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JULY2013 SKIBA ET AL. 1301
On the Resonance and Shelf/Open-Ocean Coupling of the Global Diurnal Tides
AARONW.SKIBA*
DepartmentofAerospaceEngineering,UniversityofMichigan,AnnArbor,Michigan
LIBOZENG*
DepartmentofPhysics,UniversityofMichigan,AnnArbor,Michigan
BRIANK.ARBIC
DepartmentofEarthandEnvironmentalSciences,UniversityofMichigan,AnnArbor,Michigan
1
MALTEMU¨LLER
SchoolofEarthandOceanSciences,UniversityofVictoria,Victoria,BritishColumbia,Canada
WILLIAMJ.GODWIN#
DepartmentofPhysicsandCenterforOcean–AtmosphericPredictionStudies,TheFloridaStateUniversity,
Tallahassee,Florida
(Manuscriptreceived17March2012,infinalform24November2012)
ABSTRACT
Theresonanceofdiurnaltidalelevationsisinvestigatedwithaforwardoceantidemodelruninarealistic
near-globaldomainandasynthesisoffreeoscillations(normalmodes)computedforrealisticglobalocean
geometriesandoceanphysics.Asapreludetoperformingtheforwardoceantidesimulations,thetopographic
wavedrag,whichisnowcommonlyemployedinforwardoceantidemodels,istunedspecificallyfordiurnal
tides.Thesynthesisofglobalfreeoscillationspredictsreasonablywelltheforwardoceandiurnaltidemodel
sensitivitytochangesinthefrequency,zonalstructure,andmeridionalstructureoftheastronomicaldiurnal
tidalforcing.Threeglobalfreeoscillationsthatareimportantforunderstandingdiurnaltidesasasuperposition
offorced-damped,resonant,freeoscillationsareidentified.Anadmittanceanalysisofthefrequencysweep
experimentsdemonstratesthatsomecoastallocationssuchastheSeaofOkhotskareresonanttodiurnaltidal
forcing.Asinearlierworkdonewithsemidiurnaltides,aseriesofsimulationsareperformedinwhichregions
possessingsignificantcoastaldiurnaltidesareblockedout.Thelargestperturbationstotheopen-oceandiurnal
tidestakeplaceinBlockedSeaofOkhotskexperiments.Lesserbutstillsignificantperturbationsalsoarisefrom
theblockingoutofotherregionsoflargediurnaltidalelevationsordissipation.Interpretationoftheresultsis
mademorecomplex,however,bythefactthatsubstantialperturbationsalsoarisefromblockingoutregions
where neither tidal elevations nor dissipation are large. The ‘‘blocking’’ experiments are relevant to un-
derstandingtidesoftheiceage,duringwhichlowersealevelsentailareducedareaofcontinentalshelves.
1. Introduction
*Theseauthorscontributedequallytothiswork.
1Current affiliation: Research and Development Department, Thispaperexaminestheresonanceofdiurnaltidesin
NorwegianMeteorologicalInstitute,Oslo,Norway.
the open and coastal oceans and the coupling between
#Currentaffiliation:DepartmentofNuclearandRadiological
theopen-oceanandcoastaldiurnaltides.Wearemoti-
Engineering,UniversityofFlorida,Gainesville,Florida.
vated by the inherent interest in understanding the
global tidal system and by recent studies which
Correspondingauthoraddress:Dr.BrianK.Arbic,Department
stronglysuggestthattidesoftheiceage,duringwhich
ofEarthandEnvironmentalSciences,UniversityofMichigan,
lower sea levels implied a much reduced area of
AnnArbor,MI48109-1005.
E-mail:[email protected] continental shelves, were much larger than those of
DOI:10.1175/JPO-D-12-054.1
(cid:1)2013AmericanMeteorologicalSociety
1302 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME43
today (e.g., Thomas and Su¨ndermann 1999; Egbert vL
pffiffiffiffiffiffiffi5np, (1)
et al. 2004; Arbic et al. 2004b, 2008; Uehara et al. gH
2006; Griffiths and Peltier 2008, 2009; Green 2010;
Hilletal.2011). wherenisanintegerandvisthetidalforcingfrequency
The topic of tidal resonance has a long and rich his- (e.g., Proudman 1953; Defant 1961; Godin 1988). A
tory.Theopen-oceantideshavebeenarguedtobenear simple model that has been used to explain regions of
resonant in both admittance studies of observations large resonant coastal tides consists of a small non-
(Wunsch 1972; Garrett and Greenberg 1977; Heath rotating basin with one end closed (the continent) and
1981)andinstudiesnotingthesimilaritiesbetweenfree the other end open to, and forced by, the open ocean.
