diff --git a/docs/images/background_varying.png b/docs/images/background_varying.png new file mode 100644 index 0000000000..44a65175a0 Binary files /dev/null and b/docs/images/background_varying.png differ diff --git a/docs/parameterizations_vertical.rst b/docs/parameterizations_vertical.rst index 0d22787294..c9404c5088 100644 --- a/docs/parameterizations_vertical.rst +++ b/docs/parameterizations_vertical.rst @@ -21,9 +21,12 @@ Interior and bottom-driven mixing --------------------------------- Kappa-shear - MOM_kappa_shear implement the shear-driven mixing of :cite:`jackson2008`. + MOM_kappa_shear implements the shear-driven mixing of :cite:`jackson2008`. + + :ref:`Internal_Shear_Mixing` 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` @@ -33,6 +36,8 @@ Vertical friction Vertical viscosity is implemented in MOM_vert_frict and coefficient computed in MOM_set_viscosity, although contributions to viscosity from other parameterizations are calculated in those respective modules (e.g. MOM_kappa_shear, MOM_KPP, MOM_energetic_PBL). + :ref:`Vertical_Viscosity` + Vertical diffusion ------------------ diff --git a/docs/zotero.bib b/docs/zotero.bib index 957097f217..a00fe569bd 100644 --- a/docs/zotero.bib +++ b/docs/zotero.bib @@ -655,6 +655,30 @@ @article{killworth1992 pages = {1379--1387} } +@article{killworth1999, + doi = {10.1175/1520-0485(1999)029<1221:atbblc>2.0.co;2}, + year = 1999, + publisher = {American Meteorological Society}, + volume = {29}, + number = {6}, + pages = {1221--1238}, + author = {P. D. Killworth and N. R. Edwards}, + title = {A Turbulent Bottom Boundary Layer Code for Use in Numerical Ocean Models}, + journal = {J. Phys. Oceanography} +} + +@article{zilitinkevich1996, + doi = {10.1007/bf02430334}, + year = 1996, + publisher = {Springer Science and Business Media {LLC}}, + volume = {81}, + number = {3-4}, + pages = {325--351}, + author = {S. Zilitinkevich and D. V. Mironov}, + title = {A multi-limit formulation for the equilibrium depth of a stably stratified boundary layer}, + journal = {Boundary-Layer Meteorology} +} + @article{gent1995, title = {Parameterizing {Eddy}-{Induced} {Tracer} {Transports} in {Ocean} {Circulation} {Models}}, volume = {25}, @@ -800,6 +824,18 @@ @article{jackson2008 pages = {1033--1053} } +@article{turner1986, + doi = {10.1017/s0022112086001222}, + year = 1986, + publisher = {Cambridge University Press ({CUP})}, + volume = {173}, + pages = {431--471}, + author = {J. S. Turner}, + title = {Turbulent entrainment: the development of the entrainment assumption, and its application to geophysical flows}, + journal = {J. Fluid Mech.} +} + + @article{reichl2018, title = {A simplified energetics based planetary boundary layer ({ePBL}) approach for ocean climate simulations.}, volume = {132}, @@ -1426,6 +1462,18 @@ @article{harrison2008 pages = {1894--1912} } +@article{danabasoglu2012, + doi = {10.1175/jcli-d-11-00091.1}, + year = 2012, + publisher = {American Meteorological Society}, + volume = {25}, + number = {5}, + pages = {1361--1389}, + author = {G. Danabasoglu and S. C. Bates and B. P. Briegleb and S. R. Jayne and M. Jochum and W. G. Large and S. Peacock and S. G. Yeager}, + title = {The {CCSM}4 Ocean Component}, + journal = {J. Climate} +} + @article{henyey1986, title = {Energy and action flow through the internal wave field: {An} eikonal approach}, volume = {91}, @@ -1761,6 +1809,18 @@ @article{large1994 pages = {363--403} } +@article{pacanowski1981, + doi = {10.1175/1520-0485(1981)011<1443:povmin>2.0.co;2}, + year = 1981, + publisher = {American Meteorological Society}, + volume = {11}, + number = {11}, + pages = {1443--1451}, + author = {R. C. Pacanowski and