diff --git a/components/omega/doc/design/OmegaV1GoverningEqns.md b/components/omega/doc/design/OmegaV1GoverningEqns.md
index 0e7946cd38fa..2a972016e58d 100644
--- a/components/omega/doc/design/OmegaV1GoverningEqns.md
+++ b/components/omega/doc/design/OmegaV1GoverningEqns.md
@@ -73,7 +73,7 @@ $$
+ \int_{\partial V(t)}\rho({\bf x},t)\, {\bf v}({\bf x},t) \left({\bf v}({\bf x},t) - {\bf v}_r \right) \cdot {\bf n} \, dA
\\ & \; \; \; =
\int_{V(t)} \rho({\bf x},t) \, {\bf b}({\bf x},t)\, dV
-+ \int_{\partial V(t)} {\bf \tau}({\bf n},{\bf x},t) \, dA
++ \int_{\partial V(t)} {\bf \tau}({\bf n},{\bf x},t) \, dA
$$ (continuous-momentum)
The operator $\frac{d}{dt}$ used here denotes the rate of change within a moving control volume, sometimes referred to as a Reynolds transport derivative. It differs from the partial derivative $\frac{\partial}{\partial t}$, which represents the local rate of change at a fixed point in space (Eulerian frame), and from the material derivative $\frac{D}{Dt}$, which follows an individual fluid parcel (Lagrangian frame). The use of $\frac{d}{dt}$ allows for conservation laws to be expressed in a general framework that includes both stationary and moving control volumes, consistent with the Reynolds transport theorem.
@@ -198,7 +198,7 @@ We now express the top and bottom surfaces using the pseudo-height variable $\ti
$$
{\bf n}^{\text{top}} \approx (-\nabla \tilde{z}^{\text{top}}, 1), \quad
-{\bf n}^{\text{bot}} \approx (-\nabla \tilde{z}^{\text{bot}}, 1)
+{\bf n}^{\text{bot}} \approx (\nabla \tilde{z}^{\text{bot}}, -1)
$$ (top-bot-normal-pseudo)
Here we apply a small-slope approximation, assuming $|\nabla \tilde{z}| \ll 1$, and omit normalization factors for clarity. This approximation retains leading-order effects of slope while simplifying the surface geometry.
@@ -295,7 +295,7 @@ $$
&=
\int_A \int_{\tilde{z}^{\text{bot}}}^{\tilde{z}^{\text{top}}} {\bf b}_{\perp} \, d\tilde{z} \, dA
+
-\int_{\partial A} \left( \int_{\tilde{z}^{\text{bot}}}^{\tilde{z}^{\text{top}}} \frac{{\bf \tau}}{\rho} \, d\tilde{z} \right) dl \\
+\int_{\partial A} \left( \int_{\tilde{z}^{\text{bot}}}^{\tilde{z}^{\text{top}}} \frac{{\bf \tau}}{\rho} \, d\tilde{z} \right) {\bf n}_\perp dl \\
&\quad
+ \int_A \left[ \frac{{\bf \tau}}{\rho} \right]_{\tilde{z} = \tilde{z}^{\text{top}}} \, dA
- \int_A \left[ \frac{{\bf \tau}}{\rho} \right]_{\tilde{z} = \tilde{z}^{\text{bot}}} \, dA
@@ -501,7 +501,7 @@ $$
& + \int_A \rho_0 \, {\bf u} \left[ \tilde{w}_{tr} - \tilde{\bf u} \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA
- \int_A \rho_0 \, {\bf u} \left[ \tilde{w}_{tr} - \tilde{\bf u} \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA \\
& = -\int_A \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left({\bf f} \times \mathbf{u} + \nabla \Phi \right) \, d\tilde{z} \, dA
-- \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \frac{\rho_0 p}{\rho} \, d\tilde{z} \right) dl \\
+- \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \frac{\rho_0 p}{\rho} \, d\tilde{z} \right) {\bf n}_\perp dl \\
& - \int_A \left[ \frac{\rho_0 p}{\rho} \nabla \tilde{z}_k^{\text{top}} \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA
+ \int_A \left[ \frac{\rho_0 p}{\rho} \nabla \tilde{z}_k^{\text{bot}} \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
