generation of the m4 tide in cook strait, new zealand

Generation of the M4 Tide in Cook Strait, New Zealand

Ronald Allan H e a t h U D C 551.466.72 : 551.465 A1 ; Southwest Pacific, Marsden Square 462

Summary Analytical models are used to examine the generation of the M4 tide in Cook Strait,

New Zealand. The largest contribution to the M4 tidal elevation, comes from the acceleration of the constricted M2 flow - the M4 elevation reaches about 7 % of the small M2 elevation and is therefore of importance in determining the nature of the sympathetic tide in the adjacent Marlborough Sounds. The M4 tidal flow is generally small compared to the large M2 flow.

Erzeugung der M4-Tide in der Cook-Strafle yon Neuseeland (Zusammenfassung)

Um die Erzeugung der M4-Tide in der Cook-StraBe von Neuseeland untersuchen zu k6nnen, werden analytische Modelle benutzt. Der gr6Bte Beitrag zu der M4-Ge- zeit kommt yon der Beschleunigung des eingeengten M2-Stromes. Die M4-Tide erreicht etwa 7% der kleinen M2-Tide und ist daher f/Jr den Charakter der Mit- schwingungsgezeit in den angrenzenden Marlborough-Sunden von Bedeutung. Im allgemeinen ist der M4-Strom klein gegen den grogen M2-Strom.

Formation de l'onde de mar~e M4 dans le D6troit de Cook, Nouvelle-Z~lande (R~sum~)

On utilise des mod61es analytiques pour examiner la formation de l 'onde M4 dans le D6troit de Cook, Nouvelle-Z61ande. La plus importante contribution l 'amplitude de l 'onde M4 provient de l'acc616ration de l '6coulement resserr6 de M2 - l 'amplitude de M4 atteint environ 7% de la petite amplitude de Mz et est, par cons6quent, importante pour d6terminer la nature de l 'onde en r6sonnance dans les Marlborough Sounds adjacents. Le courant de l 'onde M4 est g6n6ralement petit compar6 ~ l ' important courant de M2.

List of symbols

al, b i ,d ,

A 2

A1 2

B 2

e , f , m , n , p , q

Fourier coefficients of the series expansion of the bottom stress

The amplitude of the M2 tide advancing into Cook Strait from the south- east at x = 0 in the case with a variable cross-sectional area

The amplitude of the M2 tide advancing into Cook Strait from the south- east, in the constant width case

The amplitude of the M2 tide advancing into Cook Strait from the north- west at x = L in the case with a variable cross-sectional area

The amplitude of the M2 tide advancing into Cook Strait from the north- west, in the constant width case

The coefficients in the development of M4 tidal elevations and phases arising primarily from the Mz overlapping standing wave

The acceleration of the earth's gravity

262 Dt. hydrogr. Z. 35, 1982. H. 6. H e a t h, Generation of the M4 Tide

h

.i

Ji

k

Ki, ~i, 7i

F

t

Lt

Uo

blO0

Ul

" i1 (1 q- qr) (X))

122

W

W0, h0

WL, hL

X

x = 0

x = L ~ # A a b 2 2

ai

fi = (02/22 gh

fii = 4092/~i 2 gh

~,, (1 + ~(x))

~2

~22~ /'/22

~3, 1A3

~4 , hi4

~5 ~ tA5

0

The undisturbed water depth

An integer

The amplitudes in the development of u5

The M: tide wave number

The amplitudes temporal phase and spatial phase respectively in the development of ~s

The quadratic frictional coefficient

The time

The water speed associated with the M2 tide, in the constant width case, positive in the positive x direction

The time-averaged mean water speed

The mean flow associated with u5

The water speed associated with the M2 tide in the case with a variable cross-sectional area

The amplitude of the water speed associated with the M2 tide in the case with a variable cross-sectional area

