1
Classical Tsunami Theory - a la Ward
Steven N. Ward 3/6/03
1. Formulation
1.1. Fluid dynamics starts with Euler's equations
(r, t)
and the continuity equation
Dv(r, t)
   t(r, t) + (r, t)F(r, t)
Dt
D(r,t)
 (r, t) v(r, t) = 0
Dt
to be solved in fluid volume V.
(1.1.1)
(1.1.2)
Here (r, t) is density, v(r, t) =
u(r, t)
t
displacement, t(r, t) is the stress tensor, F(r, t) is body force per unit mass, and
is velocity, u(r, t) is
D

  v(r, t)  
Dt 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
(r, t)
Dv(r, t)
 p(r, t) + (r, t)F(r,t)
Dt
(1.2.2)
1.3. If the motions and body forces are irrotational
Dv(r, t) v(r, t)
v(r, t) 1 2

 v(r, t)  v(r, t) =
 v (r, t)  v(r, t)    v(r, t)
Dt
t
t
2
v(r, t) 1 2
=
 v (r, t)
t
2
(1.3.1)
and
F(r, t) = -(r, t) = g(r, t)
(1.3.2)
then (1.2.2) become the Bernoulli equations
(r, t)
v(r, t)
1
2
 p(r, t) - (r, t)v (r, t)- (r, t)(r, t)
t
2
(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
(r, t) =  0 (r) + 1 (r, t)
p(r, t) = p0 (r) + p1 (r, t)
(r, t) = 0 (r) + 1 (r, t)
(1.4.1)
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
 0 (r) Ý
uÝ(r, t)  p1 (r, t)+  [0 (r)u(r, t)] 0 (r) -  0 (r)1 (r,t)
(1.4.2)
2
1 (r, t)
   [ 0 (r)v(r, t)] = 0  1 (r,t)    [ 0 (r)u(r, t)]
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
(1.4.4)
Pressure increment p1(r, t) consists of an elastic term and an advected term
p 0 (r) = - 0 (r) 0 (r) =  0 (r)g0 (r)
(1.4.5)
The (r) is fluid incompressibility. Equations (1.4.2), (1.4.5) together with
p1 (r, t) = -(r) u(r, t) - u(r, t)  p 0 (r)
 2 1(r, t) = -4G [0 (r)u(r, t)]
(1.4.6)
represent five equations for five unknown functions u(r, t) , p1(r, t), 1(r, 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
(r) = , (r)  0
and gravity to be constant and unchanging
 0 (r) = -g ˆz, 1 (r, t)  0
The four equations of interest now are
Ý
Ý(r, t)  p1 (r, t)- g 0 zˆ  u(r, t)
0u
p1 (r, t) = -  u(r, t) - g 0 uz (r, t)
or
with
(2.1.1)
0Ý
uÝ(r, t)  p e (r, t)+ 0 g[u z (r, t) - zˆ  u(r, t)]
pe (r, t) = -  u(r, t)
(2.1.2)
(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.
pe (r, t) = p1 (r, t)+ u(r, t)  p 0 (r) = -  u(r, t)
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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.
u z (r, t) pe (r, 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
Ý
Ý(r, t)  p e (r, t)
0u
pe (r, t) = -  u(r, t)
(3.1.3a,b)
with more complex boundary conditions
u z (r, t) p1(r, t)    p e (r, t) +  0 gu z (r, 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 zˆ , uy and all /y =0. Equations (3.1.3a,b) become
0u
Ý
Ýz (x, z, t)  p e (x, z, t)/z
Ý
Ýx (x, z, t)  p e (x, z, t)/x
0u
(4.1.1)
pe (x, z, t) = -[u x (x, z, t)/x + uz (x, z, t)/z]
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, )   dx  dt f(x, z, t)e -i(kx-t) ;
(4.1.2a)
 
These are reconstituted by
f(x, z, t) 
1


i(kx-t)
(4.1.2b)
 dk  d f(k, z, )e
4 2  
With this convention in (4.1.1)  / t  i and  / x  ik
 uz (k, z, )

z 
pe (k, z, )

 0

2
 0 
2 / 0 2 u z (k, z, ) 

