Numerical solution of combined models of weathering-mediated erosion
& bedrock channel.
Class that provides numerical solution of a combination model 1d
weathering-mediated erosion and 1+1d bedrock channel cross-section
(at the channel wall).
Methods:
Source code in wmbe/channel1d.py
| def __init__(self, eqns: Equations, physical_parameters: dict) -> None:
"""
Initialize class instance.
"""
# Attributes:
# physical_parameters (:obj:`dict`) :
# model parameters dictionary
# w0_eqn_z_calibrated (:class:`sympy.Eq <sympy.core.relational.Equality>`) :
# surface weakness calibrated using model parameters:
# :math:`w_0 = w_r H_s[z,z_\\mathrm{wc},k_\\mathrm{w}]`
# where
# :math:`H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_w(z-z_\\mathrm{wc}]}\\right)`
# v0_eqn_z_calibrated (:class:`sympy.Eq <sympy.core.relational.Equality>`) :
# erosion rate calibrated using model parameters:
# :math:`v_0 = v_r \\left\{ (h-H_s[z,z_\\mathrm{vc},k_\\mathrm{v}])(1-v_b)+v_b \\right\}`
# where
# :math:`H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_v(z-z_\\mathrm{vc}]}\\right)`
# vs_eqn (:class:`sympy.Eq <sympy.core.relational.Equality>`) :
# surface-normal erosion rate (uncalibrated)
# :math:`v_0 = v_r \\left\{ (h-H_s[z,z_\\mathrm{vc},k_\\mathrm{v}])(1-v_b)+v_b \\right\}`
# where
# :math:`H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_v(z-z_\\mathrm{vc}]}\\right)`
# W_eqn_z_calibrated (:class:`sympy.Eq <sympy.core.relational.Equality>`) :
# weathering number calibrated using model parameters:
# :math:`W = \\dfrac{w_0}{k v_0}`
# vs_eqn_z_calibrated (:class:`sympy.Eq <sympy.core.relational.Equality>`) :
# surface-normal erosion rate calibrated using model parameters:
# :math:`v_0 = v_r \\left\{ (h-H_s[z,z_\\mathrm{vc},k_\\mathrm{v}])(1-v_b)+v_b \\right\}`
# where
# :math:`H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_v(z-z_\\mathrm{vc}]}\\right)`
self.physical_parameters = physical_parameters
self.w0_eqn_z_calibrated \
= eqns.w0_eqn_wr_z.subs(self.physical_parameters)
self.v0_eqn_z_calibrated \
= eqns.v0_eqn_vr_h_z.subs(self.physical_parameters)
self.W_eqn_z_calibrated \
= eqns.W_eqn.subs(self.physical_parameters)
self.vs_eqn \
= eqns.vs_eqn_w0_v0.subs({v_0:eqns.v0_eqn_vr_h_z.rhs})
self.vs_eqn_z_calibrated \
= (
self.vs_eqn
.subs({w_0:self.w0_eqn_z_calibrated.rhs})
.subs(self.physical_parameters)
)
|
compute_cross_section() -> None
Compute dependence of various properties with height above channel base.
Source code in wmbe/channel1d.py
| def compute_cross_section(self) -> None:
"""
Compute dependence of various properties with height above channel base.
"""
# Attributes:
# vs_array (:class:`numpy.ndarray`) :
# weathering-mediated surface-normal erosion rate along vertical profile
# (at steady state)
# dzdy_array (:class:`numpy.ndarray`) :
# weathering-mediated vertical erosion rate along vertical profile
# (at steady state)
# y_array (:class:`numpy.ndarray`) :
# horizontal positions of sample points along vertica profile
# ch_y_array (:class:`numpy.ndarray`) :
# horizontal positions of sample points along channel boundary
# ch_z_array (:class:`numpy.ndarray`) :
# vertical positions of sample points along channel boundary
self.vs_array = np.array([
np.float64(self.vs_eqn_z_calibrated.rhs.subs(z,z__))
for z__ in self.z_array
])
self.dzdy_array \
= 1/np.sqrt(((np.max(self.vs_array)+0.01)/self.vs_array)**2-1)
self.y_array \
= integrate.cumulative_trapezoid(self.dzdy_array,x=self.z_array, initial=0)
self.ch_y_array = np.concatenate(
(np.linspace(-np.max(self.z_array)*0.3,0,self.z_array.shape[0]),self.y_array))
self.ch_z_array \
= np.concatenate(( np.zeros(self.y_array.shape[0]),self.z_array ))
|
compute_vertical_profiles(n_pts: int = 100) -> None
Compute dependence of various properties with height above channel base.
Source code in wmbe/channel1d.py
| def compute_vertical_profiles(self, n_pts: int=100,) -> None:
"""
Compute dependence of various properties with height above channel base.
"""
# Args:
# n_pts (int):
# number of sampling points along vertical profile
# Attributes:
# n_pts (:obj:`int`):
# record of number of sampling points
# z_array (:class:`numpy.ndarray`) :
# sample points along vertical profile
# vs_eqn_z_calibrated (:class:`sympy.Eq <sympy.core.relational.Equality>`) :
# weathering-mediated surface-normal erosion rate at steady state
# calibrated by model parameters supplied in :attr:`wme.pdict`
# w0_array (:class:`numpy.ndarray`) :
# surface weakness along vertical profile (at steady state)
# v0_array (:class:`numpy.ndarray`) :
# baseline (absent weathering-driven weakening) surface-normal erosion rate
# along vertical profile (at steady state)
# vs_array (:class:`numpy.ndarray`) :
# weathering-mediated surface-normal erosion rate along vertical profile
# (at steady state)
# eta0_array (:class:`numpy.ndarray`) :
# surface weakness along vertical profile (at steady state)
# W_array (:class:`numpy.ndarray`) :
# weathering number along vertical profile
self.n_pts = n_pts
self.z_array = np.linspace(0,1,self.n_pts)
self.w0_array = np.array([
np.float64(self.w0_eqn_z_calibrated.rhs.subs(z,z__)) for z__ in self.z_array])
self.v0_array = np.array([
np.float64(self.v0_eqn_z_calibrated.rhs.subs(z,z__)) for z__ in self.z_array])
self.vs_array = np.array([
np.float64(self.vs_eqn_z_calibrated.rhs.subs(z,z__)) for z__ in self.z_array])
self.eta0_array = self.vs_array/self.v0_array
self.W_array = np.array([
np.float64( (self.W_eqn_z_calibrated.rhs.subs(z,z__))
.subs({v_0:self.v0_array[idx]})
.subs({w_0:self.w0_array[idx]}) )
for idx,z__ in enumerate(self.z_array)
])
self.vs_eqn_z_calibrated \
= self.vs_eqn.subs({w_0:self.w0_eqn_z_calibrated.rhs}) \
.subs(self.physical_parameters)
|