Environmental Fluid Mechanics - Chapter 9 pot

2.9.26, involves a balance between the pressure and Coriolis terms in the equations of motion, that is,... nor-Figure 9.2 Surface tilt as a result of geostrophic balance flow is into the

of characteristic velocity to the product of characteristic length and rotationrate This is derived from the ratio of the relative magnitude of the nonlinearacceleration terms to the Coriolis terms in the equations of motion, as shown

in Sec 2.9 Here we modify this definition slightly to be more consistent withthe literature in this field, using

Ro D U

where Ro D Rossby number, U D characteristic velocity, L D characteristic

length, and f D Coriolis parameter, or planetary vorticity D20sin ,where 0 is the angular rotation rate of the earth and is the latitude Themagnitude of f varies between 1.45 ð 10
4 s
1 at the poles to zero at theequator Referring back to Table 2.1, Rossby numbers that are sufficientlysmall that rotation effects become important are associated with large lakes,estuaries, coastal regions, and oceanic currents Atmospheric motions alsoare subject to Coriolis effects, but the present discussion focuses on aqueoussystems

In addition to the inclusion of the Coriolis terms, an interesting feature

of the analysis of fluid motions in very large systems is the relative portance of solid boundaries This sometimes poses difficulties in specifyingboundary conditions, since the location of the boundaries is not well-defined

unim-The only clear boundary in the deep oceans, for instance, is the air/water face Boundaries do become important, however, in developing descriptions

inter-of general circulation

The general equations of motion for geophysical flows consist of the continuityand Navier–Stokes equations for incompressible flow introduced inChap 2.For convenience, these are repeated here:

whereV D u,v, wis the velocity vector,is the rotation rate of the earth, g

is gravity, p is pressure,
0is a reference density, and is kinematic viscosity.Usually the main concern is with horizontal or two-dimensional motions, sothat w will usually be assumed to be zero for the present discussion With

w D0, Eq (9.2.2) in component form appears as

9.2.1 Geostrophic Balance

Geostrophic flow, as introduced in Eq (2.9.26), involves a balance between

the pressure and Coriolis terms in the equations of motion, that is,

that neglect the nonlinear accelerations are sometimes referred to as Ekman

models They are mostly appropriate for relatively small values of Ro, which

as noted previously expresses the ratio of the magnitudes of the accelerationand Coriolis terms It should be noted that the geostrophic balance is notvalid near the equator, within a latitude of about š3°, where f becomes verysmall

An interesting result of Eq (9.2.6) is that the flow direction is dicular to the pressure gradient Therefore, on a weather map, isobars areapproximately the same as streamlines of the flow, and the streamlines arelines of constant pressure Also, the quantity p/f
0 can be regarded as

perpen-a streperpen-am function Figure 9.1 shows a schematic description of flow along

an isobar in the northern hemisphere, around centers of high and low sure The Coriolis force and the pressure gradient are colinear, with oppositedirections The velocity vector is perpendicular to those vectors and creates acounterclockwise angle of 90° with the Coriolis force In the southern hemi-sphere, the velocity acts 90°to the left of the Coriolis force, due to the oppositesense of rotation

pres-Figure 9.1 Relationship between isobars and streamlines in atmospheric flows thern hemisphere).

(nor-Figure 9.2 Surface tilt as a result of geostrophic balance (flow is into the page).

Another way of looking at the geostrophic balance is to consider ahomogeneous fluid with a surface at an angle to the horizontal direction, asshown inFig 9.2.The pressure along the surface is the atmospheric pressure,

pa The acceleration generated as a result of this angle is

where the negative sign arises because x is positive to the right If the fluid

is moving at velocity U into the plane ofFig 9.2,there will be a horizontal

component of the Coriolis acceleration with magnitude fU in the positive x

direction (northern hemishere) If these two accelerations are in balance, then

gtan D fU ) U D gtan

This gives the expected geostrophic velocity, for a given surface slope or,conversely, the expected surface slope for a given flow velocity Note thatthe flow is in a direction normal to the pressure gradient, as was shown in

Eq (9.2.6) Recall that in order for the balance that leads to Eq (9.2.8) toexist, the flow must be uniform and in a straight direction, since no otheraccelerations are assumed than the pressure gradient and Coriolis terms

It is somewhat surprising to consider the magnitude of the sea surfacetilt angle that corresponds to expected velocities in the ocean For example,

if U D 1 m/s, and we assume a latitude of 45°, then tan ¾D 10
5, or about

1 cm/km This is much too small to be measureable However, measurements

in a stratified ocean are much easier

There is almost always some density stratification in the oceans, due

to temperature or salinity variations or both Issues related to stratificationare discussed inChap 13, but for now consider that the stratification can beidealized as a two-layer system as sketched in Fig 9.3, which shows a fluid

of density
1 flowing over a stagnant layer of density
2 Since fluid 2 is atrest, its free surface must be horizontal, while the free surface of fluid 1 istilted due to its motion, making an angle with the horizontal direction, aspreviously described Consider that the interface between the two fluids lies

at some angle i, as shown in the figure

Figure 9.3 Surface and interface tilt for geostrophic flow in a two-layer ocean.

