def _setup_solver(self):
state = self.state # just cutting down line length a bit
dt = state.timestepping.dt
beta = dt*state.timestepping.alpha
mu = state.mu
# Split up the rhs vector (symbolically)
u_in, p_in, b_in = split(state.xrhs)
# Build the reduced function space for u,p
M = MixedFunctionSpace((state.V[0], state.V[1]))
w, phi = TestFunctions(M)
u, p = TrialFunctions(M)
# Get background fields
bbar = state.bbar
# Analytical (approximate) elimination of theta
k = state.k # Upward pointing unit vector
b = -dot(k,u)*dot(k,grad(bbar))*beta + b_in
# vertical projection
def V(u):
return k*inner(u,k)
eqn = (
inner(w, (u - u_in))*dx
- beta*div(w)*p*dx
- beta*inner(w,k)*b*dx
+ phi*div(u)*dx
)
if mu is not None:
eqn += dt*mu*inner(w,k)*inner(u,k)*dx
aeqn = lhs(eqn)
Leqn = rhs(eqn)
# Place to put result of u p solver
self.up = Function(M)
# Boundary conditions (assumes extruded mesh)
dim = M.sub(0).ufl_element().value_shape()[0]
bc = ("0.0",)*dim
bcs = [DirichletBC(M.sub(0), Expression(bc), "bottom"),
DirichletBC(M.sub(0), Expression(bc), "top")]
# preconditioner equation
L = self.L
Ap = (
inner(w,u) + L*L*div(w)*div(u) +
phi*p/L/L
)*dx
# Solver for u, p
up_problem = LinearVariationalProblem(
aeqn, Leqn, self.up, bcs=bcs, aP=Ap)
nullspace = MixedVectorSpaceBasis(M,
[M.sub(0),
VectorSpaceBasis(constant=True)])
self.up_solver = LinearVariationalSolver(up_problem,
solver_parameters=self.params,
nullspace=nullspace)
# Reconstruction of b
b = TrialFunction(state.V[2])
gamma = TestFunction(state.V[2])
u, p = self.up.split()
self.b = Function(state.V[2])
b_eqn = gamma*(b - b_in +
dot(k,u)*dot(k,grad(bbar))*beta)*dx
b_problem = LinearVariationalProblem(lhs(b_eqn),
rhs(b_eqn),
self.b)
self.b_solver = LinearVariationalSolver(b_problem)
def _setup_solver(self):
state = self.state # just cutting down line length a bit
dt = state.timestepping.dt
beta = dt*state.timestepping.alpha
cp = state.parameters.cp
mu = state.mu
# Split up the rhs vector (symbolically)
u_in, rho_in, theta_in = split(state.xrhs)
# Build the reduced function space for u,rho
M = MixedFunctionSpace((state.V[0], state.V[1]))
w, phi = TestFunctions(M)
u, rho = TrialFunctions(M)
n = FacetNormal(state.mesh)
# Get background fields
thetabar = state.thetabar
rhobar = state.rhobar
pibar = exner(thetabar, rhobar, state)
pibar_rho = exner_rho(thetabar, rhobar, state)
pibar_theta = exner_theta(thetabar, rhobar, state)
# Analytical (approximate) elimination of theta
k = state.k # Upward pointing unit vector
theta = -dot(k,u)*dot(k,grad(thetabar))*beta + theta_in
# Only include theta' (rather than pi') in the vertical
# component of the gradient
# the pi prime term (here, bars are for mean and no bars are
# for linear perturbations)
pi = pibar_theta*theta + pibar_rho*rho
# vertical projection
def V(u):
return k*inner(u,k)
