Preliminary reference for The 8th JCOMM/TCP Workshop on Storm Surge and Wave Forecasting Nairobi, Kenya 19 – 23/Nov/2012
Basics about wind waves, storm surges, and tsunamis -A brief introduction to linear external gravity waves-
Nadao Kohno Office of Marine Prediction, Global Environment and Marine Department, JMA
[email protected]
N um be r of Affe cte d Pe ople by N atural Disaste rs (1975-2002) Eu rope 1%
Ocean ia 0%
Afri ca 7%
Americas 3%
Afri ca Am e ri ca s As i a Eu rope Oce a n i a
As ia 89%
The scale of waves in oceans Storm surges
Ocean waves
（Munk,W.H. 1951）
Comparison of storm surges, tsunamis and ocean waves Same cause ⇒ simultaneously occurs
Storm surges
Ocean waves
Tsunamis
Cause
Strong winds and pressure depressions ( by TC etc )
(strong) winds
Crust movement （earthquakes, eruptions）
Property of waves
Long wave (shallow water)
Short wave （deep water）
Long wave (shallow water)
Horizontal scales (m)
105*
102
105～6
Time scales (s)
103～5
101
103～5
Character is different ⇒ neediness of individual dealing
*The horizontal scale of storm surges is assumed as TC scale.
Storm surges
The damage of the storm surges by Typhoon Bart (9918) in Japan
Storm surges
Stormvideo.com
Storm surges by Hurricane katrina in 2005
Track of Hur. Katrina
The flooded New Orleans
The Flooded area and the highest tides
The highest tides (m) The flooded area
Floods by Hurricane Katrina(2005)
Landsat 7 Image of New Orleans Left: 24/Apr/2005 Right: 30/Aug/2005
Orange Beach, Alabama (USGS)
Simulated storm surge 00UTC – 22UTC 29th/Aug/2005
Tsunamis
Thailand 2004
The Indian ocean tsunami in 2004
USC Tsunami research group
Sea topography of the Indian Ocean
the movement of fault
Simulated tsunami
Ocean waves
North Pacific storm waves (NOAA)
High waves in Toyama (Mainichi Newspa.)
High waves and stranded ship (Mainichi Newspa.)
High waves hit the breakwater (Yamagata Newspa.)
Gravity waves Wave motion: periodic motion around equilibrium line disturbing force, restoring force, and inertia. disturbing force static state (equilibrium level ) inertia restoring force (=gravity)
Gravity wave : the restoring force is the gravitational force.
Gravity waves External or internal? External: the wave at the surface boundary by the deformation of shape etc. not propagate in the medium. ○ ×
( usually dumps exponentially)
Internal: the wave by the difference of density etc in the medium. propagate in the medium.
ρ = ρ (z )
Ocean waves, storm surges, and tsunamis are classified to the external gravity waves
The dynamics of surface waves （linear theory) Consider a vertical cross section (x,z) (wave is supposed to be homogeneous in y direction) With the assumption of ・Fluid is to be inviscid ・surface tension is negligible ・Waves are generated from static state by gravity effect ⇒ irrotational (no vorticity)
z x 0 y
Governing equations We can define the velocity potential φ ∂φ ∂φ
u = ∇φ .
u=
∂x
, w=
∂z
(1)
In the incompressible fluid like water, the continuity equation is
∇ ⋅ u = 0.
