clean up docs

Author: szy21

docs/src/equations.md

@@ -79,19 +79,14 @@ We make use of the following operators
   ```
 
 * ``\Phi = g z`` is the geopotential, where ``g`` is the gravitational acceleration rate and ``z`` is altitude above the mean sea level.
-* ``\boldsymbol{b}`` is the reduced gravitational acceleration
-  ```math
-  \boldsymbol{b} = - \frac{\rho - \rho_{\text{ref}}}{\rho} \nabla \Phi
-  ```
-* ``\rho_{\text{ref}}`` is the reference state density
 * ``K = \tfrac{1}{2} \|\boldsymbol{u}\|^2 `` is the specific kinetic energy (J/kg), reconstructed at cell centers by
   ```math
   K = \tfrac{1}{2} (\boldsymbol{u}_{h} \cdot \boldsymbol{u}_{h} + 2 \boldsymbol{u}_{h} \cdot I_{c} (\boldsymbol{u}_{v}) + I_{c}(\boldsymbol{u}_{v} \cdot \boldsymbol{u}_{v})),
   ```
   where ``\boldsymbol{u}_{h}`` is defined on cell-centers, ``\boldsymbol{u}_{v}`` is defined on cell-faces, and ``I_{c} (\boldsymbol{u}_{v})`` is interpolated using covariant components.
 
 * ``p`` is air pressure, derived from the thermodynamic state, reconstructed at cell centers.
-* ``p_{\text{ref}}`` is the reference state pressure. It is related to the reference state density by analytical hydrostatic balance: ``\nabla p_{\text{ref}} = - \rho_{\text{ref}} \nabla \Phi``.
+* ``\Pi = (\frac{p}{p_0})^{\frac{R_d}{c_{pd}}}`` is the Exner function evaluated with dry-air constants.
 * ``\boldsymbol{F}_R`` are the radiative fluxes: these are assumed to align vertically (i.e. the horizontal contravariant components are zero), and are constructed at cell faces from [RRTMGP.jl](https://github.com/CliMA/RRTMGP.jl).
 
 * ``\nu_u``, ``\nu_h``, and ``\nu_\chi`` are hyperdiffusion coefficients, and ``c`` is the divergence damping factor.
@@ -127,32 +122,29 @@ term treated implicitly (check this)
 
 Uses the advective form equation
 ```math
-\frac{\partial}{\partial t} \boldsymbol{u}  = - (2 \boldsymbol{\Omega} + \nabla \times \boldsymbol{u}) \times \boldsymbol{u} - \frac{1}{\rho} \nabla p  + \nabla \Phi - \nabla K. 
+\frac{\partial}{\partial t} \boldsymbol{u}  = - (2 \boldsymbol{\Omega} + \nabla \times \boldsymbol{u}) \times \boldsymbol{u} - c_{pd} (\theta_v - \theta_{v, r}) \nabla_h \Pi  - \nabla_h [(\Phi - \Phi_r) + K]. 
+```
+Here, we use the Exner function to compute pressure gradients and are subtracting a hydrostatic reference state
+```math
+- \frac{1}{\rho} \nabla p = - c_{pd} \theta_v \Pi
 ```
+where ``\theta_v`` is the virtual potential temperature. ``\theta_{v,r} = T_r / \Pi`` is a reference virtual potential temperature (with reference temperature ``T_r``), and 
+```math
+\Phi_r = -c_{pd} \left[ T_\text{min} \log(\Pi) + \frac{(T_\text{sfc} - T_\text{min})}{n_s} (\Pi^{n_s} - 1) \right],
+```
+is a reference geopotential, which satisfies the hydrostatic balance equation $c_{pd} \theta_{v,r} \nabla \Pi + \nabla \Phi_r = 0$ for any $\Pi$.
+We use the reference temperature profile ``T_r = T_\text{min} + (T_\text{sfc} - T_\text{min}) \Pi^{n_s}``, with constants ``T_\text{min} = 215\,K``, ``T_\text{sfc}= 288\,K``, and ``n_s = 7``.
 
 #### Horizontal momentum
 
-By breaking the curl and cross product terms into horizontal and vertical contributions, and removing zero terms (e.g. ``\nabla_v  \times \boldsymbol{u}_v = 0``), we obtain
-
-```math
-\frac{\partial}{\partial t} \boldsymbol{u}_h  =
