The K-Profile-Parameterization (KPP)

The K-Profile-Parameterization, or "KPP", is proposed by Large et al (1994) as a model for convective and wind-driven mixing in the ocean surface boundary layer. In KPP, vertical turbulent fluxes of a quantity $\phi$ are parameterized as

where $\Phi$ is the resolved or horizontally-averaged quantity, $K_\phi$ is a turbulent diffusivity, and $NL_\phi$ is a 'non-local' flux.

The non-local flux and turbulent diffusivity are defined to vanish at the surface, and at the bottom of the 'mixing layer' $h$, which roughly corresponds to the depth at which turbulent fluxes and turbulent kinetic energy decay to zero.

The various numerical implementations of 'KPP' have effectively resulted in the proliferation of practically distinct KPP models in different codes. Here we describe the implementation of KPP in OceanTurb, which is meant to follow the version of KPP described by the Community Vertical Mixing Project (CVMix).

The KPP model has three distinct parts:

1. A model for the mixing layer depth, $h$.
2. A model for the non-local flux, $\NL_\Phi$.
3. A model for the local diffusivity, $K_\Phi$.

Below, we denote 'model parameters' as $\C{\mathrm{label}}{\mathrm{var}}$, where 'label' describes the parameter and 'var' is a variable like $U, V, T$ or $S$.

Mixing depth model in CVMix KPP

The mixing layer depth $h$ is defined implicitly via the bulk Richardson number criterion

where the critical $\Ri$ is $\C{\Ri}{} = 0.3$. The operator $\Delta$ is defined

where $\C{\SL}{} = 0.1$ is the surface layer fraction. The function $\K(-h)$ parameterizes unresolved kinetic energy associated with convective plumes,

The unresolved kinetic energy constant is $\C{\K}{} = 4.32$ and the minimum unresolved kinetic energy is $\C{\K_0}{} = 10^{-11}$. To solve \eqref{bulk_ri} for $h$, we evaluate the right side of \eqref{bulk_ri} for $z < 0$ at increasing depths until the right side rises above the critical $\C{\Ri}{}$. We then linearly interpolate to find $h$.

'Countergradient' non-local flux model in CVMix KPP

The non-local flux is defined only for $T$ and $S$, and is

where $d = -z/h$ is a non-dimensional depth coordinate and $\C{\NL}{} = 6.33$.

$K$-Profile model in CVMix KPP

The KPP diffusivity is

where $\W_\Phi(d)$ is the turbulent velocity scale for variable $\phi$.

We define the ratios

In wind-driven turbulence under stable buoyancy forcing such that $Q_b < 0$, the turbulent velocity scale is

where $\C{\tau}{} = 0.4$ is the Von Karman constant, $\C{\mathrm{stab}}{} = 2.0$, and $\C{n}{}=1$.

In wind-driven turbulence but under destabilizing buoyancy forcing, when $\min \left [ \C{\SL}{}, d \right ] < \C{d}{\phi} r_\tau$, the turbulent velocity scale is

where $\C{m\tau}{U}= 1/4$, $\C{m\tau}{T} = 1/2$, $\C{\mathrm{unst}}{} = 6.4$, the transition parameter for velocities is $\C{d}{U} = 0.5$, and the transition parameter for tracers is $\C{d}{T} = 2.5$.

In convection-driven turbulence affected by wind mixing, when $\min \left [ \C{\SL}{}, d \right ] >= \C{d}{\phi} r_\tau$, the turbulent velocity scale is

where $\C{b}{U} = \C{b}{V} = 0.215$, $\C{b}{T} = \C{b}{S} = 2.53$, $\C{mb}{U} = \C{mb}{T}=1/3$, $\C{\tau b}{U} = \C{\tau b}{V} = 0.374$, and $\C{\tau b}{T} = \C{\tau b}{S} = -0.717$.

Selected tests

See /test/test_kpp.jl for more tests.

Linear temperature profile and no velocity field

Some simple tests can be defined when the model state is $U=V=S=0$ and $T = \gamma z$. In this case the buoyancy becomes $B = g \alpha \gamma z$ and the buoyancy gradient is $B_z = g \alpha \gamma$. If we further take $\C{\SL}{} \to 0$ and $Q_b > 0$, and note that the value of $T$ in the top grid cell is $T_N = -\gamma \Delta z / 2$, where $N$ is the number of grid points, we find that

and

The unresolved kinetic energy function is

The bulk Richardson number criterion then becomes

Modifying the temperature profile so that $T_N = B_N = 0$ allows us to analytically calculate the mixed layer depth:

Linear temperature profile and linear shear

Consider a piecewise-constant temperature profile

and a velocity profile

so that $T(z=-h) = U(z=-h) = 0$. We then have $\Delta T(-h) = T_0$ and $\Delta U^2(-h) = U_0^2$, so that with $g=\alpha=1$,

setting $h = 9$, $\C{\Ri}{}=1$, $T_0 = 1$, and $U_0=3$ yields a consistent solution.

Limiting cases for turbulent velocity scales

Under zero momentum forcing, the turbulent vertical velocity scale is

We write the test in \eqref{buoyscaletest} using the depth in \eqref{analyticaldepth}

Under zero buoyancy forcing, the turbulent velocity scale is

Table of model parameters

The default values for adjustable model parameters in KPP are

Note: all parameters are greater than 0, and $0 \ge \C{\SL}{} \ge 1$.

The default values for 'non-adjustable' parameters in KPP are

References

• Large et al (1994)
• CVMix documentation
• Van Roekel et al (2018)