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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