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J. Fluid Mech. (2018), vol. 850, R2, doi:10.1017/jfm.2018.510

journals.cambridge.org/rapids

Robustness of nearshore vortices
James C. McWilliams1, †, Cigdem Akan2 and Yusuke Uchiyama3
1 Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles,

CA 90095-1565, USA
2 College of Computing, Engineering and Construction, University of North Florida, Jacksonville,
FL, USA
3 Department of Civil Engineering, Kobe University, Kobe, Japan(Received 21 December 2017; revised 14 May 2018; accepted 19 June 2018)

Coherent vortices with horizontal swirl arise spontaneously in the wave-driven
nearshore surf zone. Here, a demonstration is made of the much greater robustness ofcoherent barotropic dipole vortices on a sloping beach in a 2D shallow-water modelcompared with fully 3D models either without or with stable density stratification.
The explanation is that active vortex tilting and stretching or instability in 3D disrupt
an initially barotropic dipole vortex. Without stratification in 3D, the vorticity retains
a dipole envelope structure but is internally fragmented. With stratification in 3D, thedisrupted vortex reforms as a coherent but weaker surface-intensified baroclinic dipole
vortex. An implication is that barotropic or depth-integrated dynamical models of thewave-driven surf zone misrepresent an important aspect of surf-eddy behaviour.
Key words: topographic effects, vortex dynamics, vortex flows

1. Introduction
The nearshore littoral zone in the ocean has a unique fluid dynamics. Incident
surface gravity waves generated by marine winds offshore break with depth shoaling,
and the momentum they lose through breaking is transferred to a substantial
degree into wave-averaged alongshore currents and cross-shore sea-level andpressure gradients (often called wave set-up) (Bowen, Inman & Simmons 1968;
Longuet-Higgins 1970). If there are alongshore bathymetric variations, the alongshore
currents can develop stationary recirculation patterns called rip currents. Furthermore,these alongshore and rip currents often exhibit a horizontal-shear instability that leads
to a fully developed littoral turbulence comprising transient horizontal-shear waves
and horizontally recirculating surf vortices with vertical vorticity (figure 1) (e.g.
Oltman-Shay, Howd & Birkemeier 1989; Haller & Dalrymple 2001; Marchesielloet al. 2015). This is often referred to as very-low-frequency (VLF) variability with† Email address for correspondence: jcm@atmos.ucla.edu
c Cambridge University Press 2018

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

0

100 m

13.8

15.7

F IGURE 1. Infrared image of surface temperature showing a surf vortex off Huntington
Beach, CA (Marmorino, Smith & Miller 2013). The wave swash zone is at the top of theimage and deeper water is towards the bottom. The vortex swirl pattern is also evident inan accompanying visible light image, where suspended sand is approximately coincident
with the colder temperatures here. The crenellated edges of the swirl pattern are suggestiveof an incipient microscopic shear instability, and its amplitude is also seen to amplify insuccessive images over the next tens of minutes.

periods of the order of 10 min. Variations in the incident waves (i.e. waves in groups
or with finite alongshore length) provide a further source of transient vertical vorticity
that further feeds into the littoral turbulence and vortices (Feddersen 2014).
There is a substantial literature on surf-zone phenomena and their modelling(n.b. the review in Dalyrimple et al. (2011)). With respect to the ‘macroscopic’currents (i.e. with a scale comparable to the surf-zone width and averaged over thewaves), much of the modelling has been performed with wave- and depth-averaged
models in two horizontal dimensions (2D), i.e. essentially the shallow-water equations
with appropriate parameterizations for the forcing by breaking waves and the damping
by bottom-boundary drag, and sometimes with additional conservative wave-averaged
dynamical effects (McWilliams, Restrepo & Lane 2004). This formulation also
excludes the influence of stable density stratification that is often present justoutside the breaking zone. Two-dimensional models have seemed satisfactory formany surf phenomena, including many with observational validations. More recently,
however, 3D wave-averaged models have begun to be been deployed for the surf zone
(Newberger & Allen 2007; Uchiyama, McWilliams & Shchepetkin 2010; Kumar et al.
2012; Marchesiello et al. 2015), with various quantitative differences in their solutions
compared with 2D models. In particular, in the regime of wave-driven littoral currents
with spontaneously arising surf vortices, the 2D vortices seem to be longer-lived andless deformed than the 3D vortices; i.e. the 2D vortices are more robust (Uchiyama,
McWilliams & Akan 2017). There are, of course, also active ‘microscopic’ currents
in the surf zone, and they certainly require fully 3D non-hydrostatic models (Kirby& Derakhti 2018).
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Nearshore vortices
The purpose of this paper is to test nearshore vortex robustness in 2D and 3D, as
well as in 3D with density stratification. For simplicity and a focus on the vortex
behaviour by itself, wave effects are not included here. While this idealized problem
is only one aspect of the surf-zone phenomena described above, it is relevant bothto the nearshore dispersion of material concentrations (deemed to be dominated by
the horizontal vortical flows in Spydell & Feddersen (2009)) and to the question ofwhich aspects of the surf-zone and inner-shelf (lying outside the wave-breaking zone)
dynamics are well characterized with a depth-averaged model and which require afully 3D model.
The model is the hydrostatic Regional Oceanic Modeling System (ROMS;
Shchepetkin & McWilliams 2009; Uchiyama et al. 2010), which is generally
usable for realistic simulations of macroscopic shelf and surf currents under the
influences of many contributing processes. The idealized problem has an initial pair of
opposite-signed vortices (i.e. a dipole) located in shallow water on a uniformly sloping
beach and oriented so that their mutual advective tendency induces propagation into
deeper water. The resulting depth change in turn induces change in their strengthand shape, and thus provides a mildly disruptive perturbation (mimicking, e.g.,
the disruption of surf vortices in littoral turbulence). The robustness question is
how the dipole responds to this perturbation. To isolate the essential dynamics, no
incident surface gravity wave interaction effects are included here, apart from the longhydrostatic (i.e. shallow-water) gravity waves associated with the variable free surface
coupled to the depth-averaged currents in the ROMS. Thus, the problem here is more
precisely posed as vortices in the inner shelf (i.e. without waves), and as such is bothincomplete with respect to surf vortices and more generally relevant to inner-shelf
vortices generated by other processes. Dipole vortices have been extensively studiedin other contexts (some cited in § 3), although as yet without the combination ofslope and stratification. Our interest is not in dipole vortices per se, but as a meansof illustrating the robustness question.
2. Problem posing
The geometry of the initial conditions and domain is sketched in figure 2. Here,h is the inshore minimum depth, which we choose to be 0.25 m; H is a variable
outer depth that controls the steepness of the bottom slope; we have obtained a setof solutions with H = 5, 10 and 20 m at an offshore distance of Lx = 600 m. The
alongshore width is Ly = 200 m. For each H, a set of three alternative model solutions
is obtained: shallow water (case 2DU, with the U denoting unstratified), 3D with nodensity stratification (case 3DU) and 3D with a linear density stratification (case 3DS).
For the 3DS case, the surface temperature T = 21 ◦ C and the bottom √temperature T =20.4 ◦ C at x = Lx ; e.g. the value of the buoyancy frequency is N = αg∂z T = 1.8 ×
10−2 s−1 when H = 10 m (α = 2 × 10−4 C−1 is the thermal expansion coefficient for
sea water).
All solutions start with the same initial condition for the flow, i.e. a pair ofbarotropic vortices (figure 2). Their currents are independent of depth, and hence
consistent with an initial 2D flow. Their vorticity centres are located at x = 100 m
offshore and mutually separated by 1y = 10 m in the alongshore centre of thedomain. The initial vertical vorticity field is a superposition of individual axisymmetricmonopoles of opposite sign,ζ z (r) =ζ0,(1 + (r/r0 )2 )2(2.1)
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J. C. McWilliams, C. Akan and Y. Uchiyama

h

600

m

H

F IGURE 2. Sketch of the initial configuration for a barotropic vortex dipole aimed offshore
in shallow water over a uniformly sloping bathymetry. Here, h = 0.25 m and H are
the resting depths at cross-shore locations of x = 0 and 600 m. (Offshore, alongshore) =(x, y) = (east, north) coordinates respectively.

with pamplitude ζ0 = ±0.9 × 10−2 s−1 and radial size r0 = 10 m. The coordinater = (x − xc )2 + (y − yc )2 is the radial distance from the vortex centre located at(xc , yc ). The accompanying azimuthal velocity profile V(r) and surface elevation η(r)(as determined by cyclostrophic balance) are

1
v0 r0
r0 ζ0 z
ζ = ∂r [rV] ⇒ V(r) =, v0 =,rr(1 + (r/r0 )2 )
2 
(2.2)
r02 ζ02 
η0
V2


