ESTUARINE CURRENTS AND TIDAL STREAMS

Fresh water spreading out from the mouth of a river as it enters a salt sea may preserve its identity for a considerable distance on the surface if wind-generated waves, longshore currents and tidal streams are capable of producing only weak mixing. Fig. 1 shows the threedimensional shape of a fresh-water tongue overlying more dense salt water, derived by Takano (1954) on the assumption of constant eddy viscosity and constant density in the fresh water layer, below which the density increases according to an assumed law, making an asymptotic approach to the density of salt water. Takano's model is thus a water jet entraining salt from around and below it.


INTRODUCTION
Fresh water spreading out from the mouth of a river as it enters a salt sea may preserve its identity for a considerable distance on the surface if wind-generated waves, longshore currents and tidal streams are capable of producing only weak mixing.
Fig. 1 shows the threedimensional shape of a fresh-water tongue overlying more dense salt water, derived by Takano (1954) on the assumption of constant eddy viscosity and constant density in the fresh water layer, below which the density increases according to an assumed law, making an asymptotic approach to the density of salt water.
Takano's model is thus a water jet entraining salt from around and below it.
Salt or brackish water may penetrate along the deep channels of an estuary in the shape of a wedge complementary to the fresh water tongue, the salt wedge retreating seawards as heavy rainfall increases the river discharge, and advancing in dry weather intervals.Tidal streams cause a regular oscillation of both fresh and braok water in flood and ebb directions but the seasonal movements of the sloping boundary between fresh and salt water may still be important in low-lying delta regions.Strong tidal streams lead to intense mixing, when neither a fresh water tongue nor a salt wedge can be distinguished, but the isohalines (salinity contours) preserve the general wedge pattern -see Figs. 3 to 6.
In the upper reaches of an estuary it is possible to study the effect of the tidal motion by treating it as a simple harmonic perturbation of the uni-directional river flow.
Even in the middle portion of the estuary where there is reversal of the horizontal motion, one may seek a "quasi steady" solution for the net effect (seaward movement of fresh water) while allowing for the increased turbulence due to the tidal action.
At the seaward end of the estuary there is little deviation from the astronomical tidal rhythm, so the problem reduces to simple harmonic oscillations of salt water.Higher harmonics may be introduced as an extension of the simple solution.
For a first approximation it is sufficient to consider flow in the longitudinal vertical plane, to assume that the pressure distribution is hydrostatic as in long wave theory, and even to neglect inertia terms when investigating net effects.
RgJ. Tongue of fresh water in shape of hyperbolic paraboloid (after TakanoJ.

Origin ofx
EJSLL-Longitudinal section of estuary, to exaggerated vertical scale, showing the circulation pattern.

DENSITY DISTRIBUTION
For engineering purposes we may regard the sea as an infinite reservoir of salt water, its edges being diluted by fresh river water and rain while the balance is maintained by evaporation from its surface to the atmosphere.
We postulate a slow current of seawater landwards along the bed, its density being decreased by vertical diffusion and mixing, and a surface flow seawards of fresh water being gradually rendered brackish by salt rising from below.
This circulation is illustrated in Fig. 2 by full-drawn streamlines, the broken lines being profiles of longitudinal velocity.
The ehain dotted line indicates the surface of zero net motion in the longitudinal direction, and is obviously a place where high shear stresses may be expected, even exceeding the bed shear stress.This is a valid picture even when tidal motion is superimposed, although then the surface of zero net motion must be defined by averaging over a tidal period; it has a real existence only near the instants of "slack water".

O'Brien (1952) suggested that landward velocity near bed is approximately C.7H7ST, where C = Chezy coefficient and D =•&• §£
In many estuaries it is observed that the average salinity over a cross-section, and hence the average density if temperature differences can be neglected, increases from river to sea in nearly linear fashion in the middle reaches (Fig. 7), the rate of increase being smaller near the river (x = 0) and the sea (x * I).This linear increase is related to the_fact that the maximum velocity of the tidal stream U scarcely changes along the estuary* so the intensity of mixing a small fraction of the length L" For example, if ff = 1.4 m/sec.or approximately 3 knots, and the angular velocity of the lunar semi-diurnal tidal stream is w = 0.00014 rad/sec, then X = 10 km, whereas 1 s=w 80 km for a typical estuary.This density distribution fits the Thames data quoted by Inglis and Allen (1957), but other analytical curves such as the Gaussian integral and the hyperbolic tangent would also be suitable.
There is also an increase of density from surface to bed, which is nearly linear in a well-mixed estuary, as indicated by the isohalines in Pigs.

