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GFD 2007Boundary Layers: Homogeneous Ocean Circulation
One of the most significant applications of boundary layer
theory occurs in the treatment of the oceanic general
circulation
Stommel, H. 1948 The westward intensification of wind-driven
ocean currents. Trans. Amer. Geophys.Union,29, 202-206. 1 Munk,
W.H. 1950. On the wind-driven ocean circulation, J.Meteor.,7,
79-93
Munk W.H. and G.F. Carrier, 1950 The wind-driven circulation in
basins of various shapes. Tellus, 2,158-167.
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Henry Stommel
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Holly Pedlosky
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The homogeneous model
Simplest model
H
hb
δ
f/2τ
z
y
x
ρ= constant
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The model
Upper Ekman layer provides a pumping
w=we =k̂g∇×(rτ /ρf) z = H+hb
w=
δ2
ζ + rug∇hb =δ2
vx −uy⎡⎣ ⎤⎦+uhbx +vhby z = hb
β planeΩ
R
zy
θf = fo +βy, β=
∂f∂y
=2Ωcosθ
R
fo =2Ω sinθ
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Model equations of motion (1)
∂∂t
+u∂∂x
+v∂∂y
⎛⎝⎜
⎞⎠⎟ζ+βv= f
∂w∂z
+A∂2
∂x2+
∂2
∂y2⎛⎝⎜
⎞⎠⎟ζ
Vorticity equation
Integrating vertically,
∂∂t
+u∂∂x
+v∂∂y
⎛⎝⎜
⎞⎠⎟ζ+βv+
fruH
g∇hb =fweH
−fδ2H
ζ+A∂2
∂x2+
∂2
∂y2⎛⎝⎜
⎞⎠⎟ζ
u = −∂ψ∂y
, v =∂ψ∂x
, ψ = pρfoGeostrophic stream function
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Equations of motion (2) and scaling
∂∂t
∇2ψ+J(ψ,∇2ψ)+βψx +fH
J(ψ,hb)=fweH
−fδ2H
∇2ψ+A∇4ψ
U , L for velocity and length, UL for ψ
(βL)-1 �or time. Choose U as U =τ o
ρ H o L β
Ekman pumping scales with We =τ o
ρ fL
∂∂t
∇2ψ+δI2J(ψ,∇2ψ)+ψx +ηJ(ψ,hb)=we −δs∇
2ψ+δm3∇4ψ
u = −∂ψ∂y
, v =∂ψ∂x
, ψ = pρfo
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∂∂t
∇2ψ+δI2J(ψ,∇2ψ)+ψx +ηJ(ψ,hb)=we −δs∇
2ψ+δm3∇4ψ
δI =U/β( )1/2
L , η=
fΔhbHoβL
, δs =fδ
2HoβL, δm =
A/β( )1/3
L
Governing equation and boundary layer scales
Inertial
scaleStommel Munk
Relative strength of bottom topography to β effect
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The singular perturbation problem
δΙ , δΜ , and δS are all small. Boundary layer scales are much
less than the full basin width. They multiply the higher order
derivatives.
x =0, y =0 x =xe
y =1 L is the north-south extent of the basin.
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The interior problem
For a flat bottom interior, when all the boundary layer scales
are small, the governing equation is :
ψx =we(x,y) Sverdrup relation. 1st order ode in x alone.
Only
determines interior meridional velocity.
Can’t satisfy no slip and can satisfy no normal flow, or ψ =0,
only on one boundary, east or west but not both.
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Two possible (at least) interior solutions
1)Satisfy ψ =0 on western boundary ψ = we(x ', y)dx '0
x
∫
or 2) on eastern boundary
example
ψ = − we(x ',yx
xe
∫ )dx '
we =−sinπy=vIψ1 = (xw − x)sinπy,u1 = −(xw − x)π cosπy
ψ2 = (xe − x)sinπyu2 = −(xe − x)πcosπy
τx
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An Integral constraint (1)
drs
n̂
C
C a steady (closed ) streamline.
