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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 humidity-precipitation relationship?
• T gradients weak
• Simple theoretical models
• Convective parameterizations
• Potential useful analogies w/other complex systems
• Most rising parcels strongly diluted by mixing w/environmental air (entrainment)
Lag (days)Lag (days)
Over tropical oceans, moisture strongly affects stability & rainfall
Lag (days)Lag (days)
From KWAJEXFrom KWAJEX
BrethertonBretherton
L. BackL. Back
L. BackL. Back
L. BackL. Back
See also Holloway & Neelin (2009) for similar analysis
=
WVP / Saturation WVP (WVP if atmosphere were
fully saturated)
SSMI daily 2 x 2 degree averaged data
Universal moisture-precipitation relationship (depends on
temperature)
WVP Column (bulk) rel. humidity
From Bretherton, Peters & Back (2004)
Interpretation: combination of cause & effect
Pre
cipi
tatio
n [m
m/d
ay]
Universal relationship self-organized criticality?
• “…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)
Key features supporting interpretation:1. universal relationship2. power-law fit3. max variance near “critical
point”4. spatial scaling (hard to test)5. consistency w/QE postulate
TMI instantaneous 24x24 km
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• Assumptions:
– Independent Gaussian distributions of boundary layer and free trop. humidity (each contribute half to total WVP)
– rainfall only occurs when lower layer humidity exceeds threshold (stability threshold)
– Rainfall increases w/humidity (when rain is occurring)
Rainfall-humidity relationship works out to a convolution of these functions
# o
ccur
renc
es
WVP
lower RH
Rai
ning
?
no
yes
“Pot
entia
l”ra
infa
ll
WVP
Linear=null hypothesis
p(w)
• Gaussian distributions of humidity are not bad first order approximations in RCE
From RCE CRM run w/no large-scale forcing
€
P(w) = E P w=(b+t ) / 2{ }
€
=P(b, t) f (b) f (t) w= b+tdb∫f (b) f (t) w= b+tdb∫
€
P(b, t) = H(b−bt )p(b+ t)
€
P(w) = p(b+ t)erfcbt - w
σ
⎛
⎝ ⎜
⎞
⎠ ⎟
If
Precip.
Also tested more broadly non-analytically€
f (x) = exp(−(x −μ)2 /(2σ )2)
b
t
boundary layer wvp
Free trop wvp
€
f (x) = Probability distribution fctn
€
var P(w)( ) = wP(w) − P2(w)gaussian
Model description
Model results/test:
From Peters & Neelin (2006)
FromMuller et al. (2009)
Compares well with obs. -sharp increase, then leveling -max variance near threshold -power-law-like fit above
threshold
Temperature dependence of relationship
• If we assume boundary layer rel. hum. threshold, constant for different temperatures– pickup depends on
boundary layer
saturation WVP
€
P(w) = p(b+ t)erfcbt - w
σ
⎛
⎝ ⎜
⎞
⎠ ⎟
Location of pickup depends only on threshold BL water vapor
Neelin et al. 2009
Does our model describe a self-organized 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:
– 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
evaporation Convection/cold pools
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 scale-free 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 humidity-precipitation 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)
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