The water speed associated with the M4 tide

The variable width of Cook Strait which is a function of x

The width, and depth respectively of Cook Strait at x = 0

The width, and depth respectively of Cook Strait at x = L

The coordinate aligned through Cook Strait

The position where the M2 tide advancing fiom the south-east is reflected

The position where the M2 tide advancing from the north-west is reflected

(i = 3, 4)

The wave numbers in the spatial development of ~s and u5

The M2 tidal elevation in the constant width case

The Mz tidal elevation in the case with variable width

The M: tidal elevation in the case with a variable width cross-sectional area

The M4 tidal elevation

The M4 tidal elevation and water speed respectively, arising primarily from the pattern of M2 overlapping standing wave ~u, Uu

The M4 tidal elevation and water speed, respectively arising primarily from the influence of the variation in cross-sectional area on the M2 tide

T h e M4 tidal elevation and water speed, respectively, arising primarily from the influence of the variable cross-sectional area on the field accelerations

The M4 tidal elevation and water speed, respectively, arising primarily from the influence of the variable cross-sectional area on continuity

The phase lag of the wave advancing from the north-west at x = L behind the wave advancing from the south-east at x = 0

Dt.hydrogr. Z. 35, 1982. H. 6. H e a t h, Generation of the M4 Tide 263

20 The overall scale through Cook Strait

The density of seawater

r The bottom stress 1 _!

1 + ~' (x) The variation of (w (x)~ 2 (h (X)X~ 4 \ w0 / \ h0 /

l + q ) ( x ) = l + b c o s 2 ( x - e ) ; uo=

f, The phase of the M4 tidal flow

co Angular frequency of the M2 tide

Introduction In New Zealand waters the largest tidal constituent is the lunar semi-diurnal tide M2. On the

New Zealand continental shelf the M2 tide exists as a predominantly trapped wave progressing anti-clockwise around the coast (B ye and H e a th [1975], H e ath [1981a]). Cook Strait, situated approximately midway along the New Zealand landmass, provides an opening to the anti- clockwise progressing M2 tide and tidal energy enters the narrows of Cook Strait from both the east and west coasts. Within the narrows of Cook Strait the phase of the M2 tide changes rapidly by 118 ~ over a distance of 28 km (equivalent to a change of 4.1 hours in the time of high tide) and the amplitude decreases from about 0.5 m at the extremities of the narrows to only 0.18 m mlfway through the narrows. The associated amplitude of the Me tidal flow is about 1.1 m s -~ .dth a phase of 191 ~ ( G i l m o u r [1960], H e a t h [1980a]).

With strong tidal flows (Fig. 1) and rapid spatial changes in the amplitude and phase of the tidal elevation in the narrows of Cook Strait, non-linear effects will be strong and we might expect significant over-tide generation. This paper considers the mode of generation of the M4 tide in Cook Strait.

The observed M4 tide in Cook Strait The observed M4 tidal elevations in Cook Strait have a nearly constant phase of about 200 ~

with the largest observed amplitude of 0.05 m in the narrows (Fig. 2). Taking flow data from the hydrographic chart (Hydrographic Branch [1960]) (Fig. 1) and assuming the asymmetry in the tidal flow results from the M4 and $4 flows being in phase with the M2 and $2 flows to the north and hence out of phase with the semi-diurnal flow to the south) gives estimates of the M2, $2, M4

md $4, tidal-flow amplitudes of 0.91, 0.30, 0.06 and 0.02 m s -1 respectively. There are obviously many limitations on these estimates but they do serve to give a feeling for the magnitude of the M4 flOW.

Although the M4 tidal elevation is small in the Cook Strait, it is important because the M2 tidal elevation is also small. The relatively large M4/M2 tidal elevation ratio leads to appreciable tidal elevation asymmetry. In turn this elevation asymmetry leads to considerable tidal elevation and flow asymmetry in the adjacent Marlborough Sound ( H e a t h [1981b]).