I
With specified velocity and density distributions, Taylor showed that stable internal waves are possible when Ri > 0.25, but no waves can exist if Hi <. 0.25.Other workers have derived different stability criteria.
Let us assume that the vertical distributions of ^Momentum and H Salt are similar » heace tae ratio f does not vary with z.
The advection term UQ£-in (2) obviously changes sign in the vertical.
To get an idea of its effect, assume zero net velocity at mid-depth and surface velocity TJ H equal and opposite to velocity near the bed, thus U « U H J(2|>-1). Substitution in equation ( 2

V-itt***-
Hence the vertical density difference is greatest in the middle reaches.
It is zero at the river but not quite zero at the seaward end of the estuary, so that (4j is not reliable when x-*»L.
If P, « mean density of water above surface of zero li net motibd and P ?« mean density of water below surface of zero net motion, at section x, then the density difference between the two layers is p 2 -p, ^55$ .&>.
Typical values of the longitudinal and vertical densities are * ^"^toOO "^ o* w "to00 ln a well ~mixed estuary.

MEAN VELOCITIES OF CIRCULATION
The equation of continuity for quasi-steady conditions states that the volume of water between two crosssections is constant, neglecting precipitation, evaporation, and tidal motion.
Referring to Pig. 8, let 516 Between 0 and x the continuity requirement gives Turning again to Fig. 8, we can draw up the salt balance between sections 0 and x.

Application of the above theorem gives ^ --^
On eliminating Q 2 by means of equation ( 5) and neglecting small quantities, we get p-ft This pair of equations implies an infinite discharge at x = L if fPp -p,N is zero there, hence the restrictive clause after equation ( 4).
Nevertheless it is instructive to use the density distributions (l) and ( 4) to calculate the variation of discharge with x.
If we identify p i with the mean density p , and take fc-f.*iW 8in ¥--W'ft-"'* ' substitution in equation ( 7

} IT I (8)
Curves of discharge 'as a function of x are clotted in Figure 9 for the typical density ratio Ap * ^'^Max They show the enormous increase in the volume of water moving seawards, and the consequent counter drift, in a similar fashion to the curves deduced by Ketchum (1952) from a slightly different salinity distribution.
If the cross-sectional areas A, and A ? (Fig. 8) were known, we could immediately calculate the mean velocities V^ and I? 2 as sketched in Fig. 10, but unfortunately we know only the sum A, + Ap » A as a geometrical function of x.
The available facts are A, * A R and A" = 0 at x = 0, and A, «A« at x = L.The total depth H is known but the height h of the surface of zero net motion is determined by the bed roughness z Q and the intensity of mixing, which depends on the tidal streams.
Both if, and Ug are of order U R .
If b is known, the mean velocity of upwelling (¥ ) is readily determined from equation (6).
It is of order Ji...-£s£-.\JL«50.001 U R , say 1 mm/sec, which exceeds the maximum rate of rise and fall of the water surface due to normal tidal motion, but is an order of magnitude below the root mean square value of the vertical fluctuations w.

