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SIO 212B Spring 2007Problem set #2 .
ANSWERS.
1. If
€
∂ψ /∂z = N 2 = 0, eqn (2) dissolves. Then integrating (1) from the bottom at
€
z = −H + d x,y( ) to the lid at z=0, we obtain
€
H − d x,y( )( ) ∂∂t
∇2ψ + f( ) + J ψ,∇2ψ + f( )
= f0 w z= 0 − w z=−H +d( )
Since d is much smaller than H, this may be written
€
∂∂t
∇2ψ + f( ) + J ψ,∇2ψ + f( )
=f0H
w z= 0 − w z=−H +d( )
The boundary conditions are
€
w z= 0 = 0 and
€
w z=−H +d = u ∂d /∂x + v ∂d /∂y = J(ψ,d) . Thuswe have
€
∂∂t
∇2ψ + f( ) + J ψ,∇2ψ + f( )
= −
f0HJ ψ,d( )
which is equivalent to
€
∂q∂t
+ J ψ,q( ) = 0
with
€
q =∇2ψ + f +f0Hd x,y( ).
2. For arbitrary new coordinate
€
ˆ z = ˆ z (z) , (2) becomes
€
dˆ z dz
∂∂t
∂ψ∂ˆ z
+ J ψ,∂ψ
∂ˆ z
= −
N z( )2
f0
w
Therefore the choice
€
dˆ z dz
=N z( )2
N02
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converts this to
€
∂∂t
∂ψ∂ˆ z
+ J ψ,∂ψ
∂ˆ z
= −
N02
f0
w .
The vorticity equation becomes
€
∂∂t
∇2ψ + f( ) + J ψ,∇2ψ + f( ) = f0N z( )2
N02∂w∂ˆ z
The derivation then proceeds as on pp. 110-112, except that the finite-differencing occurswith respect to the
€
ˆ z coordinate. At the top level,
€
∂∂t
∇2ψ1 + f( ) + J ψ1,∇2ψ1 + f( ) = f0
N12
N02
0 − wm( )H /2
At the mid-level,
€
∂∂t
ψ1 −ψ2
H /2
+ J
ψ1 +ψ2
2,ψ1 −ψ2
H /2
= −
N02
f0wm
At the bottom level,
€
∂∂t
∇2ψ2 + f( ) + J ψ2,∇2ψ2 + f( ) = f0
N22
N02
wm − 0( )H /2
Eliminating
€
wm we obtain
€
∂q1∂t
+ J ψ1,q1( ) = 0,
€
∂q2∂t
+ J ψ2,q2( ) = 0
where
€
q1 =∇2ψ1 + f + F1 ψ2 −ψ1( ) ,
€
q2 =∇2ψ2 + f + F2 ψ1 −ψ2( )
and
€
F1 =4 f0
2N12
H 2N04 , F2 =
4 f02N2
2
H 2N04
These equations are similar to those derived for the case of two immiscible layers, where
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€
F1 =f02
′ g H1
, F2 =f02
′ g H2
Thus, stronger stratification near the top corresponds to a thinner upper layer.
3. Applying
€
dx−∞
+∞
∫ to the advection-diffusion equation and assuming
€
θ → 0 as
€
x→ ±∞
we obtain
(1)
€
∂M0
∂t=κ
∂ 2M0
∂y 2
By a similar procedure:
(2)
€
∂M1
∂t− uM0 =κ
∂ 2M1
∂y 2
(3)
€
∂M2
∂t− 2uM1 = 2κM0 +κ
∂ 2M2
∂y 2
(4)
€
∂M3
∂t− 3uM2 = 6κM1 +κ
∂ 2M3
∂y 2
Eqns. (1-4) are a closed hierarchy of equations for the moments
€
Mn y, t( ). Since theinitial conditions are y-independent, all the
€
Mn y,0( ) are constants. Then (1) implies that
€
M0 remains a constant for all time. The
€
M1-equation (2) becomes
(5)
€
∂M1
∂t−κ
∂ 2M1
∂y 2= M0U0 cosωtsinmy ≡ F y,t( )
which is the diffusion equation with a forcing term on the right-hand side. The solutionis
(6)
€
M1 y, t( ) = M1 y,0( ) + f t( )sinmy
where
€
M1 y,0( ) is a constant determined by the initial conditions, and f(t) obeys
(7)
€
dfdt
+κm2 f = M0U0 cosωt≡ F t( )
with initial condition
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(8)
€
f 0( ) = 0.
The general solution of (7) is
(9)
€
f t( ) = f 0( )e−κm 2t + F ′ t ( )0
t
∫ e−κm 2 t− ′ t ( )d ′ t
At large times this becomes
(10)
€
f t( ) = F ′ t ( )−∞
t
∫ e−κm 2 t− ′ t ( )d ′ t =M0U0
κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )
Therefore
(11)
€
M1 y, t( ) = M1 y,0( ) +M0U0
κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )sinmy
Thus the “center of mass” oscillates about its initial value. With (11) the equation (3) forthe variance becomes
(12)
€
∂M2
∂t−κ
∂ 2M2
∂y 2= 2κM0 + 2U0 sin my( ) cos ω t( )M1 y,0( )
+2M0U0
2
κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )cosωt sin2my
Using the identity
€
sin2 x = 12 1− cos2x( ) , we see that the solution takes the form
(13)
€
M2 y, t( ) = A t( ) + B t( )sinmy + C t( )cos2my
where
(14)
€
dAdt
= 2κM0 +M0U0
2
κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )cosωt
(15)
€
dBdt
+κm2B = 2U0 cos ω t( )M1 y,0( )
and
(16)
€
dCdt
+ 4κm2C = −M0U0
2
κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )cosωt
The initial conditions are
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(17)
€
A 0( ) = M2 y,0( ), B 0( ) = C 0( ) = 0
Each of (14-16) fits the form of (7). From (9) we see that, as
€
t→∞ , B contributes onlyoscillatory terms, C contributes only constant and oscillatory terms, and A contributesoscillatory and secular terms. We obtain the secular terms by integrating
(18)
€
dAdt
= 2κM0 +M0U0
2
κ 2m4 +ω 2 κm2 12
to obtain
(19)
€
M2 y, t( ) ~ 2κM0 +M0U0
2κm2
2 κ 2m4 +ω 2( )
t
Thus the variance
(20)
€
x 2 ≡x 2θ∫∫θ∫∫~ 2 κ +
U02κm2
4 κ 2m4 +ω 2( )
t ≡ 2κeff t
where
(21)
€
κeff =κ +14U0
2
ωκ*
1+κ*2 , κ* ≡
κm2
ω
is the effective diffusion in the x-direction. We see that the dispersion of
€
θ increaseslinearly with time. If no velocity field is present, then
€
x 2 ~ 2κt because of moleculardiffusion in the x-direction. If the velocity field is present, then there is an additionaldispersion, which arises from the combination of advection in the x-direction anddiffusion in the y-direction. This shear dispersion vanishes as
€
κ → 0 and also as
€
κ →∞.The first of these limits is easy to understand; if the diffusion coefficient vanishes, theoscillating velocity field periodically restores
€
θ to its initial value. The other limit ismore subtle. If
€
κ is too large, then the “wiggles” in the
€
θ diffuse away before the shearcan make them very wide. For fixed
€
U0 and
€
ω , the effects of shear dispersion aregreatest when
(22)
€
κ* =κm2
ω=1,
that is, when the diffusion time
€
Km2( )−1
just equals the advection time
€
ω−1. At the value(22), the horizontal shear and vertical diffusion are optimally tuned to cause the greatesteffective horizontal diffusion.
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