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Classical Tsunami Theory - a la WardSteven N. Ward 3/6/03

1. Formulation

1.1. Fluid dynamics starts with Euler's equations

ρ(r, t)Dv(r, t)

Dt= ∇ • t(r, t) + ρ(r, t)F(r, t) (1.1.1)

and the continuity equation

Dρ(ρ,t)Dt

+ ρ(ρ, t)∇• v(ρ, t)=0 (1.1.2)

to be solved in fluid volume V. Here ρ(r, t) is density, v(r, t) =∂u(ρ, t)

∂t is velocity, u(r, t) is

displacement, t(r, t) is the stress tensor, F(r, t) is body force per unit mass, and DDt

=∂∂t

+ v(ρ, t)• ∇

1.2. If stress linearly relates to strain and the fluid is inviscid, then the non-zero stress tensor elements are pressure p,

t(r, t) = -p(r, t)I (1.2.1)and (1.1.1) become the Navier-Stokes equations

ρ(r, t)Dv(r, t)

Dt= −∇p(r, t) + ρ(r, t)F(r,t) (1.2.2)

1.3. If the motions and body forces are irrotational

Dv(r, t)Dt

=∂v(ρ, t)∂t

+v(ρ, t)• ∇v(ρ, t)=∂v(ρ, t)∂t

+ 12∇v2 (ρ, t)−v(ρ, t)× ∇×v(ρ, t)

=∂v(ρ, t)

∂t+12∇v2(ρ, t)

(1.3.1)

andF(r, t) = -∇φ(ρ, t)=g(ρ, t) (1.3.2)

then (1.2.2) become the Bernoulli equations

ρ(r, t)∂v(r, t)

∂t= −∇p(r, t) -

12

ρ(r, t)∇v2 (r, t)- ρ(r, t)∇φ(r, t) (1.3.3)

1.4. Although (1.3.3) already uses a linear constitutive law, we carry the linearization of (1.3.3) and (1.1.2) all the way through. Following seismological procedures (because I'm a seismologist) let

ρ(r, t) = ρ0(r) + ρ1(r, t)p(r, t) = p0 (r) + p1(r, t)φ(r, t) = φ0 (r) + φ1(r, t)

(1.4.1)

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where all the sub-0 quantities refer to the undisturbed state and the sub-1 quantities are small perturbations about the initial state. Placing (1,4.1) into (1.3.3) and (1.1.2) and dropping products of sub-1 quantities (velocity v is assumed be of sub-1 size) gives

ρ0(r)˙ ̇ u (r, t) = −∇p1(r, t)+ ∇ • [ρ0 (r)u(r, t)]∇φ0(r) - ρ0(r)∇φ1(r,t) (1.4.2)

∂ρ1(r, t)∂t

+∇ • [ρ0(r)v(r, t)] = 0 ⇒ ρ1(r,t) = −∇• [ρ0 (r)u(r, t)] (1.4.3)

The equations now are expressed in displacement u instead of velocity v. In obtaining (1.4.2) we used (1.4.3) and assumed that the initial state was hydrostatic equilibrium

∇p0(r) = -ρ0(r)∇φ0(r) = ρ0(r)g0 (r) (1.4.4)Pressure increment p1(r, t) consists of an elastic term and an advected term

p1(r, t) = -κ(ρ)∇• u(ρ, t)- u(ρ, t)• ∇p0(ρ) (1.4.5)The κ(r) is fluid incompressibility. Equations (1.4.2), (1.4.5) together with

∇2φ1(r, t) = -4πG∇ • [ρ0 (r)u(r, t)] (1.4.6)represent five equations for five unknown functions u(r, t) , p1(r, t), φ1(ρ, t) to be found in V.

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2. Further Simplifications

2.1. Usually for tsunami calculations we take the media to be homogeneous

κ(r) = κ, ρ(r) = ρ0and gravity to be constant and unchanging

∇φ0 (r) = -gˆ z , φ1(r, t) = 0The four equations of interest now are

ρ0˙ ̇ u (r, t) = −∇p1(r, t)- gρ0 ˆ z ∇• u(r, t)p1(r, t) = -κ∇ • u(r, t) - gρ0uz (r, t) (2.1.1)

orρ0˙ ̇ u (r, t) = −∇pe (r, t)+ ρ0g[∇uz (r, t) - ˆ z ∇• u(r, t)]pe (r, t) = -κ∇ • u(r, t) (2.1.2)

withpe (r, t) = p1(r, t)+ u(r, t) • ∇p0(ρ)=-κ∇• u(ρ, t) (2.1.3)

Four equations (2.1.2a,b) and two seafloor/sea surface boundary conditions are the basis for rigorous (SORT OF…) tsunami calculations.

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3. Boundary Conditions-Classical Approach

In the linearization above, boundary conditions on deformed surfaces are evaluated on undeformed surfaces S0. For inviscid fluids, uz(r,t) and pe(r,t) are continuous across originally flat laying boundaries between homogenous layers, i.e.

uz (r, t)[ ]−+ pe(ρ, t)[ ]−

+ on S0 (3.1.1-2)Classical tsunami theory however, instead of solving equations (2.1.2) with simple boundary conditions (3.1.1-2) rather solves a simpler set of equations

ρ0˙ ̇ u (r, t) = −∇pe (r, t)pe (r, t) = -κ∇ • u(r, t) (3.1.3a,b)

with more complex boundary conditions

uz (r, t)[ ]−+ p1(ρ, t) [ ]−

+ = pe(ρ, t)+ρ0guz(ρ, t)[ ]−+

on S0 (3.1.4-5)This approximation takes all of gravity's effects in the body of the fluid (note that g does not appear in 3.1.3a,b) and "compresses" them onto boundaries of fluid layers of different density. The effectiveness of the classical approach can be gauged later by comparing the analytical solutions to (3.1.3a,b) and (3.1.4-5) to numerical solutions of (2.1.2) and (3.1.1-2).

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4. Two dimensional solutions.

4.1 Let's first solve some tsunami problems in two dimensions. Extensions to three dimensions are straightforward and make heavy use the 2-D results. Let coordinate x be horizontal, coordinate z be directed downward, g=g ˆ z , uy and all /y =0. Equations (3.1.3a,b) become

ρ0˙ ̇ u z (x, z, t) = −∂pe (x, z, t)/∂z

ρ0˙ ̇ u x(x, z, t) = −∂pe (x, z, t)/∂xpe (x, z, t) = -κ[∂ux(x, z, t)/∂x + ∂uz (x, z, t)/∂z]

(4.1.1)

Because we are working with linear equations, we can make use of superposition both in frequency and wavenumber. Let new wavenumber-frequency variables be transforms of space-time variables like

f(k, z,ω) = dx−∞

∞∫ dt f(x,z,t)e-i(κx-ωt)

−∞

∞∫ ; (4.1.2a)

These are reconstituted by

f(x, z, t) =1

4p2dκ dω

−∞

∞∫ f(κ,z,ω)ei(κx-ωt)

−∞

∞∫ (4.1.2b)

With this convention in (4.1.1) ∂ / ∂t ⇒ −iω and ∂ / ∂x⇒ ik

∂∂z

uz (k,z,ω )pe (k,z,ω ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

0 η2/ρ0ω2

ρ0ω 2 0

⎡

⎣ ⎢

⎤

⎦ ⎥uz (k,z,ω)pe (k,z,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (4.1.3)

where η2 = η2(ω, k) = k2 - ρ0ω 2/κ2 and use was made of

ux (k,z,ω) =iκpe(κ,z,ω)/ρ0ω2 (4.1.4)

In linear theory, horizontal tsunami motions are not independent, but can be found from p1 and uz

once they are known.