eqn = (
inner(w, (u - u_in))*dx
- beta*cp*div(theta*V(w))*pibar*dx
# following does nothing but is preserved in the comments
# to remind us why (because V(w) is purely vertical.
# + beta*cp*jump(theta*V(w),n)*avg(pibar)*dS_v
- beta*cp*div(thetabar*w)*pi*dx
+ beta*cp*jump(thetabar*w,n)*avg(pi)*dS_v
+ (phi*(rho - rho_in) - beta*inner(grad(phi), u)*rhobar)*dx
+ beta*jump(phi*u, n)*avg(rhobar)*(dS_v + dS_h)
)
if mu is not None:
eqn += dt*mu*inner(w,k)*inner(u,k)*dx
aeqn = lhs(eqn)
Leqn = rhs(eqn)
# Place to put result of u rho solver
self.urho = Function(M)
# Boundary conditions (assumes extruded mesh)
dim = M.sub(0).ufl_element().value_shape()[0]
bc = ("0.0",)*dim
bcs = [DirichletBC(M.sub(0), Expression(bc), "bottom"),
DirichletBC(M.sub(0), Expression(bc), "top")]
# Solver for u, rho
urho_problem = LinearVariationalProblem(
aeqn, Leqn, self.urho, bcs=bcs)
self.urho_solver = LinearVariationalSolver(urho_problem,
solver_parameters=self.params,
options_prefix='ImplicitSolver')
# Reconstruction of theta
theta = TrialFunction(state.V[2])
gamma = TestFunction(state.V[2])
u, rho = self.urho.split()
self.theta = Function(state.V[2])
theta_eqn = gamma*(theta - theta_in +
dot(k,u)*dot(k,grad(thetabar))*beta)*dx
theta_problem = LinearVariationalProblem(lhs(theta_eqn),
rhs(theta_eqn),
self.theta)
self.theta_solver = LinearVariationalSolver(theta_problem,
options_prefix='thetabacksubstitution')
def compressible_hydrostatic_balance(state, theta0, rho0, pi0=None,
top=False, pi_boundary=Constant(1.0),
solve_for_rho=False,
params=None):
"""
Compute a hydrostatically balanced density given a potential temperature
profile.
:arg state: The :class:`State` object.
:arg theta0: :class:`.Function`containing the potential temperature.
:arg rho0: :class:`.Function` to write the initial density into.
:arg top: If True, set a boundary condition at the top. Otherwise, set
it at the bottom.
:arg pi_boundary: a field or expression to use as boundary data for pi on
the top or bottom as specified.
"""
# Calculate hydrostatic Pi
W = MixedFunctionSpace((state.Vv,state.V[1]))
v, pi = TrialFunctions(W)
dv, dpi = TestFunctions(W)
n = FacetNormal(state.mesh)
cp = state.parameters.cp
alhs = (
(cp*inner(v,dv) - cp*div(dv*theta0)*pi)*dx
+ dpi*div(theta0*v)*dx
)
if top:
bmeasure = ds_t
bstring = "bottom"
else:
bmeasure = ds_b
bstring = "top"
arhs = -cp*inner(dv,n)*theta0*pi_boundary*bmeasure
if state.parameters.geopotential:
Phi = state.Phi
arhs += div(dv)*Phi*dx - inner(dv,n)*Phi*bmeasure
else:
g = state.parameters.g
arhs -= g*inner(dv,state.k)*dx
if(state.mesh.geometric_dimension() == 2):
bcs = [DirichletBC(W.sub(0), Expression(("0.", "0.")), bstring)]
elif(state.mesh.geometric_dimension() == 3):
bcs = [DirichletBC(W.sub(0), Expression(("0.", "0.", "0.")), bstring)]
w = Function(W)
PiProblem = LinearVariationalProblem(alhs, arhs, w, bcs=bcs)
if(params is None):
params = {'pc_type': 'fieldsplit',
'pc_fieldsplit_type': 'schur',
'ksp_type': 'gmres',
'ksp_monitor_true_residual': True,
'ksp_max_it': 100,
'ksp_gmres_restart': 50,
'pc_fieldsplit_schur_fact_type': 'FULL',
'pc_fieldsplit_schur_precondition': 'selfp',
