that the horizontal wavelength is long compared to the vertical
wavelength. The advective term in thc dcnsity equation is writtcn in a
linearized form w(db/dz) = -poN2w/g. Thus the rate of change of dcnsity
at a point is assumed to be due only to the vert

was first derived by Obukhov in 1949. Comparing with Eq. (13.40), it is
apparent that the spectra of both velocity and temperature fluctuations
in the inertial subrange have the same K5i3 form. Th;: spcclrum bcyond
the inertial subrange depends on whether

Fluid Mechanics, Cmhridgc, MA: MIT Prcss. Pwofsky, H. A. and J. A.
Dutlon (1 984). Atmospheric 7lrrbulence. New Yo& Wilcy. fillips. 0. M.
(1977). The Llynumics ofrhe Upper Oceun, London: Cambridge
Univcrsity Ress. Smith: 1,. M. and W. C. Rcynolds (lW2). O

subharmonjc, thcn R, convergcs in a geometric serics with That is, thc
horizontal gap bctween two bifurcation points is about a fifth of the
previous gap. The vcrtical gap betwcen he branchcs of the bifurcation
diagram also dccrcaqes, with each gap about

Because of this slowness the time derivative terms are an order of
magnitude smaller than the Coriolis forces and the pressure gradients in
the horizontal Figure 14.28 Ohscrved hcight (in decamckm) of tbe 50
kF'a prcrsure surface in thc norzhcrn hemisphcr

Sophisticated asymptotic tcchniques are therefore nwded to treat these
boundary layers. Alteinativcl y, solutions can be obtained numerically.
For our purposes, we shall discuss only ccrlain Featurcs of these
calculations. Additional information can be fo

satisfy the continuity equation. Thus a correction to the pressure field is
sought to modify the pressure p"+l = p* + pc, (11.119) I I I I I I I I
Figure 11.5 Staggered grid and different control volumes: (a) around the
pressure or the main variables; (b)

temperature fluctuations Tz can be obtained in a mer identical to that
used for obtaining the turbulcnt kinctic energy. The procedure is
therefore to obtain an equation for DT/Dt by subtracting those Tor
DI/Dt and DT/Dt, and then to multiply the resulting

differcntiating Eq. (14.88) with respect to y, Eq. (14.89) with rcspcct to
x, and subtracting. The pressure is climinated, and we obtain h I
( 14.91) Fsure 14.20 Shallow layer of instanmwus &plh h(x, y3 I).
Following the customary #?-planc approximation,

a rotational gravity wave is no different than that for ordinary gravity
waves, namely oscillatory in the direction of propagation and invariant
in the perpendicular direction. Inertial Motion Considcr the limit o + f,
that is when the particle paths are

reprcsents he direct viscous dkvipofion of mean kinetic mergy. Thc
energy is lost to thc agency that generates the viscous stress, and so
reappears as the kinetic energy of molecular modon (hcat). The fifth
term is analogous to the fourth tcrm. It can be

density ficld across the point. Thc continuity equation can be satisfied by
defining a streamfunction through a* a* az ax u=-, w = -. Equations
(1 2.52) and (1 2.53) thcn become (12.54) where subscripts denote
parlial derivutives. As the coefficients of Q

(,13.18)andtakingtheaverageoftheequations.Thethreeequationstransform
as follows. Mean Continuity Equation Avcraging the continuity equation
(13.1 7), we obtain where we have used thc commutation rule (13.5).
Using Ui = 0, we obtain (1 3.20) which is the c

real. The corresponding velocity components can be found by
multiplying Eq. (14.80) by exp(ikx - iwt) and taking the real part of both
sides. This gives 4 fri u = - COS(kx - ut), kH kH v = - sin(kx - ut).
(14.83) To find he particle paths, take x = 0 and

parl or thc wall layer in thc rangc S -= J+ e 70 is not a(: all passivc, as
onc might think. In Fact, it is perhaps dynamically thc most active, in
spite of thc fact hat it occupies only about 1 % of thc total thickness of
the boundary laycr. Figure 13.22

briefly here. (1) Trunsition through subharmonic cascude: As R is
increased, a typical nonlinear system develops a limit cycle of a certain
frequency w. With further increase of R, several systems are found to
generate additional hquencies 42, w/4, w/8, .

small-scale eddies. The whole cycle is called bursting, or eruption, and
is essentially an ejection of slower fluid into the flow above. The flow
into which the ejection occurs decelerates, causing a point of inflection
in the profile u(y) (Figure 13.23).

