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 Bueler_profile(mesh, R):
x, y = firedrake.SpatialCoordinate(mesh)
r = firedrake.sqrt(x**2 + y**2)
h_divide = (2 * R * (alpha/A0)**(1/n) * (n-1)/n)**(n/(2*n+2))
h_part2 = (n+1)*(r/R) - n*(r/R)**((n+1)/n) + n*(max_value(1-(r/R),0))**((n+1)/n) - 1
h_expr = (h_divide/((n-1)**(n/(2*n+2)))) * (max_value(h_part2,0))**(n/(2*n+2))
return h_expr
def advective_flux(self, **kwargs):
keys = (
"energy",
"velocity",
"vertical_velocity",
"thickness",
"energy_inflow",
"energy_surface",
)
E, u, w, h, E_inflow, E_surface = itemgetter(*keys)(kwargs)
Q = E.function_space()
ψ = firedrake.TestFunction(Q)
# NOTE: Be careful here going to xz! You might have to separate this into
# the sum of a horizontal and vertical flux if we're shadowing Firedrake's
# grad operator with out own specialized one.
U = firedrake.as_vector((u[0], u[1], w))
flux_cells = -E * inner(U, grad(ψ)) * h * dx
ν = FacetNormal(Q.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 _advect(self, dt, E, u, w, h, s, E_inflow, E_surface):
Q = E.function_space()
φ, ψ = firedrake.TrialFunction(Q), firedrake.TestFunction(Q)
U = firedrake.as_vector((u[0], u[1], w))
flux_cells = -φ * 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 = φ * ψ * outflow * h * ds_v + \
φ * ψ * firedrake.max_value(-w, 0) * h * ds_b + \
φ * ψ * firedrake.max_value(+w, 0) * h * ds_t
F = φ * ψ * h * dx + dt * (flux_cells + flux_outflow)
flux_inflow = -E_inflow * ψ * inflow * h * ds_v \
-E_surface * ψ * firedrake.min_value(-w, 0) * h * ds_b \
-E_surface * ψ * firedrake.min_value(+w, 0) * h * ds_t
A = E * ψ * h * dx + dt * flux_inflow
solver_parameters = {'ksp_type': 'preonly', 'pc_type': 'lu'}
degree_E = E.ufl_element().degree()
degree_u = u.ufl_element().degree()
degree = (3 * degree_E[0] + degree_u[0], 2 * degree_E[1] + degree_u[1])
form_compiler_parameters = {'quadrature_degree': degree}
firedrake.solve(F == A,
E,
solver_parameters=solver_parameters,
form_compiler_parameters=form_compiler_parameters)
def getModelVelocity(baseName, Q, V, minSigma=5, maxSigma=100):
"""Read in a tiff velocity data set and return
firedrake interpolate functions.
Parameters
----------
baseName : str
baseName should be of the form pattern.*.abc or pattern
The wildcard (*) will be filled with the suffixes (vx, vy.)
e.g.,pattern.vx.abc.tif, pattern.vy.abc.tif.
Q : firedrake function space
function space
V : firedrake vector space
vector space
Returns
-------
uObs firedrake interp function on V
velocity (m/yr)
speed firedrake interp function on Q
speed in (m)
sigmaX firedrake interp function on Q
vx error (m)
sigmaY firedrake interp function on Q
vy error (m)
"""
# suffixes for products used
suffixes = ['vx', 'vy', 'ex', 'ey']
rasters = {}
# prep baseName - baseName.*.xyz.tif or baseName.*
if '*' not in baseName:
baseName += '.*'
if '.tif' not in baseName:
baseName += '.tif'
# read data
for suffix in suffixes:
myBand = baseName.replace('*', suffix)
if not os.path.exists(myBand):
u.myerror(f'Velocity/error file - {myBand} - does not exist')
rasters[suffix] = rasterio.open(myBand, 'r')
# Firedrake interpolators
uObs = icepack.interpolate((rasters['vx'], rasters['vy']), V)
