KUNDU Fluid Mechanics 2 Episode 14 pdf

In fact, the continuity equation suggests that the scale of the vertical velocity W is related to that or the horizontal velocity U by u L ' where H is the depth scale and L is the hor

with height is due to the absorption of solar radiation within the upper layer of the occan Thc dcnsity distribution in the ocean is also affected by the salinity However, there is no characteristic variation of salinity with depth, and a decrease with depth

is found to be as common as an increase with depth In most cases, however, the vertical siructurc of density in the ocean is determined mainly by that of temperature, the salinity effects being secondary The upper 50-200m of ocean is well-mixed, due to thc turbulence generated by the wind, waves, current shear, and the convcctive ovcrturning caused by surface cooling The temperature gradients decrease with depth, becoming quite small below a depth of 1500 m There is usually a large temperature gradient in the depth range of 100-500 m This layer of high stability is called the thermocline Figure 14.2 also shows the profilc of buoyancy frequency N, defined by

where p of course stands for the potential density and po is a constant reference density The buoyancy frequency reaches a typical maximum value of Nmax - 0.01 s-l (period

- lOmin) in the thermocline and decreases both upward and downward

In this section we shall review the relevant equations of motion, which are derived and discussed in Chapter 4 The equations of motion for a stratified medium, observed in

a system of coordinates rotating at an angular velocity P with respect to the “ k e d stars,” are

of sound This assumption is very good in the ocean, in which c 2 / g - 200lan In the atmosphere it is less applicable, because c 2 / g - 1Okm Under the Boussinesq approximation, the principle of mass conservation is expressed by V u = 0 In

contrast, the density equation DplDt = 0 follows fromthe nondiffusive heat equation

D T I D t = 0 and an incompressible equation of state of the form Splpo = -cwST (If the density is determined by the concentration S of a constituent, say the water vapor in the atmosphere or the salinity in the ocean, then D p l D t = 0 follows from thc nondflusive conservation equation for the constituent in the form DS/ Dt = 0,

plus the incompressible equation of state Splpo = BSS.)

The equations can be written in tcrms of the pressure and density perturbations from a state of rest In thc abscnce of any motion, suppose the density and pressure have the vertical distributions P(z) and P(z), where the z-axis is taken vertically upward As this state is hydrostatic, we must have

Subtracting the hydrostatic state (14.3), this bccomes

which shows that we can replace p and p in Eq (14.2) by the perturbation quantities

pl and p'

Formulation of the Frictional Term

The friction Iorce per unit mass F in Eq (14.2) needs to be related to the velocity field From Chapter 4, Section 7, the friction force is givcn by

wherc t i j is the viscous stress tensor The stress in a laminar flow is caused by thc molecular exchanges of momcntum From Eq (4.41), the viscous stress tensor in an isotropic incompressible medium in laminar flow is given by

In large-scale geophysical flows, however, the frictional E0n.c~ are provided by turbu- lent mixing, and Ihe molecular exchanges are negligible The complexity a€ turbulent behavior makes it impossible to relatc the stress to the velocity field in a simple way

To proceed, then, wc adopt the eddy viscosity hypothesis, assuming that thc turbulent stress is proportional to the velocity gradient field

Geophysical mcdia arc in thc form of shallow stratified layers, in which the

vertical velocities are much smaller than horizontal velocities This means that thc cxchange of momentum across a horizontal surfacc is much weaker than that across a vcrtical surface We expect then that the vedcal eddy viscosity u, is much smallcr than

the horizontal eddy viscosity UH, and we assume that the turbulent stress components have the form

(1 4.5)

The difficulty with set (14.5) is that the exprcssions for txz and tJZ depend on the fluid

rotution in the vertical plane and not just the deformation In Chaptcr 4, Section 10, we

saw hat a requirement for a constitutive equation is that the stresses should be inde- pendent of fluid rotation and should dcpend only on thc deformation Therefom, rxz

should depend only on the combination ( a i r / & + a w / a x ) , whcreas thc expression

in Eq (14.5) depends on both deformation and rotation A tensorially correct gco- physical treatment of the frictional terms is discussed, for example, in Kamenkovich (1967) However, the assumed form (14.5) lcads to a simple fornulation for viscous

effects, as we shall see shortly As the eddy viscosity assumption is of questionable

validity (which Pedlosky (197 1 ) describes as a "rather disreputable and dcsperak atlcmpt"), there does not secm to be any purposc in formulating the stress-strain relation in more complicated ways merely to obey h e requirement of invariance with respcct to rotation

