The generation of internal tides by flat-bump topography

Deep-Sea Research, 1973, Vol. 20, pp. 179 to 205. PergamonPress. Printed in Great Britain.

The generation of internal tides by fiat-bump topography P. G. BAINES* (Received 11 February 1972; in revisedform 30 August 1972; accepted 14 September 1972) Abstraet--A general procedure for the calculation of the oceanic internal tides generated by the interaction of the surface tide with bottom topography is derived, and applied to typical cases. The formalism is restricted here to essentially two-dimensional topography whose surface is never tangential to the local direction of internal tidal energy propagation, but has otherwise arbitrary shape. The theory is applicable to a wide range of density stratifications which permits a two-parameter fit to any oceanic case. Results have been calculated for certain representative topographic shapes, most notably continental slopes, and these indicate that the rate of conversion of surface tide energy into internal tide energy generally increases rapidly with topographic height, but also depends strongly o n the geometry. For instance, for this model, continental slopes are far more effective internal tide generators than, say, the Mid-Atlantic Ridge. For a model ocean with constant Brunt-V/iis~il~i frequency, N, the internal wave energy fluxes on each side of a continental slope (satisfying the above criteria) are approximately equal, but for an ocean with a realistic density profile the energy flux and energy density are larger on the shallow continental shelf than in the deep ocean. 1. I N T R O D U C T I O N

AND SUMMARY

INTERNALtides, namely internal waves of tidal period, were first observed in the ocean by HELLAND-HANSENand NANSEN (1909) in the Norwegian Sea by means of hydrographic observations using Nansen bottles repeated at short intervals. They have since been observed in many places in the world's oceans~ both in mid-ocean and near coasts, over continental shelves and slopes. Some salient features of the observations to date are (i) the semi-diurnal frequency is usually more prominent than the diurnal, as is the case for surface tides; (ii) the semi-diurnal component has been observed to vary in amplitude in sympathy with spring and neap tides at a nearby coast (REID, 1956, 1958; HECHTand HUGHES, 1971); (iii) they have been observed to be prominent over a continental slope, but incoherent and barely discernible further out to sea (REID, 1956; WUNSCHand HENDRY, private communication), and (iv) they have been observed on a continental shelf propagating shoreward with crests parallel to the coast (SUMMERSand EMERY, 1963). One mechanism which seems capable of generating these waves in a manner which is consistent with the observations, and in particular with the properties mentioned above, is that of the interaction of the surface tide with bottom topography, as first suggested by ZEILON (1912) in connection with seiches and internal waves in Gullmarfjord. The motion of the surface tide over bottom topography creates horizontal density gradients in the stratified ocean which are unbalanced, and therefore act to generate internal waves of the same frequency. Cox and SANDSTROM(1962) considered the interaction of the surface tide with linearized (small amplitude) topography using a modal approach, and step-like topography has been studied by RATTRAY(1960) for a two-layer model ocean and by RATTRAY, DWORSKI and KOVALA(1969), and *C.S.I.R.O., Division of Atmospheric Physics, Aspendale, Victoria, 3195, Australia. tSome details of the early observations are given in DEFANT(1961).

179

180

P.G. BAn,ms

PRINSENBERG (1971) for a continuously stratified ocean, again using a procedure based on internal wave modes. The present theory is based on the theory of wave-characteristics, and is applicable to a wide range of density stratifications and arbitrary two-dimensional bottom topography, with the restriction here that the slope of the topography be everywhere less than the local slope of the direction of internal tidal energy propagation--' flatbump' topography in the terminology of BAINES(1971). It is well-known that for a fluid of constant Brunt-Viiisiilii frequency, N, the equations governing linearized internal wave motions may be simply solved in terms of space-characteristic variables (GORTLER, 1944, MAGAARD, 1962). The effect of the surface tide with bottom topography may be represented by a body force field for the internal wave motions, and, by use of radiation conditions with the characteristic variables, the problem of determining the internal tidal wave field may be reduced to that of solving a single Fredholm integral equation, following the procedure in BAINES (1971). The theory has been extended here to characteristics which are hyperbolic so that realistic oceanic stratification may be represented. The kernel of the integral equation depends on the form of the bottom topography, and this equation has been solved numerically for ridge and continental slope topographies. Attention in this paper is concentrated on results for constant N, although some results with oceanic N profiles are presented for comparison and to give some indication of the nature of the results. As one would expect, the nature of the internal wave field generated depends very much on the topographic shape. For the topographies investigated in this paper, namely a cosine ridge and a continental slope or depth change, the internal wave profile (viewed in terms of characteristic variables) consisted of one major peak and one major trough in all cases, with occasional small secondary peaks. However, their amplitude and relative shape each depends on the shape and amplitude of the topography, and each also varies in time with the phase of the wave motion. The energy flux and energy density of the internal wave motion generally increase rapidly as the height of the topography is increased, until the surface is reached or the slope of the topography becomes equal to the characteristic slope at some point. In many eases this increase is exponential in character, and there is usually a similar increase if the slope (of, say, a continental slope) only is increased. However, if a characteristic, continued by reflections from the boundaries, spans the topography exactly, the energy fluxes and densities may be rather small. This is most marked for continental slopes, where they may vanish exactly when N is constant and this condition is satisfied, regardless of topographic height. In general, when this condition of exact periodicity is approached, the energy fluxes and densities decrease. In oceanic situations, the energy flux generated at a typical continental slope (satisfying the flat-bump criterion) may be as large as 10 -4 Po Q2/sec/cm of slope, where Q = depth x maximum barotropic velocity normal to the slope, and Po is the density. A typical value for Q may be 105 cm2/sec, so that the loss of energy from the surface tide will be of the order 106 erg/sec/cm of slope length. For continental slopes adjacent to wide (> 100 km) continental shelves, Q may be much larger than the above value, and it seems probable that very large internal tides will be generated at the edges of many of the ' acknowledged' tidal dissipation regions such as the Argentine shelf, Gulf of Carpentaria and the Bering Sea.

The generation of internal tides by fiat-bump topography

181

2. BASIC EQUATIONS We take the equations governing the internal tides to be those of a rotating stratified inviscid fluid. Relative to axes rotating with the solid Earth, these are:

(D° )

(Po + P) ~-f + f x u

}

+ V (po + p) + (po + p)g~. = O,

~t (Po + P) + V'(po + p)u = 0,

(2.1)

D Dt (Po + P) = 0, where D / D t denotes the total derivative, u the fluid velocity, f the vector Coriolis parameter, y the acceleration due to gravity, 9. the unit vector in the vertical direction, Po, Po the pressure and density in static equilibrium, and p, p the perturbations from this state due to the wave motion, both barotropic and baroclinic. Subtracting out the static case, and assuming that the tides are essentially linear phenomena, yields V'u=0, On 0p dpo P o ~ 7 + P o f X u + Vp + py~. = O, - ~ + w --~z = 0 .