# force error to be at least 1 to avoid 0 or negatives.
sigmaX = icepack.interpolate(rasters['ex'], Q)
sigmaX = icepack.interpolate(firedrake.max_value(sigmaX, minSigma), Q)
sigmaX = icepack.interpolate(firedrake.min_value(sigmaX, maxSigma), Q)
sigmaY = icepack.interpolate(rasters['ey'], Q)
sigmaY = icepack.interpolate(firedrake.max_value(sigmaY, minSigma), Q)
sigmaY = icepack.interpolate(firedrake.min_value(sigmaY, maxSigma), Q)
speed = icepack.interpolate(firedrake.sqrt(inner(uObs, uObs)), Q)
# return results
return uObs, speed, sigmaX, sigmaY
def test_shallow_ice_prognostic_solve():
R = Constant(500e3)
num_refinements = 4
mesh = firedrake.UnitDiskMesh(num_refinements)
mesh.coordinates.dat.data[:] *= float(R)
T = Constant(254.15)
A = icepack.rate_factor(T)
Q = firedrake.FunctionSpace(mesh, "CG", 2)
V = firedrake.VectorFunctionSpace(mesh, "CG", 2)
x, y = firedrake.SpatialCoordinate(mesh)
r = firedrake.sqrt(x**2 + y**2)
β = Constant(0.5)
h_divide = Constant(4e3)
h_expr = h_divide * firedrake.max_value(0, 1 - (r / (β * R))**2)
h_0 = interpolate(h_expr, Q)
h = h_0.copy(deepcopy=True)
u = firedrake.Function(V)
b = Constant(0.0)
s = interpolate(b + h, Q)
a = Constant(0.0)
model = icepack.models.ShallowIce()
solver = icepack.solvers.FlowSolver(model)
final_time = 100.0
dt = 1.0
num_steps = int(final_time / dt)
for step in range(num_steps):
u = solver.diagnostic_solve(velocity=u,
thickness=h,
surface=s,
fluidity=A)
h = solver.prognostic_solve(dt,
thickness=h,
velocity=u,
accumulation=a)
h.interpolate(firedrake.max_value(0, h))
s.assign(b + h)
error = abs(assemble(h * dx) / assemble(h_0 * dx) - 1)
assert error < 1 / 2**(num_refinements + 1)
def terminus(u, h, s):
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
"""
mesh = u.ufl_domain()
zdegree = u.ufl_element().degree()[1]
x, y, ζ = firedrake.SpatialCoordinate(mesh)
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 solve(self, dt, **kwargs):
if not hasattr(self, '_solvers'):
self._setup(**kwargs)
else:
for name, field in kwargs.items():
if isinstance(field, firedrake.Function):
self.fields[name].assign(field)
δt = self._timestep
δt.assign(dt)
D = self.fields.get('damage', self.fields.get('D'))
solver1, solver2, solver3 = self._solvers
D1, D2 = self._stages
dD = self._damage_change
solver1.solve()
D1.assign(D + dD)
solver2.solve()
D2.assign(3 / 4 * D + 1 / 4 * (D1 + dD))
solver3.solve()
D.assign(1 / 3 * D + 2 / 3 * (D2 + dD))
S = self.model.sources(**self.fields)
D.project(min_value(max_value(D + δt * S, 0), 1))
return D.copy(deepcopy=True)
def flux(self, **kwargs):
keys = ("damage", "velocity", "damage_inflow")
D, u, D_inflow = itemgetter(*keys)(kwargs)
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 reduceNearGLBeta(s, sOrig, zF, grounded, Q, thresh, limit=False):
"""Compute beta reduction in area near grounding line where the height
above floation is less than thresh.