With the assumed form for the turbulent strcss, the components of the frictional

force fi = a t i j / i 1 x j become

Estimates of the eddy cocfficients vary greatly Typical suggcsted values are

v, - 10m2/s and vH - lo5 m2/s for thc lower atmosphere, and u, - 0.01 m2/s

and VH - 100 m2/s for the uppcr ocean In comparison, thc molecular values are

u = 1.5 x m2/s for air and u = 1W6 m2/s for water

4 Appmrimatc! LiipationsJor a Thin Layer on

The atmosphere and the Ocean are very thin layers in which the depth scale of flow

is a few kilometers, whereas the horizontal scale is of the order of hundreds, or even thousands, of kilometers The trajectories of fluid elements are very shallow and the vertical velocities are much smaller than the horizontal vclocities In fact, the continuity equation suggests that the scale of the vertical velocity W is related to that

or the horizontal velocity U by

u L '

where H is the depth scale and L is the horizontal length scale Stratification and

Coriolis effects usually constrain the vertical velocity to be even smaller than U H / L

Large-scale geophysical flow problems should be solved using spherical polar coordinates If, however, the horizontal length scales are much smaller than the radius

of the earth (= 6371 km), then the curvature of the earth can be ignored, and the

motion can be studied by adopting a bcul Cartesian system on a tangent plane

(Figure 14.3) On this plane we take an x y z coordinate system, with x increasing

eastward, y northward, and z upward The corresponding velocity components are u

(eastward), v (northward), and w (upward)

a Rotaling Sphem

-

The earth rotates a1 a rate

!J = 2~ rad/day = 0.73 x s-', around the polar axis, in an counterclockwise sense looking from above the north

pole From Figure 14.3, the components of angular velocity of thc carh in the local

figure 143 Local Cartesian coordinates Thc x-axis is inm h e plane of the pnpcr

to be twice the vertical component of 8 As vorticity is twice the angular velocity,

f is called the pluncrary vorticity More commonly, f is referred to as the Coriolis

purumeier, or thc Curiolisfkequency It is posilivc in the northern hemisphere and

negative in the southern hcmispherc, varying from f1.45 x lo4 s-' at the poles to zero at the equator This makes sense, since a person standing at the north pole spins around himself in an counterclockwise sense at a rate S2, whereas a person standing

at the equator does not spin around himsclf but simply translates The quantity

Ti = 27c/f,

is called the incrtilrlperiud, for reasons that will bc clear in Section I 1

The vertical componcnt of the Coriolis force, namely -2Ru cos 8, is generally negligiblc compared to the dominant terms in the vertical equation of motion, namely

gp'/fi and p;'(ap'/az) Using Eqs (14.6) and (14.7), the equations of motion (14.2)

These are the equations of motion for a thin shell on a rotating earth Note that only

the vertical component of the earth's angular velocity appears as a consequence of thc flatness of the fluid trajectories

f-Plane Model

The Coriolis parameter f = 2S2 sin 0 varics with latitude 0 However, we shall see later that this variation is important only for phenomena having very long timc scales

(several weeks) or very long length scales (thousands of kilometers) For many pur-

poses we can assume f to be a constant, say fo = 2S2 sin&, where & is the central latitude of the region under study A model using a conqtant Coriolis parameter is

called an.f-pZane model

/?-Plane Model

The variation of f with latitude can bc approximatcly represented by expanding .f in

a Taylor series about the central latitude 00:

Here we have neglccted the nonlinear acceleration terms, which are of order U 2 / L ,

in comparison to the Coriolis force -f U (U is the horizontal velocity scale, and L

is the horizonla] length scale.) The ratio of the nonlincar term to thc Coriolis term is callcd the Rossby number:

a latitude belt of f3’), whcre f becomes small It also brcaks down if the frictional cffects or unsteadiness bccome important

Vclocities in a geostrophic flow arc perpcndicular to the horizontal pressure gradient This is becausc Eq (14.11) implies that

