Add Pressure Gradient design document

components/omega/doc/design/OmegaV1GoverningEqns.md

@@ -529,6 +529,7 @@ $$ (vh-momentum-reynolds1)
 
 Here we have also moved the Reynolds' average through the spatial integrals given the properties of the averaging.  Next we do a Reynolds' decomposition, this yields
 
+(omega-v1-vh-momentum-reynolds2)=
 $$
 \frac{d}{dt} \int_A \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \left< {\bf u} \right> \, d\tilde{z} \, dA
 & + \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}^{\text{top}}} \left( \left< {\bf u} \right> \otimes \left< {\bf u} \right> + \left< {\bf u}^\prime \otimes {\bf u}^\prime \right> \right) \, d\tilde{z} \right) \cdot {\bf n}_\perp \, dl \\
@@ -727,6 +728,7 @@ In the tracer equation, we note that surface fluxes (e.g. latent heat fluxes) wi
 
 Omega will only predict the layer average normal velocity, so we drop the bold face on the $u$ terms except for the product of primes, which is specified in the next section.
 
+(omega-v1-momentum-eq)=
 $$
 \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  \\

components/omega/doc/design/PGrad.md

@@ -0,0 +1,251 @@
+(omega-design-pressure-grad)=
+# Pressure Gradient
+
+
+## 1 Overview
+The pressure gradient will be responsible for computing the horizontal gradients of both the pressure and geopotential terms for the non-Boussinesq primitive equations implemented in Omega.
+In the non-Boussinesq model, the conserved quantity is mass rather than volume.
+In Omega the prognostic variable $\tilde{h}$ is a pseudo thickness, rather than geometric thickness as in a Boussinesq model.
+ Some non-Boussinesq models are written in pressure coordinates (e.g. [de Szoeke and Samelson 2002](https://journals.ametsoc.org/view/journals/phoc/32/7/1520-0485_2002_032_2194_tdbtba_2.0.co_2.xml).
+ However, Omega is written in general vertical coordinates and can reference either pressure $p$ or distance $z$ in the vertical.
+In a pure pressure coordinate the pressure gradient term disappears (since the pressure does not vary along lines of constant pressure), just as how the geopotential term disappears in a pure z coordinate model.
+However, similar to MPAS-Ocean's support for tilted height coordinates, Omega will allow for tilted pressure coordinates.
+This means that Omega will need to compute both the pressure and geopotential gradients.
+
+## 2 Requirements
+
+### 2.1 Requirement: Support for tilted pressure coordinates for the non-Boussinesq primitive equations
+
+The pressure gradient will compute the horizontal gradients of both the pressure and geopotential to support tilted pressure coordinates in the non-Boussinesq model.
+This will allow for the use of a $p^\star$ coordinate, which functions similarly to the $z^\star$ in the Boussinesq MPAS-Ocean model.
+Note that we use the name "$p^\star$" to refer to the vertical coordinate, in Omega it will be expressed in terms of the pseudo-height, $\tilde{z}$, as opposed to pressure directly.
+
+### 2.2 Requirement: Initial support for a simple centered pressure gradient
+
+For initial global cases without ice shelf cavities, the pressure and geopotential gradients will be computed with a simple centered difference approximation. In later versions of Omega, one or more  high-order pressure gradients will be implemented and will replace the centered approach in production runs.
+However, the centered pressure gradient will remain an option for use in idealized testing.
+
+### 2.3 Requirement: Flexibility to support a high-order pressure gradient
+The centered pressure gradient will be insufficient for future versions of Omega that include ice shelf cavities and high resolution shelf breaks.
+The pressure gradient framework should be flexible enough to support a high-order pressure gradient in the future.
+The high order pressure gradient will be similar to [Adcroft et al. 2008](https://doi.org/10.1016/j.ocemod.2008.02.001).
