Climate Modeling InClass Discussion Legendre Polynomials Legendre Polynomials

Climate Modeling In-Class Discussion: Legendre Polynomials

Legendre Polynomials 0 - 6 P 0(x) = 1 P 1(x) = x P 2(x) = (3 x 2 - 1)/2 P 3(x) = (5 x 3 - 3 x)/2 P 4(x) = (35 x 4 - 30 x 2 + 3)/8 P 5(x) = (63 x 5 - 70 x 3 + 15 x)/8 P 6(x) = (231 x 6 - 315 x 4 + 105 x 2 - 5)/16

Plots: Even Polynomials P 0(x) = 1 P 2(x) = (3 x 2 - 1)/2 P 4(x) = (35 x 4 - 30 x 2 + 3)/8 P 6(x) = (231 x 6 - 315 x 4 + 105 x 2 - 5)/16

Plots: Odd Polynomials P 1(x) = x P 3(x) = (5 x 3 - 3 x)/2 P 5(x) = (63 x 5 - 70 x 3 + 15 x)/8

Basis Functions: Legendre Polynomials (1) 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 (2) Why? Convenient properties on the sphere when using x = sin(lat) (b) Eigenfunctions of this operator on the sphere. Simplifies evaluation of the derivatives (calculus becomes algebra).

Basis Functions: Legendre Polynomials (3) Why? Convenient properties on the sphere when using x = sin(lat) (c) Polynomials of different degrees are orthogonal. NOTE: The integral above is like taking the dot product with vectors: (A 1, B 1). (A 2, B 2) = A 1 A 2 + B 1 B 2 = 0 if the vectors are orthogonal The “components” of Pn are its values at each x.

In-Class Discussion Legendre Polynomials ~ End ~