fig. 4a), which motivates a discussion of the relation between vorticity fluxes contributing to the AC and to the slope current formation (section 3d). The anticyclone is stable and long lived, in the sense that it is little changed in form or amplitude from its formation, around day 150, until the end of the present experiment, at day 1000. The same applies to the slope current. AC longevity is further investigated in section 3c.

### b. Regime diagram

#### 1) Nonlinearity parameter

*ϵ*is defined by a vorticity magnitude

*V*/

*L*relative to

*fH*

_{b}/

*H*

_{0}, the value of topographically induced vorticity due to hypothetical crossing of the bowl slope:

*L*is an eddy length scale. The late-time radii of eddies (within and outside of the bowl) are of order 50 km in all cases, despite starting from different initial length scales. That is partially since topography limits the progression of the inverse cascade. Hence we set

*L*= 50 km. The choice is preferable to using the final AC radius

*r*

_{e}since

*L*(like

*V*) is a “coarse-grained” parameter. However, we find that both choices result in similar

*ϵ*values.

In Fig. 5a we plot the distance *D* of the AC from the center of the bowl (section 2d), averaged over the last 100 days of each experiment. This shows that trapped ACs form within the bowl in all instances with small enough *ϵ*, i.e., *ϵ* = 0.5–1. However, advection is necessary to the formation, i.e., no mean anticyclonic flow emerges inside the bowl in the limit *ϵ* → 0, and neither does the slope current. In appendix B we verify this using a linear simulation. This is also consistent with the bowl vorticity budget (section 3d).

Significant variation in *D* occurs in the range *ϵ* = 0.5–1, as discussed in section 3b(2) below. At higher values of *ϵ* may be interpreted as determining the dominance of eddy–eddy versus eddy–topography effects. However, the crossover does not occur sharply at a single *ϵ* value across different experiment batches, as there is substantial scatter in the diagnostics due to interexperiment variations in the random initialization. Note that the slope current, predicted by topographic turbulence theories (section 1), persists in all cases.

Finally, we quantify the bowl-trapped AC strength, i.e., integrated vorticity in its core, *r*_{e} is the diagnosed AC radius (section 2d). We pose a scaling for the core-averaged vorticity: *ζ*_{0} ~ *V*_{e}/*r*_{e}. Anticipating that *V*_{e} should scale with initial kinetic energy *E*, we then pose *a* in all experiments where a bowl trapped AC occurs. The result (mean ± standard deviation) is *a* = −3 ± 1 across three orders of magnitude of *E*. Thus the AC is anomalously strong relative to scaling with domain-mean kinetic energy value, i.e., |*a*| is ≥1. That is consistent with the accumulation of anomalously low PV within the AC through repeated mergers (section 3a).

#### 2) Background PV homogenization

For *ϵ* ~ 0.5–1, the trapped AC typically drifts azimuthally around the bowl center, at a radius which tends to increase with *ϵ*. These states can occur even with AC close to the slope peak (*r* = *R*_{b}). We find that these AC states near the topographic slope occur in cases where “background” PV is homogenized within the bowl. “Background” here refers to the area outside of (excluding) the trapped AC, as defined by a PV inhomogeneity parameter below. The topographic beta-drift tendency, including in the cross-slope direction (Carnevale et al. 1991), is negated due to the homogenization of background PV. An illustrative example (experiment B6E7) is shown in Fig. 6. The particular experiment was chosen for visual clarity since PV homogenization is almost complete and the eddy drifts quite close to the slope center, but is otherwise typical of this regime.

In the quasigeostrophic (QG) approximation, vortices within jets with homogenized PV propagate with the local-mean (eddy PV-weighted) velocity (Marcus 1990). We show in appendix A that in shallow water equations (SWE) there is an additional along-topography drift, not present in the QG approximation. This additional drift is related to topographic stretching of relative vorticity, and does not vanish with homogenized background PV. In the experiment B6E7 shown in Fig. 6, the AC drifts with propagation speed ~0.82 m s^{−1}. Calculation of the vortex propagation speed formula (22) in appendix A gives in this case a theoretical drift speed of 0.9–1 m s^{−1}, of which 0.24–0.34 are due to the additional SWE term (relative vorticity stretching), and the remainder is due to vortex advection by the mean velocity. The 10%–20% deviation is potentially due to vortex effects on the slope current which advects it, since the vortex is of significant magnitude and since rigid-lid barotropic vortices have long-range velocity tails (McWilliams 2006).

The vortex in B6E7 is long-lived despite residing in a region of azimuthal-mean strain induced by the slope current. It was found by Marcus (1990) that a vortex can be stable within a large-scale mean current *υ*(*r*) (flowing in the azimuthal direction, and varying in the radial direction) when its vorticity *ζ* has the same sign as the large-scale current radial strain, i.e., *σ* = *r*∂_{r}*υ*/*r*, and if *σ*/*ζ* is of *O*(1) magnitude or smaller. Indeed in the present case (B6E7) the mean radial strain is about −0.04*f* in the radial position of the AC, while AC vorticity ≈ −0.5*f*, fulfilling both the sign and magnitude requirements of Marcus (1990). This implies that bowl-trapped ACs should be able to reside (while moving azimuthally) at any bowl radial position *D* within the homogenized area at which the mean strain is sufficiently small, i.e., a continuum (in *D*) of stable steady states (or limit cycles, to be precise) is available. Multiple different steady states were indeed found when repeating, e.g., experiment B6E7 several times with different random choices of initial phases (*θ*_{k}, section 2b) in each repeat experiment. The trapped AC radial position *D* (averaged over the last 100 days, as before) has a high relative variation between these (six) repeat experiments of B6E7, with a (cross-experiment) mean and standard deviation *D*/*R*_{b} = 0.55 ± 0.25.^{6} Such repeat experiments were conducted for all batch 6 (B6) experiments. Similarly to the repeats of experiment B6E7 (not shown), large variations in *D*/*R*_{b} at constant *ϵ* are found to be the rule for 0.5 < *ϵ* < 1, but are several times smaller for *ϵ* < 0.5. The (partial) PV homogenization thus explains the possibility for trapped AC states at larger *D*/*R*_{b} values at 0.5 < *ϵ* < 1 (compared with smaller *ϵ*), and possibly also explains the substantial scatter in *D*/*R*_{b} over the same *ϵ* range in Fig. 5a.