(iu + j v ) V p = - Po 1 .f ( -i- E + j- ic) (i$+.i$)=u

Thus, the horizontal velocity is along, and not across, the lines of constant pressure

If f is rcgarded as constant, then thc geostrophic balance (14.1 1) shows that p / f p o

can bc regarded as a smamfunction The isobars on a weather map are therefore nearly the slrcamlines of the flow

Figure 14.4 shows the geostrophic flow around low and high prcssure centers

in thc northern hemisphcre Herc the Coriolis force acts to thc right of the velocity vcctor This requircs the flow to be counterclockwise (viewed from above) around

a low prcssure region and clockwise around a high pressure region The scnse of

circulation is opposite in the southern hemispherc, where the Coriolis force acts to

the left of the velocity vector (Frictional forces bccome important at lower levels in

the atmosphere and rcsult in a flow partially acmss the isobars This will be discussed

in Section 7, where we will see that the Bow around a low pressure center spirals

inwurd due to frictional effects.)

The flow along isobars at first surprises a reader unfamiliar with the cffects

of Ihc Coriolis force A question commonly asked is: How is such a motion seL up?

A typical manner of establishmcnt of such aflow is as follows Consider a horizontally converging flow in thc surface laycr of the occan The convergent flow sets up the

sea surface in the form of a gentle “hill:’ with the sea surfacc dropping away from the ccnter of the hill A fluid particle starting to move down the “hill” is deflected to

the right in the northern hemisphere, and a steady statc is reachcd when thc particle finally movcs along thc isobars

Thermal Wind

In thc presence of a horizontal gradient of density, thc geostrophic velocily devclops

a vertical shear Consider a situation in which the density contours slope downward

Figure 14.4 Gcustrophic flow murid lour and high prcssure centers Thc pressure force ( - V p ) is indi-

cated by a thin wow, and hc Coriolis f m c is indicated by a thick m w

Figure 145

indicated by solid lincs; and contours of constant dcnsiiy tlre indicated bjf dashed lincs

with x, the contours at lower levcls represenling higher density (Figure 14.5) This implies that ijp/ax is negativc, so lhal the density along Section 1 is larger than that along Section 2 Hydrostatic equilibrium requires that thc weights of columns Szr

and Sz2 are equal, so that h e separation across two isobars increases with x, that is

, % e d wind, indicated by heavy m w s pointing into the plane of papcr Isohm arc

8z2 > dz, Consequently, the isobaric surfaces must slope upward with x , with the

slopc increa$ing with height, rcsulting in a positive a p / a x whose magnitude increases

with height Since the geostrophic wind is to thc right of the horizontal pressure force (in the northern hemisphere), it follows that the geostrophic velocity is into the planc

of the paper, and its magnitude increases with height

This is casy to demonstrate from an analysis of the geostrophic and hydrostatic

Eliminating p between Eqs (14.12) and (14.14), and also between Eqs (14.13) and

(1 4.14), we obtain, respectively,

(14.15)

Metcomlogisls call these the thermal wind equations because they give the vertical

variation cd wind from measurements of horizontal tcrnperature gradients The ther-

mal wind is a baroclinic phcnomenon, because the surfaces of constant p and p do not coincide (Figure 14.5)

Taylor-Proudman Theorem

A striking phenomenon occurs in the geosmphic 00w of a homogeneous Ruid It can

only be observed in a laboratory experiment because stratification effects cannot be avoided in natural flows Consider then a laboratory experiment in which a tank of

fluid is steadily rotated at a high angular speed S2 and a solid body is movcd slowly along the bottom of the tank The purpose of making large and the movcment of the solid body slow is to make the Coriolis force much largcr than the acceleration terms, which must be made negligible for geostrophic equilibrium Away from the frictional effects of boundaries, the balancc is therefore geostrophic in the horizonta1 and hydrostatic in the vertical:

It is useful to define an Elanan number as the ratio of viscous to Coriolis forces (per unit volume):

upward extension of the cylinder, and flows around this imaginary cylinder, called the Taylor column Dye releascd from a point B within the Taylor column remained there

and moved with the cylinder The conclusion was that the flow outside h e upward cxtension of the cylinder is the same as if the cylinder extended across the entire water depth and that a column of water directly above the cylinder moves with it The motion is two dimensional, although the solid body does no1 extend across the

e n h e water depth Taylor did a second experiment, in which he dragged a solid body

puraZleZ to the axis of rotation In accordance with awl& = 0, he observed that a column of fluid is pushed ahead The lateral velocity components u and v were zero

