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Dry Boundary Layer Dynamics

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

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Dry Boundary Layer Dynamics

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

  2. 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

  3. Rayleigh vs. Reynolds number • Laminar case • Re = Ra /  • Turbulent case • Re2 = (Fr)(Ra) / 

  4. 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

  5. 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

  6. Laboratory explorations… up to Ra = 1011

  7. Lessons & Limitations • Potential for convective regime shifts & nonlinear transitions. • Atmosphere is Ra ~ 1017-1020 • Lab results only go so far • Appropriate surface BC for idealized ABL theory is constant flux (not constant temperature)

  8. 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)

  9. Thermally driven setup Volume-integrated buoyancy sink Q Statistical steady state… w’B’ Buoyancy flux What can dimensionalanalysis tell us? T = T0

  10. Mechanically driven setup Volume-integrated momentum sink M Statistical steady state… w’u’ Convective momentum flux (J/s/m2) What can dimensionalanalysis tell us? T = T0

  11. Joint setup Volume-integrated momentum sink Volume-integrated buoyancy sink M Q w’u’ w’B’ Momentum flux Buoyancy flux T = T0

  12. Whiteboard interlude…

  13. Hybrid idealized model resultsafter asymptotic matching… Theory: Obs:

  14. Summary of theoretical results • Thermally driven • Convective velocity scales as z1/3 • 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

  15. 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

  16. Typical observed properties of a dry convecting boundary layer

  17. 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

  18. Next week….? • Adding moisture to equilibrium BL theory • Ch. 13.2 • Adding phase changes • Stratocumulus-topped mixed layer models • Ch 13.3

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