Esmg docs

docs/discrete_space.rst

@@ -13,7 +13,6 @@ algorithm.
     :maxdepth: 2
 
     api/generated/pages/Discrete_Grids
-    api/generated/pages/Finite_Difference_Operators
     api/generated/pages/PPM
     api/generated/pages/Discrete_Coriolis
     api/generated/pages/Discrete_PG

docs/parameterizations_vertical.rst

@@ -26,6 +26,8 @@ Kappa-shear
 Internal-tide driven mixing
   The schemes of :cite:`st_laurent2002`, :cite:`polzin2009`, and :cite:`melet2012`, are all implemented through MOM_set_diffusivity and MOM_diabatic_driver.
 
+   :ref:`Internal_Tidal_Mixing`
+
 Vertical friction
 -----------------
 

docs/tracers.rst

@@ -9,4 +9,5 @@ Tracers in MOM6
     api/generated/pages/Horizontal_Diffusion.rst
     api/generated/pages/Vertical_Diffusion.rst
     api/generated/pages/Passive_Tracers
+    api/generated/pages/Frazil_Ice
 

docs/zotero.bib

@@ -847,6 +847,16 @@ @article{melet2012
 	pages = {602--615}
 }
 
+@article{simmons2004,
+	title = {Tidally driven mixing in a numerical model of the ocean general circulation},
+	volume = {6},
+	author = {Simmons, H. L. and S. R. Jayne and L. C. St. Laurent and A. J. Weaver},
+	year = {2004},
+	journal = {Ocean Modell.},
+	pages = {245--263},
+	doi = {10.1016/S1463-5003(03)00011-8}
+}
+
 @article{polzin2009,
 	title = {An abyssal recipe},
 	volume = {30},
@@ -863,6 +873,15 @@ @article{polzin2009
 	pages = {298--309}
 }
 
+@article{polzin2004,
+	title = {Idealized solutions for the energy balance of the finescale internal wave field},
+	volume = {34},
+	journal = {J. Phys. Oceanogr.},
+	author = {Polzin, Kurt L.},
+	year = {2004},
+	pages = {231--246}
+}
+
 @article{white2009,
 	title = {High-order regridding-remapping schemes for continuous isopycnal and generalized coordinates in ocean models},
 	volume = {228},
@@ -2524,3 +2543,13 @@ @article{hallberg2005
 	volume = {8},
 	doi = {10.1016/j.ocemod.2004.01.001}
 }
+
+@article{bell1975,
+	author = {T. H. Bell},
+	year = {1975},
+	title = {Lee wavews in stratified flows with simple harmonic time dependence"},
+	journal = {J. Fluid Mech.},
+	volume = {67},
+	pages = {705--722}
+}
+

src/core/_Sea_ice.dox

@@ -1,5 +1,11 @@
 /*! \page Sea_Ice Sea Ice Considerations
 
-\section Frazil Ice Formation
+\section section_seaice Sea Ice Considerations
+
+For realistic domains, it is assumed that MOM6 will be run in a coupled mode, such that either the
+sea-ice model or the coupler will be computing atmospheric bulk fluxes and passing them to the ocean.
+Likewise, MOM6 can compute the frazil ice formation as described in \ref section_frazil, which it
+then passes to the sea-ice model, expecting to get back the rejected brine or melted fresh water in
+return.
 
 */

src/equation_of_state/_Equation_of_State.dox

@@ -36,7 +36,7 @@ Compute the required quantities using the equation of state from \cite jackett19
 Compute the required quantities using the equation of state from
 [TEOS-10](http://www.teos-10.org/).
 
-\section TFREEZE Freezing Temperature of Sea Water
+\section section_TFREEZE Freezing Temperature of Sea Water
 
 There are three choices for computing the freezing point of sea water:
 

src/parameterizations/lateral/MOM_internal_tides.F90

@@ -2313,7 +2313,7 @@ subroutine internal_tides_init(Time, G, GV, US, param_file, diag, CS)
                "The default is 2pi/10 km, as in St.Laurent et al. 2002.", &
                units="m-1", default=8.e-4*atan(1.0), scale=US%L_to_m)
   call get_param(param_file, mdl, "KAPPA_H2_FACTOR", kappa_h2_factor, &
-               "A scaling factor for the roughness amplitude with n"//&
+               "A scaling factor for the roughness amplitude with "//&
                "INT_TIDE_DISSIPATION.",  units="nondim", default=1.0)
 
