Modeling and energetics of tidally generated wave trains in the Lombok Strait

17 Pages·2011·3.38 MB·English

JOURNAL OFGEOPHYSICAL RESEARCH, VOL.116, C03023,doi:10.1029/2010JC006589,2011
Modeling and energetics of tidally generated wave trains
in the Lombok Strait: Impact of the Indonesian Throughflow
H. Aiki,1 J. P. Matthews,2,3 and K. G. Lamb4
Received14August2010;revised1November2010;accepted2December2010;published22March2011.
[1] This study investigates the possible impact of the Indonesian Throughflow (ITF) on
tidally generated internal waves in Lombok Strait and examines the energetics of these
disturbances. Using a two‐dimensional nonhydrostatic numerical model which takes
into account the variable width of the strait region, two main experiments have been
performed, one without and one with an idealized ITF component in the upper layer
flowing southward toward the Indian Ocean. These correspond to conditions in boreal
winter and summer, respectively. Both experiments show trains of internal solitary‐like
gravity waves (ISWs). Overall, ISWs are more numerous on the north side of the sill
wherethenarrowerchannelineffectamplifiesthedisturbances.Inbothexperimentsabout
3.9 GW of energy is injected into barotropic and baroclinic tidal currents, of which
about 2.6 GW is radiated away by internal gravity waves. The ITF regulates the way that
the radiated energy is partitioned between the two sides of the sill. Without the ITF
(boreal winter), the northward radiated energy flux is greater in magnitude than that
radiated to the south. However, when the ITF is present (boreal summer), the northward
radiated energy flux is smaller in magnitude than that radiated to the south. This result
is obtained by diagnosing the flux of the Montgomery potential which can take into
account the effect of finite amplitude waves and also offers a simple and robust energy
diagnosis in the presence of time mean flows.
Citation: Aiki,H., J.P. Matthews, andK.G.Lamb(2011), Modeling andenergetics of tidallygenerated wave trains inthe
LombokStrait: Impact of the IndonesianThroughflow, J.Geophys. Res.,116,C03023,doi:10.1029/2010JC006589.
1. Introduction Arief, 1988]. Figure 1a depicts an ERS‐1 Synthetic Aper-
ture Radar (SAR) image over the LS region, acquired at
[2] Tidalmixingintheoceancanbeattributedtoaseries 0232 UTC on 23 April 1996 (as used by Susanto et al.
of nonhydrostatic processes, such as Kelvin‐Helmholtz
[2005]), which shows two sets of wave arcs (A and B) in
instability,hydraulic‐jump‐likeoverturning,internalswash/
the Java Sea and one wave arc (C) in the Indian Ocean, all
surfactiononaslopingbottom,andtheshoalingofinternal
ofwhicharegeneratedatthesillbytheM tide.Theleading
solitary‐like gravity waves (ISWs) [cf. Moum et al., 2008]. wave arc is followed by a train of waves22–7 km in length,
Recent oceanographic studies have paid considerable
which are the surface expressions of ISWs. Wave trains of
attention to tidal mixing in the Indonesian Seas, as this
this type represent some of the largest nonhydrostatic phe-
process changes the vertical structure of the Indonesian
nomena in the ocean [cf. Osborne and Burch, 1980;
Throughflow (ITF) and thereby affects heat transport
Helfrich and Melville, 2006] and highlight the differences
between the Pacific and Indian Oceans [cf. Ffield and
between the dispersion relations of hydrostatic and non-
Gordon, 1996; Koch‐Larrouy et al., 2008; Robertson and
hydrostatic internal gravity waves (Figure 2) [Lamb, 1994].
Ffield, 2008; Robertson, 2010].
[4] Recently Matthews et al. [2011, hereafter Ma11]
[3] Lombok Strait (LS) is one of the major passages for analyzed the statistics of satellite SAR images gathered
the ITF, and is known to have a tall, sharply defined sill
over the LS region, and compared the occurrences of wave
which induces strong tidal flows of ±3.5m/s [Murray and
trains propagating northward and southward from the sill.
They found it necessary to recognize two main classes of
high‐frequency nonlinear internal waves in the LS region.
1ResearchInstituteforGlobalChange,JapanAgencyforMarine‐Earth The first of these represents ISWs generated as a result of
ScienceandTechnology,Yokohama,Japan. flowenhancementcausedbythetopographyofthesill,and
2EnvironmentalSatelliteApplications,Anglesey,Wales,UK. which radiate outward in uniform concentric arcs, whereas
3DepartmentofGeophysics,KyotoUniversity,Kyoto,Japan. thesecondclassincludesirregular,braideddisturbancesthat
4Department of Applied Mathematics, University of Waterloo,
are closely associated with and may possibly be released
Waterloo,Ontario,Canada.
from the frontal boundaries of warm water intrusions that
Copyright2011bytheAmericanGeophysicalUnion. advance southward into the Indian Ocean. In the present
0148‐0227/11/2010JC006589
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C03023 AIKI ET AL.:ENERGY FLUX OFLOMBOK INTERNAL WAVES C03023
independently confirmed by comparison, for example,with
detailed in situ observations gathered by moorings. One of
the aims of our study, therefore, is to examine the internal
wave seasonality obtained by Ma11 in the light of results
derivedfromournumericalstudyandtosuggestwherefur-
therexperimentalworkisrequired.
