107

Mass Transport in Rotatory Tidal Currents

By BRYAN JOHNS 1)

Summary- A p e r t u r b a t i o n m e t h o d is used to calculate s t e a d y non- l inear effects in ro t a to ry t ida l cur ren ts . The inclus ion of t he Coriolis accelera t ion leads to a un ique ly d e t e r m i n e d second order cu r r en t s y s t e m in t he t ida l s t r eam. Th i s is shown to be a b s e n t in h igh f r equency g r a v i t y waves . I t is found t h a t par t ic le drift , or m a s s t r anspor t , is cr i t ical ly d e p e n d e n t on th is m e a n cur- rent , a n d conseque n t l y on t he va lue of f/a, f being t he Coriolis pa rame te r , and a t he angu la r f r equency of t he oscillation.

1. Introduction

Second order effects in finite amplitude water waves were considered by STOKES [6] 2) and have since received the attention of many authors. Apart from a modification to the oscillatory motion, it was found that there also existed beneath a progressive wave system a steady particle or Lagrangian drift (mass transport). This drift is uniquely determined only if the net horizontal transport is prescribed, and, if this be

zero, the transport current U is given by

- - 1 a2a {cothkh__khCOSh2k(z+h)} V - - 2 h sinh2k h '

where a is the wave amplitude, a the angular frequency of the oscillation, k the wave number, and h the undisturbed water depth.

If the oscillatory motion has standing wave characteristics, the same theory predicts that, when the net horizontal transport is zero, the transport current van- ishes identically.

We see, therefore, in STOKES' theory the particle drift is an essentially indeter- minate quantity.

The modification to the predicted mass transport introduced by frictional effects at rigid boundaries has been discussed by LONGIJET-HIGGINS [3], and the extension of these boundary layer arguments to oscillations of a tidal scale has been considered by HUNT and JoI~NS [2]. The result of both investigations is that the mass transport current cannot be arbitrarily assigned near rigid boundaries (contrary to STOKES' theory), but is there a uniquely determined quantity. The relevance of the above

i) Department of Applied Mathematics, University College of North Wales, Bangor, U. K. 2) N u m b e r s in b racke t s refer to References , page 116.

108 B. Johns (Pageoph,

ideas to oceanic circulatory systems was indicated in ~21 by an application of the theory to co-oscillating tides in shMlow seas, and a particle drift was deduced, which, it was suggested, could play an important role in the movement of loose bed material.

I t is, however, a mat ter of some conjecture as to whether frictional effects will have a significant influence on particle drift in oscillations of a tidal scale. On the high frequency gravity wave scale, experimental work by several authors (e. g. BAGNOLD [1 has furnished an affirmative answer to this question. The absence of detailed observa- tional data from oceanic regions precludes an answer being given for low frequency oscillations, but it is desirable to have a frictionless theory to accompany !21 when comparisons with experimental work are possible.

I t is therefore the purpose of the present paper to conduct an investigation into the nature of steady second order effects in frictionless oscillations of a tidal scale, and to compare the results with the beforementioned theory of STOKES, and the boundary layer predictions of [2 i .

Far removed from rigid lateral boundaries, it is known that tidal currents are frequently rotatory in structure (PRouDI~AN ~4~, p. 264), and it is in such a situation that the results of this paper are established. A most significant feature is the pre- dicted existence of a uniquely determined mean Eulerian current in the tidal stream which renders the mass transport identically zero. This result is quite different from the corresponding predictions in [21. There, the mean Eulerian current near a rigid boundary was identically zero, but the mass transport induced by a progressive Sverdrup wave system was non-vanishing in the direction of wave propagation. I t is therefore strongly suggested that mass transport in rotatory tidal currents must result from the action of viscosity at rigid boundaries, or neglected external circum- stances.

The vanishing of the mass transport in frictionless rota tory tidal currents has an interesting consequence in the rate of energy transmission in the system. In the con- cluding section, it is established that the mean energy flux is generally dependent on the atmospheric pressure (frequently neglected), and that the only exception is when the net horizontal transport is zero. In ro ta tory tidal currents, therefore, the mean flux is automatically independent of the atmospheric pressure, but this does not necessarily apply to the situation in high frequency oscillations.