Note that (4.1.3) are simpler than the original equations (2.1.2) which are

∂∂z

uz (k, z,ω )pe (k, z,ω ) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

−k2g/ω 2 η2/ρ0ω 2

ρ0ω 2 − k2ρ0g2/ω2 k2g/ω 2 ⎡

⎣ ⎢

⎤

⎦ ⎥uz (k, z,ω)pe (k, z,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (4.1.5)

with ux (k,z,ω) =iκ[pe(κ,z,ω)- ρ0guz(κ,z,ω)] /ρ0ω2 (4.1.6)

Given uz and p1 and any depth z0, equations (4.1.3) tell us how to find uz and p1 and any other depth z

uz (k, z,ω)pe(κ,z,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

C(z,z0 ) ηS(z,z0 ) /ρ0ω2

ρ0ω2S(z,z0) / η C(z,z0 )

⎡ ⎣ ⎢

⎤ ⎦ ⎥uz(κ,z0 ,ω)pe(κ,z0 ,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (4.1.7)

where C(z,z0)=cosh[η(z-z0)] and S(z,z0)=sinh[η(z-z0)]. We can also write the solutions (4.1.7) in terms of uz and p1(k,z,ω) =pe(κ,z,ω)+ uz(κ,z,ω)ρ0g that are continuous across the undeformed surfaces

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uz (k, z,ω)p1(κ,z,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

C(z,z0 )−gηS(z,z0) / ω2 ηS(z,z0 )/ ρ0ω

2

ρ0S(z,z0 )η

ω2 −g2η2 / ω2( ) C(z,z0 )+ gηS(z,z0 ) / ω

2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥uz(κ,z0 ,ω)p1(κ,z0, ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (4.1.8)

4.2 In all of the cases considered here, we employ the "decoupled" approach that assumes that the tsunami motions do not reach into the elastic space below the ocean. That is, the vertical displacement at the sea floor

uz(k,H,ω) = always specified (4.2.1)zero or otherwise. At the sea surface, the linearized boundary condition (3.1.5) says that

p1(k,0,ω)=0 (4.2.2)With p1 at the sea surface and uz at the sea floor specified, we are ready to use (4.1.8) to solve some tsunami excitation problems.

4.3 Tsunami Dispersion relation. In an ocean of depth H, consider the second equation in (4.1.8) at z=0 with (4.2.2) and a rigid bottom condition (4.2.1)

0 =[cosη(ηH)−gηsinη(ηH)/ ω2 ]p1(κ,H,ω) (4.3.1)The only way this can hold is if

ω2(k) = gη(ω(k)) tanh[η(ω(k))H]

ω 2 = gη(k(ω)) tanh[η(k(ω ))H] (4.3.2)

The frequency ω(k) for a given wavenumber k, or the wavenumber k(ω) at a given frequency form the tsunami dispersion relationship in a compressible ocean of depth H. Although it is not a necessary assumption in our theory, often we take the ocean as incompressible. In this case

κ ⇒ ∞ and η = k2 - ρ0ω 2/κ ⇒ k (4.3.3).

4.4 Tsunami Eigenfunctions. Supposing |uz|=1 at the sea surface z=0 and conditions (4.2.2) and (4.3.2), the displacements and pressures at any depth z are from (4.1.8) are

uz (z,ω)= η(κ(ω))g

ω2sinη(η(κ(ω))(H−z))cosη(η(κ(ω))H)

ux (z,ω) =- iκ(ω)g

ω2cosη(η(κ(ω))(H−z)cosη(η(κ(ω))H)

pe(z,ω)=-ρ0gcosη(η(κ(ω))(H−z)cosη(η(κ(ω))H)

p1(z,ω)=-ρ0gsinη(η(κ(ω))z))

cosη(η(κ(ω))H)sinη(η(κ(ω))H)

(4.4.1)

Clearly, uz(H,ω)=p1(0,ω)=0 as required and pe(0,ω)=-ρ0guz(0,ω). In terms of tsunami "eigenmodes"

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uz (x,z, t,ω) =η(κ(ω))g

ω2sinη(η(κ(ω))(H−z))

cosη(η(κ(ω))H)ei(κ(ω)x−ωt)

ux (x,z,t,ω)=- iκ(ω)gω2

cosη(η(κ(ω))(H −z)cosη(η(κ(ω))H)

ei(κ(ω)x−ωt)

pe(x,z,t,ω) =-ρ0gcosη(η(κ(ω))(H −z)cosη(η(κ(ω))H)

ei(κ(ω)x−ωt)

p1(x,z,t,ω) =-ρ0gsinη(η(κ(ω))z))

cosη(η(κ(ω))H)sinη(η(κ(ω))H)ei(κ(ω)x−ωt)

(4.4.2)

Taking the real part of (4.4.2): uz ~ cos(k(ω)x-ωt) and ux~ sin(k(ω)x-ωt). You can see that tsunami motion is a prograde ellipse.

To compare classical solutions (4.4.1) with solutions to the original equations, you would for a fixed ω integrate (4.1.5) numerically from z=0 to z=H starting with

uz (k,z,ω)pe(κ,z,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

10 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ (4.4.5)

per (3.1.1), while incrementally adjusting k to a k(ω) such that

uz (H,ω)pe(H,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

0pe(κ(ω),H,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (4.4.6)

For example, for long waves kH>>1 (4.1.5) and (4.4.5) integrate to

uz (k,z,ω)pe(κ,z,ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

(1−zκ2g/ω2 )−ρ0z(κ

2g2/ω2 −ω2 )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

To satisfy (4.4.6) ω2(k)=gk2H or k2(ω)=ω2/gH , whence the rigorous solutions

uz (z,ω)=(1−z/H)

ux (z,ω) =−iκ(ω)g

ω2 (1−zω2/g)

pe(z,ω)=−ρ0gzH

(1-Hω2 /g)

For long waves, (4.3.2) isω2(k) = gη2 (ω(k))H ; ω2 = gη2(k(ω))H and the classical solutions (4.4.1) become

uz (z,ω)=(1−z/H)

ux (z,ω) =- iκ(ω)g

ω2

pe(z,ω)=-ρ0g

p1(z,ω)=-ρ0gzH

So for long waves at least, if p1 plays the role of pe, the differences between the rigorous solutions and the classical ones might be O(Hω2/g). Course if you really were going to integrate the rigorous equations you would not make the simplifications of homogeneous media and gravity constant and unchanging gravity to begin with.

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5 Specific 2-D Problems.

5.1 Initial Value Problems at the sea surface. For an asteroid impact tsunami, you might select sea surface displacement to reproduce initial transient cavity shapes given by experiment or by full-blown hydrodynamic simulations of impacts. If so, we specify an initial vertical surface displacement condition like

uz (x,0, t = 0) = uztop (x) (5.1.1)

and its transform

uztop (k) = dx

−∞

∞∫ uz

top(x) e-iκx (5.1.2)

In this case, the eigenmodes (4.4.2) give the evolved tsunami straight away [From now on, I assume an incompressible fluid so η(k(ω)) = k(ω) ]

u(x,z, t) = Re dk−∞

∞∫

uztop(κ) 2p

ˆ z sinη(κ(H−z))

sinη(κH)−iˆ x

cosη(κ(H−z))sinη(κH)

⎡ ⎣ ⎢

⎤ ⎦ ⎥ei(κx−ω(κ)t)

oρ

u(x,z,t)=Re dω−∞

∞∫

uztop(κ(ω)) 2pu(ω)

ˆ z sinη(κ(ω)(H−z))

sinη(κ(ω)H)−iˆ x

cosη(κ(ω)(H−z))sinη(κ(ω)H)

⎡ ⎣ ⎢

⎤ ⎦ ⎥ei(κ(ω)x−ωt)

(5.1.3a,b)In (5.1.3b), u(ω)=dω/dk(ω), the tsunami group velocity. You can see in (5.1.3a) that at the surface z=0 for t=0

Figure 1. Equation (5.1.3a) evaluated with a parabolic initial displacement of the sea surface. This is my concept of asteroid impact tsunami.

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uz (x,0, t) = dk−∞

∞∫

uztop(κ) 2p

eiκx (5.1.4)

This is just the inverse transform of the forward transform so clearly (5.1.1) is satisfied. An initial vertical displacement is always associated with an initial horizontal field and visa-versa, in this case

ux (x,0, 0) = dk−∞

∞∫

−i uztop(κ) 2p

cosη(κH)sinη(κH)

eiκx

You can not specify vertical and horizontal displacements separately.

