Predictability aspects of global aqua-planet simulations with explicit
convection
B RIAN M APES1 , S TEFAN T ULICH2 , T OMOE NASUNO3 , M ASAKI S ATOH4 ,
1 Rosenstiel
2 Cooperative
School of Marine and Atmospheric Sciences, University of Miami, Miami, FL
Institute for Research in Environmental Sciences (CIRES), University of Colorado
3 Frontier
Research Center for Global Change (FRCGC), Yokohama, Japan
4 Center
for Climate System Research (CCSR), University of Tokyo
Revised for JMSJ, March 2008
Corresponding author address:
Dr.
Brian Mapes, RSMAS/MPO, 4600 Rickenbacker Cswy, Miami, FL 33149-1098
([email protected])
2
ABSTRACT
High-resolution global simulations over zonally symmetric aqua planets are examined using
Fourier analysis in the zonal direction. We highlight the tropics, where the large-scale weather
consists of convectively-coupled waves and explicit convection is an important novelty.
Squared differences between pairs of runs grow from initially tiny values to saturation at
twice the climatological variance. For wavelengths shorter than 103 km, differences saturate
within about a day. For tropical long waves, the time to saturation indicates predictability for least
2 weeks. This time scale is similar in middle latitude flow, which interacts with tropical waves in
the 3D model, but it is also similar in 2D pseudo-equatorial plane simulations of pure convectively
coupled gravity waves. As a result, no simple conclusions can be drawn about whether tropical
predictability is limited by tropical chaos or tropical-extratropical interactions.
Difference growth appears to fill out the saturation energy spectrum in a “vertical”
(up-magnitude) rather than “horizontal” (up-scale) manner. Up-scale growth thus occurs as a
continuing amplification of large scales after small scales saturate, which begs the question of
what determines the shape of the saturation (climatological) power spectra. Wind spectra have a
slope near -5/3 in the free troposphere, remarkably so in the 2D runs and clearly distinct from -2
(a null hypothesis of spectrally white wind divergence). A common interpretation of this slope –
as indicative of a cascade, a steady conservative transfer of energy from source to sink scales by
interactions that are local in log-wavelength space – is hard to apply to these moist tropical waves.
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1. Introduction
In principle, anything in the atmosphere can affect anything else, from butterfly wings to
planetary waves. But influences travel along pathways, many of which lead only to dissipation.
Scientific study of predictability seeks some robust knowledge or understanding about pathways
that lead to important events.
Although the skill of any given weather prediction is usually limited in some way by practical
shortcomings, studies of error growth in models (Lorenz 1969) have revealed more fundamental,
intrinsic limits to predictability. In cascading turbulence, for example, a finite time horizon for the
predictability of large eddies emerges from the ever-shorter turnover times of ever-smaller eddies,
unless the energy spectrum is extremely impoverished at small scales (logarithmic slope steeper
than -3, Rotunno and Snyder (2008)).
The use of imperfect models in estimating predictability always leaves open questions. The
predictabiliy properties of global flows containing deep moist convection have been especially
difficult to assess, because of the shortcomings and uncertainties of cumulus parameterization
schemes. Specifically, it is unclear whether models with parameterized convection might overstate
predictability (because of the excessively deterministic nature of schemes) or understate it (by
failing to simulate potentially robust and predictable convectively-coupled weather systems).
The predictability limit for convective cells is mere minutes, set by the growth time of vertical
instability, as opposed to days or weeks for midlatitude synoptic weather, set by the time scale of
baroclinic instability (Hohenegger and Schar 2007). With computing advances, experiments on
multi-scale flows with explicit convection are becoming feasible, allowing linkages between these
two error-amplifying instabilities to be glimpsed.
For example, Zhang et al. (2007) analyzed difference growth in an idealized midlatitude
baroclinic wave, with explicit convection in the warm sector, and interpreted the results in terms
of a three-stage error growth process. Initially tiny white noise grows rapidly in the convecting
region, until the small-scale (wavelength L<200 km) differences saturate (i.e., structure on these
scales has been fully scrambled by the noise) within 6h. Large-scale (L>1000 km) differences
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grow with the background baroclinic instability, and don’t fully saturate in a day. In the mesoscale
range between, saturation takes a time comparable to the inertial period, and geostrophic
adjustment processes (gravity wave radiation, growth of vorticity relative to divergence) can be
seen in the fields. The three stages were presented as a sequence – convective heating, adjustment
to balance, balanced growth – although the authors note that ”the characteristic physical processes
of different stages of error growth can coexist at the same time.” In fact, their figures show that
exponential growth rates are comparable for differences in all 3 scale bands even in the initial
rapid growth phase: small scales simply saturate sooner, rather than actually amplifying faster.
We return to this point in the discussion section.