[5] A number of numerical studies for ISW trains have
been presented for various regions of the global ocean,
including Georges Bank, the Strait of Messina, the Gulf of
Oman,theSouthChinaSea,andtheYellowSeaetc.[Lamb,
1994; Brandt et al., 1997; Small and Martin, 2002; Warn‐
Varnas et al., 2005, 2010; Chao et al., 2007; Zheng et al.,
2007; Shen et al., 2009; Buijsman et al., 2010]. These
studies used a two‐dimensional nonhydrostatic model,
except for Shen et al. [2009] who use a three‐dimensional
model, in the vertical plane and assume no variation in the
direction normal to the plane. This assumption was also
usedbyNingsihetal.[2008]whopresentedmodelresultsin
anearlyattempttosimulatetidallygeneratedwavetrainsin
LS. Although they used the World Ocean Atlas to demon-
strate the impact of the seasonal variability of the thermo-
cline on tidal internal waves, the lack of detailed
observations of the spatial distributions of temperature and
salinity in this region makes it difficult to pursue this
approachfurther.Ontheotherhand,toourknowledge,there
has been no numerical study which (1) takes into account
the variable width of the LS region in a two‐dimensional
nonhydrostaticmodel,(2)investigatestheimpactoftheITF
on tidally generated wave trains, and (3) considers the
Figure 1. (a) ERS‐1 SAR image over Lombok Strait energetics oftidal internal wavesin LS. Theseare themain
acquired at 0232 UTC on 23 April 1996 (adapted from subjects of the present study. We also address fundamental
Susanto et al. [2005]). (b) Bottom topography (m) with
the solid box representing the region covered by Figure 1a
and the dashed line representing the main axis of the strait
along which the numerical model is configured.
numericalstudy,wefocusonthefirstclassofinternalwaves
(namely the arc‐like ISWs radiating uniformly away from
the sill) and do not consider the second class. From the
analysisofamultiyearmeandatasetbasedon932archived
SAR scenes, Ma11 found that the ISW trains of the first
class were detected more frequently on the Java Sea side,
propagating northward. This directional asymmetry was
particularly well defined during the boreal winter period of
the northwest monsoon (roughly December–May). During
boreal summer, the period of the southeast monsoon
(roughlyJune–November),theirregularwavesofthesecond
classwereidentifiedinabundanceontheIndianOceanside,
propagating to the south and west. Ma11 considered this
seasonal variability to reflect the seasonal behavior of the
ITFitself,whichhasatendencytoweakenduringtheboreal
winter and flow strongly southward during the boreal
Figure 2. Hydrostatic (blue) and nonhydrostatic (red) dis-
summer [cf. Masumoto and Yamagata, 1996; Sprintall
persion relations of internal gravity waves in linear theory
et al., 2009]. A related characteristic of the ITF is that with the symbols w, N, and f representing wave, buoyancy,
its southward transport is weakened during El Niño years
and Coriolis frequencies and k and m representing the hori-
and enhanced during La Niña years [England and Huang,
zontal and vertical wavenumbers. The dashed red line near
2005]. We recognize that the satellite‐based LS internal
the origin indicates the dispersion relation of waves in a
wave results reported by Ma11 have not to date been
nonrotating frame.
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C03023 AIKI ET AL.:ENERGY FLUX OFLOMBOK INTERNAL WAVES C03023
p[x, z, t] ≡ ps[x, t] + ph[x, z, t] + pn [x, z, t] is the sum of
Rsea surface pressure (SP), hydrostatic pressure (HP, ph ≡
0 rgdz with sea surface being at z = 0 and g = 9.8m/s2
z
being the gravitational acceleration), and nonhydrostatic
pressure (NP). The last of these quantities is necessary to
producetrainsofISWs.TheD(mixing)termsarediscussed
twoparagraphsbelow.Thecontinuityequation(3)takesinto
account the variable width of the channel, so that our strat-
egy is analogous to that used by Brandt et al. [1997] who
presentedatwo‐layernonhydrostaticmodelfortheStraitof
Messina. As with Brandt et al. [1997], we do not consider
the rotation of the Earth, which is a limitation arising from
setting the channel width in the two‐dimensional model in
the vertical plane. All equations in the present manuscript
assumearigidseasurface,whichfacilitatesthederivationof
theenergyequations(thefactthatthemodelweuseforour
numerical experiments has a free surface does not compro-
mise the results of our energy diagnosis, see Appendix B).
[8] The set of equations (1)–(4) is discretized in (x, z)
coordinates and stepped forward using the numerical code
of Aiki and Yamagata [2004] and Menesguen et al. [2009].
Thehorizontalgridspacingis140m.Atotalwaterdepthof
Figure 3. Curvilinear orthogonal coordinates for the water 4200miscoveredby80depthlevelswithauniformvertical
regions of Lombok Strait with idealized coastlines. grid spacing of 10 m for the top 600 m. Grid spacings at
greater depth increase monotonically.
