RESIDENCE TIMES OF WATERS BEHIND BARRIER ISLANDS

COASTAL AND OCEANOGRAPHIC ENGINEERING

C~ge of Engineering ~iversity of Florida

Gainesvilie, Florida

Technical Report No. 7

RESIDENCE TIMES OF

WATERS BEHIND BARRIER ISLANDS by

T. Y. Chiu, J. Van de Kreeke and

R. G. Dean, Principal Investigator

Prepar ed under

OWRR Project No~ A-007-~LA Office of Water Resources Research

Department of the Interior . Washington, D. C.

FOREWORD

Estuaries represent valuable natural recreational and ecological

resources of the State and Nation. The industrial and municipal waste load on these water systems has, in some cases, increased to the level of serious degradation of water quality and reduction of their natural resource values.

An

improved understanding of estuary mechanics is necessary in order to assess the capability of these systems to assimilate waste loads, to renew their waters by flushing due to fresh water inflow or by exchange with oceanic waters and the possible improvement due to engineering structures, such as inlets. Estuaries separated from the ocean by barrier islands form a distinct type that is present along significant portions of the Atlantic and Culf of Mexico coastlines in general and the Florida shoreline in particular. The present study is directed toward an irnprovedunderstanding of the exchange processes of waters behind barrier islands.

Part I of the report examines the non-dispersive aspects of the water motion as affected by tides and freshwater discharges. A cornputational procedure is developed and discussed with reference to field measurernents conducted in Lake Worth aridSarasota Bay, Florida. Special ernphasis is directed toward the tide-induced net flow whLch can be viewed as a "rnass-transport" effect. This net flow can occur when a waterway is connected to the ocean by two or more inlets. It is concluded that this net flow can be significant in water exchange considerations and that this feature could be enhanced by the proper design of inlets. Field and/or model corroboration of this effect which was not obtained in the present study, is recommended.

literature is reviewed. Based on the assumption of quasi-steady state

conditions, dispersion coefficients are presented, based on field measurements in Lake Worth and Sarasota Bay. The measurements are also interpreted in terms of residence time for the northern and southern portions of Lake Worth, Florida.

The non-dispersive and dispersive components of constituent transport were examined separately in the present study in order to emphasize the mechanics associated with each of these modes of transport and exchange. In predicting pollutant transport and exchange in a particular estuary system, it would be expedient to incorporate the mechanics of both of these modes of transport into a single computational procedure such as those developed by Leendertse (27), Shankar and Masch (28), Fischer (25), and others.

-

,---

-,---, J. _

TABLE OF CONTENTS Part I

NON-DISPERSIVE TRANSPORT IN LAGOONS

Page LIST OF FIGURES

LIST OF STIfBOLS

1. INTRODUCTION 1

2. THE TIDAL EQUATIONS FOR THE LAGOON AND THE INLETS 3

3. A NUMERICAL MODEL FOR THE TIDAL FLOW 10

4. DETERMINATION OF THE FRICTION FACTOR F 23

5. MASS TRANSPORT IN LAGOONS DUE TO TIDAL ACTION 28

a. Analytic Solution; Physical Concepts 28

b. Influence of Inlets on the Mass Transport

40

6. APPLICATION

44

a. Lake Worth

44

b. Sarasota Bay

51

7. SUMMARY AND CONCLUSIONS 56

APPENDIX A

A-I

Part 11

DISPERSIVE TRANSPORT IN LAGOONS LIST OF FIGURES LIST OF STIfBOLS 1. INTRODUCTION 58 2. EQUATIONS OF MIXING

59

a. Three-Dimensional Equation

59

b. One-Dimensional Equation 60 3. MIXING CHARACTERISTICS 60 a. General 60

TABLE OF CONTENTS Ccont'd)

b. One-Dimensiona1 Ana1ysis Ci) General

Cii) Idea1ized case

c. Idea1ized Case for Waters Behind Barrier Is1ands

4.

FIELD MEASUREMENTS OF SALINITY

a. Lake Worth b. Sarasota Bay

5.

CORRELATION OF FIELD MEASUREMENTS

AND

THEORETICAL ANALJSIS a. Lake Horth

b. Sarasota Bay

6. RESIDENCE TIMES BASED ON MIXING. a. Lake Worth In1et

b. South Lake Worth In1et c. Sarasota Bay

7.

CONCLUSION, DISCUSSION

AND

RECOMMENDATION APPENDIX B APPENDIX C APPENDIX D REFERENCES Page 61 61 63 66 79 79 79 82 82 85 89 90 90 90 91 B-1 C-1 D-1 R-1

PART I

NON-DISPERSIVE TRANSPORT IN LAGOONS

by

FIGURE 1 2 3 4 5 6 7· 8 9 10 11 12 13 14a 14b 14c 14d 15 16 17 Reference frame. 3

In1et schematization. 6

Location of variables in the numerical grid. 11 Grid scheme for different in1et configurations. 16 Curves representedby Equations (3.6) and (3.7). 19

Tide in a sea level cana1. 26

Mean water level in infinite long channe1. 33

Effect of higher order terms on q*. 36

Effect of "linearization"of the friction term on q*. 37 Effect of convective acce1eration and non-1inear part of the surface gradient on the net discharge q*. 39 Lagoon connected to the ocean by in1ets. 40 Net displacement of a f1uid partiele in 1agoons versus

width of in1et 11. 42

Schematizationof Lake Worth. 45

Ocean tide measured at Lake Worth pier. 47 Computed and measured tide at Station8. 48 Computed and measured tide at Station 11. 49

Discharge at Station 8. 50

Sarasota Bay 52

Tida1 curves, Sarasota Bay. 53

LIST OF SYMBOLS

a tidal amplitude

c wave celerity in case of no friction o

f Darcy-Weisbachresistance coefficient g gravitationalacceleration

h mean depth k wave number

k wave number in case of no friction o

m velocity distribution coefficient q discharge per unit width

q* net discharge per unit width t time

u velocity in x direction v velocity in y direction w velocity in z direction

x horizontal cartesian coordinate y horizontal cartesian coordinate

z vertical cartesian coordinate,positive upward A cross-sectionalarea

~ bay area

B width of the water body D total depth

F resistance coefficientused when considering quadratic friction

Ft

resistance coefficientused when considering linear friction L length of the water body

M lateralfLow , rainfall P wetted perimeter

Q

total discharge R hydraulic radius

S slope of the water surface

w

free surface stressj(rnass density of water)

/

/

a phase angle E relative error

n water surface elevation

n*

mean water surf ace elevation ~ factor related to the resistance p density of water

o angular frequency of the tide T shear stress

w

angular frequency of the earth rotation ~ latitude, positive for northern hernisphere nCoriolis factor

_"" __ .'_ - '11

--I. NON-DISPERSIVE TP~SPORT IN LAGOONS

1. INTRODUCTION

This portio~ of the report deals with the convective, or non-dispersive, transport of constituents in lagoons. A lagoon is defined here as a body of water connected to the ocean by two or more narrow constricted inlets. In general a tidal wave when propagating through such a system will be modified by both inlets and the main water body. The special case for

which the tidal propagation in the main water body may be neglected 'HilI not be discussed here. The reader is referred to Keulegan (2), Van de Kreeke (19), Shemdin (30) and Mota Oliveira (31). Of ten the convective transport in

lagoon systems is solely attributed to fresh water inflow, while it is generally regarded that the only effect of the tidal motion is to generate the turbulence associated with the mixing process. It will be shown, based on theoretical considerations, that for shallow lagoons the convective

transport due to the tidal motion only is an important factor in the renewal of water in the lagoon. The magnitude of the convective transport is mainly governed by the location and the dimensions of the different inlets. By

properly designing the inlets a considerable increase in convective transport can be obtained. This concept of using tidal inlets for environmental

controls has been discussed in a wider context by Lo ckwoo d and Carothers (18). In describing the flow in the lagoon, different sets of equations are used for the flow in the lagoon proper and the flow in the inlets. The reason for this is that certain assumptions made in deriving the equations for the lagoon do not hold for the flow in the inlets. In both thé equations for the lagoon and the inlets the non-linear terms must be retained because in a mathematical sense the convective transport results from these non-linear terms. The inlet equations used here are algebraic equations while the flow in the lagoon is described by the well-known long wave equations which are

partial differential equations. In the solution, the inlet equations are regarded as implici t boundary eondi tions for the long wave equations. This implies that it is not necessary to extend the numerical grid used for the main water body to the inlets which, because of their relatively small widths, would lead to small grid sizes and consequently a considerable

increase in computational effort. At first the method of Leendertse (17) was considered for solution of the governing equations. This idea was

abandoned because the implicit boundary conditions would require major

modifications in his model. The numerical model ultimately used is based on a space and time staggered explicit finite difference scheme similar to the one described by Reid and Bedine (21).

