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LiteratureIntroduction to Geophysical Fluid

DynamicsPhysical and Numerical Aspects

- Benoit Cushman-Roisin
- Jean-Marie Beckers

Objectives of Geophysical Fluid Dynamics

- Study of naturally occuring large scale flows on

Earth - Although the disciplines encomopasses the motion

of both fluid phases- liquid(waters in the

ocean, molten rock in the outer core) and gases

(air in our atmosphere, atmospheres of other

planets, ionized gases in stars) a restriction

is placed on the scale of these motions - For example, problems related to river flow,

microturbulence in the upper ocean, and

convection in clouds are traditionally viewed as

topics specific to hydrology, oceanography, and

meteorology, respectively - Most of geophysical problems are at the

large-scale end, where either the ambient

rotation (of Earth, planet or star) or density

differences (warm and cold air masses, fresh and

saline waters) or both assume some importance - In this respect, geophysical fluid dynamics

comprises - rotating-stratified fluid dynamics

Importance of geophysical fluid dynamics

- Thanks in large part to advances in geophysical

fluid dynamics, the ability to predict with some

confidence the paths of hurricanes has led to the

establishment of a warning system that, no doubt,

has saved numerous lives at sea and in coastal

areas - Warning systems, however, are only useful if

sufficiently dense observing systems are

implemented, fast prediction capabilities are

available and efficient flow of information is

ensured. A dreadful example of a situation in

which a warning system was not yet adequate to

save lives was the earthquake off Indonesias

Sumatra Island on 26 December 2004. The tsunami

generated by the earthquake was not detected, its

consequences not assessed and authorities not

alerted within the two hours needed for the wave

to reach beaches in the region - On a larger scale, the passage every 3 to 5 years

of an anomalously warm water mass along the

tropical Pacific Ocean and the western coast of

South America, known as the El-Nino event, has

long been blamed for serious ecological damage

and disastrous economical consequences in some

countries. Now, thanks to increased understanding

of long oceanic waves, atmospheric convection,

and natural oscillations in air sea

interactions, scientists have uccessfully removed

the veil of mystery on this complex event, and

numerical models offer reliable predictions with

at least one year of lead time, i.e., there is a

year between the moment the prediction is made

and the time to which it applies.

Importance of geophysical fluid dynamics

- Having acknowledged that our industrial society

is placing a tremendous burden on the planetary

atmosphere and consequently on all of us,

scientists, engineers, and the public are

becoming increasingly concerned about the fate of

pollutants and greenhouse gases dispersed in the

environment and especially about their cumulative

effect - Will the accumulation of greenhouse gases in the

atmosphere lead to global climatic changes that,

in turn, will affect our lives and societies? - What are the various roles played by the oceans

in maintaining our present climate? - What are the various roles played by the oceans

in maintaining our present climate? - Is it possible to reverse the trend toward

depletion of the ozone in the upper atmosphere? - Is it safe to deposit hazardous wastes on the

ocean floor? - Such pressing questions cannot find answers

without, first, an in-depth understanding of

atmospheric and oceanic dynamics and, second, the

development of predictive models.

Hurricane Frances during her passage over Florida

on 5 September 2004. The diameter of the storm is

about 830 km and its top wind speed approaches

200 km per hour

Computer prediction of the path of Hurricane

Frances. The calculations were performed on

Friday 3 September 2004 to predict the hurricane

path and characteristics over the next 5 days

(until Wednesday 8 September). The outline

surrounding the trajectory indicates the level of

uncertainty. Compare the position predicted for

Sunday 5 September with the actual position

Distinguishing attributes of geophysical flows

- Two main ingredients distinguish the discipline

from traditional fluid mechanics the effects of

rotation and those of stratification. The

controlling influence of one, the other, or both

leads to peculiarities exhibited only by

geophysical flows - The presence of an ambient rotation, such as that

due to the earths spin about its axis,

introduces in the equations of motion two

acceleration terms that, in the rotating

framework, can be interpreted as forces. They are

the Coriolis force and the centrifugal force.

