Journal of Coastal Research 79-89 Fort Lauderdale, Florida Winter 1996

Subtidal Frequency Fluctuations in Coastal SeaLevel in the Mid and South AtlanticBights: A Prognostic For Coastal Flooding

Gerald S. Janowitz and Leonard J. Pietrafesa

Department of Marine, Earth and Atmospheric SciencesBox 8208

North Carolina State UniversityRaleigh, NC 27695, U.S.A.

ABSTRACT ---------

.tlllllllll.~~.

----; !W... ;s-=

JANOWITZ, G. S. and PIETRAFESA, L. J., 1996. Subtidal frequency fluctuations in coastal sea levelin the Mid and South Atlantic Bights: A prognostic for coastal flooding. Journal of Coastal Research,12(1),79-89. Fort Lauderdale (Florida). ISSN 0749-0208.

An analytical model is used to determine spatial and temporal variations in coastal sea level. The modelis developed for subtidal frequency motions in the viscous parameter regime, where advection of relativevorticity can be neglected with respect to production of relative vorticity by bottom Ekman layer pumping.The latter is then balanced by the topographically induced vertical velocity. The effects of the atmosphericwind stress on coastal sea level are assessed; and it is found that an upwelling (downwelling) favorablewind stress causes a continual drop (rise) in coastal sea level downstream from an initial cross-shelflocation, with modifying effects of cross-shelf profile and initial location on the downstream variation incoastal sea level also being important. The model is applied over the Mid Atlantic Bight from WoodsHole, Massachusetts to Cape Hatteras, North Carolina and then to Charleston, South Carolina. Reasonableagreement exists between observations and model results which suggests that a predictive capability hasbeen established; dimensionless mean squared errors range from 0.122 to 0.294 with a mean of 0.181 overthe six test cases. The model can be driven by several days of forecast winds to determine the timing ofcoastal flooding, with linear superposition of location specific predicted astronomical tides onto the subtidalfreq uency model predictions.

ADDITIONAL INDEX WORDS: Sea-level change, coastal flooding, shelf dynamics, Ekman pumping.

INTRODUCTION

The response of continental shelf waters to thewind and outer boundary forcing is of consider­able interest to those concerned with nearshoreor estuarine processes or with setting coastalboundary conditions on numerical models. Forexample, it has been shown that wind inducedvariations dominate the subtidal frequency fluc­tuations of coastal sea level along the east coastof the U.S. (WANG, 1979; CHAO and PIETRAFESA,1980), have a significant impact on the transportof particulate matter and fish larvae throughcoastal inlets (PIETRAFESA and JANOWITZ, 1988),or cause significant coastal ft.ooding (NUMMENDALet al., 1987). Thus, an understanding of the de­terminants of coastal sea level variation is of im­portance and will be considered here.

A steady state model is first developed and theresults examined. The model is then extended to

93087 received 27 August 1993; accepted in revision 29 September 1994.

include time varying forcing and finally a com­parison between observations and model predic­tions is undertaken, specifically for the Mid-At­lantic and the South Atlantic Bights as a test. Itis anticipated that the results of this study willbe of use to the U.S. National Weather Servicein its attempts to predict coastal flooding events.

The literature on slope and shelf dynamics isextensive and growing. Here we discuss a fewworksrelevant to the development of this paper. Severalother papers are discussed further in the text.SCHWING et al. (1985) in their study of the dy­namics of the South Carolina shelf found littlephase lag between the wind and sea level in shal­low water. They later found (1988) that for a pe­riod when the wind field was weak and disorgan­ized, continental shelf waves may occur in theSouth Atlantic Bight. WANG (1982) and MID­

DLETON (1987) showed that for wide shelves coast­al sea level is relatively insulated from offshorepressure fields. BURRAGE et al. (1991) in a studyof the central Great Barrier Reef showed that thealongshore flow is in geostrophic balance for pe-

80 Janowitz and Pietrafesa

L

1 » E Z; == (Ajfoh~) '" » ( == U/ fo L, (1)

The surface and bottom Ekman layers are thusassumed thin compared to the total depth, andthe motion is linear. We take u, v, and w to bethe components of the geost rophic velocity fieldto which corrections are added in the frictionalboundary layers . Neglecting term s of relative or­der e and E; the hor izontal momentum balancebecomes

a1/-fou = gay (2a)

in the offshore direction , and

a1/fov=gax (2b)

As th e wind stress is uniform, the surface Ekmanlayer is non-divergent and we obtain for the geo­strophic vertical velocity at the surface, z = 0,

alongshore.The con tinuity equation, with vanishing hori­

zontal divergence from equat ion (2) through orderE/ ' (>E) or equivalently the vor ticity equationwith advective terms neglected (O( E) ), is

By virtue of equations (3) and (4) the geostrophicvertical velocity throughout th e column and atthe bottom, z = - (he + a x), must vanish. Anexpression for the vertical velocity at the bottom,given in PEDLOSK Y (1987, p. 226, equation 4.9.36)after redimension alizing is

w(x, y, - he) = - au + 0.\, = 0 (5)

where o. = (Av/2fo) '" and r, = av/ax - aulay. Wenote that o. is one half the e-folding depth of aconstant eddy viscosity bottom Ekman layer anda factor of 271'" sma ller than the conventional Ek­man depth. Using Equations (2) and (5) we findthat the equation governing sea level under theforegoing assumptions is as follows:

{a21/ a21/ } ~o. - + - + a - = O. (6)ax2 ay2 ay

The first term in this equation represents Ek ­man pumping and the second the topographiceffect.

