Coastal Oceanography, Geology, and Biology

62 Pages·2017·2.63 MB·English

Chapter 2
Coastal Oceanography, Geology, and Biology
2.1 The Coast as Acoustic Waveguide
From the viewpoint of low- and mid-frequency shallow water acoustics, the ocean
shelf is a waveguide, limited by a pressure release boundary above (the ocean’s
surface)andanabsorbingboundarybelow(theoceanbottom/seabed).Indescribing
thesoundfieldradiatedfromasource(whetherpointordistributed),boththewave-
guide’sinteriorandboundariesplayanessentialrole.Todescribethiswaveguide’s
interiorandboundaries,wemustdelveintotheareasofcoastalphysicaloceanogra-
phy, geology and geophysics, and biology. These areas are vast scientific fields on
theirown,andsowelimitourselvesheretotouchingononlytheelementsofthemthat
aremostgermanetoshallowwateracoustics.Therestweleavetothereadertoexplore
independently. Toward this, we initially provide three physical oceanography
references,whichshouldgivethereaderastartingpoint:PondandPickard(1983);
Bowden(1983);andCushman-Roisin(1994).Wewillstartbylookinginthefirstsix
sections of this chapter (Sects. 2.1–2.6) at physical oceanography, then devoting
Sects. 2.7–2.10 to marine geology, and finally ending in Sect. 2.11 with some
discussionofmarinebiology.
Inlookingattheshallowwateracousticwaveguideingeneral,thesoundspeed
inboththewaterandthebottomdependsmainlyondepth(thezcoordinate)andas
a “beginners rule” varies slowly in the horizontal coordinates and in time (more
precisely,geotime).Thus,itiscommonlytakenthatthesound-speedfieldc(x,y,z)
is to first order only a function of z. This allows usto look initially at the vertical
variability of the sound speed at different geographical locations and seasons and
makebothsensibleintercomparisonsanddistributionsofvariability.
In that vein, distilled results of different observations and measurements of
sound-speed profiles in various areas of the world ocean are shown in Fig. 2.1.
We can see that the water column sound-speed profiles differ depending on
geographical location and season, whereas the seabed sound-speed profiles are
moresimplydeterminedbydepth(overburdenpressure)andmaterialcomposition.
Wewillattempttoexplainthedetailsoftheseprofilesinthesectionsthatfollow.
B.Katsnelsonetal.,FundamentalsofShallowWaterAcoustics, 17
TheUnderwaterAcousticsSeries,DOI10.1007/978-1-4419-9777-7_2,
#SpringerScience+BusinessMedia,LLC2012
18 2 CoastalOceanography,Geology,andBiology
Fig. 2.1 Sound-speed profiles in different areas of the world’s coastal ocean and for different
seasons.1–higherlatitude,wintersound-speedprofiles,2–lowerlatitudeyearroundandsummer
mid-latitudeprofiles.3–bottomsoundspeed.Thenonmonotoniccurveincludedshowsasound-
speedprofilewithapossiblepurerefractionduct
For analytical calculations and estimation, we can select a small numbers of
“canonical”(i.e.,genericortypical)sound-speedprofilesthatcanbeusedtocharac-
terizealargefractionoftheshallowwaterenvironment(KupermanandLynch2004).
However, these profiles, shown in Fig. 2.2, are merely first-order simplified
representationsoftherealityoftheshallowwaterseas(asrepresentedbyFig.2.1).
Inordertoknowwhentheycanbeusedorwhentheyshouldbeaugmented,wemust
first discuss the real environment. We will do that next, approaching the coastal
oceanographyfirst,andthenmovingtothecoastalmarinegeology.
2.2 Properties of Sea Water: Vertical Stratification
and Its Seasonal Variability
Toagoodapproximation,wecanconsiderseawaterasatwo-componentmedium,
consistingofasolvent(water)andvarioussalts.Theconcentrationofsalt(salinity)
isusuallydenotedasSandismeasuredinpro-mille(denotedas‰)orinpartsper
thousand(ppt).Theaveragevalueofsalinityindeepoceanseawaterisabout35‰.
Variationsofsalinitywithintheshelfzoneareintheorderof20–35ppt.
