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GarrettMunkSpectrum.m
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GarrettMunkSpectrum.m
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classdef GarrettMunkSpectrum < handle
properties (Access = public)
latitude % Latitude for which the modes are being computed.
f0 % Coriolis parameter at the above latitude.
Lz % Depth of the ocean.
rho0 % Density at the surface of the ocean.
z_in
rho
j_star = 3;
N_max
B
B_int
H
userSpecifiedN2 = 0
N2 % *function handle* of z
zInternal % A full depth coordinate used to precompute Phi & Gamma
N2internal % N2 at zInternal
didPrecomputePhiAndGammaForOmega = 0
omega % size(omega) = nOmega
F_omega % size(F_omega) = [nZ,nOmega,nModes]
G_omega % size(G_omega) = [nZ,nOmega,nModes]
h_omega % size(h_omega) = [nOmega,nModes]
k_omega % size(k_omega) = [nOmega,nModes]
F2_omega % size(F2_omega) = [nK,nModes]---\int F^2 dz
N2G2_omega % size(N2G2_omega) = [nK,nModes]---\int N^2 G^2 dz
didPrecomputePhiAndGammaForK = 0
k % size(k) = nK
F_k % size(F_k) = [nZ,nK,nModes]
G_k % size(G_k) = [nZ,nK,nModes]
h_k % size(h_k) = [nK,nModes]
omega_k % size(omega_k) = [nK,nModes]
F2_k % size(F2_kj) = [nK,nModes]---\int F^2 dz
N2G2_k % size(N2G2_k) = [nK,nModes]---\int N^2 G^2 dz
% We also store a copy of the various wavenumber spectra
Suv_k % size(Suv_k) = [Nz,nK];
Szeta_k % size(Szeta_k) = [Nz,nK];
nModes = 64
nEVPMin = 256 % assumed minimum, can be overriden by the user
nEVPMax = 1024
end
properties (Dependent)
Phi_omega % size(Phi_omega) = [nZ,nOmega,nModes]
Gamma_omega % size(Gamma_omega) = [nZ,nOmega,nModes]
Phi_k % size(Phi_k) = [nZ,nK,nModes]
Gamma_k % size(Gamma_k) = [nZ,nK,nModes]
end
properties (Constant)
g = 9.81;
L_gm = 1.3e3; % thermocline exponential scale, meters
invT_gm = 5.2e-3; % reference buoyancy frequency, radians/seconds
E_gm = 6.3e-5; % non-dimensional energy parameter
E = (1.3e3)*(1.3e3)*(1.3e3)*(5.2e-3)*(5.2e-3)*(6.3e-5);
end
methods
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Initialization
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function self = GarrettMunkSpectrum(rho, z_in, latitude, varargin)
if isa(rho,'char') == true
% filepath = sprintf('../PrecomputedProfiles/%s.mat',rho);
filepath = sprintf('%s.mat',rho);
if ~exist(filepath,'file')
error('Cannot find precomputed file at path %s\n',filepath);
end
file = load(filepath);
self.z_in = file.zIn;
self.latitude = self.latitude;
self.zInternal = file.zInternal;
self.N2internal = file.N2internal;
self.didPrecomputePhiAndGammaForOmega = 1;
self.omega = file.omega;
self.F_omega = file.F_omega;
self.G_omega = file.G_omega;
self.h_omega = file.h_omega;
self.k_omega = file.k_omega;
self.didPrecomputePhiAndGammaForK = 1;
self.k = file.k;
self.F_k = file.F_k;
self.G_k = file.G_k;
self.h_k = file.h_k;
self.omega_k = file.omega_k;
self.Lz = max(self.z_in) - min(self.z_in);
self.latitude = file.latitude;
self.f0 = 2*(7.2921e-5)*sin(file.latitude*pi/180);
self.N_max = file.N_max;
self.nModes = size(self.F_omega,3);
self.rho = file.rho;
self.N2 = file.N2;
else
% Make everything a column vector
if isrow(z_in)
z_in = z_in.';
end
self.Lz = max(z_in) - min(z_in);
self.latitude = latitude;
self.f0 = 2*(7.2921e-5)*sin(latitude*pi/180);
self.z_in = z_in;
% Set properties supplied as name,value pairs
userSpecifiedRho0 = 0;
self.userSpecifiedN2 = 0;
nargs = length(varargin);
if mod(nargs,2) ~= 0
error('Arguments must be given as name/value pairs');
end
for k = 1:2:length(varargin)
if strcmp(varargin{k}, 'rho0')
userSpecifiedRho0 = 1;
end
if strcmp(varargin{k}, 'N2')
self.userSpecifiedN2 = 1;
continue; % no property to set
end
self.(varargin{k}) = varargin{k+1};
end
if self.userSpecifiedN2 == 1 && userSpecifiedRho0 == 0
error('If you pass N2 instead of rho, you must also provide a rho0')
end
% Is density specified as a function handle or as a grid of
% values?
