Long term, climatologically significant statistical properties of the atmospheric component of the climate system are examined in an effort to correctly and efficiently incorporate this component into larger models for climate that include oceanic and cryospheric influences and deal realistically with climate replication. The magnetic of the unpredictable (noise) component of long term atmospheric behavior is also studied in order to develop the potential for viable climatological forecasting

Laurmann, J. A.

English

NASA Technical Reports Server

Paper No. 58
STOCHASTIC MODELING OF THE TIME-AVERAGED EQUATIONS
FOR CLIMATE DYNAMICS
OR
J. A. Laurmann, Div. of AppL Mech., Stanford Univ.,
Long Term Objectives
climatologically significant statistical properties of
the atmospheric component of the climate system.
Understanding should lead to an ability to correctly
and efficiently incorporate this component into larger
models for climate that include oceanic and cryospheric
influences and deal realistically with climate replica-
tion. An immediately usefully and critically important
consequence of successful completion of the study would
be the ability to assess the magnitude of the unpredict-
able (noise) component of long term atmospheric behav-
ior, and hence the potential for viable climatological
forecasting.
Immediate Program Content And Objectives
The Stanford program in climate is new and in proc-
ess of development. Both because of the relative nov-
elty of our approach to the problem and because our
students have minimal backgrounds in meteorology and
atmospheric sciences, we have elected to start at a
simple but fundamental level of analysis. Thus, the
initial effort is concentrating on the use of Lorenz's
"minimum hydrodynamic equations" as a prototype for
describing at least some of the non-llnear characteris,
tics of long term atmospheric behavior that we believe
are important for climate. We have two main thoughts
in mind:
a) that techniques for closure of the moment equa-
_tions that arise from treatingtime averaged
climatic variables can be tested with the mini-
mum equations. These do indeed exhibit the
counter-gradlent momentum transfer and vacilla-
347
https://ntrs.nasa.gov/search.jsp?R=19790012520 2020-03-20T18:29:29+00:00Z
and
tory behaviors that are known as essential
features on thg non-linear behavior of the
atmosphere;
b) in view of the fact that temporal response
characteristics need to be known for developing
atmospheric climate models, it is hoped that
study of such properties of the minimum equa-
tions (hopefully with only minor modifications)
will lead to a rationale for simplified atmos-
pheric climate models via a subdivision into
more easily treated sub-components. For exam-
ple, we might consider long term, large scale
changes to be describable in terms of a baro-
tropic model that is driven baroclinically, per-
haps by stochastic forcing via quasi-geostroph-
ic turbulence, as well as by deterministic
terms. The minimum equations would represent
the simplest possible description for the baro-
tropic component of such a model.
Accomplishments
We are advocating the introduction of time
averaging for the treatment of climate questions. The
approximations that then become possible (and, we
suggest, necessary for practical viability of answering
climate problems) will vary with the averaging time
selected, which in term will vary with the particular
climate question being addressed. In every case, how-
ever, the averaging process, when applied to the (non-
linear) differential equations governing atmospheric
motions, introduces new, statistically defined varia-
bles (moments and correlations) which are formally
governed by an infinite set of equations. For large
scale motions conventional turbulence theory approaches
for closing these equations do not apply, and for the
climatological case we must resort to other techniques,
the nature of which we expect to discover only from
better understanding of the atmospheric circulation
system itself. The simplest model that incorporates
the important counter-gradient transport of momentum,
necessary for the maintenance of the atmospheric energy
cycle, is represented in Lorenz's introduction of the
"minimum hydrodynamic" system of equations. These re-
present the first three Fourier modes of a standing
wave system for a rectangular domain, with B effect
neglected, and their solution (available analytically)
describes the temporal non-linear oscillations of the
standing waves. The equations governing the time-
348
variation of the amplitudes of the three wave compon-
ents, AI, A2, A3, are
dA I dA 2 dA 3
-- = CIAIA 2 ; - C2AIA 3 ; - C3A2A I ; (i)dt dt dt
where CI . _ [_(_3 + i)]-i _ C2 = 2( 2 + i)-i ,
C3 = _ ( 2 _ 1)/2a , and a = k/Z , the ratio of zonal
to meridional wave number. Jacobian elliptic functions
constitute the analytic solution to (1):
* *, 2)
A 1 = A 1 dn(ht - t h °
* 6 2) (2)
A 2 = A 2 sn(ht - t , ho
* * 2
A 3 = A 3 cn(ht - t , ho )
9
for a > 1 and 2k_E/V < 1 , where
enstrophy and energy of the system.
A 3 are defined in terms ofA1 , A2 , , t , h and k°
E and V and a phase angle.
The description and analysis can be simplified by
a scaling transformation:
X 1 = AI/A I , X2 = A2/A 2 , X 3 = A3/A 3 ,
t, ht = A1 t (3)
E and V are the
The parameters
to give
X1 = - k2o X2X3 XI = dn(t - t , k )
X2 ' '* _ (4)
= X3X 1 X 2 = sn(t - t , k )
X3 = - XIX2 X3 = cn(tl- t , k )
Provided we modify the conventional definition for
the moment of a time-averaged variable from
