## 1. Introduction

Rough bottom topography in the abyssal ocean contributes significantly to enhancement of drag and turbulent dissipation. When abyssal flow encounters rough topography, energy is lost in two ways: 1) skin friction resulting from tangential stress at the boundary and 2) via pressure/form drag resulting from normal stress. Recent studies of Zhang and Nikurashin (2020) and Klymak (2018) highlight the crucial role played by multiscale topography in extracting momentum (through topographic form stress) from the background flow and maintaining a dynamic balance in the abyssal ocean. Form drag is often the primary mechanism of energy extraction from the barotropic tide, especially at steeper topographies (McCabe et al. 2006; Horwitz et al. 2021).

In situ measurements show that the loss of momentum associated with form drag is enhanced by obstacles including coastal headlands (Edwards et al. 2004; Magaldi et al. 2008; Warner et al. 2012; Warner and MacCready 2014) and continental shelf banks (Nash and Moum 2001; Wijesekera et al. 2014). However, observational studies of form drag estimates from the abyssal ocean are limited. Lack of information on the magnitude and spatial distribution of form drag presents a challenge for form drag parameterizations in global climate models (GCMs). Numerical studies can play an important role in bridging this gap. Warner and MacCready (2009) performed numerical simulations with the hydrostatic ROMS model of a nonrotating tidal flow past a Gaussian headland to examine different components of form drag. They showed that while the normalized separation drag (the average drag coefficient) increased with an increase in the aspect ratio of the headland, it does not depend on the tidal excursion or the headland size. While these results may explain the momentum loss at regions where the tidal constituents dominate (e.g., Clément et al. 2017), the behavior in regions where a strong mean flow occurs in conjunction with tidal oscillations remains to be explored. In the present work, we examine form drag in tidally modulated flow past an underwater obstacle. Turbulence resolving simulations enable us to study the time varying flow past obstacles without compromising on the accuracy of their representation. The characterization of flow separation and pressure distribution on the obstacle allows us to link underlying physical mechanisms to changes in the observed form drag.

Form drag on an obstacle is dependent on the ambient stratification. When a steady current encounters a 3D ridge, the flow transitions to a state of high drag when the Froude number reduces below 1 (e.g., Epifanio and Durran 2001; Vosper et al. 1999). In situ tidal measurements of form drag are challenging and there are few such observations, e.g., Voet et al. (2020), who infer pressure from density measurements using the hydrostatic approximation. Additionally, tide induced unsteadiness may lead to changes in flow separation and distribution of lee vorticity. For example, tidal currents create transient lee eddies (or lee vortices) in wakes behind headlands (Pawlak et al. 2003; Callendar et al. 2011; MacKinnon et al. 2019) and submerged topography (Girton et al. 2019). Complexity in the impinging flow, variable stratification and irregular bathymetry at these sites present a challenge in elucidating the role of lee eddies. To examine the role of tides in flow separation and form drag, we perform turbulence-resolving simulations of an oceanic wake past a conical hill generated by a tidally modulated flow. The background flow may be expressed as *U _{b}* =

*U*+

_{c}*U*sin(Ω

_{t}*), where*

_{t}t*U*and

_{c}*U*are the mean and tidal components and Ω

_{t}*= 2*

_{t}*πf*is the tidal frequency (in rad s

_{t}^{−1}).

*), the tidal excursion number (Ex*

_{c}*), the Rossby number (Ro*

_{t}*), and velocity ratio (*

_{c}*R*):

*N*is the background buoyancy frequency,

*f*is the inertial frequency,

*h*is the height of the obstacle, and

*D*is the obstacle base diameter. In unstratified environments, a hill with a 3D geometry does not shed vortices similar to the vertically coherent lee eddies which are observed in the ocean. Instead a standing horseshoe vortex and periodic hairpin vortices are observed downstream at low Reynolds number (Re

*) (see Acarlar and Smith 1987), which become indistinct at higher Reynolds number (Garcia-Villalba et al. 2009). A low Fr*

_{D}*(Fr*

_{c}*≪ 1) flow is constrained to move around rather than over the obstacle, owing to the large potential energy barrier. This leads to roll-up of the lateral shear layer into lee vortices (Hunt and Snyder 1980).*

_{c}Topographic wakes are also affected by planetary rotation. A relatively large planetary rotation rate (small Ro* _{c}*) induces asymmetry in the strength of cyclonic and anticyclonic eddies shed from the topography (Dietrich et al. 1996). Dong et al. (2006) attributed the loss in symmetry to centrifugal instabilities in the wake. Perfect et al. (2018) and Srinivasan et al. (2018) showed that the change in vertical structure of wake vortices is governed by the Burger number Bu, defined as (Ro

*/Fr*

_{c}*)*

_{c}^{2}. Their idealized simulations show decoupling of vortices along their vertical extent owing to loss of geostrophic balance, when Bu > 12. Recently, Jagannathan et al. (2021) and Srinivasan et al. (2021) investigated the generation mechanisms and spatial organization of lee vorticity in topographic wakes. Other outcomes of flow–topography interactions such as generation of lee waves, upstream blocking, and hydraulic jumps have also received attention in literature (e.g., Epifanio and Rotunno 2005; Wright et al. 2014; Jagannathan et al. 2019; Perfect et al. 2020).

The regime of weak rotation and strong stratification (or equivalently, large Bu) applies to wakes behind abyssal hills. For example, consider the abyssal hills in the Brazil Basin (Ledwell et al. 2000; Nikurashin and Legg 2011). The bottom topographic roughness is *D* = 1.5 km, buoyancy period of 1 h and *U _{c}* =

*U*= 10 cm s

_{t}^{−1}, the tidal excursion number is Ex

*≈ 0.5 for the M*

_{t}_{2}tide and the average value of Fr

*lies close to 0.2. The value of Rossby number is Ro*

_{c}*≈ 3, at 15°S latitude. The resolution of GCMs is insufficient to resolve these hills. Thus, parameterization of the wake dynamics at these length scales is critical.*

_{c}Owing to numerical constraints, idealized simulations often ignore tidal forcing. Yet, in situ observations affirm that tides can significantly influence flow separation at islands, continental slopes, and submerged topography. Observations by Black and Gay (1987) showed the formation of “phase” eddies in the continental shelf of Great Barrier Reef. Denniss et al. (1995) and Chang et al. (2019) reported lee eddies shed past islands at the dominant tidal frequency. This phase-locking phenomenon is observed even when the tidal velocity amplitude is not large relative to the mean flow.