} (2.2)

We next write u = u, + us, P =P~ + P , , where u~, p~ denote the flow field for an unstratified ocean of density Po which has the same surface displacement as the one to be considered. Po is the mean vertical value of po(z), and the latter for the ocean has a vertical variation of about 2%. Hence, for all practical purposes, the velocity field u t will contain all the barotropic mass flux associated with the surface displacements of the stratified ocean. The problem of determining the internal tides due to the interaction of the surface tide with a localized region of bottom topography may be split into two: first, the determination of the field of motion of the barotropic tide ul, pt of an ' equivalent' unstratified ocean, and, secondly, the determination of the additional motion in the stratified ocean which is required to satisfy the dynamical equations and the radiation conditions. The first problem will be discussed in the next section, and for the moment we assume that u~, p~ have been determined by one means or another. The equations for us, Pi, Pt are, with the Boussinesq approximation,

~u~

--+ ~t

1

pg~.

f x ui + _--Vpt+ =0, Po #o (2.3)

V "ui = 0,

dp ° = dpo. + wi dz - w l d----z

We may therefore regard the internal or baroclinic tide as being forced by the vertical motion associated with the surface or barotropic tide due to the bottom topography. We choose Cartesian co-ordinates x, y, z with z increasing vertically upwards, and assume that ul, pl and the bottom topography are two-dimensional so that there is

182

P . G . BAINES

no y dependence. We may then define a stream function @ such that u, = - ~z,

wi = ~x,

(2.4)

where ui, v,, wi are the velocity components in the directions of x, y, z increasing, respectively, and the suffices denote derivatives. Equations (2.3) then yield the field equation for V2~tt d" N2(z)~xx - b f 2 ~ z z -- - N 2 ( Z ) W l x . (2.5) The free surface of the ocean has a typical vertical displacement of 2-3 m whereas the depth ranges typically from 200 to 4,000 m (for the cases to be considered in this paper), so that to a good approximation we may regard the surface as stationary for the purposes of determining the internal wave motion. Furthermore, since the barotropic motion contains all the mass flux we may take as boundary conditions on -- 0 = 0

on on

z = 0, z = -h(x),

(2.6)

where the origin is at the surface and z = - h (x) is the equation of the bottom. 3. T H E B A R O T R O P I C

TIDE

Most of the research on tides to date, both observational and theoretical, has been concerned with the barotropic tide with the main variable being the elevation of the free surface. The wealth of tide gauge data shows that the wavelength of the surface tide is very large (typically 10,000 km) at the oceanic boundaries. Theoretical calculations based on the numerical solution of Laplace's tidal equations for the M 2 tide by PEKERIS and ACCAD (1969) and Hendershott (HENDERSHOTTand MUNK, 1970), show that, in general, the tidal amplitude is a maximum at the oceanic boundaries. (These two solutions do not agree in many ways, and Hendershott's results indicate that the Indian Ocean may be an exception to this rule, with a tidal maximum near its centre). Since the major variation in the depth of the ocean bottom also occurs near the oceanic boundaries in the form of continental slopes it may be expected that this region will be a major source of internal tidal energy, and this type of bottom topography will be considered in detail From equation (2.5) we see that the forcing term for the internal tide depends on the vertical velocity in the equivalent 'barotropic' tide, and in this section we investigate its form for general topographies. We consider a region of bottom topography which is sufficiently small such that the surface of the Earth may be regarded as flat and f as constant. The relevant equations governing fluid motions of tidal character in a homogeneous ocean in this region are then u~t+Ul"VUl+fx

ul+ _---Vp+g~=F, Po

V'u~ =0, with boundary conditions Ul • V[z

+

h(x)]

D [z Dt

~/(x,t)]

=

1

(3.1)

h(x), "1

0,

on

z =

-

0,

on

z

r/(x, t), I

(3.2)

where F is the tide-generating force and ~/is the free surface height. Because of the

The generation of internal tides by flat-bump topography

183

great length of the surface tide, the relevant horizontal length scale, L, of the motion will be that of the bottom topography, which for the continental slopes is of the order of 50 km. We assume that the bottom topography is two-dimensional (no y-dependence) which is a reasonable approximation for the large-scale features of most continental slopes. Since the phase of the surface tide varies only slightly over distances of the order of hundreds of kilometres, it may be assumed that the barotropic motion also has no y-dependence. Writing (3.3)

x = LYe, z = H2, ul = US, w~ = W~, 1

t = -- L Pt = #ocoULfi, co

where ~, etc. denote dimensionless variables, co is the tidal frequency, His the vertical length scale ( ~ 4 km) and U, W, are typical velocities, equations (3.1) give \

Ou

U

0.6

f f~ = Fx

~-7 + L-Z ~" v c, + ~-~ - ? o UL

+ ~

fi " V ~

+

]

'

!

~-~ + g/coU

Of;

U fi f 0-~+~-~ " V ~ + - - ~ = f y , c o

and

=0,

(3.4) i

W~UH/L, /

where Fx and Py are the components of the tide-generating force. Hence, the vertical balance is hydrostatic with an error which is of order (H/L) 2, in this case ½~. The wave steepness parameter e = U/coL varies from about 5 ~ on the continental shelf to much smaller values in the deep ocean, so we shall neglect the non-linear terms here. The second of equations (3.4) yields that P = - g [~ +O(x, t)]/cov + O(H2/L2),

(3.5)

where ~ = tl/H, and the first and third then give (3.6)

fi = •(x, t) + O(H2/L2),

so that the horizontal motion has very little vertical variation for this type of topography. Hence, integrating the continuity equation we have 0 ~ul wl(x, z, t) = wi(x, - h , t) ~ dz, (3.7)

f

~-

-h

WI(X , - h , t) - (h + z) Ox"

(3.8)

The boundary conditions in dimensionalized linear form are on on

w 1 = -ulh ~

and

w I = rh

z = -h(x), z = 0,

(3.9)

where q is the free surface displacement, and in dimensionless form, equation (3.8) for z = 0 becomes

--~ (~) 22

~o2L2~O +

- -

gH

--

Of

=

o.

(3.1o)

184

P . G . BAINES

For semi-diurnal tides on continental slopes, we have c o 2 L 2 / g H ,-, 2 x 10 -4, so that, effectively,

(hu,) Writing

= 0.

