Cyclone and Storm Surge - Iczmpwb.org

4.21
G
B
2 2
= - 5 c f F S + c f ρv(
u + v ) 2
2
1
(17)
It may appear surprising that the surface stress is also included in (17). This is because sea -bed friction
is largely determined by the current profile above the sea-bed which, in turn, is dependent on the vertical
gradient of the current near the surface.
However, substituting the values from (14) and (16) into the equations (9) and (10), we get (the over bars
have been dropped for convenience)
∂u
∂t
∂u
∂u
∂ζ
1 ∂p
u + v −f
v = −g
−
∂x
∂y
∂x
ρ ∂x
1 ⎡FS
+ ⎢ −c
( ζ + h)
⎣ ρ
1
2 2
⎤
u(
u + v ) ⎥
⎦
a
+ f
2
(18)
∂v
∂t
∂v
∂v
∂ζ
1 ∂p
u + v + f u = −g
−
∂x
∂y
∂y
ρ ∂y
1 ⎡GS
+ ⎢ −c
( ζ + h)
⎣ ρ
1
2 2
⎤
v(
u + v ) ⎥
⎦
a
+ f
2
(19)
For numerical treatment, it is convenient to express equations (18) and (19) in flux form as
∂
∂t
∂
∂
[ ( ζ + h)
u] + [( ζ + h)
uu] + [( ζ + h)
uv]
− f ( ζ + h)
v =
∂x
∂y
∂ζ
1 ∂ p
1
a F S
2 2
- g ( ζ + h)
− ( ζ + h)
+ − c f u ( u + v ) 2
∂x
ρ ∂x
ρ
(20)
∂
∂t
∂
∂
[ ( ζ + h)
v]+
[( ζ + h)
uv]+
[( ζ + h)
vv]+
f ( ζ + h)
u =
∂x
∂y
∂ζ
1 ∂ p
1
a GS
2 2
− g ( ζ + h)
− ( ζ + h)
+ − c v ( u + v ) 2
f
∂y
ρ ∂y
ρ
(21)
The equations (8), (20) and (21) can be written for the sake of simplicity as
∂ζ ∂u~
∂v~
+ + = 0
∂t
∂x
∂y
(22)