Generation of the M4 tide The distribution of the M2 tide in Cook Strait is reasonably well fitted by the analytical solu-

tion of two overlapping standing waves (He a th [1974, 1978]). The tidal elevation ~ and water s~eed u on a line through tl)e Strait are given by

= A1 cos cot sin kx + B1 cos (cot - 0) sin k (L - x) (1)

U = [-- A1 sin cot cos kx + Bj sin (cot - 0) cos k (L - x)] (2)

where the x axis is directed southwards through the Strait; o) the M2 tidal angular frequency,

264 Dt . hydrogr . Z .35 , 1982. H .6 . H e a t h , G e n e r a t i o n o f the M4 Tide

2.4 ~ kn 2.0

16

1.2

0.8

I 0.4

c~ 0.4

0.8

12

1"6

'2.0

2-4

0 ~ T Station C 2-4

2'0

1.6

1.2

0-8

04 ~ Time--

80~

350 o T Station B

, , ~ j T ime~ ,

_ 6 4 / 2 0"~-2 -4h/~6

Mean

170~

Fig. 1. Hourly surface current components directed through Cook Strait over a spring tidal cycle as given in Chart NZ 46 of the Hydrographic Branch [1960]. Station C, latitude 41 ~ 13.9' S, longitude 174 ~ 29.6' E;

Station B, 40 ~ 56.0' S, 174 ~ 25.0' E. Station positions are shown in the inset. Times are referred to high tide at Westport

X

1'0 210 30 40 50 6tO km |

| (P

| 6)

|

| G 8 |

Q

|

X ~ X X X | |

|

| Makara Oteranga Bay I II I I I

Plimmerton Tory Lucky Lyall Channel Bay Bay

| Phase M 4 - 2 M 2 x Phase M 4

7'O 22 x l0 -2

i 14 ~ - ~

10 >~ : ; _~

8 r~

4 |

0-06 .~ m E 0.04 ~

.o 0-02 N>

G _.e LLI

360~

270 ~ -=~

| 180 ~ x ~>

90 ~ _~ ud

| Cape Campbell

Fig. 2. Relative M4/M2 tidal elevational amplitude, M4 elevational ampli tude, Me elevational phase, and M4 - 2 N M2 elevational phase, with meridional distance through Cook Strait.

Tide gauge locations are shown in the insert of Fig. 1

Dt. hydrogr. Z. 35, 1982. H. 6. H e a t h , Generation Of the M4 Tide 265

co A1 k = ~ the wave number where h is taken as a mean depth for Cook Strait. ~ - is the amplitude

of the progressive wave advancing into Cook Strait f rom the south-east. This wave is reflected where the Strait widens out towards the nor th-west at x = 0 with an open mouth boundary con- dition whereby the elevation is zero and the water speed associated with the wave is at a maxi-

B1 mum. Similarly ~ - is the ampli tude of the progressive wave advancing into Cook Strait f rom the

north-west which is reflected where the Strait widens out towards the south-east (at x = L). The wave advancing from the north-west at x = L lags 0 ~ behind the wave advancing f rom the south- .';?st at x = 0.

The above solution to the momen tum and continuity equations is for a channel of constant depth and width. For a wave travelling in a channel of variable cross-section without reflection, the rate of propagat ion of energy is conserved. In Cook Strait the scale of the variation in depth and width is smaller than the wavelength of the waves incident from the east or west. We might therefore expect considerable reflection f rom all locations in the Strait. However the observed tidal distribution is consistent with reflection predominant ly from where Cook Strait widens out to the south-east and north-west. We can conveniently then assume that the rate of propagat ion of energy in the incident and reflected waves is conserved and the elevation ~ and flow u satisfy

1 _1 -- w 2 h 4 (3)

_1 _3 u -- w 2 h 4 (4)

vhere w is the width and h the depth (see e.g. L a m b [1953]). In seeking an analytical solution for the M4 tide we will therefore take the M2 tide to be represented by

r = (A cos cot sin kx + B cos (cot - 0) sin k (L - x)) (1 + �9 (x))

= ~I1 (1 @ ~ (X))

where the variation in wave ampli tude due to the variable cross-sectional area is given by 1 _1 (w

\ w0 / \ h0 / = A ( I + ( P ( x ) )

