Tidal energy loss in coastal embayments

Estuarine, Coastal and Shelf Science (xO8x) x2, 279-290

Tidal Energy Loss in Coastal Embayments

R. A. Heath New Zealand Oceanographic Institute, Department of Scientific and Industrial Research, IVellington, Nero Zealand Received

20

21farch x98o

Keywords: bays; tidal friction; energy balance; friction coefficients; phase; amplitude; Bristol Channel The tidal energy balance for an emba~ra'nent over the spring neap cycle is used to obtain a relationship between the phase of the Mt and S~ tidal flow at the entrance using solely tidal elevation harmonic constants as inputs. For possible values of the phase of the flow, the calculated frictional drag coefficient (k), or comparison with observation of the variation of the phase of the flow through the spring neap cycle, may be used to allow the frictional energy dissipation to be estimated. Application to the Bristol Channel for values of k which are consistent with the generally accepted range of values used in two-dimensional numerical models give estimates of the phase of the flow within the range of those observed and of the tidal energy dissipation consistent with previous estimates. The method should allow estimates of the energy dissipation per unit mass to be made in embayments where tidal elevation harmonic constants alone are available and thus provide an insight into the potential mixing.

Introduction The tides play a major role in the dynamics of coastal embayments providing as they do energy for mixing waters of differing densities, for moving sediments and in many cases being the main agent by which embayment waters are exchanged with those further offshore. Tidal energy loss can be taken as a measure of the energy available for mixing of embayment waters and therefore in association with the freshwater inflow has a strong influence on the stratification. The tidal energy loss in an embayment can be calculated directly if the amplitudes and phases of the tidal elevation and flow fields in the entrance are known. However, only in a very few embayments are there adequate flow data to allow such a calculation to be made. Tidal data from most embayments consist at best of harmonic tidal constants of the surface elevation at several positions. It is the purpose of the paper to evaluate how the tidal energy loss might be estimated from tidal elevation harmonic constants with few additional data, with the overall aim in the future of allowing evaluation of a potential mixing parameter involving the tidal mixing and freshwater inflows estimated from data which are generally readily available. This latter parameter is related to the Estuarine Richardson Number (see e.g. Fischer, ,976 ) and analogous to the parameter used by Simpson & Hunter (1974) in the continental shelf studies where the buoyancy input is from solar radiation. They found that the reciprocal of the tidal energy loss per unit mass, S, expressed as the ratio of the water depth to the product of the current speed cubed and the bottom drag coefficient, can be used to indicate z79

o27z-77,4/Sx/o3o279+ x2 $o2.0o/0

9 x9$t Academic Press Inc. (London) Ltd.

280

R. A . tleath

where frontal regions may form (Simpson & Hunter, 1974) between stratified (log S > x . 5 c.g.s.) and unstratified (log S < 1.5) waters. Frictional dissipation of tidal energy within a coastal embayment leads to an inercasing delay in the time of high tide landwards from the entrance. The power needed to compensate for the energy loss is provided by a delay in the tidal flow relative to that in a frietionless system, the delay in the flow leading to the observed increase in phase of the elevation upstream. Under the assumption that the frictional bottom stress is linearly dependent on the water speed, it is possible to estimate the size of the frictional coefficient, and hence the energy loss, by matching the observed change in phase and amplitude of a specific tidal constituent with that given by a simple superposition of damped progressive waves (see e.g. Ippen, 1966 ). Use of a linear frictional law is not very satisfactory. For example it leads to an equal phase change of all the tidal constituents (assuming the same frictional coefficient) at any location within an embayment relative to that outside whereas there is usually a noticeably larger phase shift in the smaller amplitude constituents than in the major constituents within a tidal band. This effect is the result of the non-linear velocity dependence of the bottom stress, the dissipation on the smaller constituents being greater than on the larger. Alternatively if an embayment is near resonance in a tidal band the observed tidal constituents can be fitted to a damped harmonic oscillator model to allow estimation of the ertergy loss, as given by the Q, and resonant period in the manner used by Garrett (1972) for the Bay of Fundy. In its most simple form dissipation is taken as linear in the oscillator with differences in response between constituents in the same tidal band being assumed to be mainly due to a frequency dependent resonance. Allowance can be made for different values of frictional dissipation by having different values of the dissipative contribution to the Q of the system. In using models which give the elevations and phases there is some difficulty in incorporating the effect of the direct astronomical forcing. The direct astronomical power input is generally small compared to the power input through the entrance in the co-oscillating tide. However models of the tidal response of embayments often use the change in the amplitude ratio and phase difference of the tidal constituents as input data and the effect of the direct astronomical power input on these input data may not always be small. For example when the observed co-oscillating tide and the equilibrium tide are in phase or completely out of phase there may be an observable shift in phase of the tidal constituents inside the embayment due to the astronomical input. If the direct astronomical forcing is to be taken into account in calculations of the energy dissipation it would appear to be necessary to work directly with the energy equation. If there are many sets of harmonic constants evenly spatially distributed inside the embayment, the phase of the flow, and hence the energy inflow, could be evaluated using continuity considerations. However the spatial coverage of harmonic constants in most embayments is generally not sufficient to allow accurate estimates of the flow phase although they do allow reasonable estimates of the flow amplitude. The magnitude of the frictional power loss within an embayment changes substantially through a spring neap tidal cycle. Considering the frictional power loss to be proportional to the flow cubed, in an embayment where the principal lunar semi-diurnal tide ~1"2) has an amplitude couble that of th~ principal solar semi-diurnal tide ($2) the power loss at spring tide is 27 times that at neap tide. T o compensate for this change in power loss there must be a change in the phase of the tidal flow relative to the tidal elevation at the entrance. In turn this change in phase in the energy balance over a spring neap cycle is reflected in the