(3.1 l)

hu 1 = Q cos cot,

(3.12)

equation (3.10) shows that the barotropic mass flux Q is effectively constant over the region of bottom topography being considered. From equations (3.8) and (3.9) we have 8u~ w l ( x , z, t) = - ~x (hUl) - z ~ox " (3.13) The first term gives the vertical motion caused by the divergence associated with the true surface motion, which is negligible compared with that caused by the bottom topography in the second term. Hence, we may take w x ( x , z, t) = - Q z

(1) ~

cos cot.

(3.14)

X

We now consider continental slope topography of the form shown in Fig. 1. In Z

y

Fig. 1. Continental slope topography, with the y-axis as coastline. The dashed line at the coast represents the amplitude of a tidal Kelvin wave (not drawn to scale)propagating in the direction of the arrow, and the corresponding dashed line at the continental slope represents the associated horizontal mass flux. order to determine the quantity Q for a given slope in the ocean we may employ global solutions such as those of PEKEalS and ACCAD (1969) or Hendershott directly. Alternatively, we may use a ' local' theory after the manner of MLrNK, SNODGRASSand WI~BUSH (1970) to relate the mass flux across the slope to the amplitude of the observed tide at the adjacent coast. Clearly this may be done with varying degrees of sophistication depending on the accuracy required. From equations (3.1) and (3.2), with the linearized hydrostatic approximation, we obtain, for the free surface elevation, r/, rht, + f 2 r l , = gh[(rl - rT)~ + (~l - ~)yy], + gh~(rl - ~)x, + fghx(rl - ¢i)y,

(3.15)

(LAMa, 1945), where ~ is the effective height of the equilibrium tide (MLrNK, SNODORAS$

The generation of internal tides by flat-bump topography

185

a n d W|MBUSH, 1970), h is a function of x only, and the variation o f f is assumed

negligible for present purposes. Considering only the region of the continental shelf where h is effectively constant, this equation becomes: r/, + f 2 r / = #h(r/xx + r/yy)- #h(%~ + %y).

(3.16)

In order to give an estimate for a typical value of Q we consider a simple example of a tidal component which may be modelled by a Kelvin (or Poincar6) type wave on the shelf. Taking x = 0 as the coastline, we may write:

~l -- rh(x)exp[i(klY + tot)], where 2~/kl ~ R, the radius of the Earth (or more exactly the circumference of an ocean basin, which we take to be comparable with R). From equation (3.16), neglecting the forcing terms we have, on the shelf, ql = ao cos kx + a 1 sin kx,

(3.17)

where ao is the amplitude of the tide at the coastline and k2

(.02 _ f 2

-- - -

kl 2.

gh

(3.18)

The velocity normal to the coast is given by

u= ~

ig

(klfil + torlx),

(3.19)

and u --- 0 at the coastline yields al =

klf a ~

o"

(3.20)

Hence

ghtokao (

Q = humax = ~

(g_.hk12'~

= aotod 1

k l 2f2~

1 + o--~k2] sin kd,

2] 1 + ~---5-'fff] [1 + 0(kd)2],

(k12f

092 _ f 2 ]

(3.21)

where dis the width of the shelf. In general, kd.~ 1, and i f k 2 >0, Ikl/k[ < 1 we have Q "~ a o tod.

(3.22)

This is simply a measure of the quantity of water which must flow on to and off the shelf each tidal period (Fig. 1), and is proportional to the height of the tide and the width of the shelf. This simple picture may be refined for any h(x) and the variation of Q with x across the slope calculated. This variation may also be included in the internal wave theory given below, if desired. The result that Q oc a o d is expected to be valid for most continental shelves. 4. THE I N T E R N A L T I D E - - F O R M U L A T I O N From equations (2.5) and (3.14) we see that the field equation governing the internal tide with two-dimensional bottom topography is:

(1)

V2~k,t + N2(z)O:,x + f 2 0 = = QzN2(z) ~ xxCOScot.

(4.1)

186

P.G. BAIN~

We consider solutions which have the tidal frequency only, so we write: (4.2)

= U/(x, z)e-to, t,

and we obtain Wxx -- C2(z)W~ = 1

Q

-

-

¢02/N 2

z 1

h xx,

c2 = ~ .

N 2 - 0)2

(4.3)

To solve this equation for given h, we first obtain a particular solution W2, and then the corresponding solution to the homogeneous equation (L.H.S.* = 0) such that the sum of the two satisfies the boundary conditions and the appropriate radiation conditions. If N 2 is constant, a suitable particular solution is

Q

z (4.4)

ud2 = 1 - 0 ) 2 / N 2 h(x) '

and since co/N ,~ 1/10 it may readily be shown that this is also a good approximation when N 2 varies over the range of values found in the ocean. Writing: (4.5) kIJ = k~ 2 -~- klJ3, we have ~P3,,,, - c2~P3,, = O, (4.6) where W3 -- 0 on z = 0, (4.7) tIJ 3 = Q N 2 / ( N 2 - ¢o2)

on

z = -h(x),

(4.8)

from equation (2.6). We now define the characteristic variables for the hyperbolic equation (4.7) to be

r'd: = Jo c ( z ) + x ,

'1 = s o c ( z ) - x ,

(4.9)

so that (4.7) becomes 8~ St/

4 dz

+ --~-q] = O.

(4.10)

We look for solutions of this equation of the form

% = b(z)[f(0 + a(n)],

(4.11)

where, for some function B, b(z) = B(~ + n).

(4.12)

Substituting (4.11) into (4.10) shows that f ( 0 and #(t/) may be arbitrary functions provided that b(z) satisfies both bz~ = 0, bz-

~c b = O .

(4.13)

b(z) = c(z) 1/2,

(4.14)

c(z) = co(l + clz) 2,

(4.15)

This will be so provided and *Throughout the paper, L.H.S. denotes ' left-hand side', R.H.S. denotes ' fight-hand side'.

The generation of internal tides by flat-bump topography

187

so that b is a linear function ofz. Hence, we may take (4.11) to be the general solution of (4.10), provided the stratification of the fluid is such that c(z) may be written in the form (4.15). We may, therefore, use the theory developed here with a two-parameter fit to any density profile. The characteristics ~ = constant, r / = constant are sections of hyperbolae, reducing to straight lines as c~ ~ 0. Some typical shapes are illustrated in Fig. 2. N 2 (z) has the form Z

t

1.0 1

-- I£ °--

X

,'~

Co

0

I

-I (a) °!

