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)