In the tropics, there are convectively-coupled large-scale waves (Takayabu 1994; Wheeler
and Kiladis 1999), which in certain cases can last a surprisingly long time, even circling the
Earth entirely (Straub et al. 2006). Is there an intrinsic predictability limit on these waves? If
so, is this limit determined by tropical chaos from embedded convection, or by midlatitude
disruptions? Unfortunately, since many models with parameterized convection simulate tropical
wave variability poorly, they cannot be trusted to reliably estimate its predictability characteristics.
The first global explicit-convection simulations have recently been performed, initially on
a zonally symmetric aqua planet. These simulations produced a hierarchy of tropical convective
variability, including long convectively coupled Kelvin waves (Nasuno et al. 2007, 2008).
Here we study the suite of spinup integrations done along the way toward the groundbreaking
global simulation with a 3.5 km mesh. While the resolutions used here are merely ”cloud
system permitting,” the results may be more reliable than those from models with cumulus
parameterizations. This study is an investigation of the growth (in both amplitude and scale)
of initially small differences in these simulations, motivated by the predictability issues and
questions above.
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2. Fraternal vs. identical model twin comparisons
This predictability estimation exercise is based solely on differences between pairs of
model runs. Furthermore, the 3D model differences examined here are from what Tribbia
and Baumhefner (2004) called “fraternal” twin experiments, in which the model’s resolution
(rather than its state) is changed at t = 0. The reason for this strategy is that the unprecedented
computations here were motivated by other goals; this study is opportunistic. Interpretation is
importantly aided by some “identical” model twin experiments, but only in two dimensions, for
computational reasons.
Use of the term predictability in connection with fraternal-twin results is based on
assumptions that need to be reviewed. Suppose the weather in a pair of simulations diverges from
a nearly identical initial state. If the two are from an identical model, with differences only in the
initial state, both have equal validity and relevance and one can only speak of differences. If the
two models are of different quality, the inferior one can be thought of as a flawed forecast model
for the superior one. (We will use the term difference rather than error, but note the connection to
“error growth” usage in the literature).
For fraternal twin models, growing differences may have both random (weather trajectory
evolution) and systematic (“climate drift”) components. Climate drifts can begin directly at large
scales, and might exhibit no dependence on the small initial condition differences. If climate
drift dominates the differences, fraternal-twin experiments may reveal only shortcomings of the
inferior model, not the intrinsic predictability of the flow in the superior model (much less in
nature).
In this study, we examine differences between pairs of explicit-convection simulations,
after resolution doublings. When we interpret these differences in terms of the predictability of
(aqua-planet) weather, we are tacitly asserting that systematic (climate) differences between the
different resolutions are sufficiently small. Moreover, systematic differences in variability as
well as mean flow should be small. For example, if there were a simple resolution dependence
of wave speed, Eulerian squared differences might grow quickly, even though useful information
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(a ”prediction”) about one model’s weather state could still be gleaned from the other, through
clever postprocessing (wave phase adjustment). In short, claims about predictability limits based
on simple metrics in imperfect models must always be viewed with appropriate caution.
Our predictability interpretations thus hinge on the claim that computations of the convecting
flow on an aqua planet have (nearly) converged at 7-14 km mesh spacings, not just the flagship
3.5 km. This will surely bother some readers, since such coarse resolutions would perform
poorly at predicting individual convective clouds and storms (Bryan et al. 2003; Weisman et al.
1997). Still, the global near-convergence of large-scale moist flow is a somewhat separate
question (Hamilton 2008; Williamson 2008; Yamada et al. 2005). A comprehensive assessment
of resolution dependence in the NICAM simulations used here is beyond the current scope: The
reader is referred to the other literature describing (Satoh et al. 2005; Tomita et al. 2005) and
further analyzing (Nasuno et al. 2007, 2008; Satoh et al. 2008) these simulations. We consider
just a few concerns here.
At the level of zonal means, the clearest resolution dependence is that the tropical convection
belt on the equator is wider in the higher resolution runs (Fig. 4 of Tomita et al. 2005). Somehow
the energy in small scale degrees of freedom helps to limit the frontogenetic effects of mean
convergence into the equatorial convection zone. Interestingly, broadening of too-collapsed
tropical convergence zones is also one of the main large-scale impacts of a stochastic backscatter
scheme under development at the European Center for Medium-Range Weather Forecasting
(ECMWF) (Tim Palmer and Judith Berner, personal communication 2007). Still, since the width
of the equatorial rain belt is much less than an equatorial deformation radius in all cases, this mild
resolution dependence may not have drastic consequences for large scale dynamics. The zonal
mean zonal wind [u] has no systematic resolution dependence that can be established with the
10-30 day data records examined here. For example, the latitudinal profile of [u] in a 14 km mesh
simulation examined here exhibits slow vacillations from one 10-day segment to the next which
are as large as the difference between these and the 10-day zonal mean from the 3.5 km mesh
simulation.