TURBULENCE MEASUREMENTS
Complete understanding of an estuary would need records of the variation of bed contours, water density, tide, shear stress, time-mean velocities U, V and W, and the velocity fluctuations u, v and w, over shoals and channels throughout several tidal cycles.
Civil engineers take many observations in nature, and in hydraulic models, to deal with the problems of dredging away sand bars and stabilising the flood and ebb channels along the approaches to a port.
Oceanographers, meteorologists, and coastal engineers have amassed considerable data on water density, sediment concentration, tidal levels and discharges, and the influence of fresh water flow, wind action, and atmospheric pressure irregularities on long term averages.
Increasing attention is now being paid to the rapid fluctuations of velocity and other elements in tidal streams, as a measure of fluid turbulence, but no investigator has yet measured simultaneously the velocity components in the three co-ordinate directions over the cross-section of a tidal channel.
Several investigators have measured the longitudinal component (U+u) and a few the vertical fluctuation w at fixed positions.
Prior to, and in the first years of the Second World War, German oceanographers were obtaining velocity records from a paddlewheel current meter anchored on the sea bed.
In Norway, about 1947, experiments on bottom friction were conducted by the University of Bergen, using sensitive cup or bucket type current meters attached to a tripod resting on the sea bed.
In the United Kingdom, about 1948, a team from Liverpool University made observations of U and u with Doodson pressure-operated current meters mounted in a stand or suspended from a boat in the Mersey, an example of a well-mixed estuary; here the R.M.S. value of u averaged 0.05 U to 0.10 U, without any clear trend in the vertical.
Further observations, reported by Bowden and Pairbairn (1956) were carried out by the Liverpool team in the Irish Sea near the coast of Anglesey, using electromagnetic flow meters fixed to a tripod on the sea bed, to record u and w accurately, and U with less precision; the Reynolds stress -u.w.varied from 1 to 4 dyne/cmr near the strength of flood or ebb.
The large amount of numerical data on turbulent velocities resulting from later experiments with the electro-magnetic flow meters off Anglesey is being analysed by the DEUCE computer.
Measurements at the same site by Bowden et al. (1959) of the time-mean velocity U throughout the depth, at half-hourly intervals through the tidal cycle, using a Doodson meter suspended from the research vessel and cup-wheel meters on a tripod for the velocities just above the bed, indicated a systematic departure from the logarithmic profile due to phase differences between the velocities at each measuring point.
However, this effect of tidal inertia was negligible close to the bed, say for the bottom 2m. in a total depth of 22 m., and here the logarithmic U = % In * was obeyed, with von Karman's constant K » 0.4 and the roughness height z = 0.16 cm., corresponding to k_ G& 5 cm.
This equivalent sand roughness may be interpreted as due to ripples, for the bed consists of firm sand with small fragments of shell.
The maximum » value of bed shear stress "C 0 =* P-VL w * s about 8 dyne/cm .The eddy viscosity varied in space and time; it was somewhat higher at mid-depth than nearer the surface or bed, and tended to maximum values when the tidal stream was at a maximum, numerical values of ^Momentum ^>e i n S °£ the order of 270 cm./sec.near strength of flood, and 130 cm./sec.near strength of ebb, when the depth-mean velocities were tf » 45 cm./sec, and U* « 39 cm./sec.respectively.
In the U.S.A., experiments by Lesser (1951) in which U was measured by four Ekman current meters suspended from a tripod in the lowest 2m. of water 45 m. deep off the coast of California gave logarithmic velocity profiles with z = 0.1 cm.over sand which was hydrodynamically rough, with maximum XQ *» 5 dyne/cm., and SB© = 0.02 cm.over mud which behaved as a smooth boundary, with a ?maximum shear stress at the bed of only 0.2 dyne/cm .
In 1952, workers at Woods Hole Oceanographic Institution measured U, u, and w (with less certainty) in the Kennebec estuary.
Their turbulenee meter was suspended from the research vessel, so observations near the water surface may have been distorted by its proximity.
The R.M.S. values of u and w were of the same order, about 0.05 2 U.
High Reynolds stresses, of the order 10 to 30 dyne/cmr, were associated with large velocity gradients 3JL in this stratified estuary, which has a well-marked salt wedge below the fast moving upper layer.
Turbulence measurements have been made for some years in the rivers and estuaries of the Netherlands.While working with the fiijkswaterstaat in 1958, the author was able to measure the longitudinal velocity simultaneously with temperature and salinity, using an instrument designed by the Technical Physics Department (T.N.O.) for the Hijkswaterstaat, which is being des-cribed at the present Conference.
Essentially this turbulence meter is a sensitive impeller, which responds rapidly to changes in water velocity, mounted on the streamlined body of an Ott current meter weighing 100 kg.
It was suspended from a davit on the vessel "Christiaan Brunings H anchored at different positions in the Haringvliet.
Observations were taken in one of two ways: either the instrument was steadily lowered to the bed, then winched to the surface, thus getting the vertical distribution over a short time interval, as in Pig.11, or it was lowered in steps of 1 or 2 m. and held for 2 Minutes at each depth, giving a record of velocity against time at each depth, from which the i-x^u.time-mean velocity U and the standard deviation <rs=(u) n could be estimated.
Results of the latter method of observation are plotted in Pig. 12 for flood and Pig. 13 for ebb streams.It will be noted that while the curves of U(z) are markedly different due to the net seaward flow near the surfaoe, the curves of cr(z) are alike, with peak values of 0*«0»i TJ near the bed, similar to a river or other open channel with steady flow.
The vertical fluctuations were not measured, so the Reynolds stresses could not be determined.
However, if u and w are of the same order, we can estimate the mixing length P from the formula ft , *•« * v except where U(z) passes through a maximum.There is not sufficient information to deduce the vertical distribution of JL , but it seems to have a magnitude of order one tenth of the water depth.
Por purposes of calculation we will assume the mixing length distribution plotted in Pig. 14 is invariable throughout the tidal period.
The curve in Pig.14-has the equation ( 9) where von Karman's constant K « 0.4 and the figure is plotted for the case h » 0.5 H.