In both of these experiments, there are shear layers at the edge of the Taylor column

Figun! 14.6 Taylor's cxperimenr in a shngly r o t a h flow of a homogcncous fluid

I n surnmuiy, Taylor's cxperiment established the following striking fact for steady inviscid motion of homogcncous fluid in a strongly rotating system: Bodies moving either parallcl or perpendicular to the axis of rotation carry along with their motion

a so-called Taylor column of fluid, oriented parallel to the axis The phenomenon is analogous LO lhe horizontal blocking caused by a solid body (say a mountain) in a strongly stratified system, shown i n Figure 7.33

In the preccding section, we discussed a steady h e a r inviscid motion expected to be valid away from frictional boundary layers We shall now examine the motion within frictional layers ovcr horizontal surfaces In viscous flows unaffected by Coriolis

forccs and pressure gradients, the only tcnn which can balancc the viscous force is

either the limc dcrivative au/i)r or the advection u -Vu Thc balance of au/ar and the viscous force givcs rise to a viscous layer whose thickness increases with time,

as in the suddenly accderated plate discusscd in Chapter 9, Section 7 The balance

conditions (14.24) and (14.25) can be combined as p v , ( d V / d z ) = t at z = 0, from whicb Eq (14.28) givcs

t J ( 1 - i )

2PVv -

A =

Substitution of this into EQ (14.28) givcs the vclocity components

Thc Swedish oceanogaphcr Ekman worked out this solution in 1905 The solu-

tion is shown in Figure 14.7 for the case of Ihc northern hemisphere, in which f

is positive The vclocities at various depths are plotted in Figure 14.74 where cach arrow represents the velocity vector at a certain depth Such a plot of L’ vs u is some- times called a “hodograph” plot The vertical distributions of u and u are shown

in Figure 14.7b The hodograph shows that thc surface velocity is dcflected 45‘: to

thc right or Ihc applied wind stress (In the southern hemisphere the dcflection is to

thc left of thc surface strcss.) The vclocity vector rotates clockwise (looking down)

with depth, and the rna,onitude exponentially decays with an e-folding scale of 8 , which is called the Ekman Xuyer thickness Thc tips of the velocity vcctor at various

depths form a spiral, called the E&n~zn spiral

Figure 14.7 Ekman layer a1 a I-nx surlwc The left pancl shows velocity a1 vurious dcpths; values of -z/S are indicalcd along the curve heed out by the tip of Ihc vclocity veckm Thc right panel shows

vcrlical diGtributionu oTu

The components of the volume transport in the Ekman layer arc

This shows that the net transport is to the right of the applied stress and is independent

of LJ, Tn fact, the result $u dz = -t/fr, follows directly from avertical integration of

the equation of motion in the form -pf 1: = d(stress)/dz, so that the result does not

depend on the eddy viscosity assumption Thc fact that the transport is to the right

of the applied stress makes scnse, because then the net (depth-integratcd) Coriolis force, dirccted to the right of the depth-integrated transport, can balance the wind

thc sea surface slopes down to the north, so that there is a pressure force acting north-

w a d throughout the Ekman layer and below (Figure 14.8) This means that at thc bottom of the Ekman Iaycr (z/6 + -XI) there is a geostrophic velocity U to the right of the pressure force The surface Ekman spiral forced by the wind stress joins

smoothly to this geostrophic velocity as z / 6 + -m

F i y e 14.8 Ekman layer at a Free surface in h e presence of a pressure gradient The geostrophic vclwily li~rrcd by Ihc prcssurc gradient is 0

Pure Ekman spirals are not obscrved in the surface layer of the ocean, mainly

because the assumptions of constant eddy viscosity and steadiness are particularly restrictive When the flow is averaged over a few days, however, several instances have been found in which the current does look likc a spiral One such example is shown in Figure 14.9

N

Figurc

-20 0 1 0 c d s

v (crn/s)

