8.4.2. KPP: Nonlocal KProfile Parameterization for Vertical Mixing¶
Authors: Dimitris Menemenlis and Patrick Heimbach
8.4.2.1. Introduction¶
The nonlocal KProfile Parameterization (KPP) scheme of [lareta: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 doublediffusion (not implemented here).
 a boundary layer depth \(h\) 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 boundarylayer 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 “nonlocal” term \(\hat{\gamma}\) 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. [lareta:97]) and is now common in many ocean models.
The current code originates in the NCAR NCOM 1D 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 KPPderived diffusivities, viscosities and boundarylayer/mixedlayer depths ([sec:pkg:kpp:diagnostics]).
8.4.2.2. 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 addingkpp
to it,  or using
genmake2
addingenable=kpp
ordisable=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
Table 8.5 summarizes them.
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 
8.4.2.3. Runtime parameters¶
Runtime parameters are set in files data.pkg
and data.kpp
which
are read in kpp_readparms.F
. Runtime parameters may be broken into
3 categories: (i) switching on/off the package at runtime, (ii) required
MITgcm flags, (iii) package flags and parameters.
8.4.2.3.1. Enabling the package¶
The KPP package is switched on at runtime by setting useKPP = .TRUE.
in data.pkg
.
8.4.2.3.2. 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 
8.4.2.3.3. Package flags and parameters¶
Table 8.6 summarizes the runtime flags
that are set in data.pkg
, and their default values.
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 nonlocal 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  5.2E5 s^{–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 MoninObukhov 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  nondim. coeff. for velocity scale of turbulant velocity shear ( = function of concv,concs,epsilon,vonk,Ricr)  
Boundary layer mixing parameters (S/R blmix)  
cstar  proportionality coefficient for nonlocal transport  
cg  nondimensional coefficient for countergradient 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  0.2E4 s^{–2}  BruntVäisalä squared 
difm0  0.005 m^{2} s^{–1}  viscosity max. due to shear instability 
difs0  0.005 m\(^2\)/s  tracer diffusivity max. due to shear instability 
dift0  0.005 m\(^2\)/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 
8.4.2.4. 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 [lareta:94].
8.4.2.4.1. KPP_CALC:¶
Toplevel routine.
8.4.2.4.2. KPP_MIX:¶
Intermediatelevel routine
8.4.2.4.3. BLMIX: Mixing in the boundary layer¶
The vertical fluxes \(\overline{wx}\) of momentum and tracer properties \(X\) is composed of a gradientflux term (proportional to the vertical property divergence \(\partial_z X\)), and a “nonlocal” term \(\gamma_x\) that enhances the gradientflux mixing coefficient \(K_x\)
Boundary layer mixing profile It is expressed as the product of the boundary layer depth \(h\), a depthdependent turbulent velocity scale \(w_x(\sigma)\) and a nondimensional shape function \(G(\sigma)\)
\[K_x(\sigma) \, = \, h \, w_x(\sigma) \, G(\sigma)\]with dimensionless vertical coordinate \(\sigma = d/h\). For details of :math:` w_x(sigma)` and \(G(\sigma)\) we refer to .
Nonlocal mixing term The nonlocal transport term \(\gamma\) is nonzero only for tracers in unstable (convective) forcing conditions. Thus, depending on the stability parameter \(\zeta = d/L\) (with depth \(d\), MoninObukhov length scale \(L\)) it has the following form:
\[\begin{split}\begin{aligned} \begin{array}{cl} \gamma_x \, = \, 0 & \zeta \, \ge \, 0 \\ ~ & ~ \\ \left. \begin{array}{c} \gamma_m \, = \, 0 \\ ~ \\ \gamma_s \, = \, C_s \frac{\overline{w s_0}}{w_s(\sigma) h} \\ ~ \\ \gamma_{\theta} \, = \, C_s \frac{\overline{w \theta_0}+\overline{w \theta_R}}{w_s(\sigma) h} \\ \end{array} \right\} & \zeta \, < \, 0 \\ \end{array}\end{aligned}\end{split}\]
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 kbl1 grid level
8.4.2.4.4. 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 \(<\) 0).
TO BE CONTINUED.
8.4.2.4.5. 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(k1).
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.
8.4.2.4.6. KPP_CALC_DIFF_T/_S, KPP_CALC_VISC:¶
Add contribution to net diffusivity/viscosity from KPP diffusivity/viscosity.
TO BE CONTINUED.
8.4.2.4.7. 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.
8.4.2.4.8. Implicit time integration¶
TO BE CONTINUED.
8.4.2.4.9. Penetration of shortwave radiation¶
TO BE CONTINUED.
8.4.2.5. Flow chart¶
C !CALLING SEQUENCE:
c ...
c kpp_calc (TOP LEVEL ROUTINE)
c 
c  statekpp: o compute all EOS/densityrelated 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
8.4.2.6. KPP diagnostics¶
Diagnostics output is available via the diagnostics package (see Section [sec:pkg:diagnostics]). Available output fields are summarized here:

<Name>Levsgrid< 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  Shortwave flux fraction penetrating mixing layer
8.4.2.8. References¶
8.4.2.9. Experiments and tutorials that use kpp¶
 Labrador Sea experiment, in lab_sea verification directory