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## Shallow Water Waves 2: Tsunamis and Tides

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**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**0 = pressure gradient + horizontal tide generating force**0 = g ∂ht/∂s + hTGF Equilibrium Tide ht gives ht (“bulge”) of water-covered earth, no accelerations**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**Resonance:**response of an oscillatory system forced close to its natural frequency d damping parameter (friction) Response Forcing Frequency/Natural Frequency**0 = pressure gradient + horizontal tide generating force**0 = g ∂ht/∂s + hTGF Equilibrium Tide ht gives ht (“bulge”) of water-covered earth, no accelerations 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 = - 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**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.**L**H u/t = - g h/x 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) ---> T=4L(gH)-1/2 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 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**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**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**COAST**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**High**High convergence convergence Low Low u u u u divergence divergence High High Kelvin wave: geostrophic across the shore COAST COAST Time t Time t+dt convergence y,v y,v convergence x,u x,u**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 Subject to the dispersion relation or c2 = gH w2 = k2 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**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 COAST ∂(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 w>f or c2 = w2/k2 = gh + f2/k2**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 U/T U/T 1 uu/x+vu/y U2/L U2h0/(UHT) e= h0/H << 1 fv fU 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 ~ 12-24 hours, hence Coriolis acceleration contributes as fT~1