A toy model for understanding
the observed relationship
between column-integrated
water vapor and tropical
precipitation
Larissa Back*, Caroline Muller,
Paul O’Gorman, Kerry Emanuel
*Blame LB for interpretation
given here
Why care about humidityprecipitation relationship?
•
•
•
•
T gradients weak
Simple theoretical models
Convective parameterizations
Potential useful analogies w/other
complex systems
Over tropical oceans, moisture strongly
affects stability & rainfall
From KWAJEX
L. Back
Bretherton
Lag (days)
Most rising parcels strongly diluted by mixing
w/environmental air (entrainment)
L. Back
L. Back
Lag (days)
See also Holloway & Neelin
(2009) for similar analysis
Precipitation [mm/day]
Universal moisture-precipitation
relationship (depends on
daily 2 x 2
temperature) SSMI
degree averaged
data
WVP
From Bretherton, Peters & Back (2004)
Interpretation: combination of cause &
effect
Column (bulk) rel. humidity
=
WVP / Saturation WVP
(WVP if atmosphere were
fully saturated)
Universal relationship  selforganized criticality?
Key features supporting
interpretation:
1. universal relationship
2. power-law fit
3. max variance near “critical
point”
4. spatial scaling (hard to
TMI instantaneous
test)
24x24 km
5. consistency w/QE postulate
“…the attractive QE (quasi-equilibrium)
state… is the critical point of a continuous
phase transition and is thus an instance of
SOC (self-organized criticality)”
Peters & Neelin (2006)
Goal:
• Develop simple physically based model
to explain observations of water-vapor
precipitation relationship
– Focus on reproducing:
• Sharp increase, then slower leveling
• Peak variance near sharp increase
Model description
– Rainfall increases w/humidity
(when rain is occurring)
Rainfall-humidity relationship
works out to a convolution of
these functions
Raining?
– rainfall only occurs when lower
layer humidity exceeds
threshold (stability threshold)
WVP
yes
no
lower RH
“Potential”
rainfall
– Independent Gaussian
distributions of boundary layer
and free trop. humidity (each
contribute half to total WVP)
# occurrences
• Assumptions:
p(w)
WVP
Linear=null
hypothesis
• Gaussian distributions of humidity are not bad
first order approximations in RCE
From RCE CRM run w/no large-scale forcing
pressure
Model
description
Precip.
If
Free trop wvp
b
boundary layer wvp
P( w )  E P
P( b, t )  H( b  b t ) p( b  t )
f (x) 
t

Probability distribution fctn

f ( x )  exp( ( x   ) /( 2 ) ) 
2
2
gaussian
Also tested more broadly

non-analytically
w (b  t )/ 2
 P (b, t ) f (b) f (t )
 f (b) f (t )

w b t
w b t
db
db
b t - w 
P ( w )  p( b  t ) erfc 

  
var P( w )   w P ( w )  P ( w )
2
Model results/test:
From
Muller et
al. (2009)
Compares well with obs.
-sharp increase, then leveling
-max variance near threshold
-power-law-like fit above
threshold
From Peters &
Neelin (2006)
Temperature dependence of
relationship
b t - w 
P ( w )  p( b  t ) erfc 

  
Location of pickup depends
only on threshold BL water
vapor
• If we assume boundary layer rel. hum.
threshold, constant for different temperatures
– pickup depends on
boundary layer
saturation WVP
Neelin et
al. 2009
Does our model describe a selforganized critical (SOC) system?
• Short answer: maybe, maybe not
– An SOC system “self-organizes” toward the
critical point of a continuous phase transition
– continuous phase transition= scale-free
behavior, “long-range” correlations in
time/space or another variable (“long-range”
correlations fall off with a power law, so mean
is not useful a descriptor)
Self-organized criticality?
• Mechanisms for self-organization towards
threshold boundary layer water vapor is
implicit in model:
evaporation
Convection/cold pools
– BL moisture above threshold for rainfall
convection, decreased BL moisture
– BL moisture below threshold for rainfall
evaporation, increased BL moisture
– Similar idea to boundary layer quasi-equilibrium
Is our model (Muller et al.) consistent with
criticality/continuous phase transition?
– Gaussians  no long-range correlations
• But tails aren’t really Gaussian…
– Heaviside function  transition physics
unimportant (in that part of model)
– No explicit interactions between “columns”… but
simplest percolation model with critical behavior
(scale-free cluster size) doesn’t have that either…
• See Peters, Neelin, Nesbitt ‘09 for evidence of scalefree behavior in convective cluster size in rainfall
– Criticality could enter in P vs. wvp relationship,
when raining? E.g. dependent on microphysics in
CRM’s?
Conclusions:
• Simple, two-level physically based model
can explain observed relationship
between WVP & rainfall
– Stability threshold determines when it rains
– Amount of rain determined by WVP
– Model is agnostic about stat. phys. analogies
Open questions:
• Time/space scaling properties of
rainfall/humidity like “critical point” in
stat. phys. sense?
– (.e.g. long-range correlations)
Model:
Why care about humidityprecipitation relationship?
• In tropics, temperature profile varies little-->
convection/instability strongly affected by
moisture profile (maybe show from KWAJEX?)
• Relationship is a key part of simple theoretical
models (e.g. Raymond, Emanuel, Kuang, Neelin,
Mapes)
• Understanding relationship --> convective
parameterization tests or development
(particularly stochastic)
• Analogies with statistical physics or other
complex systems may lead to new insight or
analysis techniques (e.g. Peters and Neelin
2006)