oscillations computed in both idealized and realistic Theclassicalquarter-wavelengthresonancearisesinthe
geometries(Rao1966;Longuet-HigginsandPond1970; lattercasewhen
Platzmanetal.1981;Platzman1991;ZahelandMu¨ller
2005;Mu¨ller2007)andtheactualoceantides.Anumber pvffiffiLffiffiffiffiffi5(2n21)p, (2)
of studies have argued that selected coastal regions gH 2
aroundtheglobeareresonanttosemidiurnalordiurnal
tidalforcing(e.g.,Garrett1972;Clarke1991;Sutherland (e.g.,Proudman1953;Defant1961;Godin1988).Allof
etal.2005;Arbicetal.2007;Cumminsetal.2010,among thisimpliesthatg,theforcingfrequencyv,waterdepth
others). H,andbasin(orcoastal)scaleLarecriticalcontrolson
Here, as in two previous papers, we add to the dis- tides. A more complete description of tidal resonance
cussion of tidal resonance through experimentation must account for the effects of rotation as well as the
with global forward tide models. Arbic et al. (2009, values of g, v, H, and L. Gravity waves propagating
hereafter AKG) and Arbic and Garrett (2010, here- along boundaries in a rotating basin take the form of
after AG) explored the resonant attributes of the Kelvinwaves,whicharehighlyprominentinmapsand
globalsemidiurnal tides.Followinguponourworkin animationsofbothsemidiurnalanddiurnaltides.Inthe
AKG andAG,hereweconductasystematicexplora- presence of rotation, the resonantnormal modesof an
tionofresonanceinaforwardnear-globalmodelofthe idealizedbasinarequitedifferentfromthoseinanon-
diurnal tides. The forward ocean tide model simula- rotatingbasin(Rao1966).Ofcourse,thenormalmodes
tions are interpreted primarily with a model of free of the ocean computed under realistic geometries, and
oscillationscomputedforrealisticglobaloceangeom- utilizedinthispaper,accountfortheCoriolisforcedue
etriesandoceanphysicsandincludingfrictionalterms totheearth’srotation.
and the full self-attraction and loading effect (Mu¨ller Thestructureofthetidesisalsodependentuponthe
2007). Note that the set of free oscillations are often spatial structure in the astronomical tidal forcing. The
called ‘‘normal modes’’, and we will at times refer to equilibriumtidalforcingofthelargestsemidiurnaltides
the globalfreeoscillationsmodelasa‘‘modalsynthe- intheocean(M ,S ,N ,andK )isproportionaltothe
2 2 2 2
sis’’model.Notealsothatboththeforwardoceantide sphericalharmonicY2(f, l), wherefislatitudeandl
2
modelandthesetoffreeoscillationsare‘‘numerical.’’ is longitude (Laplace 1775, 1776; Cartwright 1977;
Toavoidconfusion,inthispapertheterm‘‘numerical’’ Hendershott1981;ArfkenandWeber2001).Theequi-
will not generally be used in the description of either librium tidal forcing of the largest diurnal tides in the
model. ocean(K ,O ,P ,andQ )isproportionaltothespherical
1 1 1 1
Ocean tides are impacted by several factors. For harmonic Y1(f, l) (Laplace 1775, 1776; Cartwright
2
instance, they are influenced by solid-earth body 1977;Hendershott1981;ArfkenandWeber2001).Res-
tides and the self-attraction and loading (SAL) term onantfreeoscillationsoftheoceanhavecharacteristic
(Hendershott 1972), as well as by damping (quadratic spatial structures and frequencies (Platzman 1991;
bottom boundary layer drag and topographic internal Mu¨ller2009)whosealignmentwiththespatialstructure
wavedrag).Thesefactorswillbedescribedlaterinthe and frequency of the astronomical tidal forcing de-
paper.Belowwedescribetheimpactofoceaniclength terminestheoceanicresponsetotidalforcing.
anddepthscales. Thefactorsthatcontroldiurnaltideswillbeexamined
Ocean depths and length scales exert a critical in- indetailhere.Intheforwardoceantidemodelwevary
fluenceonoceantides.ThewatercolumndepthHand thefrequencyv,zonalspatialstructure,andmeridional
gprffiaffiffivffiffiffiiffitational acceleration g set the tidal phase speed spatial structure of the astronomical diurnal tidal forc-
gH.AnonrotatingbasinoflengthLthatisclosedat ing.Thesetofglobalfreeoscillationsofferspredictions
both ends experiences the classical half-wavelength of tidal sensitivity to forcing frequency. The spatial
resonancewhen structures associated with the global free oscillations
JULY2013 SKIBA ET AL. 1303
alsopermitanexaminationofthesensitivitiesofdiurnal However, we will not utilize the simple interpretive
tides to both zonal and meridional spatial structure in modelsinthispapernearlyasextensivelyasinAKGand
theastronomicaltidalforcing. AG, in part for the sake of brevity, in part owing to
Becausetidesstronglydependuponwaterdepthsand areviewerwhoobjectedtothelackofrotationeffectsin
basinlengthscales,theyrespondsensitivelytochanges eithersimplemodel,andinpartduetothefactthateven
in coastal ocean geometry. An additional factor in the removal of regions of small coastal diurnal tidal eleva-
sensitivityofopen-oceantidestochangesincoastalre- tionsanddissipationcansignificantlyimpacttheopen-
gions is the substantial fraction of globally integrated oceantides(apredictionnotconsistentwiththesimple
tidaldissipationtakingplaceinsuchregions(Egbertand models). Note that recomputing the normal modes of
Ray 2003). As previously noted, forward tide models Mu¨ller (2009) under various blocking scenarios or
suggestthattidesoftheiceagesweremuchlargerthan other altered ocean geometries would be far too time
thoseof today.Duringice ages,lowersealevelsled to consumingtobepracticalforthepresentpaper.