S. G. H. Philander}, + title = {Parameterization of Vertical Mixing in Numerical Models of Tropical Oceans}, + journal = {J. Phys. Oceanography} +} + @article{van_roekel2018, title = {The {KPP} {Boundary} {Layer} {Scheme} for the {Ocean}: {Revisiting} {Its} {Formulation} and {Benchmarking} {One}-{Dimensional} {Simulations} {Relative} to {LES}}, volume = {10}, @@ -2343,6 +2403,19 @@ @article{hallberg2000 pages = {1402--1419} } +@article{umlauf2005, + doi = {10.1016/j.csr.2004.08.004}, + year = 2005, + publisher = {Elsevier {BV}}, + volume = {25}, + number = {7-8}, + pages = {795--827}, + author = {L. Umlauf and H. Burchard}, + title = {Second-order turbulence closure models for geophysical boundary layers. A review of recent work}, + journal = {Continental Shelf Res.} +} + + @article{easter1993, title = {Two Modified Versions of Bott's Positive-Definite Numerical Advection Scheme}, @@ -2545,11 +2618,60 @@ @article{hallberg2005 } @article{bell1975, - author = {T. H. Bell}, - year = {1975}, - title = {Lee wavews in stratified flows with simple harmonic time dependence"}, - journal = {J. Fluid Mech.}, + doi = {10.1017/s0022112075000560}, + year = 1975, + publisher = {Cambridge University Press ({CUP})}, volume = {67}, - pages = {705--722} + number = {4}, + pages = {705--722}, + author = {T. H. Bell}, + title = {Lee waves in stratified flows with simple harmonic time dependence}, + journal = {J. Fluid Mech.} +} + +@article{nikurashin2010a, + doi = {10.1175/2009jpo4199.1}, + year = 2010, + publisher = {American Meteorological Society}, + volume = {40}, + number = {5}, + pages = {1055--1074}, + author = {M. Nikurashin and R. Ferrari}, + title = {Radiation and Dissipation of Internal Waves Generated by Geostrophic Motions Impinging on Small-Scale Topography: Theory}, + journal = {J. Phys. Oceanography} +} + +@article{nikurashin2010b, + doi = {10.1175/2010jpo4315.1}, + year = 2010, + publisher = {American Meteorological Society}, + volume = {40}, + number = {9}, + pages = {2025--2042}, + author = {M. Nikurashin and R. Ferrari}, + title = {Radiation and Dissipation of Internal Waves Generated by Geostrophic Motions Impinging on Small-Scale Topography: Application to the Southern Ocean}, + journal = {J. Phys. Oceanography} +} + +@article{miles1961, + title = {On the stability of heterogeneous shear flows}, + author = {JW Miles}, + year = {1961}, + journal = {J. of Fluid Mech.}, + volume = {10}, + pages = {496--508}, + doi = {10.1017/S0022112061000305} +} + +@article{bryan1979, + doi = {10.1029/jc084ic05p02503}, + year = 1979, + publisher = {American Geophysical Union ({AGU})}, + volume = {84}, + number = {C5}, + pages = {2503}, + author = {K. Bryan and L. J. Lewis}, + title = {A water mass model of the World Ocean}, + journal = {J. Geophys. Res.} } diff --git a/src/parameterizations/vertical/MOM_set_diffusivity.F90 b/src/parameterizations/vertical/MOM_set_diffusivity.F90 index f4874252f4..0d07f0fea4 100644 --- a/src/parameterizations/vertical/MOM_set_diffusivity.F90 +++ b/src/parameterizations/vertical/MOM_set_diffusivity.F90 @@ -198,16 +198,6 @@ module MOM_set_diffusivity contains -!> Sets the interior vertical diffusion of scalars due to the following processes: -!! 1. Shear-driven mixing: two options, Jackson et at. and KPP interior; -!! 2. Background mixing via CVMix (Bryan-Lewis profile) or the scheme described by -!! Harrison & Hallberg, JPO 2008; -!! 3. Double-diffusion, old method and new method via CVMix; -!! 4. Tidal mixing: many options available, see MOM_tidal_mixing.F90; -!! In addition, this subroutine has the option to set the interior vertical -!! viscosity associated with processes 1,2 and 4 listed above, which is stored in -!! visc%Kv_slow. Vertical viscosity due to shear-driven mixing is passed via -!! visc%Kv_shear subroutine set_diffusivity(u, v, h, u_h, v_h, tv, fluxes, optics, visc, dt, & G, GV, US, CS, Kd_lay, Kd_int, Kd_extra_T, Kd_extra_S) type(ocean_grid_type), intent(in) :: G !< The ocean's grid structure. diff --git a/src/parameterizations/vertical/MOM_set_viscosity.F90 b/src/parameterizations/vertical/MOM_set_viscosity.F90 index 99bd91d8f8..9a2680ecc1 100644 --- a/src/parameterizations/vertical/MOM_set_viscosity.F90 +++ b/src/parameterizations/vertical/MOM_set_viscosity.F90 @@ -115,80 +115,6 @@ module MOM_set_visc contains !> Calculates the thickness of the bottom boundary layer and the viscosity within that layer. -!! -!! A drag law is used, either linearized about an assumed bottom velocity or using the -!! actual near-bottom velocities combined with an assumed unresolved velocity. The bottom -!! boundary layer thickness is limited by a combination of stratification and rotation, as -!! in the paper of Killworth and Edwards, JPO 1999. It is not necessary to calculate the -!! thickness and viscosity every time step; instead previous values may be used. -!! -!! \section set_viscous_BBL Viscous Bottom Boundary Layer -!! -!! If set_visc_cs.bottomdraglaw is True then a bottom boundary layer viscosity and thickness -!! are calculated so that the bottom stress is -!! \f[ -!! \mathbf{\tau}_b = C_d | U_{bbl} | \mathbf{u}_{bbl} -!! \f] -!! If set_visc_cs.bottomdraglaw is True then the term \f$|U_{bbl}|\f$ is set equal to the -!! value in set_visc_cs.drag_bg_vel so that \f$C_d |U_{bbl}|\f$ becomes a Rayleigh bottom drag. -!! Otherwise \f$|U_{bbl}|\f$ is found by averaging the flow over the bottom set_visc_cs.hbbl -!! of the model, adding the amplitude of tides set_visc_cs.tideamp and a constant -!! set_visc_cs.drag_bg_vel. For these calculations the vertical grid at the velocity -!! component locations is found by -!! \f[ -!! \begin{array}{ll} -!! \frac{2 h^- h^+}{h^- + h^+} & u \left( h^+ - h^-\right) >= 0 -!! \\ -!! \frac{1}{2} \left( h^- + h^+ \right) & u \left( h^+ - h^-\right) < 0 -!! \end{array} -!! \f] -!! which biases towards the thin cell if the thin cell is upwind. Biasing the grid toward -!! thin upwind cells helps increase the effect of viscosity and inhibits flow out of these -!! thin cells. -!! -!! After diagnosing \f$|U_{bbl}|\f$ over a fixed depth an active viscous boundary layer -!! thickness is found using the ideas of Killworth and Edwards, 1999 (hereafter KW99). -!! KW99 solve the equation -!! \f[ -!! \left( \frac{h_{bbl}}{h_f} \right)^2 + \frac{h_{bbl}}{h_N} = 1 -!! \f] -!! for the boundary layer depth \f$h_{bbl}\f$. Here -!! \f[ -!! h_f = \frac{C_n u_*}{f} -!! \f] -!! is the rotation controlled boundary layer depth in the absence of stratification. -!! \f$u_*\f$ is the surface friction speed given by -!! \f[ -!! u_*^2 = C_d |U_{bbl}|^2 -!! \f] -!! and is a function of near bottom model flow. -!! \f[ -!! h_N = \frac{C_i u_*}{N} = \frac{ (C_i u_* )^2 }{g^\prime} -!! \f] -!! is the stratification controlled boundary layer depth. The non-dimensional parameters -!! \f$C_n=0.5\f$ and \f$C_i=20\f$ are suggested by Zilitinkevich and Mironov, 1996. -!! -!! If a Richardson number dependent mixing scheme is being used, as indicated by -!! set_visc_cs.rino_mix, then the boundary layer thickness is bounded to be no larger -!! than a half of set_visc_cs.hbbl . -!! -!! \todo Channel drag needs to be explained -!! -!! A BBL viscosity is calculated so that the no-slip boundary condition in the vertical -!! viscosity solver implies the stress \f$\mathbf{\tau}_b\f$. -!! -!! \subsection set_viscous_BBL_ref