$$ (vh-momentum-forces)
@@ -521,7 +521,7 @@ $$
& + \int_A \rho_0 \,\left< {\bf u} \left[ \tilde{w}_{tr} - \tilde{\bf u} \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \right> \, dA
- \int_A \rho_0 \, \left< {\bf u} \left[ \tilde{w}_{tr} - \tilde{\bf u} \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \right> \, dA \\
& = -\int_A \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \, \left<\left( {\bf f} \times \mathbf{u} + \nabla \Phi \right) \right> \, d\tilde{z} \, dA
-- \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left< \frac{p}{\rho} \right> \, d\tilde{z} \right) dl \\
+- \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left< \frac{p}{\rho} \right> \, d\tilde{z} \right) {\bf n}_\perp dl \\
& - \int_A \rho_0 \left[ \left< \frac{p}{\rho} \nabla \tilde{z}_k^{\text{top}} \right> \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA
+ \int_A \rho_0 \left[ \left< \frac{p}{\rho} \nabla \tilde{z}_k^{\text{bot}} \right> \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
$$ (vh-momentum-reynolds1)
@@ -534,7 +534,7 @@ $$
& + \int_A \rho_0 \, \left[ \left(\left< {\bf u} \right> \left< \tilde{w}_{tr} \right> + \left<{\bf u}^\prime \tilde{w}_{tr}^\prime \right> \right) - \left(\left<{\bf u}\right> \left<\tilde{\bf u}\right> + \left< {\bf u}^\prime \tilde{\bf u}^\prime \right> \right) \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
& - \int_A \rho_0 \, \left[ \left(\left< {\bf u} \right> \left< \tilde{w}_{tr} \right> + \left<{\bf u}^\prime \tilde{w}_{tr}^\prime \right> \right) - \left(\left<{\bf u}\right> \left<\tilde{\bf u}\right> + \left< {\bf u}^\prime \tilde{\bf u}^\prime \right> \right) \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA \\
& = - \int_A \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \, \left( {\bf f} \times \left<\mathbf{u}\right> + \nabla \left<\Phi\right> \right) \, d\tilde{z} \, dA \\
-& - \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left(\left< \alpha \right> \left + \left<\alpha^\prime p^\prime\right> \right) \, d\tilde{z} \right) dl \\
+& - \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left(\left< \alpha \right> \left
+ \left<\alpha^\prime p^\prime\right> \right) \, d\tilde{z} \right) {\bf n}_\perp dl \\
& - \int_A \rho_0 \left[ \left< \alpha \right> \left
+ \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{top}}\right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
& + \int_A \rho_0 \left[ \left< \alpha \right> \left
+ \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{bot}}\right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
$$ (vh-momentum-reynolds2)
@@ -547,7 +547,7 @@ $$
& + \int_A \rho_0 \, \left[ \left(\left< {\bf u} \right> \left< \tilde{w}_{tr} \right> + \left<{\bf u}^\prime \tilde{w}_{tr}^\prime \right> \right) - \left(\left<{\bf u}\right> \left<\tilde{\bf u}\right> + \left< {\bf u}^\prime \tilde{\bf u}^\prime \right> \right) \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
& - \int_A \rho_0 \, \left[ \left(\left< {\bf u} \right> \left< \tilde{w}_{tr} \right> + \left<{\bf u}^\prime \tilde{w}_{tr}^\prime \right> \right) - \left(\left<{\bf u}\right> \left<\tilde{\bf u}\right> + \left< {\bf u}^\prime \tilde{\bf u}^\prime \right> \right) \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA \\