[9] The terms DU, Dw, and Dr in (1)–(4) represent
aspects of the energy diagnosis of internal waves in the the subgrid‐scale mixing of U, w, and r, respectively. The
presence of time mean flows. subgrid‐scale horizontal mixing is parameterized by the
standard Smagorinsky [1963] scheme with a nondimen-
sional coefficient of 0.1 after squaring, which is a value
2. Model often used in high‐resolution coastal ocean models. The
Prandtl number is set to unity as the horizontal mixing
[6] We consider fluid motion in a vertical plane aligned parameter. The subgrid‐scale vertical mixing is parameter-
along the main axis of LS (the dashed line in Figure 1b). ized by setting the hyperviscosity to −1.0m4/s and the
The model strait (hereafter channel) consists of a realistic hyperdiffusivity to−0.01m4/s,whichsuggest10−4(1/s)and
bottomprofileandanidealizedwidthprofileasfollows.The 10−6(1/s)forthedampingratesofgridnoiseinvelocityand
bottom depth is given by the General Bathymetric Chart of
tracer fields, respectively. The bottom friction is parame-
the Ocean. The channel width L[x](>0) is idealized in the
terized using a quadratic stress law with a standard nondi-
form shown in Figure 3. The sill is located at x = 0 km
mensional drag coefficient of 0.0025.
wherethewaterdepthandthechannelwidthare270mand
30 km, respectively.
[7] Themodelisbasedonthetwo‐dimensionalequations 2.1. Forcing
intheverticalplanederivedbyassumingradialpropagation [10] The model is forced by the barotropic volume fluxes
ofwavesawayfromthesillandtakingthewidthaverageof through the open boundaries located at x = ±200 km. The
the three‐dimensional nonhydrostatic incompressible Bous- boundaryfluxesoscillateattheM tidalperiod(12.42h≡t).
2
sinesqequations(AppendixA).Theequationscanbewritten Theamplitudeofthefluxesisbasedontheoutputofa1/24°×
intheform 1/24° resolution barotropic inversion model for the North
Australian region (Egbert and Erofeeva, http://volkov.oce.
(cid:1)0ðUtþUUxþwUzÞ¼(cid:2)pxþDU; ð1Þ orst.edu/tides/region.html). This inversion model gives bar-
otropicvolumetransportsof8.0,3.7,2.7,and1.6Sv(1Sv=
106 m3/s) for the M , S , K , O tides through LS. The
(cid:1)0ðwtþUwxþwwzÞ¼(cid:2)pnz þDw; ð2Þ simplest way to repr2odu2ce 81.0Sv1for the barotropic tidal
transport“atthesill”inourmodelistosettheopenboundary
ðLUÞ þðLwÞ¼0; ð3Þ fluxesat8.0Svwithnophasedifferencebetweenthefluxes
x z
on the two sides of the sill. This approach has been con-
ventionally adopted in the modeling of tidal internal waves
(cid:1)tþU(cid:1)xþw(cid:1)z¼D(cid:1); ð4Þ with a rigid sea surface. Although our model has a free sea
surface, we follow the above approach for simplicity. The
whereU[x,z,t]andw[x,z,t]arethenorthwardandupward
barotropictidalforcingof8.0Svresultsinabarotropic(i.e.,
componentsofvelocityandr andr[x,z,t]arethereference
0 depthaveraged)velocityoverthesillof0.99m/s.Aradiation
and model density of sea water. See Table 1 for an expla-
boundary condition for the baroclinic part of the flow is
nationof the symbols used in the present paper. The term
given attheopenboundaries.
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Table1. Listof Symbols 2.02 m/s on the Indian Ocean side where the maximum
bottom depth is 4200 m. These values are close to an
Symbol Description observed propagationspeed of2.01m/s = 90 km/t whichis
(x,y) Cartesiancoordinatesinthehorizontalplane estimatedfromtheseparationdistanceofwavearcsAandB
(x,h) curvilinearorthogonalcoordinatesinthe inFigure1a.Theobservedpropagationspeedisanonlinear
horizontalplane
(hx,hh) positivescalefactorsin(x,h)coordinates propagation speed and hence larger than the linear long
xs positionofthesillofLombokStrait wave speed. There are buffer zones next to the open
hl positionofthecoastlineoftheLombokisland boundaries where the model density r is restored to the
hb R positionofthecoastlineoftheBaliisland initialdensityriwithinatimescaleoftthroughoutthetime
L≡ (cid:3)b hhdh channelwidth
(cid:3)l integration.
