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+
+Wetting and drying in coastal and ice-shelf settings
+====================================================
+
+date: 2022/02/16 
+
+Contributors: Carolyn Begeman, Darren Engwirda, Xylar Asay-Davis
+
+
+
+Summary
+-------
+
+"Wetting and drying" refers to the capability to have cells switch between
+wet and dry in time corresponding to changes in ocean extent. This
+implementation uses a "thin film" approach in which cells with water column
+thickness below a threshold are considered dry. Instabilities have been noted
+when MPAS-Ocean is run with shallow water columns, precluding stable wetting
+and drying. We aim to produce stable and accurate thin-film wetting and drying
+for both coastal and ice-shelf settings.
+
+
+
+Requirements
+------------
+
+Requirement: Monotonic thickness advection scheme
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+The thickness advection scheme needs to be monotonic to prevent layer
+thicknesses from becoming negative. 
+
+
+Requirement: Maximize energy conservation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+The advection scheme should still approximate energy conservation.
+
+
+Requirement: Maximize mass conservation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+The advection scheme should still approximate mass conservation.
+
+
+Requirement: Stability under existing time-stepping algorithms
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+The wetting and drying algorithm should be stable for both RK4 and split-
+explicit time stepping.
+
+
+Requirement: Avoid new constraints on the time step
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Wetting and drying must not significantly decrease the time step required for
+stability beyond constraint already present in MPAS-O.
+Ensure that it is impossible to completely empty a cell within one time step.
+
+
+Requirement: Avoid significant depreciation in performance
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Wetting and drying algorithm should be scalable and not increase run time by
+more than about 5%.
+
+
+Requirement: Allow baroclinic velocities in shallow, wet cells
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+The wetting and drying algorithm should be compatible with baroclinic velocities
+as cells become wet on the barotropic time step.
+
+
+Requirement: Allow tracer diffusion within the thin film
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Tracer diffusion should be permitted in the thin film to avoid large changes in
+tracer concentration during wetting. Other tracer tendencies will be set to
+zero.
+
+
+Requirement: Algorithm is appropriate for grounding line motion
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Stability and consistency of continuity and momentum equations is preserved in
+land ice settings where gradients in sea surface pressure (land ice pressure)
+are much larger than coastal settings.
+
+
+Requirement: Water is evacuated from isolated lakes when drying occurs
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Ideally, when isolated lakes occur after a cycle of wetting and drying, the
+water in those lakes is evacuated. It would also be best for mass conservation
+if that water were returned to the ocean. This is particularly desirable for
+grounding line migration.
+
+
+Algorithm Design (optional)
+---------------------------
+
+Algorithm Design: Maximize energy conservation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Energy conservation in the advection scheme is only guaranteed for centered
+thicknesses, not upwinded thicknesses. Thus, we only conduct upwinding of layer
+thicknesses in the shallow water regions where it is necessary to preserve
+stability, as determined by a water column thickness threshold. In order to
+transition gradually between upwinded and centered solutions, we implement a
+ramping function with water column thickness as an input.
+
+.. math::
+
+   \beta_{e} = 1 - \min\left(1,\, \max\left(0,\,
+   \frac{H_{\text{edge}}^{*}}{H_{\text{upwind}}} - 1\right)\right)
+
+where :math:`H_{\text{upwind}}` is the water column thickness at which only
+upwinding is used, with a ramp between upwinding and centered thicknesses from
+:math:`H_{\text{upwind}}` to :math:`2H_{\text{upwind}}`.
+:math:`H_{\text{edge}}^{*}` is the harmonic mean of the water column thickness
+of the adjacent cells:
+
+.. math::
+
+   H_{\text{edge}}^{*} = \frac{2\, H_{c_1} H_{c_2}}{H_{c_1} + H_{c_2}}
+
+Additionally, any enhancement of drag in cells with a thin water column needed
+for stability should not dissipate excessive amounts of energy (e.g., tidal
+energy).
+
+
+Algorithm design: Stability under existing time-stepping algorithms
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Stability of the wet/dry interface is achieved through a damping function on the
+volume fluxes and velocities out of a dry cell
+
+.. math::
+
+   \alpha_{e} = \min\left(1,\, \max\left(0,\,
+   \frac{H_{\text{edge}}^{*}}{H_{\text{thin}}} - 1\right)\right)
+
+An alternative to damping the fluxes and velocities at dry cells is removing
+terms from the momentum equation.
+
+
+Algorithm design: Allow baroclinic velocities in shallow, wet cells
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+We follow O'Dea et al. (2020) [NEMO] in scaling baroclinic fluxes and
+velocities by the mean of the factor \alpha over the barotropic substeps. Thus,
+baroclinic velocities are permitted as the cell transitions from wet to dry.
+
+
+Implementation
+--------------
+
+Implementation: Monotonic thickness advection scheme
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+First-order thickness upwinding is sufficient to produce monotonicity.
+The option to choose upwinded thickness advection rather than centered
+thickness advection is already implemented with flag 
+`config_thickness_flux_type`. 
+
+
+Implementation: Maximize energy conservation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+In addition to the option to have centered edge thicknesses and upwinded edge
+thicknesses, we add a third option which is either centered, upwinded, or a
+blend of the two as a function of water column thickness. This is 
+`config_thickness_flux_type = 'thickness-dependent'`.