& = - \int_A \rho_0 \, \tilde{h}_k \overline{\left( {\bf f} \times \left<\mathbf{u}\right> + \nabla \left<\Phi\right> \right)}^{\tilde{z}}_k \, dA \\
-& - \int_{\partial A} \rho_0 \tilde{h}_k \left( \overline{\left< \alpha \right> \left
}^{\tilde{z}}_k + \overline{\left<\alpha^\prime p^\prime\right>}^{\tilde{z}}_k \right) dl \\
+& - \int_{\partial A} \rho_0 \tilde{h}_k \left( \overline{\left< \alpha \right> \left
}^{\tilde{z}}_k + \overline{\left<\alpha^\prime p^\prime\right>}^{\tilde{z}}_k \right) {\bf n}_\perp dl \\
& - \int_A \rho_0 \left[ \left< \alpha \right> \left
+ \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{top}} \right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
& + \int_A \rho_0 \left[ \left< \alpha \right> \left
+ \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{bot}} \right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
$$ (vh-momentum-reynolds-lay-avg)
@@ -560,7 +560,7 @@ $$
& + \int_A \rho_0 \, \left[ \left(\left< {\bf u} \right> \left< \tilde{w}_{tr} \right> + \left<{\bf u}^\prime \tilde{w}_{tr}^\prime \right> \right) - \left(\left<{\bf u}\right> \left<\tilde{\bf u}\right> + \left< {\bf u}^\prime \tilde{\bf u}^\prime \right> \right) \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
& - \int_A \rho_0 \, \left[ \left(\left< {\bf u} \right> \left< \tilde{w}_{tr} \right> + \left<{\bf u}^\prime \tilde{w}_{tr}^\prime \right> \right) - \left(\left<{\bf u}\right> \left<\tilde{\bf u}\right> + \left< {\bf u}^\prime \tilde{\bf u}^\prime \right> \right) \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA \\
& = - \int_A \rho_0 \, \tilde{h}_k \overline{\left( {\bf f} \times \left<\mathbf{u}\right> + \nabla \left<\Phi\right> \right)}^{\tilde{z}}_k \, dA \\
-& - \int_{\partial A} \rho_0 \tilde{h}_k \left( \overline{\left< \alpha \right>}^{\tilde{z}}_k \overline{\left
}^{\tilde{z}}_k + \overline{\delta \alpha \delta p}^{\tilde{z}}_k+ \overline{\left<\alpha^\prime p^\prime\right>}^{\tilde{z}}_k \right) dl \\
+& - \int_{\partial A} \rho_0 \tilde{h}_k \left( \overline{\left< \alpha \right>}^{\tilde{z}}_k \overline{\left
}^{\tilde{z}}_k + \overline{\delta \alpha \delta p}^{\tilde{z}}_k+ \overline{\left<\alpha^\prime p^\prime\right>}^{\tilde{z}}_k \right) {\bf n}_\perp dl \\
& - \int_A \rho_0 \left[\left< \alpha \right> \left
+ \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{top}} \right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
& + \int_A \rho_0 \left[\left< \alpha \right> \left
+ \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{bot}} \right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
$$ (vh-momentum-reynolds-lay-avg2)
@@ -711,7 +711,7 @@ $$
= 0.
$$ (discrete-mass)
-In this equation, mass source term ($Q$), such as sources like river runoff, sea ice freshwater fluxes, precipitation, and evaporation are bound up in the surface value of $\tilde{W}_tr^{\text{top}}$. The mass sources will be normalized by $\rho_0$ to achieve consistent units.
+In this equation, mass source term ($Q$), such as sources like river runoff, sea ice freshwater fluxes, precipitation, and evaporation are bound up in the surface value of $\tilde{W}_{tr}^{\text{top}}$. The mass sources will be normalized by $\rho_0$ to achieve consistent units.