,
η
=
−
⇒ η(r) =.
−g∂r η = −0r
1 + (r/r0 )28g
From the preceding values for ζ0 and r0 , accompanying values here are a swirl velocity
v0 = 0.05 m s−1 and a surface elevation η0 = 10−4 m. Both are somewhat weaker thanis likely to be true for the situation in figure 1, but we would expect an approximate
invariance of the advectively controlled behaviour with proportionally increased ζ0and evolutionary rate, and have confirmed this expectation with a modest survey ofsolutions with different r0 and ζ0 values. Such a monopole vortex by itself is aninviscid stationary state over a flat bottom, and it is linearly stable with respect to
2D horizontal-shear instability because it has no inflection point.
The nearby superposition of these two vortices is not necessarily a stableconfiguration, although our solutions in § 3 do not show any obvious patterns of
2D shear instability. The vortex with positive ζ0 – by convention called cyclonicbecause it has the same sign as the Coriolis frequency f = 10−4 s−1 – is located to
the north (yc > 0) of the negative (anticyclonic) vortex (see the arrows in figure 2).
This orientation gives the dipole an x propagation tendency in the eastward offshore
direction towards deeper water, with a behaviour similar to the uniformly propagating
strictly 2D dipole solution in (Lamb 1932, § 155). The choice of the monopole profilesin (2.1)–(2.2) is somewhat arbitrary, and no special care is taken in modifying theirshape in forming the dipole (n.b., in Lamb’s dipole there is an elongation of thevorticity isolines perpendicular to the centre separation line). As a consequence, there
is an early evolutionary period of cyclostrophic adjustment, as the two monopoles
slightly modify their ζ z and η shapes into a more slowly evolving propagating dipole
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Nearshore vortices
vortex. During this early period, shallow-water gravity waves are radiated away fromthe adjusting vortex.
Conservation of potential vorticity in its rotating shallow-water approximation is
Dqsw
≈ 0,
Dt

qsw =

f + ζz ζz
≈ .
h+η
h

(2.3)

This implies that as the dipole moves into deeper water (larger h), carrying some of its
core fluid with it, the vorticity in the dipole centres will increase in magnitude (Arthur
1962). In the 2DU cases, we can anticipate that this will be a prominent effect, but
it is not known a priori how much this behaviour will carry over into the 3DU and3DS cases.
The problem posing is completed with f = 10−4 s−1 (i.e. small compared withthe peak vorticity), K-profile parameterization for the vertical eddy diffusion, a dragcoefficient of CD = 3 × 10−3 used in a quadratic bottom stress law, and weak lateral
diffusion as part of the upstream-biased advection operator in the ROMS (Shchepetkin& McWilliams 2009). None of these influences have an especially large effect overthe multihour period of model integration, but the bottom drag influence is evident
in the weak decay of the ζ z extrema and the volume-integrated energy.
With respect to the numerical discretization, the horizontal grid size is dx = dy =1 m, the number of vertical sigma levels is N = 20 for the 3D solutions, uniformly
distributed in depth, and the barotropic-mode time-step size is dt = 0.05
√ s, consistent
with resolving shallow-water gravity waves where the phase speed ≈ gh is largest.
Solution behaviours are not sensitive to further refinements of these discretization
values.
3. Results
To establish a reference standard, we first consider the set of three alternativemodel cases for the dipole vortex over a flat bottom, which for definiteness wetake as having a uniform depth of 10 m. In an integration over a period of 4 h,
the dipole propagates almost directly offshore at an average speed of approximately
0.01 m s−1 . Over this period, the loss of volume-integrated kinetic energy KE in (3.1)
is approximately 5 % of the initial value and the decrease in the peak magnitude ofζ z at the surface is approximately 14 % (in contrast to the prediction of no change
from (2.3)), in association with minor changes in its pattern shape and bottom drag.
These quantitative statements apply equally to each of cases 2DU, 3DU and 3DS.
Thus, in the absence of disruption (e.g. by depth change), they all exhibit a weakly
decaying approximation to a uniformly propagating barotropic dipole vortex over aflat bottom.
ZZZ
ZZZ
KE(t) = 21
u2 dx dy dz, Ens(t) =|∇ × u|2 dx dy dz.
(3.1a,b)
The cases of more interest are those with a sloping bottom because of the disruption
this provides to the propagating dipole vortex (Centurioni 2002). Our standard
integration period is tmax = 4 h, which is sufficient to establish the differences inbehaviour among the three dynamical models. We made a complete set of dynamicalcases for both the uniform-depth case described in the previous paragraph and thethree different slope values with H = 5, 10 and 20 m at x = Lx (figure 2). From these,
we conclude that the qualitative differences among the different dynamical cases over
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J. C. McWilliams, C. Akan and Y. Uchiyama