self.rho = rho;
if isa(rho,'function_handle') == true
if numel(z_in) ~= 2
error('When using a function handle, z_domain must be an array with two values: z_domain = [z_bottom z_surface];')
end
if userSpecifiedRho0 == 0
self.rho0 = rho(max(z_in));
end
if self.userSpecifiedN2 == 1
self.N2 = rho;
im = InternalModesAdaptiveSpectral(self.N2,self.z_in,self.z_in,self.latitude,'N2',1,'rho0',self.rho0);
self.rho = im.rho_function;
else
im = InternalModesAdaptiveSpectral(self.rho,self.z_in,self.z_in,self.latitude);
end
elseif isa(rho,'numeric') == true
if numel(rho) ~= length(rho) || length(rho) ~= length(z_in)
error('rho must be 1 dimensional and z must have the same length');
end
if isrow(rho)
rho = rho.';
end
self.rho0 = min(rho);
im = InternalModesAdaptiveSpectral(self.rho,self.z_in,self.z_in,self.latitude);
else
error('rho must be a function handle or an array.');
end
self.N_max = max(sqrt(im.N2_xLobatto));
self.zInternal = im.z_xLobatto;
self.N2internal = im.N2_xLobatto;
if self.userSpecifiedN2 == 0
self.N2 = @(z) interp1(self.zInternal,self.N2internal,z,'linear');
end
end
H1 = (self.j_star+(1:3000)).^(-5/2);
H_norm = 1/sum(H1);
self.H = @(j) H_norm*(self.j_star + j).^(-5/2);
f = self.f0;
Nmax = self.N_max;
B_norm = 1/acos(f/Nmax);
self.B_int = @(omega0,omega1) B_norm*(atan(f./sqrt(omega0.*omega0-f*f)) - atan(f./sqrt(omega1.*omega1-f*f)));
self.B = @(omega0,omega1) (omega1<f | omega1 > Nmax).*zeros(size(omega0)) + (omega0<f & omega1>f).*self.B_int(f*ones(size(omega0)),omega1) + (omega0>=f & omega1 <= Nmax).*self.B_int(omega0,omega1) + (omega0<Nmax & omega1 > Nmax).*self.B_int(omega0,Nmax*ones(size(omega1)));
self.PrecomputeComputeInternalModesForOmega();
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Optimized setters and getters
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function Phi = get.Phi_omega(self)
self.PrecomputeComputeInternalModesForOmega;
Phi = nan(length(self.zInternal),length(self.omega),self.nModes);
for i=1:length(self.omega)
j0 = sum(~isnan(self.F_omega(1,i,:)));
Phi(:,i,1:j0) = (squeeze(self.F_omega(:,i,1:j0).^2)) .* (1./squeeze(self.h_omega(i,1:j0)) .* self.H(1:j0));
end
end
function Gamma = get.Gamma_omega(self)
self.PrecomputeComputeInternalModesForOmega;
Gamma = nan(length(self.zInternal),length(self.omega),self.nModes);
for i=1:length(self.omega)
j0 = sum(~isnan(self.G_omega(1,i,:)));
Gamma(:,i,1:j0) = (1/self.g)*(squeeze(self.G_omega(:,i,1:j0).^2)) .* self.H(1:j0);
end
end
function Phi = get.Phi_k(self)
self.PrecomputeComputeInternalModesForK;
Phi = nan(length(self.zInternal),length(self.k),self.nModes);
for i=1:length(self.k)
j0 = sum(~isnan(self.F_k(1,i,:)));
Phi(:,i,1:j0) = (squeeze(self.F_k(:,i,1:j0).^2)) .* (1./squeeze(self.h_k(i,1:j0)) .* self.H(1:j0));
end
end
function Gamma = get.Gamma_k(self)
self.PrecomputeComputeInternalModesForK;
Gamma = nan(length(self.zInternal),length(self.k),self.nModes);
for i=1:length(self.k)
j0 = sum(~isnan(self.G_k(1,i,:)));
Gamma(:,i,1:j0) = (1/self.g)*(squeeze(self.G_k(:,i,1:j0)).^2) .* self.H(1:j0);
end
end
function [omega2, k2] = SquaredFrequencyForWavenumber(self,k)
self.PrecomputeComputeInternalModesForK;
k = reshape(k,[],1);
j = reshape(1:self.nModes,1,[]);
h = interpn(self.k,j,self.h_k,k,j);
k2 = k.*k;
omega2 = self.g * h .* k2 + self.f0*self.f0;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Horizontal Velocity Spectra
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function E = HorizontalVelocityVariance(self, z, varargin)
% Returns the total horizontal velocity variance at a given depth. [m^2/s^s]
%
% z array of depths, in meters
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
[z,approximation] = self.validateVarianceArguments(z,varargin{:});
% Compute the total variance by grabbing the one-sided
% spectrum, and summing over all frequencies.
om = linspace(0,self.N_max,2000);
S = self.HorizontalVelocitySpectrumAtFrequencies(z,om,approximation,'one-sided');
E = sum(S,2)*(om(2)-om(1));
end
function S = HorizontalVelocitySpectrumAtFrequencies(self,z,omega,varargin)
% The horizontal velocity frequency spectrum at given depths
% and frequencies. [m^2/s]
%
% z array of depths, in meters
% omega array of frequencies, in radians/second
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
% spectrumType (optional) 'one-sided', or 'two-sided'.
