Background (see also David Randall

1
Convective quasi-equilibrium (QE) Arakawas
vision upheld and extended
J. David Neelin1, Ole Peters1,2, ( Chris
Holloway1, Katrina Hales1)
1Dept. of Atmospheric Sciences Inst. of
Geophysics and Planetary Physics, U.C.L.A. 2Santa
Fe Institute

• Background (see also David Randalls talk)
• A sample of recent topics with a basis in Akios
  work
• - QE as seen in vertical T structure
• ( implications for the large
  scale flow)
• - QE and stochastic parameterization
• - QE and the onset of strong convection
  regime as a continuous phase transition with
  critical phenomena

2
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)

3
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 BM86BettsMiller 1986

4
Background Convective Quasi-equilibrium contd
Manabe et al 1965 Arakawa Schubert 1974
Moorthi Suarez 1992 Randall Pan 1993
Emanuel 1991 Raymond 1997

• 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 tc 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

5
1. Tropical vertical Temperature structure

• 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)?

and thus a gross moist stability,
simplifications to large-scale dynamics,
(Neelin 1997 N Zeng 2000)
6
Vertical Temperature structure
(Daily, as function of spatial scale)

• AIRS daily T
• Regression of T at each level on
• 850-200mb avg T
• For 4 spatial averages,
• from all-tropics to 2.5 degree box
• Red curve corresp to moist adiabat.

(b) Correlation of T(p) to 850-200mb avg T
Holloway Neelin, JAS, 2007 ( Chriss AMS talk
Thursday)
AIRS lev2 v4 daily avg 11/03-11/05
7
Vertical Temperature structure
(Rawinsondes avgd for 3 trop W Pacific stations)

• Monthly T regression coeff. of each level on
  850-200mb avg T.

Correlation coeff.

• CARDS monthly 1953-1999 anomalies, shading lt 5
  signif.
• Curve for moist adiabatic vertical structure in
  red.

Holloway Neelin, JAS, 2007
8
QE in climate models (HadCM3, ECHAM5, GFDL CM2.1)
Monthly T anoms regressed on 850-200mb T vs.
moist adiabat.
Model global warming T profile response

• Regression on 1970-1994 of IPCC AR4 20thC runs,
  markers signif. at 5. Pac. Warm pool 10S-10N,
  140-180E. Response to SRES A2 for 2070-2094 minus
  1970-1994 (htpps//esg.llnl.gov).

9
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

10
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)

11
Xu, Arakawa and Krueger 1992Cumulus Ensemble
Model (2-D)
Precipitation rates (domain avg) Note large
variations Imposed large-scale forcing (cooling
moistening)
Experiments Q03 512 km domain, no shear Q02 512
km domain, shear Q04 1024 km domain, shear
12
Xu et al (1992) Cumulus Ensemble Model Mesoscale
organization
No shear
With shear
Cloud-top temperatures
13
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 tA c -MbF Closure tA c -t -1A
ð Mb
A(tF)-1 (for Mb gt 0) Stochastic modification
Mb (A x)(tF)-1 ð tA c -t -1( A
x) , (A x gt 0) i.e., stochastic
effect in cloud base mass flux Mb modifies decay
of CAPE (convective available potential energy) x
Gaussian, specified autocorrelation time, e.g.
1day Quasi-equilibrium Tropical Circulation Model
14
Impact of CAPE stochastic convective
parameterization on tropical intraseasonal
variability in QTCM
Lin Neelin 2000
15
CCM3 variance of daily precipitation
Control run
CAPE-Mb scheme (60000 vs 20000)
Observed (MSU)
Lin Neelin 2002
16
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 SystemsTRMM
  radar 2A25 in progress)
• Analysed in tropics 20N-20S

Peters Neelin, Nature Phys. (2006) ongoing
work .
17
Background

• 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 - wc( T))/tc (if
  positive)
• w vertical int. water vapor
• wc convective threshold, dependent on
  temperature T
• tc time scale of convective adjustment

18
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 104 to 107 counts per bin in 5 yrs)
• 0.25 degree resolution
• No explicit time averaging

Western Pacific
Eastern Pacific
Peters Neelin, 2006
19
Oslo model (stochastic lattice model motivated
by rice pile avalanches)
Power law fit OP(z)a(z-zc)b

• Frette et al (Nature, 1996)
• Christensen et al (Phys. Res. Lett., 1996 Phys.
  Rev. E. 2004)

20
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-wc)b above
  transition
• exponent b 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.

21
Things to expect (cont.)

• Precip variance sP(w) should become large at
  critical point.
• For susceptibility c(w,L) L2 sP(w,L),
• expect c (w,L) µ Lg/n 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 sP(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)

22
log-log Precip. vs (w-wc)

• Slope of each line (b) 0.215

shifted for clarity
Eastern Pacific
Western Pacific
Atlantic ocean
Indian ocean
(individual fits to b within 0.02)
23
How well do the curves collapse when rescaled?

• Original (seen above)

24
How well do the curves collapse when rescaled?

• Rescale w and P by factors fp, fw for each region
  i

i
i
25
Collapse of Precip. Precip. variance for
different regions

• Slope of each line (b) 0.215

Variance
Eastern Pacific
Western Pacific
Precip
Atlantic ocean
Indian ocean
Western Pacific
Eastern Pacific
Peters Neelin, 2006
26
Precip variance collapse for different averaging
scales
Rescaled by L2
Rescaled by L0.42
27
TMI column water vapor and PrecipitationWestern
Pacific example
28
TMI column water vapor and PrecipitationAtlantic
example
29
Dependence on Tropospheric temperature

• Averages conditioned on vert. avg. temp. T, as
  well as w (T 200-1000mb from ERA40 reanalysis)
• Power law fits above critical wc changes, same ?
• note more data points at 270, 271

30
Dependence on Tropospheric temperature

• Find critical water vapor wc 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

31
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 wc

32
Frequency of occurrence. drops above critical
Western Pacific for SST within 1C bin of 30C
Frequency of occurrence (all points)
Precip
Frequency of occurrence Precipitating
33
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)?

34
Extending QE

• Recall Critical water vapor wc empirically
  determined for each vert. avg. temp. T
• Here use to schematize relationship ( extension
  of QE) to continuous phase transition/SOC
  properties

35
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

36
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

37
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

38
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.
• Arakawas 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