Vertical Mixing Parameterizations

Vertical Mixing Parameterizations

ROMS contains a variety of methods for setting the vertical viscous and diffusive coefficients. The choices range from simply choosing fixed values to the KPP, generic lengthscale (GLS) and Mellor-Yamada turbulence closure schemes. See Large (1998) for a review of surface ocean mixing schemes. Many schemes have a background molecular value which is used when the turbulent processes are assumed to be small (such as in the interior).

K-Profile Parameterization

The vertical mixing parameterization introduced by Large, McWilliams and Doney (1994) is a versatile first order scheme which has been shown to perform well in open ocean settings. Its design facilitates experimentation with additional or modified representations of specific turbulent processes.

Surface boundary layer

The Large, McWilliams and Doney scheme (LMD) matches separate parameterizations for vertical mixing of the surface boundary layer and the ocean interior. A formulation based on boundary layer similarity theory is applied in the water column above a calculated boundary layer depth $h_{sbl}$. This parameterization is then matched at the interior with schemes to account for local shear, internal wave and double diffusive mixing effects.

Viscosity and diffusivities at model levels above a calculated surface boundary layer depth ($h_{sbl}$ ) are expressed as the product of the length scale $h_{sbl}$, a turbulent velocity scale $w_{x}$ and a non-dimensional shape function.

$$\nu _{x}=h_{sbl}w_{x}(\sigma )G_{x}(\sigma )$$

(1)

where $\sigma$ is a non-dimensional coordinate ranging from 0 to 1 indicating depth within the surface boundary layer. The $x$ subscript stands for one of momentum, temperature and salinity.

Surface Boundary layer depth

The boundary layer depth $h_{sbl}$ is calculated as the minimum of the Ekman depth, estimated as,

$$h_{e}=0.7u_{*}/f$$

(2)

(where $u_{*}$ is the friction velocity $u_{*}={\sqrt {\tau _{x}^{2}+\tau _{y}^{2}}}/\rho$ ), the Monin-Obukhov depth:

$$L=u_{*}^{3}/(\kappa B_{f})$$

(3)

(where $\kappa =0.4$ is von Karman's contant and $B_{f}$ is the surface buoyancy flux), and the shallowest depth at which a critical bulk Richardson number is reached. The critical bulk Richardson number ($Ri_{c}$) is typically in the range 0.25--0.5. The bulk Richardson number ($Ri_{b}$) is calculated as:

$$Ri_{b}(z)={\frac {(B_{r}-B(d))d}{|{\vec {V}}_{r}-{\vec {V}}(d)|^{2}+{V_{t}}^{2}(d)}}$$

(4)

where $d$ is distance down from the surface, $B$ is the buoyancy, $B_{r}$ is the buoyancy at a near surface reference depth, ${\vec {V}}$ is the mean horizontal velocity, ${\vec {V}}_{r}$ the velocity at the near surface reference depth and $V_{t}$ is an estimate of the turbulent velocity contribution to velocity shear.

The turbulent velocity shear term in this equation is given by LMD as,

$$V_{t}^{2}(d)={\frac {C_{v}(-\beta _{T})^{1/2}}{Ri_{c}\kappa }}(c_{s}\epsilon )^{-1/2}dNw_{s}$$

(5)

where $C_{v}$ is the ratio of interior $N$ to $N$ at the entrainment depth, $\beta _{T}$ is ratio of entrainment flux to surface buoyancy flux, $c_{s}$ and $\epsilon$ are constants, and $w_{s}$ is the turbulent velocity scale for scalars. LMD derive (5) based on the expected behavior in the pure convective limit. The empirical rule of convection states that the ratio of the surface buoyancy flux to that at the entrainment depth be a constant. Thus the entrainment flux at the bottom of the boundary layer under such conditions should be independent of the stratification at that depth. Without a turbulent shear term in the denominator of the bulk Richardson number calculation, the estimated boundary layer depth is too shallow and the diffusivity at the entrainment depth is too low to obtain the necessary entrainment flux. Thus by adding a turbulent shear term proportional to the stratification in the denominator, the calculated boundary layer depth will be deeper and will lead to a high enough diffusivity to satisfy the empirical rule of convection.