The thermodynamic state of sea water is characterized by temperature T(cid:1),
densityr,salinityS,andpressurePandisdescribedbythestateequation
r¼rðT(cid:1);S;PÞ: (2.1)
2.2 PropertiesofSeaWater:VerticalStratificationandItsSeasonalVariability 19
Mixed or mid-latitude Typical 3-layer mid-latitude
winter profile (Pekeris) summer profile
1) 2)
Surface mixed layer
C(z) C(z)
Isovelocity water Thermocline
Bottom mixed layer
Isovelocity Bottom Isovelocity Bottom
Halfspace Halfspace
Coastal Front
3)
C (z)
1 C (z)
2
Isovelocity Bottom
Halfspace
Fig.2.2 Threesimplesound-speedprofileswhichcharacterizemuchofshallowwateracoustics.
Thefirstprofileistheso-called“Pekeriswaveguide,”typicalofmixedwatercolumnconditions
over a simple bottom. The second profile is the “three-layer model,” typical of low-latitude or
summer mid-latitude waters over a simple bottom. The third profile is that of an enormously
simplifiedcoastalfront,againoverasimplebottom
Due to the complexity of the composition of seawater, this equation can be
written only empirically. In physical oceanography, the parameters temperature,
salinity, and pressure are basic, whereas other observed characteristics (sound
speed, heat capacity, conductivity, etc.) are expressed through these three
parameters. As is well known, these parameters depend on both space and time
(wewill denotethis“slow” timeasgeotime, T, incontrastwith the “fast” time,t,
usedinformingexpressionsforthehigh-frequencyoscillationsofthewavefields).
Oneofthemaincharacteristicsofseawaterisitsbulkadiabaticcompressibility
(definedbytherelativevariationofthedensitywithpressure)
(cid:1) (cid:3)
1 @r
b ¼ (2.2)
a r @P
a
which also depends on the basic thermodynamic parameters and, in relating
it to acoustics, generally depends on the frequency of the pressu(cid:5)rpe ffioffiffisffifficffiffiffiillations
(of sound) as well. The sound speed is connected to b by c¼1 rb . In our
a a
case, b (cid:3)4(cid:4)5(cid:5)10(cid:4)10Pa(cid:4)1. b depends on T(cid:1) and S, and according to
a a
experimental data (e.g., Babii 1983), compressibility decreases with both increas-
ing temperature and salinity, whereas sound speed correspondingly increases.
20 2 CoastalOceanography,Geology,andBiology
Table2.1 Coefficientsofequationofstate
b [Pa(cid:4)1] b [(cid:1)K(cid:4)1] b ð(cid:1)=(cid:1)(cid:1)Þ(cid:4)1 @c=@T[ms(cid:4)1(cid:1)K(cid:4)1] N[cph]
a T S
4(cid:4)5(cid:5)10(cid:4)10 1:3(cid:5)10(cid:4)4 8(cid:5)10(cid:4)4 4.6 <10
This variability of b determines the dependence of the sound speed upon the
a
thermodynamicparameters;ofparticularimportance,thedependenceofthesound
speedontemperaturegoesas @c (cid:3)4:6m/(sdegree),asarough“ruleofthumb.”
@T(cid:1)
Wealsoneedtodefinethethermalexpansioncoefficient
(cid:1) (cid:3)
1 @r
b ¼(cid:4) (2.3)
T r @T(cid:1)
S;P
andthecoefficientofvariationofthedensityasafunctionofsalinity
(cid:1) (cid:3)
1 @r
b ¼ : (2.4)
S r @S
P;T
Withthesequantitiesdefined,wecanthenwritedownthedifferentialequation
ofstateofseawateras
dr
¼b dT(cid:1)þb dPþb dS (2.5)
r T a S
where(typical)numericalvaluesofthecoefficientsareshownintheTable2.1and
NistheBrunt–Vaisalafrequency(seeAppendixA).Thisresultwillbeusedinour
furtherwork.
