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equations.py

RichterSlopeTheory dataclass 

RichterSlopeTheory(
    hillslope_tan_erosion_eq: Eq | None = None,
    hillslope_erosion_eq: Eq | None = None,
    xih_pxpz_eq: Eq | None = None,
    hillslope_pxpz_eq: Eq | None = None,
    Fstar_hillslope_pxpz_eq: Eq | None = None,
    H_hillslope_pxpz_eq: Eq | None = None,
    dHdpx: Mul | None = None,
    dHdpz: Mul | None = None,
    v_pxpz_eq: Eq | None = None,
    v_p_beta_eq: Eq | None = None,
    tanalpha_eq: Eq | None = None,
    d2Hdpx2: Mul | None = None,
    d2Hdpxpz: Mul | None = None,
    d2Hdpz2: Mul | None = None,
    gstar_eq: Eq | None = None,
    fgtx_raw_eq: Eq | None = None,
    fgtx_p_beta_eq: Eq | None = None,
    p_covec_fn: MutableDenseMatrix | None = None,
    p_fn: exp | Any | None = None,
    p_conjugate_beta_eq: Eq | None = None,
    v_vec_fn: ImmutableDenseMatrix | None = None,
)

Development of Hamiltonian rock slope erosion model using Sympy.

Attributes

hillslope_tan_erosion_eq: Eq | None = None hillslope_erosion_eq: Eq | None = None xih_pxpz_eq: Eq | None = None hillslope_pxpz_eq: Eq | None = None Fstar_hillslope_pxpz_eq: Eq | None = None H_hillslope_pxpz_eq: Eq | None = None dHdpx: Mul | None = None dHdpz: Mul | None = None v_pxpz_eq: Eq | None = None v_p_beta_eq: Eq | None = None tanalpha_eq: Eq | None = None d2Hdpx2: Mul | None = None d2Hdpxpz: Mul | None = None d2Hdpz2: Mul | None = None gstar_eq: Eq | None = None fgtx_raw_eq: Eq | None = None fgtx_p_beta_eq: Eq | None = None p_covec_fn: MutableDenseMatrix | None = None p_fn: exp | Any | None = None p_conjugate_beta_eq: Eq | None = None v_vec_fn: ImmutableDenseMatrix | None = None

Used by:

Methods:

  • __post_init__ –

    Solve Hamiltonian rock slope erosion model equations using Sympy.

__post_init__ 

__post_init__() -> None

Solve Hamiltonian rock slope erosion model equations using Sympy.

Attributes

hillslope_tan_erosion_eq: Eq | None = None hillslope_erosion_eq: Eq | None = None xih_pxpz_eq: Eq | None = None hillslope_pxpz_eq: Eq | None = None Fstar_hillslope_pxpz_eq: Eq | None = None H_hillslope_pxpz_eq: Eq | None = None dHdpx: Mul | None = None dHdpz: Mul | None = None v_pxpz_eq: Eq | None = None v_p_beta_eq: Eq | None = None tanalpha_eq: Eq | None = None d2Hdpx2: Mul | None = None d2Hdpxpz: Mul | None = None d2Hdpz2: Mul | None = None gstar_eq: Eq | None = None fgtx_raw_eq: Eq | None = None fgtx_p_beta_eq: Eq | None = None p_covec_fn: MutableDenseMatrix | None = None p_fn: exp | Any | None = None p_conjugate_beta_eq: Eq | None = None v_vec_fn: ImmutableDenseMatrix | None = None

Source code in erosionfront/theory/equations.py 
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def __post_init__(self) -> None:
    """
    Solve Hamiltonian rock slope erosion model equations using Sympy.