Suppose instead we have some initial vertical surface velocity condition like

˙ u z (x,0, t = 0) = ˙ u ztop (x) (5.1.5)

with its transform

˙ u ztop (k) = dx

−∞

∞∫ ̇ u z

top(x) e-iκx (5.1.6)

The same reasoning suggests that tsunami the velocity field would be

˙ u (x,z, t) = Re dω−∞

∞∫

˙ u ztop(κ(ω)) 2pu(ω)

ˆ z sinη(κ(ω)(H−z))



⎡ ⎣ ⎢


oρ

˙ u (x,z,t)=Re dκ−∞

∞∫

˙ u ztop(κ) 2p


sinη(κH)−iˆ x


⎡ ⎣ ⎢


(5.1.7)

and that the tsunami displacement field is

u(x,z, t) = Re dω−∞

∞∫

˙ u ztop(κ(ω))

−iω2pu(ω)ˆ z sinη(κ(ω)(H−z))



⎡ ⎣ ⎢


oρ

u(x,z,t)=Re dκ−∞

∞∫

˙ u ztop(κ)

−iω(κ)2pˆ z sinη(κ(H−z))sinη(κH)

−iˆ x cosη(κ(H −z))

sinη(κH) ⎡ ⎣ ⎢


(5.1.8)

(5.1.8) follows from (5.1.7) because application of (-iω)-1 in the frequency domain is the same as integration in the time domain. (5.1.3) and (5.1.8) are in fact independent solutions so that they may be combined like

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u(x,0, t) = Re dω−∞

∞∫

uztop(κ(ω))+ i˙ u z

top(κ(ω))/ω{ }

2pu(ω)ˆ z sinη(κ(ω)(H−z))



⎡ ⎣ ⎢


oρ

u(x,0, t)=Re dκ−∞

∞∫

uztop(κ)+ i˙ u z

top(κ)/ω(κ){ }

2pˆ z sinη(κ(ω)(H−z))



⎡ ⎣ ⎢


and

˙ u (x,0, t)=Re dω−∞

∞∫

−iωuztop(κ(ω))+ ˙ u z

top(κ(ω)){ }

2pu(ω)ˆ z sinη(κ(ω)(H−z))



⎡ ⎣ ⎢


oρ

˙ u (x,0, t)=Re dκ−∞

∞∫

−iω(κ)uztop(κ)+ ˙ u z

top(κ){ }

2pˆ z sinη(κ(ω)(H−z))

sinη(κ(ω)H)−iˆ x cosη(κ(ω)(H−z))

sinη(κ(ω)H) ⎡ ⎣ ⎢


(5.1.9) (5.1.9) states the tsunami initial value problem. It says "Given the vertical displacement and vertical velocity OF THE SEA SURFACE AT ANY ONE TIME, (5.1.9) can be used to find the displacement and velocity of the sea AT ANY DEPTH AT ANY TIME LATER. This is quite important. For landslide sources for instance, if you can specify sea surface conditions just once after the slide, then you can use (5.1.9) to propagate the waves anytime further. We have been using this idea in matrix form to construct a "time stepping" tsunami propagation model. Too (5.1.9) explains why workers who employ "initial static lumps of water" as tsunami sources can't correctly model many situations. Given a fixed lump, different selections of initial velocity can give totally different tsunami motions.

5.2 Finite duration sources. Suppose now that we have some vertical surface displacement condition that takes place over a finite period of time t>0 like

uz (x,0, t) = uztop (x, t) H(t) (5.2.1)

The convolution theorem tells us how to form the tsunami fields given (5.1.3)

u(x, z, t) = Re dω−∞

∞∫

uztop(κ(ω)) 2pu(ω)

ˆ z sinη(κ(ω)(H−z))sinη(κ(ω)H)

−iˆ x cosη(κ(ω)(H−z))sinη(κ(ω)H)

⎡ ⎣ ⎢


× dx0−∞

∞∫ dt0

0

t∫ ˙ u z

top(x0 , t0 ) e−i(κ(ω)x0−ωt0)

oρ

u(x,z,t)=Re dκ−∞

∞∫

uztop(κ) 2p


sinη(κH)−iˆ x


⎡ ⎣ ⎢


× dx0−∞

∞∫ dt0

0

t∫ ˙ u z

top(x0 , t0 ) e−i(κx0−ω(κ)t0)

(5.2.2)

Be aware of the time differentiation of the surface condition in (5.2.2). Sometimes, uztop (x, t) can be

simplified such that one or both of the sub-0 integrals above can be done by hand. For instance, for a propagating source all the time histories of uplift at different points might be the same within a constant factor, only delayed in time.

uztop (x, t) = u z

top(x)S(t - t(x)); S(t) = 0 if t < 0

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dx0−∞

∞∫ dt0

0

t∫ ˙ u z

top(x0 , t0 ) e−i(κ(ω)x0−ωt0) = dx0

−∞

∞∫ uz

top(x0 )e−i(κ(ω)x0−ωt(x0 )) dt0

0

t−t(x0 )∫ ˙ S (t0 ) e

iωt0 (5.2.3)

If S(t) was a step function, the last integral would equal 1 for t>t(x0) and 0 for t<t(x0). If S(t) was a ramp function, the last integral would equal min[1, (t- t(x0))/TR] for t>t(x0) and 0 for t<t(x0).

5.3 Initial Value Problems at the seafloor. To model a submarine landslide, you might select seafloor vertical displacement to follow a certain uplift history. In this case we'd like the tsunami from an initial vertical bottom displacement condition like

uz (x,H, t = 0) = uzbot (x) (5.3.1)

and its transform

uzbot (k) = dx

−∞

∞∫ uz

bot(x) e-iκx (5.3.2)

In this problem, you can't just plug in the eigenmodes (4.4.2) like we did for asteroid impacts because uz in the eigenmodes vanish at the seafloor. There is no way to match (5.3.2). To solve this problem, we have to go all the way back to (4.1.8), now with (5.3.2) and (3.1.5)

uz (k,0,ω)0

⎡ ⎣ ⎢

⎤ ⎦ ⎥=

C(0,H)−gηS(0,H)/ ω2 ηS(0,H)/ρ0ω2

ρ0S(0,H)η

ω2 −g2η2 /ω2( ) C(0,H)+gηS(0,H)/ ω2

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥uzbot(κ,ω)

p1(κ,H,ω)

⎡ ⎣ ⎢

⎤ ⎦ ⎥ (5.3.3)

where C(z,z1) = cosh[k(z-z1)], T(z,z1) = tanh[k(z-z1)], etc. Solve the second equation of (5.3.3) first for pressure at the seafloor

P1(k, H,ω)=-ρ0gω

2 1−ω2T(H,0)/κg( )

ω2 −ω2 (κ)uzbot(κ,ω) (5.3.4)

then substitute into the first equation of (5.3.30 at any depth z to find displacement

uz (k,z,ω) =

ω2 C(H,0)C(z,H)+ S(z,H)S(H,0)( )

−gκ C(z, H)S(H,0)+C(H,0)S(z,H)( )

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

C(H,0)[ω2 −ω2 (κ)]uzbot(κ,ω) (5.3.5)

Take care here to distinguish general frequency ω from the characteristic frequency ω(k). (5.3.5) is our first landslide tsunami. All we need to do is transform it back to time and space by (4.1.2b). As formulated above, u z

bot (k,ω) actually is any function of time. If we want it to be a fixed initial uplift then

uzbot (k,ω) =

uzbot(κ)

-iω (5.3.6)

The (-iω)-1 is the time transform of a step function, thus

uz (k,z,ω) =

iω C(H,0)C(z,H)+ S(z,H)S(H,0)( )

−(igκ/ω) C(z,H)S(H,0)+C(H,0)S(z,H)( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

C(H,0)[ω2 −ω2 (κ)]uzbot(κ) (5.3.7)

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We apply the inverse time transform, making use of the residue theorem to evaluate the simple poles that lay at ω=0, ω=ω(k). Here's a useful table of transforms.

iωω2 −ω2 (κ)

⇔ cos[ω(κ)t]H(t)

ωω2 −ω2 (κ)

⇔ −icos[ω(κ)t]H(t)

1ω(ω2 −ω2(κ))

⇔ −i cos[ω(κ)t]H(t)ω2 (κ)

+ iH(t)ω2 (κ)

iω2 −ω2 (κ)