H R bottomdepth
Ae≡ (cid:3)b Ahhdh/L widthaveragealongthelineofafixedx
(cid:3)l
(Aisanarbitraryquantity)
A″≡A−Ae deviationfromthewidthaverage 3. Experiments
(comparedatfixedx,Af′′≡0) [12] We have performed two main experiments, one
A Euleriantimemean
A′≡A−A deviationfromtheEuleriantimemean(A′≡0) without and one with an idealized ITF, corresponding to
(u,v) horizontalcomponentsofvelocityin(x,y) conditions in boreal winter and summer (or in El Niño and
coordinates La Niña years), respectively. Such sensitivity experiments
(U,V) horizontalcomponentsofvelocityin(x,h) havebeguntoattractattentioninrecentnumericalstudiesof
coordinates ISWsintheotherregionsoftheworld’soceans,suchasthat
w verticalcomponentofvelocity
(Ue;we) componentsofwidth‐averagedvelocityin ofWarn‐Varnasetal.[2010]whoinvestigatedthepossible
(x,z)and(x,z)coordinates impact of the variability of the Kuroshio current on tidally
(Uebt,weRbt) barotropiccomponentof(Ue;we) generatedISWsintheSouthChinaSea.Foreachofourtwo
wUeebbtt≡≡(z(cid:2)/0HH)UUeedHz/H bbaarroottrrooppiicc(cio.em.,pdoenpetnhtaovfewera:g(eLdUe)bcto)m+po(nLewenbtt)of=Ue0 experimentsthemodelwasintegratedfor25M2cyclesuntil
(Uebc,webc) x barocliniccomponentof(Ue;we) x z a statistically equilibriated state was achieved, although the
Uebc≡Ue −Uebt baroclinic(i.e.,residual)componentofUe basicfeaturesofthetidalwaveswereobtainedwithinafew
webc≡we −webt barocliniccomponentofwe:(LUebc) +(Lwebc) =0 M cycles of integration.
x z 2
r totaldensityofseawater
r referencedensityofseawater 3.1. Case Without the ITF (Boreal Winter)
0
g gravityacceleration
ps R seasurfacepressure(SP) [13] Thisexperimentwasstartedfromaninitialcondition
ph≡ 0 grdz hydrostaticpressure(HP) ofnomotion.ThebarotropiccomponentofUoverthesillis
pn z nonhydrostaticpressure(NP) indicatedbytheblacklineinFigure4a,wherethemaximum
p≡ps+ph+pn totalpressure southward and northward tides occur at t = 24.25t and
M≡ps+ph+rgz Montgomerypotential(MP) 24.75t, respectively.
rz densityofthebackgroundstratificationrz=rz=0
ra≡ra−rz densityanomaly x t [14] Snapshots of r and U at maximum southward and
ragz R potentialenergyanomaly(PEA) northward barotropic tidal currents are plotted in Figure 5.
phz≡ 0 grzdz hydrostaticpressurebasedonbackground OntheJavaSeasidethethermoclinehasdepressionsAand
z stratificationphz=phz=0
x t
pha≡ph−phz hydrostaticpressureanomaly(HPA)pha=−gra
z
ri initialdensity
H referencethicknessofupperlayer
1
t≡12.42h periodofM tide
2
2.2. Initial Condition
[11] Theinitialdensityfieldissetbyaquarticexponential
vertical profile: ri ≡ r − Dr exp[−0.7(z/H )4] whereDr is
0 1
the density difference across the thermocline, and H > 0 is
1
the upper layer reference thickness. (Although the standard
Gaussian function (i.e., exp[−z2]) might be closer to the
observed density profile, we used the quartic Gaussian
function (i.e., exp[−z4]) so that the stratification can be
regarded as a two‐layer model, which helps simplify the
vertical profile of the model ITF (i.e., a vertically uniform
southward flow in the upper layer and no flow in the lower
layer, see section 3.2). The qualitative nature of the model
result will be unchanged whichever of the standard and
quartic Gaussian functions is used.) We set H = 150 m
1
based on the observed velocity profile of the ITF passing
Figure4. TimeseriesofthecombinedwidthRanddepthinte-
through LS [Sprintall et al., 2009] and the thermocline gralofthepotentialenergyanomaly(PEA)L 0 (r−ri)gzdz
depth intheWorld Ocean Atlas of2005. Wethen choose a (color,1012J/m)andthedepRthaverageofthen(cid:2)orHthwardcom-
valueofDr=3.50kg/m3sothatthelinearlongwavespeed 0
ponentofvelocityatthesill Udz/Hatx=0km(blackline,
of the first baroclinic mode becomes 1.93 m/s on the Java (cid:2)H
m/s)duringthe25thM cycleintheexperiments(a)without
Sea side where the maximum bottom depth is 1400 m and 2
theITFand(b)withit.
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Figure 5. The result of an experiment without the ITF. Distributions of the northward component of
velocity(color)anddensity(contours)atthemaximum(a)southwardand(b)northwardbarotropictides
in the 25th M cycle (i.e., t = 24.25 and t = 24.75 in Figure 4, respectively).
2
Bwhicharegeneratedbyanorthwardtideoverthesill.The uniformbottomdepthof1400minregionsofx>100kmin
leadingedgeofeachdepressionisfollowedbyawavetrain ordertoeliminatesouthwardwavereflectionscausedbythe
withwavesabout4kminwavelengthwhichisintherange northward shoaling of high‐amplitude waves.) In this
oftheobservedwavelengths.Thethermoclinedisplacement experiment only the asymmetric bottom topography of LS
gradually decreases as waves propagate away from the sill.
