Changeset 11678 for NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex
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 20191011T00:15:35+02:00 (5 years ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex
r11677 r11678 555 555 The OSMOSIS model is fundamentally based on results of Large Eddy 556 556 Simulations (LES) of Langmuir turbulence and aims to fully describe 557 this Langmuir regime. 557 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. 558 558 559 559 The OSMOSIS turbulent closure scheme is a similarityscale scheme in … … 593 593 \end{center} 594 594 \end{figure} 595 The pycnocline i s 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 th e Langmuirvelocity scale:595 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 that the Langmuir turbulence has velocity scale: 598 598 \begin{equation}\label{eq:w_La} 599 w_{*L}= \left(u_*^2 u_{s 0}\right)^{1/3};599 w_{*L}= \left(u_*^2 u_{s\,0}\right)^{1/3}; 600 600 \end{equation} 601 this is the 602 Where the mixedlayer is stable, a composite velocity scale is assumed: 601 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: 603 602 \begin{equation}\label{eq:compositenu} 604 \nu_{\ast}= \left{}u_*^3 \left[\right]1exp(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} 605 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: 606 \begin{equation}\label{eq:velscaleunstable} 607 \omega_* = \left(\nu_*^3 + 0.5 w_{*C}^3\right)^{1/3}. 608 \end{equation} 609 606 610 \subsubsection{The flux gradient model} 607 The fluxgradient relationships used in the OSMOSIS scheme take the form, 611 The fluxgradient relationships used in the OSMOSIS scheme take the form: 612 % 608 613 \begin{equation}\label{eq:fluxgradgen} 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 % 611 617 where $\chi$ is a general variable and $N_{\chi,s}, N_{\chi,b} \mathrm{and} N_{\chi,t}$ are the nongradient terms, and represent the effects of the different terms in the turbulent fluxbudget 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 fluxgradient relationship is used to parametrize the transports of momentum, heat and salinity. 618 619 In terms of the nondimensionalized 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 % 625 in unstable conditions the eddy diffusivity ($K_d$) and eddy viscosity ($K_\nu$) profiles are parametrized as: 626 % 627 \begin{align}\label{eq:diffunstable} 628 K_d=&0.8\, \omega_*\, h_{\mathrm{ml}} \, \sigma_{\mathrm{ml}} \left(1\beta_d \sigma_{\mathrm{ml}}\right)^{3/2} 629 \\\label{eq:viscunstable} 630 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) 631 \end{align} 632 % 633 where $\beta_d$ and $\beta_\nu$ are parameters that are determined by matching Eqs \ref{eq:diffunstable} and \ref{eq:viscunstable} to the eddy diffusivity and viscosity at the base of the wellmixed layer, given by 634 % 635 \begin{equation}\label{eq:diffwmlbase} 636 K_{d,\mathrm{ml}}=K_{\nu,\mathrm{ml}}=\,0.16\,\omega_* \Delta h. 637 \end{equation} 638 % 639 For stable conditions the eddy diffusivity/viscosity profiles are given by: 640 % 641 \begin{align}\label{diffstable} 642 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:viscstable} 643 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). 644 \end{align} 645 % 646 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: 647 \begin{equation}\label{eq:diffwmlbase} 648 L_L=w_{*L}^3/\left<\overline{w^\prime b^\prime}\right>_L, 649 \end{equation} 650 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 651 \begin{equation} \label{eq:stableavbuoyflux} 652 \left<\overline{w^\prime b^\prime}\right>_L = \tfrac{1}{2} {\overline{w^\prime b^\prime}}_0g\alpha_E\left[\tfrac{1}{2}(I(0)+I(h))\left<I\right>\right]. 653 \end{equation} 654 % 655 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 $w_{*C}$. 656 657 Details of the nongradient terms in \eqref{eq:fluxgradgen} 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} 660 The prognostic equation for the depth of the neutral/unstable boundary layer is given by \citep{grant+etal18}, 661 662 \begin{equation} \label{eq:dhdtunstable} 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} 665 where $h_\mathrm{bl}$ is the horizontallyvarying 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 mixedlayer 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:dhdtunstable} 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 $\sim20$\%. 666 667 The entrainment rate for the combination of convective and Langmuir turbulence is given by , 668 612 669 613 670
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