   ! Allocate various arrays needed for loss rates

src/parameterizations/vertical/MOM_diabatic_aux.F90

@@ -45,7 +45,7 @@ module MOM_diabatic_aux
   real    :: rivermix_depth = 0.0  !< The depth to which rivers are mixed if do_rivermix = T [Z ~> m].
   logical :: reclaim_frazil  !<   If true, try to use any frazil heat deficit to
                              !! to cool the topmost layer down to the freezing
-                             !! point.  The default is false.
+                             !! point.  The default is true.
   logical :: pressure_dependent_frazil  !< If true, use a pressure dependent
                              !! freezing temperature when making frazil.  The
                              !! default is false, which will be faster but is

src/parameterizations/vertical/_Frazil.dox

@@ -0,0 +1,33 @@
+/*! \page Frazil_Ice Frazil Ice Formation
+
+\section section_frazil Frazil Ice Formation
+
+Frazil ice forms in the model when the in situ temperature drops below
+the local freezing point, taking into account the in situ salinity and
+pressure. Starting at the bottom and working up through the water column,
+if the water is below freezing, set it to freezing and add the heat
+required to the heat deficit. If the water above is warmer than freezing,
+use that heat to take away the heat deficit and to cool the water. If
+you get all the way to the surface with a heat deficit, that quantity
+is passed to the ice model as a heat flux it will need to provide to
+the ocean.
+
+The local freezing point code is provided by the equation of state being
+used by MOM6. See \ref section_TFREEZE for the MOM6 options.
+
+The salinity is adjusted only at the surface when frazil ice is
+formed. This happens when the ice model creates ice with the heat deficit,
+taking salt out of the surface waters. We inherit this behavior from
+older versions of MOM, but the effect of not adjusting the in situ
+salinity is thought to be small.
+
+Note that versions simply whisking all the heat deficit to the surface
+without checking for warm water above tended to produce rapidly-melting
+ice floes in warm waters. This was deemed unphysical and was corrected.
+
+A similar process that we are also omitting is the formation of salt
+crystals when the salinity becomes too high. The salt crystals should
+form and sink, leaving a layer on the bed that will be diluted when the
+salinity drops again. This process can be seen in a lake in Death Valley.
+
+*/