 (4.1.3)
0
p e (k, z, ) 
where 2 = 2 (, k) = k2 - 0  2 / 2 and use was made of
u x (k, z, )  ikpe (k, z,  )/0  2
(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
 uz (k, z,  )

z 
pe (k, z,  )


k2 g/ 2

2
2
2 2
 0   k 0 g /
2 /0  2 u z (k, z, ) 
(4.1.5)

k2 g/ 2 
p e (k, z, ) 

with u x (k, z, )  ik[p e (k, z, ) -  0 guz (k, z,  )] / 0  2 (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,  ) 
S (z, z 0 ) /  0 2 uz (k, z 0 , )
p (k, z,  )   C(z, z 0 )

 e
   2 S(z, z ) / 

C(z, z 0 )
 0
pe (k, z 0 , )
0
(4.1.7)
where C(z,z0)=cosh[(z-z0)] and S(z,z0)=sinh[(z-z0)]. We can also write the solutions (4.1.7) in
terms of uz and p1(k, z, ) = p e (k, z, ) + u z (k, z, )0g that are continuous across the undeformed
surfaces
uz (k, z,  )  C(z, z 0 )  gS(z, z 0 ) / 2


p1 (k, z, ) 
  0 S(z, z 0 ) 2  g 2 2 /  2





u z (k, z 0 ,) 
2
C(z, z 0 )  gS(z, z 0 ) /  
p1 (k, z 0 , ) 


S(z, z 0 )/  0 2
(4.1.8)
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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,) = always specified (4.2.1)
zero or otherwise. At the sea surface, the linearized boundary condition (3.1.5) says that
p1(k,0,)=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  [cosh(H)  gs inh(H) / 2 ]p1 (k, H, )
(4.3.1)
The only way this can hold is if
 2 (k)  g((k)) tanh[((k))H]
 2  g(k()) tanh[(k())H]
(4.3.2)
The frequency (k) for a given wavenumber k, or the wavenumber k() 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
   and  = k2 - 0 2/  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
(k())g s inh((k())(H  z))
cosh((k())H)
2
ik( )g cosh((k( ))(H  z)
u x (z, ) = cosh((k( ))H)
2
(4.4.1)
cosh((k( ))(H  z)
pe (z,  ) = - 0 g
cosh((k( ))H)
s inh((k())z))
p1 (z,  ) = - 0 g
cosh((k( ))H) s inh((k())H)
u z (z,  ) =
Clearly, uz(H,)=p1(0,)=0 as required and pe(0,)=-0guz(0,). In terms of tsunami "eigenmodes"
(k( ))g s inh((k())(H  z)) i(k()x t)
e
cosh((k())H)
2
ik()g cosh((k())(H  z) i(k()x t)
u x (x, z, t, ) = e
cosh((k())H)
2
(4.4.2)
cosh((k())(H  z) i(k()x t)
pe (x, z, t, ) = -0 g
e
cosh((k())H)
s inh((k())z))
p1 (x, z, t, ) = -0 g
e i(k()x t)
cosh((k())H)s inh((k())H)
u z (x, z, t, ) =
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Taking the real part of (4.4.2): uz ~ cos(k()x-t) and ux~ sin(k()x-t). 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  integrate (4.1.5) numerically from z=0 to z=H starting with
uz (k, z, ) 1
p (k, z, )  0 (4.4.5)
 e
  
per (3.1.1), while incrementally adjusting k to a k() such that
0
uz (H, ) 

p (H, )  p (k(), H,) (4.4.6)
 e
  e

For example, for long waves kH>>1 (4.1.5) and (4.4.5) integrate to

uz (k, z,  )  (1 zk 2 g/ 2 )
 



pe (k, z,  ) 0 z(k2 g 2 / 2  2 )
To satisfy (4.4.6) 2(k)=gk2H or k2()=2/gH , whence the rigorous solutions
u z (z,  ) = (1 z/H)
ik( )g
2
u x (z, )  
(1 z /g)
2
z
pe (z,  ) =  0 g (1- H2 / g)
H
For long waves, (4.3.2) is  2 (k)  g2 ((k))H ; 2  g2 (k())H and the classical solutions (4.4.1)
become
u z (z,  ) = (1  z/H)
u x (z, ) = -
ik( )g

pe (z,  ) = - 0 g
2
p1 (z,  ) = - 0 g
z
H
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(H2/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
u z (x, 0, t = 0) = u top
z (x)
(5.1.1)
and its transform

top
-ikx
(5.1.2)
u top
z (k)   dx u z (x) e

In this case, the eigenmodes (4.4.2)
give the evolved tsunami straight away
[From now on, I assume an
incompressible fluid so (k())  k() ] Figure 1. Equation (5.1.3a) evaluated with a parabolic initial
displacement of the sea surface. This is my concept of asteroid impact

tsunami.
u top(k)  s inh(k(H  z))
cosh
(k(H  z))  i(kx (k)t)
u(x, z, t) = Re  dk z
zˆ
 ixˆ

e
2
s
inh(kH)
s
inh(kH)



or

u top(k())
u(x, z, t) = Re  d z
2 u()

 s inh(k()(H  z))
cosh (k()(H  z))  i(k()x t)
zˆ
 iˆx
e

s inh(k( )H) 
 s inh(k()H)