∂u ∂w + =0 ∂x ∂z
∂ρ = 0 Θ ∂t
(2)
Therefore it follows Laplace’s equation:
∇ φ = 0. 2
(Θ
∂ 2φ ∂ 2φ + 2 = 0. 2 ∂x ∂z
∇ ⋅ u = ∇ ⋅ ∇φ = ∇ 2φ
)
(3)
Boundary conditions At the bottom floor, water never flows into，the vertical component vanishes (4) (w)z =− d = ∂φ = 0. ∂ z z =− d At the free surface z = η(x,z) ，water parcels keep the position within water, if the wave height is not large Dη ∂η ∂η w= = +u⋅ . (5) D z ∂t ∂x Using the velocity potential, the kinematic boundary condition is expressed as ∂φ ∂η ∂φ ∂η = + ⋅ . ∂ z z =η ∂t ∂ x z =η ∂ x
(6)
The pressure p at any point is expressed by the Bernoulli equation 2 2 p0 ∂φ 1 ∂φ ∂φ = − − + − gz+ F(t), ρ ρ ∂t 2 ∂x ∂z
p
(7)
whereρ is water density. The atmospheric pressure p0 is assumed as constant and F(t) is combined with φ, the dynamical boundary condition is derived as 2 2 ∂φ 1 ∂φ ∂φ + gη + + = 0, 2 ∂ x ∂ z ∂t z =η z =η
(8)
The 4 equations of Laplace equation (3)， boundary conditions (4)，(6)，and (8) are the governing equations about the gravity waves.
Linearization It is difficult to solve these equations rigorously, since the non-linear terms are involved in the equations. For simplicity, we suppose that the wave amplitude is smaller than the wave length (this assumption is relatively good approximation in the real ocean waves). The equation is expressed by the Taylor series expansions about z = 0. For example, the kinematic boundary condition at free surface:
∂ ∂φ ∂φ ∂φ 1 ∂2 ∂φ 2 = + η + 2 η +Λ ∂z z=η ∂z z=0 ∂z ∂z z=0 2 ∂z ∂z z=0 =
∂η ∂φ ∂η + ⋅ ∂t ∂x z=η ∂x
∂φ ∂η = . ∂ z z = 0 ∂t
(9)
In same way, we can derive the linearized equation set: １．Laplace equation
∂ 2φ ∂ 2 φ + 2 = 0. 2 ∂x ∂z
(10)
２．The boundary condition at bottom ∂φ = 0. ∂ z z =− d
(11)
３．The kinematic boundary condition at surface ∂φ ∂η = . ∂ z z = 0 ∂t
(12)
４． The dynamic boundary condition at surface ∂φ + gη = 0 ∂t z = 0
(13)
Using equations (12) and (13), ηis eliminated as, ∂φ ∂ ∂φ ∂ ∂η = − gη = − g = − g ∂t ∂t z =0 ∂t ∂t ∂ z z =0
∂ 2φ ∂φ 2 = − g ∂t z = 0 ∂ z z =0
(14)
Now consider the wave-like solution toward x direction. The velocity potential is expressed as
φ = Z ( z ) ⋅ e i ( kx −ωt )
(15)
Putting it into the Laplace equation (10) d 2φ d 2φ d 2 Z i ( kx −ωt ) d 2e i ( kx −ωt ) + 2 = e +Z 2 2 dz dx dz d x2 d 2 Z i ( kx −ωt ) = e − k 2 Ze i ( kx −ωt ) 2 dz d 2Z = − k 2Z = 0 2 dz
(16)
The general solution of (16) has the form of
Z = Ae kz + Be − kz
(17)
which leads to
φ = (Ae kz + Be − kz )⋅ e i ( kx −ωt )
(18)
Putting (18) into (11) yields ∂φ ∂ Ae kz + Be − kz ⋅ ei ( kx −ωt ) = ∂ z z =− d ∂ z z =− d
(
)
(
)