References -!! -!! \arg Killworth, P. D., and N. R. Edwards, 1999: -!! A Turbulent Bottom Boundary Layer Code for Use in Numerical Ocean Models. -!! J. Phys. Oceanogr., 29, 1221-1238, -!! doi:10.1175/1520-0485(1999)029<1221:ATBBLC>2.0.CO;2 -!! \arg Zilitinkevich, S., Mironov, D.V., 1996: -!! A multi-limit formulation for the equilibrium depth of a stably stratified boundary layer. -!! Boundary-Layer Meteorology 81, 325-351. -!! doi:10.1007/BF02430334 -!! subroutine set_viscous_BBL(u, v, h, tv, visc, G, GV, US, CS, symmetrize) type(ocean_grid_type), intent(inout) :: G !< The ocean's grid structure. type(verticalGrid_type), intent(in) :: GV !< The ocean's vertical grid structure. diff --git a/src/parameterizations/vertical/_Internal_tides.dox b/src/parameterizations/vertical/_Internal_tides.dox index 882b73dd1b..a07663d4a1 100644 --- a/src/parameterizations/vertical/_Internal_tides.dox +++ b/src/parameterizations/vertical/_Internal_tides.dox @@ -4,7 +4,7 @@ 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 +parameters 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. @@ -69,7 +69,7 @@ case the maximum of all the contributions is used. 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 +which links the dissipation profile associated 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 @@ -135,9 +135,9 @@ at the ocean floor, so that in both formulations: \int_{0}^{H} \epsilon (z) dz = \frac{qE}{\rho} . \f] -Whereas \cite polzin2009 assumed tthat the total dissipation was locally in balance with the +Whereas \cite polzin2009 assumed that 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. +of \f$q=1/3\f$ to retain as much consistency as possible between both parameterizations. \subsection subsection_vertical_decay_scale Vertical decay-scale reformulation @@ -212,5 +212,16 @@ 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. +\section Nikurashin Lee Wave Mixing + +If one has the INT_TIDE_DISSIPATION flag on, there is an option to also turn on +LEE_WAVE_DISSIPATION. The theory is presented in \cite nikurashin2010a +while the application of it is presented in \cite nikurashin2010b. For +the implementation in MOM6, it is required that you provide an estimate +of the TKE loss due to the Lee waves which is then applied with either +the St. Laurent or the Polzin vertical profile. + +\todo Is there a script to produce this somewhere or what??? + */ diff --git a/src/parameterizations/vertical/_V_diffusivity.dox b/src/parameterizations/vertical/_V_diffusivity.dox new file mode 100644 index 0000000000..4d671fec88 --- /dev/null +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -0,0 +1,284 @@ +/*! \page Internal_Shear_Mixing Internal Vertical Mixing + +Sets the interior vertical diffusion of scalars due to the following processes: + +-# Shear-driven mixing: two options, \cite jackson2008 and KPP interior; +-# Background mixing via CVMix (Bryan-Lewis profile), the scheme described by + \cite harrison2008, or that in \cite danabasoglu2012. +-# Double-diffusion, old method and new method via CVMix; +-# Tidal mixing: many options available, see \ref Internal_Tidal_Mixing. + +In addition, the MOM_set_diffusivity has the option to set the interior vertical +viscosity associated with processes 1,2 and 4 listed above, which is stored in +visc\%Kv\_slow. Vertical viscosity due to shear-driven mixing is passed via +visc\%Kv\_shear + +The resulting diffusivity, \f$K_d\f$, is the sum of all the contributions +unless you set BBL_MIXING_AS_MAX to True, in which case the maximum of +all the contributions is used. + +In addition, \f$K_d\f$ is multiplied by the term: + +\f[ + \frac{N^2}{N^2 + \Omega^2} +\f] + +where \f$N\f$ is the buoyancy frequency and \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. + +\section section_Shear Shear-driven Mixing + +Below the surface mixed layer, there are places in the world's oceans +where shear mixing is known to take place. This shear-driven mixing can +be represented in MOM6 through either CVMix or the parameterization of +\cite jackson2008. + +\subsection subsection_CVMix_shear Shear-driven mixing in CVMix + +The community vertical mixing (CVMix) code contains options for shear +mixing from either \cite large1994 or from \cite pacanowski1981. In MOM6, +CVMix is included via a git submodule which loads the external CVMix +package. The shear mixing routine in CVMix was developed to reproduce the +observed mixing of the equatorial undercurrent in the Pacific. + +We first compute the gradient Richardson number \f$\mbox{Ri} = N^2 / S^2\f$, +where \f$S\f$ is the vertical shear (\f$S = ||\bf{u}_z ||\f$) and \f$N\f$ +is the buoyancy frequency (\f$N^2 = -g \rho_z / \rho_0\f$). The +parameterization of \cite large1994 is as follows, where the diffusivity \f$\kappa\f$ +is given by + +\f[ + \kappa = \kappa_0 \left[ 1 - \min \left( 1, \frac{\mbox{Ri}}{\mbox{Ri}_c} \right) ^2 \right] ^3 ,\ +\f] + +with \f$\kappa_0 = 5 \times 10^{-3}\, \mbox{m}^2 \,\mbox{s}^{-1}\f$ and \f$\mbox{Ri}_c = 0.7\f$. + +One can instead select the \cite pacanowski1981 scheme within CVMix. Unlike +the \cite large1994 scheme, they propose that the\ vertical shear +viscosity \f$\nu_{\mbox{shear}}\f$ be different from the vertical shear +diffusivity \f$\kappa_{\mbox{shear}}\f$. For gravitationally stable +profiles (i.e., \f$N^2 > 0\f$), they chose + +\f[ + \nu_{\mbox{shear}} = \frac{\nu_0}{(1 + a \mbox{Ri})^n} +\f] + +\f[ + \kappa_{\mbox{shear}} = \frac{\nu_0}{(1 + a \mbox{Ri})^{n+1}} +\f] + +where \f$\nu_0\f$, \f$a\f$ and \f$n\f$ are adjustable parameters. Common settings are \f$a = 5\f$ +and \f$n = 2\f$. + +For both CVMix shear mixing schemes, the mixing coefficients are set to +a large value for gravitationally unstable profiles. + +\subsection subsection_kappa_shear Shear-driven mixing in Jackson + +While the above parameterization works well enough in the equatorial +Pacific, another place one can expect shear-mixing to matter is +in overflows of dense water. \cite jackson2008 proposes a new shear +parameterization with the goal of working in both the equatorial undercurrent +and for overflows, also to have smooth transitions between unstable and +stable regions. Their scheme looks like: + +\f{eqnarray} + \frac{\partial^2 \kappa}{\partial z^2} - \frac{\kappa}{L^2_d} &= - 2 SF(\mbox{Ri}) . + \label{eq:Jackson_10} +\f} + +This is similar to the locally constant stratification limit of +\cite turner1986, but with the addition of a decay length scale +\f$L_d = \lambda L_b\f$. Here \f$L_b = Q^{1/2} / N\f$ is the buoyancy +length scale where \f$Q\f$ is the turbulent kinetic energy (TKE) per +unit mass, and \f$\lambda\f$ is a nondimensional constant. The function +\f$F(\mbox{Ri})\f$ is a function of the Richardson number that remains +to be determined. As in \cite turner1986, there must be a critical +value of \f$\mbox{Ri}\f$ above which \f$F(\mbox{Ri}) = 0\f$. +For better agreement with observations in a law-of-the-wall configuration, +we modify \f$L_d\f$ to be \f$\min (\lambda L_b, L_z)\f$, where \f$L_z\f$ +is the distance to the nearest solid boundary. This can be understood by +considering \f$L_d\f$ to be the size of the largest turbulent eddies, +whether they are constrained by the stratification (through \f$L_b\f$) +or through the geometry (through \f$L_z\f$). + +There are two length scales: the width of the low Richardson