(5.1.3a,b)
In (5.1.3b), u()=d/dk(), the tsunami group velocity. You can see in (5.1.3a) that at the surface
z=0 for t=0
 u top (k)
u z (x, 0, t) =  dk z
eikx
2

(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

u x (x, 0, 0) =  dk

i u top
z (k) cosh(kH) e ikx
2
sinh(kH)
You can not specify vertical and horizontal displacements separately.
Suppose instead we have some initial vertical surface velocity condition like
uÝz (x, 0, t = 0) = uÝtop
z (x)
with its transform
(5.1.5)

top
-ikx
(5.1.6)
uÝtop
z (k)   dx uÝz (x) e

The same reasoning suggests that tsunami the velocity field would be
9

uÝtop(k())
uÝ(x, z, t) = Re  d z
2 u()

 s inh(k()(H  z))
cosh (k()(H  z))  i(k()x t)
zˆ s inh(k()H)  iˆx s inh(k( )H) e


(5.1.7)
or
 uÝtop(k)  s inh(k(H  z))
cosh (k(H  z))  i(kx (k)t)
uÝ(x, z, t) = Re  dk z
zˆ
 ixˆ
e

2
s
inh(kH)
s inh(kH) 



and that the tsunami displacement field is

uÝtop(k())  s inh(k()(H  z))
cosh (k()(H  z))  i(k()x t)
u(x, z, t) = Re  d z
zˆ s inh(k()H)  iˆx s inh(k( )H) e
i2u()



(5.1.8)
or

uÝtop (k)  s inh(k(H  z))
cosh (k (H  z))  i(kx(k)t)
ˆz
u(x, z, t) = Re  dk z
 ixˆ
e

i(k )2   s inh(kH)
s inh(kH) 


(5.1.8) follows from (5.1.7) because application of (-i)-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

u(x, 0, t) = Re  d

u top
utop
z (k()) + iÝ
z (k( ))/ s inh(k()(H  z))
cosh(k()(H  z))  i(k()x t)
zˆ
 iˆx
e


2u( )
s inh(k()H)
s inh(k( )H)


or
uztop(k) + iÝ
utop

z (k)/(k)  s inh(k( )(H  z))
cosh(k( )(H  z))  i(kx (k)t)
ˆz
u(x, 0, t) = Re  dk
 ixˆ
e


2


s inh(k( )H)
s inh(k()H)


and
top
iu top

z (k()) + uÝz (k( )) s inh(k()(H  z))
cosh(k()(H  z))  i(k()x t)
uÝ(x, 0, t) = Re  d
zˆ
 ixˆ
e
(5.1.9)


or

uÝ(x, 0, t) = Re  dk



2u()
s inh(k()H)
s inh(k()H)


top
i(k)u top
z (k) + uÝz (k) s inh(k()(H  z))
cosh(k()(H  z))  i(kx ( k)t)
zˆ
 ixˆ
e
2


s inh(k()H)
s inh(k( )H)


(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
u z (x, 0, t) = utop
z (x, t) H(t)
(5.2.1)
The convolution theorem tells us how to form the tsunami fields given (5.1.3)
10

top
 s inh(k()(H  z))
cosh (k()(H  z))  i(k()x t)
 iˆx
zˆ
e
 s inh(k()H)
s inh(k( )H) 
u (k())
u(x, z, t) = Re  d z
2 u()


t

0
i(k()x 0 t 0 )
  dx 0  dt 0 uÝtop
z (x 0 , t0 ) e
(5.2.2)
or
 u top(k)
u(x, z, t) = Re  dk z
2


t

0
 s inh(k(H  z))
cosh (k(H  z))  i(kx (k)t)
zˆ
 ixˆ
e

s inh(kH) 
 s inh(kH)

i(kx0 (k)t 0 )
  dx 0  dt 0 uÝtop
z (x 0 , t0 ) e
Be aware of the time differentiation of the surface condition in (5.2.2). Sometimes, u top
z (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.
top
u top
z (x, t) = u z (x)S(t - t(x)); S(t) = 0 if t < 0

t

t t(x0 )

0

0
top
i(k()x 0 t 0 )
i(k( )x 0  t(x0 ))
  dx0 utop
 dx 0  dt 0 uÝz (x 0 , t0 ) e
z (x 0 )e
it
 dt 0 SÝ(t 0 ) e 0 (5.2.3)
If S(t) was a step function, the last integral would equal 1 for t>t(x 0) 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
u z (x, H, t = 0) = ubot
z (x)
and its transform
(5.3.1)

bot
-ikx
u bot
z (k)   dx u z (x) e
(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, ) 


0


 C(0, H)  gS(0, H) / 2
 S(0, H)
 0
2  g2 2 /  2
 

 bot
 u z (k,  ) 