Although the latter is the more palpable of the

two, it plays no role in geophysical flows,

however surprising this may be. The former and

less intuitive of the two turns out to be a

crucial factor in geophysical motions - a major effect of the Coriolis force is to impart

a certain vertical rigidity to the fluid. In

rapidly rotating, homogeneous fluids, this effect

can be so strong that the flow displays strict

columnar motions that is, all particles along

the same vertical evolve in concert, thus

retaining their vertical alignment over long

periods of time

Experimental evidence of the rigidity of a

rapidly rotating, homogeneous fluid. In a

spinning vessel filled with clear water, an

initially amorphous cloud of aqueous dye is

transformed in the course of several rotations

into perfectly vertical sheets, known as Taylor

curtains

Scales of motions

- Example hurricane Frances 2004
- Length scale satellite pictures revealed nearly

circular feature spanning approximately 7.5 º of

latitude (830 km) - L800 km
- Velocity scale - Sustained surface wind speeds of

a category-4 hurricane such as Frances range from

59 to 69 m/s - U60 m/s
- Time scale - In general, hurricane tracks display

appreciable change in direction and speed of

propagation over 2-day intervals - T2x105 s (55.6 h)
- geophysical fluids generally exhibit a certain

degree of density heterogeneity, called

stratification. The important parameters are - the average density ?0,
- the range of density variations ? ?
- the height H over which such density

variations occur - As the person new to geophysical fluid dynamics

has already realized, the selection of scales for

any given problem is more an art than a science.

Choices are rather subjective. The trick is to

choose quantities that are relevant to the

problem, yet simple to establish. There is

freedom. Fortunately, small inaccuracies are

inconsequential because the scales are meant only

to guide in the clarification of the problem,

whereas grossly inappropriate scales will usually

lead to flagrant contradictions. Practice, which

forms intuition, is necessary to build

confidence.

Importance of rotation

- Ambient rotation rate

Let us define the dimensionless quantity

If fluid motions evolve on a time scale

comparable to or longer than the time of one

rotation, we anticipate that the fluid does feel

the effect of the ambient rotation

- LENGTH AND VELOCITY SCALES OF MOTIONS IN WHICH

ROTATION EFFECTS ARE IMPORTANT

L 1 m U 0.012 mm/s L 10 m U 0.12 mm/s L

100 m U 1.2 mm/s L 1 km U 1.2 cm/s L 10

km U 12 cm/s L 100 km U 1.2 m/s L 1000 km

U 12 m/s L Earth radius 6371 km U 74 m/s

Our criterion is as follows If ? is on the order

of or less than unity (? lt 1), rotation effects

should be considered. On Earth, this occurs when

T exceeds 24 hours.

A second and usually more useful criterion

results from considering the velocity and length

scales of the motion. Let us denote these by U

and L, respectively. Naturally, if a particle

traveling at the speed U covers the distance L in

a time longer than or comparable to a rotation

period, we expect the trajectory to be influenced

by the ambient rotation, and so we write

- Examples
- The flow of water at a speed of 5 m/s in a

turbine 1 m in diameter - the air flow past a 5-m wing on an airplane

flying at 100 m/s - an ocean current flowing at 10 cm/s and

meandering over a distance of 10 km - a wind blowing at 10 m/s in a 1000-km-wide

anticyclonic formation

Importance of stratification

- If ?? is the scale of density variations in the

fluid and H is its height scale, a prototypical

perturbation to the stratification consists in

raising a fluid element of density ?0 ?? over

the height H and, in order to conserve volume,

lowering a lighter fluid element of density ?0

over the same height. The corresponding change in

potential energy, per unit volume, is - (?0??) gH -?0gH ??gH. With a typical fluid

velocity U, the kinetic energy available per unit

volume is 2?0U2. Accordingly, we construct the

comparative energy ratio

- If s is on the order of unity a typical

potential-energy increase necessary to perturb

the stratification consumes a sizable portion of

the available kinetic energy, thereby modifying

the flow field substantially - If s is much less than unity, there is

insufficient kinetic energy to perturb

significantly the stratification, and the latter

greatly constrains the flow - if s is much greater than unity potential-energy

modifications occur at very little cost to the

kinetic energy, and stratification hardly affects

the flow.

