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Stabilizing effect of random waves on rip currents

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Stabilizing effect of random waves on rip currents
Haider Hasan,
1,2
Nicholas Dodd,
1
and Roland Garnier
1,3
Received 18 July 2008; revised 16 February 2009; accepted 16 April 2009; published 9 July 2009.
[
1
]
The instability leading to the formation of rip currents in the nearshore for normalwaves on a nonbarred, nonerodible beach is examined with a comprehensive linear stability numerical model. In contrast to previous studies, the hypothesis of regular waveshas been relaxed. The results obtained here point to the existence of a purelyhydrodynamical positive feedback mechanism that can drive rip cells, which is consistent with previous studies. This mechanism is physically interpreted and is due to refractionand shoaling. However, this mechanism does not exist when the surf zone is not saturated because negative feedback provided by increased (decreased) breaking for positive(negative) wave energy perturbations overwhelms the shoaling/refraction mechanism.Moreover, turbulent Reynolds stress and bottom friction also cause damping of the ripcurrent growth. All the nonregular wave dissipations examined give rise to thesehydrodynamical instabilities when feedback onto dissipation is neglected. When thisfeedback is included, the dominant effect that destroys these hydrodynamical instabilitiesis the feedback of the wave energy onto the dissipation. It turns out that this effect isstrong and does not allow hydrodynamical instabilities on a planar beach to grow for random seas.
Citation:
Hasan, H., N. Dodd, and R. Garnier (2009), Stabilizing effect of random waves on rip currents,
J. Geophys. Res.
,
114
,C07010, doi:10.1029/2008JC005031.
1. Introduction
[
2
] Normally incident waves approaching a straight shoreline shoal and eventually break. This generates set-up(increased elevation in mean free surface), through radiationstresses [
Longuet-Higgins and Stewart
, 1962, 1964], andthereforealsooffshoredirectedhorizontalpressuregradients.In theory this dynamical balance can pertain along analongshore uniform shore with correspondingly uniformwave conditions. In reality water is often seen to recirculate back to sea at certain locations, in rip currents, which are, inturn, fed by alongshore flowing currents on either side.[
3
] Studies have shown that rip currents may approachspeeds of up to 2 m s
1
[see
MacMahan et al.
, 2006],although the average strength is often less than that. Thesecirculations therefore can become an issue for beach safety[see
Short and Hogan
, 1994]. Nearshore circulation alsoresults in the circulation of cleaner water from the offshoreinto the nearshore region [see
Inman et al.
, 1971]. Impor-tantly, nearshore circulation together with waves also trans- port beach sediment, and indeed the rips themselvesapparently erode rip channels. Hence it is vital to understandthe physical mechanisms of the generation of rip currents if wearetounderstandthenearshoresedimentbudget.Itshould be noted, however, that it is not clear whether these channelsarepassively erodedbythecurrents,orif thechannels appear in combination with the currents as part of a dynamicalinteraction. We return to this point later.[
4
] On a long straight coast, rip currents can occur at quite regular intervals so one can often assign an alongshorespacing to these features, which ranges from 50 to 1000 m[
Short
, 1999], and as reported by
Short
[1999], rip currentsare most readily formed on intermediate energetic beachescharacterized by slopes of 1:30 to 1:10. This quasi-regularityinspacing ofripcurrents is intriguing, and hasreceived someattention in previous studies.[
5
] Early observational studies of rip currents in the fieldsuch as that made by
Shepard et al.
[1941] and
Shepard and Inman
[1950], and more recent ones by
Brander and Short
[2000] and
MacMahan et al.
[2005], have found rip currentsto occur where there are alongshore variations in the bathymetry with rip channels cut through the alongshore bar. This alongshore variation in the bathymetry results in anaccompanying variation in wave height. These wave height variations generate alongshore variable pressure gradientsthat drive nearshore circulation cells. Experimental studies inlaboratories have also supported this idea where measure-ments were conducted for a barred beach profile with incisedrip channels [see
Haller et al.