The boundary conditions on this equation are:

(4)

(3)aw = o.az

w(x, y, 0) = o.

A Steady State Model

Although we are ultimately concerned with thetime varying response of coastal sea level to windand outer boundary forcing, we first consider thesteady state case. Consider a portion of the con ­tinental shelf bounded by a st raight coastline witha depth that increa ses linearly with the offshorecoordinate x, i.e., h = he + ax. The coastal depthis he and a is the diabathic bottom slope. Theorigin of th e x axis is the coast , positive offshore,with the y coordinate taken in the alongshore di ­rection ninety degrees to the left of the x axis, ortowards higher latitude, see Figure 1. We considerthe region 0 -s x < 00 and - L ::; y ::; O. As wewill show, the alongshore length L can be takento be arbitrarily large . A uniform alongshore windstress T~ is applied at the free surface and cross­shelf profiles of the sea level etx , y) are specifiedat y = 0, - L. Far offshore, 1) approaches zero . Thedensity field is taken to be homogeneous and theCoriolis parameter fo is assumed constant. Ver ­tical mixing is parameterized by a constant eddyviscosity Av and U and L, are taken to be theoffshore velocity and length scales respectively.The model is in the viscous domain, i.e., it is re­quired that

Figure 1. Model geometry.

riods exceeding 50 hours and that much of themotion of the shelf is wind driven. The elementsof geostrophy, steadiness, wind-forcing and theneglect of offshore forcing all playa role in thepresent work .

Journal of Coasta l Research, Vol. 12. No. 1, 1996

Prognostic for Coastal Floodings 81

(10)

(11)

at y = 0, 1](x, 0) = 1]o(x), 1]0(' := 1]() (0), (7)

at y = -L, 1](x, - L) = 1]1 (X), (8)

and

as x~oo, 1](X, y) ~ o. (9)

Note that n: is simply the initial coastal sea level.At the coast, the net transport in the x directionis taken to vanish. This is composed of the surfaceEkman transport (T~/pfJ, the geostrophic trans­port (uh.), and the bottom Ekman layer transport(-vDJ. Using the geostrophic velocity from equa­tion (2), we find

at x = 0 i oY

-. h f aTJ - (j f a1] = 0'pt C(dy et ax .

In the Appendix, a constant eddy viscosity for­mulation (following WELANDER, 1957), of the gov­erning equation and boundary conditions in waterdepths varying from small to large, compared tobe, is outlined. It is shown that, condition (10) isa valid approximation to utilizing a zero cross­shelf transport at vanishing depth; equation (6)is the deep water form of the general constanteddy viscosity governing equation; and the deepwater form is valid in waters as shallow as h =5De, where De is typically 2 m in deep water.

CSANADY (1978) is considered a similar problembut applied his coastal boundary condition at van­ishing depth; in essence dropping the middle termin Equation (10). CSANADY reached this conditionfor this shallow depth by assuming that the along­shore wind stress balances the bottom alongshorestress as the depth vanishes. He then used thegeostrophic balance in the cross shelf direction torelate the alongshore bottom stress to the crossshelf pressure gradient. However, if the geo­strophic relation utilized by CSANADY is violatedin shallow water, i.e., if the offshore pressure gra­dient balances the vertical gradient of offshorestress in these shallow depths rather than the Cor­iolis force, then this boundary condition is notvalid. By applying our boundary condition in suf­ficiently deep water, this problem is avoided.Moreover, with differing boundary conditions ourresults differ significantly from those of CSANADY(1978). The Appendix provides a further discus­sion of these points.

Consider Equation 6. Note that the term oerJyyis important only if the alongshore length scale isless than or equal to De/a C-J 1 kilometer). Sincewe are considering alongshore length scales of or-

der greater than 10 km, then we drop this termwhich decreases the order of the equation in yandeliminates the need for a downstream boundarycondition (Equation 11). For solutions with a ylength scale order De/a, and with larger offshore

• iJ2rJ arJscales, Equation 6 reduces to 0e- + a- = O. A

iJy2 iJyboundary layer correction near y = - L of the formA(x)ea(y+L)/oe can be added to the solution dis-cussed below to satisfy any specified forcing at y= - L. The function A(x) would equal the spec­ified forcing 1]1 (X), less the solution obtained be­low, evaluated at y = - L. The forcing near X =

- L extends only a distance oe/a upstream. Drop-iJ2rJ

ping the iJy

2 term, Equation 6 then becomes

a2rJ dl10-+ a- = 0

e dX2 iJy .