+
+### 2.4 Requirement: Flexibility to support tidal forcing and sea level change
+In later versions of Omega, the pressure gradient will need to be able to include tidal forcing in the geopotential term.
+These tidal forcings include both the tidal potential and the self attraction and loading terms.
+Additionally, other long-term changes to the geoid can be included in the geopotential.
+This requirement is satisfied by tidal forcing terms being included in the geopotential calculation in the `VertCoord` class.
+
+### 2.5 Desired: Pressure gradient for barotropic mode
+
+For split barotropic-baroclinic timestepping, the pressure gradient should provide the bottom pressure gradient tendency in the barotropic mode.
+The details of the barotropic pressure gradient will be added in a future design document for split time stepping.
+
+## 3 Algorithmic Formulation
+### 3.1 Centered Pressure Gradient
+In the layered non-Boussinesq  {ref}`momentum equation <omega-v1-momentum-eq>` solved in Omega, the pressure gradient tendency term for edge $e$ and layer $k$, $T^p_{e,k}$, includes the gradient of the geopotential, the gradient of a term involving pressure, and two terms evaluated at the cell interface:
+
+$$
+T^p_{e,k} = - \left(\nabla \Phi \right)_{e,k} - \frac{1}{\left[\tilde{h}_k\right]_e} \nabla \left( \tilde{h}_k \alpha_k p_k \right) + \frac{1}{\left[\tilde{h}_k\right]_e} \left\{ \left[ \alpha p \nabla \tilde{z}\right]_{e,k}^\text{top} -  \left[ \alpha p \nabla \tilde{z}\right]_{e,k+1}^\text{top} \right\}.
+$$
+
+The geopotential and interface terms are necessary to account for tilted layers that occur when using a general vertical coordinate, where the gradient operator is taken along layers.
+In this equation, $\alpha_{i,k}$ specific volume, $p_{i,k}$ is the pressure, and $\Phi_{i,k}$ is the geopotential.
+These three quantities are evaluated at the mid-point of layer $k$ of cell $i$ in the first two terms and at the cell interfaces in the third term along with the interface psudo-height, \tilde{z}.
+The discrete gradient operator at an edge is:
+
+$$
+ \nabla {(\cdot)} = \frac{1}{d_e} \sum_{i\in CE(e)} -n_{e,i} (\cdot)_i
+$$
+
+where $d_e$ is the distance between cell centers, $CE(e)$ are the cells on edge $e$, and $n_{e,i}$ is the sign of the edge normal with respect to cell $i$.
+The (cell-to-edge) horizontal averaging operator is:
+
+$$
+ [\cdot]_e = \frac{1}{2}\sum_{i\in CE(e)} (\cdot)_i
+$$
+
+Therefore, the centered pressure gradient will be calculated as:
+
+$$
+ T^p_{e,k} = \frac{1}{d_e}\sum_{i\in CE(e)} n_{e,i} \Phi_{i,k} + \frac{2}{d_e\sum_{i \in CE(e)}\tilde{h}_{i,k}}\left(\sum_{i \in CE(e)} n_{e,i} \tilde{h}_{i,k}\alpha_{i,k}p_{i,k}\right. \\ \left. - \frac{1}{2} \sum_{i\in CE(e)} \alpha_{i,k-1/2} p_{i,k-1/2}\sum_{i\in CE(e)} n_{e,i}\tilde{z}_{i,k-1/2} \right.\\ \left.+ \frac{1}{2} \sum_{i\in CE(e)} \alpha_{i,k+1/2} p_{i,k+1/2}\sum_{i\in CE(e)} n_{e,i}\tilde{z}_{i,k+1/2}\right),
+$$
+where $k-1/2$ and $k+1/2$ refer to the top and bottom of layer $k$, respectively.
+
+
+### 3.2 High-order Pressure Gradient
+The high order pressure gradient will be based on the {ref}`full volume integral form <omega-v1-vh-momentum-reynolds2>` of the geopotential and pressure terms:
+$$
+T^p &= - \int_A \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \, \left( \nabla \left<\Phi\right> \right) \, d\tilde{z} \, dA \\
+& - \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left(\left< \alpha \right> \left<p \right> + \left<\alpha^\prime p^\prime\right> \right) \, d\tilde{z} \right) dl \\
+& - \int_A \rho_0 \left[ \left< \alpha \right> \left<p \nabla \tilde{z}_k^{\text{top}} \right> + \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{top}}\right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
