WVTransformHydrostatic

A class for disentangling hydrostatic waves and vortices in variable stratification

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Declaration

classdef WVTransformHydrostatic < WVTransform

Overview

To initialization an instance of the WVTransformHydrostatic class you must specific the domain size, the number of grid points and either the density profile or the stratification profile.

N0 = 3*2*pi/3600;
L_gm = 1300;
N2 = @(z) N0*N0*exp(2*z/L_gm);
wvt = WVTransformHydrostatic([100e3, 100e3, 4000],[64, 64, 65], N2=N2,latitude=30);

Topics

• Initialization
  • WVTransformHydrostatic create a wave-vortex transform for variable stratification
• Primary flow components
  • geostrophicComponent returns the geostrophic flow component
  • waveComponent returns the internal gravity wave flow component
  • inertialComponent returns the inertial oscillation flow component
  • mdaComponent returns the mean density anomaly component
• Stratification
  • rho_nm no-motion density profile
  • N2 \(N^2(z)\), squared buoyancy frequency of the no-motion density, \(N^2\equiv - \frac{g}{\rho_0} \frac{\partial \rho_\textrm{nm}}{\partial z}\)
  • dLnN2 \(\frac{\partial \ln N^2}{\partial z}\), vertical variation of the log of the squared buoyancy frequency
  • verticalModes instance of the InternalModes class
  • effectiveVerticalGridResolution returns the effective vertical grid resolution in meters
  • Vertical modes
    • FMatrix transformation matrix \(F_g\)
    • FinvMatrix transformation matrix \(F_g^{-1}\)
    • GMatrix transformation matrix \(G_g\)
    • GinvMatrix transformation matrix \(G_g^{-1}\)
  • Validation
    • isDensityInValidRange checks if the density field is a valid adiabatic re-arrangement of the base state
• Initial conditions
  • Waves
    • addGMSpectrum add waves following a Garrett-Munk spectrum
    • addWaveModes add amplitudes of the given wave modes
    • addWavesWithFrequencySpectrum add waves with a specified frequency spectrum
    • initWavesWithFrequencySpectrum initialize with waves of a specified frequency spectrum
    • initWithAlternativeSpectrum initialize with an alternative formulation of the GM spectrum in the wavenumber domain.
    • initWithGMSpectrum initialize the wave field following a Garrett-Munk spectrum
    • initWithWaveModes initialize with the given wave modes
    • removeAllWaves removes all wave from the model, including inertial oscillations
    • setWaveModes set amplitudes of the given wave modes
  • Geostrophic Motions
    • initWithGeostrophicStreamfunction initialize with a geostrophic streamfunction
    • setGeostrophicStreamfunction set a geostrophic streamfunction
    • addGeostrophicStreamfunction add a geostrophic streamfunction to existing geostrophic motions
    • setGeostrophicModes set amplitudes of the given geostrophic modes
    • addGeostrophicModes add amplitudes of the given geostrophic modes
    • removeAllGeostrophicMotions remove all geostrophic motions
  • Inertial Oscillations
    • addInertialMotions add inertial motions to existing inertial motions
    • initWithInertialMotions initialize with inertial motions
    • removeAllInertialMotions remove all inertial motions
    • setInertialMotions set inertial motions
  • Mean density anomaly
    • addMeanDensityAnomaly add inertial motions to existing inertial motions
    • initWithMeanDensityAnomaly initialize with inertial motions
    • removeAllMeanDensityAnomaly remove all mean density anomalies
    • setMeanDensityAnomaly set inertial motions
• Energetics of flow components
  • geostrophicEnergy total energy, geostrophic
• Operations
  • Grid transformation
    • transformFromDFTGridToWVGrid convert from DFT to WV grid
    • transformFromWVGridToDFTGrid convert from a WV to DFT grid
  • Fourier transformation
    • transformFromSpatialDomainToDFTGrid transform from \((x,y,z)\) to \((k,l,z)\) on the DFT grid
    • transformToSpatialDomainFromDFTGrid transform from \((k,l,z)\) on the DFT grid to \((x,y,z)\)
    • transformToSpatialDomainFromDFTGridAtPosition transform from \((k,l)\) on the DFT grid to \((x,y)\) at any position
  • Transformations
    • transformToKLAxes transforms in the spectral domain from (j,kl) to (kAxis,lAxis,j)
    • transformToOmegaAxis transforms in the from (j,kRadial) to omegaAxis