⇔ cos[ω(κ)t]H(t)

ω2 (κ)− H(t)

ω2 (κ)

(5.3.8)

In the time domain (5.3.7) is

uz (k, z, t) =

C(H, 0)C(z, H)+ S(z,H)S(H,0)( )cos[ω(κ)t]

+C(H,0)S(H,0)

C(z,H)S(H,0)+ C(H,0)S(z,H)( ) −cos[ω(κ)t]+1[ ]

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

C(H,0) uzbot(κ)H(t)

=−S(z,H)

C(H,0)S(H,0) uzbot(κ)cos[ω(κ)t]H(t)+ C(z,H)+

S(z,H)T(H,0)

S(z,H) ⎛ ⎝ ⎜

⎞ ⎠ ⎟u zbot(κ)H(t)

(5.3.9)

All we have to do now is the inverse wavenumber transform. For t>0 we have

€

u(x,z,t) = Re dk−∞

∞

∫ uzbot (k)

2π cosh(kH)ˆ z

sinh(k(H − z))sinh(kH)

− iˆ x cosh(k(H − z))

sinh(kH) ⎡ ⎣ ⎢

⎤ ⎦ ⎥e i(kx−ω (k)t)

+ dk−∞

∞

∫ uzbot (k) 2π

ˆ z cosh(k(H - z)) −sinh(k(H - z))

tanh(kH) ⎧ ⎨ ⎩

⎫ ⎬ ⎭− i ˆ k sinh(k(H - z)) −

cosh(k(H - z))tanh(kH)

⎧ ⎨ ⎩

⎫ ⎬ ⎭

⎡ ⎣ ⎢

⎤ ⎦ ⎥e ikx

(5.3.10a,b)Equation (5.3.10) is the full 2-D tsunami wave solution for an instantaneous uplift of the sea bottom. It looks messy, but it is not so bad. Consider integral (5.3.10b), see that time does not appear. This is a static field that does not propagate. Too, at the surface z=0 the vertical component of the static field vanishes. At z=H the static term reduces to the vertical displacement at the sea floor as it should. Recall that the propagating eigenmodes in (5.3.10a) have zero vertical motion at the sea floor. In a word, (5.3.10b), is needed to match the boundary conditions, but it is usually not of much interest.

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It is remarkable to me that the propagating tsunami from a bottom displacement is identical to the tsunami from the same surface displacement (5.1.3) except for the cosh(kH) term in the former. This term acts as a low pass filter. Short wavelength elements in the bottom uplift history, don't show up in the tsunami field at the surface. This helps us in the modeling because small details in landslides usually are not important. Consider the surface vertical displacement of the propagating tsunami from an instantaneous bottom vertical bottom disturbance

uz (x,0, t) = Re dk−∞

∞∫

uzbot(κ)ei(κx−ω(κ)t)

2pcosη(κH) (5.3.11)

You can't get much simpler than this! Well,.... you can actually. Let's ignore dispersion for the moment, then ω(k)= |k| gH , and so (5.3.11) becomes

uz (x,0, t) = Re dk−∞

∞∫

uzbot(κ)ei(κx−|κ| gHt)

2pcosη(κH)=12G (x − gHt)+G (x + gHt)[ ]

with G(x) = dk−∞

∞∫

uzbot(κ)eiκx 2pcosη(κH) If the bottom uplift uz

bot (x) = constant U over some small

width -W/2<x<W/2 then

G(x) ≈UW dκ0

∞∫

cosκx pcosη(κH)

=UW2H

secηpx2H ⎛ ⎝ ⎜

⎞ ⎠ ⎟

So, you can consider the "2-D, non-dispersive landslide greens function" to be

Figure 2. Equations (5.3.10a,b) evaluated for a sliding block landslide model. As the block slides by, water moves up then back down. Surface waves run ahead.

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g(x, t, x0, t 0) =UW4H

secηp[(x −x0 )− gH(t- t0 )]

2H

⎛ ⎝ ⎜

⎞ ⎠ ⎟+ secη

p[(x −x0 )+ gH(t- t0 )]2H

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥ (5.3.11b)

It's fun to program (5.3.11b) in a little "do-loop" to sum up a sequence of bottom uplifts shifted in time and space to get a feeling for how landslide tsunami operate on the simplest level.

If the landslide uplift has an arbitrary history, then the landslide tsunami (5.3.11) at the surface would be

uz (x, 0, t) = Re dk−∞

∞∫

ei(κx−ω(κ)t)

2pcosη(κH)dx0

−∞

∞∫ dt0

0

t∫ ˙ u z

bot(x0, t0) e−i(κx0−ω(κ)t0) (5.3.12)

(5.3.12) is still pretty straightforward considering that it handles any landslide that you care to name and it includes all propagation, dispersion, and ocean filtering effects. (5.3.12) is the way to go as far as I am concerned. But in using such a nice formula, I've lost the bragging rights to all those fancy "non-linear behaviors".

All that remains is to take (5.3.12) to 3-D and include the effects of a variable depth ocean.

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6. Passage to 3-D

6.1 Once we have the 2-D tsunami fields, the passage to 3-D is fairly trivial. Here are the steps,

1) position x, and wave number k to go vectors k=kx ˆ x +ky ˆ y , k=|k|, r=x ˆ x +y ˆ y 2) product kx goes to kr 3) 1-D integrals over wave number and position now cover all 2-D space 4) the unit vector ˆ x goes to ˆ k

With these, the propagating 2-D tsunami from an arbitrary bottom landslide

u(x, z, t) = dk−∞

∞∫

eiκx 2pcosη(κH)

ˆ z sinη(κ(H−z))sinη(κH)

−iˆ x cosη(κ(H −z))sinη(κH)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

× dx0 e−iκx0−∞

∞∫ dt0

0

t∫ ˙ u z

bot(x0 , t0 ) cos[ω(κ)(t- t0 )] (6.1.1)

becomes in 3-D

u(x,y, z, t) = dkk∫

eiκ•ρ

4p2 cosη(κH)ˆ z sinη(κ(H−z))sinη(κH)

−iˆ κ cosη(κ(H−z))

sinη(κH) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

× dρ0 e−iκ• ρ0ρ0∫ dt0

0

t∫ ˙ u z

bot(ρ0, t0) cos[ω(κ)(t- t0 )] (6.1.2)

where dk=dkxdky, dr0=dx0dy0. There are millions of ways we can recast (6.1.2). One is

u(r, t) =κ dκ

2pcosη(κη)0

∞∫ dρ0 ˆ z J0 (κR )

sinη(κ(η- z))sinη(κη)

+ ˆ R J1(κR)cosη(κ(η- z))

sinη(κη) ⎡ ⎣ ⎢

⎤ ⎦ ⎥ρ0

∫

× dt0 ˙ u zbot(ρ0 ,t0 )cos[ω(κ)(t−t0 )]

0

t∫

(6.1.3)

where R=|r-r0|, ˆ R = (r-r0)/R and J0 and J1 are cylindrical Bessel functions. In (6.1.3) everything is real and only positive wavenumbers appear. This might give some computational advantage.

We can speed the calculation still more if the observation point r is not too close to any point r0 on the slide, then we can pass the r0 integral through

u(r, t) ≈κ dκ

2p cosη(κη)0

∞∫ ˆ z J0 (κRc)

sinη(κ(η- z))sinη(κη)

+ ˆ R cJ1(κRc)cosη(κ(η- z))sinη(κη)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

× dρ0ρ0∫ dt0 ˙ u z

bot(ρ0, t0)cos[ω(κ)(t−t0) +κ(ˆ R c • ρ0)]0

t∫

(6.1.4)

where Rc=|r-rc|, ˆ R c = (r-rc)/Rc with rc being some fixed reference location on the slide. The added term in the cosine argument effectively delays the start time for sources further ˆ R c r0<0 from the observation point than rc and advances the start time of sources closer ˆ R c r0>0 to the observation point than rc. Equation (6.1.4) is good to O(kW) or better, with W being the half-dimension of the source. Although approximate, (6.1.4) has considerable advantage over (6.1.3) in that the last three integrals in (6.1.4) depend only on the direction to the receiver ˆ R c from the reference location and not the receiver distance Rc itself. Thus, for simple sources, these integrals can be evaluated

15

analytically and we are left with a single integral to perform numerically. Equation (6.1.4) is pretty much the one I use for most of my landslide tsunami models (see Figure 3). All that need be done now is to adjust for variable depth oceans.