**Tracer:**
@@ -730,7 +730,7 @@ $$
\frac{\partial u_{e,k}}{\partial t}
& + \left[ {\bf k} \cdot \nabla \times u_{e,k} +f_v\right]_e\left(u_{e,k}^{\perp}\right) + \left[\nabla K\right]_e \\
& + \frac{\rho_0}{\left[\tilde{h}_{i,k}\right]_e} \left\{ \left[\left(u - u_k\right) \left\{\tilde{W}_{tr} \right\} \right]_{e,k}^\text{top} - \left[ \left(u - u_k\right) \left\{\tilde{W}_{tr} \right\} \right]_{e,k+1}^\text{top} \right\} \\
-& = - \left(\nabla \Phi \right)_{e,k} - \frac{1}{\left[\tilde{h}_k\right]_e} \nabla \left( \tilde{h}_k \alpha_k p_k \right) - \frac{1}{\left[\tilde{h}_k\right]_e} \left\{ \left[ \alpha p \nabla \tilde{z}^{\text{top}}\right]_{e,k}^\text{top} - \left[ \alpha p \nabla \tilde{z}^{\text{bot}}\right]_{e,k+1}^\text{top} \right\} \\
+& = - \left(\nabla \Phi \right)_{e,k} - \frac{1}{\left[\tilde{h}_k\right]_e} \nabla \left( \tilde{h}_k \alpha_k p_k \right) - \frac{1}{\left[\tilde{h}_k\right]_e} \left\{ \left[ \alpha p \nabla \tilde{z}^{\text{top}}\right]_{e,k}^\text{top} - \left[ \alpha p \nabla \tilde{z}^{\text{top}}\right]_{e,k+1}^\text{top} \right\} \\
& - \frac{1}{\left[\tilde{h}_{i,k}\right]_e} \nabla \cdot \left( \tilde{h}_k \left< {\bf u}^\prime \otimes {\bf u}^\prime \right>_k \right) - \frac{\rho_0}{\left[\tilde{h}_{i,k}\right]_e^\text{top}} \left\{ \left[ \left<\mathbf{u}^\prime \tilde{w}_{tr}^\prime \right> - \left< \mathbf{u}^\prime \tilde{ u}^\prime \right> \right]_{e,k}^\text{top} - \left[ \left<\mathbf{u}^\prime \tilde{w}_{tr}^\prime \right> - \left< \mathbf{u}^\prime \tilde{ u}^\prime \right> \right]_{e,k+1}^\text{top} \right\}.
$$ (discrete-momentum)
@@ -859,7 +859,7 @@ $$ (weak-derivative)
If [](#weak-derivative) is averaged between $\tilde{z}_{k-1/2}^{\text{top}}$ and $\tilde{z}_{k+1/2}^{\text{top}}$, we arrive at the expected form of the gradient centered on layer interfaces
$$
-\left[\frac{\partial \varphi}{\partial z}\right]_{avg} = \frac{\left(\varphi_k - \varphi_{k+1}\right)}{0.5 \left(\tilde{h}_k + \tilde{h}_{k+1}\right)}
+\left[\frac{\partial \varphi}{\partial \tilde{z}}\right]_{avg} = \frac{\left(\varphi_k - \varphi_{k+1}\right)}{0.5 \left(\tilde{h}_k + \tilde{h}_{k+1}\right)}
$$ (weak-deriv-final)
where $0.5 \left(\tilde{h}_k + \tilde{h}_{k+1}\right)$ is the distance between $\tilde{z}_{k-1/2}^{\text{top}}$ and $\tilde{z}_{k+1/2}^{\text{top}}$. A similar derivation can be followed to compute gradients across layer centers. This will form discrete derivatives in Omega.
@@ -867,7 +867,7 @@ where $0.5 \left(\tilde{h}_k + \tilde{h}_{k+1}\right)$ is the distance between $
With this, we can now fully discretize [](#discrete-mom-vert-diff) as
$$
--\frac{\rho_0}{\left[\tilde{h}_k\right]_e} \left\{ \left[\left \right]_{e,k} - \left[\left \right]_{e,k+1} \right\} = -\frac{\rho_0}{\left[\tilde{h}_k\right]_e} \left\{ \frac{\left(u_{e,k-1} - u_{e,k}\right)}{0.5 \left(\tilde{h}_{k-1} + \tilde{h}_{k}\right)} - \frac{\left(u_{e,k} - u_{e,k+1}\right)}{0.5 \left(\tilde{h}_k + \tilde{h}_{k+1}\right)} \right\}.
+-\frac{1}{\left[\tilde{h}_k\right]_e} \left\{ \left[\left \right]_{e,k} - \left[\left \right]_{e,k+1} \right\} = -\frac{1}{\left[\tilde{h}_k\right]_e} \left\{ \frac{\left(u_{e,k-1} - u_{e,k}\right)}{0.5 \left(\tilde{h}_{k-1} + \tilde{h}_{k}\right)} - \frac{\left(u_{e,k} - u_{e,k+1}\right)}{0.5 \left(\tilde{h}_k + \tilde{h}_{k+1}\right)} \right\}.
$$ (final-vert-vel-dissipation)
This form can be interfaced with the Omega [tridiagonal solver](TridiagonalSolver.md) routine.
@@ -881,20 +881,20 @@ The discretized momentum and tracer forcing appear as the surface value of the v
The wind forcing is applied as a top boundary condition during implicit vertical mixing as
$$
-\frac{\tau_{e}}{[ \tilde{h}_{i,k}]_e}
+\frac{\tau_{e}}{\rho_0 [ \tilde{h}_{i,k}]_e}
$$
-where $\tau$ is the wind stress in Pa. Since the mass-thickness $\tilde{h}$ is in kg/s/m$^2$, this results in the desired units of m/s$^2$ for a momentum tendency term.
+where $\tau$ is the wind stress in [Pa]. Since the pseudo-thickness $\tilde{h}$ is in [m], this results in the desired units of [m s$^{-2}$] for a momentum tendency term.
#### Bottom Drag
Bottom Drag is applied as a bottom boundary condition during implicit vertical mixing as
$$
-- C_D \frac{u_{e,k}\left|u_{e,k}\right|}{[\alpha_{i,k}\tilde{h}_{i,k}]_e}.
+- C_D \frac{u_{e,k}\left|u_{e,k}\right|}{\rho_0[\alpha_{i,k}\tilde{h}_{i,k}]_e}.
$$ (discrete-mom-bottom)
-The units of specific volume times mass-thickness $\alpha h$ are length (m), so that the full term has units of m/s$^2$.
+The units of the term in the denominator is length [m], so that the full term has units of [m s$^{-2}$].
#### Rayleigh Drag
@@ -912,7 +912,7 @@ $$
\frac{LHF}{C_p \rho_1}
$$
-where $\rho_1$ is the density in the top layer of Omega. This gives units of mK/s.
+where $\rho_1$ is the density in the top layer of Omega. This gives units of [m K s$^-1$].
#### Freshwater forcing
@@ -920,7 +920,7 @@ Since Omega is a non-Boussinesq ocean, surface sources of water will be mass flu
### Horizontal Tracer Diffusion
-As with momentum dissipation, the horizontal tracer diffusion arises from the $\left<\mathbf{u}_k^\prime \varphi_k \right>$ and $\left< \tilde{u}^\prime \varphi^\prime \right>$. As in MPAS-Ocean, the former term can be parameterized either as Laplacian or Biharmonic diffusion,
+As with momentum dissipation, the horizontal tracer diffusion arises from the $\left<\mathbf{u}_k^\prime \varphi_{k}^\prime \right>$ and $\left< \tilde{u}^\prime \varphi^\prime \right>$. As in MPAS-Ocean, the former term can be parameterized either as Laplacian or Biharmonic diffusion,
$$
D^\varphi_{i,k} = \kappa_{2,e} \nabla^2 \varphi_{i,k} - \kappa_{4,e} \nabla^4 \varphi_{i,k}.
