-
Notifications
You must be signed in to change notification settings - Fork 1
/
j2011_hydro_FC_nrho3_onset.py
275 lines (229 loc) · 9.7 KB
/
j2011_hydro_FC_nrho3_onset.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
"""
Dedalus script for shell anelastic convection,
based on Jones et al 2011 convective benchmark case 0.
Usage:
j2011_hydro_FC.py [options]
Options:
--Ntheta=<Ntheta> Latitude coeffs [default: 64]
--Nr=<Nr> Radial coeffs [default: 64]
--Mach=<Ma> Mach number [default: 1e-2]
--gamma=<gamma> Ideal gas gamma [default: 5/3]
--Rayeigh_init=<Ra> Initial guess for Rayleigh [default: 3e4]
--Legendre Use Legendre polynomials in radius
--jones Use the Jones polytrope
--tol=<tol> Tolerance for opitimization loop [default: 1e-5]
--eigs=<eigs> Target number of eigenvalues to search for [default: 20]
--ncc_cutoff=<ncc> Amplitude cutoff for NCCs [default: 1e-8]
--label=<label> Additional label for run output directory
"""
import sys
import numpy as np
import dedalus.public as de
import logging
logger = logging.getLogger(__name__)
from mpi4py import MPI
ncpu = MPI.COMM_WORLD.size
from docopt import docopt
args = docopt(__doc__)
from fractions import Fraction
ncc_cutoff = float(args['--ncc_cutoff'])
# parameters
beta = 0.35
Ekman = 2e-3
Prandtl = 1
#Rayleigh = 61621.682
n = 2
Nrho = nρ = 3
m_crit = 10
Ma = float(args['--Mach'])
Ma2 = Ma*Ma
γ = gamma = float(Fraction(args['--gamma']))
N_eigs = int(float(args['--eigs']))
tol = float(args['--tol'])
Nr = int(args['--Nr'])
Ntheta = int(args['--Ntheta'])
Nphi = 2*Ntheta
dealias = 3/2
dtype = np.complex128
data_dir = sys.argv[0].split('.py')[0]
data_dir += '_Ma{}'.format(args['--Mach'])
data_dir += '_Th{}_R{}'.format(args['--Ntheta'], args['--Nr'])
if args['--label']:
data_dir += '_{:s}'.format(args['--label'])
logger.info("saving data in {}".format(data_dir))
import dedalus.tools.logging as dedalus_logging
dedalus_logging.add_file_handler(data_dir+'/logs/dedalus_log', 'DEBUG')
Ro = r_outer = 1/(1-beta)
Ri = r_inner = Ro - 1
logger.info('Ri = {:}, Ro = {:}'.format(Ri, Ro))
if args['--jones']:
zeta_out = (beta + 1) / ( beta*np.exp(Nrho/n) + 1 )
zeta_in = (1 + beta - zeta_out) / beta
c0 = (2*zeta_out - beta - 1) / (1 - beta)
c1 = (1 + beta)*(1 - zeta_out) / (1 - beta)**2
else:
nh = nρ/n
c0 = -(Ri-Ro*np.exp(-nh))/(Ro-Ri)
c1 = Ri*Ro/(Ro-Ri)*(1-np.exp(-nh))
# Bases
coords = de.SphericalCoordinates('phi', 'theta', 'r')
dist = de.Distributor(coords, dtype=dtype)
if args['--Legendre']:
basis = de.ShellBasis(coords, alpha=(0,0), shape=(Nphi, Ntheta, Nr), radii=(Ri, Ro), dtype=dtype)
else:
basis = de.ShellBasis(coords, shape=(Nphi, Ntheta, Nr), radii=(Ri, Ro), dtype=dtype)
basis_ncc = basis.meridional_basis
b_inner = basis.S2_basis(radius=r_inner)
b_outer = basis.S2_basis(radius=r_outer)
s2_basis = basis.S2_basis()
# Fields
u = dist.VectorField(coords, name='u', bases=basis)
θ = dist.Field(name='θ', bases=basis)
Υ = dist.Field(name='Υ', bases=basis)
S = dist.Field(name='S', bases=basis)
τ_p = dist.Field(name='τ_p')
τ_S1 = dist.Field(name='τ_T1', bases=b_outer)
τ_S2 = dist.Field(name='τ_T2', bases=b_inner)
τ_u1 = dist.VectorField(coords, name='τ_u1', bases=b_outer)
τ_u2 = dist.VectorField(coords, name='τ_u2', bases=b_inner)
grad = lambda A: de.Gradient(A, coords)