∂ru
∂tgdrs
C—∫ + ruδI2ζ+y+ηhb⎡⎣ ⎤⎦ĝn
C—∫ ds=
rτgdrs
C—∫ −δs rugdrs
C—∫ +δm3 ∇ζĝn
C—∫ ds
= 0
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An Integral Constraint (2)
0=
rτgdrs
C—∫ −δs rugdrs
C—∫ +δm3 ∇ζĝn
C—∫ ds
The net input of vorticity on each streamline must be balanced
by bottom friction and lateral friction (for steady flow)
If there are eddies that flux vorticity integral must include
that effect.
0=
rτgdrs
C—∫ −δs rugdrs
C—∫ +δm3 ∇ζĝn
C—∫ ds−δI2 ru'ζ '
C—∫ ĝnds
but this last term must be zero for the streamline coincident
with boundary.
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The Energy constraint
Multiplying by ψ and integrating over the closed basin for
steady flow:
weψ =−δs |∇ψ |2 −δm
3 ∇2ψ2
So that ψ and we must be negatively correlated. On the whole
this implies a circulation in the direction of the wind stress.
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The linear boundary layer problem
δ I
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The Stommel model
In the boundary layer, keeping only x derivatives and letting x
= δξ
After a single integration in x
φa}
= −(δs /δ)∂φ∂ξ
b6 74 84
+ δm3
δ 3⎛⎝⎜
⎞⎠⎟
∂3φ∂ξ3
c6 744 84 4Consider case δm
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The Stommel solution
φ = A ( y )e − ξ To satisfy no normal flow condition on x =0
A = −ψ I (0, y)If we try to do the same on the eastern boundary
ξ ' = (xe − x) / δ s
φ =∂ φ∂ ξ '
bl correction function grows exponentially. No boundary layer
possible on eastern boundary. Hence,Ψ(y) =0 in interior
solution
ψ (x, y) = ψ I (x, y) + ψ I (0, y)e− x /δ s ,
ψ I (x, y) = − we (x ', y)dx 'x
xe
∫
δs=fδ/2ΗβL
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The western intensification
Stommel’s original explanation of western intensification and
the existence of the Gulf Stream due to βeffect.
Controlled by boundary layer
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The no slip condition and the sublayer
•Need to satisfy no slip condition .So far ignored.
•The vorticity balance of the whole basin depends on the lateral
diffusion term if no slip condition applies. So far ignored.
To preserve the total order of the system and to satisfy the no
slip condition we need to include terms b and c in boundary layer
equation. Now, x= δsξ
φa}
= −∂φ∂ξ
b}+ δ m
3
δ s3
⎛⎝⎜
⎞⎠⎟
∂ 3φ∂ξ 3
c6 744 84 4For sub layer define ξ = l η
l =δ mδ s
⎛⎝⎜
⎞⎠⎟
3 / 2
δ s u b = δ s l
=δ m
3 / 2
δ s1 / 2
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Correction function in sublayer
χ(η) χηη −χ=0Essentially, the Stewartson E1/4 layer. Independent
of β, symmetric east west.
