12. Shallow water equations with rotation – Poincaré waves
Considering now motions with L<<R, we can write the equations of motion in
Cartesian coordinate:
1
px
ρo
1.
ut - fv =
2.
v t + fu = −
3.
0=−
4.
ux+vy+wz = 0
1
py
ρo
1
gρ
pz −
ρo
ρo
ρt+wρ = 0
oz
In the general case:
Again, the vertical momentum equation 3, and the adiabatic equation 5 can be
combined:
1
p zt + N 2w = 0
ρo
Consider first a homogeneous fluid for which ρtotal = ρo + ρ = ρo the hydrostatic equation.
Then poz = -gρo is the total pressure. Integrating from z to the free surface η:
p|η - p(z) = -gρo(η-z) or
p(z) = patm + gρo (η-z) => p(z) = gρo (η-z) as patm = 0
Then the horizontal pressure gradient ∇p = gρ∇η is independent of z. The
equations of motion can be written:
ut-fv = -gηx
(there is no adiabatic equation as there
vt+fu = -gηy
is no perturbation density)
ux+uy+wz = 0
1
Notice that the right-hand side of the horizontal momentum equations is independent of
z: (u,v) are independent of z
η
Integrate the continuity equation ∫ dz
−D
η
∫ (u x + v y )dz + w |z=η −w |z=−D = 0
−D
Top and bottom b.c. are (including non-linearities)
We have: w =
− dD
dη
at z = η; w =
= −u H • ∇ D at z = −D
dt
dt
As:
wzz = 0
wz = a(t)
at z = -D
w = a(t)z + b(t)
w=0
-Da+b = 0 b = aD
w = a(t)(z+D)
at z = 0
w=
In general
at z = o a =
w=
w=
∂η
∂t
1 ∂η
(z + D)
D ∂t
Da(t) =
∂η
∂t
linearized version
w = a(x,y,t)(z+D)
dη 1
dt D
1 dη
(z + D)
D dt
dη ∂η
=
+ u H • ∇η
dt ∂t
2
and we obtain
(u x + v y )(η + D) +
dη
+ u H • ∇D = 0
dt
(u x + v y )(η + D) +
dη
+ u H • ∇(η + D) = 0
dt
ηt+[u(η+D)]x + [v(η+D)]y
or
exactly
Assuming η <<D i.e. linearizing, we have
ut-fv = -gηx
vt+fu = -gηy
ηt + (uD)x + (VD)y = 0
These are the linear shallow water equations for a homogenous fluid with rotation.
Now return to the equations with stratification and separate variables
u = U(x,y,t)F(z)
v = V(x,y,t)F(z)
p = P(x,y,t)H(z)
w = W(x,y,t)G(z)
We get
( Ut-fV)F =
-1
PxH
ρo
(Vt+fU)F = −
N2WG = −
1
PyH
ρo
1
Pt H z
ρo
(Ux+Vy)F + WGz = 0
Choose W = Pt
(as we derived in LTE)
3
H = gρoF
Gz = F/D
We have
Ut-fV = -gPx
Horizontal structure equations
Vt+fV = -gPy
Pt+Dn(Ux+Vy) = 0
Compare with the homogeneous layer equations with D=constant.
They are the same with P = η. The pressure plays the part of the sea/surface elevation.
The vertical structure equation is again:
G zz +
Gz -
N 2 (z)
G=o
gD n
1
G = 0 at z=o
Dn
The same identical as for LTE with hn = Dn
G=o at z = -D
The hydrostatic approximation we have made assumes wt<<g which is equivalent to
assuming
ω2<<N2 ->
In a flat-bottom ocean stratification makes possible an infinite sequence of internal
replicas of the barotropic, long, shallow water gravity waves. We shall study the latter
first.
