Ocean Modelling 147 (2020) 101565

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Ocean Modelling
journal homepage: www.elsevier.com/locate/ocemod

Topographic and coastline influences on surf EddiesÇiğdem Akan a ,∗, James C. McWilliams b , Yusuke Uchiyama cabcCollege of Computing, Engineering, & Construction, University of North Florida, Jacksonville, FL, USA
Department of Atmospheric and Oceanic Sciences, University of California Los Angeles, Los Angeles, CA, USA
Department of Civil Engineering, Kobe University, Kobe, Japan

ARTICLEINFOABSTRACT

Keywords:
Rip current
Surf zone
Instability
Topographic effects

Rip currents arise from the momentum flux associated with depth-induced breaking of incident surface
gravity waves on beaches with irregular bottom topography. In 3D simulations with an idealized topographic
configuration, uniform density, and a steady incident wave field, a striking enhancement of transient surf
eddy variability occurs over irregular topography compared to smooth topography, especially at nearly normal
incident wave angles. Alternatively, with highly oblique incident waves, transient shear instability is observed
over smooth topography whereas for irregular topography, standing rip eddies are generated. In the presence oflarger-scale coastline variations, mega-rip circulations emerge, and they are especially strong for embayments.
In all cases with significant transient surf-eddy activity, the eddy-induced mean horizontal transport is a nontrivial component of the total mean transport, and sometimes it is much larger than the gravity-wave Stokes
drift. Typically the transport by the eddy-induced flow partially cancels the Eulerian transport by the timeand/or alongshore-averaged currents, but not by enough to reverse the ‘‘residual’’ pattern of total transport.

1. Introduction

Rip channel spacing and depth has been shown to play a significant
role in the strength of the rip currents and on the instabilities (Yuand Slinn, 2003; Castelle et al., 2014). Castelle et al., 2014 showeda decrease in surf zone retention with increasing rip spacing. The
incident wave angle is another important factor that with increasing
obliqueness intensifies the alongshore current and suppresses the rip
current formation. As a result, the surf eddies die out, and the flow
becomes more nearly steady when the incident wave field is steady(Yu and Slinn, 2003; Weir et al., 2011). Moreover, field and laboratory
measurements (e.g., Haller et al., 2014; MacMahan et al., 2004) reveal
energetic very low frequency (VLF) motions correlated with the ripchannels. The relation between the topographic irregularities and rip
current instabilities (VLF motions) is poorly understood, and this paper
further explores it. Another perspective on the origins of VLF currents isthat they are associated with alongshore and temporal variability of theincident surface wave field (Feddersen et al., 2011), but this mechanism
is not explored here.
Most earlier studies were performed using depth-averaged (2-D)
models, but in recent years the importance of the three-dimensionality
in modeling rip current systems started to receive more attention(Marchesiello et al., 2015; also see the companion paper by Uchiyama
et al. (2017) for a detailed analysis of 2-D versus 3-D simulations for the
DUCK experimental site). Study by Uchiyama et al. (2017) also shows
vigorous surf eddy activity associated with the topographic irregularities. This motivates the present, more systematic study of this influence

More than 1/3 of the U. S. population lives in the coastal regions(Crossett et al., 2013). These areas are important for recreational
activities, and their popularity is accompanied by beach hazards. Oneof the leading coastal hazards resulting in surf zone fatalities is rip
currents (National Weather Service, 2016). Rip currents are commonly
observed as strong and narrow currents moving offshore from the coastand exist generally on beaches with strong surf. They are also important
for transporting dissolved and suspended materials across the surf zone.
Alongshore currents and sea-level set-up (i.e., increasing surface
height approaching the shore) arise from depth-induced wave breaking (Longuet-Higgins and Stewart, 1962), and rip currents arise due
to either current instability, incident wave variability, or topographic
variations. Many previous studies concerning rip currents attempt tounderstand the mechanisms behind their generation and dynamics,e.g., Dalyrmple et al. (2011) or Castelle et al. (2016) are literature reviews on mechanisms for rip current generation. Rip current modelingstudies can be divided into two categories: modeling studies accompanied with measurements or laboratory experiments (e.g.,Bruneau et al.,2011; Haas et al., 2003; Marchesiello et al., 2015, 2016; Uchiyama
et al., 2017); idealized modeling studies (e.g.,Yu and Slinn, 2003; Özkan
Haller and Li, 2003; Uchiyama et al., 2009; Weir et al., 2011). Idealized rip current studies are mostly performed with alongshore-uniform
bathymetries or with added periodic alongshore oscillations, termed ripchannels for the offshore flow that forms where the depth is larger.
∗ Corresponding author.

E-mail address: cigdem.akan@unf.edu (Ç. Akan).
https://doi.org/10.1016/j.ocemod.2019.101565
Received 12 March 2019; Received in revised form 16 December 2019; Accepted 26 December 2019
Available online 16 January 20201463-5003/Published by Elsevier Ltd.

Ç. Akan, J.C. McWilliams and Y. Uchiyama

Ocean Modelling 147 (2020) 101565
Table 2
Case list for variable coastlines. 𝜃 is the incident wave angle. For example, H5 would
denote the headland case with 𝜃 = 5◦ whereas B5 would denote the bay case with
𝜃 = 5◦ .

Table 1Case list for straight coastlines. 𝜃 is the incident wave angle. Note that the case names
are coded based on the topography type, smoothness and wave angle. For example,
SB3 would denote the smooth barred topography case with 𝜃 = 3◦ whereas IB3 would
denote the irregular barred topography case with 𝜃 = 3◦ .

SB0
SB3
SB5
SB10
SB15
SB20
SB25
SB30
SB35
SB40
IB0
IB3
IB5
IB10
IB15
IB20
IB25
IB30
IB35
IB40

Topo

𝜃

Smooth/irregular

BarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarredBarred

0◦
3◦
5◦
10◦
15◦
20◦
25◦
30◦
35◦
40◦
0◦
3◦
5◦
10◦
15◦
20◦
25◦
30◦
35◦
40◦

SmoothSmoothSmoothSmoothSmoothSmoothSmoothSmoothSmoothSmoothIrregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular

SP0
SP3
SP5
SP10
SP15
SP20
SP25
SP30
SP35
SP40
IP0
IP3
IP5
IP10
IP15
IP20
IP25
IP30
IP35
IP40

Topo

𝜃

Bathymetry type

PlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanarPlanar

0◦
3◦
5◦
10◦
15◦
20◦
25◦
30◦
35◦
40◦
0◦
3◦
5◦
10◦
15◦
20◦
25◦
30◦
35◦
40◦

SmoothSmoothsmoothSmoothSmoothSmoothSmoothSmoothSmoothSmoothIrregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular
Irregular

H0
H5
H10
H20
H30
H40

Coast shape𝜃HeadlandHeadlandHeadlandHeadlandHeadlandHeadland

0◦
5◦
10◦
20◦
30◦
40◦

B0
B5
B10
B20
B30
B40

Coast shape

𝜃

Bay
Bay
Bay
Bay
Bay
Bay

0◦
5◦
10◦
20◦
30◦
40◦

For the primary cases in Tables 1–2, the only forcing is by incident
steady, monochromatic waves specified at the offshore boundary witha wave amplitude of 𝑎 = 0.5 m and a peak wave period 𝑇𝑝 = 10 s;i.e., there is no wind forcing, no heating, and no river inflow. The
incident wave angle 𝜃 is varied among cases. Additional sensitivity tests
are performed varying 𝑎 while keeping the offshore incident angle 𝜃 andwave period 𝑇𝑝 constant Appendix C. The depth-induced wave-breaking
parameterization proposed by Thornton and Guza (1983) is used:(√] )−5∕2 ⎞−1[
5⎛
3 𝜋 𝜌𝑔𝑓𝑝 𝛽 3 𝐻rms
𝐻𝑟𝑚𝑠 2
⎟ ,
⎜
𝜖𝑏 =
1− 1+
⎟
⎜
16
𝛾ℎ
𝛾 2 ℎ3⎝⎠(1)