Parameters
----------
s : firedrake function
Current surface.
sOrig : firedrake function
Original surface.
zF : firedrake function
Flotation height.
grounded : firedrake function
grounde mask
Q : firedrake function space
scaler function space for model
thresh : float
Threshold to determine where to reduce beta (zAbove < thresh)
"""
# compute original height above flotation for grounded region
sAboveOrig = (sOrig - zF) * grounded
# avoid negative/zero values
sAboveOrig = firedrake.max_value(sAboveOrig, 0.0)
# Current height above flotation with negative values zeroed out.
sAbove = firedrake.max_value((s - zF) * grounded, 0)
# mask so only areas less than thresh but grounded
sMask = (sAbove < thresh) * grounded
# print(f'{sAbove.dat.data_ro.min()}, {sAbove.dat.data_ro.max()} {thresh}')
# Inverse mask
sMaskInv = sMask < 1
# scale = fraction of original height above flotation
# Use 5 to avoid potentially large ratio at small values.
scaleBeta = sAbove / \
firedrake.max_value(firedrake.min_value(thresh, sAboveOrig), 3)
if limit:
scaleBeta = firedrake.min_value(3, scaleBeta)
scaleBeta = scaleBeta * sMask + sMaskInv
# scaleBeta = icepack.interpolate(firedrake.min_value(scaleBeta,1.),Q)
# sqrt so tau = scale * beta^2
# scaleBeta = icepack.interpolate(firedrake.sqrt(scaleBeta) * grounded, Q)
# Removed grounded above because grounded is always applied in friction
scaleBeta = icepack.interpolate(firedrake.sqrt(scaleBeta), Q)
# print(f'{scaleBeta.dat.data_ro.min()}, {scaleBeta.dat.data_ro.max()}')
return scaleBeta
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 readSMB(SMBfile, Q):
''' Read SMB file an limit values to +/- 6 to avoid no data values
Returns water equivalent values.
'''
if not os.path.exists:
myerror(f'readSMB: SMB file ({SMBfile}) does not exist')
SMB = mf.getModelVarFromTiff(SMBfile, Q)
# avoid any unreasonably large value
SMB = icepack.interpolate(
firedrake.max_value(firedrake.min_value(SMB, 6), -6), Q)
return SMB
def solve(self, dt, h0, a, u, h_inflow=None):
r"""Propagate the thickness forward by one timestep
This function uses the implicit Euler timestepping scheme to avoid
the stability issues associated to using continuous finite elements
for advection-type equations. The implicit Euler scheme is stable
for any timestep; you do not need to ensure that the CFL condition
is satisfied in order to get an answer. Nonetheless, keeping the
timestep within the CFL bound is a good idea for accuracy.
Parameters
----------
dt : float
Timestep
h0 : firedrake.Function
Initial ice thickness
a : firedrake.Function
Sum of accumulation and melt rates
u : firedrake.Function
Ice velocity
h_inflow : firedrake.Function
Thickness of the upstream ice that advects into the domain
Returns
-------
h : firedrake.Function
Ice thickness at `t + dt`
"""
grad, ds = self.grad, self.ds
h_inflow = h_inflow if h_inflow is not None else h0
Q = h0.function_space()
h, φ = firedrake.TrialFunction(Q), firedrake.TestFunction(Q)
n = self.facet_normal(Q.mesh())
outflow = firedrake.max_value(inner(u, n), 0)
inflow = firedrake.min_value(inner(u, n), 0)
flux_cells = -h * inner(u, grad(φ)) * dx
flux_out = h * φ * outflow * ds
F = h * φ * dx + dt * (flux_cells + flux_out)
accumulation = a * φ * dx
flux_in = -h_inflow * φ * inflow * ds
A = h0 * φ * dx + dt * (accumulation + flux_in)
h = h0.copy(deepcopy=True)
solver_parameters = {'ksp_type': 'preonly', 'pc_type': 'lu'}
firedrake.solve(F == A, h, solver_parameters=solver_parameters)
return h
def reduceNearGLBeta(s, sOrig, zF, grounded, Q, thresh, limit=False):
"""Compute beta reduction in area near grounding line where the height
above floation is less than thresh.