= Ake −kd − Bke kd ⋅ ei ( kx −ωt ) = 0 Ae − kd − Be kd = 0
(19)
From the substitution (14) by (18)
∂ 2φ ∂φ 2 + g ∂ z z =0 ∂t z =0 2 i ( kx−ωt ) kz −kz ∂ e i ( kx−ωt ) ∂ kz −kz = Ae + Be + ge Ae + Be 2 ∂ ∂ t z z =0
(
)
[ (
(
)
(
)
)
= ω 2 Aekz + Be−kz ei ( kx−ωt ) + g kAekz − kBe−kz ei ( kx−ωt ) = ω 2 ( A + B)ei ( kx−ωt ) + g (kA − kB)ei ( kx−ωt ) = 0 (ω 2 + gk) A + (ω 2 − gk) B = 0
]
z =0
(20)
Finally the equations (19) and (20) for A, B is derived as,
Ae − kd − Be kd = 0 2 2 ( ω + gk ) A + ( ω − gk ) B = 0 e − kd − e kd A 2 ω + gk ω 2 + gk B = 0
− e kd
ω + gk ω − gk 2
2
=0
e − kd (ω 2 − gk ) − (−e kd )(ω 2 + gk ) = ω 2 (e − kd + e kd ) − gk (e − kd − e kd ) = 0
(21)
If A and B have a non trivial solution, it needs to be e − kd
From this determinant
(22)
kd − kd e − e ω 2 = gk kd − kd e +e = gk tanh kd
Therefore the dispersion relation about the ocean waves is
ω 2 = gk tanh kd ω = gk tanh kd
(23)
phase speed: ω 1 c=
k
=
k
group velocity: 1 dω d (gk tanh kd )2 = dk dk 1 d −1 = (gk tanh kd ) 2 ⋅ g tanh kd + gk 2 2 cosh kd
gk tanh kd
=
g tanh kd k
=
gL 2πd tanh , Θ k = 2π 2π L L
cg =
1 g d 2 gk 2 = tanh kd + 2 2 k tanh kd ⋅ cosh kd 1
(24)
1
1 g kd 2 = c + tanh kd 2 2 k tanh kd ⋅ cosh kd 1 kd = c + c 2 sinh ⋅ cosh kd kd 1
wave length: gL 2πd tanh ⋅T 2π L gL 2πd 2 L2 = tanh ⋅T 2π L gT 2 2πd L= tanh 2π L
L = cT =
2kd c = 1 + , 2 sinh 2kd
(25)
(26)
Shallow water waves When the water depth d is smaller compared to the wave length L(1/k), taking the limit of d 0, tanh kd / kd 1 and sinh kd / kd 1, the relations become c =
g tanh kd → gd ⋅1 = gd tanh kd = gd k kd L = cT = T gd , c 2kd c c g = 1 + → (1 + 1) = c 2 sinh 2kd 2
This approximation is quite accurate if d / L < 1/25 In real ocean, the mean sea depth : about 3km the horizontal scale of storm surge or tsunami : more than 100km d / L = 3/100 < 1/25
We can assume storm surges and tsunamis, as pure shallow water waves
(27)
Deep water waves When the water depth d is larger compared to the wave length L, taking the limit of d ∞, tanh kd 1, sinh kd / kd infinity etc. c =
g tanh kd = k
gT 2 gT gT tanh kd tanh kd = tanh kd → 2π 2π 2π gT 2 , L = cT = 2π 2 kd c c c c g = 1 + → (1 + 0 ) = 2 sinh 2 kd 2 2
gL ⋅ tanh kd = 2π
g 2π
This is a good approximation since the error will be under 1% if d / L < 1/2 The mean sea depth is about 3km, and the horizontal scale of ocean waves is only 100m, d / L = 3000(m) / 100(m) >> 1/2 We can assume ocean wave as deep water waves in the offshore. However, in a beach where water depth is shallower than 50m, d / L = 50(m) / 100(m) = 1/2 We need to consider shallow water effects into ocean waves.