Turbulent velocity scale

To estimate $w_{x}$ (where $x$ is $m$ - momentum or $s$ - any scalar) throughout the boundary layer, surface layer similarity theory is utilized. Following an argument by Troen and Mahrt (1986), Large et al. estimate the velocity scale as

$$we_{x}={\frac {\kappa u_{*}}{\phi _{x}(\zeta )}}$$

(6)

where $\zeta$ is the surface layer stability parameter defined as $z/L$. $\phi _{x}$ is a non-dimensional flux profile which varies based on the stability of the boundary layer forcing. The stability parameter used in this equation is assumed to vary over the entire depth of the boundary layer in stable and neutral conditions. In unstable conditions it is assumed only to vary through the surface layer which is defined as $\epsilon h_{sbl}$ (where $\epsilon$ is set at 0.10) . Beyond this depth $\zeta$ is set equal to its value at $\epsilon h_{sbl}$.

The flux profiles are expressed as analytical fits to atmospheric surface boundary layer data. In stable conditions they vary linearly with the stability parameter $\zeta$ as

$$\phi _{x}=1+5\zeta$$

(7)

In near-neutral unstable conditions common Businger-Dyer forms are used which match with the formulation for stable conditions at $\zeta =0$. Near neutral conditions are defined as

$$-0.2\leq \zeta <0$$

for momentum and,

$$-1.0\leq \zeta <0$$

for scalars. The non dimensional flux profiles in this regime are,

$${\begin{aligned}\phi _{m}&=(1-16\zeta )^{1/4}\\\phi _{s}&=(1-16\zeta )^{1/2}\end{aligned}}$$

(8)

In more unstable conditions $\phi _{x}$ is chosen to match the Businger-Dyer forms and with the free convective limit. Here the flux profiles are

$${\begin{aligned}\phi _{m}&=(1.26-8.38\zeta )^{1/3}\\\phi _{s}&=(-28.86-98.96\zeta )^{1/3}\end{aligned}}$$

(9)

The shape function

The non-dimensional shape function $G(\sigma )$ is a third order polynomial with coefficients chosen to match the interior viscosity at the bottom of the boundary layer and Monin-Obukhov similarity theory approaching the surface. This function is defined as a 3rd order polynomial.

$$G(\sigma )=a_{o}+a_{1}\sigma +a_{2}\sigma ^{2}+a_{3}\sigma ^{3}$$

(10)

with the coefficients specified to match surface boundary conditions and to smoothly blend with the interior,

$${\begin{aligned}a_{o}&=0\\a_{1}&=1\\a_{2}&=-2+3{\frac {\nu _{x}(h_{sbl})}{hw_{x}(1)}}+{\frac {\partial _{x}\nu _{x}(h)}{w_{x}(1)}}+{\frac {\nu _{x}(h)\partial _{\sigma }w_{x}(1)}{hw_{x}^{2}(1)}}\\a_{3}&=1-2{\frac {\nu _{x}(h_{sbl})}{hw_{x}(1)}}-{\frac {\partial _{x}\nu _{x}(h)}{w_{x}(1)}}-{\frac {\nu _{x}(h)\partial _{\sigma }w_{x}(1)}{hw_{x}^{2}(1)}}\end{aligned}}$$

(11)

where $\nu _{x}(h)$ is the viscosity calculated by the interior parameterization at the boundary layer depth.

Countergradient flux term

The second term of the LMD scheme's surface boundary layer formulation is the non-local transport term $\gamma$ which can play a significant role in mixing during surface cooling events. This is a redistribution term included in the tracer equation separate from the diffusion term and is written as

$$-{\frac {\partial }{\partial z}}K\gamma \,.$$

(12)

LMD base their formulation for non-local scalar transport on a parameterization for pure free convection from Mailhôt and Benoit (1982). They extend this parameterization to cover any unstable surface forcing conditions to give

$$\gamma _{T}=C_{s}{\frac {{\overline {wT_{0}}}+{\overline {wT_{R}}}}{w_{T}(\sigma )h}}$$

(13)

for temperature and

$$\gamma _{S}=C_{s}{\frac {\overline {wS_{0}}}{w_{S}(\sigma )h}}$$

(14)

for salinity (other scalar quantities with surface fluxes can be treated similarly). LMD argue that although there is evidence of non-local transport of momentum as well, the form the term would take is unclear so they simply specify $\gamma _{m}=0$.

