## 1. Introduction

An emerging body of evidence from realistic modeling studies suggests that topographic interactions are a significant source of vertical vorticity generation in the ocean. Among regions where this is seen are the Gulf Stream (Gula et al. 2015), California Undercurrent (Molemaker et al. 2015), Solomon Sea in the southwestern Pacific (Srinivasan et al. 2017), the Gulf of Oman (Vic et al. 2015), and the Alboran Sea region of the western Mediterranean (Capó et al. 2021). In all these studies, the vorticity generated on the topographic slopes evolves, through current separation and shear, centrifugal, or symmetric instability mechanisms, to form a turbulent wake populated by submesoscale coherent vortices (SCVs). Oceanic observations of SCVs with a putative topographic origin include, the Beaufort Sea anticyclones (D’Asaro 1988), the eddying wake past the northern end of Palau (MacKinnon et al. 2019), and most recently, a deep, intense cyclonic SCV in the Arabian Sea (De Marez et al. 2020). SCVs are dynamically important because they can transport mass and dissolved materials over long distances in the ocean (Armi and Stommel 1983; Armi and Zenk 1984; McWilliams 1985; Riser et al. 1986; McCoy et al. 2020) and enhance rates of diapycnal mixing in the thermocline (Dewar et al. 2015; Zhang et al. 2019).

The phenomenology underlying vertical vorticity generation in flow past topography is still being unraveled. D’Asaro (1988) proposed, on the basis of observations of potential vorticity (PV) anomalies in the Beaufort Sea SCVs, that frictional torques which arise during flow–topography interactions have an important role in the generation process. Molemaker et al. (2015) provide a geometric argument, subsequently verified in Srinivasan et al. (2019), to describe how bottom drag acting on slope currents produces a horizontal shear, i.e., vertical vorticity. Employing the framework of the barotropic vorticity equation, defined as the curl of the vertically integrated horizontal momentum equations, Molemaker et al. (2015) and Gula et al. (2015) further show that barotropic vorticity is generated primarily through the action of the bottom pressure torque (BPT) (see also Hughes and De Cuevas 2001; Jackson et al. 2006), with the bottom stress curl not contributing significantly in an integral sense. The apparently contradictory roles of bottom friction and BPT in generating vertical vorticity over slopes remains to be reconciled and will be examined in this study.

In developing a mechanistic understanding of vorticity generation on topographic slopes, we seek to elucidate and quantify how the bottom stress mediates this process. A putative role for the bottom stress needs to in turn be reconciled with the expected occurrence of Ekman arrest on slopes, following boundary stress collapse (MacCready and Rhines 1991). Pursuing an integrated vorticity balance analysis, we explore the dynamics of vorticity generation in flow past an elongated ridge using solutions from idealized, fully three-dimensional ROMS simulations. ROMS is the Regional Oceanic Modeling System. The model setup is detailed in section 2. Figure 1 provides a glimpse of the essential dynamics. Barotropically forced flow past an elongated ridge leads to vorticity generation along the slopes, culminating in the shedding of vertically coherent vortices into the wake. In section 3, we derive an integral formulation of the vertical vorticity equation that explicitly connects BPT with bottom frictional effects, allowing for a quantification of the quasi-Lagrangian vorticity evolution along barotropic streamlines. We shall demonstrate in section 4 that while the stress does weaken substantially on the slopes as the flow evolves downstream, significant vorticity generation (e.g., Fig. 1) occurs during the early flow encounter with the ridge, as a result of the bottom stress divergence torque (BSDT), a source term in the integrated vorticity equation.

The central role of BSDT raises questions about previous studies that have demonstrated vorticity generation without bottom drag. Among the earliest such studies are the numerical experiments of Smolarkiewicz and Rotunno (1989). In their free-slip simulations of for nonrotating, low-Froude-number flows past topography, a symmetric pair of vertically oriented lee vortices was observed to form in the wake. Using asymptotic arguments, the authors demonstrated that the vertical vorticity was created purely through the tilting of baroclinically generated horizontal vorticity. Since then, lee vortices have been reported in several other studies of nonrotating flows employing zero-stress or free-slip bottom boundary conditions (e.g., Ólafsson and Bougeault 1996; Jagannathan et al. 2019; Puthan et al. 2020). However, to our knowledge there are no studies documenting vorticity generation without bottom drag using ROMS or other realistic ocean models that include the effect of rotation, nor are there any studies making a quantitative comparison between drag and no-drag solutions. These questions will be addressed in section 5, both theoretically and numerically, with a bottom-drag-free ROMS configuration.

## 2. Model setup

The simulations are performed using ROMS (Shchepetkin and McWilliams 2005), a split-explicit, terrain-following (*σ* coordinate) ocean model that solves the primitive, hydrostatic equations of motion, with a nonlinear equation of state for seawater (Jackett and McDougall 1995).