Tidal energyloss in coastal embayments

28x

spatial changes in the phase between the $2 and tXI=tides, a difference which needs fewer data to define than that needed to evaluate the phase of the tidal flow from continuity considerations.

Energy balance The tidal energy balance within an embayment (see e.g. Itendershott, x972; Garrett, x975) may be written as

9

~

dA -~ fw

gh--U~'ndl-l-5o gh U'V~EdA

(I)

where the overbar represents a mean over a tidal cycle, E is the energy density E=~hOZ+ 89 ~ and U the tidal elevation and tidal flow amplitudes respectively, ~n the equilibrium tidal elevation, h the mean water depth, g the acceleration of gravity, F the frictional bottom stress, ~ the surface area, n the unit vector perpendicular to the width at the entrance, zo, positive into the embayment. Long-term measurements of the flow and the tidal elevation at entrances are needed to allow estimation of the net energy input in the co-oscillating tide f gh U~.n dl available to compensate foi" frictional dissipation. Measurements over just a few tidal cycles are not always sufficient for the energy denslt3' contains a contribution associated with the neapspring cycle modulation of the tides. This is illustrated by considering an embayment where the principal lunar semi-diurnal tide (M2) has an amplitude double that of the principal solar semi-diurnal tide ($2). The energy density in the seven days between neap and spring tides varies by a factor of 9 or an average of o.6 per cycle. With non-linear frictional dissipation there must also be a change in relative phase between the tidal flow and elevation at the entrance over a spring-neap tidal cycle. There is therefore a need for accurate estimation of the tidal phases which short-term records, in which variable non-tldal flows influence the tidal estimates, do not provide. Current and sea-surface elevation measurements over a period of at least 2 weeks are needed at several positions across an embayment with a wide entrance. These types of measurement are seldom available. In the following development of the energy equation the phase of the flow at the entrance is therefore taken as the unknown. Let the x axis lie along the main axis of the embayment perpendicular to the entrance with x = o at the head. Assuming the tidal elevation is controlled mainly by two tidal constituents, taken here to be the Mz and So constituents, with respective amplitudes and phases AI, B1, 91, tPz the tidal elevation on the line perpendicular to x=xx is given by ~t=Ax cos (o91t--gx)+Bx cos

(w.,t--92)

where cox, coz are the respective frequencies of the ~'Io and S z tidal constituents. Defining Aco=coz--wx the tidal eleva{ion is given by ct = C~ cos ( c o , t - e )

(2)

C I = V'./112+Bt"+zAzBI cos (Acot+9,--~a=)

(3)

where

{ At sin ~,+B, sin (~z--Aoot) } V = tan-X

A 1 cos ~0t+B 1 cos (~oz--Acot) "