Recently Puthan et al. (2021) found tidal synchronization in a study of flow past a conical hill where the frequency of the far-wake lee vortices locked to a subharmonic (depending on the value of *f ^{*}*) is linearly related to Ex

*as*

_{t}*is the vortex shedding Strouhal number in a steady background flow. This relation simplifies to*

_{c}*= 0.265) when*

_{c}*R*= 1. However, the near wake characteristics such as flow separation at the hill and the attendant form stress were not studied by Puthan et al. (2021) and have not received adequate attention in other previous studies. Moreover, the effect of tidal modulations on flow separation is often ignored in literature (e.g., Puthan et al. 2020).

In this work, we address the questions pertaining to momentum loss of tidally modulated abyssal currents during flow–topography interactions at the obstacle and the associated wake-vorticity distribution. The motivation for this work is twofold. We explore possible states of large form drag owing to changes in pressure distribution in the lee and determine the qualitative changes to the vorticity distribution in the near wake in each state. The numerical formulation is detailed in section 2. Section 3 introduces the parameter space and lists the cases performed in the study. A brief introduction to form stress and an overview of previous literature related to form drag is provided in section 4. Section 5 elucidates the changes in form drag on varying

## 2. Computational model

The computational domain is 9.5 km in the streamwise (*x*) direction, 3.8 km in the spanwise (*y*), and 2 km in the vertical (*z*) direction. The conical obstacle of height *h* and base diameter *D* is placed at the origin. For convenience, the horizontal and vertical distances are normalized by *D* and *h* such that *U _{b}* =

*U*+

_{c}*U*sin(2

_{t}*πf*) encounters a conical obstacle in a uniformly stratified environment.

_{t}t*f*plane are given below in tensor notation:

*u*= (

_{m}*u*

_{1},

*u*

_{2},

*u*

_{3}) = (

*u*,

*υ*,

*w*) denotes the velocity components and

*ρ*is the density field. Here,

*ρ*′ represents deviation of density from its background value and

*p*is the deviation from the mean pressure imposed by geostrophic and hydrostatic balance. The pressure deviation

*p*may be represented as

*p*

_{∞}/∂

*x*= −

*ρ*

_{0}

*dU*/

_{b}*dt*= −

*ρ*

_{0}

*U*Ω

_{t}*cos(Ω*

_{t}*) is the pressure gradient driving the barotropic current*

_{t}t*U*and

_{b}*p*is the dynamic pressure. The stress tensor

_{d}*τ*and density flux vector Λ

_{mn}*are computed using the approach of Puthan et al. (2020). The grid parameters, time-advancement scheme and boundary conditions are adopted from Puthan et al. (2021).*

_{n}## 3. Simulation parameters

A regime of weak rotation and strong stratification is considered in the study. Inertial effects on the wake are weak in the lee of abyssal hills and headlands at length scales of * _{c}* > 1). The inertial frequency is set to its value at 15°N such that Ro

*= 5.5. Much of the topography in the abyssal ocean is subject to flow conditions with Fr*

_{c}*≪ 1, e.g., Nikurashin and Ferrari (2010). To this end, we consider a topographic Froude number Fr*

_{c}*of 0.15 where the flow is predominantly around the obstacle, creating coherent vortices as observed in geophysical wakes (Perfect et al. 2018).*

_{c}Tidal modulations are added to the mean flow. We consider tidal velocities of amplitude equal to the mean current, so that *R* = *U _{t}*/

*U*= 1. The relative frequency

_{c}*varying from 0.06 to 0.6 (assuming St*

_{t}*= 0.265), values of relevance in the ocean (Signell and Geyer 1991; Edwards et al. 2004; Musgrave et al. 2016). Therefore,*

_{c}*U*/

_{t}*f*to the obstacle diameter

_{t}*D*. At a constant tidal frequency

*f*, e.g., the M

_{t}_{2}tide, larger values of the nondimensional parameter

*f*in the numerical setup. Therefore, in this parametric study, we modify

_{t}*f*to explore the influence of varying

_{t}Different regimes of tidal synchronization were observed by Puthan et al. (2021), wherein the lee vortices in the far wake were found at frequencies *f _{s}*

_{,}

*,*

_{c}*f*/4, or

_{t}*f*/2. These regimes are listed in Table 1. Regime 1 consists of a single case (

_{t}*R*= 0). In regimes 2–4,

*R*equals 1 and multiple cases are explored within each regime at discrete

Different cases in this study: the relative frequency * _{D}*, Fr

*, and Ro*

_{c}*are fixed at 20 000, 0.15, and 5.5, respectively.*

_{c}## 4. Form drag in oscillatory flows

*C*is the added mass coefficient. The second term on the right-hand side of Eq. (6) represents the Froude–Krylov force (

_{a}*p*

_{∞}(Yu et al. 2018). The sum of the added mass force (

Warner and MacCready (2009) concluded that the net contribution of inertial drag vanishes when averaged over a tidal cycle. Therefore, its contribution is estimated using Eq. (8), and removed from *F _{D}* [computed using Eq. (6)] to compute the separation drag. The separation drag is assumed to be in phase with

*U*while the inertial drag variability has a phase difference of

_{b}*π*/2 with respect to

*U*(Morison et al. 1950).

_{b}A similar drag force decomposition was proposed by Lighthill (1986), wherein the inviscid inertial drag is estimated from potential flow theory. However, the force decomposition of Lighthill (1986) has limitations. Sarpkaya (2001) argued that the force decomposition of Lighthill (1986) excludes the effects of viscosity on the added mass. His results also demonstrate that the added mass coefficient *C _{a}* is time dependent. He asserted that the subtraction of the ideal inertial force (calculated from potential flow approximation) from the total form drag, leaves behind a force that consists of both the separation drag

*and*an “acceleration-dependent” inertial force. However, the contribution from inertial drag is negligible in the time-averaged value of

*F*, when the average is computed over a large time duration (spanning multiple tidal cycles) in a statistically stationary flow (Sarpkaya 2001). Thus the approximations of Warner and MacCready (2009) can serve as a theoretical basis to aid the interpretation of the present results by providing an adequate estimate of the separation drag. In other words, the time-mean of the form drag is representative of the separation drag

_{D}*F*relative to the tidal phase. These diagnostic tools, which will be utilized in section 5, are also useful for comparing the energy loss associated with form drag across the regimes.