*h*

_{m}and Gaussian half-width

*a*, elongated in the flow direction

*y*. Mathematically the ridge elevation is given by

*σ*

_{y}represents the extent of the initial encounter region over which the ridge height changes rapidly. One of the motivations for considering an elongated ridge is that it allows for longer downstream development of the cross-slope Ekman dynamics and is thus well suited for studying the departure from one-dimensional and doubly periodic models of slope BBLs and Ekman arrest. The Ekman adjustment problem will be separately considered in a forthcoming study.

In all our simulations, we set *h*_{m} = 400 m, the half-width *a* = 3.5 km, length *b* = *y*_{2} − *y*_{1} = 144 km, and the encounter length *σ*_{y} = 12 km. Note that these choices imply *b* ≫ *a* and

*σ*levels. To resolve BBL dynamics, the ROMS grid is stretched at the bottom so that the vertical resolution ranges from 1.1 m over the flat bottom to 0.9 m at the ridge crest. The turbulent bottom drag is parameterized using a quadratic drag law,

*ρ*

_{0}is a constant reference density, and

*C*

_{d}is the drag constant given by

*κ*= 0.4 is the von Karman constant, Δ

*z*

_{b}is the thickness of the bottommost

*σ*layer and

*z*

_{ob}is the roughness length, set to 1 cm. Vertical mixing in the BBL is parameterized using KPP (Large et al. 1994; McWilliams et al. 2009). It is pertinent to note that, in addition to parameterized vertical mixing, both in the BBL and interior, the third-order upwind-biased scheme used for computing the nonlinear advective terms additionally introduces horizontal hyperdiffusive terms (Shchepetkin and McWilliams 2003, 2005).

*V*

_{0}= 0.1 m s

^{−1}, geostrophically balanced by a zonal gradient in the sea surface elevation, along with a linear vertical profile of potential temperature

*θ*. With the nonlinear equation of state, this produces an approximately uniform background stratification

*N*, permitting the definition of a nondimensional ridge height

*f*is the Coriolis frequency. Note that

The Coriolis frequency *f* is fixed at a typical midlatitude value of 7 × 10^{−5} s^{−1} and *N* alone. The flow variables are held constant at the inflow, with open, radiative conditions (Marchesiello et al. 2001) applied at the other boundaries. We consider four values of *N* ranges from 5 × 10^{−4} s^{−1} to 4 × 10^{−3} s^{−1}. In this parameter space *H*, provided it is larger than the ridge height itself (Srinivasan et al. 2019). Here, *H* is set to 1000 m. Two sets of solutions are examined—one with, and the other, without bottom drag. All the simulations are run for four months of physical time. Flow variables and momentum diagnostics are output twice daily and temporal averaging is performed over the last two months of the model output so as to exclude the spinup time.

## 3. Theoretical formulation

### a. An integrated vorticity balance

We develop a vertically integrated vorticity formulation to analyze the vorticity balances in our solutions. The central question is, what causes vorticity generation when a current encounters sloping bathymetry. The hitherto overlooked role of the bottom stress divergence torque (BSDT), which appears as one of the boundary terms in this formulation, will be demonstrated in section 4.

**u**is the horizontal velocity vector, ∇

_{H}is the horizontal gradient operator,

**/∂**

*τ**z*is the vertical stress divergence and other symbols have their usual meaning. Note that ROMS also has horizontal hyperviscosity through the third-order upwind biased scheme, but this is a negligible term in the vorticity balances for our simulations.

*f*is constant in our simulations, we neglect the

*β*effect. To obtain the integrated vorticity equation, vertically integrate Eq. (6) from

*z*= −

*H*+

*h*(

*x*,

*y*) to

*z*=

*η*, where

*H*is the constant water depth away from the topography and

*η*is the sea surface elevation,

*b*denotes “bottom.” Note that the first term on the RHS is the familiar bottom stress curl (BSC). The second term −({∂

**/∂**

*τ**z*}|

_{b}/

*ρ*

_{0}) × ∇

_{H}

*h*appears as a result of interchanging the curl and integral operators. We refer to this as the bottom stress divergence torque (BSDT). It is the twisting force produced due to the vertical divergence of stress in the direction orthogonal to the topographic gradient ∇

_{H}

*h*. Equation (7) now becomes

_{H}

*h*. This yields the relation

Equations (11) and (12) underscore the direct relationship between BPT and BSDT. As we will further see in section 4d, this resolves the apparently contradictory explanations for vorticity generation provided here and in previous studies such as Molemaker et al. (2015) and Gula et al. (2015).

### b. Quasi-Lagrangian analysis

We now develop a quasi-Lagrangian technique for analyzing the integrated vorticity equation, Eq. (9). We call it quasi-Lagrangian as opposed to Lagrangian to emphasize the fact that we will be tracking the evolution of source terms on mean barotropic streamlines and not individual particle trajectories.

*η*. Now, denoting time averages by

**u**

_{s}⋅ ∇

*η*=

*w*

_{s}and

**u**

_{b}⋅ ∇

*h*=

*w*

_{b}, respectively, after cancellation of the boundary terms we have

*D*= (

*H*−

*h*+

*η*) is the local water column depth.

