MAR 555 Lecture 20: Coastal Tides

Tides caused by the resulting gravitational and centrifugal forces is called“equilibrium tide”. They can be expressed using the harmonic functions as

�

! e (semi ) = " i

i=1

Nsemi

# Aei

cos2$cos(%

ei

t + 2&)

�

! e (diurnal ) = " i

i =1

Ndiurnal

# ˆ A ei cos 2$cos( ˆ % ei t + &)

�

Ae i

and ˆ A e i the amplitudes of the ith semidiurnal and diurnal equilibrium tidal elevations

�

!ei

and ˆ ! ei the frequencies of the ith semidiurnal and diurnal equilibrium tidal elevations

�

! and " latitude and longitude

�

Nsemi

and Ndiurnal

the total number of semidiurnal and diurnal tidal constituents

�

!i the parameters that are different for different tidal constituents

In the Gulf of Maine, the semi-diurnal equilibrium tidal amplitude ~0.38 m

M2 co-tidal chartSolid line: amplitude (m)Dashed line: phase (oG)

Characteristics:

1. Irregular coastlines, particularly in the Canadian Archipelago;2. Near-resonance tidal oscillation in bays;3. Steep bottom slope and deep ridges between the basins

Gree

nland

Cana

dian A

rchipe

lago

Hudson Bay/Strait

White Sea

Iceland

2-4 m

0.2-1.0 m 1-2 m

>1 m

The tides observed in the coastal region consists of twoparts: equilibrium tides and tidal waves propagating fromthe open ocean.

• Strong near the coast and weak in the open ocean;• Vary over topography in the shallow water and estuaries• Resonance when the tidal frequency match the local geometric frequency

The Tidal Bore

u1h1

u2h2

�

h1u1

= h2u2

�

u1 = (h2

h1

)u2 >> u2

�

Cb = (1+h1 ! h2

h1 + h2) g

h1 + h2

2The phase speed of tidal bore

~1 m in 10 second!

Tidal Resonance

Consider a closed narrow tank:

LGiven any kind of fluctuation, the body of water can oscillate. The simplestmode is the one shown above:

at A and E: up and down (anti-nodes)at C: forth and back with no vertical motion (node)at B and D: up-down and also forth-back

In this special case, the length of the tank L = 2λ, λ is the wave length.

h

A ECB D

If L = λ, one will find two nodes in the tank.

L=λ

h

Node Node

This problem can happen when two sets of wave traveling in opposite directionsor a progressive wave propagates towards the wall and the reflect to produce a“standing wave”.

The time for a wave to travel fore and back between two ends is

�

2L /C C: the wave propagation speed. which equals to nT: n is a positive integer, T is the period of oscillation;

�

2L /C = nT ! T =2L

nC=

2L

n ghMerian’s period

For the first case, L=2λ and n = 1

�

T =2L

gh

Let us consider the semi-enclosed bay in which the water flow induring the flood tide and flow out during the ebb tide. The length ofthe bay Lb= L/2

�

T =4Lb

gh: the natural period of the bay

If the period of the progressive wave entering thisbay is equal to this period, the oscillation willbecome “resonant”

Lb

h

Node

Lb

h

Lb

h

Slightly < Lb < < Lb

> 2 m with a maximum tidal range of ~ 8 mThe lowest natural surfacegravity wave mode in theGulf of Maine (GOM) andBay of Fundy (BF) region is

12.8 hours

The M2 tidal wave (12.42hours) enters the GOM/BFis near the resonanceperiod:

High tidal elevation !

Garrett (1972)

Hudson Bay (Hudson Strait)

The White Sea

English Channel/Strait ofDover

Denmark Strait

Internal TidesCauses:

When a surface tidal wave propagates onto the slope, the interaction betweenthe tidal currents and bottom topography can lead to the vertical oscillation ofpycnoclines and hence produce internal waves with the tidal period. Such atidal-induced internal wave is called the “internal tide”.

Surface tidal wave

ρ1ρ2ρ3ρ4

ρ5

ρ6

ρ7

• Large amplitudes but slow phase speed • Intensifies near the bottom and decreases upward

20

40

60

80

100

Dep

th (m

)

0 24 48 72Time (hours)

5

0

Surface Tide (m)

11o10o9.5o

9o

8o

7o

6o

Internal tides in Bute Inlet, British Columbia, Canada in July 1953.Note that the surface tide is magnified by a factor of 4.

Stellwagan Bank off Mass Bay

Download from Rich Signell’s personal website

Strong tidal forcing case:

Horizontal length scale is meter, vertical length scale is enlarged 100 times

Mean tidal forcing case:

Horizontal length scale is meter, vertical length scale is enlarged 100 times

Weak tidal forcing case:

Horizontal length scale is meter, vertical length scale is enlarged 100 times

Internal Waves over Stellwagan Bank