# Quasi-Geostrophic Model
# =======================
#
# .. rst-class:: emphasis
#
# This tutorial was contributed by `Francis Poulin
# `__, based on code from `Colin Cotter
# `__.
#
# The Quasi-Geostrophic (QG) model is very important in geophysical fluid
# dynamics as it describes some aspects of large-scale flows in the oceans
# and atmosphere very well. The interested reader can find derivations in
# :cite:`QG-Pedlosky:1992` and :cite:`QG-Vallis:2006`.
#
# In these notes we present the nonlinear equations for the one-layer QG
# model with a free-surface. Then, the weak form will be derived as is
# needed for Firedrake.
#
# Governing Equations
# -------------------
#
# The Quasi-Geostrophic (QG) model is very similar to the 2D vorticity
# equation. Since the leading order geostrophic velocity is incompressible
# in the horizontal, the governing equations can be written as
#
# .. math::
#
# \begin{aligned}
# \partial_t q + \vec \nabla \cdot \left( \vec u q \right) + \beta v &= 0, \\
# \vec u & = \vec\nabla^\perp \psi, \\
# \nabla^2 \psi - \frac{1}{L_d^2} \psi &= q. \end{aligned}
#
# where the :math:`\psi` and :math:`q` are the streamfunction and
# Potential Vorticity (PV). The Laplacian is 2D since we are only in the
# horizontal plane and we defined
#
# .. math:: \vec\nabla^\perp = \hat e_z \times \vec\nabla.
#
# The first equation above states that the PV is conserved following the
# flow. The second equation forces the leading order velocity to be
# geostrophic and the third equation is the definition for the QG PV for
# this barotropic model. To solve this using Finite Elements it is
# necessary to establish the weak form of the model, which is done in the
# next subsection.
#
# Weak Form
# ---------
#
# Evolving the nonlinear equations consists of two steps. First, the
# elliptic problem must be solved to compute the streamfunction given the
# PV. Second, the PV equation must be integrated forward in time. This is
# done using a strong stability preserving Runge Kutta 3 (SSPRK3) method.
#
# Elliptic Equation
# ~~~~~~~~~~~~~~~~~
#
# First, we focus on the elliptic inversion in the case of a flat bottom.
# If we compute the inner product of the equation with the test function
# :math:`\phi` we obtain,
#
# .. math::
#
# \begin{aligned}
# \langle \nabla^2 \psi, \phi \rangle - \frac{1}{L_d^2} \langle \psi, \phi \rangle &= \langle q, \phi \rangle, \\
# \langle \nabla \psi, \nabla \phi \rangle + \frac{1}{L_d^2} \langle \psi, \phi \rangle &= -\langle q, \phi \rangle,\end{aligned}
#
# where in the second equation we used the divergence theorem and the
# homogeneous Dirichlet boundary conditions on the test function.
#
# Evolution Equation
# ~~~~~~~~~~~~~~~~~~
#
# The SSPRK3 method used as explained in :cite:`QG-Gottlieb:2005` can be written as
#
# .. math::
#
# \begin{aligned}
# q^{(1)} &= q^n - \Delta t \left[ \vec \nabla \cdot \left( \vec u^n q^n \right) + \beta v^n \right] , \\
# q^{(2)} &= \frac34 q^n + \frac14 \left[ q^{(1)} - \Delta t \vec \nabla \cdot \left( \vec u^{(1)} q^{(1)} \right)
# - \Delta t \beta v^{(1)}\right], \\
# q^{n+1} &= \frac13 q^n + \frac23 \left[ q^{(2)} - \Delta t \vec \nabla \cdot \left( \vec u^{(2)} q^{(2)} \right) - \Delta t \beta v^{(1)} \right].\end{aligned}
#
# To get the weak form we need to introduce a test function, :math:`p`,
# and take the inner product of the first equation with :math:`p`.
#
# .. math::
#
# \begin{aligned}
# \langle q^{(1)}, p \rangle &= \langle q^n, p \rangle - \Delta t \langle \vec \nabla \cdot \left( \vec u^n q^n \right), p \rangle