2. Methods

𝐻 = ℎ + 𝜂𝑐 is the local water-column thickness, where ℎ is the stillwater depth and 𝜂𝑐 = 𝜂 + 𝜂̂ is the wave-averaged surface elevationthat is the sum of the dynamic sea level and the quasi-static response
due to the conservative wave effects on the currents (WEC) (Uchiyama
et al., 2010), 𝑔 is gravitational acceleration, 𝜌 is the oceanic density, 𝛾
is the maximum ratio of wave height to water depth, 𝑓𝑝 is the peakfrequency of the narrow band wave energy spectrum, 𝛽 is a factor√
representative of the level of energy dissipation, and 𝐻rms = 𝐻∕ 2
is the root-mean-square wave height.
Previous studies show that different wave breaking parameterizations can give different results, and they are sensitive to changes in thesignificant wave height, 𝐻𝑠 . In realistic applications the parameters 𝛾
and 𝛽 are tuned in to fit the observations. In our idealized cases, weuse a universal form of 𝛾 as proposed by Apotsos et al. (2008) whilechoosing 𝛽 = 1,

2.1. The ROMS-WEC model𝛾 = 𝐴 + 𝐵[tanh(𝐶𝐻𝑜 ) ] ,on surf eddies. Here we present results from idealized simulations using
the three dimensional (3-D) ROMS-WEC model (Uchiyama et al., 2010)
to explore the role of complex, small-scale bathymetry in instigating very low frequency (VLF) currents that are highly turbulent andeddy-like, in contrast to currents over a smooth, alongshore-uniform
bathymetry. Further, we examine the effects of shoreline bays andheadlands on wave-driven currents.
In Section 2 we describe the modeling approach and the stochastic
perturbation method used to generate different topographies. Resultsfrom numerical simulations with and without complex bathymetry arepresented in Section 3. In Section 4, the role of the coastline shape andits interaction with rip currents are explored. Summary and discussionof the results are in Section 5.

(2)

where 𝐻𝑜 is the offshore wave height, 𝐴 = 0.18, 𝐵 = 0.40, and 𝐶 = 0.90
for the parameterization (1). The bottom stress is an important elementof a surf-zone model, and there is a wide variety of representations
used in the literature. For our idealized study, we choose a particular
representation and keep it the same as the topography is varied. The
nonlinear parameterization of bottom stress proposed by Feddersen
et al. (2000) is used:[) ]1∕2(|𝐮𝐛 | 2𝑢𝑤𝑣
(3)
𝝉 𝑏 = 𝜌𝑐𝐷 (1.16)2 +𝑟𝑚𝑠 𝐮𝐛 ,𝑢𝑤𝑣
𝑟𝑚𝑠

A total of 52 cases (plus additional sensitivity tests in the Appendices) are performed using the coupled ROMS-WEC model, i.e., theRegional Oceanic Modeling System with Wave Effects on Currents(Uchiyama et al., 2010). Lists of the cases are in Tables 1–2 andthe Appendices. The model consists of the circulation model ROMS
and a spectrum-peak, finite-depth wave model (i.e., WKB ray theory
with a depth- and current-induced refraction, and a depth-induced
breaking, and a bottom-drag wave dissipation parameterization). The
Stokes drift is calculated from the WKB wave solution, and the current acceleration is parameterized from the wave dissipation rates
by surface breaking and by bottom drag. The model is fully-coupledwith surface gravity wave effects on currents (WEC) and current effecton the waves (CEW) based on the conservative asymptotic theory
developed by McWilliams et al. (2004) and the non-conservative model
implementation by Uchiyama et al. (2010).
Periodicity is assumed in the alongshore direction. At the offshore
open boundary, we assume no net outflow by the sum of the Euleriancurrent and Stokes drift. At the western shoreline a wall boundary
condition is applied with a minimum depth of 0.1 m, and wetting
and drying behaviors are excluded. In practice bottom drag and depthinduced breaking make the waves and currents nearly zero in suchshallow water near the shoreline. The Coriolis force is neglected because the Rossby number is very large for littoral currents and surf
eddies. The water density is uniform.

where 𝐶𝐷 is the bottom friction coefficient equal to 0.01, 𝑢𝑤𝑣
𝑟𝑚𝑠 is theroot mean square wave orbital velocity, and 𝐮𝐛 = (𝑢𝑏 , 𝑣𝑏 ) is the bottom
current velocity. We tested different bottom stress parameterizations
found in the literature including linear, quadratic, logarithmic, andone proposed by Soulsby (1995) that also includes combined effects ofwaves and currents. The linear and quadratic bottom stress parameterizations generate more instabilities compared to the logarithmic bottom
stress parameterization and parameterizations proposed by Soulsby(1995) and Feddersen et al. (2000). We also found that Soulsby (1995)
suppresses littoral current instabilities much more than (3) does. Lateral
momentum diffusion is included with 𝜈 = 1 m2 s−1 (but see Appendix E).
All cases are initialized at rest with a flat sea level and no waves
inside the domain. They are integrated forward in time to a statisticalsteady state that is then analyzed for the results in Sections 3–4.
2

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Ocean Modelling 147 (2020) 101565

Fig. 1. Maps of the irregular (a) planar, (b) barred, (c) headland, and (d) bay topographies. Colormaps show the depth, and contour lines accentuate the variations. In all cases
the topographic perturbation height 𝑝𝐿𝑧 is the same, but the domains are wider and longer in (c) and (d), and the total depth range is larger.
Table 3
Perturbation cross-shore width (𝑝𝐿𝑥 ), alongshore length (𝑝𝐿𝑦 ), amplitude (𝑝𝐻) and theoffshore limit of the perturbations 𝑥𝑜𝑠𝑑 for different cases.

2.2. Topographies
Most idealized rip current studies are carried out with alongshoreuniform or -periodic topographies where parameters such as incident
wave angle and bottom friction coefficient, are varied to identify therelevant alongshore flow and rip current generation mechanisms. Commonly used topographies are alongshore-uniform barred topography(e.g., Özkan Haller and Li, 2003; Uchiyama et al., 2009) and barred
topography with uniformly spaced channels (e.g., Yu and Slinn, 2003;
Weir et al., 2011; Xie, 2011). Here, we make a more complete examination of the role of the more complex alongshore irregularities in the
topography in rip currents.
The first set of simulations listed in Table 1 is divided into two maingroups; ‘‘smooth" and ‘‘irregular" where ‘‘smooth" means alongshoreuniform topography and ‘‘irregular" means smooth topography combined with stochastic depth variations, as explained below. The alongshore averages of the smooth and irregular topographies are the same.
To illustrate the irregularities, we show two simple domains withstraight coastlines in the top panels of Fig. 1: panel (a) is an irregularplanar beach topography (i.e., with a uniform bottom slope in thealongshore average), and panel (b) is the irregular barred topography(i.e., with an isolated bump at an intermediate distance from the shoreline), whose smooth counterpart is the same as in the study by Özkan
Haller and Li (2003). The size of these domains is 512 m by 768 m
with 𝑑𝑥 = 𝑑𝑦 = 4 m horizontal resolution, and there are 20 terrainfollowing layers in the vertical direction. Cases with these topographies
are analyzed in Section 3.
In the second set of simulations in Table 2, we explore the influenceof the shape of the coastline in generating and interacting with the currents (Section 4). Two different coastline configurations are generated
with either a headland or a bay, as illustrated in Fig. 1c, d. The sizeof the headland and bay domains is larger to accommodate the range

