These beginning articles may seem quite simplistic, though most readers are probably unaware of some subtle details of these simplistic topics. Because I desire a series of articles which progress from A to W, as it were, the simple foundations of our endeavor must first be discussed.
Elevation angle is angle above the local horizontal plane (related with distance of objects
in our atmosphere), and azimuth angle is direction. Please notice that my specification
differs from the standard of 0° being North at any location. My primary reason for this is
not confusing people - rather that time is determined using solar noon as a reference, which is
the time the solar disc crosses the local meridian, which should be considered 0 similarly
as 0 separates positive & negative numbers on the number line :
Solar transit of the local meridian is most often directly equatorward at any location not at it, which is 0° azimuth in the illustrations above.
Day of month is chosen as the abscissa and temperature as the ordinate. Though such is arbitrary, it is more appealing as shown. Data was obtained from NCDC's CLIMVIS. A polar coordinate example is the hodograph (upper left section) from an atmospheric sounding for Topeka, KS, from UCAR's Realtime Weather Data site. The reference axis is North, and wind direction during balloon ascent is indicated as its angle from it, positive clockwise (standardly-defined azimuth). Distance from the origin represents wind speed. 3-dimensional plots using spherical coordinates or 3-dimensional Cartesian coordinates can be made, though representation in 2 dimensions is difficult. Such plots can be extended for representing physical systems, e.g., ocean surface oscillations, from NOAA-CIRES CLimate Diagnostics Center. Plots and mathematics for coordinate systems greater than 3 dimensions are made, some with practical applications. Time can be considered a 4th dimension, though not a physical dimension.
.625 = 625/1000 = 5/8, 6 = 6/1, etc.
Calculations are only exact when such ratios are retained. E.g., suppose you wish calculating :
(5/7)(3/26 + 17/9)
(A product and sum of rational numbers). You can write approximate decimal numbers as :
5/7 = .71429, 3/26 = .11538, 17/9 = 1.88889
Thus :
(.71429)(.11538 + 1.88889) = (.71429)(2.00421) = 1.4315871609
But retaining rational numbers, you must use the least common denominator of added terms, which is 26 × 9 = 234. Thus :
(5/7)((27 + 442)/234)) = (5/7)(469/234) = 2345/1638
which is the (only) correct answer. Expressed as decimals, this is :
1.4316239316239...
... indicating that the sequence continues (rather than its usual indication of a
smart-alecky comment Notice that the approximate answer above only agrees with the exact rational answer (if you
round it) for the least number of approximating digits used (5), which is typical.
Among the worst mistakes that people as a group ever made was deciding to represent numbers
with our decimal system. That's an exaggeration, but with a point (perhaps also words with
multiple meanings, such as point). Most things have a binary character. E.g., either a
branch is attached to a tree or separated, either you catch a fish or don't, a musical scale
consists of repeating octaves (though the number of notes in such is arbitrary), either you
like a someone or hate him Whichever number system is used, it represents counting. Joseph means he shall add -
so I know about this. I'm not sure regarding accuracy or significance of such meanings, but some
people worked determining those I suppose. Thus, you should not disregard alternative methods of
forecasting events (e.g., psychic powers, premonitions, dreams, etc.), even if that event is
weather. Previous societies probably did not consider such things so much because short-term
weather did not affect them nearly so much as us (e.g., aviation), though long-term weather
(e.g., drought) did more so. Another Joseph is recorded as
effectively
dealing with a future drought (I am the probably "other" one actually - being named after
him - quite literally). A main problem with such things is false prophets though, which exist
regarding most any endeavor. People will claim that a person "has salvation", etc., when they
are really unsure. If a situation occurs such as the drought mentioned above, the person who
knowledge has been revealed to is perhaps as certain of it as you are that when you drop a ball
it will fall to the ground. That is a slight digression.
Addition is the only computational operation.
Every other - subtraction, division, derivatives, etc. - are (some quite complicated)
combinations of addition. An integral is analogous with multiplication (a different form of it),
and a derivative analogous with division; which are all consequences of addition.
