From b039e62c320d01963ae5df0cee44ad1a89578207 Mon Sep 17 00:00:00 2001 From: Kate Hedstrom Date: Fri, 27 Aug 2021 17:52:15 -0800 Subject: [PATCH 1/6] More documentation - Jackson start. --- docs/parameterizations_vertical.rst | 7 +- docs/zotero.bib | 107 ++++++++++++- .../vertical/MOM_set_diffusivity.F90 | 10 -- .../vertical/MOM_set_viscosity.F90 | 74 --------- .../vertical/_Internal_tides.dox | 11 ++ .../vertical/_V_diffusivity.dox | 150 ++++++++++++++++++ .../vertical/_V_viscosity.dox | 64 ++++++++ 7 files changed, 333 insertions(+), 90 deletions(-) create mode 100644 src/parameterizations/vertical/_V_diffusivity.dox create mode 100644 src/parameterizations/vertical/_V_viscosity.dox 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..8a5c14dcfa 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}, @@ -1761,6 +1797,18 @@ @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 +2391,19 @@ @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 +2606,47 @@ @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} +} 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..bf6e16ea5c 100644 --- a/src/parameterizations/vertical/_Internal_tides.dox +++ b/src/parameterizations/vertical/_Internal_tides.dox @@ -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. + +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..5c40768eaf --- /dev/null +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -0,0 +1,150 @@ +/*! \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) or the scheme described by + \cite harrison2008. +-# 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$\mbox{Ri}_c = 0.7\f$. + +\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$F(\mbox{Ri}) = 0\f$. 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 assumtion. 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 frequeny (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 contitions 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. + +\section section_Background Background 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$\mathbf{\tau}_b\f$. + +*/ From 4f707ce546be8850d0469de824d310b948af2f78 Mon Sep 17 00:00:00 2001 From: Kate Hedstrom Date: Tue, 31 Aug 2021 17:17:14 -0800 Subject: [PATCH 2/6] Some background diffusivity text --- docs/zotero.bib | 25 ++++ .../vertical/_V_diffusivity.dox | 107 +++++++++++++++++- 2 files changed, 130 insertions(+), 2 deletions(-) diff --git a/docs/zotero.bib b/docs/zotero.bib index 8a5c14dcfa..a00fe569bd 100644 --- a/docs/zotero.bib +++ b/docs/zotero.bib @@ -1462,6 +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}, @@ -2650,3 +2662,16 @@ @article{miles1961 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/_V_diffusivity.dox b/src/parameterizations/vertical/_V_diffusivity.dox index 5c40768eaf..1d79f58997 100644 --- a/src/parameterizations/vertical/_V_diffusivity.dox +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -76,8 +76,15 @@ 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$F(\mbox{Ri}) = 0\f$. There -are two length scales: the width of the low Richardson number region +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 @@ -143,8 +150,104 @@ 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 contants 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$10^{-4}\, \mbox{m}^2 +\, \mbox{s}^{-1}\f$. + +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 + \section section_Double_Diff Double Diffusion */ From 9ee07718a66e064ecf4ca2cd941d707aaa2cd9ff Mon Sep 17 00:00:00 2001 From: Kate Hedstrom Date: Wed, 1 Sep 2021 17:18:17 -0800 Subject: [PATCH 3/6] More on background mixing. --- docs/images/background_varying.png | Bin 0 -> 51342 bytes .../vertical/_Internal_tides.dox | 10 ++-- .../vertical/_V_diffusivity.dox | 47 +++++++++++++++--- 3 files changed, 44 insertions(+), 13 deletions(-) create mode 100644 docs/images/background_varying.png diff --git a/docs/images/background_varying.png b/docs/images/background_varying.png new file mode 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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 @@ -221,7 +221,7 @@ 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. -IS THERE A SCRIPT to produce this somewhere or what??? +\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 index 1d79f58997..3e0ca7d4e1 100644 --- a/src/parameterizations/vertical/_V_diffusivity.dox +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -3,8 +3,8 @@ 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) or the scheme described by - \cite harrison2008. +-# 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. @@ -50,11 +50,31 @@ parameterization of \cite large1994 is as follows, where the