lecture 6: boundary layers - github pages · 2020-07-10 · turbulence what are boundary layers? i...

The role of boundary layers in air–sea interactionFluid dynamics for boundary layers

Boundary layers in the atmosphere and ocean

Lecture 6:Boundary Layers

Jonathon S. Wright

[email protected]

28 March 2016



The role of boundary layers in air–sea interactionWhat are boundary layers?The sensible and latent heat fluxesTurbulence

Fluid dynamics for boundary layersAccounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers

Boundary layers in the atmosphere and oceanThe atmospheric boundary layerThe ocean mixed layer



What are boundary layers?The sensible and latent heat fluxesTurbulence

What are boundary layers?

I The lowermost part of the atmosphere and the uppermost part of the ocean

I Control the exchange of heat, moisture, and momentum across the air–sea interface

I Boundary layer processes directly affect SSTs, which affects interactions between the atmosphereand ocean at a wide range of space and time scales

I Affect surface exchange of aerosols and chemical constituents

I Strong turbulence means that boundary layers are well-mixed and respond rapidly to changes insurface conditions

I The atmospheric boundary layer and the ocean mixed layer are similar in many ways, but alsoprofoundly different




100∼1000 m

10∼100 m

free atmosphere

atmospheric boundary layer

ocean mixed layer

thermocline

turbulentmixing

momentumflux

watervapor

potential temperature

What are boundary layers?




QNET = QSW −QLW −QLH −QSH

ocean surface

Surface fluxes




QSH = cpρawθ

Sensible heat flux

I Direct heating of air in contact with the surface

I Transmitted through the boundary layer by turbulence




QLH = Lvρawqv

Latent heat flux

I Energy exchanged in the form of evaporation

I Transmitted through the boundary layer by turbulence




B ≡ QSH

QLS≤ Be =

(L

cp

∂q∗

∂T

)−1Theoretical maximum over a wet surface (when RH ≈ 1)

The Bowen ratio

I Ratio of sensible heat flux to latent heat flux

I Over the ocean, the Bowen ratio is almost always small




Reynolds averaging: w = w + w′ θ = θ + θ′

wθ = (w + w′)(θ + θ′) = wθ + w′θ + wθ′ + w′θ′

wθ = wθ + w′θ′

Time average turbulent fluxes

Accounting for turbulence: Reynolds averagingWe can only use wθ if w and θ are sampled often enough to account for turbulent variations...




QSH = cpρa(wθ + w′θ′

)= cpρaw′θ′

QLH = Lvρa(wqv + w′q′v

)= Lvρaw′q′v

Accounting for turbulence: Reynolds averagingThe primary contributors to the sensible and latent heat fluxes are the time mean turbulent terms:




mechanical turbulence

(momentum conversion)

convective turbulence

(buoyancy conversion)

ρθ

wind stress

shear–flow mixing

ρθ

buoyancy gain

convective mixing

Sources of boundary layer turbulence




ρθ

wind stress

shear–flow mixing

ρθ

stronger wind stress

stronger mixing

entrainment

EntrainmentIf the forcing increases, turbulence deepens and draws fluid into the well-mixed layer from below




ρθ

detrainment

dm

ρθ

entrainment

dm

d (dm)

dt= w − wechanges in mixed layer depth:

dm ≡mixed layer depth

w ≡ vertical velocity (upwelling or downwelling)

we ≡ the flux of denser fluid into the mixed layer

Entrainment




TKE ≡ u′2 + v′2 + w′2

2

d(TKE)

dt= MP + BPL + TR− ε

Mechanical Production

Buoyancy Production & LossTransport

frictional dissipation

Turbulent kinetic energy




the source of turbulent kinetic energy

ρθ

the entrainment rate

dm

the depth (or height)

of the boundary layer

the magnitude of the inversion

Turbulence and boundary layer propertiesEven without knowing the details of boundary layer turbulence, the bulk properties of boundary layerscan be summarized and their evolution tracked (or parameterized for use in models)



Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers

buoyancy b ≡ g ρ− ρ0ρ0

Boussinesq continuity equation:

(∇ · v = 0)

changes in buoyancy (ρ) due to changes in θ or composition

The Boussinesq equations

Density constant except where coupled to gravity

∂u

∂t+ (v · ∇)u− fv = − 1

ρ0

∂p

∂x+ Fx

∂v

∂t+ (v · ∇) v + fu = − 1

ρ0

∂p

∂y+ Fy

∂w

∂t+ (v · ∇)w = − 1

ρ0

∂p

∂z− gρ− ρ0

ρ0+ Fz

∂u

∂x+∂v

∂y+∂w

∂z= 0

∂b

∂t+ (v · ∇) b = b




Boussinesq continuity equation:

(∇ · v = 0)


Using the Boussinesq approximation to account for turbulence

∂u

∂t+ (v · ∇)u =

∂u

∂t+ (v · ∇)u+ u

(∂u

∂x+∂v

∂y+∂w

∂z

)=∂u

∂t+ u

∂u

∂x+ v

∂u

∂y+ w

∂u

∂z+ u

(∂u

∂x+∂v

∂y+∂w

∂z

)=∂u

∂t+∂u2

∂x+∂uv

∂y+∂uw

∂z






∂u

∂t+ (v · ∇)u =

∂u

∂t+∂u2

∂x+∂uv

∂y+∂uw

∂z

∂u

∂t+ (v · ∇)u =

∂u

∂t+

∂

∂x

(uu+ u′u′

)+

∂

∂y

(uv + u′v′

)+

∂

∂z

(uw + u′w′

)=∂u

∂t+ (v · ∇)u+

∂

∂x

(u′u′

)+

∂

∂y

(u′v′

)+

∂

∂z

(u′w′

)