χ = Ce−η δ sub = AL2H o2ν f
⎡
⎣⎢
⎤
⎦⎥
1/ 2
ψ=ψI(x,y)+A(y)e−x/δs +C(y)e−x/δsub
matching ψ (0,y) = 0 = ψ I (0,y) + A(y) + C(y)
ψ x(0,y) = 0 = ψ Ix (0,y) −Aδs
−C
δsub
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Total solution (linear)
δs
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Velocity profile in boundary layer near western boundary
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The dissipation balance (1)
Integrate across basin. Ignore y derivatives in dissipation
terms. For Ekman pumping independent of x,
0=xewe +δsψx(0)−δm3ψxxx(0)
=0 for no slip
Boundary current has no net vorticity
The contribution of the sublayer to the final term is:
−xeweδm3 δm
δs
⎛
⎝⎜⎞
⎠⎟
3/21
δs3 δm
δs( )9/2 = −xewe
Balances input of vorticity
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The dissipation balance (2)
Most of the fluid flowing south in the interior returns in the
Stommel layer and not the sublayer. On those streamlines always
outside the sublayer the dissipation balance only involves bottom
friction,
Integrating across the basin from just outside the sublayer to
the eastern boundary, the total mass flux balances and:
0 = xewe(0+ , y) + δ sφx (0+ , y),
⇒0 = xewe(0+ , y) − δ s xewe(0, y) / δ s
Vorticity balance on streamlines through Stommel layer
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An integral balance for the boundary layer
R
y2
y1
x = 0
δI2∇gruζdA
R∫ + vdA
R∫ +η ∇g
ruhbdAR∫ =−δs ζdA
R∫ + δm3∇2ζdA
R∫
Ignoring vorticity input by wind in the boundary layer
region
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The integral balance with bl approximations
δI2 12
v2(0,y1)−v2(0,y2)⎡⎣ ⎤⎦+ ψI(0,y)dy
y1
y2
∫ −η ψ∂hb∂x y=y2
−ψ∂hb∂x y=y1
⎡
⎣⎢
⎤
⎦⎥
=δs v(0,y)dyy1
y2
∫ −δm3 ζx(0,)dyy1
y2
∫
ζ ≈ vxused
If the bottom is flat and the no slip condition applies
ψ I (0,y)dyy1
y2
∫ = −δm3 ζx(0,y)dyy1
y2
∫the vorticity put into the fluid along latitude y must be
dissipated in the boundary layer at that latitude to obtain a
steady state balance. In the presence of an uneven bottom the
pressure drag can locally enter the balance but when integrated
along a closed streamline the topographic term can give no net
contribution (just as the planetary or relative vorticity
advection)
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See Hughes C. W. and B. de Cuevos. 2001 Why western boundary
currents in realistic oceans are inviscid: A link between form
stress and bottom pressure torques. J.Phys.
Ocean. 31, 2871-2885.
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Inertial boundary layers
δI >>δm>>δsMost fluid will go through inertial layer
but there is not enough dissipation in the layer to satisfy the
vorticity balance on those streamlines
ξ = x /δ I
ψξψξξy−ψyψξξξ+ψξ =0 ψξξ + y = Q(ψ)Total vorticity conserved on
streamline
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Inertial boundary layer:Example
δI
U = constant ψξξ + y = Q(ψ)
Far from the boundary the relative vorticity is negligible
so
ξ ∞ Q(ψ ) y
ψ y
Q(ψ )=ψ On all streamlines connected to far field
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Inertial layer (2)
ψ ξξ + y = ψ ψ = y 1− e−ξ⎡⎣ ⎤⎦In non-dimensionless units
ψ* =Uy 1− exp −xβ
U( )1/2⎧
⎨⎩
⎫⎬⎭
⎡
⎣⎢
⎤
⎦⎥
Interior flow needs to be westward.
Greenspan H.P. 1962 A criterion for the existence of inertial
boundary layers in the oceanic circulation. Proc. Nat. Acad.Sci,,
48, 2034-2039.
Pedlosky, J. 1965 A note on the western intensification of the
oceanic circulation. J. Marine Res. , 23, 207-209.
Can’t close circulation
or satisfy no slip.
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Inertial sublayer
δd =Aδ*IU
⎛⎝⎜
⎞⎠⎟
1/2
= δ*IδmδI
⎛⎝⎜
⎞⎠⎟
3/2vorticity flux through sublayercould balance vorticity input
by the wind.
Most streamlines don’t go through sublayer.
In Stommel model the streamlines that did not go through the
sublayer still had a proper vorticity balance. This is no longer
true.
Re =δ Iδ m
⎛⎝⎜
⎞⎠⎟
3Boundary layer Reynolds number
Inertial/viscous in inertial layer>> 1
What happens?