However, Figure 4a shows that the combined width and
depth integral of thepotential energy anomaly (PEA = (r−
ri)gz) for each wave increases slightly during propagation,
suggestingthatenergyisstillbeingtransferredfrominternal
tides to ISWs. (This transfer from low‐ to high‐frequency
internal waves is not isolated in the energy diagnosis in
section4.)TheleadingwaveofeachwavetrainontheJava
Sea side propagates at 1.97 m/s (in regions between x =
40kmandx=160km),whilethatontheIndianOceanside
propagatesat2.11m/s.Thesespeedsareslightlyfasterthan
the linear long wave speed. The slight deceleration in the
regions of x > 160 km in Figure 4a reflects the shoaling of
waves as they approach the northern end of the basin
(Figure 1b).
[15] These topographically generated wave trains are
more pronounced on the Java Sea side of the sill. A large
number of ISWs form in the northward propagating wave
trains, which is consistent with observations [Mitnik et al.,
2000; Susanto et al., 2005; Ma11]. In the south only a
singleISWisgenerated,whichisgenerallymoredifficultto
identify as an ISW event in remote sensing imagery. In
order to examine the reason for this directional asymmetry, Figure6. (a)SameasFigure4aand(b)sameasFigure5a
we performed a supplementary experiment with exactly the except for the result of an experiment using a uniform
same settings as the original experiment except that a uni- channel width of 30 km (also a uniform bottom depth of
form channel width of 30 km was used. (We also used a 1400 m has been used in regions of x > 100 km).
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Figure 7. Same as Figure 5 except for the result of an experiment with the ITF.
is capable of causing an asymmetry between wave trains [17] Thenorthwardpropagatingwavetrainsaresimulated
generated north and south of the sill. The result in this case again but with considerably smaller amplitudes (trough‐to‐
is that more than ten ISWs develop within each wave train peakheightisabout20minregionsaroundx=150km,see
on both the Indian Ocean side and the Java Sea side A′ in Figure 7a) than the corresponding events in the
(Figure6).Theamplitudeofthedepressionsisabout100m experiment without the ITF where the trough‐to‐peak
(trough‐to‐peak) on both sides of the sill, indicating that heightisabout30minregionsaroundx=150km(seeBin
flows and waves are enhanced in this uniformly narrow Figure5a).Thedecreaseofwaveamplitudescanbeseenin
channel. On the other hand, in spite of the increased wave the distributions of thePEA (Figure 4)and, as Ma10 show,
amplitude,thewidthintegralofthePEAshowninFigure6a in the horizontal divergence of velocity, which has clear
ismuchsmallerthanthatinFigure4a.Theseresultssuggest implications for the detection of the surface signals by sat-
theimportance of taking into account the variable width of ellite‐borne SARs. In this “ITF on” experiment, the wave
the LS region for reproducing the observed asymmetry of trains have an increased wave length of about 6 km. The
the wave trains. The narrow channel width in the regions depressionsA′andB′arelocatedclosertothesill,owingto
between x = 0 and 50 km (Figure 3) of the original experi- the strong counter advection by the ITF near the sill. Once
ment induces a megaphone‐like amplification of the north- the northward propagating waves reach regions that are
ward propagating depressions, resulting in the formation sufficiently removed from the sill (x > 80 km), they prop-
of many ISWs in each wave train. agate at speeds that are somewhat slower than those in the
first experiment without the ITF. On the other hand, the
3.2. Case With the ITF (Boreal Summer) presence of the ITF speeds up the southward propagating
wavesontheIndianOceansideofthesill(Figure4b).Two
[16] This experiment introduces a vertically uniform
to three ISWs are formed in each wave train.
southward ITF in the upper 150 m layer, by changing both
the initial and open boundary conditions. This current lies [18] Concerning the Doppler shift by the ITF, below we
compare two types of length scale of tidal internal waves.
above the sill which is at a depth of 270 m. The magnitude
The first length scale is the separation distance between the
of the ITF is set to 5.2 Sv, which is twice the observed
leading edge of depressions (about 90 km), which can be
multiyear mean volume transport of 2.6 Sv through LS
explained by the Doppler shift. In fact our experiment with
between 2004 and 2006 [Sprintall et al., 2009]. The sea-
the ITF shows that the separation distance is decreased
sonaldifferenceintheobservedLombokITFisabout5.0Sv
upstream the sill (the Java Sea side) and increased down-
in2005and2006,butissmallerin2004[seeSprintalletal.,
stream the sill (the Indian Ocean side). The second length
2009, Figure 9a]. The phase of the barotropic tide is
scale is the wavelength of each ISW (about 5 km), which
unchanged as shown by the black line in Figure 4b.
cannot be explained by the Doppler shift. In fact our
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Figure 8. Energy diagram based on (C7) and (C12)–(C15) with the overbar symbol representing the
Euleriantimeaverage.Thetriplelinesindicateenergyfluxesthroughthelateralboundariesoftheanalysis
domain. The red and green shaded boxes represent the definitions of barotropic and baroclinic energies
used in previous studies. See section C1.