To put Equation 11 in more standard form, weredefine the alongshore variable and non-dimen­sionalize the variables as

Lx = he/a, L, = Lxhc/oe, l1ref = ioYLy/pghc'

X = x/Lx, y = -y/Ly, i] = rr/1]ref' (12)

This scaling results from giving equal weight togeostrophic and bottom layer cross-shelf trans­port in Equation 10 and to the balance requiredby Equation 11. The cross-shelf scale, L; is thenatural geometric scale over which the coastaldepth doubles. In the model of CSANADY (1978),he is set equal to zero and no natural scales occur.Next substitute the redefined variables in (12)into (7)-(11) and drop the over bars on dimen­sionless variables. Hereafter, dimensional quan­tities will be denoted by an asterisk. The followinggoverning equations and boundary conditions nowresult:

iJ2rr drJ(13a)

dX2 ay,

at x = 0,af] iJrr

(13b)---=Tax iJy ,

as x ----+ 00, 1] ---+ 0, (13c)

at y = 0, rr = f]o(x). (13d)

Equation 13a is the one-dimensional diffusionequation with y interpreted as the timelike vari­able. We note that in (13b) T = 1, but retain thissymbol to track the wind stress response. This

Journal of Coastal Research, Vol. 12, No.1, 1996

82 .lanowitz and Pietrafesa

(14)

Ntx, s) ~ 0, as x ~ 00. (13h)

problem, defined by 13a-13d, may be solved viaLaplace Transform techniques. Let

N(x, s) = l'~ e-- SY7] (x, y) dy (13e)o

If the alongshore stress varies with y then T /sin (13g) becomes T'(s) and (15) becomes

1/(0,y) = - IY T(y - y')eY'Erfc(vY') dy'

+ 1]m.eyErfc(vY)

+ 1= 1/o(X') dx

x ErfcC~ + vY) dx' (15)

2vY1/1 (x, y) = ~T y!; (e r' - Y!;jErfc(W».

1/(O,y) = -T{~Y1/"- 1 + eYErfC(vY)}

+ 1]o('eYErfc(vY)

+ lex) 1](x')ex' ; V

o

and hence an alongshore change in water level, tobalance the bottom friction-induced vertical ve­locity. The distinct origin of the final two forcingfunctions TJ(l(, and 7]0 (x) can be seen as follows. Twodifferent initial profiles with the same value of TJocwill differ solely in the profile term, and two initialprofiles which are identical except in the imme­diate vicinity of x = 0, where n.; is different, willdiffer solely in the 1]0(' term in Equation 14.

Our interest in this paper lies in the source andmanifestation of subinertial frequency variationsin coastal sea level. Before we turn to this, let usconsider the spatial location of the majority of sealevel variation occurring in the cross-shelf direc­tion. For large values of y, the second term inEquation 14 becomes 112 = 7]O{.e (/./yry. The ratioTJ2(X, y)/TJ2(0, y) is e \.1. When x = 1.04vY (~1),

this ratio is 0.75. Thus most of the sea level changeobserved at the coast has occurred in deep water.For large values of y, the first term in Equation14 is

The ratio of offshore to coastal sea level 1]1 (x, y)/TJI(O, y) is e tLY!;sErfc(W) When x = 0.5vY (~1)

this ratio is -0.63. Thus, again, most of the sealevel change has occurred in deep water. This sug­gests that the choice of oe should reflect deep watervalues. We now return to coastal sea level.

Coastal sea level is given by Equation 14 whenevaluated at x = 0, i.e.,

(13f)

(1~3g)

a2

N _ sN = -7]o(x),ax2

and

so

aN /ax (0, s) - sN(O, s) = T s - 7]oe'

where j = x/2vY, and Erfc(z) = J; f= e t' dt.

As we can see from Equation 14, there are threedistinct sources of sea level variation. The firsttwo terms in this equation arise from the coastalboundary conditions (13b) and (13d) and moreclearly in the transformation of this conditionEquation 13g. Physically the first term resultsfrom the fact that the alongshore wind causescross-shelf flow. The second term is associatedwith the production of coastal pressure gradientsdue to rJoc. The pressure gradient driven flowscaused by 'rJoc must produce no cross-shelf trans­port at the coast. The third term in Equation 14,which we shall refer to as the profile term, arisesfrom the fact that any vorticity present at theupstream boundary requires a cross-shelf motion,

The solution to 13e-13h may be found via vari­ation of parameters and then inverted to yield thefollowing

1/(x, y) = -T{~Yl/2e r' - (1 + x)ErfcW

+ e(x+YIErfc(j + vY)}

+ 1Jocex+YErfc(~ + vY)

+ 1= 1/o(X') dx'

{ (x ·+ x' )

x e+x+YErfc 2vY + vY

+2~ (e t x x')'/4y - e-{'+""/4Y)}

-Iournal of Coastal Research, Vol. 12, No.1, 1996

Prognostic for Coas ta l Flood ings 83

(15a )

1.60

Figur e 2. Alongshore variation of coastal sea level to initialcoastal sea level (A(y» and alongsho re wind (- B(y» .

Equation 15a allows us to conside r cases of largealongshore exten t over wh ich the alo ngs hore windmay var y subs ta nti a lly.