'fieldsplit_0_ksp_type': 'richardson',
'fieldsplit_0_ksp_max_it': 5,
'fieldsplit_0_pc_type': 'gamg',
'fieldsplit_1_pc_gamg_sym_graph': True,
'fieldsplit_1_mg_levels_ksp_type': 'chebyshev',
'fieldsplit_1_mg_levels_ksp_chebyshev_estimate_eigenvalues': True,
'fieldsplit_1_mg_levels_ksp_chebyshev_estimate_eigenvalues_random': True,
'fieldsplit_1_mg_levels_ksp_max_it': 5,
'fieldsplit_1_mg_levels_pc_type': 'bjacobi',
'fieldsplit_1_mg_levels_sub_pc_type': 'ilu'}
PiSolver = LinearVariationalSolver(PiProblem,
solver_parameters=params)
PiSolver.solve()
v, Pi = w.split()
if pi0 is not None:
pi0.assign(Pi)
kappa = state.parameters.kappa
R_d = state.parameters.R_d
p_0 = state.parameters.p_0
if solve_for_rho:
w1 = Function(W)
v, rho = w1.split()
rho.interpolate(p_0*(Pi**((1-kappa)/kappa))/R_d/theta0)
v, rho = split(w1)
dv, dpi = TestFunctions(W)
pi = ((R_d/p_0)*rho*theta0)**(kappa/(1.-kappa))
F = (
(cp*inner(v,dv) - cp*div(dv*theta0)*pi)*dx
+ dpi*div(theta0*v)*dx
+ cp*inner(dv,n)*theta0*pi_boundary*bmeasure
)
if state.parameters.geopotential:
F += - div(dv)*Phi*dx + inner(dv,n)*Phi*bmeasure
else:
F += g*inner(dv,state.k)*dx
rhoproblem = NonlinearVariationalProblem(F, w1, bcs=bcs)
rhosolver = NonlinearVariationalSolver(rhoproblem, solver_parameters=params)
rhosolver.solve()
v, rho_ = w1.split()
rho0.assign(rho_)
else:
rho0.interpolate(p_0*(Pi**((1-kappa)/kappa))/R_d/theta0)
def _setup_solver(self):
state = self.state # just cutting down line length a bit
Dt = state.timestepping.dt
beta_ = Dt*state.timestepping.alpha
cp = state.parameters.cp
mu = state.mu
Vu = state.spaces("HDiv")
Vtheta = state.spaces("HDiv_v")
Vrho = state.spaces("DG")
# Store time-stepping coefficients as UFL Constants
dt = Constant(Dt)
beta = Constant(beta_)
beta_cp = Constant(beta_ * cp)
# Split up the rhs vector (symbolically)
u_in, rho_in, theta_in = split(state.xrhs)
# Build the reduced function space for u,rho
M = MixedFunctionSpace((Vu, Vrho))
w, phi = TestFunctions(M)
u, rho = TrialFunctions(M)
n = FacetNormal(state.mesh)
# Get background fields
thetabar = state.fields("thetabar")
rhobar = state.fields("rhobar")
pibar = thermodynamics.pi(state.parameters, rhobar, thetabar)
pibar_rho = thermodynamics.pi_rho(state.parameters, rhobar, thetabar)
pibar_theta = thermodynamics.pi_theta(state.parameters, rhobar, thetabar)
# Analytical (approximate) elimination of theta
k = state.k # Upward pointing unit vector
theta = -dot(k, u)*dot(k, grad(thetabar))*beta + theta_in
# Only include theta' (rather than pi') in the vertical
# component of the gradient
# the pi prime term (here, bars are for mean and no bars are
# for linear perturbations)
pi = pibar_theta*theta + pibar_rho*rho
# vertical projection
def V(u):
return k*inner(u, k)
# specify degree for some terms as estimated degree is too large
dxp = dx(degree=(self.quadrature_degree))
dS_vp = dS_v(degree=(self.quadrature_degree))
# add effect of density of water upon theta
if self.moisture is not None:
water_t = Function(Vtheta).assign(0.0)
for water in self.moisture:
water_t += self.state.fields(water)
theta_w = theta / (1 + water_t)
thetabar_w = thetabar / (1 + water_t)
else:
theta_w = theta
thetabar_w = thetabar
eqn = (
inner(w, (state.h_project(u) - u_in))*dx
- beta_cp*div(theta_w*V(w))*pibar*dxp
# following does nothing but is preserved in the comments
# to remind us why (because V(w) is purely vertical).
# + beta_cp*jump(theta*V(w), n)*avg(pibar)*dS_v
- beta_cp*div(thetabar_w*w)*pi*dxp
+ beta_cp*jump(thetabar_w*w, n)*avg(pi)*dS_vp
+ (phi*(rho - rho_in) - beta*inner(grad(phi), u)*rhobar)*dx
+ beta*jump(phi*u, n)*avg(rhobar)*(dS_v + dS_h)
)
if mu is not None:
eqn += dt*mu*inner(w, k)*inner(u, k)*dx
aeqn = lhs(eqn)
Leqn = rhs(eqn)
# Place to put result of u rho solver
self.urho = Function(M)
# Boundary conditions (assumes extruded mesh)
bcs = [DirichletBC(M.sub(0), 0.0, "bottom"),
DirichletBC(M.sub(0), 0.0, "top")]
# Solver for u, rho
urho_problem = LinearVariationalProblem(
aeqn, Leqn, self.urho, bcs=bcs)
self.urho_solver = LinearVariationalSolver(urho_problem,
solver_parameters=self.solver_parameters,
options_prefix='ImplicitSolver')
# Reconstruction of theta
theta = TrialFunction(Vtheta)
gamma = TestFunction(Vtheta)
u, rho = self.urho.split()
self.theta = Function(Vtheta)
theta_eqn = gamma*(theta - theta_in
+ dot(k, u)*dot(k, grad(thetabar))*beta)*dx
theta_problem = LinearVariationalProblem(lhs(theta_eqn),
rhs(theta_eqn),
self.theta)
self.theta_solver = LinearVariationalSolver(theta_problem,
options_prefix='thetabacksubstitution')
def _setup_solver(self):
state = self.state # just cutting down line length a bit
dt = state.dt
beta_ = dt * self.alpha
Vu = state.spaces("HDiv")
Vb = state.spaces("theta")
Vp = state.spaces("DG")
# Store time-stepping coefficients as UFL Constants
beta = Constant(beta_)
# Split up the rhs vector (symbolically)
self.xrhs = Function(self.equations.function_space)
u_in, p_in, b_in = split(self.xrhs)
# Build the reduced function space for u,p
M = MixedFunctionSpace((Vu, Vp))
w, phi = TestFunctions(M)
u, p = TrialFunctions(M)
# Get background fields
bbar = state.fields("bbar")
# Analytical (approximate) elimination of theta
k = state.k # Upward pointing unit vector
b = -dot(k, u) * dot(k, grad(bbar)) * beta + b_in
# vertical projection
def V(u):
return k * inner(u, k)
eqn = (inner(w, (u - u_in)) * dx - beta * div(w) * p * dx -
beta * inner(w, k) * b * dx + phi * div(u) * dx)
if hasattr(self.equations, "mu"):
eqn += dt * self.equations.mu * inner(w, k) * inner(u, k) * dx
aeqn = lhs(eqn)
Leqn = rhs(eqn)
# Place to put result of u p solver
self.up = Function(M)