    Attributes
    ----------
    hillslope_tan_erosion_eq: Eq | None = None
    hillslope_erosion_eq: Eq | None = None
    xih_pxpz_eq: Eq | None = None
    hillslope_pxpz_eq: Eq | None = None
    Fstar_hillslope_pxpz_eq: Eq | None = None
    H_hillslope_pxpz_eq: Eq | None = None
    dHdpx: Mul | None = None
    dHdpz: Mul | None = None
    v_pxpz_eq: Eq | None = None
    v_p_beta_eq: Eq | None = None
    tanalpha_eq: Eq | None = None
    d2Hdpx2: Mul | None = None
    d2Hdpxpz: Mul | None = None
    d2Hdpz2: Mul | None = None
    gstar_eq: Eq | None = None
    fgtx_raw_eq: Eq | None = None
    fgtx_p_beta_eq: Eq | None = None
    p_covec_fn: MutableDenseMatrix | None = None
    p_fn: exp | Any | None = None
    p_conjugate_beta_eq: Eq | None = None
    v_vec_fn: ImmutableDenseMatrix | None = None
    """
    self.hillslope_tan_erosion_eq = Eq(
        xi_h(beta), 
        xi_h0 * (
            0 +
            (
                (exp(-(tan(phi*1) / tan(beta))**n) ) 
                +
                (exp(-(tan(phi*2*Rational(4,5)) / tan(beta))**n) ) * 0
            )
        )/1
    )
    self.hillslope_erosion_eq = self.hillslope_tan_erosion_eq

    self.xih_pxpz_eq = Eq(xi_h(beta), 1/sqrt(px**2+pz**2))

    self.hillslope_pxpz_eq = (
        self.hillslope_erosion_eq
            .subs(e2d(self.xih_pxpz_eq))
            .subs({tan(beta):(-px/pz)})
            .subs({tan(beta/2):sin(beta)/(1+cos(beta))})
            .subs({sin(beta):(px/sqrt(px**2+pz**2))})
            .subs({cos(beta):(-pz/sqrt(px**2+pz**2))})
    )

    self.Fstar_hillslope_pxpz_eq = Eq(
        Fstar,
        solve(simplify(
                self.hillslope_pxpz_eq.subs({px: px/Fstar, pz: pz/Fstar})
            ), Fstar)[
            0
        ],
    )

    self.H_hillslope_pxpz_eq \
        = Eq(H, self.Fstar_hillslope_pxpz_eq.rhs**2 / 2)

    self.dHdpx = simplify(diff(self.H_hillslope_pxpz_eq.rhs,px))
    self.dHdpz = simplify(diff(self.H_hillslope_pxpz_eq.rhs,pz))

    self.v_pxpz_eq = Eq(
        Matrix([rdotx, rdotz]),
        Matrix(
            [
                factor(self.dHdpx),
                factor(self.dHdpz),
            ]
        )
    )

    self.v_p_beta_eq = (
        trigsimp( self.v_pxpz_eq.subs({px:p*sin(beta), pz:-p*cos(beta)}) )
    )

    self.tanalpha_eq = simplify(Eq(
        tan(alpha),
        trigsimp( self.v_p_beta_eq.rhs[1]/self.v_p_beta_eq.rhs[0] )
    ))

    self.d2Hdpx2 = simplify(diff(self.dHdpx,px))
    self.d2Hdpxpz = simplify(diff(self.dHdpx,pz))
    self.d2Hdpz2 = simplify(diff(self.dHdpz,pz))

    self.gstar_eq = Eq(
        gstar, 
        Matrix([
            [self.d2Hdpx2, self.d2Hdpxpz], 
            [self.d2Hdpxpz,self.d2Hdpz2]
        ])
    )

    self.fgtx_raw_eq = simplify(
        self.Fstar_hillslope_pxpz_eq
            .subs({Fstar:1})
            .subs({px:p*sin(beta), pz:-p*cos(beta)})
    )

    self.fgtx_p_beta_eq = Eq(
        p,
        solve( self.fgtx_raw_eq, p )[0]
    )

    self.p_covec_fn = Matrix([
        (self.fgtx_p_beta_eq.rhs*sin(beta)),
        (-self.fgtx_p_beta_eq.rhs*cos(beta))
    ])
    self.p_fn = self.fgtx_p_beta_eq.rhs

    self.p_conjugate_beta_eq = Eq(
        p,
        powsimp(solve(
            simplify(
                Eq(
                    (self.v_p_beta_eq.rhs[0] * px 
                     + self.v_p_beta_eq.rhs[1] * pz)
                     .subs({px: p*sin(beta), pz: -p*cos(beta)}
                    ),
                    1,
                )
            ),
            p,
        )[1]),
    )

    self.v_vec_fn =  (self.v_p_beta_eq.rhs)