training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic behaviour and free convection scaling The outer layer

training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic behaviour and free convection scaling The outer layer

training course: boundary layer II Similarity theory Motivation: Closure problem requires empirical expressions for turbulent diffusion coefficients (which include dependency on flow characteristics). Number of independent parameters has to be limited. Procedure: Select relevant parameters and plot dimensionless functions. Use constraints from asymptotic cases. Apply empirical functions as turbulence closure. Similarity theory is an intelligent way of organizing data e.g. from field experiments or large eddy simulations. Note: turbulence closure is based on observations not theory.

training course: boundary layer II Buckinham Pi dimensional analysis (Stull, 1988): example 2. Count number of fundamental dimensions. 1. Define relevant variables and their dimensions.

training course: boundary layer II Buckinham Pi dimensional analysis: example 3. 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 4. Measure as a function of 5. Further simplification; assume: i.e.

training course: boundary layer II Weight as a function of cruising speed ( “The simple science of flight” by Tennekes, 1997, MIT press) Weight (Newtons) Flying objects range from small insects to Boeing 747 W~U 6 Speed (m/s) The great flight diagram

training course: boundary layer II Dimensional analysis example: windmill/anemometer 1. Define relevant variables and their dimensions. 2. Count number of fundamental dimensions.

training course: boundary layer II Dimensional analysis example: windmill/anemometer 3. Form n dimensionless groups where n is the number of variables minus the number of fundamental dimensions. produced power wind power speed of rotor tip wind speed 5. For an anemometer: 4. Measure as a function ofby changing load.

training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic behaviour and free convection scaling The outer layer

training course: boundary layer II Flux profile For z/h << 1 flux is approximately equal to surface flux. Relevant parameters: Say we are interested in wind shear: Surface layer similarity (Monin Obukhov similarity) Considerations about the nature of the process: z/z o >> 1 distance to surface determines turbulence length scale shear scales with surface friction rather than with z o

training course: boundary layer II MO similarity Four variables and two basic units result in two dimensionless numbers, e.g.: and The standard way of formulating this is by defining: Resulting in: dimensionless shearStability parameter

training course: boundary layer II Observations of as a function of z/L, with Empirical gradient functions to describe these observations: Note that eddy diffusion coefficients and gradient functions are related: then if stableunstable MO gradient functions

training course: boundary layer II MO-similarity applied to other quantities Quantityscaling parameter dimensionless function

training course: boundary layer II Integral profile functions Dimensionless wind gradient (shear) or temperature gradient functions can be integrated to profile functions: with: integration constant (roughness length for momentum) wind profile function, related to gradient function: Profile functions for temperature and moisture can be obtained in similar way.

training course: boundary layer II MO wind profile functions applied to observations Stable Unstable Limit of stable layer

training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic behaviour and free convection scaling The outer layer

training course: boundary layer II Asymptotic behaviour Limiting cases can help to constrain functions e.g.: Therefore e.g.: free convection scaling

training course: boundary layer II Example of free convection scaling

training course: boundary layer II Similarity theory: Outline Goals, Buckingham Pi Theorem and examples Surface layer (Monin Obukhov) similarity Asymptotic behaviour and free convection scaling The outer layer

training course: boundary layer II Above the surface layer (z/h>0.1) Fluxes are not constant but decrease monotonically with height Boundary layer height h and Coriolis parameter f are additional scales. Neutral PBL; velocity defect law:

training course: boundary layer II Mixed layer Convective boundary layer (mixed layer scaling): Effects of friction can often be neglected. Profiles well mixed, so gradient functions become less important w * is important turbulence velocity scale

training course: boundary layer II Stable boundary layer Local scaling extension of surface layer scaling, where surface fluxes are replaced by local fluxes, in other words: Surface layer closure applies to outer layer as well Z-less scaling far away from the surface, z should drop:

training course: boundary layer II Local scaling Z-less regime

training course: boundary layer II scaling regions for the unstable BL Holstlag and Nieuwstadt, 1986: BLM, 36,

training course: boundary layer II scaling regions for the stable BL Holstlag and Nieuwstadt, 1986: BLM, 36,

training course: boundary layer II Geostrophic drag law Match surface layer and outer layer to obtain relation between surface drag and geostrophic wind. drag ageostrophic angle Plot A and B as a function of stability parameter. Wangara data