_{D}Observations of Edwards et al. (2004), Warner et al. (2012), and Voet et al. (2020) show that obstacles in the ocean extract energy from the mean and tidal components of the background flow. The rate of work done by form drag (*W _{D}*) is estimated as

*F*. Its time average, 〈

_{D}U_{b}*F*〉, is representative of the average kinetic energy of the flow lost to form drag over a tidal cycle including local dissipation and internal wave energy flux (Egbert and Ray 2000). The rate of energy loss associated with the

_{D}U_{b}*mean*form drag is computed as 〈

*F*〉〈

_{D}*U*〉 = 〈

_{b}*F*〉

_{D}*U*. This includes energy loss due to flow separation and eddies. Since

_{c}*F*and

_{D}*U*are often time varying, the mean tidal energy loss 〈

_{b}*F*〉 is not the same as the energy loss rate driven by the time-averaged form drag 〈

_{D}U_{b}*F*〉

_{D}*U*.

_{c}*F*defined below:

_{D}*T*is the tidal period,

*ϕ*= Ω

*is the tidal phase and*

_{t}t*n*is the number of tidal cycles. The value of

*n*is larger than 6 for all cases. These diagnostic parameters are then employed to explain the variability of 〈

*F*〉 and 〈

_{D}U_{b}*F*〉

_{D}*U*across the cases.

_{c}## 5. Pressure anomalies in the wake

In a low Fr* _{c}* environment, the encounter of a steady current with a 3D obstacle forces a significant volume of fluid to navigate laterally. This laterally driven flow separates, and wake eddies are formed in the lee. The form stress increases as a result. To examine the origin of form stress, a thorough characterization of the dynamic pressure

*p*is crucial. The form drag is computed as the sum of the surface integral of

_{d}*p*and the Froude–Krylov force, as shown in Eq. (6). Note that the surface integral of

_{d}*p*is inclusive of the added-mass component of inertial drag. In section 5a, the dynamic pressure field is examined in the wake and on the obstacle. Changes in eddy shedding and form stress are illustrated and quantified over the flow regimes listed in Table 1. Estimates of

_{d}*F*and their values relative to

_{D}*F*and its relative magnitude with respect to frictional drag are discussed in sections 5c and 5d, respectively.

_{D}### a. Mean pressure distribution

Figures 2a, 2c, 2e, and 2g show qualitative differences in the vortex shedding patterns among the cases *f*, which is in agreement with observations of high Ro* _{c}* wakes (Chang et al. 2019; MacKinnon et al. 2019).

Instantaneous contours of normalized vertical vorticity (*ω _{z}*/

*f*) and time-averaged pressure (

*;*(c),(d)

*;*(e),(f)

*=*

_{t}t*π*/2).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Instantaneous contours of normalized vertical vorticity (*ω _{z}*/

*f*) and time-averaged pressure (

*;*(c),(d)

*;*(e),(f)

*=*

_{t}t*π*/2).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Instantaneous contours of normalized vertical vorticity (*ω _{z}*/

*f*) and time-averaged pressure (

*;*(c),(d)

*;*(e),(f)

*=*

_{t}t*π*/2).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

When *ω _{z}*) vortex remains attached to the obstacle as the cyclonic (positive

*ω*) vortex moves into the wake (see VortXY_JPO_fstar_2_15.mov in the online supplemental material). The anticyclonic vortex grows in size from repeated small pulses of vorticity (green box) created every tidal cycle. These pulses form on the lateral sides of the obstacle at the location of flow separation. The attached vortex is shed at a slower frequency of

_{z}*f*

_{s}_{,}

*(Puthan et al. 2021). Since*

_{c}*f*≈ 7.5

_{t}*f*

_{s}_{,}

*, the eddy remains attached while its circulation increases as the small pulses coalesce over 7.5 tidal cycles. Beyond*

_{c}*p*〉 is similar between the

_{d}The organization of vertical vorticity (*ω _{z}*) and mean pressure (〈

*p*〉) for

_{d}Figure 2g provides a snapshot of the wake when *p _{d}*〉

*on*the hill is plotted in Fig. 3.

Distribution of mean pressure (*;* (d),(e)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Distribution of mean pressure (*;* (d),(e)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Distribution of mean pressure (*;* (d),(e)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Figures 3a, 3d, 3g and 3b, 3e, 3h show the pressure distribution on the upstream and downstream faces of the obstacle, respectively. Each row in Fig. 3 corresponds to a different case. The panels of Figs. 3c, 3f, and 3i contrast the difference between the upstream and downstream pressure values (Δ〈*p _{d}*〉) integrated along

In all cases, Δ〈*p _{d}*〉 is bottom intensified. In their steady-current simulations, MacCready and Pawlak (2001) noted “a tendency for drag on the lower half of the ridge to be greater than that on the upper half.” In the present tidally modulated cases, the bottom intensification progressively increases from

*p*〉 is up to 5 times larger than its value at

_{d}The eddy shedding (Fig. 2) and pressure anomalies (Fig. 3) in the

Figure 4 shows snapshots of instantaneous normalized pressure *T*_{1}, *T*_{2}, *T*_{3}, and *T*_{4} are chosen at tidal phases Ω* _{t}t* = 0,

*π*/2,

*π*, and 3

*π*/2, respectively, for the snapshots. The variation of the phase-averaged form drag 〈

*F*〉

_{D}*and the barotropic current*

_{ϕ}*U*are plotted in the header.

_{b}(top) Variation of normalized phase-averaged drag force (〈*F _{D}*〉

*) with tidal phase Ω*

_{ϕ}*, plotted for*

_{t}t*U*is represented by the dotted line in the top panel. (bottom) The instantaneous contours of dynamic pressure

_{b}*p*are shown at four phases

_{d}*T*

_{1},

*T*

_{2},

*T*

_{3}, and

*T*

_{4}(marked in the top panel) for cases (a)–(d)

*A*.

_{cs}Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

(top) Variation of normalized phase-averaged drag force (〈*F _{D}*〉

*) with tidal phase Ω*

_{ϕ}*, plotted for*

_{t}t*U*is represented by the dotted line in the top panel. (bottom) The instantaneous contours of dynamic pressure

_{b}*p*are shown at four phases

_{d}*T*

_{1},

*T*

_{2},

*T*

_{3}, and

*T*

_{4}(marked in the top panel) for cases (a)–(d)

*A*.

_{cs}Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

(top) Variation of normalized phase-averaged drag force (〈*F _{D}*〉

*) with tidal phase Ω*

_{ϕ}*, plotted for*

_{t}t*U*is represented by the dotted line in the top panel. (bottom) The instantaneous contours of dynamic pressure

_{b}*p*are shown at four phases

_{d}*T*

_{1},

*T*

_{2},

*T*

_{3}, and

*T*

_{4}(marked in the top panel) for cases (a)–(d)

*A*.