experiment with the ITF shows that the wavelengths of the By using (1)–(3) and (5) one can derive equations for the
ISWs are increased upstream of the sill (the Java Sea side). barotropicandbaroclinicKEsandPEA.Theresultingenergy
ThisisnotnecessarilysurprisingasthewavelengthofISWs budget is illustrated by the diagram in Figure 8 where the
is set by the wave amplitude, stratification, and the vertical buoyancy conversion term graw is decomposed into terms
shear of the ITF [cf. Miyata, 1985; Brandt et al., 1997]. associated with the barotropic and baroclinic components of
velocity defined by wbt ≡ UbtRHxz/H and wbc = w − wbt
(sectionC1).Thesymbolsphz(≡ 0grzdz)andpha(≡ph−phz)
z
4. Energy Diagnosis represent the hydrostatic pressure (HP) associated with the
background stratification and that related to the anomaly,
[19] The purposes of the energy diagnosis are to (1)
respectively (the latter of which is hereafter abbreviated as
quantify the rate at which wave energy is radiated from the
HPA).Theenergybudgetisdiagnosedbysettingrz=riand
LS region and (2) distinguish the energetics of waves and
takingavolumeintegralintheregionbetweenx=±120km.
time mean flows.
In the following we use the word ‘flux’ to indicate the net
[20] The barotroRpic and baroclinic components of U are
defined by Ubt ≡ 0 Udz/H and Ubc ≡ U − Ubt, respec- energy flux through the lateral boundaries at x = ±120 km.
(cid:2)H Thediagnosisisalsobasedonatimeaverageforthelastfive
tively.Thenthekineticenergy(KE)ineachverticalcoluRmn
can be separated into (cid:1)0H∣Ubt∣2 (barotropic KE) and (cid:1)0 0 M2 cycles of the integration period. We use the overbar and
2 2 (cid:2)H prime symbols to represent the Eulerian time mean and
∣Ubc∣2 + ∣w∣2dz (baroclinic KE) where H[x] is the bottom
perturbation components, respectively, so that
depth [Holland, 1975; Baines, 1982; Cummins and Oey,
1997; Niwa and Hibiya, 2004; Carter et al., 2008; Aiki A½x;z;t(cid:3)¼A½x;z(cid:3)þA′½x;z;t(cid:3); ð6Þ
et al., 2011]. For simplicity we do not separate ∣w∣2 into
barotropic and baroclinic component as our numerical for an arbitrary quantity A[x, z, t].
results indicate that ∣w∣2 is 2 orders of magnitude smaller
4.1. Case Without the ITF (Boreal Summer)
than the other kinetic energy terms and hence insignificant.
[21] In order to derive an equation for the PEA, we [22] The energy cycle starts from the combined baro-
decompose densityinto that associated with the background tropic SP and HPA flux through the lateral boundaries at
stratificationandwiththeanomaly:r[x,z,t]≡rz[z]+ra[x,z,t] x = ±120 km. This represents some 3.86 GW (Figure 9a,
wheretheprofileofrzisarbitrary.Thedensityequation(4) 1GW=109W).Aboutthreequartersofthisisconvertedinto
becomes PEA(2.96GW),withtherestusedtomaintainthebudgetof
barotropic KE (0.89 GW). The barotropic and baroclinic
(cid:1)aþU(cid:1)aþw(cid:1)aþw(cid:1)z ¼D(cid:1): ð5Þ buoyancy terms roughly offset each other (+2.96 GW and
t x z z
7 of 17
C03023 AIKI ET AL.:ENERGY FLUX OFLOMBOK INTERNAL WAVES C03023
Figure 9. Energy conversion rates (GW) based on the energy diagram of Figure 8 in the experiments
(a) without the ITF and (b) with it. The values are obtained by taking a volume integral in the region
betweenx=±120kmwithatimeaverageforthelastfiveM cyclesoftheintegrationperiod.Theresidual
2
oftheenergybudgetofeachboxrepresentsthetendency.ForeachofthecombinedbarotropicSPandHPA
flux, the baroclinic HPA flux, and the PEA flux, contributions from the mean correlation and the wave
correlationarecomputed(i.e.,UA=U A+U′A′),asshownbythefirstandsecondvaluesineachbracket,
respectively, and are eventually rearranged in Figure 11. The mean correlation components of quantities
denotedbybluepathsaresensitivetothechoiceofthebackgroundstratificationasarguedinsectionC1.
−2.56 GW with the sign referring to the PEA budget). The wavenumbers (e.g., generation of ISWs and wave breaking
energycycleisterminatedbytwomajorsinks.Thefirstsink overthesill)andhenceleadtomorehorizontaldissipation.
is the baroclinic HPA flux (2.01 GW) which is due to the
radiation of internal gravity waves through the lateral 4.2. Case With the ITF (Boreal Summer)
boundariesatx=±120km.Thesecondsinkisthedissipation [25] Nonlinearandnonhydrostaticeffectsareenhancedin
ofbaroclinicKEbythehorizontalviscosityterm(0.77GW). this experiment (Figure 9b). The advective conversion
[23] Itisworthnotingthesignificanceofnonlineareffects
as revealed by two separate quantities. The first nonlinear Table 2. Comparisons of the Energy Budgets of Tidal Flows in
quantity is the advective conversion between the barotropic Lombok StraitandHawaiian Islands WithaUnit of109Wa
and baroclinic KEs (0.49 GW), which originates from the
LombokStrait HawaiianIslands
advection term of the horizontal momentum equation (1).