1 I

B (w(x)l 2(h(x)~ 4 = B ( I + ~ ( x ) ) \ WL / \ hL /

Now

ul = ( - A sin cot cos kx + B sin (cot - O) cos k (L - x)) ~ / ~ ( 1 + q~ (x))

= ul~ (1 + q~ (x ) )

Production and dissipation of M4 tidal energy also arises from friction on the combined Ma and M 4 tides. The quadratic frictional bot tom stress (r) may be expanded into a Four ier Series

T = o r [ No q- Ul q- IA2 I (Uo -[- u1 -1- u2) ~ 0 r (al cos o t q- bl cos (2cot - ~O) + d l u0)

where a,, bl, dl, are the Fourier coefficients which themselves depend on the temporal integration limits in the Four ier expansion and therefore on the water velocity; 0 the water density, r the ;)ottom drag coefficient, arid u0 the mean water speed. The M4 frictional product ion term in the

, r a l ] g l ] . rbi momentum equat ion is then given Dy ~ anu the dissipation term by ~ - ( H e a t h [1980b]).

bl 2h ( 4 lUl l lU2[ al--~ u-~-3) we see that only in shallow From the ratio of these terms, ~ [TT bl = z~

water will frictional production exceed dissipation (using estimates of] ~11 = 0.2 m, [u~ I = 1 m s 1,

266 Dt. hydrogr. Z.35, 1982. H. 6. H e a t h, Generation of the M4 Tide

l u21 = 0.05 m s -1, h = 100 m indicates that M4 dissipation exceeds M4 frictional production by a factor of 67). We will therefore neglect friction in the present development with M4 production taking place via other non-linear mechanisms and M4 energy loss via radiation.

The momentum and continuity equations connecting the M2 and M4 tides are

3r b/l . ~/g2 3~2 ~x +hG-x +--g/=o (5)

3U 2 3bt2 8~- + ul ~ x + g = 0 (6)

where the M2 tidal flow speed and elevation are indicated by subscript 1 and the M4 tide by subscript 2. The influence of the variation of the cross-sectional area on the generated M4 tide has been assumed small and therefore neglected. The M4 tide will be expressed as a component (r u22) arising primarily from the pattern of the M4 overlapping standing wave (~11, u11) and a component (r u3) arising from the variation in cross-sectional area, i.e.

r = r (1 + ~ (X)) 2 q- r

/A 2 = /A22 (1 "Jr- (/)(X)) 2 Jr /23

Equations (5), (6) give

~ F3r UI 3 ~ ) g ~ (lq-~D(X)) 2 V 77-~-7-'-X q- r q- h (1 q-e(x))23u228x

3 ~ / / ~ 3/A3 2 3r 3r q- U22 ~X (h , , , (1 + ~ (x)) 2) + h -~x + (1 + q~ (x)) ~ 7 - + ~ - = 0

~hg~ ~ " ~ F 3u223~22 i.e. ~ x ( r V h 8x + - & - = 0 (7)

r ( ( 1 + ~ ) 2 g + u 2 2 ~ x ( h ( l + ~ ( x ) ) 2 ) + h ~ - x + = 0 (8)

3U22 3/X3 g~hg~ ~u11 u21 3 V ~ - and (1 + q ~ ( x ) ) ~ - + ~ - + ( (1 +qS(x)))2ui~ 8~--+~-~-x( ( l+q~ (x))) 2

8r 8 8~2 + g (1 + q~ (x))- --57- + g r ~X (1 -}- (i~ (X)) 2 Jr g~-x = 0

-~ F3/A22 -}- g 3/211 ~r /All g = 0 (9) i.e. V h 8t ~ - x + ~ 7 -

3b/3 /Xll 2 3 ,,It /~-~ 3 3r - - g h (1 + �9 g r ~xx g ~-x ~t q- ~ - -~X ( (X)))2 + (i + q~ (x)) 2 + = 0 (10)

M4 t ide a r i s ing p r i m a r i l y f r o m the spa t i a l v a r i a t i o n of the M2 o v e r l a p p i n g s t a n d i n g waves

We look first for solutions r u22 to equations (7), (9). The non-linear field acceleration 8Ul

term ul ~-x acts as a source of transfer of energy from the M2 to the M4 tide. We therefore

expect the M4 tide to grow in space.

