Similarity theory 1. Buckingham Pi Theorem and examples Turbulent closure problem requires empirical expressions for determining turbulent eddy diffusion coefficients. The development of turbulence closure is based on observations not theory. We need to find an intelligent way of organizing observational data. Similarity theory is a method to find relationships among variables based on observations. 1. Buckingham Pi Theorem and examples We want to find the relationship between the cruising speed and the weight of airplane. a. Define relevant variables and their dimensions. b. Count number of fundamental dimensions.

c. Form n dimensionless groups where n is the number of variables minus the number of fundamental dimensions. gravitational force lift force mass of airplane mass of displaced air d. Measure as a function of Lift force = pressure*area = (rho*U^2) * (l^2) e. Further simplification; assume: i.e.

The great flight diagram Weight (Newtons) Weight as a function of cruising speed ( “The simple science of flight” by Tennekes, 1997, MIT press) W~U6 Flying objects range from small insects to Boeing 747 Speed (m/s)

Procedure of Buckingham Pi Analysis Step 1, Hypothesize which variables could be important to the flow. e.g., stress, density, viscosity, velocity, ….. Step 2, Find the dimensions of each of the variables in terms of the fundamental dimensions. Fundamental dimensions are: L=length M=mass T=time K=temperature Dimensions of any other variables can be represented by these fundamental dimensions. Example

Step 3, Count the number of fundamental dimensions in the problem there are 3 dimensions in this example: L, M, T Step 4, Pick up a subset of original variables to become “key variables”, subject to the following restrictions: The number of key variables must equal the number of fundamental dimensions. All fundamental dimensions must be represented in terms of key variables. No dimensionless group is allowed from any combination of key variables. e.g. Pick up 3 variables: Invalid set: Step 5, Form dimensionless equations of the remaining variables in terms of the key variables. e.g. Step 6, Solve for the unknowns a, b, c, d, e, f, g, h, i e.g.

Next, find relations between PIs through experiments. Step 7, Form dimensionless (PI) groups. e.g. Step 8, Form other PI groups if you want as long as the total number is the same. e.g. Which PI groups are right? They are all right, but some groups are more commonly used and follow Conventions. Next, find relations between PIs through experiments.

Surface layer similarity (Monin Obukhov similarity) Surface layer: turbulent fluxes are nearly constant. 20-30 m Relevant parameters: Say we are interested in wind shear: Four variables and two basic units result in two dimensionless numbers, e.g.: The standard way of formulating this is by defining: Monin-Oubkhov length

PI relation stable unstable Empirical gradient functions to describe these observations: Note that eddy diffusion coefficients and gradient functions are related:

Now we are interested in the vertical gradient of virtual potential temperature. We can form a new variable Again, four variables and two basic units result in two dimensionless numbers, PI relation Similarly, we have Normally,

Surface wind profile 1. Neutral condition Aerodynamic roughness length Kondo and Yamazawa (1986) Over land Over water

Displacement distance If you have observations at three levels, you may determine displacement as, d 2. Non-neutral condition

Integral form of wind and temperature profiles in the surface layer

Integral form of wind and temperature profiles in the surface layer Similarly,

Bulk transfer relations How to estimate surface fluxes using conventional surface observations, surface winds (10m), surface temperature (2m),…? Drag coefficient of momentum, heat, and moisture.

1.5 1.5 1.0 1.0 -0.5 0.5 -0.5 0.5

Over Land

A new perspective on MOS Surface layer (constant flux layer) : 1. Steady neutral condition :

Kolmogorov -5/3 power law: Example of spectrum of energy density from the SCOPE data

Strong wind Weak wind After Hunt and Carlotti (2001) ’staircase’-like η estimated from the SCOPE data Best nonlinear fitting curve Strong wind Weak wind After Hunt and Carlotti (2001) ’staircase’-like ’elevator’-like

2. Steady non-neutral condition : Unstable condition (ς<0): Stable condition (ς<0):

Flux footprint General concept of the flux footprint. The darker the red color, the more contribution that is coming from the surface area certain distance away for the instrument. Relative contribution of the land surface area to the flux for two different measurement heights at near-neutral stability.

Relative contribution of the land surface area to the flux for two different surface roughnesses at near-neutral stability. Relative contribution of the land surface area to the flux for two different cases of thermal stability.

Problem: Assuming we have wind observations but no temperature observations at two levels, say, 5 m and 10 m, in the surface layer, can we estimate surface roughness and stability?