Because of analogies mentioned, I include basic calculus as part of algebra in my list
of topics. For theoretical and thorough meteorological studies, a thorough knowledge of calculus
basics is necessary; but as I've previously mentioned, not for accomplishing the mission here.
Now I mention the concept gradient, being quite important for us. If you examine almost
any weather map, you'll see gradients with various magnitude. Because large gradients are
locations of greatest change, such are often locations of weather which most interests us -
especially regarding temperature :
). Notice that the sequence 316239
continues repeating. The decimal sequence representing any rational number either ends or repeats.
The sequence of an irrational number such as p = 3.14159265... (circumference of a circle with diameter 1) does neither.
A great majority of real numbers are actually irrational - if points of the number line above
were white if rational & black if irrational, it would appear almost same as above - only a
very slight shade of grey. Perhaps the term real numbers is a misnomer though, because
irrational numbers may describe few if any real objects. I.e., even if you think something
is circular and a physical law implies it should be, close examination would likely reveal
a collection of objects comprising it in positions which only tend toward circularity. But even
so, the law and consequential tendency would be real !
, etc. Thus things are best
represented with some form of binary numbers - base 2, 4, 8, 16, 32, etc. Computers use binary
number representations for calculation,
storage (bytes & bits), etc., because such are most efficient. For a computer, converting among
binary and decimal number representations is an extra task which slightly decreases efficiency.
If people have 10 fingers & 10 toes though, counting to 20 is natural I suppose
Fronts are regions of large temperature gradient - strongest being those with largest temperature gradient. A front is not a border between warm and cold air, but a transition zone between generally cold and warm air masses, where most active weather often occurs. I plan discussion of this much more later.
Related with gradients is interpolation. This refers to estimating magnitude of a parameter at a location between those at which the value is known or measured. Simple linear interpolation (assuming the values vary @ a constant rate between points) is often used, but regarding map analysis, can be quite incorrect in regions where large gradients exist.
I obviously cannot discuss this completely here, nor even close. Basics of these topics should be known also. A good example of probability is a 6-sided dice. If obtaining each number is as likely as any other, then the probability of obtaining each is 1/6 - 6 possibilities, only 1 of which satisfies each of them. Notice that this is the reciprocal of 6/1, mentioned above - both rational numbers. For reasons I mentioned, rational numbers should be retained when possible while doing statistical calculations involving them.
Clearly, the dice example involves rational numbers. What about meteorological variables though ? If you read a thermometer as 48.1 °F, that is quite approximate. No matter how accurate your thermometer and how careful your reading (if not digital), the actual temperature may differ. Thus I would not consider this a rational number, per se; even if the physical system determining temperature actually is (temperature is the root-mean-square kinetic energy of molecules in an ideal substance, which I plan discussion of later). Thus using 48.1 with other decimal numbers if computing monthly averages (for example) is preferable than using 48 1/10 with rational numbers. Precipitation likelihood is often expressed as a probability. Among other interpretations, it is an expression of a forecaster's uncertainty. Similar as the dice example, if (measurable) precipitation occurred approximately 4 of 10 times a weather forecaster previously felt as confident of its likelihood as a current situation, (s)he may express this as "a 40 % chance of (measurable) precipitation". I.e., 10 events, during 4 of which (measurable) precipitation occurred. Probability obviously becomes much more complicated, including conditional probabilities, combinations and permutations, functional distributions, and many other analysis techniques. Try understanding binomial trials and related distributions (particularly normal) if you can - many applications use them (perhaps I'll offer explanations later).
Statistics is closely related with probability, also much too complicated for complete discussion (if I could provide it). Ideas most commonly used for meteorology are average, mean, median, extremes (maximum & minimum), and variance (e.g., standard deviation). Standard deviation is an indicator of variance - how much values of a statistical sample vary, but assumes a normal distribution. Though many things can be approximated with statistical distributions, I'm not sure if any can be exactly described using such. Note that the idea of climate normals likely derives from the normal distribution.
Text and images are copyright of Joseph Bartlo, though may be used with proper crediting.