2. Formulatio~ o/" problem

We employ Cartesian coordinates (x, y, z) fixed with respect to a plane boundary z = 0 in the undisturbed level of the free surface. The x-axis is directed at an angle V to the south of east, the y-axis at an angle 7 to the east of the north, and the z-axis vertically upwards.

In problems hitherto considered in which account has been taken of the Coriolis. acceleration, it has been usual to make a hydrostatic pressure approximation. The objection to this approach in the present investigation is the uncertainty as to whether the non-linear effects which we propose to determine are of a greater or lesser magni- tude than the error incurred in making an assumption concerning the pressure dis- tribution. I t is therefore considered desirable to retain the unsimplified equations of a frictionless incompressible fluid; indeed, the structure of the results in section 5 is a consequence of this.

Vol. 60, 1965/1) Mass Transport in Rotatory Tidal Currents 109

Denoting the velocity field by u = (u, v, w), the fluid pressure by p, density by ~, and writing the Coriolis vector as

] = ( - - f ' s i n 7 , f ' cos 7, f ) , (2.1)

the equation of motion has the form

b~/-+ ( t t - V ) u - - u • T + g z , (2.2)

where f = 2 s Q s i n r f ' = 2 s Q c o s r

~Q being the earth's angular velocity and r the latitude (assumed uniform throughout a region). The equation expressing continuity of mass is

V - u = O, (2.3)

and, if the undisturbed water depth is constant, being denoted by h, the condition expressing zero flux of water through the rigid bot tom is

w = 0 at z = - - h . (2.4)

Finally, if the equation of the free surface is z = r the kinematical condition yields

at 3T + ( u ' v ) ( g - z ) = 0 at z = ~ , (2.5)

and if the atmospheric pressure be constant, and represented by p0, it is necessary that

p = P o at z = r (2.6)

3. Method of solution

Following STOKES [6], the dependent variablesa) of the problem will be developed in terms of a small quanti ty e (essentially a measure of the wave slope, but used here only as an ordering parameter),

r = e ~(1) + e2 r + . . . , ]

u = e u (1) + e 2 u (~) + "'" ' / (3.1)

L (p _ P0) + g z = ~ p(1) + e~ p(~) + . - . . o

In the related problem of [2], it was found convenient to specify the order of the linear Coriolis terms in the equations of motion in terms of the non-dimensional quanti ty e. More precisely, to consider separately the cases f /a = 0(e i) (j >~ 1), 0(1), where a is the angular frequency of the motion. In the present paper, these cases are again considered separately, and, with reference to steady second order effects, it is found that one is not a limiting case of the other. That is, in the solution for ] /a = 0(1)

a) If the solution is required as far as terms 0(e3), it is also necessary to develop a in terms of e,

110 B. Johns (Pageoph,

it is ~aot permissibIe to a l lowf/a -~ 0. (This s tatement will later receive certain quali- fication.)

Before proceeding to the details of the solution, the boundary conditions (2.5) and (2.6) must be expressed as conditions at the undisturbed surface z = 0. This, in the usual manner, is readily effected by expanding the equations about z = 0 (c. f. STOK~ER [51, p. 19). I t is found that (2.5) and (2.6) become

and Op

P,=0 -[- : (~-z),=0 + . . . . po. (3.3)

4. The solution for ] /a -~ O(eJ), ~ >~ 1

The present section is concerned with high frequency oscillations, generally classi- fied as gravi ty waves, in which the Coriolis terms are at least an order of magnitude smaller than the linear terms.

The first order equations are obtained by introducing (3.1) into (2.2) and (2.3) and identifying terms 0(s), whence

bU(1) - - - V p(~), bt

V " U(1) = O.

Likewise, the first order boundary conditions are obtained from (3.2) and (3.3),

(4.1)

(4.2)

w ( 1 ) = 0 at z = - - h , br

Ot w o ) = O at z = O ,

p o ) - - g ~ ( 1 ) : O at z : O .

(4.3)

A solution of these equations representing wave propagation in the x-direction (all conditions then being independent of y) is weli known, and is not given here.