4.9 An observed velocity distribution near the coast of Oregon ~ilocity is average lver 7 I ays

Wind s l m s h d a magniludc ol' 1 I dyn/crn2 and was dircclcd narly soulhward, as indicatcd at thc top of the figure Theupper panel shows v d c a l distributions of u and L', and the lowerpwcl shows thc hodogmph

in which dcpths are indicated in meters The hodograph is similar to that of a surface Ekman layer (of

dcplh 16 m ) lying ovcr lhc bollom Fkmao laycr (cxlcndiog liom a dcpth ol' 16 rn 10 tbc ocean bottom)

F? Kundu, in l3oiIom Tubu/mce, I C J Kihoul, cd., Elscvicr, 1977 and rcprinlcd wilh Ihc permission ol'

Jacqucs C J Nihoul

Explanation in Terms of Vortex Tilting

We have seen in prcvious chapters that the thickness of a viscous layer usually grows

in a nomtating flow, either in time or in the direction of flow The Ekman solution,

in contrast, results in a viscous layer that does not grow either in time or space This

can be explained by examining the vorticity equation (Pedlosky, 1987) The vorticity components in the x - and y-directions are

on the left-hand sidcs of Eqs (1 4.3 l), then causes a rate of changc of horizontal component or v0aicit.y hat just cancels the diffusion term

7 Ekman I q v r on a Rigid S u r f i e

Consider now a horizontally independent and steady viscous laycr on a solid surface

in a rotating Bow This can be the atmospheric boundary layer over the solid earth or the boundary layer over the ocean bottom We assume that at large distances from the surface the velocity is toward the x-direction and has a magnitude U Viscous forces

are negligible far from the wall, so that the Coriolis force can be balanced only by a pressure gradicnt:

(1 4.32)

This simply states that the flow outside the viscous layer is in geostrophic balance,

U being the geostrophic vclwity For our assumed case of positive U and f, we

must have d p l d y e 0, so that the pressure falls with y-that is, the pressure force is

directed along the positive y direction, resulting in a geostrophic flow U to the right

of the pressure force in the northern hemisphere The hori7antal prcssure gradient

remains constant within the thin boundary layer

Near lhe solid surface thc viscous forces are important, so that the balance within the boundary layer is

According to Eq (14.41), the tip of the velocity vector describes a spiral for various

values of z (Figure 14.10a) As with the Ekman layer at a free surface, the frictional effects are confined within a layer of lhickncss S = Jm, which increases with v,

and decreases with thc rotation rate f Interestingly, the layer thickness is indcpendent

of the magnitude of the frcc-stream velocity U ; this behavior is quite diffemnt from that: of a steady nonrotating boundary layer on a semi-infinite plate (the Blasius solution of Section 10.5) in which the thickness is proportional to 1 /a

Figure 14.10b shows the vertical distribution of the velocity components Far

from the wall the velocity is cntirely in the x-direction, and the Coriolis force balances the pressure gradient As thc wall is approached, retarding effccts decrease u and the associated Coriolis force, so that thc pressure gradient (which is indcpendent of L)

(a) Hodograph (b) Profiles of u and u

Figure 14.10 Ekman layer at a rigid surtirce The left panel shows velocity vccton at various heights; vdw of z/S are indicated along the curvc trxcd OUL by thc lip or h c vclocity vectors Thc right pancl shows vertical distributions or u and u

forces a component v in the direction of the pressure force Using Eq (14.41), the net transport in the Ekman layer normal to the uniform stream outside the layer is

which is directed to the le# of the free-stream velocity, in the direction of the pressure force

If the atmosphere were in laminar motion, q would be equal to its molecular value for air, and the Ekman layer thickness at a latitude of45O (where f 21 lo4 s-')

would be M 6 - 0.4 m The observed thickness of the atmospheric boundary layer

is of order 1 km, which implies an eddy viscosity of order u,, - 50m2/s In fact,

Taylor (1915) tried to estimate the eddy viscosity by matching the predicted velocity

distributions (14.41) with the observed wind at various heights

The Ekman layer solution on a solid surfacc dcrnonstrates that the three-way balance among the Coriolis force, the pressure force, and the frictional forcc within the boundary layer results in a component of flow directed toward the lower pressure The balance of forces within the boundary layer is illustrated in Figure 14.1 1 The net frictional force on an element is oricntcd approximately opposite to the velocity

vector u It is clear that a balance of forces is possible only if the velocity vcctor has a component from high to low pressure, as shown Frictional forces therefore cause the

flow around a low-pressure center to spiral inward Mass conservation requires that

the inward converging flow should rise over a low-pressure system, resulting in cloud

