def advective_flux(self, **kwargs):
keys = ('energy', 'velocity', 'vertical_velocity', 'thickness',
'energy_inflow', 'energy_surface')
keys_alt = ('E', 'u', 'w', 'h', 'E_inflow', 'E_surface')
E, u, w, h, E_inflow, E_surface = get_kwargs_alt(
kwargs, keys, keys_alt)
Q = E.function_space()
ψ = firedrake.TestFunction(Q)
U = firedrake.as_vector((u[0], u[1], w))
flux_cells = -E * inner(U, grad(ψ)) * h * dx
mesh = Q.mesh()
ν = facet_normal_2(mesh)
outflow = firedrake.max_value(inner(u, ν), 0)
inflow = firedrake.min_value(inner(u, ν), 0)
flux_outflow = (E * outflow * ψ * h * ds_v +
E * firedrake.max_value(-w, 0) * ψ * h * ds_b +
E * firedrake.max_value(+w, 0) * ψ * h * ds_t)
flux_inflow = (E_inflow * inflow * ψ * h * ds_v +
E_surface * firedrake.min_value(-w, 0) * ψ * h * ds_b +
E_surface * firedrake.min_value(+w, 0) * ψ * h * ds_t)
return flux_cells + flux_outflow + flux_inflow
def side_friction(**kwargs):
r"""Return the side wall friction part of the action functional
The component of the action functional due to friction along the side
walls of the domain is
.. math::
E(u) = -\frac{m}{m + 1}\int_\Gamma h\tau(u, C_s)\cdot u\; ds
where :math:`\tau(u, C_s)` is the side wall shear stress, :math:`ds`
is the element of surface area and :math:`\Gamma` are the side walls.
Side wall friction is relevant for glaciers that flow through a fjord
with rock walls on either side.
"""
u, h = get_kwargs_alt(kwargs, ('velocity', 'thickness'), ('u', 'h'))
Cs = kwargs.get('side_friction', kwargs.get('Cs', firedrake.Constant(0.)))
mesh = u.ufl_domain()
if mesh.geometric_dimension() == 2:
ν = firedrake.FacetNormal(mesh)
else:
ν = facet_normal_2(mesh)
u_t = u - inner(u, ν) * ν
τ = friction_stress(u_t, Cs)
return -m / (m + 1) * h * inner(τ, u_t)
def normal_flow_penalty(**kwargs):
r"""Return the penalty for flow normal to the domain boundary
For problems where a glacier flows along some boundary, e.g. a fjord
wall, the velocity has to be parallel to this boundary. Rather than
enforce this boundary condition directly, we add a penalty for normal
flow to the action functional.
"""
u, = get_kwargs_alt(kwargs, ('velocity', ), ('u', ))
scale = kwargs.get('scale', firedrake.Constant(1.))
mesh = u.ufl_domain()
if mesh.geometric_dimension() == 2:
ν = firedrake.FacetNormal(mesh)
elif mesh.geometric_dimension() == 3:
ν = facet_normal_2(mesh)
L = diameter(mesh)
δx = firedrake.FacetArea(mesh)
# Get the polynomial degree in the horizontal direction of the velocity
# field -- if extruded, the element degree is a tuple of the horizontal
# and vertical degrees.
degree = u.ufl_function_space().ufl_element().degree()
if isinstance(degree, tuple):
d = degree[0]
else:
d = degree
exponent = kwargs.get('exponent', d + 1)
penalty = scale * (L / δx)**exponent
return 0.5 * penalty * inner(u, ν)**2
def terminus(**kwargs):
r"""Return the terminal stress part of the ice stream action functional
The power exerted due to stress at the ice calving terminus :math:`\Gamma`
is
.. math::
E(u) = \frac{1}{2}\int_\Gamma\left(\rho_Igh^2 - \rho_Wgd^2\right)
u\cdot \nu\, ds
where :math:`d` is the water depth at the terminus. We assume that sea
level is at :math:`z = 0` for purposes of calculating the water depth.