# Boundary conditions (assumes extruded mesh)
# BCs are declared for the plain velocity space. As we need them in
# a mixed problem, we replicate the BCs but for subspace of M
bcs = [
DirichletBC(M.sub(0), bc.function_arg, bc.sub_domain)
for bc in self.equations.bcs['u']
]
# Solver for u, p
up_problem = LinearVariationalProblem(aeqn, Leqn, self.up, bcs=bcs)
# Provide callback for the nullspace of the trace system
def trace_nullsp(T):
return VectorSpaceBasis(constant=True)
appctx = {"trace_nullspace": trace_nullsp}
self.up_solver = LinearVariationalSolver(
up_problem,
solver_parameters=self.solver_parameters,
appctx=appctx)
# Reconstruction of b
b = TrialFunction(Vb)
gamma = TestFunction(Vb)
u, p = self.up.split()
self.b = Function(Vb)
b_eqn = gamma * (b - b_in + dot(k, u) * dot(k, grad(bbar)) * beta) * dx
b_problem = LinearVariationalProblem(lhs(b_eqn), rhs(b_eqn), self.b)
self.b_solver = LinearVariationalSolver(b_problem)
def compressible_hydrostatic_balance(state,
theta0,
rho0,
exner0=None,
top=False,
exner_boundary=Constant(1.0),
mr_t=None,
solve_for_rho=False,
params=None):
"""
Compute a hydrostatically balanced density given a potential temperature
profile. By default, this uses a vertically-oriented hybridization
procedure for solving the resulting discrete systems.
:arg state: The :class:`State` object.
:arg theta0: :class:`.Function`containing the potential temperature.
:arg rho0: :class:`.Function` to write the initial density into.
:arg top: If True, set a boundary condition at the top. Otherwise, set
it at the bottom.
:arg exner_boundary: a field or expression to use as boundary data for exner
on the top or bottom as specified.
:arg mr_t: the initial total water mixing ratio field.
"""
# Calculate hydrostatic Pi
VDG = state.spaces("DG")
Vu = state.spaces("HDiv")
Vv = FunctionSpace(state.mesh, Vu.ufl_element()._elements[-1])
W = MixedFunctionSpace((Vv, VDG))
v, exner = TrialFunctions(W)
dv, dexner = TestFunctions(W)
n = FacetNormal(state.mesh)
cp = state.parameters.cp
# add effect of density of water upon theta
theta = theta0
if mr_t is not None:
theta = theta0 / (1 + mr_t)
alhs = ((cp * inner(v, dv) - cp * div(dv * theta) * exner) * dx +
dexner * div(theta * v) * dx)
if top:
bmeasure = ds_t
bstring = "bottom"
else:
bmeasure = ds_b
bstring = "top"
arhs = -cp * inner(dv, n) * theta * exner_boundary * bmeasure
# Possibly make g vary with spatial coordinates?
g = state.parameters.g
arhs -= g * inner(dv, state.k) * dx
bcs = [DirichletBC(W.sub(0), zero(), bstring)]
w = Function(W)
exner_problem = LinearVariationalProblem(alhs, arhs, w, bcs=bcs)
if params is None:
