meteorology Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agricultur ...

, absolute angular momentum is the

angular momentum Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...

in an 'absolute'

coordinate system In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are ...

(

absolute time and space Absolute space and time is a concept in physics and philosophy about the properties of the universe. In physics, absolute space and time may be a preferred frame. Early concept A version of the concept of absolute space (in the sense of a prefe ...

Introduction

Angular momentum equates with the

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

of the

position (vector) In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents a point ''P'' in space. Its length represents the distance in relation to an arbitrary reference origin ''O'', and ...

of a particle (or

fluid parcel In fluid dynamics, a fluid parcel, also known as a fluid element or material element, is an infinitesimal volume of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel rema ...

) and its absolute linear

momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...

, equal to , the product of mass and velocity. Mathematically, :

\mathbf = \mathbf \times m \mathbf

Definition

Absolute angular momentum sums the angular momentum of a particle or fluid parcel in a relative coordinate system and the angular momentum of that relative coordinate system. Meteorologists typically express the three vector components of velocity (eastward, northward, and upward). The magnitude of the absolute angular momentum per unit mass :

\left, \frac\ = M = u r \cos (\phi) + \Omega r^2 \cos^2(\phi)

where * represents absolute angular momentum per unit mass of the fluid parcel (in ), * represents distance from the center of the Earth to the fluid parcel (in ), * represents earth-relative eastward component of velocity of the fluid parcel (in ), * represents

latitude In geography, latitude is a geographic coordinate system, geographic 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 t ...

(in ), and * represents angular rate of

Earth's rotation Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in progra ...

(in , usually ). The first term represents the angular momentum of the parcel with respect to the surface of the Earth, which depends strongly on weather. The second term represents the angular momentum of the Earth itself at a particular latitude (essentially constant at least on non-geological timescales).

Applications

In the shallow

troposphere The troposphere is the lowest layer of the atmosphere of Earth. It contains 80% of the total mass of the Atmosphere, planetary atmosphere and 99% of the total mass of water vapor and aerosols, and is where most weather phenomena occur. From the ...

of the Earth, humans can approximate , the distance between the fluid parcel and the center of the Earth approximately equal to the mean

Earth radius Earth radius (denoted as ''R''🜨 or ''R''E) is the distance from the center of Earth to a point on or near its surface. Approximating the figure of Earth by an Earth spheroid (an oblate ellipsoid), the radius ranges from a maximum (equato ...

: :

M \approx u a \cos (\varphi) + \Omega a^2 \cos^2(\varphi)

where * represents

(in , usually ) * represents absolute angular momentum per unit mass of the fluid parcel (in ), * represents Earth-relative eastward component of velocity of the fluid parcel (in ), * represents

(in ), and * represents angular rate of

(in , usually ). At the North Pole and South Pole (latitude ), no absolute angular momentum can exist ( because ). If a fluid parcel with no eastward wind speed () originating at the equator ( so ) conserves its angular momentum () as it moves poleward, then its eastward wind speed increases dramatically: . After those substitutions, , or after further simplification, . Solution for gives . If (), then . The zonal

pressure gradient In hydrodynamics and hydrostatics, the pressure gradient (typically of air but more generally of any fluid) is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular locat ...

and eddy stresses cause

torque In physics and mechanics, torque is the rotational analogue of linear force. It is also referred to as the moment of force (also abbreviated to moment). The symbol for torque is typically \boldsymbol\tau, the lowercase Greek letter ''tau''. Wh ...

that changes the absolute angular momentum of fluid parcels.

References

{{Citation , last=Holton , first=James R. , last2=Hakim , first2=Gregory J. , year=2012 , title=An introduction to dynamic meteorology , version=5 , publisher=

Academic Press Academic Press (AP) is an academic book publisher founded in 1941. It launched a British division in the 1950s. Academic Press was acquired by Harcourt, Brace & World in 1969. Reed Elsevier said in 2000 it would buy Harcourt, a deal complete ...

, publication-place=Waltham, Massachusetts , isbn=978-0-12-384866-6 , pages=342–343 Angular momentum Meteorological concepts Rotation