The Gravity Field of the Earth and Coriolis Effects.

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OCEAN 510
The Gravity Field of the Earth and Coriolis
Effects.
The gravity field Stationary particles on a
rotating Earth Moving particles on a rotating
Earth.Coriolis effects Geostrophic motion
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Gravity.
Newtons Universal Law of Gravitation The
force between any two particles having masses m1
and m2 separated by a distance r is an attraction
along the line joining the particles, and has a
magnitude given by
where G is a universal constant having the same
value for all pairs of particles (G 6.673?10-11
newton-meter2/kilogram2).
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Gravity.
In vector form,
21 ? force on 2 due to 1
or,
unit vector
This works OK for point masses what about finite
masses such as the Earth?
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Gravity.
?
?
A point outside the Earth
A point inside the Earth
Gravity goes as r?2
Gravity goes as r
rE
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Gravity.
Gravity in the ocean, bottom to top.
Note h ltlt a
For most applications in physical oceanography,
we can take g to be a constant over the ocean.
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Gravity.
Stationary particles on a rotating Earth
uniform circular motion
P
r
?
v
v?
v
?
?
?v? ?t
?v
P?
v?
Centripetal acceleration ? ? ? v ? ? ? ? ? r
v ???r, a ???2r v2/r in scalar form
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Gravity.
Coordinate systems.Newtons Laws are only true
in an inertial coordinate frame (ie, one fixed
with respect to distant, fixed stars). This is
not usually convenient. Generally we would like
to use an Earth-based coordinate system instead
(ie, east/north/up or longitude/latitude/altitude)
.
Centripetal acceleration/centrifugal force
questions arise. If Newtons Laws are to be used
in an Earth-based coordinate system, they must be
modified.
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Gravity and centrifugal force.
?
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Gravity and centrifugal force.
k
j
?
?
unit vector
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Gravity and centrifugal force.
force of attraction
gravity does not point down
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Gravity and potential.
The concept of a potential.
It is often convenient to define a vector force F
in terms of the gradient of a scalar potential ?,
so that
This can be done as long as the force in question
is a conservative force. In this type of field,
if a particle is displaced along a path from its
initial position and returned to its initial
position, no work is done. Examples gravity
EM not friction.
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The geopotential.
if values of ? at 1 and 2 are identical
Thus, no work is done on a particle along
contours of constant values of ?.
geopotential
This function yields a bulge at the Equator in
equilibrium this would be the shape of the sea
surface on a water covered sphere.
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Geopotential.
?
Earth
?1
?2
On a water-covered sphere, the sea surface would
have the shape of the geoid ?. The ocean would
be deeper at the Equator than at the poles.
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Moving particles on a rotating Earth.
Simple idea from classical mechanics Launch a
particle in some direction, and we get x(t)
x0 Vxt y(t) y0 Vyt z(t) z0
Vzt ? (1/2) gt2
Suppose we did this on the Earth, with Vx 0, Vy
lt0 (a launch to the south). What happens?
Recall, tangential velocity of a point on the
Earth is Vtan ???R cos ?
particles on the surface of the Earth are moving
faster at low latitudes
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Moving particles on the Earth.
Because of the initial conditions, an observer
south of the launch will outrun the particle.
This is confusing if the observer tries to apply
Newtons Laws. Result the Coriolis effect.
Newtons Laws cannot apply in their purest form,
and must be repaired.
Vtan ???R cos ?
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Coriolis effects.
A particle moves from A to B (distance L) with
speed Uo. The time required to do this is
(L/Uo).
The characteristic time of the Earths rotation
is tE , and tE ??1 . The ratio of the
times is
Ro gtgt 1 ? event short compared to rotation no
apparent Coriolis effect Ro ltlt 1 ? event long
compared to rotation Coriolis effect important
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Coriolis effects.
i, j, k east, north, up unit vectors
Coriolis acceleration 2? ? u u (u, v,
w) Coriolis force ? 2? ? u
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Coriolis effect (continued).
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Coriolis effect, continued.
Initial velocity
Coriolis force
(?2??u) north (0,v,0), vgt0
east south (0,v,0),
vlt0 west east
(u,0,0), ugt0 south,
up west (u,0,0), ult0
north, down up (0,0,w), wgt0
west down (0,0,w), wlt0
east
The Coriolis effect is due to rotation
spherical geometry
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Coriolis effect (continued).
Newtons Laws with rotation
or
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Coriolis effect, continued.
add the pressure gradient as a force
(horizontal equations)
steady or nearly steady motion
geostrophic balance
geostrophic flow horizontal pressure gradient
balances the Coriolis force
y
x
flow is along lines of constant pressure
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The Coriolis effect and geostrophic balance.
Recall the equation for the Coriolis acceleration
Note in the i term that (w cos ?)/(v sin?)
(w/v) cot ? ltlt 1, except near the Equator.
So, neglect the w term with respect to
the v term. Also, note that we have already
examined hydrostatic balance and found that
vertical accelerations are small compared to
horizontal. Thus, ignore the k term here.
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Coriolis effect and geostrophic motion.
With these simplifications the geostrophic
equations become
x pressure gradient northward flow in the N.
hemisphere
y pressure gradient westward flow in the N.
hemisphere
These equations provide a simple diagnostic tool
for examining the circulation of the atmosphere
or the ocean.a right turn in the N. hemisphere.
not like flow in a pipe
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Coriolis effect (continued).
H
H
L
H
L
Weather map shows geostrophic flow
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Geostrophic motion.
L
L
H
H
H
H
Global mean sea level from Topex/Poseidon,
1992-2001
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Satellites, sea level, etc.
microwave pulse travel time
geoid varies by 100 m
topography varies by ? 1 m
the geoid is not presently well-known
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TOPEX ground tracks.10 day repeat
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Sea level variability.
the weather of the ocean
SSH (cm)
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Sea level from satellites.
W
U
S
4
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ring S
ring 4
Sea surface height along satellite tracks shows
warm and cold-core rings near the Gulf Stream
ring W
ring 4
ring W
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Measured sea level over many tracks shows the
Gulf Stream and the subtropical gyre of the
western N. Atlantic.
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Geostrophic balance.
Sea level across the Gulf Stream or Kuroshio
?p/?y gt 0
?p/?y lt 0
p p0 ? g ?h
The distribution of sea level can be used to
estimate the geostrophic flow at the sea surface.
This is valuable, but it cannot tell us anything
about the internal flow field.
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Internal structure.
N
S
The Gulf Stream
The Gulf Stream can be seen as a region where the
isotherm and isohaline (hence, isopycnal)
surfaces show maximum slope. This is internal
structure in the density field that might have no
signature in sea level.
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The dynamic method.internal geostrophic flow
Consider 2 columns of fluid (seawater) of similar
height
?2 ?1 ?? h3 h1 ?h h4 h2 ? ?h
The pressure at the bottom of the left column
is pL g(?1h1 ?2h2) g?2 ( ?1/?2 h1
h2) The pressure at the bottom of the right
column is pR g?2 ( ?1/?2h3 h4)
? ?p pL?pR g?2?h (1? ?1/?2 ) g?? ?h
left is heavier because of more dense water
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The dynamic method.
?p pL?pR g?2?h (1? ?1/?2 ) g?? ?h
Example 1 atmosphere and ocean. ?? 1 g/cm3
if ?h 1 meter, then ?p 1 dbar . Example 2
the Gulf Stream. ?? 0.01 g/cm3 if ?h 100
meter, then ?p 1 dbar . Result large changes
in the height of a density surface (?h) have the
same dynamical effects as large changes in the
density contrast (??) between layers.
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Geostrophic motion with stratification.
the dynamic method
Recall the geostrophic equations
f 2 ? sin ?
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The dynamic method.
geostrophic hydrostatic
ctd
the thermal wind equations
light
ctd
dense
??/?y gt 0 ? ?u/?z gt 0 westward velocity
increases upwards
many possibilities ?
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The dynamic method.
Problem depth is easy to visualize but nearly
impossible to measure. Pressure is relatively
easy to measure. So, change to pressure
coordinates.
constant of integration
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The dynamic method.
geostrophic velocity relative to some level zo
change to pressure coordinates
change from density to specific volume
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The dynamic method.
dynamic height
reference velocities
? 1/?
constant part variable part
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The dynamic method.
no ?p on z surfaces, no geostrophic flow
a reference level
nonzero ?p on z surfaces, nonzero flow
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The dynamic method.
N
S
??/?y gt 0
??/?y lt 0
N
S
??/?y gt 0
??/?y lt 0
(a)
(b)
Problem in general we cannot discern the
difference between (a) and (b).
dont know the constants of integration
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The dynamic method.
Circulation inferred from hydrographic data
alone (1000 dbar reference)
Circulation inferred from combining hydrographic
data with altimetry data (sea surface referenced
to altimetry)
Inverse methods
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Estimating the ocean circulation.
Profiling floats can measure the state variables
of the ocean circulation..T, S, p, velocity
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A hypothetical array of 3000 profiling floats (
300 km resolution)
grid
The Argo project
random
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Argo array as of 10/15/07..2905 floats
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An example of Argo-like estimation of the
circulation in the Japan Sea, using 35 profiling
floats deployed over 4 years. The absolute
geostrophic velocity field at 0 m is shown.
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The absolute geostrophic velocity at 800 m.