*r*<

*R*

_{b}), excluding the trapped AC core, and PPV IQR is the IQR of the “planetary PV,” PPV =

*f*/

*H*, in the same area. For partially or fully homogenized background PV cases, PVI < 1. Complete homogenization results in PVI = 0. We plot PVI against

*ϵ*and against

*D*in Figs. 5b and c, respectively. As expected, we find that PVI decreases as

*ϵ*increases, and in most cases approaches zero for

*D*occurs when PVI < 0.2. As discussed above, PV homogenization results in a multiplicity of possible “steady states” (limit cycles) that produces the scatter in the

*ϵ*–

*D*and PVI–

*D*relations, visible in Fig. 5.

### c. Long-term evolution

Here we explore the long term evolution of bowl-trapped ACs in our experiments. This is motivated by the persistence of ocean ACs over depressions (section 1), and also in comparison with topographic turbulence theories (section 6). In all experiments in which a bowl AC formed (deduced by its persistence for

To further diagnose long term evolution of the bowl-trapped AC and its dynamical causes, we reran experiment B1E3 (hereby B1E3L) for 5000 days, with daily mean outputs including online momentum equation diagnostics. The experiment develops a trapped AC close to the bowl center, as in experiment B1E5 discussed in section 3a. The spatial resolution was ~2 km, twice as coarse as experiment B1E3. Although the trapped AC core circulation at, e.g., day 500 is 25% higher in the higher-resolution experiment, the end states are qualitatively similar and well resolved in each simulation. Thus the coarser simulation diagnostics are informative about the late time evolution of the trapped AC.

The evolution of the B1E3L AC between days 500 and 5000 is illustrated in Fig. 7. The AC central (peak) vorticity decays over time, by ~50%. However, the integrated circulation of the contiguous patch of negative vorticity around the AC center does not change by more than 1%. The lack of appreciable circulation decay is not inconsistent with the peak vorticity decay, as the AC becomes wider with time (Figs. 7a,b). The integrated circulation in the core region, as defined in section 2d, actually grows by 40% during this period.^{7}

We also diagnose the AC circulation evolution in an Eulerian frame. We calculate the circulation tendency at each time step due to each term in the momentum equation. We perform this analysis around the perimeter of a 50 km × 50 km square in the center of the bowl, which encloses the AC core. The perimeter is aligned with grid axes, to avoid introducing discretization errors. In Fig. 7c we show the circulation tendency due to the sum of the inviscid terms (nonzero contributions are due to advective terms) versus that due to the viscous term. As expected, the AC is initially spun up by the inviscid terms. Similar to the Lagrangian analysis of the eddy core discussed above, this Eulerian analysis shows no clear decay in AC circulation after its formation. Circulation oscillations are of order 2%. The viscous term causes a slight ≈2% decay in circulation between days 500 and 5000. We found that doubling the size of the perimeter used for this calculation did not qualitatively change the result. Therefore the AC does not decay directly by inviscid terms in a symmetric bowl, within at least a scale of thousands of days. The implications of these results are discussed in section 6.

### d. Spinup of the anticyclone and slope current: A vorticity budget

Here we show that the spinup of the bowl anticyclone necessarily implies a tendency for cyclonic vorticity accumulation between the anticyclone and the topographic slope. We further suggest a scaling relation for determining the relative circulation magnitudes of the bowl anticyclone and of the slope current a priori.

*r*denotes the distance from bowl center is denoted by

*r*,

*u*denotes the radial velocity,

*I*(

*r*,

*t*), i.e., cross-bathymetry eddy vorticity flux. Thus we hereafter neglect the viscous terms in (7a).

*r*

_{AC}as the radial position of maximal magnitude in retrograde (anticyclonic) final-state azimuthal-mean velocity within the bowl. Likewise, we choose

*r*

_{SC}as the radial position of maximal magnitude in prograde (cyclonic) final-state azimuthal-mean velocity over the topographic slope. In our simulations the latter occurs around the position of maximal topographic slope,

*r*=

*R*

_{b}. The equations for

*C*

_{AC}≡

*C*(

*r*

_{AC}) and for

*C*

_{SC}≡

*C*(

*r*

_{SC}) then follow from (7a) and (7b):

*r*

_{AC}<

*r*<

*r*

_{SC}) the late-time vorticity is positive (Figs. 3 and 7), principally due to the slope current. The trapped AC in our experiments typically has only a weak

^{8}“shield” of positive vorticity surrounding it (McWilliams 2006). The quantity

*M*

_{s}measures the slope current strength removing the AC integrated vorticity [by the Stokes theorem, Eq. (7a)], and thus is a preferred slope current metric compared with

*C*

_{SC}. Therefore, Eq. (8c) shows that the slope current evolution (∂

_{t}

*M*

_{s}) has positive contributions from two flux integrals, i.e., at the slope region [−

*I*(

*r*

_{SC})] and in the bowl interior [−

*I*(

*r*

_{AC})]. Since

*C*

_{AC}decreases over time (as the anticyclone forms), we diagnose

*I*(

*r*

_{AC}) > 0 for the

*C*

_{SC}increases over time (as the cyclonic slope flow forms), we diagnose

*I*(

*r*

_{SC}) < 0. We conclude that the spinup of the retrograde bowl anticyclone is necessarily associated with an eddy vorticity flux of equivalent magnitude at

*r*

_{AC}driving the slope current. That is in addition to spinup due to contributions of eddy fluxes at the slope region (

*r*

_{SC}).