(4)

282

R. A. tteath

The corresponding tidal flow at x = x , is given by U1 = Dj cos (oo,t--Oa)+G1 cos (oo.d--02) = H, cos (o:,t--y) where I11 = v ' D t -o + G l 2 +2G, D 1 cos (Aoot-{-O,--O.:)

(5)

{D, sinO,+G,sin(O.,.--Acot)} and

;' : tan-1

D 1 eo~ 01+GI cos (0,--Acot) "

(6)

Energy input to the co-oscillating tide The energy input to the co-oscillating tide is then given by ~2h UI~1 = rc~h

fT

tlIC ~ cos (o~d--V) cos (ood--y) dt

o

= wgh HIC1 ~ cos ( ~ , - y ) 2

where over the M2 period T the amplitudes tIx, C 1 .and phases V, Y have been taken as constant. Only in a frictionless system or when the direct astronomical power input matches the frictional loss is there no net energy input through the entrance (i.e. V=Zr/2+y). The time t,, of the maximum elevation amplitude in the spring-neap tidal cycle at position x=x~ is given from equation (3) by Ao~q+~o1--~oo = o

(7)

when from equation (4) =

(8)

We now introduce a small phase angle, --A01, which is the argument in the amplitude of the flow [equation (5)] at the time of maximum tidal elevation amplitude -

A0t

=

Acot,+Ox--02

(9 a)

or alternatively the difference in the phase difference of the S o--~I 2 tidal elevations from the corresponding phase of the flows --A0a = (~oz--~x)--(02--0~).

(9b)

The phase of the flow is given at t = t t by

/ D_D_ssin O,+G~ s i n (0x+A0z) } Ys =

tan-X [ Dt cos 0 t + G t cos (0t+A0x)_"

0o)

The time of mid-tidal elevation amplitude at position x = x t is given from equation (3) by 7~

Acot=+gt--9 a = 4 - - .

(II)

2

Taking the case of art increasing tidal amplitude (--~/2) equation (4) gives the phase of the elevation under mid-tidal conditions, VM, as {//M = 9 t + tan-1 B'--~t At

(12)

Tidal energy loss in coastal embayments

283

and from equation (6) with O..--Acota--Oi--AOl=~r[2 the corresponding phase of the flow is Yt*t:" tan-X

{ DzsinOt+Gze~ D t cos Ot--Gx sin (01+A01) "

(z3)

For the present we will neglect the direct astronomical power input.

Energy balance mM-Hde Under mid-tidal conditions (of the spring-neap cycle) at time t=to. the energy balance landward from x=x I is given by

,rgh - -

cos (VM--Yst) =

eodA+

~

d.z/

(x4)

with to the power loss per unit area. The rate of change of the energy density dE/dt landwards from x=xt is given by 0E x = - {h DzGo Am sin (04--Oa--Acot)+g AaBo Am sin (94--ga--Ao~t)} ~gt 2 " -

-

05)

where tl z, B 2, D 2, G 2, go4, ~a.~,0~, 04 are the average values of the respective amplitudes and phases inside the embayment landwards from x = x t. Under mid-tidal conditions t=ta, the arguments in equation (i5) both have values near rr/2 and therefore the exact values are not critical to the estimates. At the head of the embayment spring tidal conditions for the elevation and flow coincide and A0a is zero. The argument 04--O3--Aogt2 is therefore replaced by ~ + A0t, i.e. 2

2

f 9 c~EM ~ dA=

z -2

r?Am{hD~Gacos A012 +g .ad o cos

06)

Energy balance spring tide Under spring tidal conditions the energy balance is given by

wgh ttsCs cos

--

2

f o

(Us~ 3 d t+ f

OEs d a o

07)

Ot

with t=tx=~z--~dAa~. The usual assumption of the bottom frictional stress depending on the square of the water speed has been made giving F " U=t o (Us/Uo)3 the frictional energy per unit area at spring tides when the amplitude of the tidal flow is Us, the amplitude of the flow under mid-tidal conditions being U0. The spatial structure of Us and U0 will be approximately the same and therefore

From equations (i4) and (I7) we have

/Uo \z

tlsC s

( dEs

C ~gE~t

R. A. Heath

284

Estimation of the tidal parameters Values of the M z and So tidal elevation constants are known at x=x~ and the tidal elevation constants within the embayment allow estimates of Ao-,B2, tp3, 9a. The tidal flow amplitudes of the tidal constituents at x = x i , D x, G 1 can be estimated from continuity requirements on the associated average elevation amplitude Ao-, Bo- landwards from xl, e.g.

