Changeset 1841

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2010-04-15T09:32:00+02:00 (11 years ago)
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DEV_r1826_DOC : bottom friction (bfr), see ticket: #658

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branches/DEV_r1826_DOC/DOC/TexFiles
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• branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex

 r1831 %-------------------------------------------------------------------------------------------------------------- %>>>>>>>>>>>>>>>>>>>>>>>>>>>> \begin{figure}[!htb] \label{Fig_npc}   \begin{center} \label{ZDF_ddm} %-------------------------------------------namddm-------------------------------------------------------- %-------------------------------------------namzdf_ddm------------------------------------------------- \namdisplay{namzdf_ddm} %-------------------------------------------------------------------------------------------------------------- % Bottom Friction % ================================================================ \section  [Bottom Friction (\textit{zdfbfr})] {Bottom Friction (\mdl{zdfbfr} module)} \section  [Bottom Friction (\textit{zdfbfr})]   {Bottom Friction (\mdl{zdfbfr} module)} \label{ZDF_bfr} %--------------------------------------------nambfr-------------------------------------------------------- \namdisplay{namzdf_bfr} \namdisplay{nambfr} %-------------------------------------------------------------------------------------------------------------- diffusive flux. For the bottom boundary layer, one has: \label{Eq_zdfbfr_flux} A^{vm} \left( \partial \textbf{U}_h / \partial z \right) = \textbf{F}_h where $\textbf{F}_h$ is supposed to represent the horizontal momentum flux A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum outside the logarithmic turbulent boundary layer (thickness of the order of 1~m in the ocean). How $\textbf{F}_h$ influences the interior depends on the 1~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the vertical resolution of the model near the bottom relative to the Ekman layer depth. For example, in order to obtain an Ekman layer depth this, consider the equation for $u$ at $k$, the last ocean level: \label{Eq_zdfbfr_flux2} \frac{\partial u \; (k)}{\partial t} = \frac{1}{e_{3u}} \left[ A^{vm} \; (k) \frac{U \; (k-1) - U \; (k)}{e_{3uw} \; (k-1)} - F_u \right] \approx - \frac{F_u}{e_{3u}} For example, if the bottom layer thickness is 200~m, the Ekman transport will \frac{\partial u_k}{\partial t} = \frac{1}{e_{3u}} \left[ \frac{A_{uw}^{vm}}{e_{3uw}} \delta_{k+1/2}\;[u] - {\cal F}^u_h \right] \approx - \frac{{\cal F}^u_{h}}{e_{3u}} If the bottom layer thickness is 200~m, the Ekman transport will be distributed over that depth. On the other hand, if the vertical resolution is high (1~m or less) and a turbulent closure model is used, the turbulent Ekman layer will be represented explicitly by the model. However, the logarithmic layer is never represented in current primitive equation model applications: it is \emph{necessary} to parameterize the flux $\textbf{F}_h$. applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h$. Two choices are available in \NEMO: a linear and a quadratic bottom friction. Note that in both cases, the rotation between the interior velocity and the bottom friction is neglected in the present release of \NEMO. In the code, the bottom friction is imposed by adding the trend due to the bottom friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface pressure gradient algorithm, the momentum trend due to the barotropic component needs to be handled separately. For this purpose it is convenient to compute and store coefficients which can be simply combined with bottom velocities and geometric values to provide the momentum trend due to bottom friction. These coefficients are computed in \mdl{zdfbfr} and generally take the form $c_b^{\textbf U}$ where: \label{Eq_zdfbfr_bdef} \frac{\partial {\textbf U_h}}{\partial t} = - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity. . % ------------------------------------------------------------------------------------------------------------- %       Linear Bottom Friction % ------------------------------------------------------------------------------------------------------------- \subsection{Linear Bottom Friction (\np{nbotfr} = 1) } \subsection{Linear Bottom Friction (\np{nn\_botfr} = 0 or 1) } \label{ZDF_bfr_linear} The linear bottom friction parameterisation assumes that the bottom friction is proportional to the interior velocity (i.e. the velocity of the last model level): The linear bottom friction parameterisation (including the special case of a free-slip condition) assumes that the bottom friction is proportional to the interior velocity (i.e. the velocity of the last model level): \label{Eq_zdfbfr_linear} \textbf{F}_h = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b where $\textbf{U}_h^b$ is the horizontal velocity vector of the bottom ocean layer and $r$ is a friction coefficient expressed in m.s$^{-1}$. This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean, {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b where $r$ is a friction coefficient expressed in ms$^{-1}$. This coefficient is generally estimated by setting a typical decay time $\tau$ in the deep ocean, and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}. values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly1984}. A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used in quasi-geostrophic models. One may consider the linear friction as an approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982}, Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed of tidal currents of $U_{av} =0.1$~m.s$^{-1}$, and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m.s$^{-1}$. of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$. This is the default value used in \NEMO. It corresponds to a decay time scale of 115~days. It can be changed by specifying \np{bfric1} (namelist parameter). In the code, the bottom friction is imposed by updating the value of the vertical eddy coefficient at the bottom level. Indeed, the discrete formulation of (\ref{Eq_zdfbfr_linear}) at the last ocean $T-$level, using the fact that $\textbf {U}_h =0$ below the ocean floor, leads to \label{Eq_zdfbfr_linKz} of 115~days. It can be changed by specifying \np{rn\_bfric1} (namelist parameter). For the linear friction case the coefficients defined in the general expression \eqref{Eq_zdfbfr_bdef} are: \label{Eq_zdfbfr_linbfr_b} \begin{split} A_u^{vm} &= r\;e_{3uw}\\ A_v^{vm} &= r\;e_{3vw}\\ c_b^u &= - r\\ c_b^v &= - r\\ \end{split} This update is done in \mdl{zdfbfr} when \np{nbotfr}=1. The value of $r$ used is \np{bfric1}. Setting \np{nbotfr}=3 is equivalent to setting $r=0$ and leads to a free-slip bottom boundary condition. Setting \np{nbotfr}=0 sets $r=2\;A_{vb}^{\rm {\bf U}}$, where $A_{vb}^{\rm {\bf U}}$ is the background vertical eddy coefficient, and a no-slip boundary condition is imposed. Note that this latter choice generally leads to an underestimation of the bottom friction: for example with a deepest level thickness of $200~m$ and $A_{vb}^{\rm {\bf U}} =10^{-4}$m$^2$.s$^{-1}$, the friction coefficient is only $r=10^{-6}$m.s$^{-1}$. When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfric1}. Setting \np{nn\_botfr}=0 is equivalent to setting $r=0$ and leads to a free-slip bottom boundary condition. These values are assigned in \mdl{zdfbfr}. From v3.2 onwards there is support for local enhancement of these values via an externally defined 2D mask array (\np{ln\_bfr2d}=\np{.true.}) given in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1. Locations with a non-zero mask value will have the friction coefficient increased by \np{ mask\_value}$*$\np{rn\_bfrien}$*$\np{r}. % ------------------------------------------------------------------------------------------------------------- %       Non-Linear Bottom Friction % ------------------------------------------------------------------------------------------------------------- \subsection{Non-Linear Bottom Friction (\np{nbotfr} = 2)} \subsection{Non-Linear Bottom Friction (\np{nn\_botfr} = 2)} \label{ZDF_bfr_nonlinear} friction is quadratic: \label{Eq_zdfbfr_nonlinear} \textbf {F}_h = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h {\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h }{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b with $\textbf{U}_h^b = (u_b\;,\;v_b)$ the horizontal interior velocity ($i.e.$ the horizontal velocity of the bottom ocean layer), $C_D$ a drag coefficient, and $e_b$ a bottom turbulent kinetic energy due to tides, internal waves breaking and other short time scale currents. A typical value of the drag coefficient is $C_D = 10^{-3}$. As an example, the CME experiment \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$.s$^{-2}$, while the FRAM experiment \citep{Killworth1992} uses $e_b =0$ and $e_b =2.5\;\;10^{-3}$m$^2$.s$^{-2}$. The FRAM choices have been set as default values (\np{bfric2} and \np{bfeb2} namelist parameters). where $C_D$ is a drag coefficient, and $e_b$ a bottom turbulent kinetic energy due to tides, internal waves breaking and other short time scale currents. A typical value of the drag coefficient is $C_D = 10^{-3}$. As an example, the CME experiment \citep{Treguier1992} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992} uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$. The CME choices have been set as default values (\np{rn\_bfric2} and \np{rn\_bfeb2} namelist parameters). As for the linear case, the bottom friction is imposed in the code by updating the value of the vertical eddy coefficient at the bottom level: \label{Eq_zdfbfr_nonlinKz} adding the trend due to the bottom friction to the general momentum trend in \mdl{dynbfr}. For the non-linear friction case the terms computed in \mdl{zdfbfr}  are: \label{Eq_zdfbfr_nonlinbfr} \begin{split} A_u^{vm} &=C_D\; e_{3uw} \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^ {1/2}\\ A_v^{vm} &=C_D\; e_{3uw} \left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ c_b^u &= - \; C_D\; \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ c_b^v &= - \; C_D\; \left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ \end{split} This update is done in \mdl{zdfbfr}. The coefficients that control the strength of the non-linear bottom friction are initialized as namelist parameters: $C_D$= \np{bfri2}, and $e_b$ =\np{bfeb2}. % ================================================================ The coefficients that control the strength of the non-linear bottom friction are initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}. Note for applications which treat tides explicitly a low or even zero value of \np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$ is possible via an externally defined 2D mask array (\np{ln\_bfr2d}=\np{.true.