EQUATIONS OP MOTIOK z direction.
Neglecting viscosity and the attractions of sun and moon, the vertical forces on a fluid element are due to hydrostatic pressure, gravity, and upwelling, the latter being very small.
Application of Newton's x direction.

The longitudinal equation of motion, including friction but neglecting the tide-generating forces, is
On expanding the first term, substituting equation ( 12) in the second term, and neglecting the variation of density with time, equation ( 13) may be re-arranged thus 3f +u .'£+4 ^£-4.'£-^*(14) If the bed of the estuary is horizontal, the term on the right hand side represents the gravity component^• parallel to the water surface.
We will write I • -«g for the water surface slope, downwards from river to sea.
This must equal the density-induced slope along the estuary, denoted by D, if all motion ceases; then equation ( 14 The order of magnitude of each term in a typical estuary is quoted.The surfaee slope I«#D + I .coswt is balanced by components due to inertia, kinetic head, density gradient, and friction. She case where D = 0 and the kinetic head is negligible has been solved.For laminar oscillations, Lamb (1932, p.622)  where the boundary layer thickness is -&=./Ax., the most noteworthy feature being a phase advance near the bed.
Longuet-Higgins (19537 has adapted this to find the mass transport under waves of finite amplitude, so correcting the frictionless theory of Stokes, and Abbott has applied this to a tidal estuary. For turbulent oscillations in water of constant depth, theoretical and experimental investigations by Schdnfeld (1948), McDowell (1955) and the author show that the velocity distribution is qualitatively like that predicted by Lamb.
In order to study the effects of a density gradient along the estuary, we seek a quasi-steady solution, i.e. neglect the inertia term in (15) and assume the variation of kinetic head along the estuary is solely a density effect.
As a first approximation take ^-("VS")* 0 nave a constant value F, = ^LLHl_J above the surface of .t v zero net motion, and another^constant value F 2 » ^5L.f jC j below the surface of zero net motion, at any cross-section x. "We will moreover assume that the Froude number U |r is small, so the remaining part of equation ( 15) is This is an ordinary differential equation which can be integrated for the vertical distribution of shear stress.
Using subscripts 1 and 2 for the layers, the boundary conditions to be satisfied are X^** 0 at z = H, U.• T 2 at z = h, whence

}
At the bed, A^ *» I -D -F^^ + (^ -P 2 ^,"H" !B S » say * Thus t ( =p .g.H.S and we see that the direction of the net bed shear stress is very sensitive to the values of F,,F 2 , and h, for I»D at slack water.Fig. 10 shows that F, and F« may be of opposite sign in the upper reach. Writing h = oL.H, it will be shown that the fraction eC is determined by F,, F" and tne roughness height z .In anticipation, we have plotted Fig. 15 for a crosssection where ot = 0.5, ?, « -J,6 D, F g * + *.0 D, and I = D.(1 + 10.cos cot), at intervals of one lunar hour (<*»t = 30 ).
As a second approximation, the discontinuity in the stress gradient 4£-at z = h could he smoothed toy making F variable in each layer, with F, = Fp at z = h.Note the asymmetry in Fig. 15 despite the neglect of tidal inertia.