*s*now denotes the distance along the characteristics, i.e., barotropic streamlines of the flow. Equation (17) can then be integrated to determine the evolution of the depth averaged vorticity ⟨

*ζ*⟩ and hence also the vertically integrated vorticity

In section 4b we will use Eq. (18) to identify which terms are responsible for vorticity generation as a current encounters topography and advects along its slopes.

## 4. Frictional vorticity generation

### a. Vertical structure of the solutions

We briefly discuss the vertical flow structure in our solutions before proceeding to examine the balances in the integrated vorticity equation. In the remainder of the paper, we refer to the side where uphill is to the left (right) of the along-slope flow as the cyclonic (anticyclonic) side, consistent with a Northern Hemisphere orientation.

Figure 2 displays vertical sections of the mean flow structure at successive downstream locations starting from the encounter region, for

Vertical sections of normalized, time-mean along-slope velocity *υ*/*V*_{0} overlain by isopycnals at successive downstream locations: (top)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Vertical sections of normalized, time-mean along-slope velocity *υ*/*V*_{0} overlain by isopycnals at successive downstream locations: (top)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Vertical sections of normalized, time-mean along-slope velocity *υ*/*V*_{0} overlain by isopycnals at successive downstream locations: (top)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

At

Vertical sections of normalized, time-mean vertical vorticity *ζ*/*f* overlain by isopycnals at successive downstream locations: (top)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Vertical sections of normalized, time-mean vertical vorticity *ζ*/*f* overlain by isopycnals at successive downstream locations: (top)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Vertical sections of normalized, time-mean vertical vorticity *ζ*/*f* overlain by isopycnals at successive downstream locations: (top)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Figure 3 depicts the along-slope evolution of vorticity. The topographic interaction produces strong vertical vorticity [*y*/*a* = 17, 34), vertical alignment of the vorticity occurs and a distinct columnar structure emerges. This is more pronounced on the cyclonic side and at *y*/*a* = 51), advection of eddy vorticity, encapsulated by the term EA in Eq. (9), causes the time-mean vorticity to decrease.

### b. The role of the BBL in topographic vorticity generation

The advantage of the integrated vorticity formulation in Eq. (9) is that BSC and BSDT expressly illuminate the role of the bottom stress in the vorticity generation process. These terms represent nonconservative, viscous torques. By contrast, BPT, as it appears in the barotropic vorticity equation, can be difficult to interpret in ocean models, which rely on turbulent BBL parameterizations rather than an explicitly enforced no-slip condition.

The one-dimensional theory of boundary currents (MacCready and Rhines 1991) predicts a slow temporal evolution toward bottom stress collapse and hence boundary layer shutdown on slopes. However, on realistic topography, Ekman adjustment is a primarily downstream rather than temporally evolving process. Moreover, flow separation and secondary instabilities will alter the leading-order cross-slope momentum balance and a departure from the steady state one-dimensional prediction is to be expected. Indeed, while the bottom stress (Fig. 4) on the higher reaches of the ridge exhibits substantial weakening downstream with increasing

Normalized, time-averaged boundary stress. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Normalized, time-averaged boundary stress. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Normalized, time-averaged boundary stress. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

For finite *N*, the BBL height *h*_{bbl} on a flat bottom follows the empirical scaling *h*_{bbl} as a function of the cross-slope coordinate *x* at two downstream locations, one in the encounter region and the other roughly halfway along the ridge. The BBL height *h*_{bbl} as defined here is the depth over which active shear-driven entrainment and mixing occur. It is computed in ROMS using KPP, which parameterizes the effects of stratified Ekman layer turbulence. Note that *h*_{bbl} is different from the mixed layer depth which is the quantity of interest in the Ekman adjustment problem. The dimensional *h*_{bbl} have been normalized by *h*_{bbl} in the encounter region (*y*/*a* = 10) is larger on the anticyclonic side—a consequence of the along-slope flow being faster there. Downwelling on the anticyclonic side transports heavier fluid under lighter fluid, making the flow convectively unstable. Parameterized vertical mixing in ROMS then leads to the formation of a bottom mixed layer which continues to deepen moving downstream.

The time-mean BBL height *h*_{bbl} normalized by *y*/*a* = 10) and the other further downstream (*y*/*a* = 34). Values of the parameter *h*_{bbl} are smaller for larger

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

The time-mean BBL height *h*_{bbl} normalized by *y*/*a* = 10) and the other further downstream (*y*/*a* = 34). Values of the parameter *h*_{bbl} are smaller for larger

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

The time-mean BBL height *h*_{bbl} normalized by *y*/*a* = 10) and the other further downstream (*y*/*a* = 34). Values of the parameter *h*_{bbl} are smaller for larger