DISCUSSION OF THE STEADY STATERESULTS AND MODEL SENSITIVITY

In this sect ion, we di scu ss t he varia tion of coast­al sea level with y for each of the forcin g fun cti on sT~.1)oc . and 1)o(x). We first note t hat when y « 1,

eyErfc(vY) = 1 - .];yl/2 + Y + 0(y 3/2),

and when y :::l> 1,

1 4 0

30 4 0

1 100

1)(0, y) - . r- 1)o(x') dx',V7rY 0

11)(0, y) - 1)oc. r-

V7rY

and the effect drops off more slowly. T he coeffi­cient of 1)0< in Eq uation 15 is plot ted as t he func ­t ion A(y) in Figure 2.

T he effect of the ini t ial cross-she lf profile on1) (0, y) is somewhat more complex. If t he ini tialp rofile is confined to t he region 0 .s x ;5; x., andif y :::l> 1 and xm , then,

If Xm < 1 and if l1)o(x) I ;5; 11)0< I the n t he coastaleffect will dom inate the profile effect. N ow con­side r the case where 1)o(x) = o(x - x.), i.e., a con­centra ted offshore sp ike in the initial cross shelfprofile. Then

1) (0, y) = e' · +YE rfcC~ + vY). (16)

Since 1) (0, y) vani sh es as y - 0 and y - 00 , th eeffec t of this spike on coas ta l sea level will rea cha maximum, 1)M, a t some value y, say YM' Differ­en t ia ting Equation 16 with re spect to y and set­ting t he result equal to zero allows us to solve forYM as a function of x. , We find that for x, « 1,

YM = x.l2

(17)1)m = 1 - V 2x) 7r,

wh ile for Xo :::l> 1,

and

v*(o, y) = TVpf/I.,

so that the offshore sur face layer flux is balancedby onshore bottom Ekman layer flow and u(o, y)- O. At Y = 0.77 geost rophic ons ho re flow andbottom Ekman layer flow each take up half of th eoffshore surface layer flow. The term multiplying- T is plot ted as the func t ion B(y) in Figure 2.

We next cons ide r the effect of in it ial coasta l sealevel, 1)0<, on 1)(0, y) . For y « 1,

1)(0, y) - 1)0« 1 - .];y'l2)and hence initially t his effect drops off rapidly.Ne xt, for y :::l> 1,

21)(0, y) - - v;y,n

So, sea level conti nues to drop in the down streamdirect ion alt hough at less than th e linear rate.Finally, as y - 00,

1e>E rfc(vY) = - - + 0(y- 3/2).v;y

We firs t discuss the variation in 1)(0, y) due tot he wind st ress, i.e., the term mul tiplying T inEquation 15. Fo r y « 1,

1)(0, y) ;" -yo

Thus, an upwelling favorable wind causes sea levelto d rop in th e downstream dir ecti on. More gen­erally, in th e N hem isphere, sea level d rops in thedownwind d irect ion. Also u *(o, y) = TVpf.h" sofor sma ll y, t he offsh ore flux in t he surface E kmanlaye r is balanced by geostrophic onshore flow.However , for y :::l> 1,

Jo urnal of Coas ta l Research, Vol. 12, No. I , 1996

84 Janowitz and Pietrafesa

response is proportional to he and (aa.) - "'. Forsmall y, this response is proportional to 1 ­2/ViYY. We use A(y) in Figure 2 to evaluate thisresponse. If we start with y = 1, and double (aa.)to increase y to 2, the response is decreased by afactor of 0.81. If we double h., y is decreased from1 to 0.25, and the response is increased by a factorof 1.54.

The uncertainty of ao., due to the uncertaintyin a., is larger than that in he' The percentagechange in the response is less than or equal to onehalf the percentage change in aa•. Thus the modelis not very sensitive to the most poorly knownparameter.

We now turn to the effects of time varying forc­ing functions with an inertial cutoff.

41°N

33 ltN

37"N

3S"N

35"N

G6·W7B"W

I :;::;:::::;;, i , llL< I 't==LI=7t iii I I 43°N

82 "

(19)

Figure 3. The U.S. east coast between Woods Hole, Massa­chusetts to Charleston, South Carolina.

and

11m = '0e- 1/2/xo ' (18)

Hence, the influence of a spike in the initial profilenear the shelf break will be felt far downstreamalthough the magnitude of the effect is inverselyproportional to the distance offshore. HSUEH etat. (1976) found a similar result as they examinedthe effect of an upstream profile in a region offthe Oregon shelf.

We now examine the sensitivity of the dimen­sional model predictions to the parameters aa.and hewhile holding other parameters fixed. Notethat 1/,.r is proportional to hJaa. (L, = h;/aoe) andy is proportional to (aa.)/h~.We consider the sen­sitivity of the first two terms in Equation 15 asthese will be utilized in Section 5.

The wind stress term is proportional to (1/,.r) y,for small y, and (1/,.r) s", for large y. Thus, forsmall y, with y* fixed, the wind stress dimensionalresponse goes as h;'and is independent of (aa.).For large y the dimensional response goes as(ao.) - Y, and is independent of he. For intermediatevalues of y, we can use B(y) in Figure 2. If forsome choice of parameters, y = 1, then doublingthe assumed value of ao., would produce a 9 per­cent decrease in the response. Doubling the valueof he would decrease the response by 13 percent.