As stated before, the non-linear terms in the equations, which include the friction term, are important when regarding.convective transport. An accurate value of the friction factor therefore is one of the first requirements to prediction of reliable values for the convective transport. In view of this a special discussion is devoted to the determination of the friction factor.

Finally, an attempt is made to apply the results of the study to Lake Worth .and Sarasota Bay, both located in Florida.

I'

/ I

3

2. THE TIDAL EQUATIONS FOR THE LAGOON AND THE INLETS

In the Eulerian frame, the veloeities and water surfaee elevations are related to time and the various loeations in geometrie spaee by two types of equations, one expressing the eonservation of mass, the other expressing the eonservation of momentum. The equations are derived in Appendix A. The

diseharge per unit width and the water levels rather than the veloeity and the water levels are ehosen as the dependent variables beeause this leads to a simpIer

form of the eonservation of mass equation.

From the Appendix, it may be seen that the equatibn of cbhservation of mass takes the form

êq êq

!!l +

_x

+

_J__

ê t êx êy M (2.1)

in whieh x,y are horizontal eartesian eoordinates, see Figure 1.

n

=

water surface elevation q diseharge per unit width

M = net LnfLow per unit surfaee area due to rainfall, lateral inflow, ete.

The variables q and nare mean values over a time interval, which is large compared to the turbulenee time scale and small compared to the period of the tide.

In deriving the above equation, the fluid was assumed to be incompressible.

___

_jlt:-)_/_~_(X_.Y_._t

)--=:::_ -::- _

__.iÎ/~_~x--L-C-

STILL WATER LEVEL

~

J

•

h (x.y )

The equations for the conservation of momentum in the x and y directions are, respectively - F (h

+

n)

+

Wx

(2.2)

(2.3)

in which h mean depth

S1 2w sin <p

=

Coriolis factor

w

= angular frequency of the earth rotation <p latitude, positive for northern hemisphere F

=

f/8

=

resistance coefficient

f Darcy-Weisbach resistance coefficient

W free surface stress/(mass density of water)

The ma in assumptions made in deriving Equations

(2.2)

and

(2.3)

are - incompressible fluid .

- vertical accelerations are negligible - lateral stresses are negligible

When assuming the tidal flow to be one dimensional, the equations of conservation of mass and momentum may be written as

(2.4)

19_

+

.!. ~

+

g(h

+

n) ~ = -

Fq Iq I

+

W

5

The.assumptions made in deriving the long wave equations, in general, hold very weIl for large bodies of water but are less justified when dealing with transitions and regions in which the flow is restricted. For example in

inlets, the lateral stresses can no longer be neglected. Unfortunately, the complete equations which include these stresses are still very complex and difficult to solve. Therefore, for inlet flows, recourse is taken to the following simplified representation.

It is assumed that the flow in the inlet region is one dimensional and storage is neglected. The equation of conservation of mass then becomes

(2.6)

in which

Q

=

total discharge.

To describe the dynamics of the flow, the inlet is divided into three regions as indicated in Figure 2. During flood, the flow in Region I is governed by the convective acceleration and the pressure forces; bottom friction and local

acceleration are considered small. Integrating the dynamic equation and neglecting the velocity head in the ocean then yields the Bernoulli equation

(2.7)

in which m velocity distribution coefficient

P

=

wetted perimeter of inlet cross-section measured at Mean Ocean Level B

=

the width of the inlet, which is assumed constant

R hydraulic radius for the inlet cross-section measured at Mean Ocean Level It is assumed that during ebb, that portion of the kinetîc energy at location 1 is lost which is not required for the velocity head at location O. As stated

~)

z 0 C) w a:: z W.J <t > LU LU :iE .J z 0 ï= 0 w Ul Ul Ul 0 _j a::0 .J .al <t Z 0 :::> t-5 z z 0 .J 0 j: ct l-C/) z o o C) <t .J

z

o

~ N ~ ~ LU ::I: U

en

z Q C) w a:: ~ w > z o ~ LU a:: Z <t .J Q. t-LU __J Z Z <t LU o o

7

before, the velocity head in the ocean is neg1igib1e and therefore all the kinetic energy present at 10cation 1 is lost. Neg1ecting bottom friction

it then fo110ws that for ebb

(2.8)

J

O

The flow conditions in Region 111 are the reverse of those in Region 1.

I

I Bernou11i's equation ho1ds during ebb whi1e energy dissipation takes p1ace during f100d. The on1y difference with Region 111 is that, in general, the

velocity head at 10cation i can no longer be neg1ected as was the case for the

corresponding 10cation 0 in Region I~ For ebb f10ws, the equation for Region 111

now reads m2Q2 m.Q2

+

+

1 (2.9) n2 n. -2 _{- B 2} 1 -2 B. 2g P (R

+

=-

n2) 2g P. (R.

+

_1 _n.)2 P 1 1 _Pi 1

Again neg1ecting bottom friction in Region 111, the equation describing the flow

during f100d becomes

n

.

1 (2.10)

--In Region 11, the actual restricted part of the in1et, on1y pressure forces and

bot tom friction are taken into account and thus

A

l!l

= _ P 'to

dX (2.11)

Pg

in which A A

+

Bn cross-sectiona1 area p

=

p

+

2n wetted perimeter 'to

=

bot tom shear stress

For most in1ets 2n « Pand thus P ~ P. The bottom shear stress is re1ated to the discharge and cross-sectiona1 area by the empirica1 re1ation

Furthermore A

=

R P.

Substituting these expressions for P, T and

A

in Equation (2.11) and o

rearranging slight1y yie1ds

g (R

+

!

n) dn -=-FQ~I~Q'-I-I_

p

dX = - p2

[R

+

!

n]

2

p

(2.12)

This equation ho1ds for both ebb and f1ood. Integrating Equation (2.12) with respect to x between Stations 1 and 2, the resu1ting expression can be written, to

[- B gR+= P

+

FQI~ ~ P (2.13)

This particu1ar form of Equation (2.13) was chosen to be comparable with Equation (2.16) which is common1y used to describe the f Low in in1ets. Equations (2.7), (2.10) and

(2.13)'are then combined. It is assumed that the velocity head in Equation (2.7) is of Oen). When neglecting terms of 0(n3) and higher, the fol1owing equation describing the dynamies of the flow in the inlet during flood, Q < 0, is obtained.

- B n +n. FQ2L mlQ2R g(R + = o 1) (ni-no) -:::2

(

R'

+ ~ I1 +n, 2- _{mlQ 2} p 2 p o 1) _2 - B _(n --2-2.)] 2 2P [R + = P _P 0 2gP -R (2.14) IDIQ2 m Q2 B _+~ 3 B 1

+

_-2.-2 ( -2 - n n. +"2= _2_.2 )

-2

P

R- P a

-

P 1 P 2gP R

/

I

/

9

Similarly for ebb, Q> 0, it fo1lows by combining Equations (2.8), (2.9) and

(2.13) B

n +n .