, 2002;
Haas and Svendsen
,2002]. Using the concept of radiation stresses,
Bowen
[1969]showedthatnearshorecirculationcells,includingrips,canbeforcedbyimposingalongshorevariationsinwaveheightona plane sloping beach for waves normal to the shoreline. Thealongshore variation in wave height results in corresponding
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, C07010, doi:10.1029/2008JC005031, 2009
Click Here
for
Full Article
1
Environmental Fluid Mechanics Research Centre, Process andEnvironmental Division, Faculty of Engineering, University of Nottingham, Nottingham, UK.
2
Now at Department of Mathematics and Basic Sciences, NED
University of Engineering and Technology, Karachi,
Pakistan.
3
Now at Applied Physics Department, Universitat Politecnica deCatalunya, Barcelona, Spain.Copyright 2009 by the American Geophysical Union.0148-0227/09/2008JC005031$09.00
C07010
1 of 14
variations in radiation stresses, thus driving the circulationcells as areas of large set-up (large wave height) drive acurrent alongshore into the rips.[
6
] The model of
Bowen
[1969], however, does not explain the quasiperiodicity itself, and nor does it consider an erodible beach. Attention has therefore subsequently been focused on examining the stability to periodic pertur- bations of alongshore uniform setup on an erodible beach,namely, the morphodynamical stability. The pioneeringstudy of
Hino
[1974] identified such an instability, and,fornormalincidentwaves,ripcurrents/channelswereformedtogether with cuspate morphological features when an along-shore periodic perturbation was imposed. This theory wassubstantially progressed by
Deigaard et al.
[1999] and aninstability mechanism identified by
Falque´s et al.
[2000],who noted that the potential stirring (depth-averaged con-centration) gradient governed the positive feedback mech-anism by which the rips and their associated morphologyevolve. Thus, for an offshore flowing current (the rip), anynegative perturbation (the incipient rip channel) will beenhanced (positive feedback, and therefore instability)where the concentration is increasing offshore, becausethe current flows from regions of lower to greater concen-tration,so thaterosionmust takeplace fortheconcentration profile to be maintained. Studies with more comprehensivemodels have been presented by
Damgaard et al.
[2002],
Caballeria et al.
[2002],
Calvete et al.
[2005], and
Garnier et al.
[2006].[
7
] Theseexplanationshavefocusedonmorphodynamicalinstability. However, it has also been suggested that along-shorevariationsinwaveheightmayariseasaresultofwave-current interaction, which poses the question about theformation of rip channels alluded to earlier: do channelsand currents form together as a morphodynamical instabil-ity, or do rip currents form as a hydrodynamical instabilityand then erode the channels. The episodic nature of ripcurrents observed by
Smith and Largier
[1995] at Scripps beach possibly indicates that rip currents are a result of ahydrodynamic instability, but these events could also bedue to temporal variations in the incoming wavefield[
Reniersetal.
,2004].Fieldevidenceofripcurrentswithout channels is hard to find.
MacMahan et al.
[2006] note that all currents are accompanied by perturbations in the bedlevel, albeit small ones at some locations. The laboratoryinvestigation by
Bowen and Inman
[1969] on a plane sloping beach found rip currents generated as a result of the interac-tion between edge waves and incoming incident waves, withripspacingequalto(andthereforedictatedby)thealongshorewavelength of the edge wave. Nevertheless, some other studies have continued to consider the purely hydrodynam-ical instability (or something akin to this) to investigatewhether rip currents can be formed without perturbations inthe bed.[
8
] The analytical study of
Dalrymple and Lozano
[1978],who extended the work of
LeBlond and Tang
[1974],suggested that wave refraction and shoaling (as opposed toshoaling alone) was responsible for the existence of ripcurrents. The authors considered normally incident waveson a planar foreshore of constant slope and a flat offshore bathymetry, on which they imposed an alongshore periodic perturbation (neither growing nor decaying), and noted that theinteractionofthewavefieldwiththeoffshorecurrentsactsto intensify them.[
9
] For incoming waves normal to the shoreline,
Falque´set al.