theremoval ofsignificantareasofpresent-dayshelves. This paper is organized as follows. In section 2, we
Motivated by this reduction in shelf area, and by the discuss the modal synthesis (global free oscillation)
inherentinterestinunderstandingthecouplingofopen- model. Details of the forward ocean tide model, the
ocean and shelf tides, AKG and AG examined the re- tuning required to optimize its topographic internal
sponseofopen-oceansemidiurnaltidestotheblocking wavedragfordiurnaltides,anditsaccuracywithrespect
out of regions of significant coastal semidiurnal tides. to satellite-altimetry-constrained tide models, are pre-
Here we perform experiments in which regions of sig- sented in section 3. Section 4 presents the sensitivity
nificantcoastaldiurnaltidesareblockedout. teststothefrequenciesandspatialstructuresinthetidal
Itcanbedifficulttoascertainwhetherchangesingeo- forcing, conducted with the forward ocean tide model
metry or in dissipation lie behind the impacts of the and the set of global free oscillations. Section 4 also
blocking experiments. The blocking experiments for displaystheresultsoftheblockingexperiments.Finally,
semidiurnal tides in AKG and AG were interpreted section4demonstratesthattheSeaofOkhotskisreso-
with two highly simplified analytical models of open nant to diurnal tidal forcing and that blocking out the
ocean–coastal tidal coupling. The coupled oscillator SeaofOkhotskperturbstheopen-oceantidesmorethan
modelofAGconsiderstheshelfandtheopenoceanto blocking out other regions of similar area. In section 5
eachbeadampedsinglespringsystem;thetwoarethen wesummarizeourresults.
coupled together. The one-dimensional nonrotating
shallowwatermodelofAKGconsiderstheopenocean
andshelftobothbeboxes.Theirdynamicsarecoupled 2. Themodalsynthesismodel
becausethesmallerboxisopentothelargerboxonone
Oceantideshavebeendescribed(e.g.,Platzman1991)
side. In both of these analytical models, if the natural
as a superposition of global free oscillations. Here we
periods of the shelf and open ocean are both set to be
useasetofbarotropicglobalfreeoscillationsandtheir
near the forcing period, removal of the shelf leads to
adjoint counterparts computed on a 18 horizontal grid
substantial perturbations—generally, increases—in the
includingtheNorthPole,linearizedfrictionterms,and
open-ocean tides. This suggests that even a very small
fullself-attractionandloadingeffects(Mu¨ller2007).
oscillatorcanhaveaprofoundeffectonanearbymuch
In the synthesis procedure we write the kth free os-
larger oscillator it is coupled to, especially if both are
cillationasavector
near resonance. The AG and AKG analytical models
allow for both the effects of coastal resonance and x 5[z (f,l),u (f,l),y (f,l)], (3)
coastaltidaldissipationontheopen-oceantides;seefor k k k k
instanceEqs.(12)–(18)ofAG.
wherefislatitude,lislongitude,z istheseasurface
InthispaperwewillshowthattheSeaofOkhotskis k
elevation,u isthezonalvelocity,andy isthemeridi-
resonanttodiurnaltidalforcingandthatremovalofthe k k
onal velocity. The free oscillations are the solutions of
Sea of Okhotsk leads to larger perturbations to the
the homogeneous tidal equations, which are written in
open-ocean diurnal tides than does removal of other
generaloperatornotationequationas
coastal regions of similar area. This result is consistent
with the simple interpretive models of AKG and AG, ›
which continue to predict a large ‘‘back effect’’ of res- ›txke2ivkt1Lxke2ivkt50/(L2ivk)xk50, (4)
onant coastal tides upon the open ocean when their
governing parameters are changed from those suitable wheretheoperatorLrepresentsthelinearshallowwater
forsemidiurnaltidestothosesuitablefordiurnaltides. dynamicswithrotationeffectsincludedandv represents
k
1304 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME43
thekthcomplexeigenvalue,iv 5v 1iv ,withthe counterparts fx^ g every forced barotropic oscilla-
k k,1 k,2 k k51,‘
oscillatorypartv andthedampingpartv . tion can be obtained by superposition via Eq. (6).
k,2 k,1
The ocean tides are the solutions of the inhomo- However, a subset of oscillations in the period range
geneoustidalequations;thatis, from10to80 hissufficientforsynthesisofthediurnal
andsemidiurnaltides.Experimentsinwhichwevarythe
›x e2ivt1Lx e2ivt5F e2ivt forcing frequency v and the zonal and meridional
›t tide tide tide wavelengthsoftheforcingF arediscussedinsections
tide
/(L2iv)x 5F , (5) 4a,4b,and4c,respectively.
tide tide
whereF e2ivtrepresentsastronomicaltidalforcingof
tide
frequencyv.Generally,everytidalsolutionx canbe 3. Forwardoceantidemodel
tide
expressedbyasuperpositionoffreeoscillationsx :
k This section discusses the forward ocean tide model.
