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SIO 212B Spring 2007Problem set #2 .

ANSWERS.

1. If

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∂ψ /∂z = N 2 = 0, eqn (2) dissolves. Then integrating (1) from the bottom at

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z = −H + d x,y( ) to the lid at z=0, we obtain

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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

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∂∂t

∇2ψ + f( ) + J ψ,∇2ψ + f( )

=f0H

w z= 0 − w z=−H +d( )

The boundary conditions are

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w z= 0 = 0 and

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w z=−H +d = u ∂d /∂x + v ∂d /∂y = J(ψ,d) . Thuswe have

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∂∂t

∇2ψ + f( ) + J ψ,∇2ψ + f( )

= −

f0HJ ψ,d( )

which is equivalent to

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∂q∂t

+ J ψ,q( ) = 0

with

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q =∇2ψ + f +f0Hd x,y( ).

2. For arbitrary new coordinate

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ˆ z = ˆ z (z) , (2) becomes

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dˆ z dz

∂∂t

∂ψ∂ˆ z

+ J ψ,∂ψ

∂ˆ z

= −

N z( )2

f0

w

Therefore the choice

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dˆ z dz

=N z( )2

N02

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converts this to

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∂∂t

∂ψ∂ˆ z

+ J ψ,∂ψ

∂ˆ z

= −

N02

f0

w .

The vorticity equation becomes

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∂∂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

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ˆ z coordinate. At the top level,

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∂∂t

∇2ψ1 + f( ) + J ψ1,∇2ψ1 + f( ) = f0

N12

N02

0 − wm( )H /2

At the mid-level,

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∂∂t

ψ1 −ψ2

H /2

+ J

ψ1 +ψ2

2,ψ1 −ψ2

H /2

= −

N02

f0wm

At the bottom level,

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∂∂t

∇2ψ2 + f( ) + J ψ2,∇2ψ2 + f( ) = f0

N22

N02

wm − 0( )H /2

Eliminating

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wm we obtain

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∂q1∂t

+ J ψ1,q1( ) = 0,

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∂q2∂t

+ J ψ2,q2( ) = 0

where

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q1 =∇2ψ1 + f + F1 ψ2 −ψ1( ) ,

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q2 =∇2ψ2 + f + F2 ψ1 −ψ2( )

and

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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

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θ → 0 as

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x→ ±∞

we obtain

(1)

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∂M0

∂t=κ

∂ 2M0

∂y 2

By a similar procedure:

(2)

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∂M1

∂t− uM0 =κ

∂ 2M1

∂y 2

(3)

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∂M2

∂t− 2uM1 = 2κM0 +κ

∂ 2M2

∂y 2

(4)

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∂M3

∂t− 3uM2 = 6κM1 +κ

∂ 2M3

∂y 2

Eqns. (1-4) are a closed hierarchy of equations for the moments

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Mn y, t( ). Since theinitial conditions are y-independent, all the

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Mn y,0( ) are constants. Then (1) implies that

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M0 remains a constant for all time. The

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M1-equation (2) becomes

(5)

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∂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)

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M1 y, t( ) = M1 y,0( ) + f t( )sinmy

where

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M1 y,0( ) is a constant determined by the initial conditions, and f(t) obeys

(7)

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dfdt

+κm2 f = M0U0 cosωt≡ F t( )

with initial condition

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(8)

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f 0( ) = 0.

The general solution of (7) is

(9)

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f t( ) = f 0( )e−κm 2t + F ′ t ( )0

t

∫ e−κm 2 t− ′ t ( )d ′ t

At large times this becomes

(10)

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f t( ) = F ′ t ( )−∞

t

∫ e−κm 2 t− ′ t ( )d ′ t =M0U0

κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )

Therefore

(11)

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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)

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∂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

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sin2 x = 12 1− cos2x( ) , we see that the solution takes the form

(13)

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M2 y, t( ) = A t( ) + B t( )sinmy + C t( )cos2my

where

(14)

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dAdt

= 2κM0 +M0U0

2

κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )cosωt

(15)

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dBdt

+κm2B = 2U0 cos ω t( )M1 y,0( )

and

(16)

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dCdt

+ 4κm2C = −M0U0

2

κ 2m4 +ω 2 κm2 cosωt +ωsinωt( )cosωt

The initial conditions are

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(17)

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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

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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)

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dAdt

= 2κM0 +M0U0

2

κ 2m4 +ω 2 κm2 12

to obtain

(19)

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M2 y, t( ) ~ 2κM0 +M0U0

2κm2

2 κ 2m4 +ω 2( )

t

Thus the variance

(20)

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x 2 ≡x 2θ∫∫θ∫∫~ 2 κ +

U02κm2

4 κ 2m4 +ω 2( )

t ≡ 2κeff t

where

(21)

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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

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θ increaseslinearly with time. If no velocity field is present, then

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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

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κ → 0 and also as

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κ →∞.The first of these limits is easy to understand; if the diffusion coefficient vanishes, theoscillating velocity field periodically restores

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θ to its initial value. The other limit ismore subtle. If

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κ is too large, then the “wiggles” in the

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θ diffuse away before the shearcan make them very wide. For fixed

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U0 and

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ω , the effects of shear dispersion aregreatest when

(22)

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κ* =κm2

ω=1,

that is, when the diffusion time

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Km2( )−1

just equals the advection time

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ω−1. At the value(22), the horizontal shear and vertical diffusion are optimally tuned to cause the greatesteffective horizontal diffusion.