19

P  =  Po  +P  =-PogZ  +  Pog i l

cosh  L/27t(z  +  h 0 )

cosh  L/2^ 0
p'  differs  from  p' h  by  a  factor  approaching  1  for  small  L/2^ 0 .
Boussinesq  equations:
T7  -  r,  du  du  dp  dw  dp
V-v  =  0,  p Q —  +  p 0 u—  =  ——  and  p 0  —  =  ——~p 0 g  with  boundary  conditions
dt  dx  dx  dt  dz
dr/  dr/  ,
w  =  —-  +  u—-,  p  =  0  at  z  =  r/  and  w  =  0  at  z  =  -h 0 .
dt  dx

1*  1*  1*  O  u
It  follows  from  V  ■  v  =  0  that  w--\  —dz  ,  from  irrotationality  u  =  u(z  =  0)  +  —-/dz"dz'.

o  z'  -\2

d  H  j  //  j  /

~K

dx

z  -An

dx 2

Expressed  by  the  approximation  for  w,  the  pressure  is:
0  z  02^
P  =  Po+Pb  =-PogZ  +  PogP-Po$  j  ~^f Z " dz '  ■

z  -h,

dw

p' B  differs  from  p' h  by  a  summand  vanishing  with  >  0.  A  solution  to  the  equations  is
dt
possible  by  assuming  =  anc |  ¡t  ¡ s  determined  by  the  choice  of  u(z-  0).  The
dx  dx
additional  terms  in  this  simple  version  of  the  Boussinesq  equations  as  compared  to  nonlinear ­
  hydrostatic  equations  can  be  deduced  from  the  local  temporal  derivative  of  u  and
from  the  horizontal  pressure  gradient  (cf.  Peregine  1972).

5.1.3  Applicability  of  the  equations  to  tsunami  events
The  various  approximations  of  equations  2  apply  only  to  a  certain  region  of  the  parameters
wave  height,  wave  length,  and  undisturbed  water  depth.  In  section  5.1.1,  a  solitary  wave  was
stated  to  be  a  possible  way  of  describing  a  tsunami.  It  is  the  solution  to  simple  Boussinesq
equations.
Let  a  tsunami  be  a  simple  wave  characterised  by  the  parameters  H,L,  and  h.  With  suitable
scaling  of  the  equations  (Voit  1978),  the  orders  of  magnitude  of  the  individual  terms  in  (2)
can  be  estimated  through  these  parameters,  and  simplifications  evaluated.  The  non-linear
terms  in  the  horizontal  momentum  equations  and  in  the  surface  boundary  conditions  are  on
the  order  of  0.5H/h,  in  the  vertical  momentum  equation  0.5H/h  ■  h 2 /1}  and  in  the  bottom
boundary  conditions  1.  The  order  of  magnitude  of  the  local  temporal  change  of  horizontal
motion  with  this  scaling  is  1,  that  of  vertical  motion  h 2 /L 2 .  Komar  (1976)  accordingly
determined  the  range  of  validity  of  different  wave  theories  in  dependence  on  the  two
important  parameters  h/L  and  H/h  (Fig.  5.1.3).  Compare  also  Mader  (2004,  Table  1.1);
Frohle  et  al.  (2002,  Table  A.3.1);  and  Peregine  (1972).
Consequently,  /z 2 /l 2  toward  zero  (i.e.  h/L  <0,05)  marks  the  hydrostatic  limit  case  of
equations  2,  0.5H/h  toward  zero  (i.e.  H/h«  1)  the  linear  limit  case.  However,  both
parameters  alone  are  not  sufficient  to  determine  the  range  of  validity  of  individual  wave
theories.  According  to  Ursell  (1953),  it  is  the  relative  order  of  magnitude  of  non-hydrostatics
(local  temporal  change  of  vertical  velocity)  h 2 /l}  and  non-linearity  0.5H/h,  the  so-called
Ursell  parameter  U,  which  determines  the  necessary  generality  of  the  equations.