AsinAKGandAG,theforwardmodelisthetidemodel
‘
x 5 (cid:1) a x , (6) ofArbicetal.(2004a,hereafterAGHS).Afewdetailsof
tide k51 k k the model are given here; AGHS can be consulted for
full descriptions of model details including the prepa-
where a are the complex weighting coefficients. A di- ration of topography. In this paper we use a latitude–
k
rect computation of the weighting coefficients is only longitudegridrunningfrom868Sto828N,asinAGHS,
possiblewiththeknowledgeoftheeigenvectors^x ofthe AKG,andAG.Incontrasttomostofthesimulationsin
k
adjointeigenproblem AGHS, AKG, and AG, the simulations here use 1/88
horizontal resolution instead of ½8 resolution. Consis-
(Lb2iv )^x 50. (7) tentwithEgbertetal.(2004),inArbicetal.(2008)and
k k Mu¨lleretal.(2011)wefoundthatthisreducedtheerror
oftheforwardmodelmeasuredwithrespecttoaltimetry-
If dissipative termsareneglectedin thetidal dynamics
constrained models.To allow forspinup, all ofthesim-
the adjoint eigenvectors ^x are simply the complex
k ulations in this paper are run for at least 60 days, and
conjugates of their corresponding eigenvectors. This
analysis is undertaken on the latter part of the record
relationshipwasutilizedbyPlatzman(1984)foratidal
(last1–4days).Notethat,sincealloftheexperimentsin
synthesis procedure. However, in the present study
thispaperareforcedbyjustonefrequency,thereareno
dissipativetermsareconsidered,andthustheoperator
difficultiesinseparatingnearbyfrequenciesasthereare
L becomes nonself-adjoint and the adjoint eigensolu-
intheactualocean.
tions have to be determined separately. These pairs of
The governing equations for the forward ocean tide
eigensolutions are, when properly normalized, bio-
model are given in AGHS. The astronomical diurnal
rthonormal (hx,^xi50 when i 6¼ j), where the angle
i j equilibriumtidalforcingh is
brackets represent an arbitrary scalar product. Com- EQ
biningEqs.(4),(5),(6),and(7)yields
h 5(11k 2h )A sin(2f)cos(vt1l),
EQ 2 2 astronomical
1 (9)
a 5 (cid:2)hF ,^x i. (8)
k i(v 2v) tide k
k
where k and h are Love numbers accounting for ef-
2 2
In the following we use the biorthonormal systems of fects of the solid-earth body tide (Hendershott 1972),
eigenfunctions described by Mu¨ller (2008, 2009). We A is the astronomical forcing amplitude, f
astronomical
refertothesepublicationsforamoredetaileddiscussion is latitude, v is the forcing frequency, t is time, and l
of the adjoint solutions and the definition of the scalar is longitude. Except where noted, we set A ,
astronomical
product. 11k 2h ,andvtotheK valuesof14.1565 cm,0.736,
2 2 1
AsdescribedinEq.(8),thesensitivityoftheocean’s and 0.729 211 7 3 1024 s21, respectively, in all of the
response to the tidal forcing is determined by the dis- forwarddiurnaltidemodelsimulationsinthispaper.
tancebetweentheforcingandfreeoscillationfrequency Our forward ocean tide model incorporates a pa-
jv2v j,thefreeoscillationdecaytime1/(2v )and rameterized topographic internal wave drag scheme
k,2 k,1
the shape factor hF ,^x i of the free oscillation. The (Garner 2005), which is motivated by inferences from
tide k
latterdescribesthespatialcoherenceofthetidalforcing bothinsitumicrostructuredata(Polzinetal.1997)and
and the adjoint free oscillation. With the knowledge satellite-altimetry-constrainedtidemodels(Egbertand
of the free oscillations fxkgk51,‘ and their adjoint Ray2000,2001,2003)ofenhanceddissipationinregions
JULY2013 SKIBA ET AL. 1305
of rough topography. Parameterized topographic in- too strongly. Free internal waves are not generated by
ternalwavedragimprovestheaccuracyofforwardtide diurnal tides at topography poleward of about308 lati-
models(JayneandSt.Laurent2001;CarrereandLyard tude because of the requirement that the Coriolis pa-
2003; Egbert et al. 2004; AGHS; Lyard et al. 2006; rameterfbelessthantheforcingfrequency(Gill1982).
Uehara et al. 2006; Griffiths and Peltier 2008, 2009; Asnotedby,forinstance,JayneandSt.Laurent(2001),
Green2010)becausetidalamplitudesarecontrolledby topographicinternalwavedrapgffiffisffihffiffiffioffiffiffiuffiffilffiffidffiffibecorrectedby
thestrengthofthetopographicdrag(Egbertetal.2004; afrequency-dependentfactor v22f2/v.Wetherefore
AGHS).InAGHSthewavedragwastunedtominimize testedasolutioninwhichthewavedragsetupforM and
2
thediscrepancyinopen-oceanM2elevationsbetweenthe with a multiplicativqeffiffiffiffifffiffiaffiffifficffiffiffitffioffiffiffiffirffiffiffiffiffioffi f 3 qisffiffiffitffiffihffiffiffieffiffiffinffiffiffiffiffiffimffiffiffiffiffiuffiffiltiplied by
forward model and the satellite-altimetry-constrained acorrectionfactor v2 2f2v / v2 2f2v ,where
GOT99modelofRay(1999).Asdiscussedbelow,inthe v andv arethefoKr1cingfreMq2uenciMe2sforKK1andM ,
K1 M2 1 2
present paper we retain the same topographic internal respectively. This correction factor is the ratio of the
wavedrag scheme, but optimized for K tides.Simula- frequency-dependentfactorforK dividedbythefactor
1 1
tionsforthepresentpaperareconductedwithacutoff forM —thelatterisemployedinthewavedragscheme
2
depth for activating topographic internal wave drag of tuned for M . This method is referred to in Table 1 as
2
100m,ratherthan1000 masinAGHS.AsnotedinArbic ‘‘K correctionmethod 1.’’ We alsoexamined chang-
1
et al. (2008) this improves the accuracy of the forward ing the frequency-dependent factor to the K value
1
model and allows for a smaller value of the multipli- (‘‘K correction method 2’’ in Table 1), which is not
1
cativefactordiscussedinAGHS.Forallofthediurnal exactlythesameqaffiffisffiffiffifficffiffiffioffiffiffirffiffiffirffiffieffiffifficffiffiffitingtqhffieffiffiffiffifffiffiuffiffiffiffilffilffiffiffiffidffiffiffirffiffiffiaffiffigusedinM2
simulations presented in thispaper, the multiplicative simulations by v2 2f2v / v2 2f2v because
K1 M2 M2 K1
factor is set to 3. It should be noted that, despite the the Garner (2005) drag scheme includes nonlinear as
differenttuningfactorsintheAGHSschemeandsay,for wellaslinearterms.
