How we know what the weather was like
How do you represent variables in
spectral coordinates?
A generic variable, call it "A" in an atmospheric model can change with space and time. Space coordinates (instead of x, y and z) are θ (latitude), λ (longitude), and σ (height above the surface). Spectral coordinates are functions of latitude and longitude, so to represent a variable there must be another function of height and time:
where A is whatever variable we are looking at, Y is
a spherical harmonic function, and M and N are constraints on how we truncate the infinite spectrum. Because we
know that the harmonic functions are orthogonal, we can use that
constraint to narrow down what A has to be. To give specific
examples, the representation of the wind relative vorticity and
divergence,
C1 and C2 are constants, ψ is the stream function,
and x is the
potential velocity, all written in terms of horizontal wavenumbers.
Design by JeremyD