PlanarBarredHeadland
Bay

𝑝𝐿𝑥

𝑝𝐿𝑦

𝑝𝐻

𝑥𝑜𝑠𝑑 [m]

80 m
80 m
100 m
100 m

80 m
80 m
100 m
100 m

0.5
0.5
0.5
0.5

Lx/4
Lx/4
Lx/3
Lx/3

of coastline orientations, i.e., 5000 m by 5000 m with 𝑑𝑥 = 𝑑𝑦 = 10 m
grid resolution and 20 terrain-following layers.
Irregular topography is composed of an alongshore-uniform topography superimposed with random depth perturbations. To generatestochastic perturbations, the procedure described by Evensen (2007)
is followed. This method generates random depth fields ℎ′ (𝑥, 𝑦) withprescribed characteristics: 𝜇 = 0 is the mean, 𝜎 2 = 𝑝𝐻 2 is the variance,
and the horizontal covariance function is𝐶ℎℎ (𝛥𝑥, 𝛥𝑦) = 𝑝𝐻 2 exp(−

𝛥𝑦2
𝛥𝑥2
−
),
𝑝𝐿2𝑥 𝑝𝐿2𝑦(4)

where 𝛥𝑥 and 𝛥𝑦 are spatial separation distances, 𝑝𝐿𝑥 is the spatial
scale of the perturbations in the cross-shore direction, 𝑝𝐿𝑦 is the spatial
scale of the perturbations in the alongshore direction, and 𝑝𝐻 is theperturbation amplitude. The size and amplitude of the perturbations(𝑝𝐿𝑥 , 𝑝𝐿𝑦 , 𝑝𝐻) are to be specified. For the present study, we havechosen 𝑝𝐿𝑥 , 𝑝𝐿𝑦 and 𝑝𝐻 values such that random rip channels are
generated at realistic vertical and lateral scales (see Table 3 for thevalues used).
Thus, the perturbed topographic depth is defined by[]ℎ(𝑥, 𝑦) = ℎ0 (𝑥, 𝑦) 1 + 𝑠0 ℎ′ (𝑥, 𝑦) ,(5)
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Ocean Modelling 147 (2020) 101565

Fig. 2. Map illustrating the topographic components [m]: (a) smooth barred topography ℎ0 (𝑦), (b) random perturbations ℎ′ (𝑥, 𝑦), and (c) irregular total topography ℎ(𝑥, 𝑦). The
dashed line in panel (c) indicates the location of 𝑥𝑜𝑠𝑑 .

where ℎ0 (𝑥, 𝑦) is the smooth topography, 𝑠0 is a roughness control
parameter (𝑠0 ∈ [0, 1]) whose sensitivity is examined in Appendix C, andℎ′ (𝑥, 𝑦) is the non-dimensional random depth field. The sensitivity toparticular realizations for ℎ′ is examined in Appendix A. The offshore
extent of the perturbations is limited on a scale that is slightly larger
than the surfzone width, 𝑥𝑜𝑠𝑑 , to prevent artificial effects close to theoffshore boundary; its value for different cases is also listed in Table 3.
Then local smoothing is applied near 𝑥 = 𝑥𝑜𝑠𝑑 to assure that ℎ iscontinuous. The factor 𝑟𝑚𝑎𝑥 , maximum grid stiffness factor, is less than
0.35 for all the perturbed cases. Also, the topography close to theshoreline is kept the same for both smooth and irregular cases. Thisis determined by setting up a distance limit similar to the offshore
limit, but for a much shorter distance, i.e. the depth is set to minimum
depth from shoreline to 𝑥 = 16 m. This prevented the occurrence ofsteep bathymetry close to the shoreline. All the runs listed in Tables 1
and 2 are carried out with 𝑠0 = 1. We illustrate the composition ofirregular topography using the preceding method in Fig. 2. Panel (a) is
the smooth barred topography ℎ0 (𝑦), panel (b) is a random realization
for ℎ′ (𝑥, 𝑦), and panel c is the combined total topography ℎ(𝑥, 𝑦).
For headland and bay cases, we first examined the bathymetryaround several headlands and bays found in the nature. In our observations the topography around most headlands deepens sharply comparedto the beach bathymetries adjacent to it whereas the topography withinmost bays has a milder slope. Moreover, the study by George (2016)
investigated circulation and sediment transport at headlands and by
doing so he categorized different headlands in the California coast.
Appendix 1 of that study presents a database of classified headlands.
The size of the headland (perimeter) used in the present modelingwork is consistent with most the headlands listed in the aforementioned
database. The bay used in the study is just the symmetric inverse of theheadland, but as will be explained in Section 4, it generates mega-rips
similar to the one observed around a submarine canyon off La Jolla,CA, near the Scripps Research Pier (Long and Özkan Haller, 2005).
Thus, we construct smooth topographies ℎ0 (𝑥, 𝑦) of the headland andbay domains accordingly, then add the random field ℎ′ (𝑥, 𝑦), and again
limit its offshore extent. The smooth topography in these cases does nothave an offshore bar.

the alongshore-uniformity of ℎ is broken, e.g.,by adding rip channels,
then offshore-directed rip currents will form, both steady and transient(Section 1). To trigger flow instabilities, small perturbations (i.e., random noise in the order of 10−5 [m s−1 ]) are sometimes added to thevelocity field at an early time but after currents have arisen. In ourirregular bathymetry cases, we do not need to add perturbations to
the velocity field because the instabilities develop almost from the
beginning with all incident wave angles. However, with increasing 𝜃,
the instabilities are suppressed. We define a surf eddy as a deviationof vorticity from an alongshore-average within the nearshore region,and eddies can be either time-varying (transient) or steady in time(standing). Note that identification of the eddies was done visually
where we consider an eddy as a closed circulation.
As an illustration of the current field, Fig. 3 shows the instantaneous
surface vertical vorticity for smooth and irregular barred cases withalternative wave incident angles 𝜃 = 0◦ (top panels) and 𝜃 = 40◦ bottom
panels) at the end of the simulation (at the 12th hour of integration).
Surf eddies do not develop in the smooth case when the waves arenormally incident; in other words, having a smooth alongshore-uniform
topography does not generate any appreciable currents, so there isno possible instability for their generation. With irregular topography,
however, currents do develop and become unstable due to their horizontal shear, and the resulting surf eddies are nearly isotropic in theirhorizontal pattern (Fig. 3b). If the waves are obliquely incident, thenan alongshore current arises from the breaking-wave acceleration andhorizontal shear instabilities ensue (Fig. 3c). With irregular topography
the surface vertical vorticity is enhanced for obliquely incident waves,
but the current instability is suppressed: the surf eddies are now almost
entirely steady in time, i.e., standing eddies rather than transient ones
(Fig. 3d). The sensitivity test in Appendix B shows that with a somewhat
lowered bar height, transient eddies still occur with irregular topography and large 𝜃, albeit still weaker than the standing eddies in that
regime.