_{cs}Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Consider the case *t* = *T*_{1}, the accelerating fluid impinges on the obstacle, creating high pressure upstream and a low pressure zone downstream (Fig. 4i). Owing to the asymmetry in flow separation, the low pressure region is dominant between *T*_{2} and *T*_{3}, the upstream pressure recedes concomitantly while the low pressure in the lee is sustained at *t* = *T*_{3} (Fig. 4k). The lee eddy remains attached at this instant preserving the low pressure zone. The low pressure region continues to exist at the downstream face between *T*_{4} (Fig. 4l). With the formation of persistent anticyclonic and cyclonic vortices on opposite sides during successive tidal cycles, the low pressure in the lee is maintained, albeit subject to lateral oscillations (see Pressure_JPO_fstar_5_6.mov in the online supplemental material). The outcome is an elevated mean drag.

For case *t* = *T*_{1} on both sides of the obstacle lee (Fig. 4f). While the barotropic current remains above its mean value *U _{c}* between

*t*=

*T*

_{1}and

*T*

_{3}, the vortices continue to grow larger on either side of the hill (elaborated further in section 6 and Fig. 8). The pressure drop across the hill increases concurrently in Figs. 4f and 4g. At the zero velocity phase (

*t*=

*T*

_{4}), the recirculating fluid at low pressure accelerates upstream relative to the barotropic flow. A region of positive dynamic pressure (Fig. 4h) is identified at this instant near

The instantaneous dynamic pressure *p _{d}* has contributions from two sources: the added mass and flow separation [see Eq. (6)]. When the tidal frequency is larger, the added mass component increases owing to its dependence on the tidal acceleration magnitude

*U*Ω

_{t}*. This is likely to occur when*

_{t}*p*in Figs. 4a–d. At

_{d}*T*

_{1}, the tidal acceleration reaches its maximum value and the dynamic pressure

*t*=

*T*

_{3}),

*t*=

*T*

_{2}and

*T*

_{4}), the dynamic pressure is purely associated with flow separation. At

*T*

_{2}, the low pressure region in the lee (Fig. 4b) is generated due to the attached eddy in Fig. 2c, while the high

*p*upstream is generated by the impinging current. On the contrary, at

_{d}*T*

_{4}, the pressure anomalies in the obstacle center plane are marginal (Fig. 4d). At the lateral sides of the obstacle, a weak pressure drop is observed, attributed to the vortex pulses discussed earlier. The tidal period is smaller than the natural eddy shedding time scale by a factor of 7.5 for this case. This constrains the size of the attached eddy to a small vortex pulse, which has a correspondingly weak effect on the pressure drop.

*p*directly affects the variation in form drag as illustrated by the phase-averaged form drag 〈

_{d}*F*〉

_{D}*in the header of Fig. 4. The form drag exhibits large modulation in the*

_{ϕ}*F*〉

_{D}*and*

_{ϕ}*U*exceeds

_{b}*π*/4. Recall that the inertial drag has a phase difference of

*π*/2 with

*U*. Therefore, it is possible that instantaneous inertial drag contributions are important in this case. To confirm this, we follow the procedure of Warner and MacCready (2009) to estimate the ratio of inertial drag to separation drag for our obstacle geometry. We assume that the magnitude of

_{b}*C*= 1 in Eq. (7) [for explanation, see Warner and MacCready (2009, p. 2979)]. For the conical obstacle, and taking

_{a}*R*=

*U*/

_{t}*U*= 1,

_{c}*F*〉

_{D}*and*

_{ϕ}*U*is smaller than

_{b}*π*/2 at

### b. States of high drag

From the preceding text, it is clear that the pressure anomalies on the obstacle change significantly owing to tidal oscillations. Here we demonstrate that tidal oscillations lead to high levels of instantaneous and mean drag. The ratio of inertial drag to separation drag is inversely related to *F _{D}*〉

*at*

_{ϕ}*t*=

*T*

_{1}and dropped rapidly below −10 near

*T*

_{3}. On the other hand, 〈

*F*〉

_{D}*has a smaller variance over the tidal cycle in the cases with*

_{ϕ}*F*〉

_{D}*from its cycle-averaged mean value is also much smaller. The values of 〈*

_{ϕ}*F*〉

_{D}*lie between −5 and 8 for this case.*

_{ϕ}Quantifying the mean and RMS values of *F _{D}* is key in this comparative analysis. The time averaged value, namely 〈

*F*〉, offers a diagnosis of high-drag states by nearly eliminating the contribution from inertial drag and providing an estimate of the separation drag

_{D}*F*〉 and 〈

_{D}*F*〉

_{D}_{rms}are plotted in Fig. 5 as a function of

*A*is the projected obstacle area in the streamwise direction. The value of form drag coefficient in the

_{cs}*F*〉 (relative to 〈

_{D}*F*

_{D}_{∞}〉) is observed in regimes 3 and 4. The mean drag coefficient exceeds 3.5 in regime 4, and its average value over the cases of regime 4 is 3, signifying a twofold increase in form drag with respect to the steady case. In regime 3, there is a 60% increase in average form drag relative to 〈

*F*

_{D}_{∞}〉. In comparison, only a marginal increase of 〈

*F*〉 is seen in regime 2. On the other hand, the RMS drag, 〈

_{D}*F*〉

_{D}_{rms}, is strongly influenced by inertial forces. The RMS drag decays rapidly as

Variation of mean drag force (〈*F _{D}*〉) and its root-mean-square value (〈

*F*〉

_{D}_{rms}) with

*F*

_{D}_{∞}〉).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Variation of mean drag force (〈*F _{D}*〉) and its root-mean-square value (〈

*F*〉

_{D}_{rms}) with

*F*

_{D}_{∞}〉).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Variation of mean drag force (〈*F _{D}*〉) and its root-mean-square value (〈

*F*〉

_{D}_{rms}) with

*F*

_{D}_{∞}〉).