src/parameterizations/vertical/_Internal_tides.dox

@@ -0,0 +1,216 @@
+/*! \page Internal_Tidal_Mixing Internal Tidal Mixing
+
+Two parameterizations of vertical mixing due to internal tides are
+available with the option INT_TIDE_DISSIPATION. The first is that of
+\cite st_laurent2002 while the second is that of \cite polzin2009. Choose
+between them with the INT_TIDE_PROFILE option. There are other relevant
+paramters which can be seen in MOM_parameter_doc.all once the main tidal
+dissipation switch is turned on.
+
+\section section_st_laurent St Laurent et al.
+
+The estimated turbulent dissipation rate of
+internal tide energy \f$\epsilon\f$ is:
+
+\f[
+   \epsilon = \frac{q E(x,y)}{\rho} F(z).
+\f]
+
+where \f$\rho\f$ is the reference density of seawater, \f$E(x,y)\f$ is
+the energy flux per unit area transferred from barotropic to baroclinic
+tides, \f$q\f$ is the fraction of the internal-tide energy dissipated
+locally, and \f$F(z)\f$ is the vertical structure of the dissipation.
+This \f$q\f$ is estimated to be roughly 0.3 based on observations. The
+term \f$E(x,y)\f$ is given by \cite st_laurent2002 as:
+
+\f[
+   E(x,y) \simeq \frac{1}{2} \rho N_b \kappa h^2 \langle U^2 \rangle
+\f]
+
+where \f$\rho\f$ is the reference density of seawater, \f$N_b\f$ is
+the buoyancy frequency along the seafloor, and \f$(\kappa, h)\f$ are
+the wavenumber and amplitude scales for the topographic roughness, and
+\f$\langle U^2 \rangle\f$ is the barotropic tide variance. It is assumed
+that the model will read in topographic roughness squared \f$h^2\f$
+from a file (the variable must be named "h2").
+
+To convert from energy dissipation to vertical diffusion \f$K_d\f$,
+the simple estimate is:
+
+\f[
+   K_d \approx \frac{\Gamma q E(x,y) F(z)}{\rho N^2}
+\f]
+
+where \f$\Gamma\f$ is the mixing efficiency, generally set to 0.2
+and \f$F(z)\f$ is a vertical structure function with exponential decay
+from the bottom:
+
+\f[
+   F(z) = \frac{e^{-(H+z)/\zeta}}{\zeta (1 - e^{H/\zeta}}.
+\f]
+
+Here, \f$\zeta\f$ is a vertical decay scale with a default of 500 meters.
+One change in MOM6 from the St. Laurent scheme is to use this form of \f$\Gamma\f$:
+
+\f[
+   \Gamma = 0.2 \frac{N^2}{N^2 + \Omega^2}
+\f]
+
+instead of \f$\Gamma = 0.2\f$, where \f$\Omega\f$ is the angular velocity
+of the Earth. This allows the buoyancy fluxes to tend to zero in regions
+of very weak stratification, allowing a no-flux bottom boundary condition
+to be satisfied.
+
+This \f$K_d\f$ gets added to the diffusivity due to the background and
+other contributions unless you set BBL_MIXING_AS_MAX to True, in which
+case the maximum of all the contributions is used.
+
+\section section_polzin Polzin
+
+The vertical diffusion profile of \cite polzin2009 is a WKB-stretched
+algebraic decay profile. It is based on a radiation balance equation,
+which links the dissipation profile associtated with internal breaking to
+the finescale internal wave shear producing that dissipation. The vertical
+profile of internal-tide driven energy dissipation can then vary in time
+and space, and evolve in a changing climate (\cite melet2012). \cite melet2012
+describes how the Polzin scheme is implemented in MOM6,
+copied here.
+
+The parameterization of \cite polzin2009 links the energy dissipation
+profile to the finescale internal wave shear producing that
+dissipation, using an idealized vertical wavenumber energy spectrum
+to identify analytic solutions to a radiation balance equation
+(\cite polzin2004). These solutions yield a dissipation profile
+\f$\epsilon(z)\f$:
+
+\f[
+   \epsilon = \frac{\epsilon_0}{[1 + (z/z_p)]^2},
+\f]
+
+where the magnitude \f$\epsilon_0\f$ and scale height \f$z_p\f$ can be expressed in terms of the
+spectral amplitude and bandwidth of the idealized vertical wavenumber energy spectrum in uniform
+stratification (\cite polzin2009).
+
+To take into account the nonuniform stratification, \cite polzin2009 applied a buoyancy scaling
+using the Wentzel-Kramers-Brillouin (WKB) approximation. As a result, the vertical wavenumber of a
+wave packet varies in proportion to the buoyancy frequency \f$N\f$, which in turn implies an
+additional transport of energy to smaller scales, and thus a possible enhanced mixing in regions of
+strong stratification. Such effects can be described by buoyancy scaling the vertical coordinate
+\f$z\f$ as
+
+\f[
+   z^{\ast}(z) = \int_{0}^{z} \left[ \frac{N^2 (z^\prime )}{N_b^2} \right] dz^{\prime} ,