number region +as in \cite turner1986, and the buoyancy length scale, which is the +length scale over which the TKE is affected by the stratification (see +\cite jackson2008 for more details). In particular, the inclusion of a +decay length scale means that the diffusivity decays exponentially away +from the mixing region with a length scale of \f$L_d\f$. This is important +since turbulent eddies generated in the low \f$\mbox{Ri}\f$ layer can +be vertically self-advected and mix nearby regions. This method yields +a smoother diffusivity than that in \cite hallberg2000, especially in +areas where the Richardson number is noisy. + +This parameterization predicts the turbulent eddy diffusivity in terms +of the vertical profiles of velocity and density, providing that the +TKE is known. To complete the parameterization we use a TKE \f$Q\f$ +budget such as that used in second-order turbulence closure models +(\cite umlauf2005). We make a few additional assumptions, however, +and use the simplified form + +\f{eqnarray} + \frac{\partial}{\partial z} \left[ (\kappa + \nu_0) \frac{\partial Q} + {\partial z} \right] + \kappa (S^2 - N^2) - Q(c_N N + c_S S) &= 0. + \label{eq:Jackson_11} +\f} + +The system is therefore in balance between a vertical diffusion of +TKE caused by both the eddy and molecular viscosity \f$(\nu_0)\f$, +the production of TKE by shear, a sink due to stratification, and the +dissipation. Note that we are assuming a Prandtl number of 1, although a +parameterization for the Prandtl number could be added. We have assumed +that the TKE reaches a quasi-steady state faster than the flow is evolving +and faster than it can be affected by mean-flow advection so that \f$DQ/Dt = +0\f$. Since this parameterization is meant to be used in climate models +with low horizontal resolution and large time steps compared to the +mixing time scales, this is a reasonable assumption. The most tenuous +assumption is in the form of the dissipation \f$\epsilon = Q(C_N N + +c_S S)\f$ (where \f$c_N\f$ and \f$c_S\f$ are to be determined), +which is assumed to be dependent on the buoyancy frequency (through loss +of energy to internal waves) and the velocity shear (through the energy +cascade to smaller scales). + +We can rewrite \eqref{eq:Jackson_10} as the steady "transport" equation +for the turbulent diffusivity (i.e., with \f$D\kappa/Dt = 0\f$), + +\f[ + \frac{\partial}{\partial z} \left( \kappa \frac{\partial \kappa}{\partial z} + \right) + 2\kappa SF(\mbox{Ri}) - \left( \frac{\kappa}{L_d} \right)^2 - + \left( \frac{\partial \kappa}{\partial z} \right) ^2 = 0 . +\f] + +The first term on the left can be regarded as a vertical transport of +diffusivity, the second term as a source, and the final two as sinks. +This equation with \eqref{eq:Jackson_11} are simple enough to solve quickly +using an iterative technique. + +We also need boundary conditions for \eqref{eq:Jackson_10} +and \eqref{eq:Jackson_11}. For the turbulent diffusivity we use +\f$\kappa = 0\f$ since our diffusivity is numerically defined on +layer interfaces. This ensures that there is no turbulent flux across +boundaries. For the TKE we use boundary conditions of \f$Q = Q_0\f$ where +\f$Q_0\f$ is a constant value of TKE, used to prevent a singularity +in \eqref{eq:Jackson_10}, that is chosen to be small enough to not +influence results. Note that the value of \f$\kappa\f$ calculated here +reflects shear-driven turbulent mixing only; the total diffusivity would +be this value plus any diffusivities due to other turbulent processes +or a background value. + +Based on \cite turner1986, we choose \f$F(\mbox{Ri})\f$ of the form + +\f[ + F(\mbox{Ri}) = F_0 \left( \frac{1 - \mbox{Ri} / \mbox{Ri}_c} + {1 + \alpha \mbox{Ri} / \mbox{Ri}_c} \right) , +\f] + +where \f$\alpha\f$ is the curvature parameter. This table shows the default +values of the relevant parameters: + + + +