In Fig. 8a we compare the time evolution of the circulations *C*_{AC} and *C*_{SC} in experiment B1E5 (cf. with Fig. 4a). These are equivalent to the two cumulative eddy-flux tendencies forcing the slope current strength metric *M*_{s}. First, in Fig. 8a it is seen that both eddy fluxes have similar evolutionary time scales. Second, the exterior (*r*_{SC}) fluxes are larger in magnitude (see below). These qualitative observations occur across all the experiments (Table 1). The role of eddy fluxes as a function of position [*I*(*r*), Eq. (7b)] is examined in Fig. 8b. This panel shows cumulative change in circulation over days 150–250 [*C*_{AC}| gains most of its amplitude during this time, as a function of *r*. It is seen that the locations *r*_{AC} and *r*_{SC} are indeed associated with approximately the largest cumulative eddy fluxes. Over this period the eddy vorticity fluxes at *r*_{AC} and at *r*_{SC} have similar sized contributions to the slope current metric *M*_{s}.

Figure 8a shows that the magnitude of the late-time bowl AC circulation is around 1/3 of that of the slope current. Therefore, by the above analysis inner bowl fluxes *I*(*r*_{AC}) supply approximately 1/4 of the cyclonic vorticity that accumulates on the inner bowl slope *M*_{s}. The remainder is contributed by slope-region fluxes *I*(*r*_{SC}). We suggest that the larger contribution from external fluxes *I*(*r*_{SC}) relative to internal fluxes *I*(*r*_{AC}) is largely a geometrical effect, due to larger area outside of the bowl. If this is the case then the slope current circulation should equal the anticyclone circulation, scaled by the appropriate area from which external fluxes are sourced: *B* = 1000 km is the domain length. Indeed, in the figure *C*_{SC} and C_{SC,theory} agree to within 3% at later times. In other experiments (Table 1), the agreement is typically within ~20%.

## 4. Cross-slope motion and PV segregation

Our findings in section 3 suggest that material transport is fundamental to the formation of the bowl central anticyclonic circulation. Indeed, the AC grows in our experiments through eddy mergers, and nonlinearity appears to be necessary (section 3a and appendix B). Furthermore, PV (a materially conserved quantity) is anomalously low within the emergent AC (sections 3a and 3b). Here we address the following questions: Are all ACs that participate in central AC formation initiated within the bowl at *t* = 0, or do some enter from outside of the bowl? If the latter, what are the mechanisms via which they transition from outside to inside the bowl?

### a. Cross-bowl material transport

First, we examine the amount of cross-bowl material transport which occurs in the process of AC evolution using passive tracer deployments. The tracer is initialized (section 2b) with a radial tanh profile, with values close to 1 (0) inside (outside) of the bowl. Therefore a final average value *c* = *c*_{b} < 1 within the bowl center (diagnosed at *r* ≤ *R*_{b}/2), implies that the fraction of material in the same area which originated outside the bowl is *c*_{o} = (1 − *c*_{b}).

An anticyclone can maintain its coherence while crossing the slope into the bowl only if its vorticity *ζ* is high enough to avoid destruction by vortex stretching. Therefore crossing requires *δ* = *H*_{0}|*ζ*|/*fH*_{B} ≥ 1. We test this in Fig. 9, which shows that the amount of bowl anticyclone material in the final state originating from outside the bowl is approximately proportional to *F*_{δ}, the fraction of material with *δ* ≥ 1 at *t* = 0. For small enough *F*_{δ} virtually no material is exchanged across the slope, and the AC forms only from material originally present within the bowl. At higher *F*_{δ} values, a substantial fraction of the late-time material in the AC originates outside the bowl.

The interpretation of the late-time tracer concentration is complicated by the initial *O*(50)-km-wide tracer transition area on the bowl slope, where 0 ≤ *c* ≤ 1. Due to this transition region, in a hypothetical end state in which no material is exchanged across *r* = *R*_{b} + 2*W*_{b} (at the 98th depth percentile) and all material in *r* < *R*_{b} + 2*W*_{b} is well mixed, the bowl tracer concentration would be *c* ≈ 0.75. Hence we estimate the maximal possible bias from the tracer transition region via an added dilution of 25%, shown by the gray area in Fig. 9. Another complication is that in a finite domain only a finite maximal dilution can occur; this is also marked in Fig. 9 by the dotted line.

In cases with small *F*_{δ}, the anticyclonic vortex growth still occurs by repeated vortex mergers, but only emerges from AC eddies that are already in the bowl at *t* = 0. We observe (not shown) that anomalously high-PV material (i.e., cyclones) initially inside the bowl ultimately transit to the slope region, either as part of the cyclonic slope current, or as a coherent cyclonic eddy embedded within the slope current. Stability of cyclonic eddies within the inner part of the slope current (for small *F*_{δ}) is generally consistent with the results of Marcus (1990), as we diagnose strain *σ* > 0 in the inner slope region in these cases. The sign is opposite to that reported for experiment B6E7 [section 3b(2)]. To see why, note that cyclonic slope current velocity peaks close to the slope center. Hence strain *σ* = *r*∂_{r}(*υ*/*r*) is (negative) positive in the inner slope region if the velocity peak is locally (less) more sharp than a linear profile. Weak *ϵ* (and *F*_{δ}) results in weak PV homogenization, and hence in sharp slope currents, and vice versa, consistent with the diagnosed signs in each case.