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

The dominant tendency terms in the vertical vorticity equation, Eq. (6) are displayed in Fig. 6. The stress divergence curl within the BBL initiates vorticity generation during the early encounter, with advective processes being a secondary source in the flow interior. Vortex stretching occurs in response to Ekman upwelling and downwelling in the BBL. Further aloft, the oscillatory structure of VS + VT is due to vertical internal wave modes that are launched when the flow encounters the ridge. Assuming rotational effects are small, the vertical wavelength of these waves is proportional to *V*_{0}/*N* (e.g., Baines 1998). Hence for fixed *V*_{0}, it scales inversely with

Time-averaged tendency terms in the 3D vertical vorticity equation, Eq. (6), normalized by *fV*_{0}*a*^{−1} and overlain by flow isopycnals for (top) *y*/*a* = 10 and (right) averaged across the straight section of the ridge.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Time-averaged tendency terms in the 3D vertical vorticity equation, Eq. (6), normalized by *fV*_{0}*a*^{−1} and overlain by flow isopycnals for (top) *y*/*a* = 10 and (right) averaged across the straight section of the ridge.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Time-averaged tendency terms in the 3D vertical vorticity equation, Eq. (6), normalized by *fV*_{0}*a*^{−1} and overlain by flow isopycnals for (top) *y*/*a* = 10 and (right) averaged across the straight section of the ridge.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

With a view to quantifying precisely how drag against the ocean bottom injects vertical vorticity into the flow, we now examine the integrated vorticity balances in our solutions. Using the momentum diagnostics directly from ROMS, the various source terms in the integrated vorticity equation, Eq. (9) are computed to the level of ROMS accuracy. Snapshots and time averages of the vertically integrated vorticity are displayed in Figs. 1 and 7 respectively. Also displayed in Fig. 7 are the streamlines of the mean barotropic transport. The rotation-induced asymmetry is clearly visible in the streamline patterns.

Time averages of vertically integrated vorticity, normalized by *fh*_{m} for the ridge solutions. The black and green contour lines are, respectively, streamlines of the barotropic transport **U** and bathymetry contours at *z* = 0.14*h*_{m}, *z* = 0.37*h*_{m}, and *z* = 0.9*h*_{m}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Time averages of vertically integrated vorticity, normalized by *fh*_{m} for the ridge solutions. The black and green contour lines are, respectively, streamlines of the barotropic transport **U** and bathymetry contours at *z* = 0.14*h*_{m}, *z* = 0.37*h*_{m}, and *z* = 0.9*h*_{m}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Time averages of vertically integrated vorticity, normalized by *fh*_{m} for the ridge solutions. The black and green contour lines are, respectively, streamlines of the barotropic transport **U** and bathymetry contours at *z* = 0.14*h*_{m}, *z* = 0.37*h*_{m}, and *z* = 0.9*h*_{m}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

In a Lagrangian reference frame, water columns on the cyclonic (anticyclonic) side acquire positive (negative) vorticity as they advect downstream along mean transport streamlines. As the flow separates from the slopes, vorticity generation is followed by rapid merger events where smaller like-signed vortices roll up to form larger ones (Srinivasan et al. (2019)) that eventually separate further downstream as submesoscale coherent vortices. The prominent small-scale structures seen on the anticyclonic side are manifestations of hybrid centrifugal/symmetric instability of the flow (e.g., Wenegrat and Thomas 2020). This aspect of the solutions will be further explored in a follow up study.

In Fig. 8 we plot each of the tendency terms of the integrated vorticity equation as they appear on the RHS of Eq. (9). Interestingly, the BSC is of minor importance, and further, is a sink rather than a source of vorticity on both sides of the ridge, regardless of the value of

Time-averaged generation terms on the RHS of the integrated vorticity equation, Eq. (9). VSVT is the sum of the vortex stretching (VS) and tilting (VT) terms. Each term has been normalized by *fh*_{m}*V*_{0}*a*^{−1}: (a)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Time-averaged generation terms on the RHS of the integrated vorticity equation, Eq. (9). VSVT is the sum of the vortex stretching (VS) and tilting (VT) terms. Each term has been normalized by *fh*_{m}*V*_{0}*a*^{−1}: (a)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Time-averaged generation terms on the RHS of the integrated vorticity equation, Eq. (9). VSVT is the sum of the vortex stretching (VS) and tilting (VT) terms. Each term has been normalized by *fh*_{m}*V*_{0}*a*^{−1}: (a)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

To gain further insight into the interplay of BSDT and VSVT as a water column advects along a topographic slope, we take recourse to the quasi-Lagrangian technique described in section 3b. Partial cumulative integrals of the source terms of

Partial cumulative integrations of source terms on the RHS of Eq. (18) to *a* is the ridge half-width and *s*/*a* is the normalized distance along the streamline. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Partial cumulative integrations of source terms on the RHS of Eq. (18) to *a* is the ridge half-width and *s*/*a* is the normalized distance along the streamline. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Partial cumulative integrations of source terms on the RHS of Eq. (18) to *a* is the ridge half-width and *s*/*a* is the normalized distance along the streamline. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

