1. Climate Modeling In-Class Discussion: Legendre Polynomials

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

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

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

5. Why? Convenient properties on the sphere when using x = sin(lat) Some examples: (a) Even P n (e.g., above) satisfy boundary conditions 1 & 2 All = 0 at x = 0. All are finite at x = 1. Basis Functions: Legendre Polynomials (1)

6. 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)

7. Why? Convenient properties on the sphere when using x = sin(lat) Polynomials of different degrees are orthogonal. (c) Basis Functions: Legendre Polynomials (3) 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 P n are its values at each x.

8. In-Class Discussion Legendre Polynomials ~ End ~