Temperaturevariabilityisagoodfirst-orderproxyforsound-speedvariabilityin
shallow water, since the well-known equation of state (Urick 1975) gives that the
soundspeedisapproximately
c¼1449:2þ4:623T(cid:1)(cid:4)0:0546ðT(cid:1)Þ2þ1:391ðS(cid:4)35Þþ0:017z (2.6)
where c is in m/s, T(cid:1) is in degrees Celsius, S is salinity in ppt or psu, and z is in
meters.Sincedeptheffectsonlyaddabout0.017s(cid:4)1tothisequation,theyareoften
ignoredtofirstorderinshallowwater.Salinityalsoisagenerallyweaksound-speed
signal(thoughnotalways,ascoastalwatersshowthelargestvariationinS).Forthe
range of T(cid:1) and S typically found in coastal waters (T(cid:1) ~ 6–25(cid:1)C and
S ~ 20–35 ppt), it is easily seen that temperature effects dominate. However, to
get at the temperature field theoretically, the physical oceanographer has to con-
siderthesalinityjustasmuchasthetemperature,sincethedensityfieldisthemost
critical parameter in ocean flows and structure, and density is about equally
dependent on temperature and salinity effects. Thus, (2.5) is the critical state
equation for physical oceanography, whereas (2.6) is the important state equation
foroceanacoustics.
2.2 PropertiesofSeaWater:VerticalStratificationandItsSeasonalVariability 21
Thetemperaturedistributionintheshallowwatercolumnisdeterminedbyboth
the large-scale water masses found at a particular location and the competition
betweensolarheating,mediumstratificationandverticalmixing,thelatterofwhich
is inducedby wind stress on the surface as well aswaveand current stress on the
bottom.Inmodelingthe vertical temperature distribution,we will consider here a
simple yet physical and useful model, published by James (1977), which is also
describedinthecoastaloceanographytextbookbyBowden(1983).Thisvariability
intheverticaltemperatureprofilewithgeotime,i.e.,thefunctionT(cid:1)ðz;TÞ,iscentral
toshallowwateracoustics.
Webeginwiththeone-dimensionalverticaldiffusionequationfortemperature,
(cid:1) (cid:3)
@T(cid:1) @ @T(cid:1)
¼ K ; (2.7)
@T @z z @z
where T is the geotime and K is the vertical eddy diffusion coefficient of heat,
z
dependent ondepth andgeotime. Totake intoaccountthatmixingis inhibited by
densitystratification/stability,K ismodifiedto
z
K ¼K ð1þsRiÞ(cid:4)p; (2.8)
z 0
whereg0 ¼K gisthecoefficientforneutralstability(nostratificationpresent)and
0
RiisthegradientRichardsonnumber,givenby
,
(cid:1) (cid:3)
@T(cid:1) @U 2
Ri¼gb : (2.9)
T @z @z
In (2.8) and (2.9), s; p are empirically determined constants, on the order of
s¼0:3andp¼1:5 respectively;Uisthehorizontalcurrentmagnitude;andb is
, T
thecoefficientofthermalexpansionofwater.
Thesolutionof(2.7)issubjecttosurfaceandbottomboundaryconditions.
Atthesurface,
(cid:6) (cid:7)
@T(cid:1) Q~
K ¼ ; (2.10a)
z @z rc
z¼0
~
whereQisthefluxofheatdownwardthroughthesurface(inparticular,theaverage
heatfluxfromtheSunisabout0.14W/cm2).Atthebottom,theturbulentdiffusion
ofheatisinhibited,so
(cid:6) (cid:7)
@T(cid:1)
K ¼0: (2.10b)
z @z
z¼H
Thetemperaturegradient,@T(cid:1)=@z,isthendeterminedviacalculation.Theheat
~
input Q and the wind speed are the meteorological inputs to the model (the wind
22 2 CoastalOceanography,Geology,andBiology
speed being one of the factors in K, as we will see). The term ð@U=@zÞ2 in (2.9)
z
isobtainedbyassumingthatthebottomandsurfaceboundarylayershavelogarith-
mic velocity profiles, the surface one caused by the wind stress and the bottom
onecausedbythetidalcurrents.Forinstance,thenear-bottomvelocityprofilewill
looklike
u z
u¼ (cid:6) ln ; (2.11)
k z
0 0
where z is the distance from the bottom, u ¼ðt=rÞ1=2 is the so-called friction
(cid:6)
velocity(tbeingthestressduetothenear-bottomtidalcurrent),z isaroughness
0
parameter,andk isvonKarman’sconstant,approximatelyequalto0.4.Asimilar
0
formiseasilyfoundintheliteratureforthesurfaceboundarylayer.Thisformwill
befamiliartomanyreadersasthe“lawofthewall.”
