Xin Xi

1946: Obukhov Length, as a universal length scale for exchange processes in surface layer.1954: Monin-Obukhov Similarity Theory, as a starting point for modern micrometeorology.

Interpretation of Obukhov Length

In 1946, Obukhov length was first proposed as a length scale (of order one to tens of meters) to characterize the dynamical surface layer. It gives a relation between the parameters that describe the dynamic, thermal and buoyant processes.

Constant momentum/temperature flux (with height) assumption in the surface layer:

NOT potential T

A dimensionless length scale z/L is further used to describe the stability of surface layer

z/L=0 for statically neutral stratification

z/L<0 for unstable stratification

z/L>0 for stable stratification

Dimensional analysis of TKE budget equation:

Surface layer turbulent kinetic energy (TKE) budget equation:

' '' ' ' '

' ' ' '1 i i

j j

u ue e U we w p gU u w wT

t x z z z T x x

1 2 3 4 5 6

fluctuating wind velocity x-component virtual temperature=(1+0.61q)T

1: rate of change of TKE due to mean wind advection2: TKE production/loss due to wind shear. Usually a positive contribution to TKE3: flux divergence of TKE; represents the turbulent transport of TKE by w’4: divergence of pressure flux; describe the redistribution of TKE by pressure perturbation (e.g., buoyancy, gravity waves)5: TKE production/loss due to buoyancy; depends on the sign of the heat flux6: loss of TKE due to viscosity; or dissipation (e.g., conversion of TKE to heat)

TKE tendency

Assuming constant flux (with height) in surface layer (variation within 10%), one can use the surface values of heat and momentum fluxes to define turbulence scales and nondimensionalize the TKE equation.

The momentum flux at the surface (shear stress):

*1

kz U

u z

' ' *2u w u

The logarithmic velocity law in the constant-flux layer:

Therefore, the dimensionless stability parameter is given by:

' '' '

*3' '

shear induced TKE/

buoyancy induced TKE

gwT g z wTT

z LU T uu wz

*3

' '

T uL

g wT

Obukhov length:

Monin-Obukhov Similarity Theory

Similarity theory is based on the assumption that dimensionless groups of variables (V’, T’) may be arranged in terms of functional relationships to the flow field, and where the number of variables is reduced to a closed set for easy application.

Note: zero-plane displacement is the height above the ground where the wind approaches zero due to flow obstacles, e.g., building, vegetation, etc

Monin-Obukhov similarity theory is developed based on the following findings:

The similarity theory gives the profile of any bulk quantity (X) which satisfies the assumptions in the theory:

For wind speed and temperature:Universal functions of the dimensionless stability parameter, z/L

ORDimensionless wind and temperature gradients

Integration of these two formulae between the surface and a certain reference layer (observational level or the first model level) gives:

Logarithmic wind profile

Z0 and Z0T are the surface roughness length for momentum and heat, respectively

Universal stability functions (of z/L);Have different forms in different situations (stable/unstable)

Rearrangement of the two equations gives:

CD: bulk transfer coefficient for momentum, or drag coefficientCH: bulk transfer coefficient for heat

Questions for discussion

1. Stability parameter z/L = Richardson Number?

2. Monin-Obukhov similarity theory

the large eddy simulation method

3. (scale issue / assumption breakdown) how is vegetation considered currently in applying the similarity theory in vegetated area (in terms of surface roughness, stability function, etc)? And how is this combined with other parts, such as soil thermodynamics, plant evapotranspiration (hydrological cycle).