.. _sub_phys_pkg_kpp: KPP: Nonlocal K-Profile Parameterization for Vertical Mixing ------------------------------------------------------------ Authors: Dimitris Menemenlis and Patrick Heimbach .. _ssub_phys_pkg_kpp_intro: Introduction ++++++++++++ The nonlocal K-Profile Parameterization (KPP) scheme of :cite:`lar-eta:94` unifies the treatment of a variety of unresolved processes involved in vertical mixing. To consider it as one mixing scheme is, in the view of the authors, somewhat misleading since it consists of several entities to deal with distinct mixing processes in the ocean’s surface boundary layer, and the interior: #. mixing in the interior is goverened by shear instability (modeled as function of the local gradient Richardson number), internal wave activity (assumed constant), and double-diffusion (not implemented here). #. a boundary layer depth :math:`h` or ``hbl`` is determined at each grid point, based on a critical value of turbulent processes parameterized by a bulk Richardson number; #. mixing is strongly enhanced in the boundary layer under the stabilizing or destabilizing influence of surface forcing (buoyancy and momentum) enabling boundary layer properties to penetrate well into the thermocline; mixing is represented through a polynomial profile whose coefficients are determined subject to several contraints; #. the boundary-layer profile is made to agree with similarity theory of turbulence and is matched, in the asymptotic sense (function and derivative agree at the boundary), to the interior thus fixing the polynomial coefficients; matching allows for some fraction of the boundary layer mixing to affect the interior, and vice versa; #. a “non-local” term :math:`\hat{\gamma}` or ``ghat`` which is independent of the vertical property gradient further enhances mixing where the water column is unstable The scheme has been extensively compared to observations (see e.g. :cite:`lar-eta:97`) and is now common in many ocean models. The current code originates in the NCAR NCOM 1-D code and was kindly provided by Bill Large and Jan Morzel. It has been adapted first to the MITgcm vector code and subsequently to the current parallel code. Adjustment were mainly in conjunction with WRAPPER requirements (domain decomposition and threading capability), to enable automatic differentiation of tangent linear and adjoint code via TAMC. The following sections will describe the KPP package configuration and compiling ([sec:pkg:kpp:comp]), the settings and choices of runtime parameters ([sec:pkg:kpp:runtime]), more detailed description of equations to which these parameters relate ([sec:pkg:kpp:equations]), and key subroutines where they are used ([sec:pkg:kpp:flowchart]), and diagnostics output of KPP-derived diffusivities, viscosities and boundary-layer/mixed-layer depths ([sec:pkg:kpp:diagnostics]). .. _ssub_phys_pkg_kpp_comp: KPP configuration and compiling +++++++++++++++++++++++++++++++ As with all MITgcm packages, KPP can be turned on or off at compile time - using the ``packages.conf`` file by adding ``kpp`` to it, - or using ``genmake2`` adding ``-enable=kpp`` or ``-disable=kpp`` switches - *Required packages and CPP options:* No additional packages are required, but the MITgcm kernel flag enabling the penetration of shortwave radiation below the surface layer needs to be set in ``CPP_OPTIONS.h`` as follows: ``#define SHORTWAVE_HEATING`` (see Section [sec:buildingCode]). Parts of the KPP code can be enabled or disabled at compile time via CPP preprocessor flags. These options are set in ``KPP_OPTIONS.h``. Table :numref:`tab_phys_pkg_kpp_cpp_options` summarizes them. .. table:: CPP flags for KPP :name: tab_phys_pkg_kpp_cpp_options +------------------------------------------+-------------------+ | **CPP option** | **Description** | +==========================================+===================+ | ``_KPP_RL`` |   | +------------------------------------------+-------------------+ | ``FRUGAL_KPP`` |   | +------------------------------------------+-------------------+ | ``KPP_SMOOTH_SHSQ`` |   | +------------------------------------------+-------------------+ | ``KPP_SMOOTH_DVSQ`` |   | +------------------------------------------+-------------------+ | ``KPP_SMOOTH_DENS`` |   | +------------------------------------------+-------------------+ | ``KPP_SMOOTH_VISC`` |   | +------------------------------------------+-------------------+ | ``KPP_SMOOTH_DIFF`` |   | +------------------------------------------+-------------------+ | ``KPP_ESTIMATE_UREF`` |   | +------------------------------------------+-------------------+ | ``INCLUDE_DIAGNOSTICS_INTERFACE_CODE`` |   | +------------------------------------------+-------------------+ | ``KPP_GHAT`` |   | +------------------------------------------+-------------------+ | ``EXCLUDE_KPP_SHEAR_MIX`` |   | +------------------------------------------+-------------------+ .. _ssub_phys_pkg_kpp_runtime: Run-time parameters +++++++++++++++++++ Run-time parameters are set in files ``data.pkg`` and ``data.kpp`` which are read in ``kpp_readparms.F``. Run-time parameters may be broken into 3 categories: (i) switching on/off the package at runtime, (ii) required MITgcm flags, (iii) package flags and parameters. Enabling the package #################### The KPP package is switched on at runtime by setting ``useKPP = .TRUE.