Coriolis and Geostrophy

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The Coriolis Force and Geostrophy Corinne Le
Quéré for Karen J. Heywood (Convenor) Room
01.37a in ENV c.lequere_at_uea.ac.uk
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Forces and motion
Forces acting on parcels of air or water
Variations in sea level (Lecture 10)

• Pressure gradient force (lect.5)
• Gravitational force (lect. 5)

?
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Forces and motion
Forces acting on parcels of air or water
Variations in sea level (Lecture 10)

• Pressure gradient force (lect. 5)
• Gravitational force (lect. 5)
• Coriolis force (today)
• Centrifugal force (today)
• Friction (Friday)

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Coriolis force

• an apparent force due to the rotation of the
  earth.
• invoked since we observe the world from a
  reference frame that is rotating with the earth.
• only acts on objects that are moving over the
  earths surface.

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Centrifugal force

• experienced by objects on a rotating body. e.g.
  the earth, or a roundabout.
• an apparent force, acting outwards.
• balances the inward-acting centripetal
  acceleration needed to keep object going in a
  circle.
• Earth rotates with angular speed 2? radians per
  day 7.29 x 10-5 radians per second.
• 360 2? radians, so 1 radian is 57.3.

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v
Conker would go in a straight line if you let go.
m
r
You pull on the string to provide the centripetal
acceleration needed to draw the conker round in a
circle. You call the apparent outward force you
can feel, the centrifugal force.
The force you need to apply is m v2 / r.
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v
Centrifugal force
?
r
X
v
If an object rotates about point X with an
angular speed ? radians per second, and a radius
r, then Angular speed ?
radians/second. Radial
acceleration m v2 / r m (v2 / r2 ) r m
?2 r So centrifugal force per unit mass is ?2 r.
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• Now consider an
• object sitting on
• the earths
• surface.
• The centrifugal force per unit mass can be split
  into 2 components
• vertical ?2 r cos ?
• southward ?2 r sin ?
• The earth has changed shape to be fatter at the
  equator, so the equatorward force is now zero.
• The vertical force is centrifugal force.

?
?2 r cos ?
r
?
?2 r
?
?2 r sin ?
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• Now imagine an object moving east on the earths
  surface with speed u.
• It feels a bigger centrifugal force per unit mass
  of
• We call the extra 2?u the Coriolis force.
• As before, we can split it into a vertical and a
  horizontal component.
• The horizontal component is 2 ? sin? u.

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• Now consider an object moving north on earths
  surface.
• If no net (real) force, then angular momentum
  must be conserved. Angular momentum m v r m
  ? r2.
• As object moves north, radius of the earth, and
  so the radius of rotation, decreases.
• To compensate, angular speed ? must increase to
  be greater than ?, the earths angular speed.
• So, viewed from the earth, the object starts to
  move east.
• We call this effect the Coriolis force.
• The force per unit mass is 2 ? sin? u.

V and ? must increase
Radius r smaller
Radius r greater
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Coriolis force

• an apparent force due to the earths rotation.
• invoked since we observe the world from a
  reference frame that is rotating with the earth.
• only acts on objects that are moving over the
  earths surface.

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A ping pong ball on a non rotating table.
A ping pong ball on a rotating table.
The camera is fixed to the table.
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• The Coriolis force per unit mass on a parcel of
  air or water moving with speed v is given by f v,
  where f 2 ? sin ?.
• ? is latitude and ? is the rate of rotation of
  the earth, 7.29 x 10-5 radians per second (2?
  radians 360?).
• Coriolis force is perpendicular to direction of
  motion of object.
• Coriolis force acts to right in northern
  hemi-sphere, and to left in southern hemisphere.

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f 2 ? sin ? f is proportional to the sine of
the latitude the line is a sine curve. At the
equator, f is zero. In the northern hemisphere,
f is positive because latitude is positive. In
the southern hemisphere, f is negative because
latitude is negative.
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• How big is the effect of the Coriolis force?
• Assume 52N. f 2 ? sin 52? 1.14 x
  10-4 s-1.
• Bullet
• Bullet travelling 1 km at 1000 m s-1.
• Coriolis force per unit mass acceleration.
• f v 1.14 x 10-4 s-1 x 1000 m s-1.
• 1.14 x 10-1 m s-2.
• Now displacement s after time t, speed u and
  constant acceleration a is given by
• So bullet is displaced to right after 1 second by
  
• s 0 0.5 x 1.14 x 10-1 x 12 5 cm.

5 cm
1 km
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• Water in bath
• Water travels 50 cm at 10 cm s-1, taking 5
  seconds.
• Acceleration to right due to Coriolis force
• 1.14 x 10-4 s-1 x 0.1 m s-1
• 1.14 x 10-5 m s-2.
• s 0 0.5 x 1.14 x 10-5 x 52
• 1.4 x 10-4 m.
• Coriolis force is negligible compared with other
  forces on water in bath.
• So sadly it does not affect the direction the
  water goes down the plug hole..

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• Water in North Atlantic Current
• Water travels 1 km at 10 cm s-1 taking 1 x 104
  seconds (2.8 hours).
• Acceleration to right due to Coriolis force
• 1.14 x 10-4 s-1 x 0.1 m s-1
• 1.14 x 10-5 m s-2.
• s 0 0.5 x 1.14 x 10-5 x (1 x 104)2
• 570 m.
• Coriolis force is significant in the ocean and
  atmosphere.

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• Geostrophic Currents and Winds
• In geostrophic equilibrium, the horizontal
  pressure gradient force balances the Coriolis
  force.
• In the vertical, the vertical pressure gradient
  force balances gravity.
• Say the sea surface has a slope with an angle i,
  creating a pressure gradient force.
• Then if current has a constant speed V, we can
  balance all the forces to obtain
• which is the geostrophic equation.

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Vertical component of pressure gradient force
Horizontal pressure gradient force
Coriolis force
x
Gravitational force
Geostrophic flow is into the page. f V g tan
i tan i h / d
h
i
d
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• By how much does the sea surface height change
  across the Gulf Stream?
• At 40?N, f 2 x 7.29 x 10-5 x sin (40)
• 9.4 x 10-5 s-1.
• g 9.81 m s-2.
• If geostrophic current at the ocean surface is
  0.8 m s-1, then
• If Gulf Stream is 100 km wide, then
• height difference tan i x 100 x 103
• 0.76 m.
• So sea level is 76 cm higher on southeast than
  northwest side of Gulf Stream.

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Looking from above, the Coriolis force is equal
and opposite to the horizontal pressure gradient
force. The resulting geostrophic wind or
geostrophic current is perpendicular to
both. This is why the wind blows roughly along
the isobars of weather systems. The wind starts
off blowing from high to low pressure, but once
it moves, Coriolis force acts, and equilibrium is
reached when the forces balance.
Northern hemisphere case
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Variations in the height of the sea surface,
leading to geostrophic currents.
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0 m