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Eq. (27) in an initial value problem shows that T spreads energy to both small and

large k, but primarily to small k. The converse is true for k

EðkÞ. However, for

decaying turbulence, the only consistent constant ﬂux inertial range is a forward

ﬂuxed enstrophy inertial range. Then deﬁning

Z ¼dE=dt ð28Þ

Eq. (28) gives

EðkÞZ

2=3

3

=Oðln

1=3

ðkÞÞ ð29Þ

On the other hand, we may verify by direct substitution that for (27),

Tðk

5=3

Þ¼0. These remarks apply to freely decaying turbu lence, and the situation

changes if we consider turbulence forced at some intermediate wavenumber

½FðkÞdðk  k

Þ, for example]. Then we may solve Eqs. (23) to (27) outside

the dissipation range to ﬁnd:

EðkÞZ

2=3

3

ðk

=kÞ

5=3

; k  k

;  Z

2=3

3

=ð1 þ2lnðk=k

ÞÞ

1=3

; k > k

ð30Þ

Thus, for stationary two-dimensional turbulence, energy is ﬂuxed to small k, and

enstrophy to large k. However, for a steady state to exist, the energy ﬂuxed toward

k ! 0 must somehow be dissipated by friction or some other large-scale dissipat ion.

The ideas of a dual cascade car ries over to QG ﬂow in which the continuous

degrees of freedom are permitted in the vertical direction. First, we may note that the

equipartit ioning formalism implies in this case an isotropic form for cðkÞ¼cðjkjÞ.

This, combined with the form of the enstrophy inertial range sketched above

comprises the basis for Charney’s (1971) discussion of QG turbulence in the atmo-

sphere. The problem has subsequently been examined via the statistical theory

of turbulence (Herring, 1980) and high-resolution direct numerical simulation

(McWilliams et al., 1994). Both studies examined decaying QG ﬂows. In general,

scales larger than the energy peak developed into strong two-dimensional ﬂow with

scales smaller than the energy peak mixed barotropic and baroclinic. Above k

peak

the statistical theory’s prediction for Q(k) displayed weak baroclinic anisotropy just

beyond the energy peak, which weakens into isotropy as k !1. The simulation, on

the other hand, had a much more pronounced baroclinicity beyond the energy peak

that persisted over the entire range kf > k

peak

3 COHERENT STRUCTURES

Our considerations of spectra have said nothing of the structures imbedded in the

turbulence, and in fact two-point covariances cannot distinguish between strain and

vorticity, except to say that their root-mean-square (rms) values are equal. Thus,

ﬂows with intense, isolated vortices surrounded by expansive strain ﬁelds would

have the same spectra as ﬂows in which the vortex regions are spatially merged,

76 TURBULENCE

inseparable from the strain regions. The latter situation could be obtained by picking

the amplitudes, Q(k) according to a Gaussian distribution. Such a ﬁeld would have

T(k) ¼0, thereby excluding energy transfer, which gives rise to inertial ranges. This

suggests that some sim ple measure of non-Gaussianity would be an indicator of the

presence of structures, although the converse is not the case. The simplest such

measure is the kurtosis,

KhQ

ðrÞi=hQ

ðrÞi

ð31Þ

Here, the angular brackets indicate an average over the spatial domain of the ﬂow.

For the case in which Q(r) consists of isolated pulses of width w separated by empty

areas, of area A, K¼A=w, while for Gaussian ﬁelds, K¼3.

Numerical simulations of large R

two-dimensional decaying ﬂows (McWilliams,

1984) indicate that ﬂows that start with Gaussian initial conditions develop intense,

well-separated vortices, and that KðtÞ systematically increases during the ﬂows

development with values of K¼60 typical at late stages. The same comment

may be made of homogeneous quasi-geostrophi c ﬂows. For comparison, similar

studies of three-dimensional ﬂows have values of K6. As the ﬂow develops,

the vortices become more isolated and more circular, and the vortex cores probably

do not participate in the cascade. Thus regions contributing to enstrophy cascade are

conﬁned to thin regions surrounding vortex cores, so that the turbulence lives on a

subspace available to it, with transfer to small scales that is progressively reduced

during the decay.

One simple consequence derived from Eqs. (23) to (27) is that the decay of

enstrophy is EðtÞt

p

, p ¼2, ﬁrst proposed by Batchelor (1969). Such conclusions

are also reached by more complete statistical theories, which are reviewed by Lesieur

(1990, p. 269 et seq.). But high-resolution numerical simulations show a much

slower decay. Thus Carnavale et al. (1991) ﬁnd p ¼0.5, while Chasnov (1997)

ﬁnds from his 4096 resolution simulation that p ¼0.8. However, these results for

decaying turbulence appear not to bring into question the nature of the inverse

cascade discussed elsewhere [see, e.g., Gotoh (1998) or Foias and Chae (1993)].

4 STRATIFIED THREE-DIMENSIONAL TURBULENCE AND WAVES

Stably stratiﬁed turbulence shares certain features of two-dimensional turbulence;

indeed it has been proposed that motion in the larger scale s of the mesoscale are

explained as inverse cascading quasi two-dimensional turbulence (Gage, 1978; Lilly,

1983). To explore this issue, we write the Boussinesq equations in a convenient

nondimensional form:

ð@

 H

Þu ¼Hp  u  Hu þ

ggNy  2V  u ð32Þ

ð@

 sH

Þy ¼Nw  u  Hy ð33Þ

H  u ¼ 0 ð34Þ

4 STRATIFIED THREE-DIMENSIONAL TURBULENCE AND WAVES 77

Here, N ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

bbT

. We take the stratiﬁcation to be constant and stable N > 0. Now

introduce, in Fourier space, the representation

u ¼ e

þ e

; e

ðkÞ¼k 

gg=jk 

ggj; e

ðkÞ¼k  e

=jk  e

jð35Þ

where f

describes horizontal motion (and hence its name ‘‘vortical mode’’), while

describes gravity waves (for the case O ¼0). In terms of ðf

; yÞ (33) may be

written:

¼ M

þ NLff; ygð36Þ

Nonlinearities are indicated symbolically here by ‘NL’, and

M ¼

02O cos j 0

2O cos j 0 N sin j

0 N sin j 0

ð37Þ

Here,

gg 

kk ¼ cos j. For simplicity, we take

ggkV. The eigenvalues of M are

l ¼ð0; iRÞ; R¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

cos

j þ N

sin

ð38Þ

Some insight may be gained from supposing that the nonlinear terms are weak and

act as random agitators of ðf

; f

; yÞ. Such a linearization of (36) is called rapid-

distortion theory. Then for a given N, O the dominant structures activated would be

those that minimize the frequency, l.ForN = O  1, gravity waves having W  0

dominate, and for N =O  1, inertial waves with W  p=2 dominating. The former

condition is satisﬁed by shear layers (‘‘pancakes’’), while the latter by vertically

oriented columns. Of course, simple linear reasoning cannot resolve structural

issues, for example, whether the shear layers are organized into circular pancakes

or the vertical spacing between ‘‘pancakes.’’

Turbulence and waves coexist in stably stratiﬁed ﬂows, but their time scales may

be disparate: The time scale for waves is 1=N, while the eddy turn over time scale

is 1=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

EðkÞ

. Thus, if the scale-size distribution is less steep than k

3

, waves will

dominate the large scales, and turbulence the small. If small scales are isotropic

[with spect ra (22)], the scale at which these two time scales are equal is

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

e=N

, known as the Ozmidov scale (Ozmidov 1965). Eddies whose k > k

are unstable to vertical overturning; and vice versa: at scales smaller than k

,anrms

Richardson number R

 N

=j@

 1. We should mention here that this picture

may be a gross oversimpliﬁcation of the physics: There ha s yet to emerge a direct

veriﬁcation of it from either numerical simulations or experiments.

Stable stratiﬁcation severely suppresses vertical eddy diffusion of scalar ﬁelds.

Neglecting molecular processes, a vertically moving parcel of ﬂuid exhausts its

78 TURBULENCE

kinetic energy and stops. In reality the parcel’s kinetic energy is refreshed through

thermal exchanges with neighboring parcels and can continue its vertical migration.

In terms of turbulence concepts, we may argue that the gravity wave par t of the

spectrum (k k

) contributes little to eddy diffusion, with only scales with k > k

active in vertical diffusion. Eddy diffusion may be estimated by treating each Fourier

mode of the vertical velocity as a random component contributing to the parcel’s

velocity,

ðtÞ=dt ¼ wðk; tÞ; k

eddy

=dti

dkEðkÞtðkÞð39Þ

where t(k) is the correlation time associated with hwðk; tÞwðk; t

Þi  1=

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

EðkÞ

Here, wðk; tÞ is the k th component of the Lagrangian vertical velocity ﬁeld. The

equation for k

eddy

may be found in Lesieur (1990, p. 285). If in the sum in (39) we

suppress scales larger than 1=k

eddy

 e=N

ð40Þ

This represents a reduction by a factor (L

=L)

4=3

of the unstratiﬁed value o f eddy

diffusivity, u

rms

L e

1=3

4=3

. Although the above estimate has been given a more

quantitative form by the theory of Weinstock (1978), numerical simulations (Kimura

and Herring, 1996) as well as observational studies (Britter et al., 1985) suggest

much smaller eddy diffusivity than (40). It is interesting to note that rapid distortion

theory is able to account qualitatively for the suppression of eddy diffusivity

(Kaneda and Ishida, 2000).

The search for the ‘‘balanced’’ dynamics of (32) and (34) in the limit V ! 0,

N !1 is of vital interest. As noted above, it was thought early on that the ‘‘reduced

system’’ would sufﬁciently resemble two-dimensional turbulence so that the latter

paradigm would be useful in understanding the large horizontal scales of mesoscale

variability [for which E(k) k

5=3

is observed]. Early analysis suggested two-

dimensional turbulence in horizontal, independent layers (Riley et al., 1982; Lilly,

1983). Current mathematical reformulation of the reduced system (Majda and Grote,

1997) now seems able to explain how ‘‘pancakes’’ form, as well as their vertical

spacing. Of course, with rotation present, quasi-geostrophy may be invoked as an

explanation, but the difﬁculty is that as the smaller scales of mesoscale variability are

approached, R

[see (9)] becomes 1 with the approximate 

range still present.

The issue has been examined numerically by Herring and Me

tais (1989), who

concluded that forced ﬂow did indeed become two dimensionally layered with

horizontal motion that varies from layer to layer. But layer edges are rough, and

frictional effects associated with roughness tends to seriously suppress inverse

cascade. A recent study of the mesoscale variability by Lindborg (1999) indicates

that the dynamics of the k

5=3

range (thought by Gage and Lilly to be inverse

cascading two-dimensional turbulence) does not conform to that of inverse casca-

ding two-dimensional turbulence. His analysis compared theoretical estimates of the

4 STRATIFIED THREE-DIMENSIONAL TURBULENCE AND WAVES 79

third-order structure function [which gives a measure of T(k, t) as in (23)] to that

gathered from an ensemble of aircraft observations. It remains to be seen if the

theory of Majda and Grote will be able to give insight into the issue of inverse

cascade.

REFERENCES

Batchelor, G. K. (1969). Computation of the energy spectrum in homogeneous two-

dimensional turbulence, Phys. Fluids Suppl. 12(II), 233–239.

Britter, R. E., J. C. R. Hunt, G. L. Marsh, and W. H. Snyder (1983). The effects of stable

stratiﬁcation on the turbulent diffusion and the decay of grid turbulence, J. Fluid Mech. 127,

27–44.

Carnavale, G. F., J. C. McWilliams, Y. Pomeau, J. B. Weiss, and W. R. Young (1991). Evolution

of vortex statistics in two-dimensional turbulence, Phys. Rev. Lett. 66, 2735–2737.

Charney, J. G. (1971). Quasigeostrophic turbulence, J. Atmos. Sci. 28, 1087–1095.

Chasnov, J. R. (1997). On the decay of two-dimensional homogeneous turbulence, Phys. Fluids

9, 171–180.

Foias, C., and D. Chae (1993). A probability measure representing 2-D homogeneous

turbulence, in S. Kida (Ed.), Unstable and Turbulent Motion of Fluid, World Scientiﬁc,

Singapore, pp. 131–140.

Gage, K. S. (1979). Evidence for a k

5=3

law inertial range in mesoscale two-dimensional

turbulence, J. Atmos. Sci. 36, 1950–1954.

Galperin, B., S. Sukoriansky, and H.-P. Huang (2001). Universal n

5

spectrum of zonal ﬂows

on giant planets, Phys. Fluids 13, 1545–1548.

Gotoh, T. (1998). Energy spectrum in the inertial and dissipation ranges of the two-dimensional

steady turbulence, Phys. Rev. E 57, 2984–2991.

Griffa, A., and R. Salmon (1989). Wind-driven ocean circulation and equilibrium statistical

mechanics, J. Marine Res. 47, 457–492.

Herring, J. R. (1980). Statistical theory of quasi-geostrophic turbulence, J. Atmos. Sci. 37,

969–977.

Herring, J. R. and O. Me

tais (1989). Numerical experiments in forced stably stratiﬁed

turbulence. J. Fluid Mech. 202, 97–115.

Holloway, G. (1992). Representing topographic stress for large-scale ocean models, J. Phys.

Ocean 22, 1033–1046.

Holloway, G., and M. C. Hendershott (1977). Stochastic modeling for non-linear Rossby

waves, J. Fluid Mech. 82, 747–765.

Kaneda, Y., and T. Ishida (2000). Suppression of vertical diffusion in strongly stratiﬁed

turbulence, J. Fluid Mech. 402, 311–327.

Kimura, Y., and J. R. Herring (1996). Diffusion in stably stratiﬁed turbulence, J. Fluid Mech.

328, 253–269.

Kraichnan, R. H. (1968). Small-scale structure convected by turbulence, Phys. Fluids 11,

945–953.

Kraichnan, R. H. (1975). Statistical dynamics of two dimensional turbulence, J. Fluid Mech.

67, 155–175.

TURBULENCE

Leith, C. E. (1971). Atmospheric predictability and two-dimensional turbulence, J. Atmos. Sci.

28, 145–161.

Lesieur, M. (1990). Turbulence in Fluids, 2nd ed., Dordrecht, Kluwer Academic.

Lilly, D. G. (1983). Stratiﬁed turbulence and the mesoscale variability of the atmosphere,

J. Atmos. Sci. 40, 749–761.

Lindborg, E. (1999). Can the atmospheric energy spectrum be explained by two-dimensional

turbulence? J. Fluid Mech. 388, 259–288.

Majda, A. J., and M. J. Grote (1997). Model dynamics and vertical collapse in decaying

strongly stratiﬁed ﬂows, Phys. Fluids 9(10), 2932–2940.

McWilliams, J. C. (1984). The emergence of isolated vortices in turbulent ﬂows, J. Fluid Mech.

146, 21–43.

McWilliams, J. C., J. B. Weiss, and I. Yavneh (1994). Anisotropy and coherent vortex structures

in planetary turbulence, Science 264, 410–413.

Ozmidov, R. V. (1965). On the turbulent exchange in a stably stratiﬁed ocean. Izu. Accad. Sci.

USSR Atmos. Oceanic Phys. 1, 493–497.

Pedlosky, J. (1987). Geophysical Fluid Dynamics, 2nd ed., New York, Springer.

Pouquet, A., M. Lesieur, J. C. Andre, and C. Basedevant (1975). Evolution of high Reynolds

number two-dimensional turbulence, J. Fluid Mech. 72, 305–319.

Riley, J. J., R. W. Metcalfe, and M. A. Weissman (1982). Direct numerical simulations of

homogeneous turbulence in density stratiﬁed ﬂuids, Proc. AIP Conf. On Nonlinear

Properties of Internal Waves, (AIP, Woodbury, 1981) pp. 79–112.

Salmon, R., G. Holloway, and M. C. Hendershott (1978). The equilibrium statistical mechanics

of simple quasi-geostrophic models, J. Fluid Mech. 75, 691–703.

Weinstock, J. (1978). Vertical turbulent diffusion in a stably stratiﬁed ﬂuid, J. Atmos. Sci. 35,

1022–1027.

REFERENCES 81

CHAPTER 7

PREDICTABILITY AND CHAOS

JEFFREY B. WEISS

Attempts to predict the weather are probably as old as humanity itself. One can

imagine our earliest ancestors anxiously examining a stormy sky trying to decide

whether to stay in their caves or venture out to hunt and gather. By the early nine-

teenth century, science viewed the universe as a deter ministic and predictable clock-

works, a view concisely expressed by Laplace (1825):

An intelligence that, at a given instant, could comprehend all the forces by which nature

is animated and the respective situation of the beings that make it up, if moreover it

were vast enough to submit these data to analysis, would encompass in the same

formula the movements of the greatest bodies of the universe and those of the lightest

atoms. For such an intelligence nothing would be uncertain, and the future, like the

past, would be open to its eyes.

Laplace’s viewpoint reduces the problem of weather prediction to that of ﬁnding

the right equations of motion, Laplace’s forces, and the exact initial state of the

appropriate system, Laplace’s respective positions of the beings. Apart from the

inherent randomness arising from the quantum nature of the universe, Laplace’s

view is still correct, but of limited utility. The problem is one of approximations.

We will never know the exact equations for the entire universe and the exact posi-

tions and velocities of all its particles. For linear systems, such as an archetypical

clock, approximate knowledge of only a small portion of the universe, the current

location of the clock’s hands to within some uncertainty, allows prediction of its

entire future to within a similar uncertainty. However, for nonlinear dynamical

systems, such as the atmosphere, the question is: How accurately do we need to

know the equations and the initial state to make a prediction within a desired error?

Handbook of Weather, Climate, and Water: Dynamics, Climate, Physical Meteorology, Weather Systems,

and Measurements, Edited by Thomas D. Potter and Bradley R. Colman.

ISBN 0-471-21490-6 # 2003 John Wiley & Sons, Inc.

In what follows, the focus will be on the atmosphere, but the issues are similar for

any other component of the climate system and for its entirety.

Consider the one-dimensional linear dynamical system

¼ ax ð1Þ

which has a solution

xðtÞ¼e

xð0Þð2Þ

If a > 0, the system is unstable and x(t) !1, while if a < 0 the system is stable,

x(t) !0. When a < 0, the point x ¼0 is said to be the attractor of the dynamical

system in that all initial conditions are attracted to x ¼0ast !1.

Suppose we know the initial condition x(0) and the parameter a to within (posi-

tive) uncer tainties dx(0) and da, respec tively. Then, if those uncertainties are small,

the uncertainty, or error, dx(t) in the prediction x(t) for t > 0is

dxðtÞ¼e

jxð0Þjdatþ e

dxð0Þð3Þ

When the system is unstable, the error, as well as the system itself, grows exponen-

tially to inﬁnity; but, when the system is stable, the error eventually decays to zero.

Assume now that we know the dynamics exactly, da ¼0. Since the error growth is

exponential, it is convenient to deﬁne the mean rate of exponential error growth l as

l ¼

dxðtÞ

dxð0Þ

¼ a ð4Þ

So stability corresponds with errors shrinking and negative l, while instability

causes error growth and positive l.

Most natural systems are, however, not linear. The atmosphere, ocean, and

climate system are presum ed to be extremely high-dimensional deterministic dissi-

pative dynamical systems. They are dynamical systems in that their state at a future

time is a function of their state at a previous time. They are diss ipative in that energy

is not co nserved but is input through forcing, e.g., shortwave solar radiation, and is

drained away through damping, e.g., viscosity and outgoing longwave radiation. The

dimensionality of a dynamical system refers to the number of variables, or degrees

of freedo m, needed to fully describe its state. The atmosphere is a continuous ﬂuid

described by partial differential equations and its dimensionality is inﬁnite. Discre-

tization of the continuum, necessary for numerical modeling, renders the dimen-

sionality ﬁnite, but any reasonable discretization results in a high dimensionality.

Cur rent numerical weather prediction models have O(10

) degrees of freedom.

Lorenz (1975) classiﬁed predictability into the ﬁrst and second kinds. Predictions

of the ﬁrst kind are deterministic predictions, predictions of the speciﬁc state of a

system evolving from an initial condition. This is the kind of predictability addressed

84 PREDICTABILITY AND CHAOS

by Laplace: Given an initial state with some uncertainty how well can we predict the

future? Predictability of the second kind takes over after predictability of the ﬁrst

kind is no longer possible. Once the errors in an initial value problem have grown so

large that a prediction of the ﬁrst kind is useless, all that can be predicted are the

statistics of the system. Those statistics will depend on the properties of the external

forces. Some of the external forces for the atmosphere are incoming solar radiation,

sea surface temperature (SST), and CO

concentration. Prediction of the second

kind, statistical prediction, attempts to predict how the structure of the attractor

and its resulting statistics respond to changes in the forcing.

1 NONLINEAR DYNAMICS AND CHAOS

Dynamical systems theory discusses the time evolution of a physical system in terms

of its phase space, which is deﬁned by the collection of all the variables needed to

determine the state of the system at a given time (Lichtenberg and Lieberman, 1992).

A single state is then a point in phase space. For a simple pendulum, the state is

deﬁned by its position and velocity, and its phase space is two dimensional. The state

of the atmosphere is deﬁned by the temperature, wind velocity, humidity, etc. at

every point in space. Determining the appropriate state variables of a complex

system is often extremely difﬁcult. As a continuous dynamical system evolves in

time, the state changes continuously, and the point representing the system’s state

moves through phase space tracing out a one-dimensional curve called the p hase

space trajector y.

If one starts a dissipative dynamical system at an arbitrary initial condition, it will

typically display some transient behavior and then eventually settle down to its long-

term behavior. This long-term behavior is the system’s attractor and can take several

forms. If the system sits at an equilibrium state, such as the pendulum hanging

straight down, then the attractor is a single point in phase space, called a stable

ﬁxed point. If the system undergoes regular oscillations, the attractor is a closed

curve. If the system undergoes quasiperiodic oscillations, i.e., the superposition of

oscillations with irrationally related frequencies, the attractor covers a torus. Motion

on all of these attractors is regular and predictable.

Most nonlinear dynamical systems, however, do not display any of these beha-

viors. First, the attractors are typically fractals, i.e., complex sets with fractional

dimension. Attractors with fractal dimension are said to be strange attractors.

Second, even though the dynamics is restricted to evolve on an attractor, two

nearby states will rapidly separate. Dynamics with this sensitive dependence on

initial conditions is said to be chaotic. While there are examples of chaotic

nonstrange attractors, as well as nonchaotic strange attractors, it seems that dissipa-

tive nonlinear systems typically have chaotic strange attractors and the two words are

often used interchangeably. Sensitive dependence on initial conditions refers to

predictions of the ﬁrst kind, while the properties of the attractor as a whole are

predictions of the second kind.

1 NONLINEAR DYNAMICS AND CHAOS 85