Larissa Back*, Caroline Muller, Paul O’Gorman, Kerry Emanuel

1. 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

2. Why care about humidity-precipitation relationship? • T gradients weak • Simple theoretical models • Convective parameterizations • Potential useful analogies w/other complex systems

3. 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

4. Universal moisture-precipitation relationship (depends on temperature) SSMI daily 2 x 2 degree averaged data = WVP / Saturation WVP (WVP if atmosphere were fully saturated) Precipitation [mm/day] WVP Column (bulk) rel. humidity From Bretherton, Peters & Back (2004) Interpretation: combination of cause & effect

5. Universal relationship  self-organized criticality? Key features supporting interpretation: 1. universal relationship 2. power-law fit 3. max variance near “critical point” 4. spatial scaling (hard to test) 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)” TMI instantaneous 24x24 km Peters & Neelin (2006)

6. 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

7. Model description # occurrences • 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 WVP yes Raining? no lower RH p(w) “Potential” rainfall Linear=null hypothesis WVP

8. Gaussian distributions of humidity are not bad first order approximations in RCE From RCE CRM run w/no large-scale forcing

9. pressure Free trop wvp Model description t boundary layer wvp b Precip. Probability distribution fctn If gaussian Also tested more broadly non-analytically

10. 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)

11. Temperature dependence of relationship 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

12. 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)

13. 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

14. 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?

15. 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

17. Open questions: • Time/space scaling properties of rainfall/humidity like “critical point” in stat. phys. sense? • (.e.g. long-range correlations)

18. Model:

19. 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)