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1 The Spectral Method
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2 Definition where (e m,e n )=δ m,n e n = basis of a Hilbert space (.,.): scalar product in this space In L 2 space where f * : complex conjugate of f Discretization: limit the sum to a finite number of terms (consistent if the e m ’s are appropriately ordered)
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3 The Galerkin procedure Linear case If e m ’s are eigenfunctions of H: He m =λ m e m H: linear space operator R: discretization error we assume R to be a function of the omitted e m ’s only therefore R is orthogonal to e m (m ≤ M) (alternatively we minimize ||R|| 2 ) || δ m,n || 0 analytical solution (no need to discretize in t) compute once and store
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4 One-dimensional linear advection equation periodic boundary conditions Analytical solution phase speed: γ=cte (rad/s) Basis functions: Fourier functions e imλ (eigenfunctions of ∂ / ∂ λ) ω being a real field ==> ω -m (t)= ω m * (t); we need to solve only for 0≤m ≤M Galerkin procedure this is a system of 2M+1 equations (decoupled) for the (complex) ω n coefficients
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5 One-dimensional linear advection equation (2) Exact solution ω n (0) nγ/2π - in physical space the same form as the analytical solution no dispersion due to the space discretization because the derivatives are computed analytically
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6 Calculation of the initial conditions computation of ω m (0) given ω(λ,0) - Direct Fourier transform where A m : normalization factor - Inverse Fourier transform B m : normalization factors Discrete Fourier transforms - Direct : - Inverse : Transformations are exact if K 2M+1 Procedure: Fast Fourier Transform (FFT) algorithm Products of two functions have no aliassing if K 3M+1
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7 The linear grid Unfitted function Fitted with quadratic grid Fitted with linear grid 3M+1 points in λ ensure no aliassing in computations of quadratic terms (case of Eulerian advection) Quadratic grid 2M+1 points in λ ensure exact transforms of linear terms to grid-point and back Linear grid
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8 Stability analysis Leapfrog scheme no need to discretize in time if we do not have other terms in the equation Substituting and dividing by W (n-1) conditionally stable and neutral - Comparison with finite differences U 0 =Rγ M~N/2 Δx=2πR/N if using the quadratic grid M~N/3 ---->
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9 Graphical representation
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10 Non-linear advection equation = there are more wavenumbers on the r.h.s. than in the original function Galerkin procedure k=0 …. M therefore F m m>M are not used but no aliassing produced because of misrepresentation
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11 Non-linear advection equation (cont) Calculation of F k Interaction coefficients I jnk ---> interaction coeff. matrix I is not a sparse matrix Transform method I. FFT f(λ l ); l=1, … L g(λ l ); l=1, … L I. FFT F(λ l ) = - f(λ l ) g(λ l ) D. FFT can be shown to have no aliassing if L 3M+1
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12 One-dimensional gravity-wave equations Galerkin procedure no need to discretize in time
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13 One-dimensional gravity-wave equations (cont) Explicit time stepping (leapfrog) no need to transform to grid-point space Stability and dispersion (von Neumann method) assume substituting: smaller than with finite differences dispersion due solely to the time discretization
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14 One-dimensional gravity-wave equations (cont) Implicit time stepping substituting Decoupled set of equations because the basis functions e imλ are eigenfunctions of the space operator ∂ / ∂ λ with eigenvalues im Stability and dispersion using von Neumann we get stable dispersion larger than in leapfrog scheme
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15 Shallow water equations Linearize about a basic state U 0, V 0, Φ 0 and assume f=cte=f 0 (f-plane approx) substitute and neglect products of perturbations
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16 Leapfrog (explicit) time scheme Stability according to von Neumann assume and substitute
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17 Leapfrog (explicit) time scheme stability (cont) or, calling where
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18 Leapfrog (explicit) time scheme stability (cont) calling this system has non-trivial solutions if for α to have a real solution The most restrictive case is when which gives
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19 Semi-implicit time scheme stability Following the same steps as in the explicit scheme we arrive at: therefore if
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20 Spherical harmonics Orthogonal basis for spherical geometry m: zonal wavenumber n: total wavenumber λ= longitude μ= sin(θ) θ: latitude P n m : Associated Legendre functions of the first kind
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21 Spectral representation n-|m|: effective meridional wavenumber spectral transform Since X is a real field, X n -m =(X n m )* Fourier coefficients direct Fourier transform or inverse Legendre transform
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22 Some spherical harmonics (n=5)
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23 Spherical harmonics (cont) Properties of the spherical harmonics eigenfunctions of the operator ∂ / ∂ λ eigenfunctions of the laplacian operator semi-implicit method leads to a decoupled set of equations latitudinal derivatives easy to compute although the spherical harmonics are not eigenfunctions of the latitudinal derivative operator
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24 Spherical harmonics (cont.) Usual truncations N=min(|m|+J, K) pentagonal truncation M=J=K triangular K=J+M rhomboidal K=J>M trapezoidal m n M m n M m n M m n M pentagonal triangular rhomboidaltrapezoidal K=J=M K K K=J J J
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25 Gaussian grid Use of the transform method for non-linear terms Integrals with respect to λ ---> 3M+1 points equally spaced in λ Integrals with respect to μ computed exactly by means of Gaussian quadrature using the values at the points where Gaussian latitudes In triangular truncation NG (3N+1)/2 Gaussian latitudes are approximately equally spaced same spacing as for λ
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26 The reduced Gaussian grid Full gridReduced grid Triangular truncation is isotropic Associated Legendre functions are very small when m is large and |μ| near 1
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27 The linear Gaussian grid
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28 Two resolutions using the same Gaussian grid T213 orographyT L 319 orography
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29 Diffusion very simple to apply - Leapfrog Stability physical solution 0≤λ≤1 stable computational solution λ≤-1 unstable - Forward - Backward (implicit) decoupled system of equations 0 ≤ λ ≤ 1 Stable
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