In cases where *F*_{δ} is large enough, cross-slope motion may be induced by at least two processes in the present experiments: 1) monopole vortex (topographic) beta drift (Carnevale et al. 1991) and 2) dipole propagation. Rather than attempting to determine the fraction of monopole and dipole (or multipole) interactions contributing to the AC formation in our turbulent experiments, we concentrate on a more tractable task. We present results from topographic bowl experiments initialized with a single monopole or dipole (section 2c), and compare the cross-topography propagation speed of each. For brevity, in each case a single illustrative experiment is presented, along with formulae for propagation speeds in the general case.

### b. Monopole topographic beta-drift

*β*) drift meridionally (equivalently, cross-slope on a topographic

*β*plane) with speed

*r*

_{0}is the vortex radius (or, more generally, the pressure

*e*-folding scale). It is not clear to what extent these results should hold for a finite-width and curved bottom slope. In this case we define a cross-slope vortex (monopole) propagation speed (

*υ*

_{m,1}) predicted from the local topographic beta value,

*β*(

**x**) = −(

*f*/

*H*)|∇

*H*(

**x**)|:

*f*/

*H*contours. Therefore, it may be that the relevant

*β*value for vortex drift is to be evaluated in a region around the vortex rather than at its center. Thus we also test a second hypothetical speed,

*r*

_{0}≤

*r*

_{υ}≤ 2

*r*

_{0}, with

*r*

_{υ}= distance from vortex center.

Expressions *υ*_{m,1} and *υ*_{m,2} are compared with the diagnosed vortex downslope propagation speed, *υ*_{m,e} for a vortex of radius *r*_{0} = 45 km initialized outside the bowl in Fig. 10a. Indeed, the approximate nonlocal generalization of the beta drift term, *υ*_{m,2}, compares quite well with the diagnosed speed up until the vortex arrives at the middle of the slope. In comparison, *υ*_{m,1} is considerably lower outside the bowl. After the vortex reaches the middle of the slope region, both of the above formulae fail to reproduce its subsequent propagation speed. The vortex acquires a weak dipole component as it propagates across the slope (McWilliams and Flierl 1979), which survives into the flat interior region. The companion is largely responsible for the deviation between the theoretical predictions and the diagnosed propagation speed, as are the presence of a slope current and topographic Rossby waves induced by the passage of the vortex.

### c. Dipole propagation

Dipoles propagate perpendicular to their eddy-separation axis. Topography causes an asymmetry in their dynamics such that propagation in the downslope direction often results in the anticyclone shedding into the depression while the cyclone is repelled outside (Carnevale et al. 1988).

The trajectory of the anticyclone from a representative dipole experiment (see section 2c) is shown in Fig. 10b. Here the initial mean vorticity in each vortex is 0.25*f*, and the dipole orientation was chosen such that the dipole propagates directly toward the bowl. The dipole propagates toward the bowl until the cyclone is shed upon arrival at the bowl slope (Carnevale et al. 1988) between days 15–20, after which the anticyclone continues downslope and the cyclone is eventually ejected upslope. It is seen that away from the bowl rim, the dipole travels considerably faster than the monopole in Fig. 10a. For example at 1.5 bowl radii away (*r* = 1.5*R*_{b}) from its center (with present slope half-width parameter *W*_{b} ≈ *R*_{b}/6) the dipole is an order of magnitude faster than the monopole. The dipole is initialized at a greater distance than the anticyclone to illustrate this difference further.

The dipole speed can be predicted based on a theoretical model of point vortices (McWilliams 2006; Kloosterziel et al. 1993). The point model predicts a dipole speed *υ*_{d} = *C*/(2*πd*), where *C* is the strength (peak circulation) of each vortex, and *d* the pair separation distance. The value of *υ*_{d} is estimated based on diagnosis of these parameters from the dipole experiment. Its value generally agrees well in order of magnitude with the diagnosed anticyclone downslope drift speed before the cyclone is shed. The dipole begins somewhat slower than *υ*_{d} and overshoots its value slightly. This may be explained by that the initialized state is comprised of two superposed monopole vorticity fields, and hence some initial adjustment occurs. The adjustment process results in a smaller *d*, which explains the faster velocity at later times.

In summary, in this subsection it is shown that a bowl trapped anticyclone can form either locally from material initially present within the bowl, or also by sourcing material from outside the bowl. The percentage of externally sourced material depends largely on *F*_{δ}, the fraction of the initial anticyclonic vorticity that exceeds the topographically imposed vorticity change. Anticyclones that enter the bowl can do so via either monopolar or dipolar propagation. It is shown that dipole cross-slope propagation can be considerably faster than monopole topographic beta drift former under certain conditions. For the monopole case, due to the nonuniform bottom slope, an approximate generalization the constant-slope formula is suggested, and reasonably matches the diagnosed speed.

## 5. Two-layer experiments

In this section we report the results of our two-layer experiments—the minimal configuration that permits baroclinicity. We examine whether a bowl AC forms in baroclinic conditions, and diagnose its vertical structure as a function of system parameters. The very long lived ACs observed in the ocean above depressions (section 1) are surface intensified (note their velocity maximum commonly occurs at depths of ~500 m), but a nonnegligible barotropic component is observed as well (Mann 1967; Willis and Fu 2008; Köhl 2007; Fer et al. 2018; Bosse et al. 2019; Le Corre et al. 2019). In our idealized barotropic experiments AC formation depends strongly on topographic effects, and it is unclear whether this should favor a surface-intensified trapped AC structure.