Combined effect of rotationa nd stratification

- The most interesting situation arises in

geophyscial flows when rotation and

stratification are simultaneously important

Elimination of the velocity scale yields the

fundamental length sacle

typical conditions in the atmosphere (?0 1.2

kg/m3, ?? 0.03 kg/m3, H 5000 m)

Typical conditions in the ocean (?0 1028 kg/m3,

? 2 kg/m3, H 1000 m)

Distinction between the atmosphere and oceans

Moving-rotating coordinate systems

?

z

? ? ?

x a1 a2 a3

y ß1 ß2 ß3

z ?1 ?2 ?3

?

r0

M

O

A

y

?

x

(1)

Relation between acceleration in both coordinate

systems

Definitions Absolute acceleration acceleration

with respect to the fixed coordinate

system Relative acceleration - acceleration

with respect to the moving

coordinate system Acceleration of transport -

acceleration of that point attached to moving

frame which coincides with point A at a given

moment

Absolute acceleration

(2)

(3)

Relative acceleration

Projections of the realtive acceleration on the

axes of the fixes reference frame

(4)

- Accelaration of transport we obtain assuming

that the point A is rigidly attached to the

moving frame that means ?, ? and ? are fixed.

(5)

If angles a1, a2, a3,, ?3 are constant then

(6)

Therefore if a frame moves in advancing motion

then accelaration of transport of all points

equals to the acceleration of the origin of the

moving frame

Let us differentiate eq.(1) twice

(7)

In virtue of eqs (4) and (5) the absolute

acceleration can be expressed as

(8)

Where

Coriolis acceleration is

(9)

z, ?

P

?

r

?

ß2

(10)

f

a0

y

a1

x

?

(11)

(12)

(13)

Centripetal acceleration

(14)

(15)

(No Transcript)

Navier-Stokes equations in the rotating reference

frame

(16)

(17)

Non-dimensional parameters characterising the

influence of the Coriolis effect

(18)

(19)

Geostrophic flow

- Consider a steady flow in which the Coriolis

effect is large compared with both the inertia of

the relative motion and viscous forces (Roltlt1,

Ekltlt1)

(20)

Flows in which this balance of forces between

Corriolis effects and prssure gradinet forces is

dominant are called Geostrophic Flows

(21)

(22)

y

y1

x1

UG

x

ß

Pconst

Northern hemisphere

(22)

Southern hemisphere

(23)

(24)

Velocity of geostrophic wind grows for higer

temperature and lower pressure

Scale of motion

- It is generally not required to discriminate

between the two horizontal directions, and we

assign the same length scale L to both

coordinates and the same velocity scale U to both

velocity components. The same, however, cannot be

said of the vertical direction. Geophysical flows

are typically confined to domains that are much

wider than they are thick, and the aspect ratio

H/L is small. The atmospheric layer that

determines our weather is only about 10 km thick,

yet cyclones and anticyclones spread over

thousands of kilometers. Similarly, ocean

currents are generally confined to the upper

hundred meters of the water column but extend

over tens of kilometers or more, up to the width

of the ocean basin.

For large scale motion

Hence we expect that Uz is much different from Ux

and Uy

The continuity equation contains three terms, of

respective orders of magnitude

We ought to examine three cases W/H is much less

than, on the order of, or much greater than U/L.

Scales of motion

Flow is divergence free in horizontal plane -

possible

possible

In summary, the vertical-velocity scale must be

constrained by

Vertical flow rigity

This reduced set of equations has a number of

surprising properties. First, if we take the

vertical derivative of the first equation, we

obtain, successively,

TaylorProudman theorem

Taylor-Proudman theorem

- Physically, it means that the horizontal

velocity field has no vertical shear and that all

particles on the same vertical move in concert.

Such vertical rigidity is a fundamental property

of Rotating homogeneous fluids.

Homogeneous geostrophic flows over an irregular

bottom

This property has profound implications. In

particular, if the topography consists of an

isolated bump (or dip) in an otherwise flat

bottom, the fluid on the flat bottom cannot rise

onto the bump, even partially, but must instead

go around it. Because of the vertical rigidity of

the flow, the fluid parcels at all levels

including levels above the bump elevation must

likewise go around. Similarly, the fluid over the

bump cannot leave the bump but must remain there.

Such permanent tubes of fluids trapped above

bumps or cavities are called Taylor columns

(Taylor, 1923).

Geostrophic flow in a closed domain and over

irregular topography. Solid lines are isobaths

(contours of equal depth). Flow is permitted only

along closed isobaths.

Gradient wind

- Geostrophic winds exist in locations where there

are no frictional forces and the isobars are

straight. However, such locations are quite rare.