Along any horizontal line drawn in fluid 2, the pressure must be constant.For such a line drawn at depth h, the pressure is given by the hydrostaticrelation

Now, with x drawn so that x D 0 at the point at which the interface betweenthe two fluids meets the surface, the pressure along the line drawn at depth hcan be written as

phD paC gf
1tan
tan ix C
2h C xtan ig 9.2.10Equating Eqs (9.2.9) and (9.2.10) then results in

tan iD
tan
1

This shows that the slope of the interface can have a much greater magnitudethan the surface slope, depending on the relative values of
1 and
2 Forexample, if fresh water (specific gravity D 1) flows over sea water (specificgravity D 1.025), then the interface slope is approximately 40 times as great

as the surface slope In most cases the density difference is less than this, sothat the interface slope would likely be even greater

When surfaces of constant density are parallel to surfaces of constant

pressure, the system is said to be in a barotropic state When these surfaces intersect, the field is baroclinic It is possible for a barotropic system to be

statically stable, but in a baroclinic system there must be motion The layer ocean considered above is an example of a baroclinic field, since motion

two-was required to generate the Coriolis force to oppose the pressure gradientforce.

The geostrophic flow that results from a horizontal density gradient is

also called a thermal wind This terminology stems from the usual situation

in which the density differences are generated as a result of temperature ations When there is a horizontal density gradient, the geostrophic flow alsodevelops a vertical shear This can be seen by considering the system shown

vari-inFig 9.4, which shows several contours of constant density and contours ofcontant pressure Assuming that ∂
/∂x < 0, then the density along section 1

is greater than that along section 2 In order to maintain hydrostatic rium, the weight of columns υz1 and υz2 must be equal Therefore the intervalbetween the two isobars increases with x, or υz1 < υz2 The isobars, as shown

equilib-inFig 9.4,then must be consistent with ∂p/∂x > 0, and their slope increaseswith increasing z Following the same arguments as before (coming from thegeostrophic balance), the thermal wind is thus seen to be into the plane ofFig 9.4,and its magnitude increases with z

This phenomenon is clearly demonstrated using Eq (9.2.6), along withthe hydrostatic balance equation in the z-direction Differentiating equation

Figure 9.4 Baroclinic field, showing several contours of constant density and sure.

pres-(9.2.6a) with respect to z and using Eq (9.2.5) to substitute for the verticalpressure gradient, we obtain

These equations are called the thermal wind equations They provide the

vertical variation of velocities from measurements of the horizontal ture (density) gradients The thermal wind, as indicated inFig 9.4,is associ-ated with systems in which surfaces of constant pressure and constant densityintersect, i.e., the baroclinic case

tempera-9.2.2 Potential Vorticity

Another concept useful in the study of large-scale flows is that of conservation ofpotential vorticity This is demonstrated by first writing the momentum equationsfor horizontal motion, neglecting the friction terms From Eq (9.2.3),

where (D/Dt) is taken here as the two-dimensional or horizontal material

derivative operator We now differentiate Eq (9.2.13) with respect to y andsubtract the result from the derivative of Eq (9.2.14) with respect to x, givingD

Eq (9.2.15) may be rewritten as

v∂f

∂y D Df

Substituting Eq (9.2.16), and rearranging the terms of Eq (9.2.15) thenleads to

DDt

The sum of the relative vorticity  and the planetary vorticity f is called

the absolute vorticity Equation (9.2.19) states that the absolute vorticity is

conserved, following a fluid particle along its path line

If the flow field is required to satisfy the full continuity constraint, then

The right-hand side of this equation can be related to the rate of stretching of

a column of fluid of thickness H, by

∂w

∂z D 1H

DH

Dividing both sides by H, we obtain

1H



 C fH



where the quantity  C f/H is called the potential vorticity This last result

shows that, for frictionless incompressible flow, potential vorticity is conservedfollowing a fluid particle For steady flows the particle paths are the same asthe streamlines, so potential vorticity is thus conserved along streamlines Use

of this concept, along with geostrophic flow assumptions and other results

such as the Bernoulli equation (Chap 2) has formed the basis for a number

of theoretical models of ocean currents

A number of laboratory experiments have been performed, usually usingrotating tables, to simulate different aspects of low-Rossby-number flows.One of the more interesting experiments of this type involves simulation ofgeostrophic flow of a homogeneous fluid and produces direct observations ofTaylor columns, as explained below This experiment involves a tank of fluidthat is rotated at a steady angular speed  The rotation speed is sufficientlyhigh that the Coriolis force is much larger than the acceleration terms, andconditions of geostrophic equilibrium may be assumed