= FQ2L m2Q2R g(R

+

=

0 1.) (n.-n) .=...::l<_--=-

+

---~-P 2 1. 0 _2 n +rï . 2 _2 B m.Q2 m2Q2 2 P (R+B 0 1.) 2P[R+=(ni+ 1._L7- :::2::::::21 P 2 P 2gP. R. 2gP R -1. 1. -tn._Q2"R m2Q2 _..:::1.'--

+

(

L_L 2 B. 2 2P R

2

P

.

(R:.

+

2. n1.') 1. 1. P. 1. miQ2 B 2 2) (-

=

n_o 2P. R. P 1. 1. rri.Q2 1. ) _ 2_ 2 2gP. R. 1. 1. (2.15)

Equations (2.14) and (2.15) are the same as those used by Keu1egan (2) when

neg1ecting the second order terms and also the velocity head in the l&goon and

assuming mI

=

m2

=

m. The equations then read,

2 - 2 (F Q L

+

m Rj )

±

=Z-~- ::::::2 P R 2P R (2.16) + sign for

Q

> 0 - sign for

Q

< 0

Finally, it is noted that in deriving Equations (2.14) and (2.15) the actual

restricted part of the inlet, Region 11, is assumed to be a prismatic channe1. Many

inlet channels do not fu1fil1 this condition, because the width and depth vary from

point to point. In that case, the irregu1ar channel may be replaced by a prismatic

one having the same discharge for a given head difference. For a detailed outline

/

I

/

3. A NUMERICAL HODEL FOR THE TIDAL FLOW

The numerical procedure presented here is based upon an explicit

finite difference scheme; the unknowns depend only on values previously

computed at a lower time level. This, in general, leads to a more stringent

stability requirement (see for instance Vreugdenhil (

14 ))

as compared to implicit methods but has the advantage that the different steps in the

computation are easier to trace.

The numerical scheme is space and time staggered.Water levels are

computed at n.bt and discharges at (n + ~)bt. The ,,,,ratenlevelsapply at the I

center of the grid blocks and the discharges are computed at the gridlines.

See Figure 3. The values of Wand Ware given at the location and time

x y

level of respzctively qx and qy' the mean depth hand the lateral inflow or

rainfall s are given at the time level and location of

n

.

The basic recurrence equations are

~ ..(i,j) 1 {qx (i,j) + gM [D(i,j) + D (i-1, j)] [n (i-1 ,j)

-G1(i-1,j) 2~s (3.1 ) n(i,j)] + W (i,j)bt} x (3.2) n(i,j)] + W (i,j)bt} Y

n"(i,j)

=

n(i,j) + bt [q (i,j) + q .:.(i,j) - q" (i+1,j) - q ..(i,j+1)]

bs x y x y

(3.3) +l1(i,j)M

'(

1

q (i,j

+i)

x

11

óS

I

I

-

~

+

7J(i,j

+

l)

-

r-I I

Iqy(i, j

+

j)

Ir

_qy

'

_I

₊

_I,J

₊

_I

)

-

r-+7J(i,j)

-

~q

_x

(i

+

I

_'

n

+

_("

_'

₎

-

....

h (" ')

_{1, J}

7J

I

+I

,J

su.n

I I I

..

_I

qy

(I,J)

W

y

«.n

x

óS

FIG

3:

LOC

AT

IO

N

O

F

V

AR

I

ABLE

S

IN

T

HE

NU

MERI

CAL

in which

D(i,j)

=

n(i,j) + h(i,j)

Gl(i-1,j)

=

1 + F~t{[4q (i,j)]2 + [q (i-1,j) + q (i,j) + q (i-1,j+1) x Y y. Y k +q (i,j+1)]2}2 [D(i-1,j) + D(i,j)]-2 Y

/

/ / 1 + F~t{[4q (i,j)]2 + [q (i,j-1) + q (i+1,j-1) + q (i,j) Y x x x 1

- + q (i+1,j)J2}~ [D(i,j-1) + D(i,j)]-2

x

Primed symbols denote va1ues of the variables at time step ~t later,

Equations (3.1), (3.2) and (3.3) are based respective1y upon the differentia1 equations (2.2), (2.3) and (2.1), neg1ecting the convective acce1erations and

the effect of the earth's rotation. The differentia1 quotients in these

equations are rep1aced by difference quotients using central differences. The difference quotients for Equation (2.2) are centered about time level n~t and are centered in space about the location of q. The difference

x

quotients for Equation (3.2) are centered about n~t and the location of qy, and the difference quotients for Equation (2.1) are centered about (n + ~)~t

and the location of n. Starting from the initia1 conditions, _{all the 'Lx}'s are

computed from Equation (3.1), then the q 's are computed from Equation (3.2)

. y.

and fina1ly the n's from Equation (3.3). For a more detailed outline of the

numerical scheme, the reader is referred to Reid and Bodine(21) and Verma and Dean(2 The numerical procedure for solving the one-dimensional tidal Equations

(2.4) and (2.5) is similar to the one presented for two=ddraensLona L tidal f Low,

The convective acce1eration now is taken into account because it can be

relative1yeasily incorporated in the numerical scheme. The recurrence formu1ae

/

I

13

q"(i) G(i~l) [q(i) + ~~: (D(i) + D(i-I) Cn(i-I) - n(i»

+ W(i)lIt].

(3.4)

n~(i) n(i) +

ss.

(q~(i) - q'"(i+l») + H(i)lIt

lis (3.5)

in which

D(i) n(i) + hei)

G(i-l) 4F lIt Iq(i)1 lIt(q(i+l) - g(i-I»

I +(D(i) + D (i-I»)2+ 2 lis (h(i) + h(i-I»)

Again starting from the initial conditions, all the q's are computed for the next time level by means of Equation (3.4), then the n's are computed using Equation (3.5). It is noted that because of the introduction of the

convective acceleration, the recurrence formula (3.4) includes va lues as far apart as 2 space steps, see expression for G(i-I). This leads to difficulties

when the boundary conditions at open boundaries are given as water levels. In

that case, the convective acceleration has to be taken off center for the grids

adjacent to those boundaries.

Because of the non-linearities in the differential equations, the existing

mathematical theory is inadequate to determine the exact criteria for stability

and convergence of the numerical scheme. Ho,"ever, same insight might be

obtained by regarding a simplified set of equations. The equations describing

the tidal fLow, Equations (2.1), (2.2) and (2.3) may be replaced by three wave

equations in respectively n, qx and qy, when neglecting the non-linear terms,

/

I

a horizontal bottom and irrotational motion. It then can be shown, starting

from the recurrence formulae for the simplified set of equations, that for

this case, the difference scheme is equivalent to solving the wave equation

on the mesh ~s, ~t. The stability and convergence criteria then are, see

Platzman ( 4)

~s r=r:

- > v'gh

~t for the one-dimensional wave equation

bs r;:;--;-- > v2gh

~t for the two-dimensional wave equation

When regarding the solution of the tidal equations as a sum of Fourier

components, stability as used here implies that there should be a limit to

the extent to which any component of the solution is amplified in the numerical

procedure. Platzman ( 4 ) showed that even before the stability criterion is

reached, the amplitude of especially the highest modes might be greatly

magnified and might obscure the true solution. To study the eventual

magnification of higher harmonies, a few computations were carried out for a 17

mile long and 7 ft. deep channel, open at one end and closed at the other. In

addition, these computations provided some insight as to whether the terms

neglected when discussing the stability criteria in the previous paragraph give

rise to instabilities. The tide at the open end of the channel was sinusoidal

with an amplitude of 1.5 ft. and a period of 12.5 h. The computations were

carried out for values of the friction factor F

=

0.002 and F

=

0.004 and for

the time steps ~t

=

100 sec., ~t

=

200 sec. and ~t

=

300 sec. The space-step

~s

=

5000 ft. was the same for all computations. The stability criterion for the case discussed here and when using the simplified equations is

~s

15

were judged visually; no overflow occurred, and amplification of higher modes

was not apparent. The source term and the free surface forces were not

included in the computations.

