Modeling the Atmospheric Boundary Layer (2)

Review of last lecture• Reynolds averaging: Separation of mean and turbulent components

u = U + u’, < u’ > = 0• Intensity of turbulence: turbulent kinetic energy (TKE)

• Eddy fluxes Fx = - <u’w’>/z

TKE = ‹ u’ 2 + v’ 2 + w’ 2 › /2

The turbulence closure problem

Governing equations for the mean turbulent flow

p̂pp ,)(T ),r0.608(1T ,T Rp 0C/R

p1000

vvddd +==+== θρ

0z w

y v

xu =++ ∂

∂∂∂

∂∂

vz

vy

vx

τ

τ

τ

ρρ

ρρ

ρρρ

ρρ

ρρ

ρρ

ρρρ

ρρ

ρρ

ρρρ

+−−−−−=+++

+−−−−−=+++

+−−−+−=+++

∂′′∂

∂′′∂

∂′′∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂′′∂

∂′′∂

∂′′∂

∂∂

∂∂

∂∂

∂∂

∂∂

∂′′∂

∂′′∂

∂′′∂

∂∂

∂∂

∂∂

∂∂

∂∂

z)ww(1

y)vw(1

x)uw(1ˆ

zp̂1

zw

yw

xw

τw

z)wv(1

y)vv(1

x)uv(1

yp̂1

zv

yv

xv

τv

z)wu(1

y)vu(1

x)uu(1

xp̂1

zu

yu

xu

τu

g wvu

uf wvu

vf wvu

tz)wC(

C1

y)vC(

C1

x)uC(

C1

zyxtp

p

p

p

p

pwvu Q+−−−=+++ ∂

′′∂∂

′′∂∂

′′∂∂∂

∂∂

∂∂

∂∂ θρ

ρθρ

ρθρ

ρθθθθ

rz)rwL(

L1

y)rvL(

L1

x)ruL(

L1

zr

yr

xr

tr v

v

v

v

v

vwvu Q+−−−=+++ ∂

′′∂∂

′′∂∂

′′∂∂∂

∂∂

∂∂

∂∂ ρ

ρρ

ρρ

ρ

Where

The turbulence closure problem

• For large-scale atmospheric circulation, we have six fundamental equations (conservation of mass, momentum, heat and water vapor) and six unknowns (p, u, v, w, T, q). So we can solve the equations to get the unknowns.

• When considering turbulent motions, we have five more unknowns (eddy fluxes of u, v, w, T, q)

• We have fewer fundamental equations than unknowns when dealing with turbulent motions. The search for additional laws to match the number of equations with the number of unknowns is commonly labeled the turbulence closure problem.

The current status of the turbulent closure problem

• For surface layer (surface flux): nearly solved except for some extreme conditions (e.g. huricane’s boundary layer)

• For mixed layer: not solved. There is a variety of approaches available:

(1) Local theories (2) Non-local theories

However, these have not converged towards a commonly accepted BL theory, and they often show some biases when comparing with observations.

Surface layerEddy flux is assumed to be proportional to the vertical

gradience of the mean state variable

• Sensible heat flux <w’h’> = Qh = ρ Cd Cp V (Tsurface - Tair) • Latent heat flux L <w’q’> = Qe = ρ Cd L V (qsurface - qair)

Where ρ is the air density, Cd is flux transfer coefficient, Cp is specific heat of air, V is surface wind speed, Tsurface is surface temperature, Tair is air temperature, qsurface is surface specific humidity, qair is surface air specific humidity

Mixed layer theory I: Local theories• K-theory: In eddy-diffusivity (often called K-theory)

models, the turbulent flux of an adiabatically conserved quantity a (such as θ in the absence of saturation, but not temperature T, which decreases when an air parcel is adiabatically lifted) is related to its gradient:

< w’a’ > = - Ka dA/dz• The local effect is always down-gradient (i.e. from high

value to low value)• The key question is how to specify Ka in terms of known

quantities. Three commonly used approaches: (1) First-order closure (2) 1.5-order closure or TKE closure (3) K-profile