The interior scheme

The interior scheme of Large, McWilliams and Doney estimates the viscosity coefficient by adding the effects of several generating mechanisms: shear mixing, double-diffusive mixing and internal wave generated mixing.

$$\nu _{x}(d)=\nu _{x}^{s}+\nu _{x}^{d}+\nu _{x}^{w}$$

(15)

Shear generated mixing

The shear mixing term is calculated using a gradient Richardson number formulation, with viscosity estimated as:

$$\nu _{x}^{s}={\begin{cases}\nu _{0}&Ri_{g}<0\,,\\\nu _{0}[1-(Ri_{g}/Ri_{0})^{2}]^{3}&0<Ri_{g}<Ri_{0}\,,\\0&Ri_{g}>Ri_{0}\,.\end{cases}}$$

(16)

where $\nu _{0}$ is $5.0\times 10^{-3}$, $Ri_{0}=0.7$.

Double diffusive processes

The second component of the interior mixing parameterization represents double diffusive mixing. From limited sources of laboratory and field data LMD parameterize the salt fingering case ($R_{\rho }>1.0$)

$${\begin{aligned}\nu _{s}^{d}(R_{\rho })&={\begin{cases}1\times 10^{-4}[1-({\frac {(R_{\rho }-1}{R_{\rho }^{0}-1}})^{2})^{3}&{\text{for}}\;1.0<R_{\rho }<R_{\rho }^{0}=1.9,\\0&{\text{otherwise.}}\end{cases}}\\\\\nu _{\theta }^{d}(R_{\rho })&=0.7\nu _{s}^{d}\end{aligned}}$$

(17)

For diffusive convection ($0<R_{\rho }<1.0$) LMD suggest several formulations from the literature and choose the one with the most significant impact on mixing (Fedorov 1988).

$$\nu _{\theta }^{d}=(1.5\times 10^{-6})(0.909\exp(4.6\exp[-0.54(R_{\rho }^{-1}-1)])$$

(18)

for temperature. For other scalars,

$$\nu _{s}^{d}={\begin{cases}\nu _{\theta }^{d}(1.85-0.85R_{\rho }^{-1})R_{\rho }&{\text{for}}0.5<=R_{\rho }<1.0,\\\nu _{\theta }^{d}0.15R_{\rho }&{\text{otherwise.}}\end{cases}}$$ !--\eqno{(19)}-->

(19)

Internal wave generated mixing

Internal wave generated mixing serves as the background mixing in the LMD scheme. It is specified as a constant for both scalars and momentum. Eddy diffusivity is estimated based on the data of Ledwell et al. (1993), while Peters et al. (1988) suggest eddy viscosity should be 7 to 10 times larger than diffusivity for gradient Richardson numbers below approximately 0.7. Therefore LMD use

$${\begin{aligned}\nu _{m}^{w}&=1.0\times 10^{-4}m^{2}s^{-1}\\\nu _{s}^{w}&=1.0\times 10^{-5}m^{2}s^{-1}\end{aligned}}$$

(20)

Mellor-Yamada 2.5

One of the more popular closure schemes is that of Mellor and Yamada (1982). They actually present a hierarchy of closures of increasing complexity. ROMS provides only the "Level 2.5" closure with the Galperin et al. (1988) modifications as described in Allen et al. (1995). This closure scheme adds two prognostic equations, one for the turbulent kinetic energy (${1 \over 2}q^{2}$) and one for the turbulent kinetic energy times a length scale ($q^{2}l$).

The turbulent kinetic energy equation is:

$${D \over Dt}\left({q^{2} \over 2}\right)-{\partial \over \partial z}\left[K_{q}{\partial \over \partial z}\left({q^{2} \over 2}\right)\right]=P_{s}+P_{b}-\xi _{d}$$

(21)

where $P_{s}$ is the shear production, $P_{b}$ is the buoyant production and $\xi _{d}$ is the dissipation of turbulent kinetic energy. These terms are given by

$${\begin{aligned}P_{s}&=K_{m}\left[\left({\partial u \over \partial z}\right)^{2}+\left({\partial v \over \partial z}\right)^{2}\right],\\P_{b}&=-K_{s}N^{2},\\\xi _{d}&={q^{3} \over B_{1}l}\end{aligned}}$$

