Absolute angular momentum

Last updated

In meteorology, absolute angular momentum refers to the angular momentum in an 'absolute' coordinate system (absolute time and space).

Meteorology Interdisciplinary scientific study of the atmosphere focusing on weather forecasting

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.

Angular momentum measure of the extent to which an object will continue to rotate in the absence of an applied torque

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.

Coordinate system System for uniquely determining the position of a point in a space of dimension n, by an n-tuplet of scalars

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.

Contents

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,

Cross product mathematical operation on two vectors

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.

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

where

Latitude The angle between zenith at a point and the plane of the equator

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.

Earths rotation rotation of the solid Earth around its own axis

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.

Radian SI derived unit of angle

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:

Troposphere The lowest layer of the atmosphere

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 mean distance from the Earths center to its surface

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.

where

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.

Zonal and meridional

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

Eddy (fluid dynamics) The swirling of a fluid and the reverse current created when the fluid is in a turbulent flow regime

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.

Related Research Articles

Coriolis force A force on objects moving within a reference frame that rotates with respect to an inertial frame.

In physics, the Coriolis force is an inertial or fictitious force that seems to act on objects that are in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motion of the object. In one with anticlockwise rotation, the force acts to the right. Deflection of an object due to the Coriolis force is called the Coriolis effect. Though recognized previously by others, the mathematical expression for the Coriolis force appeared in an 1835 paper by French scientist Gaspard-Gustave de Coriolis, in connection with the theory of water wheels. Early in the 20th century, the term Coriolis force began to be used in connection with meteorology.

Nutation A component of the motion of a freely rotating rigid object. To be distinguished from Q190009 Nutation (astronomy).

Nutation is a rocking, swaying, or nodding motion in the axis of rotation of a largely axially symmetric object, such as a gyroscope, planet, or bullet in flight, or as an intended behaviour of a mechanism. In an appropriate reference frame it can be defined as a change in the second Euler angle. If it is not caused by forces external to the body, it is called free nutation or Euler nutation. A pure nutation is a movement of a rotational axis such that the first Euler angle is constant. In spacecraft dynamics, precession is sometimes referred to as nutation.

Plane wave Type of wave propagating in 3 dimensions

In the physics of wave propagation, a plane wave is a wave whose wavefronts are infinite parallel planes. Mathematically a plane wave takes the form

In physics, the angular velocity of a particle is the rate at which it rotates around a chosen center point: that is, the time rate of change of its angular displacement relative to the origin. It is measured in angle per unit time, radians per second in SI units, and is usually represented by the symbol omega. By convention, positive angular velocity indicates counter-clockwise rotation, while negative is clockwise.

Differential rotation is seen when different parts of a rotating object move with different angular velocities at different latitudes and/or depths of the body and/or in time. This indicates that the object is not solid. In fluid objects, such as accretion disks, this leads to shearing. Galaxies and protostars usually show differential rotation; examples in the Solar System include the Sun, Jupiter and Saturn.

Spherical harmonics special functions defined on the surface of a sphere

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations that commonly occur in science. The spherical harmonics are a complete set of orthogonal functions on the sphere, and thus may be used to represent functions defined on the surface of a sphere, just as circular functions are used to represent functions on a circle via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

Particle velocity is the velocity of a particle in a medium as it transmits a wave. The SI unit of particle velocity is the metre per second (m/s). In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse wave as with the vibration of a taut string.

In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave, and its direction is ordinarily the direction of wave propagation.

Elliptic orbit Kepler orbit with an eccentricity of less than 1

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

The rigid rotor is a mechanical model of rotating systems. An arbitrary rigid rotor is a 3-dimensional rigid object, such as a top. To orient such an object in space requires three angles, known as Euler angles. A special rigid rotor is the linear rotor requiring only two angles to describe, for example of a diatomic molecule. More general molecules are 3-dimensional, such as water, ammonia, or methane.

Stokes flow

Stokes flow, also named creeping flow or creeping motion, is a type of fluid flow where advective inertial forces are small compared with viscous forces. The Reynolds number is low, i.e. . This is a typical situation in flows where the fluid velocities are very slow, the viscosities are very large, or the length-scales of the flow are very small. Creeping flow was first studied to understand lubrication. In nature this type of flow occurs in the swimming of microorganisms and sperm and the flow of lava. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.

The Coriolis frequencyƒ, 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 φ.

The Eötvös effect is the change in perceived gravitational force caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity. When moving eastbound, the object's angular velocity is increased, and thus the centrifugal force also increases, causing a perceived reduction in gravitational force.

Position angle measurement relating to observed visual binary stars

Position angle, usually abbreviated PA, is the convention for measuring angles on the sky in astronomy. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images this is a counterclockwise measure relative to the axis into the direction of positive declination.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics determined by Maxwell's equations, as an analogue for all types of particles.

Rossby-gravity waves are equatorially trapped waves, meaning that they rapidly decay as their distance increases away from the equator. These waves have the same trapping scale as Kelvin waves, more commonly known as the equatorial Rossby deformation radius. They always carry energy eastward, but their 'crests' and 'troughs' may propagate westward if their periods are long enough.

Equatorial Rossby waves, often called planetary waves, are very long, low frequency waves found near the equator and are derived using the equatorial beta plane approximation.

In classical mechanics, the central-force problem is to determine the motion of a particle under the influence of a single central force. A central force is a force that points from the particle directly towards a fixed point in space, the center, and whose magnitude only depends on the distance of the object to the center. In many important cases, the problem can be solved analytically, i.e., in terms of well-studied functions such as trigonometric functions.

Blade element momentum theory is a theory that combines both blade element theory and momentum theory. It is used to calculate the local forces on a propeller or wind-turbine blade. Blade element theory is combined with momentum theory to alleviate some of the difficulties in calculating the induced velocities at the rotor.

References

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