diffusivity \f$\kap is given by \f[ - \kappa = \kappa_0 \left[ 1 - \min \left( 1, \frac{\mbox{Ri}}{\mbox{Ri}_c} \right) ^2 \right] ^3 , + \kappa = \kappa_0 \left[ 1 - \min \left( 1, \frac{\mbox{Ri}}{\mbox{Ri}_c} \right) ^2 \right] ^3 ,\ \f] with \f$\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$\nu_{\mbox{shear}}\nf$ 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 @@ -117,10 +137,10 @@ 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 assumtion. The most tenuous +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 frequeny (through loss +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). @@ -138,7 +158,7 @@ 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 contitions for \eqref{eq:Jackson_10} +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 @@ -189,7 +209,7 @@ The background vertical mixing in \cite bryan1979 is of the form: \kappa = C_1 + C_2 \mbox{atan} [ C_3 ( |z| - C_4 )] \f] -where the contants are runtime parameters as shown here: +where the constants are runtime parameters as shown here: @@ -227,7 +247,10 @@ the diffusivity is 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$10^{-4}\, \mbox{m}^2 -\, \mbox{s}^{-1}\f$. +\, \mbox{s}^{-1}\f$. Note that this is the vertical structure described +in \cite harrison2008, but that isn't what is in the code. Instead, the surface +value is propagated down, with the assumption that the tidal mixing parameterization +will provide the deep mixing: \ref Internal_Tidal_Mixing. There is also a "new" Henyey version, taking into account the effect of stratification on TKE dissipation, @@ -248,6 +271,14 @@ The original version concentrates buoyancy work in regions of strong stratificat \subsection subsection_danabasoglu_back Danabasoglu background mixing +The shape of the \cite danabasoglu background mixing has a uniform background value, with a dip +at the equator and a bump at \f$\pm 30^{\circ}$ 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 */ From db3fd1cdf77abe27c38e96a168d6aba35f300090 Mon Sep 17 00:00:00 2001 From: Kate Hedstrom Date: Wed, 1 Sep 2021 18:29:44 -0800 Subject: [PATCH 4/6] Fixed forgotten \f$ --- src/parameterizations/vertical/_V_diffusivity.dox | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/parameterizations/vertical/_V_diffusivity.dox b/src/parameterizations/vertical/_V_diffusivity.dox index 3e0ca7d4e1..e6f0139661 100644 --- a/src/parameterizations/vertical/_V_diffusivity.dox +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -272,7 +272,7 @@ The original version concentrates buoyancy work in regions of strong stratificat \subsection subsection_danabasoglu_back Danabasoglu background mixing The shape of the \cite danabasoglu background mixing has a uniform background value, with a dip -at the equator and a bump at \f$\pm 30^{\circ}$ degrees latitude. The form is shown in this figure +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]} From 9c2b4e4b5de6423b63aac0bc30f9d318a04fe593 Mon Sep 17 00:00:00 2001 From: Kate Hedstrom Date: Wed, 1 Sep 2021 18:38:13 -0800 Subject: [PATCH 5/6] Fixing a citation --- src/parameterizations/vertical/_V_diffusivity.dox | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/parameterizations/vertical/_V_diffusivity.dox b/src/parameterizations/vertical/_V_diffusivity.dox index e6f0139661..cdb927f5ef 100644 --- a/src/parameterizations/vertical/_V_diffusivity.dox +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -271,7 +271,7 @@ The original version concentrates buoyancy work in regions of strong stratificat \subsection subsection_danabasoglu_back Danabasoglu background mixing -The shape of the \cite danabasoglu background mixing has a uniform background value, with a dip +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." From 6d13b1890651d974ff4f95d0a86366bd0b044fae Mon Sep 17 00:00:00 2001 From: Kate Hedstrom Date: Wed, 1 Sep 2021 20:09:25 -0800 Subject: [PATCH 6/6] Still more syntax issues --- src/parameterizations/vertical/_V_diffusivity.dox | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/parameterizations/vertical/_V_diffusivity.dox b/src/parameterizations/vertical/_V_diffusivity.dox index cdb927f5ef..4d671fec88 100644 --- a/src/parameterizations/vertical/_V_diffusivity.dox +++ b/src/parameterizations/vertical/_V_diffusivity.dox @@ -57,7 +57,7 @@ with \f$\kappa_0 = 5 \times 10^{-3}\, \mbox{m}^2 \,\mbox{s}^{-1}\f$ and \f$\mbox One can instead select the \cite pacanowski1981 scheme within CVMix. Unlike the \cite large1994 scheme, they propose that the\ vertical shear -viscosity \f$\nu_{\mbox{shear}}\nf$ be different from 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
Bryan Lewis parameters