Meteorology/Dynamics/Kinematics

Kinematic Structures

The kinematics branch of dynamics describes the properties of pure motion without regard to force, momentum, or energy--topics discussed in later subchapters. Kinematic variables include translation, advection, and deformation.

Meteorologists use various conventions to apply the laws of physics to the atmosphere. The first set of conventions involves the coordinate system. Each application may imply or call for a different set of conventions. Meteorologists use some conventions more commonly than others and sometimes choose conventions for the problem at hand or to comport with those utilized in available tools. In confusing situations, meteorologists use multiple incompatible conventions simultaneously.

Coordinate systems

A coordinate system or reference frame uniquely identifies each point of the atmosphere (or any other continuum). The atmosphere of the earth mostly concentrates in a very thin shell around an almost spherical planet earth, making traditional inertial reference frames mathematically difficult to apply to the equations of atmospheric dynamics.

Measuring distance: meters

The position vector leads from the origin of the coordinate system to a point in space, thus specifying the position of this point relative to the chosen coordinate system. In any coordinate system, at least one specified component of the position vector must have dimensions of length and hence fundamental units of meters in International System of Units (le Système international d'unités, SI).

The Revolutionary French originally devised the International System of Units, hence its French name and acronym. The French Academy of Sciences in 1791 defined the meter such that ten million meters along a meridian of constant longitude equal the length of one quadrant, the shortest distance along the surface of the earth from the Equator to the North Pole through Paris, France. According to this definition, the circumference of the spherical earth--four times the distance along the surface of the earth from the Equator to the North Pole--equals 40 million meters (40×10⁶ m or 40,000 km). Because the circumference of a circle (a slice through the center of the earth) equals $2\pi$ times the radius of the earth $R_{earth}$ would equal ${\frac {40\times 10^{6}{\mbox{ m}}}{2\pi }}\approx 6,366,197{\mbox{ m}}$.

To enable the use of the meter in practical applications, the French scientists who defined the meter made a particular metal bar exactly one meter in length. A slight error in their estimate of meridional circumference of the earth left the bar 0.2 mm too short for the corresponding definition of the meter. The meridional circumference of the earth, the original basis for the old definition of the meter, measures 40,008.00 km. The earth does not take the form of a perfect sphere; its equatorial circumference measures a slightly wider 40,075.16 km. The radius of the earth actually varies with latitude from 6356.8 km at the poles to 6378.1 km at the equator. The official bar changed from a brass prototype in 1795 to a platinum one in 1799. In 1889, the first General Conference on Weights and Measures defined the meter officially as the distance between two lines on a particular bar of an alloy of platinum with ten percent iridium, measured at the melting point of ice.

In 1960, this official definition changed to make one meter equal to 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. In 1983, the conference redefined the meter to make the speed of light in a vacuum exactly equal to 299,792,458 m/s (meters per second). The speed of light in a vacuum thus now provides the official definition of the meter.

Prefixes

To express very large and very small lengths, scientists use scientific notation and prefixes. These prefixes also apply to any other units within the International System of Units. In the United States of America, the Constitution gives the Congress the power to "fix the standard of weights and measures", and National Institute of Standards and Technology implements various acts of Congress dealing with weights and measures. The units used in this book conform to this implementation.

SI prefixes

Prefix	Symbol	1000m	10n	Decimal	American word
yotta-	Y	10008	1024	1 000 000 000 000 000 000 000 000	septillion
zetta-	Z	10007	1021	1 000 000 000 000 000 000 000	sextillion
exa-	E	10006	1018	1 000 000 000 000 000 000	quintillion
peta-	P	10005	1015	1 000 000 000 000 000	quadrillion
tera-	T	10004	1012	1 000 000 000 000	trillion
giga-	G	10003	109	1 000 000 000	billion
mega-	M	10002	106	1 000 000	million
kilo-	k	10001	103	1 000	thousand
hecto-	h	10002/3	102	100	hundred
deca-	da	10001/3	101	10	ten
		10000	100	1	one
deci-	d	1000−1/3	10−1	0.1	tenth
centi-	c	1000−2/3	10−2	0.01	hundredth
milli-	m	1000−1	10−3	0.001	thousandth
micro-	μ	1000−2	10−6	0.000 001	millionth
nano-	n	1000−3	10−9	0.000 000 001	billionth
pico-	p	1000−4	10−12	0.000 000 000 001	trillionth
femto-	f	1000−5	10−15	0.000 000 000 000 001	quadrillionth
atto-	a	1000−6	10−18	0.000 000 000 000 000 001	quintillionth
zepto-	z	1000−7	10−21	0.000 000 000 000 000 000 001	sextillionth
yocto-	y	1000−8	10−24	0.000 000 000 000 000 000 000 001	septillionth

Other acceptable units

Scientists occasionally use alternative units beyond those in the International System of Units. For length, these include the astronomical unit (ua), approximately the mean distance between the earth and the sun. Technically defined as the radius of an unperturbed circular Newtonian orbit about the Sun of a particle of infinitesimal mass, moving with a mean motion of 0.017 202 098 95 radians per day (Gaussian constant), one astronomical unit approximately equals 149.597 870 691 × 10⁹ m.

When describing the atomic unit, scientists occasionally use the Bohr radius ($a_{0}$) as a unit of length

Meteorologists (and mariners) frequently use the nautical mile, traditionally defined as the distance along the surface of the earth corresponding to one arc minute of latitude. The actual distance so corresponding varies somewhat with latitude. International agreement defined the nautical mile as exactly 1852 m in 1929; the United States of America adopted this definition in 1954; earlier documents may use a slightly different interpretation. Sixty arc minute of latitude correspond to one arc degree of latitude; ninety arc degrees of latitude constitute the quarter-circumference of the earth originally used to define the meter. Mathematically, 1852 m × 90 × 60 × 4 = 40,003,200 m, slightly more than the actual meridional circumference of the earth.

Unacceptable units

Unacceptable units of length, sometimes encountered in publications, include fermi for 10^-15 m = 1 fm (femtometer) and the micron (μ) for 10^-6 m = 1 μm (micrometer).

Customary units

One meter equals 3.2808399 feet or 39.370079 inches.

Curvilinear coordinate systems

A spherical coordinate system, approximating the earth as a sphere, represents every point according to its latitude φ, longitude λ, and distance r from the center of the earth.

Cartographers use maps to represent the surface of the spherical planet on a flat sheet of paper. Meteorologists likewise can adapt the equations of motion to Cartesian coordinates used in maps.

Eulerian coordinates

An Eulerian coordinate system refers to points in physical space.

Lagrangian coordinates

A Lagrangian coordinate system follows a parcel of moving air.