VELOCITY DISTRIBUTION IN DENSITY CURRENT
By definition, t = f»Jt *jfar fz ' exce P' t very ol °se to the bed.
Substitution in equation ( 18) with the mixing length distributions assumed in equation ( 9
Note that it is impossible to have simple harmonic motion of both surface slope and tidal stream when the flow is turbulent.Also it is impossible to have S.H.M. of both tide and stream, even in a rectangular channel, so to this extent all solutions in this paper must be regarded as approximate.
Retention of the first term in equation ( 15) gives

11U
1_ 2JL « T A trial solution is U 2 (z,t) = U Q .cos (art -<(>), where U Q and 6 are functions of z.The velocity amplitude U Q is well approximated by a logarithmic expression, so take u 0 * ^g,HtI 0 ) . in | .
The phase lag f will be assumed K O to be a linear function of z, although observations (e.g.Proudman, 1953, p.313, at Smith's Knoll) indicate that <6(z) is more nearly parabolic.
If <p is the phase difference between velocities at surface and bed, we make the simple assumption that fl5=-£..aS, and we get the order H of magnitude of <f> from the value of 0 .= phase difference between mean velocity and bed shear stress or velocity gradient, as follows.Schflnfeld (1948) applied the mixing length theory to compute the time difference t, between u and tL.
His result, for a rough bed, is This correction has been applied to the velocities in (21) to estimate the velocity distributions in the presence of both density and inertia effects, using the linear phase distribution 0*g.^o with p Q =» 3°, and the previously assumed values of slope components and roughness ratio, giving Nikuradse kg «*33 cm. with H = 10 m., corresponding to a tidal channel with large sand ripples on the bed.Fig. 17 shows the resulting profiles.
Although the velocity gradients near the bed, hence XL* are increased after reversal of the tidal stream, they are correspondingly decreased before reversal, and the maximum value of € appears to be the same as that obtained when inertia is neglected, with this "slowly varied" flow; only at very short tidal periods, as in hydraulic models, is there a measurable increase in the maximum bed shear stress.
However, the phase difference between velocity and water surface slope is by no means negligible, so linear superposition of the quasi-steady and inertial solutions of the dynamic equation cannot produce very reliable results.
But here we are interested in the general behaviour of estuarine water, so further refinement in the correction AU will not be attempted.
Before leaving this subject, it should be emphasized that the phase differences inside the fluid, although small, may be important for the proper operation of tidal models with movable bed material, since grains set in motion relatively early in the tidal cycle may continue moving with the main stream even when the bed shear stress has fallen below the value required to initiate movement.For reproduction of inertia effects the Iamb number should be the same in model and prototype (unless the friction coefficient differs), calling for models without vertical exaggeration if the Proude scale law is followed.

Fig
Fig. 7 shows the average density distributions at slack water after high water (change from flood to ebb), and slack water after low water (change from ebb to flood), on the assumption that f = p, + Ap. $ln z JLg-(l) channel through tidal flats there is little variation of density in the transverse direction.The effect of the Earth's rotation is to tilt the isohalines (Pig.4) much more than the water surface.Assuming longitudinal translation of the isohalines without rotation, we now seek for a suitable law describing variation of the vertical density difference along the estuary.Defining the salt concentration e » Itss of waier at a point, the density is P = (l+c).p,and the vertioal flux or transfer of salt per unit area is cp37 » "P^salt 'that Y • Ri if the work doi by fluid turbulence is wholly devoted to mixing (i.e.increasing potential energy of variable density fluid), and Y > R i if some energy is dissipated by fluid viscosity.

)
Considering the space between sections x and (x + Sx) above the surface of zero net motion in Fig.8, the water .entering per unit time is Q 1 + W.b.Sx, and the water leaving per unit time is Q-, + •{«* • ooc , whence mass of salt in any part of the estuary, and by definition, the mass of salt is c.(Mass of water), « Ve.o.Q.St = constant, therefore 21c.p.Q ) gives & * l *&*• ft and a* &rf • tan TT.3C ) gives the velocity gradient in each layer:g.t Vt»fr*i;*»w ^^ m and ^..^^H-IS-^-t -P'frl fr, k>z> J where P « %I -D -#F 2 , R * I -D -F x , and S = I -D -(1 -cO.F, -oC.F 9 , as above.Re-writing (19) with the dimensioniess elevation H = # gives and K^.ifVU^p.n-Dta*!.(20)Theseare standard integrals but their solutions are too complicated for normal use.However, they may be simplified to yield the following approximations:-) to about 1.2 under unstable conditions (Hi negative).If m, n are not functions of z, integrate for the velocity profiles:-This may he integrated twice, assuming that mixing length jt = K .z.(l - §) » "to give the logarithmic velocity profile U(z) = I|2LL..^n •£ • Hence the simplest approximation to the velocity distribution in a tidal stream, where surface slope I « I .coswt, is