ThesecondnonlinearquantityisthefluxofPEA(0.45GW), Model,resolution 2‐D,dx=140m 3‐D,dx’1000m
which originates from the advection term of the density Waterarea 0.9×1011m2 1.2×1011m2
Typeoftide M2 M2
anomaly equation (5).
[24] The above results can be compared with the energy BarotropicEnergyBudget
budgetofM2tidesintheHawaiianIslandsregionasestimated Tendency 0.00 −0.006
byCarteretal.[2008,hereafterCM08]usingacomprehen- r·Flux +3.86 +2.733
sive three‐dimensional hydrostatic model (Table 2). Their Advection −0.49 +0.005
Conversion −2.96 −2.286
experiment has no geostrophic background current, thus
Dissipation (−0.26−0.13=)−0.39 −0.163
couldbecomparedwithourexperimentwithouttheITF.The Nonhydrostatic −0.01 None
barotropic energy input to the Hawaiian Islands region is Error 0.00 −0.296
2.733 GW of which 2.286 GW is converted to PEA, and
BaroclinicEnergybudget
then radiated away by internal gravity waves at a rate of
Tendency 0.00 +0.054
1.701 GW. These values are comparable to our estimates in r·Flux −2.01 −1.701
theLSregion.CM08commentthattheeffectofadvectionis Advection (0.49+0.03–0.45=)+0.07 −0.016
underestimatedintheirmodelbecauselittleenergyispresent Conversion +2.96 +2.340
inotherconstituents.Thebaroclinicadvectioninourmodelis Dissipation (−0.17−0.77−0.03+0.02=) −0.445
−0.95
alsosmall(0.07GW),howeverhereitisanartifactresulting
Nonhydrostatic +0.01 None
from the cancelation of the advective conversion (between Error −0.08 −0.125
barotropicandbaroclinicKEs,0.49GWinFigure9a)andthe
PEAflux(−0.45GW).Ontheotherhand,whileitisnotclear aEnergy budgets of tidal flows in Lombok Strait are adapted from
Figure 9a of this paper, and those in the Hawaiian Islands are adapted
whetherhorizontal orverticalmixing is dominantinCM08,
fromTable2ofCarteretal.[2008].Thenamesoftheconversionfluxterms
the dissipation of baroclinic KE in our model is caused followthatusedbyCarteretal.[2008],whodefinedthebarotropicenergy
mainlybythehorizontalmixingterm.Nonhydrostaticmodels asthesumofbarotropicKEandthePEassociatedwithseasurfaceheight
can more accurately resolve signals at higher horizontal (thelatterofwhichisabsentinourformulation),withthebaroclinicenergy
beingthesumofbaroclinicKEandPEA.
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C03023 AIKI ET AL.:ENERGY FLUX OFLOMBOK INTERNAL WAVES C03023
Figure10. SameasFigure8exceptthatitisbasedon(C7),(C14),and(C17)–(C18).TheMontgomery
potential (MP) is defined by M ≡ ps + ph + rgz. See section C2.
betweenbarotropicandbaroclinicKEsis0.66GWwiththe Figures9aand9b)issensitivetochoicesofthebackground
ITF in comparison to 0.49 GW without it. The nonhydro- stratification. There is no such sensitivity in the wave cor-
static pressure (NP) conversion between barotropic and relation component, because A′(cid:1)a′ = A′(cid:1)′ for an arbitrary
baroclinic KEs is 0.12 GW with the ITF (0.01 GW without quantity A. Thesesensitivity issues are fundamental aspects
it). These increased inputs into baroclinic KE are canceled oftheenergydiagnosisofinternalwavesinthepresenceof
byincreasesin(1)dissipationbythehorizontalandvertical time mean flows (section C1).
viscosity term (1.29 GW and 0.42 GW with the ITF in [28] In order to avoid the above complexity (i.e., sensi-
contrast to 0.77 GW and 0.17 GW without it, respectively) tivity to the background stratification) and also to focus on
and(2)thefluxofbaroclinic KE(0.17GWwith theITFin estimating the radiation rate of the overall internal wave
contrast to −0.03 GW without it). fluxes, below we use another energy diagram which is
[26] We find a decrease in the baroclinic HPA flux simple and robust in that each conversion/flux term is
(1.28GWwith,incontrastto2.01GWwithout).However,in independent of the definitions of the background stratifica-
the presence of time mean flows such as the ITF, it is nec- tion and the anomaly (Figure 10 and section C2). The PEA
essarytodistinguishcontributionsfromthemeancorrelation budget is given by the flux divergence of Montgomery
component and the wave correlation component for each potential (MP) defined by M ≡ ps + ph + rgz. The MP has
flux/conversionterm.NamelyAB=A B+A′B′isconsidered beenusedinmanystudiesoflarge‐scaleorfiniteamplitude
for arbitrary quantities A and B. For simplicity such a geophysical flows [Montgomery, 1938; Andrews, 1983;
diagnosis has been done for only three separate quantities: Pedlosky, 1996; Aiki and Richards, 2008].
