1. Dynamical Balance in the Earth’s Atmosphere
Lisa Goddard
15 Sept 2005
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2. Outline Newton’s laws of motion
Pressure gradients and hydrostatic balance
Coriolis force
Equations of large scale horizontal motion
Geostrophic balance
Surface friction
Vertical motion
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3. Synoptic Map
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4. Sir Isaac Newton 9/18/201815 Sept. 2005
Born: 4 Jan 1643, Lincolnshire, England
Died: 31 March 1727, London, England
“A plague closed the University in the summer of 1665 and he had to return to Lincolnshire. There, in a period of less than two years, while Newton was still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy ...”
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5. Newton’s Laws of motion
A mass in uniform motion – relative to a coordinate system fixed in space – will remain in uniform motion in the absence of any forces
The rate of change of momentum of an object – relative to a coordinate system fixed in space – equals the sum of all the forces acting
... these two laws, together with conservation of mass and heat, form the basis of general circulation models of the atmosphere and ocean ... using the differential calculus!
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6. ... consider forces acting on a small (”differential”) volume of fluid
Atmospheric forces
pressure gradient force
gravity
Coriolis/centrifugal force
friction
height
north
east
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... consider forces acting on a small (”differential”) volume of fluid

7. Vertical pressure gradient force
Due to random molecular motions, momentum is continually imparted to the walls of the volume element by the surrounding air.
The momentum transfer per unit time, per unit area, is the pressure
In the absence of atmospheric motions the gravity force must be exactly balanced by the vertical component of the pressure gradient force.
“Hydrostatic Balance”
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8. Sea-level pressure
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9. Horizontal pressure gradient force
... eastward pressure-gradient force per unit mass
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10. Sea Breeze
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13. Newton’s Laws
A mass in uniform motion – relative to a coordinate system fixed in space – will remain in uniform motion in the absence of any forces
The rate of change of momentum of an object – relative to a coordinate system fixed in space – equals the sum of all the forces acting
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14. Deflection due to the Earth’s rotation: The Coriolis Force
Newton’s laws can only be applied in a rotating frame if the acceleration of the coordinates is taken into account
Most satisfactory way of including coordinate acceleration is to include “apparent” forces into the statement of Newton’s 2nd law: the Coriolis force
Pierre Simon Laplace (1778); Gaspard Gustave de Coriolis (1835)
In 1848, Jean Foucault discovered that when a large pendulum swings, the earth appears to "move under it.”
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15. East-west motion Centrifugal forceΩ
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16. North-south motion Conservation of angular momentumΩ Ω
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17. ... more on the Coriolis Force
Fco vanishes at equator
Fco is proportional to velocity of parcel
Fco is negligible for motions with timescales very short compared to the period of Earth’s rotation
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18. ... back to Newton’s 2nd Law following our fluid element ...
acceleration = sum of forces acting per unit mass
height
north
east
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19. Atmospheric scale analysis
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20. Large-scale dynamical balance
To within about 15% in mid & high lats
The horizontal pressure gradient force is in approximate balance with the Coriolis force for synoptic-scale motions at mid and high latitudes. The result is that winds tend to blow parallel to the isobars.
“the geostrophic approximation”
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22. Surface friction 9/18/201815 Sept. 2005
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23. Vertical motion
Vertical scales are much smaller than horizontal ones; the atmosphere is “shallow.”
For synoptic-scale motions, the pressure field is in hydrostatic balance to a very high degree of accuracy.
Vertical velocity cannot be determined from the vertical momentum equation. But it can be determined indirectly.
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24. Summary
The vertical component of the pressure gradient force is in hydrostatic balance with the gravity force to a very high degree of accuracy.
On synoptic scales, the horizontal component of the pressure gradient force is in approximate geostrophic balance with the Coriolis force.
Friction makes an important contribution near the earth’s surface, to give a 3-way balance
Scale is key: “synoptic” means ~day, ~1000km
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