Merge pull request #83 from NumericalEarth/ncc/latex-render

Tweak some latex rendering in the docs

Author: bgroenks96

docs/src/physics/soil_energy_water.md

@@ -24,7 +24,7 @@ where $\mathbf{j}_\text{h}$ (W/m²) is the diffusive heat flux vector and $\math
 Since ground materials are often porous, i.e., there exists void space between the solid particles, it is necessary to consider the potential presence of water and/or ice in this void space, which is hereafter referred to as pore space, or simply, soil pores. The thermal effects of water and ice can be accounted for by considering not only the temperature of the material but rather the total internal energy of the elementary volume. Combining the diffusive flux with a potential advective heat flux $j_z^{\text{w}}$ due to water flow yields the energy conservation law,
 ```math
 \begin{equation}
-\frac{\partial U(T,\theta)}{\partial t} - \nabla \cdot \left(\mathbf{j}_\text{h} + \mathbf{j}_h^{\text{w}}\right) - F_h(z,t) = 0\,,
+\frac{\partial U(T,\theta)}{\partial t} - \boldsymbol{\nabla} \cdot \left(\mathbf{j}_\text{h} + \mathbf{j}_h^{\text{w}}\right) - F_h(z,t) = 0\,,
 \end{equation}
 ```
 where $U(T,\theta)$ (J/m³) is the volumetric internal energy as a function of temperature and total water/ice content $\theta$ (m³/m³), and $F_h(z,t)$ is an inhomogeneous heat source/sink (forcing) term.
@@ -42,7 +42,7 @@ where $L_{\text{sl}}$ and $c_{\text{w}}$ (J/kg) represent the specific latent he
 The constitutive relationship between energy and temperature plays a key role in characterizing the subsurface energy balance. This relation can be defined in integral form as
 ```math
 \begin{equation}
-    U(T,\theta) = \int_{T_{\text{ref}}}^T \tilde{C}(x,\theta) \, dx\,,
+    U(T,\theta) = \int_{T_{\text{ref}}}^T \tilde{C}(x,\theta) \, \mathrm{d}x\,,
     %= \overbrace{\HC(\thetaw,\thetai)\left[T-T_{\text{ref}}\right]}^{\text{Sensible}} + \overbrace{\densityw \LHF\thetaw(T,\thetawi)}^{\text{Latent}},
 \end{equation}
 ```
@@ -84,7 +84,7 @@ where $C = C(\theta_{\text{w}},\theta)$ is the volumetric heat capacity (J/K/m³
 
 The vertical flow of water in porous media, such as soils, can be formulated as following the conservation law
 ```math
-    \phi\frac{\partial\vartheta(\psi)}{\partial t} - \nabla \cdot \textbf{j}_{\text{w}} - F_{\text{w}}(z,t) = 0,
+    \phi\frac{\partial\vartheta(\psi)}{\partial t} - \boldsymbol{\nabla} \cdot \textbf{j}_{\text{w}} - F_{\text{w}}(z,t) = 0,
 ```
 where $\phi$ is the natural porosity (or saturated water content) of the soil volume and $F_{\text{w}}(z,t)$ (m/s) is an inhomogeneous source/sink (forcing) term.
 

src/processes/soil/energy/soil_energy.jl

@@ -4,7 +4,7 @@
 Represents an explicit formulation of the two-phase heat conduction operator in 1D:
 
 ```math
-\\frac{\\partial U(T,\\phi)}{\\partial t} = \\nabla \\cdot \\kappa(T)\\nabla_x T(x,t)
+\\frac{\\partial U(T,\\phi)}{\\partial t} = \\boldsymbol{\\nabla} \\cdot \\left[ \\kappa(T) \\boldsymbol{\\nabla}_x T(x,t) \\right]
 ```
 where \$T\$ is temperature [K], \$U\$ is internal energy [J m⁻³], and \$\\kappa\$ is the thermal conductivity [W m K⁻¹].
 """

src/processes/soil/energy/soil_thermal_properties.jl

@@ -100,7 +100,7 @@ end
 The inverse quadratic (or "quadratic parallel") bulk thermal conductivity formula (Cosenza et al. 2003):
 
 ```math
-k = [\\sum_{i=1}^N θᵢ\\sqrt{kᵢ}]^2
+k = \\left[\\sum_{i=1}^N θᵢ\\sqrt{kᵢ}\\right]^2
 ```
 