params = {
'ksp_type': 'preonly',
'pc_type': 'python',
'mat_type': 'matfree',
'pc_python_type': 'gusto.VerticalHybridizationPC',
# Vertical trace system is only coupled vertically in columns
# block ILU is a direct solver!
'vert_hybridization': {
'ksp_type': 'preonly',
'pc_type': 'bjacobi',
'sub_pc_type': 'ilu'
}
}
exner_solver = LinearVariationalSolver(exner_problem,
solver_parameters=params,
options_prefix="exner_solver")
exner_solver.solve()
v, exner = w.split()
if exner0 is not None:
exner0.assign(exner)
if solve_for_rho:
w1 = Function(W)
v, rho = w1.split()
rho.interpolate(thermodynamics.rho(state.parameters, theta0, exner))
v, rho = split(w1)
dv, dexner = TestFunctions(W)
exner = thermodynamics.exner_pressure(state.parameters, rho, theta0)
F = ((cp * inner(v, dv) - cp * div(dv * theta) * exner) * dx +
dexner * div(theta0 * v) * dx +
cp * inner(dv, n) * theta * exner_boundary * bmeasure)
F += g * inner(dv, state.k) * dx
rhoproblem = NonlinearVariationalProblem(F, w1, bcs=bcs)
rhosolver = NonlinearVariationalSolver(rhoproblem,
solver_parameters=params,
options_prefix="rhosolver")
rhosolver.solve()
v, rho_ = w1.split()
rho0.assign(rho_)
else:
rho0.interpolate(thermodynamics.rho(state.parameters, theta0, exner))
def compressible_hydrostatic_balance(state, theta0, rho0, pi0=None,
top=False, pi_boundary=Constant(1.0),
water_t=None,
solve_for_rho=False,
params=None):
"""
Compute a hydrostatically balanced density given a potential temperature
profile.
:arg state: The :class:`State` object.
:arg theta0: :class:`.Function`containing the potential temperature.
:arg rho0: :class:`.Function` to write the initial density into.
:arg top: If True, set a boundary condition at the top. Otherwise, set
it at the bottom.
:arg pi_boundary: a field or expression to use as boundary data for pi on
the top or bottom as specified.
:arg water_t: the initial total water mixing ratio field.
"""
# Calculate hydrostatic Pi
VDG = state.spaces("DG")
Vv = state.spaces("Vv")
W = MixedFunctionSpace((Vv, VDG))
v, pi = TrialFunctions(W)
dv, dpi = TestFunctions(W)
n = FacetNormal(state.mesh)
cp = state.parameters.cp
# add effect of density of water upon theta
theta = theta0
if water_t is not None:
theta = theta0 / (1 + water_t)
alhs = (
(cp*inner(v, dv) - cp*div(dv*theta)*pi)*dx
+ dpi*div(theta*v)*dx
)
if top:
bmeasure = ds_t
bstring = "bottom"
else:
bmeasure = ds_b
bstring = "top"
arhs = -cp*inner(dv, n)*theta*pi_boundary*bmeasure
g = state.parameters.g
arhs -= g*inner(dv, state.k)*dx
bcs = [DirichletBC(W.sub(0), 0.0, bstring)]
w = Function(W)
PiProblem = LinearVariationalProblem(alhs, arhs, w, bcs=bcs)
if params is None:
params = {'pc_type': 'fieldsplit',
'pc_fieldsplit_type': 'schur',
'ksp_type': 'gmres',
'ksp_monitor_true_residual': True,
'ksp_max_it': 100,
'ksp_gmres_restart': 50,
'pc_fieldsplit_schur_fact_type': 'FULL',
'pc_fieldsplit_schur_precondition': 'selfp',
'fieldsplit_0_ksp_type': 'richardson',
'fieldsplit_0_ksp_max_it': 5,
'fieldsplit_0_pc_type': 'gamg',
'fieldsplit_1_pc_gamg_sym_graph': True,
'fieldsplit_1_mg_levels_ksp_type': 'chebyshev',
'fieldsplit_1_mg_levels_ksp_chebyshev_esteig': True,
'fieldsplit_1_mg_levels_ksp_max_it': 5,
'fieldsplit_1_mg_levels_pc_type': 'bjacobi',
'fieldsplit_1_mg_levels_sub_pc_type': 'ilu'}
PiSolver = LinearVariationalSolver(PiProblem,
solver_parameters=params)
PiSolver.solve()
v, Pi = w.split()
if pi0 is not None:
pi0.assign(Pi)
if solve_for_rho:
w1 = Function(W)
v, rho = w1.split()
rho.interpolate(thermodynamics.rho(state.parameters, theta0, Pi))
v, rho = split(w1)
dv, dpi = TestFunctions(W)
pi = thermodynamics.pi(state.parameters, rho, theta0)
F = (
(cp*inner(v, dv) - cp*div(dv*theta)*pi)*dx
+ dpi*div(theta0*v)*dx
+ cp*inner(dv, n)*theta*pi_boundary*bmeasure
)
F += g*inner(dv, state.k)*dx
rhoproblem = NonlinearVariationalProblem(F, w1, bcs=bcs)
rhosolver = NonlinearVariationalSolver(rhoproblem, solver_parameters=params)
rhosolver.solve()
v, rho_ = w1.split()
rho0.assign(rho_)
else:
rho0.interpolate(thermodynamics.rho(state.parameters, theta0, Pi))
def moist_hydrostatic_balance(state, theta_e, water_t, pi_boundary=Constant(1.0)):
"""
Given a wet equivalent potential temperature, theta_e, and the total moisture
content, water_t, compute a hydrostatically balance virtual potential temperature,
dry density and water vapour profile.
