November
Journal
of the
2008
Meteorological Society of Japan, Vol.B.86A,
MAPES
pp. 175−185,
et al. 2008
175
Predictability Aspects of Global Aqua-planet Simulations
with Explicit Convection
Brian MAPES
Rosenstiel School of Marine and Atmospheric Sciences, University of Miami, Miami, Florida, USA
Stefan TULICH
Cooperative Institute for Research in Environmental Sciences (CIRES), University of Colorado,
Boulder, Colorado, USA
Tomoe NASUNO
Frontier Research Center for Global Change (FRCGC), JAMSTEC, Yokohama, Japan
and
Masaki SATOH
Center for Climate System Research (CCSR), The University of Tokyo, Kashiwa, Japan
(Manuscript received 30 August 2007, in final form 30 May 2008 )
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 so that explicit convection is an especially topical 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 at 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 vertical plane simulations of pure convectively coupled gravity waves. As a result, no
simple conclusions can be drawn about whether tropical predictability is limited more by tropical chaos or
by 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 sets the shape of the saturation
(climatological) power spectra. Wind spectra are nearly power-law with a logarithmic slope of about −5/3
in the free troposphere, remarkably so in the 2D runs and clearly distinct from slope −2 (a null hypothesis of spectrally white wind divergence). A common interpretation of −5/3 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
have revealed more fundamental, intrinsic limits
to predictability (Lorenz 1969). In cascading turbulence, for example, a finite time horizon for the
predictability of large eddies is implied by the evershorter 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.
They interpreted the results in terms of a threestage error growth process. Initially tiny white
noise grows rapidly in the convecting region, until
Corresponding author: Brian Mapes, RSMAS/MPO,
4600 Rickenbacker Cswy, Miami, FL 33149-1098,
USA
E-mail: [email protected]
©2008, Meteorological Society of Japan
Vol. 86A
the small-scale (wavelength L < 200 km) differences saturate (i.e., structure on these scales has been
fully scrambled by the noise) within 6h. Largescale (L > 1000 km) differences 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 amplifying
faster.
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 Madden-Julian oscillation may
have even longer predictability, but is not evident
in these aqua planet simulations and so is not discussed further here.
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,” not fully
cloud resolving, the results are still probably 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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B. MAPES et al.
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 to wave speed, Eulerian squared differences might grow quickly, even
though useful information (a “prediction”) about
one model’s weather state could still be gleaned
from the other, through clever postprocessing
177
(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.
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,
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Journal of the Meteorological Society of Japan
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
a simple 2-category bulk water + ice scheme, with
2-stream radiative transfer. 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-hour averaged outputs were 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 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
Vol. 86A
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 interpolation consisting of 90-minute averages every 3
hours rather than 3-hour averages. Along with the
nearest-neighbor interpolation scheme, this shorter
averaging interval makes the N3.5 dataset 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 nearlyidentical initial state. Some detailed characteristics
of N3.5 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−1,
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 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).
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B. MAPES et al.
4. Tropical OLR difference growth
a. In physical space
Figure 1 shows time-longitude sections of the
tropical (10N−10S) OLR from experiment N7+2K
and N14+2K. 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 show N14+2K and
N7+2K because they are independent of those published figures, from a different spinup history.
179
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 (cf., 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 Fig. 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.
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-
Fig. 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.
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Journal of the Meteorological Society of Japan
Vol. 86A
The great similarity of Figs. 2a and 2b 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 30°, 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 40° and long waves at about
20−30°. 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 60−70°.
Fig. 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).
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+2K: the most different of the resolutions,
with different SST, and with totally independent
weather histories.
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.
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,
and are not shown. This growth of large scales almost doubles the difference variance (area under
curve).
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B. MAPES et al.
Fig. 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.
5. Is tropical predictability limited by mid­
latitude 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 differences of meridional wind v12km.
This depiction was developed based on physicalspace impressions of model data: specifically, that
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
181
Fig. 4. RMS differences in wavenumbers 3
−6 of OLR (shading) and 12 km meridional wind (contours), for a) N7−N14 runs;
and b) N7+2K−N14+2K runs.
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 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.
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Journal of the Meteorological Society of Japan
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 sometimes stated, might the
predictability of large scales be protected through
the use of some sort of filtering, as we have heard
suggested? The typical basis for discussions of predictability − mutually advecting vortices in quasihorizontal and mostly-rotational turbulence (e.g.,
chapter 8 of Vallis (2006)) − seems 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, exclusively through the interaction of vortices of comparable scale. Energy “saturation” at
each scale is defined by the onset of this nonlinear
interaction process that leaks energy to adjacent
scales. When noise is added, squared differences
similarly cascade upscale, filling out the saturation
curve quasi-horizontally in spectrum space. Intrinsic predictability is finite, because initial smallscale 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.
In Fig. 5a, the energy spectrum is much steeper
(−3), so that the turnover time of small eddies is
no shorter than that of 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
Vol. 86A
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, neglecting
the small vertical velocity contribution) 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 imply that a “cascade” interpretation is indicated 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, spectra of 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 might be expected to depend upon having underlying total flow amplitude sufficient to
excite the nonlinear advection terms that define
saturation and transfer energy upscale. The usual
caveats of Fourier analysis should be borne in
mind however: random spikes with no horizontal
correlation still exhibit long wavelength power for
example, and it may be misguided to refer to “large
scales growing” within that view of the vertical filling of the error spectrum.
In physical space, the upscale growth mechanisms in the 2D cloud model involve merging
“rivers” of amplitude, in a field of 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. The −5/3
slope prevails across the entire scale range, and
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B. MAPES et al.
183
Fig. 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 12 km 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).
source terms aren’t localized in scale: With moist
convection generating buoyancy b, the KE equation has important spectrally broad sources in its
w ′b ′ 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 of power-law form
with −5/3 slope, it is close enough to suggest that
panel (d) is not just some strange artifact of strict
2-dimensionality. The two models thus comple-
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Journal of the Meteorological Society of Japan
ment 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. More
incisive diagnostics are needed.
In summary, these explicit-convection simulations suggest that substantial untapped mediumrange 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 if these
results could be used to build confidence in one
or more GCMs 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).
Acknowledgements
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, Harm Jonker, and anonymous reviewers are gratefully acknowledged.
Vol. 86A
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, 7−22. Wataru
Ohfuchi and Kevin Hamilton, Eds., ISBN: 9780387366715, 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. Meteor. 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.
86A,
Khairoutdinov, M.F. and D.A. Randall, 2003: Cloudresolving 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,
available through.
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.
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.
November 2008
B. MAPES et al.
Neale, R. and B.J. Hoskins, 2001: A standard test for
AGCMs including their physical parameterizations. ii: Results for the Met Office model. Atmos.
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, 99−112. Wataru Ohfuchi and Kevin Hamilton, Eds., 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:10.1029/2005
GL022459.
Tribbia, J.J. and P.D. Baumhefner, 2004: Scale interactions and atmospheric predictability: an updated
perspective. Mon. Wea. Rev., 132, 703−713.
185
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 largescale 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, 745 pp.
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, andW.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,
(submitted).
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.