Equations of motion

equations of motion

A set of hydrodynamical equations representing the application of Newton's second law of motion to a fluid system.

The total acceleration on an individual fluid particle is equated to the sum of the forces acting on the particle within the fluid. Written for a unit mass of fluid in motion in a coordinate system fixed with respect to the earth, the vector equation of motion for the atmosphere is

where u is the three-dimensional velocity vector, Ω the angular velocity of the earth, k a unit vector directed upward, ρ the density, p the pressure, g the acceleration of gravity, and F the frictional force per unit mass. The usual form for the scalar equations of motion in spherical coordinates (λ, φ, r), with λ the longitude, φ the latitude, and r the radius from the center of the earth, is as follows:

Most global numerical weather prediction models and general circulation models use an approximate version of the above nonhydrostatic primitive equations. This version involves the approximation of the vertical equation of motion by the hydrostatic equation and the selective approximation of r = a + z by r ≈ a, where a is the constant radius to mean sea level and z is the height above mean sea level. These approximations result in the quasi-static primitive equations

See Newton's laws of motion, vorticity equation.

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