:arg state: The :class:`State` object.
:arg theta_e: The initial wet equivalent potential temperature profile.
:arg water_t: The total water pseudo-mixing ratio profile.
:arg pi_boundary: the value of pi on the lower boundary of the domain.
"""
theta0 = state.fields('theta')
rho0 = state.fields('rho')
water_v0 = state.fields('water_v')
# Calculate hydrostatic Pi
Vt = theta0.function_space()
Vr = rho0.function_space()
Vv = state.fields('u').function_space()
n = FacetNormal(state.mesh)
g = state.parameters.g
cp = state.parameters.cp
R_d = state.parameters.R_d
p_0 = state.parameters.p_0
VDG = state.spaces("DG")
if any(deg > 2 for deg in VDG.ufl_element().degree()):
state.logger.warning("default quadrature degree most likely not sufficient for this degree element")
quadrature_degree = (5, 5)
params = {'ksp_type': 'preonly',
'ksp_monitor_true_residual': True,
'ksp_converged_reason': True,
'snes_converged_reason': True,
'ksp_max_it': 100,
'mat_type': 'aij',
'pc_type': 'lu',
'pc_factor_mat_solver_type': 'mumps'}
theta0.interpolate(theta_e)
water_v0.interpolate(water_t)
Pi = Function(Vr)
epsilon = 0.9 # relaxation constant
# set up mixed space
Z = MixedFunctionSpace((Vt, Vt))
z = Function(Z)
gamma, phi = TestFunctions(Z)
theta_v, w_v = z.split()
# give first guesses for trial functions
theta_v.assign(theta0)
w_v.assign(water_v0)
theta_v, w_v = split(z)
# define variables
T = thermodynamics.T(state.parameters, theta_v, Pi, r_v=w_v)
p = thermodynamics.p(state.parameters, Pi)
w_sat = thermodynamics.r_sat(state.parameters, T, p)
dxp = dx(degree=(quadrature_degree))
# set up weak form of theta_e and w_sat equations
F = (-gamma * theta_e * dxp
+ gamma * thermodynamics.theta_e(state.parameters, T, p, w_v, water_t) * dxp
- phi * w_v * dxp
+ phi * w_sat * dxp)
problem = NonlinearVariationalProblem(F, z)
solver = NonlinearVariationalSolver(problem, solver_parameters=params)
theta_v, w_v = z.split()
Pi_h = Function(Vr).interpolate((p / p_0) ** (R_d / cp))
# solve for Pi with theta_v and w_v constant
# then solve for theta_v and w_v with Pi constant
for i in range(5):
compressible_hydrostatic_balance(state, theta0, rho0, pi0=Pi_h, water_t=water_t)
Pi.assign(Pi * (1 - epsilon) + epsilon * Pi_h)
solver.solve()
theta0.assign(theta0 * (1 - epsilon) + epsilon * theta_v)
water_v0.assign(water_v0 * (1 - epsilon) + epsilon * w_v)
# now begin on Newton solver, setup up new mixed space
Z = MixedFunctionSpace((Vt, Vt, Vr, Vv))
z = Function(Z)
gamma, phi, psi, w = TestFunctions(Z)
theta_v, w_v, pi, v = z.split()
# use previous values as first guesses for newton solver
theta_v.assign(theta0)
w_v.assign(water_v0)
pi.assign(Pi)
