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L96.py
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L96.py
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"""
Definition of the Lorenz96 model.
Created on 2019-04-16-12-28
Author: Stephan Rasp, raspstephan@gmail.com
"""
import sys
def in_notebook():
"""
Returns ``True`` if the module is running in IPython kernel,
``False`` if in IPython shell or other Python shell.
"""
return 'ipykernel' in sys.modules
import numpy as np
import xarray as xr
if in_notebook():
from tqdm import tqdm_notebook as tqdm
else:
from tqdm import tqdm
class L96OneLevel(object):
def __init__(self, K=36, J=10, h=1, F=10, c=10, b=10, dt=0.001,
X_init=None, noprog=False):
self.K, self.J, self.h, self.F, self.c, self.b, self.dt = K, J, h, F, c, b, dt
self.params = [self.F]
self.noprog = noprog
self.X = np.random.rand(self.K) if X_init is None else X_init.copy()
self.Y = np.zeros((self.K, self.J))
self._history_X = [self.X.copy()]
def _rhs(self, X):
"""Compute the right hand side of the ODE."""
dXdt = (
-np.roll(X, -1) * (np.roll(X, -2) - np.roll(X, 1)) -
X + self.F
)
return dXdt
def step(self):
"""Step forward one time step with RK4."""
k1 = self.dt * self._rhs(self.X)
k2 = self.dt * self._rhs(self.X + k1 / 2)
k3 = self.dt * self._rhs(self.X + k2 / 2)
k4 = self.dt * self._rhs(self.X + k3)
self.X += 1 / 6 * (k1 + 2 * k2 + 2 * k3 + k4)
self._history_X.append(self.X.copy())
def iterate(self, time):
steps = int(time / self.dt)
for n in tqdm(range(steps), disable=self.noprog):
self.step()
@property
def state(self):
return self.X
def set_state(self, x):
self.X = x
@property
def parameters(self):
return np.atleast_1d(self.F)
def erase_history(self):
self._history_X = []
@property
def history(self):
da = xr.DataArray(self._history_X, dims=['time', 'x'], name='X')
return xr.Dataset(
{'X': da},
coords={'time': np.arange(len(self._history_X)) * self.dt, 'x': np.arange(self.K)}
)
def mean_stats(self, ax=None, fn=np.mean):
h = self.history
return np.concatenate([
np.atleast_1d(fn(h.X, ax)),
np.atleast_1d(fn((h.X**2), ax)),
])
class L96TwoLevel(object):
def __init__(self, K=36, J=10, h=1, F=10, c=10, b=10, dt=0.001,
X_init=None, Y_init=None, noprog=False, noYhist=False, save_dt=0.1,
integration_type='uncoupled', parameterization=None):
# Model parameters
self.K, self.J, self.h, self.F, self.c, self.b, self.dt = K, J, h, F, c, b, dt
self.noprog, self.noYhist, self.integration_type = noprog, noYhist, integration_type
self.step_count = 0
self.save_dt = save_dt
self.parameterization = parameterization
if self.parameterization is not None: self.integration_type = 'parameterization'
self.save_steps = int(save_dt / dt)
self.X = np.random.rand(self.K) if X_init is None else X_init.copy()
self.Y = np.zeros(self.K * self.J) if Y_init is None else Y_init.copy()
self._history_X = [self.X.copy()]
self._history_Y_mean = [self.Y.reshape(self.K, self.J).mean(1).copy()]
self._history_Y2_mean = [(self.Y.reshape(self.K, self.J)**2).mean(1).copy()]
self._history_B = [-self.h * self.c * self.Y.reshape(self.K, self.J).mean(1)]
if not self.noYhist:
self._history_Y = [self.Y.copy()]
def _rhs_X_dt(self, X, Y=None, B=None):
"""Compute the right hand side of the X-ODE."""
if Y is None:
dXdt = (
-np.roll(X, -1) * (np.roll(X, -2) - np.roll(X, 1)) -
X + self.F + B
)
else:
dXdt = (
-np.roll(X, -1) * (np.roll(X, -2) - np.roll(X, 1)) -
X + self.F - self.h * self.c * Y.reshape(self.K, self.J).mean(1)
)
return self.dt * dXdt
def _rhs_Y_dt(self, X, Y):
"""Compute the right hand side of the Y-ODE."""
