InternalModesSpectral

Solve the vertical internal-mode EVP with Chebyshev collocation.

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Declaration

classdef InternalModesSpectral < InternalModesBase

Overview

InternalModesSpectral is the core numerical solver described in Sections 4 and 5 of Early, Lelong, and Smith (2020). It represents the vertical structure on a Gauss-Lobatto grid and solves the manuscript eigenvalue problems

\[\partial_{zz} G_j - K^2 G_j = \frac{f_0^2 - N^2}{g h_j} G_j\]

for fixed \(K\), or

\[\partial_{zz} G_j = \frac{\omega^2 - N^2}{g h_j} G_j\]

for fixed \(\omega\), together with

\[F_j = h_j \partial_z G_j.\]

The solver stores background fields on the user-facing output grid z, but internally works with a spectral grid and exact integrals of Chebyshev polynomials for mode normalization. Subclasses reuse this machinery in stretched coordinates while preserving the same public mode API.

im = InternalModesSpectral(rho=rho, zIn=zIn, zOut=zOut, latitude=latitude, nEVP=257);
[F, G, h, omega] = im.modesAtWavenumber(2*pi/1000);

Topics

• Create and initialize modes
  • InternalModesSpectral Initialize the spectral solver on depth coordinates.
  • nEVP Number of collocation points used for the generalized EVP.
• Compute modes
  • bottomModesAtWavenumber Return the bottom SQG mode at fixed horizontal wavenumber.
  • boundaryModesAtWavenumber Return either the surface or bottom boundary mode at fixed wavenumber.
  • eigenmatricesForFrequency Assemble the fixed-\(\omega\) generalized EVP on the spectral grid.
  • eigenmatricesForWavenumber Assemble the fixed-\(K\) generalized EVP on the spectral grid.
  • modesAtQuadraturePoints Return resolved modes on mode-adapted quadrature points.
  • surfaceModesAtWavenumber Return the surface SQG mode at fixed horizontal wavenumber.

Developer Topics

These items document internal implementation details and are not part of the primary public API.

• Configure normalization and boundaries
  • ApplyBoundaryConditions Apply the active surface and bottom conditions to an EVP pair.
• Developer topics
  • ChebyshevDifferentiationMatrix Return the matrix that differentiates Chebyshev coefficients.
  • ChebyshevInterpolationDerivative Return the Lobatto-grid differentiation matrix for interpolation values.
  • ChebyshevPolynomialsOnGrid Evaluate Chebyshev polynomials and derivatives on an arbitrary grid.
  • ChebyshevTransformForGrid Build a spectral interpolation map from a Lobatto grid to an output grid.
  • CheckIfReasonablyMonotonic Test whether a gridded density profile is monotonic enough for stretched coordinates.
  • Diff1_xCheb First-derivative operator in Chebyshev coefficient space.
  • DifferentiateChebyshevVector Differentiate a Chebyshev series in coefficient space.
  • EigenmatricesForGeostrophicFModes Assemble the geostrophic interior F-mode EVP.
  • EigenmatricesForGeostrophicGModes Assemble the geostrophic interior G-mode EVP.
  • EigenmatricesForGeostrophicRigidLidGModes Assemble the rigid-lid geostrophic EVP with displaced boundaries.
  • EigenmatricesForMDAModes Assemble the MDA diagnostic EVP.
  • EigenmatricesForRigidLidGModes Assemble the rigid-lid geostrophic G-mode EVP.
  • EigenmatricesForSmithVannesteModes Assemble the Smith-Vanneste SQG-like EVP used by a helper workflow.
  • FFromVCheb Map Chebyshev coefficients to \(F_j\) on the active internal grid.
  • FNorm \(\omega\)-constant normalization functional.
  • FOutFromVCheb Map Chebyshev coefficients to \(F_j(z)\) on the public output grid.
  • FindRootsFromChebyshevVector Find physical-domain roots of a Chebyshev series.
  • FindSmallestChebyshevGridWithNoGaps Find the coarsest Lobatto grid that covers a target grid without gaps.
  • FindTurningPointBoundariesAtFrequency Find turning points and region signs for \(N^2(z)-\omega^2\).
  • GFromVCheb Map Chebyshev coefficients to \(G_j\) on the active internal grid.
  • GNorm \(K\)-constant normalization functional.
  • GOutFromVCheb Map Chebyshev coefficients to \(G_j(z)\) on the public output grid.
  • GaussQuadraturePointsForEigenmatrices Return quadrature points inferred from a generalized EVP.
  • GaussQuadraturePointsForMDAModes Return quadrature points tailored to the diagnostic MDA modes.
  • GaussQuadraturePointsForModesAtFrequency Return quadrature points tailored to fixed-\(\omega\) modes.
  • GaussQuadraturePointsForModesAtWavenumber Return quadrature points tailored to fixed-\(K\) modes.
  • GeostrophicFModesAtWavenumber Return the geostrophic interior F-modes at fixed horizontal wavenumber.
  • GeostrophicModesAtWavenumber Return the geostrophic interior G-modes at fixed horizontal wavenumber.
  • GeostrophicNorm Geostrophic normalization functional.
  • GeostrophicRigidLidModes Return rigid-lid geostrophic modes with displaced boundaries.
  • GeostrophicSmithVannesteModesAtWavenumber Return Smith-Vanneste modes at fixed horizontal wavenumber.
  • Int_xCheb Exact integral weights for Chebyshev coefficients on the active domain.
  • IntegrateChebyshevVector Integrate a Chebyshev series in coefficient space.
  • IntegrateChebyshevVectorWithLimits Integrate a Chebyshev series between two physical limits.
  • IsChebyshevGrid Test whether a grid is a Chebyshev-Lobatto grid up to tolerance.
  • Lx Total span of the active spectral coordinate.
  • MDAModes Return the diagnostic MDA modes.
  • N2_function Buoyancy-frequency profile represented as a function of depth.
  • N2_xLobatto Buoyancy frequency squared sampled on xLobatto.
  • ProjectOntoChebyshevPolynomialsWithTolerance Project a profile onto Chebyshev coefficients to a requested tolerance.
  • T_xCheb_zOut Transform from Chebyshev coefficients to the public output grid.
  • T_xLobatto Chebyshev basis evaluated on the Lobatto grid.
  • Tx_xLobatto First derivatives of the Chebyshev basis on the Lobatto grid.
  • Txx_xLobatto Second derivatives of the Chebyshev basis on the Lobatto grid.
  • ValueOfFunctionAtPointOnGrid Evaluate a Chebyshev series at arbitrary points in its physical domain.
  • fInverseBisection Invert a monotonic function by vectorized bisection.
  • fct Compute the fast Chebyshev transform of values on a Lobatto grid.
  • hFromLambda Set on initialization by the subclass, these transformations are
  • ifct Compute the inverse fast Chebyshev transform.
  • rho_function Density profile represented as a spline or functional object.
  • standardChop Choose a truncation index for a Chebyshev coefficient sequence.
  • xDomain Bounds of the active spectral coordinate.
  • xLobatto The ‘x’ refers to the stretched coordinate being used.
  • xMax Maximum value of the active spectral coordinate.
  • xMin Minimum value of the active spectral coordinate.
  • xOut Output locations mapped into the active spectral coordinate.
  • x_function Active stretched-coordinate map \(x(z)\) used by the solver.
  • z_xLobatto Physical depths associated with xLobatto.

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