50
0

– 50

0

100

200

300

400

500

600

x (m)

F IGURE 3. Trajectory of the maximum (red) and minimum (black) in surface vorticity
ζ z (x, y, 0, t) for the 2DU case with H = 20 m over an integration period of 16 h, withdots at 2 h intervals. Grey contours indicate isobaths with an interval of dh = 2.5 m.

a sloping bottom are essentially similar for all values of H. Therefore, for the mostpart, we will report only on the H = 10 m solutions.
However, to first illustrate the grossest aspect of the depth-induced disruption,
namely the change in vortex strength as argued in (2.3) and its consequence on the
dipole propagation, we show a long-time (16 h) trajectory for the case 2DU vortex as
it moves offshore over the case with the steepest slope, i.e. H = 20 m (figure 3). The
initial adjustment causes a modest reshaping of the vortex, with the centres increasing
their separation distance while propagating offshore. The vorticity extrema increasein magnitude over the first 104 s, as anticipated in (2.3), during which the centres
move closer together. After this period, however, there is an oscillatory ‘instability’ inthe trajectory that causes the dipole vortex to meander in its trajectory over a longer
period. It can be understood by a point-vortex argument based on (2.3). If a small
perturbation causes one centre to become stronger (here the cyclone), it will tend
to advect the weaker centre (anticyclone) around it (in an anticlockwise direction)
by pushing it into deeper water, where the weaker centre will then strengthen, andthe vortex will come back into a more equal balance of mutual advection along a
straight trajectory oriented now in a different direction that is no longer perpendicular
to the slope. With the previously weaker anticyclonic centre now leading on the
deeper-water side, it will in turn strengthen and pull the cyclonic centre in theclockwise direction, thus strengthening it, etc., with the long-time outcome becoming
an oscillatory and wandering trajectory. While all this meandering is going on, thedisrupted vortices begin to weaken, with decreasing peak vorticity amplitudes after
their early maximum around t = 104 s, and they are reduced to approximately half of
the initial amplitude in |ζ z | by the end time of t = 5.8 × 104 s. With respect to theintegral variance measures (3.1), the meandering dipole also exhibits a monotonicallydecreasing kinetic energy KE (down to approximately 43 % by the end time) as well
as a first increasing and then decreasing enstrophy Ens (down to 55 % by the end).
These changes are much greater than in the flat-bottom cases over an equivalent
integration time, for case 2DU in particular. Of course, in these 2DU cases, because∇ is purely horizontal and w is nearly zero (in association with the small η changes),Ens measures only the variance of the vertical vorticity ζ z = ẑ · ∇ × u. Thus, although
the dipole vortex is substantially disrupted as its traverses the sloping bottom, it
retains its general dipole shape and propagation behaviour, although the expectationof bulk qsw conservation (2.3) does not hold. Similar behaviour occurs for the other
2DU cases with their different H values (i.e. different slopes).
Now, we return to our primary experiments: a comparison of the different dynamicalcases for H = 10 over the shorter integration period of tmax = 1.44 × 104 s = 4 h, which
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Nearshore vortices(a) 0.04
0.03(b) 0.4

2DU
3DU
3DS

0.3

0.02

0.2

0.01

0.1

0

(c) 0.3

5

10

t (s)

15

0

0.2
0.1
5

t (s)

10

15

0

5

10

15

t (s)

F IGURE 4. Time series for the three different dynamical cases with H = 10 m over a
period of 4 h: surface vorticity magnitude max |ζ z (x, y, 0)| (a), KE (b) and Ens (c). Theline colours for the cases are indicated in the inset in (a).