`` in ``data.pkg``. Required MITgcm flags ##################### The following flags/parameters of the MITgcm dynamical kernel need to be set in conjunction with KPP: +----------------------------------+--------------------------------------+ | ``implicitViscosity = .TRUE.`` | enable implicit vertical viscosity | +----------------------------------+--------------------------------------+ | ``implicitDiffusion = .TRUE.`` | enable implicit vertical diffusion | +----------------------------------+--------------------------------------+ Package flags and parameters ############################ :numref:`tab_phys_pkg_kpp_runtime_flags` summarizes the runtime flags that are set in ``data.pkg``, and their default values. .. table:: Runtime flags for KPP :name: tab_phys_pkg_kpp_runtime_flags +------------------------+--------------------------------+--------------------------------------------------+ | **Flag/parameter** | **default** | **Description** | +========================+================================+==================================================+ | *I/O related parameters* | +------------------------+--------------------------------+--------------------------------------------------+ | kpp\_freq | ``deltaTClock`` | Recomputation frequency for KPP fields | +------------------------+--------------------------------+--------------------------------------------------+ | kpp\_dumpFreq | ``dumpFreq`` | Dump frequency of KPP field snapshots | +------------------------+--------------------------------+--------------------------------------------------+ | kpp\_taveFreq | ``taveFreq`` | Averaging and dump frequency of KPP fields | +------------------------+--------------------------------+--------------------------------------------------+ | KPPmixingMaps | ``.FALSE.`` | include KPP diagnostic maps in STDOUT | +------------------------+--------------------------------+--------------------------------------------------+ | KPPwriteState | ``.FALSE.`` | write KPP state to file | +------------------------+--------------------------------+--------------------------------------------------+ | KPP_ghatUseTotalDiffus | ``.FALSE.`` | if ``.T.`` compute non-local term using | +------------------------+--------------------------------+--------------------------------------------------+ | | | total vertical diffusivity | +------------------------+--------------------------------+--------------------------------------------------+ |   |   | if ``.F.`` use KPP vertical diffusivity | +------------------------+--------------------------------+--------------------------------------------------+ | *General KPP parameters* | +------------------------+--------------------------------+--------------------------------------------------+ | minKPPhbl | ``delRc(1)`` | Minimum boundary layer depth | +------------------------+--------------------------------+--------------------------------------------------+ | epsilon | 0.1 | nondimensional extent of the surface layer | +------------------------+--------------------------------+--------------------------------------------------+ | vonk | 0.4 | von Karman constant | +------------------------+--------------------------------+--------------------------------------------------+ | dB_dz | 5.2E-5 s\ :sup:`--2` | maximum dB/dz in mixed layer hMix | +------------------------+--------------------------------+--------------------------------------------------+ | concs | 98.96 |   | +------------------------+--------------------------------+--------------------------------------------------+ | concv | 1.8 |   | +------------------------+--------------------------------+--------------------------------------------------+ | *Boundary layer parameters (S/R bldepth)* | +------------------------+--------------------------------+--------------------------------------------------+ | Ricr | 0.3 | critical bulk Richardson number | +------------------------+--------------------------------+--------------------------------------------------+ | cekman | 0.7 | coefficient for Ekman depth | +------------------------+--------------------------------+--------------------------------------------------+ | cmonob | 1.0 | coefficient for Monin-Obukhov depth | +------------------------+--------------------------------+--------------------------------------------------+ | concv | 1.8 | ratio of interior to entrainment depth | | | | buoyancy frequency | +------------------------+--------------------------------+--------------------------------------------------+ | hbf | 1.0 | fraction of depth to which absorbed solar | | | | radiation contributes | |   |   | to surface buoyancy forcing | +------------------------+--------------------------------+--------------------------------------------------+ | Vtc | | non-dim. coeff. for velocity scale of | | | | turbulant velocity shear ( = function | |   |   | of concv,concs,epsilon,vonk,Ricr) | +------------------------+--------------------------------+--------------------------------------------------+ | *Boundary layer mixing parameters (S/R blmix)* | +------------------------+--------------------------------+--------------------------------------------------+ | cstar | 10. | proportionality coefficient for nonlocal | | | | transport | +------------------------+--------------------------------+--------------------------------------------------+ | cg |   | non-dimensional coefficient for counter-gradient | | | | term | |   |   | ( = function of cstar,vonk,concs,epsilon) | +------------------------+--------------------------------+--------------------------------------------------+ | *Interior mixing parameters (S/R Ri_iwmix)* | +------------------------+--------------------------------+--------------------------------------------------+ | Riinfty | 0.7 | gradient Richardson number limit for shear | | | | instability | +------------------------+--------------------------------+--------------------------------------------------+ | BVDQcon | -0.2E-4 s\ :sup:`--2` | Brunt-Väisalä squared | +------------------------+--------------------------------+--------------------------------------------------+ | difm0 |0.005 m\ :sup:`2` s\ :sup:`--1` | viscosity max. due to shear instability | +------------------------+--------------------------------+--------------------------------------------------+ | difs0 | 0.005 m\ :math:`^2`/s | tracer diffusivity max. due to shear instability | +------------------------+--------------------------------+--------------------------------------------------+ | dift0 | 0.005 m\ :math:`^2`/s | heat diffusivity max. due to shear instability | +------------------------+--------------------------------+--------------------------------------------------+ | difmcon | 0.1 | viscosity due to convective instability | +------------------------+--------------------------------+--------------------------------------------------+ | difscon | 0.1 | tracer diffusivity due to convective instability | +------------------------+--------------------------------+--------------------------------------------------+ | diftcon | 0.1 | heat diffusivity due to convective instability | +------------------------+--------------------------------+--------------------------------------------------+ | Rrho0 | not used | limit for double diffusive density ratio | +------------------------+--------------------------------+--------------------------------------------------+ | dsfmax | not used | maximum diffusivity in case of salt fingering | +------------------------+--------------------------------+--------------------------------------------------+ .. _ssub_phys_pkg_kpp_eqns_key_routines: Equations and key routines ++++++++++++++++++++++++++++++++++++++++++++++++++ We restrict ourselves to writing out only the essential equations that relate to main processes and parameters mentioned above. We closely follow the notation of :cite:`lar-eta:94`. KPP_CALC: ######### Top-level routine.   KPP_MIX: ######## Intermediate-level routine   BLMIX: Mixing in the boundary layer ################################### The vertical fluxes :math:`\overline{wx}` of momentum and tracer properties :math:`X` is composed of a gradient-flux term (proportional to the vertical property divergence :math:`\partial_z X`), and a “nonlocal” term :math:`\gamma_x` that enhances the gradient-flux mixing coefficient :math:`K_x` .. math:: \overline{wx}(d) \, = \, -K_x \left( \frac{\partial X}{\partial z} \, - \, \gamma_x \right) - *Boundary layer mixing profile* It is expressed as the product of the boundary layer depth :math:`h`, a depth-dependent turbulent velocity scale :math:`w_x(\sigma)` and a non-dimensional shape function :math:`G(\sigma)` .. math:: K_x(\sigma) \, = \, h \, w_x(\sigma) \, G(\sigma) with dimensionless vertical coordinate :math:`\sigma = d/h`. For details