A few more Math, Physics & Related Topics

This feature and the next are required before I begin discussion of topics relating more specifically with weather; and even then, some fundamental physics & chemistry concepts must be mentioned.

Force balance should be described with more detail, being the basis of much of what we observe in the physical world. When previously discussing Earth's gravitational acceleration, it was presented as a sum of vectors. Newton's Second Law of motion can be written F = m a, meaning that acceleration occurs to the specific direction a force is applied, which can be a sum of several forces :

Though objects considered "particles" such as crudely depicted above do not often interest us, these forces apply at each point in the primarily gaseous atmosphere. Here is another interesting discussion regarding Newton's Laws.

Projectile motion is relevant because some objects in our atmosphere either are or can be treated as projectiles. E.g., falling hailstones, raindrops, or even clouds droplets.

Applying Netwon's Laws, equations describing this are found from simple integration. Neglecting air friction for simplicity, Netwon's 1^st Law implies horizontal speed (u = dx/dt) won't change (i.e., no horizontal forces applied). Vertical acceleration of the object is simply that because of gravity, which is approximately 9.81 m/sec² as indicated above. From simple integration, equations describing vertical motion of the projectile are :

Acceleration : a = g = 9.81 m/sec²
Speed : w = a t = 9.81 t m/sec
Position : z = ½ a t² = 4.905 t² m

I conveniently chose my coordinate axes such that (x,z) = (0,0) at the point of release (thus integration constants are 0 above), the positive z-axis pointing down to simplify calculation (thus gravitational acceleration is to the positive z direction), and mks units are used. Please be aware that almost everything subsequently written will use z as positive to the vertical - upward in our atmosphere. I think this illustration is relevant because it illustrates a technique of simplifying a problem - a very useful skill for practical application of things learned !

Quite often only an additional consideration of air drag is sufficient for analyzing forces on such an object. Though this is not extremely necessary for accomplishing the mission discussed, such analyses improve a person's perceptions of forces influencing weather they observe.

Chemical Elements : Matter consists of combinations of chemical elements. E.g., water's chemical formula is H₂O, meaning a each water molecule consists of 2 hydrogen (H) and 1 oxygen (O) atom. Protons, neutrons, and electrons are the fundamental components of matter, masses of which are :

Proton : 1.67265 × 10^-27 kg
Neutron : 1.67495 × 10^-27 kg
Electron : 9.10953 × 10^-31 kg

As the name implies, electrons have a (electronic) charge. This has been chosen as being a negative (-) charge. Protons have an opposite (positive (+)) charge, and as the name implies, neutrons no (neutral) charge. Elements are often displayed in a periodic table, which among other things indicates :

Atomic number - Number of protons, which characterizes a distinct element. Number of neutrons in an element can vary (isotopes), but adding or eliminating a proton changes element type. E.g., 6 protons are in each carbon atom, 7 in Nitrogen.
Atomic mass - Mass of a mole of atoms of an element, with a standard distribution of isotopes assumed.

Organization of the table, isotopes, ions, etc. can be topics of very lengthy discussion. Many of these features are shown & explained at the site linked to. Thus, I don't mention any more, though a weather forecaster should know basic atomic theory. Chemical reactions are not very meteorologically important, unless environmental issues concern you, but phase changes are - particularly for water. This link is from HyperPhysics, an excellent online resource !

Phases of Matter Matter exists in 4 basic phases with which we are all familiar - solid, liquid, gas, and plasma - the first 3 of which are common on Earth. Substances can acquire each of the phases, which depends on temperature and pressure. A phase diagram can be determined for each substance, as I illustrated for water. At the triple point for water, all 3 phases (ice, liquid, and gas) co-exist in equilibrium (meaning that some ice molecules become liquid, some liquid molecules freeze, etc., but total number of each type remains same). Note that such occurs at a ridiculously small pressure of 6.11 mb though, something we don't see in our ~ 1000 mb atmosphere. Similarly, a critical point exists for which liquid water and gas cannot be distinguished - with temperature > 647 K and pressure > 2.22 × 10⁷ Pa (219 × standard sea level pressure). The phase diagram is very relevant for meteorology though, because it indicates how much water vapor air can exist at specific temperatures, and evaporation/condensation, melting/freezing, and sublimation/deposition processes which are important regarding cloud and frost formation, cooling via evaporation, and other things.

Text and images are copyright of Joseph Bartlo, though may be used with proper crediting.

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