C(0, H)  gS(0, H) / 2 
p (k, H,  )
 1
S(0, H) /  0  2

(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, ) = -

0 g 2 1 2 T (H, 0)/kg
2
2
   (k)
u bot(k,  )
z
(5.3.4)
then substitute into the first equation of (5.3.30 at any depth z to find displacement
11
  2 C(H, 0)C(z, H)  S(z, H)S(H, 0) 



gk C(z, H)S(H, 0)  C(H, 0)S(z, H) 
 bot
u z (k, z, ) =
u (k, )
2
2
z
C(H, 0)[   (k)]
(5.3.5)
Take care here to distinguish general frequency  from the characteristic frequency (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 bot (k,) actually is any function of time. If we want it to be a fixed initial uplift
z
then
u
bot
z
(k, ) =
u bot (k)
z
-i
(5.3.6)
The (-i)-1 is the time transform of a step function, thus
 iC(H, 0)C(z, H)  S(z, H)S(H, 0) 


(igk/)C(z, H)S(H, 0)  C(H, 0)S(z, H) bot
u z (k, z, ) =
u (k)
z
C(H, 0)[ 2  2 (k)]
(5.3.7)
We apply the inverse time transform, making use of the residue theorem to evaluate the simple
poles that lay at =0, =(k). Here's a useful table of transforms.
i
 co s[ (k)t ]H(t)
  2 (k)

 i co s[(k)t]H(t)
2
  2 (k)
(5.3.8)
1
i co s[ (k )t ]H(t) iH(t)


(2   2 (k ))
 2 (k)
2 (k)
i
co s[ (k )t ]H(t)
H(t)

 2
2
2
2
   (k)
 (k )
 (k)
2
In the time domain (5.3.7) is
 C(H, 0)C(z, H)  S(z, H)S(H, 0) cos[(k)t]



 C(H, 0) C(z, H)S(H, 0)  C(H, 0)S(z, H)  cos[(k)t]+ 1


 S(H, 0)
 bot
u z (k, z, t) =
u (k)H(t) (5.3.9)
z
C(H, 0)
=
S(z, H)

S(z, H)
 bot
bot
u (k) cos[(k)t]H(t) + C(z, H) 
S(z, H)u (k)H(t)
C(H, 0)S(H, 0) z
T
(H,
0)

 z
All we have to do now is the inverse wavenumber transform. For t>0 we have
 sinh( k(H  z))
u bot
cosh(k(H  z))  i(kx  (k)t)
z (k)
u(x, z, t) = Re  dk
 ixˆ
zˆ
e
2 cosh(kH)  sinh( kH)
sinh( kH) 


 
u bot
sinh( k(H - z))  ˆ 
cosh(k(H - z))  ikx
z (k)
 iksinh( k(H - z)) 
e
zˆ cosh(k(H - z)) 
2  
tanh( kH)  
tanh( kH) 

(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

+  dk

12
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.
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
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.
 u bot (k) e i(kx ( k)t)
u z (x, 0, t) = Re  dk z
2 cosh(kH )

(5.3.11)
You can't get much simpler than this! Well,.... you can actually. Let's ignore dispersion for the
moment, then (k)= |k| gH , and so (5.3.11) becomes
 u bot (k) e i(kx |k| gHt)
1
u z (x, 0, t) = Re  dk z
 G(x  gHt) + G(x  gHt)
2 cosh(kH)
2


with
 u bot (k) e ikx
G(x) =  dk z
2 cosh(kH)

-W/2<x<W/2 then

If the bottom uplift u bot
z (x) = constant U over some small width

G(x)  UW  dk
0
coskx
UW
 x 

sech 
2H 
 cosh(kH ) 2H
So, you can consider the "2-D, non-dispersive landslide greens function" to be
g(x, t, x 0 , t 0 ) 
UW
4H

[(x  x0 )  gH(t - t 0 )] 
[(x  x0 )  gH(t - t0 )] 
sech
+ sech

2H
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
13

u z (x, 0, t) = Re  dk

t
e i(kx(k)t) 
bot
i(kx 0 (k)t 0 )
 dx 0  dt 0 uÝz (x 0, t 0) e
2 cosh(kH)  0
(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.
14
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 xˆ +ky yˆ , k=|k|, r=x xˆ +y yˆ
2) product kx goes to kr
3) 1-D integrals over wave number and position now cover all 2-D space
4) the unit vector xˆ goes to kˆ
With these, the propagating 2-D tsunami from an arbitrary bottom landslide

ikx
 sinh(k(H  z))
e
cosh(k (H  z)) 
zˆ
 ixˆ

2

cosh(kH)

sinh(kH)
sinh(kH) 


u(x, z, t) =  dk

  dx 0 e
ikx 0

becomes in 3-D
u(x, y, z, t) =  dk
k
t
(6.1.1)
bot
 dt 0 uÝz (x 0 , t0 ) cos[(k)(t - t 0 )]
0
e ik r
 s inh(k(H  z)) ˆ cosh(k(H  z)) 
 ik
zˆ
s inh(kH) 