instance,thewavedragschemeinJayneandSt.Laurent Tocomparetheforwardoceantidemodel(andmodal
(2001), the spatially averaged drag strengths of the two synthesis model) with satellite-altimetry-constrained
schemesareverysimilar,aspointedoutbyAGHS.Thus, tide models we compute several diagnostics. We mea-
thesuggestionisthatacertainaveragestrengthofdragis sure the rms area-averaged magnitude of tidal eleva-
needed to dissipate the proper amount of energy, re- tionsbythediagnostic
gardless of the exact details of the wave drag scheme sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(cid:3)ðð (cid:4)(cid:5)ðð
used.
h 5 h2dA dA, (10)
The effects of deformation of the solid earth by the rms
loading of ocean tides and the perturbations in the
gravitationalpotentialduetotheself-gravitationofboth where h is the sea surface elevation, square brackets
ocean tides and the load-deformed solid earth are col-
denotetimeaveragingoveronetidalperiod,anddAis
lectivelyknownastheself-attractionandloading(SAL)
an element of area. We measure the discrepancy be-
term(Hendershott1972).AsinAGHS(seealsoEgbert
tween the forward or modal synthesis models and
etal.2004),theSALtermiscomputedherewithanit-
a benchmark altimetry-constrained model having ele-
erativemethod.AsinAGHSandAKG,wedisplayre- vationsh via
BENCHMARK
sultsfromthethirditerationoftheSALterm.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
(cid:3)ðð (cid:4)(cid:5)ðð
TuningtheforwardmodelforK tides
1 discrepancy5 (h2h )2dA dA.
BENCHMARK
This subsection discusses the tuning of the forward
model’s parameterized topographic internal wave drag (11)
schemeforK tides.Ourgoalistominimizediscrepancies
1
between the forward model and satellite-altimetry- As in AGHS, the percent variance captured is then
constrainedtidemodels.Hereweexaminekineticen- computedby100[12(discrepancy/h )2].AsinAGHS
rms
ergiesaswellaselevations.TheK kineticenergiesin and AKG, the h , discrepancy, and percent variance
1 rms
forwardglobaltidemodelswithdragschemesoptimally captured values are computed over latitudes equator-
tuned for M elevations are weaker than K kinetic ward of 668, the latitudes covered by the Ocean To-
2 1
energies found in either current meter observations or pography Experiment (TOPEX)/Poseidon altimeter,
altimetry-constrainedmodels(P.Timko2011,personal and, unless otherwise noted, are computed only over
communication;Timkoetal.2012).Weconjecturethat locations where the water depths exceed 1000 m (i.e.,
thelowdiurnalkineticenergiesresultfromthefactthat shallow regions are excluded). The grid points over
modelsemployingawavedragtunedforM willdampK which water depths exceed 1000 m will vary slightly in
2 1
1306 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME43
TABLE1.Firstcolumn:ListofdiagnosticsusedtomeasureperformanceofforwardmodelsimulationsoftheK1tide.Secondcolumn:
diagnosticscomputedfromaforwardK simulationwith‘‘originaldrag,’’thatis,adragschemeoptimallytunedforM elevations.Third
1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2
cFooluurmthnc:osalummena:sssaemcoenadssceoclounmdncoonlulymwniothnltyhewMith2tdhreagvaslcuheeomfevcionrtrheectferdeqbuyetnhceyf-adcetpoerndven2Kt1f2acft2ovrMof2/thevd2Mr2a2gsfc2hveKm1edcishcaunsgseedditnostehcetvioanlu3e.
correspondingtotheK tide.Fifthcolumn:diagnosticscomputedfromTPXO(Egbertetal.1994)output.Valuesfor‘‘Dissipation’’and
1
‘‘Percentdissipationdeep’’wereretrievedfromTable1ofEgbertandRay(2003).Sixthcolumn:sameasfifthcolumnonlywithdatafrom
GOT99(Ray1999andTable1ofEgbertandRay2003).‘‘Percentvariancecaptured’’oftheforwardoceantidemodeliscomputedwith
respecttoGOT99(Ray1999)inwatersdeeperthan1000mequatorwardof668.
Diagnostics Originaldrag K correctionmethod1 K correctionmethod2 TPXO GOT99
1 1
h (cm) 9.44 9.51 9.42 9.46 9.54
rms
Percentvariancecaptured 97.2 95.0 94.5 — —
Kineticenergy(1015J) 29.4 37.9 37.3 41.0 —
Percentkineticenergyshallow 36.3 44.4 44.4 45.1 —
Potentialenergy(1015J) 22.8 24.3 23.9 22.6 22.4
Powerinput(TW) 0.334 0.313 0.316 0.340 0.347
Dissipation(TW) 0.337 0.312 0.315 0.343 —
Percentdissipationdeep 26.1 16.8 19.2 11.3 16.9
some of the blocking experiments, in which a small elevations–h , percent variance captured, potential
rms
number of gridpoints deeper than 1000 m are blocked energy,and‘‘Powerinput’’–comparewelltoTPXOand
out.Theh valuescomputedfromtheforwardocean GOT99forallthreeforwardsimulations.‘‘Dissipation’’
rms
tide model simulations are compared against those matches ‘‘Power input’’ well, as it must in energy bal-
fromGOT99inTable1.Wecomputethearea-integrated ance. In the simulation utilizing wave drag tuned for
kinetic energy (KE) and available potential energy M (‘‘Originaldrag’’),theK kineticenergyis,indeed,
2 1
(APE)by muchweakerthaninTPXO,asanticipated.Thepercent
ðð ofkineticenergyinshallowwatersisalsomuchsmaller
1
KE5 r (H1h)u(cid:2)udA and (12) than the value in TPXO. In addition, the percent of
2 0 dissipation in deep waters is larger than in TPXO or
ðð
GOT99. The two solutions with drag corrected for K
1 1
APE5 r g h2dA, (13) match the TPXO and GOT99 values more closely
2 0
overall, although the ‘‘Percent variance captured,’’
wherer 51035 kg m23isanaverageseawaterdensity ‘‘Power input,’’ and ‘‘Dissipation’’ diagnostics actually
0
compare somewhat less well than in the solution with
andHistherestingwaterdepth.Themetricdenotedby
M drag. The ‘‘K correction method 1’’ matches the
‘‘Powerinput’’iscomputedfrom 2 1
percentage of deep dissipation better than ‘‘K correc-
ðð(cid:6) (cid:7) 1
›h tionmethod2’’;‘‘K correctionmethod1’’istherefore
P 5r g h dA, (14) 1
input 0 EQ›t taken as the baseline, or nominal, K1 solution in the
remainderofthepaper.
(Egbert and Ray 2001 and references therein), where The K elevation amplitudes from GOT99 (Ray
1
h is the equilibrium tide. The metric denoted by 1999), the optimally tuned forward ocean tide model
EQ
‘‘Dissipation’’ is computed from the dissipation sum- (‘‘K correction method 1’’), and the modal synthesis
1
med over contributions due to the topographic wave modelaredisplayedinFig.1.Thethreemodels match
drag,quadraticbottomboundarylayerdrag,andeddy each other fairly well, although differences are clearly
viscosity, using code developed in Simmons et al. visible.Note,forinstance,theoverlystrongtidesinthe
(2004). southernSeaofOkhotskintheforwardoceantidemodel.
AsummaryofthesediagnosticscomputedfromaK The h and percent variance captured diagnostics for
1 rms
simulationwiththeoriginalwavedrag(tunedforM ,as themodalsynthesismodelaregiveninTable2.
2
inAGHS),K simulationswithK correctionmethods1 The dissipation map from our optimally tuned for-
1 1
and2,andoutputfromthealtimetry-constrainedmodels ward model of K is shown in Fig. 2. It looks qualita-
1
TPXO (Egbert et al. 1994) and GOT99 (Ray 1999) is tively similar to the K dissipation map in Egbert and
1
provided in Table 1. Some of the TPXO and GOT99 Ray (2003), but there are notable differences in some
valuesaretakenfromTable1inEgbertandRay(2003). locations, for example, Patagonia. Dissipation from
The quantities which can be computed strictly from quadratic bottom boundary layer drag is substantial in
JULY2013 SKIBA ET AL. 1307
TABLE2.Secondcolumn:diagnosticscomputedfromthemodal
synthesis model’s simulation of the K tide. Third column: di-
1
agnostics computed from the satellite-altimetry-constrained tide
modelGOT99(Ray1999)afterinterpolationtoa18grid.
Diagnostics Modalsynthesismodel GOT99
h (cm) 9.36 9.55
rms
Percentvariancecaptured 61.8 —
geometryandtopographyasthenominalsolution,butin
which the forcing frequency v in the astronomical di-
urnaltidalforcingEq.(9)isallowedtovary.Forthesake
of brevity, global amplitude maps of these simulations
are not shown. Instead, in Fig. 3a, we plot global h
rms
values from the forward ocean tide model and modal
synthesis model (averaged over latitudes equatorward
of668andinwaterdepthsexceeding1000m),asafunction
of the forcing period. Both the forward and modal syn-
thesis models display clear peaks near 22 and 33h, and
alessprominentpeaknear26h.Toassesstherealismof
thefrequencysweepsconductedwiththeforwardmodel,
Fig.3aincludesresultsonthediurnalconstituentsQ ,O ,
1 1
P , and K from the satellite-altimetry-constrained tide
1 1
models TPXO7.2 (Egbert et al. 1994) and GOT99 (Ray
1999). The TPXO7.2 and GOT99 results have been re-
scaledbytheratioofA(11k 2h )evaluatedforK to
2 2 1
FIG.1.AmplitudeofK1elevation(m)in(a)GOT99(Ray1999), A(11k22h2)evaluatedfortheconstituentinquestion
a highly accurate satellite-altimetry-constrained tide model, (b) (see Table 3), the underlying assumption being that
optimally tuned forward ocean tide simulation (K1 correction tidesarelineartofirstorder.Discrepanciesbetweenthe
method1),referredtoasthe‘‘nominal’’simulationinthetext,and
forwardoceantidemodelandthealtimetry-constrained
(c)themodalsynthesis(globalfreeoscillation)model.
models (and between the two altimetry-constrained
models themselves) are clearly seen. The agreement
theSeaofOkhotskandothershallowsearegions.The
between the forward and modal synthesis models and
topographic internal wave drag dissipates substantial
the altimetry-constrained models is close for K and
1
energyintheopenPacificandIndianOceansinlatitudes
equatorwardof308.
