WVGeometryDoublyPeriodic

A domain periodic in both x and y.

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Overview

The WVGeometryDoublyPeriodic encapsulates the transformations and logic necessary for computing Fourier transforms and spectral derivatives in a doubly periodic domain.

The class primarily converts between two different data structures for representing functions of (k,l): what we call the “DFT” grid, and the “WV” grid. The DFT grid is appropriate for many FFT algorithms, while the WV grid is ideal for the vertical mode matrix multiplications in the WaveVortexModel, as it contains no redundant coefficients.

The DFT layout is the the matrix structure used by many modern FFT algorithms. For real-valued functions, this format contains twice as much information as necessary, due to Hermitian conjugacy. Additionally, we often want to restrict ourselves to wavenumbers that do not alias with quadratic multiplication (the two-thirds rule). In two dimensions, this means that 4/9ths of the available wavenumbers are aliased. The WV layout thus does not include either the Hermitian conjugates nor the aliased modes.

The basic usage of the indices is as follows: assume wvMatrix and dftMatrix are shaped as

    size(wvMatrix) == [Nkl_wv 1];
    size(dftMatrix) == [Nk_dft Nl_dft]; % (equivalently [Nx Ny]

then to transform data from the DFT matrix to the WV matrix,

    wvMatrix = dftMatrix(dftPrimaryIndices);

and the reverse is

    dftMatrix(dftPrimaryIndices) = wvMatrix;
    dftMatrix(dftConjugateIndices) = conj(wvMatrix(wvConjugateIndex));

Topics

• Initialization
  • WVGeometryDoublyPeriodic create a geometry for a doubly periodic domain
• Domain attributes
  • Spatial grid
    • Lx length of the x-dimension
    • Ly length of the y-dimension
    • Nx number of grid points in the x-dimension
    • Ny number of grid points in the y-dimension
    • x dimension
    • y dimension
  • DFT grid
    • Nk_dft length of the k-wavenumber dimension on the DFT grid
    • Nl_dft length of the l-wavenumber dimension on the DFT grid
    • conjugateDimension assumed conjugate dimension
    • kMode_dft k mode-number on the DFT grid
    • k_dft k wavenumber dimension on the DFT grid
    • lMode_dft l mode-number on the DFT grid
    • l_dft l wavenumber dimension on the DFT grid
  • WV grid
    • Nkl_wv length of the combined kl-wavenumber dimension on the WV grid
    • dftConjugateIndices index into the DFT grid of the conjugate of each WV mode
    • dftPrimaryIndices index into the DFT grid of each WV mode
    • kMode_wv k mode number on the WV grid
    • kRadial_wv radial (k,l) wavenumber on the WV grid
    • k_wv k-wavenumber dimension on the WV grid
    • lMode_wv l mode number on the WV grid
    • l_wv l-wavenumber dimension on the WV grid
    • shouldAntialias whether the WV grid includes quadratically aliased wavenumbers
    • shouldExcludeNyquist whether the WV grid includes Nyquist wavenumbers
    • shouldExludeConjugates whether the WV grid includes wavenumbers that are Hermitian conjugates
• Operations
  • Grid transformation
    • transformFromDFTGridToWVGrid convert from DFT to WV grid
    • transformFromWVGridToDFTGrid convert from a WV to DFT grid
  • Fourier transformation
    • transformFromSpatialDomain transform from \((x,y,z)\) to \((k,l,z)\) on the DFT grid
    • transformToSpatialDomain transform from \((k,l,z)\) on the DFT grid to \((x,y,z)\)
    • transformToSpatialDomainAtPosition transform from \((k,l)\) on the DFT grid to \((x,y)\) at any position
  • Differentiation
    • diffX differentiate a spatial variable in the x-direction
    • diffY differentiate a spatial variable in the y-direction
• Index gymnastics
  • indicesFromDFTGridToWVGrid indices to convert from DFT to WV grid
  • indicesFromWVGridToDFTGrid indices to convert from WV to DFT grid
  • isValidConjugateWVModeNumber return a boolean indicating whether (k,l) is a valid conjugate WV mode number
  • isValidPrimaryWVModeNumber return a boolean indicating whether (k,l) is a valid primary (non-conjugate) WV mode number
  • isValidWVModeNumber return a boolean indicating whether (k,l) is a valid WV mode number
  • modeNumberFromWVIndex return mode number from a linear index into a WV matrix
  • primaryModeNumberFromWVModeNumber takes any valid WV mode number and returns the primary mode number
  • wvIndexFromModeNumber return the linear index into k_wv and l_wv from a mode number
• Masks
  • maskForAliasedModes returns a mask with locations of modes that will alias with a quadratic multiplication.
  • maskForConjugateFourierCoefficients a mask indicate the components that are redundant conjugates
  • maskForNyquistModes returns a mask with locations of modes that are not fully resolved
• Utility function
  • degreesOfFreedomForComplexMatrix a matrix with the number of degrees-of-freedom at each entry
  • degreesOfFreedomForRealMatrix a matrix with the number of degrees-of-freedom at each entry
  • indicesOfFourierConjugates a matrix of linear indices of the conjugate
  • isHermitian Check if the matrix is Hermitian. Report errors.
  • setConjugateToUnity set the conjugate of the wavenumber (iK,iL) to 1

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