Background: precipitation moist convection & its parameterization; Arakawa’s Quasi-Equilibrium postulate (QE); + reasons to careBackground: precipitation moist convection & its parameterization; Arakawa’s Quasi-Equilibrium postulate (QE); + reasons to care QE in vertical structure The onset of strong convection regime as a continuous phase transition with critical phenomena J. David Neelin 1, Ole Peters 1,2, Chris Holloway 1, Katrina Hales 1, Steve Nesbitt 3 Chris Holloway 1, Katrina Hales 1, Steve Nesbitt 3 1 Dept. of Atmospheric Sciences & Inst. of Geophysics and Planetary Physics, U.C.L.A. 2 Santa Fe Institute (& Los Alamos National Lab) 3 U of Illinois at Urbana-Champaign The transition to strong convection

Background: precipitation, moist convection and its parameterization; Arakawa’s Quasi-Equilibrium postulate (QE); + reasons to care QE in vertical structure QE in vertical structure The onset of strong convection regime as a continuous phase transition with critical phenomena J. David Neelin 1, Ole Peters 1,2, Chris Holloway 1, Katrina Hales 1, Steve Nesbitt 3 Chris Holloway 1, Katrina Hales 1, Steve Nesbitt 3 The transition to strong convection 1 Dept. of Atmospheric Sciences & Inst. of Geophysics and Planetary Physics, U.C.L.A. 2 Santa Fe Institute (& Los Alamos National Lab) 3 U of Illinois at Urbana-Champaign

Background: precipitation, moist convection and its parameterization; Arakawa’s Quasi-Equilibrium postulate (QE); + reasons to care QE in vertical structure The onset of strong convection regimeThe onset of strong convection regime as a continuous phase transition as a continuous phase transition with critical phenomena with critical phenomena The transition to strong convection J. David Neelin 1, Ole Peters 1,2,*, Chris Holloway 1, Katrina Hales 1, Steve Nesbitt 3 Chris Holloway 1, Katrina Hales 1, Steve Nesbitt 3 1 Dept. of Atmospheric Sciences & Inst. of Geophysics and Planetary Physics, U.C.L.A. 2 Santa Fe Institute (& Los Alamos National Lab) 3 U of Illinois at Urbana-Champaign * + thanks to Didier Sornette for connecting the authors & Matt Munnich & Joyce Meyerson for terabytes of help

July January Background: Precipitation climatology mm/day Note intense tropical moist convection zones (intertropical convergence zones)

Rainfall at shorter time scales Weekly accumulation Rain rate from a 3-hourly period within the week shown above (mm/hr) From TRMM-based merged data (3B42RT)

Convective quasi-equilibrium (Arakawa & Schubert 1974) Convection acts to reduce buoyancy (cloud work function A) on fast time scale, vs. slow drive from large-scale forcing (cooling troposphere, warming & moistening boundary layer, …) M65= Manabe et al 1965; BM86=Betts&Miller 1986 parameterizns Modified from Arakawa (1997, 2004)

Background: Convective Quasi-equilibrium cont’d Slow driving (moisture convergence & evaporation, radiative cooling, …) by large scales generates conditional instability Fast removal of buoyancy by moist convective up/down-drafts Above onset threshold, strong convection/precip. increase to keep system close to onset Thus tends to establish statistical equilibrium among buoyancy-related fields – temperature T & moisture, including constraining vertical structure using a finite adjustment time scale  c makes a difference Betts & Miller 1986; Moorthi & Suarez 1992; Randall & Pan 1993; Zhang & McFarlane 1995; Emanuel 1993; Emanuel et al 1994; Yu and Neelin 1994; … Arakawa & Schubert 1974 Manabe et al 1965; Arakawa & Schubert 1974 ; Moorthi & Suarez 1992; Randall & Pan 1993; Emanuel 1991; Raymond 1997; …

Xu, Arakawa and Krueger 1992 Cumulus Ensemble Model (2-D) Note large variations Precipitation rates (domain avg): Note large variations Imposed large-scale forcing (cooling & moistening) Experiments:Q03512 km domain,no shear Q02512 km domain,shear Q km domain,shear

Departures from QE and stochastic parameterization In practice, ensemble size of deep convective elements in O(200km) 2 grid box x 10minute time increment is not large Expect variance in such an avg about ensemble mean This can drive large-scale variability –(even more so in presence of mesoscale organization) Have to resolve convection?! (costs *10 9 ) or –stochastic parameterization? [Buizza et al 1999; Lin and Neelin 2000, 2002; Craig and Cohen 2006; Teixeira et al 2007] –superparameterization? with embedded cloud model (Grabowski et al 2000; Khairoutdinov & Randall 2001; Randall et al 2002)