Parameters
----------
s : firedrake function
Current surface.
sOrig : firedrake function
Original surface.
zF : firedrake function
Flotation height.
grounded : firedrake function
grounde mask
Q : firedrake function space
scaler function space for model
thresh : float
Threshold to determine where to reduce beta (zAbove < thresh)
"""
# compute original height above flotation for grounded region
sAboveOrig = (sOrig - zF) * grounded
# avoid negative/zero values
sAboveOrig = firedrake.max_value(sAboveOrig, 0.001)
# Current height above flotation with negative values zeroed out.
sAbove = firedrake.max_value((s - zF) * grounded, 0)
# mask so only areas less than thresh but grounded
sMask = (sAbove <= thresh) * grounded
# Inverse mask
sMaskInv = sMask < 1
# scale = fraction of of original height above flotation
scaleBeta = sAbove / \
firedrake.max_value(firedrake.min_value(thresh, sAboveOrig), 5)
if limit:
scaleBeta = firedrake.min_value(3, scaleBeta)
scaleBeta = scaleBeta * sMask + sMaskInv
# scaleBeta = icepack.interpolate(firedrake.min_value(scaleBeta,1.),Q)
# sqrt so tau = scale * beta^2
scaleBeta = icepack.interpolate(firedrake.sqrt(scaleBeta) * grounded, Q)
return scaleBeta
def friction(**kwargs):
keys = ('velocity', 'thickness', 'surface', 'friction')
u, h, s, C = map(kwargs.get, keys)
p_W = ρ_W * g * max_value(0, -(s - h))
p_I = ρ_I * g * h
N = p_I - p_W
τ_c = N / 2
u_c = (τ_c / C)**m
u_b = sqrt(inner(u, u))
return τ_c * ((u_c**(1 / m + 1) + u_b**(1 / m + 1))**(m / (m + 1)) - u_c)
def solve(self, dt, h0, a, u, h_inflow=None):
r"""Propagate the thickness forward by one timestep
This function uses an implicit second-order Taylor-Galerkin (also
known as Lax-Wendroff) scheme to solve the conservative advection
equation for ice thickness.
Parameters
----------
dt : float
Timestep
h0 : firedrake.Function
Initial ice thickness
a : firedrake.Function
Sum of accumulation and melt rates
u : firedrake.Function
Ice velocity
h_inflow : firedrake.Function
Thickness of the upstream ice that advects into the domain
Returns
-------
h : firedrake.Function
Ice thickness at `t + dt`
"""
grad, div, ds = self.grad, self.div, self.ds
h_inflow = h_inflow if h_inflow is not None else h0
Q = h0.function_space()
h, φ = firedrake.TrialFunction(Q), firedrake.TestFunction(Q)
n = self.facet_normal(Q.mesh())
outflow = firedrake.max_value(inner(u, n), 0)
inflow = firedrake.min_value(inner(u, n), 0)
flux_cells = -h * inner(u, grad(φ)) * dx
flux_cells_lax = 0.5 * dt * div(h * u) * inner(u, grad(φ)) * dx
flux_out = (h - 0.5 * dt * div(h * u)) * φ * outflow * ds
F = h * φ * dx + dt * (flux_cells + flux_cells_lax + flux_out)
accumulation = a * φ * dx
flux_in = -(h_inflow - 0.5 * dt * div(h0 * u)) * φ * inflow * ds
A = h0 * φ * dx + dt * (accumulation + flux_in)
h = h0.copy(deepcopy=True)
solver_parameters = {'ksp_type': 'preonly', 'pc_type': 'lu'}
firedrake.solve(F == A, h, solver_parameters=solver_parameters)
return h
def test_mass_transport_solver_convergence(solver_type):
Lx, Ly = 1.0, 1.0
u0 = 1.0
h_in, dh = 1.0, 0.2
delta_x, error = [], []
model = icepack.models.IceShelf()
for N in range(24, 97, 4):
delta_x.append(Lx / N)
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
degree = 1
V = firedrake.VectorFunctionSpace(mesh, family='CG', degree=degree)
Q = firedrake.FunctionSpace(mesh, family='CG', degree=degree)
solver = icepack.solvers.FlowSolver(
model, prognostic_solver_type=solver_type
)
h0 = interpolate(h_in - dh * x / Lx, Q)
a = firedrake.Function(Q)
u = interpolate(firedrake.as_vector((u0, 0)), V)
T = 0.5
δx = 1.0 / N
δt = δx / u0
num_timesteps = int(T / δt)
h = h0.copy(deepcopy=True)
for step in range(num_timesteps):
h = solver.prognostic_solve(
δt,
thickness=h,
velocity=u,
accumulation=a,
thickness_inflow=h0
)
z = x - u0 * num_timesteps * δt
h_exact = interpolate(h_in - dh/Lx * firedrake.max_value(0, z), Q)
error.append(norm(h - h_exact) / norm(h_exact))
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
print('log(error) ~= {:g} * log(dx) + {:g}'.format(slope, intercept))
assert slope > degree - 0.1
def compute_surface(h, b):
r"""Return the ice surface elevation consistent with a given
thickness and bathymetry
If the bathymetry beneath a tidewater glacier is too low, the ice
will go afloat. The surface elevation of a floating ice shelf is
.. math::
s = (1 - \rho_I / \rho_W)h,
provided everything is in hydrostatic balance.
"""
Q = h.ufl_function_space()
s_expr = firedrake.max_value(h + b, (1 - ρ_I / ρ_W) * h)
return firedrake.interpolate(s_expr, Q)
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 test_converting_fields():
δT = 5.0
T_surface = Tm - δT
T_expr = firedrake.min_value(Tm, T_surface + 2 * δT * (1 - ζ))
f_expr = firedrake.max_value(0, 0.0033 * (1 - 2 * ζ))
model = icepack.models.HeatTransport3D()
E = firedrake.project(model.energy_density(T_expr, f_expr), Q)
f = firedrake.project(model.meltwater_fraction(E), Q)
T = firedrake.project(model.temperature(E), Q)
avg_meltwater = firedrake.assemble(f * ds_b) / (Lx * Ly)
assert avg_meltwater > 0
avg_temp = firedrake.assemble(T * h * dx) / firedrake.assemble(h * dx)
assert (avg_temp > T_surface) and (avg_temp < Tm)
def equation(q):
Q = q.function_space()
φ = firedrake.TestFunction(Q)
F = q * u
mesh = Q.mesh()
n = firedrake.FacetNormal(mesh)
c = abs(inner(u, n))
sources = s * φ * dx
fluxes = (forms.cell_flux(F, φ) + forms.central_facet_flux(F, φ) +
forms.lax_friedrichs_facet_flux(q, c, φ) +
q * max_value(0, inner(u, n)) * φ * ds +
q_in * min_value(0, inner(u, n)) * φ * ds)
return sources - fluxes
def setup(self, **kwargs):
r"""Create the internal data structures that help reuse information
from past prognostic solves"""
for name, field in kwargs.items():
if name in self._fields.keys():
self._fields[name].assign(field)
else:
if isinstance(field, firedrake.Constant):
self._fields[name] = firedrake.Constant(field)
elif isinstance(field, firedrake.Function):
self._fields[name] = field.copy(deepcopy=True)
else:
raise TypeError(
"Input %s field has type %s, must be Constant or Function!"