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

### c. Energy loss to form drag

Figure 6 shows the variation of the tidally averaged power loss 〈*F _{D}U_{b}*〉 normalized by

*F*〉 is close to 5 across the tidally modulated cases and the deviations from this magnitude are minimal. This value is at least 3 times larger than the value of 1.5 computed for

_{D}U_{b}*R*= 1) cases. The energy loss driven by the time-averaged form drag 〈

*F*〉〈

_{D}*U*〉 = 〈

_{b}*F*〉

_{D}*U*follows a similar variability depicted by 〈

_{c}*F*〉 in Fig. 5. This quantity is smaller than the mean tidal energy loss 〈

_{D}*F*〉 and the difference stems from the energy loss associated with the oscillatory part of form drag that is in phase with the tidal modulation. This difference is larger when

_{D}U_{b}Variation of rate of work done by drag (*W _{D}*) normalized by

*F*〉 and the blue dashed line represents normalized 〈

_{D}U_{b}*F*〉〈

_{D}*U*〉 = 〈

_{b}*F*〉

_{D}*U*. The difference between these two quantities is represented by the dotted purple line. The gray dashed line denotes the rate of work done in the

_{c}Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Variation of rate of work done by drag (*W _{D}*) normalized by

*F*〉 and the blue dashed line represents normalized 〈

_{D}U_{b}*F*〉〈

_{D}*U*〉 = 〈

_{b}*F*〉

_{D}*U*. The difference between these two quantities is represented by the dotted purple line. The gray dashed line denotes the rate of work done in the

_{c}Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Variation of rate of work done by drag (*W _{D}*) normalized by

*F*〉 and the blue dashed line represents normalized 〈

_{D}U_{b}*F*〉〈

_{D}*U*〉 = 〈

_{b}*F*〉

_{D}*U*. The difference between these two quantities is represented by the dotted purple line. The gray dashed line denotes the rate of work done in the

_{c}Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

### d. Frictional drag versus form drag

*C*) of approximately 0.002 based on previous studies (e.g., McCabe et al. 2006), the mean frictional drag (〈

_{f}*F*

^{BBL}〉) is of the order

*A*is the projected area of the hill on the horizontal plane. Therefore, the ratio of mean form drag to frictional drag is

_{b}## 6. Vortex dynamics

Investigations of cylinder wakes created by homogeneous nonrotating flow, have often been used to add to our understanding of vortex dynamics in oceanic wakes (Chang et al. 2019). However, density stratification and planetary rotation influence the wake significantly. For example, Lin and Pao (1979) presented a detailed review of internal wave radiation and the emergence of thin lee vortices when Fr* _{c}* < 1. With the addition of rotation, Dong et al. (2006) showed manifestations of barotropic, baroclinic and centrifugal instabilities in the wake. The recent study of Puthan et al. (2021) showed that vortices in the far wake occur at frequencies coinciding with tidal subharmonics. Form drag is directly related to flow separation and near wake vortices. We discuss these facets and also the vertical organization of lee vortices in this section.

### a. Vertical structure of lee vortices

Figures 7a and 7c show contours of *ω _{z}* on horizontal planes at four different heights in the wake for

Eddy formation is depicted by the normalized vertical vorticity (*ω _{z}*/

*f*) in four horizontal (

*x*–

*y*) planes: (a)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Eddy formation is depicted by the normalized vertical vorticity (*ω _{z}*/

*f*) in four horizontal (

*x*–

*y*) planes: (a)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Eddy formation is depicted by the normalized vertical vorticity (*ω _{z}*/

*f*) in four horizontal (

*x*–

*y*) planes: (a)

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

For

For the case of *f _{t}*/4 in

*f*/2 in

_{t}*z*direction up to a vertical length scale permitted by the background stratification. At elevations above

### b. Symmetric eddy dipoles

The symmetric twin dipoles shown previously in Fig. 2e are created through a sequence of events presented in Fig. 8. A locally adverse streamwise pressure gradient (shown in Figs. 4e,f) develops on either lateral side of the obstacle. While the barotropic flow remains positive, as in Fig. 8a, two opposite-signed vortices form in the recirculation zone and grow in size until the velocity approaches zero. As the tide-associated pressure gradient changes sign at the zero velocity phase, the high vorticity fluid accelerates upstream relative to the background flow on both lateral sides. During this event, additional vorticity of opposite sign is generated from shear when this fluid is near the obstacle. For example, the attached anticyclonic eddy in Fig. 8a accrues positive vorticity in Fig. 8b during its deflection to the +*y* direction. Thus, twin dipoles form symmetrically on either lateral side of the obstacle. During the subsequent acceleration phase, the dipoles gain enough momentum to advect downstream (Fig. 8c). At the same instant, a new vortex pair starts to grow near the separation points, completing a full cycle of oscillation.

Normalized vertical vorticity (*ω _{z}*/

*f*) at three different phases of a tidally perturbed wake at

*t*/

*T*= 18.52, (b)

*t*/

*T*= 18.85, and (c)

*t*/

*T*= 19.19.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Normalized vertical vorticity (*ω _{z}*/

*f*) at three different phases of a tidally perturbed wake at

*t*/

*T*= 18.52, (b)

*t*/

*T*= 18.85, and (c)

*t*/

*T*= 19.19.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Normalized vertical vorticity (*ω _{z}*/

*f*) at three different phases of a tidally perturbed wake at

*t*/

*T*= 18.52, (b)

*t*/

*T*= 18.85, and (c)

*t*/

*T*= 19.19.

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

The formation of the symmetric vortices is phase locked to the tidal cycle. Note that the positive vorticity generated by the upstream flow (directed to the +*y* direction) in the previous tidal cycle (located at *U _{t}*/

*f*, or equivalently,

_{t}*f*from the tidal frequency

_{s}*f*occurs in this region, as elaborated in the next section.

_{t}### c. Temporal variation of vorticity

Hovmöller diagrams of normalized vorticity in Fig. 9 demonstrate clearly that transitions of the vortex frequency from the near to the far wake vary among regimes 2–4. Space–time (*y*–*t*) contours of vertical vorticity are plotted at two stations: S1 in the near wake and S2 in the far wake (see dashed lines in Figs. 2c,e,g). In Fig. 9a, the signature of the tidal frequency is observed in the vortex pulses spaced over one tidal period (*T*) for *T _{s}*. The vortex period (

*T*) in the far wake aligns with the natural shedding period of the obstacle wake,

_{s}*T*

_{s}_{,}

*.*

_{c}Time evolution of vertical vorticity (*ω _{z}*/

*f*) is depicted by a

*y*–

*t*Hovmöller diagram, at stations S1 (at

*T*is the tidal period and

*T*is the time period of far wake vortices. Note that the range of

_{s}*f*decreases from the top to the bottom row. The stations S1 and S2 are indicated by brown dashed lines in Fig. 2.

_{t}tCitation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Time evolution of vertical vorticity (*ω _{z}*/

*f*) is depicted by a

*y*–

*t*Hovmöller diagram, at stations S1 (at

*T*is the tidal period and

*T*is the time period of far wake vortices. Note that the range of

_{s}*f*decreases from the top to the bottom row. The stations S1 and S2 are indicated by brown dashed lines in Fig. 2.