Variations about QE: Stochastic convection scheme (CCM3 * & similar in QTCM ** ) Mass flux closure in Zhang - McFarlane (1995) scheme Evolution of CAPE, A, due to large-scale forcing, F  t A c = -M b F Closure:  t A c = -  - 1 ( A +  ), (A +  > 0) i.e.  M b = (A +  )(  F) -1 (for M b > 0) Stochastic modification  in cloud base mass flux M b modifies decay of CAPE (convective available potential energy)  Gaussian, specified autocorrelation time, e.g. 1 day *Community Climate Model 3 **Quasi-equilibrium Tropical Circulation Model

Impact of CAPE stochastic convective parameterization on tropical intraseasonal variability in QTCM Lin &Neelin 2000

CCM3 variance of daily precipitation Control run CAPE-M b scheme (60000 vs 20000) Observed (MSU) Lin &Neelin 2002

Background cont’d: Reasons to care Besides curiosity… Model sensitivity of simulated precipitation to differences in model parameterizations –Interannual teleconnections, e.g. from ENSO –Global warming simulations* *models do have some agreement on process & amplitude if you look hard enough (IGPP talk, May 2006; Neelin et al 2006, PNAS)

Precipitation change in global warming simulations Fourth Assessment Report models: LLNL Prog. on Model Diagnostics & Intercomparison; SRES A2 scenario (heterogeneous world, growing population,…) for greenhouse gases, aerosol forcing Dec.-Feb., avg minus avg. 4 mm/day model climatology black contour for reference mm/day Neelin, Munnich, Su, Meyerson and Holloway, 2006, PNAS

GFDL_CM2.0 DJF Prec. Anom.

CCCMA

CNRM_CM3

CSIRO_MK3

NCAR_CCSM3

GFDL_CM2.1

UKMO_HadCM3

MIROC_3.2

MRI_CGCM2

NCAR_PCM1

MPI_ECHAM5

QE postulates deep convection constrains vertical structure of temperature through troposphere near convection If so, gives vertical str. of baroclinic geopotential variations, baroclinic wind ** Conflicting indications from prev. studies (e.g., Xu and Emanuel 1989; Brown & Bretherton 1997; Straub and Kiladis 2002) On what space/time scales does this hold well? Relationship to atmospheric boundary layer (ABL)? 1. Tropical vertical structure (temperature & moisture) associated with convection 1. Tropical vertical structure (temperature & moisture) associated with convection ** and thus a gross moist stability, simplifications to large-scale dynamics, … (Neelin 1997; N & Zeng 2000)

Vertical Temperature structure Monthly T regression coeff. of each level on mb avg T. CARDS Rawinsondes avgd for 3 trop Western Pacific stations, shading < 5% signif. Curve for moist adiabatic vertical structure in red. AIRS monthly (avg for similar Western Pacific box, ) Holloway & Neelin, JAS, 2007 (& Chris’s talk March 14 AOS)

Vertical Temperature structure AIRS daily T (a)Regression of T at each level on mb avg T For 4 spatial averages, from all-tropics to 2.5 degree box Red curve corresp to moist adiabat. (Daily, as function of spatial scale) [AIRS lev2 v4 daily avg 11/03-11/05] (b) Correlation of T(p) to mb avg T

Vertical Temperature structure Monthly T regression coeff. of each level on mb avg T. (Rawinsondes avgd for 3 trop W Pacific stations) CARDS monthly anomalies, shading < 5% signif. Curve for moist adiabatic vertical structure in red. Correlation coeff. Holloway & Neelin, JAS, 2007

QE in climate models (HadCM3, ECHAM5, GFDL CM2.1) Monthly T anoms regressed on mb T vs. moist adiabat. Model global warming T profile response Regression on of IPCC AR4 20 th C runs, markers signif. at 5%. Pac. Warm pool= 10S-10N, E. Response to SRES A2 for minus (htpps://esg.llnl.gov).

