Merge branch 'development' into JAMESinputs

Docs/sphinx_doc/theory/DryEquations.rst

@@ -19,11 +19,11 @@ In compressible mode, in the absence of moisture, ERF solves the following parti
 expressing conservation of mass :math:`(\rho)`, momentum :math:`(\rho \mathbf{u})`, potential temperature :math:`(\rho \theta_{d})`, and scalars :math:`(\rho \mathbf{\phi})`:
 
 .. math::
-   \frac{\partial \rho_d}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u}),
+   \frac{\partial \rho_d}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u}),
 
-   \frac{\partial (\rho_d \mathbf{u})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} ( \nabla p^\prime  - \delta_{i,3}\mathbf{B} ) - \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{u},
+   \frac{\partial (\rho_d \mathbf{u})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} ( \nabla p^\prime  - \delta_{i,3}\mathbf{B} ) - \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{u},
 
-   \frac{\partial (\rho_d \theta_d)}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \theta_d) + \nabla \cdot ( \rho_d \alpha_{\theta}\ \nabla \theta_d) + F_{\theta},
+   \frac{\partial (\rho_d \theta_d)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \theta_d) + \nabla \cdot ( \rho_d \alpha_{\theta}\ \nabla \theta_d) + F_{\theta},
 
    \frac{\partial (\rho_d \boldsymbol{\phi})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \boldsymbol{\phi}) + \nabla \cdot ( \rho_d \alpha_{\phi}\ \nabla \boldsymbol{\phi}) + \mathbf{F}_{\phi}.
 

Docs/sphinx_doc/theory/Microphysics.rst

@@ -30,15 +30,15 @@ where :math:`C_c` is the rate of condensation of water vapor to cloud water, :ma
 rain to water vapor and :math:`F_r = \rho_{d} w_{t} q_p` is the sedimentation flux. The source terms that enter into the governing equations are then given by:
 
 .. math::
-   \mathbf{F_{n}} \equiv [F_{q_v}, F_{q_c}] = \left[ -C_c, \;\; C_c \right],
+   \mathbf{F_{n}} &\equiv [F_{q_v}, F_{q_c}] = \left[ -C_c, \;\; C_c \right],
 
-   \mathbf{G_{p}} = \left[ E_r, \;\; -A_c - K_c \right],
+   \mathbf{G_{p}} &= \left[ E_r, \;\; -A_c - K_c \right],
 
-   H_{n} = \rho_d \frac{L_v}{c_p} \frac{\theta_d}{T} C_c,
+   H_{n} &= \rho_d \frac{L_v}{c_p} \frac{\theta_d}{T} C_c,
 
-   F_{p} = A_c + K_c - E_c,
+   F_{p} &= A_c + K_c - E_c,
 
-   H_{p} = -\rho_d \frac{L_v}{c_p} \frac{\theta_d}{T} E_r.
+   H_{p} &= -\rho_d \frac{L_v}{c_p} \frac{\theta_d}{T} E_r.
 
 The parametrizations provided in `klemp1978simulation`_ are given below for each term.
 Note that in all the equations, :math:`p` is specified in millibars and :math:`\overline{\rho}` is specified in g cm :math:`^{-3}`. The parametrization

Docs/sphinx_doc/theory/WetEquations.rst

@@ -35,15 +35,15 @@ Governing Equations
 The governing equations without precipitating moisture variables are
 
 .. math::
-   \frac{\partial \rho_d}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u}),
+   \frac{\partial \rho_d}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u}),
 
-   \frac{\partial (\rho_d \mathbf{u})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} ( \nabla p^{\prime}  - \delta_{i,3}\mathbf{B} ) - \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{u},
+   \frac{\partial (\rho_d \mathbf{u})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} ( \nabla p^{\prime}  - \delta_{i,3}\mathbf{B} ) - \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{u},
 
-   \frac{\partial (\rho_d \theta_d)}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \theta_d) + \nabla \cdot ( \rho_d \alpha_{\theta}\ \nabla \theta_d) + F_{\theta} + H_{n},
+   \frac{\partial (\rho_d \theta_d)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \theta_d) + \nabla \cdot ( \rho_d \alpha_{\theta}\ \nabla \theta_d) + F_{\theta} + H_{n},
 
-   \frac{\partial (\rho_d \boldsymbol{\phi})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \boldsymbol{\phi}) + \nabla \cdot ( \rho_d \alpha_{\phi}\ \nabla \boldsymbol{\phi}) + \mathbf{F}_{\phi},
+   \frac{\partial (\rho_d \boldsymbol{\phi})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \boldsymbol{\phi}) + \nabla \cdot ( \rho_d \alpha_{\phi}\ \nabla \boldsymbol{\phi}) + \mathbf{F}_{\phi},
 
