Lecture 6: Boundary Layers - GitHub Pages · 2020-07-10 · Turbulence What are boundary layers? I...
Transcript of 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
28 March 2016
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
100∼1000 m
10∼100 m
free atmosphere
atmospheric boundary layer
ocean mixed layer
thermocline
turbulentmixing
momentumflux
watervapor
potential temperature
What are boundary layers?
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
QNET = QSW −QLW −QLH −QSH
ocean surface
Surface fluxes
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
QSH = cpρawθ
Sensible heat flux
I Direct heating of air in contact with the surface
I Transmitted through the boundary layer by turbulence
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
QLH = Lvρawqv
Latent heat flux
I Energy exchanged in the form of evaporation
I Transmitted through the boundary layer by turbulence
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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...
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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:
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
mechanical turbulence
(momentum conversion)
convective turbulence
(buoyancy conversion)
ρθ
wind stress
shear–flow mixing
ρθ
buoyancy gain
convective mixing
Sources of boundary layer turbulence
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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 role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
What are boundary layers?The sensible and latent heat fluxesTurbulence
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)
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
Boussinesq continuity equation:
(∇ · v = 0)
The Boussinesq equations
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
The Boussinesq equations
Using the Boussinesq approximation to account for turbulence
∂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′
)
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
Turbulent fluxes
b ≡ g ρ− ρ0ρ0
The Boussinesq equations
Using the Boussinesq approximation to account for turbulence
∂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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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 role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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...
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
Accounting for turbulence in the momentum equationsnHeat and momentum fluxesEkman layers
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 role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
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 role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
The atmospheric boundary layerThe ocean mixed layer
The Atmospheric Boundary Layer
entrainment zone
free atmosphere
mixed layer
surface layer
✓v q u
The atmospheric boundary layer
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
The atmospheric boundary layerThe ocean mixed 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 role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
The atmospheric boundary layerThe ocean mixed layer
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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
The atmospheric boundary layerThe 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
The role of boundary layers in air–sea interactionFluid dynamics for boundary layers
Boundary layers in the atmosphere and ocean
The atmospheric boundary layerThe ocean mixed layer
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: