Literature: Introduction to Geophysical Fluid Dynamics Physical and Numerical Aspects

1
LiteratureIntroduction to Geophysical Fluid
DynamicsPhysical and Numerical Aspects

• Benoit Cushman-Roisin
• Jean-Marie Beckers

2
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

3
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.

4
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.

5
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
6
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

7
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
8
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.

9
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
10

• 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.
11
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

12
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.

13
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)
14
Distinction between the atmosphere and oceans
15
Moving-rotating coordinate systems
?
z
? ? ?
x a1 a2 a3
y ß1 ß2 ß3
z ?1 ?2 ?3
?
r0
M
O
A
y
?
x
(1)
16
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)
17

• 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
18
Let us differentiate eq.(1) twice
(7)
19
In virtue of eqs (4) and (5) the absolute
acceleration can be expressed as
(8)
Where
Coriolis acceleration is
(9)
20
z, ?
P
?
r
?
ß2
(10)
f
a0
y
a1
x
?
(11)
21
(12)
(13)
Centripetal acceleration
(14)
(15)
22
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23
Navier-Stokes equations in the rotating reference
frame
(16)
24
(17)
Non-dimensional parameters characterising the
influence of the Coriolis effect
(18)
(19)
25
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)
26
(22)
27
y
y1
x1
UG
x
ß
Pconst
28
Northern hemisphere
(22)
Southern hemisphere
(23)
(24)
Velocity of geostrophic wind grows for higer
temperature and lower pressure
29
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.
30
Scales of motion
Flow is divergence free in horizontal plane -
possible
possible
In summary, the vertical-velocity scale must be
constrained by
31
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
32
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).
33
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.
34
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
35
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)
36
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

37
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)
38
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
39
Gradient wind direction

• Northern Hemisphere

Low pressure system
L
High pressure system
H
40

Gradient wind direction
Northern Hemisphere
Low pressure system
L
High pressure system
H
41
Gradient wind magnitude
(28)
(29a)
(29b)
Conclusion With the same horizontal pressure
gradient the cyclonic gradient wind speed is
lower than anticyclonic one
42
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)
43
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
44
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)
45
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)
46
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.
47
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)
48
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)
49
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)
50
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)
51
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)
52
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.
53
Ekman layer
Interaction of wind with the ocean
54
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
55
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)
56
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)
57
Stratification
Typical profile of potential temperature in the
lower atmosphere above warm ground
58
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)
59
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)
60
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.
61
Combination of rotation and stratification
From geostrophic balance we have
(71)
The ratio of the vertical to horizontal
convergence then becomes
(72)
62
Combination of rotation and stratification