Lecture 4
Effects of Cumulus on
Large Scale Heat Budgets
Yanai, Esbensen and Chu (1972)
The key paper outlining a
framework for diagnostic studies
of convective scale interaction
Why?
• We can take observations at scales of 100’s
of km only
• We can notice that dynamic and
thermodynamic evolution of the flow is
occurring due to the impacts of cumulus scale
features that we cannot directly see
• We need a consistent framework to assess
this “apparent” effect
Define energy
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Relationship with 1st Law
Note that:
  s , i.e. dry static energy is the hydrostatic equivalent of s, i.e.
assuming that - dp  gdz in the 1st Law.
 e  h , i.e. dry static energy is the hydrostatic equivalent of s, i.e.
assuming that - dp  gdz in the 1st Law.
or
dT g
ds
 dz 
T T
T
dT g
dh
dsm'  c p d ln   c p
 dz  Ldqv 
T T
T
where I am defining s'm = moist (liquid phase only) specific entropy.
ds'  c p d ln   c p
Mean Governing Equations
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Average product => includes
subgrid scale correlations,
i.e. convection effects
condensation
radiation
evaporation
We can separate convection
effects from effects of mena
flow
Apparent Heat Source
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Apparent Moisture Sink
Large Scale Response
Can be Measured
Convective Scale Forcing
Can be inferred
Apparent Moist Static Energy Source
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Integrate Q1
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Integrate Q2
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Check on Accuracy
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Cloud Model
(Use Top Hat Averaging)
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Average includes cloud and environment
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Average includes cloud and environment
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Diagnose vertical mass flux
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Generalize a bit more
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Note the observable Q1, Q2
and QR imply effect of cloud
ensemble
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Cloud Model
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Idealization of Clouds
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Conservation Eqs. for
individual cloud “i”
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Saturation Moist Static Energy
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Simplify
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Assumptions
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Assumptions
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Assumptions
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Summary
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Schematic of Model
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Relationship between clouds and Q1, Q2
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Total Ensemble Effect
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Conditions at Cloud Base
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Cloud Base (Continued)
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Iterative Solution
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Diagnostic flow from data
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Subgrid scale forcing Implied
from large scale flow
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Subgrid scale forcing Implied
from large scale flow
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