-   \frac{\partial (\rho_d \mathbf{q_{n}})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \mathbf{q_{n}}) + \nabla \cdot (\rho_d \alpha_{q} \nabla \mathbf{q_{n}}) + \mathbf{F_{n}},
+   \frac{\partial (\rho_d \mathbf{q_{n}})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{q_{n}}) + \nabla \cdot (\rho_d \alpha_{q} \nabla \mathbf{q_{n}}) + \mathbf{F_{n}},
 
 the non-precipitating water mixing ratio vector :math:`\mathbf{q_{n}} = \left[ q_v \;\; q_c \;\; q_i \right]` includes water vapor, :math:`q_v`, cloud water, :math:`q_c`, and cloud ice, :math:`q_i`, although some models may not include cloud ice. The source terms for moisture variables, :math:`\mathbf{F_{n}}`, and their corresponding impact on potential temperature, :math:`H_{n}` are specific to the employed model. For the Kessler microphysics scheme, these terms are detailed in :ref:`sec:Kessler Microphysics model <Microphysics>`.
 
@@ -119,16 +119,16 @@ Governing Equations
 The governing equations with precipitating moisture components are
 
 .. math::
-   \frac{\partial \rho_d}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u}),
+   \frac{\partial \rho_d}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u}),
 
-   \frac{\partial (\rho_d \mathbf{u})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} ( \nabla p^{\prime}  - \delta_{i,3}\mathbf{B} ) - \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{u},
+   \frac{\partial (\rho_d \mathbf{u})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{u}) - \frac{1}{1 + q_v + q_c} ( \nabla p^{\prime}  - \delta_{i,3}\mathbf{B} ) - \nabla \cdot \boldsymbol{\tau} + \mathbf{F}_{u},
 
-   \frac{\partial (\rho_d \theta_d)}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u}        \theta_d) + \nabla \cdot ( \rho_d \alpha_{\theta}\ \nabla \theta_d) + F_{\theta} + H_{n} + H_{p},
+   \frac{\partial (\rho_d \theta_d)}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u}        \theta_d) + \nabla \cdot ( \rho_d \alpha_{\theta}\ \nabla \theta_d) + F_{\theta} + H_{n} + H_{p},
 
-   \frac{\partial (\rho_d \boldsymbol{\phi})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \boldsymbol{\phi}) + \nabla \cdot ( \rho_d \alpha_{\phi}\ \nabla \boldsymbol{\phi}) + \mathbf{F}_{\phi},
+   \frac{\partial (\rho_d \boldsymbol{\phi})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \boldsymbol{\phi}) + \nabla \cdot ( \rho_d \alpha_{\phi}\ \nabla \boldsymbol{\phi}) + \mathbf{F}_{\phi},
 
-   \frac{\partial (\rho_d \mathbf{q_{n}})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \mathbf{q_{n}}) + \nabla \cdot (\rho_d \alpha_{q} \nabla \mathbf{q_{n}}) + \mathbf{F_{n}} + \mathbf{G_{p}},
+   \frac{\partial (\rho_d \mathbf{q_{n}})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{q_{n}}) + \nabla \cdot (\rho_d \alpha_{q} \nabla \mathbf{q_{n}}) + \mathbf{F_{n}} + \mathbf{G_{p}},
 
-   \frac{\partial (\rho_d \mathbf{q_{p}})}{\partial t} = - \nabla \cdot (\rho_d \mathbf{u} \mathbf{q_{p}}) + \partial_{z} \left( \rho_d \mathbf{w_{t}} \mathbf{q_{p}} \right) + \mathbf{F_{p}}.
+   \frac{\partial (\rho_d \mathbf{q_{p}})}{\partial t} &= - \nabla \cdot (\rho_d \mathbf{u} \mathbf{q_{p}}) + \partial_{z} \left( \rho_d \mathbf{w_{t}} \mathbf{q_{p}} \right) + \mathbf{F_{p}}.
 
 the non-precipitating water mixing ratio vector :math:`\mathbf{q_{n}} = \left[ q_v \;\; q_c \;\; q_i \right]` includes water vapor, :math:`q_v`, cloud water, :math:`q_c`, and cloud ice, :math:`q_i`, although some models may not include cloud ice; similarly, the precipitating water mixing ratio vector :math:`\mathbf{q_{p}} = \left[ q_r \;\; q_s \;\; q_g \right]` involves rain, :math:`q_r`, snow, :math:`q_s`, and graupel, :math:`q_g`, though some models may not include these terms. The source terms for moisture variables, :math:`\mathbf{F_{p}}`, :math:`\mathbf{F_{n}}`, :math:`\mathbf{G_{p}}`, and their corresponding impact on potential temperature, :math:`H_{n}` and :math:`H_{p}`, and the terminal velocity, :math:`\mathbf{w_{t}}` are specific to the employed model. For the Kessler microphysics scheme, these terms are detailed in :ref:`sec:Kessler Microphysics model <Microphysics>`.