@@ -942,16 +942,16 @@ The biharmonic is a Laplacian operator applied twice,
$$
- \nabla \cdot \left( \kappa_{4,e} \nabla
-\right[
+\left[
\nabla \cdot \left( \tilde{h}_{i,k} \nabla \varphi_{i,k} \right)
-\left]
+\right]
\right).
$$ (discrete-tracer-del4)
Each of these operators are written as horizontal stencils in the {ref}`Omega V0 Operator Formulation Section <33-operator-formulation>`. Again we note that the variables in these equations are the layer average.
#### Horizontal tracer diffusion across a sloping surface
-As with horizontal momentum dissipation, there is a turbulent flux of tracer across a sloping $\tilde{z}$ interface. We interpret the $\left< \tilde{z}^\prime \varphi^\prime \right>$ as the projection of the horizontal turbulent flux across the sloping interface. The form of the diffusion is similar, taking Laplacian diffusion as an example
+As with horizontal momentum dissipation, there is a turbulent flux of tracer across a sloping $\tilde{z}$ interface. We interpret the $\left< \tilde{u}^\prime \varphi^\prime \right>$ as the projection of the horizontal turbulent flux across the sloping interface. The form of the diffusion is similar, taking Laplacian diffusion as an example
$$
\nabla \cdot \left( \tilde{h}_{i} \kappa_{2,e} \nabla \varphi_{i} \right)_k.
@@ -1020,51 +1020,54 @@ Table 1. Definition of variables. Geometric variables may be found in the {ref}`
| symbol | name | units | location | name in code | notes |
|---------------------|-----------------------------|----------|-|---------|-------------------------------------------------------|
-|$D_{i,k}$ | divergence | 1/s | cell | Divergence |$D=\nabla\cdot\bf u$ |
-|$f_v$ | Coriolis parameter| 1/s | vertex | FVertex | $f = 2\Omega sin(\phi)$, $\Omega$ rotation rate, $\phi$ latitude|
+|$D_{i,k}$ | divergence | s$^{-1}$ | cell | Divergence |$D=\nabla\cdot\bf u$ |
+|$f_v$ | Coriolis parameter| s$^{-1}$ | vertex | FVertex | $f = 2\Omega sin(\phi)$, $\Omega$ rotation rate, $\phi$ latitude|
|$f_{eos}$ | equation of state | - | any | function call | |
-|$g$ | gravitational acceleration | m/s$^2$ | constant | Gravity |
-|$\tilde{h}_{i,k}$ | pseudo-thickness | m | cell | PseudoThickness | |
+|$g$ | gravitational acceleration | m s$^{-2}$ | constant | Gravity |
+|$\tilde{h}_{i,k}$ | pseudo-thickness | m | cell | LayerThickness | $\tilde{h} = (\rho/\rho_0) h$ |
+|$h_{i,k}$ | geometric layer thickness | m | cell | GeometricThickness | |
|$k$ | vertical index | |
|${\bf k}$ | vertical unit vector | |
|$K_{min}$ | shallowest active layer | |
|$K_{max}$ | deepest active layer | |
-|$K_{i,k}$ | kinetic energy | m$^2$/s$^2$ | cell | KineticEnergyCell |$K = \left\| {\bf u} \right\|^2 / 2$ |
+|$K_{i,k}$ | kinetic energy | m$^2$ s$^{-2}$ | cell | KineticEnergyCell |$K = \left\| {\bf u} \right\|^2 / 2$ |
|$p_{i,k}$ | pressure | Pa | cell | Pressure | see [](discrete-pressure) |