The tidal computations discussed here will be applied for lagoons

connected to the ocean by one or more inlets. In the computational procedure

the inlet equations, Equations (2.14) and (2.15), may be regarded as the

implicit boundary conditions for the flow in the lagoon. The way in which

these boundary conditions are incorporated in the numerical scheme depends on

the inlet configuration and whether the flow in the lagoon is considered one

-or two- dimensional. The following three cases are considered:

_ An

inlet connecting a lagoon with two-dimensional flow to the ocean.

See Figure 4a.

- An inlet connecting a lagoon to the ocean such that no branching of

the inlet flmv occurs; the flow in the lagoon is considered to be one

-dimensional. See Figure 4b.

_ An

inlet connecting a lagoon to the ocean such that branching of the

inlet flow can occur; the flow in the lagoon again is considered to

be one-dimensional. See Figure 4c.

For the case of t\vo-dimensionalfLow in the lagoon, see Figure 4a, the

axis of the inlet is parallel to either the i or j axis and passes through the

location of the water levels~ The auxiliary water level n. is introduced

~

which is computed at the same time level as the water levels in the lagoon.

Starting from the initial conditions, all the discharges in the lagoon except

q (i, j+1) can be computed using the procedure described before. The value of

y

q

"(i,

j+l) is then computed as foLl.ows. The discharge q (i, j+l) ..6s= Q. is

y . . y ~

,

I

l

~

.

-I

~

.~

+

z

-

~. -<! -+ f:"" w u

₎

~se 0 er-, ~ ~

/

~

-

-

-

.1.. -f:"" .- ~ 0 ~ ~ >-~

*

-

_'\

-!l-

'

l

~

er f:"" 0- ~

y-=

~ .-K ~ 0- l/)

I

I

~

"'1

1

S'i7 z o o Cl « ..J

z

r<) I-Z + w 3: <! 'f:"" _J 0 w

4-z _J U IJ.. 0 j;)+ Z

j

'

-

0 ~ ~ _J N 0 0 <! .i l?

z

_f:""-+

<! 0 _J C/) ~ Z Z W ~ 0 !: 0 f:"" 0

*

_J I.L.. 0 I-,3: _J w I- <! _J Z Z :::r:: 0 l-C/) _' 3: ~

+

~l

z

l? Z W Z I- 0 ~ I W 0 1= 0 U _J (.9 0 Z

-Z <! ...!. w <[ _J

+

Z Cl:: ~ 0

'.

CO Z 0 0 Cl « ..J 18 l-c:( 0::

::::>

(.!) L1.. Z

o

U

I-W ...J Z

I-Z W 0:: W LL LL

o

0::

o

L1..

w

~

w

:::c

U

,

en

o

0:: (.!) (.!) u,

17

inlet equations. The difference form of the inlet equations used here is

based on the following simplified form of the Equations(2.l4) and (2.15). ± ~~

FQ2L

-~~--- ± 2 B T)

+

T).

P

(R: +

p

0 2 ~)2

/

( / It is assumed that mI

=

m2

=

m.

Af ter some algebraic manipulation, the following results is obtained.

in which ± DD

/

IT). - _T)ol q = t,s ~

+

sign for T)• > _no ~ - sign for T). < T) ~ 0 q ~(i,j

+

1)

+

q (i, j

+

1) q = 2 ..3. ·2 T)o

+

T). 2 DD 2gP

(

R

+

B

~)

2FL

+

mR P 2 (3.6)

It is noted that for many practical cases the modifications madè in Equations .(2.14)and (2.15) are justified. They were introduced here to simplify the

algebra, however, the computational procedure applies equally weIl when using the complete Equations (2.14) and (2.15) A second equation relating

q (i, j

+

1) and T). is found by applying the dynamic equation, in this case

y ~

Equation (2.3) between the discharge stations q (i, j

+

1) and q (i, j). Note

y y

that when computing the flow in the lagoon, the dynamic equations are applied

between two water level stations. In determining the difference form, the terms in both the inlet and dynamic equation are centered about nt,t. The

difference form of the dynamic equation, when neglecting the convective acceleration, the Coriolis force, and the free surface stresses, then yields the following relation between q (i, j+l) and n.

y 1 q AA [no - n ] + BB + AA n 1 0 0 (3.7) in which

/

/ / q q --(i,j+l) + q (i, j+l) y y 2 AA -gD(_Gi,j) 6₆t_s BB 6t [q (i, j+l) + q (i,j) - q --(i,j)• G + gD(i,j) "s Y Y Y u (n(i,j) + n(i, j-l»] / + qy(ii j+l)

/2

G

G F6t

V

[q (i,j) + q (i+l,j)]2 + [qy(i, j+l) + q (i,j)]2 1.

+

.

x x -

Y

2. D(i,j)2

The general shapes of the curves q

=

fen. - n ) represented by the Equations

1 0

(3.6)

and

(3.7)

are indicated in Figure

5.

Equation

(3.7)

represents a

straight 1ine. The slope of this line, AA, is a Lways negative. Therefore, the

sign of (BB + AAn ), which is a known quantity, determines the sign of

. 0

(n. - n ) which in turn determines the sign to be used in Equation

(

3

.6).

1 0

Eliminating (ni - no) between Equations

(3

.6

)

and

(3.7)

yields

1 - J

l -

4

~2 (AAno

+

BB) • 6s2

2

~2 • 6s2 (3.8) q for (BB + AAn ) > 0 o

19

o

z

<{

--

Cf)

z

o

I-<{ :::l

o

W

>-CD

o

w

I-Z

w

Cf) W 0::: n,

w

0::: Cf)

w

>

0::: :::l (.) l()

and q 1 -

VI

+

4 ~2 (AAn_o + BB) • ~s2 (3.9) AA -2 -2 DD for (BB + AAn ) < 0 o

--:-Note that only the first order terms in (n{ - n ) are eliminated because

.L 0 .

second order terms are still present in the factor DD. Equations (3.8) and (3.9)

therefore may be regarded as being quasi-linear, which suggests finding a

solution by means of a perturbation metbod. First the value of (ni - nol

in DD is taken equal to the value at the previous time step. The value of q

can then be found from Equation (3.8) or (3.9) depending on the sign of (BB + AAn). Knowing q, the value of (n. - n ) is determined from either

010

Equation (3.6) or (3.7). This value of (n. -₁ n₀) is substituted in DD. The

procedure then is repeated until the difference between the computed and

previously computed value of q is within certain limits.

When dealing with one-dimensional tidal flow, the total discharge Q

rather than the discharge per unit width q is used as a dependent variabie.

Consider the "non-branching inlet", see Figure 4b. Again an auxiliary water

level

n

.

is introduced. The inlet equation is similar to Equation (3.6)

1 with q·~s replaced by Q. Q ± DD

I

In

i - n

I

(3.10) 0 + sign for

n.

> n 1 0 - sign for

n

.

< n 1 0 in which Q ~(i+l) + Q(i+l) 2

The difference form of the dynamic equation applied between stations Q(i+l)

and Q(i), taking into account the convective acceleration, yields the following relation.

21 Q

AA

(

n. - n

)

+

BB

+

AA

n

1 0 0 (3.11) in which

AA

-g D(i)

BI

t.t Glls

BB

=

{Q(i+1) + Q(i) - Q~(i).G + g BI D(i) llt[n(i)

lls + n(i-1)]} / / 2G + Q(i+1) 2 G 1 + F llt IQ(i+1) + Q₂ (i) D(i)2

BI

+ 2 llt [Q(i+1) - Q(i)] llshei) BI

Equations (3.10) and (3.11) with unknowns Q and n. can be solved using the 1

procedure out1ined in the previous paragraph.