Shear mixing parameters
Parameter Default value MOM6 parameter +
\f$\mbox{Ri}_c\f$ 0.25 RINO_CRIT +
\f$\nu_0\f$ \f$1.5 \times 10^{-5}\f$ KD_KAPPA_SHEAR_0 +
\f$F_0\f$ 0.089 SHEARMIX_RATE +
\f$\alpha\f$ -0.97 FRI_CURVATURE +
\f$\lambda\f$ 0.82 KAPPA_BUOY_SCALE_COEF +
\f$c_N\f$ 0.24 TKE_N_DECAY_CONST +
\f$c_S\f$ 0.14 TKE_SHEAR_DECAY_CONST +
+ +These can all be adjusted at run time, plus some other parameters such as the maximum number of iterations +to perform. + +\section section_Background Background Mixing + +There are three choices for the vertical background mixing: that in +CVMix (\cite bryan1979), that in \cite harrison2008, and that in +\cite danabasoglu2012. + +\subsection subsection_bryan_lewis CVMix background mixing + +The background vertical mixing in \cite bryan1979 is of the form: + +\f[ + \kappa = C_1 + C_2 \mbox{atan} [ C_3 ( |z| - C_4 )] +\f] + +where the constants are runtime parameters as shown here: + + + +
Bryan Lewis parameters
Parameter Units MOM6 parameter +
\f$C_1\f$ m2 s-1 BRYAN_LEWIS_C1 +
\f$C_2\f$ m2 s-1 BRYAN_LEWIS_C2 +
\f$C_3\f$ m-1 BRYAN_LEWIS_C3 +
\f$C_4\f$ m BRYAN_LEWIS_C4 +
+ +\subsection subsection_henyey Henyey IGW background mixing + +\cite harrison2008 choose a vertical background mixing with a latitudinal +dependence based on \cite henyey1986. Specifically, theory predicts +a minimum in mixing due to wave-wave interactions at the equator and +observations support that theory. In this option, the surface background +diffusivity is + +\f[ + \kappa_s (\phi) = \max \left[ 10^{-7}, \kappa_0 \left| \frac{f}{f_{30}} \right| + \frac{ \cosh^{-1} (1/f) }{ \cosh^{-1} (1/f_{30})} \right] , +\f] + +where \f$f_{30}\f$ is the Coriolis frequency at \f$30^\circ\f$ latitude. The two-dimensional equation for +the diffusivity is + +\f[ + \kappa(\phi, z) = \kappa_s + \Gamma \mbox{atan} \left( \frac{H_t}{\delta_t} \right) + + \Gamma \mbox{atan} \left( \frac{z - H_t}{\delta_t} \right) , +\f] +\f[ + \Gamma = \frac{(\kappa_d - \kappa_s) }{\left[ 0.5 \pi + \mbox{atan} \left( \frac{H_t}{\delta_t} \right) + \right] }, +\f] + +where \f$H_t = 2500\, \mbox{m}\f$, \f$\delta_t = 222\, \mbox{m}\f$, and +\f$\kappa_d\f$ is the deep ocean diffusivity of \f$10^{-4}\, \mbox{m}^2 +\, \mbox{s}^{-1}\f$. Note that this is the vertical structure described +in \cite harrison2008, but that isn't what is in the code. Instead, the surface +value is propagated down, with the assumption that the tidal mixing parameterization +will provide the deep mixing: \ref Internal_Tidal_Mixing. + +There is also a "new" Henyey version, taking into account the effect of stratification on +TKE dissipation, + +\f[ + \epsilon = \epsilon_0 \frac{f}{f_0} \frac{\mbox{acosh} (N/f)}{\mbox{acosh} (N_0 / f_0)} +\f] + +where \f$N_0\f$ and \f$f_0\f$ are the reference buoyancy frequency and inertial frequencies, respectively +and \f$\epsilon_0\f$ is the reference dissipation at \f$(N_0, f_0)\f$. In the previous version, \f$N = +N_0\f$. Additionally, the relationship between diapycnal diffusivities and stratification is included: + +\f[ + \kappa = \frac{\epsilon}{N^2} +\f] +This approach assumes that work done against gravity is uniformly distributed throughout the water column. +The original version concentrates buoyancy work in regions of strong stratification. + +\subsection subsection_danabasoglu_back Danabasoglu background mixing + +The shape of the \cite danabasoglu2012 background mixing has a uniform background value, with a dip +at