dYdt = (
-self.b * np.roll(Y, -1) * (np.roll(Y, -2) - np.roll(Y, 1)) -
Y + self.h / self.J * np.repeat(X, self.J)
) * self.c
return self.dt * dYdt
def _rhs_dt(self, X, Y):
return self._rhs_X_dt(X, Y=Y), self._rhs_Y_dt(X, Y)
def step(self, add_B=True, B=None):
"""Integrate one time step"""
if self.parameterization is None:
B = -self.h * self.c * self.Y.reshape(self.K, self.J).mean(1) if B is None else B
if self.integration_type == 'coupled':
k1_X, k1_Y = self._rhs_dt(self.X, self.Y)
k2_X, k2_Y = self._rhs_dt(self.X + k1_X / 2, self.Y + k1_Y / 2)
k3_X, k3_Y = self._rhs_dt(self.X + k2_X / 2, self.Y + k2_Y / 2)
k4_X, k4_Y = self._rhs_dt(self.X + k3_X, self.Y + k3_Y)
elif self.integration_type == 'uncoupled':
k1_X = self._rhs_X_dt(self.X, B=B)
k2_X = self._rhs_X_dt(self.X + k1_X / 2, B=B)
k3_X = self._rhs_X_dt(self.X + k2_X / 2, B=B)
k4_X = self._rhs_X_dt(self.X + k3_X, B=B)
# Then update Y with unupdated X
k1_Y = self._rhs_Y_dt(self.X, self.Y)
k2_Y = self._rhs_Y_dt(self.X, self.Y + k1_Y / 2)
k3_Y = self._rhs_Y_dt(self.X, self.Y + k2_Y / 2)
k4_Y = self._rhs_Y_dt(self.X, self.Y + k3_Y)
self.X += 1 / 6 * (k1_X + 2 * k2_X + 2 * k3_X + k4_X)
self.Y += 1 / 6 * (k1_Y + 2 * k2_Y + 2 * k3_Y + k4_Y)
else: # Parameterization case
k1_X = self._rhs_X_dt(self.X, B=0)
k2_X = self._rhs_X_dt(self.X + k1_X / 2, B=0)
k3_X = self._rhs_X_dt(self.X + k2_X / 2, B=0)
k4_X = self._rhs_X_dt(self.X + k3_X, B=0)
B = self.parameterization(self.X) if B is None else B
self.X += 1 / 6 * (k1_X + 2 * k2_X + 2 * k3_X + k4_X)
if add_B: self.X += B * self.dt
self.step_count += 1
if self.step_count % self.save_steps == 0:
Y_mean = self.Y.reshape(self.K, self.J).mean(1)
Y2_mean = (self.Y.reshape(self.K, self.J)**2).mean(1)
self._history_X.append(self.X.copy())
self._history_Y_mean.append(Y_mean.copy())
self._history_Y2_mean.append(Y2_mean.copy())
self._history_B.append(B.copy())
if not self.noYhist:
self._history_Y.append(self.Y.copy())
def iterate(self, time):
steps = int(time / self.dt)
for n in tqdm(range(steps), disable=self.noprog):
self.step()
@property
def state(self):
return np.concatenate([self.X, self.Y])
def set_state(self, x):
self.X = x[:self.K]
self.Y = x[self.K:]
@property
def parameters(self):
return np.array([self.F, self.h, self.c, self.b])
def erase_history(self):
self._history_X = []
self._history_Y_mean = []
self._history_Y2_mean = []
self._history_B = []
if not self.noYhist:
self._history_Y = []
@property
def history(self):
dic = {}
dic['X'] = xr.DataArray(self._history_X, dims=['time', 'x'], name='X')
dic['B'] = xr.DataArray(self._history_B, dims=['time', 'x'], name='B')
dic['Y_mean'] = xr.DataArray(self._history_Y_mean, dims=['time', 'x'], name='Y_mean')
dic['Y2_mean'] = xr.DataArray(self._history_Y2_mean, dims=['time', 'x'], name='Y2_mean')
if not self.noYhist:
dic['X_repeat'] = xr.DataArray(np.repeat(self._history_X, self.J, 1),
dims=['time', 'y'], name='X_repeat')
dic['Y'] = xr.DataArray(self._history_Y, dims=['time', 'y'], name='Y')
return xr.Dataset(
dic,
coords={'time': np.arange(len(self._history_X)) * self.save_dt, 'x': np.arange(self.K),
'y': np.arange(self.K * self.J)}
)
def mean_stats(self, ax=None, fn=np.mean):
h = self.history
return np.concatenate([
np.atleast_1d(fn(h.X, ax)),
np.atleast_1d(fn(h.Y_mean, ax)),
np.atleast_1d(fn((h.X ** 2), ax)),
np.atleast_1d(fn((h.X * h.Y_mean), ax)),
np.atleast_1d(fn(h.Y2_mean, ax))
])