    • transformToPseudoRadialWavenumber transforms in the from (j,kRadial) to kPseudoRadial
    • transformToPseudoRadialWavenumberA0 transforms in the from (j,kRadial) to kPseudoRadial
    • transformToPseudoRadialWavenumberApm transforms in the from (j,kRadial) to kPseudoRadial
    • transformToRadialWavenumber transforms in the spectral domain from (j,kl) to (j,kRadial)
• Wave-vortex coefficients
  • at time \(t\)
    • A0t geostrophic coefficients time t
    • Amt negative wave coefficients at reference time t
    • Apt positive wave coefficients at reference time t
• Domain Attributes
  • f Coriolis parameter
  • g gravity of Earth
  • inertialPeriod inertial period
  • latitude central latitude of the simulation
  • planetaryRadius radius of the planetary body
  • rho0 , dLnN2
  • rotationRate rotation rate of the planetary body
  • Grid
    • Spectral
      • J j-coordinate matrix
      • K k-coordinate matrix
      • K2 squared horizontal wavenumber, \(K2=K^2+L^2\)
      • Kh horizontal wavenumber, \(Kh=\sqrt(K^2+L^2)\)
      • L l-coordinate matrix
      • Nj points in the j-coordinate, length(z)
    • Spatial
      • Nz points in the third, untransformed, dimension
      • X x-coordinate matrix
      • Y y-coordinate matrix
      • Z z-coordinate matrix
  • Spatial grid
    • x dimension
    • y dimension
    • Lx length of the x-dimension
    • Ly length of the y-dimension
    • Lz length of the z-dimension
    • Nx number of grid points in the x-dimension
    • Ny number of grid points in the y-dimension
    • fastTransform fast transform object
  • 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 length of the combined kl-wavenumber dimension on the WV grid
    • dftConjugateIndex index into the DFT grid of the conjugate of each WV mode
    • dftConjugateIndices2D index into the DFT grid of the conjugate of each WV mode
    • dftPrimaryIndex index into the DFT grid of each WV mode
    • dftPrimaryIndices2D index into the DFT grid of each WV mode
    • dk wavenumber spacing of the \(k\) axis
    • dl wavenumber spacing of the \(l\) axis
    • k wavenumber dimension on the WV grid
    • kMode_wv k mode number on the WV grid
    • kRadial radial (k,l) wavenumber on the WV grid
    • kl wavenumber dimension
    • l wavenumber dimension on the WV grid
    • lMode_wv l mode number 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
    • wvConjugateIndex index into the WV mode that matches the dftConjugateIndices
  • Stratification
    • h_0 [Nj 1]
• 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
• Properties
  • effectiveHorizontalGridResolution returns the effective grid resolution in meters
• Energetics
  • inertialEnergy total energy of the inertial flow
  • mdaEnergy total energy of the mean density anomaly
  • geostrophicKineticEnergy kinetic energy of the geostrophic flow
  • waveEnergy total energy of the geostrophic flow
  • geostrophicPotentialEnergy potential energy of the geostrophic flow
• Index gymnastics
  • indexFromKLModeNumber return the linear index into k_wv and l_wv from a mode number
  • indexFromModeNumber return the linear index into a spectral matrix given (k,l,j)
  • indicesFromDFTGridToWVGrid indices to convert from DFT to WV grid
  • indicesFromWVGridToDFTGrid indices to convert from WV to DFT grid
  • indicesFromWVGridToFFTWGrid indices to convert from WV to DFT grid
  • isValidConjugateKLModeNumber return a boolean indicating whether (k,l) is a valid conjugate WV mode number
  • isValidConjugateModeNumber returns a boolean indicating whether (k,l,j) is a valid conjugate mode number
  • isValidKLModeNumber return a boolean indicating whether (k,l) is a valid WV mode number
  • isValidModeNumber returns a boolean indicating whether (k,l,j) is a valid mode number
  • isValidPrimaryKLModeNumber return a boolean indicating whether (k,l) is a valid primary (non-conjugate) WV mode number
  • isValidPrimaryModeNumber returns a boolean indicating whether (k,l,j) is a valid primary (non-conjugate) mode number
  • klModeNumberFromIndex return mode number from a linear index into a WV matrix
  • primaryKLModeNumberFromKLModeNumber takes any valid WV mode number and returns the primary 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
• Utilities
  • placeParticlesOnIsopycnal places Lagrangian particles along a specified isopycnal
• Developer
