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Figure 6 (a) Antisymmetric (with respect to equator) observed OLR power divided by the background power. The background power is

calculated as a smoothed average of the power of the antisymmetric and symmetric components. The power is calculated for the 1979 to

1996 period and summed between 15 S and 15 N. Contour interval is 0.1, and shading begins at a value of 1.1 for which the spectral

signatures are statistically signiﬁcant above the background at the 95% level (based on 500 degrees of freedom). Contours less than 1.0 are

dashed. Superimposed are the dispersion curves of the even meridional mode-numbered equatorial waves for the three equivalent depths of

h ¼12, 25, and 50 m. (b) Same as (a) except for the symmetric component of OLR, and the corresponding odd meridional mode-numbered

equatorial waves. Frequency spectral bandwidth is 1=96 cpd (Wheeler and Kiladis, 1999).

(ER) waves, mixed Rossby gravity (MRG) waves, and westward inertiogravity

(WIG) waves. Eastward inertiogravity (EIG) waves, along with tropical depression

(TD) type and the MJO are also depicted. The mode with the most power is the

MJO, which is unique in these spectra since it does not correspond to any particular

shallow-water mode. It has most of its power in the eastward-propagating symmetric

zonal wavenumbers 1 through 6 (Fig. 6b), with a substantial contribution to the

zonal mean as well. There is also some MJO power in the same region of the

antisymmetric spectrum in Figure 6a .

The periodic and apparently linear appearance of these waves would imply that

they are predictable for a lead time of about half their period. Such relationships can

be applied to predict tropical rainfall variability on the submonthly time scale using

knowledge of the base state provided by the MJO (Lo and Hendon, 2000). The MJO

itself can provide forecast information concerning the onset of ENSO or swings in

the TBO, while the interannual time scale base states affect MJO and submonthly

convection in the tropics in a continuum of upscale and downscale interactions

involving tropical dynamics (Meehl et al., 2001).

REFERENCES

Bjerknes, J. (1969). Atmospheric teleconnections from the equatorial Paciﬁc, Mon. Wea. Rev.

97, 163–172.

Kessler, W. S., M. J. McPhaden, and K. M. Weickmann (1995). Forcing of intraseasonal Kelvin

waves in the equatorial Paciﬁc, J. Geophys. Res. 100, 10,613–10,631.

Krishnamurti, T. N. (1971). Tropical east-west circulations during the northern summer,

J. Atmos. Sci. 28, 1342–1347.

Lo, F., and H. H. Hendon (2000). Empirical extended-range prediction of the Madden-Julian

Oscillation, Mon. Wea. Rev. 128, 2528–2543.

Madden, R. A., and P. R. Julian (1972). Description of global-scale circulation cells in the

tropics with a 40–50 day period, J. Atmos. Sci. 29, 1109–1123.

Matsuno, T. (1966). Quasi-geostrophic motions in the equatorial area, J. Meteorol. Soc. Jpn. 44,

25–43.

McPhaden, M. (1999). Genesis and evolution of the 1997–98 El Nin

o, Science 283, 950–954.

Meehl, G. A. (1997). The South Asian monsoon and the tropospheric biennial oscillation,

J. Climate 10, 1921–1943.

Meehl, G. A., and J. M. Arblaster (2002). The tropospheric biennial oscillation and Asian-

Australian monsoon rainfall, J. Climate 15, 722–744.

Meehl, G. A., R. Lukas, G. N. Kiladis, M. Wheeler, A. Matthews, and K. M. Weickmann

(2001). A conceptual framework for time and space scale interactions in the climate system,

Clim. Dyn. 17, 753–775.

Rasmusson, E. M., and T. H. Carpenter (1982). Variations in tropical sea surface temperature

and surface wind ﬁelds associated with the southern oscillation=EL Nin˜o, Mon. Wea. Rev.

110, 354–384.

REFERENCES 67

Webster, P. J., V. O. Magana, T. N. Palmer, J. Shukla, R. A. Tomas, M. Yanai, and T. Yasunari

(1998). Monsoons: Processes, predictability, and the prospects for prediction, J. Geophys.

Res. 103, 14,451–14,510.

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.

DYNAMICS OF THE TROPICAL ATMOSPHERE

CHAPTER 6

TURBULENCE

JACKSON R. HERRING

1 TWO-DIMENSIONAL AND QUASI-GEOSTROPHIC TURBULENCE

Large-scale motions of Earth’s atmosphere are perforce nearly two dimensional since

the atmosphere’s thickness is thin compared to Earth’s radius. However, more

important dynamical considerations also dictate near two dimensionality: Flows

that are both stable against convection, and rapidly rotating, exhibit strong two-

dimensional motion regardless of their vertical dimensions. The atmosphere, on

average, meets these conditions and it is this latter constraint that determines the

dynamic natur e of its approximate two dimensionality. To illustrate this point, we

consider a simple model consisting of a thin layer of incompressible ﬂuid, subjected

to the gravitational force g and rotation V. The equations of motion for such a ﬂuid

are the Boussinesq approximation to the Navier–Stokes equations:

þ u  Hgu ¼

ggaT  2V u  Hp þ nH

u ð1Þ

þ u  Hgy ¼

bbw þ H uy þ kH

y ð2Þ

H  u ¼ 0 ð3Þ

In Eq. (1), u ¼(u, v, w) denotes the velocity, with w the velocity component

parallel to Earth’s radius, u the eastward component, and v the northern. The

temperature T is split into its horizontal average T

, and a ﬂuctuation about that

average y, with b

dT

=dz; a is the coefﬁcient of thermal expansion of air

( ¼1=T ). We consider these equations in a Cartesian coordinate frame tangential

to Earth’s sphere at a latitude p=2 7 W. Here, dp is the deviation of the press ure ﬁeld

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.

from that necessary to provide a balance with g ravitational and centrifugal forces. Its

role in Eq. (1) is to preserve Eq. (3).

Consider now Eqs. (1) to (3) in the limit of large g and V. Then under suitable

conditions [so as to neglect breaking gravity waves, and hence H 

uy in (2)] the ﬂow

separates into two nearly noninteracting components. The ﬁrst is comprised of

rapidly ﬂuctuating gravity and inertial waves, and the second is a slowly evolving

component in which terms in (1) 

gg and O nearly cancel. This separation is much

analogous to the separation of acoustics and incompressible ﬂow for low Mach

numbers. It is the second ‘‘balanced’’ part that constitutes the focus of this chapter.

For it, Eqs. (1) to (3) may be replaced by a simpler set, the quasi-geostrophic system

(hereafter indicated by QG), whose derivation we now sketch. The QG equations do

not contain gravity waves or other rapid ﬂuctuations, as do Eqs. (1) to (3), hence

their simplicity. Since the right-hand side of (1) contai ns individual terms (each of

which is large), we may expect, in the balanced state, a cancellation among the

largest of these. The equation for vorticity, o ¼ H u follows from (1) as

o ¼u  Ho þ o  Hu þ 2ð V  HÞu  g  Hy  bv þ nH

o ð4Þ

An independent constraint balancing the effects of rotating and gravity may be

formed from the evolution equation for g( H o):

þ2ðV  HÞo  agH

y ¼ 0 þ ð5Þ

Here b ¼ 2V sin y=R, with R the earth’s radius, and y the polar angle.

In Eq. (5), H

 @

þ @

, and þ  indicates that only leading-order terms are

recorded. Anticipating that u is nearly two dimensional, but with a possible z

dependency, we represented it by a stream function, c(x, y, z): (u, v) ¼(@

)c(x, y, z). In terms of c, Eq. (5) simpliﬁes to

agy ¼ 2O@

c þ ð6Þ

The evolution equations for QG may now be obtained by eliminating (@

w) between

(2) and the z component of (4):

ð@

þ u  HÞ o 

bbg

()

¼bv ð7Þ

In Eq. (7), we omit the viscous contribution (n) from (4). In deriving (7), we

neglect ðO

þ O

Þo as compared to O

o since the vertical extent of the

atmosphere is much smaller than its horizontal extent. The divergence of Eq. (1)

and Eq. (6) may be used to relate the stream function to the pressure dp,

dp ¼ 2Oc þ ð8Þ

70 TURBULENCE

For Eq. (7) to be a reasonable approximation, it is clear [from Eq. (4)] that the term

proportional to O on the right-hand-side of (4) should dominate the term: o Hu.

Such dominance is indicated by the smallness of the Rossby number:

¼joj=O juj=ðO‘Þð9Þ

where ‘ estimates the length scale of typical horizontal gradients of the ﬂow. For a

more comprehensive statement of conditions under which the quasi-geostrophic

equations are valid; see Pedlosky (1987).

We next make a few comments concerning the physics of the quasi-geostrophic

approximation, conﬁning our remarks to the case of inviscid ﬂows, with b ¼0.*

First note that (7) simply states that the quantity

Q  o 

abg

c þ by



; o ¼H

c ð10Þ

is conserved along streamlines. Q is the potential vorticity, and for ﬂows conﬁned to

a given spatial volume, both it and an associated kinetic energy,

E ¼

dr ðH

cÞ

bbg



()

; and EðzÞ

dxdy Q

ð11a;bÞ

are conserved. U denotes the volume to which the ﬂow is conﬁned, which, for

simplicity, we take as a recta ngular box, L

, L

. E is called the total enstrophy,

and Qðx; y; zÞ

the enstrophy density. In Eq. (11a,b), H

¼ @

þ @

Flows described by (7) are known to be subject to a variety of instabilities, which

lead to intrinsic temporal ﬂuctuations in Q. We shall not discuss these instabilities

here, except to remark that they lead, through the effects of nonlinearities, to the

existence in the ﬂow of eddies with a broad distribution of sizes. A convenient

description of this distribution stems from a Fourier analysis of Q(r):

QðrÞ¼

expðik  rÞQðkÞð12Þ

where k ¼2p (n=L

, m=L

, l=L

), and (n, m, l ) range over all positive integers.

A basic question in the study of turbulent ﬂows is the distribution of Q among

available modes k. The simples t indicators of this distribution are moments

hEðkÞi  ðk

þð4O

=ða

bbgÞÞk

ÞhjcðkÞj

EðkÞhjQðkÞj

*Two-dimensional turbulence with b 6¼0 (Rossby wave turbulence) is important in the study on large-scale

planetary ﬂows. Statistical studies of the sort described here were initiated by Holloway and Hendershott

(1977). For recent interesting advances, see Galperin et al. (2001).

1 TWO-DIMENSIONAL AND QUASI-GEOSTROPHIC TURBULENCE 71

where the angular brackets denote ensemble averages.

We now explore brieﬂy the implications of the inviscid constraints E and E

constants of motion for QG ﬂow, conﬁning our attention to the case b ¼0 in (10).

For the vertical variability, we take a simple form

cðr; z; tÞ¼

ðc

þ c

Þþ

ðc

 c

Þcosðpz=LÞð13Þ

Here r ¼(x,y). Evaluating (7) at z ¼(0,L) with (13) gives

ð@

þ u

 HÞH

 eð1Þ

ðc

 c



¼ 0; i ¼ð1; 2Þð14a,bÞ

Here, u

¼ð@

Þc

, etc., and

e ¼ 2O

=ða

bbgL

Þ1=L

ð14cÞ

In (14c), L

is the Rossby radius of deformation, which for the atmosphere is

400 km. Note that c

c

is a solution to (14). Such ﬂows are barotropic, and

strictly two dimensional. Flows for which c

7 c

6¼0 are designated as baroclinic

since in that case the pressure gradient is inclined to g, according to (8).

{

Our

discussion now focuses on the qualitative behavior of the ﬂow at horizontal scales

above and below L

. For Eqs. (14a) and (14b), conservation laws (11a) and (11b)

specialize to

¼ 0; J

ðkÞj

¼ 0; ði ¼ 1; 2Þð15aÞ

I¼0; I¼

þ u

þ eðc

 c

gð15bÞ

where

ðk Þ¼k

 e ð 1Þ

ðc

 c

Þ i ¼ 1; 2; u

¼ k

ð15cÞ

It is of interest to ask what distribution in k emerges as the most probable having

given values of total energy, I, and total enstrophy, J

þJ

. Such a distribution

ﬁxes hjc

ðkÞj

i, as well as h c

ðkÞc

ðkÞi.IfIðc

; c

Þ and J

ðc

; c

ÞþJ

ðc

; c

are, separately, constants of motion, then so is their linear combination,

{

Equations (14a) and (14b) have more respectability than just a simple vertical collocation of (7). They

describe exactly the behavior of two immiscible ﬂuids (the lighter above) under QG conditions.

72 TURBULENCE



aaIþ



bbðJ

þJ

Þ, where



aa, and



bb are arbitrary constants. We may diagonalize the

quadratic form



aaIþ



bbðJ

þJ

Þ by a change of variables:

ﬃﬃﬃ

ðf

þ f

Þ; c

ﬃﬃﬃ

ðf

 f

Þð16Þ

In terms of f

; f

2ð



aaIþ



bbðJ

þJ

ÞÞ ¼

aðkÞjf

þðaðkÞþdðkÞÞjf

ð17Þ

where

aðkÞ¼



aak



bbk

; dðkÞ¼2eð



aa þ 2eðk

þ eÞÞ ð18Þ

Here, f

is the barotropic part of the ﬂow, and f

, the baroclinic. According to

standard thermodynamics, the most probable distribution of c

ðk Þ; c

ðk Þ should be

a negative exponential of



aaIþ



bbðJ

þJ

Þ, with



aa;



bb adjusted so that the invariants

I and J

þJ

have prescribed values. This yields;

hjf

iðk Þ

aðkÞ

; hjf

iðkÞ

aðkÞþdðkÞ

ð19Þ

hjc

ðk Þj

i¼hjc

ðk Þj

i

aðkÞ

aðkÞþdðkÞ

ð20Þ

and,



ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

hjc

ihjc

dðkÞ

2aðkÞþdðkÞ

ð21Þ

Thus, the ﬂow approaches maximum two dimensionality at large scales

ðkÞ!1, k ! 0]. As k increases beyond

ﬃﬃ

, levels 1 and 2 becoming increas-

ingly decorrelated so that R

ðkÞ!4e

,ask=

ﬃﬃ

!1.

{

Distributions (19) to (21) hold for inviscid ﬂows conﬁned to a ﬁnite wavenumber

domain and have no energy or enstrophy transfer among scales of differing sizes.

They represent an end state toward which nonlinear interactions impel the system,

but which molecular dissipation prevents.

{

The analysis sketched here has been given more substantial form by Salmon et al. (1978). For an

application of equipartitioning ideas to ocean basin circulation, see Holloway (1992) and Griffa and

Salmon (1989).

1 TWO-DIMENSIONAL AND QUASI-GEOSTROPHIC TURBULENCE 73

2 ENERGY AND ENSTROPHY CASCADE

We turn next to dissipative ﬂows an d the scale-size distribution of two-dimensional

ﬂows. For simplicity, we focus on barotropic ﬂows, however much of the discussion

may be made for baroclinic ﬂows as well. We again ignore the variation of Coriolis

force with latitude.

Although equipartitioning arguments of the last section are qualitatively instruc-

tive, they are unable to incorporate dissipative effects, which are essential in deter-

mining the scale -size distribution of the ﬂow. In atmospheric ﬂows molecular

dissipation (viscous and diffusive effects) are centered at very small scales, well

below the range of most meteorological interest. However, even if dissipation is

negligible in a given spectral range of k, its presence at very small scales induces

a ﬂux of energy (or enstrophy, for QG ﬂow) which is vital in determining scale-size

distributions, such as that for E(k), or Q(k). Thus, for the viscous form of (4), the

principal range of interest is that for which inertial forces ðu  HoÞ far exceed

dissipation ðnH

oÞ, so that

¼ju  Hoj=ðnjH

ojÞ  1 ð21Þ

This equation deﬁnes a Reynolds number. It is clear that its value depends on

specifying a length scale to estimate the gradient operator.

In three-dimensional ﬂows, it is well known that as R

!1dissipation acts to

signiﬁcantly reduce the total energy:

EðtÞ¼

dkEðk; tÞ

on a time scale ð

ﬃﬃﬃﬃ

=LÞ

1

, where L pertains to the energy containing range (that

region in wavenumber that contributes most signiﬁcantly to E). Thus, the decrease of

total energy is independent of n. The underlying physics is that of a cascade. If we

discretize wavenumber space in bins Dk

, in which Dk

contains most of the total

energy, and ðDk

iþ1

=Dk

Þ¼2, then energy is passed from a given bin to its right-hand

neighbor at a rate independent of i, until a value of k is reached for which

ð1=kÞ1. The range over which this constant ﬂux is maintained is called the

inertial range, and the range for which R

ðkÞ1, the dissipation range. The wave-

number distribution of energy in the inertial range is wholly calculable from the

constant ﬂux of energy, F, and the decay time scale noted above. Thus, the energy

within Dk

 k

Eðk

Þ, and its residency time is 1=k

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Eðk

, so that

F ½k

Eðk

Þðk

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

Eðk

Þk

,or

EðkÞ¼CF

2=3

5=3

ð22Þ

74 TURBULENCE

A central question in turbulence theory is to determine the equation of motion of

Eðk; tÞ: We write this evolution equation in the form

EEðk; tÞ¼T ðk; tÞ2nk

Eðk; tÞþFðkÞð23Þ

where, for simplicity, we assume isotropy. In (23) FðkÞ is a possible external forcing

function. The energy transfer function, Tðk; tÞ represents the effects of the non-

linearity and is a funct ional of the energy spectrum, during the entire history of

the ﬂow. The idea of a constant of energy ﬂux noted above for the inertial range may

be more formal ly put in terms of (22) as

F¼

Tðp; t Þdp ð24Þ

where k is within the inertial range. Energy conservation for inviscid ﬂows is

expressed by

dk Tðk; tÞ¼0 ð25Þ

We now discuss how ideas of cascade apply to two-dimensional and QG ﬂows,

for which both the energy and enstrophy are inviscid constants of motion. In this

case, we must have in addition to (25),

dp p

Tðp; t Þ¼0 ð26Þ

We illustrate the physics of energy and enstrophy transfer by means of a simple

model of Tðk; tÞ, originally proposed by Leith (1967) and modiﬁed by Pouquet et al.

(1975). Leith proposed that T be modeled as a diffusion in wavenumber space, with

a diffusion rate ﬁxed by the large-scale strain ﬁeld. It seems a plausible suggestion,

and if we assume the straining ﬁeld acts rapidly, such approximation follows in a

nearly exact manner, if we assume the large-scale strain is independent of the

vorticity upon which it acts (Kraichnan, 1968). The form for T( k)is

TðkÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

dp EðpÞ

EðkÞ

;

ð27Þ

We note that both

dk TðkÞ, and

dk TðkÞ vanish, so that for n ¼ 0 Eq. (27)

conserves total energy and total enstrophy. In our discussion of three-dimensional

turbulence, we saw that at large R

, a cascade, together with a constant ﬂux, was

associated with a constant of motion (i.e., energy). The question we now address is

how does the existence of two constants of motion translate into possible inertial

ranges (with associated constant ﬂuxes) for energy and enstrophy? The use of

2 ENERGY AND ENSTRO PHY CASCADE 75