On the anticyclonic side, BSDT is again the dominant generation term. A notable observation is that, for the *y*/*a* ≈ 16) where VSVT and BSDT switch signs. A similar reversal is seen for

Along the straight section of the ridge and prior to flow separation, the mean value of the integrated vorticity remains nearly constant. This might be expected, for example, from the geometric argument of Molemaker et al. (2015) according to which the vertical vorticity in the BBL is given by *θ*| ≪ 1 is the slope, so that integrating over the BBL yields

### c. A heuristic explanation for BSC and BSDT patterns

Consider our geometry with a ridge of height *h*(*x*, *y*) and an inflow *V*_{0} directed northward. Assume, heuristically, that the horizontal circulation around the ridge is weak (i.e., *υ*(*x*, *y*) is accelerated on the flanks of the ridge and decelerated over the ridge top, with some broadscale return to the inflow *V*_{0} in the *x* far field.

For BSC, the left side of the ridge is positive and the right side is negative because of the sign of ∂*υ*_{b}/∂*x*; thus, it is opposite to the sense of the vorticity generation. For BSDT, the signs are the opposite due to the opposite sign of ∂*h*/∂*x* on the two sides; thus, this is a generation term. These heuristic predictions are broadly consistent with what we see in our solutions (Fig. 8).

*δ*indicates the size of the changes over the ridge. Further assuming that

*δυ*

_{b}≈

*υ*

_{b}, Eq. (21) then implies that BSC is small relative to BSDT in our solutions simply because

*h*

_{bbl}is smaller than

*h*.

### d. The connection between BSDT and BPT

*A*

_{Σ}encapsulates all the nonlinear terms and

*ω*through the bottom horizontal velocity as

*almost*identical, which is what we find in our simulations.

Recall from Eq. (11) that BPT can be written as the sum of BSDT and nonlinear bottom stretching, tilting and advective contributions. Further, the term *A*_{Σ} in Eq. (22) has embedded within it the cumulative effects of nonlinear vortex stretching and tilting in the interior. This implies that, in general, BPT and *A*_{Σ} are not necessarily mutually independent with respect to the processes they represent. A comparison of Figs. 10b and 10c, which depicts the time-mean BPT distribution over the ridge, with Fig. 8 reveals the similarity in the patterns of BPT and BSDT. However, the difference of BPT and BSDT, plotted in Figs. 10d–f shows that BPT additionally has a smaller inviscid part to it. The implication is that, when the turbulent BBL is well resolved, the dominant dynamical role of BPT is as a frictional torque, with a smaller “flow turning” component that steers the current around the topography. We shall see in section 5 that the interpretation of BPT changes completely when bottom drag is “turned off” or as may be the case, the BBL resolution is inadequate.

(top) Bottom pressure torque (BPT), defined in Eq. (10) and (bottom) difference of BPT and BSDT [defined in Eq. (9)], both time-averaged and normalized by *fh*_{m}*V*_{0}*a*^{−1}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(top) Bottom pressure torque (BPT), defined in Eq. (10) and (bottom) difference of BPT and BSDT [defined in Eq. (9)], both time-averaged and normalized by *fh*_{m}*V*_{0}*a*^{−1}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(top) Bottom pressure torque (BPT), defined in Eq. (10) and (bottom) difference of BPT and BSDT [defined in Eq. (9)], both time-averaged and normalized by *fh*_{m}*V*_{0}*a*^{−1}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

## 5. Vorticity generation without bottom drag

### a. The role of vortex stretching and tilting

We saw in section 4b that, for large

Snapshots of the vertically integrated vorticity for the no-drag solutions (Figs. 11a–c) show that, after separation, the wake vortices have a smaller horizontal scale compared to the cases with drag at the same

Snapshots of vertically integrated vorticity, normalized by *fh*_{m} for solutions with no bottom drag: (a)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Snapshots of vertically integrated vorticity, normalized by *fh*_{m} for solutions with no bottom drag: (a)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Snapshots of vertically integrated vorticity, normalized by *fh*_{m} for solutions with no bottom drag: (a)

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(top) Time-averaged source term VSVT (sum of VS and VT) in Eq. (9) normalized by *fh*_{m}*V*_{0}*a*^{−1} for the no-drag solutions. (bottom) Time-averaged bottom pressure torque (BPT), normalized by *fh*_{m}*V*_{0}*a*^{−1}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(top) Time-averaged source term VSVT (sum of VS and VT) in Eq. (9) normalized by *fh*_{m}*V*_{0}*a*^{−1} for the no-drag solutions. (bottom) Time-averaged bottom pressure torque (BPT), normalized by *fh*_{m}*V*_{0}*a*^{−1}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(top) Time-averaged source term VSVT (sum of VS and VT) in Eq. (9) normalized by *fh*_{m}*V*_{0}*a*^{−1} for the no-drag solutions. (bottom) Time-averaged bottom pressure torque (BPT), normalized by *fh*_{m}*V*_{0}*a*^{−1}. Values of the parameter