Dt.hydrogr. Z. 35, 1982. H. 6. H e a t h, Generation of the M4 Tide 267

A solution exists in the form

r = r + ~2 (1 + �9 (x)) ~

= (1 + qb (x)) (A cos cot sin k x + B cos (cot 0) sin k (L - x))

3 m 3p + x (1 + ~ (x)) 2 [cos 2cot ( - ~ --g cos 2 k x - ~ g sin 2kx )

3 f 3e + sin 2cot ( - ~ g cos 2 k x - -~g sin 2kx)]

n q ( 1 + �9 (x)) 2 [ ~ cos 2 k x - ~gkgk sin 2/~x1

/A = /A1 -{- U22 (1 "1- I~ (X) ) 2

= V g ( 1 g + ~ (x)) ( - a sin cot cos k l x + B sin (cot - O) cos k (L - x))

3 e 3 f + x (1 + �9 (x)) 2 [cos 2cot ( - 2~hgh cos 2 k x + ~ sin 2kx )

3 p 3 rn + sin 2cot (2 ~hgh cos 2 k x - 2 ~ h g h sin 2kx)]

(1 + 4~ (x)) 2 [ ~ cos 2 k x + sin 2kx]

(11)

(12)

B 2 u 2 k A B where e = -~- u~ k sin 20 cos 2 k L 2 sin 0 cos k~L

B 2 u 2 k A B f = - - - u u ~ k s i n 2 0 s i n 2 k L - - 2 s i n 0 s i n k L

1 2 1 B 2 u ~ k p = - ~ u o k A B cos k L cos 0 + ~ u 2 k A 2 + ~ c o s 20 cos2kL

1 B 2 m = ~ u~ k A B sin k L cos 0 - ~ - u 2 k cos 20 sin 2 k L

u~ k u 2 k B 2 u~ k A B A 2 _ _ _ n - - 4 4 cos2kL + ~ c o s O c o s k L

u~ k B 2 2 k A B q = ~ sin 2 k L - uo T cos 0 sin k L

with u0 = V g

Fitting the solution to the observed M2 elevation at Cape Campell A sin k L = 0.64 m and Titahi Bay B sin k L - 0.37 m (with 0 = 125 ~ L = 65 000 m, h = 175 m) gives reasonable agreement with the distribution of the M2 tide in Cook Strait (neglecting any two-dimensional effects result- ing from the earth's rotation, see H e a t h [1978]). However, the M4 elevations given by the so- ~ ' ion are an order of maghitude smaller than those observed.

We conclude that the significant M4 tidal elevation observed in Cook Strait does not result primarily from the pattern of flow and elevation associated with the two overlapping standing waves.

268 Dt. hydrogr. Z.35, 1982. H. 6. H e a t h , Generation of the M4 Tide

M4 t i d e a r i s i n g p r i m a r i l y f r o m t h e v a r i a b l e c r o s s - s e c t i o n a l a r e a s The scale of the solution to equations (8) and (10) is associated with that of the local bathy-

metry rather than that of a free wave and thus the solution need not grow in space. The solution to equations (7) and (9) indicated that u22, ~22 are small and hence will be neglected in equa t ions (8) and (10). We split ~3, u3 into two parts.

