xi.py¶
Equation definitions and derivations using SymPy.
- class gme.core.xi.XiMixin¶
Erosion model \(\xi\) eqns supplement to eqn definition class.
- define_xi_eqns() → None¶
Define eqns for surface erosion speed \(\xi\).
Also for its vertical behavior.
- define_xi_model_eqn() → None¶
Define the equation for erosion rate model.
Define the form of the surface erosion model, giving the speed of surface motion in its normal direction \(\xi^{\perp}\). For now, the model must have a separable dependence on position \(\mathbf{r}\) and surface tilt \(\beta\). The former sets the ‘flow’ dependence of the erosion model, and is given by the function \(\varphi(\mathbf{})\), which must be specified at some point. The latter is specified in self.beta_type and may be sin or tan; it is given a power exponent \(\eta\) which must take a rational value.
Define equations related to the erosion model.
Define equations related to surface erosion speed \(\xi\) and its vertical behavior. The derivations below are for an erosion model with \(\left|\tan\beta\right|^\eta\), \(\eta=3/2\).
- xiv_varphi_pxpz_eqn¶
\(\xi^{\downarrow} = - \dfrac{p_{x}^{\eta} \left(p_{x}^{2} + p_{z}^{2}\right)^{\tfrac{1}{2} - \tfrac{\eta}{2}} \varphi{\left(\mathbf{r} \right)}}{p_{z}}\)
- Type
- px_xiv_varphi_eqn¶
\(\left(\xi^{\downarrow}\right)^{2}\) \(\left(p_{x}^{2} p_{x}^{2 \eta} \left(p_{x}^{2} \ + \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta} \varphi^{2}{\left(\mathbf{r} \right)} - 1 \ + \dfrac{p_{x}^{2 \eta} \left(p_{x}^{2} \ + \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta} \varphi^{2}{\left(\mathbf{r} \right)}} {\left(\xi^{\downarrow}\right)^{2}}\right)\) \(\times\,\,\,\left(p_{x}^{2} p_{x}^{2 \eta} \left(p_{x}^{2} \ + \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta} \varphi^{2}{\left(\mathbf{r} \right)} + 1 \ + \dfrac{p_{x}^{2 \eta} \left(p_{x}^{2} \ + \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta} \varphi^{2}{\left(\mathbf{r} \right)}} {\left(\xi^{\downarrow}\right)^{2}}\right) = 0\)
- Type
Code¶
"""
Equation definitions and derivations using :mod:`SymPy <sympy>`.
---------------------------------------------------------------------
Requires Python packages/modules:
- :mod:`SymPy <sympy>`
- `GMPLib`_
- `GME`_
.. _GMPLib: https://github.com/geomorphysics/GMPLib
.. _GME: https://github.com/geomorphysics/GME
.. _Matrix:
https://docs.sympy.org/latest/modules/matrices/immutablematrices.html
---------------------------------------------------------------------
"""
# Disable these pylint errors because it doesn't understand SymPy syntax
# - notably minus signs in equations flag an error
# pylint: disable=invalid-unary-operand-type, not-callable
# Library
import warnings
import logging
# from typing import Dict, Type, Optional # , Tuple, Eq, List
# SymPy
from sympy import Eq, simplify, factor, solve, denom, sin, cos, tan, Abs
# GMPLib
from gmplib.utils import e2d
# GME
from gme.core.symbols import p, px, pz, beta, xi, xiv, rvec, varphi_r, eta
warnings.filterwarnings("ignore")
__all__ = ["XiMixin"]
class XiMixin:
r"""Erosion model :math:`\xi` eqns supplement to eqn definition class."""
# Prerequisites
beta_type: str
do_raw: bool
eta_: float
tanbeta_pxpz_eqn: Eq
cosbeta_pxpz_eqn: Eq
sinbeta_pxpz_eqn: Eq
# Definitions
xi_p_eqn: Eq
xiv_pz_eqn: Eq
p_xi_eqn: Eq
pz_xiv_eqn: Eq
xi_varphi_beta_raw_eqn: Eq
xi_varphi_beta_eqn: Eq
xiv_varphi_pxpz_eqn: Eq
px_xiv_varphi_eqn: Eq
eta__dbldenom: Eq
def define_xi_eqns(self) -> None:
r"""
Define eqns for surface erosion speed :math:`\xi`.
Also for its vertical behavior.
Attributes:
xi_p_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\xi^{\perp} := \dfrac{1}{p}`
xiv_pz_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\xi^{\downarrow} := -\dfrac{1}{p_z}`
p_xi_eqn (:class:`~sympy.core.relational.Equality`):
:math:`{p} = \dfrac{1}{\xi^{\perp}}`
pz_xiv_eqn (:class:`~sympy.core.relational.Equality`):
:math:`{p_z} = -\dfrac{1}{\xi^{\downarrow}}`
"""
logging.info("gme.core.xi.define_xi_eqns")
self.xi_p_eqn = Eq(xi, 1 / p)
self.xiv_pz_eqn = Eq(xiv, -1 / pz)
self.p_xi_eqn = Eq(p, solve(self.xi_p_eqn, p)[0])
self.pz_xiv_eqn = Eq(pz, solve(self.xiv_pz_eqn, pz)[0])
def define_xi_model_eqn(self) -> None:
r"""
Define the equation for erosion rate model.
