Oceanic Basin Modes: Quasi-Geostrophic approach

This tutorial was contributed by Christine Kaufhold and Francis Poulin.

As a continuation of the Quasi-Geostrophic (QG) model described in the other tutorial, we will now see how we can use Firedrake to compute the spatial structure and frequencies of the freely evolving modes in this system, what are referred to as basin modes. Oceanic basin modes are low frequency structures that propagate zonally in the oceans that alter the dynamics of Western Boundary Currents, such as the Gulf Stream. In this particular tutorial we will show how to solve the QG eigenvalue problem with no basic state and no dissipative forces. Unlike the other demo that integrated the equations forward in time, in this problem it is necessary to compute the eigenvalues and eigenfunctions for a particular differential operator. This requires using PETSc matrices and eigenvalue solvers in SLEPc.

This demo requires SLEPc and slepc4py to be installed. This is most easily achieved by providing the optional –slepc flag to either firedrake-install (for a new installation), or firedrake-update (to add SLEPc to an existing installation).

Governing PDE

We first briefly recap the nonlinear, one-layer QG equation that we considered previously. The interested reader can find the derivations in [Ped92] and [Val06]. This model consists of an evolution equation for the Potential Vorticity, \(q\), and an elliptic problem through which we can determine the streamfunction,

\[\begin{split}\partial_{t}q + \vec{\nabla}\cdot (\vec{u}q) + \beta v = 0 \\ q = \nabla^{2} \psi - F\psi\end{split}\]

Where \(\psi\) is the stream-function, \(\vec{u}=(u, v)\) is the velocity field, \(q\) is the Potential Vorticity (PV), \(\beta\) is the Coriolis parameter and \(F\) is the rotational Froude number. The velocity field is easily obtained using

\[\vec{u} = \vec{\nabla}^{\bot}\psi, \quad \mbox{ with } \quad \vec{\nabla}^{\bot} = \hat{e_{z}} \times \vec{\nabla}\]

We assume that the amplitude of the wave motion is very small, which allows us to linearize the equations of motion and therefore neglect the nonlinear advection,

\[\frac{\partial}{\partial t} (\nabla^{2} \psi - F\psi) = - \beta \frac{\partial \psi}{\partial x}\]

We look for wave-like solutions that are periodic in time, with a frequency of \(\omega\)

\[\psi = \hat{\psi}(x, y)e^{-i\omega t}\]

This has the advantage of removing the time derivative from the equation and replacing it with an eigenvalue, \(i \omega\). By substituting the above solution into the QG equation, we can find a complex eigenvalue problem of the form

\[i\omega (\nabla^{2} \hat{\psi} - F\hat{\psi}) = \hat{\beta} \frac{\partial \hat{\psi}}{\partial x}\]

Weak Formulation

To use a finite element method it is necessary to formulate the weak form and then we can use SLEPc in Firedrake to compute eigenvalue problems easily. To begin, we multiply this equation by a Test Function \(\phi\) and integrate over the domain \(A\).

\[i\omega \iint_{A} \Big(\phi\cdot\nabla^{2} \hat{\psi}\,dA - F\phi\hat{\psi}\,dA\Big) = \hat{\beta}\iint_{A} \phi \cdot \frac{\partial \hat{\psi}}{\partial x}\,dA\]

To remove the Laplacian operator we use integration by parts and the Divergence theorem to obtain

\[\iint_{A} \phi \cdot \nabla^{2}\hat{\psi} \,dA = - \iint_{A} \nabla\phi \cdot \nabla\hat{\psi}\,dA + \oint_{\partial A} \phi \cdot \frac{\partial \hat{\psi}}{\partial n} \,dS\]

No-normal flow boundary conditions are required and mathematically this means that the streamfunction must be a constant on the boundary. Since the test functions inherit these boundary conditions, \(\hat{\phi} = 0\) on the boundary, the boundary integral vanishes and the weak form becomes,

\[i\omega \iint_{A} \Big( \nabla\phi\cdot\nabla \hat{\psi}\,dA + F\phi\hat{\psi}\Big)\,dA = \hat{\beta}\iint_{A} \phi \cdot \frac{\partial \hat{\psi}}{\partial x}\,dA\]

Firedrake code

Using this form, we can now implement this eigenvalue problem in Firedrake. We import the Firedrake, PETSc, and SLEPc libraries.

from firedrake import *
from firedrake.petsc import PETSc
try:
    from slepc4py import SLEPc
except ImportError:
    import sys
    warning("Unable to import SLEPc, eigenvalue computation not possible (try firedrake-update --slepc)")
    sys.exit(0)

We specify the geometry to be a square geometry with \(50\) cells with length \(1\).