8 Shallow- Nblcr Equalions

Both surface and internal gravity waves were discussed in Chapter 7 The effect

of planetary rotation was assumed to be small, which is valid if the frequency w

of the wave is much larger than the Coriolis parameter f In this chapter we arc considzring phenomena slow enough for w to be comparable to f Consider surface

gravity waves in a shallow laycr of homogeneous fluid whose mean deph is H I.€ we

restrict ourselves to wavelengths A much larger than H, then the vertical velocities are much smaller than the horizontal velocities In Chapter 7, Section 6 we saw that the acceleration awlat is then negligiblc in the vertical momentum equation, so that the pressure distribution is hydrostatic Wc also demonstrated that the fluid particles execute a horizontal rectilinear motion that is independent of z When the effects

H

Figure 14.12 1-aycr or fluid on a flat bottom

of planetary rotation are included, the horizontal velocity is still depth-independent,

although the particle orbits are no longer rectilinear but elliptic on a horizontal plane,

as we shall SCC in the following section

Consider a layer of fluid over a flat horizontal bottom (Figure 14.12) Let z be

measured upward from the bottom surfacc, and q be the displacement of the free

surface The pressure at height z from the bottom, which is hydrostatic, is given by

The horizontal pressure gradients are therefore

As nulax and av/ay are independent of z, the continuity equation requires that UI

vary linearly with z, from zero at the bottom to the maximum value at the free surface

Integrating vertically across the water column from z = 0 to z = H + q , and noting that u and v are depth independcnt, we obtain

(1 4.43)

where w(q) is the vertical velocity at the surface and w(0) = 0 is the vertical velocity

at the bottom The surface velocity is given by

Drl arl all arl w(q) = - = - + u - + v -

Dt at ax ay

The continuity cquation (14.43) Lhcn becomcs

which can bc written as

the motion of a layer of fluid in which the horizontal scale is much larger than thc depth of the layer Thcse equations will be used in the following sections for studying

various types of gravity waves

Although the preceding analysis has been formulatcd for a layer of homogeneous

fluid, Eqs (14.45) arc applicable to internal wavcs in a stratified medium, if we replaced H by the equivalent depth H,, defined by

9 .!I'ormal Modex in a Cbnlinuuuly S h l i j i e d l m p r

In the prcceding section we considered a homogeneous medium and derived the governing cquations for waves of wavelength larger than the depth of thc fluid layer

Now considcr a continuously stratiEed mcdium and assume that the horizontal scale

of motion is much larger than the vertical scalc The pressure distribution is therefore

hydrostatic, and the equations of motion are

where p and p represent perfurbutions of pmssure and density from the state of

rest The advective term in the density cqusllion is written in the linearized form

w ( d p / d e ) = -poN2w/g, where N(z) is thc buoyancy frequency In this form the rate of change of density at a point is assumcd to be due only to the vertical advection

of the background density distribution p ( z ) , as discussed in Chapter 7, Section 18

In a continuously stratified medium, it is convenient to use thc mcihod of separa- tion of variables and writc q = q n ( x , y, t ) ~ , $ ~ ( z ) for some variable q The solution

is thus written as the sum of various vertical “modcs,” which are called normal modes because they turn out to be orthogonal to each other The vertical structure of a mode is described by @,, and qn describes the horizontal propagation of the mode Although

each mode propagates only horizontally, the sum of a number of modes can also

propagate vertically if the various qn are out of phase

We assume separable solutions of the form

(1 4.53)

(1 4.54)

where the amplitudes u,, v , ~ , p,, w,, and p,, are functions of (xt y, t) The z-axis

is measured from the upper free surface of the fluid layer, and z = -H rcpresents the bottom wall The rearons for assuming the various forms of z-dependence in

Eqs (14.52)+4.54) are the following: Variables u , v , and p have the same ver- tical structure in order to be consistent with Eqs (14.48) and (14.49) Continuity equation (1 4.47) requires that the vertical structure of w should be the integral of

$,, (z) Equation (1 4.50) rcquks that the vertical slructure of p must be thc e-dcrivative

of the vertical structurc of p