the equator and a bump at \f$\pm 30^{\circ}\f$ degrees latitude. The form is shown in this figure + +\image html background_varying.png "Form of the vertically uniform background mixing in \cite danabasoglu2012. The values are symmetric about the equator." +\imagelatex{background_varying.png,Form of the vertically uniform background mixing in \cite danabasoglu2012. The values are symmetric about the equator.,\includegraphics[width=\textwidth\,height=\textheight/2\,keepaspectratio=true]} + +Some parameters of this curve are set in the input file, some are hard-coded in calculate_bkgnd_mixing. + +\section section_Double_Diff Double Diffusion + +*/ diff --git a/src/parameterizations/vertical/_V_viscosity.dox b/src/parameterizations/vertical/_V_viscosity.dox new file mode 100644 index 0000000000..cc59e83457 --- /dev/null +++ b/src/parameterizations/vertical/_V_viscosity.dox @@ -0,0 +1,64 @@ +/*! \page Vertical_Viscosity Viscous Bottom Boundary Layer + +A drag law is used, either linearized about an assumed bottom velocity or using the +actual near-bottom velocities combined with an assumed unresolved velocity. The bottom +boundary layer thickness is limited by a combination of stratification and rotation, as +in the paper of \cite killworth1999. It is not necessary to calculate the +thickness and viscosity every time step; instead previous values may be used. + +\section set_viscous_BBL Viscous Bottom Boundary Layer + +If set_visc_CS\%bottomdraglaw is True then a bottom boundary layer viscosity and thickness +are calculated so that the bottom stress is +\f[ +\mathbf{\tau}_b = C_d | U_{bbl} | \mathbf{u}_{bbl} +\f] +If set_visc_CS\%bottomdraglaw is True then the term \f$|U_{bbl}|\f$ is set equal to the +value in set_visc_CS.drag_bg_vel so that \f$C_d |U_{bbl}|\f$ becomes a Rayleigh bottom drag. +Otherwise \f$|U_{bbl}|\f$ is found by averaging the flow over the bottom set_visc_CS\%hbbl +of the model, adding the amplitude of tides set_visc_CS\%tideamp and a constant +set_visc_CS\%drag_bg_vel. For these calculations the vertical grid at the velocity +component locations is found by +\f[ +\begin{array}{ll} +\frac{2 h^- h^+}{h^- + h^+} & u \left( h^+ - h^-\right) >= 0 +\\ +\frac{1}{2} \left( h^- + h^+ \right) & u \left( h^+ - h^-\right) < 0 +\end{array} +\f] +which biases towards the thin cell if the thin cell is upwind. Biasing the grid toward +thin upwind cells helps increase the effect of viscosity and inhibits flow out of these +thin cells. + +After diagnosing \f$|U_{bbl}|\f$ over a fixed depth an active viscous boundary layer +thickness is found using the ideas of Killworth and Edwards, 1999 (hereafter KW99). +KW99 solve the equation +\f[ +\left( \frac{h_{bbl}}{h_f} \right)^2 + \frac{h_{bbl}}{h_N} = 1 +\f] +for the boundary layer depth \f$h_{bbl}\f$. Here +\f[ +h_f = \frac{C_n u_*}{f} +\f] +is the rotation controlled boundary layer depth in the absence of stratification. +\f$u_*\f$ is the surface friction speed given by +\f[ +u_*^2 = C_d |U_{bbl}|^2 +\f] +and is a function of near bottom model flow. +\f[ +h_N = \frac{C_i u_*}{N} = \frac{ (C_i u_* )^2 }{g^\prime} +\f] +is the stratification controlled boundary layer depth. The non-dimensional parameters +\f$C_n=0.5\f$ and \f$C_i=20\f$ are suggested by \cite zilitinkevich1996. + +If a Richardson number dependent mixing scheme is being used, as indicated by +set_visc_CS\%rino_mix, then the boundary layer thickness is bounded to be no larger +than a half of set_visc_CS\%hbbl . + +\todo Channel drag needs to be explained + +A BBL viscosity is calculated so that the no-slip boundary condition in the vertical +viscosity solver implies the stress \f$\mathbf{\tau}_b\f$. + +*/