theta_v, w_v, pi, v = split(z)
# define variables
T = thermodynamics.T(state.parameters, theta_v, pi, r_v=w_v)
p = thermodynamics.p(state.parameters, pi)
w_sat = thermodynamics.r_sat(state.parameters, T, p)
F = (-gamma * theta_e * dxp
+ gamma * thermodynamics.theta_e(state.parameters, T, p, w_v, water_t) * dxp
- phi * w_v * dxp
+ phi * w_sat * dxp
+ cp * inner(v, w) * dxp
- cp * div(w * theta_v / (1.0 + water_t)) * pi * dxp
+ psi * div(theta_v * v / (1.0 + water_t)) * dxp
+ cp * inner(w, n) * pi_boundary * theta_v / (1.0 + water_t) * ds_b
+ g * inner(w, state.k) * dxp)
bcs = [DirichletBC(Z.sub(3), 0.0, "top")]
problem = NonlinearVariationalProblem(F, z, bcs=bcs)
solver = NonlinearVariationalSolver(problem, solver_parameters=params)
solver.solve()
theta_v, w_v, pi, v = z.split()
# assign final values
theta0.assign(theta_v)
water_v0.assign(w_v)
# find rho
compressible_hydrostatic_balance(state, theta0, rho0, water_t=water_t, solve_for_rho=True)
def _setup_solver(self):
state = self.state # just cutting down line length a bit
dt = state.timestepping.dt
beta = dt*state.timestepping.alpha
mu = state.mu
Vu = state.spaces("HDiv")
Vb = state.spaces("HDiv_v")
Vp = state.spaces("DG")
# Split up the rhs vector (symbolically)
u_in, p_in, b_in = split(state.xrhs)
# Build the reduced function space for u,p
M = MixedFunctionSpace((Vu, Vp))
w, phi = TestFunctions(M)
u, p = TrialFunctions(M)
# Get background fields
bbar = state.fields("bbar")
# Analytical (approximate) elimination of theta
k = state.k # Upward pointing unit vector
b = -dot(k, u)*dot(k, grad(bbar))*beta + b_in
# vertical projection
def V(u):
return k*inner(u, k)
eqn = (
inner(w, (u - u_in))*dx
- beta*div(w)*p*dx
- beta*inner(w, k)*b*dx
+ phi*div(u)*dx
)
if mu is not None:
eqn += dt*mu*inner(w, k)*inner(u, k)*dx
aeqn = lhs(eqn)
Leqn = rhs(eqn)
# Place to put result of u p solver
self.up = Function(M)
# Boundary conditions (assumes extruded mesh)
bcs = [DirichletBC(M.sub(0), 0.0, "bottom"),
DirichletBC(M.sub(0), 0.0, "top")]
# Solver for u, p
up_problem = LinearVariationalProblem(aeqn, Leqn, self.up, bcs=bcs)
# Provide callback for the nullspace of the trace system
def trace_nullsp(T):
return VectorSpaceBasis(constant=True)
appctx = {"trace_nullspace": trace_nullsp}
self.up_solver = LinearVariationalSolver(up_problem,
solver_parameters=self.solver_parameters,
appctx=appctx)
# Reconstruction of b
b = TrialFunction(Vb)
gamma = TestFunction(Vb)
u, p = self.up.split()
self.b = Function(Vb)
b_eqn = gamma*(b - b_in +
dot(k, u)*dot(k, grad(bbar))*beta)*dx
b_problem = LinearVariationalProblem(lhs(b_eqn),
rhs(b_eqn),
self.b)
self.b_solver = LinearVariationalSolver(b_problem)
