TURBULENCE KINETIC ENERGY BUDGET

• Turbulence Kinetic Energy (TKE) is one of the more important variables in boundary layer meteorology since it is a measure of turbulence intensity.
  • It is directly related to the transport of heat, moisture, and momentum through the boundary layer.
  • The tendency of TKE is also important to understand:
    • if TKE decreases with time, the boundary layer is becoming less turbulent with time
    • if TKE increases with time, the boundary layer is becoming more turbulent with time
  • So, what processes create and destroy TKE?
  • These processes are summarized in the TKE budget equation to be discussed next.

• TKE Budget Equation

Recall from Section 3, that TKE is defined as:

or the mean kinetic energy (averaged over some period of time) is equal to the sum of the average (again in time) velocity variances divided by 2.

Now, the tendency of TKE can be expressed as:

I            II                 III             IV                  V              VI             VII

OK, this is an ugly equation..., but we can deduce what physical process each of the terms represent:

1. Physical explanation of term #1: Answer
2. Physical explanation of term #2: Answer
3. Physical explanation of term #3: Answer
4. Physical explanation of term #4: Answer
5. Physical explanation of term #5: Answer
6. Physical explanation of term #6: Answer
7. Physical explanation of term #7: Answer

Let's now look at each of these terms in more detail.

TERM I - STORAGE/TENDENCY TERM  

• The diurnal evolution of TKE is shown in Fig. 5.1.  Notice:
  • the rapid increase of TKE beginning at about 11 AM.
  • also notice the rapid decrease of TKE beginning sometime after 2PM.
  • also notice that during the evening hours, TKE is much smaller in magnitude and is confined to near the ground.
  • Q:  When during the course of the 24 hour day is term I positive?
  • Q:  When during the course of the 24 hour day is term I negative.
  • Q:  Is there any time(s) when term I is 0?
• Q:  How would this plot of the TKE tendency be different for the marine boundary layer?  Answer

TERM II - ADVECTION

• Believe it or not, little is known about this term
• However, when averaged over a 10x10 km area or larger, it is often assumed that there is little horizontal variation of TKE and therefore, the advection term is 0.
• Q:  What is an example(s) of a real-world situation where this assumption would break down at the advection of TKE would be important?

TERM III - BUOYANT PRODUCTION/CONSUMPTION  

Production of TKE:

• Term III represents the generation/destruction of TKE by buoyancy, i.e., the vertical flux of virtual potential temperature.
• Recall from the last section and also shown in the figure to the right, this flux is positive and decreases approximately linearly with height up to the top of the mixed layer.
• This term represents the effect of thermals on the generation of TKE
  • Hence, this term will be larger on sunny days, and weaker on cloudy days.

Consumption of TKE:

• Since a stable layer of air tends to suppress convective motions, static stability tends to suppress, or consume TKE.
• Q:  Where or when will this commonly occur in the boundary layer?
  • Answer
  • Answer

TERM IV - MECHANICAL (SHEAR) PRODUCTION 

• As can be seen in this term, it is the interaction of the turbulent momentum flux with the mean vertical wind shear that generates turbulence.
• Notice from Fig. 5.4 above, that this term is very large and positive near the ground.  Why is this the case?
• This term is obviously going to be larger on a windy day (larger vertical shear) and smaller on a calm day.
• So this term would be large in the surface layer associated with a synoptic-scale cyclone, and would be small in the surface layer associated with an area of high pressure.
• Notice that this term contains w', i.e., eddy vertical motion
  • this vertical motion is not due to convection, but due to the production of eddies by shear.  These shear-generated eddies can in turn generate clouds.  This mechanism for cloud generation is called "forced convection".
  • Forced convection would be more common with a synoptic-scale low than a high.

TERM V - TURBULENT TRANSPORT

• The fifth term represents the vertical transport of TKE
• It does not create or destroy TKE, but simply transports it vertically!   
• More specifically, this term represents the flux divergence of TKE for a layer since it depends on the vertical gradient of vertical flux.
• So, for a given layer, if more flux in entering the layer than leaving it, there is a net convergence of the vertical flux, and therefore, the TKE of the layer will increase, as illustrated in the figure to the right.

Now, take a look at Fig. 5.4.  Consider the layer between 0.2-0.4.  Is the turbulent flux divergence positive or negative in this layer?  Answer

TERM VI - PRESSURE CORRELATION TERM

• This term is hard to measure since p' is often quite small, therefore the w'p' term is usually in the noise level.
• Waves in the boundary layer are also thought to contribute to this term.

TERM VII - DISSIPATION

• The last term in the TKE budget equation is the dissipation term. 
• The largest dissipation rates are near the ground
  • This is also where the buoyancy and shear source terms are largest (see Fig. 5.4)
• The dissipation rate decreases rapidly once you get above the mixed layer.
• Note that the dissipation term arises as the smallest eddies give up their energy in the form of viscous heating.
• The small eddies gain their energy from larger eddies.  Recall that this is called the energy cascade - energy is transferred from the large eddies to the smaller ones, and eventually into heat at the molecular level.