3. Effects of irregular topography

We quantify surf eddy activity with the time-averaged eddy kinetic energy 𝐸𝐾𝐸. We decompose the fields into time-averaged andtransient-eddy components, 𝜙 = 𝜙̄ + 𝜙′ , where an overbar denotes
time averaging, a prime denotes deviation from the time average. A

3.1. Eddy kinetic energy (𝐸𝐾𝐸)This section analyzes the cases with a straight shoreline and demonstrates the differences between smooth and irregular topographies. If
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Ocean Modelling 147 (2020) 101565

Fig. 3. Instantaneous surface vertical vorticity 𝜁𝑧𝑠 (𝑥, 𝑦) [s−1 ] at the 12th hour of the simulation for (a) smooth barred topography with 𝜃 = 0◦ (SB0), (b) irregular barred topography
with 𝜃 = 0◦ (IB0), (c) smooth barred topography with 𝜃 = 40◦ (SB40), and (d) irregular barred topography with 𝜃 = 40◦ (IB40). Parameters for the cases are in Table 1. The
currents in panel (a) are too weak to be seen with this colorbar.

further decomposition of 𝜙̄ can be made into an alongshore-averaged
̄ and its standing-eddy deviation, 𝜙𝑠 = 𝜙̄ − ⟨𝜙⟩,̄ wherecomponent, ⟨𝜙⟩,angle brackets denote the alongshore average.1 Thus, the total field is̄ + 𝜙𝑠 + 𝜙′ ,𝜙 = ⟨𝜙⟩Fig. 4 shows clearly that irregular barred topography is highly energetic even when the waves are normally incident (solid black line).2
For normally and slightly oblique waves, the transient eddy component
makes up about one-half of the total 𝐸𝐾𝐸. This indicates generationof energetic eddies due to the irregularities in the topography. As theincident wave angle increases, transient eddies start to disappear andstanding eddies dominate (compare the solid blue line versus solid redline).
In contrast, because there is not a mean current to generate instabilities, the 𝐸𝐾𝐸 is almost zero for smooth topography for normal andnear normal incident wave angles (dashed black line). With increasing
incident wave angle, the transient eddy component increases due toshear instabilities (dashed red line) while the standing eddy component
is zero apart from finite-time sampling error. Thus, we conclude thatthe dominant 𝐸𝐾𝐸 components change between alongshore-uniform
and irregular topographies.
Similar behaviors occur in the planar beach cases (Fig. 5). Even
though depth-induced breaking occurs with the same total wave dissipation, hence with the same total current acceleration, it is more

(6)

and 𝜙𝑠 + 𝜙′ is the total eddy component. Furthermore, the time- andalongshore-averaged variance is̄ 2 + ⟨𝜙𝑠 2 ⟩ + ⟨𝜙′2 ⟩ .
⟨𝜙2 ⟩ = ⟨𝜙⟩(7)

The first right-side term in (7) is the variance of the mean, and thesecond and third terms are the variances of the standing and transient
eddies, respectively.
This decomposition is applied to the velocity and then depthintegrated and multiplied by 21 to calculate mean, standing-eddy, andtransient eddy kinetic energies. 3-D velocity vectors are (𝐮, 𝑤), where𝐮 = (𝑢, 𝑣) is the velocity vector in the horizontal direction and 𝑤 is thevelocity in the vertical direction. The total eddy kinetic energy 𝐸𝐾𝐸 is
the sum of the standing and transient components. Steady rip currents
would contribute through the standing 𝐸𝐾𝐸.

2
For EKE and the shear production terms 𝑃 in (9)–(10), we compute
3D fields with time averages, then depth integrate using the mean surface
height for wave-averaged currents, 𝜂̄𝑐 (𝑥, 𝑦), and, where indicated, take an
additional alongshore- or area-average. This procedure overlooks an additional
nonlinearity in averages associated with the variable 𝜂𝑐′ = 𝜂𝑐 −𝜂̄𝑐 . This neglected
effect is small in both quantities.

1
With a variable ℎ the only meaningful alongshore average of a 3D field
is one at constant (𝑥, 𝑦), the terrain — a free-surface following coordinate in
ROMS, rather than one at constant 𝑧 that would multiply intersect the bottom.

5

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Ocean Modelling 147 (2020) 101565

Fig. 4. Total (black), standing (blue), and transient (red) 𝐸𝐾𝐸 components [m3 s−1 ] for smooth (dashed lines) and irregular (solid lines) barred topographies as a function of
offshore wave incident angles. 𝐸𝐾𝐸 is depth-integrated and then horizontally averaged. The EKE for smooth topography and small 𝜃 is ≈ 10−5 [m3 s−1 ]. Because the dashed blackand the dashed red lines are indistinguishably close to each other, a small displacement is added to make both lines noticeable. The standing 𝐸𝐾𝐸 for smooth topography (bluedashed line) is zero. (Cases SB1-10 and IB1-20 in Table 1).

Fig. 5. Total (black), standing (blue), and transient (red) 𝐸𝐾𝐸 components [m3 s−1 ] for smooth (dashed lines) and irregular (solid lines) planar beach topographies as a functionof offshore wave incident angles. 𝐸𝐾𝐸 is depth-integrated and then horizontally averaged. The EKE for smooth topography and small 𝜃 is ≈ 10−5 [m3 s−1 ]. Because the dashedblack and the dashed red lines are indistinguishably close to each other, a small displacement is added to make both lines noticeable. The standing 𝐸𝐾𝐸 for smooth topography(blue dashed line) is zero. (Cases SB1-10 and IB1-20 in Table 1).

spatially diffused on a planar beach, so the currents are broader andweaker, compared to the depth-induced breaking with smooth barred
topography. Thus, the kinetic energy generated due to depth-induced
breaking is also much smaller. The 𝐸𝐾𝐸 values for the irregular planar
beach topography is similar to the one for irregular barred topography
but somewhat weaker. Notice that the planar smooth topography does
not generate shear waves.
In all these solutions there is an important dependency on the bottom drag, both its functional form and the amplitude of its coefficient,and thus the drag parameterization is an important element of surf zone
modeling. We chose not to make this a focus of this paper; refer to
Uchiyama et al. (2009, Sec. 4) to learn more about the bottom-dragparameterization dependency.

3.2. Alongshore-averaged current
Previous studies have shown that the Stokes drift 𝐮𝑠𝑡 tends to cancel
the Eulerian velocity in the inner shelf (Fewings et al., 2008). However,this is not always true, resulting in a non-zero Lagrangian velocity,𝐮𝐿 = 𝐮 + 𝐮𝑠𝑡 .

(8)

Cross-shore profiles of the depth- and time- and alongshore-averaged
Eulerian and Lagrangian velocities and Stokes drift are in Fig. 6.
These include both irregular (solid lines) and smooth (dashed lines)
barred topographies. The Eulerian cross-shore velocities ⟨𝑢⟩̄ are close
to each other for all cases shown with the exception of SB40 andIB40 where 𝜃 = 40◦ , the maximum angle examined. Eulerian velocities
6Ç. Akan, J.C. McWilliams and Y. Uchiyama

Ocean Modelling 147 (2020) 101565

Fig. 6. Cross-shore profiles of depth-, time-, and alongshore-averaged velocities [m s−1 ]: (a) ⟨𝑢⟩,
̄ (b) ⟨𝑣⟩,
̄ (c) ⟨𝑢̄ 𝑠𝑡 ⟩, (d) ⟨𝑣̄ 𝑠𝑡 ⟩, (e) ⟨𝑢̄ 𝐿 ⟩, and (f) ⟨𝑣̄ 𝐿 ⟩. These are all for barred
topographies; the smooth topography cases have solid lines, and the irregular have dashed ones (Table 1). The nonzero ⟨𝑢𝐿 ⟩ is balanced by the eddy-induced velocity to have zero
total mean Lagrangian cross-shore transport (Section 3.5).