~3 = ~4 q" ~5 U3 = /24 q- b/5

with ~4, u4 generated by the influence of {he variable cross-sectional area on the field accelerations, being solutions to

ul~ 3 Du4 2 ax (! +4>(x) ) 2 = g ~ V ; - = 0 (13)

au4+ ~;4 h Sx ~ - = 0 (14)

and ~5, u~ generated by the influence of the variable cross-sectional area on the continuity, being

~11 /Xll ~XX (1 + (/))2 --t- h ~ - x -{- ~ - = 0

Du5 o 3~5 = 0 -g +o ~- 2

solutions to

(15)

(16)

There is little spatial variation in the M2 flow, u~i, through Cook Strait and we therefore set

l l l i = a COS (co/ -- ~ ) (17)

_• _2 3 ~ The main variation in the functions, h 4 w 2 and h 4 w 2 arises from the variat ion in the width

of Cook Strait. We therefore neglect the spatial variation of in equat ions (13), (15) and

represent the variation of 1 + q~(x) by 1 + b cos 2 (x - F) (see H e a t h [1978], Fig. 3). The solution to equations (13), (14) is then

~4 = E1 COS 2 (cot -- ~ ) c o s ~. (x - ~')

+ E2 cos 2 (cot - ~p) cos 22 (x - co)

u4 = F1 sin 2 (cot - W) sin 2 (x - ~e)

+ F2 sin 2 (cot - ~0) sin 22 (x = ( )

a 2 b a 2 b 2

where E1 - h (4fl 2 - 1)' E2 -

(,O 2 w i t h / ~ 2 _ •2 gh

2coa 2 b

4h (/3 2 - 1)' F1 - ~h 2 (4/~ 2 __ 1)'

a)a 2 b

/72 - 2h 2 4 (fi2 _ 1)

Values of the parameters appropriate to Cook Strait are

~o = 11 ~ (from the observations of G i l m o u r [1960],

L = ~- = 32.5 x 103 m, b = 0.7, ZL = ~ (from H e a t h [1978], fig. 3),

s -1 ( H e a t h [1980a]). a = l . l m

Dt. hydrogr. Z. 35, 1982. H. 6. H e a t h , Genera t ion of the M4 Tide 269

The solut ion to equa t ions (13), (14) is

~4 = cos 2 (cot - 0.19) { - 0 .073 cos 2 (x - ~) - 0.012,,cos 22 (x - t~)} (18)

u4 = sin 2 (cot - 0.19) { - 0 .0024 sin 2 (x - F) - 0 .00021 sin 22 (x - F) } (19)

The ampl i tude of the e leva t ion given by equa t ion (18) is of the same order as tha t obse rved and the e levat ion phase of 202 ~ (equa t ion 18) is cons is tent with the observa t ions (Fig. 2).

The M4 t idal flow associated with this solut ion is very small, a consequence of the small hor i - 7~ontal scale of the M4 t ide which is d e t e r m i n e d by the scale of the const r ic t ion of Cook Strai t -

( F1 f!~e rat io of the flow to e leva t iona l ampl i tude e.g. E~ ~ - is p ropor t iona l to the hor izon ta l

Aga in neglect ing the spat ial va r ia t ion in [ Ul~ [ t h rough Cook Strait , the solut ion to equa- t ions (15) and (16) is

8

~5 = ~ g i sin (2cot - ~0i) cos (aix - Yi) i = 1

4

U5 = UO0 q- E Ji c o s (2cot - ~0) sin (aix - 7i) i = 1

Zi :-~ere Ji - h),i (,82 _ 1) Ki = 2coJi/gai fil = 4co2/ai 2 gh

al = k - 2 71 = - ) - ~ ~Pi=~P f o r i = 1 ~ 4

"I~ F A a b 2 a2 = k + ,'~ Y2 : .~,~ Z i = - - V ~ ~ i = 1 , 2

ff a 3 = k - 2 2 y3 = - 2 2 ~ z i = V h 4 i = 3 , 4

a 4 = k + 22 ~4 = 22F

: : : : = - ; ~ - k 7 5 = - ) - ( - k L g t i = 0 + ~ 0 f o r i = 5 ~ 8

t-~ Bab2 a6 = 2 - k 76 = 2~ - k L ~i = - ~ i = 5, 6

fl~ Bab 2"~ a7 = - k - 22 77 = - k L - 2d, F ~i = 2 i = 7, 8

a8 = 22 - k 78 = - k L + 2 ) ~

a V r ' ~ { - A sin kx cos ~p - B sin k (L - x) cos 0P - 0)} u00 - 2h '

For values appropr ia t e to the b r o a d t r end of Cook Strait typical values of the coefficients are:

J1 = - 0 . 0 0 7 m s -~ Ks = 0.005 m

J 2 ~ 0.006 m s -1 K2 = 0 .004 m

with a t x = 0 u00 = 4.9 x 10 =6 m s -~

270 Dt. hydrogr. Z. 35, 1982. H f6. H e a t h , Generation of the M4 Tide

Both the elevational and flow amplitudes are smaller than those observed. However , in contrast

( Ji to the previous solution, the ratio of the flow to elevational amplitudes e.g. ~ i = 2o)J for)~ >> k

is inversely proport ional to the horizontal scale 2. This would suggest that the main M4 flow would be generated in shallow water (the elevational and flow amplitudes both depend inversely on the depth h), where the scale of the local variations of the function q) is significant smaller than

the overall trend of 20 = 130 km 2 -- used above, via the influence of the constriction of

Cook Strait on the contribution to the total depth of the tidal elevation.

Conclusion The present analysis indicates that the observed M4 tidal elevation is largely generated by

the non-l inear field accelerations associated with the constriction of the flow in Cook Strait. Only a small M4 flow is associated with this generat ional mechanism.

The main M4 flow probably arises from the non-linearity associated with the contribution to the total depth of the tidal elevation, in combination with the constriction of the flow - this contribution depends inversely on the horizontal scale of the constriction and is therefore most significant near small-scale features. Further detailed analysis of the M4 flow will have to await detailed current observations which would allow determinat ion of the phase of the M4 tidal flow.

The M4 flow is of minor significance compared to the large semi-diurnal tidal flow. How- ever, with the M2 tidal elevations small in Cook Strait, the M4 tidal elevation is significant. For example, the M4 tidal elevation in Cook Strait has a strong influence in determining the tidal asymmetry within the adjacent Marlborough Sounds ( H e a t h [1981b]).

Acknowledgements Thanks are expressed to Mrs. H. P. Newpor t for drawing the figures and Miss G. Marsden

for typing the text.

References Bye, J. A.T. and R. A. Hea th , 1975: The New Zea-

land semi-diurnal tide. J. mar. Res. 33,423 -442. G i lmour , A. E., 1960: Currents in Cook Strait,

New Zealand. N.Z.J. Geol. Geophys. 3, 410431. Hea th , R. A., 1974: The lunar semi-diurnal tide in

Cook Strait, New Zealand. Dt. hydrogr. Z. 27, 214-224.

Hea th , R. A., 1978: Semi-diurnal tides in Cook Strait, New Zealand. J. mar. Freshw. Res. 12, 87-97.

Hea th , R. A., 1980a: Current measurements derived from trajectories of Cook Strait Swim- mers. N.Z.J. mar. Freshw. Res. 14, 183-188.

H e a t h, R. A., 1980b: Phase relations between the

over- and fundamental-tides. Dt. hydrogr. Z. 33, 177-191.

Hea th , R. A., 1981a: Variations of the semi-diurnal tidal admittance near New Zealand. Deep-Sea Res. 28, 847-858.

Heath , R. A., 1981b: Tidal asymmetry on the New Zealand coast and its implications for the net transport of sediment. N.Z.J. Geol. Geophys. 24, 361-372.

Hydrographic Branch, New Zealand, 1960: Chart NZ 46: Wellington to Patea including Cook Strait. 1 : 200 000.

Lamb, H., 1953: Hydrodynamics. Cambridge: University Press. 738 p.

Eingegangen am 23. August 1982 Angenommen am 20. Januar 1983 Anschrift des Verfassers: Dr. R. A. Heath, Department of Scientific and industrial Research, New Zealand Oceanographic Institute, P.O. Box 12-346, Wellington North, New Zealand

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generation of the m4 tide in cook strait, new zealand

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