+
+The ramping function is implemented in `ocn_diagnostic_solve_layerThickEdge`
+in order to weight both centered and upwinded layerThickEdge. The water column
+thickness input to the ramping function is the harmonic mean of water column
+thicknesses in the two cells adjacent to the edge at which layerThickEdge is
+computed.
+
+We need to ensure that the upwinded or hybrid layerThickEdge is appropriate
+everywhere it is used. In most terms it is appropriate as it is linked with the
+volumetric flux through edges.
+
+Is is appropriate to use upwinded layerThickEdge in the computation of the 
+horizontal gradient of density at the top of edges in mpas_ocn_gm.F_?
+
+.. _mpas_ocn_gm.F: https://github.com/E3SM-Project/E3SM/blob/460ef4af4b91d01213ea0d00290236c996d100f2/components/mpas-ocean/src/shared/mpas_ocn_gm.F#L455-L493
+
+
+Implementation: Stability under existing time-stepping algorithms
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Add additional options for damping thickness fluxes and velocities to the
+existing `ocn_wetting_drying` module. 
+
+Existing options for damping thickness fluxes and velocities:
+
+If `config_zero_drying_velocity` is true, every cell that would reach
+the minimum thin film thickness (plus an optional tolerance
+`config_drying_safety_height`) through the outgoing flux alone in that 
+timestep has `normalVelocity` and `normalTransportVelocity` set to zero. 
+
+If `config_zero_drying_velocity` is false, for only edges where the flux is
+outgoing:
+
+.. math::
+
+   u_{\text{Trans}} = u_{\text{Trans}} + u_{\text{Wet}}
+   u = u + u_{\text{Wet}}
+   u_{\text{Wet}} = - u * min(max(0,\\
+                                 (1 - \nabla \cdot F_{\text{out}})^2)\\
+                              1)
+   u_{\text{tend}} = u_{\text{tend}} * (1 - u_{\text{Wet}})
+   u = u * (1 - u_{\text{Wet}}) \text{(diagnostic update)}
+
+In addition, `normalVelocity` and `normalTransportVelocity` are set to zero
+when :math:`|u_{\text{Trans}} + u_{\text{Wet}}| < \varepsilon` where
+:math:`\varepsilon` is some small tolerance, designed to prevent spurious fluxes.
+
+Each of these updates are applied on each RK4 iteration.
+
+
+Implementation: Avoid new constraints on the time step
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+We hope that in most physical conditions the velocities are insufficiently
+high to evacuate all of the water from a cell in one barotropic timestep. One
+way to mitigate this possibility is to either increase the thin film thickness
+or to increase the width of the water column thickness transition between wet
+and dry dynamics (i.e., damping fluxes and velocity in cells close to the thin
+film threshold).
+
+
+Implementation: Water is evacuated from isolated lakes when drying occurs
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Evaluate when wet/dry interface makes a closed loop. Reduce water column
+thickness to the threshold thin film thickness. Set a flux equivalent to that
+evacuated volume using existing distributed river flux implementation.
+
+Testing
+-------
+
+Testing and Validation: Monotonic thickness advection scheme
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+No testing is needed as this is an existing capability.
+
+
+Testing and Validation: Maximize energy conservation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Ensure that the solution with `config_thickness_flux_type = 'thickness-dependent'`
+is identical to the solution with `config_thickness_flux_type = 'centered'`
+when all cells in the domain have water column thicknesses greater than the
+threshold at which ramping to upwinded thicknesses begins. This should be
+tested in the global ocean configuration with ecosystem tracers.
+
+
+Testing and Validation: Maximize mass conservation
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Compute the total mass evolution in the uniform bed slope case.
+
+
+Testing and Validation: Stability under existing time-stepping algorithms
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+We test both the uniformly sloping bed test case and the dam break test case
+with RK4 and split-explicit time-stepping schemes. 
+
+
+Testing and Validation: Avoid new constraints on the time step
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Verify that no cells drop below the minimum thickness in either uniform sloping
+bottom or dam break test cases.
+
+
+Testing and Validation: Avoid significant depreciation in performance
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Test scalability of wetting and drying through the uniform bed slope test
+cases. Test performance relative to non-wetting and drying cases by comparing
+the former test case to the same test case with wetting and drying disabled.
+
+
+Testing and Validation: Allow baroclinic velocities in shallow, wet cells
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+TBD
+
+
+Testing and Validation: Allow tracer diffusion within the thin film
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+Verify that tracers are diffused into the thin film by initializing the uniform
+bed slope simulation with a discontinuity in tracer concentrations across the
+wet/dry interface.
+
+
+Testing and Validation: Algorithm is appropriate for grounding line motion
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+TBD: need to develop new test case for ice shelf cavity. We could add a tidal
+signal at the open boundary of ISOMIP+ Ocean0.
+
+
+Testing and Validation: Water is evacuated from isolated lakes
+^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
+
+Date last modified: YYYY/MM/DD
+
+Contributors: (add your name to this list if it does not appear)
+
+The uniformly sloping bed test case can be modified to include a depression that
+would fill with water to ensure that the emptying algorithm functions as
+expected.