# Changeset 11678

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Timestamp:
2019-10-11T00:15:35+02:00 (11 months ago)
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 r11677 The OSMOSIS model is fundamentally based on results of Large Eddy Simulations (LES) of Langmuir turbulence and aims to fully describe this Langmuir regime. this Langmuir regime. The description in this section is of necessity incomplete and further details are available in the manuscript The OSMOSIS scheme'', Grant. A (2019); in prep. The OSMOSIS turbulent closure scheme is a similarity-scale scheme in \end{center} \end{figure} The pycnocline is shallow but important, since here the turbulent OSBL interacts with the underlying ocean. In a finite difference model the pycnocline must be at least one model level thick. The pycnocline in the OSMOSIS scheme is assumed to have a finite thickness, and may include a number of model levels. This means that the OSMOSIS scheme must parametrize both the thickness of the pycnocline, and the turbulent fluxes within the pycnocline. Consideration of the power input by wind acting on the Stokes drift suggests the Langmuir velocity scale: The pycnocline in the OSMOSIS scheme is assumed to have a finite thickness, and may include a number of model levels. This means that the OSMOSIS scheme must parametrize both the thickness of the pycnocline, and the turbulent fluxes within the pycnocline. Consideration of the power input by wind acting on the Stokes drift suggests that the Langmuir turbulence has velocity scale: \label{eq:w_La} w_{*L}= \left(u_*^2 u_{s0}\right)^{1/3}; w_{*L}= \left(u_*^2 u_{s\,0}\right)^{1/3}; this is the Where the mixed-layer is stable, a composite velocity scale is assumed: but at times the Stokes drift may be weak due to e.g.\ ice cover, short fetch, misalignment with the surface stress, etc.\ so  a composite velocity scale is assumed for the stable (warming) boundary layer: \label{eq:composite-nu} \nu_{\ast}= \left{}u_*^3 \left[\right]1-exp(-1.5 \mathrm{La}_t^2})\right]+w_{*L}^3\right}^{1/3} \nu_{\ast}= \left\{ u_*^3 \left[1-\exp(-1.5 \mathrm{La}_t^2})\right]+w_{*L}^3\right\}^{1/3}. For the unstable boundary layer this is merged with the standard convective velocity scale $w_{*C}=\left(\overline{w^\prime b^\prime}_0 \,h_\mathrm{ml}\right)^{1/3}$, where $\overline{w^\prime b^\prime}_0$ is the upwards surface buoyancy flux to give: \label{eq:vel-scale-unstable} \omega_* = \left(\nu_*^3 + 0.5 w_{*C}^3\right)^{1/3}. \subsubsection{The flux gradient model} The flux-gradient relationships used in the OSMOSIS scheme take the form, The flux-gradient relationships used in the OSMOSIS scheme take the form: % \label{eq:flux-grad-gen} \overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t} \overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t}, % where $\chi$ is a general variable and $N_{\chi,s}, N_{\chi,b} \mathrm{and} N_{\chi,t}$  are the non-gradient terms, and represent the effects of the different terms in the turbulent flux-budget on the transport of $\chi$. $N_{\chi,s}$ represents the effects that the Stokes shear has on the transport of $\chi$, $N_{\chi,b}$  the effect of buoyancy, and $N_{\chi,t}$ the effect of the turbulent transport.  The same general form for the flux-gradient relationship is used to parametrize the transports of momentum, heat and salinity. In terms of the non-dimensionalized depth variables % \label{eq:sigma} \sigma_{\mathrm{ml}}= -z/h_{\mathrm{ml}}; \;\sigma_{\mathrm{bl}}= -z/h_{\mathrm{bl}}, % in unstable conditions the eddy diffusivity ($K_d$) and eddy viscosity ($K_\nu$) profiles are parametrized as: % \begin{align}\label{eq:diff-unstable} K_d=&0.8\, \omega_*\, h_{\mathrm{ml}} \, \sigma_{\mathrm{ml}} \left(1-\beta_d \sigma_{\mathrm{ml}}\right)^{3/2} \\\label{eq:visc-unstable} K_\nu =& 0.3\, \omega_* \,h_{\mathrm{ml}}\, \sigma_{\mathrm{ml}} \left(1-\beta_\nu \sigma_{\mathrm{ml}}\right)\left(1-\tfrac{1}{2}\sigma_{\mathrm{ml}}^2\right) \end{align} % where $\beta_d$ and $\beta_\nu$ are parameters that are determined by matching Eqs \ref{eq:diff-unstable} and \ref{eq:visc-unstable} to the eddy diffusivity and viscosity at the base of the well-mixed layer, given by % \label{eq:diff-wml-base} K_{d,\mathrm{ml}}=K_{\nu,\mathrm{ml}}=\,0.16\,\omega_* \Delta h. % For stable conditions the eddy diffusivity/viscosity profiles are given by: % \begin{align}\label{diff-stable} K_d= & 0.75\,\, \nu_*\, h_{\mathrm{ml}}\,\,  \exp\left[-2.8 \left(h_{\mathrm{bl}}/L_L\right)^2\right]\sigma_{\mathrm{ml}} \left(1-\sigma_{\mathrm{ml}}\right)^{3/2} \\\label{eq:visc-stable} K_\nu = & 0.375\,\,  \nu_*\, h_{\mathrm{ml}} \,\, \exp\left[-2.8 \left(h_{\mathrm{bl}}/L_L\right)^2\right] \sigma_{\mathrm{ml}} \left(1-\sigma_{\mathrm{ml}}\right)\left(1-\tfrac{1}{2}\sigma_{\mathrm{ml}}^2\right). \end{align} % The shape of the eddy viscosity and diffusivity profiles is the same as the shape in the unstable OSBL. The eddy diffusivity/viscosity depends on the stability parameter $h_{\mathrm{bl}}/{L_L}$ where $L_L$ is analogous to the Obukhov length, but for Langmuir turbulence: \label{eq:diff-wml-base} L_L=-w_{*L}^3/\left<\overline{w^\prime b^\prime}\right>_L, with the mean turbulent buoyancy flux averaged over the boundary layer given in terms of its surface value $\overline{w^\prime b^\prime}}_0$ and (downwards) )solar irradiance $I(z)$ by \label{eq:stable-av-buoy-flux} \left<\overline{w^\prime b^\prime}\right>_L = \tfrac{1}{2} {\overline{w^\prime b^\prime}}_0-g\alpha_E\left[\tfrac{1}{2}(I(0)+I(-h))-\left\right]. % In unstable conditions the eddy diffusivity and viscosity depend on stability through the velocity scale $\omega_*$, which depends on the two velocity scales $\nu_*$ and $w_{*C}$. Details of the non-gradient terms in \eqref{eq:flux-grad-gen} and of the fluxes within the pycnocline \$-h_{\mathrm{bl}}