tu the lowest order. The small departures from gcostrophy, however, arc
important because they determine the evolution of the flow with time. We
start with tbc shallow-water potential vorticity equation which can bc
written as We now expand the matcrial d

we obtain 3- Figurc 13.28 Small and large valucs or time on a plot of
the correlatioii runclion. which shows that thc ms displxcment increases
linearly with time and is proportional to the intensity of turbulent
fluctuations in the medium. Belzawinrjur la

shearproduction of turbulence by the interaction of Reynolds stresses
and the mcan shear. The sixth term represents the work done by gravity
on the mean vertical motion. For example, an upward mean motion
results in a loss of mean kinetic energy, which is

critical Reynolds number; for example, for a jct of the form zi =
Usech'(y/L), it is Re, = 4. These wall-he shear flows therefore become
unstable very quickly, and the inviscid criterion hat these flows are
always unstable is a fairly good description. Th

viscous. Y turbulent , \ Y 1- 6 1 Vigure 13.17 Variation orshcslr stress
across a channel and a boundary layer: (a) chwncl; wcl (b) houndary
layer. In a boundary layer on a flat plate there is no pressurc gradient
and the mean flow equation is av av at PU

energy to the shear production is called theJlux Richardson number: -gawT' buoyant destruction -Z(dU/dz) shear production ' Rf = - (1 3.68)
where we have oriented the x-axis in the direction of flow. As the shear
production is positive, the sign of Rf dep

outside the viscous sublayer but still near the wall, the Reynolds stress
can be taken equal to the wall stress pui. This gives which can be written
as (13.66) This intcgrates to u1 - = - Iny + const. us. k In recent years
the mixing length theory has fal

dissipation drains enstrophy out of the system. At later stagcs in the
evolution, thcn, Eq. (1 4.144) may not be a good assumption. However,
it can be shown (see Pedlosky, 1987) that the dissipation of enstrophy
actually inlensi$es thc process of energy t

of mass, so that thcsc two variablcs must always occur togethcr in any
nondimensional pup. The important ratio (13.48) 530 li~uit!Ilm~ has
thc dimension of velocity and is called thefiction velocity. Equation
(13.47) can then be written as u = U(U*, V1 y)

wing tip (Figure 15.1). joined to the main wing by a hinged connection,
as shown in Figure 15.4. They move differentially in the sense that one
moves up while the other moves down. A depressed aileron incrcases the
lift, and a raised aileron decreases the

downswam distances. As the mean field is affected by the Reynolds
stress through thc equations of motion, this means that the various
turbulent quantities (such as Reynolds stress) also must reach selfsimilar states. This is indeed found to be approximate

momentum equations (1 1.1 17) and (1 1.1 1 8). Next, it solves a discrete
Poisson equation (1 1.1 23) for the pressure correction. This pressure
con-ection is then used to modify the prcssure using Eq.
(11.119),andtoupdatcthevelocityatthenewcimcstepusin~E

= U cos ot is itself periodic. 4. Calculate the zero-lag cross-correlation
u(t)u(t) between two periodic series u(t) = cos ut and u(t) = cos (or +
4). For values of q5 = 0, n/4, and n/2, plot the scatter diagrams of u vs u
at different times, as in Figure

breaking up the spanwise coherence of vortex shedding fmm a
cywcalrod. While an eddy on one side is shed, that on the other side
forms, resulting in an unsteady flow near the cylinder. As vortices of
opposite circulations are shed off alternately from the

equivalent temperature. Undcr the Boussinesq approximation, the
equations of motion for the instantaneous variables are ai, - =o, ax, ( 1
3.1 7) (13.18) As in the preceding chapter, we are denoting the
instantaneous quantitics by a tilde (-). Let the vari

the location of column By and integrating over halfthe wavelength. This
is because an interchange of A and B over half a wavclength
automatically forms a complete wavelength of the deformed surface.
The mass of column A is pq dx and the center of gravity

are not orthogonal as the opcrdtors are not self-adjoint. Results for
Poiseuillc pipe flow and compressible blunt body flows arc given. 1.2.
Ciommmts on Aonlinear I@?ch To this point we have discussed only
linear slability theory, which considers infinite

as simply the product of typical velocity and length scales of large
eddies. Consider the thermal convection between two horizontal plates
in air. The walls are separated by a distance L = 3 m, and the lower
layer is warmer by AT = 1 C. The equation of mo