fixes geopotential gradient term in eqs and typo in \nabla def in not…

…ation.

docs/equations/ALE-algorithm.rst

@@ -5,7 +5,7 @@ The semi-discrete, vertically integrated, Boussinesq hydrostatic equations of
 motion in general-coordinate :math:`r` are
 
 .. math::
-  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdot \boldsymbol{\underline{\tau}} ,\\
+  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdot \boldsymbol{\underline{\tau}} ,\\
   \rho \delta_k \Phi + \delta_k p &= 0 ,\\
   \partial_t h + \nabla_r \cdot ( h \boldsymbol{u} ) + \delta_k ( z_r \dot{r} ) &= 0 ,\\
   \partial_t (h \theta) + \nabla_r \cdot ( h \boldsymbol{u} \theta ) + \delta_k ( z_r \dot{r} \theta ) &= \boldsymbol{\nabla} \cdot \boldsymbol{Q}_\theta ,\\
@@ -17,7 +17,7 @@ that in the first (Lagrangian) phase, regardless of the current mesh (or coordin
 :math:`r`) we integrate the equations forward with :math:`\dot{r}=0`, i.e.:
 
 .. math::
-  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdot \boldsymbol{\underline{\tau}} ,\\
+  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdot \boldsymbol{\underline{\tau}} ,\\
   \rho \delta_k \Phi + \delta_k p &= 0 ,\\
   \partial_t h + \nabla_r \cdot ( h \boldsymbol{u} ) &= 0 ,\\
   \partial_t (h \theta) + \nabla_r \cdot ( h \boldsymbol{u} \theta ) &= \boldsymbol{\nabla} \cdot \boldsymbol{Q}_\theta ,\\

docs/equations/general_coordinate.rst

@@ -9,7 +9,7 @@ The Boussinesq hydrostatic equations of motion in general-coordinate
 :math:`r` are
 
 .. math::
-  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdotp \boldsymbol{\underline{\tau}} ,\\
+  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdotp \boldsymbol{\underline{\tau}} ,\\
   \rho \partial_z \Phi + \partial_z p &= 0 ,\\
   \partial_t z_r + \boldsymbol{\nabla}_r \cdotp ( z_r \boldsymbol{u} ) + \partial_r ( z_r \dot{r} ) &= 0 ,\\
   \partial_t (z_r \theta) + \boldsymbol{\nabla}_r \cdotp ( z_r \boldsymbol{u} \theta ) + \partial_r ( z_r \dot{r} \theta ) &= \boldsymbol{\nabla} \cdotp \boldsymbol{Q}_\theta ,\\

docs/equations/governing.rst

@@ -33,7 +33,7 @@ The non-divergence of flow allows a total derivative to be re-written in flux fo
 The above equations of motion can thus be written as:
 
 .. math::
-  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdotp \boldsymbol{\underline{\tau}} ,\\
+  D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdotp \boldsymbol{\underline{\tau}} ,\\
   \rho \partial_z \Phi + \partial_z p &= 0 ,\\
   \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \partial_z w &= 0 ,\\
   \partial_t \theta + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \partial_z ( w \theta ) &= \boldsymbol{\nabla} \cdotp \boldsymbol{Q}_\theta ,\\

docs/equations/notation.rst

@@ -35,7 +35,7 @@ component of velocity normal to the vertical.
 The gradient operator without a suffix is three dimensional:
 
 .. math::
-  \boldsymbol{\nabla} = ( \boldsymbol{\nabla}_z, \boldsymbol{\nabla}_z  ) .
+  \boldsymbol{\nabla} = ( \boldsymbol{\nabla}_z, \partial_z  ) .
 
 but a suffix indicates a lateral gradient along a surface of constant property indicated by the suffix: