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AndreyAPopovAndreyAPopov
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initial commit of QG docs
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‎toolbox/+otp/+quasigeostrophic/+presets/Canonical.m‎

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classdef Canonical < otp.quasigeostrophic.QuasiGeostrophicProblem
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% A preset of the Quasi-geostrophic equations starting at the unstable
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% state $ψ_0 = 0$.
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%
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% The value of the Reynolds number is $Re= 450$, the Rossby number is
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% $Ro=0.0036$, and the spatial discretization is $255$ interior units in
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% the $x$ direction and $511$ interior units in the $y$ direction.
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%
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% The timespan starts at $t=0$ and ends at $t=100$, which is
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% roughly equivalent to $25.13$ years.
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%
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methods
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function obj = Canonical(varargin)
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% Create the Canonical Quasi-geostrophic problem object.
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%
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% Parameters
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% ----------
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% varargin
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% A variable number of name-value pairs. The accepted names are
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%
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% - ``ReynoldsNumber`` – Value of $Re$.
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% - ``RossbyNumber`` – Value of $Ro$.
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% - ``Size`` – Two-tuple of the spatial discretization, $[nx, ny]$.
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%
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Re = 450;
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Ro = 0.0036;

toolbox/+otp/+quasigeostrophic/+presets/PopovMouIliescuSandu.m renamed to toolbox/+otp/+quasigeostrophic/+presets/PopovMouSanduIliescu.m

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classdef PopovMouIliescuSandu < otp.quasigeostrophic.QuasiGeostrophicProblem
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classdef PopovMouSanduIliescu < otp.quasigeostrophic.QuasiGeostrophicProblem
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% A preset for the Quasi-geostrophic equations created for
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% CITE ME HERE.
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%
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% The initial condition is given by an internal file and was created by
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% integrating the canonical preset until time $t=100$.
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%
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% The value of the Reynolds number is $Re= 450$, the Rossby number is
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% $Ro=0.0036$, and the spatial discretization is $63$ interior units in
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% the $x$ direction and $127$ interior units in the $y$ direction.
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%
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% The timespan starts at $t=100$ and ends at $t=100.0109$, which is
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% roughly equivalent to one day in model time.
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methods
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function obj = PopovMouIliescuSandu(varargin)
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function obj = PopovMouSanduIliescu(varargin)
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% Create the PopovMouSanduIliescu Quasi-geostrophic problem object.
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%
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% Parameters
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% ----------
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% varargin
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% A variable number of name-value pairs. The accepted names are
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%
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% - ``ReynoldsNumber`` – Value of $Re$.
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% - ``RossbyNumber`` – Value of $Ro$.
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% - ``Size`` – Two-tuple of the spatial discretization, $[nx, ny]$.
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%
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defaultsize = [255, 511];
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defaultsize = [63, 127];
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Re = 450;
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Ro = 0.0036;
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%% Do the rest
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sixHours = 6*80/176251.2;
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oneday = 24*80/176251.2;
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tspan = [100, 100 + sixHours];
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tspan = [100, 100 + oneday];
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obj = obj@otp.quasigeostrophic.QuasiGeostrophicProblem(tspan, ...
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psi0, params);

‎toolbox/+otp/+quasigeostrophic/QuasiGeostrophicProblem.m‎

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classdef QuasiGeostrophicProblem < otp.Problem
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% A chaotic PDE modeling the flow of a fluid on the earth.
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%
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% The governing partial differential equation that is discretized is,
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%
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% $$
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% Δψ_t = -J(ψ,ω) - {Ro}^{-1} ψ_x -{Re}^{-1} Δω - {Ro}^{-1} F, \\
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% $$
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% where the Jacobian term is a quadratic function,
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% $$
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% J(ψ,ω) \equiv ψ_x ω_y - ψ_y ω_x,
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% $$
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% the relationship between the vorticity $ω$ and the stream function $ψ$ is
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% $$
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% ω = -Δψ,
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% $$
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% the term $Δ$ is the two dimensional Laplacian over the
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% discretization, $Ro$ is the Rossby number, $Re$ is the Reynolds
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% number, and $F$ is a forcing term.
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% The spatial domain is fixed to $x ∈ [0, 1]$ and $y ∈ [0, 2]$, and
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% the boundary conditions of the PDE are assumed to be zero dirichlet
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% everywhere.
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%
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% A second order finite difference approximation is performed on the
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% grid to create the first derivative operators and the Laplacian
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% operator. The Jacobian is discretized using the Arakawa approximation
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% CITEME
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% $$
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% J(ψ,ω) = \frac{1}{3}[ψ_x ω_y - ψ_y ω_x + (ψ ω_y)_x - (ψ ω_x)_y + (ψ_x ω)_y - (ψ_y ω)_x],
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% $$
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% in order for the system to not become unstable.
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%
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% The Poisson equation is solved by the eigenvalue sylvester method for
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% computational efficiency.
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%
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% A ADLES MORE HERE.
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%
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% Notes
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% -----
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% +---------------------+-----------------------------------------------------------+
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% | Type | ODE |
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% +---------------------+-----------------------------------------------------------+
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% | Number of Variables | $nx \times ny$ |
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% +---------------------+-----------------------------------------------------------+
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% | Stiff | not typically, depending on $Re$, $Ro$, $Nx$, and $Ny$ |
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% +---------------------+-----------------------------------------------------------+
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%
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% Example
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% -------
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% >>> problem = otp.quasigeostrophic.presets.PopovMouSanduIliescu;
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% >>> sol = problem.solve();
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% >>> problem.movie(sol);
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%
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methods
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function obj = QuasiGeostrophicProblem(timeSpan, y0, parameters)
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methods (Static)
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function u = resize(u, newsize)
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% resize uses interpolation to resize states
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s = size(u);
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function psi = resize(psi, newsize)
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% Resizes the state onto a new grid by performing interpolation
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%
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% Parameters
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% ----------
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% psi : numeric(nx, ny)
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% the old state on the $x \times y$ grid.
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% newsize : numeric(1, 2)
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% the new size as a two-tuple $[nx, ny]$ indicating the new state of the system.
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%
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s = size(psi);
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X = linspace(0, 1, s(1) + 2);
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Y = linspace(0, 2, s(2) + 2).';
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Xnew = Xnew(2:end-1);
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Ynew = Ynew(2:end-1);
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u = interp2(Y, X, u, Ynew, Xnew);
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psi = interp2(Y, X, psi, Ynew, Xnew);
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end
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