[1999] examined the growth of nearshore circulationcells (rip cells) using linear stability analysis (thus allowinggrowing modes) on a non-erodible, plane beach. Two caseswere considered. The first was set-up in isolation; that is,waves were normally incident on a monotonic beach andwave refraction because of current neglected. An analyticalanalysis showed that the set-up did not induce instability.The subsequent inclusion of wave refraction on current, thesecond case, led to instability. Here
Falque´s et al.
[1999]considered a simplistic situation in which waves approachedover a flat bathymetry, and perturbations in wave energydissipation due to wave breaking were neglected. The flat bed was chosen so as to avoid dealing with wave breakingand the discontinuities at the breaking line, and to allow theapplication of shallow water theory throughout.[
10
] An alternative hydrodynamical rip current study wasmade by
Murray and Reydellet
[2001]. The focus of their studywasonself-organized ripcurrentsdrivenbyafeedback involving a newly hypothesized interaction between wavesand currents, in which the rip current causes a diminution inwave height, through turbulence generated by shears inorbital velocities. This would then lead to reduced shorelineset-up, now no longer balanced because of decrease in waveheight, which then leads to alongshore flows into the rip. Asimple model with effects of currents on wave number fieldnot included was developed on the basis of cellular automatawhere each variable defined in a particular cell in a grid of cells interacted according to rules encapsulating the above physics, which then indeed led to a positive feedback that intensified the current [see
Murray and Reydellet
, 2001].[
11
]
Yu
[2006] examined the growth of rip current cellsdue to the inclusion of wave-current interactions. This wasmotivated by the uncertainties in the results obtained by
Dalrymple and Lozano
[1978].[
12
] Assuming monochromatic waves,
Yu
[2006] split thecross-shore domain into an offshore part (prior to breaking)
Figure 1.
Sketch of the nearshore coordinate system andthe general bathymetry.
C07010
HASAN ET AL.: INSTABILITIES LEADING TO RIP CURRENTS2 of 14
C07010
mostly on constant depth and therefore allowing shallowwater theory to be used in the offshore portion of thedomain, and a part in the surf zone (plane beach). Before breaking a wave energy equation is implemented to trans-form wave height; within the surf zone the wave height iscontrolled by the local water depth. The break point occurson the slope, and a moving shoreline is implemented.[
13
] For an offshore wave height
H
1
of 1.88 m and period
T
= 8.17 s
Yu
[2006] predicts a growing ‘‘rip cell’’mode with
e
-folding time 14.83 s. This increased rapidly for smaller wave heights:
H
1
= 1.12 m possessed an
e
-foldingtime of 97.20 s. The comparison of predicted rip-spacingwith field observations showed fairly good agreement for large wave breaker heights. However, for smaller breaker heights the agreement was poor. For more details, see
Yu
[2006].[
14
] The idealized nature of the study of
Falque´s et al.
[1999] and the more sophisticated but still restricted (e.g.,shallow water theory and the use of regular waves) study of
Yu
[2006], thus leads us to re-examine the hydrodynamicalinstability to alongshore periodic disturbances of normallyincident waves on a plane beach, but this time using acomprehensive model incorporating finite depth wave propagation and random waves. To this end we investigatethe possible formation of rip cells under more realisticconditions (random waves) using the model of
Calvete et al.
[2005], which also includes other effects (e.g., turbu-lent Reynolds stresses) not considered by the earlier authors.[
15
] In section 2, we describe the model used in the present study. In section 3, we show results for the numer-ical investigations. Thereafter we draw some conclusions.
2. Model Description
[
16
] The model used here is described in detail by
Calveteet al.
[2005]. It is based on the depth and time averagedmass (1) and momentum (2) equations and wave energy (3),
Figure 2.
Basic state cross-shore profiles for different types of waves, i.e., intermediate and random(both
Thornton and Guza
[1983] (TG) and
Church and Thornton
[1993] (CT)) on a plane sloping beach profile with beach slope
b
= 0.07,
H
rmso
1
= 1.5 m, and
T
= 10 s: (a) mean surface elevation, (b) bed level,(c) root mean square wave height, (d) wave energy dissipation, and (e) wave number.