Several batches of experiments were conducted. Stratification parameters were varied between batches, and the initial vertical structure was varied within each batch. For brevity we report mainly on a reference batch (hereafter *B*_{BC1}, BC standing for “baroclinic”) of experiments with *λ*_{0} = 90 km, upper-layer rest thickness *H*_{1} = 1000 m, and topographic parameters *R*_{b} = 300, *W*_{b} = 50, *H* = 2, and *H*_{b} = 0.5 km. The reduced gravity is set to *g*′ = 10^{−2} m s^{2}, resulting in a baroclinic Rossby deformation radius of *H*_{eq} = *H*_{1}*H*_{2}/*H* is used, with *H*_{2} = *H* − *H*_{1}. These parameters are loosely motivated by the Mann Eddy basin (see also section 2b). The Rossby deformation radius in the top (bottom) layer is

The initial kinetic energy densities of the top and bottom layers [*E*_{1} and *E*_{2} (m^{2} s^{−2}), respectively] were varied between the *B*_{BC1} batch experiments, taking the following relative values: (*E*_{1}, *E*_{2})/*E*_{0} = (0,1), (0.25,1), (0.5,1), (0.75,1), (1,1), (1,0.75), (1,0.5), (1,0.25), (1,0). These experiments were repeated with two different kinetic energy densities in the dominant layer: *E*_{0} = 0.01 or *E*_{0} = 0.1 (m^{2} s^{−2}). In each experiment, the random initial phases *θ*_{k} were generated independently in each layer.

We find that a bowl-trapped AC forms in all *B*_{BC1} experiments. A specific example of the evolution is shown in Fig. 11, from the *B*_{BC1} experiment with *E*_{2} = 0 and *E*_{1} = 0.1 [m^{2} s^{−2}]. By day 30 the circulation has largely barotropized as is generally expected for circulation features larger than the deformation radius (Salmon 1998). By day 250 a single coherent AC is formed near the bowl center. It is top intensified but has a substantial barotropic component. Topographic Rossby waves (TRW), straining of eddies on the slope, and the emergent slope current all appear to be bottom intensified.

More generally, we find that the trapped AC tends to be top (bottom) intensified for top (bottom) intensified (random) initial conditions. Vertical structure results are presented in Fig. 12, in terms of a surface intensification metric: *S* ≡ log(*υ*_{1}/*υ*_{2}), where *υ*_{i} is a velocity magnitude in layer *i*. For the initial *S* metric, *S*_{0}, *υ*_{i} is defined as the initial time spatial-RMS velocity value. For the final *S* metric, *S*_{f}, *υ*_{i} is defined as an average over the last 100 days of azimuthally averaged azimuthal velocity at the radius of peak eddy velocity (for the AC), or at midslope (for the slope current). In contrast to the AC, the slope current is consistently bottom intensified, with a similar value of final *S* value in all *B*_{BC1} experiments. Following these diagnosed trends, we suggest simple scaling rules for top intensification of the AC and slope current.

^{9}

*i*= 1 and

*i*= 2 subscripts, respectively. For any wave mode (with nonzero frequency) the expression within the parentheses need be identically zero. Therefore,

*W*

_{b}(section 2b). Hence

*S*

_{0}= ±∞). Suppose conditions are initially surface intensified, i.e.,

*ψ*

_{2}(

*t*= 0) = 0 and

*S*

_{0}= ∞. Then initially the lower layer evolution is approximately described by the linearized QG equation,

*ψ*

_{1}(

*t*= 0) ≈ 0 at the bowl center. We also assume that the topographic term will be initially negligible since

*ψ*

_{2}(

*t*= 0) ≈ 0 and since the topography is weak at bowl center. Therefore, in this case the AC amplitude in the lower layer follows from

*r*

_{i}is the AC core radius in layer

*i*.

^{10}The eddy core radii

*r*

_{i}are diagnosed in each case (section 2d), and are typically 35–60 km in these simulations. Therefore

*S*

_{0}= −∞ the late-time scaling resulting from similar reasoning is

*B*

_{BC1}numerical experiments results in Fig. 12. The scaling laws predict the right sign of

*S*

_{f}, i.e., top or bottom intensification. The slope–current prediction [Eq. (13)] is indeed very close to the numerical final state

*S*

_{f}, for all examined

*S*

_{0}values. The AC

*S*scaling relations, (16) and (17), are of the right

*S*

_{f}sign and order of magnitude (within 7%–40%) in all applicable cases (i.e.,

*S*

_{0}= ±∞).

The scalings (13)–(17) all share a dependency of the form *L*_{i}/*l*, where *l* is the relevant circulation feature length scale. Hence we conducted additional experiment batches varying the Rossby radii *L*_{d,i} (e.g., by changing stratification). Cases with much higher Rossby radii of order 100–150 km were attempted. The AC core radii *r*_{i} were <90 km in all cases. We find that trapped anticyclones still form in cases with *S*_{0} < ∞, and in some cases with *S*_{0} = ∞.^{11} The trend predicted by the scaling relations is correct in these cases, i.e., top or bottom intensification is more acute relative to *B*_{BC1}, by as much as on order of magnitude. The scaling relations also predict the right *S* order of magnitude in each applicable case. The scatter in *S* is however relatively larger than in *B*_{BC1}, and we do not attempt a further systematic exploration.

