Climate Modeling
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Transcript of Climate Modeling
Climate ModelingClimate ModelingIn-Class Discussion:In-Class Discussion:
Legendre PolynomialsLegendre Polynomials
P0(x) = 1
P1(x) = x
P2(x) = (3x2 - 1)/2
P3(x) = (5x3 - 3x)/2
P4(x) = (35x4 - 30x2 + 3)/8
P5(x) = (63x5 - 70x3 + 15x)/8
P6(x) = (231x6 - 315x4 + 105x2 - 5)/16
Legendre Polynomials 0 - 6 Legendre Polynomials 0 - 6
P0(x) = 1P2(x) = (3x2 - 1)/2P4(x) = (35x4 - 30x2 + 3)/8P6(x) = (231x6 - 315x4 + 105x2 - 5)/16
Plots: Even PolynomialsPlots: Even Polynomials
P1(x) = xP3(x) = (5x3 - 3x)/2P5(x) = (63x5 - 70x3 + 15x)/8
Plots: Odd Polynomials Plots: Odd Polynomials
Why? Convenient properties on the sphere when using x = sin(lat)
Some examples:
(a) Even Pn (e.g., above) satisfy boundary conditions 1 & 2
All = 0 at x = 0. All are finite at x = 1.
Basis Functions: Legendre Polynomials (1)Basis Functions: Legendre Polynomials (1)
Why? Convenient properties on the sphere when using x = sin(lat)
Eigenfunctions of this operator on the sphere.
Simplifies evaluation of the derivatives (calculus becomes algebra).
(b)
Basis Functions: Legendre Polynomials (2)Basis Functions: Legendre Polynomials (2)
Why? Convenient properties on the sphere when using x = sin(lat)
Polynomials of different degrees are orthogonal.
(c)
Basis Functions: Legendre Polynomials (3)Basis Functions: Legendre Polynomials (3)
NOTE: The integral above is like taking the dot product with vectors:(A1,B1).(A2,B2) = A1A2 + B1B2
= 0 if the vectors are orthogonalThe “components” of Pn are its values at each x.
In-Class DiscussionIn-Class DiscussionLegendre PolynomialsLegendre Polynomials
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