Fig. 7. Cross-shore sections of time- and alongshore averaged Lagrangian velocity in the cross-shore direction, ⟨𝑢̄ 𝑙 ⟩ [m s−1 ]. Smooth barred topography cases are in the left column
and irregular ones are in the right column (Table 1).

in the alongshore direction ⟨𝑣⟩
̄ differ significantly between irregularand smooth cases with 𝜃 < 10◦ , but then became more similar at
highly oblique incident angles. In the 𝑥-far field ⟨𝑣⟩
̄ decreases somewhat
slowly and has an appreciable amplitude for oblique waves. This is dueto the bottom-drag wave dissipation that in turn provides an alongwave acceleration of the current at the bottom (Uchiyama et al., 2010);
it persists as far offshore as the wave orbital velocity reaches to thebottom. We performed sensitivity studies (not shown) and found that7

Ç. Akan, J.C. McWilliams and Y. Uchiyama

Ocean Modelling 147 (2020) 101565

Fig. 8. Maps of time-averaged and depth-integrated horizontal Reynolds stress transient kinetic energy production 𝑃𝑀+𝑆→𝑇 [m3 s−3 ] for smooth (top panels) and irregular (bottom
panels) barred topographies. The upper-left case has no 𝑃𝑀+𝑆→𝑇 because it has no transient-eddy 𝐸𝐾𝐸 (Fig. 4).

above and just outside the bar. This primary overturning circulationis strongest for normal and slightly oblique angles in both smooth andirregular topographies. Near the bar there is a weak reverse overturning
circulation that is stronger with smooth topography and smaller 𝜃. In
the offshore region, the bottom flow is shoreward because of bottom
streaming associated with the shoreward waves (Section 3.2), with acompensating offshore flow above to achieve a zero depth integral.

the alongshore current in the offshore region is quite sensitive to theparticular functional form of this wave bottom-drag parameterization,even though the behavior around the surf zone is robust. This parameterization gives rise to ‘‘bottom streaming’’ currents. More research isneeded on whether its prescriptions are accurate.
Stokes drift in the cross-shore direction ⟨𝑢̄ 𝑠𝑡 ⟩ has the same order ofmagnitude as the mean Eulerian velocity ⟨𝑢⟩.
̄ However, the resulting⟨𝑢̄ 𝐿 ⟩ is almost as large in the smooth topography cases, but smaller
than the irregular ones. ⟨𝑣𝑠𝑡 ⟩ is relatively small though variable among
the cases, in particular between irregular and smooth topographies, so⟨𝑣̄ 𝐿 ⟩ ≈ ⟨𝑣⟩,
̄ which itself is quite different among the cases. Although
not shown, the mean Lagrangian cross-shore depth-integrated transport ⟨𝑈̄ 𝐿 ⟩, defined in (12), is essentially zero, as it should be in an
equilibrium state.
Cross-shore sections of ⟨𝑢̄ 𝐿 ⟩ for irregular and smooth barred topographies are in Fig. 7. The pattern of 𝑢𝐿 is dominated by the Eulerian
rip currents. In all cases the dominant pattern is a shoreward surface current accompanied by an offshore flow at depth (undertow)

3.3. Eddy-mean kinetic energy conversion
The overall energy cycle for surf currents is a kinetic source fromthe acceleration due to wave breaking and a sink primarily due tobottom drag, although lateral momentum diffusion also contributes.
There is an exchange with potential energy as well due to the freesurface elevation, but it is secondary compared to the kinetic energy
cycle. The mean currents can be unstable and convert mean to eddy
kinetic energy, and then the eddies can be an important route to total
current dissipation.
8Ç. Akan, J.C. McWilliams and Y. Uchiyama

Ocean Modelling 147 (2020) 101565

Fig. 9. Rip transport: alongshore-average of the offshore-directed parts of the time-averaged and depth-integrated cross-shore transport, (𝑈 𝑚 + 𝑈 𝑒 )+ [m2 s−1 ] for barred topographies:
smooth (solid lines) and dashed (dashed). Only the cases with angles of 𝜃 = 0◦ , 20◦ , 40◦ are included here (Table 1). Notice that the smooth cases SB0 and SB20 have very weak
offshore-directed mean flows.
Table 4
Horizontal Reynolds stress (HRS) (9)–(10) [m5 s−3 ] integrated over depth and area,
and averaged in time for smooth and irregular barred topographies.

To assess the energy conversion from time-mean kinetic energy
to transient 𝐸𝐾𝐸, we calculate the horizontal shear production term
𝑃𝑀+𝑆→𝑇 , whose integrand is the product of eddy Reynolds stress andmean horizontal shear,
𝜕 𝑢̄
𝜕 𝑣̄
𝜕 𝑣̄
𝜕 𝑢̄
𝑃𝑀+𝑆→𝑇 = − 𝑢′ 𝑢′− 𝑢′ 𝑣′− 𝑣′ 𝑣′− 𝑢 ′ 𝑣′
,
(9)
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦

𝑃𝑀+𝑆→𝑇
𝑃𝑀→𝑆+𝑇
𝑃𝑀→𝑆
𝑃𝑆→𝑇

with a further integration over the depth [m3 s−3 ]. The subscript
notation for 𝑃 in (9) is based on the energy decomposition implied in(7): it is a conversion from the total time-averaged flow (i.e., the sumof the alongshore mean (M) and standing-eddy (S) components) to thetransient-eddy (T) component. A positive sign of 𝑃 means kinetic energy conversion from the time-mean flow to the transient fluctuations,
i.e., transient eddy generation. A two-signed pattern indicates mixed
eddy-mean energy conversion with an implication of spatial transport
connecting the two regions. The horizontal average of 𝑃 determines thenet conversion [m5 s−3 ] (Table 4).
The 𝑃𝑀+𝑆→𝑇 term for different cases is shown in Fig. 8. Recall
from Fig. 1, which shows the horizontal view of the domains used,that for this particular irregular topographic realization there are three
rip channels with varying widths located at around 80 m, 200 m and
600 m, respectively. The largest 𝑃𝑀+𝑆→𝑇 conversion coincides withthese channels (Fig. 8d) at 𝜃 = 0◦ . As anticipated, with rip currents
there are regions of mixed sign for 𝑃 . As the incident wave angle
increases, the magnitude of the 𝑃𝑀+𝑆→𝑇 decreases, although it remains
significant for all 𝜃. In contrast, with smooth topography 𝑃𝑀+𝑆→𝑇 isnearly zero for normal-incident waves and it increases with 𝜃. 𝑃𝑀+𝑆→𝑇
reaches a similar magnitude at 𝜃 = 40◦ as in the irregular cases, eventhough its total 𝐸𝐾𝐸 is less (Fig. 4). This is the expected signature oftransient shear instability of the alongshore current. It is nearly uniform
alongshore. In all cases 𝑃 is mostly confined to the bar region, although
it does reach inside the bar as well in the case with normal incidence
over irregular topography. After taking a 𝑦-average, 𝑃𝑀+𝑆→𝑇 is positive
for almost all 𝑥 values in all cases; this is reflected in the positivearea-integrated values in the first row of Table 4.
We can also consider other mean-eddy conversions as defined inthese alternative shear production terms:
𝜕⟨𝑣⟩
̄
𝜕⟨𝑢⟩
̄
− 𝑢𝑠 𝑣𝑠
𝜕𝑥
𝜕𝑥
𝜕⟨𝑢⟩
̄
𝜕⟨𝑣⟩
̄
𝑃𝑀→𝑆+𝑇 = − (𝑢′ 𝑢′ + 𝑢𝑠 𝑢𝑠 )− (𝑢′ 𝑣′ + 𝑢𝑠 𝑣𝑠 )
𝜕𝑥
𝜕𝑥
𝜕𝑢𝑠
𝜕𝑣𝑠
𝜕𝑣𝑠
𝜕𝑢𝑠
𝑃𝑆→𝑇 = − 𝑢′ 𝑢′− 𝑢 ′ 𝑣′− 𝑣′ 𝑣′− 𝑢′ 𝑣′
,
𝜕𝑥
𝜕𝑥
𝜕𝑦
𝜕𝑦

SB0 (0◦ )SB20 (20◦ )SB40 (40◦ )IB0 (0◦ )IB20 (20◦ )IB40 (40◦ )

