1. Shallow Water Waves 2: Tsunamis and Tides
MAST-602 Lecture Oct.-16, 2008 (Andreas Muenchow)
Knauss (1997):
p (tsunamis and seiches)
p tides
p Kelvin waves
Descriptions: Tsunamis, tides, bores
Tide Generating Force
Equilibrium tide
Co-oscillating basins
Kelvin and Poincare waved

2. Equilibrium Tide ht
gives ht (“bulge”) of water-covered earth, no accelerations
0 = pressure gradient + horizontal tide generating force
0 = g ∂ht/∂s + hTGF

3. Tidal Breaking:
Friction between the ocean’s bulge and solid earth drags the bulge in the direction of the earth’s rotation.
This frictional effect removes rotational kinetic energy from the earth, thus increasing the length of the day by about seconds in 100 years.
It also implies a net forward acceleration of the moon that moves it about 3.8 cm/year away from earth (lunar recession).
© Richard Pogge

4. Tidal Locking of the Moon
The early moon rotated much faster:
As earth does now, it rotated under its tidal bulge;
internal friction resulted, which slowed the moon's rotation.
the Moon's rotation slowed until it matched its orbital period around the earth (29 days), and the friction stopped.
The end result is that the moon became Tidally Locked in synchronous Rotation.Therefore the moon keeps the same face towards the earth. Its rotational and orbital periods are the same:
---> the moon is tidally locked to the earth.

5. Resonance: response of an oscillatory system
forced close to its natural frequency
Breaking wine glass ---> 1-min movie
Tacoma Bridge > 4-minutes movie
Forced string ---> Java applet

6. Resonance: response of an oscillatory system
forced close to its natural frequency
d damping parameter
(friction)
Response
Forcing Frequency/Natural Frequency

7. Equilibrium Tide ht
gives ht (“bulge”) of water-covered earth, no accelerations
0 = pressure gradient + horizontal tide generating force
0 = g ∂ht/∂s + hTGF
Unrealistic as water must “instantaneously” adjust to changing forcing
but known ht provides useful in the dynamics of tides

8. Dynamics of Tides ht is a known forcing function of (x,y,t):
acceleration + coriolis = pressure gradient
u/t - fv = - g (h-ht)/x x-momentum
v/t + fv = - g (h-ht)/y y-momentum
H(u/x + v/y) + h/t = 0 continuity
used p(x,y,z,t) = gr[h(x,y,t)-z]
from z-momentum p/z = - rg
to convert 1/r p/x ---> g h/x

9. These are Kelvin and Poincare Waves.
Ocean Basin responding to tidal forcing ht under the influence of the earth’s rotation (Coriolis);
Apparent standing wave rotating around the basins (Atlantic or Pacific Oceans);
These are Kelvin and Poincare Waves.

10. L H u/t = - g h/x Tidal Co-oscillation (without Coriolis):
H u/x + h/t = 0
Tidal Co-oscillation (without Coriolis):
Standing wave due to perfect reflection at wall
c=l/T=(gH)1/2 and L=l/4 (quarter wavelength resonator) L H
---> T=4L(gH)-1/2
Tomzcak web-site
deep ocean tide forced by
hTGF due to moon/sun
shelf tide forced by
small h(t) at seaward boundary

11. Currents
Sealevel
Time

13. L Example of quarter-wavelength resonator: Cook Inlet, Alaska h0~5m
tidal bore
forms: L
L ~ 290 km
H ~ 50 m
T=12.42 hours
c=(gh)1/2~22 m/s
l=c*T~12.42 hrs*22 m/s=990km

14. Dynamics of Tides ht is a known forcing function of (x,y,t):
acceleration + coriolis = pressure gradient
u/t - fv = - g (h-ht)/x x-momentum
v/t + fv = - g (h-ht)/y y-momentum
(uh)/x + (vh)/y + h/t = 0 continuity
used p(x,y,z,t) = gr[h(x,y,t)-z]
from z-momentum p/z = - rg
to convert 1/r p/x ---> g h/x

15. Kelvin Wave:
peculiar balance of acceleration, Coriolis, and p-grad
in the presence of a coast
∂u/∂t - fv = -g∂h∂x along-shore force balance
∂v/∂t + fu = -g∂h∂y across-shore force balance
COAST
∂(uh)/∂x + ∂(vh)/∂y = -∂h∂t mass balance
Assume v=0 everywhere
y,v
x,u

16. Kelvin Wave:
peculiar balance of acceleration, Coriolis, and p-grad
in the presence of a coast
∂u/∂t - fv = -g∂h∂x along-shore force balance
x,u
y,v
COAST
∂v/∂t + fu = -g∂h∂y across-shore force balance
∂(uh)/∂x + ∂(vh)/∂y = -∂h∂t mass balance
Assume v=0 everywhere

