The barotropic vorticity equation assumes the atmosphere is nearly

barotropic In fluid dynamics, a barotropic fluid is a fluid whose density is a function of pressure only. The barotropic fluid is a useful model of fluid behavior in a wide variety of scientific fields, from meteorology to astrophysics. The density of most ...

, which means that the direction and speed of the

geostrophic wind In atmospheric science, geostrophic flow () is the theoretical wind that would result from an exact balance between the Coriolis force and the pressure gradient force. This condition is called '' geostrophic equilibrium'' or ''geostrophic balanc ...

are independent of height. In other words, there is no vertical

wind shear Wind shear (or windshear), sometimes referred to as wind gradient, is a difference in wind speed and/or direction over a relatively short distance in the atmosphere. Atmospheric wind shear is normally described as either vertical or horizontal ...

of the geostrophic wind. It also implies that thickness contours (a proxy for temperature) are parallel to upper level height contours. In this type of atmosphere, high and low pressure areas are centers of warm and cold temperature anomalies. Warm-core highs (such as the

subtropical ridge The horse latitudes are the latitudes about 30 degrees north and south of the Equator. They are characterized by sunny skies, calm winds, and very little precipitation. They are also known as Subtropics, subtropical ridges, or highs. It is a h ...

and the Bermuda-Azores high) and

cold-core low A cold-core low, also known as an upper level low or cold-core cyclone, is a cyclone aloft which has an associated cold pool of air residing at high altitude within the Earth's troposphere, without a frontal structure. It is a low pressure syst ...

s have strengthening winds with height, with the reverse true for cold-core highs (shallow Arctic highs) and warm-core lows (such as

tropical cyclone A tropical cyclone is a rapidly rotating storm system characterized by a low-pressure center, a closed low-level atmospheric circulation, strong winds, and a spiral arrangement of thunderstorms that produce heavy rain and squalls. Depend ...

s). A simplified form of the vorticity equation for an inviscid,

divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ...

-free flow (

solenoidal In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: \nabla \cdot \mathbf ...

velocity field), the barotropic vorticity equation can simply be stated as :

\frac = 0,

where is the

material derivative In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...

and :

\eta = \zeta + f

is ''absolute

vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wit ...

'', with ''ζ'' being ''relative vorticity'', defined as the vertical component of the

curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was fi ...

of the fluid velocity and ''f'' is the ''

Coriolis parameter The Coriolis frequency ''ƒ'', also called the Coriolis parameter or Coriolis coefficient, is equal to twice the rotation rate ''Ω'' of the Earth multiplied by the sine of the latitude \varphi. :f = 2 \Omega \sin \varphi.\, The rotation rate ...

'' :

f = 2 \Omega \sin \varphi,

where Ω is the

angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...

of the planet's rotation (Ω = for the earth) and ''φ'' is

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north pol ...

. In terms of ''relative vorticity'', the equation can be rewritten as :

\frac = -v \beta,

where ''β'' = is the variation of the ''Coriolis parameter'' with distance ''y'' in the north–south direction and ''v'' is the component of velocity in this direction. In 1950, Charney, Fjørtoft, and von Neumann integrated this equation (with an added

diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...

term on the

right-hand side In mathematics, LHS is informal shorthand for the left-hand side of an equation. Similarly, RHS is the right-hand side. The two sides have the same value, expressed differently, since equality is symmetric.computer A computer is a machine that can be programmed to Execution (computing), carry out sequences of arithmetic or logical operations (computation) automatically. Modern digital electronic computers can perform generic sets of operations known as C ...

for the first time, using an observed field of 500

hPa HPA may refer to: Organizations * Harry Potter Alliance, a charity * Halifax Port Authority, Canada * Hamburg Port Authority, Germany * Hawaii Preparatory Academy, a school in Hawaii, US * Health Protection Agency, UK * Heerespersonalamt, the Ger ...

geopotential height Geopotential height or geopotential altitude is a vertical coordinate referenced to Earth's mean sea level, an adjustment to geometric height (altitude above mean sea level) that accounts for the variation of gravity with latitude and altitude. Thu ...

for the first timestep. This was one of the first successful instances of

numerical weather prediction Numerical weather prediction (NWP) uses mathematical models of the atmosphere and oceans to predict the weather based on current weather conditions. Though first attempted in the 1920s, it was not until the advent of computer simulation in th ...

References

External links

* http://www.met.reading.ac.uk/~ross/Science/BarVor.html Equations of fluid dynamics {{math-physics-stub

See also

References

External links