Docs

Docs/sphinx_doc/Numerical_Solution_Technique.rst

@@ -99,9 +99,9 @@ Grid Variables
 +-------------------------------------------------+-----------------------------+------------------------------------------------------------------+-------------------+-------------------+
 | :math:`h`                                       | ``vec_hOfTheConfusingName`` | bathymetry (always positive)                                     |                   |                   |
 +-------------------------------------------------+-----------------------------+------------------------------------------------------------------+-------------------+-------------------+
-| :math:`z_{r\left(i,j,k\right)}`                 |                             | depth (negative below sea level)                                 | center            |                   |
+| :math:`z_{r}`                                   |                             | depth (negative below sea level)                                 | center            |                   |
 +-------------------------------------------------+-----------------------------+------------------------------------------------------------------+-------------------+-------------------+
-| :math:`z_{w\left(i,j,k\right)}`                 |                             | depth (negative below sea level)                                 | bottom/top faces  |                   |
+| :math:`z_{w}`                                   |                             | depth (negative below sea level)                                 | bottom/top faces  |                   |
 +-------------------------------------------------+-----------------------------+------------------------------------------------------------------+-------------------+-------------------+
 | :math:`T`                                       | ``vec_t3``                  | temperature                                                      |                   |                   |
 +-------------------------------------------------+-----------------------------+------------------------------------------------------------------+-------------------+-------------------+
@@ -334,6 +334,18 @@ where we have introduced the advective fluxes:
    F^{\eta}&=\frac{H_zvC}{m}\\
    F^{\sigma}&=\frac{H_z\Omega C}{mn}.
 
+Third-order upwinding is available for both momentum and tracer advection. The second-order centered scheme is available for momentum advection. Fourth-order centered advection is available for tracer advection.
+
+Second-order Centered
+~~~~~~~~~~~~~~~~~~~~~
+In the second-order centered scheme, fluxes are calculated as:
+
+.. math::
+   F^{\xi}&=\frac{\overline{H_z}^{\xi}u\overline{C}^{\xi}}{\overline{n}^{\xi}}\\
+   F^{\eta}&=\frac{\overline{H_z}^{\eta}v\overline{C}^{\eta}}{\overline{m}^{\eta}}\\
+   F^{\sigma}&=\frac{\overline{H_z}^{\sigma} \Omega \overline{C}^{\sigma}}{mn}.
+
+
 Fourth-order Centered
 ~~~~~~~~~~~~~~~~~~~~~
 The barotropic advection is centered fourth-order. Create gradient terms:
@@ -407,6 +419,9 @@ The integral is actually computed as a sum from the bottom upwards and also as a
 
 Equation of State
 -----------------
-The density is obtained from temperature and salinity via an equation of state. REMORA provides a choice of a nonlinear equation of state :math:`\rho=\rho\left(T,S,z\right)` or a linear equation of state :math:`\rho=\rho\left(T\right)`. The nonlinear equation of state has been modified and now corresponds to the UNESCO equation of state as derived by ``Jackett and McDougall (1995)``. It computes `in situ` density as a function of potential temperature, salinity and pressure.
+The density is obtained from temperature :math:`\left(T\right)` and salinity :math:`\left(S\right)` via a linear equation of state:
+
+.. math::
+   \rho\left(T,S\right) = R_0 - R_0 T_{\mathrm{coef}} (T - T_0) + R_0 S_{\mathrm{coef}} (S-S_0).
 
-Warning: although we have used it quite extensively in the past, McDougall (personal communication) claims that the single-variable :math:`\left(\rho=\rho\left(T\right)\right)` equation of state is not dynamically appropriate as is. He has worked out the extra source and sink terms required, arising from vertical motions and the compressibility of water. They are quite complicated and we have not implemented them to see if they alter the flow.
+The constants :math:`R_0`, :math:`T_0`, :math:`S_0`, :math:`T_{\mathrm{coef}}`, and :math:`S_{\mathrm{coef}}` are specified in the :ref:`inputs`<list-of-parameters-15>` file.

Docs/sphinx_doc/Plotfiles.rst

@@ -34,7 +34,7 @@ List of Parameters
 |                                | or one file for  |                       |            |
 |                                | all timesteps?   |                       |            |
 +--------------------------------+------------------+-----------------------+------------+
-| **remora.plot_file**           | prefix for       | String                | “*plt”     |
+| **remora.plot_file**           | prefix for       | String                | “plt”      |
 |                                | plotfiles        |                       |            |
 +--------------------------------+------------------+-----------------------+------------+
 | **remora.plot_int**            | how often (by    | Integer               | -1         |

Docs/sphinx_doc/Verification.rst

@@ -45,9 +45,9 @@ The basic version of this test simulates a flow with a depth-dependent horizonta
 Ideal Mini Grid
 ---------------
 
-This small idealized grid is used to test netCDF-provided initial and boundary conditions. The ocean is initializaed with zero velocity and a constant temperature and salinity. Time-varying boundary conditions are then applied for velocity, temperature, or salinity (provided by netCDF file). The default is to used a clamped boundary condition for all quantities, but options for Chapman-Flather and radiation conditions are available. This test also verifies correct behavior with land-sea masking when using the ``_masked`` grid file.
+This small idealized grid is used to test netCDF-provided initial and boundary conditions. The ocean is initialized with zero velocity and a constant temperature and salinity. Time-varying boundary conditions are then applied for velocity, temperature, or salinity (provided by netCDF file). The default is to used a clamped boundary condition for all quantities, but options for Chapman-Flather and radiation conditions are available. This test also verifies correct behavior with land-sea masking when using the ``_masked`` grid file.
 
-The netCDF files needed to run these tests can be found in the `remora-data`<https://github.com/seahorce-scidac/remora-data` repository under the ``IdealMiniGrid`` directory.
+The netCDF files needed to run these tests can be found in the `remora-data <https://github.com/seahorce-scidac/remora-data>`_ repository under the ``IdealMiniGrid`` directory.
 
 .. _particlesseamount:
 
@@ -61,12 +61,12 @@ This problem tests advection of tracer particles on a flat domain.
 Seamount
 --------
 
-The `Seamount<https://www.myroms.org/wiki/SEAMOUNT_CASE>` problem involves an (analytically) stably stratified fluid at rest over a seamount. In the absence of numerical errors, the fluid will remain at rest. However, this may not occur due to numerical errors in the calculation of the horizontal pressure gradient when the vertical coordinates are misaligned with the geopotential surfaces, as is the case in problems with spatially-varying bathymetry in ROMS/REMORA.
+The `Seamount <https://www.myroms.org/wiki/SEAMOUNT_CASE>`_ problem involves an (analytically) stably stratified fluid at rest over a seamount. In the absence of numerical errors, the fluid will remain at rest. However, this may not occur due to numerical errors in the calculation of the horizontal pressure gradient when the vertical coordinates are misaligned with the geopotential surfaces, as is the case in problems with spatially-varying bathymetry in ROMS/REMORA.
 
 
 .. _upwelling:
 
 Upwelling
 ---------
 
-The `Upwelling<https://www.myroms.org/wiki/UPWELLING_CASE>` demonstrates wind-driven upwelling over a perioidc channel. It closely matches the test problem by the same name in ROMS.
+The `Upwelling <https://www.myroms.org/wiki/UPWELLING_CASE>`_ demonstrates wind-driven upwelling over a perioidc channel. It closely matches the test problem by the same name in ROMS.