The numerical scheme for the "branching in1et", see Figure 4c, invo1ves four unknowns Q~(i+1), Q~(i+2), Q~. and n. as compared to on1y two, Q(i+1) and

1 1

n., for the "non-branching in1et". The four unknowns are re1ated by the 1

fo11owing four equations

- the in1et equation which takes the form of Equation (3.10) with

Q

=

Q~.

+

Q

1 i

2

- the dynarnic equation app1ied be tweeri the locations of Q(i) and Q(i+1); this equation takes the form of Equation (3.11)

- the dynarnic equation applied b etween the locations of Q(i+2) and Q(i+3); this equation takes the farm of Equation (3.11) with Q(i)

replaced by Q(i+3), Q(i+l) replaced by Q(i+2), n(i) replaced by n(i+2),

n(i-l) replaced by n(i+3), and hei) replaced by h(i+2), llsreplaced by

-llsand BI replaced by B2.

the continui ty condition which, when assuming that the Hater level in

the hatched area is the same everywhere, takes the form

Q(i+l)

=

Q. + Q(i+2)

Elimination of Q(i+l) and Q(i+2) between the two dynamic equations and the

continuity equation yields arelation b etwe en Q. and TI. similar to Equation

1. 1.

(3.11). This equation together with the inlet equation then can be solved following the procedure described before.

Finally, it is noted that in one-dimensional flow computations, it is often necessary to divide the lagoon in parts with different widths, see

Figure 4d. The flow at the boundary of two such parts may be computed

following a procedure similar to the one applied to the "non-branching inlet", replacing the inlet equation by a second dynamic equation between the location of Q(i+l) and Q(i+2).

23

4. DETERMINATION OF THE FRICTION FACTOR F

The factor F in the friction term of the conservation of moroentum

equation is difficult to estiroatein advance; its value has to be derived

from measurements. In this chapter two methods for deterroining F are

compared. One method is based on matching computed and measured water

levels, the other is based on matching computed and measured discharges.

Special attention is given to the influence of errors in the measured water

levels and discharges on the value of F. The discussion is restricted to

one-dimensional flow.

To calibrate a tidal model, that is to determine the value of F, the

area covered by the model is divided into sections each with approximately

uniform geometry for which the values of F may be assumed constant. It

is assurned that the boundary conditions at both ends of each section are

known and are given as water levels. Given these water levels and assuming

a value of F, the water levels and discharges within each section can be

cornputed. The cornputations are carried out for different values o~ F until

a good match exists between cornputed and rneasuredwater levels or discharges.

The F value obtained in this manner reflects, in addition to friction, the

effects of schematization, neglected terms in the equations and measuring

errors. These effects may cause large variations in the value of F during

a tidal cycle especially during periods of slack tide when the inertia

forces are predominant. Therefore in determining F these periods should

be disregarded. Furtherroore,arealistic value of F, that is a value mainly

determined by friction and more or less constant during a tidal cycle

(See Dronkers (9) , page 425),can only be obtained in cases for which the

flow is friction dorninated during the larger part of the tidal cycle.

As stated before measurement errors result in errors in F. In this

forward. In the next paragraph an example will be presented sho\ving

the order of magnitude of the errors in F resulting from errors-in the

"calibration curves", (a calibration curve is defined as measured water

levels or discharges within a section and used to determine F by matching

with computed water levels or discharges). The water surface in each section may, as a first approximation, be regarded as a straight line because in practice the length of the sections are small compared to the

length of the tidal wave. The end points of the straight line are

determined by the water levels at the boundaries. Therefore a change in F will hardly affect the computed water level in the section. From this it may be inferred that,when determining F by matching computed and measured

water levels, an error in the latter or in the boundary conditions will lead to a large change in F. ~fuen calibrating on discharges the effect of

measuring errors can be expressed as follows. During periods of friction

dominated flow a first approximation of the equation of conservation of

momentum is h dn g dX

FQIQI

h and thus €(F)

in which €

=

relative error.

The error in F thus is of the same order as the error in the boundary

conditions (~:) and the calibration curve

(Q).

The following example serves to illustrate the order of magnitude of errors in F due to errors in the calibration curve. Consider a section with a constant depth h = 7 ft. and a length

L

= 50,000 ft. The boundary

25

n

2

= 1.3 sin (2n(t-2700)/45000), see Figure ·6a. The water levels and discharges in the middle of the section are computed for F

=

0.002, F

=

0.0025 and

F

=

0.003. The results are plotted in Figures 6b and 6c. The differences between the water levels for the different values of F were too small to be reproduced in the figure. In addition, for the same

location and for F 0.002, the values of the different terms in the equation of conservation of momentum are plotted, see Figure 6d. The latter results provide some insight in the relative magnitude of the forces and accelerations represented by these terms and their phase relationship with water levels and discharges. It may be seen from Figure 6d that the flow is friction dominated during the larger part of the tidal cycle, which as stated before is a condition to arrive at a realistic value of F. When using a measured water level in the section as calibration curve, the effect of an error

in'this curve on the F value can be demonstrated as folLows , Let the curve corresponding with F 0.002 in Figure 6b represent the correct water level

(calibration curve) in the section. Let the curve corresponding with

F

=

0.003 in the same figure be the measured calibration curve. This curve deviates from the correct curve at the most by 0.03 ft.*; the relative error

(maximum deviation divided by tidal amplitude times hundred) is 2%. An errór of 2% in the water level (calibration curve) would thus lead to an

(0.003 - 0.000.002 2 • 100%•

=

) 5

o

%

.

error in F and as may be seen from Figure 6c an error of 20% in

Q.

The influence of errors in the calibration curve on the F value when using discharges can be demonstrated in a similar way.

Let the curve in Figure 6c corresponding with F

=

0.002 be the correct discharge curve. Consider the curve corresponding with F

=

0.0025 in the

*This value cannot be read from Figure 6b, it was taken directly from the computer output.

o

-1

.

I-LL.

=

0 LLJ = I--1

10

5

o

-5

-10

o

-1.10-3

=

X=O FT.

>'_/

I- ",,'" LL.

_,

,

,

LLJ

=

-J>- TIME IN HOURS ...._.... (a) TIOE AT X= 0 FT. AND X= 50,000 FT. 6 -J=- TIME IN HOURS (b) TIDE AT X=25,OOO FT. ~ LLJ cr.> <,

""

I-LL.

=

=

I-=

3::

I-=

0

=

=

LLJ c.... LLJ

=

= oe<:

=

_~ en

=

_. - ._. ~ g(h-h])Or] »:' . .,

ot

o

q

<

.

_"

/

~

--.

,

"

-<,

_.

_...

..."'--__

-_--~--'

F=0.0025 __ - - - - -_ ~ F= O. 002

_--

....

_

_

'

_'-'-'_ '...

J

.

.

...

"

. " " F=0003

_.

.,,', '~,:, __._ T IME IN HOURS ...

_

.

_

.

(c ) DISCHARGE AT X=25, 000 FT. F=0.002 ...

...

...

"

6 ,,"

/YI- TIME IN HOURS

o

/

...

...

""

_"

_{-_}

_...

/ /

-

.

_

"

--

.

-

'

...-(d) MAGNITUDE OF TERMS IN ~ONSERVATION OF MOMENTUM EDUATION

AT X=25,OOO FT.

27

same figure to be the measured discharge curve. The re1ative error

in the measured discharge is approximately 10%. The resulting error in

F is 25% and-the error in the water level appeared to be 1.2%.

To summarize for the example presented here, when ca1ibrating on

the water level a maximum error of 2% in the water level leads to an error

of 50% in F and a maximum error in

Q

of 20%. When ca1ibrating on a

discharge curve an error of 10% in the ca1ibration curve leads to an error

of 25% in F and a maximum error in the water level of 1.2%.

