diff --git a/docs/source/simulator/mass_point_model.rst b/docs/source/simulator/mass_point_model.rst index a4ee7e3d070b7e761c4f67856eb6c9cc9b427808..e56f07ddf74b9ec09ae6d59316c70be30b45e41b 100644 --- a/docs/source/simulator/mass_point_model.rst +++ b/docs/source/simulator/mass_point_model.rst @@ -4,9 +4,8 @@ Mass Point Model Theory of mass point model -------------------------- -A `mass point model` is an extremely simplified model to simulate the movement -of some mass on a given topography. It assumes that the flow mass is condensed -to a single point. +A `mass point model` is an extremely simplified model to simulate mass movement +on a given topography. It assumes that the flow mass is condensed to a single point. Let :math:`Z(x,y)` define a topography in a Cartesian coordinate system :math:`\{x, y, z\}`. It induces a local non-orthogonal @@ -38,13 +37,13 @@ given by \frac{g}{\xi}\|\boldsymbol{U}\|^2\right) :label: dUTydt -where :math:`\mu` and :math:`\xi` are dry-Coulomb and turbulent friction coefficients -respectively. :math:`\mathbf{K}` is the curvature tensor :cite:p:`Fischer2012`. -:math:`\mathbf{U}` prepresents the masspoint's velocity. :math:`U_{T_x}` and -:math:`U_{T_y}` are the velocity components along :math:`T_x` and :math:`T_y` direction -respectively. +where :math:`\mu` and :math:`\xi` are dry-Coulomb and turbulent friction coefficient +respectively (Voellmy friction model is used here). :math:`\mathbf{K}` is the +curvature tensor :cite:p:`Fischer2012`. :math:`\mathbf{U}` represents the masspoint's +velocity. :math:`U_{T_x}` and :math:`U_{T_y}` are the velocity components along :math:`T_x` +and :math:`T_y` direction respectively. -Equations :eq:`dxdt` to :eq:`dUTydt` can be rewritten in vector format as +Equations :eq:`dxdt` to :eq:`dUTydt` can be rewritten in the vector format as .. math:: \frac{d \boldsymbol{\alpha}}{d t}=\boldsymbol{f}(t, \boldsymbol{\alpha}) :label: dalphadt @@ -67,7 +66,7 @@ Equations :eq:`dxdt` to :eq:`dUTydt` can be rewritten in vector format as \end{array}\right] :label: vector-f -Equation :eq:`dalphadt` define an initial value problem. Given initial +Equation :eq:`dalphadt` defines an initial value problem. Given initial :math:`\boldsymbol{\alpha}_0`, the system can be solved forward in time using numerical schemes such as the runge-kutta method. Class :class:`.MassPointModel` utilizes the explicit runge-kutta method ``dopri5`` provided by