4. Resultsofsensitivityexperiments
Inthissection,wediscussthesensitivityoftheforward
oceantidemodelandtheglobalfreeoscillationmodelto
changesinthefrequency,zonalstructure,andmeridio-
nal structure in the astronomical diurnal tidal forcing.
Our initial emphasis is on the open ocean, but later in
the section we present an analysis of the frequency
sweepintheSeaofOkhotsk,whichisshowntoberes-
onanttodiurnaltidalforcing.Weconcludethissection
byexaminingthesensitivityofopenoceandiurnaltides
totheremovalofregionsofstrongcoastaldiurnaltides
suchastheSeaofOkhotsk.
a. Sensitivitytoforcingfrequency
This subsection focuses on simulations of the for-
FIG.2.Dissipationmap(milliwattspersquaredmeter)ofK1from
ward and modal synthesis models with the same basin our‘‘nominal’’simulation.
1308 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME43
TABLE3.Second,third,andfourthcolumns:Periods,astronom-
icalforcingamplitudes,andLovenumbercombination11k 2h,
2 2
respectively,forthefourlargestdiurnalconstituentslistedinthe
firstcolumn.Fifthcolumn:scalefactors[theratioofthevalueof
A(11k 2h)forK tothatoftheconstituentinquestion]usedto
2 2 1
plottheTPXO7.2andGOT99resultsondiurnalconstituentsin
Figs.3,7,and8.
Period Astronomicalforcing Lovenumber Scale
Constituent (h) amplitude(cm) combination factor
K 23.9345 14.1565 0.736 1
1
P 24.0678 4.6848 0.706 3.1502
1
O 25.8192 10.0661 0.695 1.4893
1
Q 26.8684 1.9273 0.695 7.7786
1
32.64-hmodehasthelargestdecaytimeandshapefac-
tor,explainingitsdominanceinthefrequencysweep.
The amplitudes and phases of the 21.97-, 26.20-,
and 32.64-h normal modes are displayed in Fig. 4. The
21.97- and 26.20-h modes display a strong signature in
the North Pacific, while the 32.64-h mode displays a
strongsignatureofanAntarcticKelvinwave.
The largest peak in Fig. 3a corresponds to the free
oscillationwiththe32.64-hperiod,whichrepresentsthe
first-order Antarctic Kelvin wave and Kelvin wave
propagatingalongtheNorthPacificcoasts(Figs.4eand
4f).Thisfreeoscillationisoneofthemajorcontributors
to the main diurnal tides (Mu¨ller 2008), although the
FIG.3.(a)Globallyandtemporallyaveragedhrmsvalues(cm) 32.64-h mode lies farther from the diurnal band than
computedoverthedeepocean(waterdepths1000mandgreater)
equatorwardof668insimulationsoftheforwardandmodalsyn- many other modes shown in Figs. 3b and 3c. The im-
thesismodelswithK-likeastronomicalforcingandrealisticocean portance of this free oscillation stems from its large
1
geometrybutwithavaryingforcingperiod(equivalently,varying damping time of about 54 h and its large shape factor.
frequency). Additionally, results from the satellite-altimetry- Thetwomodesat21.97and26.20 h,seenaspeaksinthe
constrainedmodelsTPXO7.2andGOT99,inwhichQ ,O ,and
1 1 frequency sweep, are also important in the modal syn-
P resultsarerescaledasiftheirastronomicalforcingamplitude
w1ereequaltothatofK1,areshown.(b)ShapefactorhFtide,^xkiand thesisofdiurnaltides.AllthreemodesshowninFig.4
(c)thedecaytimeT 51/(2v )offreeoscillationsinthediurnal haveasignatureoftheAntarcticKelvinwave,butonly
D 1
periodrange.[SeeZahelandMu¨ller(2005),andnotethatinthat the26.20-and32.64-hmodessignificantlycontributeto
paper modal frequencies are denoted by the symbol s instead thetidallyforcedone.The32.64-hmodehas49%ofits
ofv].
total energy in the Southern Ocean, while the 21.97-
and26.20-hmodeshave17%and24%oftheirenergyin
theAtlanticOcean,respectively,and58%oftheirtotal
(scaled) P , but is not as close for (scaled) O and energyinthePacificOcean.Adetaileddiscussionofthe
1 1
(scaled)Q . featuresofallmodesusedforthemodalsynthesismodel
1
InFigs.3band3cweshowtheshapefactorsanddecay andtheircontributiontothediurnaltidescanbefound
timesofallfreeoscillationsinthediurnalspectrum,as inMu¨ller(2008,2009).