is long enough to demonstrate differences among the cases. They all exhibit the grossbehaviour of offshore propagation of the dipole with relatively little meandering during
this shorter period, but they differ considerably in their pattern evolutions.
Starting with time series, the different model solutions are compared in figure 4.
The barotropic solution in case 2DU shows a monotonic increase in the peak vorticity
at the surface, as anticipated with potential vorticity conservation (2.3). Similarly, its
total enstrophy Ens(t) increases monotonically, due almost entirely to the verticalcomponent, ζ z2 , because there are no vertical derivatives in this model and the w
associated with surface elevation changes is small; hence, the horizontal componentof Ens is small. (All of the solutions exhibit propagating shallow-water gravity waves
throughout their integrations, especially during the initial adjustment phase, but they
are quite weak compared with the vortex currents.) The kinetic energy KE(t) decays
monotonically due to lateral diffusion and bottom drag. Its change over 4 h is adecrease by 17 %, which is only a modest amount but still is much larger thanthe comparable case over a flat bottom (i.e. 5 %, as stated above), indicating thedisruptive effect of the changing water depth as the vortex propagates offshore.
The unstratified case 3DU solution starts by following the evolutionary path ofthe case 2DU solution. However, at a time of around 2.5 h (104 s), it shows an
explosive growth in the peak amplitude of the surface ζ z and in Ens(t). For thelatter, it is the horizontal vorticity components, (ζ x , ζ y ) ≈ (−∂z v, ∂z u), that cause
the largest enstrophy growth, with the vertical component ζ z enhanced over its case
2DU value only by approximately a factor of 2. When this explosion occurs, theKE(t) decreases rapidly, indicating an enhanced dissipation by a cascade of energy
down to smaller scales. While we cannot give a precise explanation of this apparent
vortex instability that amplifies into subvortex-scale turbulence, we heuristically see
it as consistent with the instability of an elliptically deformed monopole vortex thatin its axisymmetric state is linearly stable (e.g. Pierrehumbert 1986). The adjacentdipole vorticity centres induce such an elliptical deformation on one another, in
addition to whatever other asymmetric flow patterns may arise. This instability grows
most rapidly for perturbations with the smallest inviscid scales, consistent with thesmall-scale patterns that are shown in figure 6. Even in the 3DU case with a flatbottom, discussed at the start of the section, there is a discernible departure of Ens(t)from its decreasing vertical component around the same time of 2.5 h, leading to a netincrease in Ens of approximately 6 % over the 4 h integration period, indicating themilder onset of a similar instability even in that case (but much less than the increase
in Ens that occurs over a slope). In other studies of unstratified dipole vortices over
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(c)

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

125

150

175

200

–25
100

225
0.02

20

(d)

125

150

175

200

225
0.02

20

10

0.01

10

0.01

0

0

0

0

– 10

–0.01

– 20

–0.02
155

165

175

x (m)

185

195

– 10

–0.01

– 20

–0.02
155

165

175

185

195

x (m)

F IGURE 5. (a,b) Four hour trajectories for the horizontal maximum (red) and minimum(black) vorticity centres for case 3DS, with dots every two hours, and (c,d) ζ z (x, y, z0 , t0 )at the middle time, t0 = 7200 s = 2 h. Panels (a,c) are at the bottom (z0 = −h(x)) and (b,d)
are at the surface (z0 = 0). Here, H = 10 m and the grey contours in (a,b) are isobaths
with dh = 2.5 m.

a flat bottom, similar 3D breakdowns are sometimes seen (Duran-Matute et al. 2010;
van Heijst 2014).
Another perspective is that vortex stretching and tilting (i.e. [(∇ × u) · ∇]u) areenabled in the 3D vorticity equation but absent in 2D; their occurrence is demonstrated
by the Ens explosion in figure 4 and their rates are explicitly diagnosed in Uchiyama
et al. (2017). This process allows 3D vorticity perturbations to amplify even astransient growth, if not literally as an instability (Trefethan et al. 1993). In the 3DUcase, this amplification is unimpeded by the gravitational force, whereas in the 3DS
case, it is modified by the stable density stratification, as discussed next.
The stratified case 3DS solution behaves quite differently from either case 2DU or
3DU. From the outset, its surface vorticity maximum does not increase as anticipatedfrom shallow-water qsw conservation (2.3) even as it moves offshore into deeper
water; its KE(t) more rapidly decreases and its Ens(t) rapidly increases due toincreasing vertical shear of u, hence increasing the variance of its horizontal vorticity
components. Furthermore, a topological transformation in the dipole vortex occurs
starting at around t = 5 × 103 s = 1.5 h: in the lower-half of the water column, thecyclonic vorticity centre strengthens with an increased depression in its mid-depth
isothermal surfaces, implying an amplified central hydrostatic pressure in cyclostrophic
balance, while the isothermal surfaces lift over the anticyclone, implying a weaker
vortical flow in cyclostrophic balance. This causes an anticlockwise turning of the
deeper vortex pair towards the northeast, and the stronger cyclone pulls the weaker
anticyclone around it in an anticlockwise direction and deforms its shape whileremaining nearly stationary in space and circular in shape over the rest of theintegration period (figure 5). Meanwhile, near the surface, the vorticity centres remain
approximately equal in strength and continue to propagate eastward in cyclostrophic
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0.02