The re1ative errors in the measured water levels and discharges of

respectively 2% and 10% used in the previous examp1e are realistic va1ues

for conventiona1 types of equipment and measuring procedures. Also the

examp1e presented here is a typical case for flo,",in lagoons of the type

considered in this report. Therefore it may be safe1y concluded that, when

calibrating on water levels and assuming that the boundary conditions for

a section are given as water levels, small inaccuracies in the measured

calibration curve leads to large errors in F and

Q

.

When calibrating on

discharges, the errors resulting from inaccuracies in the calibration

5. MASS TRANSPORT IN LAGOONS DUE TO TIDAL ACTION

In this chapter an analytic solution is presented for the mass transport due to tidal waves in lagoons. The analytic approach is restricted to

lagoons and inlets of relatively simple geometry, which prohibits its application to the of ten irregular shaped tidal waters in nature. In that

case recourse has to be taken to numerical techniques. Therefore the

analytic model cannot replace the numeri cal procedure; it merely serves to

build faith in the numerical results and to obtain a better understanding

of the physics underlying the phenomenon of mass transport. In the final

part of the chapter the effect of inlets on the net transp~rt is discussed.

Consider a straight lagoon of finite length ~ in which the ocean

tide can freely enter at both ends. The depth and width of the lagoon are

uniform. The conservation of mass equation neglecting the source term is

2!l+lg_

o

t

ox

o

(5.1)

The conservation of momentum equation, when neglecting the non-linear terms

except the friction term, reduces to

Fq!q!

(h

+

n)2 (5.2)

The term on the righthand side of Equation (5.2) was derived (see

Appendix A),by relating the bottom shear stress and the mean velocity by

means of a Darcy-Heisbach type friction Law,

S = f

.L

u

!u I

4R 2g

(5.3)

in which S slope of the water level

29

u

=

mean velocity in the vertical

R

hydraulic radius Furthermore

p - gRS (5.4) in which L o bottom shear stress. From Equations (5.3) and (5.4), it then follows ~=-i_ulul p 8 (5.5)

The friction term in Equation

(

5.2)

will now be re-derived by replacing

Equation (5.5) by its linearized version. Assume that as a first approximation

the velocity u in the lagoon is simple harmonic in t

u = u cos (ot

+

a) (5.6)

in which a

=

a phase angle. Substituting this expression for u in Equation

(5.5)

and expanding in a Fourier series yields

l'

o _ i_ ~ uu

+

third order term

+

p 8 3n

The second, fourth etc. harmoni~are zero. Thus neglecting third order terms

and higher yields L o f 8 8 ~ (j; u)q h

+

n (5.7) p When approximating 1 1 n ~ h -

h7

h

+

n and introducing f

8

F = _- u R, 8 3n 8 ~ F - U 3n (5.8)

Equation (5.7) may be written as

p (5.9)

1:

o

-

=-With this expression for the bottom friction; the conservation of mornentum

equation becomes

19..

+

h dn

at

g dX

(5.10)

Equations

(5.1)

and

(5.10)

are used to describe the flow in the lagoon.

Equation

(5.1)

is a linear equation while Equation

(5.10)

may be regarded as

quasi-linear because the non-linear term is of a second order nature. The

oeean tides at both ends of the lagoon serve as the boundary eonditions. To

avoid lengthy computations, the amplitude and phase of the oeean tides are

taken sueh that when negleeting the non-linear term in Equation

(5.10)

the

resulting solution represents a damped progressive wave. The water level

and discharges for such a wave as a function of x and t are,

n a e-l1Xcos (at- kx) (5.11)

(5.12)

in whieh 11,k and a are defined by

a

=

tan-1 (l:!..)

k

with k

.31

Thus when

n

=

a cos ert

o

(5.13)

is the boundary condition for x

=

0, it then follows from Equation (5.11) that

the boundary condition at x

=

L should be

-ilL

nL

=

a e cos (ert- kL) (5.14)

A trial solution for Equations (5.1) and (5.10) when including the non-linear

term in the last equation and considering the boundary conditions (5.13) and (5.14)

is n*(x)

+

a -ilX eert- kx)

+

(5.15) n e cos

..

.

k q*(x)

+

a 0 + trx (at - kx

+

a)

+

(5.16) q

=

c e cos

...

0

_lil

2

+

k2

Substituting this trial solution in Equation (5.1) and averaging over the tidal

period yields

(5.17)

Substituting in Equation (5.10) and neglecting higher order terms yields

after time averaging

-2ilx

e (5.18)

Because of the imposed boundary conditions, _n*

=

o

for x =0 and x

=

L. With

these boundary values o_{f n*} the solution o_{f q* f}rom Equations(5.l7) and (5.18)

is

a2c

1 k k 1 _{(1 _ e-2ilL)}

0 0 _(5.19)

q* =-2h- 2].1 ].12

+

k2 L

This is the expression for the net discharge in a straight 1agoon when imposing

a2c

the boundary conditions (5.13) and (5.14). Note that the factor 2h 0 is the

~_t.,- I ₎ #4./ '

+

'.( '.!::,

6

_~

_i,

/

32

The fo11owing serves as an explanation of the physics under1ying the

expression for the net transport, Equation (5.19). Again consider a damped

progressive wave but now in an infinite long channe1. The mathematica1

expression for such a wave in terms of n and u, assuming the friction to be

1inear and neg1ecting second order terms in the field equations, is

/

I / n ae-l.lxcos (ot - kx) (5.20) u a c o ~ (ot - kx

+

Cl) (5.21)

See, for examp1e, Ippen (12), page 507.

The mass transport accompanying such a wave, q*w 'may be found from

Tt

T

!

q·w ~

V

df

.

u dz I

r

<:

-

(:,

, - (5.22) ::; I ti. I ./ F/( o -·h "T

-

"

~I---...

~t(f;u!l

+

₂1,a.e

f<_,

OM ol!

I ,..} ____../

Substituting the expression for u in this/equation and integrating yie1ds

w

k k

o

(5.23)

This expression shows the mass transport to decrease with increasing x.

Therefore, in order to satisfy the continuity condition, which states that the net flow in any cross-section must be the same, the mean water level

must vary with x such that the flow resulting from the slope of the mean level q* ' compensates for the decrease in net fLow due to the waves. Thus

s

Const

(5.24)

in which Const is positive or zero. The expression for q* in terms of the s

s

(

5

.

25

)

Substituting the expressions for q* w

and q* in Equation

(

5

.

2

4

)

yie1ds s 2 k k ~o 2h -1l";;2'---+-k"7're-211x 2 Const

(5.26)

Integration of Equation

(5.

2

6)

yie1ds

/

/

(5.27)

/

From Equation

(5.27)

it may be seen that in order to maintain a finite mean water level in àn infinite1y long channe1, then Const

=

O.

Setting arbitrari1y n*

=

0 at x

0,

it then fo110ws

(5.28)

Equation

.

(5.

2

8)

is the expression for the mean water level in an infinite long

channe1 of finite depth in the presence of a free damped progressive wave. See

a1so Figure

7

_F_t_1__1__a_2_c__0k-;;.o_k_~ h gh 211 2h 112

+

k2 o "- mean level in the presence of a free damped

progressive wave

777777 77777777777777 77 /77 7 7 77

Fig. 7 Hean Water Level in Infinite Long Channe1

When regarding a 1agoon of finite 1ength L, the mean water level at x

=

L is suppressed by an amount of n*(L) c~mpared to the water level at x

=

L in case of the free wave in the infinite long channe1 because of the

34

imposed boundary conditions. The latter are assumed to fluctuate about the same mean level. By suppressing the mean level, a slope of S is

m introduced which, to a first approximation,is equal to

S m

(5.29)

The discharge q resulting from this slope is, see Equation (5.25)

q _.8È_

n*

(

L)

r;/

h L

(5.30)

Substituting the expression for

n*

(

L

)

as given by Equation (5.28) in Equation (5.30) yields

a2c k k

o

1

0

1

-

2~L

q

=

Zb

2i1

~2

+

k2 L (1 _ e ) (5.31)

which is the same as the expression derived for the net transport q*, see Equation (5.19). The net transport therefore may be regarded as the result of forcing the mean level at the boundaries to be different from the mean level at these boundaries in the case of a free wave.