_{t}tCitation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Time evolution of vertical vorticity (*ω _{z}*/

*f*) is depicted by a

*y*–

*t*Hovmöller diagram, at stations S1 (at

*T*is the tidal period and

*T*is the time period of far wake vortices. Note that the range of

_{s}*f*decreases from the top to the bottom row. The stations S1 and S2 are indicated by brown dashed lines in Fig. 2.

_{t}tCitation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Case *T _{s}* = 4

*T*, demonstrating the tidal synchronization noted in Puthan et al. (2021). Thus the wake vortex frequency is modified to

*f*/4. In other words, during every tidal cycle the symmetric vortices feed their vorticity into a larger vortex downstream, formed every four tidal cycles.

_{t}In the *D* (extending from *D* and 2.2*D* for *D* from the center plane. The space–time plot reveals thin filaments of positive vorticity in the temporal frame, in the far wake of *f _{t}*/2 for

To explore the time evolution of eddy vorticity injected into the wake, the absolute value of *ω _{z}* is volume averaged over a domain encompassing the hill and extending to a distance of 8

*D*into the wake. Figure 10a shows the temporal evolution of normalized volume-averaged vorticity

*U*. The pulses of vorticity which form during the maximum velocity phase of every tidal cycle are likely responsible for the small increases of

_{b}(a) Volume-averaged vertical vorticity *ω*_{z}|〉* _{V}* shown as a function of

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

(a) Volume-averaged vertical vorticity *ω*_{z}|〉* _{V}* shown as a function of

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

(a) Volume-averaged vertical vorticity *ω*_{z}|〉* _{V}* shown as a function of

Citation: Journal of Physical Oceanography 52, 6; 10.1175/JPO-D-21-0178.1

Time averages of *U _{c}T*. This fluid migrates into the wake permanently if

*U*exceeds

_{c}T## 7. Discussion and conclusions

LES were undertaken to examine the wake created by a stratified tidally modulated flow (*U _{b}*) in the presence of weak background rotation. The barotropic flow

*U*has a mean component

_{b}*U*, and a tidal component

_{c}*U*of equal strength. Since the Froude number Fr

_{t}*is small, the near-bottom flow is forced to separate laterally from the obstacle. As a result, large coherent eddies such as those observed in the ocean (e.g., Pawlak et al. 2003; MacKinnon et al. 2019), form in the lee. We find that the flow exhibits four regimes, based on vortex patterns, as summarized in Table 1. In the first regime where tides are absent, lee vortices separate from the obstacle at a constant frequency*

_{c}*f*

_{s}_{,}

*and form a Kármán vortex street downstream. In the next three regimes, the arrangement of vortices is altered by the tidal flow. Changes in flow separation, accompanied by bottom-intensified pressure differences on the obstacle, are responsible for states of high drag in the tidally modulated cases, especially regime 4.*

_{c}The effect of tidal oscillations is characterized by varying the relative frequency parameter (*U _{t}*/

*f*. Beyond this location, the vortices partially break down or merge to create a disorganized wake downstream. This event is accompanied by a change in the wake vortex frequency from

_{t}*f*to

_{t}*f*/4. At

_{t}*f*/2 is observed in the entire wake. The wake is laterally wider in comparison to the other regimes.

_{t}The timing of the shed vortices is strongly influenced by the barotropic tidal oscillation. When |*ω _{z}*| is volume averaged over the wake, its temporal variation is affected by the tidal oscillation. At

Changes in flow separation also lead to variations in pressure along the streamwise direction of the obstacle. The difference between the mean pressure field fore and aft of the hill (Δ〈*p _{d}*〉) is bottom intensified in cases

The normalized form drag, i.e., drag coefficient, obtained by integrating the pressure field over the obstacle surface area, varies among these cases. Form drag has two components, namely, the inertial and the separation drag. The separation drag is the dissipative part of form drag. The mean drag (averaged over several cycles) which is associated with the separation drag (Sarpkaya 2004; Warner and MacCready 2009), generally increases with increasing *p _{d}*〉 (associated with large eddies) and the longer eddy residence time in the lee contribute to approximately a twofold increase of mean drag coefficient in regime 4 relative to its value in the no-tide case. High drag states are also present in regime 3, wherein approximately a 60% increase is observed in mean drag coefficient relative to the no-tide case. Therefore, for a continuous distribution of topographic scales in the abyssal ocean, obstacles with

*F*〉 does not vary significantly on changing

_{D}U_{b}Warner and MacCready (2009) investigated a coastal headland in a purely tidal flow and did not find a significant effect of the excursion number (equivalently * _{t}* of 0.25. Deviations of the present findings (regarding excursion number effects) from their results may be attributed to a difference in geometry (submerged conical obstacle) or the presence of a mean current in this study. The drag coefficient associated with mean form drag normalized using the obstacle frontal area in regime 4 is 3. Converting this to a value based on the obstacle plan area, the drag coefficient takes a value of approximately 0.6, substantially larger than the drag coefficient associated with the frictional bottom boundary layer of

## Acknowledgments.

We gratefully acknowledge the funding provided by NSF OCE-1737367.

## Data availability statement.

The data used in this study is available in the repository https://doi.org/10.5281/zenodo.6506920.

## REFERENCES

Acarlar, M. S., and C. R. Smith, 1987: A study of hairpin vortices in a laminar boundary layer. Part 1. Hairpin vortices generated by a hemisphere protuberance.

,*J. Fluid Mech.***175**, 1–41, https://doi.org/10.1017/S0022112087000272.Black, K., and S. Gay, 1987: Eddy formation in unsteady flows.

,*J. Geophys. Res.***92**, 9514–9522, https://doi.org/10.1029/JC092iC09p09514.Callendar, W., J. M. Klymak, and M. G. G. Foreman, 2011: Tidal generation of large sub-mesoscale eddy dipoles.

,*Ocean Sci.***7**, 487–502, https://doi.org/10.5194/os-7-487-2011.Chang, M., S. Jan, C. Liu, Y. Cheng, and V. Mehsah, 2019: Observations of island wakes at high Rossby numbers: Evolution of submesoscale vortices and free shear layers.

,*J. Phys. Oceanogr.***49**, 2997–3016, https://doi.org/10.1175/JPO-D-19-0035.1.Clément, L., A. M. Thurnherr, and L. C. S. Laurent, 2017: Turbulent mixing in a deep fracture zone on the Mid-Atlantic Ridge.

,*J. Phys. Oceanogr.***47**, 1873–1896, https://doi.org/10.1175/JPO-D-16-0264.1.Denniss, T., J. H. Middleton, and R. Manasseh, 1995: Recirculation in the lee of complicated headlands: A case study of Bass Point.