  • propertyAnnotationsForGeometry return array of CAPropertyAnnotations initialized by default
• State Variables
  • psi geostrophic streamfunction
  • ssu x-component of the fluid velocity at the surface
  • ssv y-component of the fluid velocity at the surface
• Potential Vorticity & Enstrophy
  • qgpv quasigeostrophic potential vorticity
• Internal
  • quadraturePointsForStratifiedFlow return the quadrature points for a given stratification
  • verticalProjectionOperatorsWithRigidLid return the normalized projection operators with prefactors
• Other
  • A0N matrix component that multiplies \(\tilde{\eta}\) to compute \(A_0\).
  • A0U matrix component that multiplies \(\tilde{u}\) to compute \(A_0\).
  • A0V matrix component that multiplies \(\tilde{v}\) to compute \(A_0\).
  • A0Z
  • ApmD
  • ApmN
  • Feta
  • Fu
  • Fv
  • Lr2 squared Rossby radius
  • N2Function takes \(z\) values and returns the squared buoyancy frequency of the no-motion density.
  • NA0 matrix component that multiplies \(A_0\) to compute \(\tilde{\eta}\).
  • NAm
  • NAp
  • Omega
  • P0 Preconditioner for F, size(P)=[Nj 1]. Fu = uhat, (PF)u = Puhat, so ubar==Puhat
  • PA0
  • PF0 size(PF,PG)=[Nj x Nz]
  • PF0inv Transformation matrices
  • Q0 Preconditioner for G, size(Q)=[Nj 1]. Geta = etahat, (QG)eta = Qetahat, so etabar==Qetahat.
  • QG0 Preconditioned G-mode forward transformation
  • QG0inv Preconditioned G-mode inverse transformation
  • UA0 matrix component that multiplies \(A_0\) to compute \(\tilde{u}\).
  • UAm
  • UAp
  • VA0 matrix component that multiplies \(A_0\) to compute \(\tilde{v}\).
  • VAm
  • VAp
  • WAm
  • WAp
  • beta
  • boussinesqTransform
  • buoyancyPeriod
  • chebfunForZArray
  • classRequiredPropertyNames
  • conjPhase phase of the Am wave modes
  • crossSpectrumWithFgTransform
  • crossSpectrumWithGgTransform
  • diffX
  • diffY
  • diffZF
  • diffZG
  • effectiveJMax
  • enstrophyFluxFromF0
  • eta approximate isopycnal deviation
  • exactPotentialEnstrophy
  • exactTotalEnergy
  • fluxAtTimeCellArray y0 is a 3x1 cell array
  • fluxForForcing
  • geometryFromFile
  • geometryFromGroup
  • h_pm
  • iOmega
  • intZF
  • intZG
  • j vertical mode number
  • kAxis k coordinate
  • kPseudoRadial
  • kljGrid
  • lAxis l coordinate
  • maxFg
  • maxFw
  • modeNumberFromIndex
  • namesOfRequiredPropertiesForGeometry
  • namesOfRequiredPropertiesForRotatingFPlane
  • namesOfRequiredPropertiesForTransform
  • namesOfTransformVariables
  • newNonrequiredPropertyNames
  • newRequiredPropertyNames
  • p pressure anomaly
  • phase phase of the Ap wave modes
  • pi height anomaly
  • propertyAnnotationsForRotatingFPlane
  • qgpvFluxFromF0
  • requiredPropertiesForGeometryFromGroup
  • requiredPropertiesForRotatingFPlaneFromGroup
  • requiredPropertiesForTransformFromGroup
  • rhoFunction eta_true operation needs rhoFunction
  • rho_bar mean density
  • rho_e excess density
  • rho_nm0 \(\rho_\textrm{nm}(z)\), no-motion density at time t0
  • rho_total total potential density
  • rk4FluxForForcing
  • shouldUseTrueNoMotionProfile whether eta_true uses rho_nm instead of rho_nm0
  • spatialFluxForForcingWithName
  • spatialMatrixSize
  • spectralMatrixSize
  • spectrumWithFgTransform
  • spectrumWithGgTransform
  • ssh sea-surface height
  • sumFluxDictionary
  • throwErrorIfDensityViolation checks if the proposed coefficients are a valid adiabatic re-arrangement of the base state
  • totalEnstrophy
  • totalEnstrophySpatiallyIntegrated
  • transformFromGroup
  • transformFromSpatialDomainWithFio
  • transformFromSpatialDomainWithFourier
  • transformToSpatialDomainWithFourier
  • transformToSpatialDomainWithFourierAtPosition
  • transformWithG_wg
  • u x-component of the fluid velocity
  • uvMax max horizontal fluid speed
  • v y-component of the fluid velocity
  • verticalProjectionOperatorsWithFreeSurface
  • volumeIntegral
  • w z-component of the fluid velocity
  • wMax max vertical fluid speed
  • waveCoefficientsAtTimeT
  • waveVortexTransformWithExplicitAntialiasing
  • xyzGrid
  • z z coordinate
  • z_int Quadrature weights for the vertical grid
  • zeta_x x-component component of relative vorticity
  • zeta_y y-component component of relative vorticity
  • zeta_z vertical component of relative vorticity

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