Parameters
----------
velocity : firedrake.Function
thickness : firedrake.Function
surface : firedrake.Function
"""
keys = ('velocity', 'thickness', 'surface')
keys_alt = ('u', 'h', 's')
u, h, s = get_kwargs_alt(kwargs, keys, keys_alt)
d = firedrake.min_value(s - h, 0)
τ_I = ρ_I * g * h**2 / 2
τ_W = ρ_W * g * d**2 / 2
ν = firedrake.FacetNormal(u.ufl_domain())
return (τ_I - τ_W) * inner(u, ν)
def sources(self, **kwargs):
keys = ('energy', 'thickness', 'heat', 'heat_bed')
keys_alt = ('E', 'h', 'q', 'q_bed')
E, h, q, q_bed = get_kwargs_alt(kwargs, keys, keys_alt)
Q = E.function_space()
ψ = firedrake.TestFunction(Q)
internal_sources = q * ψ * h * dx
boundary_sources = q_bed * ψ * ds_b
return internal_sources + boundary_sources
def diffusive_flux(self, **kwargs):
keys = ('energy', 'thickness', 'energy_surface')
keys_alt = ('E', 'h', 'E_surface')
E, h, E_surface = get_kwargs_alt(kwargs, keys, keys_alt)
Q = E.function_space()
ψ = firedrake.TestFunction(Q)
κ = self.surface_exchange_coefficient
cell_flux = α * E.dx(2) * ψ.dx(2) / h * dx
surface_flux = κ * α * (E - E_surface) * ψ / h * ds_t
return cell_flux + surface_flux
def __call__(self, dt, **kwargs):
keys = ('thickness', 'velocity', 'accumulation')
keys_alt = ('h', 'u', 'a')
h, u, a = utilities.get_kwargs_alt(kwargs, keys, keys_alt)
h_inflow = kwargs.get('thickness_inflow', kwargs.get('h_inflow', h))
Q = h.function_space()
q = firedrake.TestFunction(Q)
grad, ds, n = self.grad, self.ds, self.facet_normal(Q.mesh())
u_n = inner(u, n)
flux_cells = -inner(h * u, grad(q)) * dx
flux_out = h * firedrake.max_value(u_n, 0) * q * ds
flux_in = h_inflow * firedrake.min_value(u_n, 0) * q * ds
accumulation = a * q * dx
return accumulation - (flux_in + flux_out + flux_cells)
def terminus(**kwargs):
r"""Return the terminus stress part of the ice shelf action functional
The power exerted due to stress at the calving terminus :math:`\Gamma` is
.. math::
E(u) = \int_\Gamma\varrho gh^2u\cdot\nu\; ds
We assume that sea level is at :math:`z = 0` for calculating the water
depth.
"""
u, h = get_kwargs_alt(kwargs, ('velocity', 'thickness'), ('u', 'h'))
mesh = u.ufl_domain()
ν = firedrake.FacetNormal(mesh)
ρ = ρ_I * (1 - ρ_I / ρ_W)
return 0.5 * ρ * g * h**2 * inner(u, ν)
def sources(self, **kwargs):
keys = ('damage', 'velocity', 'strain_rate', 'membrane_stress')
keys_alt = ('D', 'u', 'ε', 'M')
D, u, ε, M = get_kwargs_alt(kwargs, keys, keys_alt)
# Increase/decrease damage depending on stress and strain rates
ε_1 = eigenvalues(ε)[0]
σ_e = sqrt(inner(M, M) - det(M))
ε_h = firedrake.Constant(self.healing_strain_rate)
σ_d = firedrake.Constant(self.damage_stress)
γ_h = firedrake.Constant(self.healing_rate)
γ_d = firedrake.Constant(self.damage_rate)
healing = γ_h * min_value(ε_1 - ε_h, 0)
fracture = γ_d * conditional(σ_e - σ_d > 0, ε_1, 0.) * (1 - D)
return healing + fracture
def flux(self, **kwargs):
keys = ('damage', 'velocity', 'damage_inflow')
keys_alt = ('D', 'u', 'D_inflow')
D, u, D_inflow = get_kwargs_alt(kwargs, keys, keys_alt)
Q = D.function_space()
φ = firedrake.TestFunction(Q)
mesh = Q.mesh()
n = firedrake.FacetNormal(mesh)
u_n = max_value(0, inner(u, n))
f = D * u_n
flux_faces = (f('+') - f('-')) * (φ('+') - φ('-')) * dS
flux_cells = -D * div(u * φ) * dx
flux_out = D * max_value(0, inner(u, n)) * φ * ds
flux_in = D_inflow * min_value(0, inner(u, n)) * φ * ds
return flux_faces + flux_cells + flux_out + flux_in
def penalty(**kwargs):
r"""Return the penalty of the shallow ice action functional
The penalty for the shallow ice action functional is
.. math::
E(u) = \frac{1}{2}\int_\Omega l^2\nabla u\cdot \nabla u\; dx
Parameters
----------
velocity : firedrake.Function
thickness : firedrake.Function
Returns
-------
firedrake.Form
"""
u, h = get_kwargs_alt(kwargs, ('velocity', 'thickness'), ('u', 'h'))
l = 2 * firedrake.max_value(firedrake.CellDiameter(u.ufl_domain()), 5 * h)
return .5 * l**2 * inner(grad(u), grad(u))
def sources(self, **kwargs):
keys = ('damage', 'velocity', 'fluidity')
keys_alt = ('D', 'u', 'A')
D, u, A = get_kwargs_alt(kwargs, keys, keys_alt)
# Increase/decrease damage depending on stress and strain rates
ε = sym(grad(u))
ε_1 = eigenvalues(ε)[0]
σ = M(ε, A)
σ_e = sqrt(inner(σ, σ) - det(σ))
ε_h = firedrake.Constant(self.healing_strain_rate)
σ_d = firedrake.Constant(self.damage_stress)
γ_h = firedrake.Constant(self.healing_rate)
γ_d = firedrake.Constant(self.damage_rate)
healing = γ_h * min_value(ε_1 - ε_h, 0)
fracture = γ_d * conditional(σ_e - σ_d > 0, ε_1, 0.) * (1 - D)
return healing + fracture
def terminus(**kwargs):
r"""Return the terminal stress part of the hybrid model action functional
The power exerted due to stress at the calving terminus :math:`\Gamma` is
.. math::
E(u) = \int_\Gamma\int_0^1\left(\rho_Ig(1 - \zeta) -
\rho_Wg(\zeta_{\text{sl}} - \zeta)_+\right)u\cdot\nu\; h\, d\zeta\; ds
where :math:`\zeta_\text{sl}` is the relative depth to sea level and the
:math:`(\zeta_\text{sl} - \zeta)_+` denotes only the positive part.