D l wh

f

Tlt

cos ogxt dt = 29 Ao-.

d r14

The tidal speeds D2, Go- are difficult to determine. In a bay with a narrow entrance opening out inside the heads, typified by a bay where the tidal flow controls the cross-sectional area at the entrance, the maximum flow is concentrated in the entrance and therefore estimates of Do, Gz are not critical to the estimate of OE/Ot. In a bay with a more uniform crosssectional area Do-, Go- may more properly be estimated from continuity. The ratio of the mean spring to mid-tidal flow in the embayment can be taken as the ratio of the corresponding tidal flows at the entrance

(vs

= iits Vt

,]

9

There remain three unknowns in equations (x4) and 08). eo the energy dissipation under mid-tidal conditions, A0x, the difference between the Sa--M2 phase difference for the tidal elevations from that of the corresponding flows, and 01 the phase of the l~,{o-flow at x = x 1. Equation (18) thus allows computation of relations between A0t and 0t, with 0t then remaining to be determined.

dstronomical power input The power input from direct astronomical forcing can be related to the equilibrium tidal elevation rather than the gradient of the equilibrium tide by

hg

U~e'ndl+g

U-V~E dA = - - g h O~

w

~e ~ ~

dA.

09)

t~

However, then the astronomical input is calculated as the difference between two large numbers. The last term in equation (I9) involves spatial integration of the equilibrium and observed tide and cannot generally be calculated accurately. With generally available data the best estimate of the direct astronomical input is from U ' V ~ dA using means of U and V~E. Considering just the ~'I 2 and Sa equilibrium tidal constituents with respective amplitudes and phases l, m, t , e.the equilibrium tidal elevation is given by

~E = l cos (coat--fl--kx)+m cos (co.,t--e--kx) where for a zonally oriented bay, fix gives the spatial variation of the equilibrium tide, k=2n]zrR cos ~a, R the earth's radius, ~ the latitude. We then have

Tidal energy loss in coastal cmbayments

O~E

~x = K sin

where

K =

~r

Z = tan -1

285

(cod--Z)

(zo)

0-cos (Acot) { lk sin kx +mk sin (kx--Acot) } lk cos kx-Flllk cos (kx--Acot)

the time t being taken as zero at a sysygy (fl=e=o). The direct astronomical input is then given approximately by

ght FzK 2

t? sin (yt--Z)

where hx, Fx, Yl are mean values of the depth, and amplitude and phase of the flow inside the embayment. The power inputs given by equation (i9) under mid and spring tidal conditions are then added to the left-hand sides of the power input equations (i4) and (x7) respectively. Application to the Bristol Channel The tidal regime of the Bristol Channel on the south-west coast of England has recently been studied quite intensively, (see e.g. Owen, irt press) and provides a convenient location for comparing the energy loss estimates based mainly on the tidal elevation, as indicated above, with existing estimates based mainly on numerical model studies. Plots of the amplitude of the $2 and M o tidal elevation and the difference in phase between them along the Bristol Channel are shown in Figure z. Calculations have been made for the Bristol Channel, both with and without direct astronomical forcing for the channel

9.?.::.:::)i:):.:)i:)i:i:::.:.:.:.:.:.:.. i:..... -51~

Lundy I

!i

i

"

::':~i~i~:~i~:~::.i[.~::i~i:.~i.~J:~::g::.::!:.i!:i!:.:!:?i::::::::::.::.:.:i:i;i?ili:;::i?i~ili~i:i~!i?~::;~!.i!:i!~:;:i~;~:~i~i~;~::::::: .?i~:i?:~::i:~::~~:~:~:;::~:~;::;:?:?~:.;~;/ F i g u r e z. L o c a t i o n o f t h e s i t e s i n t h e B r i s t o l C h a n n e l f r o m w h i c h t i d a l h a r m o m c constants are used.