0.027
0.027
4.02 × 10−6
−1.74 × 10−6

13.40
13.40
0.0009−0.005

27.38
27.70
0.53
0.31

22.65
−9.73
−10.72
21.66

4.17
2.77
1.20
2.57

4.93
19.49
16.58
2.03

• For normal incident waves and smooth topography, there is little
instability and little 𝐸𝐾𝐸.
• For normal incident waves and irregular topography, the standing
eddies 𝑆 are unstable and transfer energy to both the alongshore
mean flow 𝑀 and the transient eddies 𝑇 .
• For oblique waves and smooth topography, the instability is from𝑀 to 𝑇 with little exchange with 𝑆.
• For oblique waves and irregular topography, 𝑀 is strongly unstable to 𝑆, and the conversions to 𝑇 are moderate.
3.4. Wave energy
Because of CEW the wave field is modulated by interactions with thecurrent as well as with the variable topography. Previous rip-current Yu
and Slinn (2003) and Weir et al. (2011) show how CEW weakens therip currents and their instability especially for smaller values of 𝜃 bycomparing with solutions in which CEW is neglected. Here all solutions
include CEW through full wave–current coupling (Section 2.1). The
results discussed below are not shown since the all the cases (smoothand irregular barred topography cases) exhibit similar behavior.
We first discuss the alongshore- and time-averaged sea-level andsignificant wave height. Crossing the bar the waves begin to break anddissipate, and the sea-level rises shoreward as wave setup (LonguetHiggins and Stewart, 1962). There is a modest indication of wave
setdown and amplification just offshore of the bar. A secondary roundof breaking and setup occurs close to the shore at 𝑥 ≈ 1 m depth. In
a broad view all cases have similar behavior, although the irregular
topographies induce somewhat greater dissipation and setup.
Next, we discuss the wave energy for all the smooth and irregularbarred topographies. Wave energy is calculated from the linear wave
theory at leading order in wave steepness. The time averaged wave
energy for all barred topography cases shows a broad pattern of 𝐸𝑤
decreasing shoreward from the bar. There is also evident modulationof the waves in the irregular topography cases. For 𝜃 ≥ 20◦ , themagnitude of 𝐸𝑤 is similar for both smooth and irregular cases. For

𝑃𝑀→𝑆 = − 𝑢𝑠 𝑢𝑠(10)whose meanings are evident in their different subscripts.
The total magnitudes of these various shear production terms arelisted in Table 4. Using the language of horizontal shear instability, they
indicate the following energy pathways:
9Ç. Akan, J.C. McWilliams and Y. Uchiyama

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Fig. 10. Maps of time-averaged and depth-integrated transport components [m2 s−1 ]: eddy-induced 𝐔𝑒 (left panels), mean 𝐔𝑚 (middle panels), and total Lagrangian mean 𝐔𝐿 (rightpanels) for irregular barred topography with 𝜃 = 0◦ (case IB0; top panels) and 𝜃 = 40◦ (case R20; bottom panels). Arrows denote transport vectors, and colors indicate magnitudes.

𝜃 < 20◦ , highest wave energy is seen where the rip channels arelocated (i.e., in this ℎ′ realization, at ≈ 80 m, 200 m, and 600 m in thealongshore and ≈ 120 m in the cross-shore direction). These are sites ofwave amplification and enhanced dissipation partly through the CEW
associated with the offshore rip currents there.

where the first right-side term is the mean eddy-induced Euleriantransport, the second term is the Eulerian transport by the mean flow,
and the last term is the Stokes transport. The second right-side linein (12) is a more compact notation for the terms in the first line.
Notice that the occurrence of surf eddies allows for there to be ‘‘eddy’’
contributions to eddy-induced mean flow, the familiar one from surface
gravity waves, 𝐔̄ 𝑠𝑡 , and another from the rectified transport by surf
eddies, 𝐔̄ 𝑒 .
In the context of oceanic mesoscale eddy material fluxes, theisopycnal-layer thickness-weighted eddy-induced transports are a significant element in general circulation dynamics (Gent et al., 1995).
The dynamical processes that generate these different types of eddyinduced transport are different, of course, and as yet there is no clear
explanation for this surf eddy manifestation.3

3.5. Horizontal transport
The Lagrangian horizontal volume transport (or mass flux when
multiplied by 𝜌0 in this Boussinesq model) is defined as
𝐔𝐿 =

𝜂𝑐

∫−ℎ

( 𝐮 + 𝐮𝑠𝑡 ) 𝑑𝑧 = 𝐻𝐮𝑏𝑡 + 𝐔𝑠𝑡 .

(11)

𝐮𝑏𝑡 is the depth-averaged (barotropic) velocity, and 𝐔𝑠𝑡 is the depthintegrated Stokes transport (Uchiyama et al., 2010). Whentime-averaged, the Lagrangian mean transport is3
It is well known that shallow-water gravity waves generate a non-zero
Stokes transport, but in our incident-wave-averaged ROMS-WEC model, thesolutions in our present cases exhibit relatively weak infra-gravity wave
behavior, so here 𝐔𝑒 ≠ 0 is due mainly to surf eddy dynamics.

𝐔𝐿 = 𝐻 ′ 𝐮′𝑏𝑡 + 𝐻̄ 𝐮̄ 𝑏𝑡 + 𝐔̄ 𝑠𝑡=𝐔̄ 𝑒+ 𝐔̄ 𝑚+ 𝐔̄ 𝑠𝑡 ,(12)10Ç. Akan, J.C. McWilliams and Y. Uchiyama

Ocean Modelling 147 (2020) 101565

Fig. 11. Instantaneous surface vertical vorticity 𝜁𝑧𝑠 (𝑥, 𝑦) [𝑠−1 ] in the headland cases with irregular topography and incident wave angles of (a) 𝜃 = 0◦ (case H0), (b) 𝜃 = 5◦ (caseH5), (c) 𝜃 = 20◦ (case H20), and (d) 𝜃 = 40◦ (case H40) at the 24th hour of the simulation.