[z,omega,approximation,spectrumType] = self.validateSpectrumArguments(z,omega,varargin{:});
% Make sure it's a column vector
z = reshape(z,[],1);
% Choose a small increment
dOmega = omega(2)-omega(1);
dOmega = min( [self.f0/2,dOmega]);
% Create the function that converts to energy
f = self.f0;
Nmax = self.N_max;
if strcmp(spectrumType,'two-sided')
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1-f/omega)*(1-f/omega) )*0.5;
else
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1+f*f/(omega*omega)) );
end
S = zeros(length(z),length(omega));
for i=1:length(omega)
Bomega = self.B( abs( omega(i) ) - dOmega/2, abs( omega(i) ) + dOmega/2 )/dOmega;
S(:,i) = self.E* ( Bomega .* C(omega(i)) );
end
S(isnan(S))=0;
if strcmp(approximation,'exact')
self.PrecomputeComputeInternalModesForOmega();
Phi = interpn(self.zInternal,self.omega,sum(self.Phi_omega,3,'omitnan'),z,abs(omega),'linear',0); % 0 to everything outside
elseif strcmp(approximation,'wkb') || strcmp(approximation,'wkb-hydrostatic')
Phi = self.PhiForOmegaWKBApproximation(z, omega, approximation);
elseif strcmp(approximation,'gm')
Phi = self.PhiForOmegaGM(z, omega);
end
S = S.*Phi;
end
function S = HorizontalVelocitySpectrumAtWavenumbers(self,z,k)
self.PrecomputeComputeInternalModesForK();
if isrow(k)
k = k.';
end
if isempty(self.Suv_k)
f = self.f0;
Nmax = self.N_max;
% We are using the one-sided version of the spectrum
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax).*( (1+(f./omega).^2) );
% Integrate B across the frequency bands, \int B domega
omegaMid = self.omega_k(1:end-1,:) + diff(self.omega_k,1,1)/2;
omegaLeft = cat(1,self.omega_k(1,:),omegaMid);
omegaRight = cat(1, omegaMid, self.omega_k(end,:));
BofK = self.B(omegaLeft,omegaRight);
% Now divide by dk. This is essentially applying the Jacobian.
kMid = self.k(1:end-1) + diff(self.k)/2;
kLeft = cat(2,self.k(1),kMid);
kRight = cat(2,kMid,self.k(end));
dk = (kRight-kLeft).';
BofK = BofK./dk;
self.Suv_k = sum(self.E * self.Phi_k .* shiftdim( C(self.omega_k) .* BofK, -1),3);
end
S = interpn(self.zInternal,self.k,self.Suv_k,z,k,'linear');
end
function [Suv_kj,j] = HorizontalVelocitySpectrumAtWavenumberAndMode(self,k)
% returns the horizontal velocity spectrum as a function of
% horizontal wavenumber (k) and mode (j).
self.PrecomputeComputeInternalModesForK();
if isrow(k)
k = k.';
end
j = reshape(1:self.nModes,1,[]);
f = self.f0;
Nmax = self.N_max;
% We are using the one-sided version of the spectrum
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax).*( (1+(f./omega).^2) );
% Integrate B across the frequency bands, \int B domega
omegaMid = self.omega_k(1:end-1,:) + diff(self.omega_k,1,1)/2;
omegaLeft = cat(1,self.omega_k(1,:),omegaMid);
omegaRight = cat(1, omegaMid, self.omega_k(end,:));
BofK = self.B(omegaLeft,omegaRight);
% Now divide by dk. This is essentially applying the Jacobian.
kMid = self.k(1:end-1) + diff(self.k)/2;
kLeft = cat(2,self.k(1),kMid);
kRight = cat(2,kMid,self.k(end));
dk = (kRight-kLeft).';
BofK = BofK./dk; % m, or 1/(radians/m)
% depth integrated, so unitless
Phi = nan(length(self.k),self.nModes);
for i=1:length(self.k)
j0 = sum(~isnan(self.F2_k(i,:)));
Phi(i,1:j0) = (squeeze(self.F2_k(i,1:j0))) .* (1./squeeze(self.h_k(i,1:j0)) .* self.H(1:j0));
end
S = self.E * Phi .* C(self.omega_k) .* BofK; % m^3/s^2/(radians/m)
Suv_kj = interpn(self.k,j,S,k,j,'linear');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Horizontal Isopycnal Spectra
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function E = IsopycnalVariance(self, z, varargin)
% Returns the total isopycnal variance at a given depth. [m^2]
%
% z array of depths, in meters
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
[z,approximation] = self.validateVarianceArguments(z,varargin{:});
om = linspace(0,self.N_max,2000);
S = self.IsopycnalSpectrumAtFrequencies(z,om,approximation,'one-sided');
E = sum(S,2)*(om(2)-om(1));
end
function S = IsopycnalSpectrumAtFrequencies(self,z,omega,varargin)
% The isopycnal variance frequency spectrum at given depths
% and frequencies. [m^2 s]
%
% z array of depths, in meters
% omega array of frequencies, in radians/second
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
% spectrumType (optional) 'one-sided', or 'two-sided'.
