- 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,
- It is easy to write down three prognostic equations of motion, one each
for U, V,
- 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,
- In the boundary layer, in addition to the above variables, we also need to
consider variables such as:
- u', v', w', w'q',
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
- 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
- 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,
- 2nd order terms - u'u',
u'w', ... etc.
- This closure scheme ignores 3rd order and higher terms (e.g., u'u'u',
- 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:
- horizontally homogeneous
- no subsidence
- Then, the prognostic equations become:
unknown (having no prognostic equations) variables are: u'w',
"close" this set of equations, we must approximate the unknown
variables (u'w', v'w',
in terms of the known variables (U,
How does one do this?????
- A: Recall that the variables u'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:
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.
Can you draw a diagram illustrating the above statement? Please do so.
closure approximation is often called gradient transport theory, or K-theory.
It is one of the more simple turbulence closure techniques.
is often called:
this closure technique, equation set (1) can be written as:
set (3) can now be solved for U,
. Problem solved.
not really, you still need to figure out appropriate values for KH
and KM. This can be very difficult and will not be