/

-0.2

0 y/ -0,6

-0"8

, /C,=-l.O

.,.o

/

/

C,=0

/_o., (b)

Fig. 2. (a) Some characteristic shapes for various values of ct. The horizontal scale is determined bye.. (b) Corresponding N2(z) profiles, based on equation (4.16). N2(z) = (02.4_ ((02 _f2)/c02(1 + ClZ)4, (z < 0, ct < 0 for the cases considered here), and

(4.16)

188

P.G. B~a~ms

po(z) = po(0)exp -

1 + co2( 1 + clz) 2 1 + c12z2/(1 + clz)

(4.17)

We assume that the bottom surface becomes horizontal for Ixl large (though not necessarily at the same level on each side), so that the region being considered is localized, and that the bottom slope is everywhere less than the characteristic slope at the same depth [' flat' bumps, in the terminology of BAINZS (1971)]. Hence, the equation of the bottom surface z = -h(x) may be written: ¢ = - K(t/)

or

r / = - H(~),

(4.18)

and these equations will be single-valued and monotonic. The functionsf(~), #(r/) will also be single-valued. Equations (4.8) and (4.11) together give f(~) + g ( - ¢ ) = 0,

(4.19)

f(~) + g [ - H ( ~ ) ] = Q/c(z)l/2[1 -- to2/N2(z)], which yield f(¢) - f [ H ( ~ ) ] = Q/c(z)l/2[1 - o92/N2(z)] = A(~),

(4.20)

where the value of z required is that of the point on the bottom surface where the latter intersects the ~ - characteristic. To obtain the wave field it will suffice to determine the functionf(~); g(t/) is then given by the first of equations (4.19). We* let the depth on the L.H.S. of the bottom variations be hr and on the R.H.S. hR, and define (0 dz f o dz hR ?L = ./ - - , ?R . . . . (4.21) -hL C -hR C C0(1- clhR) Equation (4.20) then yields that on the L.H.S. a

f(x - ?L) - f ( x + ?L) - c(_hL)l/2[1 _ o2/N2(_hL)] = AL,

(4.22)

and similarly for the R.H.S. Hence on the L.H.S. the functionf(~) may be written in the form

f(~) = ~-ALf + ~ aL,,exp(imr~/yL) ' 2~)L

(4.23)

n= --co

and on the R.H.S.

AR~

f(~) = ~YR +,,=-~ ~ aR"exp(inn~/?~)'

(4.24)

where the aL,, a~, are constants. The terms where n = 0 represent constant terms in f(~), which will have no significance for fluid motion and may, therefore, be taken as zero. The exponential terms represent modes propagating horizontally which have the form ¢ . = Co,/2( 1 + clz)si n [nz(1 - ~)]exp[i(nrcco(1 _ cl)x/?L - a~t)]

L hL(1 -S-

(4.25)

*It is suggested that readers who are not particularly interested in the details of the analysis may omit the remainder of this section and refer to it when necessary, noting in particular equations (4.43)-(4.47).

The generation of internal tides by flat-bump topography

189

on the L.H.S., with h~, 7R replacing hL, 7L for the R.H.S. It has been shown by LIrnTmLL (1965) for the case of constant N 2 (cl = 0) for an infinite fluid that all the constituent plane waves of any small disturbance must have their group velocities directed away from the source, and this result is readily generalized to modes in a horizontal channel (BAINES, 1969). The more general modes of equation (4.25) form a complete set (since ~ varies monotonically with z), and we make the assumption (plausible after some reflection) that the same radiation condition applies to these modes also * (i.e. the condition of no incoming energy from infinity implies that, when a wavefield is decomposed into these modes, they will all propagate away from the source). This implies that in equations (4.23) and (4.24) the coefficients aL, all vanish for n > 0 and as, vanish for n < 0, respectively. It may then be shown by a procedure analogous to that for the infinite depth case (BAINES, 1971) that fp(O = -I- ~i P f~_, f,(s) cot ~ (s - ~)ds,

(4.26)

wherefp(O denotes the periodic part o f f ( i ) and the + sign with 7 = 7L applies on the L.H.S., - , ), = 7R for the R.H.S. The only change in the proof involves replacing Fourier integrals by Fourier series. P denotes a Cauchy principal value integral. We next consider geometry of the form (Fig. 3), where all the variation in topoZ=01

\

,,/ '~/"

Z - - hL

"9-o / J

~%~'=o

~ _~X%"~""

% ""~.x..___'v

\\\\\\\\x\""

"

\

/ /

~

Z = - hn

Fig. 3. An example of single hop topography--a cosine ridge. Z=O \

Z = -h L

~

~ /

\

\

" ~ - , 3 , , ~ ,'" " v , - ,

/

,,",$,, ~

I

/

.( i . . - . . . " " -"~, ~"'Hs ~{ r,',','"

z=-,.

Fig. 4. Multiple hop topography.

graphy may be subtended by two characteristics emanating from a single point on the top surface, termed ' single-hop' topography. We place the origin of the co-ordinate system at such a point, and for ~ in the r a n g e - 27L < ~ < 0, we have correspondingly 0 < H({) < 27R. *It seems probable that this radiation condition applies to the appropriate Sturm-Liouvillemodes for any stable density distribution po(Z),but to the author's knowledgeno proof is yet available.

190

P . G . BAINES

Applying (4.26) tof(~), f[H(~)] on their respective sides of the bumpy region, we have

AL~ f({) + 2YL

i

=

[

f(s) +

2YLP f-2,L k

f[H({)] + +

o

2y LJ

AR H ( ~ ) - i

2~,---~ = 2~'R P

A.H(s)] ~

ALS]

. zt

; cot ~

cot

f;'"{

zc ~r

(s - ¢)ds,

(4.27)

f[H(s)] (4.28)

[n(s) - Hf¢)ldU(s),

and substituting (4.20) in (4.28) gives: f(¢)=

AsH(i) + A(~)

i P f°

[ ARH(s)] n Lf(s)-A(s)+ ~ J cot~[H(s)

-

- H(~)] ~ds H ds.

(4.29)

Adding (4.27) and (4.29) and defining: /~ = ¢/'~L, M(~) = n(~)/y R, F(~r) =f({), X(~) = A,(O,

(4.30)

we obtain

l o d fsin lr/2 tM(s) - M(()]~ ds, F(~) = G(~) + ~ f-2 F(s) -d-slog ~ sin ~/2 (s ~') ')

(4.31)

where AL( 4

a(O =

M(O AR .... ~ + i { AL ~ log (2 sin ~_~ ) _ ~AR log [2 sin nM(O] }

0 + -As(~) T - + i 2, f_AXs) cot 2 [M(s)

M(0] ~--- ds.

(4.32)

Equation (4.32) is a regular Fredholm integral equation of the second kind, analogous to the equation governing reflection of internal waves from bumpy surfaces (BAINES, 1971), and is to be solved for the function F(0 in the range - 2 < ( <0. The relations Pf

0

scot~(s-()ds=41og

(

2sin

,P

scot~[s-M(0]ds

-2

have been employed in obtaining the function G(~). The terms involving As(0 are due to the variation in stratification in the range of bottom depths,* and if N 2 = constant in this range, we have AL = AR = As, (4.34) so that As may be taken as zero in (4.33) in this case. In the event that the barotropic forcing motion (Section 3) does contain some horizontal divergence, (i.e. u t is a function of x in the region of bottom topography), this may be incorporated into the present formalism via the function A,(0, which will vary in accordance with such divergence (together with the stratification). This has not *And also the variation in Q, if any.