286

R. A . Iteath

inland from the natural entrance at Lundy (Figure 2), and inland from Barry near where there is a rapid change in phase difference bctwecn the S 2 and g I 2 tidal constituents. The vector averages of tile tidal elevation data used are listed in Table I. The tidal flow in the estimates has been taken as uniform in the channel and equal to that at the entrance calculated from continuity. Plots of the difference in phase between the S z and g I z tidal flow and elevation (A01 equations 9 a, 9 b) vs. the phase of the b l 2 tidal flow at the entrance to the

16~I 9

I

9

%

*

XX

X

X

9

o9

9

9

4

56 I

7 I

8 1

9

4O

2O

4

X X

X

9

9

9

I I

2 l

3

3 o=

2 F-

0 Lundy

I

I

~rry

I 50

I I00

Distance (kin)

Figure 2. Plots of the amplitude of the M t ( x ) and S, (O) tidal constituents and the phase difference bet~veen them from west to east along the Bristol Channel 9

T~mL]z x. Bristol Channel tidal elevation constants used in the tidal energy equation Case I. Entrance near Lundy (Figure x). Entrance: width 69"5 km, mean depth 5 x m Inland: mean depth 27 m, surface area 5500 "kins

l~I=

Entrance (Lundy) Vector mean inland from L u n d y N u m b e r of sets of constants

S.

Amplitude (m)

Phase (deg)

~Mnplitude (m)

Phase (deg)

2"58 3"66 7

x6z'6 x89"9 7

0"93 x'29 7

207"6 245"4 7

Case I1. Entrance near Barry (Figure z). Entrance: width 2x'7 km, depth x 4 m Irdand:.mean depth 5 m, surface area 56x krn t

M, 9 Amplitude (m) Entrance (Barry) Vector mean inland from Barry" N u m b e r of sets of constants Equilibrium tide

3"82 4.x5 4 0-096

S~ Phase (deg)

Amplitude (m)

Phase (deg)

x86-o 2ox.8 4 o

x'43 I"46 4 0"0449

238"0 258"0 4 o

z87

Tidal energy loss in coastal enzbayments

x

"-

/ U

6 ._

,zJ

~o o~

F ~

~ ~ o

/

50 ~ .~ ~ Q.

n,ega'l'ive/

2~

~'~

20 ~ ~o

,eL /I,"

.--

2 .~ .:-

-

~

/

h

o

I

80

",,ix I 90

I0 ~

posifive

-0 E v

I

I00

0,

-'~F /

z~O Q"

-50

t f, I

80

90

O,

I00

/ko /~-Inpuf fhrough

x

/

f

entrance

._U . ~

tn

E o~

-I

Inpuf fhrough sea surf0c3

I

I

I

80

90

I00

0,

Figure 3. Plot of the possible values of the phase of the M2 tidal flow at the entrance (0t) for values of the difference in the phase difference be~veen the $2 to M , flow and elevation phases at the entrance (A01). T h e entrance is taken in a line across the Bristol Channel near L u n d y Island. Also shown are the possible values of the drag coefficient (k) and the power dissipation per unit mass (P), the power input in the spring co-oscillating tide, the astronomical power inputs, all under spring tidal conditions and the phase difference of the tidal flow at mid-tidal conditions relative to that at spring tide.

system are shown in Figures 3 and 4. The energy plots are shown only for the case inincluding direct astronomical forcing, where astronomical power input (of order 3 x xo4 k~,V in the Bristol Channel under spring tides) is small compared to that ia the co-oscillating tide and the plots of the A01 vs. 0x are substantially the same with. or without astronomical forcing included. Negative values of A0x have been included in the plots for they satisfy the energy balance. In the Bristol Channel the difference in phase between the $2 and gI2 tidal elevations increases towards the head, which from equation (3) implies that the phase of the spring tide elevation also increases towards the head of the channel. For continuity to be satisfied the time of maximum flow at any location must be delayed behind that of the maximum elevation in the spring-neap cycle. Therefore A0x must be positive. The frictional power dissipation estimates clearly depend strongly on the phase of the flow. Harmonic analysis of current meter records at depths of 25 and 45 m in 6o m of water at latitude 5I~ longitude 4~ north of Lundy Island (Figure x, Howarth, personal communication) gives phases of the IV[2 flow on a line perpendleular to the entrance to the Bristol Channel of 89 ~ and 83 ~ respectively. These phases are in the range of those given by the energy balance (Figure 3) but do not really allow better definition of the frictional dissipation than the energy balance itself.