4  cosh(kH)  s inh(kH)
2
  dr0 e
 ikr0
r0
t
(6.1.2)
bot
 dt 0 uÝz (r0 , t 0 ) cos[(k)(t - t0 )]
0
where dk=dkxdky, dr0=dx0dy0. There are millions of ways we can recast (6.1.2). One is

k dk

s inh(k(h - z)) ˆ
cosh(k(h - z)) 
u(r, t)  
dr0 zˆJ 0 (kR)
 RJ1 (kR)

2 cosh(kh) r
s inh(kh)
s inh(kh) 


0
0
t
(6.1.3)
  dt0 uÝbot
z (r0 ,t 0 )cos[(k)(t  t0 )]
0
where R=|r-r0|, Rˆ = (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

k dk

sinh(k(h - z)) ˆ
cosh(k(h - z)) 
u(r, t)  
zˆ J0 (kR c )
 R cJ1 (kR c )

2
cosh(kh)
sinh(kh)
sinh(kh) 


0
(6.1.4)
t
ˆ
  dr0  dt 0 uÝbot
z (r0 , t 0 ) cos[(k)(t  t 0 ) + k(R c  r0 )]
r0
0
where Rc=|r-rc|, Rˆ 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 Rˆ  r0<0 from the
c
observation point than rc and advances the start time of sources closer Rˆ 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 Rˆ c from the reference location and
not the receiver distance Rc itself. Thus, for simple sources, these integrals can be evaluated
analytically and we are left with a single integral to perform numerically. Equation (6.1.4) is pretty
15
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

k()d
u(r, t)  
2u(
)cosh(k()H)
0

s inh(k(
zˆ J0 (T(,r,rc ))

s inh(

t
Figure 3. Equation (6.1.4) evaluated for surface vertical tsunami motion from a
3-D sliding block landslide source.
ˆ
  dr0  dt0 uÝbot
z (r0 , t0 )cos[(t  t 0 )  k( )(R c
r0
0
(6.2.1)
In (6.2.1) we identified k()Rc=Rc/c()=T(,r,rc). c()=/k() is the tsunami phase velocity
and T(,r,rc) is the travel time of a tsunami wave of frequency  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

k c ( )d
u(r, t)  
2u c () cosh(k c ()H c )
0
1/ 2
u c ( ) 

 u() 


s inh(k()(H - z)) ˆ
cosh(k( )(H - z)) 
zˆ J0 (T(,r, rc ))
 Rc J1 (T(, r, rc ))

s inh(k()H)
s inh(k( )H) 


(6.2.2)
t
ˆ
G(r,rc )   dr0  dt0 Ý
ubot
z (r0 , t0 ) cos[(t  t0 ) + k c ()( R c  r0 ) ]
r0
0
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(,r,rc)=R/ c( ) now is computed in a variable depth
ocean along ray paths with c( ) being the mean phase velocity along the path. OK, so how do I
compute T(,r,rc)?
1) Evaluate a zero frequency travel time
T(  0,r,rc ) 
 dp / gH (p)
ray pat h
2) Find a “mean” ocean depth H(r,rc ) associated with the ray path of length P(r,rc),
2
2
H(r,rc )  P (r,rc )/gT (0, r,rc )
3) Determine a “mean” wavenumber k( ) associated with the ray path by solving
2
  gk()tanh[k()H(r,rc )]
c( ) , the mean phase velocity along the path is c( ) =  / k ( ) . You can compute a mean group
velocity along the path u( ) using the standard uniform depth formulas with k( ) and H(r,rc ) in
place of k() and H. I use u( ) and the other path averaged quantities later in these notes.
16
Also new in (6,2,2) is G(r,rc), a ray geometrical spreading factor, and SL=[uc()/u()]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 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
Figure 4. Equation (6.3.3) evaluated with fifteen 2x2km subblocks that each are small, you can assign to
host two simple slides that uplift and fall in a way to mimic a block sliding them simple shapes (square)
down the side of a v-shaped channel. The ocean here has depths variable from 0
and
behaviors
(uniform
to 700m.
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 u z 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
17
t
bot
ˆ r )]
 dr0  dt 0 uÝz (r0 , t 0 )cos[(t  t 0 ) + k c ()( R
c 0
r0
0
-W/2 +W(t)
I n (,t) = u z

