Changeset 11678


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2019-10-11T00:15:35+02:00 (11 months ago)
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agn
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  • NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex

    r11677 r11678  
    555555The OSMOSIS model is fundamentally based on results of Large Eddy 
    556556Simulations (LES) of Langmuir turbulence and aims to fully describe 
    557 this Langmuir regime. 
     557this 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. 
    558558 
    559559The OSMOSIS turbulent closure scheme is a similarity-scale scheme in 
     
    593593  \end{center} 
    594594\end{figure} 
    595 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. 
    596  
    597 Consideration of the power input by wind acting on the Stokes drift suggests the Langmuir velocity scale: 
     595The 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. 
     596 
     597Consideration of the power input by wind acting on the Stokes drift suggests that the Langmuir turbulence has velocity scale: 
    598598\begin{equation}\label{eq:w_La} 
    599 w_{*L}= \left(u_*^2 u_{s0}\right)^{1/3}; 
     599w_{*L}= \left(u_*^2 u_{s\,0}\right)^{1/3}; 
    600600\end{equation}  
    601 this is the  
    602 Where the mixed-layer is stable, a composite velocity scale is assumed: 
     601but 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: 
    603602\begin{equation}\label{eq:composite-nu} 
    604 \nu_{\ast}= \left{}u_*^3 \left[\right]1-exp(-1.5 \mathrm{La}_t^2})\right]+w_{*L}^3\right}^{1/3} 
    605 \end{equation}  
     603  \nu_{\ast}= \left\{ u_*^3 \left[1-\exp(-1.5 \mathrm{La}_t^2})\right]+w_{*L}^3\right\}^{1/3}. 
     604\end{equation} 
     605For 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: 
     606\begin{equation}\label{eq:vel-scale-unstable} 
     607\omega_* = \left(\nu_*^3 + 0.5 w_{*C}^3\right)^{1/3}. 
     608\end{equation} 
     609 
    606610\subsubsection{The flux gradient model} 
    607 The flux-gradient relationships used in the OSMOSIS scheme take the form, 
     611The flux-gradient relationships used in the OSMOSIS scheme take the form: 
     612% 
    608613\begin{equation}\label{eq:flux-grad-gen} 
    609 \overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t} 
    610 \end{equation}  
     614\overline{w^\prime\chi^\prime}=-K\frac{\partial\overline{\chi}}{\partial z} + N_{\chi,s} +N_{\chi,b} +N_{\chi,t}, 
     615\end{equation} 
     616% 
    611617where $\chi$ 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. 
     618 
     619In terms of the non-dimensionalized depth variables 
     620% 
     621\begin{equation}\label{eq:sigma} 
     622\sigma_{\mathrm{ml}}= -z/h_{\mathrm{ml}}; \;\sigma_{\mathrm{bl}}= -z/h_{\mathrm{bl}}, 
     623\end{equation} 
     624% 
     625in unstable conditions the eddy diffusivity ($K_d$) and eddy viscosity ($K_\nu$) profiles are parametrized as: 
     626% 
     627\begin{align}\label{eq:diff-unstable} 
     628K_d=&0.8\, \omega_*\, h_{\mathrm{ml}} \, \sigma_{\mathrm{ml}} \left(1-\beta_d \sigma_{\mathrm{ml}}\right)^{3/2} 
     629\\\label{eq:visc-unstable} 
     630K_\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) 
     631\end{align} 
     632% 
     633where $\beta_d$ 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 
     634% 
     635\begin{equation}\label{eq:diff-wml-base}  
     636K_{d,\mathrm{ml}}=K_{\nu,\mathrm{ml}}=\,0.16\,\omega_* \Delta h. 
     637\end{equation} 
     638% 
     639For stable conditions the eddy diffusivity/viscosity profiles are given by: 
     640% 
     641\begin{align}\label{diff-stable} 
     642K_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} 
     643K_\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). 
     644\end{align} 
     645% 
     646The 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: 
     647\begin{equation}\label{eq:diff-wml-base}  
     648  L_L=-w_{*L}^3/\left<\overline{w^\prime b^\prime}\right>_L, 
     649\end{equation} 
     650with 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 
     651\begin{equation} \label{eq:stable-av-buoy-flux} 
     652\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<I\right>\right]. 
     653\end{equation} 
     654% 
     655In unstable conditions the eddy diffusivity and viscosity depend on stability through the velocity scale $\omega_*$, which depends on the two velocity scales $\nu_*$ and $w_{*C}$. 
     656 
     657Details of the non-gradient terms in \eqref{eq:flux-grad-gen} and of the fluxes within the pycnocline $-h_{\mathrm{bl}}<z<h_{\mathrm{ml}}$ can be found in Grant (2019). 
     658 
     659\subsubsection{Evolution of the boundary layer depth} 
     660The prognostic equation for the depth of the neutral/unstable boundary layer is given by \citep{grant+etal18}, 
     661 
     662\begin{equation} \label{eq:dhdt-unstable} 
     663\frac{\partial h_\mathrm{bl}}{\partial t} + \mathbf{U}_b\nabla h_\mathrm{bl}= W_b - \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}} 
     664\end{equation} 
     665where $h_\mathrm{bl}$ is the horizontally-varying depth of the OSBL, $\mathbf{U}_b$ and $W_b$ are the mean horizontal and vertical velocities at the base of the OSBL, ${\overline{w^\prime b^\prime}}_\mathrm{ent}$ is the buoyancy flux due to entrainment and $\Delta B_\mathrm{bl}$ is the difference between the buoyancy, averaged over the depth of the OSBL, and the buoyancy just below the base of the OSBL. This equation is similar to that used in mixed-layer models \cite[e.g.][]{kraus+turner67}, in which the thickness of the pycnocline is taken to be zero. \cite{grant+etal18} show that this equation for $\partial h_\mathrm{bl}/\partial t$ can be obtained from the potential energy budget of the OSBL when the pycnocline has a finite thickness. Equation \ref{eq:dhdt-unstable} is the leading term in the parametrization.%The full equation obtained by \cite{grant+etal18} includes additional terms that depend on the thickness of the pycnocline, and increase the rate of deepening of the entraining OSBL by less than $\sim20$\%.  
     666 
     667The entrainment rate for the combination of convective and Langmuir turbulence is given by ,  
     668 
    612669 
    613670 
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