Dynamical Balance in the Earth’s Atmosphere

Dynamical Balancein the Earth’s Atmosphere Lisa Goddard goddard@iri.columbia.edu 15 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

Synoptic Map

Sir Isaac Newton 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 ...”

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!

height north east Atmospheric forces • pressure gradient force • gravity • Coriolis/centrifugal force • friction ... 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”

Sea-level pressure

... eastward pressure-gradient force per unit mass Horizontal pressure gradient force

Sea Breeze

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

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.”

East-west motionCentrifugal force Ω

North-south motionConservation of angular momentum Ω Ω

... more on the Coriolis Force • Fco vanishes at equator • Fco is proportional to velocity of parcel • Fco is negligible for motions with timescales very shortcompared to the period ofEarth’s rotation

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

Atmospheric scale analysis

Large-scale dynamical balance “the geostrophic approximation”

Surface friction http://ww2010.atmos.uiuc.edu/(Gh)/guides/mtr/fw/bndy.rxml 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.

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

Dynamical Balance in the Earth’s Atmosphere