-W/2
W/2
 dy 0 cos[ (t  t u (x 0 )) + k c ()(y 0 sin  x 0 cos) ]
dx 0
-W/2
(6.3.1-4)
sinY(k c (), )W  -W/2 +W(t)
I n (,t) = u z
 dx 0 cos[(t  tu (x 0 )) + x0 k c () cos ]
Y(k c (), )
-W/2
I n (,t) = u z
sinY(k c (), )W 
Y(k c (), )
min[t,t f ]
 dt0 v r (t 0 ) cos[(t  t0 ) + x0 (t 0 )k c () cos ]
tu
The tu= tu(-W/2) and tf= tu(W/2) are the times that the simple slide started and finished. The vr(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 x0 or t0 such
that the change in phase of the cosine function is small. For slides starting and slowing to zero
velocity, tu(x0) 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
I n (,t) = u z
s inY(k c (), )W  s inX(k c (), )W(t)
cos {(t  t u - W/2v r )  X(k c (),)[W(t) - W]}
Y(k c (), )
X(k c (), )
(6.3.5)
with W(t)=min[(t-tu)vr,W]. In 6.3.2-5, =-0 is the angle between the slide direction and the
receiver direction and
Y(k , )  ksin / 2 ; X(k, )  kcos  /v r / 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



sinY(k c (), )W  sinX(k c (), )W(t) sin T /2
I n (,t) = u z
Y(k c (), )
X(k c (), )
TR /2
(6.3.7)
 cos{(t  t u - W/2v r )  X(k c (), )[W(t) - W]- T  /2}
where T*=min(TR,t-tu). The -T*/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=∞
I n (,t) = u z

cos{(t  t u ) - T/2}

sinkc ()sin W/2 sink c ()cosW/2 sin T /2
k c ()sin/2
k c ()cos/2
TR /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.
18
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)
u z (r, t) 



all
0
sublock s
n
J0 (T(, r, rc )) k c ( )d
G(r, rc ) I n (, t)
2 cosh(k c ( )H c ) u c ( )u( )
(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.
19
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
u z (x, y, 0, t) =  dk
k

e ikr

top
Ý
uz (r0 )
ikr0
 u top
s in[(k)t]
z (r0 ) cos[(k)t] +
2  dr0 e
 (k)
4 r0




or

 k dk
top

uÝ (r )
top
 dr0 J 0 (kR)  uz (r0 ) cos[(k)t] + z 0 s in[(k)t]  (7.1.1)
2 r
(k)




0
0
u z (x, y, 0, t) = 
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
 k dk

J0 (kRc )  utop
z (k) cos[(k)t] +
2

0
u z (x, y, 0, t)  
with
ˆ r ) top
ik(R
c 0 u
u top
z (k) = Re  dr0 e
z (r0 ) ;
r0

Ý
utop
z (k) s in[(k)t]

(k)

(7.1.2)
ˆ r ) top
ik(R
c 0 Ý
uÝtop
uz (r0 )
z (k) = Re  dr0 e
r0
and Rc=|r-rc|, Rˆ c = (r-rc)/Rc. The stationary phase approximation to (7.1.1) and (7.1.2) are
u z (x,y,0,t)   dr0
r0
1
2R
u( s)  R / t; 
u z (x, y, 0, t) 
1
2R c

k( s )
 u( )

u( s)

 top
uÝtopk(r0 )

u z (r0 ) cos  - z
sin


s

 (7.1.3)
  s
  s
 0;    s t - k( s )R  [c(s ) - u( s )]k( s )t
 top

uÝtop (k( s ))
s in 
 u z (k( s )) cos  - z
s

 (7.1.4)
k(s )
u()

 
s
u( s )
u( s)  Rc / t; 
 0;    s t - k( s )Rc  [c( s ) - u( s)]k(s )t
   
s
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 s 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() is a smooth function, you always have a good starting
guess for s in the value for the previous time or position. With a good starting guess, you can
"zero in" on the new value of s quickly. For R<t gH ,  [c( s )-u( s )]k( s )t is clearly
positive since phase velocity always exceeds group velocity. Where R approaches t gH , k() is
vanishingly small, so from the dispersion relation (4.3.2) in general
 k( ) 2 H 2 
;
6


  k( ) gh 1-
and setting u()=u(s)=Rc/t
 k( ) 2 H2  u( )
 u( )
2
2
;
u( )  gh 1 k( )H gH ;
 k( )H
2

k(

)



(7.1.5)
20
1/ 2
 2

k( s) =  2

h t gH 
t
gH - R c

1 /2
1/ 2

k
2  2
;  s = s R c  2t gH ;  =
2

3t
3 
H t gH 


t
gH - R c

3 /2
So, phase  goes from positive to imaginary at the turning point Rc=t
2/3
3 

2 
 

1/ 3
 2

 2


H t gH 

t

gH  Rc
gH
(7.1.6)
, however
is always real with a sign change from positive to negative at
the turning point. For R c < t gH , in place of the cos term, we are suggested of solutions like
1/4
N() ( a )
Ai( a ) with
3
2 /3
 a    (   / 4)
2