17. Kelvin wave: geostrophic across the shoreu
High
Low
divergence
convergence u
High
Low
divergence
convergence
COAST
COAST
Time t
Time t+dt
convergence
y,v
y,v
convergence
x,u
x,u

18. Wave Equation ∂2u/∂t2 = c2 ∂2u/∂x2
EQ-1 ∂(x-mom)/∂t: ∂2u/∂t2 = -g ∂(∂h/∂x)/∂t = -g ∂(∂h/∂t)/∂x
EQ-2 ∂(continuity)/∂x: H∂2u/∂x2 = -∂(∂h/∂t)/∂x
Insert EQ-2 into EQ-1: ∂2u/∂t2 = gH ∂2u/∂t2
Wave Equation
∂2u/∂t2 = c2 ∂2u/∂x2
Subject to the dispersion relation
w2 = k2 gH
or c2 = gH

19. Wave Equation ∂2u/∂t2 = c2 ∂2u/∂x2y
Try solutions
u = Y(y)*cos(kx-ct)
c/f is the lateral
decay scale
(Rossby radius)
to find that
Y(y) = A e-fy/c

20. Tidal co-oscillation with Coriolis (Taylor, 1922)
(u,v)
head
head

21. The progress of the tidal wave from the Atlantic Ocean into the North Sea is clearly demonstrated by the co-phase lines. The wave enters from the north and propagates along the British coast; it then proceeds around two amphidromic points along the Dutch, German and Danish coastline. Another wave enters from the south west, through the English Channel. In the Irish Sea the wave enters from the south.The influence of the Coriolis force is demonstrated by the co-range lines, which show large tidal range along the British coast and small tidal range along the German, Danish and Norwegian coast. The same effect (amplification on the right side of the wave) is seen in the English Channel, where the tidal range along the French coast is as high as 11†m compared with 3†m on the English coast, and in the Irish Sea, where 8†m on the English coast compare with 2†m on the Irish coast.
© 1996 M. Tomczak

22. Internal Kelvin Wave in a closed basin
layer-1
Layer-3
h layer-2
(u,v) layer-2
h layer-3
(u,v) layer-3
from Dr. Antenucci

23. Inertia Gravity (Poincare)Wave:
balance of acceleration, Coriolis, and p-grad
∂u/∂t - fv = -g∂h∂x along-shore force balance
∂v/∂t + fu = -g∂h∂y across-shore force balance
COAST
∂(uh)/∂x + ∂(vh)/∂y = -∂h∂t mass balance
No assumption on v
y,v
x,u

24. Wave Equation ∂2u/∂t2 = c2 ∂2u/∂x2
subject to the dispersion relation
w2 = k2 gh + f2
w>f or
c2 = w2/k2 = gh + f2/k2

25. Progressive Poincare Wave in a Channel
Sea level
Horizontal
velocity
from Dr. Antenucci

26. Progressive Poincare Wave in a Channel
Mode-2
Sea level
Horizontal
velocity
from Dr. Antenucci

27. Standing Poincare Wave in a Channel
Mode-1
Sea level
Horizontal
velocity
from Dr. Antenucci

28. Standing Poincare Wave in a Channel
Mode-2
Sea level
Horizontal
velocity
from Dr. Antenucci

29. Internal Poincare Wave in a closed basin
(vertical mode-1)
layer-1
(u,v) layer-1
layer-3
h layer-2
(u,v) layer-2
h layer-2
(u,v) layer-3
from Dr. Antenucci

30. Internal Poincare Wave in a closed basin
(vertical mode-1, horizontal mode-2)
layer-1
(u,v) layer-1
layer-3
h layer-2
(u,v) layer-2
h layer-2
(u,v) layer-3
from Dr. Antenucci

31. Tidal Dynamics: Scaling
Depth-integrated (averaged) continuity (mass) balance:
(uh)/x + (vh)/y + h/t = 0
UH/L UH/L h0/T
---> U ~ (h0/H) (L/T)
or L ~ UHT/h0
Velocity scale U
Vertical length scales H (depth) and h0(sealevel amplitude)
Hirozontal length scale L
Time scale T

32. Depth-integrated (averaged) force (momentum) balance:
Acceleration + nonlinear advection + Coriolis = pressure gradient
u/t
U/T 1
uu/x+vu/y
U2/L
U2h0/(UHT)
e= h0/H
<< 1 fv fU fT
~ 1
g h/x
gh0/L
gH(h0 /H)2/UT
(ec/U)2
~ 1
L ~ UHT/h0
h0 ~1m, H~100m --> e~0.01<<1
--> (gH)1/2~30 m/s
--> U~0.3 m/s
2p/f ~ hours, hence Coriolis acceleration contributes as fT~1