[z,omega,approximation,spectrumType] = self.validateSpectrumArguments(z,omega,varargin{:});
% Make sure it's a column vector
z = reshape(z,[],1);
% Choose a small increment
dOmega = omega(2)-omega(1);
dOmega = min( [self.f0/2,dOmega]);
% Create the function that converts to energy
f = self.f0;
Nmax = self.N_max;
if strcmp(spectrumType,'two-sided')
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1-f*f/(omega*omega)) )*0.5;
else
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1-f*f/(omega*omega)) );
end
S = zeros(length(z),length(omega));
for i=1:length(omega)
Bomega = self.B( abs( omega(i) ) - dOmega/2, abs( omega(i) ) + dOmega/2 )/dOmega;
S(:,i) = self.E* ( Bomega .* C(omega(i)) );
end
S(isnan(S))=0;
if strcmp(approximation,'exact')
self.PrecomputeComputeInternalModesForOmega();
Gamma = interpn(self.zInternal,self.omega,sum(self.Gamma_omega,3,'omitnan'),z,abs(omega),'linear',0); % 0 to everything outside
elseif strcmp(approximation,'wkb') || strcmp(approximation,'wkb-hydrostatic')
Gamma = self.GammaForOmegaWKBApproximation(z, omega, approximation);
elseif strcmp(approximation,'gm')
Gamma = self.GammaForOmegaGM(z, omega);
end
S = S.*Gamma;
end
function [S, m, s_grid] = IsopycnalSpectrumAtVerticalWavenumbers(self,varargin)
% Isopycnal vertical wavenumber spectrum. Because the domain is
% finite, the vertical wavenumbers are pre-determined and are
% returned as m.
if length(varargin) == 1
shouldStretch = varargin{1};
else
shouldStretch = 1;
end
% Create the function that converts to energy
f = self.f0;
Nmax = self.N_max;
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1-f*f/(omega*omega)) );
om = linspace(0,self.N_max,2000);
if shouldStretch == 1
zHD = linspace(min(self.z_in),max(self.z_in),2^14 + 1).';
xi = cumtrapz(zHD,sqrt(self.N2(zHD)));
Lxi = max(xi)-min(xi);
s = (self.L_gm/Lxi)*xi;
s_grid = linspace(min(s),max(s),513).';
dgrid = s_grid(2)-s_grid(1);
z = interp1(s,zHD,s_grid); % positions in z, of the evenly spaced stretched coordinate
else
z = linspace(min(self.z_in),max(self.z_in),513).';
dgrid = z(2)-z(1);
end
z(end) = [];
dOmegaVector = diff(om);
if any(dOmegaVector<0)
error('omega must be strictly monotonically increasing.')
end
dOmega = unique(dOmegaVector);
if max(abs(diff(dOmega))) > 1e-7
error('omega must be an evenly spaced grid');
end
dOmega = min( [self.f0/2,dOmega]);
M = zeros(length(om),self.nModes);
for i=1:length(om)
Bomega = self.B( abs( om(i) ) - dOmega/2, abs( om(i) ) + dOmega/2 );
M(i,:) = self.E* ( Bomega .* C(om(i)) ) * self.H(1:self.nModes);
end
M(isnan(M))=0;
M = sqrt(M/9.81); % Units of sqrt(self.E/g) is meters
[Z,OMEGA,J] = ndgrid(reshape(self.zInternal,1,[]),reshape(self.omega,1,[]),reshape(1:self.nModes,1,[]));
[Zo,OMEGAo,Jo] = ndgrid(reshape(z,1,[]),reshape(om,1,[]),reshape(1:self.nModes,1,[]));
G = interpn(Z,OMEGA,J,self.G_omega,Zo,OMEGAo,Jo,'linear',0);
N = length(z);
if shouldStretch == 1
rescale = sqrt(Lxi/self.invT_gm/self.invT_gm/self.Lz)*(self.N2(z)).^(1/4);
iso = rescale .* G .* shiftdim(M,-1);
dm = pi/self.L_gm;
% This gives us 13.8
% trapz(s_grid(1:512),sum(sum(iso.^2,3),2))/5000
else
dm = pi/(N*dgrid);
iso = G .* shiftdim(M,-1);
% This gives 13.7, as it should
% trapz(z(1:512),(self.N2(z)/self.invT_gm/self.invT_gm).*sum(sum(iso.^2,3),2))/5000
end
m = ((1:N)*dm)';
% compute the discrete sine transform (DST), [f(1:N), 0, -f(N:-1:2)]
dstScratch = ifft(cat(1,iso,shiftdim(zeros(length(om),self.nModes),-1),-iso(N:-1:2,:,:)),2*N,1);
isobar = 2*imag(dstScratch(2:N+1,:,:));
% This definition of the spectrum means that,
% (1/L)*sum(iso.*iso)*dz== sum(S)*dm
% The units of S are then in meters^3.