288

E~

R. A. Heath

~o

.~,

x

j I~

o~ ~,~

..

o_ ~

~o

~ = 2-

~,

pha~e'~-o 2o0 ~

o

=

X

negative

~

~ ~

~

~,_.-- '~

o ~ .I-.

o

n~

~hose

/\ A0 positive

I

I00

-a o ._ I0 ~ "o

~

I10

120

8, -I

~ "o ~ Z

= =.

J

positive AO ~ n e g a t i v e

J Input through

I

I-se~ surface .c_ _1/ I I

I00

I10

120

100

0,

J

I

I10

I I

120

O,

F i g u r e 4. As for F i g u r e 3 b u t for the entrance near Barry.

The frictional drag coefficient k has been calculated as the frictional power dissipation divided by the product of the tidal velocity cubed ,and the sea-surface area upstream from Lundy Island or Barry (Figures 3, 4); the estimate is obviously not that accurate for the flow field inside the channel is not well defined. Values of k for which taro-dimensional numerical tidal models closely fit to observations range from o.ooz5 to o.oo3, the 'standard' value being o.oo26, these values being appropriate for comparison with the present calculations for both the two-dimensional numerical model and the energy calculation involve depth mean flows. For k=o.oo26 the frictional energy dissipation at spring tide, which is equal to the power input in the co-oscillating spring tide, in the Bristol Channel upstream from Lundy is estimated at z.65 x xo 7 kW with 0=83 ~ A0=z3 ~ (Figure 3). Previous estimates of the frictional energy dissipation have been made for the M2 tidal constituent alone. Scaling the present spring tide estimate by the cube of the M z to M z plus Sz tidal elevation amplitude at the entrance to the Bristol Charmel gives an estimate for the M2 frictional energy dissipation ,alone of o.4x x.65 x xo7=o.66• xo 7 kW. This is of the same order as the estimate of x.x • io 7 kW of Owen (in press), based on a numerical model, x.z X xo 7 kW of Bennett (x975), based on an analytical model, and o.7x zo 7 kW of Robinson (z979) based on the current observations near Lundy Island referred to above. Comparison of the power input for the Bristol Channel inland from Lundy (Figure 3) and Barry (Figure 4) clearly indicates that most of the total frictional energy loss occurs bet~veen Lundy and Barry; this agrees with the recent numerical model results of Owen (in press) who shows that the main energy loss occurs over the shallows immediately seawards from Barry.

Tidal energy loss in coastal embayments

289

The direct astronomical power input into the Bristol Channel is small (of order 3 • Io4 kW) compared to that in the co-oscillating tide. Estimates of the components of the astronomical power input due to the flux of 'equilibrium' energy through the entrance [first term on the right-hand side of equation (I9) ] and to the direct vertical working [last term equation (x9)] have been made (Figures 3, 4) and illustrate the difficulty in estimating the astronomical power input from these components (see e.g. Garrett, I975). The energy input to the spring-neap modulation cycle at mid tide is estimated as 0"7 • xo6 kW or about 4% of the energy input in the spring co-oscillating tide.