Define the form of the surface erosion model,
giving the speed of surface motion in its normal direction
:math:`\xi^{\perp}``.
For now, the model must have a separable dependence on
position :math:`\mathbf{r}` and surface tilt :math:`\beta`.
The former sets the 'flow' dependence of the erosion model,
and is given by
the function :math:`\varphi(\mathbf{})`,
which must be specified at some point.
The latter is specified in `self.beta_type` and may be `sin` or `tan`;
it is given a power exponent :math:`\eta` which must
take a rational value.
Attributes:
xi_varphi_beta_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\xi^{\perp}
= \varphi(\mathbf{r}) \, \left| \sin\beta \right|^\eta`
"""
logging.info("gme.core.xi.define_xi_model_eqn")
if self.beta_type == "sin":
xi_model = varphi_r(rvec) * abs(sin(beta)) ** eta
else:
xi_model = varphi_r(rvec) * abs(tan(beta)) ** eta
self.xi_varphi_beta_raw_eqn = Eq(xi, xi_model)
# self.xi_varphi_beta_eqn = Eq(xi, xi_model)
if self.do_raw:
self.xi_varphi_beta_eqn = self.xi_varphi_beta_raw_eqn
else:
self.xi_varphi_beta_eqn = Eq(
self.xi_varphi_beta_raw_eqn.lhs,
self.xi_varphi_beta_raw_eqn.rhs.subs({eta: self.eta_}),
)
def define_xi_related_eqns(self) -> None:
r"""
Define equations related to the erosion model.
Define equations related to surface erosion speed :math:`\xi`
and its vertical behavior.
The derivations below are for an erosion model with
:math:`\left|\tan\beta\right|^\eta`, :math:`\eta=3/2`.
Attributes:
xiv_varphi_pxpz_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\xi^{\downarrow}
= - \dfrac{p_{x}^{\eta}
\left(p_{x}^{2} + p_{z}^{2}\right)^{\tfrac{1}{2}
- \tfrac{\eta}{2}} \varphi{\left(\mathbf{r} \right)}}{p_{z}}`
px_xiv_varphi_eqn (:class:`~sympy.core.relational.Equality`):
:math:`\left(\xi^{\downarrow}\right)^{2}`
:math:`\left(p_{x}^{2} p_{x}^{2 \eta} \left(p_{x}^{2} \
+ \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta}
\varphi^{2}{\left(\mathbf{r} \right)} - 1 \
+ \dfrac{p_{x}^{2 \eta} \left(p_{x}^{2} \
+ \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta}
\varphi^{2}{\left(\mathbf{r} \right)}}
{\left(\xi^{\downarrow}\right)^{2}}\right)`
:math:`\times\,\,\,\left(p_{x}^{2} p_{x}^{2 \eta}
\left(p_{x}^{2} \
+ \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta}
\varphi^{2}{\left(\mathbf{r} \right)} + 1 \
+ \dfrac{p_{x}^{2 \eta} \left(p_{x}^{2} \
+ \frac{1}{\left(\xi^{\downarrow}\right)^{2}}\right)^{- \eta}
\varphi^{2}{\left(\mathbf{r} \right)}}
{\left(\xi^{\downarrow}\right)^{2}}\right) = 0`
eta_dbldenom (:class:`~sympy.core.numbers.Integer`) :
a convenience variable, recording double the denominator of
:math:`\eta`, which must itself be a rational number
"""
logging.info("gme.core.xi.define_xi_related_eqns")
eta_dbldenom = 2 * denom(self.eta_)
self.xiv_varphi_pxpz_eqn = simplify(
Eq(
xiv,
(self.xi_varphi_beta_eqn.rhs / cos(beta))
.subs(e2d(self.tanbeta_pxpz_eqn))
.subs(e2d(self.cosbeta_pxpz_eqn))
.subs(e2d(self.sinbeta_pxpz_eqn))
.subs({Abs(px): px}),
)
)
xiv_eqn = self.xiv_varphi_pxpz_eqn
px_xiv_varphi_eqn = simplify(
Eq(
(xiv_eqn.subs({Abs(px): px})).rhs ** eta_dbldenom
- xiv_eqn.lhs ** eta_dbldenom,
0,
).subs(e2d(self.pz_xiv_eqn))
)
# HACK!!
# Get rid of xiv**2 multiplier...
# should be a cleaner way of doing this
self.px_xiv_varphi_eqn = factor(Eq(px_xiv_varphi_eqn.lhs / xiv ** 2, 0))
self.eta__dbldenom = eta_dbldenom
#