The generationof internal tides by flat-bump topography

191

been discussed here because the divergence is probably negligible for the tides, but it should be included in such applications as internal waves generated by swell passing over shallow ridges where the length of the surface wave is comparable with the width of the ridge. The modifications required involve obtaining the appropriate form of w1 and a suitable solution ~2For the case of flat-bump bottom topography of considerable extent, as shown for instance in Fig. 4, we apply the appropriate radiation condition at each end of the bottom topography as before, and relate these two equations via the boundary conditions through each successive reflection of the family of characteristic curves. This yields a Fredholm integral equation of the same form as (4.31), as shown below, although for complicated topography the procedure may become rather involved. We consider in Fig. 4 the continued reflections of a single characteristic ~1 which has successive values ~2, ~3, • • •, ~. such that all the bottom variations are contained between ~1 and ~., and denotef(~), H(¢), A(~) in the various ranges byfl(~), HI(~), A 1(~), f2(~) H2(~), A2(¢) etc. Then, from the general equation (4.20), we may write: f,(~) -f2[Hl(~)] = AI(~), f2(~) -fa[H2(~)] = A2(~), f~- ~(~) - f , [ n , - a(¢)] = A._~(~),

~1 < ~ < ~2, 42 < ~ < ~3, ~.-1 < ¢ < ~,,

(4.35)

since H~(~) lies in the range ~i < Hi(l) < ~i+ 1 for all relevant i. We then have:

fa(~) = f3{H2[HI(¢)]} + AI(~) + Az[HI(~)],

(4.36)

which relates f l a n d f 3 if n = 3. In general, equations (4.35) yield fx(~) = f , ( H , _ , ( H , _ 2 . . . H2[HI(~)]}) + AI(~) + A2[HI(~)]

(4.37) + "'" + A,-I{H,-z[... H1(¢)...l}, ¢1 < ~ <~2.

This may be written: f1(¢) =f,[J/g(¢)] + s¢(~), ~1 < ¢ <~2.

(4.38)

=~/YL, #-(~) =fx(¢), Jt'(~) = ~(~)/~R, S¢~(~) = Sd(~),

(4.39)

Defining and applying (4.26) to both fl(~) and f,(~) as appropriate and invoking (4.38) and (4.39) we obtain _1 (~ d fsin re/2 [~¢/(s) - J[(¢)]~ ~-(~) = ff(~) + 2hi ~¢, ~-(s) ~ss log ~- ~ ~-~ ~ ~ -j ds, (4.40) where f¢(~) has the same form as equation (4.32) with .//(~), ~¢~(~) replacing M(~), A,(~). Equation (4.40) is essentially the same equation as (4.31), with the effect of successive reflections of the waves from bottom topography contained in the function M(~). The above suggests that the geometry of the bottom topography vis-d~-vis the characteristics is crucial in determining the internal waves generated, and this is borne out in the following sections. Equations (4.31) and (4.40) may be solved by the well-known iteration procedure /71(~) = G(~), F2(~) 2nd approx., etc. =

- 1st approximation,

1 f° d{sinrc/2[M(s)-M(~)]~ds, G(~) + ~ _2Fl(s) dss log sin ~z/2(s - 0 J

(4.41) (4.42)

192

P.G. BAINKS

This has been done numerically for various examples in the following sections, and in every case convergence to within 1% was obtained within three iterations (F4)./:2 was a good approximation in many cases. The total velocity field may be written u = u l + u2 + u 3 ,

(4.43)

where u 1 is the barotropic motion given by equations (3.12) and (3.14), u 2 is given by the stream function W2 [equation (4.5)], and u 3 by W3 [(4.1 l) and (4.15)]. If o92]N2 .~. l, which is the case for the oceanic tides, the motions u~, u2 very nearly cancel, so that the total motion is effectively given by u3. $/3 :

~Z {Co1/2(1 "1- cl z)[f(~) -'[-g(?])]},

= - Co- 1/2(1 + cl z)- l[f'(O + g'0/)] w3 = Col/2(1 +

-

Col/2el[f(~) "[-if(n)],

clz)[f'(~) -- g'(r/)],

(4.44) (4.45)

where dashes denote derivatives. On, say, the left-hand side of the topography, these become Q(1 - ClZ/Col/2hL) 1 U3 [1 - o92/N2(-hL)](1 + clz ) Co1/2(1 + clz) [f p'(O + gp'(r/)] (4.46)

-- C01/2Cl[fp(~ ) "q- gp(t/)], w3 = Col/Z(1 +

clz)[fp'(O - gp'(r/)].

(4.47)

The first term of (4.46) represents the horizontal mass flux, modified in the vertical by the variable stratification. However, if c 1 = 0, this term cancels exactly with u 2 . In the following sections, the velocity field will be presented in terms of the function fp(0, from which the complete picture may easily be seen. With equations (4.46) and (4.47) the total internal wave energy flux across a vertical section, 0 Eflux = f 1/2(pu* + p*u)dz, (4.48) --h where the asterisk denotes complex conjugate, when averaged over x and t becomes, on the L.H.S., Eflux---- Po (092 ~ f 2 )

fo2~,Lfpr(~)fp,(~)d~,_

(4.49)

where the suffices r and i denote real and imaginary parts. The vertically integrated energy density is given by o g 2 Eo=4f_h[u.u'+(~oN ) pp*Jdz, (4.50) and when averaged over x and t this becomes

,o( so--hi c(z)2dz/hL );

E D = -~ 1 +

[fp,'(~)2 +fp{(~)2]d~ 271

The generation of internal tides by flat-bump topography

+ (1 +f2/oj2)CoCl2 + P4 - - (1 + f2/~o2) f

hL Po ~ 0

f-2r, 0

[fp,(~)2 + fpf(~)2]d ~

193 (4.51)

"dK" 2

~7~ ~ozl

dz,

~ hL

where 1 '2 r z d z

K(z) =

~ _1 -~ I _f2/•2j

-

c(_hL)I/2TL ~1 + 1 --f2/c02]"

For typical oceanic conditions, the first of the three terms in equation (4.51) is the largest. Analogous expressions may be written down for the R.H.S. 5. I N T E R N A L TIDES FROM A SYMMETRIC COSINE R I D G E