(22)

where $B_{1}$ is a constant. One can also add a traditional horizontal Laplacian or biharmonic diffusion (${\cal {D}}_{q}$) to the turbulent kinetic energy equation. The form of this equation in the model coordinates becomes

$${\partial \over \partial t}\left({H_{z}q^{2} \over mn}\right)+{\partial \over \partial \xi }\left({H_{z}uq^{2} \over n}\right)+{\partial \over \partial \eta }\left({H_{z}vq^{2} \over m}\right)+{\partial \over \partial s}\left({H_{z}\Omega q^{2} \over mn}\right)-{\partial \over \partial s}\left({K_{q} \over mnH_{z}}{\partial q^{2} \over \partial s}\right)=$$

$${2H_{z}K_{m} \over mn}\left[\left({\partial u \over \partial z}\right)^{2}+\left({\partial v \over \partial z}\right)^{2}\right]+{2H_{z}K_{s} \over mn}N^{2}-{2H_{z}q^{3} \over mnB_{1}l}+{H_{z} \over mn}{\cal {D}}_{q}\,.$$

(23)

The vertical boundary conditions are:

top ($z=\zeta (x,y,t))$:

$${\begin{aligned}&{H_{z}\Omega \over mn}=0\\&{K_{q} \over mnH_{z}}\,{\frac {\partial q^{2}}{\partial s}}={B_{1}^{2/3} \over \rho _{o}}\left[\left(\tau _{s}^{\xi }\right)^{2}+\left(\tau _{s}^{\eta }\right)^{2}\right]\\&H_{z}K_{m}\left({\partial u \over \partial z},{\partial v \over \partial z}\right)={1 \over \rho _{o}}\left(\tau _{s}^{\xi },\tau _{s}^{\eta }\right)\\&H_{z}K_{s}N^{2}={Q \over \rho _{o}c_{P}}\end{aligned}}$$

and bottom ($z=-h(x,y)$):

$${\begin{aligned}&{H_{z}\Omega \over mn}=0\\&{K_{q} \over mnH_{z}}{\partial q^{2} \over \partial s}={B_{1}^{2/3} \over \rho _{o}}\left[\left(\tau _{b}^{\xi }\right)^{2}+\left(\tau _{b}^{\eta }\right)^{2}\right]\\&H_{z}K_{m}\left({\partial u \over \partial z},{\partial v \over \partial z}\right)={1 \over \rho _{o}}\left(\tau _{b}^{\xi },\tau _{b}^{\eta }\right)\\&H_{z}K_{s}N^{2}=0\end{aligned}}$$

There is also an equation for the turbulent length scale $l$:

$${D \over Dt}\left({lq^{2}}\right)-{\partial \over \partial z}\left[K_{l}{\partial lq^{2} \over \partial z}\right]=lE_{1}(P_{s}+P_{b})-{q^{3} \over B_{1}}{\tilde {W}}$$

(24)

where ${\tilde {W}}$ is the wall proximity function:

$${\begin{aligned}{\tilde {W}}&=1+E_{2}\left({l \over kL}\right)^{2}\\L^{-1}&={1 \over \zeta -z}+{1 \over H+z}\end{aligned}}$$

(25)

The form of this equation in the model coordinates becomes

$${\partial \over \partial t}\left({H_{z}q^{2}l \over mn}\right)+{\partial \over \partial \xi }\left({H_{z}uq^{2}l \over n}\right)+{\partial \over \partial \eta }\left({H_{z}vq^{2}l \over m}\right)+{\partial \over \partial s}\left({H_{z}\Omega q^{2}l \over mn}\right)-{\partial \over \partial s}\left({K_{q} \over mnH_{z}}{\partial q^{2}l \over \partial s}\right)=$$

$${H_{z} \over mn}lE_{1}(P_{s}+P_{b})-{H_{z}q^{3} \over mnB_{1}}{\tilde {W}}+{H_{z} \over mn}{\cal {D}}_{ql}\,.$$

(26)

where ${\cal {D}}_{ql}$ is the horizontal diffusion of the quantity $q^{2}l$. Equations (23) and (26) are timestepped much like the model tracer equations, including an implicit solve for the vertical operations and options for centered second or fourth-order advection. They are timestepped with a predictor-corrector scheme in which the predictor step is only computing the advection.