The dimensional response to the second termin Equation 15 is proportional to the dimensionalvalue of 1/00 times a function of y. For large valuesof y, this is y-"'. Thus for large y, the dimensional

A MODEL FOR SUBINERTIALSEA-LEVEL VARIATIONS

We now consider motions with time scales inexcess of f;'. Under the assumptions of Equation1, vorticity Equation 3 is replaced with

ar,* aw*at* = f az* .

Using Equation 5, then (19) becomes

as,* _ f * * ()at* - - h*(x) {a.s, - au }. 20

We now consider the case when the forcing func­tions (TY, 1/00' 1/o(x» are slowly varying functions oftime. The effects of Ty(t) and 1/oo(t) are applied atthe coast where the coastal spin up time hJfa. istypically about one half a day . If the time scalesof these coastal forcing functions are considerablylarger than one day we might expect coastal sealevel to respond in a quasi -steady manner to theseforcing functions. The conclusion was also reachedby WRIGHT (1986). The response to the initialprofile in general would be quite complex as thespin up time over the outer shelf can be severaldays to a week. If the cross shelf profile at y = 0is confined to the region 0 .s x .s Km « 1, we canneglect the profile term and take

T *1/*(0, y, t*) = 1/00*(t*)A(y) - ...L1/",rB(y) (21a)Tyo

where, as previously denoted,

A(y) = eyErfc(vY) (21b)

2B(y) = vY y1l2 - 1 + eyErfc(vY). (21c)

Journal of Coastal Research, Vol. 12, No.1, 1996

Prognostic for Coastal Floodings 85

Figure 4. Pr edicted (. ----) and observed (- - ) sea level atMontauk, New York, and Sandy Hook, New Je rsey.

cause of the relatively small value of L, in thisregion, the response to the 1)oc forcing drops rap­idly in the alongshore directions. The values forA(y) drop to 0040 and 0.27 at Montauk and SandyHook respectively. The response of sea level isthus primarily wind driven. If we consider theobserved spike at March 29, 60 percent of themodel prediction of sea level is wind-driven atMontauk while fully 81 percent is wind driven atSandy Hook.

Our second comparison utilizes data from Jul­ian days 50 to 150 of 1988. The section consideredis the coast of New Jersey from Sandy Hook toLewes, Delaware, just across the mouth of Dela­ware Bay south of Cape May, New Jersey. Wetake y = 0 at Sandy Hook and predict sea levelat Ventnor City, New Jersey and Lewes, Delawareapproximately 130and 200 kilometer s to the south.We again take h, = 10 m, O. = 2 m and the ex =

0.5 X 10- 3• This leads to L, of 100 kilometers.

2MAY

,2

MAY

, i

12 22APR

12 22AP R

23

196 4

19 6 4

13MAR

SAN OY HOOK , NJ. _- -- - PRED ICTED

- OBSERVED

MONTAUK POINT, NY

---- PREDICTED- OBSERVED

22FEB

BO

100

120

100

COMPARISON WITH OBSERVATIONS

In this section we utilize Equation 21 to predictalongshore sea level variations in two sections ofthe Mid-Atlantic Bight and one sect ion of theSouth Atlantic Bight for an independen t evalu­ation of the technique. In utilizing thi s equation,we neglect the contribution of the profile term.We do this not becau se the term is not significant ,as it may well be for small y, but rather becau seinitial profile data are not available.

We first compare predictions of Equation 21with observations in the Mid-Atlantic Bight. For­ty hour low passed data is utilized in all compar­isons. First, consider a region of shelf in the Mid­Atlantic Bight ranging from Woods Hole, Mas­sachusetts, to Sandy Hook, New Jersey, see Fig­ure 3. Sea level at Woods Hole and the alongshorewind at the Brookhaven National Laboratorytower are taken as the forcing functions and sealevel at Montauk, (Long Island) New York , andSandy Hook are computed and compared withobservations. We take h, = 10 m, O. = 2 m, andex = 0.62 x 10- 3 ; we then find that L, is 80 km(= h~/exoe). We note , while our assumed value ofO. appears small, the equivalent value of O. utilizedby Hickey and Pola (1983) in th eir study of westcoast sea level changes is even smaller, i.e., ap­proximately 1 meter. Their linear bottom coeffi­cient X (= 0.01 em/sec) is equivalent to a value ofO. equal to X/f.This of course does not justify theuse of O. = 2 m. As discussed earlier, O. shouldassume a deep water value where the bottom tur­bulence is driven by the overlying geostrophic cur­rents. Following CSANADY (1982), we take Av =

u~2/200f which yields O. = u~20f, where u~ is thebottom frictional velocity . If we now take U~2 =

Coq~ where q. is the overlying geostrophic current,then taking Co = 1.6 X 10- 3 and q. = 10 em/secwe find O. = 2 m. Rational arguments can be madefor using larger values of CDand/or smaller valuesof q. so that 2 m appears a reasonable choice. Thesensitivity to the choice of O. has been discussedpreviously and is not large. Comparisons are givenin Figure 4. There is an increase in amplitude ofboth observed and predicted sea level in goingdown the coast and agreement is quite good. Be-

We now apply Equation 21 to the Middle AtlanticBight and then to the South Atlantic Bight (bothshown in Figure 3), by driving the model withactual wind data and then make a comparison ofmodel results to observed coastal sea level, to as­sess the applicability of the model in thes e regions.