We first consider a ridge in an ocean (or laboratory channel) of unit depth where the equation for the bottom surface is z=-l+d z= -1,

l+cos

,

Ixl

(5.1)

a/c,

and N 2 is constant, with c = 0.15 (see Fig. 3). Equation (4.32) was solved numerically for various values of d and a by the iteration procedure with 81 grid points evenly spaced in ~ from ~ = - 2 to ~ = 0. The adequacy of this number of grid-points was checked by repeating some calculations with 41 grid points, and these yielded essentially the same results. M(O was obtained from equation (5.1), and Az, AR were taken to be unity with As = 0. Centre differencing was used for the derivatives. Care was required in specifying the topographic function r/= - H ( ~ ) near the points corresponding to x = +_a/c, as any spurious jumps or discontinuities produced comparatively large oscillations in f(~) and its derivatives (see Appendix). Figure 5 shows velocity profiles f,'(~) (in phase with the barotropic motion) and f{(~) (out of phase) for two ridges, both of width 10 units but with heights 0.0955 and 0.286. For ridges of the shape given by (5.1) and very small height& the profile f,'(¢) is very nearly sinusoidal [with its Hilbert transform f~'(~) antisymmetric], and the distortion of the waveform with increasing height is evident in Fig. 5. As the slope of the ridge approaches that of the characteristics, the energy density becomes concentrated in a narrow region radiating up-slope on each side of the ridge. Figure 6 shows the energy flux away from the ridge (on each side) for three different bump widths as a function of height. Clearly, the height is a dominant parameter for energy conversion (surface --, internal wave) for configurations of this kind. However, topography with large height does not necessarily imply a large energy conversion, as is shown in the next section. 6. CONTINENTAL SLOPE TOPOGRAPHY--CONSTANT N As a model for a continental slope we take a linear variation of depth from the

194

P . G . BAINES

,,

-

'(~)

0"05 f~

(~)

0"04

0.03

O'OZ 0"01 -I'0

-1.5 0

-

f/y<_

0"01

-0.02

-0.05

-0-04

(a)

0 "35

/'P(~}

-.

,,-.

0.3

f~(~) 0-25

0-2

0.15

~1(¢)

0"1

0'05 -1"5 0

f -I-0 .___._.._d_.

f/)'L

- 0.05 -0"1

- 0"15

(b) Fig. 5. Velocity profiles for the ~:-motion for two cosine ridges in an ocean of unit depth with c = 0-15, (a) width = 10, height = 0.096; (b) width = 10, height = 0.286. f / ( ~ ) is the velocity profile when the barotropic motion is at a maximum and from left to right, as shown in the inset, whilst ~" (~) is the profile ~-period later. The magnitudes are expressed as a fraction of the maximum barotropic velocity in constant depth.

The generation of internal tides by fiat-bump topography

195

I

10"4x 1"4

.-!

~-

,O-4kl.O

~

bJ

=

0

.

7

5

Io-ex 6

IOV6x2

O: I

O

0"5

I

!

0-2

Oi Height

0'3

(~'4

O"5

of r i d g e

Fig. 6. The time averaged energy flux on either side of a ' single hop' cosine ridge as a function of its height,with c = 0.15. a is the ratio of the total width of the ridge to the horizontal characteristic period. The curves terminate when the maximum slope approaches c, and the units of energy flux are ¼poQ2/sec. shallow to the deep level--

z= --hL, z = -h z-

--oo
sx,

"]

O
z = --1,

(6.1)

where the depth he on the deep side is taken as the unit of length. Single, double, triple and quadruple hop conditions wcrc considered, as discussed at the end of Section 4, and the appropriate function ~(~) was obtained in accordance with equations (4.37) and (4.38). The procedure is quite simple, and in each case the function X~'(~)is made up of 3 sections,all linear in ~ except for smoothing at their junctions. The conditions determining the number of characteristichops required to span the slope are shown in Fig. 7, and depend only on the ratios sic and hL/h R .

Steep slopes 1.0

,

,

,

0"8 i ~

,

,

,

,

~

,

SingleHop

% 0.6

~

0.4

0.2

0"0 Fig. 7.

t

0:2

i

04"

06" 08"

HL/H.

I'0

The criteria for single hop, double hop, etc. continental slopes, hL/hR is the ratio of the depths on each side, sic is the ratio of the topographic and characteristic slopes.

196

P.G. B,Jda~_s

In general, n hops are required if

s/c~ ~ [1 - s/e~n_ , 1 + sic] < hL/hg < ~1 + sic] ' 1 -

(6.2)

and the number of hops needed increases indefinitely if either hL/h ~ --) 0 or s/c --) O. For the co-ordinate system as given in equation (6.1), ~ for equation (4.40) ranges from - 2 to 0, and 81 grid points evenly spaced in ~ were again used for the numerical solution, s was given the value 0.1, with o~ as semi-diurnal frequency 1.45 x 10-4 rad/sec, N--- 1.26 x 10 -3 rad/sec and fcorresponding to a latitude of 30 °. To avoid singularities in the solution due to the sharp corners (HURLEY, 1970), the topographic function o~t°(~)[and.A'(0] was smoothed over 5 grid points at each change in gradient. The details and effects of this process are sketched in the appendix, and although the solution was altered locally there was little difference between the total energy fluxes and densities for the smoothed and unsmoothed cases. In the function . g ( 0 [equation (4.39)] c, s, hR and hz only appear in the ratios s/c and hL/hg (=hL in this case). From equation (4.31) this must also apply to the function ~ ( 0 (except for multiplicative factors) so that the form of the solution ~-(0 also depends only on these ratios for topography given by 6.1 when c is constant. Equation (4.31) [or (4.40), as appropriate] was solved by the iteration method for large numbers of values of these ratios, and the corresponding velocity profiles and total energy fluxes and densities on each side of the topography were calculated therefrom. In every case, the velocity profiles on each side of the topography, taken as a function of either characteristic variable have the same structure (with differing proportions), namely, an oscillating rectangular wave-form in phase with the motion of the surface tide, together with its Hilbert transform displaced in time by ¼-period which is anti-symmetric about the rectangular wave and has peak velocities where the in-phase motion changes direction. Velocity profiles for two cases, both single hop, are shown in Fig. 8. This motion, together with the associated motion on the other set of characteristics, constitutes the only motion present in addition to the surface tide. On each side of the topography, the two profiles combine to give a phase propagation which is in the same direction as that of plane waves whose group velocities are directed away from the topography. Also, as s/c -) 1, the velocity profile on the shallow side becomes concentrated into an intense narrow region. This limit will be discussed further in Section 8. The energy flux and energy density on each side of the slope were calculated using equations (4.49), and (4.51), with 40 grid points in the vertical and Cl = 0. The energy flux on the left-hand (shallow) side as a function of sic and h)./h R is shown in Fig. 9 (where the base is Fig. 7) on a logarithmic scale. The corresponding values for the fight-hand side are very nearly the same and would not justify drawing a separate figure, although some differences are indicated in the cross-section for hz/hR = 0.05 illustrated in Fig. 10, where the scale is linear. The near-equality of these fluxes is to be expected on physical grounds and these are discussed in Section 8. A salient feature of Fig. 9 is that the energy flux vanishes whenever 'inequality' (6.2) becomes an equality so that a reflected characteristic exactly spans the slope. In these cases, no internal waves are generated, regardless of the height of the topography. The energy flux from surface tide to internal tide increases rapidly as s/c--) 1 - and hL/h R