6.2 Variable depth oceans. In variable depth oceans, the wavelength of tsunami change with changing of depth. The frequency of the waves however, is always conserved (Hooray for linear theory!). The frequency version of (6.1.4) is

u(r, t) ≈κ(ω)dω

2pu(ω)cosη(κ(ω)H)0

∞∫ ˆ z J0 (ωT(ω,ρ,ρc))

sinη(κ(ω)(H- z))sinη(κ(ω)H)

+ ˆ R cJ1(ωT(ω, ρ, ρc))cosη(κ(ω)(H- z))sinη(κ(ω)H)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

× dρ0ρ0∫ dt0 ˙ u z

bot(ρ0 , t0 )cos[ω(t−t0 )+ κ(ω)(ˆ R c • ρ0 )]0

t∫

(6.2.1)In (6.2.1) we identified k(ω)Rc=ωRc/c(ω)=ωT(ω,r,rc). c(ω)=ω/k(ω) is the tsunami phase velocity and T(ω,r,rc) is the travel time of a tsunami wave of frequency ω from the reference location on the slide rc to the observation point r. The last step in the process allows for a variable depth ocean

u(r, t) ≈κc(ω)dω

2puc(ω)cosη(κc(ω)Hc)0

∞∫ ˆ z J0 (ωT(ω,ρ, ρc))

sinη(κ(ω)(H- z))sinη(κ(ω)H)

+ ˆ R cJ1(ωT(ω, ρ, ρc))cosη(κ(ω)(H- z))sinη(κ(ω)H)

⎡ ⎣ ⎢

⎤ ⎦ ⎥

uc(ω)u(ω) ⎡ ⎣ ⎢

⎤ ⎦ ⎥1/ 2

G(ρ,ρc) × dρ0ρ0∫ dt0 ˙ u z

bot(ρ0 , t0 )cos[ω(t−t0 )+ κc(ω)( ˆ R c • ρ0 ) ]0

t∫

(6.2.2)There's not much difference really between (6.2.1) and (6.2.2). The sub-c quantities in (6.2.2) refer to the water depth at the reference site. The T(ω,r,rc)=R/ c (ω ) now is computed in a variable depth ocean along ray paths with c (ω ) being the mean phase velocity along the path. OK, so how do I compute T(ω,r,rc)?

Figure 3. Equation (6.1.4) evaluated for surface vertical tsunami motion from a 3-D sliding block landslide source.

16

1) Evaluate a zero frequency travel time T(ω =0,ρ,ρc)= dp/ gH(p)ρaypatη

∫

2) Find a “mean” ocean depth H (r,rc) associated with the ray path of length P(r,rc),

H (r,rc ) =P2 (ρ,ρc )/gT2(0,ρ,ρc )

3) Determine a “mean” wavenumber k (ω) associated with the ray path by solving

ω2 = gk (ω)tanh[k (ω)H (r,rc)]c (ω ) , the mean phase velocity along the path is c (ω )=ω / κ (ω) . You can compute a mean group velocity along the path u (ω) using the standard uniform depth formulas with k (ω) and H (r,rc) in place of k(ω) and H. I use u (ω) and the other path averaged quantities later in these notes.

Also new in (6,2,2) is G(r,rc), a ray geometrical spreading factor, and SL=[uc(ω)/u(ω)]1/2, the linear shoaling term. SL acts to grow the waves as they slow down at the beach. The amount of shoaling growth depends on the water depth at the wave generation site and wave frequency. Long waves generally grow more than short waves. Waves from deeper water sources tend to grow more than waves from shallow water sources. For G(r,rc) you can find the expressions in my papers. So far, I continue to employ straight rays in this work even for variable depth oceans because tsunami ray tracing is not very stable and too the results look fine. For straight rays, G(r,rc)=1 for flat oceans and path length P(r,rc) is just Rc.

Because landslide sources now lay under water of variable depth, complex extended slides have to be broken up into small subblocks centered on many reference positions rc and integral (6.2.2) has

Figure 4. Equation (6.3.3) evaluated with fifteen 2x2km subblocks that each host two simple slides that uplift and fall in a way to mimic a block sliding down the side of a v-shaped channel. The ocean here has depths variable from 0 to 700m.

17

to be evaluated for each block. It sounds like a lot of work, but by this "Green's Function" approach you can construct any shape slide, moving in any direction and any speed. There’s no other way to model realistic slides that I can see. To reduce calculation time, because the subblocks are small, you can assign to them simple shapes (square) and behaviors (uniform rupture velocity and direction) such that the final two integrals in (6.2.2) can be done analytically. I call these building blocks "simple slides".

6.3 Simple Slide. To simulate large landslides, I shingle the landslide source with many square cells. These cells need not have parallel sides (landslides do curve deflecting off channel walls, etc.) nor even be fully overlapping (small spatial details don't matter remember due to the low pass filtering of the ocean layer). For instance, let the slide disturbance in the n-th cell, be a constant uplift uz that starts along one width of a square element and runs down its length at variable velocity vr(x0) reaching points x0 at times tu(x0). Then

dr0r0∫ dt0 ˙ u z

bot(ρ0, t0 )cos[ω(t−t0) +κc(ω)(ˆ R c • ρ0 ) ]0

t∫

In (ω,t) = uz dx0 dy0-W/2

W/2∫ cos[ω(t−tu(x0 ))+ κc(ω)(y0sinφ +x0cosφ) ]

-W/2

-W/2 +W(t)∫

In (ω,t)= uzsinY(κc(ω), φ)W[ ]

Y(κc(ω), φ)dx0 cos[ω(t−tu(x0 ))+ x0κc(ω) cosφ ]

-W/2

-W/2 +W(t)∫


Y(κc(ω), φ)dt0 vρ(t0 )cos[ω(t−t0 )+ x0 (t0 )κc(ω) cosφ ]

tu

min[t,tf]

∫

(6.3.1-4)

The tu= tu(-W/2) and tf= tu(W/2) are the times that the simple slide started and finished. The v r(t0) and -W/2<x0(t0)<W/2 are the velocity and position of the slide front at time t0 and W(t)=x(t)+W/2. If you are going to integrate (6.3.3 or 6.3.4) numerically, you need to take steps Dx0 or Dt0 such that the change in phase of the cosine function is small. For slides starting and slowing to zero velocity, tu(Dx0) can be quite long even for very small distance steps. The time version looks better behaved.

If the velocity on the simple slide is constant then, tu(x0)=tu+(x0+W/2)/vr, x0(t0)= vr (t0-tu)-W/2, tf=tu+W/vr then (6.3.3) or (6.3.4) become


Y(κc(ω), φ)sinX(κc(ω), φ)W(t)[ ]

X(κ c(ω), φ) cos{ω(t−tu - W/2vρ)+ X(κc(ω),φ)[W(t)-W]}

(6.3.5)

with W(t)=min[(t-tu)vr,W]. In 6.3.2-5, f=-0 is the angle between the slide direction and the receiver direction and

Y(k,φ) =κsinφ/2 ; X(k,φ) = κcosφ−ω/vρ( ) / 2 . (6.3.6)

If you'd rather make a simple slide with an uplift that "ramps up" over time TR rather than stepping up instantly, then expression (6.3.5) would be

18


Y(κc(ω), φ)sinX(κc(ω), φ)W(t)[ ]

X(κ c(ω), φ)

sinωT∗/2[ ]ωTR /2

× cos{ω(t−tu - W/2vρ)+ X(κ c(ω), φ)[W(t)- W] - ωT∗/2}

(6.3.7)

where T*=min(TR,t-tu). The -ωT*/2 in the cos term adds a time delay to the waves of T*/2 and the sin term is yet another low pass filter that eliminates tsunami waves with periods less than 2T*. Naturally as TR gets small, (6.3.7) reverts to (6.3.5). If fact, (6.3.7) opens a whole new way to think about landslide sources. Instead of sliding blocks think of each "cell" as a square walled cylinder that fills or empties over an interval TR starting at tu. By firing these cylinders off in sequence with some law for selecting the fill or drain time TR, (maybe TR is proportional to the landslide thickness at each point) you can make a very plausible landslide model. The slide velocity in this case would not be all that important, in fact you can set vr=∞

In (ω,t)= uzsinκc(ω)sinφW/2[ ]

κc(ω)sinφ/2sinκc(ω)cosφW/2[ ]

κc(ω)cosφ/2

sinωT∗/2[ ]ωTR /2

cos{ω(t−tu) - ωT∗/2} (6.3.8)

Course any cell could have more than one episode of filling and draining, e.g. to simulate a passing by of a "pile" of landslide material.