% (name, type(field))
)
dt = firedrake.Constant(1.0)
h = self._fields["thickness"]
u = self._fields["velocity"]
h_0 = h.copy(deepcopy=True)
Q = h.function_space()
mesh = Q.mesh()
n = FacetNormal(mesh)
outflow = firedrake.max_value(0, inner(u, n))
inflow = firedrake.min_value(0, inner(u, n))
# Additional streamlining terms that give 2nd-order accuracy
q = firedrake.TestFunction(Q)
ds = firedrake.ds if mesh.layers is None else firedrake.ds_v
flux_cells = -div(h * u) * inner(u, grad(q)) * dx
flux_out = div(h * u) * q * outflow * ds
flux_in = div(h_0 * u) * q * inflow * ds
d2h_dt2 = flux_cells + flux_out + flux_in
dh_dt = self._continuity(dt, **self._fields)
F = (h - h_0) * q * dx - dt * (dh_dt + 0.5 * dt * d2h_dt2)
problem = firedrake.NonlinearVariationalProblem(F, h)
self._solver = firedrake.NonlinearVariationalSolver(
problem, solver_parameters=self._solver_parameters
)
self._thickness_old = h_0
self._timestep = dt
def __call__(self, dt, **kwargs):
keys = ("thickness", "velocity", "accumulation")
h, u, a = itemgetter(*keys)(kwargs)
h_inflow = kwargs.get("thickness_inflow", h)
Q = h.function_space()
q = firedrake.TestFunction(Q)
mesh = Q.mesh()
n = FacetNormal(mesh)
ds = firedrake.ds if mesh.layers is None else firedrake.ds_v
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 compute_surface(**kwargs):
r"""Return the ice surface elevation consistent with a given
thickness and bathymetry
If the bathymetry beneath a tidewater glacier is too low, the ice
will go afloat. The surface elevation of a floating ice shelf is
.. math::
s = (1 - \rho_I / \rho_W)h,
provided everything is in hydrostatic balance.
"""
# TODO: Remove the 'h' and 'b' arguments once these are deprecated.
h = kwargs.get('thickness', kwargs.get('h'))
b = kwargs.get('bed', kwargs.get('b'))
Q = h.ufl_function_space()
s_expr = firedrake.max_value(h + b, (1 - ρ_I / ρ_W) * h)
return firedrake.interpolate(s_expr, Q)
def setup(self, **kwargs):
r"""Create the internal data structures that help reuse information
from past prognostic solves"""
for name, field in kwargs.items():
if name in self._fields.keys():
self._fields[name].assign(field)
else:
if isinstance(field, firedrake.Constant):
self._fields[name] = firedrake.Constant(field)
elif isinstance(field, firedrake.Function):
self._fields[name] = field.copy(deepcopy=True)
else:
raise TypeError(
'Input fields must be Constant or Function!')
dt = firedrake.Constant(1.)
h = self._fields.get('thickness', self._fields.get('h'))
u = self._fields.get('velocity', self._fields.get('u'))
h_0 = h.copy(deepcopy=True)
Q = h.function_space()
model = self._continuity
n = model.facet_normal(Q.mesh())
outflow = firedrake.max_value(0, inner(u, n))
inflow = firedrake.min_value(0, inner(u, n))
# Additional streamlining terms that give 2nd-order accuracy
q = firedrake.TestFunction(Q)
div, grad, ds = model.div, model.grad, model.ds
flux_cells = -div(h * u) * inner(u, grad(q)) * dx
flux_out = div(h * u) * q * outflow * ds
flux_in = div(h_0 * u) * q * inflow * ds
d2h_dt2 = flux_cells + flux_out + flux_in
dh_dt = model(dt, **self._fields)
F = (h - h_0) * q * dx - dt * (dh_dt + 0.5 * dt * d2h_dt2)
problem = firedrake.NonlinearVariationalProblem(F, h)
self._solver = firedrake.NonlinearVariationalSolver(
problem, solver_parameters=self._solver_parameters)
self._thickness_old = h_0
self._timestep = dt
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 betaInit(s, h, speed, V, Q, Q1, grounded, inversionParams):
"""Compute intitial beta using 0.5 taud.
Parameters
----------
s : firedrake function
model surface elevation
h : firedrake function
model thickness
speed : firedrake function
modelled speed
V : firedrake vector function space
vector function space
Q : firedrake function space
scalar function space
grounded : firedrake function
Mask with 1s for grounded 0 for floating.