(7.1.7)
since Ai( ) behaves like cos for   0 {R c << t gH } . (Use Ai ( ) for the sin term.) For
R c > t gH , in place of cos I suggest
2/ 3
 3 
Ai( b );  b   
 2 

1/ 3
2 /3 

2
 
  
  2

4 
H t gH 


Rc  t
2/3
 
gH   
4 

(7.1.8)
since Ai( ) behaves like e
for   0 {R c >> t gH } . To blend (7.1.7) to (7.1.8), I choose function
N()=1+(N0-1)exp(-) with N0 and  taken such that solutions (7.1.7) and (7.1.8) and their
derivatives equal at =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)

u z (r, t)  u z 
J 0 (T(, r, rc )) k c ()d 
0 2 cosh(k c ()H c ) u c ()u()
 G(r,rc )
cos {(t  t u - W/2v r )  X(k c (), )[W(t) - W]}
s inY(k c ( ), )W  s inX(k c (), )W(t)
Y(k c ( ),)
X(k c ( ),)
is
u N( )  ( a )1/4 Ai( a ) k c ( s )
u z (x, y, 0, t)  z
2Rc cosh(kc (s )H c )
 G(r, rc )
1
u()
 k( s )
 
1/ 2
uc (s ) 
 u( ) 

s 
s
s inY(k c (s ),)W  s inX(k c (s ), )W(t)
Y(k c (s ), )
u( s )  R c /(t - t u );
(7.1.9)
X(k c (s ), )
2/3
u(s )
3


 0; a    (   / 4) 
2

  
s
   s (t  tu - W/2v r  T( s , r, rc ))  X(k c (s ), )[W(t) - W]
where again the sub-c variables use the water depth at the source. The c( ) , u( ) , k() 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
21
u WW(t)Ai(
u z (x, y, 0, t)  z
2R c H
(7.1.10)
 2
1/ 3
 R c  (t - t u )
 b   2

H t gH 


Now, let’s see
how well these
approximations
work.
7.2
Stationary
phase example.
Take
initial
surface
displacement of
the form
top
2
2 1/2
u z (r0 ) = A(1-r /W )
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=2/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.

u z (x,y,0, t) = 
0
k dk
A cos[ (k)t] dr0 J 0 (kR )(1- R2 /W 2 )1/2
2
r
0

k dk
u z (x,y,0, t) = 
J o (kr) A F(k)cos[ (k)t]F(k)
0 2
F (k) 
centered about
the origin rc=0.
The full tsunami
solution (7.1.1)
is
2 sin( kW)
 cos( kW); r 

k 2  kW
(7.2.1)
x2  y2
The approximate tsunami solution (7.1.2) is
 k dk
u z (x,y,0, t) = 
0

2
A cos[ (k)t]J 0 ( kr )2  dr0 r0 (1- r0 2 /W 2 )1/2
r0
k dk
W
u z (x,y,0, t) = 
J o (kr) A cos[ (k)t]F(k); F( k)  2  

2

3 
0
(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=2/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 cos term. As you can see it works
very well for distances r>tU(k=2/W) (dashed diagonal line) and even closer.
r <W
22
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= 1/ 4 h( r)/h( r0 ) . 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 h(rc) at position rc
 is a constant parameter. It might
depend on the slope on the beach for
instance. Experiments suggest that ~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
h(rc) as the eventual tsunami runup
height R. So, if A(r) is the offshore Figure 6. Cartoon of wave shoaling and run up. Between
point rc and the ultimate run-in position, complex non-linear
wave amplitude in water of depth h(r), processes govern tsunami behavior. An empirically based law
the depth h(rc) at which A(rc)= h(rc) is R=1/5A(r)4/5H(r)1/5 estimates wave height on the beach from
h(rc)=-4/5(r)A(r)4/5h(r)1/5, and I claim the height of a pack of waves offshore in the linear domain.
run up would be
R =
1 /5
4 /5
A(r)
1 /5
h(r)
(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 nonbreaking waves might stay below R until the final
surge up the beach (see Figure 6). Real breaking
waves might exceed R for a while the plunge back
to
that
level.
Run
up
laws
like

[R/h(r)]=[A(r)/h(r)] are common in the literature;
in fact, (8.1.1) with =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
Figure 6a. Comparison of laboratory runup data
amplitude A(r) through all of the unmodeled
and empirical law 8.1.1 with =1. Although it
is simply based, formula 8.1.1 does a reasonable processes to a final height on the beach R as
(perhaps even conservative) job of estimating
runup from offshore heights.
A(r)T(,h(r),A(r))=R
T(,h(r),A(r))= [h(r)/A(r)]
(8.1.2)
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
23
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 =1 then the run up amplification factor T is 4.0 to 1.6.
24
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