That's pretty much the whole Classical Tsunami Theory - a la Ward --- just evaluate (6.3.9) a bunch of times over the landslide area (see Figure 4)

uz (r, t) ≈allsublocκsn

∑J0 (ωT(ω, ρ, ρc)) κc(ω)dω

2pcosη(κc(ω)Hc) uc(ω)u(ω)0

∞∫ G (ρ, ρc) In (ω, t)

(6.3.9)

The In would be (6.3.5) or (6.3.7) or (6.3.8) if uniform velocity subblocks are adequate, otherwise use (6.3.4). The remaining issues are sampling (How many subblocks do you need?) and efficient evaluation of the integrals.

In many ways, I have found that the effort to prescribe the kinematics of a complex landslide (shape, thickness, velocity, direction etc. for a 100 subblocks, say) is more consuming than running the tsunami calculation itself. In my opinion, prescribing landslide kinematics is where the lion's share all the uncertainty lies in landslide tsunami work, not in the theory, linear or otherwise. Modelers need good geological input.

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7. Applications of Stationary Phase Approximations to Tsunami Calculations

7.1 Consider surface vertical motions in 3-D for the initial value problem in a uniform depth ocean

uz (x,y, 0, t) = dkk∫

eiκ•ρ

4p2dρ0 e−iκ•ρ0

ρ0∫ uz

top(ρ0 ) cos[ω(κ)t]+˙ u ztop(ρ0 )ω(κ)

sin[ω(κ)t] ⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

or

uz (x,y, 0, t) =k dk2p0

∞∫ dρ0 J0(κR )

ρ0∫ uz

top(ρ0 ) cos[ω(κ)t]+˙ u ztop(ρ0)ω(κ)

sin[ω(κ)t] ⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥ (7.1.1)

with R=|r-r0|. Suppose that initial disturbance is confined near reference location rc and that the observation point r is not “too close” to the source (I define too close below), then

uz (x,y, 0, t) ≈κ dκ2p0

∞∫ J0 (κRc) uz

top(κ) cos[ω(κ)t]+˙ u ztop(κ)ω(κ)

sin[ω(κ)t] ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (7.1.2)

with uz

top(k) = Re dr0 e−iκ(ˆ R c•ρ0)uz

top(ρ0 ) ;ρ0∫ ̇ u z

top(κ)=Re dρ0 e−iκ(ˆ R c•ρ0)˙ u z

top(ρ0 ) ρ0∫

and Rc=|r-rc|, ˆ R c = (r-rc)/Rc. The stationary phase approximation to (7.1.1) and (7.1.2) are

u z(x,y, 0, t)≈ dρ0ρ0∫

12pR

κ(ωs)

−∂u(ω)∂ω ω=ωs

uztop(ρ0) cosy -

˙ u ztopκ(ρ0 )ωs

siny ⎡ ⎣ ⎢

⎤ ⎦ ⎥ (7.1.3)

u(ωs)=R / t; −∂u(ωs)∂ω ω=ω s

> 0; y =ωst- κ(ωs)R =[c(ωs)- u(ωs)]κ(ωs)t

uz (x,y, 0, t) ≈ 12pR c

κ(ωs)

−∂u(ω)∂ω ω=ωs

uztop(κ(ωs)) cosy -

˙ u ztop(κ(ωs))

ωs siny

⎡ ⎣ ⎢

⎤ ⎦ ⎥ (7.1.4)

u(ωs)=Rc / t; −∂u(ωs)∂ω ω=ωs

> 0; y =ωst- κ(ωs)Rc =[c(ωs)-u(ωs)]κ(ωs)t

Where it can be used, (7.1.4) offers considerable computational advantage over (7.1.1) as no integration whatsoever is required. The only computational issues are: a) how quickly ωs can be found; and b), what to do if distance R approaches turning point t gH . (Stationary phase is not valid here.) Usually you evaluate (7.1.4) for a fixed x, and many nearby values of t; or, for a fixed t, at many nearby values of x. Because u(ω) is a smooth function, you always have a good starting guess for ωs in the value for the previous time or position. With a good starting guess, you can "zero in" on the new value of ωs quickly. For R<t gH , y =[c(ωs ) -u(ωs )]k(ωs )t is clearly positive since phase velocity always exceeds group velocity.

Where R approaches t gH , k(ω) is vanishingly small, so from the dispersion relation (4.3.2) in general

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ω ≈k(ω) gh 1-k(ω) 2H 2

6 ⎛ ⎝ ⎜ ⎞

⎠ ⎟; u(ω) ≈ gh 1-

k(ω) 2 H2

2 ⎛ ⎝ ⎜ ⎞

⎠ ⎟; ∂u(ω)

∂k(ω)≈ −k(ω)H2 gH;

∂u(ω)∂ω

≈ −k(ω )H2

(7.1.5)and setting u(ω)=u(ωs)=Rc/t

k(ωs)=2

η2 t gH

⎡ ⎣ ⎢

⎤ ⎦ ⎥1/ 2

t gH- R c[ ]1 / 2

; ωs=κs3t

R c +2t gH[ ]; y = 23

2H 2t gH

⎡ ⎣ ⎢

⎤ ⎦ ⎥1/ 2

t gH -R c[ ]3 / 2

(7.1.6)So, phase y goes from positive to imaginary at the turning point Rc=t gH , however

32y ⎡ ⎣ ⎢

⎤ ⎦ ⎥2 / 3

=2

H2t gH

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

1/ 3

t gH −Rc[ ] is always real with a sign change from positive to negative at

the turning point. For R c < t gH , in place of the cosy term, we are suggested of solutions like

N(y) p(−Ja)1/4Ai(Ja) ωitη Ja=−

32(y +p / 4) ⎡

⎣ ⎢ ⎤ ⎦ ⎥2 /3

(7.1.7)

since Ai(J ) behaves like cosy for J << 0 {R c << t gH} . (Use A ′ i (J ) for the siny term.) For R c > t gH , in place of cosy I suggest

Ai(J b); J b = −32y ⎡

⎣ ⎢ ⎤ ⎦ ⎥2/ 3

−p4 ⎡ ⎣ ⎢

⎤ ⎦ ⎥2 /3

=2

H2t gH

⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

1/ 3

Rc−t gH[ ] −p4 ⎡ ⎣ ⎢

⎤ ⎦ ⎥2 / 3

(7.1.8)

since Ai(J ) behaves like e−y for J >> 0 {R c >> t gH} . To blend (7.1.7) to (7.1.8), I choose function N(y)=1+(N0-1)exp(-ly) with N0 and l taken such that solutions (7.1.7) and (7.1.8) and their derivatives equal at y=0. Be advised that this patching of solutions across the turning point is not strictly correct. It is just a way to extend the domain of usefulness of (7.1.3) and (7.1.4) near the front of the tsunami. However, the approach seems to work OK (see Figure 5) and because it can really speed the calculation of (6.3.5) when hundreds of integrations might be needed, I take a blind eye to small discrepancies near the turning point.