Discussion In using just the elevation tidal constants in the energy balance there remains one unknown. This may be determined from various types of observations. In view of the spatial variability of the phase of the flow in the entrance to an embayment and the sensitivity of the energy estimate to this phase, extensive current measurements would be needed in a wide-entranced embayment. A simpler measurement to obtain might be the difference in phase of the tidal flow at one location in the entrance between mid and spring tide for there is a strong dependence of this phase difference on the phase of the ~,{2 tide (Figures 3, 4). If such measurements are not possible comparison of the calculated frictional drag coefficient with the range of values from o.oo25 to o.oo3, with a realistic approximate flow field, should provide good estimates. In terms of the energy available for mixing the power loss per unit mass is the significant quantity. The dissipative power loss per unit mass under spring tidal conditions (P) has been calculated as the net energy input in the co-oscillating tide and direct astronomical forcing minus the energy going into the spring-neap cycle, divided by the mass of water within the embayment, i.e. P = P(~,n+j'/'~ V ' V ~ a a - y s ) l V p where p is the water density. Within the Bristol Channel inland from Lundy Island P has a value of about i • IO- 7 kx,V kg -x, whereas inland from Barry in the inland end of the channel P is about 8 • to - ~ka,V kg -t. Freshwater inflow acts generally as a buoyancy input to an estuary with the density of the freshwater subsequently being changed in the estuary, the energy for mixing being provided by the tides and wind. Of the total tidal energy loss only a fraction goes into mixing [o.26% is quoted by Garrett et al. (1978) for the Bay of Fundy]. Several authors have defined parameters linking the buoyancy input to the energy dissipation and given values of these parameters for which estuaries will be in transition from stratified to well mixed (see e.g. Fischer, x976 ). In these parameters the tidal energy dissipation is generally taken as proportional to the root mean square speed (Us) cubed. For example the estuarine Richardson number R e is defined as Re=ApgO/pbUs 3, where p is the ocean water density outside the estuary. Ap the difference in density of the freshwater inflow from p, O the rate of freshwater inflow and b the width of the estuary. Rc represents the ratio of the input of buoyancy per unit width to the effect of the tidal current (see e.g. Fischer, x976 ). Transition from strongly stratified to well-mixed estuaries corresponds to values of o.o8
336

ft. R. West ~ A . P. Cottott

7. The vertical diffusion coefficient was larger for the flood tides than for the ebb tides, the dimensionless parameter if3 ( = e3/DU,) being of the order of o.oz 5 and o.oz respectively. These values are smaller than for homogeneous flow but are compatible with the small vertical salinity gradients (,-,o.z kg m -4) subsequently observed in the study reach for similar flow conditions. References Bowden, K. F. & Gilligan, R. M. x97x Characteristic features of estuarine circulation as represented in the Mersey estuary. Limnology and Oceanography xt, 49o-5oz. Bowden, K. F. & Lewis, R. E. x973 Dispersion in flow from a continuous source at sea. Water Research 7~ x7os-17zz. 9Carter, H. x974 T h e measurement of Rhodamine tracers in natural systems by fluorescence.ffournal du Conseil, Coltreil international pour l'Exploration de la ~ler z67, x93-zoo. Cotton, A. P. I978 On mixing coefficients in an urban stream and a tidal river. Ph.D. Thesis, University of Birmingham, England. Csanady, G. T. 1973 Turbulent Diffusion b~ the Environment. D. Reldel Publishing Co., Derdrecht, Holland. Fischer, H. B. I976 Mixing and dispersion in estuaries. Annual Review of Fluid J$lechanics 8~ xo7-s33. Frenkiel, F. N. x953 Turbulent diffusion: mean concentration distribution in a flow field of homogeneous turbulence. I n Advmzes in Applied 2~Iechanies Vol. 3 yon lklises, R. & yon Karman, T , eds). Academic Press, New York, pp. 6x-so7. Kent, R. E. & Pritchard, D. W. x959 A test of mixing length theories in a coastal plain estuary.ffournal of ~]larine Research x8, 62--72. Knight, D. W. (x98I) Some field measurements concerned with the behaviour of resistance coefficients in a tidal channel. Estuarine, Coastal and Shelf Science x2~ 3 o 3 - 3 z z . Munk, W. H. & Anderson, E. R. x948 Notes on a theory of the thermocline.ffournal of 2~IarineResearch 7J z76-z95. Odd, N. V. M. & Rodger, J. G. x978 Vertical mixing in stratified tidal floxvs, ffournal of Hydraulics Division, A.8.C.E. xo4 (HY3), March x978, 337-35x. Sumer, S. 1~,I.& Fischer, H. B. x977 Transverse mixing in partially stratified flow. ffournal of Itydraulic Division, A.S.C.E. xo3, June x977, 587--600Talbot, J. W. & Talbot, G. A. x974 Diffusion in shallow seas and in English coastal and estuarine waters. Rapports et Procgs-Verbaux des Reunions, Conseil International pour l'Exploration de la 2$Ier I67~ 93-x to. West, J. R. & Cotton, A. P. (x98o) Transverse diffusion for unidirected flow in wide open channels. Proceedings of the Institution of Civil Et~ineers, Part 2 69~ 49x-498.