C07010
HASAN ET AL.: INSTABILITIES LEADING TO RIP CURRENTS3 of 14
C07010
wave phase (4), and sediment conservation equations. Sincewe are examining hydrodynamical instabilities we do not consider the sediment conservation equation here. Thegoverning equations are
@
D
@
t
þ
@
Dv
i
@
x
i
¼
0
;
ð
1
Þ
@
v
i
@
t
þ
v
j
@
v
i
@
x
j
¼
g
@
z
s
@
x
i
1
r
D
@ @
x
j
S
0
ij
S
00
ij
t
bi
r
D
;
ð
2
Þ
@
E
@
t
þ
@ @
x
i
v
i
þ
c
gi
E
þ
S
0
ij
@
v
j
@
x
i
¼ D
;
ð
3
Þ
@
F
@
t
þ
s
þ
v
i
@
F
@
x
i
¼
0
;
ð
4
Þ
where
i
,
j
= 1, 2, so
x
i
= (
x
1
,
x
2
) = (
x
,
y
), where
x
and
y
arecross-shore and alongshore coordinates. Horizontal veloci-tiesare
v
i
=(
v
1
,
v
2
)=(
u
,
v
),
g
isaccelerationduetogravityand
r
is water density. Total water depth is denoted
D
=
z
s
z
b
,where
z
s
is the mean surface elevation and
z
b
is the (fixed) bed level: see Figure 1. Further,
E
=
18
r
gH
rms
2
is wave energydensity, where
H
rms
is root mean squared wave height,
F
isthe wave phase,
S
0
ij
are the components of the radiationstress tensor,
S
00
ij
are the Reynolds stress tensor compo-nents,
t
bi
is the
i
th component of the bottom friction,
D
isthe dissipation due to wave breaking, and
c
gi
are the groupvelocity vector components.
s
is the intrinsic frequencygiven by
s
¼
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
gK
tanh
KD
ð Þ
p
;
ð
5
Þ
where
K
isthewavenumber(
K
=
j
~
K
j
).Thewavevector
~
K
isgiven by
~
K
=
~
r
F
.[
17
] The expression for the Reynolds’ stresses appliedhere is [see, e.g.,
Svendsen
, 2006]
S
00
ij
¼
rn
t
D
@
v
i
@
x
j
þ
@
v
j
@
x
i
;
ð
6
Þ
where the
Battjes
[1975] parameterization for horizontaleddy viscosity has been implemented
n
t
¼
M
D
r
13
H
rms
;
ð
7
Þ
where
M
is a parameter that characterizes turbulence and isof
O
(1).
Calvete et al.
[2005] choose
M
= 1 [
Battjes
, 1975].However,
Svendsen et al.
[2002] recommended 0.05 <
M
<0.1. Here we take a default value
M
= 0.5. Bed shear stressis parameterized using a linear friction law:
t
bi
¼
r
2
p
C
D
u
rms
v
i
;
ð
8
Þ
Figure 3.
Rip current growth rate curves for different wave types, i.e., intermediate and random (both
Thornton and Guza
[1983] (TG) and
Church and Thornton
[1993] (CT) cases) with no feedback of perturbations on
D
for
H
rmso
1
= 1.5 m,
T
= 10 s,
b
= 0.07, and
z
o
= 0.001 m and when Reynolds Stressterms are turned on and off, i.e.,
M
= 0 and
M
= 0.5.