,*J. Geophys. Res.***100**, 16 087–16 101, https://doi.org/10.1029/95JC01279.Dewey, R. K., and W. R. Crawford, 1988: Bottom stress estimates from vertical dissipation rate profiles on the continental shelf.

,*J. Phys. Oceanogr.***18**, 1167–1177, https://doi.org/10.1175/1520-0485(1988)018<1167:BSEFVD>2.0.CO;2.Dietrich, D. E., M. J. Bowman, C. A. Lin, and A. Mestasnunez, 1996: Numerical studies of small island wakes in the ocean.

,*Geophys. Astrophys. Fluid Dyn.***83**, 195–231, https://doi.org/10.1080/03091929608208966.Dong, C., J. C. McWilliams, and A. F. Shchepetkin, 2006: Island wakes in deep water.

,*J. Phys. Oceanogr.***37**, 962–981, https://doi.org/10.1175/JPO3047.1.Drazin, P., 1961: On the steady flow of a fluid of variable density past an object.

,*Tellus***13**, 239–251, https://doi.org/10.3402/tellusa.v13i2.9451.Edwards, K. A., P. MacCready, J. M. Moum, G. Pawlak, J. M. Klymak, and A. Perlin, 2004: Form drag and mixing due to tidal flow past a sharp point.

,*J. Phys. Oceanogr.***34**, 1297–1312, https://doi.org/10.1175/1520-0485(2004)034<1297:FDAMDT>2.0.CO;2.Egbert, G. D., and R. D. Ray, 2000: Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data.

,*Nature***405**, 775–778, https://doi.org/10.1038/35015531.Epifanio, C. C., and D. R. Durran, 2001: Three-dimensional effects in high-drag-state flows over long ridges.

,*J. Atmos. Sci.***58**, 1051–1065, https://doi.org/10.1175/1520-0469(2001)058<1051:TDEIHD>2.0.CO;2.Epifanio, C. C., and R. Rotunno, 2005: The dynamics of orographic wake formation in flows with upstream blocking.

,*J. Atmos. Sci.***62**, 3127–3150, https://doi.org/10.1175/JAS3523.1.Garcia-Villalba, M., N. Li, W. Rodi, and M. Leschziner, 2009: Large eddy simulation of separated flow over a three-dimensional axisymmetric hill.

,*J. Fluid Mech.***627**, 55–96, https://doi.org/10.1017/S0022112008005661.Girton, J., and Coauthors, 2019: Flow-topography interactions in the Samoan Passage.

,*Oceanography***32**, 184–193, https://doi.org/10.5670/oceanog.2019.424.Horwitz, R. W., S. Taylor, Y. Lu, J. Paquin, D. Schillinger, and D. A. Greenberg, 2021: Rapid reduction of tidal amplitude due to form Drag in a narrow channel.

,*Cont. Shelf Res.***213**, 104299, https://doi.org/10.1016/j.csr.2020.104299.Hunt, J., and W. Snyder, 1980: Experiments on stably and neutrally stratified flow over a model three-dimensional hill.

,*J. Fluid Mech.***96**, 671–704, https://doi.org/10.1017/S0022112080002303.Jagannathan, A., K. Winters, and L. Armi, 2019: Stratified flows over and around long dynamically tall mountain ridges.

,*J. Atmos. Sci.***76**, 1265–1287, https://doi.org/10.1175/JAS-D-18-0145.1.Jagannathan, A., K. Srinivasan, J. C. McWilliams, M. J. Molemaker, and A. L. Stewart, 2021: Boundary-layer-mediated vorticity generation in currents over sloping bathymetry.

,*J. Phys. Oceanogr.***51**, 1757–1778, https://doi.org/10.1175/JPO-D-20-0253.1.Keulegan, G. H., and L. H. Carpenter, 1958: Forces on cylinders and plates in an oscillating fluid.

,*Amer. Soc. Mech. Eng.***60**, 423–440, https://doi.org/10.6028/jres.060.043.Klymak, J. M., 2018: Nonpropagating form Drag and turbulence due to stratified flow over large scale abyssal hill topography.

,*J. Phys. Oceanogr.***48**, 2383–2395, https://doi.org/10.1175/JPO-D-17-0225.1.Lamb, H., 1930:

Cambridge University Press, 738 pp.*Hydrodynamics.*Ledwell, J. R., E. T. Montgomery, K. L. Polzin, L. C. S. Laurent, R. W. Schmitt, and J. M. Toole, 2000: Evidence for enhanced mixing over rough topography in the abyssal ocean.

,*Nature***403**, 179–182, https://doi.org/10.1038/35003164.Lighthill, J., 1986: Fundamentals concerning wave loading on offshore structures.

,*J. Fluid Mech.***173**, 667–681, https://doi.org/10.1017/S0022112086001313.Lin, J. T., and Y. H. Pao, 1979: Wakes in stratified fluids: A review.

,*Annu. Rev. Fluid Mech.***11**, 317–338, https://doi.org/10.1146/annurev.fl.11.010179.001533.Liu, C., and M. Chang, 2018: Numerical studies of submesoscale island wakes in the Kuroshio.

,*J. Geophys. Res. Oceans***123**, 5669–5687, https://doi.org/10.1029/2017JC013501.MacCready, P., and G. Pawlak, 2001: Stratified flow along a corrugated slope: Separation drag and wave drag.

,*J. Phys. Oceanogr.***31**, 2824–2839, https://doi.org/10.1175/1520-0485(2001)031<2824:SFAACS>2.0.CO;2.MacKinnon, J. A., M. H. Alford, G. Voet, K. Zeiden, T. M. S. Johnston, M. Siegelman, S. Merrifield, and M. Merrifield, 2019: Eddy wake generation from broadband currents near Palau.

,*J. Geophys. Res. Oceans***124**, 4891–4903, https://doi.org/10.1029/2019JC014945.Magaldi, M. G., T. M. Ozgokmen, A. Griffa, E. P. Chassignet, M. Iskandarani, and H. Peters, 2008: Turbulent flow regimes behind a coastal cape in a stratified and rotating environment.

,*Ocean Modell.***25**, 65–82, https://doi.org/10.1016/j.ocemod.2008.06.006.McCabe, R. M., P. MacCready, and G. Pawlak, 2006: Form Drag due to flow separation at a headland.

,*J. Phys. Oceanogr.***36**, 2136–2152, https://doi.org/10.1175/JPO2966.1.Morison, J. R., J. W. Johnson, and S. A. Schaaf, 1950: The force exerted by surface waves on piles.

