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Climate Modeling

This in-class discussion focuses on Legendre polynomials, specifically their applications in climate modeling. It covers polynomial definitions from P0 to P6, highlighting their even and odd properties. Additionally, it emphasizes why Legendre polynomials are convenient for modeling on the sphere, particularly when using sin(lat) for arguments. Their integral orthogonality properties facilitate the simplification of calculus into algebra, making them ideal basis functions for representing climate data. The discussion includes visual plots and insights into the significance of these polynomials in scientific analysis.

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Climate Modeling

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  1. Climate Modeling In-Class Discussion: Legendre Polynomials

  2. Legendre Polynomials 0 - 6 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

  3. Plots: Even Polynomials P0(x) = 1 P2(x) = (3x2 - 1)/2 P4(x) = (35x4 - 30x2 + 3)/8 P6(x) = (231x6 - 315x4 + 105x2 - 5)/16

  4. Plots: Odd Polynomials P1(x) = x P3(x) = (5x3 - 3x)/2 P5(x) = (63x5 - 70x3 + 15x)/8

  5. 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.

  6. 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).

  7. 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: (A1,B1).(A2,B2) = A1A2 + B1B2 = 0 if the vectors are orthogonal The “components” of Pn are its values at each x.

  8. In-Class DiscussionLegendre Polynomials~ End ~

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