Barotropic vorticity equation
Encyclopedia
This Barotropic vorticity equation assumes the atmosphere is nearly barotropic, which means that the direction and speed of the geostrophic wind
are independent of height. In other words, no vertical wind shear
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
and Bermuda-Azores high) and cold-core low
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
s).
A simplified form of the vorticity equation
for an inviscid, divergence
-free flow (solenoidal velocity field), the barotropic vorticity equation can simply be stated as
where
is the material derivative and
is absolute vorticity, with
being relative vorticity, defined as the vertical component of the curl of the fluid
velocity and f is the Coriolis parameter
where
is the angular frequency
of the planet's
rotation (
=0.7272*10−4 s−1 for the earth) and 
is latitude
.
In terms of relative vorticity, the equation can be rewritten as
where
is the variation of the Coriolis parameter with distance 
in the north-south direction
and
is the component of velocity in this direction.
In 1950, Charney, Fjorloft, and von Neumann integrated this equation (with an added diffusion
term on the RHS) on a computer
for the first time, using an observed field of 500 hPa
geopotential height
for the first timestep. This was one of the first successful instances of numerical weather prediction
.
Geostrophic wind
The geostrophic wind is the theoretical wind that would result from an exact balance between the Coriolis effect and the pressure gradient force. This condition is called geostrophic balance. The geostrophic wind is directed parallel to isobars . This balance seldom holds exactly in nature...
are independent of height. In other words, no vertical wind shear
Wind shear
Wind shear, sometimes referred to as windshear or wind gradient, is a difference in wind speed and direction over a relatively short distance in the atmosphere...
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
Subtropical ridge
The subtropical ridge is a significant belt of high pressure situated around the latitudes of 30°N in the Northern Hemisphere and 30°S in the Southern Hemisphere. It is characterized by mostly calm winds, which acts to reduce air quality under its axis by causing fog overnight, and haze during...
and Bermuda-Azores high) and cold-core low
Cold-core low
A cold-core low, also known as a cold 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. It is a low pressure system which strengthens with height in accordance with the thermal wind relationship. These...
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
Tropical cyclone
A tropical cyclone is a storm system characterized by a large low-pressure center and numerous thunderstorms that produce strong winds and heavy rain. Tropical cyclones strengthen when water evaporated from the ocean is released as the saturated air rises, resulting in condensation of water vapor...
s).
A simplified form of the vorticity equation
Vorticity equation
The vorticity equation is an important prognostic equation in the atmospheric sciences. Vorticity is a vector, therefore, there are three components...
for an inviscid, divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
-free flow (solenoidal velocity field), the barotropic vorticity equation can simply be stated as
where
is the material derivative andis absolute vorticity, with
being relative vorticity, defined as the vertical component of the curl of the fluidvelocity and f is the Coriolis parameter
where
is the angular frequencyAngular frequency
In physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of the planet's
rotation (
=0.7272*10−4 s−1 for the earth) and is latitudeLatitude
In geography, the latitude of a location on the Earth is the angular distance of that location south or north of the Equator. The latitude is an angle, and is usually measured in degrees . The equator has a latitude of 0°, the North pole has a latitude of 90° north , and the South pole has a...
.
In terms of relative vorticity, the equation can be rewritten as
where
is the variation of the Coriolis parameter with distance in the north-south directionand
is the component of velocity in this direction.In 1950, Charney, Fjorloft, and von Neumann integrated this equation (with an added diffusion
Diffusion
Molecular diffusion, often called simply diffusion, is the thermal motion of all particles at temperatures above absolute zero. The rate of this movement is a function of temperature, viscosity of the fluid and the size of the particles...
term on the RHS) on a computer
Computer
A computer is a programmable machine designed to sequentially and automatically carry out a sequence of arithmetic or logical operations. The particular sequence of operations can be changed readily, allowing the computer to solve more than one kind of problem...
for the first time, using an observed field of 500 hPa
Pascal (unit)
The pascal is the SI derived unit of pressure, internal pressure, stress, Young's modulus and tensile strength, named after the French mathematician, physicist, inventor, writer, and philosopher Blaise Pascal. It is a measure of force per unit area, defined as one newton per square metre...
geopotential height
Geopotential height
Geopotential height is a vertical coordinate referenced to Earth's mean sea level — an adjustment to geometric height using the variation of gravity with latitude and elevation. Thus it can be considered a "gravity-adjusted height"...
for the first timestep. This was one of the first successful instances of numerical weather prediction
Numerical weather prediction
Numerical weather prediction 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 the 1950s that numerical weather predictions produced realistic...
.