 Cosenza, P., Guérin, R., and Tabbagh, A.: Relationship between thermal

src/processes/soil/hydrology/soil_hydrology_rre.jl

@@ -5,7 +5,7 @@
 of the Richardson-Richards equation:
 
 ```math
-\\phi(z) \\frac{\\partial s_{\\mathrm{wi}}(\\Psi(z,t))}{\\partial t} = \\n+\\nabla \\cdot \\bigl(K(s_{\\mathrm{wi}}, T) \\; \\n+\\nabla (\\psi_m + 1)\\bigr)
+\\phi(z) \\frac{\\partial s_{\\mathrm{wi}}(\\Psi(z,t))}{\\partial t} = \\boldsymbol{\\nabla} \\cdot \\left[ K(s_{\\mathrm{wi}}, T) + \\boldsymbol{\\nabla} (\\psi_m + 1)\\right]
 ```
 which describes the vertical movement of water according to gravity-driven
 percolation and capillary-driven diffusion.

src/processes/surface_hydrology/ground_evaporation.jl

@@ -76,11 +76,10 @@ ConstantEvaporationResistanceFactor(::Type{NF}; kwargs...) where {NF} = Constant
 Implements the soil moisture limiting resistance factor of Lee and Pielke (1992),
 
 ```math
-\\beta = \\begin{cases}
-\\frac{1}{4} \\left(1 - \\cos\\left(π \\frac{\\theta_1}{\\theta_{\\text{fc}}} \\right)\\right) & \\theta_1 < \\theta_{\\text{fc}} \\
-1 & \\text{otherwise}
-\\end{cases}
+\\beta =
+\\frac{1}{4} \\left[1 - \\cos\\left(π \\theta_1/\\theta_{\\text{fc}} \\right)\\right] \\quad \\text{for } \\theta_1 < \\theta_{\\text{fc}}
 ```
+otherwise ``\\beta=1``.
 """
 struct SoilMoistureResistanceFactor{NF} <: AbstractGroundEvaporationResistanceFactor end
 

src/processes/vegetation/plant_available_water.jl

@@ -5,10 +5,11 @@ Implementation of vegetation water availability (a.k.a "plant available water")
 that computes the wilting fraction
 
 ```math
-W_i = \\min(\\frac{\\theta_{\\text{w},i} - \\theta_{\\text{wp},i}}{\\theta_{\\text{fc},i} - \\theta_{\\text{wp},i}}}, 1)
+W_i = \\min\\left(\\frac{\\theta_{\\text{w},i} - \\theta_{\\text{wp},i}}{\\theta_{\\text{fc},i} - \\theta_{\\text{wp},i}}
+, 1\\right)
 ```
 
-where ``\\theta_{\\text{w},i}`` is the volumetric water content of the i'th soil layer, ``\\theta_{\\text{fc},i}``
+where ``\\theta_{\\text{w},i}`` is the volumetric water content of the ``i``'th soil layer, ``\\theta_{\\text{fc},i}``
 is the "field capacity", and ``\\theta_{\\text{wp},i}`` is the "wilting point". The water availability
 
 Properties:

src/processes/vegetation/root_distribution.jl

@@ -6,7 +6,7 @@ based on the scheme proposed by Zeng (2001). The PDF of the root distribution
 is modeled as
 
 ```math
-\\frac{\\partial R}{\\partial z} = \\frac{1}{2} \\left( a \\exp(a z) + b \\exp(b z) \\right)
+\\frac{\\partial R}{\\partial z} = \\frac{1}{2} \\left[ a \\exp(a z) + b \\exp(b z) \\right]
 ```
 which is then integrated over the soil column and normalized to sum to unity. Note that
 this is effectively the average of two exponential distributions with rates `a` and `b`, both