In regions that are not affected by the friction induced by the aries, the equations for geostrophic equilibrium in the horizontal directions,and hydrostatic conditions in the vertical direction, are given by Eqs (9.2.6a),(9.2.6b), and (9.2.5), respectively Note, however, that f D 2 for the condi-tions of the experiment By differentiating Eq (9.2.6a) with respect to y and

bound-Eq (9.2.6b) with respect to x and subtracting, we find

Since there is no vertical motion, the angular velocity vector is oriented

in the z-direction Equations (9.3.2) and (9.3.3) indicate that the velocity vectordoes not vary with z, so we may conclude that steady, slow motions in arotating, homogeneous, inviscid fluid are two-dimensional This result is called

the Taylor–Proudman theorem It was obtained theoretically by Proudman in

1916 Soon afterwards, Taylor proved this theorem using an experimental setup

as sketched inFig 9.5 A tank full of fluid was rotating as a solid body Asmall cylinder was slowly dragged along the bottom of the tank and dye wasreleased at point A, above the cylinder and slightly ahead of it The thread

Figure 9.5 Schematic diagram of Taylor’s experiment.

of dye divided at the point S, while appearing to belong to a column of fluidextending over the depth of the cylinder This column of fluid is called a

Taylor column Taylor’s experiment indicated that bodies moving slowly in

a strongly rotating system of homogeneous fluid carry along their motion in

a two-dimensional column of fluid

9.4 WIND-DRIVEN CURRENTS (EKMAN LAYER)

We have already considered the special case of geostrophic flow, resultingfrom a balance between the Coriolis and pressure terms in the momentumequation In that case, it was shown that the velocity vector is perpendicular

to the pressure gradient, so that the velocity follows the isobars This balancecan only exist in situations where other factors such as the acceleration andfriction terms are negligible, and therefore these flows generally occur in theupper atmosphere or deep oceans

Another interesting effect of rotation can be seen on wind-driven

cur-rents, where the velocity field is formed into the so-called Ekman spiral.

Consider a flow far from any boundaries, with a wind stress acting on thesurface and with negligible pressure gradients Here we again consider steadyhorizontal flow, in a system with z D 0 at the surface and increasing down-ward Neglecting also the acceleration terms (i.e., the Ekman model assump-tion) and assuming that the only friction effect is from vertical shear stresses,the momentum equations for the two horizontal velocity components are

This equation has a solution,

By separating the real and imaginary parts (according to the definition

of ), the velocity components are found from

u D C1exp



zp2

cos



zp

cos



zp

2C C4



9.4.9and

vD C1exp



zp2

sin



zp

sin



zp

2C C4



9.4.10

where C3 and C4 are constant phase shifts For illustration, it is assumed that

the x–y plane is oriented so that wind blows in the positive y direction The

boundary conditions include vanishing velocities when z becomes very large

z ! 1, shear stress in the x-direction is zero at the surface, i.e., xD 0, at

z D0, and shear stress in the y-direction at the surface is given by

y D yjzD0 D



Advdz

Since both velocities vanish for large z, we conclude that C1 D 0 Then,

by differentiating Eqs (9.4.9) and (9.4.10) with respect to z, using the shearboundary conditions, we obtain

dudz

To simplify the notation, define a length scale L, so that

L D 

p2

 D

2Af

1/2

9.4.14

Using this definition, the final results for the velocity components are written

by substituting for C2 and C4 into Eqs (9.4.9) and (9.4.10), respectively,giving

Eq (9.4.11), to obtain

dvdz

, vD V0e
sin



34



9.4.18

and the velocity magnitude is V D u2Cv21/2D V0e
, or about 1/23V0.Also, the direction of the velocity at this depth is exactly opposite that ofthe surface velocity For intermediate depths, the magnitude of the velocitydecreases exponentially with increasing depth and its direction turns clockwise,according to Eqs (9.4.15) and (9.4.16).Figure 9.6illustrates this result, which

is known as the Ekman spiral The depth L is considered to represent the layer

depth for which frictional force driven by surface wind shear has an influence

on the motions It is called the Ekman depth As defined in Eq (9.4.14), L is

not a function of surface shear, but it does depend on latitude, approaching

1 at the equator (where f D 0) This presents a problem, for instance, indefining V0 However, Coriolis effects are not important at the equator, sothis is not an issue of practical interest (i.e., the above derivation is not valid

at the equator)

One further observation of interest concerns the mean mass transportassociated with the wind-driven drift currents The mass flux is calculated as

bound-Eq (9. 2.6b)... as indicated inFig 9. 4,is associ-ated with systems in which surfaces of constant pressure and constant densityintersect, i.e., the baroclinic case

tempera -9 . 2.2 Potential Vorticity... paC gf
1tan
tan ix C
2h C xtan ig 9. 2.10Equating Eqs (9. 2 .9) and (9. 2.10) then results in

tan iD
tan
1

This