div = lambda A: de.Divergence(A)
cross = lambda A, B: de.CrossProduct(A, B)
trans = lambda A: de.TransposeComponents(A)
trace = lambda A: de.Trace(A)
radial = lambda A: de.RadialComponent(A)
angular = lambda A: de.AngularComponent(A, index=1)
# Substitutions
phi, theta, r = dist.local_grids(basis)
ez = dist.VectorField(coords, bases=basis_ncc, name='ez')
ez['g'][2] = np.cos(theta)
ez['g'][1] = -np.sin(theta)
f = 2*ez/Ekman
zeta = dist.Field(bases=basis_ncc, name='zeta')
zeta['g'] = c0 + c1/r
rho0 = (zeta**n).evaluate()
rho0.name='ρ0'
p0 = (zeta**(n+1)).evaluate()
p0.name='p0'
grad_log_rho0 = de.grad(np.log(rho0)).evaluate()
grad_log_rho0.name='grad_ln_ρ0'
h0 = (zeta).evaluate()
h0.name='h0'
grad_h0 = de.grad(h0)
θ0 = (np.log(h0)).evaluate()
θ0.name='θ0'
grad_θ0 = de.grad(θ0).evaluate()
grad_θ0.name='grad_θ0'
er = dist.VectorField(coords, bases=basis_ncc, name='er')
er['g'][2] = 1
lift_basis = basis.derivative_basis(1)
lift = lambda A, n: de.Lift(A, lift_basis, n)
e = grad(u) + trans(grad(u))
viscous_terms = div(e) + e@grad_log_rho0 - 2/3*grad(div(u)) - 2/3*grad_log_rho0*div(u)
trace_e = trace(e)
u_r_inner = radial(u(r=r_inner))
u_r_outer = radial(u(r=r_outer))
u_perp_inner = radial(angular(e(r=r_inner)))
u_perp_outer = radial(angular(e(r=r_outer)))
S0 = dist.Field(bases=basis_ncc, name='S0')
S0['g'] = (zeta_out**(-2) - (c0 + c1/r)**(-2)) / (zeta_out**(-2) - zeta_in**(-2))
S0['g'] *= Ma2
grad_S0 = grad(S0).evaluate()
logger.info("NCC expansions:")
for ncc in [grad_log_rho0, h0, grad_h0, grad_S0, grad_θ0]:
logger.info("{}: {}".format(ncc.evaluate(), np.where(np.abs(ncc.evaluate()['c']) >= ncc_cutoff)[0].shape))
g = dist.VectorField(coords, bases=basis_ncc, name='g')
g['g'][2] = 1/r**2
Rayleigh = dist.Field(name='Rayleigh')
Ra_c = 6.2e4/(h0(r=Ro).evaluate()['g'][0,0,0]/g(r=Ro).evaluate()['g'][2][0,0,0]).real
print('Ra = {:}'.format(Ra_c))
scrC = 1/(gamma-1)/Ma2
#scrC = 1/(gamma-1)*Co2/Ma2
#Co2 = Rayleigh*Ekman**2/Prandtl
logger.info("scrC = {:}".format(scrC))
omega = dist.Field(name='omega')
dt = lambda A: omega*A
# Problem Rayleigh/Prandtl*scrC*(h0*grad(θ) + grad_h0*θ - h0*grad(S)
print('r')
print(Ri)
print(Ro)
print('h0')
print(h0(r=Ri).evaluate()['g'][0,0,0])
print(h0(r=Ro).evaluate()['g'][0,0,0])
print('g')
print(g(r=Ri).evaluate()['g'][2][0,0,0])
print(g(r=(Ro-0.5)).evaluate()['g'][2][0,0,0])
print(g(r=Ro).evaluate()['g'][2][0,0,0])
print('grad_s0')
print(grad_S0(r=Ri).evaluate()['g'][2][0,0,0])
print(grad_S0(r=Ro).evaluate()['g'][2][0,0,0])
print('delta s0')
print(S0(r=Ro).evaluate()['g'][0,0,0]-S0(r=Ri).evaluate()['g'][0,0,0])
problem = de.EVP([u, Υ, θ, S, τ_u1, τ_u2, τ_S1, τ_S2], eigenvalue=omega, namespace=locals())
problem.add_equation("dt(Υ) + div(u) + u@grad_log_rho0 + lift(τ_u2,-1)@er = 0 ")
problem.add_equation("dt(u) + scrC*Rayleigh*h0*grad(θ) + scrC*Rayleigh*grad_h0*θ - scrC*Rayleigh*h0*grad(S) - cross(u, f) - viscous_terms + lift(τ_u1, -1) + lift(τ_u2, -2) = 0 ")
problem.add_equation("dt(S) + u@grad_S0 - (lap(θ) + 2*grad_θ0@grad(θ))/Prandtl + lift(τ_S1, -1) + lift(τ_S2, -2) = 0 ")
problem.add_equation("θ - (γ-1)*Υ - γ*S = 0")
problem.add_equation("S(r=Ri) = 0")
problem.add_equation("u_r_inner = 0")
problem.add_equation("u_perp_inner = 0")
problem.add_equation("S(r=Ro) = 0")