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At the level of cloud systems, NICAM global simulations appears to produce fairly realistic
statistics (Inoue et al. 2008). Cloud-tracking composites like Mapes et al. (2008), omitted here for
brevity, also indicate reasonable simulation of mesoscale convective system (MCS) structure and
life cycles. While the MCS life cycle is not understood as fundamentally as parcel instability and
cumulus cell growth, the relevant dynamics do appear to function on grids as coarse as 14 km.
Zhang et al. (2007) also presents evidence that 10 km grid spacing appears adequate for studies of
this type, and further reassurances will be offered below. All these similarities among resolutions
encourage us to at least offer the strong interpretation of these difference growth diagnoses: as
indicative of inherent predictability properties of convecting large-scale flows, rather than merely
drifts due to model resolution dependence.
3. Model and simulation details
The simulations examined here are described in (Satoh et al. 2005; Tomita et al. 2005). They
come from the Nonhydrostatic ICosahedral Atmosphere Model (NICAM), described further in
Satoh et al. (2008), run on the Earth Simulator supercomputer. Sea surface temperature (SST)
was specified in a zonally symmetric pattern, the control case of Neale and Hoskins (2001). No
cumulus parameterization was used. Microphysics was computed by the simple 2-category bulk
water + ice scheme of Grabowski (1998), while radiative transfer was computed by the 2-stream
method of Nakajima (2000). A Mellor-Yamada level 2 planetary boundary layer scheme was
used.
NICAM simulations were initialized from a reference state obtained using a coarse-resolution
general circulation model. This state was interpolated onto the 14 km icosahedral grid, and then
60 days of NICAM spinup integration was performed. For the subsequent 30 days, integration
was continued and 3-hourly output was saved for analysis. The resulting data set, interpolated
from nearest neighbors to a 1/4 degree grid and then further averaged to 1/2 degree for analysis
convenience, is here called N14. For these same 30 days, an integration was also performed on a
7 km grid, with the initial state interpolated from the 14 km model. The resulting data set (again
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interpolated to 1/4 degree and averaged to 1/2 degree) is called N7. N14 and N7 thus represent a
pair of 30-day integrations that began from almost identical weather states, yet ended with very
different weather states.
A second pair of 30-day data sets was obtained when the entire experiment was repeated with
SST increased by 2K everywhere (Miura et al. 2005). Since the entire spinup history was different,
these data sets (N14+2K and N7+2K ) represent completely independent weather realizations from
N14 and N7. These realizations are also drawn from a slightly different climate because of the
warmer water, but the SST dependence is, like resolution dependence, thought to be a secondary
effect for present purposes.
A final pair of diverging model weather realizations was created when the state at day 20
of the N7 simulation was interpolated to a 3.5 km grid and integrated forward for 10 days. The
resulting data set (N3.5) is again 3-hourly at 1/2 degree resolution, but was averaged from a
1/4 degree dataset consisting of 90-minute averages every 3 hours rather than 3-hour averages.
Together with the nearest-neighbor interpolation scheme, this shorter averaging interval makes
N3.5 data exhibit a bit more variance at small scales. N3.5 and the last 10 days of N7 thus
comprise a third pair of simulations in which the weather diverges from a nearly-identical initial
state. Some detailed characteristics of N3.5, the finest-mesh global simulation ever, are presented
in Nasuno et al. (2007, 2008). For present purposes, the key feature is a rich spectrum of scales,
with ∼10 km cellular clouds, ∼100 km cold-pool driven (squall type) convective systems, and
∼1000-km super cloud clusters, all in the context of convectively coupled Kelvin waves of
∼10,000-km wavelengths and speeds near 17 m/s, comparable to the observed moist Kelvin
waves in the Earth’s tropics (Straub and Kiladis 2003).
Two-dimensional (x-z) nonrotating model experiments with the System for Atmospheric
Modeling SAM (Khairoutdinov and Randall 2003) cloud model were also performed for this
study, following the methodology of Tulich and Mapes (2008); Tulich et al. (2007). The domain
was expanded to 40,000 km in x, to mimic the equatorial zonal plane and make results directly
comparable to the NICAM equatorial belt. One notable point is that free-slip lower boundary
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conditions were used. Three ensemble members were generated. At days 30 and 60 after spinup
from rest, copies of the domain were seeded with small-amplitude thermal noise and integrated.
The noise was at low altitude only, horizontally uncorrelated, and involved T and q perturbations
with zero net effect on density (virtual temperature). Squared differences from the resulting
ensemble of six noise-seeded difference comparisons were averaged for the results here (Fig. 5d
below).