}). See previous section for details. % ------------------------------------------------------------------------------------------------------------- %       Bottom Friction stability % ------------------------------------------------------------------------------------------------------------- \subsection{Bottom Friction stability considerations} \label{ZDF_bfr_stability} Some care needs to exercised over the choice of parameters to ensure that the implementation of bottom friction does not induce numerical instability. For the purposes of stability analysis, an approximation to \eqref{Eq_zdfbfr_flux2} is: \label{Eqn_bfrstab} \begin{split} \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2\Delta t\\ &= -\frac{ru}{e_{3u}}\;2\Delta t\\ \end{split} \noindent where linear bottom friction and a leapfrog timestep have been assumed. To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have: |\Delta u| < \;|u| \noindent which, using \eqref{Eqn_bfrstab}, gives: \begin{split} r\frac{2\Delta t}{e_{3u}} &< 1 \\ \Rightarrow r &< \frac{e_{3u}}{2\Delta t}\\ \end{split} This same inequality can also be derived in the non-linear bottom friction case if a velocity of 1ms$^{-1}$ is assumed. Alternatively, this criterion can be rearranged to suggest a minimum bottom box thickness to ensure stability: e_{3u} > 2\;r\;\Delta t \noindent which it may be necessary to impose if partial steps are being used. For example, if $|u| = 1$ms$^{-1}$, $\Delta t = 1800$s, $r = 10^{-3}$ then $e_{3u}$ should be greater than 3.6m. For most applications, with physically sensible parameters these restrictions should not be of concern. But caution may be necessary if attempts are made to locally enhance the bottom friction parameters. To ensure stability limits are imposed on the bottom friction coefficients both during initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr} (assuming a 1ms$^{-1}$ velocity in the non-linear case). The number of breaches of the stability criterion are reported as well as the minimum and maximum values that have been set. The criterion is also checked at each time step, using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are reduced as necessary to ensure stability; these changes are not reported. % ------------------------------------------------------------------------------------------------------------- %       Bottom Friction with split-explicit time splitting % ------------------------------------------------------------------------------------------------------------- \subsection{Bottom Friction with split-explicit time splitting} \label{ZDF_bfr_ts} When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the bottom velocity at the before time step is used. This velocity includes both the baroclinic and barotropic components which is appropriate when using either the explicit or filtered surface pressure gradient algorithms ({\bf key\_dynspg\_exp} or {\bf key\_dynspg\_flt}). Extra attention is required, however, when using split-explicit time stepping ({\bf key\_dynspg\_ts}). In this case the free surface equation is solved with a small time step \np{rdtbt}, while the three dimensional prognostic variables are solved with a longer time step that is a multiple of \np{rdtbt}. The trend in the barotropic momentum due to bottom friction appropriate to this method is that given by the selected parameterisation (i.e. linear or non-linear bottom friction) computed with the evolving velocities at each barotropic timestep. In the case of non-linear bottom friction, we have elected to partially linearise the problem by keeping the coefficients fixed throughout the barotropic time-stepping to those computed in \mdl{zdfbfr} using the now timestep. This decision allows an efficient use of the $c_b^{\textbf U}$ coefficients to: \begin{enumerate} \item On entry to {\bf dyn\_spg\_ts}, remove the contribution of the before barotropic velocity to the bottom friction component of the vertically integrated momentum trend. Note the same stability check that is carried out on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to ensure that the trend removed matches that which was added in \mdl{dynbfr}. \item At each barotropic step, compute the contribution of the current barotropic velocity to the trend due to bottom friction. Add this contribution to the vertically integrated momentum trend. This contribution is handled implicitly which eliminates the need to impose a stability criteria on the values of the bottom friction coefficient within the barotropic loop. \end{enumerate} Note that the use of an implicit formulation for the bottom friction trend means that any limiting of the bottom friction coefficient in \np{dynbfr} does not adversely affect the solution when using split-explicit time splitting. This is because the major contribution to bottom friction is likely to come from the barotropic component which uses the unrestricted value of the coefficient. The implicit formulation takes the form: \label{Eq_zdfbfr_implicitts} \bar{U}^{t+\Delta t} = \; \left [ \bar{U}^{t-\Delta t}\; + 2\Delta t\;(\;RHS\;) \right ] / \left [ 1 - 2\Delta t \;c_b^u / H_e \right ] where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height), $c_b^u$ is the bottom friction coefficient as calculated in \np{zdf\_bfr} and $RHS$ represents all the components to the vertically integrated momentum trend except for that due to bottom friction. % ================================================================
• branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambfr

 r1831 rn_bfri2    =    1.e-3  !  bottom drag coefficient (non linear case) rn_bfeb2    =    2.5e-3 !  bottom turbulent kinetic energy background  (m^2/s^2) ln_bfr2d    =   .false. !  horizontal variation of the bottom friction coef (read a 2D mask file ) rn_bfrien   =    50.    !  local multiplying factor of bfr (ln_bfr2d = .true.) /
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