C07010
HASAN ET AL.: INSTABILITIES LEADING TO RIP CURRENTS4 of 14
C07010
where
C
D
is the drag coefficient and is given as
C
D
¼
0
:
40ln
D
=
z
o
ð Þ
1
2
;
ð
9
Þ
where
z
o
is the bed roughness length (default value taken as
z
o
= 0.001 m), and
u
rms
, wave orbital velocity at the sub-stitute fortheboundary layeredge,is determined usinglinear theory:
u
rms
¼
H
rms
2
gK
s
cosh
Kz
o
cosh
KD
:
ð
10
Þ
2.1. Wave Energy Dissipation
[
18
] In real seas waves are random and so will not break at one point. As a result, the surf zone can be quite extensivecompared to that for quasi-regular waves. For randomwaves, breaking can be a result of short wave interaction,the interaction of waves with the bottom, current or wind[
Roelvink
, 1993]. Here the interaction of current and windare neglected, and wave transformation is linear. Therefore,wave breaking is dictated by the effect of changes in theseabed (for energy dissipation due to current-limited wave breaking, see
Chawla and Kirby
[2002]). Apart from thesesimplifications, which were also imposed by
Yu
[2006] and
Falque´s et al.
[1999], the effects of different types of random wave breaking are here examined in detail. Thisis motivated both by the consideration of regular waves in previous studies, and by related work [see
Van Leeuwen et al.
, 2006] that showed the importance of the type of wave breaking on the evolution of bed forms.[
19
] Three types of random waves are considered here.They are distinguished by the extent of the surf zone and thesize and shape of the dissipation profile, and are imple-mented by changing the energy dissipation term
D
in (3), sothat they describe transition between random and quasi-regular (intermediate) waves.Weusethemodelsof
Thorntonand Guza
[1983] and
Church and Thornton
[1993], the first because it is standard in the literature and the second becauseit allows for somewhat more concentrated breaking at theshore. The model of
Church and Thornton
[1993] is
D ¼
3
ﬃﬃﬃ
p
p
16
r
gB
3
f
p
H
3
rms
D
1
þ
tanh 8
H
rms
g
b
D
1
1
1
þ
H
rms
g
b
D
2
!
5
=
2
2435
;
ð
11
Þ
where
B
= 1.3 (describes the type of breaking) and
g
b
= 0.42are used, and where
f
p
=
s
/2
p
is the intrinsic peak frequency(for
Thornton and Guza
[1983],
B
= 1 and
g
b
= 0.42).[
20
] The model of
Van Leeuwen et al.
[2006], which is based on that of
Roelvink
[1993], and which allows for regular, depth-limited regular and random waves, is usedhere for what we term ‘‘intermediate’’ waves, to provide thelink between regular and fully random waves. This thusallowsustosuggesttrendsaswemovetowardregularwaves,the situation examined by earlier authors. For intermediatewaves
B
= 1 and
g
b
= 0.55. See Appendix A for thesedissipation expressions.
2.2. Linear Stability Analysis
[
21
] The standard practice of linear stability analysis is tofirst define a basic state, which is a time-invariant solutionto the equations. The stability of this basic state is thenanalyzed by superimposing periodic perturbations and thenlinearizing with respect to the perturbations. In the basicstate an alongshore uniform beach,
z
b
=
b
x
is assumed,with the basic state variables
u
o
= 0,
v
o
= 0 (since we areconsidering normal incidence),
z
s
=
z
so
(
x
),
E
=
E
o
(
x
) and
F
=
F
o
(
x
) (the subscript
o
denotes basic state terms; notethat we do not use it for
z
b
, which is kept fixed throughout).[
22
] The basic state variables are found by integratingonshore the basic state equations (1), (2), (3), and (4), thusdetermining the shoreline position with a prescribedtolerance depth of
D
o
(0) = 15 cm, which is used through-out this study to avoid numerical problems.
H
rmso
andthe wave period,
T
are prescribed in deep water (offshore)conditions.
Figure 4.
Rip current growth rate curves for different wave types, i.e., intermediate and random (both
Thorntonand Guza
[1983] (TG) and
Church and Thornton
[1993](CT) cases) with no feedback of perturbations on
D
for
H
rmso
1
= 1.5 m,
T
= 10 s,
b
= 0.07, and
M
= 0.5 and for bedroughness lengths
z
o
= 0.001 m and
z
o
= 0.01 m.
C07010
HASAN ET AL.: INSTABILITIES LEADING TO RIP CURRENTS5 of 14
C07010

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