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. Most rising parcels strongly diluted by mixing w/environmental air (entrainment) Lag (days) Over tropical oceans, moisture strongly affects stability & rainfall Lag (days) From KWAJEX Bretherton L. Back See also Holloway & Neelin (2009) for similar analysis

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

5. 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 relationship 2. power-law fit 3. max variance near “critical point” 4. spatial scaling (hard to test) 5. consistency w/QE postulate TMI instantaneous 24x24 km

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

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

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

10. Model results/test: From Peters & Neelin (2006) From Muller et al. (2009) Compares well with obs. -sharp increase, then leveling -max variance near threshold -power-law-like fit above threshold

11. Temperature dependence of relationship If we assume boundary layer rel. hum. threshold, constant for different temperatures –pickup depends on boundary layer saturation WVP Location of pickup depends only on threshold BL water vapor 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 evaporationConvection/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)