First-order closure

x

z

0, zv

zu >∂

∂∂∂

Turbulent transport

) transport turbulentoft coefficien t,coefficien (exchangesity eddy visco :

uw zu

m

m

k

k ∂∂−=′′

,rw

,w

,vw

zr

z

zv

∂∂∂∂∂∂

−=′′

−=′′

−=′′

e

h

m

k

k

k

θθ

motion. offunction (b)

;unit (a) : sm2

mk x

z 0, zr

z <∂∂

∂∂θ

Turbulent transport

mk

Z

mhe kkk 35.1==

(by Prof. Ping Zhu of FIU)

Example: The Ekman layer• Assumption Km = constant (First order closure) • Then the boundary layer momentum equations are:

• Vertical boundary conditions:

• Solutions:

• Ocean surface currents – horizontal water motions• Transfer energy and influence overlying atmosphere• Surface currents result from frictional drag caused by wind -

Ekman Spiral

General circulation of the oceans

• Water moves at a 45o angle (right) in N.H. to prevailing wind direction

• Greater angle at depth

• Surface currents mainly driven by surface winds• North/ South Equatorial Currents pile water westward, create the Equatorial Countercurrent• western ocean basins –warm poleward moving currents (example: Gulf Stream)• eastern basins –cold currents, directed equatorward

Global surface currents

Boundary layer

Pressure grad.force

Centrifugalforce

Pressure grad.force

Centrifugalforce

RV 2

Pressure grad.force

Centrifugalforce

RV 2

Frictionalforce

Inflow

Secondary circulation

Ekman pumping

(by Prof. Ping Zhu of FIU)

X

BoundaryLayer

EkmanPumping

Hurricane Vortex

DivergingSpin down

ConvergingSpin upBuoyancy

Pressure grad.force

Centrifugalforce

Coriolisforce

L

It is the convective clouds that generate spin up process to overcome the spin down process induced by the Ekman pumping

To close the system, first order turbulent closure is to use first-order moments to parameterize second-order moments.

Second order turbulent closure is to use second-order moments to parameterize third-order moments.

Third order turbulent closure is to use third-order moments to parameterizefourth-order moments.Fourth order turbulent closure is to use fourth-order moments to parameterizefifth-order moments.………

Why higher order closure is better than lower order closure?

The advantage of higher-order turbulence closure is that parameterizations of unclosed higher-order moments, e.g., fourth-order moments, might be very crude, but the prognosed third-order moments can be precise since there are enough remaining physics in their budget equations. The second-order moment equations bring in more physics, making them even more precise, and so on down to the first-order moments.

Mixed layer theory II: Non-local theories• Any eddy diffusivity approach will not be entirely accurate if most of the

turbulent fluxes are carried by organized eddies filling the entire boundary layer.

• The non-local effect could be counter-gradient. • Consequently, a variety of ‘nonlocal’ schemes which explicitly model the

effects of these boundary layer filling eddies in some way have been proposed.

• A difficulty with this approach is that the structure of the turbulence depends on the BL stability, baroclinicity, history, moist processes, etc., and no nonlocal parameterization proposed to date has comprehensively addressed the effects of all these processes on the large-eddy structure.

• Nonlocal schemes are most attractive when the vertical structure and turbulent transports in a specific type of boundary layer (i. e. neutral or convective) must be known to high accuracy.

Illustration of non-local turbulent mixing concept

4

1

3

2

4

1

3

2

4

1

3

2

4

1

3

2

(by Prof. Ping Zhu of FIU)

Summary

• The turbulent closure problem: Number of unknowns > Number of equations

• Surface layer: related to gradient• Mixed layer: Local theories (K-theory): < w’a’ >= - Ka dA/dz

always down-gradient Non-local theories: organized eddies filling the

entire BL, could be counter-gradient