To test the longevity of the trapped AC, similarly to the 1 layer experiment in section 3c, the duration of the *B*_{BC1} experiments with *S*_{0} = ±∞, 0 and with *E*_{0} = 0.1 (m^{2} s^{−2}) were extended from 500 to 3000 days. The main qualitative result (not shown) is that the AC and slope current both survive largely unchanged throughout the experiment. The results are quite similar to the single-layer case.^{12} Some diffusion of the trapped AC occurs: the trapped AC peak vorticity in each layer decreases by 10%–30% between days 500 and 3000 while its radius expands. In each layer again, the integrated circulation of the contiguous patch of negative vorticity around the AC center does not change by more than 1%, while that of the core grows by several tens of percent. Implications are discussed in the next section.

## 6. Discussion: Complex topography and topographic turbulence theories

All numerical experiments discussed above involved an isolated and functionally simple topographic feature. Real ocean topography is characterized by multiple scales and roughness. Hence, experiments with nonisolated topography were conducted as well. Complex topographic shapes were created using a similar random formula to (2) (e.g., Fig. 13). Cyclonic (anticyclonic) circulations emerge on the slopes of the topographic depressions (bumps), i.e., slope currents, consistent with previous results (e.g., Bretherton and Haidvogel 1976). However, embedded within the interiors of these large-scale circulations, we find that coherent ACs (cyclones) with anomalously low (high) PV appear at the center of some or all topographic depressions (bumps) in each experiment. The trapped coherent eddies emerge as long as *λ*_{0} is small enough (e.g., a factor of ≈5 was sufficient) relative to the analogous typical topographic wavelength *λ*_{t}. This condition, as well as numerical resolution, may explain the lack of previous reports on these vortices in similar numerical simulations. Results from one of our random topography experiments are shown in Fig. 13. We note that in some random topography experiments (including in Fig. 13) not all topographic (depressions) bumps contain a clear interior (anti)cyclonic vortex. That is to be expected since the trapped vortices have anomalous PV values (sections 3a and 3b). The limited reservoir of anomalously valued PV material did not produce a sufficient number of vortices to populate all topographic features, but we did not explore the dynamics or statistics of this partitioning in detail.

We also conducted experiments with isolated topographic features lacking a significant central flat region of size larger than an eddy size. That is of interest relative to seamount circulation (section 1). An example of an “inverted seamount,” a Gaussian topography with 50-km standard deviation, is shown in Fig. 2. Results are not shown here graphically in the interest of space. In this case the prograde slope current manifests as a cyclonic eddy (Taylor Cap), as predicted by seamount circulation theory (section 1). Additionally, we find that trapped (ACs) cyclones can emerge over the slope regions of such (inverted) seamount shapes, and propagate with the slope current at a constant radius. This occurs when partial homogenization of the PV field is achieved, i.e., the cause is similar to the

As summarized in section 1, topographic turbulence theories broadly predict that (anti)cyclonic circulation should form over (bumps) depressions. These predictions have been verified in previous idealized numerical simulations (Bretherton and Haidvogel 1976; Salmon et al. 1976; Merryfield 1998; Majda and Wang 2006; Venaille 2012) and in the present study. However, these theories do not predict a circulation of opposite polarity in the interior of closed topographic contours, i.e., they do not predict the formation of the bowl-trapped AC in our experiments.

A fundamental feature of these topographic turbulence theories is the prediction of a single-valued (i.e., monotonic) streamfunction (*ψ*) to PV (*q*) relation (Bouchet and Venaille 2012). To the contrary, we find the relation is multiple-valued in our simulations (e.g., Fig. 4b). It is clear that the multiple-valued *ψ*–*q* relation is due to the crossover from the bowl AC to the slope jet. Hence the theories cited above cannot predict these two features together. Earlier numerical random topography simulations (Bretherton and Haidvogel 1976) did exhibit differing, quasilinear, *ψ*–*q* relations over different subdomains. However, we find a qualitatively different, multivalued *ψ*–*q* relation within a single isolated and smooth topographic feature. A deviation from single-valued *ψ*–*q* relations in idealized simulations was also previously reported by Vallis and Maltrud (1993), associated with alternating jets parallel to steep topography.

A possible explanation for the disagreement of our results with topographic turbulence theories is that the AC (unlike the slope current) is a transient feature, and hence should not be predicted by equilibrium theories. However as shown in sections 3c and 5, the trapped AC circulation does not decay by a measurable amount over a time scale of at least ~10 years. In barotropic numerical simulations of a closed basin with a “continental slope” close to the basin edges and a flat bottom in the center, the spin up of a center anticyclone was previously reported (Cummins and Holloway 1994; Shchepetkin 1995). In the case of Cummins and Holloway (1994), the center anticyclone decayed over a time scale equivalent to ~100 years. Cummins and Holloway (1994) argued that the decay occurred inviscidly. We note, however, that the AC evolution and the slope–current evolution are not completely independent, as the results of section 3d show. Finally, it is possible that this local coupling is not captured in the discussed topographic turbulence theories since they impose conservation laws only in an integrated sense, or with a mean-field approximation (Bouchet and Venaille 2012; Venaille 2012).

## 7. Summary and conclusions

In several North Atlantic basins, long-lived and semistationary mesoscale anticyclonic vortices (ACs) have been observed repeatedly. These basins are characterized by bowl-like topography, i.e., slopes surrounding a much less steep central area. The ACs reoccur over periods of years to decades, and have a significant signature on mean regional properties (e.g., SSH, Fig. 1). Previous work suggested different dynamical mechanisms for AC formation and persistence in each of these cases.