When substituting the trial solutions Equations (5.15) and (5.16)

in Equation (5.10), all higher order terms were neglected. To evaluate the

effect of these terms on the net transport the q* as given by Equation (5.19) is compared with the true q*, that is wheri taking into account the third and

higher order terms. The true q* is found by integrating Equations (5.1) and (5.10) with the boundary conditions (5.11) and (5.12) numerically following the

procedure described in Chapter 111 and then averaging q over a tidal period. The computations were carried out for a lagoon with a length L = 70,000 ft. and a

depth h

=

7 ft., a tidal period T

=

45,000 sec., a time stepnt

=

100 sec., and

a space step ns

=

5,000 ft. Two values for the amplitude a and three values for

numerical (true) solution are presented in Fig. 8. From this figure, it

may be seen that neglecting the third and higher order terms introduces

only a very slight error.

The question might arise whether the "linearization" of the friction

term as outlined on pages 29and 30 is a valid procedure·when discussing

effects caused by second order terms in the equations. Ta shed some light on

this, the net discharges when using Equations (5.1) and (5.10) and the net

discharges when using Equations (5.1) and (5.2), which include the quadratic

friction, are compared. The same lagoon is considered as in the previous

paragraph. The boundary conditions again are presented by Equations (5.11)

and (5.12). The "linear" friction factor Ft and the "non-linear" fr:Lction

factor F are re1ated by Equation (5.8). The va1ue u in this equation is taken

equa1 to the average of the u's at bath ends and the u in the midd1e of the 1agoon.

The equations are integrated numerical1y using a time step ~t

=

100 sec. and a

space step~s

=

5000 ft. To find q*, q is averaged over a tida1 period. Two

different va lues for the·amplitude a and three va1ues for the friction factor F.

are considered. The resu1ts, which are p10tted in Fig. 9, compare wel1 especia11y

considering the way in which u was determined. The agreement is better

for 1arger than for smaller values of F which probab1y stems from the fact

that for 1arger va1ues of the net f Low and thus smaller va1ues of F, the

approximation of u as given by Equation (5.6) becomes 1ess justified.

It is noted that methods do exist to arrive at a better estimate

of the representative average va1ue u. Howeve r, incorporating these methods

wou1d lead to a considerable increase in computationa1 effort which is not

warranted by the purpose of the computations. The reader is referred to

36 0.15 0 Cl> Cl) N...

-Z :4:: 0-w 0.10 (!) 0:: <t :c u Cf) 0 t-W z 0.05

_"

_"

LlNEAR FRICTION 0.001 0.=0.5 ft

o

ANALYTIC (EOUATION (5.19)) NUMERICAL (EOUATIONS ( 5·1)8l5·10))

o

FACTOR Ft 0.002 0.003 1.5

~_LlNEAR

"

_{FRICTION FACTOR}

o Cl> Cl) N';:::

....

a=

1.5 ft.

o

ANALYTIC ( EQUATIO N (5.19)) NUMERICAL (EQUATIONS (5·1) 8 (5.10)) 1.0 ur(!) 0:: <t :c u Cf)

zs

t-W Z 0.5 0.001 0.002 0.003

0.3. 0 G> o, 0.5 ft. en

"'

... +-...

₀

_{EQUATIONS (5·1)}

_a

_{(5·2 )} z EQUATIONS (5·1)

a

(5·10)

*

0.2 C'"

0

W C> cr < ₀ :r:

/

'

Ul(.) ëi

₀

) 0.1 I-w Z

₀

G

"

NON LlNEAR

"

FRICTION FACTOR F 0.0 0.001 0.004 0.008

ct

1.5 ft. 0 G>

0

en _{EQUATION (5·1)}

a

_(5·2)

"'"::::

_....

e

EQUATIONS(5·1)

a

(5.10) 2 ~

*

C'"

₀

W C> cr Cl) <

..

.

:r: (.) Ul

0

0 I-W Z •• 11 _"

NON L1NEAR FRICTION FACTOR F 0

0.001 0.002 0.003

38

In deriving the expression for q*, the boundary conditions.were chosen such that to a first order of approximation, the water motion in the lagoon was a single damped progressive wave, However, expressions for q* can be derived in a similar way for arbitrary boundary conditions.· For example, consider a lagoon connected to the ocean by inlets at both ends.

/

/

/

The implicit boundary conditions, formed by the inlet Equations (2.14) and

(2.15), and the field Equation (5.2) first are made quasi-linear. This is

done by linearizing the factors Q2 and qlql in respectively the inlet and

dynamic equation. A first order solution is obtained by omitting the second

order terms (the non-linear terms) from both the field equations and the

boundary condition. The trial solution given by Equations (5.15) and

(5.16) is then substituted in the complete equations and boundary conditions

when inc1uding the non-linear terms. The resu1ting expressions are time averaged

which yie1ds two 1inear differentia1 equations in n* and q*, corresponding

to Equations (5.17) and (5.18) an

_,

d two linear boundary conditions. It is

thus possib1e to solve for n* and q*.

Also in deriving Equation (5.19) the non-1inear terms associated

with the convective acce1eration and surface gradients have been neg1ected.

These terms can easi1y be taken into account. In doing so, the on1y change in

:!

.-211X

the differentia1 Equation (5.18) wi1l be in the expression preceeding e

In the initia1 derivation these non-1inear terms were omitted to keep the algebra

as simp1e as possib1e. They shouLd , howeve r, be taken into account when

requiring quantitative accurate resu1ts. This may be seen from Fig. 10 in

which the net transport with and without the convective acce1eration and the

non-linear part of the surface gradient is presented for a 1agoon at both ends

free1y connected to the ocean. The resu1ts p10tted in Fig. 10 were obtained

0- + LL..c 0 N Cl> I I-Cl) 0- t=:- 11 Z

-

₊

~~ w

0-

0 a

_u.

..c ct

~11O

s

..._.. ... 0: NNI!) t=:- e> v _I

+

..c 11 II ..._... w b I- 0> (.) b ct·

-

_clx

+

IJ.. ,._ c: c: 0: N Cl)

en

0 roVO _No-lX ::::> ~ _{r- IO} 2 Cl) 10 _{a:: ro~}

*

Ó Ó w ..c Ol 0- 0> -I..c w 11

+

J: 11

+

I-I

I-::r: _l<1

0-1

- 0-N 0

Ul

-._ _{w ro(.() rot()} t=:- ~ IJ.. a 0 l- a 0 ₀ v / _a <0 a

<8

0 a l-a ei 0: r- ct 11 0-_j 0: ct W Z

*

_~ J_Z a z 0 Z ro <t a

<:J

0

aei _z IJ.. a i= <t

=

0: 0: W a J I- W (.) (.) ~ ct (.) _CT IJ.. <1 W Z e> a w 0: i= > ct (.) _l- J: ër (.) o IJ.. W Cl) > 0 0: N Z_a

o

0

<tw aa (.) I-?;

o

W ...J Z IJ.. Z a ~ w::r:

=

I- I-(.) W IJ.. IJ.. Z W a Q :>3S/1.:l NI *b 38 tlVH:>Sla 13N w_a:: ~ a ::::> a C> a IJ.. ei ex> _!:Q v N Q ex>_a U)_a v0 N_a a_a ei

o

o Ó o o

o

Ó ei o

/

/

40

b. lnfluenee of lnlets on the Mass Transport

Consider a lagoon of uniform depth and widthconneeted

to the ocean by inlets at each end. See Fig. 11.