S = N*dgrid*sum(sum(isobar.*conj(isobar), 3, 'omitnan'), 2, 'omitnan')/2/pi;
end
function [S, m, s_grid] = IsopycnalSpectrumAtVerticalWavenumbersSummed(self,varargin)
% Messing with collapsing the sum before doing an FFT.
if length(varargin) == 1
shouldStretch = varargin{1};
else
shouldStretch = 1;
end
% Create the function that converts to energy
f = self.f0;
Nmax = self.N_max;
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1-f*f/(omega*omega)) );
om = linspace(0,self.N_max,2000);
if shouldStretch == 1
zHD = linspace(min(self.z_in),max(self.z_in),2^14 + 1).';
xi = cumtrapz(zHD,sqrt(self.N2(zHD)));
Lxi = max(xi)-min(xi);
s = (self.L_gm/Lxi)*xi;
s_grid = linspace(min(s),max(s),513).';
dgrid = s_grid(2)-s_grid(1);
z = interp1(s,zHD,s_grid); % positions in z, of the evenly spaced stretched coordinate
else
z = linspace(min(self.z_in),max(self.z_in),513).';
dgrid = z(2)-z(1);
end
z(end) = [];
dOmegaVector = diff(om);
if any(dOmegaVector<0)
error('omega must be strictly monotonically increasing.')
end
dOmega = unique(dOmegaVector);
if max(abs(diff(dOmega))) > 1e-7
error('omega must be an evenly spaced grid');
end
dOmega = min( [self.f0/2,dOmega]);
M = zeros(length(om),self.nModes);
for i=1:length(om)
Bomega = self.B( abs( om(i) ) - dOmega/2, abs( om(i) ) + dOmega/2 );
M(i,:) = self.E* ( Bomega .* C(om(i)) ) * self.H(1:self.nModes);
end
M(isnan(M))=0;
M = sqrt(M/9.81); % Units of sqrt(self.E/g) is meters
[Z,OMEGA,J] = ndgrid(reshape(self.zInternal,1,[]),reshape(self.omega,1,[]),reshape(1:self.nModes,1,[]));
[Zo,OMEGAo,Jo] = ndgrid(reshape(z,1,[]),reshape(om,1,[]),reshape(1:self.nModes,1,[]));
G = interpn(Z,OMEGA,J,self.G_omega,Zo,OMEGAo,Jo,'linear',0);
N = length(z);
if shouldStretch == 1
rescale = sqrt(Lxi/self.invT_gm/self.invT_gm/self.Lz)*(self.N2(z)).^(1/4);
iso = rescale .* G .* shiftdim(M,-1);
dm = pi/self.L_gm;
% This gives us 13.8
% trapz(s_grid(1:512),sqrt(self.invT_gm*self.invT_gm./self.N2(z)).*sum(sum(iso.^2,3),2))/5000
else
dm = pi/(N*dgrid);
iso = G .* shiftdim(M,-1);
% This gives 13.7, as it should
% trapz(z(1:512),(self.N2(z)/self.invT_gm/self.invT_gm).*sum(sum(iso.^2,3),2))/5000
end
m = ((1:N)*dm)';
iso = sqrt(sum(sum(iso.^2, 3, 'omitnan'), 2, 'omitnan'));
% compute the discrete sine transform (DST), [f(1:N), 0, -f(N:-1:2)]
dstScratch = ifft(cat(1,iso,0,-iso(N:-1:2)),2*N,1);
isobar = 2*imag(dstScratch(2:N+1));
% This definition of the spectrum means that,
% (1/L)*sum(iso.*iso)*dz== sum(S)*dm
% The units of S are then in meters^3.