The second order equations are obtained by identifying terms 0(s~), and we find that

bt + ( u O ) . V ) u ( 1 ) - f • u( 11= --Vp(~), (4.4)

V �9 u(2) = 0 , (4.5)

with boundary conditions

w ( ~ = 0 at z = - - h ,

0~2) w(2) + ~ (~(i) u(1)) = 0 at z 0 (4.6) r)t

p(2) _ g ~(2) + ~(i~ op(i) oz = 0 at z = O .

I f f / a = O(s~), ] ) 2 the Coriolis terms will not appear in (4.4).

Vol. 60, 1968/I) Mass Transport in Rotatory Tidal Currents l 11

The structure of the above equations indicates that the second order solution will have a steady component (resulting from the nonlinear combination of first order t ime periodic terms), and this we proceed to evaluate. I t is seen that this component must satisfy

( u O ) . V ) u ~ 1 ) = - v ~ ( 2 1 , V - ~ 2 ) = O (4,7)

(the horizontal bars denoting mean values) together with the appropriate parts of (4.6). Since the first order solution has progressive wave characteristics in the x-direc-

tion, the steady component of the second order modification will be independent of x as well as t, and so the continuity equation in (4.7) reduces to

In addition, (4.6) reveals tha t

w{2) = 0

whence, from (4.8),

0~(2) --~ O. (4.8)

at z = O and z = - - h , (4.9)

w(~) = O, (4.10)

and there is no vertical mean current. Consistent with this analysis, it is apparent that the form of ~c~) is an indeterminate quanti ty C(z). If the motion is irrotational, then ~(2) is constant, say C.

5. The solutionforf/a = 0(1)

In the present circumstances, the oscillations being of a low frequency, it is neces- sary to retain the Coriolis terms in the first order equations. The resulting theory will then be applicable to tidal co-oscillations in mid-latitudes. We obtain

au/~) u(l) • V p(1) �9 V u(1) 0 (5.1) O t ' '

again subject to (4.3). The steady components of the second order modification must satisfy

u(1) ~ + v(~) + w ( ~ - - - + cosy ~(2) - -

av(1) Or<*) (1) 0~(*) fu<2 ) ~(2) @~/ u m ~ + v ( * ) ~ + w -%7- + + f ' s i n y = 0y

~w(~ O w ~ 1 ~ + w(1) ~w(l~ f ' cosy u(2) - - f ' siny ~(s) -- u(~) - 7 7 - . + v(1) - by - O~ - - bz

(5.2)

subject to

wl~)----O at z = - - h ,

u = a o (r v(*)) + -aT

az = 0 at z = 0 (if

at z = 0 ,

= o ) .

(5.3)

(5.4)

112 B. Johns (Pageoph,

As was indicated in Section 1 a solution of (5.1) will be sought which is representative of a rotatory tidal current system. The difficulties inherent in considering more general current systems will be discussed briefly in a subsequent section, and we simplify our present problem by prescribing that all conditions are independant of y. The solution of (5.1) having progressive wave characteristics in the x-direction then has the form

~(1) = a e i(kx-at)

i W'(z) e ~(kx-~) U(1} = k-

} v(1) ~__ __ W ~ / ' e~(k~_~t) /k ( z ) - i sin W(z) (5.5)

wO) == W(z) e i(k'~--~)

p(1) - ia 1 - - W'(z) + - - c o s y + i ~ - s i n y ~ - ~ ~

only real parts being of significance. The equation for W(z) is

(1 -- ~ ) W " ( z ) + 2 i k ff@ siny W ~ ( z ) - k~(1 -- ~ sin2y) W ( z ) = 0 . (5.6)

The solution satisfying W(-- h) -- 0 and the surface kinematical condition is

a i a e_i~, sinh/3(z + h) , (5.7) W(z) - - - ~nh~ h where

f f r --~- stay

f 2 f ' 2 . \ i / . ~ 1 O" 2 ~ 2 s t a y )

(5.s)

g ~ ta,h~ h (5.9) 0"2= {kfl~(1---dfff2) 4- ~-f' cosy tanhfl h} "

It is interesting to observe that, when 7 =/= 0 and ~ h is sufficiently large, the fore- going solution will possess significant oscillatory characteristics with depth; that is, whenf /s becomes numerically close to unity. This situation can be realised ultimateiy even for small values of k h; that is, even when the depth is much less than the wave- length. In such circumstances, therefore, the customary hydrostatic pressure approxi- mation will fail.