The neutral eddy diffusivity (at Ri¼0) is expressed as the sum ofsurface and
bottompartsaswell,i.e.,
K ¼K þK (2.12)
0 0W 0T
where K is proportional to the wind speed and K is proportional to the tidal
0W 0T
currentspeedandthedepthofthewater.
Withallthisinhand,themodelisrun.Whenathermoclineisformedatacertain
depth, z¼z , it is assumed that wind mixing cannot penetrate below that depth
0
and that bottom (tidal current) mixing cannot penetrate above it. Moreover, as is
also usual in mixing models, if @T(cid:1)=@z becomes positive, indicating unstable
stratification, the model mixes the temperature instantaneously over a range of
depth, thus restoring stability to the profiles. By integrating (2.7) numerically,
startingfromgiveninitialconditions,onecancomputetheevolutionofthetemper-
atureprofilewithtime,andindeedJames’resultslookexactlylikeourthree-layer
model of shallow water, i.e., a surface layer, a thermocline, and a bottom mixed
layer,asseeninFig.2.3.
Weshouldmaketwonotesaboutthetemperaturestratificationmodelpresented
above,particularlyfromtheviewpointofanacoustics“modeluser.”First,andmost
importantly, while this model is useful for understanding the physics of how the
surfacemixedlayer,bottommixedlayer,andthermoclinearise,itisrelativelyweak
for predictive use. Mixed layer models are demanding in their inputs for wind
stress,insolation,initialstratification,etc.,andtypically,thismuchdetailedinputis
hard to get. For acoustics users, climatology probably gives a good enough first-
orderestimateofstratification,andaninsituXBTorCTDgivesthebestimprove-
ment to that. Second, the model discussed above is one of the simpler ones, and
more complicated models do exist which include salinity effects, more detailed
modelsofturbulenceandmixing,andeven threedimensions.We referthereader
totheclassicarticlebyPriceetal.(1986)asanexampleofamoresophisticated1D
approach.
2.3 HorizontalStratificationandItsVariability:FrontsandEddies,SurfaceDucts... 23
Fig.2.3 Developmentofaseasonalthermocline,computedfromthe1Dmodeldiscussed.Top
panel shows temperature profiles at 28-day intervals, the bottom panel the corresponding
coefficientsofeddydiffusionatthesametimes.WithpermissionfromI.D.James,Estuar.Coastal.
Mar.Sci,7,197–202.CopyrightAcademicPress(London)Ltd
Extensionoftheshelfsound-speedprofileresultstotheslopeisstraightforward,
atleasttofirstorder.Onejustgetsridofthebottommixedlayerintheshelfprofile,
andextendsthedownwardrefractinggradientlayertothetypicaldownwardrefrac-
tiononegetsinthefirstkilometerdepthduetotheexponentialtemperaturedecrease,
using the climatology of the region as a guide. This gives the profile shown in
Fig.2.3toppanel.However,thisisagenericmeanprofileanddoesnotincludethe
effectsoffronts,eddies,IWs,finestructure,etc.Theseareoftenpresentontheslope,
andsotheabove“prescription”mustbeusedwithextremecare.
One also notes that down to the depth of about 750 m, the depth of the mid-
latitude deep sound channel axis, this profile is downward refracting. This means
that sound comingoff the shelf onto the slope will “hug the bottom,” giving high
soundlevelsnearthebottom,andashadowzonejustabovethat.However,pastthe
depthof750m(itoftencanbedownto~1km),thesoundwillnolongerbedriven
intothebottom,butratherthefieldwill“liftoff”thebottomandcontinueitstravel
inthewell-knowndeepoceanSOFARaxis.Thiscanbemodifiedbytheshelfbreak
frontandotherstructure;soagain,onemustlookatthiseffectexercisingcare.
2.3 Horizontal Stratification and Its Variability:
Fronts and Eddies, Surface Ducts, and Storm Surges
Intheabovediscussion,weconcentratedontheverticalvariabilityofthetempera-
turefieldanddiscussedasimplemodelofhowthatvariabilityarises.Thisvertical
structure is the most important feature of the shallow water column, as the water
column and bottom are approximately horizontally stratified (comprised of
24 2 CoastalOceanography,Geology,andBiology
vertically stacked layers) over the propagation scales of interest, which reach to
about 50 km in shallow water. However, horizontal stratification is a broad-brush
firstapproximationonly,andinmanyshallowwaterscenarios,thereisappreciable
sound-speed variability in the horizontal direction, as well as in the vertical.