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0.01

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–20
–40
195

235

255

0

0.02

–1

0.01

–2

–0.01

–3

–0.02

–4

275

0
–0.01

–40

–20

0

20

40

–0.02

(d) 0

40

0.02

20

0.01

0

0

0.02

–1

z (m)

y (m)

(c)

215

z (m)

(b)

0.01

–2

0

–20

–0.01

–3

–0.01

–40

–0.02

–4

–0.02

195

215

235

255

–40

275

(e)

0.006

40

0

0
–20

–0.003

0

20

40
0.006

–1

0.003

0.003

20

–20

(f) 0

z (m)

y (m)

(a)

y (m)

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Nearshore vortices

–2

0

–3

–0.003

–4

–40
195

215

235

x (m)

255

275

–0.006

–40

–20

0

20

40

–0.006y (m)F IGURE 6. Cross-sections of ζ z (s−1 ) for the three cases with H = 10 m at the time t =1.23 × 104 s = 3.4 h: case 2DU (a,b), case 3DU (c,d) and case 3DS (e,f ). Panels (a,c,e)
are for a horizontal section, ζ z (x, y, 0). Panels (b,d,f ) are for a vertical section ζ z (x0 , y, z)
at the offshore location x0 that passes through the surface vorticity cyclonic extremum at
this time: x0 = 262 m (2DU), 257 m (3DU) and 205 m (3DS). In the vertical section, thelocal bottom depth D(x0 ) is indicated by a dark horizontal line at z = −D with shading
below it. One should notice the different colourbar ranges for the different cases. In the
cross-section for case 3DS ( f ), the isotherms are the dashed-grey lines with dT = 0.03 ◦ C;
it should be recalled that cases 2DU and 3DU are isothermal.