Lx   = 1.
Ly   = 1.
n0   = 50
mesh = RectangleMesh(n0, n0, Lx, Ly, reorder=None)

Next we define the function spaces within which our solution will reside.

Vcg  = FunctionSpace(mesh,'CG',3)

We impose zero Dirichlet boundary conditions, in a strong sense, which guarantee that we have no-normal flow at the boundary walls.

bc = DirichletBC(Vcg, 0.0, "on_boundary")

The two non-dimensional parameters are the \(\beta\) parameter, set by the sphericity of the Earth, and the Froude number, the relative importance of rotation to stratification.

beta = Constant('1.0')
F    = Constant('1.0')

Additionally, we can create some Functions to store the eigenmodes.

eigenmodes_real, eigenmodes_imag = Function(Vcg), Function(Vcg)

We define the Test Function \(\phi\) and the Trial Function \(\psi\) in our function space.

phi, psi = TestFunction(Vcg), TrialFunction(Vcg)

To build the weak formulation of our equation we need to build two PETSc matrices in the form of a generalized eigenvalue problem, \(A\psi = \lambda M\psi\). We impose the boundary conditions on the mass matrix \(M\), since that is where we used integration by parts.

a =  beta*phi*psi.dx(0)*dx
m = -inner(grad(psi), grad(phi))*dx - F*psi*phi*dx
petsc_a = assemble(a).M.handle
petsc_m = assemble(m, bcs=bc).M.handle

We can declare how many eigenpairs, eigenfunctions and eigenvalues, we want to find

num_eigenvalues = 1

Next we will impose parameters onto our eigenvalue solver. The first is specifying that we have an generalized eigenvalue problem that is nonhermitian. The second specifies the spectral transform shift factor to be non-zero. The third requires we use a Krylov-Schur method, which is the default so this is not strictly necessary. Then, we ask for the eigenvalues with the largest imaginary part. Finally, we specify the tolerance.

opts = PETSc.Options()
opts.setValue("eps_gen_non_hermitian", None)
opts.setValue("st_pc_factor_shift_type", "NONZERO")
opts.setValue("eps_type", "krylovschur")
opts.setValue("eps_largest_imaginary", None)
opts.setValue("eps_tol", 1e-10)

Finally, we build our eigenvalue solver using SLEPc. We add our PETSc matrices into the solver as operators and use setFromOptions() to call the PETSc parameters we previously declared.

es = SLEPc.EPS().create(comm=COMM_WORLD)
es.setDimensions(num_eigenvalues)
es.setOperators(petsc_a, petsc_m)
es.setFromOptions()
es.solve()

Additionally we can find the number of converged eigenvalues.

nconv = es.getConverged()

We now get the real and imaginary parts of the eigenvalue and eigenvector for the leading eigenpair (that with the largest in magnitude imaginary part). First we check if we actually managed to converge any eigenvalues at all.

if nconv == 0:
    import sys
    warning("Did not converge any eigenvalues")
    sys.exit(0)

If we did, we go ahead and extract them from the SLEPc eigenvalue solver:

vr, vi = petsc_a.getVecs()

lam = es.getEigenpair(0, vr, vi)

and we gather the final eigenfunctions

eigenmodes_real.vector()[:], eigenmodes_imag.vector()[:] = vr, vi

We can now list and show plots for the eigenvalues and eigenfunctions that were found.

print("Leading eigenvalue is:", lam)

try:
    from matplotlib import pyplot
    plot(eigenmodes_real)
    pyplot.gcf().show()
    plot(eigenmodes_imag)
    pyplot.gcf().show()
except ImportError:
    warning("Matplotlib not available, not plotting eigemodes")

Below is a plot of the spatial structure of the real part of one of the eigenmodes computed above.

../_images/eigenmode_real.png

Below is a plot of the spatial structure of the imaginary part of one of the eigenmodes computed above.

../_images/eigenmode_imag.png

This demo can be found as a Python script in qgbasinmodes.py.

References

[Ped92]Joseph Pedlosky. Geophysical Fluid Dynamics. Springer study edition. Springer New York, 1992. ISBN 9780387963877.
[Val06]Geoffrey K. Vallis. Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, Cambridge, U.K., 2006.