Skip to content

theory.py

Equations 

Equations()

Weathering-mediated erosion theory in various forms

Attributes:

  • W_eqn ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : weathering number: \(W = \dfrac{w_0}{k v_0}\)

  • nus_eqn_W ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : dimensionless steady-state erosion rate: \(\nu_s = \dfrac{1}{2}(1+\sqrt{1+4W})\)

  • nus_eqn_w0_v0 ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : dimensionless steady-state erosion rate: \(\nu_s = \frac{\sqrt{1 + \frac{4 w_{0}}{k v_{0}}}}{2} + \frac{1}{2}\)

  • etas0_eqn_W ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : steady-state surface weakness: \(\eta_{s0} = \dfrac{\sqrt{4 W + 1}}{2} + \frac{1}{2}\)

  • etas0_eqn_w0_v0 –

    class:sympy.Eq <sympy.core.relational.Equality>) : steady-state surface weakness: \(\eta_{s0} = \dfrac{\sqrt{1 + \frac{4 w_{0}}{k v_{0}}}}{2} + \frac{1}{2}\)

  • nus_eqn_etas0 ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : dimensionless steady-state erosion rate: \(\nu_s = \eta_{s}(0)\)

  • vs_eqn_etas0_v0 –

    class:sympy.Eq <sympy.core.relational.Equality>) : steady-state erosion rate: \(v_{s} = \eta_{s}(0) v_{0}\)

  • v0_eqn_etas0_vs –

    class:sympy.Eq <sympy.core.relational.Equality>) : erosion rate: \(v_{0} = \dfrac{k v_{s}^{2}}{k v_{s} + w_{0}}\)

  • vs_eqn_w0_v0 ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : steady-state erosion rate: \(v_{s} = \eta_{s}(0) v_{0}\)

  • v0_eqn_vs_w0 ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : erosion rate: \(v_{0} = \dfrac{k v_{s}^{2}}{k v_{s} + w_{0}}\)

  • v0_eqn_vr_h_z ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : erosion rate: \(v_0 = v_r \left\{ (h-H_s[z,z_\mathrm{vc},\kappa_\mathrm{v}])(1-v_b)+v_b \right\}\)

  • w0_eqn_wr_z ( ) –

    class:sympy.Eq <sympy.core.relational.Equality>) : surface weakness: \(w_0 = w_r H_s[z,z_\mathrm{wc},\kappa_w]\)

Used by:

Methods:

  • step –

    Step function at \(z=z_0\) and sharpness \(k\).

Source code in wmbe/theory.py 
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
def __init__(self):
    """
    Initialize class instance.

    Attributes:
        W_eqn           (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            weathering number: 
            $W = \\dfrac{w_0}{k v_0}$
        nus_eqn_W       (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            dimensionless steady-state erosion rate: 
            $\\nu_s = \\dfrac{1}{2}(1+\\sqrt{1+4W})$
        nus_eqn_w0_v0   (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            dimensionless steady-state erosion rate:
            $\\nu_s = \\frac{\\sqrt{1 + \\frac{4 w_{0}}{k v_{0}}}}{2} + \\frac{1}{2}$
        etas0_eqn_W     (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            steady-state surface weakness:
            $\\eta_{s0} = \\dfrac{\\sqrt{4 W + 1}}{2} + \\frac{1}{2}$
        etas0_eqn_w0_v0 (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            steady-state surface weakness:
            $\\eta_{s0} = \\dfrac{\\sqrt{1 + \\frac{4 w_{0}}{k v_{0}}}}{2} + \\frac{1}{2}$
        nus_eqn_etas0   (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            dimensionless steady-state erosion rate:
            $\\nu_s =  \\eta_{s}(0)$
        vs_eqn_etas0_v0 (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            steady-state erosion rate:
            $v_{s} = \\eta_{s}(0) v_{0}$
        v0_eqn_etas0_vs (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            erosion rate:
            $v_{0} = \\dfrac{k v_{s}^{2}}{k v_{s} + w_{0}}$
        vs_eqn_w0_v0    (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            steady-state erosion rate:
            $v_{s} = \\eta_{s}(0) v_{0}$
        v0_eqn_vs_w0    (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            erosion rate:
            $v_{0} = \\dfrac{k v_{s}^{2}}{k v_{s} + w_{0}}$
        v0_eqn_vr_h_z   (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            erosion rate: 
            $v_0 =  v_r \\left\\{ (h-H_s[z,z_\\mathrm{vc},\\kappa_\\mathrm{v}])(1-v_b)+v_b \\right\\}$
        w0_eqn_wr_z     (:class:`sympy.Eq <sympy.core.relational.Equality>`) : 
            surface weakness:
            $w_0 =  w_r H_s[z,z_\\mathrm{wc},\\kappa_w]$

    """
    self.W_eqn           = Eq(W,w_0/(k*v_0))
    self.nus_eqn_W       = Eq(nu_s,(1+sy.sqrt(1+4*W))/2)
    self.nus_eqn_w0_v0   = self.nus_eqn_W.subs(W,self.W_eqn.rhs)
    self.etas0_eqn_W     = Eq(eta_s0, self.nus_eqn_W.rhs)
    self.etas0_eqn_w0_v0 = self.etas0_eqn_W.subs(W,self.W_eqn.rhs)
    self.nus_eqn_etas0   = Eq(nu_s, eta_s0)
    self.vs_eqn_etas0_v0 = Eq(v_s, self.nus_eqn_etas0.rhs*v_0)
    self.v0_eqn_etas0_vs = Eq(v_0, sy.solve(self.vs_eqn_etas0_v0,v_0)[0])
    self.vs_eqn_w0_v0    = Eq(v_s, self.nus_eqn_w0_v0.rhs*v_0)
    self.v0_eqn_vs_w0    = Eq(v_0, sy.solve(self.vs_eqn_w0_v0,v_0)[0])
    self.v0_eqn_vr_h_z   = Eq(
        v_0, 
        v_r*( (h-self.step(z,z_vc,kappa_v))*(1-v_b)+v_b ) 
    )
    self.w0_eqn_wr_z     = Eq(w_0, w_r*self.step(z,z_wc,kappa_w))

step staticmethod 

step(z, z0, kappa)

Step function at \(z=z_0\) and sharpness \(k\).

\(H_s(z,z_0,\kappa) = \dfrac{1}{2}\left(1 + \tanh{[\kappa(z-z_0)]}\right)\)

Source code in wmbe/theory.py 
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
@staticmethod
def step(z, z0, kappa,):
    """
    Step function at $z=z_0$ and sharpness $k$.

    $H_s(z,z_0,\\kappa) = \\dfrac{1}{2}\\left(1 + \\tanh{[\\kappa(z-z_0)]}\\right)$
    """
    # Attributes:
    #     z (sympy.float)  : abscissa $z$
    #     z0 (float) : offset $z_0$
    #     kappa (float)  : step sharpness $\\kappa$

    # Returns:
    #     float: step function $H_s$
    return (1+sy.tanh((z-z0)*kappa))/2