|$p^{floor}_i$ | bottom pressure | Pa | cell | PFloor | pressure at ocean floor
|$p^{surf}_i$ | surface pressure | Pa | cell | PSurface | due to atm. pressure, sea ice, ice shelves
-|$q_{v,k}$ | potential vorticity | 1/m/s | vertex | PotentialVorticity |$q = \left(\zeta+f\right)/h$ |
-|$Ra$ | Rayleigh drag coefficient | 1/s | constant | | |
+|$q_{v,k}$ | potential vorticity | m$^{-1}$ s$^{-1}$ | vertex | PotentialVorticity |$q = \left(\zeta+f\right)/\tilde{h}$ |
+|$Ra$ | Rayleigh drag coefficient | s$^{-1}$ | constant | | |
|$S_{i,k}$ | salinity | PSU | cell | Salinity | a tracer $\varphi$ |
|$t$ | time | s | none | | |
-|${\bf u}_k$ | velocity, vector form | m/s | - | | |
-|$u_{e,k}$ | velocity, normal to edge | m/s | edge | NormalVelocity | |
-|$u^\perp_{e,k}$ | velocity, tangential to edge | m/s | edge | TangentialVelocity |${\bf u}^\perp = {\bf k} \times {\bf u}$|
-|$\alpha_{i,k}$ | specific volume | m$^3$/kg | cell | SpecificVolume | $v = 1/\rho$ |
-|$\tilde{w}_{i,k}$ | vertical velocity across a pseudo height surface | m/s | cell | VerticalVelocity | volume transport per m$^2$ |
-|$\tilde{u}_{i,k}$ | projection of normal velocity across a pseudo height surface | m/s | cell | | |
-|$\tilde{W}_{i,k}$ | total velocity across a pseudo height surface $\tilde{W}_{i,k} \equiv $\tilde{w}_{i,k} - \tilde{u}_{i,k}$ | m/s | cell | | |
-|$\tilde{z}$ | vertical coordinate | m | - | | positive upward |
-|$\tilde{z}^{top}_{i,k}$ | layer top z-location | m | cell | ZTop | see [](discrete-z) |
-|$\tilde{z}^{mid}_{i,k}$ | layer mid-depth z-location | m | cell | ZMid |
-|$\tilde{z}^{surf}_{i}$ | ocean surface, i.e. sea surface height | m | cell | ZSurface | same as SSH in MPAS-Ocean |
-|$\tilde{z}^{floor}_{i}$ | ocean floor z-location | m | cell | ZFloor | -bottomDepth from MPAS-Ocean |
-|$\zeta_{v,k}$ | relative vorticity| 1/s | vertex | RelativeVorticity |$\zeta={\bf k} \cdot \left( \nabla \times {\bf u}\right)$ |
-|$\zeta_a$ | absolute vorticity ($\zeta + f$) | 1/s | vertex | |
+|${\bf u}_k$ | velocity, vector form | m s$^{-1}$ | - | | |
+|$u_{e,k}$ | velocity, normal to edge | m s$^{-1}$ | edge | NormalVelocity | |
+|$u^\perp_{e,k}$ | velocity, tangential to edge | m s$^{-1}$ | edge | TangentialVelocity |${\bf u}^\perp = {\bf k} \times {\bf u}$|
+|$\alpha_{i,k}$ | specific volume | m$^3$ kg$^{-1}$ | cell | SpecificVolume | $v = 1/\rho$ |
+|$\tilde{u}_{i,k}$ | projection of normal velocity across a pseudo height surface | m s$^{-1}$ | cell | | |
+|$\tilde{w}_{i,k}$ | vertical velocity across a pseudo height surface | m s$^{-1}$ | cell | VerticalVelocity | volume transport per m$^2$ |
+|$\tilde{w}_{tr\ i,k}$ | net vertical transport through a moving surface | m s$^{-1}$ | cell | NetVertTransportVelocity | volume transport per m$^2$ |
+|$\tilde{W}_{tr\ i,k}$ | total vertical velocity across a pseudo height surface | m s$^{-1}$ | cell | TotalVertTransportVelocity | $\tilde{W}_{tr} \equiv \tilde{w}_{tr} - \tilde{u}$ |