Following Yu and Slinn (2003), we examine the transport in the𝑥-direction associated with the rip currents as the alongshore averageof the positive portion of the depth- and time-averaged (𝑈 𝑚 + 𝑈 𝑒 )+which is shown in Fig. 9.4 There are striking differences between thecases. First, the net rip transport is quite small for all the smoothbarred topography cases, even though it can be locally large due totransient surf eddies. With irregular topography it is much larger, butsomewhat it decreases with 𝜃, especially in comparing normal- andoblique-incident cases. This is consistent with decreasing 𝐸𝐾𝐸 with 𝜃(Fig. 4). Yu and Slinn (2003) report a similar pattern in their simulations with uniformly-spaced rip-channel topography. In our cases, we
generated topographies that are random in the alongshore direction.
Here, we see an across-shore transport that is much stronger andextends much further than in their simulations. This indicates that
irregular topography is significant in transporting materials offshore.
However, it should be also noted that the present study does not take
into account the type of wave breaking (spilling/plunging) which also
impacts the direction of transport of materials in the surf zone.
The components of mean transport in (12) are shown in Fig. 10 for
irregular barred topography. This shows an eddy-induced transport directed primarily alongshore in the surf zone, with a direction-reversing
pattern in 𝑦. For normal-incident waves, the eddy and mean transportcomponents show some recirculations (observe the direction of thetransport vectors). On the other hand, for oblique waves the eddy andmean transport components are aligned along the shore. The magnitudeof 𝐔̄ 𝑒 is smaller than 𝐔̄ 𝑚 ; where it is largest in the surf zone, its
peak value is only 5%–10% as large. For smooth barred topography(not shown), 𝐔̄ 𝑚 is very small for normal waves (because there is
4little transient eddy activity), and it has a comparable relative sizefor oblique waves. For oblique waves the total transport offshore issouthward in the direction of the waves. The transport due to the Stokes
drift is comparable in magnitude to the eddy-induced transport in theseirregular topography cases with a straight coastline, but with a very
different spatial pattern (n.b., Fig. 6).
In an additional alongshore average, we can further decompose themean transport by separating standing and transient eddies,̄ ⟨̄𝐮𝑏𝑡 ⟩ + ⟨𝐔̄ 𝑠𝑡 ⟩⟨𝐔𝐿 ⟩ = ⟨𝐻 ′ 𝐮′𝑏𝑡 ⟩ + ⟨𝐻 𝑠 𝐮𝑠𝑏𝑡 ⟩ + ⟨𝐻⟩= ⟨𝐔𝑇 𝐸 ⟩ + ⟨𝐔𝑆𝐸 ⟩ + ⟨𝐔𝑀 ⟩ + ⟨𝐔̄ 𝑠𝑡 ⟩ ,(13)where the definitions should be evident from the symbols. For the same
solutions as in Fig. 9 with irregular topography, ⟨𝐔𝑇 𝐸 ⟩ is again about
4% of ⟨𝐔𝑀 ⟩ for normal-incident waves, while the relative magnitudeof ⟨𝐔𝑆𝐸 ⟩ is close to 35%. For oblique waves ⟨𝐔𝑇 𝐸 ⟩ and ⟨𝐔𝑆𝐸 ⟩ are eachabout 10% as large as ⟨𝐔𝑀 ⟩ within the surf zone (inner 100 m in 𝑥), but
smaller farther offshore. Thus, the surf-eddy-induced mean transports
are non-trivial in these cases, although the primary transport is by themean currents.
4. Effects of coastline shape
The usual practice in idealized rip-current studies is to have astraight coastline, and as far as we know most field measurements inthe surf have been measured for a similar geometry. In this section
we examine the influence of a variable shoreline on generating andinteracting with rip currents. As explained in Section 2, the topographies of the headland and bay cases are sloped downward and seaward
with superimposed random depth perturbations; i.e., there are no bars
in these cases. In this section we do not consider alongshore averages
because of the inhomogeneity in this direction.

Subscript ‘‘+’’ denotes average of the positive portion.
11Ç. Akan, J.C. McWilliams and Y. Uchiyama

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Fig. 12. Maps of time-averaged and depth-integrated transport components [m2 s−1 ]: eddy-induced 𝐔𝑒 (left panels), mean-current 𝐔𝑚 (middle panels), and total-Lagrangian 𝐔𝐿(right panels) for a headland with irregular topography and with 𝜃 = 0◦ (case H01; top panels) or 𝜃 = 40◦ (case H40; bottom panels). Arrows denote transport vectors and colors
indicate magnitudes.

the headland, the transport is negative (i.e., directed towards shore),
and away from the headlands there is a compensating offshore flow.
The relative size of 𝐔𝑒 compared to 𝐔𝑚 is larger in this headland casethan with a straight coastline (Section 4), i.e., as large as about 50%(based on the maximum in one computational cell) locally within thesurf zone. When the waves are obliquely incident (bottom panels), thetransport is mainly directed along the incident wave direction towards
south, with an especially strong separating flow at the headland tip.
In this case the eddy-induced flows are positioned differently and aresomewhat weaker compared to the mean-current transport.

4.1. Headlands
The surface vertical vorticity around the headland for different
incident waveangles at the 24th hour of the simulation is illustrated inFig. 11. Highly energetic small-scale surf eddies are generated along thestraight sectors of the coastline with normal and nearly normal-incident
wave angles. They persist for all incident angles but weaken andcontract shoreward with larger 𝜃. The surf eddies on both sides of theheadland are suppressed. In addition, for 𝜃 ≥ 20◦ , a strong alongshore
current develops on the wave-exposed side of the headland, with strong
nearshore negative surface vertical vorticity due to increased bottom
drag in shallow water. This flow separates at the headland, leaving thewave-sheltered side relatively inactive (not evident from the figures).
Because it is a wave-driven current shaped by the irregular domain
geometry, we refer to it as a mega-rip current. It extends seaward far
beyond the surf zone where the waves break.
To understand the transport around the headland, we again utilize
the transport decomposition (12). Components of the transport andtheir sum at different incident wave angles are shown in Fig. 12. When
the waves are normally incident (top panels), in addition to the local
irregular-topography standing eddies in the surf zone, we see two large
circulating patterns in the regions next to the headland. Right around4.2. Bays
Fig. 13 is an analogous illustration of the surface vertical vorticity
around the bay for different incident wave angles at the 24th hour ofthe simulations. Along the straight sectors of the coast the behavior of
the small surf eddies is quite similar to that for the headland (Fig. 11).
A striking feature in these cases is the mega-rips extending far offshore.
Around the bay the mega-rip pattern is broadly a dipole with inflows
at the bay edges and a central outflow. The dipole is approximately
symmetric with normal incident waves, but it is tilted alongshore withoblique waves in the wave direction. The offshore-directed flow from12Ç. Akan, J.C. McWilliams and Y. Uchiyama

Ocean Modelling 147 (2020) 101565

Fig. 13. Instantaneous surface vertical vorticity 𝜁𝑧𝑠 (𝑥, 𝑦) [𝑠−1 ] in the bay cases with irregular topography and incident wave angles of (a) 𝜃 = 0◦ (case B0), (b) 𝜃 = 5◦ (B5), (c)𝜃 = 20◦ (B20), and (d) 𝜃 = 40◦ (B40) at the 24th hour of the simulation.

5. Summary and discussion

the bay center is shear-unstable and tends to meander around its primary axis. Similar mega-rips were observed around a submarine canyon
off La Jolla, CA, near the Scripps Research Pier, which is much deeper
than our idealized bay (30–40 m); Long and Özkan Haller (2005)
studied the mega-rip generated at this location and reported that rip
current generation was very sensitive to the topographic irregularities
rather than presence of the canyon. We did not repeat our simulation
without the ℎ perturbations, but based on the results in the previous
section, we would expect a similar result that irregularness enhances
variability here. The mega-rip interacts with the smaller rip currents forall of the incident wave angles. This implies that materials transportedby the smaller rip currents would be transported even further offshore
through this interaction with the mega-rip circulation.
Fig. 14 shows the transport around the bay for different incident
wave angles. The eddy-induced flow patterns generated due to thebay are stronger, more complicated, and extend farther offshore thanin the headland cases, but again its magnitude relative to the meancurrent transport is larger than with only a straight coastline (Fig. 12).
The transport direction is affected significantly because of the megarip. The mean-current and total-Lagrangian alongshore transports areenhanced by the presence of the bay and are strongest while exiting on
the down-wave side.
In the headland and bay cases with relatively larger eddy-induced
transports compared to straight coastline cases (Fig. 12), the Stokes
transport is smaller than the eddy-induced transport. Thus, variations
in coastline shape have an important influence on wave-driven currents
and surf eddies. The variety of shapes in nature is large, and here we
only show two examples to illustrate the influences.