The generation of internal tides by fiat-bump topography

197

decreases, provided the lines [1 - s/c~ n hdhR = ~1 + s/c] '

(6.3)

n an integer, are not approached. The energy densities on the L.H.S. and R.H.S. as functions of sic and hL/hR are illustrated in Figs. l l a and lib. The total energy density on the L.H.S. is greater than that on the R.H.S., in general, by a factor of order hR/h,.. It also vanishes when (6.3) is satisfied, and in principle oscillates with infinite amplitude between these lines as hlJh R --, O. In contrast, both the energy fluxes and the energy densities appear to tend to finite values as s/c ~ 1 for any given hL/hR. Figures 9, 10 and 11 were calculated with c = 0.1 for the semi-diurnal frequency at a latitude of 30 °. Because the computations only depend on the ratios sic and h~JhR, they may also be used to obtain Eflu~, Eo for any other value of c (and hence o~, N and f ) , for the same topography. From equations (4.32), (4.49) and (4.51) we have Enu~(O~, N,f, s/c, hL/hR) = p°o2 cN2 [Ef,~(s/c, hdhR)]

4 09 I_ "c-N-2/-~ Jc,lc' Po Q2 cN 2 Eflu~(s/c, hL/hR) c,lc 4

09

1"1

(6.4)

× 10 - 3

and ED(o~,f, N, S/C, hL/hR)

Po Q2 =

(1 + c 2)

4hL c(1 -

c02/N2) 2

[ED(s/c, hL/hR) ] k(1 + c2)/¢(1 - o)2/N2) 2j oaf,'

Po Q2 (1 + c 2) . Eo(s/c, hL/hR),,lo, 4h L c(1 - co2/N2) 2 12"9

(6.5)

where the suffix 'calc' implies the values used and calculated above and graphed in Figs. 9, 10, and 11. 7.

CONTINENTAL

SLOPE

TOPOGRAPHY--VARIABLE

/V

The topography represented by equations (6.1) has been investigated for a number of cases with hyperbolic characteristics (c 1 # 0). The integral equation now depends on four parameters (hL/hR, Co, Cl, S) rather than two, and only some sample results will be given here. More complete results for oceanic situations will be presented in a later paper. For a typical 'oceanic' characteristic with Co = 0.01, c t = -2"16 (c -- 0.1 at z = - 1 ) , hL--0.4 and s = 0.013 (single hop topography), the wave-form on the L.H.S. has the form shown in Fig. 12. Instead of being rectangular (as in the previous section), the 'in-phase' wave-form tends to be somewhat triangular (depending on the relative size of ct) or more accurately, trapezoidal. The corresponding ' o u t of phase' motion is r.o longer anti-symmetric and tends to have one broad peak and one sharp peak. The .riangular wave-form with its corresponding Hilbert transform is also characteristic of double hop cases. The results for energy flux and energy density have broad similarities with those of the previous section but the details are very different. Figure 13 shows the energy fluxes and energy densities for a continental slope of slope 0.013 as a function of hr.. There is a general and (in places) very rapid increase in all four quantities with

198

P.G. BAINES

decreasing hL, with the dip at hL/hR = 0"34 being due to a change from single to double hop geometry (there are an infinite number of such changes between htJhR = 0.1 and 0-0). The energy fluxes and densities are not zero when characteristics exactly span the slope because the generating function ff(~) does not vanish, owing to the nonuniformity of the stratification. There is also a general increase in energy fluxes and

-1'5

~ / ~'t.

-0.1

1 0"15

-O'Z

(a)

0.25

O.z 0,15 0"1

0'05

0'0

-0"05

-O'i -0.15

-0.2

-0'25

(b)

The generation o f internal tides by fiat-bump topography

0.55

199

(~'l

0.5

o.4~ 0.4 0-35

0.5 0.2_5 0'2 0"15 0'1

0"05

p;ol i

0"0

-05,

-0-05' -0'1

-0.15 -0.2 - 0"25 -0"3 -0.55 -0.4

fc)

0.25

I

9i' I',;]

0'2 r

g/(~)

0.15 -

0.1 0'05 )'5

-I'O/

0.0

-1'5

vfrR

-0.05 -0,1 -0'15 ' -0.2 -0'25

-0.3 i .~0.35

(d) Fig. 8. Characteristic velocity profiles on each side o f two 'single h o p ' continental slopes with c == 0'1, hL/hR = 0"5, and the depth in deep water is taken as unity. F o r (a) and (b) s i c = 0"666 and the geometry is shown in the insertion (a). Similarly for (c) and (d), with s / c = 0.89.

200

P . G . BAINES E flux -3

-4

"6

0'00-f

0'2 0"3 0"4

0.5 0'6 0'7

0-9

0'8

I'0

hL/h R

Fig. 9. The energy flux, in units of ¼poQ~/sec, on the left-hand (shallow) side of a continental slope, as a function of the ratio of the depths hL/hR and the ratio of the topographic and characteristic slopes sic. N =is constant, with c = 0"1. The base of the diagram is Fig. 7. The corresponding figure for the total energy flux on the right-hand side would be almost identical on this logarithmic scale, and some differences on a linear scale are shown in Fig. 10. 20x I0 -4

,

17.0 xlO "-4"

15

i

hL/hR

|

I

i

I

I

I

I

"kd,j

07

0'8

0'9

= 0"05 LHS RHS

~

xlO"4

I

E flux

12.5x10 "4

IOxlO "4

7'5 x iO -4

5xlO - 4

2"15 X I0 - ~

0 Fig. 10.

I

I

t

i

0'1

0"2

0"3

0'4

~

l

, ~ , ~ f l

0"5

%

0"6

A cross-section of Fig. 9 for hL/hR = 0"05. Units here are poQ2/sec.

1,0

The generation

of internal tides by flat-bump topography

201

hL/hR

O-3 Fig. 11. The total energy densities on (a) the left-hand side and (b) the right-hand side of a continental slope for the same conditions as shown for Figs. 9 and 10, in units of #p,,Qz/hn.