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Inertial Runaway
Panel a shows the linear solution when δI is zero, panel b shows
the case for Re=1, panel c shows the flow for Re = 1.95
, while for panel d, Re=4.29,
δI =.0625, δm =.05
δI =.08125, δm =.05
Circulation intensifies until vorticity is dissipated on each
streamline.
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References
Veronis, G. 1966 Wind-driven ocean circulation-part I. Linear
theory and perturbation analysis. Deep-Sea Res. 13, 17-29Ierley,
G.R. and V.A. Sheremet. 1995 Multiple solutions and
advection-dominated flows in the wind-driven circulation. Part I:
Slip. J. Marne Res.53, 703-737,Fox-Kemper, B. 2003. Eddies and
Friction: Removal of vorticity from the wind-driven gyre. MIT/WHOI
Joint Program Ph.D. thesis
Fox-Kemper,B. and J.Pedlosky, 2004. Wind-driven barotropic gyre
I: Circulation control by eddy vorticity fluxes to an enhanced
removal region. J. Marine Res., 62, 169-193.
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The enhanced sublayer
Fox-Kemper allowed the dissipation to locally increase in a
sublayer near the western boundary. Rei =
δIδm
When we set a value of A, the momentum mixing coefficient, we
are conflating two somewhatindependent physical processes.
The first is a measure of the unresolved eddy scales and their
effect on the large scale flow in the interior and the boundary
layers.
The second is a measure of the strength of the interaction of
the fluid with the boundary.
Decreasing the single parameter, A, then reduces both processes.
If the interaction with the boundary is related to a different
physical process than the dissipation of vorticity away from the
boundary, it seems overly constraining to represent both with a
single parameterization.
⎛⎝⎜
⎞⎠⎟
3
interior
δm3 =
δI3
Rei+
δI3
Reb−
δI3
Rei
⎛⎝⎜
⎞⎠⎟
e−x /δd + e−(1−x)/δd( ) δd = δIRei
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The enhanced sublayer (2)
δ m3 =
δ I3
R e bNear the boundary
δm3 =
δI3
Rei+
δI3
Reb−
δI3
Rei
⎛
⎝⎜⎞
⎠⎟e−x/δd +e−(1−x)/δd( )
∂∂t
∇2ψ+δI2J(ψ,∇2ψ)+ψx +ηJ(ψ,hb)=we−δs∇
2ψ+∇g{δm3∇(∇2ψ)}
variable
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The turbulent boundary layer and the role of eddies.
For substantial values of the interior Reynolds numbers the
western boundary layer becomes unstable to shear instability. The
eddies in the inertial portion of the boundary layer, through which
most of the mean streamlines pass, will flux vorticity to the
sublayer where it is dissipated by the locally enhanced friction.
The process is a three step one, instability, flux and dissipation
and the student is referred to Fox-Kemper and Pedlosky for the
details of the analysis. The result though is striking and shown in
the following figure.
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The arrested runaway
compare
Control of large scale circulation by details of dissipation in
the boundary layer.
GFD 2007�Boundary Layers: Homogeneous Ocean CirculationHenry
StommelThe homogeneous modelThe model Model equations of motion (1)
Equations of motion (2) and scalingGoverning equation and boundary
layer scalesThe singular perturbation problemThe interior
problemTwo possible (at least) interior solutionsAn Integral
constraint (1)An Integral Constraint (2)The Energy constraintThe
linear boundary layer problemThe Stommel modelThe Stommel
solutionThe western intensification The no slip condition and the
sublayerCorrection function in sublayerTotal solution
(linear)Velocity profile in boundary layer near western boundaryThe
dissipation balance (1)The dissipation balance (2)An integral
balance for the boundary layerThe integral balance with bl
approximationsSee Hughes C. W. and B. de Cuevos. 2001 Why western
boundary currents in realistic oceans are inviscid: A link between
form stInertial boundary layersInertial boundary
layer:�ExampleInertial layer (2)Inertial sublayerInertial
RunawayReferencesThe enhanced sublayerThe enhanced sublayer (2)The
turbulent boundary layer and the role of eddies.The arrested
runaway