,*J. Pet. Tech.***2**, 149–154, https://doi.org/10.2118/950149-G.Musgrave, R. C., J. A. Mackinnon, R. Pinkel, and A. F. Waterhouse, 2016: Tidally driven processes leading to near-field turbulence in a channel at the crest of the Mendocino Escarpment.

,*J. Phys. Oceanogr.***46**, 1137–1155, https://doi.org/10.1175/JPO-D-15-0021.1.Nash, J. D., and J. N. Moum, 2001: Internal hydraulic flows on the continental shelf: High drag states over a small bank.

,*J. Geophys. Res.***106**, 4593–4611, https://doi.org/10.1029/1999JC000183.Nikurashin, M., and R. Ferrari, 2010: Radiation and dissipation of internal waves generated by geostrophic flows impinging on small scale topography: Application to the Southern Ocean.

,*J. Phys. Oceanogr.***40**, 2025–2042, https://doi.org/10.1175/2010JPO4315.1.Nikurashin, M., and S. Legg, 2011: A mechanism for local dissipation of internal tides generated at rough topography.

,*J. Phys. Oceanogr.***41**, 378–395, https://doi.org/10.1175/2010JPO4522.1.Pawlak, G., P. MacCready, K. A. Edwards, and R. McCabe, 2003: Observations on the evolution of tidal vorticity at a stratified deep water headland.

,*Geophys. Res. Lett.***30**, 2234, https://doi.org/10.1029/2003GL018092.Perfect, B., N. Kumar, and J. Riley, 2018: Vortex structures in the wake of an idealized seamount in rotating, stratified flow.

,*Geophys. Res. Lett.***45**, 9098–9105, https://doi.org/10.1029/2018GL078703.Perfect, B., N. Kumar, and J. Riley, 2020: Energetics of seamount wakes. Part 2: Wave fluxes.

,*J. Phys. Oceanogr.***50**, 1383–1398, https://doi.org/10.1175/JPO-D-19-0104.1.Puthan, P., M. Jalali, J. L. Ortiz-Tarin, K. Chongsiripinyo, G. Pawlak, and S. Sarkar, 2020: The wake of a three-dimensional underwater obstacle: Effect of bottom boundary conditions.

,*Ocean Modell.***149**, 101611, https://doi.org/10.1016/j.ocemod.2020.101611.Puthan, P., S. Sarkar, and G. Pawlak, 2021: Tidal synchronization of lee vortices in geophysical wakes.

,*Geophys. Res. Lett.***48**, e2020GL090905, https://doi.org/10.1029/2020GL090905.Rudnick, D. L., K. L. Zeiden, C. Y. Ou, T. M. S. Johnston, J. A. MacKinnon, M. H. Alford, and G. Voet, 2019: Understanding vorticity caused by flow passing an island.

,*Oceanography***32**, 66–73, https://doi.org/10.5670/oceanog.2019.412.Sarpkaya, T., 2001: On the force decompositions of Lighthill and Morrison.

,*J. Fluids Struct.***15**, 227–233, https://doi.org/10.1006/jfls.2000.0342.Sarpkaya, T., 2004: A critical review of the intrinsic nature of vortex-induced vibrations.

,*J. Fluids Struct.***19**, 389–447, https://doi.org/10.1016/j.jfluidstructs.2004.02.005.Sheppard, P. A., 1956: Airflow over mountains.

,*Quart. J. Roy. Meteor. Soc.***82**, 528–529, https://doi.org/10.1002/qj.49708235418.Signell, R. P., and W. R. Geyer, 1991: Transient eddy formation around headland.

,*J. Geophys. Res.***96**, 2561–2575, https://doi.org/10.1029/90JC02029.Srinivasan, K., J. C. McWilliams, J. Molemaker, and R. Barkan, 2018: Submesoscale vortical wakes in the lee of the topography.

,*J. Phys. Oceanogr.***49**, 1949–1971, https://doi.org/10.1175/JPO-D-18-0042.1.Srinivasan, K., J. C. McWilliams, and A. Jagannathan, 2021: High vertical shear and dissipation in equatorial topographic wakes.

,*J. Phys. Oceanogr.***51**, 1985–2001, https://doi.org/10.1175/JPO-D-20-0119.1.Voet, G., M. H. Alford, J. A. MacKinnon, and J. D. Nash, 2020: Eddy wake generation from broadband currents near Palau.

,*J. Phys. Oceanogr.***50**, 1489–1507, https://doi.org/10.1175/JPO-D-19-0257.1.Vosper, S., I. Castro, W. Snyder, and S. Mobbs, 1999: Experimental studies of strongly stratified flow past three-dimensional orography.

,*J. Fluid Mech.***390**, 223–249, https://doi.org/10.1017/S0022112099005133.Warner, S. J., and P. MacCready, 2009: Dissecting the pressure field in tidal flow past a headland: When is form drag “real”?

,*J. Phys. Oceanogr.***39**, 2971–2984, https://doi.org/10.1175/2009JPO4173.1.Warner, S. J., and P. MacCready, 2014: The dynamics of pressure and form drag on a sloping headland: Internal waves versus eddies.

,*J. Geophys. Res. Oceans***119**, 1554–1571, https://doi.org/10.1002/2013JC009757.Warner, S. J., P. MacCready, J. M. Moum, and J. D. Nash, 2012: Measurement of tidal form drag using seafloor pressure sensors.

,*J. Phys. Oceanogr.***43**, 1150–1172, https://doi.org/10.1175/JPO-D-12-0163.1.Wijesekera, H. W., E. Jarosz, W. J. Teague, D. W. Wang, D. B. Fribance, J. M. Moum, and S. J. Warner, 2014: Measurements of form and frictional drags over a rough topographic bank.

,*J. Phys. Oceanogr.***44**, 2409–2432, https://doi.org/10.1175/JPO-D-13-0230.1.Wright, C. J., R. B. Scott, P. Ailliot, and D. Furnival, 2014: Lee wave generation rates in the deep ocean.

,*Geophys. Res. Lett.***41**, 2434–2440, https://doi.org/10.1002/2013GL059087.Yu, X., J. H. Rosman, and J. L. Hench, 2018: Interaction of waves with idealised high-relief bottom roughness.

,*J. Geophys. Res. Oceans***123**, 3038–3059, https://doi.org/10.1029/2017JC013515.Zhang, X., and M. Nikurashin, 2020: Small scale topographic form stress and local dynamics of the Southern Ocean.

,*J. Geophys. Res. Oceans***125**, https://doi.org/10.1029/2019JC015420.