-  - (2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h +  \nabla_h \times \boldsymbol{u}_v) \times \boldsymbol{u}^v
-  - (2 \boldsymbol{\Omega}^v + \nabla_h \times \boldsymbol{u}_h) \times \boldsymbol{u}^h
-  - \frac{1}{\rho} \nabla_h p  - \nabla_h (\Phi + K),
-```
-where ``\boldsymbol{u}^h`` and ``\boldsymbol{u}^v`` are the horizontal and vertical _contravariant_ vectors. This form is implemented as 
+By breaking the curl and cross product terms into horizontal and vertical contributions, and removing zero terms (e.g. ``\nabla_v  \times \boldsymbol{u}_v = 0``), we obtain 
 ```math
 \frac{\partial}{\partial t} \boldsymbol{u}_h  =
   - (2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h +  \nabla_h \times \boldsymbol{u}_v) \times \boldsymbol{u}^v
   - (2 \boldsymbol{\Omega}^v + \nabla_h \times \boldsymbol{u}_h) \times \boldsymbol{u}^h
   - c_{pd} (\theta_v - \theta_{v, r}) \nabla_h \Pi  - \nabla_h [(\Phi - \Phi_r) + K],
 ```
-where ``\Pi`` is the Exner function evaluated with dry-air constants ``R_d`` and ``c_{pd}``, ``\theta_v`` is the virtual potential temperature, ``\theta_{v,r} = T_r / \Pi`` is a reference virtual potential temperature (with reference temperature ``T_r``), and 
-```math
-\Phi_r = -c_{pd} \left[ T_\text{min} \log(\Pi) + \frac{(T_\text{sfc} - T_\text{min})}{n_s} (\Pi^{n_s} - 1) \right],
-```
-is a reference geopotential, which satisfies the hydrostatic balance equation $c_{pd} \theta_{v,r} \nabla \Pi + \nabla \Phi_r = 0$ for any $\Pi$.
-We use the reference temperature profile ``T_r = T_\text{min} + (T_\text{sfc} - T_\text{min}) \Pi^{n_s}``, with constants ``T_\text{min} = 215\,K``, ``T_\text{sfc}= 288\,K``, and ``n_s = 7``.
+where ``\boldsymbol{u}^h`` and ``\boldsymbol{u}^v`` are the horizontal and vertical _contravariant_ vectors. 
 
 The effect of topography is accounted for through the computation of the contravariant velocity components (projections from the covariant velocity representation) prior to computing the cross-product contributions.
 
@@ -178,9 +170,9 @@ The ``(2 \boldsymbol{\Omega}^v + \nabla_h \times \boldsymbol{u}_h) \times \bolds
 ```math
 (2 \boldsymbol{\Omega}^v + \mathcal{C}_h[\boldsymbol{u}_h]) \times \boldsymbol{u}^h
 ```
-and the ``\frac{1}{\rho} \nabla_h (p - p_h)  + \nabla_h (\Phi + K)`` as
+and the ``c_{pd} (\theta_v - \theta_{v,r}) \nabla_h \Pi + \nabla_h (\Phi - \Phi_r + K)`` term is discretized as
 ```math
-\frac{1}{\rho} \mathcal{G}_h[p - p_{\text{ref}}] + \mathcal{G}_h[\Phi + K] ,
+c_{pd} (\theta_v - \theta_{v,r}) \mathcal{G}_h[\Pi] + \mathcal{G}_h[\Phi - \Phi_r + K] ,
 ```
 where all these terms are treated explicitly.
 
@@ -205,11 +197,15 @@ The ``(2 \boldsymbol{\Omega}^h + \nabla_v \times \boldsymbol{u}_h + \nabla_h \ti
 ```math
 (2 \boldsymbol{\Omega}^h + \mathcal{C}^f_v[\boldsymbol{u}_h] + \mathcal{C}_h[\boldsymbol{u}_v]) \times I^f(\boldsymbol{u}^h) ,
 ```
-and the pressure gradient and gravitational acceleration terms as
+The ``\nabla_v K`` term is discretized as
+```math
+\mathcal{G}^f_v[K],
+```
+The ``c_{pd} (\theta_v - \theta_{v,r}) \nabla_v \Pi + \nabla_v (\Phi - \Phi_r)`` term is discretized as
 ```math
 I^f[c_{pd} (\theta_v - \theta_{v, r} ) ] \mathcal{G}^f_v[\Pi] - \mathcal{G}^f_v[\Phi - \Phi_r],
 ```
-with the latter treated implicitly.
+and is treated implicitly.
 
 This is stabilized with the addition of 4th-order vector hyperviscosity
 ```math