Given these solutions for $q$ and $l$, the vertical viscosity and diffusivity coefficients are:

$${\begin{aligned}K_{m}&=qlS_{m}+K_{m_{\rm {background}}}\\K_{s}&=qlS_{h}+K_{s_{\rm {background}}}\\K_{q}&=qlS_{q}+K_{q_{\rm {background}}}\end{aligned}}$$

(27)

and the stability coefficients $S_{m}$, $S_{h}$ and $S_{q}$ are found by solving

$$S_{s}\left[1-(3A_{2}B_{2}+18A_{1}A_{2})G_{h}\right]=A_{2}\left[1-6A_{1}B_{1}^{-1}\right]$$

(28)

$$S_{m}\left[1-9A_{1}A_{2}G_{h}\right]-S_{s}\left[G_{h}(18A_{1}^{2}+9A_{1}A_{2})G_{h}\right]=A_{1}\left[1-3C_{1}-6A_{1}B_{1}^{-1}\right]$$

(29)

$$G_{h}=\min(-{l^{2}N^{2} \over q^{2}},0.028)\,.$$

(30)

$$S_{q}=0.41S_{m}$$

(31)

The constants are set to $(A_{1},A_{2},B_{1},B_{2},C_{1},E_{1},E_{2})=(0.92,0.74,16.6,10.1,0.08,1.8,1.33)$. The quantities $q^{2}$ and $q^{2}l$ are both constrained to be no smaller than $10^{-8}$ while $l$ is set to be no larger than $0.53q/N$.

Generic Length Scale

Umlauf and Burchard (2003) have come up with a generic two-equation turbulence closure scheme which can be tuned to behave like several of the traditional schemes, including that of Mellor and Yamada 2.5 (above). This is known as the Generic Length Scale, or GLS vertical mixing scheme and was introduced to ROMS in Warner et al. (2005). Its parameters are set in the ROMS input file.

The first of Warner et al.'s equations is the same as equation (21) with $k=1/2q^{2}$. Their dissipation is given by

$$\epsilon =(c_{\mu }^{0})^{3+p/n}k^{3/2+m/n}\psi ^{-1/n}$$

(32)

where $\psi$ is a generic parameter that is used to establish the turbulence length scale. The equation for $\psi$ is:

$${D\psi \over Dt}={\partial \over \partial z}\left(K_{\psi }{\partial \psi \over \partial z}\right)+{\psi \over k}(c_{1}P_{s}+c_{3}P_{b}-c_{2}\epsilon F_{wall})$$ Coefficients $c_{1}$ and $c_{2}$ are chosen to be consistent with observations of decaying homogeneous, isotropic turbulence. The parameter $c_{3}$ has differing values for stable ($c_{3}^{+}$) and unstable ($c_{3}^{-}$) stratification. Also $${\begin{aligned}\psi &=(c_{\mu }^{0})^{p}k^{m}l^{n}\\l&=(c_{\mu }^{0})^{3}k^{3/2}\epsilon {-1}\end{aligned}}$$ (34) Depending on the choice of the various parameters, these two equations can be made to solve a variety of traditional two-equation turbulence closure models. The list of parameters is shown in the following table and is also given inside the comments section of the ROMS input file. $k-kl$ $k-\epsilon$ $k-\omega$ gen $\psi =k^{1}l^{1}$ $\psi =(c_{\mu }^{0})^{3}k^{3/2}l^{-1}$ $\psi =(c_{\mu }^{0})^{-1}k^{1/2}l^{-1}$ $\psi =(c_{\mu }^{0})^{2}k^{1}l^{-2/3}$ $p$ 0.0 3.0 -1.0 2.0 $m$ 1.0 1.5 0.5 1.0 $n$ 1.0 -1.0 -1.0 -0.67 $\sigma _{k}={K_{M} \over K_{k}}$ 1.96 1.0 2.0 0.8 $\sigma _{\psi }={K_{M} \over K_{\psi }}$ 1.96 1.3 2.0 1.07 $c_{1}$ 0.9 1.44 0.555 1.0 $c_{2}$ 0.52 1.92 0.833 1.22 $c_{3}^{-}$ 2.5 -0.4 -0.6 0.1 $c_{3}^{+}$ 1.0 1.0 1.0 1.0 $k_{min}$ 5.0e-6 7.6e-6 7.6e-6 1.0e-8 $\psi _{min}$ 5.0e-6 1.0e-12 1.0e-12 1.0e-8 $c_{\mu }^{0}$ 0.5544 0.5477 0.5477 0.5544