Vertical structure of moisture Ensemble averages of moisture from rawinsonde data at Nauru*, binned by precipitation High precip assoc. with high moisture in free troposphere (consistent with Parsons et al 2000; Bretherton et al 2004; Derbyshire 2005) *Equatorial West Pacific ARM (Atmospheric Radiation Measurement) project site

Autocorrelations in time Long autocorrelation times for vertically integrated moisture (once lofted, it floats around) Nauru ARM site upward looking radiometer + optical gauge Column water vapor Cloud liquid water Precipitation

Transition probability to Precip>0 Given column water vapor w at a non-precipitating time, what is probability it will start to rain (here in next hour) Nauru ARM site upward looking radiometer + optical gauge

Processes competing in (or with) QE Links tropospheric T to ABL, moisture, surface fluxes --- although separation of time scales imperfect Convection + wave dynamics constrain T profile (incl. cold top)

2. Transition to strong convection as a continuous phase transition Convective quasi-equilibrium closure postulates (Arakawa & Schubert 1974) of slow drive, fast dissipation sound similar to self-organized criticality (SOC) postulates (Bak et al 1987; …), known in some stat. mech. models to be assoc. with continuous phase transitions (Dickman et al 1998; Sornette 1992; Christensen et al 2004) Critical phenomena at continuous phase transition well- known in equilibrium case (Privman et al 1991; Yeomans 1992) Data here: Tropical Rainfall Measuring Mission (TRMM) microwave imager (TMI) precip and water vapor estimates ( from Remote Sensing Systems;TRMM radar 2A25 in progress) Analysed in tropics 20N-20S Peters & Neelin, Nature Phys. (2006) + ongoing work ….

Precip increases with column water vapor at monthly, daily time scales (e.g., Bretherton et al 2004). What happens for strong precip/mesoscale events? (needed for stochastic parameterization) E.g. of convective closure (Betts-Miller 1996) shown for vertical integral: Precip = (w  w c ( T))/  c (if positive) w vertical int. water vapor w c convective threshold, dependent on temperature T  c time scale of convective adjustment Background Background

Western Pacific precip vs column water vapor Tropical Rainfall Measuring Mission Microwave Imager (TMI) data Wentz & Spencer (1998) algorithm Average precip P(w) in each 0.3 mm w bin (typically 10 4 to 10 7 counts per bin in 5 yrs) 0.25 degree resolution No explicit time averaging Western Pacific Eastern Pacific Peters & Neelin, 2006 Peters & Neelin, 2006

Oslo model (stochastic lattice model motivated by rice pile avalanches) Frette et al (Nature, 1996) Christensen et al (Phys. Res. Lett., 1996; Phys. Rev. E. 2004) Power law fit: OP(  )=a(  -  c ) 

Things to expect from continuous phase transition critical phenomena [NB: not suggesting Oslo model applies to moist convection. Just an example of some generic properties common to many systems.] Behavior approaches P(w)= a(w-w c )   above transition exponent  should be robust in different regions, conditions. ("universality" for given class of model, variable) critical value should depend on other conditions. In this case expect possible impacts from region, tropospheric temperature, boundary layer moist enthalpy (or SST as proxy) factor a also non-universal; re-scaling P and w should collapse curves for different regions below transition, P(w) depends on finite size effects in models where can increase degrees of freedom (L). Here spatial avg over length L increases # of degrees of freedom included in the average.

Things to expect (cont.) Precip variance  P(w) should become large at critical point. For susceptibility  (w,L)= L 2  P(w,L), expect  (w,L)  L  / near the critical region spatial correlation becomes long (power law) near crit. point Here check effects of different spatial averaging. Can one collapse curves for  P(w) in critical region? correspondence of self-organized criticality in an open (dissipative), slowly driven system, to the absorbing state phase transition of a corresponding (closed, no drive) system. residence time (frequency of occurrence) is maximum just below the phase transition Refs: e.g., Yeomans (1996; Stat. Mech. of Phase transitions, Oxford UP), Vespignani & Zapperi (Phys. Rev. Lett, 1997), Christensen et al (Phys. Rev. E, 2004)

log-log Precip. vs (w-w c ) Slope of each line (  ) = Eastern Pacific Western Pacific Atlantic ocean Indian ocean shifted for clarity (individual fits to  within ± 0.02)

How well do the curves collapse when rescaled? Original (seen above) Western Pacific Eastern Pacific

How well do the curves collapse when rescaled? Rescale w and P by factors f p, f w for each region i Western Pacific Eastern Pacific ii

Collapse of Precip. & Precip. variance for different regions Western Pacific Eastern Pacific Variance Precip Slope of each line (  ) = Eastern Pacific Western Pacific Atlantic ocean Indian ocean Peters & Neelin, 2006 Peters & Neelin, 2006