Inside the turning point R c < (t - t u ) gH , the stationary phase version of the simple slide (6.3.5)

uz (r, t) ≈uzJ0(ωT(ω, ρ, ρc)) κc(ω)dω

2pcosη(κc(ω)Hc) uc(ω)u(ω)0

∞∫ cos{ω(t−tu - W/2vρ)+ X(κc(ω), φ)[W(t)- W]}

× G (ρ,ρc)sinY(κc(ω), φ)W[ ]

Y(κc(ω),φ)sinX(κc(ω), φ)W(t)[ ]

X(κc(ω),φ)

is

uz (x,y, 0, t) ≈uzN(y) p (−Ja)

1/4Ai(Ja) κc(ωs)2pRccosη(κc(ωs)Hc)

1

−κ (ωs)∂u (ω)∂ω ω=ωs

uc(ωs)u(ωs) ⎡ ⎣ ⎢

⎤ ⎦ ⎥1/ 2

× G(ρ, ρc)sinY(κc(ωs),φ)W[ ]

Y(κc(ωs), φ)sinX(κc(ωs), φ)W(t)[ ]

X(κc(ωs), φ)

(7.1.9)

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u (ωs) =R c /(t- tu); −∂u (ωs)∂ω ω=ωs

> 0; Ja=− 32(y + p / 4) ⎡

⎣ ⎢ ⎤ ⎦ ⎥2 / 3

y =ωs(t−tu - W/2vρ−T(ωs, ρ, ρc))+ X(κc(ωs), φ)[W(t)-W]

where again the sub-c variables use the water depth at the source. The c (ω ) , u (ω) , k (ω) and H are path averaged quantities that were defined in section 6.2. Beyond the turning point R c > (t - t u ) gH , (7.1.9) becomes

uz (x,y, 0, t) ≈ uzWW(t) Ai(J b)2pRcH

HcH ⎡ ⎣ ⎢

⎤ ⎦ ⎥1/ 4

G(ρ,ρc)

(7.1.10)

J b =2

H 2t gH

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

1/ 3

R c − (t - t u ) gH [ ] −π4 ⎡ ⎣ ⎢

⎤ ⎦ ⎥2 / 3

Now, let’s see how well these approximations work.

7.2 Stationary phase example. Take initial surface displacement of the form

u ztop (r0 ) = A(1- r 2/W 2 )1/2 r < W

centered about the origin rc=0. The full tsunami solution (7.1.1) is

u z (x,y, 0, t) =k dk2p0

∞

∫ A cos[ω(κ)t][ ] dρ0 J0 (κR)(1- R2 /W 2 )1/2

ρ0∫

uz(x,y,0, t)= κ dκ2p0

∞

∫ Jo (κρ)A F(κ)cos[ω(κ)t][ ]F(κ)

F(κ)=2pκ2

sin(κW)κW

−cos(κW) ⎡ ⎣

⎤ ⎦; ρ= x2 + y2

(7.2.1)

The approximate tsunami solution (7.1.2) is

u z(x,y, 0, t) =k dk2p0

∞∫ A cos[ω(κ)t][ ]J0(κρ)2p dρ0 ρ0(1- ρ0

2/W 2 )1/2

ρ0∫

uz(x,y,0, t)= κ dκ2p0

∞∫ Jo (κρ)A cos[ω(κ)t][ ]F(κ); F(κ)=2p

W3 ⎡ ⎣

⎤ ⎦

(7.2.2)

(7.2.1) differs from (7.2.2) by O(k2W2) so "not too close” to the source means r>tU(k=2p/W). This is about the same assumption made in the stationary phase approximation (7.1.4.). Figure 5 (black traces) shows a cross section of the full solution (7.2.1) for W=10km, A=10m, B=0 out to 400 km distance and about 30 minutes time. Below as red traces are the stationary phase approximations (7.1.4) with (7.1.7) and (7.1.8) replacing the cosy term. As you can see it works very well for distances r>tU(k=2p/W) (dashed diagonal line) and even closer.

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Figure 5. Comparison of exact (black lines) and stationary phase approximations (red lines) of the tsunami from an initial circular shaped surface water pile. The approximation works well for distances r>tU(k=2p/W) (dashed diagonal line) and even less. When applied to small subblocks of a submarine landslide, short slow waves in the "invalid region" are not excited much anyway.

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8. Tsunami Run Up Estimates

In talking tsunami, everyone wants to hear the run up height on the beach. Course we all know that run up is a "highly non-linear" process, so what's a linear guy like me going to do? Here's my approach ---- Consider offshore points r, r0 and rc in moderately shallow water of depths h(r)>h(r0)>h(rc). Let the amplitude of the tsunami at point r be A(r), with A(r)<< h(r) -- nice and linear so far. In moderately shallow water, the shoaling factor is nearly independent of frequency and reduces to Green’s Law, SL= h(r)/h(r0 )1/ 4 . So by r0 the wave has grown to A(r0)=A(r)[h(r)/h(r0)]1/4. I propose that as waves come further in, they reach a critical amplitude of yh(rc) at position rc y is a constant parameter. It might depend on the slope on the beach for instance. Experiments suggest that y~1, or A(rc)~h(rc). Non-linear effects might show prior to this point, but it doesn't matter. I identify this critical amplitude yh(rc) as the eventual tsunami runup height R. So, if A(r) is the offshore wave amplitude in water of depth h(r), the depth h(rc) at which A(rc)= yh(rc) is h(rc)=y-4/5(r)A(r)4/5h(r)1/5, and I claim run up would be

R =y1/5A(r)4/5h(r)1/5 (8.1.1)

Now by this I'm not saying that real waves grow to a height equal to R and stay that large. Real non-breaking waves might stay below R until the final surge up the beach (see Figure 6). Real

Figure 6. Cartoon of wave shoaling and run up. Between point rc and the ultimate run-in position, complex non-linear processes govern tsunami behavior. An empirically based law R=y1/5A(r)4/5H(r)1/5 estimates wave height on the beach from the height of a pack of waves offshore in the linear domain.

Figure 6a. Comparison of laboratory runup data and empirical law 8.1.1 with y=1. Although it is simply based, formula 8.1.1 does a reasonable (perhaps even conservative) job of estimating runup from offshore heights.

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breaking waves might exceed R for a while the plunge back to that level. Run up laws like [R/h(r)]=a[A(r)/h(r)]b are common in the literature; in fact, (8.1.1) with y=1 fits laboratory observations of breaking and non-breaking waves within a factor of two over a large range conditions (see Figure 6a). In my view, the run up correction plays the role of a scalar transfer function T. From an offshore location r in the linear domain, T takes a wave pack of amplitude A(r) through all of the unmodeled processes to a final height on the beach R as

A(r)T(y,h(r),A(r))=R (8.1.2)T(y,h(r),A(r))= [yh(r)/A(r)]1/5 (8.1.3)

Like all transfer functions, T contains various parameters. In this case, T depends on input amplitude A(r) too, so run up is nonlinear, but only weakly so in that a proportional change in input wave height results in a largely proportional change in run up. (In experiments the proportionality constant is greater for non breaking waves than breaking ones, but still, even for breaking waves, if you increase the offshore height, you increase the runup). This behavior is consistent with what run up experts tell me (see Figure 6a again), but I'm open for suggestion. If h(r)/1000<A(r)<h(r)/10 and y=1 then the run up amplification factor T is 4.0 to 1.6.

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9. Tsunami Energy/ Tsunami Efficiency

9.1 Tsunami Energy. Consider the kinetic energy of a single tsunami eigenmode (4.4.2) per unit area of ocean surface at x,y

t n(x,y) = ρwω n2

2dz ux(ω n, x, y, z) 2 + uz (ω n, x, y, z) 2

[ ]0

H∫

=ρw2

uz (ω n , x,y, 0) 2 uz2(ω n, x, y, 0)

k2 (ω n )g2

ω n2 cosh2 (k(ω n )H)

dz sinh2 (k(ω n )(H − z)) + cosh2(k(ωn )(H − z))[ ]0

H∫

=ρw2

uz (ω n , x,y, 0) 2 k(ω n )g2

ω n2 tanh(k(ω n )H) =

gρw2

uz (ωn , x, y, 0) 2

(9.1.1)(9.1.1) says that the kinetic energy of the whole column of water is fixed by the amplitude of the mode's vertical displacement at the surface. Because of the orthonormal properties of eigenmodes, the kinetic energy at any time t of the entire sum of modes can be written in terms of the Hilbert transform of the tsunami vertical displacement at the surface like

t(t) =gρw

2dx dy Hz

2 (x, y, 0, t)wholeocean

∫ (9.1.2)The Hilbert transform of a function just adds an extra i to the spectrum of the function, effectively changing cos(ωt) to sin(ωt). For example, for the surface tsunami from 2-D bottom uplift of (5.3.1.)