Isobars are almost always curved and are very

rarely evenly spaced. This changes the

geostrophic winds so that they are no longer

geostrophic but are instead in gradient wind

balance. They still blow parallel to the isobars,

but are no longer balanced by only the pressure

gradient and Coriolis forces, and do not have the

same velocity as geostrophic winds.

cyclone

anticyclone

Centrifugal force

Centrifugal force

Pressure gradient

Ug

Coriolis force

Ug

Pressure gradient

Coriolis force

Low pressure

high pressure

Let us introduce the polar coordinate system with

the origin in the point of maximum/minimum

pressure

(25a)

(25b)

In the case when isobars have circular shape

To satisfy Eq.(25b)

(26)

Gradient wind solution

(27)

- Constraints on the gradient wind solution
- In Eq.(26), r 8 the centrifugal term

becomes negligible and geostrophic balance is

retrieved. That imposes a constraint on waht

solutions are physically acceptable they must

yield the geostrophic wind solution in the limit

of large curvature radius - The solution must be real

Gradient wind solution

Positive root

The geostrophic solution is the limiting case as

r -gt 8 only if we take the positive root, finaly

the gradient wind solution is

(27)

Gradient wind solution

- Low pressure system

No restriction on the gradient wind velocity

magnitude

High pressure system

Gradient wind balance imposes an upper limit on

the tangential wind speed around a High

Gradient wind direction

- Northern Hemisphere

Low pressure system

L

High pressure system

H

Gradient wind direction

Northern Hemisphere

Low pressure system

L

High pressure system

H

Gradient wind magnitude

(28)

(29a)

(29b)

Conclusion With the same horizontal pressure

gradient the cyclonic gradient wind speed is

lower than anticyclonic one

Ekman layer

Logarithmic profile- non-rotating flow

(30)

or

(31)

Where

(32)

Mean velocity profiles in fully developed

turbulent channel flow measured byWei

andWillmarth (1989) at various Reynolds numbers

circles Re 2970, squares Re 14914, upright

triangles Re 22776, and downright triangles Re

39582. The straight line on this log-linear

plot corresponds to the logarithmic profile of

Equation (8.2). (From Pope, 2000)

Ekman layer

- The part of the profile closer to the wall,

where logarithmic profile fails can be

approximated by the laminar solution, as

turbulent fluctuations are damped by the wall

presence

(33)

Comparing equations (30) and (33), ignoring the

reagion of transition bewteen both solutions, one

finds the relation

(34)

for the atmosphere the air viscosity at ambient

temperature and pressure is about 1.5x10-5 m2/s

and U rarely falls below 1 cm/s, giving d lt 5

cm, smaller than most irregularities on land

Ekman layer

Velocity profile in the vicinity of a rough wall.

The roughness heigh z0 is smaller than the

averaged height of the surface asperities. So,

the velocity u falls to zero somewhere within the

asperities, where local flow degenerates into

small vortices between the peaks, and the

negative values predicted by the logarithmic

profile are not physically realized.

When this is the case, the velocity profile above

the bottom asperities no longer depends on the

molecular viscosity of the fluid but on the

so-called roughness height z0, such that

(35)

Ekman layer

Eddy viscosity

In analogy with Newtons law for viscous fluids,

which has the tangential stress t proportional to

the velocity shear du/dz with the coefficient of

proportionality being the molecular viscosity ?,

we write for turbulent flow

(36)

For the logaritmic velocity profile of a flow

along the rough surface the velocity shear is

(37)

and the stress t is uniform across the flow (at

least in the vicinity of the boundary for lack of

other significant forces), hence

(38)

giving

(39)

Ekman layer

Friction and rotation

The ratio of vertical turbulent friction and

Coriolis forces is described by the

dimensionless Ekman number

(40)

Let us consider a uniform, geostrophic flow in a

homogeneous fluid over a flat bottom. This bottom

exerts a frictional stress against the flow,

bringing the velocity gradually to zero within a

thin layer above the bottom.

Frictional influence of a flat bottom on a

uniform flow in a rotating framework.

