### Apply formula symbol convention to parameterization docs

* Use regular math for scalars, bold for vectors, sans-serif for
tensors.
* Replace superscripts by subscripts.
* Replace \mathrm by \text.
* Fix typos.
parent 2b87d375
 ... ... @@ -133,8 +133,9 @@ As the *soil porosity* :math:\phi \, [-] and the *residual water content* .. math:: \Theta = \frac{\theta_w - \theta_r}{\phi - \theta_r}, \quad \Theta \in \left[ 0, 1 \right] \quad \Theta \in \left[ 0, 1 \right], where :math:\theta_w \, [-] is the volumetric soil water content. One typically assumes that the entire pore space can be filled up with water, hence we set the *saturated water content* :math:\theta_s \, [-] equal to the porosity, :math:\theta_s = \phi. ... ... @@ -158,8 +159,8 @@ porosity, :math:\theta_s = \phi. Parameters: * theta_r: Residual water content :math:\theta_r * theta_s: Saturated water content / Porosity :math:\theta_s * k0: Saturated conductivity :math:K_0 \, [\mathrm{ms}^{-1}] * alpha: Air-entry value :math:\alpha \, [\mathrm{m}^{-1}] * k0: Saturated conductivity :math:K_0 \, [\text{ms}^{-1}] * alpha: Air-entry value :math:\alpha \, [\text{m}^{-1}] * n: Retention curve shape factor :math:n * tau: Skew factor :math:\tau ... ... @@ -180,16 +181,16 @@ porosity, :math:\theta_s = \phi. Transport Parameterizations ^^^^^^^^^^^^^^^^^^^^^^^^^^^ Regarless the parameterization, the transport simulation always computes the microscopic péclet number, for which it requires the characteristic pore Regardless of the parameterization, the transport simulation always computes the microscopic péclet number, for which it requires the characteristic pore length, the molecular diffusion, and the fluid velocity. The latter is directly provided by the richards simulation while the other two have to be specified for each volume: Permanent parameters: * mol_diff: Molecular diffusion :math:\mathsf{D}^\mathsf{m} \, [\mathrm{m}^2\mathrm{s}^{-1}] * char_length: Characteristic pore length :math:\ell \, [\mathrm{m}] :math:D_m \, [\text{m}^2\text{s}^{-1}] * char_length: Characteristic pore length :math:\ell \, [\text{m}] .. note:: We support two options for specifying tensors through the YAML syntax. ... ... @@ -210,11 +211,15 @@ Permanent parameters: The sequence is interpreted as a flattened tensor with row-major layout. .. warning:: You must omit any component containing the spatial direction z in a 2D setup. Hydrodynamic Dispersion Parameterizations """"""""""""""""""""""""""""""""""""""""" The hydrodynamic dispersion tensor :math:\mathsf{D}^\mathsf{hd} \, [\mathrm{m}^2\mathrm{s}^{-1}] is the main parameter to provide in the The hydrodynamic dispersion tensor :math:\mathsf{D}_\text{hd} \, [\text{m}^2\text{s}^{-1}] is the main parameter to provide in the transport simulation. Below you will find several parameterization for this. .. object:: Constant parameterization ... ... @@ -228,17 +233,17 @@ transport simulation. Below you will find several parameterization for this. .. math:: \mathsf{D}^\mathsf{hd} = \text{const}. \mathsf{D}_\text{hd} = \text{const}. * type: Dhd_const Parameters: * hydrodynamic_disp_: :math:(i,j) Component of the * hydrodynamic_disp_: (i, j)-th component of the hydrodynamic dispersion tensor, :math:\mathsf{D}^\mathsf{hd}_{ij} \, [\mathrm{m}^2\mathrm{s}^{-1}], :math:\left( \mathsf{D}_\text{hd} \right)_{ij} \, [\text{m}^2\text{s}^{-1}], **or** * hydrodynamic_disp: Flattened hydrodynamic dispersion tensor :math:\mathsf{D}^\mathsf{hd} \, [\mathrm{m}^2\mathrm{s}^{-1}]. :math:\mathsf{D}_\text{hd} \, [\text{m}^2\text{s}^{-1}]. YAML template: ... ... @@ -270,18 +275,18 @@ transport simulation. Below you will find several parameterization for this. .. math:: \mathsf{D}^\mathsf{hd} = \gamma\mathsf{D}^\mathsf{m}\text{pe}^\alpha. D_\text{hd} = \gamma D_m \operatorname{pe}^\alpha. * type: Dhd_pl Parameters: * gamma: Scale the power law :math:[-] * alpha: Exponent of the power law :math:[-] * gamma: Scale the power law :math:\gamma \, [-] * alpha: Exponent of the power law :math:\alpha \, [-] * mol_diff: Molecular diffusion :math:\mathsf{D}^\mathsf{m} \, [\mathrm{m}^2\mathrm{s}^{-1}] The Peclét number :math:\text{pe} is specified in the :doc:config file . :math:D_m \, [\text{m}^2\text{s}^{-1}] The Peclét number :math:\operatorname{pe} is specified in the :doc:config file . YAML template: ... ... @@ -302,13 +307,13 @@ The hydrodynamic dispersion tensor is said to be the superposition of several diffusion/dispersion