Motivated by these observations, we conduct idealized numerical experiments of flow evolution over bowl-like topography to determine if and how trapped ACs evolve in a minimal complexity model. Primitive equation simulations with one or two isopycnal layers are conducted. Although many processes are neglected, the lighter computational burden facilitates multiple experiments, sweeping wide parameter ranges.

We find that a bowl-trapped AC does emerge spontaneously from random initial conditions under a wide range of circumstances. Typically this occurs through repeated mergers of ACs within the bowl interior, and the resulting trapped AC is characterized by anomalously low PV. Another general result, consistent with previous theory and simulations (the Neptune effect, section 1), is the emergence of a prograde slope current, corresponding to cyclonic circulation around a bowl.

To determine the robustness and parameter dependence of these phenomena, we conduct a large array of single-layer experiments. The initial kinetic energy and dominant initial circulation spatial scales, as well as topographic shape parameters, are varied. A nonlinearity parameter *ϵ* is identified as largely determining the formation of a trapped AC in these experiments. This parameter is the ratio of vorticity scale to topographic vorticity stretching, and describes the relative effects of eddy–eddy interactions to topographic effects. The AC typically forms for *ϵ* = 0 (i.e., with advection terms neglected, as shown in appendix B). In one regime, *r* ≤ *R*_{b}, depending on initial conditions. In the third regime,

To explain the varying radial positions of the bowl-trapped ACs, we introduced a second nondimensional parameter, PVI, a metric of PV inhomogeneity (excluding the trapped AC PV signature). At small *ϵ* the radial PV gradient is dominated by the topography, and PVI ≈ 1. However, PVI generally decreases with increasing *ϵ* values, and in most cases satisfies PVI ≪ 1 for

While it is clear that AC mergers contribute to the trapped AC formation, it is not a priori clear whether these ACs should originate from inside or outside of the bowl. We show that the origins of the ACs that contribute to the bowl-trapped AC is set by a parameter *F*_{δ} (section 4), which quantifies the fraction of the initial anticyclones that are outside of the bowl and are sufficiently strong to cross the topographic PV gradient. We show through tracer analysis that for weak vortices, i.e., small *F*_{δ}, there is negligible exchange of material across the topography. The relative exchange and amount of final “dilution” of the inner bowl material, grows approximately linearly with *F*_{δ}.

Motivated by the role of anticyclones migrating into the bowl in forming the trapped AC, we isolated and examined two different mechanisms of cross-bowl transport: eddy (topographic) beta drift (Carnevale et al. 1991), and dipole interactions (Carnevale et al. 1988; Kloosterziel et al. 1993). The former was previously suggested to be important in the case of the Lofoten AC (Köhl 2007). Using experiments with initial conditions of a single (monopole) AC, or of a close pair of opposite-signed eddies (dipole), we show that both mechanisms can allow anticyclones to enter the bowl. However, dipole propagation is typically more efficient for reasonable parameter values, especially at larger distances from the bowl slope. We proposed an approximate generalization of the previous theoretical formula for monopole beta drift, to account for the variable topographic slope.

Two-layer experiments support the one-layer results on AC formation. We focus on the characterization of vertical structure of the emergent trapped AC, as well as of the slope current. We find that the emergent slope current is bottom intensified in all cases, to a degree determined by stratification conditions, similar to a Taylor Cap (Hogg 1973). In contrast, the emergent trapped AC is bottom intensified (surface intensified) if domain-mean initial conditions are bottom intensified (surface intensified). The vertical structures of the observed ocean ACs (see section 1) are surface intensified, with maximal velocities within the upper 1 km of the water column, and velocities decaying toward smaller but finite values near the sea floor. Thus an interpretation of our two-layer experimental findings is that the general tendency for ocean mesoscale eddies to be surface intensified leads to surface-intensified trapped ACs such as the Mann and Lofoten Eddies.

We find that, in the single layer experiments, the spinup times of the bowl AC and the slope current are similar. Therefore, we examine the relation between vorticity and circulation evolution equations of both the AC and the slope current (section 3d). This relation shows that the vorticity fluxes driving the formation of the bowl AC also contribute to the spinup of the slope current. Furthermore, although the slope current is accelerated by a second source of vorticity fluxes as well (from outside the bowl), the final bowl AC strength is a good predictor of the slope current strength. These relations suggest that the vorticity fluxes inside and outside of the bowl are both mediated by similar mechanisms.

In section 6 we contrasted our results with topographic turbulence theories. These theories predict the emergence of a prograde slope current, but not a bowl-trapped AC. We find that depression-trapped ACs also occur in simulations with complex topographies, similar to experiments previously used to test topographic turbulence theories. We suggest an emergence criteria (for trapped ACs over depressions) in such simulations, based on the relative length scales of topography and circulation. The lack of bowl-trapped ACs in topographic turbulence theories is discussed. The aforementioned theories only predict strictly steady states. It is possible that the trapped AC is a transient state. However trapped ACs in our bowl simulations are stable over time scales of at least dozens of years. Additionally, it is unclear if local vorticity fluxes, which couple the slope current and AC formation tendencies (section 6), are captured by mean-field approximations in such theories.

The numerical model used here is much too simple to directly apply to the discussed ocean observations of long-lived semisteady ACs (section 1). One of the potentially important factors which are not included are more complicated stratification conditions (Bashmachnikov et al. 2017). The low stratification conditions, high *f*, and low planetary *β* values in the subpolar areas make this limitation less severe than in other areas. Indeed, Isachsen et al. (2003) found that a barotropic model explains over 50% of seasonal gyre variability in the Nordic seas (including the Lofoten Basin). As mentioned above, each of the observed quasi-stationary vortices has a significant barotropic component (section 5). However, inclusion of surface or bottom mixed layers can lead to submesoscale instability of the eddies (Brannigan et al. 2017). Trapping of near-inertial waves within a fully three-dimensional eddy can lead to additional dissipation as well (Fer et al. 2018).

