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 N semi! e(semi) = #" i A ei cos 2 $cos(% ei t + 2&) i =1 N diurnal! e(diurnal ) = #" ˆ i A ei cos 2$cos( % ˆ ei t + &) i =1 A e i and A ˆ 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 N semi and N diurnal 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
M 2 co-tidal chart Solid line: amplitude (m) Dashed line: phase ( o G)
Canadian Archipelago 0.2-1.0 m 1-2 m White Sea 2-4 m Greenland >1 m Iceland Hudson Bay/Strait 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
The tides observed in the coastal region consists of two parts: equilibrium tides and tidal waves propagating from the 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 u 1 h 1 u 2 h 2 h 1 u 1 = h 2 u 2 u 1 = ( h 2 h 1 )u 2 >> u 2 C b = (1 + h 1! h 2 ) g h 1 + h 2 h 1 + h 2 2 The phase speed of tidal bore ~1 m in 10 second!
Consider a closed narrow tank: Tidal Resonance A B C D E h L Given any kind of fluctuation, the body of water can oscillate. The simplest mode 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.
If L = λ, one will find two nodes in the tank. Node Node h L=λ This problem can happen when two sets of wave traveling in opposite directions or 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 gh Merian 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 in during the flood tide and flow out during the ebb tide. The length of the bay L b = L/2 h Node T = 4L b gh : the natural period of the bay L b If the period of the progressive wave entering this bay is equal to this period, the oscillation will become resonant
Slightly < L b < < L b h h L b L b
> 2 m with a maximum tidal range of ~ 8 m The lowest natural surface gravity wave mode in the Gulf of Maine (GOM) and Bay of Fundy (BF) region is 12.8 hours The M 2 tidal wave (12.42 hours) enters the GOM/BF is near the resonance period: High tidal elevation! Garrett (1972)
Hudson Bay (Hudson Strait) The White Sea English Channel/Strait of Dover Denmark Strait
Internal Tides Causes: When a surface tidal wave propagates onto the slope, the interaction between the tidal currents and bottom topography can lead to the vertical oscillation of pycnoclines and hence produce internal waves with the tidal period. Such a tidal-induced internal wave is called the internal tide. Surface tidal wave ρ 1 ρ 2 ρ 3 ρ 4 ρ 5 ρ 6 ρ 7 Large amplitudes but slow phase speed Intensifies near the bottom and decreases upward
Time (hours) 0 24 48 72 20 11 o 10 o 9.5 o Depth (m) 40 60 80 9 o 8 o 7 o 6 o 100 Surface Tide (m) 5 0 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