Consider if you wish, 2-dimensional coordinate axes corresponding previously discussed :
Similarly, for 3-dimensional coordinate axes :
Relations among representations of points are :
Thus, if position of a point for either system is known, it can be determined for the other. These are the 4 main coordinate systems meteorologists use, often with modifications. E.g., for atmospheric modeling, points are often described using 3-dimensional Cartesian coordinate system with the origin at some place on Earth's surface; such that equations for motion must be modified to account for the fact that the x-y "plane" is curved, etc.
As an example, consider the spherical coordinate point shown above. For Cartesian coordinates, we can calculate :
x = 2.35 cos(55°) cos(60°) = .67 For the Cartesian coordinate systems shown above, unit vectors can be defined - vectors
parallel with each axis with length 1 :
Forces, velocities, etc. can be described using unit vectors. E.g., suppose the point
described above represents a 2.35 N (Newton) force. X, Y, and Z components of this force are :
Fx = .67 N Fy = 1.17 N Fz = 1.93 N
Thus, you can see that :
y = 2.35 cos(55°) sin(60°) = 1.17
z = 2.35 sin(55°) = 1.93
F = i Fx + j Fy + k Fz
bold letters indicating vector quantities. Thus, a vector quantity can be described as a sum
of the product of unit vectors and scalar components.
2 operations often used for meteorology are scalar and cross products. Scalar product of 2
vectors is :
A · B = Ax Bx + Ay By + Az Bz = |A||B| cos(a)
for which || represents absolute value - positive value of the quantity contained (magnitude
of the vector), and "a" angle between the vectors. Note that scalar product is an expression of
how parallel vectors are. It is expressed as a number relative with the product of magnitudes of
the vectors. Cross product of 2 vectors is :
A × B = i (Ay Bz - Az By) + j (Az Bx - Ax Bz) + k (Ax By - Ay Bx) = |A||B| sin(a)
It is a vector (magnitude and direction), direction determined using the "right-hand rule"
(described below). Thus, 2 vectors for a cross product must be written in the proper order.
I.e., A × B points in the opposite direction of B × A.
As an example, suppose 2 vectors A & B (these may be forces, etc.) :
Direction for the cross product is determined using the 'right-hand rule'. I.e., open your
right palm & curl your fingers in the plane of A & B from A toward B,
closing your hand. Point your thumb up also Why mention these things ? Because meteorological analysis requires such specifications.
Consider the temperature example of gradient I previously showed :
On this diagram I now suppose a cold front just passed the location indicated. I include a
wind velocity, specified as both magnitude and direction, and temperature change components at a
location. I previously mentioned that cold fronts are regions with a large temperature gradient,
where storms often occur. If you are unfamiliar with calculus, this may not be so meaningful
to you; but even so, you can probably see what I describe. A gradient can be mathematically
expressed using the del operator d representing 'change'. Ideally, this is an infinitesimal change at the point of interest,
but approximating it with finite changes (finite differences) is okay. Because our map is
2-dimensional, only the i & j (x & y) components concern us. (If it did,
calculation of the k (z) component would also be included). Temperature gradient
How much does temperature change in the x & y directions ? On our map, we see that temperature
change from A to B is 18.8 °F, which occurs over a distance of 70711 m. Thus, temperature
gradient is
about ¼ °F/km. Expressed as components, this is :
Note that I am labeling units now because I am not using strictly MKS units.
Advection is a meteorological term used for transport of a property from one location to
another. Mathematically, for temperature it is :
The direction
your thumb points is the direction of A × B (toward you for this example).
T advection = - V · ÑT = - Vx dT/dx - Vy dT/dy
For this, we consider the teperature differences centered at the location shown (i.e., across
the point along x & y directions). If wind velocity V is as shown, temperature advection
is :
T advection = - (9.06 m/sec)(9.9°F)/(50000 m) - (-4.23 m/sec)(-11.4°F)/(50000 m)
= -.002758 °F/sec = -9.93 °F/hr
Temperature at the point indicated is decreasing .002758 °F/sec or because 3600 sec are an hour,
9.93 °F/hr (assuming conditions remained same). That is only because of advection. Other factors
influence temperature also - radiative cooling, turbulence, etc., but would be a realistic estimate
for temperature decrease during the next hour. Hopefully this illustrates usefulness of vector
representations.
For reference, the Cartesian representation of gradient using polar coordinates is :
and using spherical coordinates is :