202

P . G . BAINES

w xlO 2 In p h a s e 0.5 0'3 0.2 0,1

F~

0"0 --0"1 --0"2

-0.5 -1"0

Out of phose ~ j V Fig. 12. Typical velocity profiles for hyperbolic characteristics. These are horizontal profiles of the vertical velocity at z = hL as a function o f one characteristic variable [equation (4.47)] for single-hop geometry with co -----0.01, cl = -- 2.16, hL/hl~ = 0.4 and s = 0"013. The corresponding amplitudes at any other depth may be obtained by multiplying by the depth factor (1 + clz)/(1 + cthL).

I

10- 4

i

i

i

t

[

"

, R.H.S

...... ~

I'0

/

/ /

L.H.S

/

E flux

/ /

/ /

/

E O.

/

/

/

/

/ /

lO-O

IO-O

iO-Z

//

IO-Z

I

1.0

0'9

.... f

O.B

I

I

I

I

I

0.7"

0.6

0-5

0"4

0.5

,

I ,

02

1

O.i

~0 "3

O0

hL/hR

Fig. 13. Energy fluxes in units of poQ2 (left-hand scale) and energy densities in units of poQ2/ha (right-hand scale) as a function ofh~,/h~ for a continental slope of slope s = 0"013.

The generation of internal tides by fiat-bump topography

203

densities with increasing slope, although the picture is again complicated by the same geometric transitions just mentioned. To give a typical numerical estimate of the rate of energy generation by a 'flat' continental slope, we take a = 1 m, d = 100 km so that taking Q = toad, Q _ 2 x 105 cm2/sec corresponding to a barotropic velocity of 1 cm/sec normal to the slope in deep water, if hR = 2 km. With h L = 200 m and a slope of 0-013, we obtain, from Fig. 13, Eflux = 1"85 × 10-4po Q2 LHS + RHS

= 6.4 x 106 erg/sec/cm of slope length. 8. D I S C U S S I O N

In order to comprehend the foregoing results on a physical basis we may represent the effect of the surface tide by a virtual body force in the following manner. From equations (3.14) and the last of (2.3) we may write: P

Q dpo co dz z (1/h)x sin cot + fi,

(8.1)

so that equations (2.3) may be written ~t + w~

~ai

= 0,

a"-t" + f x u / + 1PoV p +

V.n

0,

P--gt Po =

QN 2 zh'(x) o~

h2

sin o t i,

(8.2)

where p - fi is the density perturbation caused by the surface wave motion only. From this equation we see that the effect of the surface tide is that of a coherent body force varying with position over the region where the bottom slope is not zero, and 90 ° out of phase with ul and wl. The body force leads the vertical velocity w, over the topography by 90 ° and the displacement of the density surfaces by 180 °. An analogy may be drawn with the case of a mass attached to an infinite massive spring where the former is subjected to a longitudinal oscillating force. Whether or not the body force is an effective generator of internal waves will depend on its distribution in space, i.e. on the height, shape and width of the topography involved. For instance, one might expect that continental slopes whose width is approximately one-half characteristic period would be effective, whilst those of width one characteristic period would be comparatively ineffective. This is vividly borne out in Section 6 with the large variations of energy flux in Fig. 9. Symmetric bumps, however, should be most effective when their width is approximately one characteristic period. If we consider a small region of the generating volume in isolation, as shown in Fig. 14, we would expect the energy flux radiating away from the region to be approximately the same in each of the four characteristic directions. Since the topography considered here is 'flat-bump' topography only, approximately equal energy flux from this region on either side of the topography should be observed. Hence, we might expect that when the generating region is taken as a whole, the energy fluxes on each side would be very nearly equal, although the wave motion is different, especially for simple topography like linear continental slopes. However, the overall asymmetry

204

P . G . BAtr,ms

f/If

f f f J f

Fig. 14. Directions of propagation of wave energy from a small part of the source region. o f the generating region coupled with the constraints imposed by the radiation conditions m a y still cause some inequality. F o r example, m o d e 1 (with the highest g r o u p velocity), m a y be m o r e easily generated on one side o f the slope than the other. The differences between the energy fluxes on the L.H.S. and R.H.S. shown in Fig. 13 are attributed to this p h e n o m e n o n (the effects o f shallower depth and stronger stratification on the L.H.S. tend to cancel each other). We n o w consider some limiting cases for constant N. Regarding the limit sic --, 0 for continental slopes, the energy conversion (surface ~ internal waves) vanishes as s ---, 0, but if c --, oo for given s, the factors AL, AR ~ oo (not represented in Figs. 9, 10, 11), as given by equation (4.22). The combined effect is difficult to determine f r o m the above formalism, since the n u m b e r o f hops increases without limit as the characteristics become m o r e vertical. F o r application to the oceanic tides this question is somewhat academic, since, in general, og/N < 1/4 in the ocean (the m a x i m u m Brunt-V~iisiilit period is about 3 hours except possibly in the deep trenches where sufficiently accurate observations are not available). The situation 09 ~, N may, however, be reached by some tsunami waves, and one would expect the resulting m o t i o n to consist o f large b u o y a n c y oscillations over the topography f r o m which little energy escapes. The limit sic ~ 1 is more interesting, since the equality o f these two slopes seems to be a c o m m o n occurrence for the semi-diurnal tides in the ocean. The m o s t prominent feature for sic < 1 is the sharp peaked velocity profile on the shallow side o f the slope.

Acknowledgements This work was carried out in the Department of Earth and Planetary Sciences, Massachusetts Institute of Technology, and at the C.S.I.R.O. Division of Atmospheric Physics, Aspendale. The work was supported at MIT by grant N00014-67-A-0204-0048 from the Office of Naval Research and at Aspendale by a Queen's Fellowship in Marine Science from the Commonwealth of Australia. APPENDIX It is well known that when internal waves reflect from surfaces with sharp comers, whether concave or convex, analytical singularities in the wave field arise owing to the accumulation of energy associated with small-scale waves with very low group velocity (ROBINSON,1970; HURL~Y,1970; WUNSCrl,1969). In order to avoid the effects of the corresponding singularities in the present formulation, the sharp comers in the topography were smoothed by a simple integral process, over five grid points (with eighty-one for the total range). The effect of this smoothing on velocity profiles and energy flux was investigated, and the change in the latter was found to be small, being typically of the order of 1 ~o of the total energy flux. However, for the unsmoothed topography the velocity profiles showed large oscillations at the points corresponding to the sharp comers, and as smoothing was introduced these oscillations decreased considerably in amplitude. In most cases they did not disappear completely, and they are evident in Figs. 5 and 8.

The generation of internal tides by fiat-bump topography

205

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