4. Tropical OLR difference growth
a. in physical space
Figure 1 shows time-longitude sections of the tropical (10N-10S) OLR from experiment
N7+2 and N14+2. Eastward-moving Kelvin waves are seen (Nasuno et al. 2008), with almost
identical structure initially, then differing considerably after about 2 weeks. Differences grow
because different cloud features develop or decay, within the context of a propagation speed that
is similar between the two runs. Wavenumber 1 retains its phase for the entire 30 day period,
while essentially all smaller scales become scrambled by day 30. Similar figures comparing N7
and N3.5 may be seen in Fig. 3 of Tomita et al. (2005), Fig. 3 of Nasuno et al. (2007), or Fig. 4
of Satoh et al. (2005), and were the initial impetus for this study; here we chose N14+2 and N7+2
because they are independent of those published figures, from a different spinup history.
b. in spectral space
Fourier analysis in longitude is especially appropriate for these simulations, since boundary
conditions were zonally symmetric. We do not wish to blend zonal structure with meridional
using total (spherical) wavenumber analysis (c.f. Tribbia and Baumhefner (2004)). We prefer
wavelength to wavenumber, on a logarithmic basis that conforms to normal meaning of the word
“scale”. The resulting analysis space is as shown in Figure 2. Solid curves are constant zonal
wavenumbers k = 1, 4, 40, 400, while the dotted reference curve shows a Rossby deformation
scale 2c/f with c = 50 m s−1 (gravest tropospheric vertical structure) and f the Coriolis parameter.
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Shading on Fig. 2 indicates mean spectral power (variance) density of Outgoing Longwave
Radiation (OLR). This density is on a per unit log-wavelength basis, ensuring that the amount of
ink in any given horizontal strip of Fig. 2 is proportional to OLR variance at that latitude. Variance
here is defined as squared deviations from the time-mean zonal-mean OLR at each latitude.
Zonal-mean fluctuations are mapped to k = 1/2 (L = 80,000 km at the equator) for contouring.
The shading is identical between the 2 panels, which show results for N3.5 and N14+2: the most
different of the resolutions, with different SST, and with totally independent weather histories.
The great similarity of Figs. 1a and 1b indicates that cloudy weather variations are nearly
independent of resolution, and that these 10-30 day samples are long enough to comprise a stable
climatology, within the sampling uncertainty indicated by deviations from N-S hemispheric
symmetry.
Equatorial OLR variance is spread broadly across scales, with especially prominent
variability near k = 4 (10,000 km) and k = 1. Midlatitude variance peaks near 30o , between the
deformation scale (dotted curve) and k = 4 curve (L ∼ 6000 km). This peak is part of a variance
ridge that distinctly slopes with latitude, with small-scale cloud structures indicated near 40o and
long waves at about 20-30o . These subtropical long waves have spectra similar to the equatorial
OLR (k = 1 and near 4 prominent), suggesting possible tropical-subtropical interactions on these
scales; see also the OLR spatial snapshots in Figs 5-6 of Satoh et al. (2005) and Fig. 1 of Nasuno
et al. (2007). A faint third belt of OLR variance can also be seen near 60-70o .
c. growth of differences in spectral space
For a model with stationary statistics, the mean squared difference between randomly
selected pairs of weather states is twice the climatological variance. Since this fact holds for all
wavenumbers, the power spectra of Fig. 2 (doubled) represent a saturation spectrum, to which
squared differences between diverging model runs will asymptote at long times. To reach this
saturation envelope, initially small and small-scale (say, spectrally white) errors must grow in both
magnitude and scale.
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The growth (in both senses) of squared OLR differences, averaged over the tropical belt
(5S-5N), is illustrated by Fig. 3. The two panels, for N7 - N14 (upper) and N7+2K - N14+2K ,
indicate a very similar evolution in these independent realizations. The initial growth, lowest to
dotted (3h) to dashed (6h) is rapid. At large scales, a mild overshoot (dotted above dashed) may
indicate slight resolution shock or climate drift. The succession of curves indicates time averages
over successively longer time scales (1d, 2d, 4d,...); this averaging helps reduce noise and spreads
out the curves so the sequence can be discerned better. Long wave differences (near 10,000 km,
wavenumbers 3-6) are still growing sharply from the 16d average to the 30d average. Spectra
from the 10-day N3.5 run lack this growth entirely, and are not shown. This growth of large scales
almost doubles the difference variance (area under curve).
5. Is tropical predictability limited by midlatitude interactions?
The 2-week saturation time for errors in tropical long waves is seen more clearly in
Fig. 4, a time-latitude section of root mean square (RMS) OLR differences in wavenumbers
3+4+5+6. Open contours indicate RMS v12km differences. This depiction was developed based
on physical-space impressions of model data: tropical convection is enhanced at times when its
divergent outflow at upper levels aligns with the poleward phase of quasi-barotropic extratropical
flows, which include substantial stationary waves as well as traveling components. It is noteworthy
that midlatitude v12km differences saturate on the same time scale as tropical OLR (about 2
weeks). Although creative annotation could be used to highlight apparent influences traveling
both poleward and equatorward in Fig. 4, it appears that little can be concluded robustly about
this section’s title question from these two experiments alone.
