The
Rossby number, named for Carl-Gustav Arvid Rossby, is a dimensionless number used in describing fluid flow. The Rossby number is the ratio of inertial to Coriolis force, terms
and
in the Navier–Stokes equations, respectively. It is commonly used in
geophysicalGeophysics is the physics of the Earth and its environment in space; also the study of the Earth using quantitative physical methods. The term geophysics sometimes refers only to the geological applications: Earth's shape; its gravitational and magnetic fields; its internal structure and...
phenomena in the oceans and
atmosphereThe atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
, where it characterizes the importance of
Coriolis accelerationsIn physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
arising from
planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
ary
rotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
. It is also known as the
Kibel number.
The Rossby number (Ro and not
) is defined as:
where
U and
L are, respectively, characteristic velocity and length scales of the phenomenon and
f = 2 Ω sin φ is the
Coriolis frequencyIn physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...
, where Ω is the
angular frequencyIn physics, angular frequency ω is a scalar measure of rotation rate. Angular frequency is the magnitude of the vector quantity angular velocity...
of
planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
ary
rotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
and φ the
latitudeIn 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...
.
A small Rossby number signifies a system which is strongly affected by Coriolis forces, and a large Rossby number signifies a system in which inertial and centrifugal forces dominate. For example, in
tornadoA tornado is a violent, dangerous, rotating column of air that is in contact with both the surface of the earth and a cumulonimbus cloud or, in rare cases, the base of a cumulus cloud. They are often referred to as a twister or a cyclone, although the word cyclone is used in meteorology in a wider...
es, the Rossby number is large (≈ 10
3), in low-pressure systems it is low (≈ 0.1 – 1) and in oceanic systems it is of the order of unity, but depending on the phenomena can range over several orders of magnitude (≈ 10
−2 – 10
2). As a result, in tornadoes the Coriolis force is negligible, and balance is between pressure and centrifugal forces (called
cyclostrophic balance). This balance also occurs at the outer
eyewallThe eye is a region of mostly calm weather found at the center of strong tropical cyclones. The eye of a storm is a roughly circular area and typically 30–65 km in diameter. It is surrounded by the eyewall, a ring of towering thunderstorms where the second most severe weather of a cyclone...
of a
tropical cycloneA 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...
. In low-pressure systems, centrifugal force is negligible and balance is between Coriolis and pressure forces (called
geostrophic balance). In the oceans all three forces are comparable (called
cyclogeostrophic balance). For a figure showing spatial and temporal scales of motions in the atmosphere and oceans, see Kantha and Clayson
When the Rossby number is large (either because f is small, such as in the tropics and at lower latitudes; or because L is small, that is, for small-scale motions such as flow in a bathtub; or for large speeds), the effects of
planetA planet is a celestial body orbiting a star or stellar remnant that is massive enough to be rounded by its own gravity, is not massive enough to cause thermonuclear fusion, and has cleared its neighbouring region of planetesimals.The term planet is ancient, with ties to history, science,...
ary
rotationA rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...
are unimportant and can be neglected. When the Rossby number is small, then the effects of planetary rotation are large and the net acceleration is comparably small allowing the use of the
geostrophic approximationThe 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...
.
Further reading
For more on numerical analysis and the role of the Rossby number, see:
For an historical account of Rossby's reception in the United States, see