# - \Delta t \langle \beta v, q \rangle, \\
# \langle q^{(1)}, p \rangle - \Delta t \langle \vec u^n q^n, \vec\nabla p \rangle + \Delta t \langle \beta v, q \rangle
# &= \langle q^n, p \rangle - \Delta t \langle \vec u^n q^n, p \rangle_{bdry}\end{aligned}
#
# The first and second terms on the left hand side are referred to as
# :math:`a_{mass}` and :math:`a_{int}` in the code. The first term on the
# right-hand side is referred to as :math:`a_{mass}` in the code. The
# second term on the right-hand side is the extra term due to the DG
# framework, which does not exist in the CG version of the problem and it
# is referred to as :math:`a_{flux}`. This above problem must be solved
# for :math:`q^{(1)}` and then :math:`q^{(2)}` and then these are used to
# compute the numerical approximation to the PV at the new time
# :math:`q^{n+1}`.
#
# We now move on to the implementation of the QG model for the case of a
# freely propagating Rossby wave. As ever, we begin by importing the
# Firedrake library. ::
from firedrake import *
# Next we define the domain we will solve the equations on, square
# domain with 50 cells in each direction that is periodic along the
# x-axis. ::
Lx = 2.*pi # Zonal length
Ly = 2.*pi # Meridonal length
n0 = 50 # Spatial resolution
mesh = PeriodicRectangleMesh(n0, n0, Lx, Ly, direction="x", quadrilateral=True)
# We define function spaces::
Vdg = FunctionSpace(mesh,"DG",1) # DG elements for Potential Vorticity (PV)
Vcg = FunctionSpace(mesh,"CG",1) # CG elements for Streamfunction
Vu = VectorFunctionSpace(mesh,"DG",1) # DG elements for velocity
# and initial conditions for the potential vorticity, here we use
# Firedrake's ability to :doc:`interpolate UFL expressions <../interpolation>`. ::
x = SpatialCoordinate(mesh)
q0 = Function(Vdg).interpolate(0.1*sin(x[0])*sin(x[1]))
# We define some :class:`~.Function`\s to store the fields::
dq1 = Function(Vdg) # PV fields for different time steps
qh = Function(Vdg)
q1 = Function(Vdg)
psi0 = Function(Vcg) # Streamfunctions for different time steps
psi1 = Function(Vcg)
# along with the physical parameters of the model. ::
F = Constant(1.0) # Rotational Froude number
beta = Constant(0.1) # beta plane coefficient
Dt = 0.1 # Time step
dt = Constant(Dt)
# Next, we define the variational problems. First the elliptic problem
# for the stream function. ::
psi = TrialFunction(Vcg)
phi = TestFunction(Vcg)
# Build the weak form for the inversion
Apsi = (inner(grad(psi),grad(phi)) + F*psi*phi)*dx
Lpsi = -q1*phi*dx
# We impose homogeneous dirichlet boundary conditions on the stream
# function at the top and bottom of the domain. ::
bc1 = DirichletBC(Vcg, 0., (1, 2))
psi_problem = LinearVariationalProblem(Apsi,Lpsi,psi0,bcs=bc1)
psi_solver = LinearVariationalSolver(psi_problem,
solver_parameters={
'ksp_type':'cg',
'pc_type':'sor'
})
# Next we'll set up the advection equation, for which we need an
# operator :math:`\vec\nabla^\perp`, defined as a python anonymouus
# function::
gradperp = lambda u: as_vector((-u.dx(1), u.dx(0)))
# For upwinding, we'll need a representation of the normal to a facet,
# and a way of selecting the upwind side::
n = FacetNormal(mesh)
un = 0.5*(dot(gradperp(psi0), n) + abs(dot(gradperp(psi0), n)))
# Now the variational problem for the advection equation itself. ::
q = TrialFunction(Vdg)
p = TestFunction(Vdg)
a_mass = p*q*dx
a_int = (dot(grad(p), -gradperp(psi0)*q) + beta*p*psi0.dx(0))*dx
a_flux = (dot(jump(p), un('+')*q('+') - un('-')*q('-')) )*dS
arhs = a_mass - dt*(a_int + a_flux)
q_problem = LinearVariationalProblem(a_mass, action(arhs,q1), dq1)
# Since the operator is a mass matrix in a discontinuous space, it can
# be inverted exactly using an incomplete LU factorisation with zero
# fill. ::
q_solver = LinearVariationalSolver(q_problem,
solver_parameters={
'ksp_type':'preonly',
'pc_type':'bjacobi',
'sub_pc_type': 'ilu'
})
# To visualise the output of the simulation, we create a :class:`~.File`
# object. To which we can store multiple :class:`~.Function`\s. So
# that we can distinguish between them we will give them descriptive
# names. ::
q0.rename("Potential vorticity")
psi0.rename("Stream function")
v = Function(Vu, name="gradperp(stream function)")
v.project(gradperp(psi0))
output = File("output.pvd")
output.write(q0, psi0, v)
# Now all that is left is to define the timestepping parameters and
# execute the time loop. ::
t = 0.
T = 10.
dumpfreq = 5
tdump = 0
v0 = Function(Vu)
while(t < (T-Dt/2)):
# Compute the streamfunction for the known value of q0
q1.assign(q0)
psi_solver.solve()
q_solver.solve()
# Find intermediate solution q^(1)
q1.assign(dq1)
psi_solver.solve()
q_solver.solve()
# Find intermediate solution q^(2)
q1.assign(0.75*q0 + 0.25*dq1)
psi_solver.solve()
q_solver.solve()
# Find new solution q^(n+1)
q0.assign(q0/3 + 2*dq1/3)
# Store solutions to xml and pvd
t += Dt
print(t)
tdump += 1
if tdump == dumpfreq:
tdump -= dumpfreq
v.project(gradperp(psi0))
output.write(q0, psi0, v, time=t)
# A python script version of this demo can be found `here `__.
#
# .. rubric:: References
#
# .. bibliography:: demo_references.bib
# :filter: docname in docnames
# :keyprefix: QG-