S = N*dgrid*isobar.*conj(isobar)/2/pi;
end
function [S, m, s_grid] = IsopycnalSpectrumAtVerticalWavenumbersSummedFail(self)
% You can't just take the sqrt of the isopycnal variance
% because that changes the wavenumber content (you make
% everthing positive)
zHD = linspace(min(self.z_in),max(self.z_in),2^14 + 1).';
xi = cumtrapz(zHD,sqrt(self.N2(zHD)));
Lxi = max(xi)-min(xi);
s = (self.L_gm/Lxi)*xi;
s_grid = linspace(min(s),max(s),2^14+1).';
s_grid(end) = [];
dgrid = s_grid(2)-s_grid(1);
z = interp1(s,zHD,s_grid); % positions in z, of the evenly spaced stretched coordinate
zeta2 = self.IsopycnalVariance(z);
N = length(z);
rescale = (Lxi/self.invT_gm/self.invT_gm/self.Lz)*(self.N2(z)).^(1/2);
iso = sqrt( rescale .* zeta2 );
dm = pi/self.L_gm;
m = ((1:N)*dm)';
% compute the discrete sine transform (DST), [f(1:N), 0, -f(N:-1:2)]
dstScratch = ifft(cat(1,iso,0,-iso(N:-1:2)),2*N,1);
isobar = 2*imag(dstScratch(2:N+1));
% This definition of the spectrum means that,
% (1/L)*sum(iso.*iso)*dz== sum(S)*dm
% The units of S are then in meters^3.
S = N*dgrid*isobar.*conj(isobar)/2/pi;
end
function S = IsopycnalSpectrumAtWavenumbers(self,z,k)
self.PrecomputeComputeInternalModesForK();
if isrow(k)
k = k.';
end
if isempty(self.Suv_k)
f = self.f0;
Nmax = self.N_max;
% We are using the one-sided version of the spectrum
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax).*( (1-(f./omega).^2) );
% Integrate B across the frequency bands, \int B domega
omegaMid = self.omega_k(1:end-1,:) + diff(self.omega_k,1,1)/2;
omegaLeft = cat(1,self.omega_k(1,:),omegaMid);
omegaRight = cat(1, omegaMid, self.omega_k(end,:));
BofK = self.B(omegaLeft,omegaRight);
% Now divide by dk. This is essentially applying the Jacobian.
kMid = self.k(1:end-1) + diff(self.k)/2;
kLeft = cat(2,self.k(1),kMid);
kRight = cat(2,kMid,self.k(end));
dk = (kRight-kLeft).';
BofK = BofK./dk;
self.Szeta_k = sum(self.E * self.Gamma_k .* shiftdim( C(self.omega_k) .* BofK, -1),3);
end
S = interpn(self.zInternal,self.k,self.Szeta_k,z,k,'linear');
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Vertical Velocity Spectra
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function E = VerticalVelocityVariance(self,z, varargin)
% Returns the total vertical velocity variance at a given depth. [m^2/s^s]
%
% z array of depths, in meters
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
%
% NOTE: You'll notice that there are oscillations near the
% surface. What's happening is that each peak occurs near the
% turn depth of the highest frequencies that have been
% computed. In other words, I need to compute a more dense
% frequency spectrum near the buoyancy frequency. OR, to smooth
% over this, we just coarsen the z grid near the surface to
% match those peaks
[z,approximation] = self.validateVarianceArguments(z,varargin{:});
om = linspace(0,self.N_max,2000);
S = self.VerticalVelocitySpectrumAtFrequencies(z,om,approximation,'one-sided');
E = sum(S,2)*(om(2)-om(1));
end
function S = VerticalVelocitySpectrumAtFrequencies(self,z,omega,varargin)
% The vertical velocity frequency spectrum at given depths
% and frequencies. [m^2/s]
%
% z array of depths, in meters
% omega array of frequencies, in radians/second
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
% spectrumType (optional) 'one-sided', or 'two-sided'.
[z,omega,approximation,spectrumType] = self.validateSpectrumArguments(z,omega,varargin{:});
% Make sure it's a column vector
z = reshape(z,[],1);
% Choose a small increment
dOmega = omega(2)-omega(1);
dOmega = min( [self.f0/2,dOmega]);
% Create the function that converts to energy
f = self.f0;
Nmax = self.N_max;
if strcmp(spectrumType,'two-sided')
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( omega*omega - f*f )*0.5;
else
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( omega*omega - f*f );
end
S = zeros(length(z),length(omega));
for i=1:length(omega)
Bomega = self.B( abs( omega(i) ) - dOmega/2, abs( omega(i) ) + dOmega/2 )/dOmega;
S(:,i) = self.E* ( Bomega .* C(omega(i)) );
end
S(isnan(S))=0;
if strcmp(approximation,'exact')
self.PrecomputeComputeInternalModesForOmega();
Gamma = interpn(self.zInternal,self.omega,sum(self.Gamma_omega,3,'omitnan'),z,abs(omega),'linear',0); % 0 to everything outside
elseif strcmp(approximation,'wkb') || strcmp(approximation,'wkb-hydrostatic')
Gamma = self.GammaForOmegaWKBApproximation(z, omega, approximation);
elseif strcmp(approximation,'gm')
Gamma = self.GammaForOmegaGM(z, omega);
end
S = S.*Gamma;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Sea-surface height spectra
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [S,k_] = SSHSpectrumAtFrequencies(self,omega)
% The horizontal velocity frequency spectrum at given depths
% and frequencies. [m^2/s]
%
% z array of depths, in meters
% omega array of frequencies, in radians/second
% appoximation (optional) 'exact' (default), 'wkb', 'wkb-hydrostatic', 'gm'
% spectrumType (optional) 'one-sided', or 'two-sided'.