  2H
 n (x, y)  w n  dz u x ( n , x, y, z) 2  uz ( n , x, y, z) 2
2
0


 (9.1.1)
H

k 2 ( n )g2
2
2
2
 w u z ( n , x, y, 0) u 2z ( n , x, y, 0) 2
 dz sinh (k( n )(H  z))  co sh (k( n )(H  z))
2
2
 n co sh (k(  n )H) 0
2

g
2 k( n )g
2
 w u z ( n , x, y, 0)
tan h(k( n )H)  w uz (n , x, y, 0)
2
2
2
n
(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) 
gw
2
 dx dy H z (x, y, 0, t)
2 whole
(9.1.2)
ocean
The Hilbert transform of a function just adds an extra i to the spectrum of the function, effectively
changing cos(t) to sin(t). For example, for the surface tsunami from 2-D bottom uplift of (5.3.1.)
 u bot (k) e i(kx ( k)t)
u z (x, 0, t) = Re  dk z
2 cosh(kH )

(5.3.11)
The Hilbert transform would be

u bot (k) e i(kx (k)t)
H z (x, 0, t) = Re  dk i z
2 cosh(kH)

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
E T (t) 

gw
2
2
 dx dy u z (x, y, 0, t)  H z (x, y,0, t)
2 whole

(9.1.3)
ocean
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
25


E(t)  u 2z (x, y, 0, t )  H2z (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.
Figure 7. Example of the use of the tsunami envelope (9.1.4). To the left is the Tsunami efficiency is nothing
surface vertical motion of a tsunami from a small submarine landslide with red more than the ratio of tsunami
being up and blue down. The envelope to the right matches the same heights at energy to the gravitational
the crests but varies smoothly elsewhere. You can see the shoaling of the energy released in a landslide
waves near shore as the bright spot.
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
E L  g dA(r) eff (r)u(r)h(r)
r
(9.2.1)
Here h(r) is the pre-slide elevation below sea level and u(r) is the excavation or deposition
thickness (excavation take negative). The
integral covers all of the slide area where
u(r)0. eff(r) is the effective density of
the column of slide material at r. eff(r)
equals s, the density of the slide material.
eff(r) equals (s-w) for fully submerged
columns [h(r)>|u(r)|>0]. For partly
submerged columns [|u(r)|>h(r)>0], eff(r)
equals s-wh(r)/|u(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 Figure 8. Envelope of tsunami from a 200 m diameter
depending on how it unfolds (velocity, asteroid impact. The frames show time in one hour
increments.
26
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 R I, density I and velocity VI is
EI = (1/2)I(4/3)R3IV2I (9.2.2)
As shown in Figure 1 and Figure 8, I model impact tsunami from an initial parabolic surface cavity
of the form
u impact
(r)  D c (1  r 2 / R2 ) r  2R (9.2.3)
z
c
c
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 = wg(DCRC)2/ 2IR3IV2I
(9.2.4)
Experimental and observational scaling laws give cavity Dc and Rc as a function of impactor RI, I
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.
27
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(sin(x)   cos(x))  g(tan(x)  )cos(x)
 g(dh(x)/dx  )
(10.1.1)
Here, h(x) is the slope profile shape and Figure 9. Simple sliding block with friction. I use this model to
(x) is its slope angle. For subaerial slides, first approach any new landslide.
 is the coefficient of basal friction. For
submarine slides,  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
1/ 2
 x

v(x)  2  a(xˆ )d ˆx


 0


2gh(0) - h(x)  x 
1/ 2
(10.1.2)
The slide hits peak velocity at position xp where h(xp)/x=-. Smooth exponential curves
x
h(x)  h 0e
with   tan0 / h 0
(10.1.3)
characterize many slopes with 0 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):

v(x)  2gh 0 1 e
x
x / h 0

1/ 2
;
x d ˆx
t(x)  
ˆ
0 v( x)
(10.1.4)
28
Figure 10. Landslide velocity from (10.1.4) with h0=800m, 0=12.50 and basal friction values of = 0.08 to 0.05. Slides
accelerate and de-accelerate quickly and spend much of their time at a nearly constant speed.
Final runout distance xc and slide duration are found by solving
x c 
x c = (h 0 / )
1 e
 (h0 / )
then evaluating
xc
d xˆ
.
v(
xˆ )
0
tc  
(10.1.5)
Mean velocity is defined by the ratio of xc to tc whereas peak velocity at
xp=ln(tan0/)/ is just the free fall speed times a number less than one
v(x p )  2gh 0 1 ( / tan 0 )(1 ln(tan 0 / )
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 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.