+& + \int_A \rho_0 \left[ \left< \alpha \right> \left<p \nabla \tilde{z}_k^{\text{bot}} \right> + \left<\alpha^\prime \left(p \nabla \tilde{z}_k^{\text{bot}}\right)^\prime\right> \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
+$$
+To obtain the expression that will be used, we neglect the turblent correlations and drop the Reynold's average notation for single variables:
+$$
+T^p &= - \int_A \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \, \left( \nabla \Phi \right) \, d\tilde{z} \, dA \\
+& - \int_{\partial A} \left( \int_{\tilde{z}_k^{\text{bot}}}^{\tilde{z}_k^{\text{top}}} \rho_0 \left(\alpha p \right) \, d\tilde{z} \right) dl \\
+& - \int_A \rho_0 \left[ \alpha \left<p \nabla \tilde{z}_k^{\text{top}} \right> \right]_{\tilde{z} = \tilde{z}_k^{\text{top}}} \, dA \\
+& + \int_A \rho_0 \left[ \alpha \left<p \nabla \tilde{z}_k^{\text{bot}} \right> \right]_{\tilde{z} = \tilde{z}_k^{\text{bot}}} \, dA.
+$$
+These volume and area integrals will be computed using quadrature to account for the variablity of $\alpha$ with the recontructed values of temperature, salinity, and pressure atthe quadrature points.
+The complete details for the high-order pressure gradient will be the subject of a future design document.
+
+%### 3.3 Barotropic Pressure Gradient
+%
+%When split baroclinic-barotropic time stepping is implemented in the future, the barotropic pressure gradient will be calculated by the pressure gradient class.
+%The barotropic pressure gradient is found by depth integrating the pressure gradient.
+%The pressure is
+%
+%$$
+%p(z) = p_b - g \int^z_{-h} \rho dz^\prime,
+%$$
+%
+%where $p_b$ is the bottom pressure.
+%The bottom pressure is the sum of the atmospheric surface pressure, $p_s$, and the pressure contribution of the water column:
+%
+%$$
+%p(z) &=  p_s + g\int_{-h}^\eta \rho dz - g \int^z_{-h} \rho dz^\prime, \\
+%     &=  p_s + g\rho_0\widetilde{H} - g \int^z_{-h} \rho dz^\prime,
+%$$
+%
+%where the total water column pseudo height is expressed by
+%
+%$$
+%\widetilde{H} = \int_{-h}^\eta \frac{\rho}{\rho_0} dz.
+%$$
+%
+%$\widetilde{H}$ is the prognositc variable in the barotropic continuity equation.
+%The vertical integral of the pressure gradient is
+%
+%$$
+%\frac{1}{\rho_0\widetilde{H}}\int^\eta_{-h} \nabla p dz &= \frac{1}{\rho_0\widetilde{H}}\int^\eta_{-h} \nabla \left( p_s + g\rho_0 \widetilde{H} - g \int^z_{-h} \rho dz^\prime \right) dz, \\
+%                           &= \frac{H}{\rho_0\widetilde{H}}\nabla p_s + \frac{gH}{\widetilde{H}}\nabla \widetilde{H} - \frac{g}{\rho_0\widetilde{H}} \int_{-h}^\eta \left( \nabla \int_{-h}^z \rho dz^\prime\right) dz,
+%$$
+%
+%where the height of the water column is represented by $H$.
+%The $1/\rho_0\widetilde{H}$ factor comes vertically integrating the material derivative and expressing the resulting barotropic momentum equation in non-conservative form.
+%
+%Therefore, the barotorpic pressure gradient term is discretized as:
+%
+%$$
+%\overline{T}_e^p = g\left[ \frac{H_i}{\widetilde{H}_i} \right]_e\sum_{i \in CE(e)} n_{e,i}\widetilde{H}_e
+%$$
+
+## 4 Design
+
+### 4.1 Data types and parameters
+
+The `PressureGradient` class will be used to perform the horizontal gradients of the pressure and geopotential
+```c++
+class PressureGrad{
+    public:
+    private:
+        std::unique_ptr<PressureGrad> OmegaPressureGrad;
+        PressureGradCentered CenteredPGrad;
+        PressureGradHighOrder HighOrderPGrad1; // To be implemented later
+        PressureGradHighOrder HighOrderPGrad2; // Multiple high order options are likely in the future
+        PressureGradType PressureGradChoice;
+        I4 NVertLevels;
+        I4 NChunks;
+        Array2DI4 CellsOnEdge;
+        Array1DReal DvEdge;
+        Array1DReal EdgeSignOnCell;
+}
+```
+The user will select a pressure gradient option at runtime in the input yaml file under the pressure gradient section:
+```yaml
+    PressureGrad:
+       PressureGradType: 'centered'
+```
