Modeling the atmospheric boundary layer 2
This presentation is the property of its rightful owner.
Sponsored Links
1 / 17

Modeling the Atmospheric Boundary Layer (2) PowerPoint PPT Presentation


  • 115 Views
  • Uploaded on
  • Presentation posted in: General

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.

Download Presentation

Modeling the Atmospheric Boundary Layer (2)

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Modeling the atmospheric boundary layer 2

Modeling the Atmospheric Boundary Layer (2)


Review of last lecture

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


Modeling the atmospheric boundary layer 2

The turbulence closure problem

Governing equations for the mean turbulent flow

Where


The turbulence closure problem

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

The current status of the turbulent closure problem

  • For surface layer (surface flux): nearly solvedexcept 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 layer

Surface layer

Eddy flux is assumed to be proportional to the vertical gradience of the mean state variable

  • Sensible heat flux

    <w’h’> = Qh =  Cd Cp V (Tsurface - Tair)

  • Latent heat flux

    L <w’q’> = Qe =  Cd L V (qsurface - qair)

    Where  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

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


Modeling the atmospheric boundary layer 2

First-order closure

z

Turbulent

transport

x

z

Turbulent

transport

x

Z

(by Prof. Ping Zhu of FIU)


Example the ekman layer

Example: The Ekman layer

  • Assumption Km = constant (First order closure)

  • Then the boundary layer momentum equations are:

  • Vertical boundary conditions:

  • Solutions:


General circulation of the oceans

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


Global surface currents

Global surface currents

  • 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


Modeling the atmospheric boundary layer 2

Pressure grad.

force

Centrifugal

force

Pressure grad.

force

Centrifugal

force

Secondary

circulation

Frictional

force

Inflow

Pressure grad.

force

Centrifugal

force

Boundary layer

Ekman pumping

(by Prof. Ping Zhu of FIU)


Modeling the atmospheric boundary layer 2

Hurricane Vortex

Pressure grad.

force

Coriolis

force

L

Centrifugal

force

Converging

Spin up

Diverging

Spin down

Buoyancy

X

Ekman

Pumping

Boundary

Layer

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


Modeling the atmospheric boundary layer 2

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 parameterize

fourth-order moments.

Fourth order turbulent closure is to use fourth-order moments to parameterize

fifth-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

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.


Modeling the atmospheric boundary layer 2

Illustration of non-local turbulent mixing concept

1

1

2

2

3

3

4

4

1

1

2

2

3

3

4

4

(by Prof. Ping Zhu of FIU)


Summary

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


  • Login