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Generation of the M4 tide in Cook Strait, New Zealand

Erzeugung der M4-Tide in der Cook-Straße von Neuseeland

Formation de l'onde de marée M4 dans le Détroit de Cook, Nouvelle-Zélande

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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.

Zusammenfassung

Um die Erzeugung der M4-Tide in der Cook-Straße von Neuseeland untersuchen zu können, werden analytische Modelle benutzt. Der größte Beitrag zu der M4-Gezeit kommt von der Beschleunigung des eingeengten M2-Stromes. Die M4-Tide erreicht etwa 7% der kleinen M2-Tide und ist daher für den Charakter der Mitschwingungsgezeit in den angrenzenden Marlborough-Sunden von Bedeutung. Im allgemeinen ist der M4-Strom klein gegen den großen M2-Strom.

Résumé

On utilise des modèles analytiques pour examiner la formation de l'onde M4 dans le Détroit de Cook, Nouvelle-Zélande. La plus importante contribution à l'amplitude de l'onde M4 provient de l'accélération de l'écoulement resserré de M2-l'amplitude de M4 atteint environ 7% de la petite amplitude de M2 et est, par conséquent, importante pour déterminer la nature de l'onde en résonnance dans les Marlborough Sounds adjacents. Le courant de l'onde M4 est généralement petit comparé à l'important courant de M2.

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Abbreviations

a 1, b1, d1 :

Fourier coefficients of the series expansion of the bottom stress

A/2:

The amplitude of the M2 tide advancing into Cook Strait from the southeast, atx=0 in the case with a variable cross-sectional area

A 1/2:

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

B/2:

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

B 1/2:

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

e, f, m, n, p, q :

The coefficients in the development of M4 tidal elevations and phases arising primarily from the M2 overlapping standing wave

g :

The acceleration of the earth's gravity

h :

The undisturbed water depth

i :

An integer

J i :

The amplitudes in the development ofu 5

k :

The M2 tide wave number

K i,Ψ i,γ i :

The amplitudes temporal phase and spatial phase respectively in the development of\(\zeta _s \)

r :

The quadratic frictional coefficient

t :

The time

u :

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

U 0 :

The time-averaged mean water speed

U 00 :

The mean flow associated withu 5

u 1 :

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

\(u_{11 } \sqrt {\frac{g}{h} (1 + \Phi (x} ))\) :

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

u 2 :

The water speed associated with the M4 tide

w :

The variable width of Cook Strait which is a function ofx

w 0, h0 :

The width, and depth respectively of Cook Strait atx=0

w L, hL :

The width, and depth respectively of Cook Strait atx=L

x :

The coordinate aligned through Cook Strait

x=0:

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

x=L :

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

\(Z_i = \sqrt {\frac{g}{h} } \frac{{A ab^2 \lambda }}{4}\) :

(i=3,4)

αi :

The wave numbers in the spatial development of\(\zeta _5 \) andu 5

\(\begin{gathered} \beta = \omega ^2 /\lambda ^2 gh \hfill \\ \beta i = 4\omega ^2 /a_i ^2 gh \hfill \\ \end{gathered} \) ζ:

The M2 tidal elevation in the constant width case

ζi :

The M2 tidal elevation in the case with variable width

\(\zeta _{11} (1 + \Phi (x))\) :

The M2 tidal elevation in the case with a variable width cross-sectional area

ζ2 :

The M4 tidal elevation

ζ22,u 22 :

The M4 tidal elevation and water speed respectively, arising primarily from the pattern of M2 overlapping standing wave ζ11,u 11

ζ3,u 3 :

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

ζ4,u 4 :

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

ζ5,u 5 :

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

θ:

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

λo :

The overall scale through Cook Strait

ϱ:

The density of seawater

τ:

The bottom stress

1+Φ(x):

The variation of\((\frac{{w (x)}}{{w_o }})^{\frac{1}{2}} (\frac{{h (x)}}{{h_o }})^{\frac{1}{4}} \)

\(1 + \Phi (x) = 1 + b \cos \lambda (x - \ell ); u_o = \sqrt {\frac{g}{h}} \) ψ:

The phase of the M4 tidal flow

ω:

Angular frequency of the M2 tide

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Author notes

  1. Ronald Allan Heath

    Present address: Department of Scientific and Industrial Research, New Zealand Oceanographic Institute, P.O. Box 12-346, Wellington North, New Zealand

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    1. Ronald Allan Heath
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    Heath, R.A. Generation of the M4 tide in Cook Strait, New Zealand. Deutsche Hydrographische Zeitschrift 35, 261–270 (1982). https://doi.org/10.1007/BF02317437

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    • DOI: https://doi.org/10.1007/BF02317437

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