Turbulent fluxes

b ≡ g ρ− ρ0ρ0



∂u

∂t+ (v · ∇)u+

∂

∂x

(u′u′

)+

∂

∂y

(u′v′

)+

∂

∂z

(u′w′

)= − 1

ρ0

∂p

∂x+ fv + Fx

∂v

∂t+ (v · ∇) v + ∂

∂x

(v′u′

)+

∂

∂y

(v′v′

)+

∂

∂z

(v′w′

)= − 1

ρ0

∂p

∂y− fu+ Fy

∂w

∂t+ (v · ∇)w +

∂

∂x

(w′u′

)+

∂

∂y

(w′v′

)+

∂

∂z

(w′w′

)= − 1

ρ0

∂p

∂z+ b+ Fz

∂θ

∂t+ (v · ∇) θ + ∂

∂x

(u′θ′

)+

∂

∂y

(v′θ′

)+

∂

∂z

(w′θ′

)= −w∂θ0

∂z




Treat turbulent momentum fluxes like friction

The (turbulent) Boussinesq momentum equations

∂u

∂t+ (v · ∇)u = − 1

ρ0

∂p

∂x+ fv − ∂u′w′

∂z

∂v

∂t+ (v · ∇) v = − 1

ρ0

∂p

∂y− fu− ∂v′w′

∂z




horizontal wind above ocean surface

Wind stressThe force applied by the wind on the surface of the ocean

τx = −ρa(w′u′)τy = −ρa(w′v′)

τxy = −ρa(w′v′H)




The surface stress is consistent across the ocean surface

Friction velocityThe characteristic velocity of a fluid under a given stress

u2? ≡|τxy|ρ

|τxy| =(ρu2?

)air

=(ρu2?

)ocean




u′w′ ≈ −KM

(∂u

∂z

)v′w′ ≈ −KM

(∂v

∂z

)

θ′w′ ≈ −KH

(∂θ

∂z

)eddy diffusivity of momentum

eddy diffusivity of heat

The flux–gradient approximationTreat turbulent transport like molecular diffusion




assuming vg = 0 in the Northern Hemisphere atmosphere...

u = ug[1− e−γz cos (γz)

]v = ug

[e−γz sin (γz)

]

Ekman layersThree-way balance between pressure gradient, Coriolis force, and turbulent friction:

KM∂2u

∂z2+ f (v − vg) = 0

KM∂2v

∂z2− f (u− ug) = 0




0.0 0.2 0.4 0.6 0.8 1.0u/ug

0.0

0.1

0.2

0.3

0.4

v/u g

(a) Ekman spiral in atmospheric boundary layer

0.2 0.0 0.2 0.4 0.6 0.8 1.0u/ug

0.0

0.1

0.2

0.3

0.4

v/u g

(b) Ekman spiral in ocean surface mixed layer

wind closer to surface

geostrophic wind (base of free atmosphere)

surface current

closer to mixed layer base (dm)

Ekman spiralsAssuming vg = 0 in the Northern Hemisphere...




Ekman transportVertically-integrated transport is oriented 90◦ to the right of the wind stress in the NH (90◦ to the leftin the SH), driving convergence and divergence in the mixed layer beneath synoptic circulation systems:



The atmospheric boundary layerThe ocean mixed layer

The Atmospheric Boundary Layer

entrainment zone

free atmosphere

mixed layer

surface layer

uviscous sublayermolecular diffusion

mechanical turbulence

well-mixed layer

entrainment of free tropospheric air

The atmospheric boundary layer




The Atmospheric Boundary Layer

entrainment zone

free atmosphere

mixed layer

surface layer

✓v q u

The atmospheric boundary layer




Ri =g

θ

∂θ/∂z

(∂u/∂z)2=

N2

(∂u/∂z)2

Buoyancy production and loss

Mechanical production

A smaller Richardson number means stronger turbulence

The atmospheric Richardson number

Joint measure of thermodynamic and mechanical stability:




The Ocean Mixed Layer

Altit

ude

Dept

h

solar heating

evaporation cooling, salination

convection cold, salty

stirring by surface wind

rainfall freshening

turbulent mixing

thermocline

entrainment

The ocean mixed layer




Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov DecMonth

0

20

40

60

80

100

Depth

[m

]

10

11

11

12

13141516

1718

19

Deeper when surface

heating is weak and

wind stress is strong

Shallower when surface heating

is strong and wind stress is weak

Seasonal thermocline

Permanent thermoclinedata from World Ocean Atlas 2009

The ocean mixed layer

https://www.nodc.noaa.gov/OC5/WOA09/pr_woa09.html




Ri =g

ρ

∂ρ/∂z

(∂u/∂z)2=

N2

(∂u/∂z)2

Buoyancy production and loss

Mechanical production

The Richardson number in the oceanDefined similarly to that in the atmosphere:

lecture 6: boundary layers - github pages · 2020-07-10 · turbulence what are boundary layers? i...

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