balance with their surface elevation depressions. These divergent trajectories between
the upper and lower parts of the dipole act to split the vortex vertically. The
asymmetric deeper pair of vorticity centres evolves into a weak stationary cyclonicmonopole, while the more symmetric upper pair persists as a coherent vortex. In thetime series, the spitting event is marked by a temporary increase in the ζ z (x, y, 0)
extremum and by diminished rates of KE(t) decay and Ens(t) growth. In the late
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J. C. McWilliams, C. Akan and Y. Uchiyama
stages, after Ens(t) peaks at around t = 8 × 103 s = 2.3 h, there is steady decayof the surface extremum, KE and Ens. In this later period, the dipole is surface
intensified and is no longer importantly influenced by the sloping bottom as itpropagates further offshore while approximately conserving its 3D Ertel potential
vorticity, qE = ( f ẑ + ∇ × u) × ∇T. This late-time propagation behaviour is similar tothat of a stratified dipole over a flat bottom (Voropayev, Fernando & Morrison 2008).
The flow structures during the late stage of these model integrations (t =1.23 × 104 s = 3.42 h) are illustrated in figure 6 with horizontal and verticalsections of ζ z . The barotropic vortex in case 2DU is propagating eastward withstrong symmetry between the vorticity centres, and they are independent of depth(by construction). The centres are close together and their pattern is elongated in
the propagation direction, much like in Lamb’s dipole solution. Among the threecases, it is the most coherent and it has propagated the farthest to the east. The
3D unstratified barotropic vortex in case 3DU has propagated almost as far, but it
exhibits fragmentation in both its horizontal and vertical patterns. Nevertheless, insome coarse-grained sense, it still behaves like an almost barotropic dipole vortex,
at least over this 4 h integration period. Finally, the stratified dipole vortex in case
3DS, after the vertical splitting event described above, has recovered a very coherent
pattern, but its vertically connected extent is restricted to the upper-half of thewater column. In case 3DS, it has a larger horizontal size, a wider separation of its
vorticity centres, a weaker vorticity amplitude and thus a slower eastward propagation
speed compared with the barotropic dipole in case 2DU. Its thermal structure isa cold anomaly with respect to its surroundings that weakens with depth. This isconsistent with cyclostrophic and hydrostatic balance for a surface-intensified vortex:
the sea-level depression at the vortex cores (η < 0) has a low-pressure anomaly thatweakens with depth (ρ0−1 ∂z p < 0), hence a negative thermal anomaly in the upper
levels (T = −(ρ/αρ0 ) < 0, with α the thermal expansion coefficient), hence an upliftof the isotherms, as seen in figure 6( f ).
4. Summary and discussion
Depth-averaged fluid dynamics are very useful in understanding many aspects ofnearshore currents. These include the formation of alongshore and rip currents whenwaves break and the occurrence of current instabilities that give rise to shear waves
and vortices in surf turbulence. However, we have shown here with an idealized
dipole vortex over a sloping bottom that the 2D depiction of coherent surf vortices
is incomplete and inaccurate in comparison with a 3D model. In particular, thevortex robustness is overestimated in 2D, as previously noted in more complex flow
simulations (Uchiyama et al. 2017).
This experimental result adds impetus to the evolution of nearshore modelling
towards including fully 3D dynamics. In the surf zone itself, when the density
is nearly uniform due to mixing by breaking waves, the unstratified 3D model(case 3DU) indicates a moderate degree of fragility of its barotropic vortices toelliptical instability and to vortex stretching and tilting. For surf vortices that moveoffshore into stratified water, there can be an emergence of surface-intensified vortices,
after a somewhat dissipative transition, that then achieve a degree of coherence intheir further lifetime while somewhat insulated from further depth changes.
A question arises: how well can the hydrostatic model used here represent the true
fluid dynamics of surf vortices? Depth-averaged models are inherently hydrostatic and,as noted, have been found to be useful for many surf phenomena, including vortex
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Nearshore vortices
emergence (figure 1). It is likely that much of the 3D flow structure on microscopic
scales smaller than the macroscopic vortices will be partly non-hydrostatic, namely
the 3D vortex instability in case 3DU and much of the vortex tilting and stretching
behaviours; a non-hydrostatic model would be needed to follow these processes indetail. With respect to the macroscopic behaviours of vortex propagation, decay,
incipient horizontal enstrophy generation and, in case 3DS, structural transformation,
we are confident that the hydrostatic approximation is adequate. A definitive
demonstration of this assertion could only be made with a non-hydrostatic model, but
in lieu a consistency test has been made by comparing the magnitude of ∂t w with theanomalous vertical pressure gradient, ρ0−1 ∂z p, at a middle depth level in the midst of
the most violent macroscopic event in case 3DS, namely its vertical fissioning aroundt = 2 h (figure 5). The magnitude of ∂t w is nearly three orders of magnitude smaller
than the vertical pressure gradient anomaly; i.e. the macroscopic behaviour is quite
safely hydrostatic.
A modern scientific frontier is the combined interactions among surface gravity
waves, turbulent nearshore currents and turbulent submesoscale currents in the deeper
stably stratified water over the continental shelf (Dauhajre, McWilliams & Uchiyama2017). In all of these combinations, the dynamics are often 3D with important effects
from stratification and, at least for shelf currents, from the Earth’s rotation (Lentz &Fewings 2012; Kumar & Feddersen 2017).
Acknowledgements
We appreciate technical assistance by D. Dauhajre. This research is supported by
the US Office of Naval Research (grant N00014-15-1-2645), the National Science
Foundation (grant OCE-1355970) and Japan Society for the Promotion of Science,
Grants-in-Aid for Scientific Research, grant numbers 15KK0207 and 15H04049. This
work used the Extreme Science and Engineering Discovery Environment (XSEDE),
which is supported by National Science Foundation grant number ACI-1548562.
References
A RTHUR , R. S. 1962 A note on the dynamics of rip currents. J. Geophys. Res. 67, 2777–2779.
B OWEN , A., I NMAN , D. & S IMMONS , V. 1968 Wave set-down and set-up. J. Geophys. Res. 73,
2569–2577.
C ENTURIONI , L. R. 2002 Dynamics of vortices on a uniformly shelving beach. J. Fluid Mech. 472,
211–228.
DALYRIMPLE , R. A., M AC M AHAN , J. H., R EINERS , A. J. & N ELKO , V. 2011 Rip currents. Annu.
Rev. Fluid Mech. 43, 551–581.
DAUHAJRE , D., M C W ILLIAMS , J. C. & U CHIYAMA , Y. 2017 Submesoscale coherent structures on
the continental shelf. J. Phys. Oceanogr. 47, 2949–2976.
D URAN -M ATUTE , M., A LBAGNAC , J., K AMP, L. P. J. & VAN H EIJST, G. J. F. 2010 Dynamics
and structure of decaying shallow dipolar vortices. Phys. Fluids 22, 116606.
F EDDERSEN , F. 2014 The generation of surfzone eddies in a strong alongshore current. J. Phys.
Oceanogr. 44, 600–617.
H ALLER , M. C. & DALRYMPLE , R. A. 2001 Rip current instabilities. J. Fluid Mech. 433, 161–192.
VAN H EIJST, G. J. F. 2014 Shallow flows: 2d or not 2d? Environ. Fluid Mech. 14, 945–956.
K IRBY, J. T. & D ERAKHTI , M. 2018 Short-crested wave breaking. Eur. J. Mech. B (in press).
K UMAR , N. & F EDDERSEN , F. 2017 The effect of Stokes drift and transient rip currents on the
inner shelf. Part II. With stratification. J. Phys. Oceanogr. 47, 243–260.
K UMAR , N., VOULGARIS , G., WARNER , J. C. & O LABARRIETA , M. 2012 Implementation of thevortex force formalism in the coupled ocean–atmosphere–wave–sediment transport (COAWST)
modeling system for inner shelf and surf zone applications. Ocean Model. 47, 65–95.