From now on we shall study the homogeneous one layer problem as it is equivalent to the
horizontal structure equations (P=η) for the fully stratified case. With D = constant
ut – fv = -gηx
vt + fu = -gηy
ηt + D(ux+vy) = 0
4
Form the vorticity equation ζ = vx-uy cross-differentiating the 2 horizontal momentum
equations
or
f ∂η
∂ζ
= −f(u x + v y ) =
D ∂t
∂t
Statement of conservation of potential vorticity
∂
f
(ξ- η)=0
∂t
D
for the linear, homogeneous model with f = constant
f
η
D
relative vortix
vorticity stretching
q=ζ−
For periodic motions
∂
q = −iwq = 0 q vanishes
∂t
Now we want an equation for η: take the divergence of horizontal momentum equations
∂
(u x + v y ) − fζ = −g∇ 2 η
∂t
From continuity
∂ 2η
∂
∂
1 ∂ 2η
+
+
=
⇒
+
=
−
D
(u
v
)
0
(u
v
)
y
y
∂t 2
∂t x
∂t x
D ∂t 2
−
1 ∂2η
− fζ=-g∇ 2 η
2
D ∂t
From the statement for PV
ζ=q+
-
f
η
D
1 ∂2η
f
− f(q+ η)+g∇ 2 η=0
2
D ∂t
D
∇ 2η −
1 ∂2 η f 2
f
−
η
=
+
q
gD ∂t 2 gD
g
5
Note that as
∂q
= 0 ; potential vorticity is conserved steadily.
∂t
So we can separate
η=ηsteady+ηwave=ηs+ηw.
The unsteady part of ηwave carries no potential vorticity so
∇ 2 ηw −
ηs =
1 ∂2 η w f 2
−
ηw = 0
gD
gD ∂t 2
D
D
q = qo
f
f
homogeneous equation
particular solution
Steady part is in geostrophic balance with
(u,v) and reflects the initial distribution of
∂q
PV as
= 0 ; q = qo
∂t
The wave equation is:
1 ∂ 2η f 2
∇ η− 2 2 − 2 η=0
c o ∂t
co
2
When c o = gD is the phase speed for long gravity waves.
If there were no rotation f = 0 we would get the non-dispersive wave equation (onedimension)
1
η xx − 2 ηtt = 0
co
with solution
η = F(x-cot) + G(x+cot);
F and G determined by initial condition
Taking a solution of the form:
η = Aei(kx+ly-ωt)
K= k 2 +12
we obtain ω 2 = c 2o (k 2 + l2 ) + f 2 ⇒ ω = ± c 2o (k 2 + l2 ) + f 2
These are long, shallow water gravity waves modified by rotation, often called
Poincaré waves. Visualize the particle motion:
6
K, c, cg
ph
as
es
ph
a
se
s
K, c, cg
No Rotation
Rotation
Figure 12-1
Figure by MIT OpenCourseWare.
All these waves have ω> f, f is the lowest possible frequency:
Group velocity
c gx =
k
∂ω
= c o2 = c o2
ω
∂k
f + c o2 K 2
c gy =
∂ω
= c o2 = c o2
ω
∂
f 2 + c o2 K 2
k
2
cg //K
The horizontal velocities are obtained
i)
eliminating v from momentum eqn. → eq. for u
ii)
eliminating u → eq. for v
7
∂ 2u
∂ 2η
∂η
2
+
f
u
=
−
g
− fg
2
∂t
∂x∂t
∂y
operator LHS = 0 w = ±f
inertial oscillations
∂2v
∂ 2η
∂η
2
+
f
v
=
−
g
− gf
2
∂t
∂t∂y
∂x
u = cos(ft); v = sin(ft)
If we align x with K then full solution is
η = ηocos(kx-ωt)
u=
ηo ω
cos (kx-ωt)
Dk
v=
ηo f
sin(kx-ωt)
Dk
Poincaré wave energy → concentrated at lowest possible frequency, near f
8
MIT OpenCourseWare
http://ocw.mit.edu
12.802 Wave Motion in the Ocean and the Atmosphere
Spring 2008
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