+|$\tilde{z}$ | vertical coordinate (pseudo-height) | m | - | | positive upward |
+|$\tilde{z}^{top}_{i,k}$ | layer top $\tilde{z}$-location | m | cell | ZTop | see [](discrete-z) |
+|$\tilde{z}^{mid}_{i,k}$ | layer mid-depth $\tilde{z}$-location | m | cell | ZMid |
+|$\tilde{z}^{surf}_{i}$ | ocean surface $\tilde{z}$-location | m | cell | ZSurface |
+|$\tilde{z}^{floor}_{i}$ | ocean floor $\tilde{z}$-location | m | cell | ZFloor |
+|${z}^{floor}_{i}$ | ocean floor geometric z-location | m | cell | GeometricZFloor | -bottomDepth from MPAS-Ocean |
+|$\zeta_{v,k}$ | relative vorticity| s$^{-1}$ | vertex | RelativeVorticity |$\zeta={\bf k} \cdot \left( \nabla \times {\bf u}\right)$ |
+|$\zeta_a$ | absolute vorticity ($\zeta + f$) | s$^{-1}$ | vertex | |
|$\Theta_{i,k}$ | conservative temperature | C | cell | Temperature | a tracer $\varphi$ |
-|$\kappa_2$| tracer diffusion | m$^2$/s | cell | | |
-|$\kappa_4$| biharmonic tracer diffusion | m$^4$/s | cell | | |
-|$\kappa_v$| vertical tracer diffusion | m$^2$/s | cell | | |
+|$\kappa_2$| tracer diffusion | m$^2$ s$^{-1}$ | cell | | |
+|$\kappa_4$| biharmonic tracer diffusion | m$^4$ s$^{-1}$ | cell | | |
+|$\kappa_v$| vertical tracer diffusion | m$^2$ s$^{-1}$ | cell | | |
|meshScaling | variable that holds the scaling factor for biharmonic and laplacian mixing | unitless | edge | | |
-|$\nu_{2,e}$ | horizontal del2 viscosity scaled by mesh resolution | m$^2$/s | edge | | |
-|$\nu_{4,e}$ | horizontal biharmonic (del4) viscosity scaled by mesh resolution | m$^4$/s | edge | | |
-|$\nu_v$| vertical momentum diffusion | m$^2$/s | edge | | |
-|$\varphi_{i,k}$ | tracer | kg/m$^3$ or similar | cell | | e.g. $\Theta$, $S$ |
-|$\rho_{i,k}$ | density | kg/m$^3$ | cell | Density |
-|$\rho_0$ | Reference density | kg/m$^3$ | | constant |
-|$\tau_i$ | wind stress | Pa=N/m$^2$ | edge | SurfaceStress |
-|$\Phi_{i,k}$ | geopotential| | cell | Geopotential |$\partial \Phi / \partial z = g$ for gravity |
-|$\omega$ | mass transport | kg/s/m^2 | cell | VerticalTransport |$\omega=\rho_0 w$|
+|$\nu_{2,e}$ | horizontal del2 viscosity scaled by mesh resolution | m$^2$ s$^{-1}$ | edge | | |
+|$\nu_{4,e}$ | horizontal biharmonic (del4) viscosity scaled by mesh resolution | m$^4$ s$^{-1}$ | edge | | |
+|$\nu_v$| vertical momentum diffusion | m$^2$ s$^{-1}$ | edge | | |
+|$\varphi_{i,k}$ | tracer | kg m$^{-3}$ or similar | cell | | e.g. $\Theta$, $S$ |
+|$\rho_{i,k}$ | density | kg m$^{-3}$ | cell | Density |
+|$\rho_0$ | Reference density | kg m$^{-3}$ | | constant |
+|$\tau_i$ | wind stress | Pa=N m$^{-2}$ | edge | SurfaceStress |
+|$\Phi_{i,k}$ | geopotential| m$^2$ s$^{-2}$ | cell | Geopotential |$\partial \Phi / \partial z = g$ for gravity |
+|$\omega$ | mass transport | kg s$^{-1}$ m$^{-2}$ | cell | VerticalTransport |$\omega=\rho_0 w$|
## 13. Verification and Testing