problem.add_equation("u_r_outer = 0")
problem.add_equation("u_perp_outer = 0")
# Solver
solver = problem.build_solver(ncc_cutoff=ncc_cutoff)
Jones_et_al_2011_Rayleigh = 61621.682
Jones_et_al_2011_ω = -1j*101.381 + 0 # adjusts Jones et al 2009 convention of dt = -1j*omega to ours
Jones_et_al_2011_ω /= np.sqrt(Jones_et_al_2011_Rayleigh)
target = 0.0 + -1.0*1j
def compute_eigenvalues(log_Ra_i, target_m=10):
Rayleigh['g'] = np.exp(log_Ra_i)/Prandtl
current_Ra = Rayleigh['g'][0,0,0].real
logger.info('m = {:d}, Ra = {:.6g}'.format(target_m, current_Ra))
subproblem = solver.subproblems_by_group[(target_m, None, None)]
solver.solve_sparse(subproblem, N=N_eigs, target=target, rebuild_matrices=True)
i_evals = np.argsort(solver.eigenvalues.real)
evals = solver.eigenvalues[i_evals]
evals /= np.sqrt(current_Ra)
return(evals)
def peak_growth_rate(*args):
evals = compute_eigenvalues(*args)
peak_eval = evals[-1]
logger.info('current peak eigenvalue: {:.3g}, {:.3g}i'.format(peak_eval.real, peak_eval.imag))
# flip sign so minimize finds maximum
return np.abs(peak_eval.real)
import scipy.optimize as sciop
Rayleigh_init = float(args['--Rayeigh_init'])
log_Ra_i = np.log(Rayleigh_init) # search in log Ra
#bounds = sciop.Bounds(lb=np.log(1e4), ub=np.log(1e6)) # bounds=bounds,
result = sciop.minimize(peak_growth_rate, log_Ra_i, tol=tol, method='Nelder-Mead')
logger.info(result)
logger.info('optimization complete, solving for critical modes')
evals = compute_eigenvalues(result.x[0])
critical_Ra = np.exp(result.x[0]).real
logger.info("Jones et al 2011 critical Ra: {:.6g}".format(Jones_et_al_2011_Rayleigh))
logger.info(" critical Ra: {:.6g}".format(critical_Ra))
logger.info(" fractional difference: {:.2g}".format(np.abs(critical_Ra-Jones_et_al_2011_Rayleigh)/np.abs(critical_Ra)))
logger.info("")
logger.info("Jones et al 2011 eigenvalue: {:.6g}, {:.6g}i".format(Jones_et_al_2011_ω.real, Jones_et_al_2011_ω.imag))
logger.info(" critical eigenvalue: {:.6g}, {:.6g}i".format(evals[-1].real, evals[-1].imag))
logger.info(" fractional difference: {:.2g}".format(np.abs(evals[-1]-Jones_et_al_2011_ω)/np.abs(evals[-1])))
logger.info("")
logger.info("Ra_crit: {:.6g}, Ra/g(Ri): {:.6g}, Ra/g(Ro): {:.6g}".format(critical_Ra, critical_Ra/g(r=Ri).evaluate()['g'][2][0,0,0].real, critical_Ra/g(r=Ro).evaluate()['g'][2][0,0,0].real))
logger.info("eigenvalues:\n{:}".format(evals))
target_m=10
import matplotlib.pyplot as plt
fig, ax = plt.subplots()
ax.scatter(evals.real, evals.imag, alpha=0.5, zorder=5)
ax.axhline(y=0, linestyle='dashed', color='xkcd:dark grey', alpha=0.5, zorder=0)
ax.axvline(x=0, linestyle='dashed', color='xkcd:dark grey', alpha=0.5, zorder=0)
ax.set_title('m = {:d}, '.format(target_m)+r'$N_\theta = '+'{:d}'.format(Ntheta)+r'$, $N_r = '+'{:d}'.format(Nr)+r'$')
ax.set_xlabel(r'$\omega_R$')
ax.set_ylabel(r'$\omega_I$')
ax.scatter(Jones_et_al_2011_ω.real, Jones_et_al_2011_ω.imag, marker='s', label='Jones et. al (2011)', alpha=0.2, color='xkcd:dark red', zorder=2, s=100)
ax.scatter(target.real, target.imag, marker='x', label='target', color='xkcd:dark green', alpha=0.2, zorder=1)
ax.legend()
fig_filename = 'eigenspectrum'
if args['--Legendre']:
fig_filename += '_Legendre'
fig.savefig(data_dir+'/'+fig_filename+'.png', dpi=300)