Background (see also David Randall’s talk) A sample of recent topics with a basis in Akio’s work: - QE as seen in vertical T structure - QE as seen in vertical T structure (+ implications for the large scale flow) (+ implications for the large scale flow) - QE and stochastic parameterization - QE and stochastic parameterization - QE and the onset of strong convection regime as a continuous phase transition with critical phenomena - QE and 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 ) 1 Dept. of Atmospheric Sciences & Inst. of Geophysics and Planetary Physics, U.C.L.A. 2 Santa Fe Institute Convective quasi-equilibrium (QE): Arakawa’s vision upheld and extended

Background: Arakawa and Schubert 1974: “When the time scale of the large-scale forcing, is sufficiently larger than the [convective] adjustment time, … the cumulus ensemble follows a sequence of quasi-equilibria with the current large-scale forcing. We call this … the quasi- equilibrium assumption.” “The adjustment … will be toward an equilibrium state … characterized by … balance of the cloud and large-scale terms…” Convection acts to reduce a measure of buoyancy, the cloud work function A (for a spectrum of entraining plumes)

As summarized in Arakawa 1997, 2004 (modified) : 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

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 q, 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; …

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 Temperature structure 1. Tropical vertical Temperature structure ** and thus a gross moist stability, simplifications to large-scale dynamics, … (Neelin 1997; N & Zeng 2000)

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 Holloway & Neelin, JAS, 2007 (& Chris’s AMS talk Thursday)

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).

Processes competing in (or with) QE Convection + wave dynamics constrain T profile (incl. cold top) Links tropospheric T to ABL, moisture, surface fluxes --- although separation of time scales imperfect Bretherton and Smolarkiewicz 1989; Yano and Emanuel 1991; Yu & Neelin 1994; Emanuel et al 1994; N97; Raymond 2000; Yano 2000; Zeng et al 2000; Su et al 2001; Chiang et al 2001; Chiang & Sobel 2002; Su & Neelin 2002; Fuchs and Raymond 2002

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) Can such variations about QE be represented by either –a stochastic parameterization? [Buizza et al 1999; Lin and Neelin 2000, 2002; Craig and Cohen 2006; Teixeira et al 2007;] –or superparameterization? with embedded cloud model (see talk by D. Randall)

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

Xu et al (1992) Cumulus Ensemble Model Mesoscale organization No shearWith shear Cloud-toptemperatures

Stochastic convection scheme tested in CCM3 (and 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  M b = A(  F) -1 (for M b > 0) Stochastic modification  M b = (A +  )(  F) -1   t A c = -  - 1 ( A +  ), (A +  > 0) i.e., stochastic effect in cloud base mass flux M b modifies decay of CAPE (convective available potential energy)  Gaussian, specified autocorrelation time, e.g. 1day *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

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

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

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 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) 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 (Mapes & Houze 1993; Nesbitt et al 2006;…) Mimic properties in stochastic convection schemes (Buizza et al 1999, Lin & Neelin 2000, Majda and Khouider 2002)?

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 …

QE or not QE? After 3 decades, QE remains a natural first approximation But with new emphasis on the importance of the adjustment process because: –separation of time scales does not hold uniformly –there are associated critical phenomena Although now a little “More quasi”…. Arakawa’s framework of ensembles of convecting elements acting to constrain moisture and temperature profiles by reducing the source of instability remains a pillar of convective parameterization and a powerful tool in theoretical exploration of the interaction of convection with larger scales