Changeset 1841


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Timestamp:
2010-04-15T09:32:00+02:00 (11 years ago)
Author:
gm
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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 r1841  
    394394%-------------------------------------------------------------------------------------------------------------- 
    395395 
    396  
    397396%>>>>>>>>>>>>>>>>>>>>>>>>>>>> 
    398397\begin{figure}[!htb] \label{Fig_npc}   \begin{center} 
     
    520519\label{ZDF_ddm} 
    521520 
    522 %-------------------------------------------namddm-------------------------------------------------------- 
     521%-------------------------------------------namzdf_ddm------------------------------------------------- 
    523522\namdisplay{namzdf_ddm} 
    524523%-------------------------------------------------------------------------------------------------------------- 
     
    595594% Bottom Friction 
    596595% ================================================================ 
    597 \section  [Bottom Friction (\textit{zdfbfr})] 
    598       {Bottom Friction (\mdl{zdfbfr} module)} 
     596\section  [Bottom Friction (\textit{zdfbfr})]   {Bottom Friction (\mdl{zdfbfr} module)} 
    599597\label{ZDF_bfr} 
    600598 
    601599%--------------------------------------------nambfr-------------------------------------------------------- 
    602 \namdisplay{namzdf_bfr} 
     600\namdisplay{nambfr} 
    603601%-------------------------------------------------------------------------------------------------------------- 
    604602 
     
    607605diffusive flux. For the bottom boundary layer, one has: 
    608606\begin{equation} \label{Eq_zdfbfr_flux} 
    609 A^{vm} \left( \partial \textbf{U}_h / \partial z \right) = \textbf{F}_h 
    610 \end{equation} 
    611 where $\textbf{F}_h$ is supposed to represent the horizontal momentum flux  
     607A^{vm} \left( \partial {\textbf U}_h / \partial z \right) = {{\cal F}}_h^{\textbf U} 
     608\end{equation} 
     609where ${\cal F}_h^{\textbf U}$ is represents the downward flux of horizontal momentum  
    612610outside the logarithmic turbulent boundary layer (thickness of the order of  
    613 1~m in the ocean). How $\textbf{F}_h$ influences the interior depends on the  
     6111~m in the ocean). How ${\cal F}_h^{\textbf U}$ influences the interior depends on the  
    614612vertical resolution of the model near the bottom relative to the Ekman layer  
    615613depth. For example, in order to obtain an Ekman layer depth  
     
