Dynamical Balance in the Earth’s Atmosphere Lisa Goddard goddard@iri.columbia.edu 15 Sept 2005 9/18/201815 Sept. 2005

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 9/18/201815 Sept. 2005

Synoptic Map 9/18/201815 Sept. 2005

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 ...” 9/18/201815 Sept. 2005

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! 9/18/201815 Sept. 2005

... consider forces acting on a small (”differential”) volume of fluid Atmospheric forces pressure gradient force gravity Coriolis/centrifugal force friction height north east 9/18/201815 Sept. 2005 ... consider forces acting on a small (”differential”) volume of fluid

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” 9/18/201815 Sept. 2005

Sea-level pressure 9/18/201815 Sept. 2005

Horizontal pressure gradient force ... eastward pressure-gradient force per unit mass 9/18/201815 Sept. 2005

Sea Breeze 9/18/201815 Sept. 2005

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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 9/18/201815 Sept. 2005

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.” 9/18/201815 Sept. 2005

East-west motion Centrifugal force Ω 9/18/201815 Sept. 2005

North-south motion Conservation of angular momentum Ω Ω 9/18/201815 Sept. 2005

... 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 9/18/201815 Sept. 2005

... back to Newton’s 2nd Law following our fluid element ... acceleration = sum of forces acting per unit mass height north east 9/18/201815 Sept. 2005

Atmospheric scale analysis 9/18/201815 Sept. 2005

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” 9/18/201815 Sept. 2005

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Surface friction 9/18/201815 Sept. 2005 http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/bndy.rxml 9/18/201815 Sept. 2005 http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/home.rxml

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. 9/18/201815 Sept. 2005

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 9/18/201815 Sept. 2005