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

As in the drag solutions, we perform a quasi-Lagrangian integration of Eq. (18) along barotropic streamlines and average across many streamlines on either side of the ridge (Fig. 13a). Here we show the vortex stretching (VS) and tilting (VT) contributions separately rather than as a sum (VSVT). Note that while BSC is identically zero, BSDT is also practically negligible in these no-drag solutions (it is not identically zero because of the small background viscosity in ROMS). Figures 13b and 13c reveal that vorticity generation on both sides is attributable primarily to VT during the early flow encounter with the ridge. This is to be contrasted with the drag cases (Fig. 9) where vorticity is primarily generated by BSDT during the early encounter. An asymptotic analysis of the no-drag problem along the lines of Smolarkiewicz and Rotunno (1989) (see appendix) illustrates how a rotating, stratified flow encountering bottom topography causes tilting of horizontally oriented vortex tubes, generating vertical vorticity in the process.

(a) Time average of vertically integrated vorticity, normalized by *fh*_{m} for the

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(a) Time average of vertically integrated vorticity, normalized by *fh*_{m} for the

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

(a) Time average of vertically integrated vorticity, normalized by *fh*_{m} for the

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

On the anticyclonic side, we note an abrupt reversal in the tendencies of VS and VT just ahead of the straight ridge section. However, this does not produce any discernible change in the net vorticity, suggesting it represents merely a reversible, advective flow adjustment on the slopes rather than irreversible vorticity generation. Finally, here again, as in the drag solutions, there is negligible net generation of vorticity along the straight ridge section, where VS, VT, and eddy advection are approximately in balance.

That eddying solutions (Fig. 11) are obtained without bottom drag and BBL separation may seem surprising on the face of it. However, recall that although bottom drag is set to zero, these solutions are not truly inviscid. This is because of the biharmonic horizontal dissipation and mixing (Lemarié et al. 2012) implicit in the third-order upwind-biased scheme. As we shall see below, the eddies in Fig. 11 are in fact associated with potential vorticity (PV) anomalies.

*b*= −

*gρ*/

*ρ*

_{0}is the buoyancy and

**Ω**

_{a}is the three-dimensional absolute vorticity. The PV balance equation may be written in flux-divergence form as follows (e.g., Thomas 2005; McWilliams 2016),

**J**

_{V}and

**J**

_{B}denote viscous and diabatic fluxes of PV, respectively. The nonconservative terms are expressed concisely as

*D*

_{υ}(

*b*) is vertical divergence of the turbulent buoyancy flux and

*D*

_{h}(

**u**) and

*D*

_{h}(

*b*) represent horizontal momentum and buoyancy mixing, respectively.

In Fig. 14a, we display the PV, normalized by the background value *fN*^{2} on the horizontal plane *z* = −*H* + (*h*_{m}/2) for the case *D*_{h}(*b*) of the diabatic flux **J**_{B} (Fig. 14b), with both **J**_{V} and *D*_{υ}(*b*) being negligible in comparison. A possible interpretation is that horizontal buoyancy mixing leads to vertical shear (horizontal vorticity) generation through the baroclinic torque, which is then tilted into the vertical during the topographic encounter. Hence, although frictional torques do not contribute directly to vertical vorticity generation in these no-drag solutions, the source of vorticity is ultimately nonconservative.

PV and PV flux for the *fN*^{2} on the horizontal plane *z* = −*H* + (*h*_{m}/2) and (b) normalized, time-mean diabatic bottom flux of PV *D*_{h}(*b*)**Ω**_{a}.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

PV and PV flux for the *fN*^{2} on the horizontal plane *z* = −*H* + (*h*_{m}/2) and (b) normalized, time-mean diabatic bottom flux of PV *D*_{h}(*b*)**Ω**_{a}.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

PV and PV flux for the *fN*^{2} on the horizontal plane *z* = −*H* + (*h*_{m}/2) and (b) normalized, time-mean diabatic bottom flux of PV *D*_{h}(*b*)**Ω**_{a}.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

### b. Comparison with the drag solutions

Figure 15 shows that the eddy integral scales in the no-drag cases do not depend sensitively on *L*_{I} are almost 60% smaller compared to the drag cases. Likewise, for

Downstream evolution of eddy integral length scales [defined in Eq. (28)] on the cyclonic and anticyclonic sides for the ridge solutions (a) horizontal scale *L*_{I}/*a* and (b) vertical scale *H*_{I}/*h*_{m}. The dashed vertical lines mark the approximate locations of the ridge centerline and northern edge of the straight section.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Downstream evolution of eddy integral length scales [defined in Eq. (28)] on the cyclonic and anticyclonic sides for the ridge solutions (a) horizontal scale *L*_{I}/*a* and (b) vertical scale *H*_{I}/*h*_{m}. The dashed vertical lines mark the approximate locations of the ridge centerline and northern edge of the straight section.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Downstream evolution of eddy integral length scales [defined in Eq. (28)] on the cyclonic and anticyclonic sides for the ridge solutions (a) horizontal scale *L*_{I}/*a* and (b) vertical scale *H*_{I}/*h*_{m}. The dashed vertical lines mark the approximate locations of the ridge centerline and northern edge of the straight section.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

_{x}[min

_{y}(

*ω*′)] and maxima max

_{x}[max

_{y}(

*ω*′)] of

*ω*′ in the wake region.