If midlatitude chaos were the limiter of tropical long wave predictability, then long waves in
a 2D model of the equatorial plane in isolation might be hypothesized to resist difference growth
for significantly longer than 2 weeks. To test this, the 2D experiments described in section 2 were
performed. Results (discussed more below) were negative: differences grow to saturation within
about 2 weeks, even in the pure case of 2D convectively-coupled gravity waves. This 3-way
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near-coincidence of difference growth saturation times unfortunately limits our ability to conclude
anything strongly without substantially more data or a detailed mechanistic analysis beyond the
present scope.
6. Discussion
How does the rapid growth of initially small differences in moist convection affect
the predictability of larger-scale flows? Is the problem of Hodyss and Majumdar (2007) –
contamination of synoptic data-impact signals by unintended differences induced remotely in
tropical convection – realistic and inescapable, rather than merely an artifact of parameterizations
and assimilation schemes? If convective-synoptic interactions involve an “upscale cascade”,
as is often stated, might the predictability of large scales be protected by the use of some sort
of filtering, as we have heard suggested? The typical basis for discussions of predictability –
mutually advecting vortices in quasi-horizontal and mostly-rotational flow (e.g. chapter 8 of Vallis
(2006)) – may be inadequate to address these questions fully.
To illustrate predictability considerations in the horizontal-turbulence paradigm, Fig. 5
adapts Rotunno and Snyder (2008)’s Fig. 1, inspired by Lorenz (1969); see also Fig. 1 of Tribbia
and Baumhefner (2004). In Fig. 5, unlimited (a) vs. limited (b) predictability correspond to
steeper vs. flatter energy spectra, via an eddy turnover time argument. The limited predictability
in panel (b) is thus tied explicitly to its -5/3 spectral slope. In this case, the -5/3 slope arises
as the only possibility (on dimensional grounds) for a conservative energy cascade, in which
energy flows from one scale to the next through the interaction of vortices of comparable sizes.
Energy “saturation” at each scale is defined by the onset of this nonlinear interaction process.
Squared differences similarly cascade upscale, filling out the saturation curve horizontally in
spectrum space. Intrinsic predictability is finite, because initial small-scale errors saturate almost
instantly, then creep inexorably upscale along the saturation ceiling. In this setting, there is a close
interrelationship among the physical process of the energy cascade, the meaning of saturation, the
slope of the energy spectrum, the nature of error growth, and the finite limit to predictability.
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In Fig. 5a, the energy spectrum is much steeper, so that small eddies have a turnover time no
shorter than large eddies. In this case errors grow vertically in spectrum space, by an exponential
instability (depicted as spectrally peaked at some scale). Large-scale predictions of a given
accuracy are not limited by small-scale contamination, only by the accuracy of large scale initial
conditions, which could in principle be increased without bound, so predictability is formally
unlimited in that case.
Predictability characteristics of our tropical long waves are depicted in similar diagrams in
panels d and e. Wind variance (kinetic energy) spectra in the tropical free troposphere of (c)
NICAM and (d) the 2D SAM model are shown. Saturation spectral slope is fairly close to -5/3 in
both cases, compellingly so in the 2D model. A -2 slope (dashed), representing the null hypothesis
of a white spatial spectrum of wind divergence (uncorrelated convective mass sources/sinks) is
clearly a poorer fit. Does a -5/3 slope indicate that “cascade” language is appropriate for these
multiscale flows?
The tropical difference growth lines in Fig. 5 c, d appear to fill up the saturation spectrum
vertically (up-amplitude) rather than horizontally (upscale). Large scales grow just as rapidly
as small scales, and they do so before small scales saturate. The local in scale upscale growth
implication of the word cascade may thus be inappropriate, and filtering surely couldn’t prevent
large-scale error growth. Furthermore, difference growth in runs with fully developed flow appear
very similar (not shown) to the initial spinup from rest described by Tulich and Mapes (2008).
This result appears to be unlike a 2D cascade, where scale transfers would be expected to depend
upon having underlying total flow amplitude sufficient to excite the nonlinear advection terms that
define saturation and transfer energy upscale.
In physical space, the upscale growth mechanisms in the 2D cloud model involve merging
“rivers” of amplitude in waves of a short vertical wavelength (Tulich and Mapes 2008). Even
if this merging process can be shown to be isomorphic or analogous to the mutually straining
vortices of turbulence theory, the implication of the word “cascade” as conservatively transferring
energy between a source and sink at opposite ends of the scale spectrum seems inapplicable.
