Transcript Document
Shallow Water Waves 2: Tsunamis and Tides
MAST-602 Lecture Oct.-16, 2008 (Andreas Muenchow)
Knauss (1997):
p. 218-222 (tsunamis and seiches)
p. 234-244 tides
p. 223-228 Kelvin waves
Descriptions: Tsunamis, tides, bores
Tide Generating Force
Equilibrium tide
Co-oscillating basins
Kelvin and Poincare waved
Equilibrium Tide ht
gives ht (“bulge”) of water-covered earth, no accelerations
0
=
pressure gradient
+ horizontal tide generating force
0
=
g ∂ht/∂s
+
hTGF
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 0.0023 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
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.
Resonance:
response of an oscillatory system
forced close to its natural frequency
Breaking wine glass ---> 1-min movie
Tacoma Bridge 1940 ---> 4-minutes movie
Forced string
---> Java applet
Response
Resonance:
response of an oscillatory system
forced close to its natural frequency
d damping
parameter
(friction)
Forcing Frequency/Natural Frequency
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
Dynamics of Tides
ht is a known forcing function of (x,y,t):
acceleration + coriolis = pressure gradient
u/t - fv
v/t + fv
= - g (h-ht)/x
= - g (h-ht)/y
x-momentum
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
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.
Tidal Co-oscillation (without Coriolis):
Standing wave due to perfect reflection at wall
c=l/T=(gH)1/2 and L=l/4
u/t = - g h/x
H u/x + h/t = 0
(quarter wavelength resonator)
---> T=4L(gH)-1/2
L
H
deep ocean tide forced by
hTGF due to moon/sun
shelf tide forced by
small h(t) at seaward boundary
Currents
Sealevel
Time
Example of quarter-wavelength resonator: Cook Inlet, Alaska
h0~5m
L
tidal bore
forms:
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
Dynamics of Tides
ht is a known forcing function of (x,y,t):
acceleration + coriolis = pressure gradient
u/t - fv
v/t + fv
= - g (h-ht)/x
= - g (h-ht)/y
(uh)/x + (vh)/y + h/t = 0
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
x-momentum
y-momentum
continuity
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
∂(uh)/∂x + ∂(vh)/∂y = -∂h∂t mass balance
Assume v=0 everywhere
y,v
x,u
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
∂(uh)/∂x + ∂(vh)/∂y = -∂h∂t mass balance
Assume v=0 everywhere
y,v
x,u
Kelvin wave: geostrophic across the shore
High
High
convergence
convergence
Low
u
Low
u
divergence
divergence
High
u
convergence
x,u
High
Time t
y,v
x,u
u
Time t+dt
y,v
convergence
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
Wave Equation
∂2u/∂t2 = c2 ∂2u/∂x2
x
y
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
Tidal co-oscillation with Coriolis (Taylor, 1922)
h
(u,v)
head
head
© 1996 M. Tomczak
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
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
∂(uh)/∂x + ∂(vh)/∂y = -∂h∂t mass balance
No assumption on v
y,v
x,u
Wave Equation
∂2u/∂t2 = c2 ∂2u/∂x2
subject to the dispersion relation
w2 = k2 gh + f2
or
c2 = w2/k2 = gh + f2/k2
w>f
Progressive Poincare Wave in a Channel
Sea level
Horizontal
velocity
from Dr. Antenucci
Progressive Poincare Wave in a Channel
Mode-2
Sea level
Horizontal
velocity
from Dr. Antenucci
Standing Poincare Wave in a Channel
Mode-1
Sea level
Horizontal
velocity
from Dr. Antenucci
Standing Poincare Wave in a Channel
Mode-2
Sea level
Horizontal
velocity
from Dr. Antenucci
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
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
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
Depth-integrated (averaged) force (momentum) balance:
Acceleration + nonlinear advection + Coriolis = pressure gradient
u/t
uu/x+vu/y
fv
g h/x
U/T
U/T
1
U2/L
U2h0/(UHT)
e= h0/H
<< 1
fU
fU
fT
~1
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 ~ 12-24 hours, hence Coriolis acceleration contributes as fT~1