Ekman layer

The set of equations governing the rotaitng

turbulent boundary layer

(41)

For convenience we align the x-axis with the

geostrophic flow with the velocity

The boundary conditions are then

(42)

Ekman layer

Pressure is constant accross the layer and in

the outer flow, where

The horizontal momentu equations are

(43)

Substitution of the pressure gradient components

which are valid accross the layer into the

momentum equations (41) yields

(44)

Ekman layer

Looking for the solution of Eq. (44) one can note

that

(45)

Substitution of Eq.(45) into first of Eqs (44)

leads to

(46)

Looking for a solution of Eq.(46) in the form

(47)

Substitution of the particular solution (47) into

Eq. (46) leads to condition

(48)

Ekman layer

(49)

or

(50)

Where

Restricting attention to the Northern Hemisphere

The boundary conditions (42) eliminate

exponentialy growing solutions leaving

(51)

Taking into account boundary condition

(52)

Ekman layer

(53)

Substituting second derivative (53) into Eq.45

gives

(54)

Finally using the boundary condition

(55)

One arrives at the velocity profile in the Ekman

layer in the following form

(56)

Ekman layer

The velocity spiral in the bottom Ekman layer.

The figure is drawn for the Northern Hemisphere

(f gt 0), and the deflection is to the left of the

current above the layer. The reverse holds for

the Southern Hemisphere.

This solution has a number of important

properties. First and foremost, we notice that

the distance over which it approaches the

interior solution is on the order of d. Thus d

gives the thickness of the boundary layer. For

this reason, d is called the Ekman depth. A

comparison with (40) confirms the earlier

argument that the boundary-layer thickness is the

one corresponding to a local Ekman number near

unity.

Ekman layer

Interaction of wind with the ocean

Stratification

- As stated, problems in geophysical fluid

dynamics concern fluid motions with one or both

of two attributes, namely, ambient rotation and

stratification. In the preceding chapters,

attention was devoted exclusively to the effects

of rotation, and stratification was avoided by

the systematic assumption of a homogeneous fluid.

We noted that rotation imparts to the fluid a

strong tendency to behave in a columnar fashion

to be vertically rigid. - By contrast, a stratified fluid, consisting of

fluid parcels of various densities, will tend

under gravity to arrange itself so that the

higher densities are found below lower densities.

This vertical layering introduces an obvious

gradient of properties in the vertical direction,

which affectsamong other thingsthe velocity

field. Hence, the vertical rigidity induced by

the effects of rotation will be attenuated by the

presence of stratification. In return, the

tendency of denser fluid to lie below lighter

fluid imparts a horizontal rigidity to the

system.

Static stability

When an incompressible fluid parcel of density

(z) is vertically displaced from level z to level

z h in a stratified environment, a buoyancy

force appears because of the density difference

(z)-(zh) between the particle and the ambient

fluid

Stratification

If the fluid is incompressible, our displaced

parcel retains its former density despite a

slight pressure change, and at that new level is

subject to a net downward force equal to its own

weight minus, by Archimedes buoyancy principle,

the weight of the displaced fluid, thus

(57)

where V is the volume of the parcel. As it is

written, this force is positive if it is directed

downward. Newtons law (mass times acceleration

equals upward force) yields

(58)

As the density differences in geophysical flow

are weak we can use Boussinesq approximation

(59)

Stratification

Assuming the solution with respect to time in the

form

(60)

and introducing this solution into the

differential equation one obtains

(61)

(62)

(63)

Stratification

Typical profile of potential temperature in the

lower atmosphere above warm ground

Importance of stratification

The time passed in the vicinity of the obstacle

is approximately the time spent by a fluid parcel

to cover the horizontal distance L at the speed

U, that is, T L/U. To climb a height of z, the

fluid needs to acquire a vertical velocity on the

order of

(64)

Importance of stratification

The vertical displacement is on the order of the

height of the obstacle and, in the presence of

stratification ?(z), causes a density

perturbation on the order of

(65)

In turn, this density variation gives rise to a

pressure disturbance that scales, via the

hydrostatic balance, as

(66)

By virtue of the balance of forces in the

horizontal, the pressure-gradient force must be

accompanied by a change in fluid velocity

(67)

(68)

Importance of stratification

(69)

We immediately note that if U is less than the

product NH,W/H must be less than U/L, implying

that convergence in the vertical cannot fully

meet horizontal divergence. Consequently, the

fluid is forced to be partially deflected

horizontally so that the term

The stronger the stratification, the smaller is U

compared to NH and, thus, W/H compared to U/L.

From this argument, we conclude that the ratio

(70)

is a measure of the importance of stratification.

The rule is If Fr lt 1, stratification effects

are important the smaller Fr, the more important

these effects are.

Combination of rotation and stratification

From geostrophic balance we have

(71)

The ratio of the vertical to horizontal

convergence then becomes

(72)

Combination of rotation and stratification