% [z,omega,approximation,spectrumType] = self.validateSpectrumArguments(z,omega,varargin{:});
omega = reshape(omega,[],1);
% Choose a small increment
dOmega = omega(2)-omega(1);
dOmega = min( [self.f0/2,dOmega]);
% Create the function that converts to energy
f = self.f0;
Nmax = self.N_max;
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax)*( (1-f*f/(omega*omega)) );
S = zeros(length(omega),self.nModes);
for i=1:length(omega)
Bomega = self.B( abs( omega(i) ) - dOmega/2, abs( omega(i) ) + dOmega/2 )/dOmega;
S(i,:) = self.E* ( Bomega .* C(omega(i)) );
end
S(isnan(S))=0;
k_ = interpn(self.omega.',1:self.nModes,real(self.k_omega),omega,1:self.nModes);
Phi = squeeze(self.Phi_omega(1,:,:)).*self.h_omega;
S = S.*interpn(self.omega.',1:self.nModes,Phi,omega,1:self.nModes)/self.g;
end
function [Sssh_kj, k_, j_] = SSHSpectrumAtWavenumberAndMode(self)
self.PrecomputeComputeInternalModesForK();
f = self.f0;
Nmax = self.N_max;
% We are using the one-sided version of the spectrum
C = @(omega) (abs(omega)<f | abs(omega) > Nmax)*0 + (abs(omega) >= f & abs(omega) <= Nmax).*( (1-(f./omega).^2) );
% Integrate B across the frequency bands, \int B domega
omegaMid = self.omega_k(1:end-1,:) + diff(self.omega_k,1,1)/2;
omegaLeft = cat(1,self.omega_k(1,:),omegaMid);
omegaRight = cat(1, omegaMid, self.omega_k(end,:));
BofK = self.B(omegaLeft,omegaRight);
% Now divide by dk. This is essentially applying the Jacobian.
kMid = self.k(1:end-1) + diff(self.k)/2;
kLeft = cat(2,self.k(1),kMid);
kRight = cat(2,kMid,self.k(end));
dk = (kRight-kLeft).';
BofK = BofK./dk;
Sssh_kj = squeeze(self.E * self.Phi_k(1,:,:) .* shiftdim( C(self.omega_k) .* BofK, -1)).*self.h_k;
k_ = reshape(self.k,[],1);
j_ = 1:self.nModes;
end
function [Sssh_k_omega, k_, omega_] = SSHSpectrumAtWavenumberAndFrequency(self)
% Convert wavenumber-mode to wavenumber-frequency
[Sssh_kj, k_, j_] = self.SSHSpectrumAtWavenumberAndMode();
% Choose a full range of frequency, spanning the known domain
omega_ = exp(linspace(log(self.f0),log(self.N_max),50)).';
omegaMid = omega_(1:end-1) + diff(omega_)/2;
omegaLeft = cat(1,omega_(1),omegaMid);
omegaRight = cat(1, omegaMid, self.omega_k(end));
domega = (omegaRight-omegaLeft);
Sssh_k_omega = zeros(length(k_),length(omega_));
for i=1:length(k_)
Sssh_j = Sssh_kj(i,:);
omega_j = self.omega_k(i,:);
for iOmega=1:length(omega_)
omegaIndices = omega_j >= omegaLeft(iOmega) & omega_j < omegaRight(iOmega);
Sssh_k_omega(i,iOmega) = sum( Sssh_j(omegaIndices) )/domega(iOmega);
end
end
end
end
methods (Access = protected)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Error checking and validation
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function isValid = validateOmega(~, omega)
dOmegaVector = diff(omega);
if any(dOmegaVector<0)
error('omega must be strictly monotonically increasing.')
