def lane_emden(Nr, m=1.5, n_rho=3, radius=1,
ncc_cutoff = 1e-10, tolerance = 1e-10, dtype=np.complex128, comm=None):
# TO-DO: clean this up and make work for ncc ingestion in main script in np.float64 rather than np.complex128
c = de.SphericalCoordinates('phi', 'theta', 'r')
d = de.Distributor((c,), comm=comm, dtype=dtype)
b = de.BallBasis(c, (1, 1, Nr), radius=radius, dtype=dtype)
br = b.radial_basis
phi, theta, r = b.local_grids()
# Fields
f = d.Field(name='f', bases=b)
R = d.Field(name='R')
τ = d.Field(name='τ', bases=b.S2_basis(radius=radius))
# Parameters and operators
lap = lambda A: de.Laplacian(A, c)
lift_basis = b.clone_with(k=2) # match laplacian
lift = lambda A: de.LiftTau(A, lift_basis, -1)
problem = de.NLBVP([f, R, τ])
problem.add_equation((lap(f) + lift(τ), - R**2 * f**m))
problem.add_equation((f(r=0), 1))
problem.add_equation((f(r=radius), np.exp(-n_rho/m, dtype=dtype))) # explicit typing to match domain
# Solver
solver = problem.build_solver(ncc_cutoff=ncc_cutoff)
# Initial guess
f['g'] = np.cos(np.pi/2 * r)**2
R['g'] = 5
# Iterations
logger.debug('beginning Lane-Emden NLBVP iterations')
pert_norm = np.inf
while pert_norm > tolerance:
solver.newton_iteration()
pert_norm = sum(pert.allreduce_data_norm('c', 2) for pert in solver.perturbations)
logger.debug(f'Perturbation norm: {pert_norm:.3e}')
T = d.Field(name='T', bases=br)
ρ = d.Field(name='ρ', bases=br)
lnρ = d.Field(name='lnρ', bases=br)
T['g'] = f['g']
ρ['g'] = f['g']**m
lnρ['g'] = np.log(ρ['g'])
structure = {'T':T,'lnρ':lnρ}
for key in structure:
structure[key].require_scales(1)
structure['r'] = r
structure['problem'] = {'c':c, 'b':b, 'problem':problem}
return structure
# Allow restarting via command line
restart = (len(sys.argv) > 1 and sys.argv[1] == '--restart')
# Parameters
Nphi, Ntheta, Nr = 128, 64, 96
Rayleigh = 1e6
Prandtl = 1
dealias = 3 / 2
stop_sim_time = 20 + 20 * restart
timestepper = d3.SBDF2
max_timestep = 0.05
dtype = np.float64
mesh = None
# Bases
coords = d3.SphericalCoordinates('phi', 'theta', 'r')
dist = d3.Distributor(coords, dtype=dtype, mesh=mesh)
basis = d3.BallBasis(coords,
shape=(Nphi, Ntheta, Nr),
radius=1,
dealias=dealias,
dtype=dtype)
S2_basis = basis.S2_basis()
# Fields
u = dist.VectorField(coords, name='u', bases=basis)
p = dist.Field(name='p', bases=basis)
T = dist.Field(name='T', bases=basis)
tau_p = dist.Field(name='tau_p')
tau_u = dist.VectorField(coords, name='tau u', bases=S2_basis)
tau_T = dist.Field(name='tau T', bases=S2_basis)
n_rho = float(args['--n_rho'])
radius = 1
Ek = Ekman = float(args['--Ekman'])
Co2 = ConvectiveRossbySq = float(args['--ConvectiveRossbySq'])
Pr = Prandtl = float(args['--Prandtl'])
logger.info("Ek = {}, Co2 = {}, Pr = {}".format(Ek, Co2, Pr))
import dedalus.public as de
from dedalus.extras import flow_tools
from structure import lane_emden
dealias = float(args['--dealias'])
c = de.SphericalCoordinates('phi', 'theta', 'r')
d = de.Distributor(c, mesh=mesh, dtype=np.float64)
b = de.BallBasis(c,
shape=(Nφ, Nθ, Nr),
radius=radius,
dealias=dealias,
dtype=np.float64)
b_S2 = b.S2_basis()
phi, theta, r = b.local_grids()
p = d.Field(name='p', bases=b)
s = d.Field(name='s', bases=b)
u = d.VectorField(c, name='u', bases=b)
τ_p = d.Field(name='τ_p')
τ_s = d.Field(name='τ_s', bases=b_S2)
τ_u = d.VectorField(c, name='τ_u', bases=b_S2)