To complete the details of the first order solution, it remains to satisfy the free surface pressure condition in (4.3). By substituting from (5.5) ane (5.7), it is readily seen that g and k must fulfil the relation

Vol. 60, 1965/I) Mass Transport in Rotatory Tidal Currents 113

Except for y --- ~/2, it is apparent that a cannot be replaced by -- a without changing the structure of (5.9). That is, we cannot prescribe two geometrically identical waves propagating in opposite directions with the same speed. Alternatively, there does not exist an analogy of the standing wave of classical water wave theory. The best that can be done is to superpose two geometrically identical waves propagating in opposite direction with slightly differing speeds.

Before substituting from (5.5) into (5.2) and (5.4) (again with b /by ~ 0), it must be recalled that only real parts are of significance. Appeal must therefore be made to the formula

1 (2* R e P R e Q - - , v R e P

where P and Q are any two complex quantities with a time period 2 7~/a, and the asterisk denotes the complex conjugate.

We observe that the first two conditions in (5.4) reduce to

w(21=0 at z = - - h and z = 0 , (5.10)

whilst (5.3) becomes

o--~- = o (5.11)

steady components again being independent of x. I t therefore follows that

w(~ = 0 . (5.12)

Introducing (5.12) into (5.2), and substituting from (5.5).

and

-~(2) ~ _ 2 k----~--I ~t W'(z)12 -}- { W ( z ) } * W"(z)] , ( 5 . 1 3 )

f r V(2) -- 1 ~ - sin),

f~ W(z ) {W'(z)}*, (5.14) a l - - - -

where use has been made of (5.6). When z ~- - - h,

~(2) 1 I ~ ] ~ 2 ~ ~ - - 0 (5.15) 2 k s i n h 2 f l h '

In the tidal stream, therefore, there exists a uniquely determined steady second order Eulerian current at the ocean bed in a direction opposite to that of wave propagation.

Should the value of the m e a n pressure distribution pc2) be required, this may be readily obtained by substituting from (5.13) and (5.14) into the last equation in (5.2), and integrating the resultant subject to the mean pressure condition in (5.4).

I t is important to observe that the establishment of (5.13) and (5.14) has involved the cancellation of a f a c t o r f / a . I t is not, therefore, permissible to allow f i g --> 0 in the present section and the predictions are critically dependent on the assumption that f / a = 0(1). Elementary considerations indicate however, that the structure of the foregoing results is not materially altered by changes in the orders of magnitude of f f / a siny a n d f ' / a cosy, and so all values may be assigned to y without impairing the validity of our formulae.

S PAG.EOPH 60 (196~]I)

114 B. Johns (Pageoph,

6. Mass transport

The conclusions of sections 4 and 5 are of critical importance in calculating the

wave induced particle drift or mass transport U as given by LONGUET-HmGINS [3], namely

U-~u"~)+ ( / u ( l ' d t . V ) u'l'. (6.1)

Forf/a =- 0(d), ] ~> 1, we obtain the classical result of Stokes, and choosing

1 a2a co thkh C - - 2 h

SO that the net horizontal transport is zero,

~-[ -- 21 a2c;h {cothk h - - k h coShZsinh~ki~(z h + h).} . (6.2)

F o r ] / a = 0(1), the results of section 5 show that the transport in rotatory tidal currents is quite different from (6.2). Substituting from (5.5), (5.13) and (5.14) into (6.1) we find that the mass transport vanishes identically. This is in complete contrast to the corresponding result of [2], where the particle drift in rotatory tidal currents was non-vanishing near the ocean bed.

