Absolute angular momentum

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

Introduction

Angular momentum Template:Math equates with the cross product of the position (vector) Template:Math of a particle (or fluid parcel) and its absolute linear momentum Template:Math, equal to Template:Math, the product of mass and velocity. Mathematically,

$𝐋=𝐫\times m𝐯$

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 Template:Math (eastward, northward, and upward). The magnitude of the absolute angular momentum Template:Math per unit mass Template:Math

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

where

• Template:Math represents absolute angular momentum per unit mass of the fluid parcel (in Template:Math),
• Template:Math represents distance from the center of the Earth to the fluid parcel (in Template:Math),
• Template:Math represents earth-relative eastward component of velocity of the fluid parcel (in Template:Math),
• Template:Math represents latitude (in Template:Math), and
• Template:Math represents angular rate of Earth's rotation (in Template:Math, usually Template:Math).

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

$M\approx ua\cos (\phi )+\mathrm{\Omega }{a}^2{\cos }^2(\phi )$

where

• Template:Math represents Earth radius (in Template:Math, usually Template:Math)
• Template:Math represents absolute angular momentum per unit mass of the fluid parcel (in Template:Math),
• Template:Math represents Earth-relative eastward component of velocity of the fluid parcel (in Template:Math),
• Template:Math represents latitude (in Template:Math), and
• Template:Math represents angular rate of Earth's rotation (in Template:Math, usually Template:Math).

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

The zonal pressure gradient and eddy stresses cause torque that changes the absolute angular momentum of fluid parcels.