# Absolute angular momentum

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In meteorology, absolute angular momentum refers to the angular momentum in an 'absolute' coordinate system (absolute time and space). Meteorology is a branch of the atmospheric sciences which includes atmospheric chemistry and atmospheric physics, with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not occur until the 18th century. The 19th century saw modest progress in the field after weather observation networks were formed across broad regions. Prior attempts at prediction of weather depended on historical data. It was not until after the elucidation of the laws of physics and more particularly, the development of the computer, allowing for the automated solution of a great many equations that model the weather, in the latter half of the 20th century that significant breakthroughs in weather forecasting were achieved. In physics, angular momentum is the rotational equivalent of linear momentum. It is an important quantity in physics because it is a conserved quantity—the total angular momentum of a system remains constant unless acted on by an external torque. In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is significant, and they are sometimes identified by their position in an ordered tuple and sometimes by a letter, as in "the x-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and vice versa; this is the basis of analytic geometry.

## Introduction

Angular momentum L equates with the cross product of the position (vector) r of a particle (or fluid parcel) and its absolute linear momentum p, equal to mv, the product of mass and velocity. Mathematically, In mathematics and vector algebra, the cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol . Given two linearly independent vectors and , the cross product, , is a vector that is perpendicular to both and and thus normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming. It should not be confused with the dot product.

In geometry, a position or position vector, also known as location vector or radius vector, is a Euclidean vector that represents the position of a point P in space in relation to an arbitrary reference origin O. Usually denoted x, r, or s, it corresponds to the straight line segment from O to P. In other words, it is the displacement or translation that maps the origin to P:

In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, while—in a compressible flow—its volume may change. And its shape changes due to the distortion by the flow. In an incompressible flow the volume of the fluid parcel is also a constant.

$\mathbf {L} =\mathbf {r} \times m\mathbf {v}$ ## 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 v = (u, v, w) (eastward, northward, and upward). The magnitude of the absolute angular momentum L per unit mass m

$\left|{\frac {\mathbf {L} }{m}}\right|=M=ur\cos(\phi )+\Omega r^{2}\cos ^{2}(\phi )$ where

• M represents absolute angular momentum per unit mass of the fluid parcel (in m2/s),
• r represents distance from the center of the earth to the fluid parcel (in m),
• u represents earth-relative eastward component of velocity of the fluid parcel (in m/s),
• φ represents latitude (in rad), and
• Ω represents angular rate of Earth's rotation (in rad/s, usually 2 π rad / 72.921150 × 10−6rad/s). In geography, latitude is a geographic coordinate that specifies the north–south position of a point on the Earth's surface. Latitude is an angle which ranges from 0° at the Equator to 90° at the poles. Lines of constant latitude, or parallels, run east–west as circles parallel to the equator. Latitude is used together with longitude to specify the precise location of features on the surface of the Earth. On its own, the term latitude should be taken to be the geodetic latitude as defined below. Briefly, geodetic latitude at a point is the angle formed by the vector perpendicular to the ellipsoidal surface from that point, and the equatorial plane. Also defined are six auxiliary latitudes which are used in special applications. Earth's rotation is the rotation of Planet Earth around its own axis. Earth rotates eastward, in prograde motion. As viewed from the north pole star Polaris, Earth turns counter clockwise. The radian is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees. The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.

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 of the earth, one can approximate ra, the distance between the fluid parcel and the center of the earth approximately equal to the mean Earth radius: The troposphere is the lowest layer of Earth's atmosphere, and is also where nearly all weather conditions take place. It contains approximately 75% of the atmosphere's mass and 99% of the total mass of water vapor and aerosols. The average height of the troposphere is 18 km in the tropics, 17 km in the middle latitudes, and 6 km in the polar regions in winter. The total average height of the troposphere is 13 km. Earth radius is the distance from a selected center of Earth to a point on its surface, which is often chosen to be sea level, or more commonly, the surface of an idealized ellipsoid representing the shape of Earth. Because Earth is not a perfect sphere, the determination of Earth's radius can have several values, depending on how it is measured; from its equatorial radius of about 6,378 kilometres to its polar radius of about 6,357 kilometres.

$M\approx ua\cos(\varphi )+\Omega a^{2}\cos ^{2}(\varphi )$ where

• a represents Earth radius (in m, usually 6.371009 Mm)
• M represents absolute angular momentum per unit mass of the fluid parcel (in m2/s),
• u represents earth-relative eastward component of velocity of the fluid parcel (in m/s),
• φ represents latitude (in rad), and
• Ω represents angular rate of Earth's rotation (in rad/s, usually 2 π rad / 72.921150 × 10−6rad/s).

At the North Pole and South Pole (latitude φ=±90°=π/2rad), no absolute angular momentum can exist (M=0 m2/s because cos(±90°)=0). If a fluid parcel with no eastward wind speed (u0=0m/s) originating at the equator (φ=0 rad so cos(φ)= cos(0 rad) = 1) conserves its angular momentum (M0 =M) as it moves poleward, then its eastward wind speed increases dramatically: u0a cos(φ0) + Ωa2 cos2(φ0) = ua cos(φ) + Ωa2 cos2(φ). After those substitutions, Ωa2 = ua cos(φ) + Ωa2 cos2(φ), or after further simplification, Ωa(1-cos2(φ)) = u cos(φ). Solution for u gives Ωa(1/cos(φ) - cos(φ)) = u. If φ = 15° (cos(φ)=1+3/22), then 72.921150 × 10−6rad/s× 6.371009 Mm ×(22/1+3 - 1+3/22) 32.2m/su.

The zonal pressure gradient and eddy stresses cause torque that changes the absolute angular momentum of fluid parcels. The terms zonal and meridional are used to describe directions on a globe.

In atmospheric science, the pressure gradient is a physical quantity that describes in which direction and at what rate the pressure increases the most rapidly around a particular location. The pressure gradient is a dimensional quantity expressed in units of pressure per unit length, or Pa/m In fluid dynamics, an eddy is the swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime. The moving fluid creates a space devoid of downstream-flowing fluid on the downstream side of the object. Fluid behind the obstacle flows into the void creating a swirl of fluid on each edge of the obstacle, followed by a short reverse flow of fluid behind the obstacle flowing upstream, toward the back of the obstacle. This phenomenon is naturally observed behind large emergent rocks in swift-flowing rivers.

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Holton, James R.; Hakim, Gregory J. (2012), An introduction to dynamic meteorology, 5, Waltham, Massachusetts: Academic Press, pp. 342–343, ISBN   978-0-12-384866-6