Quasi-geostrophic Model

class otp.quasigeostrophic.QuasiGeostrophicProblem

Bases: 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

\[\begin{split} Δ = \begin{bmatrix} & 1 & \\ 1 & -4 & 1\\ & 1 & \end{bmatrix}, \end{split}\]

which is scaled with respect to the square of the step size in each respective direction. The first derivatives,

\[\begin{split} \partial_x &= \begin{bmatrix} & 0 & \\ 1/2 & 0 & 1/2\\ & 0 & \end{bmatrix},\\ \partial_y &= \begin{bmatrix} & 1/2 & \\ 0 & 0 & 0\\ & 1/2 & \end{bmatrix}, \end{split}\]

are the standard second order central finite difference operators in the \(x\) and \(y\) directions.

The Jacobian is discretized using the Arakawa approximation,

\[ J(ψ,ω) = \frac{1}{3}[ψ_x ω_y - ψ_y ω_x + (ψ ω_y)_x - (ψ ω_x)_y + (ψ_x ω)_y - (ψ_y ω)_x], \]

in order for the system to not become unstable.

The Poisson equation is solved by the eigenvalue sylvester method for computational efficiency.

An Approximate deconvolution large eddy simulation closure model from [SSWI11] is also implemented to have the same level of accuracy with a coarser grid size.

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);

Method Summary

static 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.

Parameters

class otp.quasigeostrophic.QuasiGeostrophicParameters

Bases: otp.Parameters

Parameters for the Quasi-geostrophic problem.

Constructor Summary

QuasiGeostrophicParameters(varargin)

Create a quasi-geostrophic parameters object.

Parameters:: varargin – A variable number of name-value pairs. A name can be any property of this class, and the subsequent value initializes that property.

Property Summary

Nx: Nx is the number of grid points in the x direction

Ny: Ny is the number of grid points in the y direction

ReynoldsNumber: ReynoldsNumber is the Reynolds number

RossbyNumber: RossbyNumber is the Rossby number

ADLambda: ADlambda is the approximate deconvolution parameter

ADPasses: ADPasses defines how many passes of the filter to do

Presets

class otp.quasigeostrophic.presets.Canonical

Bases: 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.

Constructor Summary

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 \(Ro\).
Nx – Spatial discretization in \(x\).
Ny – Spatial discretization in \(y\).
ADLambda – Scaling factor for approximate deconvolution RHS
ADPasses – Number of AD passes

class otp.quasigeostrophic.presets.PopovMouSanduIliescu

Bases: otp.quasigeostrophic.QuasiGeostrophicProblem

A preset for the Quasi-geostrophic equations created for [PMSI21].

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.

Constructor Summary

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 \(Ro\).
Nx – Spatial discretization in \(x\).
Ny – Spatial discretization in \(y\).
ADLambda – Scaling factor for approximate deconvolution RHS
ADPasses – Number of AD passes