I T I .
Mij...l(t) = _ t£T [Xi(t ) - _i(t )]
[Xj(t') - _i(t')]...[Xz(t') - _z(t')]dt'
349
to
i .ft
Mij'" "/(t) = t-T [Xi(t') - _i(t)] [Xj(t')
_ _j(t)]...[X/(t') - u/(t)]dt' (5)
moment equations for the time averaging interval T
can be readily derived. It is this (infinite) system
of equations that we need to solve for the climate case,
and since the solutions (4) can be used to calculate
the moments (5) exactly for the minimum equations, we
are in position to evaluate the accuracy of any pro-
posed closure scheme, at least as far as the properties
of the minimum equations are concerned. Exact moments,
for arbitrary values of averaging time T can in fact
be derived in analytic, though rather cumbersome form,
and this has been done up to the third order moments
Tijk
A possible approach for achieving approximate
closure that has worked well in the case of stochastic
dynamics weather forecasting, is linearisation of the
moment equations. In our case, the linearisation would
be made about an assumed known climatic state, which,
for problems of the first kind would be based on ob-
served records over the period 0 > t > -T , and, for
prediction of the second kind, would correspond to
climatological mean estimates of statistically sta-
tionary states. Note that in the second case we need
to treat ensemble as well as temporal means, as situa-
tion which leads to complications in the development of
the moment equations which we are starting to look into.
Tests on the efficacy of this concept of a linearised
approximation are now being carried out by numerical
solution of truncated forms of the (linearised) moment
equations.
However, for the minimum equations we have discov-
ered an unexpected property that indeed seems to ob-
viate the need for devises such as linearisation for
closing the moment equations. It turns out that for
small values of the parameter
k 2 _ I A_ 2
o _ *2 (6)
A I
350
moments of the solutions (2) for the minimum equations
tend to decrease with increasing order of the moment.
In fact power series of the form
_i = _ k2n _i (n) [_i (0) = i]
n=0 o
_. k2(_li + 61J ) _=0 k2n o(n)
• = o ij (7)
mj o
k2(_li + 6ij + $ik ) _ k 2n (n)
_ijk= o n=0 o _ijk
when 6ab = 0 a _ b , _ab = i , a = b , yield series of
of differential equations for the moments which are
automatically closed. For example, the lowest order
equations in k 2 turn out to be
o
_i = 0
_2 = _3
_3 = - _2
_22 = 2_32
_33 = - 2_32
_32 = _33 - c22
which can readily be shown to have solutions identical-
ly equal to those obtained from the exact solution (4),
when the exact moments are expanded for k 2 <<l
o
Small values of k ° in fact correspond to mild vac-
illations of the zonal flow and, as discussed by Lorenz,
represent the predominant mode of atmospheric vacilla-
2 close to 1 (which is the uppertion phenomena. For k o
bound for k 2) corresponding to large oscillations in
o
the east-west wlnd, this approach does not appear to be
351
applicable.
The fact that cl_sure of the time-averaged moment
equations for the minimum equations is possible for
physically reasonable values of parameters immediately
brings to the fore the question of the applicability
of moment equation closure procedures such as (7) to
more realistic equations for atmospheric motions. It
is our intent to enter at depth into this question,
both by j_d_im_ which p_opertles of the .........LLL_*equa-
tions can be expected to hold more generally and by
modifying the minimum equations to allow for clearly
omitted features (such as B , or adding secular and
stochastic forcing terms).
One of the important results looked for in the
statistical approach we are trying to develop is the
determination of the principal tlme-scaled components
into which the atmospheric system can be subdivided.
These separate, but coupled components are distin-
guished by their disparate time scales, and their con-
tent would surely vary with climatic average of con-
cern; the aim is to obtain the simplest possible mode
of description of the atmospheric portion of the
climate system. Whether, or under what circumstances
such a form of representation is possible is still at
issue. Our investigations currently seek to evaluate
the 'response time' for the minimum equations, so as to
better understand how both long term forcing (e.g. sea-
sonal variation) and rapid fluctuations (e.g. from two-
dimensional turbulence) should be coupled to the equa-
tions. We are in process of determining how best to
proceed with this problem, and we envisage that the
coupling has at least in part to be stochastic, leading
to variability in climate evolution as prescribed by
the equations of currently unknown magnitude.
Bibliosraphy
J. A. Laurmann. A small perturbation approximation for
stochastic dynamic weather prediction. To appear
in Tellus.
E. N. Lorenz.
equations.
Maximum simplification of the dynamic
Tellus 12 243 1960
J. Paegle and E. Robl. The time dependent behavior of
the probability density function of some simpli-
fied atmospheric flows. J. Atm. Sci. 34 979 1977
G. D. Robinson. Weather and climate forecasting as
352
problems in hydrodynamics. Month. Wea. Rev. 106
448 1978. Also comments on this paper by J. A.
Laurmann, to appear in the Monthly Weather Review.
G. P. Williams. Planetary circulations: i. Barotropic
representation of Jovian and terrestrial tur-
bulence. J. Atm. Sci. 35 1399 1978
353


Stochastic modeling of the time-advanced equations for climate dynamics

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server