Parameters
----------
u : firedrake.Function
ice velocity
h : firedrake.Function
ice thickness
s : firedrake.Function
ice surface elevation
ice_front_ids : list of int
numeric IDs of the parts of the boundary corresponding to the
calving front
"""
keys = ('velocity', 'thickness', 'surface')
keys_alt = ('u', 'h', 's')
u, h, s = get_kwargs_alt(kwargs, keys, keys_alt)
mesh = u.ufl_domain()
zdegree = u.ufl_element().degree()[1]
ζ = firedrake.SpatialCoordinate(mesh)[2]
b = s - h
ζ_sl = firedrake.max_value(-b, 0) / h
p_W = ρ_W * g * h * _pressure_approx(zdegree + 1)(ζ, ζ_sl)
p_I = ρ_I * g * h * (1 - ζ)
ν = facet_normal_2(mesh)
return (p_I - p_W) * inner(u, ν) * h
def gravity(**kwargs):
r"""Return the gravitational part of the ice stream action functional
The gravitational part of the hybrid model action functional is
.. math::
E(u) = -\int_\Omega\int_0^1\rho_Ig\nabla s\cdot u\; h\, d\zeta\; dx
Parameters
----------
u : firedrake.Function
ice velocity
h : firedrake.Function
ice thickness
s : firedrake.Function
ice surface elevation
"""
keys = ('velocity', 'thickness', 'surface')
keys_alt = ('u', 'h', 's')
u, h, s = get_kwargs_alt(kwargs, keys, keys_alt)
return -ρ_I * g * inner(grad_2(s), u) * h
def gravity(**kwargs):
r"""Return the gravitational part of the ice shelf action functional
The gravitational part of the ice shelf action functional is
.. math::
E(u) = -\frac{1}{2}\int_\Omega\varrho g\nabla h^2\cdot u\; dx
Parameters
----------
u : firedrake.Function
ice velocity
h : firedrake.Function
ice thickness
Returns
-------
firedrake.Form
"""
u, h = get_kwargs_alt(kwargs, ('velocity', 'thickness'), ('u', 'h'))
ρ = ρ_I * (1 - ρ_I / ρ_W)
return -0.5 * ρ * g * inner(grad(h**2), u)
def viscosity(**kwargs):
r"""Return the viscous part of the hybrid model action functional
The viscous component of the action for the hybrid model is
.. math::
E(u) = \frac{n}{n + 1}\int_\Omega\int_0^1\left(
M : \dot\varepsilon_x + \tau_z\cdot\varepsilon_z\right)h\, d\zeta\; dx
where :math:`M(\dot\varepsilon, A)` is the membrane stress tensor and
:math:`\tau_z` is the vertical shear stress vector.
This form assumes that we're using the fluidity parameter instead
the rheology parameter, the temperature, etc. To use a different
variable, you can implement your own viscosity functional and pass it
as an argument when initializing model objects to use your functional
instead.
Keyword arguments
-----------------
velocity : firedrake.Function
surface : firedrake.Function
thickness : firedrake.Function
fluidity : firedrake.Function
`A` in Glen's flow law
Returns
-------
firedrake.Form
"""
keys = ('velocity', 'thickness', 'surface', 'fluidity')
keys_alt = ('u', 'h', 's', 'A')
u, h, s, A = get_kwargs_alt(kwargs, keys, keys_alt)
ε_x, ε_z = horizontal_strain(u, s, h), vertical_strain(u, h)
M, τ_z = stresses(ε_x, ε_z, A)
return n / (n + 1) * (inner(M, ε_x) + inner(τ_z, ε_z)) * h
def gravity(**kwargs):
r"""Return the gravity term for the shallow ice action functional
The gravity function for the shallow ice action functional is
.. math::
E(u) = \int_\Omega\frac{2A(\varrho_I g)**n}{n+2} (\nabla h^2\cdot u) h^{n+1} \nabla s^{n-1}\; dx
Parameters
----------
velocity : firedrake.Function
thickness : firedrake.Function
surface : firedrake.Function
fluidity : firedrake.Function or firedrake.Constant
Returns
-------
firedrake.Form
"""
keys = ('velocity', 'thickness', 'surface', 'fluidity')
keys_alt = ('u', 'h', 's', 'A')
u, h, s, A = get_kwargs_alt(kwargs, keys, keys_alt)
return (2 * A * (ρ_I * g)**n / (n + 2)) * h**(n + 1) * grad(s)**(n - 1) * inner(grad(s), u)
