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In meteorology, absolute angular momentum is the angular momentum in an 'absolute' coordinate system (absolute time and space).
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,
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
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).
In the shallow troposphere of the Earth, humans can approximate r ≈ a, the distance between the fluid parcel and the center of the Earth approximately equal to the mean Earth radius:
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) + Ω a2 cos2(φ0) = ua cos(φ) + Ω a2 cos2(φ). After those substitutions, Ω a2 = ua cos(φ) + Ω a2 cos2(φ), or after further simplification, Ω a(1-cos2(φ)) = u cos(φ). Solution for u gives Ω a(1/cos(φ) − cos(φ)) = u. If φ = 15° (cos(φ) = 1+√3/2√2), then 72.921150 × 10−6rad/s × 6.371009 Mm ×(2√2/1+√3 − 1+√3/2√2) ≈ 32.2m/s ≈ u.
The zonal pressure gradient and eddy stresses cause torque that changes the absolute angular momentum of fluid parcels.
In physics, angular momentum is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, flying discs, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it.
In mechanics and physics, simple harmonic motion is a special type of periodic motion an object experiences due to a restoring force whose magnitude is directly proportional to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an oscillation that is described by a sinusoid which continues indefinitely.
In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.
In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.
In fluid dynamics, Stokes' law is an empirical law for the frictional force – also called drag force – exerted on spherical objects with very small Reynolds numbers in a viscous fluid. It was derived by George Gabriel Stokes in 1851 by solving the Stokes flow limit for small Reynolds numbers of the Navier–Stokes equations.
Rossby waves, also known as planetary waves, are a type of inertial wave naturally occurring in rotating fluids. They were first identified by Sweden-born American meteorologist Carl-Gustaf Arvid Rossby in the Earth's atmosphere in 1939. They are observed in the atmospheres and oceans of Earth and other planets, owing to the rotation of Earth or of the planet involved. Atmospheric Rossby waves on Earth are giant meanders in high-altitude winds that have a major influence on weather. These waves are associated with pressure systems and the jet stream. Oceanic Rossby waves move along the thermocline: the boundary between the warm upper layer and the cold deeper part of the ocean.
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.
Particle displacement or displacement amplitude is a measurement of distance of the movement of a sound particle from its equilibrium position in a medium as it transmits a sound wave. The SI unit of particle displacement is the metre (m). In most cases this is a longitudinal wave of pressure, but it can also be a transverse wave, such as the vibration of a taut string. In the case of a sound wave travelling through air, the particle displacement is evident in the oscillations of air molecules with, and against, the direction in which the sound wave is travelling.
In physics, a wave vector is a vector used in describing a wave, with a typical unit being cycle per metre. It has a magnitude and direction. Its magnitude is the wavenumber of the wave, and its direction is perpendicular to the wavefront. In isotropic media, this is also the direction of wave propagation.
In physics, the gyromagnetic ratio of a particle or system is the ratio of its magnetic moment to its angular momentum, and it is often denoted by the symbol γ, gamma. Its SI unit is the radian per second per tesla (rad⋅s−1⋅T−1) or, equivalently, the coulomb per kilogram (C⋅kg−1).
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. In technology, it occurs in paint, MEMS devices, and in the flow of viscous polymers generally.
Rotation around a fixed axis or axial rotation is a special case of rotational motion around an axis of rotation fixed, stationary, or static in three-dimensional space. This type of motion excludes the possibility of the instantaneous axis of rotation changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rotation theorem, simultaneous rotation along a number of stationary axes at the same time is impossible; if two rotations are forced at the same time, a new axis of rotation will result.
The Eötvös effect is the change in measured Earth's gravity 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.
In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.
In general relativity, Lense–Thirring precession or the Lense–Thirring effect is a relativistic correction to the precession of a gyroscope near a large rotating mass such as the Earth. It is a gravitomagnetic frame-dragging effect. It is a prediction of general relativity consisting of secular precessions of the longitude of the ascending node and the argument of pericenter of a test particle freely orbiting a central spinning mass endowed with angular momentum .
In mathematics, vector spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions expressed in the spherical coordinate basis vectors.
In fluid dynamics, Airy wave theory gives a linearised description of the propagation of gravity waves on the surface of a homogeneous fluid layer. The theory assumes that the fluid layer has a uniform mean depth, and that the fluid flow is inviscid, incompressible and irrotational. This theory was first published, in correct form, by George Biddell Airy in the 19th century.
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.
In classical mechanics, the central-force problem is to determine the motion of a particle in a single central potential field. 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 a few 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.
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