determinedbyMu¨ller(2008).Obviouslynoteveryfree Aconclusiveresultisthatthegloballyaveragedh
rms
oscillationyieldsapeakinthefrequencysweep.Onlyif valuesdisplayedinFig.3aarequiteconsistentbetween
frictionislow(i.e.,thedecaytimeislarge)andtheshape the forward ocean tide model and the modal synthesis
factor is large (which corresponds to a high spatial co- model.Thepeaksnear22and33 h,forinstance,nearly
herence of the corresponding adjoint free oscillation coincidebetweenthetwomodels.
with the forcing field) is a peak likely to be generated.
b. Sensitivitytozonalstructureofforcing
Decay times areapparently themost importantfactor,
sincethe26.20-hmodehasafairlylowshapefactor,but Inthissubsectionweconductsimulationswiththefor-
still yields a small peak in the frequency sweep. The ward ocean tide model and the global free oscillations
JULY2013 SKIBA ET AL. 1309
FIG.4.AmplitudeandGreenwichphase(8),respectively,oftidalelevationsforthe(a),(b)21.97-hmode,forthe
(c),(d)26.20-hmode,andfor(e),(f)the32.64-hmode.AmplitudeandphasearenormalizedasinMu¨ller(2008).
model,withthesamebasingeometryandtopographyas whereN,thezonalwavenumber,takesonvaluesof0,1,
inthenominalsolutionbutinwhichthezonalstructure 2, 3, 4, 5, 8, and 16 (recall that N 5 1 is the nominal
oftheastronomicaldiurnaltidalforcingisvaried.Weset realistic case). As in section 4a, maps of the tidal am-
plitudes for these experiments are not shown for the
h 5(11k 2h )A sin(2f)cos(vt1Nl),
EQ 2 2 astronomical sakeofbrevity.Allofthemdiffer(inpatternaswellas
(15) magnitude) from the amplitude map of the nominal
1310 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME43
modelandthemodalsynthesismodel.Asinsections4a
and4b,eachsimulationinthissectionisconductedwith
realistic basin geometry and topography. The astro-
nomical forcing of the second-degree diurnal tides is
proportionaltothesphericalharmonic
Y1(f,l)5N1sin(2f)eil and (16)
2 2
rffiffiffiffiffiffi
1 15
N152 , (17)
2 4 2p
where N1 is a normalization constant (Arfken and
2
Weber2001).Thisnormalizationconstantensuresthat
integrating jY1(f,l)j2 over a unit sphere yields unity.
2
WerewritethegravitationalforcinginEq.(9)as
1
h 5(11k 2h )A <[Y1(f,vt1l)],
EQ 2 2 astronomicalN1 2
2
(18)
where <[Y1(f,vt1l)] represents the real portion of
2
Y1(f,vt1l). To vary the meridional structure of the
2
astronomical diurnal tidal forcing, we construct tidal
potentials of degree l 5 1, 2, 3, 4, and 5, using the as-
sociatedsphericalharmonicsofdegreel:
FIG.5.(a)Globallyandtemporallyaveragedhrmsvalues(cm),
computedoverlatitudesequatorwardof668inwaterdepthsex- b1
ceeding1000m(openocean)andindepthsshallowerthan1000m hEQ5(11k22h2)AastronomicalNl1<[Yl1(f,vt1l)].
(shelf),insimulationswithastronomicalforcinglikethatofK but 2
1
withavaryingzonalwavenumberNinEq.(15).Theextravertical (19)
line represents the realistic case (N 5 1). (b) Shape factors
hFtide,^xkiforthefreeoscillationsintheperiodrangefrom10to Here b1 is a normalization constant guaranteeing that
80hplottedvsN. l
thermsvalueoftheequilibriumtidalforcingaveraged
overtheoceanicareaisthesameforeachvalueoflas
forK forcing(l52).Weenforcethiscriterionsothat
simulation, with the N 5 16 experiment displaying the 1
discrepanciesbetweenthel6¼2forwardoceantide(and
largestdiscrepancies. modal synthesis) experiments and the nominal l 5 2
Theh values,intheopenoceanandintheshelf,are
rms casesareduetochangesinthespatialstructureandnot
plotted versus N in Fig. 5a. The vertical line in Fig. 5a
to the magnitude of the astronomical forcing averaged
marksthevalueofNcorrespondingtotherealisticcase
overtheareaoftheWorldOcean.Thus,weset
(N51).AscanbeseeninFig.5a,predictionsfromthe sffiffiffiffiffiffi
modalsynthesismodelmatchgeneraltrendsseeninthe F1
b15 2, (20)
forward ocean tide model results, although there are l F1
clear discrepancies. Figure 5b displays the dependence l
oftheshapefactoronthezonalwavenumberformodes whereF1 isdefinedby
l
with periods between 10 and 80 h. This figure clearly
ðð
showsclustersoflargeshapefactorsbetweenN51and
F15 jY1j2dA. (21)
4,whichalignswiththestrongoceanicresponseseenin l l
oceanarea
Fig.5a.
Thevaluesofb1deviateby4%orlessfromunityforall
c. Sensitivitytomeridionalstructureofforcing l
valuesoflandforboththeforwardandmodalsynthesis
Inthissubsectionweinvestigatethesensitivityoftidal models.
elevations to meridional structure in the astronomical AsshowninFig.6,theforwardoceantidemodeland
diurnaltidalforcing,againusingtheforwardoceantide modalsynthesismodelbothdisplayapeakinresponse