Numerical modeling of waves for a tsunami early warning system

Numerical modeling of waves for a tsunami early warning system
derivatives of u and v with respect to x and y. To change this, the chain rule
derivative operator for x to the elemental lengths dx and dy is applied,
xuux + xvvx =1
yuux + yvvx =0
this set of linear equations is easily solved for ux and vx
(B.15)
ux = 1
J yv
vx = − 1
J yu
(B.16)
Where J = xuyv − xvyu is the Jacobian of the transformation. Using the
y derivative operator aplied to dx and dy results in
uy = − 1
J xv
vy = 1
J xu
Thus in equation (B.14) some terms can be replaced as follows
u2 x + u2 y = 1
J 2 (x2 v + y2 v); v2 x + v2 y = 1
J 2 (x2 u + y2 u);
uxvx + uyvy = 1
J 2 (−yuyv)+ 1
J 2 (−xuxv) =− 1
J 2 (xuxvyuyv)
thus equation (B.14) becomes
By calling
(x 2 v + y 2 v)(ccgau) u +(x 2 u + y 2 u)(ccgav) v +
− (xuxv + yuyv)
au (ccg) v + av (ccg) u +2ccgauv +
J 2 [k 2 ccga + ccg (au∇ 2 u + av∇ 2 v)] = 0
x 2 v + y 2 v = α;
x 2 u + y 2 u = β;
xuxv + yuyv = γ;
equation (B.19) can be rewritten in a more compact way
α (ccgau) u + β (ccgav) v − γ
au (ccg) v + av (ccg) u +2ccgauv +
J 2 [k2ccga + ccg (au∇2u + av∇2v)] = 0
(B.17)
(B.18)
(B.19)
(B.20)
(B.21)
Now ∇ 2 u and ∇ 2 v do not depend on x and y, ∇ 2 u = uxx + uyy and
∇ 2 v = vxx + vyy change and depend on J, ∂
∂u
, ∂
∂v
, ∂2
∂u 2 and ∂2
∂v 2
Università degli Studi di Roma Tre - DSIC 116