Figures 17a and 17b display time averages over 50 inertial periods, *fh*_{m}*πa*^{2}, the strength of an axisymmetric columnar vortex of height *h*_{m} and radius *a*, with vorticity *f* at the center and a Gaussian radial distribution. There is only a weak

To summarize, compared to the no-drag cases, bottom-drag-mediated vorticity generation spawns SCVs that are stronger and more energetic, and larger in scale, both horizontal and vertical.

## 6. Discussion and summary

Using idealized ROMS solutions and an integrated vorticity balance analysis, we have demonstrated the role of BBLs in mediating vorticity generation on ridge slopes when the nondimensional ridge height

For all values of

When the barotropic vorticity equation Eq. (22) is employed to analyze the vorticity balances, BPT is often interpreted as the inviscid twisting force responsible for steering flow around bottom topography (e.g., Jackson et al. 2006; Molemaker et al. 2015). However, because pressure is only a Lagrange multiplier when the incompressibility constraint is enforced, there is necessarily some ambiguity in its interpretation, particularly when viscous processes are involved. In the inviscid quasigeostrophic limit, it is easily shown that BPT is exactly equal to bottom vortex stretching −*fw*_{b}. More generally, when expressed as a bottom momentum balance [Eq. (11)], BPT is seen to be directly related to both frictional (BSDT) and advective terms that account for the effects of bottom vortex stretching, tilting, and flow inertia. Indeed, as Fig. 10 demonstrates, in our solutions with a well-resolved BBL, the viscous torque BSDT is in fact the dominant component of BPT. These findings show that when BBLs are present, the apparently contradictory roles of BPT and BSDT in vorticity generation are only illusory. The advantage of the integrated vorticity formulation used here is that it explicitly eliminates the ambiguous pressure gradient term and partitions the generation into inviscid vortex stretching and tilting contributions and nonconservative boundary injection terms associated with the bottom drag.

Visually (e.g., Fig. 1), cyclones are at least as prevalent as anticyclones in our solutions, if not more so. Moreover, Fig. 17 shows that, by an average integral measure of circulation, cyclonic SCVs are in fact stronger than their anticyclonic counterparts. These results appear to contradict the fact that most observed SCVs in the ocean are anticyclonic—a theoretical puzzle that remains unresolved (McWilliams 1985, 2016). Recently, an intense cyclonic SCV has been documented in the Arabian Sea (De Marez et al. 2020), which the authors hypothesize has its origin at the mouth of the Gulf of Aden, a site of steep topography. More studies are needed to bridge the apparent gap between observations and simulations.

The alternating positive and negative patterns along the cyclonic slope in Figs. 12d and 12e for the no-drag case are reminiscent of the BPT signals around the Charleston Bump in the Gulf Stream simulation of Gula et al. (2015, their Fig. 13). This is consistent with their observation that bottom vortex stretching is locally the leading-order term in BPT around the Bump, implying a largely inviscid balance against the seaboard. In light of our results, it would appear that realistic simulations with higher BBL resolution are needed to ascertain if the western boundary current truly represents an inviscid balance.

The importance of bottom drag in vorticity generation has been recognized previously, for example by Signell and Geyer (1991). Using a simple analytical model of flow separation and 2D simulations of the linearized, depth-averaged shallow water equations, they found that the choice of the drag coefficient strongly influenced eddy formation in tidally forced flows around headlands. In their formulation, the depth-averaged drag manifests through the so-called “speed torque” and “slope torque” terms. These may be considered roughly analogous to BSC and BSDT, respectively. A key difference is that while the Signell and Geyer (1991) model is 2D and moreover, relies on empirical choices for the depth averaged drag coefficient, here we directly demonstrate the role of BSDT in vorticity generation using three-dimensional ROMS simulations that resolve the BBL using the KPP parameterization.

Vorticity generation can happen even without bottom drag. The nonrotating, free-slip solutions of Smolarkiewicz and Rotunno (1989) are the earliest modeling evidence for this phenomenon. Recent work by Jagannathan et al. (2019) and Puthan et al. (2020), again for nonrotating flows, also show lee vortex formation with a free-slip bottom boundary. The present study demonstrates that vorticity generation without drag is possible in rotating systems as well, through vortex stretching and tilting mechanisms. However, as seen in Figs. 15–17, the wake eddies tend to be substantially less robust compared to the cases with bottom drag. Hence, model simulations that lack a bottom drag parameterization and/or insufficiently resolve the BBL will often tend to underestimate the spatial scales and strength of the SCVs, and care is needed in interpreting such solutions.