J ournal of Coastal Research, Vol. 12. No. I, 1996

86 Janowitz and Pietrafesa

T26

26166

JUl

26166

JUN1919

2717

27 7 17 27 6 16 26 6 ~

APR MAY JUN JUL1979

40

~e ''''l'~Il!£15 -08 ERVEO 1\-2: ~tJ5v:~J;,e:

] -40 I I i Vi i I I j i I- 27 7

APR MAY

- 30

60

so

0

- 30

~ - 60

...J 19 29 10 20 30 9 19 29 9 19 29\oJ FEB MAR APR MAY>\oJ 19B8...J

i:.I 80, LEWES,DEIII

- - -- PREDICTEO- OBSERVED

40

o 30 9 19APR19B8

Figure 6. Pred icted (- -- --) and observed (--) sea level atSouthport, North Carolina, and Charleston, South Carolina.

Figure 5. Predicted (-- - --) and observed (--) sea level atVentnor City, New Jersey, and Lewes, Delaware .

The main component of the wind is to the north,northeast at 300 east of north. A comparison ofpredicted and observed sea levels is given in Fig­ure 5. Agreement is quite reasonable given theassumptions of uniformity in the alongshore di­rection which is not totally satisfied in the actualcase and given th e lack of initial profile data. Seebelow for a quantitative estimate of model skill.

Next, we consider a region in the South AtlanticBight from Beaufort, North Carolina, to Charles­ton, South Carolina, see Figure 3. We take y = 0to be Beaufort and we utilize Beaufort sea leveland the alongshore wind at Wilmington, NorthCarolina to predict sea level at Southport, NorthCarolina and Charleston, South Carolina. For thisshelf, we take he = 10 m, o. = 2 m, and a = 0.33x 10- 3• This leads to a value of L, of 150 km.Predicted (dashed) and observed (solid line) sealevel at Southport and Charleston are given inFigure 6. Agreement is quite reasonable given th efact that the Gulf Stream, which is present at theshelf break, frequently is present over the shelfand must impact upon coastal sea level. We notethat both predicted and observed sea level fluc­tuations increase in amplitude in going fromSouthport to Charleston.

We can quantify the skill of the model by de­fining a dimensionless mean squared error, E2, asfollows:

E2 = ~ (1]01>0 . - 1]P'.d)2/~ 1]obe.2

where the sum is over all units of observation andprediction (every six hour s) over the time periodof comparison. For Montauk, Sandy Hook , Vent­nor, Lewes, Charleston and Southport, the cal­culated values of E2are 0.173, 0.122, 0.219, 0.181,0.294, and 0.201, respectively. The model worksbest when fluctuations are most robust . For op­erational use of the model , the parameters h~/aoe

and hJaoe could be adjusted to minimize the errorin any given region.

CONCLUSIONS

A simple analytical model for predicting along­shore variations in coastal sea level has been de­veloped and applied to three US East Coast shelfregions. The three regions are from Woods Hole,Massachusetts to Sandy Hook, New Jersey, thenfrom Ventnor, New Jersey to Lewes, Delawareand finally from Cape Hatteras, North Carolinato Charleston, South Carolina.

The good agreement between model pred ic­t ions of coastal sea level and actual observationssuggest that for sub-inertial frequencies, we nowmight have the capability of predicting the timeseries of sea level from Woods Hole, Massachu­setts, to Charleston, South Carolina, simply bytaking the time series of the alongshore compo­nent of the wind and sea level at Woods Hole;preferably this could be done in segments alongwhich the alongshore wind is uniform or Equation15a with alongshore varying wind could be used .The ability to predict sea level fluctuations alongthe coastline of the eastern U.S. seab oard is par­ticularly important to set inner shelf boundary

Jo urnal of Coasta l Research, Vol. 12, No. I , 1996

Prognostic for Coastal Floodings 87

APPENDIXIn this section we utilize the constant eddy vis­

cosity model to examine the validity of some ofthe assumptions we have made. While this closurescheme is the simplest approach it does allow usto examine the governing equation and boundaryconditions in both shallow and deep waters. Sub­scripts indicate partial differentiation. FollowingWELANDER (1957) we start with the followingequations and boundary conditions.

HSUEH, Y.; PENG, C.Y., and BLUMSACK, S.L., 1976. Ageostrophic computation of currents over a continen­tal shelf. Memories Societe Royale des Saimes deLiege. 10,315-330.

MIDDLETON, J.H., 1987. Steady Coastal Circulation Dueto Oceanic Alongshore Pressure Gradients. Journalof Physical Oceanography, 604-612.

PEDLOSKY, J., 1987. Geophysical Fluid Dynamics, NewYork: Springer-Verlag, 710p.

PIETRAFESA, L.J.; MORRISON, J.M.; MCCANN, M.P.;CHURCHILL, J.; BOHM, E., and HOUGHTON, R.W., 1994.Water mass linkages between the Middle and SouthAtlantic Bights. Journal of Continental Shelf Re­search, in press.

