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Quasi-geostrophic problem documentation #121

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bcf8ee4
initial commit of QG docs
AndreyAPopov Oct 25, 2023
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added content
AndreyAPopov Nov 8, 2023
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Merge branch 'master' into QG-docs
AndreyAPopov Nov 8, 2023
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22 changes: 22 additions & 0 deletions toolbox/+otp/+quasigeostrophic/+presets/Canonical.m
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@@ -1,6 +1,28 @@
classdef Canonical < otp.quasigeostrophic.QuasiGeostrophicProblem
% A preset of the Quasi-geostrophic equations starting at the unstable
% state $ψ_0 = 0$.
%
% The value of the Reynolds number is $Re= 450$, the Rossby number is
% $Ro=0.0036$, and the spatial discretization is $255$ interior units in
% the $x$ direction and $511$ interior units in the $y$ direction.
%
% The timespan starts at $t=0$ and ends at $t=100$, which is
% roughly equivalent to $25.13$ years.
%

methods
function obj = Canonical(varargin)
% Create the Canonical Quasi-geostrophic problem object.
%
% Parameters
% ----------
% varargin
% A variable number of name-value pairs. The accepted names are
%
% - ``ReynoldsNumber`` – Value of $Re$.
% - ``RossbyNumber`` – Value of $Ro$.
% - ``Size`` – Two-tuple of the spatial discretization, $[nx, ny]$.
%

Re = 450;
Ro = 0.0036;
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34 changes: 29 additions & 5 deletions ...ostrophic/+presets/PopovMouIliescuSandu.m → ...ostrophic/+presets/PopovMouSanduIliescu.m
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@@ -1,8 +1,32 @@
classdef PopovMouIliescuSandu < otp.quasigeostrophic.QuasiGeostrophicProblem
classdef PopovMouSanduIliescu < otp.quasigeostrophic.QuasiGeostrophicProblem
% A preset for the Quasi-geostrophic equations created for
% CITE ME HERE.
%
% The initial condition is given by an internal file and was created by
% integrating the canonical preset until time $t=100$.
%
% The value of the Reynolds number is $Re= 450$, the Rossby number is
% $Ro=0.0036$, and the spatial discretization is $63$ interior units in
% the $x$ direction and $127$ interior units in the $y$ direction.
%
% The timespan starts at $t=100$ and ends at $t=100.0109$, which is
% roughly equivalent to one day in model time.

methods
function obj = PopovMouIliescuSandu(varargin)
function obj = PopovMouSanduIliescu(varargin)
% Create the PopovMouSanduIliescu Quasi-geostrophic problem object.
%
% Parameters
% ----------
% varargin
% A variable number of name-value pairs. The accepted names are
%
% - ``ReynoldsNumber`` – Value of $Re$.
% - ``RossbyNumber`` – Value of $Ro$.
% - ``Size`` – Two-tuple of the spatial discretization, $[nx, ny]$.
%

defaultsize = [255, 511];
defaultsize = [63, 127];

Re = 450;
Ro = 0.0036;
Expand Down Expand Up @@ -40,9 +64,9 @@

%% Do the rest

sixHours = 6*80/176251.2;
oneday = 24*80/176251.2;

tspan = [100, 100 + sixHours];
tspan = [100, 100 + oneday];

obj = [email protected](tspan, ...
psi0, params);
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89 changes: 83 additions & 6 deletions toolbox/+otp/+quasigeostrophic/QuasiGeostrophicProblem.m
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@@ -1,5 +1,74 @@
classdef QuasiGeostrophicProblem < otp.Problem

% A chaotic PDE modeling the flow of a fluid on the earth.
%
% The governing partial differential equation that is discretized is,
%
% $$
% Δψ_t = -J(ψ,ω) - {Ro}^{-1} \partial_x ψ -{Re}^{-1} Δω - {Ro}^{-1} F,
% $$
% where the Jacobian term is a quadratic function,
% $$
% J(ψ,ω) \equiv \partial_x ψ \partial_x ω - \partial_x ψ \partial_x ω,
% $$
% the relationship between the vorticity $ω$ and the stream function $ψ$ is
% $$
% ω = -Δψ,
% $$
% the term $Δ$ is the two dimensional Laplacian over the
% discretization, the terms $\partial_x$ and $\partial_y$ are the first derivatives
% in the $x$ and $y$ directions respectively, $Ro$ is the Rossby number, $Re$ is the Reynolds
% number, and $F$ is a forcing term.
% The spatial domain is fixed to $x ∈ [0, 1]$ and $y ∈ [0, 2]$, and
% the boundary conditions of the PDE are assumed to be zero dirichlet
% everywhere.
%
% A second order finite difference approximation is performed on the
% grid to create the first derivative operators and the Laplacian
% operator.
%
% The Laplacian is defined using the 5-point stencil
% $$
% Δ = \begin{bmatrix} & 1 & \\ 1 & -4 & 1\\ & 1 & \end{bmatrix},
% $$
% which is scaled with respect to the square of the step size in each
% respective direction.
% The first derivatives,
% $$
% \partial_x &= \begin{bmatrix} & 0 & \\ 1/2 & 0 & 1/2\\ & 0 & \end{bmatrix},\\
% \partial_y &= \begin{bmatrix} & 1/2 & \\ 0 & 0 & 0\\ & 1/2 & \end{bmatrix},
% $$
% are the standard second order central finite difference operators in
% the $x$ and $y$ directions.
%
% The Jacobian is discretized using the Arakawa approximation
% CITEME
% $$
% J(ψ,ω) = \frac{1}{3}[ψ_x ω_y - ψ_y ω_x + (ψ ω_y)_x - (ψ ω_x)_y + (ψ_x ω)_y - (ψ_y ω)_x],
% $$
% in order for the system to not become unstable.
%
% The Poisson equation is solved by the eigenvalue sylvester method for
% computational efficiency.
%
% A ADLES MORE HERE.
%
% Notes
% -----
% +---------------------+-----------------------------------------------------------+
% | Type | ODE |
% +---------------------+-----------------------------------------------------------+
% | Number of Variables | $nx \times ny$ |
% +---------------------+-----------------------------------------------------------+
% | Stiff | not typically, depending on $Re$, $Ro$, $Nx$, and $Ny$ |
% +---------------------+-----------------------------------------------------------+
%
% Example
% -------
% >>> problem = otp.quasigeostrophic.presets.PopovMouSanduIliescu;
% >>> sol = problem.solve();
% >>> problem.movie(sol);
%

methods
function obj = QuasiGeostrophicProblem(timeSpan, y0, parameters)

Expand All @@ -19,10 +88,18 @@

methods (Static)

function u = resize(u, newsize)
% resize uses interpolation to resize states

s = size(u);
function psi = resize(psi, newsize)
% Resizes the state onto a new grid by performing interpolation
%
% Parameters
% ----------
% psi : numeric(nx, ny)
% the old state on the $x \times y$ grid.
% newsize : numeric(1, 2)
% the new size as a two-tuple $[nx, ny]$ indicating the new state of the system.
%

s = size(psi);

X = linspace(0, 1, s(1) + 2);
Y = linspace(0, 2, s(2) + 2).';
Expand All @@ -34,7 +111,7 @@
Xnew = Xnew(2:end-1);
Ynew = Ynew(2:end-1);

u = interp2(Y, X, u, Ynew, Xnew);
psi = interp2(Y, X, psi, Ynew, Xnew);

end

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