~

L

1

I

Fig. 11 Lagoon Conneeted to the Oeean by lnlets

lt is assumed that the depth of the lagoon is so large that the propagation

of the tide in the lagoon itself ean be negleeted. The continuity equation for

the inlets then may be written as

an.

1

at

(5.32)

in which

~

=

lagoon area

Q

diseharge into lagoon

n

i lagoon level

Eqüation (2.16) is used to describe the dynamies of the flow in the iniet. Prom'

this equation, it f oLl.ows

Q = ± A /2=...gOL:R::.:_.__-=-_

V

"2FL

+

mR or in Keulegan's ( 2) notation Q (5.33) in which K T 2nH A

J2

g

RH

'" 2FL

+

mR

K coeffieient of repletion

2H

=

tidal range in the ocean

The solution to this problem, for a simple harmonie oeean tide is given by Keulegan (2). For an oeean tide of the form

H . 21ft

no= S1.n

T

'

in whieh H

=

tidal amplitude in the oeean , the eorresponding bay level

n

.

is

1.

/

f

/

A1 sin (

T

21ft + q)+ A3 sin 3 (21fTt

+

E:3

)

+ ... (5.34)

The amplitudes Al and A3 ... and the phase angles El' E:3 ••• are a funetion of

the eoeffieient of repletion K

=

Kl

+

K2; Kl and K2 are the eoeffieients of

repletion respeetively,for the inlets land 11. It may be seen from Equation

(5.34) that both amplitude and phase of the oeean tide are modified while in

addition, higher harmonies are generated.

The foregoing solution pertains to relative deep lagoons. No analytie solution is known in ease of a shallow lagoon. However, it is likely that eaeh inlet again will modify the oeean tide but not to the same extent, as was the ease for the deep lagoon, beeause the propagation of the tide in the

lagoon ean no longer be negleeted. In general, therefore, the amplitude and

phase of the lagoon tide at one inlet will be different from the amplitude and phase of the lagoon tide at the other inlet, even though the oeean tide is

the same for both inlets. A differenee in tide at both ends of the lagoon is

a requirement for net transport beeause in ease of equal tides, the net

transport is zero as a result of the symrnetryof the problem.

It may be inferred from the foregoing general eonsiderations that the inlets play an important role in the flushing of the lagoon. By properly

ehoosing the dimensions of the inlets, the net transport of water in the lagoon

/

/ 42 ~

-

I(') 0 C\J₀ 11 0 1=1 ::r: ei I-0 l- c, u. a lIJ w (Ij _-' 0 z +-- w

-

-' LL. a co 0 _I(')a <l: 0: C\J _<l: (f) Z > ... 0 0 " ... 0 _ü) ::r: 11 ,..._ I- I-z <.:) ::r: w w _Z I- ....J ~ 0 w z 0 -' ~ ,_. u;

....

0 .-0 :r: 0 C\J t-0 0 <D 11 10 ...3: H I C\J l- a (/) a, 0 :::> I- w ei (/) lIJ 0 ex::: ;_) 11 W Z :::-+- LL. 0

-

z 0 lIJ u. 0 10 _{0: 0} 0 +- 0 0 ... c:!} ~ I(') ₀ c:;: <.:) _C\J 0 -' u. (f) Z 11 co Z 0 11 t--t lIJ ü) ::r: w z_w I-<.:)- :r:I- w_--' (f) ~ z 0 _u 0 0 w ~

,_.

0 -' o:t z₀ I-ex::: 0 c:;: <.:) 0-<l: -'

....

_... 0,_. et:: :::> r- --' a u, LL. 11 0 ::r: 10 '-'l:: a 1-- C\J ro a, 0 lJ___ w a 0 0 _ei z 11 f-0 z 0 ~ u. w o .- ~ <l: ... w -' 0 +- _<u 0 0

....

_--' a LL. a 0-C\J _{0 Ó} a (/) co a ... 10 0 (f) C\J --_ Z 11 t-a _" w iï) ::r: ;z: z I- _:r: <.:) lIJ _Z

_ti

::!: N 0 lIJ ~ a -' .-52 _c:!} ... '-'-0 ~ 0--* ~ o:t_-. C\J

-c.O -~ ~ 0

"*

-

I(')

-

<D-: N C\J

-

-z z 0 0

-

~ I-<l: <l: ~ ::> 0 0 lIJ lIJ 0

_*'

I- l- r: lIJ lIJ U. -' -' z ~ z

-

-t::I 0

*

I-lIJ O.--J

_'-

_"*

~

_-

-u. 0 ::r: I-0

*0

:s:

1

*-0 *-0 • _1:1 NI 3-':>).:) -'VOLt ~3d lN3Y>l3::>V-'dSIO 13N

-I I I I o o o o o (Ij o o <D o o o:t o -0 C\J o o

/

I /

in Figure 11. The ocean tide is assumed simple harmonie and the same at

both inlets; the amplitude is 1.30 ft. and the tidal period is 45,000 sec. For the dimensions of the lagoon and the inlets, see Figure 12. In the same

figure, the results of the computations are presented; the net horizontal displacement per tidal cycle of a particle is plotted versus the width of Inlet 11. A definite maximum in the net displacement occurs when the width

of Inlet 11 is in the order of 300 - 400 ft. The computations were carried

out for both the complete inlet equation, Equations (2.14) and (2.15)

neglecting the velocity head in the bay and for the simplified version of

the inlet equation, Equation (2.16). The results for the two inlet equations

differ only slightly. Equation (2.5) in which the convective term and the

free surface stresses were neglected, was used to describe the dynamics of

the flow in the lagoon. The equations were solved by applying the numerical

method discussed in Chapter 111. The net discharge was found by integrating

the computed discharge over a tidal period. The time step in the computations

44

6. APPLICATION

In this section an attempt is made to apply the results presented

earlier to calculate the net transport in two lagoons (Lake Horth and

Sarasota Bay). Only the effects of tidal motion and fresh water inflow

are included in this treatment; wind stresses are not accounted for and

/

(

/

mixing is treated in Part 11 of this report.

a. Lake Horth

The Equations (2.4) and (2.5) are employed to describe the time

-varying f Low in the main water body of the lagoon system. The flow in the inlets is described by the simplified vers ion of the inlet equations given

on page 17. The equations are solved using the numerical scheme described in

Chapter 3. The schematization of Lake Worth, based on hydrographic maps, see

Figure 5 Part 11, and field data, see reference (24), is presented in Figure 13.

The lagoon is divided into three different sections for which the friction

factor F is assumed constant. Each of the sections has a conveying channel and a

storage area. The space step in the computation is 6s

=

5000 ft., the time step

is 6t

=

200 sec.

A review of the field measurements carried out in Lake Worth is

presented in reference (24). Unfortunately, the tide curves and the discharges, the latter derived from one point velocity measurements, are not sufficiently

accurate to determine a reliable value for the friction factor F. To still

obtain an idea on the order of magnitude of the mass transport computations

were carried out for F

=

0.002 and F

=

0.003, covering the range of values

of F commonly found suitable for,this type of tidal waters, In the

computations, the F value is assumed to be the same for the sections of the lagoon and the inlets. The ocean tide as measured at the ocean pier off the

000' !.30U·. 5~ 2~

A

---s

/

I / 2.000' ::z

=

._

c..;> LLJ en z 0 j:; z ~ 0 l-(/) ~ (/) .J W > W W .J C> 0:: ~ 0:: _::x: w _u l- (/) ~ 0 3:

+

:x:: I-0.:: c::>

==

._

LU _, ::z

=

._

=

=

==

=

._

=

=

en