processes. In DORiE this possible by summing several valid parameterizations types. Currently we provide parameterizations for the *effective diffusion* :math:\mathsf{D}^\mathsf{eff} and for the *effective hydromechanic tensor* :math:\mathsf{D}^\mathsf{hm} identified by the key prefixes Deff and :math:D_\text{eff} and for the *effective hydromechanic tensor* :math:\mathsf{D}_\text{hm} identified by the key prefixes Deff and Dhm respectively. .. math:: \mathsf{D}^\mathsf{hd} = \mathsf{D}^\mathsf{hm}+\mathsf{D}^\mathsf{eff}. \mathsf{D}_\text{hd} = \mathsf{D}_\text{hm} + D_\text{eff}. * type: +  ... ... @@ -316,17 +321,18 @@ Each of the types are further parameterized and can be found below. .. object:: Constant effective diffusion parameterization In this case, the effective diffusion is inserted directly. In this case, the effective diffusion is inserted directly, .. math:: \mathsf{D}^\mathsf{eff} = \text{const}. D_\text{eff} = \text{const}. * Deff_type: Deff_const Parameters: * eff_diff: Effective diffusion :math:\mathsf{D}^\mathsf{eff} \, [\mathrm{m}^2\mathrm{s}^{-1}] :math:D_\text{eff} \, [\text{m}^2\text{s}^{-1}] YAML template: ... ... @@ -346,16 +352,16 @@ Each of the types are further parameterized and can be found below. .. math:: \mathsf{D}^\mathsf{eff} = \mathsf{D}^\mathsf{m}\frac{\theta_s^{7/3}}{\phi^{2/3}}. D_\text{eff} = D_m \frac{\theta_w^{7/3}}{\phi^{2/3}}. where the volumetric water content :math:\theta_s \, [-] is automatically obtained from richards simulation. where the volumetric water content :math:\theta_w \, [-] is automatically obtained from the Richards simulation. * Deff_type: Deff_MQ1 Parameters: * mol_diff: Molecular diffusion :math:\mathsf{D}^\mathsf{m} \, [\mathrm{m}^2\mathrm{s}^{-1}] * mol_diff: Molecular diffusion :math:D_m \, [\text{m}^2\text{s}^{-1}] * phi: Soil porosity :math:\phi \, [-] YAML template: ... ... @@ -376,16 +382,16 @@ Each of the types are further parameterized and can be found below. .. math:: \mathsf{D}^\mathsf{eff} = \mathsf{D}^\mathsf{m}\frac{\theta_s}{\phi^{2/3}}. D_\text{eff} = D_m \frac{\theta_w}{\phi^{2/3}}. where the volumetric water content :math:\theta_s \, [-] is automatically obtained from richards simulation. where the volumetric water content :math:\theta_w \, [-] is automatically obtained from the Richards simulation. * Deff_type: Deff_MQ2 Parameters: * mol_diff: Molecular diffusion :math:\mathsf{D}^\mathsf{m} \, [\mathrm{m}^2\mathrm{s}^{-1}] * mol_diff: Molecular diffusion :math:D_m \, [\text{m}^2\text{s}^{-1}] * phi: Soil porosity :math:\phi \, [-] YAML template: ... ... @@ -402,22 +408,22 @@ Each of the types are further parameterized and can be found below. .. object:: Constant effective hydromechanic dispersion tensor parameterization In this case, the effective hydromechanic dispersion tensor is inserted In this case, the effective hydromechanic dispersion tensor is inserted directly. .. math:: \mathsf{D}^\mathsf{hm} = \text{const}. \mathsf{D}_\text{hm} = \text{const}. * Dhm_type: Dhm_const Parameters: * eff_hydromechanic_disp_: (i,j) Component of the hydromechanic dispersion tensor, :math:\mathsf{D}^\mathsf{hm}_{ij} \, [\mathrm{m}^2\mathrm{s}^{-1}], * eff_hydromechanic_disp_: (i, j)-th component of the hydromechanic dispersion tensor, :math:\left(\mathsf{D}_\text{hm}\right)_{ij} \, [\text{m}^2\text{s}^{-1}], **or** * eff_hydromechanic_disp: Flattened hydromechanic dispersion tensor :math:\mathsf{D}^\mathsf{hm} \, [\mathrm{m}^2\mathrm{s}^{-1}]. :math:\mathsf{D}_\text{hm} \, [\text{m}^2\text{s}^{-1}]. YAML template: ... ... @@ -451,13 +457,18 @@ Each of the types are further parameterized and can be found below. .. math:: \mathsf{D}^\mathsf{hm}_{ij} = (\lambda_l-\lambda_t)\frac{v_i v_j}{|v|} + \delta_{ij}\lambda_t |v|. \left( \mathsf{D}_\text{hm} \right)_{ij} = (\lambda_l-\lambda_t)\frac{v_i v_j}{\lvert \mathbf{v} \rvert} + \delta_{ij}\lambda_t \lvert \mathbf{v} \rvert, where :math:\mathbf{v} \, [\text{ms}^{-1}] is the local fluid velocity and :math:\delta_{ij} is the Kronecker delta. * Dhm_type: Dhm_iso Parameters: * lambda_l: Longitudinal dispersivity :math:\lambda_l \, [\mathrm{m}^2\mathrm{s}^{-1}] * lambda_t: Transverse dispersivity :math:\lambda_t \, [\mathrm{m}^2\mathrm{s}^{-1}] * lambda_l: Longitudinal dispersivity :math:\lambda_l \, [\text{m}^2\text{s}^{-1}] * lambda_t: Transverse dispersivity :math:\lambda_t \, [\text{m}^2\text{s}^{-1}] YAML template: ... ...
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