end
if max(abs(diff(unique(dOmegaVector)))) > 1e-7
error('omega must be an evenly spaced grid');
end
isValid = 1;
end
function isValid = validateSpectrumType(~, omega, spectrumType)
if (any(omega<0)) && strcmp(approximation,'one-sided')
error('omega contains negative frequencies, yet you requested a one-sided spectrum. This makes no sense. Try again.');
end
isValid = any(validatestring(spectrumType,{'one-sided','two-sided'}));
end
function isValid = validateApproximations(~, x)
isValid = any(validatestring(x,{'exact','wkb', 'wkb-hydrostatic', 'gm'}));
end
function isValid = validateZ(self, z)
isValid = all( z >= min(self.z_in) ) && all(z <= max(self.z_in));
end
function [z,approximation] = validateVarianceArguments(self,z,varargin)
p = inputParser;
addRequired(p,'z',@(x) self.validateZ(x));
addOptional(p,'approximation','exact',@(x) self.validateApproximations(x));
parse(p,z,varargin{:})
z = p.Results.z;
approximation = p.Results.approximation;
end
function [z,omega,approximation,spectrumType] = validateSpectrumArguments(self,z,omega,varargin)
if length(varargin) < 2 && all(omega>=0)
spectrumTypeDefault = 'one-sided';
else
spectrumTypeDefault = 'two-sided';
end
p = inputParser;
addRequired(p,'z',@(x) self.validateZ(x));
addRequired(p,'omega',@(x) self.validateOmega(x));
addOptional(p,'approximation','exact',@(x) self.validateApproximations(x));
addOptional(p,'spectrumType',spectrumTypeDefault,@(x) self.validateSpectrumType(omega,x));
parse(p,z,omega,varargin{:})
z = p.Results.z;
omega = p.Results.omega;
approximation = p.Results.approximation;
spectrumType = p.Results.spectrumType;
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Computation of the vertical structure functions Phi and Gamma
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [F_out,G_out,h_out,F2_out,N2G2_out] = InternalModesForCoordinate(self,x,methodName)
% This will return size(Phi) =
% [length(self.zInternal),length(x),nModes]. This is really an
% unsummed version of Phi, so Phi = sum(Phi,3) would match the
% definition in the manuscript. Entries will contain NaN where
% no mode was determined.
nX = length(x);
nEVP = self.nEVPMin;
if self.userSpecifiedN2 == 1
im = InternalModesAdaptiveSpectral(self.N2,self.z_in,self.zInternal,self.latitude, 'nEVP', nEVP, 'normalization', Normalization.kConstant,'N2',1,'rho0',self.rho0);
else
im = InternalModesAdaptiveSpectral(self.rho,self.z_in,self.zInternal,self.latitude, 'nEVP', nEVP, 'normalization', Normalization.kConstant);
end
F_out = nan(length(self.zInternal),nX,self.nModes);
G_out = nan(length(self.zInternal),nX,self.nModes);
h_out = nan(nX,self.nModes);
F2_out = nan(nX,self.nModes);
N2G2_out = nan(nX,self.nModes);
for i = 1:length(x)
[F, G, h, ~, F2, N2G2,~] = im.(methodName)(x(i));
% Increase the number of grid points until we get the
% desired number of good quality modes (or reach some max).
while( (isempty(h) || length(h) < self.nModes) && nEVP < self.nEVPMax )
nEVP = nEVP + 128;
if self.userSpecifiedN2 == 1
im = InternalModesAdaptiveSpectral(self.N2,self.z_in,self.zInternal,self.latitude, 'nEVP', nEVP, 'normalization', Normalization.kConstant,'N2',1,'rho0',self.rho0);
else
im = InternalModesAdaptiveSpectral(self.rho,self.z_in,self.zInternal,self.latitude, 'nEVP', nEVP, 'normalization', Normalization.kConstant);
end
im.normalization = Normalization.kConstant;
[F, G, h, ~, F2, N2G2,~] = im.(methodName)(x(i));
end
if length(h) < self.nModes
fprintf('Only found %d good modes (of %d requested). Proceeding anyway.\n',length(h),self.nModes);
end
j0 = min(length(h),self.nModes);
h = reshape(h,1,[]);
F_out(:,i,1:j0) = F(:,1:j0);
G_out(:,i,1:j0) = G(:,1:j0);
h_out(i,1:j0)=h(1:j0);
F2_out(i,1:j0)=F2(1:j0);
N2G2_out(i,1:j0)=N2G2(1:j0);
end
end
function PrecomputeComputeInternalModesForOmega(self)
if self.didPrecomputePhiAndGammaForOmega==0
nOmega = 25;
% self.omega = linspace(self.f0,0.99*self.N_max,nOmega);
% self.omega = exp(linspace(log(self.f0),log(0.99*self.N_max),nOmega));
% dL = zeros(size(self.omega));
% for i=1:length(self.omega)
% [zBoundariesAndTPs, thesign, boundaryIndices] = InternalModesSpectral.FindTurningPointBoundariesAtFrequency(self.N2internal, self.zInternal, self.omega(i));
% for j=1:length(thesign)
% if thesign(j)>0
% dL(i) = dL(i) + zBoundariesAndTPs(j+1)-zBoundariesAndTPs(j);
% end
% end
% end
% % first find all the extrema, including end points.
[~,indices,~] = blocknum(diff(self.N2internal)>0);
extrema = sort(sqrt([self.f0*self.f0; self.N2internal(indices); self.N2internal(end)]));
extrema(end) = 0.99*extrema(end);
% Numerically integrate N to find the thickness at each
% frequency.
[omegaAxis,intStrat] = self.integrateStratificationForEachOmega();
% now make this unique, for interpolation
% [intStratUnique,ia,~] = unique(intStrat);
% intStratUnique = flip(intStratUnique);
% omegaAxisUnique = flip(omegaAxis(ia));
%
%
% nOmega = 30;
% L = interp1(omegaAxis,intStrat,extrema);