Another limitation is that the neglect of external circulation patterns (e.g., boundary currents and eddy fluxes), and of atmospheric forcing. This limitation is very partially addressed here in the random topography experiments, as different topographic bowls (or bumps) and their emergent circulations, are not isolated from each other. The influence of regional circulation features and perpetual external variability or forcing need to be studied within intermediate complexity models or realistic regional numerical simulations.

Different formation mechanisms were previously suggested and diagnosed for observed semistationary ocean ACs (section 1). The present results suggest that whatever the source of low PV material (cf. section 1, e.g., current instability, eddy mergers, deep convection, etc.) and its exact location (cf. section 4) within or near a topographic bowl region, it can lead to bowl-trapped AC formation. A rough estimate of the nonlinearity parameter value in the discussed oceanic basins (section 1) based on bathymetry and observed eddy strengths, gives *ϵ* < 1, which in our idealized experiments predicts AC formation and topographic trapping. The vertical structure of the formed AC, also in rough agreement with observations, is a reflection of the domain (regionally) averaged vertical structure. As discussed above, in our model, trapped AC formation is related to prograde slope current formation tendency. The tendency for vorticity segregation by eddy fluxes may in principle be tested within more realistic numerical models of the North Atlantic basins.

## Acknowledgments

This material is based in part upon work supported by the National Science Foundation under Grant OCE-1751386, by the Office of Naval Research (ONR) Grant N00014-18-1-2599, and by the National Aeronautics and Space Administration ROSES Physical Oceanography program under Grant 80NSSC19K1192. This work used the Extreme Science and Engineering Discovery Environment (XSEDE; Towns et al. 2014), which is supported by National Science Foundation Grant ACI-1548562. The SSH altimeter product used is produced by Ssalto/Duacs and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS) (http://marine.copernicus.eu).

## Data availability statement

The AWSIM model code used in this study can be obtained from https://github.com/andystew7583/AWSIM.

## APPENDIX A

### Vortex propagation in homogenized ambient PV over a slope in SWE

**u**

*H*) = 0 in SWE]:

*x*′ ≡

*x*−

*X*, we finally have

**u**= 0). It operates by differential self-advection and accompanying relative vorticity stretching over a bottom slope, rather than by planetary vorticity stretching as in topographic beta drift. The latter does not occur here despite of the bottom slope since mean PV is homogenized.

*x*direction) and if the vortex has an axis of symmetry perpendicular to isobaths, i.e., (

*υ*′)

*q*′ and

*u*′ are (anti)symmetric in

*x*′, then these symmetries vanish a few of the terms which appeared above, resulting in

_{y}

*H*< 0 then ∂

_{t}

*X*< 0, since

*q*′ < 0,

*Q*< 0, and

*x*′

*υ*′ < 0.

## APPENDIX B

### Free linear evolution in a numerical model

To show explicitly that nonlinear effects are critical to the formation of the trapped anticyclone, we conduct a numerical experiment where the nonlinear (advection) terms are not included in the model equations. Since this option was not readily available in the model of Stewart and Dellar (2016), we conducted the linear experiment using the Back of Envelope Ocean Model (BEOM; St-Laurent 2018), a primitive equation layered isopycnal model. First, we ran BEOM in a fully nonlinear configuration identical to experiment B7E5, except that BEOM has a free surface rather than rigid upper lid. The BEOM experiment has very similar results to B7E5, including a trapped bowl AC and a cyclonic slope current. We conducted a second BEOM experiment identical to the previous, but without the advective terms in the momentum equation. Results are shown in Fig. B1. The circulation appears, as expected, composed of topographic Rossby waves. No tendency toward AC formation occurred over 1000 days. Indeed, linear dynamics cannot change the azimuthally averaged azimuthal velocity. That may be seen directly from Eq. (8a), as the right-hand side vanishes under linear dynamics.

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^{1}

We loosely define a bowl as a topographic depression that has a central relatively flat region of similar width or wider than the outer slope region.

^{2}

The North Atlantic is by far the most sampled of the oceans, and hence a statistical bias may be present.

^{3}

The same propagation direction as that of coastal Kelvin waves and topographic Rossby waves.

^{4}

Based on the tendency of enstrophy to dissipate faster than energy in 2D turbulence.

^{5}

Such circulations can also result from tidal rectification (e.g., Beckmann and Haidvogel 1997).

^{6}

In all of the B6E7 repeat experiments PV homogenization was near complete, i.e., the metric PVI defined in the next paragraph was <0.07. Compare with Figs. 5b and 5c.

^{7}

We do not attempt to determine here if this apparent intensification is an artifact of our definition of the core. For example the dying out of external transients may cause a larger area around the AC to be identified as part of the core.

^{8}

In the reference experiment B1E5 for example, the AC shield has an integrated vorticity just 20% the magnitude of the integrated negative vorticity of the AC interior.

^{9}

Here we linearize around the initial state of approximate zero mean flow along isobaths.

^{10}

This scaling is exact in case of a vortex core in solid body rotation, e.g., a Rankine vortex.

^{11}

Time scales to (partial) cross-layer coupling grow with *L*_{i}/*l* magnitude, and with *S*_{0} magnitude, which makes simulation more demanding and complicates analysis of the results.

^{12}

The comparison is only qualitatively informative since the parameters *H*_{0}, *E*_{0}, and *λ*_{0} differ between these experiments.