Response to comments

Author: mark-petersen

components/omega/doc/design/OmegaV1GoverningEqns.md

@@ -323,34 +323,36 @@ $$
 =  D^\varphi_k + S^\varphi_k 
 $$ (layered-tracer)
 
+mrp todo comment on omega vertical pressure advection, versus w vertical velocity.
+
 ### Discrete Equations
 
 The discretized versions of the governing equations are
 
 $$
-\frac{\partial u_e}{\partial t} + \left[ \frac{{\mathbf k} \cdot \nabla \times u_e +f_v}{[h_i]_v}\right]_e\left([h_i]_e u_e^{\perp}\right)
+\frac{\partial u_{e,k}}{\partial t} + \left[ \frac{{\mathbf k} \cdot \nabla \times u_{e,k} +f_v}{[h_{i,k}]_v}\right]_e\left([h_{i,k}]_e u_{e,k}^{\perp}\right)
 = 
-- \frac{1}{\rho_i} \nabla_r p_i - \nabla_r \Phi_i
-- \nabla K_i + \nu_2 \nabla^2 u_e - \nu_4 \nabla^4 u_e + \mathcal{D}_e + \mathcal{F}_e
+- \frac{1}{\rho_{i,k}} \nabla_r p_{i,k} - \nabla_r \Phi_{i,k}
+- \nabla K_{i,k} + \nu_2 \nabla^2 u_{e,k} - \nu_4 \nabla^4 u_{e,k} + \mathcal{D}_{e,k} + \mathcal{F}_{e,k}
 $$ (discrete-momentum)
 
 $$
-\frac{\partial h_i}{\partial t} + \nabla \cdot \left([h_i]_e u_e\right) = 0,
+\frac{\partial h_{i,k}}{\partial t} + \nabla \cdot \left([h_{i,k}]_e u_{e,k}\right) = 0,
 $$ (discrete-thickness)
 
 $$
-\frac{\partial h_i \varphi_i}{\partial t} + \nabla \cdot \left(u_e [h_i \varphi_i]_e \right) = \kappa_2 h_i \nabla^2 \varphi_i - \kappa_4 h_i \nabla^4 \varphi_i,
+\frac{\partial h_{i,k} \varphi_{i,k}}{\partial t} + \nabla \cdot \left(u_{e,k} [h_{i,k} \varphi_{i,k}]_e \right) = \kappa_2 h_{i,k} \nabla^2 \varphi_{i,k} - \kappa_4 h_{i,k} \nabla^4 \varphi_{i,k},
 $$ (discrete-tracer)
 
-where subscripts $i$, $e$, and $v$ indicate cell, edge, and vertex locations ($i$ was chosen for cell because $c$ and $e$ look similar).  Here square brackets $[\cdot]_e$ and $[\cdot]_v$ are quantities that are interpolated to edge and vertex locations. The interpolation is typically centered, but may vary by method, particularly for advection schemes. For vector quantities, $u_e$ denotes the normal component at the center of the edge, while $u_e^\perp$ denotes the tangential component. In the discrete system, the normal component $u_e$ points positively from the lower cell index to the higher cell index, while the tangential component $u_e^\perp$ points positively $90^o$ to the left of $u_e$ (for unit vectors, ${\mathbf n}_e^\perp = {\mathbf k}\times {\mathbf n}_e$).
+where subscripts $i$, $e$, and $v$ indicate cell, edge, and vertex locations ($i$ was chosen for cell because $c$ and $e$ look similar) and subscript $k$ is the layer.  Here square brackets $[\cdot]_e$ and $[\cdot]_v$ are quantities that are interpolated to edge and vertex locations. The interpolation is typically centered, but may vary by method, particularly for advection schemes. For vector quantities, $u_e$ denotes the normal component at the center of the edge, while $u_e^\perp$ denotes the tangential component. In the discrete system, the normal component $u_e$ points positively from the lower cell index to the higher cell index, while the tangential component $u_e^\perp$ points positively $90^o$ to the left of $u_e$ (for unit vectors, ${\mathbf n}_e^\perp = {\mathbf k}\times {\mathbf n}_e$).
 