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The -5/3 slope prevails across the entire scale range, and source terms simply aren’t localized in
scale: With moist convection generating buoyancy b, the KE equation has important spectrally
broad sources in its w0 b0 term, while vorticity has sources and sinks from ∂b/∂x that spoil
the enstrophy conservation principle of ordinary 2D turbulence. Might there be a deeper, less
restrictive self-similarity principle at work, of which both cascading turbulence and multi-scale
convectively-coupled gravity waves are embodiments?
The NICAM model’s equatorial belt (Fig. 5c) is similar to the 2D result in its spectral slope
and nearly vertical (up-amplitude) difference-growth process, albeit with a viscous tailing off at
small scales, more sampling noise, and some bending due to use of time-averaged outputs. While
this spectrum is less compellingly linear at -5/3, it is close enough to suggest that panel (d) is not
just some strange artifact of strict 2-dimensionality. The two models thus complement each other.
Observations indicate that -5/3 spectral slopes are indeed characteristic of the tropics (Wikle et al.
1999), while Lindborg (1999)’s negative answer to his title question lends further credence to the
suggestion that processes besides horizontal 2D inertial cascades may be important in producing
this spectral slope in nature.
In summary, these explicit-convection simulations suggest that substantial untapped
medium-range predictability may reside in the poorly understood and imperfectly parameterized
processes of large-scale wave-convection interaction. The possibility of long-range predictability
in moist Kelvin waves is supported by observations to some degree: for example the dramatic
round-the-world Kelvin wave that triggered monsoon onsets and ENSO state changes in 1998
(Straub et al. 2006). In the real world, the Madden-Julian oscillation is another source of long
potential predictability, and can apparently be captured by the explicit-convection NICAM model
(Miura et al. 2007).
Unfortunately, the role of tropical-extratropical interactions remains unclear, as the error
growth time scales in our 2D and 3D runs are too similar to allow strong deductions. The
massive computations required to gain these few realizations of coupling among convection,
tropical waves, and midlatitudes are ungainly, so it would be ideal of these results could be
15
used to build confidence in one or more GCM with parameterized convection. A GCM whose
aqua-planet climatology contains long tropical Kelvin waves is a first requirement, and at least
some parameterized models do pass that test (Neale and Hoskins 2001).
Acknowledgments.
This material is based upon work supported by the National Science Foundation under grants ATM
0097116, 0407559, 0336790, and 0112715. The simulations used for analysis in this study were done
using the Earth Simulator at the Japan Agency for Marine-Earth Science and Technology. This research
was supported by the Core Research for Evolutional Science and Technology (CREST) program of the
Japan Science and Technology Corporation (JST). The earnest puzzlement of initial seminar audiences, and
subsequent scientific discussions with Drs. Dan Hodyss, Neil Johnson, Sharan Majumdar, Ralph Milliff,
David Nolan, Rich Rotunno, Fuqing Zhang, and anonymous reviewers are gratefully acknowledged.
16
REFERENCES
Bryan, G. H., J. C. Wyngaard, and J. M. Fritsch, 2003: Resolution requirements for the simulation of deep
moist convection. Mon.Wea. Rev., 131, 23942416.
Hamilton, K., 2008: Numerical resolution and modeling of the global atmospheric circulation: A review
of our current understanding and outstanding issues. High Resolution Numerical Modelling of
the Atmosphere and Ocean, pp. 7–22. Edited by Wataru Ohfuchi and Kevin Hamilton, ISBN:
978-0387366715, 280pp.
Hodyss, D., and S. J. Majumdar, 2007: The contamination of ‘data impact’ in global models by rapidly
growing mesoscale instabilities. Quart. J. Roy. Meteor. Soc., 133, 1865–1875.
Hohenegger, C., and C. Schar, 2007: Atmospheric predictability at synoptic versus cloud-resolving scales.
Bull. Am. Meteorol. Soc., 88, 1783–1793.
Inoue, T., M. Satoh, H. Miura, and B. E. Mapes, 2008: Characteristics of cloud size of deep convection
simulated by a global cloud resolving model over the western tropical pacific. J. Meteor. Soc. Japan.
This issue.
Khairoutdinov, M. F., and D. A. Randall, 2003: Cloud-resolving modeling of the ARM summer 1997 IOP:
Model formulation, results, uncertainties, and sensitivities. J. Atmos. Sci., 60, 607–625.
Lindborg, E., 1999: Can the atmospheric energy spectrum be explained by two-dimensional turbulence? J.
Fluid Mech., 338, 259–288.
Lorenz, E. N., 1969: The predictability of a flow which possesses many scales of motion. Tellus, 21,
289–307.
Mapes, B. E., J. Morzel, and R. F. Milliff, 2008: Composite life cycle of maritime tropical mesoscale
convective systems in scatterometer and microwave satellite observations. J. Atmos. Sci.. Submitted
Dec. 2007, available through http://www.rsmas.miami.edu/users/bmapes.
Miura, H., H. Tomita, T. Nasuno, S. Iga, , M. Satoh, and T. Matsuno, 2005: A climate sensitiviy test using
a global cloud resolving model under an aqua planet condition. Geophys. Res. Lett., 32. L19717,
doi:1029/2005GL023672.
17
Miura, H., M. Satoh, T. Nasuno, A. T. Noda, and K. Oouchi, 2007: A madden-julian oscillation event
simulated using a global cloud-resolving model. Science, 318, 1763–1765.
Nasuno, T., H. Tomita, S. Iga, H. Miura, and M. Satoh, 2007: Multi-scale organization of convection
simulated with explicit cloud processes on an aqua planet. J. Atmos. Sci., 64, 1902–1921.
Nasuno, T., H. Tomita, S. Iga, H. Miura, and M. Satoh, 2008: Convectively coupled equatorial waves
simulated by a global nonhydrostatic experiment on an aquaplanet. J. Atmos. Sci., 65, 1246–1265.
Neale, R., and B. J. Hoskins, 2001: A standard test for AGCMs including their physical parameterizations.
ii: Results for the Met Office model. Atm. Sci. Letters, 1, 108–115.
Rotunno, R., and C. Snyder, 2008: A generalization of lorenz’s model for the predictability of flows with
many scales of motion. J. Atmos. Sci.. In press.
Satoh, M., H. Tomita, H. Miura, S. Iga, and T. Nasuno, 2005: Development of a global cloud resolving
model – a multi-scale structure of tropical convections. J. Earth Simulator, 3, 11–19.
Satoh, M., T. Nasuno, H. Miura, H. Tomita, S. Iga, and Y. Takayabu, 2008: Precipitation statistics
comparison between global cloud resolving simulation with nicam and trmm pr data. High
Resolution Numerical Modelling of the Atmosphere and Ocean, pp. 99–112. Edited by Wataru
Ohfuchi and Kevin Hamilton, ISBN: 978-0387366715, 280pp.
Straub, K. H., and G. N. Kiladis, 2003: The observed structure of convectively coupled kelvin waves:
Comparison with simple models of coupled wave instability. J. Atmos. Sci., 60, 1655–1668.
Straub, K. H., G. N. Kiladis, and P. E. Ciesielski, 2006: The role of equatorial waves in the onset of the
South China Sea summer monsoon and the demise of El Niño during 1998. Dyn. Atmos. Oceans,
42, 216–238.
Takayabu, Y. N., 1994: Large-scale cloud disturbances associated with equatorial waves. Part I: Spectral
features of the cloud disturbances. J. Meteor. Soc. Japan, 72, 433–448.
Tomita, H., H. Miura, S. Iga, T. Nasuno, and M. Satoh, 2005: A global cloud-resolving simulation:
Preliminary results from an aqua planet experiment. Geophys. Res. Lett., 32. L08805,
doi:10.1029/2005GL022459.
18
Tribbia, J. J., and P. D. Baumhefner, 2004: Scale interactions and atmospheric predictability: an updated
perspective. Mon. Wea. Rev., 132, 703–713.
Tulich, S. N., and B. E. Mapes, 2008: Multiscale convective wave disturbances in the tropics: Insights from
a two-dimensional cloud-resolving model. J. Atmos. Sci., 65, 140–155.
Tulich, S. N., D. A. Randall, and B. E. Mapes, 2007: Vertical-mode and cloud decomposition of large-scale
convectively coupled gravity waves in a two-dimensional cloud resolving model. J. Atmos. Sci., 64,
1210–1229.
Vallis, G., 2006: Atmosphere-Ocean Fluid Dynamics: Fundamentals and large-scale circulation..
Cambridge University Press.
Weisman, M. L., W. C. Skamarock, and J. B. Klemp, 1997: The resolution dependence of explicitly
modeled convective systems. Mon. Wea. Rev., 125, 527–548.
Wheeler, M., and G. N. Kiladis, 1999: Convectively coupled equatorial waves: Analysis of clouds and
temperature in the wavenumber-frequency domain. J. Atmos. Sci., 56, 374–399.
Wikle, C. K., R. F. Milliff, and W. G. Large, 1999: Surface wind variability on spatial scales from 1 to 1000
km observed during toga coare. J. Atmos. Sci., 56, 2222–2231.
Williamson, D. L., 2008: Convergence of aqua-planet simulations with increasing resolution in the
community atmospheric model, version 3. Tellus. In review.
Yamada, Y., T. Sampe, Y. Takahashi, M. Yoshioka, W. Ohfuchi, M. Ishiwatari, K. Nakajima, and Y.-Y.
Hayashi, 2005: A resolution dependence of equatorial precipitation activities represented in a
general circulation model. Theor. Appl. Mech. Jpn., 54, 289–297.