uz (x,0, t) = Re dk−∞

∞∫


2pcosη(κH) (5.3.11)

The Hilbert transform would be

Hz (x,0, t) = Re dk−∞

∞∫ i


2pcosη(κH)

Generally where uz is large, Hz is small and visa versa. You expect kinetic energy to be large where velocities are large, this is where displacements are small, hence the Hz in (9.1.2). On the other hand, you'd expect gravitational potential energy to be large where displacements are large, so in fact, the total energy ET(t) of a tsunami is

ET (t) =gρω2

dx dy uz2(x,y,0, t)+ Hz

2(x, y,0, t)[ ]ωηoleocean

∫ (9.1.3)When averaged over many wavelengths, both terms in (9.1.3) are equal. Because I ignore dissipation, once the impact or landslide is over, ET(t) should be constant so long as none of the waves have run all the way to shore. The function

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E(t) = uz2 (x,y,0, t)+ Hz

2 (x,y,0, t)[ ]1/ 2

(9.1.4)I call the tsunami "envelope". It is always positive, spatially smooth and it has units of meters. The

envelope acts like the surface of a tight sheet placed over the tsunami wave train. I often plot the tsunami envelope instead of the vertical displacement itself because E(t) shows well the size and shoaling of the waves with out all the distractions of sign changes (see Figure 7).

9.2 Tsunami Efficiency. Tsunami efficiency is nothing more than the ratio of tsunami energy to the gravitational energy released in a landslide or the kinentic impact energy of asteroids. Tsunami efficiency has been a topic of interest over the years. We have tsunami energy already (that's the hard part!) all we need is the energy of the respective sources.

For landslides into the ocean, the released gravitational energy available for wave generation is

EL =g dA(ρ)ρeff(ρ)Du(ρ)η(ρ) ρ∫ (9.2.1)

Figure 7. Example of the use of the tsunami envelope (9.1.4). To the left is the surface vertical motion of a tsunami from a small submarine landslide with red being up and blue down. The envelope to the right matches the same heights at the crests but varies smoothly elsewhere. You can see the shoaling of the waves near shore as the bright spot.

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Here h(r) is the pre-slide elevation below sea level and Du(r) is the excavation or deposition thickness (excavation take negative). The integral covers all of the slide area where Du(r)0. ρeff(r) is the effective density of the column of slide material at r. ρeff(r) equals ρs, the density of the slide material. ρeff(r) equals (ρs-ρw) for fully submerged columns [h(r)>|Du(r)|>0]. For partly submerged columns [|Du(r)|>h(r)>0], ρeff(r) equals ρs-ρwh(r)/|Du(r)|. EL fixes the energy budget of all landslide processes. The energy of radiated tsunami and all manner of frictional losses draw from this pool (tsunami generated during landslides can be considered a frictional loss). Note that EL depends only on the initial and final state of the slide. Tsunami energy on the other hand depends on the entire kinematic history of the slide (e.g. equation 5.3.12). So depending on how it unfolds (velocity, shape of flow, etc.) any given slide with the same initial and final state could have very high tsunami efficiency down to zero tsunami efficiency. In my models most landslides give 0.5% (small submarine events) to maybe 10% (stratovolcano collapses) of their energy to tsunami.

Asteroid Impact kinetic energy for an asteroid of radius RI, density ρI and velocity VI is

EI = (1/2)ρI(4p/3)R3IV2

I (9.2.2)As shown in Figure 1 and Figure 8, I model impact tsunami from an initial parabolic surface cavity of the form

uzimpact (r) =D c(1 −ρ

2 / Rc2 ) ρ< 2R

c (9.2.3)where Dc and Rc are the depth and radius of the initial cavity. At t=0, only the first term in (9.1.3) counts in this case and we can integrate (9.2.3) squared to obtain tsunami efficiency as

ET/EI = ρwg(DCRC)2/ 2ρIR3IV2

I (9.2.4)Experimental and observational scaling laws give cavity Dc and Rc as a function of impactor RI, ρI

and VI. The result is that impactors <500m diameter have tsunami efficiency of about 15% -- much greater than most landslides. What did you expect? These rocks are moving at 20 km/s! Still larger impactors "bottom out" and blow away the entire thickness of the ocean. Tsunami efficiency falls for those big guys even though their waves are larger.

Figure 8. Envelope of tsunami from a 200 m diameter asteroid impact. The frames show time in one hour increments.

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10. Simple Landslide Dynamics

10.1 The formulations in these notes tell you how to compute tsunami from any landslide once you prescribe its kinematics. Outside of wave tank experiments however, landslides are messy business so you don't want to characterize them with more parameters than you can constrain. For historical landslides, you usually know where the material ended up and maybe where it came from. For future events you usually know where the material will come from and maybe where it will go. So after the shape of the seafloor itself, perhaps the volume of landslides is something that you know the best and with luck, its initial area and thickness. After these, the next likely parameters to be fixed are drop height h0 and run out distance xc. For historical events you can just measure these. For future events, you still can make a good guess drop height = height of slope etc. I suggest that if you want to build a dynamic landslide model you'd best couch it in terms of these few things that you can likely measure.

I'd say that the simplest dynamic model that has relevance is the sliding block with basal friction (Figure 9). This model fixes block acceleration a(x) as the gravitational acceleration less the frictional acceleration

a(x) =g(sinb(x) −m cosb(x))=g(tanb(x)−m)cosb(x)≈−g(dη(x)/dx+m) (10.1.1)

Here, h(x) is the slope profile shape and b(x) is its slope angle. For subaerial slides, m is the coefficient of basal friction. For submarine slides, m can be considered an "effective" coefficient of friction that includes true basal friction plus all other loss mechanisms (viscous dissipation, energy transfer to waves, etc.). Integrating the approximate version of formula (10.1.1) yields block velocity directly as a function of the slope profile

v(x) = 2 a(̂ x )dˆ x 0

x∫ ⎡ ⎣ ⎢ ⎢

⎤ ⎦ ⎥ ⎥

1/ 2

= 2g η(0)- η(x)−mx[ ]1/ 2 (10.1.2)

The slide hits peak velocity at position xp where h(xp)/x=-m. Smooth exponential curves

h(x) =η0e−xx with x =tanβ0 / h0 (10.1.3)

characterize many slopes with b0 being the initial inclination and h0 the drop height. Shape (10.1.3) yields landslide velocity and travel time as a function of run out distance (Figure 10):

Figure 9. Simple sliding block with friction. I use this model to first approach any new landslide.

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v(x) = 2gη0 1−e−xx −mx / η0[ ]1/ 2

; t(x) =dˆ x v(̂ x )0

x∫ (10.1.4)

Final runout distance xc and slide duration are found by solving

xc = (h0 / m) 1−e−xxc ⎡ ⎣

⎤ ⎦≈(η0 / m) (10.1.5)

then evaluating tc =d ˆ x v(̂ x )0

xc∫ . Mean velocity is defined by the ratio of xc to tc whereas peak velocity

at xp=ln(tanb0/m)/x is just the free fall speed times a number less than one

v(x p) = 2gη0 1−(m / tanb0 )(1+ln(tanb0 /m)[ ]1/ 2

So with a known slide volume, area and thickness together with the slope shape (10.1.3) and runout distance or friction estimate (10.1.5) you can use (10.1.4) to get a velocity history. This is all you need to get a plausible first model of the tsunami from any landslide that you might approach, say Figure 11. Course, if you know more about the slide, then you can pose fancier and fancier kinematics, but everything is laid out here for you to do "Tsunami - a la Ward".

GOOD LUCK.

Figure 10. Landslide velocity from (10.1.4) with h0=800m, b0=12.50 and basal friction values of m= 0.08 to 0.05. Slidesaccelerate and de-accelerate quickly and spend much of their time at a nearly constant speed.

Figure 11. Landslide recently imaged by multi-scan sonar on the flanks of a seamount SE of Coast Rica. I used theories in Chapter 10 and elsewhere in these notes to guess the nature of tsunami hazard posed by this event and the other sea mounts out there.

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