TURBULENCE CLOSURE TECHNIQUES

• In dynamics, you will learn about the equations of motion and thermodynamic equations that can be used to calculate the total changes, and therefore predict, values of U, V, W, and q.
• It is easy to write down three prognostic equations of motion, one each for U, V, and W.
• It is also easy to write down a  prognostic equation for q.
• So, we have four equations for four unknowns..., we can solve this set of equations to compute future values of U,V,W, and q.

• In the boundary layer, in addition to the above variables, we also need to consider variables such as:
  • u', v', w', w'q', u'w', v'w', e', ....., etc.
• Q:  Do we have prognostic equations for all of these variables? ANSWER
• Hence, for boundary layer flow, there are more unknowns than equations.  Hence you can not solve the system of equations for boundary layer flow.
• This is one of the classic, unsolved problems in geophysical fluid dynamics and meteorology.
• It is also called the closure problem.
• Q:  How do we get around this????

• We need to "approximate" the unknown variables like w'q' in terms of variables that are known like W and q.
• Such approximations are called closure approximations/assumptions and are essential in order to write out a complete set of equations to study the boundary layer.
• There are a number of turbulence closure schemes of varying complexity...., we will only cover one of many schemes.

FIRST ORDER LOCAL CLOSURE

• This closure scheme keeps the following types of variables:
  • 1st order terms - U, V, W, q, etc.
  • 2nd order terms - u'u', v'v', w'w', w'q', u'w', ... etc.
• This closure scheme ignores 3rd order and higher terms (e.g., u'u'u', u'u'v', etc.)
• More generally, an nth order closure scheme will keep only n+1 and smaller order terms.
  • For example, a 0th order scheme will only keep 1st order terms, 2nd and higher order terms are ignored.

EXAMPLE OF FIRST ORDER LOCAL CLOSURE

• Consider the following environment:
  • dry
  • horizontally homogeneous
  • no subsidence
• Then, the prognostic equations become:

     (1)

• The unknown (having no prognostic equations) variables are: u'w', v'w', q'w'  
• To "close" this set of equations, we must approximate the unknown variables (u'w', v'w', q'w') in terms of the known variables (U, V, W, q ).
• Q:  How does one do this?????

• A:  Recall that the variables u'w', v'w', q'w' can be interpreted as fluxes, or the transport of some quantity through some area over a period of time.
• For any arbitrary variable, z', the flux of that variable can be expressed as:

      (2)

where K is a scaler with units of m²s^-1.  K is not a constant!!  For positive K, (2) implies that the flux of u'z' flows down the local gradient of z.  

• Q:  Can you draw a diagram illustrating the above statement?  Please do so.
• This closure approximation is often called gradient transport theory, or K-theory.  It is one of the more simple turbulence closure techniques.
• K is often called:
  • eddy viscosity
  • eddy diffusivity
  • eddy-transfer coefficient
  • turbulent-transfer coefficient
  • gradient-transfer coefficient

Using this closure technique, equation set (1) can be written as:

          (3)

• Equation set (3) can now be solved for U, V, W, and q .  Problem solved.  
• Well, not really, you still need to figure out appropriate values for K_H and K_M.  This can be very difficult and will not be discussed here.