PIETRAFESA, L.J. and JANOWITZ, G.S., 1988. Physicaloceanographic processes affecting larval transportaround and through North Carolina Inlets. AmericanFish Society Symposium, 3, 34-50.

NUMMENDAL, D., PILKEY, a.H., and HOWARD, J.D. (eds.),1987. Sea-level Fluctuation and Coastal Evolution.Special Publication No. 41. Tulsa, OK: Society ofEconomic Paleontologists and Mineralogists, 267p.

SCHWING, F.B.; OEY, L.-Y., and BLANTON, J.D., 1985.Frictional response of continental shelf water to localwind forcing. Journal of Physical Oceanography, 15,1733-1746.

SCHWING, F.B.; OEY, L.-Y., and BLANTON, J.D., 1988.Evidence for non-local forcing along the South-east­ern United States during a transitional wind regime.Journal of Geophysical Research, 93, 8221-8228.

WANG, D.P. and ELLIOT, A.J., 1978. Non-tidal variabil­ity in the Chesapeake Bay and Potomac River; evi­dence for non-local forcing. Journal Physical Ocean­ography, 8, 225-232.

WANG, D.P., 1979. Low frequency sea level variabilityon the middle Atlantic Bight. Journal of Marine Re­search, 37, 683-697.

WANG, D.P., 1982. Effects of Continental Slope on theMean Shelf Circulation. Journal of Physical Ocean­ography, 12, 1524-1526.

WELANDER, P., 1957. Wind action on a shallow sea: Somegeneralizations of Ekman's theory. Tellus 9,45-52.

WRIGHT, D.G., 1986. On quasi-steady shelf circulationdriven by along-shelf wind stress and open-oceanpressure gradients. Journal of Physical Oceanogra­phy, 16, 1712-1814.

(AI)

(A2)

LITERATURE CITED

BURRAGE, D.M.; CHURCH, J.A., and STEINBERG, C.R.,1991. Linear systems analysis of momentum on thecontinental shelf and slope of the central Great Bar­rier Reef. Journal of Geophysical Research, 96C,22,169-22,190.

CHAO, S.-Y. and PIETRAFESA, L.J., 1980. The subtidalresponse to sea level to atmospheric forcing in theCarolina Capes. Journal Physical Oceanography,10(8), 1246-1255.

ClONE, J.J.; RAMAN, S., and PIETRAFESA, L.J., 1993. Theeffect of Gulf-Stream-induced baroclinicity on U.S.East Coast winter cyclones. Monthly Weather Re­view, 121, 421-430.

CSANADY, G.T., 1978. The arrested topographic wave.Journal Physical Oceanography, 8, 47-62.

CSANADY, G.T., 1982. Circulation In the Coastal Ocean.London: D. Reidel.

HICKEY, B.M. and POLA, N.E., 1983. The seasonal along­shore pressure gradient on the West Coast of the Unit­ed States. Journal Geophysical Research, 88, 7623­7633.

conditions for numerical shelf circulation models,to determine the non-local forcing conditionswhich are so important at the mouths of the largeestuarine systems which are indigenous to the MidAtlantic Bight (WANG and ELLIOT, 1978; PIE­

TRAFESA and JANOWITZ, 1988), and to help fore­cast the sea level response to the large wintertimeatmospheric storms which buffet the entire region(ClONE et al., 1993). The general success of themodel suggests that it may be applicable to otherregions which have a significant alongshore windstress.

ACKNOWLEDGEMENTS

This study was supported by the U.S. Depart­ment of Energy under grant DE-FG09-85ER60376(L.J. Pietrafesa) and from the National Oceanicand Atmospheric Administration North CarolinaSea Grant College Program under grant NA90AA­D-SG062 (L.J. Pietrafesa and G.S. Janowitz). B.Batts and M. DeFeo did the word processing, L.Salzillo drafted the figures and C. Gabriel pro­vided programming support. The model resultswere compared to coastal sea level data collectedduring the DOE Shelf Export and Exchange Pro­cesses II field program staged in 1988 and de­scribed in Biscaye (1994) and during a NCSGCPstudy of estuary-coastal ocean coupling. Hope­fully, the results of this work will also aid S.Harned, K. Keeter, J. Pelissier and the staff ofthe N.C. National Weather Service in their effortsto better predict the time and height of offshorecoastal flooding, along the U.S. eastern seaboard.

Journal of Coastal Research, Vol. 12, No.1, 1996

88 Janowitz and Pietrafesa

where

S = u + iv, T = rylp, and TJn = TJx + iTJy.

The solution to these equations is

M = M, + iMy

= ro u dz + i 1° v dz.l., -h

The vertically integrated continuity equation pro­vides the governing equation for 1]. The real andimaginary parts of equation (A5) yields M, andMy and the continuity equation requires that

We note for future reference, from (A4) that

Av:~ I,~-h = iA, ~;: v'V2 tanh(v'V2X)

+ iT/cosh(Vf72X). (A6)

(All)

-g1]x -gT]y(h(0»)2 -5 :!: h(O) = O.3A v 15Av ot> 96 f oe4

If we now let h(O) go to zero the following bound­ary condition holds.

o1'Jax (0, y) = o.

