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