of :math:` w_x(\sigma)` and :math:`G(\sigma)` we refer to . - *Nonlocal mixing term* The nonlocal transport term :math:`\gamma` is nonzero only for tracers in unstable (convective) forcing conditions. Thus, depending on the stability parameter :math:`\zeta = d/L` (with depth :math:`d`, Monin-Obukhov length scale :math:`L`) it has the following form: .. math:: \begin{aligned} \begin{array}{cl} \gamma_x \, = \, 0 & \zeta \, \ge \, 0 \\ ~ & ~ \\ \left. \begin{array}{c} \gamma_m \, = \, 0 \\ ~ \\ \gamma_s \, = \, C_s \frac{\overline{w s_0}}{w_s(\sigma) h} \\ ~ \\ \gamma_{\theta} \, = \, C_s \frac{\overline{w \theta_0}+\overline{w \theta_R}}{w_s(\sigma) h} \\ \end{array} \right\} & \zeta \, < \, 0 \\ \end{array}\end{aligned} In practice, the routine peforms the following tasks: #. compute velocity scales at hbl #. find the interior viscosities and derivatives at hbl #. compute turbulent velocity scales on the interfaces #. compute the dimensionless shape functions at the interfaces #. compute boundary layer diffusivities at the interfaces #. compute nonlocal transport term #. find diffusivities at kbl-1 grid level RI\_IWMIX: Mixing in the interior ################################# Compute interior viscosity and diffusivity coefficients due to - shear instability (dependent on a local gradient Richardson number), - to background internal wave activity, and - to static instability (local Richardson number :math:`<` 0). TO BE CONTINUED. BLDEPTH: Boundary layer depth calculation: ########################################## The oceanic planetary boundary layer depth, ``hbl``, is determined as the shallowest depth where the bulk Richardson number is equal to the critical value, ``Ricr``. Bulk Richardson numbers are evaluated by computing velocity and buoyancy differences between values at zgrid(kl) < 0 and surface reference values. In this configuration, the reference values are equal to the values in the surface layer. When using a very fine vertical grid, these values should be computed as the vertical average of velocity and buoyancy from the surface down to epsilon\*zgrid(kl). When the bulk Richardson number at k exceeds Ricr, hbl is linearly interpolated between grid levels zgrid(k) and zgrid(k-1). The water column and the surface forcing are diagnosed for stable/ustable forcing conditions, and where hbl is relative to grid points (caseA), so that conditional branches can be avoided in later subroutines. TO BE CONTINUED. KPP\_CALC\_DIFF\_T/\_S, KPP\_CALC\_VISC: ######################################## Add contribution to net diffusivity/viscosity from KPP diffusivity/viscosity. TO BE CONTINUED. KPP\_TRANSPORT\_T/\_S/\_PTR: ############################ Add non local KPP transport term (ghat) to diffusive temperature/salinity/passive tracer flux. The nonlocal transport term is nonzero only for scalars in unstable (convective) forcing conditions. TO BE CONTINUED. Implicit time integration ######################### TO BE CONTINUED. Penetration of shortwave radiation ################################## TO BE CONTINUED. .. _ssub_phys_pkg_kpp_flowchart: Flow chart ++++++++++ :: C !CALLING SEQUENCE: c ... c kpp_calc (TOP LEVEL ROUTINE) c | c |-- statekpp: o compute all EOS/density-related arrays c | o uses S/R FIND_ALPHA, FIND_BETA, FIND_RHO c | c |-- kppmix c | |--- ri_iwmix (compute interior mixing coefficients due to constant c | | internal wave activity, static instability, c | | and local shear instability). c | | c | |--- bldepth (diagnose boundary layer depth) c | | c | |--- blmix (compute boundary layer diffusivities) c | | c | |--- enhance (enhance diffusivity at interface kbl - 1) c | o c | c |-- swfrac c o .. _ssub_phys_pkg_kpp_diagnostics: KPP diagnostics +++++++++++++++ Diagnostics output is available via the diagnostics package (see Section [sec:pkg:diagnostics]). Available output fields are summarized here: :: ------------------------------------------------------ <-Name->|Levs|grid|<-- Units -->|<- Tile (max=80c) ------------------------------------------------------ KPPviscA| 23 |SM |m^2/s |KPP vertical eddy viscosity coefficient KPPdiffS| 23 |SM |m^2/s |Vertical diffusion coefficient for salt & tracers KPPdiffT| 23 |SM |m^2/s |Vertical diffusion coefficient for heat KPPghat | 23 |SM |s/m^2 |Nonlocal transport coefficient KPPhbl | 1 |SM |m |KPP boundary layer depth, bulk Ri criterion KPPmld | 1 |SM |m |Mixed layer depth, dT=.8degC density criterion KPPfrac | 1 |SM | |Short-wave flux fraction penetrating mixing layer Reference experiments +++++++++++++++++++++ lab\_sea: natl\_box: References ++++++++++ Experiments and tutorials that use kpp ++++++++++++++++++++++++++++++++++++++ - Labrador Sea experiment, in lab\_sea verification directory