Dry Boundary Layer Dynamics Idealized theory Shamelessly ripped from Emanuel Mike Pritchard
Jan 05, 2016
Dry Boundary Layer Dynamics
Idealized theoryShamelessly 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
Re2 = (Fr)(Ra) /
The Rayleigh-Bernard problem Parallel-plate convection in the lab
Governing non-dimensional parameter is
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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 = 1011
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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)
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 = T0
QStatistical steady state…
w’B’
Buoyancy flux
Volume-integrated buoyancy sink
What can dimensionalanalysis tell us?
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Mechanically driven setup
T = T0
MStatistical steady state…
w’u’
Convective momentum flux (J/s/m2)
Volume-integrated momentum sink
What can dimensionalanalysis tell us?
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Joint setup
T = T0
M
w’u’
Momentum flux
Volume-integrated momentum sink
Q
w’B’
Buoyancy flux
Volume-integrated buoyancy sink
Whiteboard interlude…
Hybrid idealized model resultsafter asymptotic matching…
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Theory:
Obs:
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
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
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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. 13.2
Adding phase changes Stratocumulus-topped mixed layer models Ch 13.3