Dry Boundary Layer Dynamics Idealized theory Shamelessly ripped from Emanuel Mike Pritchard

Outline Highlights of Rayleigh-Bernard convection Similarity theory review (2.1) Application to semi-infinite idealized dry boundary Uniformly thermally (buoyancy) driven only Mechanically (momentum) driven only Thermally + Mechanically driven The “Monin-Obunkov” length scale Characteristics of a more realistic typical dry atmospheric boundary layer

Rayleigh vs. Reynolds number Laminar case Re = Ra /  Turbulent case Re 2 = (Fr)(Ra) / 

The Rayleigh-Bernard problem Parallel-plate convection in the lab Governing non-dimensional parameter is Linear stability analysis Critical Rayleigh number yields convection onset Steady rolls/polygons Horizontal scale ~ distance between plates

The Rayleigh-Bernard problem Linear theory succeeds near onset regime Predicts aspect ratio and critical Rayleigh number Further analysis requires lab-work or nonlinear techniques

Laboratory explorations… up to Ra = 10 11

Lessons & Limitations Potential for convective regime shifts & nonlinear transitions. Atmosphere is Ra ~ Lab results only go so far Appropriate surface BC for idealized ABL theory is constant flux (not constant temperature)

Similarity theory Applicable to steady flows only, can’t know in advance if it will work. Posit n governing dimensional parameters on physical grounds Flow can be described by n-k nondimensional parameters made out of the dimensional ones Allows powerful conclusions to be drawn (for some idealized cases)

Thermally driven setup T = T 0 Q Statistical steady state… w’B’ Buoyancy flux Volume-integrated buoyancy sink What can dimensional analysis tell us?

Mechanically driven setup T = T 0 M Statistical steady state… w’u’ Convective momentum flux (J/s/m 2 ) Volume-integrated momentum sink What can dimensional analysis tell us?

Joint setup T = T 0 M w’u’ Momentum flux Volume-integrated momentum sink Q w’B’ Buoyancy flux Volume-integrated buoyancy sink

Whiteboard interlude…

Hybrid idealized model results after asymptotic matching… Theory: Obs:

Summary of theoretical results Thermally driven Convective velocity scales as z 1/3 Mechanically driven Convective velocity independent of height Hybrid Mechanical regime overlying convective regime Separated at Monin-Obunkov length-scale Matched solution is close but not a perfect match to the real world

Things that were left out of this model Mean wind Depth-limitation of convecting layer Due to static stability of free atmosphere Height-dependent sources and sinks of buoyancy and momentum Rotation Non-equilibrium E.g. coastal areas

Typical observed properties of a dry convecting boundary layer

The Entrainment Zone Temperature inversion; boundary between convective layer and “free atmosphere” Monin-Obukov similarity relations break down Buoyancy flux changes sign Forced entrainment of free-atmosphere air I.e. boundary layer deepens unless balanced by large-scale subsidence

Next week….? Adding moisture to equilibrium BL theory Ch Adding phase changes Stratocumulus-topped mixed layer models Ch 13.3