The governing equation for shallow water x -e; 1,utilizing (A 7) and (AB) is as follows

a277 + a277 + 3h x (aT] + X2 a17 ) = ~ IT h (A12)ax2 cJy2 h ax 3 ay 2 Av x·

Both Equations All and A12 differ from thosederived in CSANADY (1978) under his assumptionthat the cross-shelf pressure gradient is balancedby the Coriolis force even in shallow water. AsEquation A9 shows, in shallow water the constanteddy viscosity model predicts that the cross-shelfpressure gradient is balanced by the cross shelfbottom stress.

The deep water limit (X ~ 1) is obtained byreplacing both tanh Vi72X and l-1/cosh Vi72Xby 1. Hence for X ~ I

-g1] goeM, = ---r(h - oe) - -f 17I + T/f (A13)

From (A6), for small X

aulAv az F-h = -g1]xh + O(h2) . (A9)

Further (AB) can be rewritten as

M = h3{-gl1x_gl1y(~)2 -5:!: ~l (AIO)I 3Ay 15A

vs, 96 f Oe4f.

At the coast, x = 0, we require MI(O) = O. If h(O)is small but finite we obtain the boundary con­dition

(A3)S\z=-h=O,

= i gh17n(1 _tanh(v'V2X»)f v'V2x

+ 7(1 -COSh(~XJ· (A5)

S = i g;n (i - cosh(v'V2z/oe)

+ cosh(v'V2X»

+ V2I~ve sinh(v'V2(z + h)/oe)

+ cosh(v'V2X) (A4)

where oe = VAv /2f and X = h/oe.Integrating thisresult from z = - h to z = 0 yields the followingcomplex volume flux vector.

(A7)

We have utilized the deep water Equation A15for depths in which X ~ 5. We note that for X =5, tanh Vi72X ~ 0.997 + O.03Ii and 1 - l/cosh

(AI6)

(A14)-g11x goe

My = -r-(h - oe)TT]y·

Utilizing (A7) we obtain

(a2

1] aZT]) ah aT]o - + - + - - = 0 (A15)e ax2 ay 2 ax ay

and for large X

M = -gT]yh -goe Tlfx f f 11I + .M - -gTJxh3 -g17yh3X2-5 ~X4 (ABa)

x - 3Av

15Ay 96 f

M - -g71)13 +g71lth3X2 + T X2. (ABb)y - 3A v 15~ 4f

cJMx + aMy = o.ax ay

This yields a linear second order partial differ­ential equation for 1]. We shall obtain the shallowwater (X -e; 1) and deep water (X ~ 1) limits tothese equations. For X -e; 1 we use the Taylorseries expansion for the hyperbolic function toobtain

Journal of Coastal Research, VoL 12, No.1, 1996

Prognostic for Coastal Floodings 89

V!2X = 1.13 + O.098i. The deep water limit isthus a reasonable choice in waters at this andgreater depths.

We have also specified that M, = 0 at our coastalboundary, where X = 5. A full determination ofthe actual transport across the X = 5 isobathwould require a solution of the full governingequation for shallow (X -e; 1) intermediate (X =

0(1», and deep waters, from some small value ofh(0) at the true coast to the very large values ofh(x). This is beyond the scope of the present work.We note, however, some conclusions from theshallow water results for cross-shelf transport. Atthe coast where h = h(O), the pressure drivencurrents totally offset the weak wind driven cross­shelf transport. If we use the shallow water resultat X = 1, the direct wind driven cross shelf trans­port is about 5 percent of its deep water value.The pressure gradients will partially offset eventhis weak transport. Thus we might expect theleakage across the X = 5 isobath to be relativelysmall.

We are also in a position to assess the differencebetween CSANADY'S (1978) solution and ours.Taking his value of r to be (\f, the dimensionlessform of his coastal boundary condition is a71/iJx(0, y) = T. The solution to Equation 13a underthis condition with 17o(X) = 0 is as follows:

1/(X, y) = - T / (,5; VYe- r' - XErfCW) (A17)

and as y ~ 00, both CSANADY'S solution and oursconverge. Our solution shows that a17/iJy (0, y)approaches zero for large y so that for large yourcoastal boundary conditions and his become thesame. However, at small and intermediate valuesof y there is a considerable difference between thetwo solutions. At the coast the ration of our so­lution to his is 0.40, 0.50, 0.58 and 0.67 and y =

0.5, 1.0, 2.0 and 4.0 respectively. Thus over scaleson which the alongshore wind might be uniformsignificant differences exist.

HICKEY and POLA (1983) utilized CSANADY'S

(1978) to predict sea level variations along thewest coast of the U.S. They considered twentyyear averages of wind and sea level data. Theirorigin was San Diego and y increases to the north.They concluded that agreement was poor for smally and improved to the north. The CSANADY so­lution differs from ours in the south due to thedifference in values in the wind driven term forsmall and intermediate values for y and in thepresence of an 1]oc term which is in our solutionbut is not in his due to differing coastal boundaryconditions. This may be the source of the poorsouthern predictions in the paper of HICKEY andPOLA.

Journal of Coastal Research, Vol. 12, No.1, 1996