the combined barotropic SP and HPA flux, the baroclinic [29] The energy input into the barotropic tides is repre-
HPA flux, and the PEA flux, as shown by the bracketed sented by the barotropic wave MP flux whose rate is
values in Figures 9a and 9b. Now the wave correlation 4.05GWwiththeITFandslightlyless,3.81GW,withoutit
componentof thebaroclinic HPAflux is1.78GW with the (Figure 11). Likewise the energy radiation by finite ampli-
ITF,andiscloseto1.92GWwithoutit.Likewisethewave tude and nonlinear internal gravity waves is represented by
correlationcomponentofthePEAfluxis0.69GWwiththe the baroclinic wave MP flux (i.e., the sum of the baroclinic
ITF,andiscloseto0.93GWwithoutit.Inbothexperiments waveHPAfluxandthebaroclinicwavePEAflux)atarate
the wave PEA flux has the same sign as and about half the of 2.42GW with theITF and a rate of 2.82 GW without it.
magnitude of the baroclinic wave HPA flux. Although the two experiments give similar rates, the dis-
[27] On the other hand the mean correlation component tributions of the baroclinic wave MP flux indicate that the
of the baroclinic HPA flux provides an energy input of presence of the ITF alters the way that the radiated wave
0.50GW with theITF.The mean correlation component of energy is partitioned between the two sides of the sill.
thePEAfluxprovides anenergyinputof0.48GW without Without the ITF (black line in Figure 12d), the baroclinic
theITF.Howeverthesevaluesareartificialandnotsuitable waveMPfluxontheJavaSeaside(1.59GWatx=120km)
for physical interpretations, because the mean correlation is greater in magnitude than that on the Indian Ocean side
component of quantities involving either the background (−1.23GWatx=−120km).WhentheITFispresent(green
stratification rz or the associated anomaly ra (i.e., five line),however,thebaroclinicwaveMPfluxontheJavaSea
separate quantities representing the blue paths in each of side (0.57 GW at x = 120 km) is smaller in magnitude than
9 of 17
C03023 AIKI ET AL.:ENERGY FLUX OFLOMBOK INTERNAL WAVES C03023
Figure 11. Same as Figure 9 except for being based on the energy diagram of Figure 10.
thatontheIndianOceanside(−1.85GWatx=−120km).It 4.3. Discussion of Time Mean Flows
is interesting that the wave energy flux is less on the Java
[31] We explain the energy input by the mean MP flux
Sea side despite the fact that there are still more ISWs on
whichis0.40GWwithouttheITFand0.73GWwithit.The
thissideinFigure7.Thisisbecausethechannelisnarrower
flux can be decomposed into two components
on the Java Sea side (Figure 3), and the energy fluxes are
Z
cross‐channel integrated values. 0
(cid:2)L U Mdz
[30] Thesecharacteristicscanberoughlyconfirmedbythe
distributionsofthebaroclinicHPAfluxinFigure12bandare (cid:2)H Z 0 Z 0 (cid:2) (cid:3)
also consistent with the result of Ma11. In the real ocean, ¼(cid:2)L U psdz(cid:2)L U phþ(cid:1)gz dz
wmaicvreosscaowpaiycmfroixmingth.eWseillexwpoeucltdsubceherveemnotutealdlyisdsiipssaitpioantetdobbye ¼(cid:2)LH(cid:2)UHbt ps(cid:2)(cid:6)L(cid:4)Z(cid:2)zHUd€z(cid:5)(cid:2)phþ(cid:1)gz(cid:3)(cid:7)(cid:8)(cid:8)(cid:8)(cid:8)z¼0
sInigdniaifnicOancteoanntshiedJeavinaStheeabsiodreeainltshuembmoreera.lFwuirnthteerramnodroe,nothuer Z 0 (cid:4)Z z (cid:5)(cid:2)(cid:2)H (cid:3) z¼(cid:2)H
findingsarerelevanttotheissueofElNiño/LaNiñamodu- þL Ud€z phþ(cid:1)gz dz
lation of the ITF and subsequent regulation of the radiated (cid:2)H (cid:2)H Z (cid:4)Z (cid:5)z
0 z
internalwavefield,asconsideredbyMa11. ¼(cid:2)LHUbt psþL Ud€z (cid:1)gzdz; ð7Þ
z
(cid:2)H (cid:2)H
Figure12. Distributionsofvariousenergyfluxes(109W,positivenorthward)intheexperimentswithout
the ITF (black) and with it (green): (a) the combined barotropic SP and HPA flux (solid line,
R R
L 0 UbtðpsþphaÞdz) and the barotropic SP flux (dashed line, L 0 Ubtpsdz), (b) the baroclinic HPA
(cid:2)H R R (cid:2)H
fluxL 0 Ubcphadz,(c)thebarotropicwaveMPflux(solidline,L 0 U′btM′dz)andthebarotropicwave
(cid:2)H R (cid:2)H R
SP flux (dashed line, L 0 U′btps′dz), and (d) the baroclinic wave MP flux L 0 U′bcM′dz, all of which
(cid:2)H (cid:2)H
are time averaged for the last five M cycles of the integration period.
2
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