+\f]
+
+with \f$z^\prime\f$ being positive upward relative to the bottom of the ocean. The turbulent
+dissipation rate then becomes
+
+\f[
+   \epsilon = \frac{\epsilon_0}{[1 + (z^{\ast} /z_p)]^2} \frac{N^2(z)}{N_b^2} .
+\f]
+
+The spectral amplitude and bandwidth of the idealized vertical wavenumber
+energy spectrum are identified after WKB scaling using a quasi-linear
+spectral model of internal-tide generation that incorporates horizontal
+advection of the barotropic tide into the momentum equation (\cite bell1975).
+As a result, Polzin's formulation leads to an expression for
+the spatially and temporally varying dissipation of internal tide energy
+at the bottom \f$\epsilon_0\f$, and the vertical scale of decay for the
+dissipation of internal tide energy \f$z_p\f$.
+
+\subsection subsection_energy_conserving Energy-conserving form
+
+To satisfy energy conservation (the integral of the vertical structure for the turbulent dissipation
+over depth should be unity), the dissipation is rewritten as
+
+\f[
+   \epsilon = \frac{\epsilon_0 z_p}{1 + (z^\ast/z_p)]^2} \frac{N^2(z)}{N^2_b} \left[
+   \frac{1}{z^{\ast(z=H)}} + \frac{1}{z_p} \right] .
+\f]
+
+In the MOM6 implementation, we use the \cite st_laurent2002 template for the vertical flux of energy
+at the ocean floor, so that in both formulations:
+
+\f[
+   \int_{0}^{H} \epsilon (z) dz = \frac{qE}{\rho} .
+\f]
+
+Whereas \cite polzin2009 assumed tthat the total dissipation was locally in balance with the
+barotropic to baroclinic energy conversion rate \f$(q=1)\f$, here we use the \cite simmons2004 value
+of \f$q=1/3\f$ to retain as much consistency as passible between both parameterizations.
+
+\subsection subsection_vertical_decay_scale Vertical decay-scale reformulation
+
+We follow the \cite polzin2009 prescription for the vertical scale of
+decay for the dissipation of internal-tide energy. However, we assume
+that the topographic power law, denoted as \f$\nu\f$ in \cite polzin2009,
+is equal to 1 (instead of 0.9) and we reformulated the expression of
+\f$z_p\f$ to put it in a more readable form:
+
+\f[
+   z_p = \frac{z_p^\mbox{ref} (\kappa^\mbox{ref})^2 (h^\mbox{ref})^2 (N_b^\mbox{ref})^3} {U^\mbox{ref}}
+   \frac{U}{h^2 \kappa^2 N_b^3} .
+\f]
+
+The superscript ref refers to reference values of the various parameters, as given by
+observations from the Brazil basin. Therefore, the above can be rewritten as
+
+\f[
+   z_p = \mu (N_b^\mbox{ref} )^2
+   \frac{U}{h^2 \kappa^2 N_b^3} .
+\f]
+
+where \f$\mu\f$ is a nondimensional constant \f$(\mu = 0.06970)\f$ and \f$N_b^\mbox{ref} = 9.6 \times
+10^{-4} s^{-1}\f$. Finally, a minimum decay scale of \f$z_p = 100 m\f$ is imposed in our
+implementation.
+
+\subsection subsection_reformulation_WKB Reformulation of the WKB scaling
+
+Since the dissipation is expressed as a function of the ratio \f$z^\ast / z_p\f$, a different WKB
+scaling can be used so long as we modify \f$z_p\f$ accordingly. In the implemented parameterization,
+we define the scaled height coordinate \f$z^\ast\f$ by
+
+\f[
+   z^\ast (z) = \frac{1}{\overline{N^2 (z)}^z} \int_{0}^{z} N^2(z^\prime ) dz ^\prime ,
+\f]
+
+with \f$z^\prime\f$ defined to be the height above the ocean bottom. By normalizing \f$N^2\f$ by its
+vertical mean \f$\overline{N^2}^z\f$, \f$z^\ast\f$ ranges from \f$0\f$ to \f$H\f$, the depth of the
+ocean.
+
+The WKB-scaled vertical decay scale for the Polzin formulation becomes
+
+\f[
+   z^\ast_p = \mu(N_b^\mbox{ref})^2 \frac{U}{h^2 \kappa^2 N_b \overline{N^2}^z} .
+\f]
+
+Unlike the \cite st_laurent2002 parameterization, the vertical decay scale now depends on physical
+variables and can evolve with a changing climate.
+
+Finally, the Polzin vertical profile of dissipation implemented in the model is given by
+
+\f[
+   \epsilon = \frac{qE(x,y)}{\rho [1 + (z^\ast/z_p^\ast)]^2} \frac{N^2(z)}{\overline{N^2}^z}
+   \left( \frac{1}{H} + \frac{1}{z_p^\ast} \right) .
+\f]
+
+In both parameterizations, turbulent diapycnal diffusivities are inferred from the dissipation
+\f$\epsilon\f$ by:
+
+\f[
+   K_d = \frac{\Gamma \epsilon}{N^2}
+\f]
+
+and using this form of \f$\Gamma\f$:
+
+\f[
+   \Gamma = 0.2 \frac{N^2}{N^2 + \Omega^2}
+\f]
+
+instead of \f$\Gamma = 0.2\f$, where \f$\Omega\f$ is the angular velocity
+of the Earth. This allows the buoyancy fluxes to tend to zero in regions
+of very weak stratification, allowing a no-flux bottom boundary condition
+to be satisfied.
+
+*/
+