Perhaps the strongest horizontal variability in shallow water is due to shallow
waterfronts,andsoweaddressthattopicnext.
There are four separate kinds of shallow water fronts which are of potential
interesttounderwateracousticians.Theyare(1)upwellingfronts,(2)tidalmixing
fronts, (3) buoyant plume fronts, and (4) shelf break fronts. These fronts all have
somewhat different dynamics and characteristics, which in turn determine what
type of horizontal temperature/sound-speed gradients they produce and where on
thecontinentalshelftheyareobserved.Wewilllookatallfourofthesefrontsand
presentexampleswherepossible.
Upwelling fronts. Upwelling of water at the coasts is a very well-known
phenomenon, particularly since it has a profound effect on fishing activity, the
Peruvian anchovy fisheries being perhaps the most famous example (Bowden
1983). Specifically, upwelling is the process whereby cool water from depths
ontheorderof50–300mispumpedupwardintothesurfacelayerduetotheeffect
ofwind,therebybringingnutrientsaltsintothephoticregionsneartheseasurface.
Fromthepointofviewofacoustics,thebiologicallyimportanttransportofsaltsto
the surface (a salinity signal) is almost irrelevant; however, the transport of cold,
deep water to the normally warm surface layer is not, as this temperature signal
providesanappreciablesound-speedsignal,andindeedanacoustic“front.”
Physically,upwellingiscausedbytwothings:thefirstisthewind-inducedEkman
transportofnear-surfacewaterawayfromthecoast(surfaceEkmantransportbeing
producedbythebalancebetweenthewindstressandtheCoriolisforce).Thesecondis
thecontinuityequationdemandingthattheamountof(warm)surfacewaterthatgets
transportedseaward mustbereplaced withanequal amountofdeeper(cold)water
beingtransportedshorewardandeventuallytothesurface,soastobalancethemasses.
EkmantransportistotherightofthewindintheNorthernHemisphereandtotheleft
ofitintheSouthernHemisphere,sothedirectionofthewindalongthecoastiscritical.
Weshouldnotethatifthewinddirectionissuchthatitispushingthewarmsurface
watersonshore,ratherthanoffshore,ananalogousprocesscalleddownwellingoccurs.
Thispushingofwarmwateronshoretendstocreateadeepersurfacewarmlayernear
thecoast,ratherthanasharpfront.ThebasicmathematicsofboththeEkmantransport
and the continuity equation are to be found in numerous elementary oceanography
textbooks, and we would refer the reader to the treatments by Bowden (1983) and
Cushman-Roisin(1994)fordetails.
Thoughoneshoulduseamoreexactdynamicalmodeltoconstructthegeometry
of an upwelling front, we can create a first-order estimate of upwelling frontal
structure useful for acoustics from rather simple dynamics. In the top and bottom
panelsofFig.2.4,weshowatwo-layeroceanbeforeandafteranupwellingevent,
withtheinitialwarmsurfacelayerbeingofdepthH,densityr ,andsoundspeedc .
1 1
This layer overlies a (mathematically) infinitely deep layer with density r and
2
2.3 HorizontalStratificationandItsVariability:FrontsandEddies,SurfaceDucts... 25
Fig.2.4 Toppanelshows
warm
thetwo-layersystemofwarm surface H C1, p1
surfacelayeroverlyingadeep layer
coolerlayernearthecoast.
Lowerpanelshowsresultof cool, thick
upwelling,wheredeepcooler subsurface
waterisbroughtuptothe layer C2, p2
surfaceduetothewarmer
surfacelayerbeingpushed
seaward.ThedistanceRover
whichthefrontextendsisa
warm
gooddynamicapproximation, surface H C1, p1
buttheexactshapeofthe layer
frontneedsamoreexact
dynamicaltreatmentto cool, thick R
determine–ourfigure subsurface
aboveisan“artist’s layer C2, p2
conception”inthatrespect
soundspeedc .Afterupwellingoccurs,thecoldlowerlayerrisesfromdepthHto
2
thesurfaceoveradistanceR,whichisgivenbyCushman-Roisin(1994)as
R¼ðg0HÞ1=2=f ; (2.13)
C
where
ðr (cid:4)r Þ
g0 ¼ 2 1 g (2.14)
r
2
isthe“reducedgravity”(seeAppendixA.3)and
f ¼2O siny (2.15)
C E
istheso-calledCoriolisparameter,whereO istherotationalrateoftheearthandy
E
is the latitude. Thus, one has a horizontal sound-speed gradient which is sharp
across the surface layer, and has a depth average value (over the layer depth) of
ðc (cid:4)c Þ=R. Here, R is typically on the order of 5–10 km, and Dc is of the order
1 2
10–50 m/s so that the average horizontal gradient is of the order 1–10 m/s/km or
10(cid:4)3(cid:4)10(cid:4)2c(cid:4)1.Thisisanappreciablehorizontalgradientforacoustics.