(28)
The relation of external (linear) gravity waves Deep water waves （ h > L/2 ）
( )
Phase speed
Ｃ Wave length
Ｌ Group velocity
Ｃg
Theoretical values
gL gT = = 1.56T m s 2π 2π
gL 2πh tanh 2π L
gT 2 = 1.56T 2 (m) 2π
gT 2 2πh tanh 2π L
1 gL C = 2 2π 2
g: gravitational acceleration h: water depth T: wave period
1 4π h / L C 1 + 2 sinh (4π h / L )
Shallow water waves （ h < L/25 ）
gh = C
T
gh
gh
Change of wave motion Velocity is independent on water depth
Cyclic motion
Slow down in shallow area
Straight motion
Contamination of water (large wave height) →wave breaking
Phase speed and group velocity as a function of water depth and wave period (wave length) Group velocity
Phase speed
15
3
10
5 10
5
15 20
0 0
20
40
60
80
100
120
140
Cg(m/s)
Cp (m/s)
20
14 12 10 8 6 4 2 0
3 5 10 15 20
0
20
40
Water depth (m)
80
100
Water depth(m)
Cp and Cg
magnitude(m/s)
60
Wave length
10 8 6 4 2 0
・period 3 (sec) ： 14.0m ・period 5 (sec) ： 39.0m ・period 10 (sec) ： 156.0m 0
20
40
60
80
100
Water depth(m) Cp
Cg
120
140
・period 15 (sec) ： 350.9m ・period 20 (sec) ： 623.9m
120
140
Image of refraction in shallow water Turn to shallow part
slow
fast Water depth
Waves propagate toward the head of peninsula
The basic equation of Fluid （Navier-Stokes equations） Consider a parcel of fluid, which will change its moving speed during t → t+Δt
u (t , x(t ), y (t ), z (t )) → u (t + ∆t , x(t + ∆t ), y (t + ∆t ), z (t + ∆t )) u(t+Δt) is expressed by the Taylor expansion as u (t + ∆ t , x (t + ∆ t ), y (t + ∆ t ), z (t + ∆ t )) ∂u ∂u ∂u ∂u ∆t + ∆x + ∆y + ∆ z + O ( ∆○ 2 ) ∂t ∂x ∂y ∂z By taking the limit of Δt→ 0 = u ( t , x (t ), y ( t ), z (t )) +
du u (t + ∆t , x(t + ∆t ), y (t + ∆t ), z (t + ∆t )) − u (t , x(t ), y (t ), z (t )) = lim dt ∆t →0 ∆t ∂u ∆x ∂u ∆y ∂u ∆z ∂u = lim + + + + O(∆t ) ∆t →0 ∂t ∆t ∂x ∆t ∂y ∆t ∂z =
∂u ∂u ∂u ∂u +u +v +w ∂t ∂x ∂y ∂z
Navier-Stokes equations If no force is added, the motion keep its status and the differential value is 0
du Du ∂u ∂u ∂u ∂u ≡ = + u + v + w =0 dt Dt ∂t ∂x ∂y ∂z (Traditionally, differential d is expressed as D) Actually outer forces such as pressure (gradient), viscosity, and gravity etc act. Du 1 = − ∇P + ν∆u + g ρ Dt
dv F Θ F = ma , a = = , m = ρ dt m
The Coriolis force should be included in the large scale phenomena in meteorology and oceanography. On the other hand, viscosity is omitted since it does not have a crucial role. ∂u ∂u ∂u ∂u 1 ∂P ∂t
Du 1 + f × u = − ∇P + g Dt ρ
+u
∂x
+v
∂y
+w
∂z
− fv = −
ρ ∂x
1 ∂P ∂v ∂v ∂v ∂v + u + v + w + fu = − ∂t ∂x ∂y ∂z ρ ∂y 1 ∂P ∂w ∂w ∂w ∂w +u +v +w =− −g ρ ∂z ∂t ∂x ∂y ∂z
Equation of continuity (conservation of mass) A local density is decided by the flow balance ρ ( x + ∆x)u ( x + ∆x) − ρ ( x)u ( x) ∂ρ = − lim ∆x →0 ∂t ∆x ∂ρ ∂ρ = − lim ρ ( x)u ( x) + ∆x + ∆x + O(∆x 2 ) − ρ ( x)u ( x) / ∆x ∆x → 0 ∂x ∂x
u(ｘ+Δx)
u(x)
∂u ∂ρ = − u ⋅ −ρ⋅ ∂x ∂x
⊿x
Consider 3 direction, ∂u ∂v ∂w ∂ρ ∂ρ ∂ρ ∂ρ = 0 −u +v +w + ρ + + ∂t ∂x ∂y ∂z ∂ x ∂ y ∂ z
∂ρ + u ⋅ ∇ρ + ρ∇ ⋅ u = 0 ∂t
1 Dρ + ∇ ⋅ u = 0 ρ Dt
In usual condition, density does not change drastically, we assume as incompressible ∇⋅u = 0
∂u ∂v ∂w + + = 0 ∂x ∂y ∂z