Downstream evolution of the normalized, vertically integrated eddy kinetic energy on the anticyclonic and cyclonic sides for the ridge solutions. The dashed vertical lines mark the approximate locations of the ridge centerline and northern edge of the straight section.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Downstream evolution of the normalized, vertically integrated eddy kinetic energy on the anticyclonic and cyclonic sides for the ridge solutions. The dashed vertical lines mark the approximate locations of the ridge centerline and northern edge of the straight section.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

Downstream evolution of the normalized, vertically integrated eddy kinetic energy on the anticyclonic and cyclonic sides for the ridge solutions. The dashed vertical lines mark the approximate locations of the ridge centerline and northern edge of the straight section.

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

The normalized, time-averaged circulation, defined in Eq. (29), of the wake SCVs for each of the drag and no-drag solutions: (a) anticyclonic circulation

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

The normalized, time-averaged circulation, defined in Eq. (29), of the wake SCVs for each of the drag and no-drag solutions: (a) anticyclonic circulation

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

The normalized, time-averaged circulation, defined in Eq. (29), of the wake SCVs for each of the drag and no-drag solutions: (a) anticyclonic circulation

Citation: Journal of Physical Oceanography 51, 6; 10.1175/JPO-D-20-0253.1

There are several outstanding issues. One question is, how do the dynamics differ for one-sided slopes vis-à-vis isolated topography, such as considered here? On isolated topography, it is conceivable that adverse pressure gradients resulting from the convex topographic curvature and horizontal around-ridge circulations influence boundary layer separation. This is certainly suggested by the analytical and two-dimensional model solutions of Signell and Geyer (1991) for flow around a headland, where the onset of flow separation is found to be controlled by a three-way balance between adverse pressure gradient, curvature, and drag effects. One-sided slopes are more directly relevant to boundary currents, and further understanding is needed there. Another pertinent question is, to what extent is Ekman arrest sensitive to ridge curvature and aspect ratio? Preliminary simulations also indicate that there is a transition from centrifugal to more strongly dissipative, hybrid centrifugal/symmetric instability as the ridge aspect ratio increases, i.e., it becomes more elongated. We will further explore some of these issues in a forthcoming paper.

## Acknowledgments

This work was made possible by the Office of Naval Research Grant N00014-18-1-2599. Computing support was provided by the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation Grant ACI-1548562. ALS was supported by the National Science Foundation under Grant OCE-1751386. We wish to thank Jacob Wenegrat for his insightful feedback on an early version of the draft.

## APPENDIX

### Asymptotic Analysis of the No-Drag Problem

*w*=

*ρ*= 0 at the upper surface. Finally, in the absence of diapycnal fluxes, we can express the vertical velocity

*w*in terms of the instantaneous isopycnal displacement field

*δ*as

*z*

_{b}= −

*H*+

*h*(

*x*,

*y*) is that of no flow into the topography

*p*is chosen so that the leading-order balance is geostrophic. The appropriate choice of scale for the vertical velocity is constrained through the continuity equation as

*w*

_{0}= 0 on the upper surface then leads to

Therefore at leading order, the streamlines and hence isopycnals lie on horizontal planes. Geostrophic balance and the hydrostatic approximation Eq. (A13d) then imply that the vertical gradients of *u*, *υ*, and *ρ* are all zero and the flow is essentially barotropic.

*x*and

*y*and subtracting Eq. (A18a) from (A18b),

Equation (A19) tells us that the vertical vorticity at

### The $O\left({\mathit{\u03f5}}^{2}\right)$ problem

*ξ*

_{1}= ∂

*υ*

_{1}/∂

*z*and

*η*

_{1}= ∂

*u*

_{1}/∂

*z*are the two components of

*z*of Eqs. (A18a) and (A18b). Recalling that (∂

*u*

_{0}/∂

*z*) = (∂

*υ*

_{0}/∂

*z*) = 0 and using the hydrostatic pressure equation Eq. (A13d) to eliminate

*p*

_{1}, we have,

*ξ*

_{1}and

*η*

_{1}can be inferred from consideration of the

*w*

_{1}can be expressed in terms of the isopycnal displacement field

*δ*

_{1}as

*w*

_{1}as

*w*

_{1}≈

**u**

_{0}⋅ ∇

_{H}

*h*. Then, Eq. (A25) can be written as

_{H}

*ρ*

_{1}near the bottom have the same parity, respectively, as those of ∇

_{H}

*h*. In the context of our ridge solutions (e.g., Fig. 11), Eq. (A24) shows that as the flow encounters the ridge,

*ξ*

_{1}< 0 (>0) on the cyclonic (anticyclonic) side, while

*η*

_{1}> 0 on both sides.

Therefore, when *ϵ* = *Nh*_{m}/(*fa*) is not asymptotically small, second-order nonlinear effects are important from the perspective of vorticity production. While the perturbation analysis above does not automatically carry over to the cases ^{−1} = *Nh*_{m}/*V*_{0}. In that case, the

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