In this paper we present results from 52 idealized cases for wavedriven surf currents over various topographies and coastline shapes
using the ROMS-WEC model (Uchiyama et al., 2010). The cases are
divided into two case groups where, in the first group, we investigate
topographic control on transient rip currents by introducing irregularities to commonly used alongshore-uniform topographies; in thesecond group idealized headland and bay domains with topographic
irregularities are examined for their influences on transient rip currents.
In both groups the role of the incident wave angle is examined, while
the other incident wave properties are held constant.
The results from the first group show that if the topography in thealongshore direction has irregular undulations, instabilities are triggered even at normally incident wave angles. With increasing incident
angle, shear instabilities dominate over topographic instabilities, andthe surf eddies transition from being mostly transient to being mostly
steady. The surf 𝐸𝐾𝐸 is enhanced by the presence of irregular topography. This enhancement is greatest for nearly normal incidence, whichis the attracting direction for depth-induced refraction over shoaling
bathymetry.
The second-group cases with idealized headland and bay configurations reveal how the flow field changes when the coastline shapechanges. The headland acts like an obstruction especially at highincident wave angles, and we observe flow separation followed byoffshore wake instabilities. The bay, however, acts like a large ripchannel generating transient mega-rips with large-scale recirculating
flow patterns that intrude out from the coast several kilometers offshore
when the waves are normally incident.
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Ocean Modelling 147 (2020) 101565

Fig. 14. Maps of time-averaged and depth-integrated transport components [m2 s−1 ]: eddy-induced 𝐔𝑒 (left panels), mean-current 𝐔𝑚 (middle panels), and total-Lagrangian 𝐔𝐿(right panels) for a headland with irregular topography and with 𝜃 = 0◦ (case B0; top panels) or 𝜃 = 40◦ (case B40; bottom panels). Arrows denote transport vectors and colors
indicate magnitudes.

In flows with significant transient eddies, an eddy-induced meantransport often occurs and can be large in relation, e.g.,to the mean
Stokes transport. It also often provides a fractional opposition to themean-current transport that dominates the total Lagrangian transport.
The eddy-induced transport is enhanced by irregular topography, just
as is the 𝐸𝐾𝐸.
These eddy-induced transports are modest fractions of the meancurrent and total transports in the idealized cases with straight coastline
and relatively small-scale topographic irregularities, and they are relatively stronger with coastline shape variations. Nevertheless, they dodemonstrate the potential importance of surf turbulence in providinga Lagrangian-mean advective material transport as well as the morefamiliar behaviors of eddy-induced dispersion and dilution of material
concentrations.
The results of this study demonstrate that topographic and coastline
variations are significant influences on surf eddy activity. Thus, for realistic surf behaviors it is important to use either measured bathymetry,
if available, or at least include a plausible degree of irregularity andcoastline variation in the simulation. Beyond the steady wave forcing
considered here, both incident wave variability and other forms ofshelf-current variability are also necessary ingredients for realistic surf
eddy simulations.
Finally, it should be noted that the currents in the surf zone canalso further be influenced by tidal forces, large (oceanic) scale thermohaline differences and river discharges. Moreover, truly irregular wave

fields with special dynamics (𝐸𝐾𝐸, surface vertical vorticity, velocity
fields) induced by realistic random wave trains could be achieved
only by coupling ROMS with proper wave models (3rd generation
or Boussinesq-type) and CFD approaches, so that other crucial wave
processes can be incorporated in simulations, such as wave triad interactions, wave reflections on the coast, surf beat, cross-shore wave
dynamics, wave run-up/run-down on the coastal boundary.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared toinfluence the work reported in this paper.
CRediT authorship contribution statementÇiğdem Akan: Conceptualization, Methodology, Software, Data curation, Writing - original draft, Visualization, Investigation. James
C. McWilliams: Supervision, Writing- reviewing & editing. Yusuke
Uchiyama: Writing - reviewing & editing.
Acknowledgments
This research is supported by the National Science Foundation(OCE-1355970) and the Office of Naval Research (N00014-14-1-0626
14

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Ocean Modelling 147 (2020) 101565

Fig. A.1. Alongshore averaged (a–d) and horizontal (f–h) views of different realizations of the barred topography generated by the method described in Section 2.2. Case v1 isthe same as the irregular barred topography used in cases IB0 through IB40. Cases v1–v3 are different random realizations with the same bar profile, and case v4 is a realization
with a bar whose top is deeper.

and N00014-15-1-2645), as well as by the Extreme Science and Engineering Discovery Environment (XSEDE) computational resources. YU
receives financial support for a sabbatical visit to UCLA from the JSPS
Grants-in-Aid for Scientific Research (Grant Numbers: 15KK0207 and
15H04049) at Kobe University.
Appendix A. Sensitivity to random realizations
We follow the procedure explained in Section 2.2 and generate
two more random realizations of the topographic perturbations ℎ′ (𝑥, 𝑦),which then are superimposed on the alongshore-uniform barred topography (Fig. A.1). Time histories of the instantaneous eddy kinetic
energy (i.e., 𝐸𝐾𝐸(𝑡), calculated using the eddy velocity 𝐮 − ⟨𝐮⟩) are
shown in Fig. A.2 for these three realizations: v1 is the ‘‘control’’ case(SB0), and v2 and v3 are two other random realizations. Otherwise,
all parameters for the three cases are the same. (Notice that, v4 is
not included in Fig. A.2 and will be discussed in Appendix B.) The
𝐸𝐾𝐸(𝑡) varies over a 50% range, but the different cases’ curves overlap
substantially with each other and have a similar time average. Thisimplies that repeating the simulations given in Table 1 with different
realizations would yield in similar statistical results. Because the sizeand location of the rip channels are different in each case, the detailed
pattern of the rip currents would be different, but the results leading
to our conclusions would be essentially similar.

Fig. A.2. Time series of 𝐸𝐾𝐸 for different realizations of the barred topographies incases v1–v3 in Fig. A.1. The incident angle is 𝜃 = 0◦ .

Appendix C. Sensitivity to roughness factor 𝒔𝟎
Here we vary 𝑠0 between 1 and 0, where 1 yields the full perturbation amplitude in (5), to assess how the topographic irregularness
amplitude controls the surf eddy activity. As expected and consistent
with the results in Uchiyama et al. (2017), smaller perturbation size
led to lower 𝐸𝐾𝐸 and therefore less energetic flow field with lessinstabilities.

Appendix B. Sensitivity to bar height

Appendix D. Sensitivity to wave amplitude 𝒂

We examine the time series of 𝐸𝐾𝐸 components (standing andtransient) for different barred topography realizations. We use fourcases where when alongshore averaging applied, the height of the barin v1, v2 and v3 is the same whereas the bar top is deepened in v4.
The components of the 𝐸𝐾𝐸 and its total decline when the bar top is
deepened. However, the transient 𝐸𝐾𝐸 component is noticeably larger
for large 𝜃 with the deepened bar and relatively weak transient eddies
are observed at large incident angle.

Now we vary the offshore wave amplitude 𝑎 and plot the resulting
𝐸𝐾𝐸 values (Fig. D.1. For small 𝑎 the rate of increase in 𝐸𝐾𝐸 is large,
but there is a saturation effect for larger 𝑎, i.e., increased incident wave
energy, hence increased wave dissipation and current acceleration,lead to increased current dissipation such that 𝐸𝐾𝐸 does not increase
further.
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Ocean Modelling 147 (2020) 101565

Fig. D.1. 𝐸𝐾𝐸 for different incident wave amplitudes. These are IB0-type cases withbarred irregular topography and 𝜃 = 0◦ .

Fig. F.1. Time and alongshore averaged and depth-integrated eddy-mean induced
transport, 𝐔̄ 𝑒 , (solid lines) and depth-, time, and alongshore-averaged Stokes drift, ⟨𝑢̄ 𝑠𝑡 ⟩,(dashed lines) for varying wave amplitudes for irregular barred topography.

For 𝑎 ≥ 0.75 the peak values of 𝐔̄ 𝑒 are indistinguishable whereas
mean Stokes drift keeps increasing with increasing wave amplitude.

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Fig. E.1. Time series of depth-integrated 𝐸𝐾𝐸 [m3 s−1 ] for different values of lateraleddy viscosity in cases with barred irregular topography and 𝜃 = 0◦ or 40◦ .

Appendix E. Sensitivity to horizontal viscosity 𝝂
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momentum eddy viscosity of 𝜈 = 1 m2 s−1 and the alternative valueof zero. In both cases the third-order upstream horizontal advection
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