850 R2-11

Downloaded from https://www.cambridge.org/core. Kobe Shosen Daigaku., on 13 Jul 2018 at 07:27:27, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/jfm.2018.510

J. C. McWilliams, C. Akan and Y. Uchiyama
L AMB , H. 1932 Hydrodynamics. Dover.
L ENTZ , S. J. & F EWINGS , M. R. 2012 The wind- and wave-driven inner-shelf circulation. Annu.
Rev. Mar. Sci. 4, 317–343.
L ONGUET-H IGGINS , M. S. 1970 Longshore currents generated by obliquely incident sea waves.
J. Geophys. Res. 75, 6778–6801.
M ARCHESIELLO , P., B ENSHILAT, R., A LMAR , R., U CHIYAMA , Y., M C W ILLIAMS , J. C. &S HCHEPETKIN , A. F. 2015 On tridimensional rip current modeling. Ocean Model. 96, 36–48.
M ARMORINO , G. O., S MITH , G. B. & M ILLER , W. D. 2013 Infrared remote sensing of surf-zone
eddies. IEEE J. Sel. Top. Appl. Earth Obs. Remote Sens. 6, 1710–1718.
M C W ILLIAMS , J. C., R ESTREPO , J. M. & L ANE , E. M. 2004 An asymptotic theory for theinteraction of waves and currents in coastal waters. J. Fluid Mech. 511, 135–178.
N EWBERGER , P. A. & A LLEN , J. S. 2007 Forcing a three-dimensional, hydrostatic, primitive-equation
model for application in the surf zone: 2. Application to Duck94. J. Geophys. Res. 112,
C08019.
O LTMAN -S HAY, J. P., H OWD , P. A. & B IRKEMEIER , W. A. 1989 Shear instabilities of the mean
longshore current: 2. Field observations. J. Geophys. Res. 94, 18031–18042.
P IERREHUMBERT, R. T. 1986 Universal short-wave instability of two-dimensional eddies in an
inviscid fluid. Phys. Rev. Lett. 57, 2157–2159.
S HCHEPETKIN , A. F. & M C W ILLIAMS , J. C. 2009 Computational kernel algorithms forfine-scale, multi-process, long-term oceanic simulations. In Handbook of Numerical Analysis:
Computational Methods for the Ocean and the Atmosphere (ed. R. Temam & J. Tribia),
vol. 14, pp. 121–183. Elsevier.
S PYDELL , M. S. & F EDDERSEN , F. 2009 Lagrangian drifter dispersion in the surf zone: directionallyspread normally incident waves. J. Phys. Oceanogr. 39, 809–830.
T REFETHAN , L. N., T REFETHAN , A. E., R EDDY, S. C. & D RISCOLL , T. A. 1993 Hydrodynamic
stability without eigenvalues. Science 261, 578–584.
U CHIYAMA , Y., M C W ILLIAMS , J. C. & A KAN , C. 2017 Three-dimensional transient wave-driven
currents: low-frequency variability in alongshore and rip currents. J. Geophys. Res. 122,
C22370.
U CHIYAMA , Y., M C W ILLIAMS , J. C. & S HCHEPETKIN , A. F. 2010 Wave–current interaction ina three-dimensional circulation model with a vortex force formalism: application to the surfzone. Ocean Model. 34, 16–35.
VOROPAYEV, S. I., F ERNANDO , H. J. S. & M ORRISON , R. 2008 Dipolar eddies in a decaying
stratified turbulent flow. Phys. Fluids 20, 026602.

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