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.

Henry Stommel

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Holly Pedlosky

The homogeneous model

Simplest model

H

hb

δ

f/2τ

z

y

x

ρ= constant

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

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

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

∂∂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

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.

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.

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

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

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.

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.

The linear boundary layer problem

δ I

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

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

The western intensification

Stommel’s original explanation of western intensification and the existence of the Gulf Stream due to βeffect.

Controlled by boundary layer

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

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

Total solution (linear)

δs

Velocity profile in boundary layer near western boundary

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

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

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

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)

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.

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

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

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.

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?

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.

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.

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

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

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.

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