Tidal mixing fronts. Whether or not a thermocline develops in the late spring or
earlysummerdependscriticallyonwhetherornotthereisenoughturbulentkinetic
energygeneratedthroughoutthewatercolumntomixheatdownwardattherateat
which it is being received at the surface. Both surface winds and tides can create
turbulence and mixing, but the winds are an intermittent mechanism, whereas the
tidesareaconstantone.Thus,wewillconsiderherethecasewherethemixingis
duetotidesalone.
26 2 CoastalOceanography,Geology,andBiology
From a rather simple derivation (Bowden 1983), one can show that there is
completeverticalmixingofthewatercolumnwhen
H
(cid:3)70(cid:4)100perm(cid:4)2s(cid:4)3; (2.16)
U3
t
whereHisthewatercolumndepthinmetersandU isthetidalcurrentspeedinm/s.
t
This form easily lets us see the depths to which one expects full mixing. Taking
U equal approximately to 0.5–0.7 m/s (about one to one and one-half knots)
t
as typical tidal current magnitudes, full mixing will be seen in water columns
with depths between ~10 and 35 m. Past this point, a two (or more)-layer system
willevolve.Forwaterdepthsgreaterthanthismixingdepth,thelowerlayerwillbe
atsomesoundspeedc ,whereasabovethemixedregion,thewatersoundspeed
mixed
willbec (inthetwo-layermodel).Thisidealizedsystemagreeswellwithreal
upper
worldmeasurementsofthiseffect,asshownbycomparingoursimplemodelwith
data,asseenbelow(Bowden1983),inFig.2.5.
Buoyant plume fronts. River outflow plumes create fronts that are well known to
even lay observers, as large river discharges after storms often contain copious
amounts of mud and fine sediment, which show the edges of the plume sharply
against the much clearer, deeper water. However, these plume fronts are not as
acousticallystrongastheotherthreetypes,forreasonswenowdiscuss.Themain
signal of a buoyant (freshwater) river plume front is salinity, which may be
~20–30 ppt compared to 35 ppt in the deep ocean. This creates up to an
~15–20m/ssound-speedcontrastbetweenthebuoyantplumeanditssurroundings,
anotinsignificantdisturbance.However,thesestrong salinity signalsonlyextend
over the top meter or two of the water column near their origin. These plumes
then travel many tens of kilometers outside the river mouth, and the initial fresh-
water pulse is spread out both horizontally (by wind and currents) and vertically
(by mixing), thus diluting it. By the time river salinity signals have traveled an
appreciabledistance,thesound-speeddisturbanceisonly5m/sorless.Thisisweak
asfarascoastalhorizontalfrontsareconcerned.However,thenear-surfaceacoustic
duct that they can create can be very effective in ducting sound at higher
frequencies,aswewilldiscusslaterinoursectiononsurfaceducts.
Shelf break fronts. If the buoyant plume fronts are the least important ones
to acoustics, their distant offshore cousins, the shelf break fronts, are perhaps the
most important. The shelf break regions of the world (the lines where the conti-
nental shelves dip abruptly into the continental slopes, usually at depths of
100–200 m) are often the boundaries between very cool (but somewhat fresher)
coastal waters and very warm (but somewhat saltier) deep waters influenced by
largeboundarycurrentsystemssuchastheGulfStreamandKuroshio.Anexample
of a well-measured shelf break front from the 1996 PRIMER experiment
(Gawarkiewicz et al. 2004; Sperry et al. 2003) is shown in Fig. 2.6. This front
obviouslyhasalotofstructure,unlikeoursimplecartooninFig.2.2(3),andserves