7. Generalisations

The foregoing sections have been concerned with non-linear effects in rotatory tidal currents. In oceanic regions there are a variety of different tidal co-oscillations present, of importance amongst which are rectilinear tidal currents in which the trans- verse current is identically zero (Kelvin waves) or in which the transverse current vanishes at rigid lateral boundaries (Poinear6 waves). To complete the present theory, therefore, it is of some importance to construct solutions of (5.1) representative of these oscillations, and to calculate the non-linear effects therein.

Unfortunately, difficulty has been encountered at this iuncture. It has not been possible to obtain solutions of (5.1) which are representative of Kelvin or Poincar6 waves by separating the variables, and further investigation is required. The present work does, however, serve to illustrate the differences which can be expected between the high and low frequency oscillation rfigimes, and viscous and non-viscous models.

8. Energy flux

The conclusions of sections 4 and 5 are of some consequence when calculating the rate of energy transfer in a wave system. By well known methods (STot~ER [51 p. 13), Equation (2.2)) may be used to show that the mean rate of energy transfer across a surface S fixed in space is given by

/I( ' ) R = p + - s e u = + g ~ z u .ds . (8.1) $

Vot. 60, 1965/1) Mass T r a n s p o r t i n R o t a t o r y T i d a l C u r r e n t s 1 1 5

The mean transfer across a vertical plane x-eonst, per unit distance in the y-direction is therefore

r

/ ( , ) R~ = p + -g e u2 + g o z u dz , (82) - h

where the mean value is to be taken after performing the integration. Equation (8.2) may be expanded about z =- 0 by TAYLOR'S theorem to give

0

- / ( , ) {( , )} R ~ = p + ~ - ~ u 2 + g e z u d z + ~ p + - ~ u 2 + g o ~ z u + ' " . Z:=O

-h

(s.s)

Substituting from (3.1), retaining only the lowest order terms, and suppressing the ordering parameter e,

0 - ]" R~ -- (po 7,(~) + e P(~ ~( ' ) dz + po ~( , u~2o. (8.4)

-h

The mean flux Rx is therefore expressible in the form

where

and

R~ = R~)+ R2) (s.5>

0

~h

(8.6)

R~> = po ~(,) dz + r u ~ 0 . (s.7)

--r The quanti ty R) represents the vertically integrated mean rate at which the dynam- ical pressure does work against the oscillatory current. To obtain an interpretation of R!~ I, we observe that

/ /l/'-' 0~ 1 ;(1) @2~ (~) w~.~o dt = u(1) ~ - 4t + -~2- = u~=o . w(1) dt dz , - h

as may be readily verified on integrating by parts. Moreover, by virtue of continuity and the progressive wave characteristics of the first order solution,

t t U(1) f 01~ OfA'(1) / ~/~(1) dt " j ~ i 2 - d t - - a ~

and (8.7) may be re-written as 0

~?) = po [ ~ d z . o

-h

(8.s)

116 B. Jolms

The quan t i ty R~}, there%re, represents the vertically integrated rate at which the excess atmospheric pressure does work against the mass t ransport current,

The dependence of the energy flux on the atmospheric pressure (frequently taken. as zero) is often overlooked. As is apparent from (8.8), the contr ibution from po vanishes only if the net horizontal mass t ransport beneath the oscillating surface is zero. This is automat ical ly the case for ro ta tory tidal currents, but not for high fre- quency gravi ty waves. This distinction may be impor tant in other investigations.

REFERENCES

~1] R. A. Ba~xoLo, Sand move~e.nt by waves, J. Inst. Civ. Eng~lrs. 27 (1947), 457. !2] J. N. HUNT and ]3. Jo~INs, Cgrre~ets i,zdzr by tides a~d gravity ~,aves, Tellus 15 (1963), 343. ~3] M. S. LONGUET-HIGGINS, ~,1ass tra*zsport i~ water waves, Phil. Trans. Roy. Soc. 245 (1953), 535. [4] J. P~OtJDMAN, Dynamical Oveanography (Methuen, London 1953). r5~ J. J. STOKER, Wate~ Waves, Intersc. Publ. Inc. {New York 1957). {6) O. G. S*O*~ES, O~ the theory of oscillatory waves, Trans. Camb, Phil. Soc. 8 (1847), 441 (Reprin-

ted in Math. and Phys. Papers 1, 314).

(Received 3rd December 1964)