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applications/channel1d.py

ChannelWall 

ChannelWall(eqns: Equations, physical_parameters: dict)

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).

Parameters:

  • eqns

    (Equations) –

    instance of 1d weathering-mediated erosion theory class

  • physical_parameters

    (dict) –

    physical parameters dictionary

Attributes:

  • physical_parameters (dict) –

    physical parameters dictionary

  • w0_eqn_z_calibrated (Eq) –

    surface weakness calibrated using model parameters: \(w_0 = w_r H_s[z,z_\mathrm{wc},k_\mathrm{w}]\) where \(H_s = \dfrac{1}{2}\left(1 + \tanh{[\kappa_w(z-z_\mathrm{wc}]}\right)\)

  • v0_eqn_z_calibrated (Eq) –

    erosion rate calibrated using model parameters: \(v_0 = v_r \left\{ (h-H_s[z,z_\mathrm{vc},k_\mathrm{v}])(1-v_b)+v_b \right\}\) where \(H_s = \dfrac{1}{2}\left(1 + \tanh{[\kappa_v(z-z_\mathrm{vc}]}\right)\)

  • vs_eqn (Eq) –

    surface-normal erosion rate (uncalibrated) \(v_0 = v_r \left\{ (h-H_s[z,z_\mathrm{vc},k_\mathrm{v}])(1-v_b)+v_b \right\}\) where \(H_s = \dfrac{1}{2}\left(1 + \tanh{[\kappa_v(z-z_\mathrm{vc}]}\right)\)

  • W_eqn_z_calibrated (Eq) –

    weathering number calibrated using model parameters: \(W = \dfrac{w_0}{k v_0}\)

  • vs_eqn_z_calibrated (Eq) –

    surface-normal erosion rate calibrated using model parameters: \(v_0 = v_r \left\{ (h-H_s[z,z_\mathrm{vc},k_\mathrm{v}])(1-v_b)+v_b \right\}\) where \(H_s = \dfrac{1}{2}\left(1 + \tanh{[\kappa_v(z-z_\mathrm{vc}]}\right)\)

Used by:

Methods:

Source code in wmbe/applications/channel1d.py 
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def __init__(self, eqns: Equations, physical_parameters: dict) -> None:
    """
    Initialize class instance.

    Attributes:
        physical_parameters (dict): physical parameters dictionary
        w0_eqn_z_calibrated (sympy.Eq):
            surface weakness calibrated using model parameters: 
            $w_0 =  w_r H_s[z,z_\\mathrm{wc},k_\\mathrm{w}]$
            where 
            $H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_w(z-z_\\mathrm{wc}]}\\right)$
        v0_eqn_z_calibrated (sympy.Eq):
            erosion rate calibrated using model parameters: 
            $v_0 =  v_r \\left\\{ (h-H_s[z,z_\\mathrm{vc},k_\\mathrm{v}])(1-v_b)+v_b \\right\\}$
            where 
            $H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_v(z-z_\\mathrm{vc}]}\\right)$
        vs_eqn (sympy.Eq):
            surface-normal erosion rate (uncalibrated)
            $v_0 =  v_r \\left\\{ (h-H_s[z,z_\\mathrm{vc},k_\\mathrm{v}])(1-v_b)+v_b \\right\\}$
            where 
            $H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_v(z-z_\\mathrm{vc}]}\\right)$
        W_eqn_z_calibrated (sympy.Eq):
            weathering number calibrated using model parameters: 
            $W = \\dfrac{w_0}{k v_0}$
        vs_eqn_z_calibrated (sympy.Eq):
            surface-normal erosion rate calibrated using model parameters: 
            $v_0 =  v_r \\left\\{ (h-H_s[z,z_\\mathrm{vc},k_\\mathrm{v}])(1-v_b)+v_b \\right\\}$
            where 
            $H_s = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa_v(z-z_\\mathrm{vc}]}\\right)$
    """
    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 

compute_cross_section() -> None

Compute dependence of various properties with height above channel base.

Attributes:

  • vs_array (NDArray) –

    weathering-mediated surface-normal erosion rate along vertical profile (at steady state)

  • dzdy_array (NDArray) –

    weathering-mediated vertical erosion rate along vertical profile (at steady state)

  • y_array (NDArray) –

    horizontal positions of sample points along vertica profile

  • ch_y_array (NDArray) –

    horizontal positions of sample points along channel boundary

  • ch_z_array (NDArray) –

    vertical positions of sample points along channel boundary

Source code in wmbe/applications/channel1d.py 
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def compute_cross_section(self) -> None:
    """
    Compute dependence of various properties with height above channel base.

    Attributes:
        vs_array (NDArray):
            weathering-mediated surface-normal erosion rate along vertical 
            profile (at steady state)                
        dzdy_array (NDArray):
            weathering-mediated vertical erosion rate along vertical profile 
            (at steady state)
        y_array (NDArray):
            horizontal positions of sample points along vertica profile
        ch_y_array (NDArray):
            horizontal positions of sample points along channel boundary
        ch_z_array (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 

compute_vertical_profiles(n_pts: int = 100) -> None

Compute dependence of various properties with height above channel base.

Parameters:

  • n_pts

    (int, default: 100 ) –

    number of sampling points along vertical profile

Attributes:

  • n_pts (int) –

    record of number of sampling points

  • z_array (NDArray) –

    sample points along vertical profile

  • vs_eqn_z_calibrated (Eq) –

    weathering-mediated surface-normal erosion rate at steady state calibrated by model parameters supplied in dict

  • w0_array (NDArray) –

    surface weakness along vertical profile (at steady state)

  • v0_array (NDArray) –

    baseline (absent weathering-driven weakening) surface-normal erosion rate along vertical profile (at steady state)

  • vs_array (NDArray) –

    weathering-mediated surface-normal erosion rate along vertical profile (at steady state)

  • eta0_array (NDArray) –

    surface weakness along vertical profile (at steady state)

  • W_array (NDArray) –

    weathering number along vertical profile

Source code in wmbe/applications/channel1d.py 
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def compute_vertical_profiles(self, n_pts: int=100,) -> None:
    """
    Compute dependence of various properties with height above channel base.

    Args:
        n_pts: number of sampling points along vertical profile  

    Attributes:
        n_pts (int): 
            record of number of sampling points
        z_array (NDArray):
            sample points along vertical profile 
        vs_eqn_z_calibrated (sympy.Eq): 
            weathering-mediated surface-normal erosion rate at steady state
            calibrated by model parameters supplied in dict
        w0_array (NDArray):
            surface weakness along vertical profile (at steady state)
        v0_array (NDArray):
            baseline (absent weathering-driven weakening) surface-normal 
            erosion rate along vertical profile  (at steady state)
        vs_array (NDArray):
            weathering-mediated surface-normal erosion rate along vertical 
            profile (at steady state)
        eta0_array (NDArray):
            surface weakness along vertical profile (at steady state)
        W_array (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)