"""
# Use a result from prior inversion
checkFile = inversionParams['initFile']
Quse = Q
if inversionParams['initWithDeg1']:
checkFile = f'{inversionParams["inversionResult"]}.deg1'
Quse = Q1
if checkFile is not None:
betaTemp = mf.getCheckPointVars(checkFile, 'betaInv', Quse)['betaInv']
beta1 = icepack.interpolate(betaTemp, Q)
return beta1
# No prior result, so use fraction of taud
tauD = firedrake.project(-rhoI * g * h * grad(s), V)
#
stress = firedrake.sqrt(firedrake.inner(tauD, tauD))
Print('stress', firedrake.assemble(stress * firedrake.dx))
fraction = firedrake.Constant(0.95)
U = max_value(speed, 1)
C = fraction * stress / U**(1/m)
if inversionParams['friction'] == 'schoof':
mExp = 1/m + 1
U0 = firedrake.Constant(inversionParams['uThresh'])
C = C * (m/(m+1)) * (U0**mExp + U**mExp)**(1/(m+1))
beta = firedrake.interpolate(firedrake.sqrt(C) * grounded, Q)
return beta
def test_mass_transport_solver_convergence():
Lx, Ly = 1.0, 1.0
u0 = 1.0
h_in, dh = 1.0, 0.2
delta_x, error = [], []
mass_transport = MassTransport()
for N in range(16, 97, 4):
delta_x.append(Lx / N)
mesh = firedrake.RectangleMesh(N, N, Lx, Ly)
x, y = firedrake.SpatialCoordinate(mesh)
degree = 1
V = firedrake.VectorFunctionSpace(mesh, family='CG', degree=degree)
Q = firedrake.FunctionSpace(mesh, family='CG', degree=degree)
h = interpolate(h_in - dh * x / Lx, Q)
a = firedrake.Function(Q)
u = interpolate(firedrake.as_vector((u0, 0)), V)
T = 0.5
num_timesteps = int(0.5 * N * u0 * T / Lx)
dt = T / num_timesteps
for k in range(num_timesteps):
h = mass_transport.solve(dt, h0=h, a=a, u=u)
z = x - u0 * T
h_exact = interpolate(h_in - dh / Lx * firedrake.max_value(0, z), Q)
error.append(norm(h - h_exact) / norm(h_exact))
print(delta_x[-1], error[-1])
log_delta_x = np.log2(np.array(delta_x))
log_error = np.log2(np.array(error))
slope, intercept = np.polyfit(log_delta_x, log_error, 1)
assert slope > degree - 0.05
print(slope, intercept)
def flotationHeight(zb, Q, rhoI=rhoI, rhoW=rhoW):
"""Given bed elevation, determine height of flotation for function space Q.
Parameters
----------
zb : firedrake interp function
bed elevation (m)
Q : firedrake function space
function space
rhoI : [type], optional
[description], by default rhoI
rhoW : [type], optional
[description], by default rhoW
Returns
-------
zF firedrake interp function
Flotation height (m)
"""
# computation for height above flotation
zF = firedrake.interpolate(firedrake.max_value(-zb * (rhoW / rhoI - 1), 0),
Q)
return zF
def terminus(u, h, s, ice_front_ids=()):
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)hd\zeta ds
where :math:`\zeta_\text{sl}` is the relative depth to sea level and the
:math:`(\zeta - \zeta_\text{sl})` 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
"""
xdegree_u, zdegree_u = u.ufl_element().degree()
degree_h = h.ufl_element().degree()[0]
degree = (xdegree_u + degree_h, 2 * zdegree_u + 1)
metadata = {'quadrature_degree': degree}
x, y, ζ = firedrake.SpatialCoordinate(u.ufl_domain())
b = s - h
ζ_sl = firedrake.max_value(-b, 0) / h
p_W = ρ_W * g * h * _pressure_approx(zdegree_u + 1)(ζ, ζ_sl)
p_I = ρ_I * g * h * (1 - ζ)
ν = facet_normal_2(u.ufl_domain())
dγ = ds_v(tuple(ice_front_ids), metadata=metadata)
return (p_I - p_W) * inner(u, ν) * h * dγ
