# Changeset 11679

Ignore:
Timestamp:
2019-10-11T00:15:41+02:00 (12 months ago)
Message:

 r11678 \begin{figure}[!t] \begin{center} \includegraphics[width=\textwidth]{Fig_ZDF_OSM_structure_of_OSBL} \includegraphics[width=0.7\textwidth]{Fig_ZDF_OSM_structure_of_OSBL} \caption{ \protect\label{fig: OSBL_structure} \label{eq:dhdt-unstable} \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}} where $h_\mathrm{bl}$ 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 $\sim20$\%. The entrainment rate for the combination of convective and Langmuir turbulence is given by , %\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}} \frac{\partial h_\mathrm{bl}}{\partial t} = W_b - \frac{{\overline{w^\prime b^\prime}}_\mathrm{ent}}{\Delta B_\mathrm{bl}} where $h_\mathrm{bl}$ 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 (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 mixed-layer models \citet[e.g.][]{kraus.turner_tellus67}, in which the thickness of the pycnocline is taken to be zero. The entrainment flux for the combination of convective and Langmuir turbulence is given by \label{eq:entrain-flux} {\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}}} + 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] where the factor $G\equiv 1 - \exp (-25\delta/h_{\mathrm{bl}})(1-4\delta/h_{\mathrm{bl}})$ takes care of the lesser efficiency of Langmuir mixing when the mboundary-layer 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 $F(x)\equiv\tanh(x^{-1}))^{0.69}$.