Changeset 11679
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 20191011T00:15:41+02:00 (12 months ago)
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NEMO/trunk/doc/latex/NEMO/subfiles/chap_ZDF.tex
r11678 r11679 586 586 \begin{figure}[!t] 587 587 \begin{center} 588 \includegraphics[width= \textwidth]{Fig_ZDF_OSM_structure_of_OSBL}588 \includegraphics[width=0.7\textwidth]{Fig_ZDF_OSM_structure_of_OSBL} 589 589 \caption{ 590 590 \protect\label{fig: OSBL_structure} … … 661 661 662 662 \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 , 668 663 %\frac{\partial h_\mathrm{bl}}{\partial t} + \mathbf{U}_b\cdot\nabla h_\mathrm{bl}= W_b  \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}} 664 \frac{\partial h_\mathrm{bl}}{\partial t} = W_b  \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}} 665 \end{equation} 666 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 (i.e.\ including the ML and pycnocline) and the buoyancy just below the base of the OSBL. This equation for the case when the pycnocline has a finite thickness, based on the potential energy budget of the OSBL, is the leading term \citep{grant+etal18} of a generalization of that used in mixedlayer models \citet[e.g.][]{kraus.turner_tellus67}, in which the thickness of the pycnocline is taken to be zero. 667 668 The entrainment flux for the combination of convective and Langmuir turbulence is given by 669 \begin{equation} \label{eq:entrainflux} 670 {\overline{w^\prime b^\prime}}_\mathrm{ent} = \alpha_{\mathrm{B}} {\overline{w^\prime b^\prime}}_0  \alpha_{\mathrm{S}} \frac{u_*^3}{h_{\mathrm{ml}}} 671 + G\left(\delta/h_{\mathrm{ml}} \right)\left[\alpha_{\mathrm{S}}e^{1.5\, \mathrm{La}_t}\alpha_{\mathrm{L}} \frac{w_{\mathrm{*L}}^3}{h_{\mathrm{ml}}}\right] 672 \end{equation} 673 where the factor $G\equiv 1  \exp (25\delta/h_{\mathrm{bl}})(14\delta/h_{\mathrm{bl}})$ takes care of the lesser efficiency of Langmuir mixing when the mboundarylayer depth is much greater than the Stokes depth, and $\alpha_{\mathrm{B}}$, $\alpha_{S}$ and $\alpha_{\mathrm{L}}$ depend on the ratio of the appropriate eddy turnover time to the inertial timescale $f^{1}$. Results from the LES suggest $\alpha_{\mathrm{B}}=0.18 F(fh_{\mathrm{bl}}/w_{*C})$, $\alpha_{S}=0.15 F(fh_{\mathrm{bl}}/u_*}$ and $\alpha_{\mathrm{L}}=0.035 F(fh_{\mathrm{bl}}/u_{*L})$, where $F(x)\equiv\tanh(x^{1}))^{0.69}$. 669 674 670 675
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