+An `enum class` will be used to specify options for the pressure gradient used for an Omega simulation:
+```c++
+enum class PressureGradType{
+   Centered,
+   HighOrder1,
+   HighOrder2
+}
+```
+The functions to compute the centered and high order pressure gradient terms will be implemented as functors and the pressure gradient class will have private instances of these classes.
+### 4.2 Methods
+
+```c++
+class PressureGrad{
+    public:
+        static PressureGrad *init();
+        static PressureGrad *get();
+        void computePressureGrad();
+    private:
+
+}
+```
+#### 4.2.1 Creation
+
+The constructor will be responsible for storing any static mesh information as private variables and handling the selection of the user-specified pressure gradient option.
+```c++
+PressureGrad::PressureGrad(const HorzMesh *Mesh, const VertCoord *VCoord, Config *Options);
+```
+
+#### 4.2.2 Initialization
+
+The init method will create the default pressure gradient and return an error code:
+```c++
+int PressureGrad::init();
+```
+It calls the constructor to create a new pressure gradient instance, producing a static managed (unique) pointer to the single instance.
+
+#### 4.2.3 Retrieval
+
+There will be a method for getting a pointer to the pressure gradient instance:
+```c++
+PressureGrad *PressureGrad::get();
+```
+
+#### 4.2.4 Computation
+
+The public `computePressureGrad` method will rely on private methods for each specific pressure gradient option (centered and high order).
+Note that the functors called by `computePressureGrad` are responsible for computing the sum of the pressure gradient and geopotential gradient accumulated in the `Tend` output array.
+```c++
+void PressureGrad::computePressureGrad(const Array2DReal &Tend,
+                                       const Array2DReal &Pressure,
+                                       const Array2DReal &Geopotential,
+                                       const Array2DReal &SpecVol) {
+OMEGA_SCOPE(LocCenteredPGrad, CenteredPGrad)
+OMEGA_SCOPE(LocHighOrderPGrad, HighOrderPGrad)
+const Array1DI4 &MinLyrEdgeBot   = VCoord->MinLayerEdgeBot;
+const Array1DI4 &MaxLyrEdgeTop   = VCoord->MaxLayerEdgeTop;
+    if (PressureGradChoice == PressureGradType::Centered){
+       parallelForOuter(
+          {NEdgesAll}, KOKKOS_LAMBDA(int IEdge, const TeamMember &Team) {
+             const int NChunks = computeNChunks(MinLyrEdgeBottom, MaxLyrEdgeTop, IEdge);
+             parallelForInner(Team, NChunks, [=](const int KChunk) {
+                LocCenteredPGrad(Tend, IEdge, KChunk, Pressure, Geopotential, SpecVol);
+             });
+       });
+    }
+    else if (PressureGradChoice == PressureGradType::HighOrder){
+       parallelForOuter(
+          {NEdgesAll}, KOKKOS_LAMBDA(int IEdge, const TeamMember &Team) {
+             const int NChunks = computeNChunks(MinLyrEdgeBottom, MaxLyrEdgeTop, IEdge);
+             parallelForInner(Team, NChunks, [=](const int KChunk) {
+                LocHighOrderPGrad(Tend, IEdge, KChunk, Pressure, Geopotential, SpecVol);
+             });
+       });
+    }
+```
+
+#### 4.2.5 Destruction and removal
+
+No operations are needed in the destructor.
+The clear method will remove the pressure gradient instance, calling the destructor in the process.
+```c++
+void PressureGrad::clear();
+```
+
+## Verification and Testing
+
+### Test: Spatial convergence to exact solution
+For given analytical functions of $\alpha$, $h$, and $z$, the spatial convergence of the pressure gradient can be assessed by computing errors on progressively finer meshes.
+
+### Test: Baroclinic gyre
+The baroclinic gyre test case will test the pressure gradient term in the full non-Boussinesq equations.

components/omega/doc/index.md

@@ -108,6 +108,7 @@ design/HorzMeshClass
 design/Logging
 design/MachEnv
 design/Metadata
+design/PGrad
 design/IO
 design/IOStreams
 design/Reductions