Precip variance collapse for different averaging scales Rescaled by L 0.42 Rescaled by L 2

TMI column water vapor and Precipitation Western Pacific example

TMI column water vapor and Precipitation Atlantic example

Check pick-up with radar precip data TRMM radar data for precipitation 4 Regions collapse again with w c scaling Power law fit above critical even has approx same exponent as from TMI microwave rain estimate (2A25 product, averaged to the TMI water vapor grid)

Mesoscale convective systems Cluster size distributions of contiguous cloud pixels in mesoscale meteorology: “almost lognormal” (Mapes & Houze 1993) since Lopez (1977) Mesoscale cluster size frequency (log-normal = straight line). From Mapes & Houze (MWR 1993)

Mesoscale cluster sizes from TRMM radar clusters of contiguous pixels with radar signal > threshold (Nesbitt et al 2006) Ranked by size Cluster size distribution alters near critical: increased probability of large clusters Note: spanning clusters not eliminated here; finite size effects in s -  G(s/s  )

Mapping water vapor to occupation probability For geometric questions, consider probability p of site precipating 2D percolation is simplest prototype process (site filled with probability p, stats on clusters of contiguous points); view as null model p incr near critical water vapor w c est from precip power law

Mean cluster size increase below critical Check how mean cluster size changes with probability p of precipitating Try against exponent and critical p for site percolation ~ consistent with this ‘null model’ in a small range below critical; but differs above (to be continued…)

Preliminary: water vapor Precip. relation temperature dependence July ERA40 reanalysis daily Temperature: Tropospheric vertical average ( mb) Average Standard deviation

Dependence on Tropospheric temperature Averages conditioned on vert. avg. temp. T, as well as w (T mb from ERA40 reanalysis) Power law fits above critical: w c changes, same  [note more data points at 270, 271] ^

Dependence on Tropospheric temperature Find critical water vapor w c for each vert. avg. temp. T (western Pacific) Compare to vert. int. saturation vapor value binned by same T Not a constant fraction of column saturation ^ ^

How much precip occurs near critical point? Contributions to Precip from each T ^ 90% of precip in the region occurs above 80% of critical (16% above critical)---even for imperfect estimate of w c 80% of critical ^ critical Water vapor scaled by w c (T)

Frequency of occurrence…. drops above critical Frequency of occurrence (all points) Frequency of occurrence Precipitating Precip Western Pacific for SST within 1C bin of 30C

Extending QE Recall: Critical water vapor w c empirically determined for each vert. avg. temp. T Here use to schematize relationship (& extension of QE) to continuous phase transition/SOC properties ^

Extending QE Above critical, large Precip yields moisture sink, (& presumably buoyancy sink) Tends to return system to below critical So frequency of occurrence decreases rapidly above critical

Extending QE Frequency of occurrence max just below critical, contribution to total precip max around & just below critical Strict QE would assume sharp max just above critical, moisture & T pinned to QE, precip det. by forcing

Extending QE “Slow” forcing eventually moves system above critical Adjustment: relatively fast but with a spectrum of event sizes, power law spatial correlations, (mesoscale) critical clusters, no single adjustment time …

Implications Transition to strong precipitation in TRMM observations conforms to a number of properties of a continuous phase transition; + evidence of self-organized criticality convective quasi-equilibrium (QE) assoc with the critical point (& most rain occurs near or above critical) but different properties of pathway to critical point than used in convective parameterizations (e.g. not exponential decay; distribution of precip events, high variance at critical,…) probing critical point dependence on water vapor, temperature: suggests nontrivial relationship (e.g. not saturation curve) spatial scale-free range in the mesoscale assoc with QE Suggests mesoscale convective systems like critical clusters in other systems; importance of excitatory short-range interactions; connection to mesocale cluster size distribution TBD: steps from the new observed properties to better representations in climate models + the temptation of even more severe regimes …

Precip pick-up & freqency of occurrence relations on a smaller ensemble Frequency of occurrence Precip Hurricane Katrina Aug. 26 to 29, 2005, over the Gulf of Mexico (100W-80W)

TMI Precip. Rate Aug. 28, 2005 TMI Precipitation Rate: August 28, millimeters/hr landno data

The transition to strong convection Background QE and Vertical structures –Temperature –Moisture Continuous phase transition to strong convection –Nature Physics –Radar & Clusters –Critical moisture as a function of temperature