Zhang, F., N. Bei, R. Rotunno, C. Snyder, and C. C. Epifano, 2007: Mesoscale predictability of moist
baroclinic waves: Convection-permitting experiments and multistage error growth dynamics. J.
Atmos. Sci., 64. 35793594.
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Figure captions
Figure 1. Time-longitude sections of OLR averaged 10N-10S, for the N7+2K and N14+2K simulation
pair. Contours are 180, 210, 240 W m−2 .
Figure 2. Time-averaged OLR zonal power spectral density at each latitude. The time mean zonal
mean has been subtracted from the data, and zonal wave k=0 is mapped to k=1/2 (80,000 km wavelength
at the equator). Solid curves indicate (right to left) k = 1, 4, 40, 400. Dotted curve indicates a Rossby
deformation diamter for the tropospheric first baroclinc mode (50 m/s wave speed). Top: N3.5 data (10
days). Bottom: N14+2K (30 days).
Figure 3. Squared differences of OLR between a) N7 - N14 and b) N7+2K - N14+2K simulations,
averaged over the 5N-5S belt, in the log-wavelength domain. The vertical axis is scaled to indicate power
per octave, while the horizontal axis covers 8 octaves exactly. The rising sequence of lines represents times
of 3h, 6h(dotted). 9h(dashed); then mean differences over 1d, 2d, 4d, 8d, 16d, 30d.
Figure 4. RMS differences in wavenumbers 3-6 of OLR (shading) and 12km meridional wind
(contours), for a) N7 - N14 runs; and b) N7+2K - N14+2K runs.
Figure 5. Climatological wind energy spectra (heavy lines), and squared-difference (error) growth
with time after initial perturbations (successions of light lines). Top panels are adapted from Rotunno and
Snyder (2008), showing predictability aspects of two idealized model paradigms for horizontal turbulence
with predominantly a) up-amplitude and b) up-scale error growth. Panel (c) is for equatorial zonal wind at
12km in the NICAM N14-N7 pair, at the same succession of times as in Fig. 3. Panel (d) is mass-weighted
vertical mean u variance in 2-dimensional periodic cloud model runs like those described in Tulich and
Mapes (2007, 2008), with error spectra at 1.5h, 3h, 6h, 12h, 1d, 2d, 4d, 8d, 16d. Reference slopes -5/3
(dotted in c, dashed in d) and -2 (dotted in d).
Figure 1. Time-longitude sections of OLR averaged 10N-10S, for the N7+2 and N14+2
simulation pair. Contours are 180, 210, 240 W m-2.
a) N3.5 OLR
b) N14+2 OLR
Figure 2. Time-averaged OLR zonal power spectral density at each latitude. The time
mean zonal mean has been subtracted from the data, and zonal wave k=0 is mapped to
k=1/2 (80,000 km wavelength at the equator). Solid curves indicate (right to left) k = 1, 4,
40, 400. Dotted curve indicates a Rossby deformation diamter for the tropospheric first
baroclinc mode (50 m/s wave speed). Top: N3.5 data (10 days). Bottom: N14+2 (30 days).
a)
b)
Figure 3. Squared differences of OLR between a) N7 - N14 and b) (N7+2) - (N14+2)
simulations, averaged over the 5N-5S belt, in the log-wavelength domain. The vertical
axis is scaled to indicate power per octave, while the horizontal axis covers 8 octaves
exactly. The rising sequence of lines represents times of 3h, 6h(dotted). 9h(dashed); then
mean differences over 1d, 2d, 4d, 8d, 16d, 30d.
a)
b)
Figure 4. RMS differences in wavenumbers 3-6 of OLR (shading) and 12km meridional
wind (contours), for a) N7 - N14 runs; and b) (N7+2K) - (N14+2K) run s.
(b) SQG: Limited
Predictability
E (k ) ~ k −3
E (k ) ~ k −5/ 3
error
error
energy
(a) 2D: Unlimited
Predictability
log wavelength →
(c) NICAM
log wavelength →
(d) 2D cloud model
Figure 5. Climatological wind energy spectra (heavy lines), and squared-difference
(error) growth with time after initial perturbations (successions of light lines). Top
panels are adapted from Rotunno and Snyder (2008), showing predictability aspects of
two idealized model paradigms for horizontal turbulence with predominantly a) upamplitude and b) up-scale error growth. Panel (c) is for equatorial zonal wind at 12km
in the NICAM N14-N7 pair, at the same succession of times as in Fig. 3. Panel (d) is
mass-weighted vertical mean u variance in 2-dimensional periodic cloud model runs
like those described in Tulich and Mapes (2008), with error spectra at 1.5h, 3h, 6h, 12h,
1d, 2d, 4d, 8d, 16d. Reference slopes -5/3 (dotted in c, dashed in d) and -2 (dotted in d).