 The discretized momentum drag and forcing terms are
 $$
-\mathcal{D}_e = - Ra \, u_e - C_D \frac{u_e\left|u_e\right|}{[h_i]_e}
+\mathcal{D}_{e,k} = - Ra \, u_{e,k} - C_D \frac{u_{e,k}\left|u_{e,k}\right|}{[h_{i,k}]_e}
 $$
 
 $$
-\mathcal{F}_e =  C_W \frac{(u_W - u_e)\left|u_W - u_e\right|}{[h_i]_e}
+\mathcal{F}_{e,k} =  C_W \frac{(u_W - u_{e,k})\left|u_W - u_{e,k}\right|}{[h_{i,k}]_e}
 $$
 
 ### Variable Definitions
@@ -365,8 +367,8 @@ Table 1. Definition of variables
 | $\mathcal{D} $      | drag                        | m/s$^2$    | edge     |   |                       |
 | $f$                 | Coriolis parameter          | 1/s      | vertex   | FVertex  |                  |
 | ${\bf F}^u $      | momentum forcing                     | m/s$^2$    | edge     |   |                       |
-|${\bf D}^u_h$, ${\bf D}^u_v$  | momentum diffusion terms | | edge | $h$ horizonal, $v$ vertical|
-|$ D^\varphi_h$, $ D^\varphi_v$ | tracer diff. terms | | cell | |
+|${\bf D}^u $, ${\bf D}^u_h$, ${\bf D}^u_v$  | momentum diffusion terms | | edge | $h$ horizonal, $v$ vertical|
+|$ D^\varphi$, $ D^\varphi_h$, $ D^\varphi_v$ | tracer diff. terms | | cell | |
 |$f$ | Coriolis parameter | | vertex |
 |$f_{eos}$ | equation of state | -  |
 |$F$ | thickness flux | | edge | $F=hu$ |
@@ -381,12 +383,14 @@ Table 1. Definition of variables
 | $q$                 | potential vorticity         | 1/m/s    | vertex   |   | $q = \eta/h = \left(\omega+f\right)/h$                                                 |
 | $Ra$                | Rayleigh drag coefficient   | 1/s      | constant |   |                                                              |
 |$S$ | salinity | | cell |  a tracer $\varphi$  |
+|$S^h$ | mass source and sink terms| | cell |    |
+|$S^\varphi$ | tracer source and sink terms| | cell |    |
 |$\mathcal{S}$  | model state |  - | |
 | $t$                 | time                        | s        | none     |   |                                                              |
 |$T^u,T^h,T^\varphi$ | tendencies | - |  |
 | ${\mathbf u}$   | velocity, vector form       | m/s      | edge     |   |                                                              |k
-| $u_e$   | velocity, normal to edge      | m/s      | edge     | NormalVelocity  |                                                              |k
-| $u^\perp_e$   | velocity, tangential to edge      | m/s      | edge     | TangentialVelocity  |                                                              |k
+| $u_{e,k}$   | velocity, normal to edge      | m/s      | edge     | NormalVelocity  |                                                              |k
+| $u^\perp_{e,k}$   | velocity, tangential to edge      | m/s      | edge     | TangentialVelocity  |                                                              |k
 | ${\mathbf u}_W$ | wind velocity               | m/s      | edge     |   |                                                              |
 | $\eta$              | absolute vorticity          | 1/s      | vertex   |   | $\eta=\omega + f$ |
 | $\kappa_2$          | tracer diffusion            | m$^2$/s    | cell     |   |                                                              |

components/omega/doc/index.md

@@ -88,6 +88,7 @@ devGuide/TridiagonalSolvers
 
 ```{toctree}
 :caption: Design documents
+:numbered:
 :maxdepth: 1
 
 design/OmegaV0ShallowWater