    623621this, consider the equation for $u$ at $k$, the last ocean level: 
    624622\begin{equation} \label{Eq_zdfbfr_flux2} 
    625 \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}} 
    626 \end{equation} 
    627 For example, if the bottom layer thickness is 200~m, the Ekman transport will  
     623\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}} 
     624\end{equation} 
     625If the bottom layer thickness is 200~m, the Ekman transport will  
    628626be distributed over that depth. On the other hand, if the vertical resolution  
    629627is high (1~m or less) and a turbulent closure model is used, the turbulent  
    630628Ekman layer will be represented explicitly by the model. However, the  
    631629logarithmic layer is never represented in current primitive equation model  
    632 applications: it is \emph{necessary} to parameterize the flux $\textbf{F}_h $.  
     630applications: it is \emph{necessary} to parameterize the flux ${\cal F}^u_h $.  
    633631Two choices are available in \NEMO: a linear and a quadratic bottom friction.  
    634632Note that in both cases, the rotation between the interior velocity and the  
    635633bottom friction is neglected in the present release of \NEMO. 
    636634 
     635In the code, the bottom friction is imposed by adding the trend due to the bottom  
     636friction to the general momentum trend in \mdl{dynbfr}. For the time-split surface  
     637pressure gradient algorithm, the momentum trend due to the barotropic component  
     638needs to be handled separately. For this purpose it is convenient to compute and  
     639store coefficients which can be simply combined with bottom velocities and geometric  
     640values to provide the momentum trend due to bottom friction.  
     641These coefficients are computed in \mdl{zdfbfr} and generally take the form  
     642$c_b^{\textbf U}$ where: 
     643\begin{equation} \label{Eq_zdfbfr_bdef} 
     644\frac{\partial {\textbf U_h}}{\partial t} =  
     645  - \frac{{\cal F}^{\textbf U}_{h}}{e_{3u}} = \frac{c_b^{\textbf U}}{e_{3u}} \;{\textbf U}_h^b 
     646\end{equation} 
     647where $\textbf{U}_h^b = (u_b\;,\;v_b)$ is the near-bottom, horizontal, ocean velocity. 
     648. 
    637649% ------------------------------------------------------------------------------------------------------------- 
    638650%       Linear Bottom Friction 
    639651% ------------------------------------------------------------------------------------------------------------- 
    640 \subsection{Linear Bottom Friction (\np{nbotfr} = 1) } 
     652\subsection{Linear Bottom Friction (\np{nn\_botfr} = 0 or 1) } 
    641653\label{ZDF_bfr_linear} 
    642654 
    643 The linear bottom friction parameterisation assumes that the bottom friction  
    644 is proportional to the interior velocity (i.e. the velocity of the last model level): 
     655The linear bottom friction parameterisation (including the special case  
     656of a free-slip condition) assumes that the bottom friction  
     657is proportional to the interior velocity (i.e. the velocity of the last  
     658model level): 
    645659\begin{equation} \label{Eq_zdfbfr_linear} 
    646 \textbf{F}_h = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 
    647 \end{equation} 
    648 where $\textbf{U}_h^b$ is the horizontal velocity vector of the bottom ocean  
    649 layer and $r$ is a friction coefficient expressed in m.s$^{-1}$. This coefficient  
    650 is generally estimated by setting a typical decay time $\tau$ in the deep ocean,  
     660{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b 
     661\end{equation} 
     662where $r$ is a friction coefficient expressed in ms$^{-1}$.  
     663This coefficient is generally estimated by setting a typical decay time  
     664$\tau$ in the deep ocean,  
    651665and setting $r = H / \tau$, where $H$ is the ocean depth. Commonly accepted  
    652 values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly_JMR84}.  
     666values of $\tau$ are of the order of 100 to 200 days \citep{Weatherly1984}.  
    653667A value $\tau^{-1} = 10^{-7}$~s$^{-1}$ equivalent to 115 days, is usually used  
    654668in quasi-geostrophic models. One may consider the linear friction as an  
    655669approximation of quadratic friction, $r \approx 2\;C_D\;U_{av}$ (\citet{Gill1982},  
    656670Eq. 9.6.6). For example, with a drag coefficient $C_D = 0.002$, a typical speed  
    657 of tidal currents of $U_{av} =0.1$~m.s$^{-1}$, and assuming an ocean depth  
    658 $H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m.s$^{-1}$.  
     671of tidal currents of $U_{av} =0.1$~m\;s$^{-1}$, and assuming an ocean depth  
     672$H = 4000$~m, the resulting friction coefficient is $r = 4\;10^{-4}$~m\;s$^{-1}$.  
    659673This is the default value used in \NEMO. It corresponds to a decay time scale  
    660 of 115~days. It can be changed by specifying \np{bfric1} (namelist parameter). 
    661  
    662 In the code, the bottom friction is imposed by updating the value of the  
    663 vertical eddy coefficient at the bottom level. Indeed, the discrete formulation  
    664 of (\ref{Eq_zdfbfr_linear}) at the last ocean $T-$level, using the fact that  
    665 $\textbf {U}_h =0$ below the ocean floor, leads to 
    666 \begin{equation} \label{Eq_zdfbfr_linKz} 
     674of 115~days. It can be changed by specifying \np{rn\_bfric1} (namelist parameter). 
     675 
     676For the linear friction case the coefficients defined in the general  
     677expression \eqref{Eq_zdfbfr_bdef} are:  
     678\begin{equation} \label{Eq_zdfbfr_linbfr_b} 
    667679\begin{split} 
    668 A_u^{vm} &= r\;e_{3uw}\\ 
    669 A_v^{vm} &= r\;e_{3vw}\\ 
     680 c_b^u &= - r\\ 
     681 c_b^v &= - r\\ 
    670682\end{split} 
    671683\end{equation} 
    672  
    673 This update is done in \mdl{zdfbfr} when \np{nbotfr}=1. The value of $r$  
    674 used is \np{bfric1}. Setting \np{nbotfr}=3 is equivalent to setting $r=0$ and  
    675 leads to a free-slip bottom boundary condition. Setting \np{nbotfr}=0 sets  
    676 $r=2\;A_{vb}^{\rm {\bf U}} $, where $A_{vb}^{\rm {\bf U}} $ is the background  
    677 vertical eddy coefficient, and a no-slip boundary condition is imposed.  
    678 Note that this latter choice generally leads to an underestimation of the  
    679 bottom friction: for example with a deepest level thickness of $200~m$  
    680 and $A_{vb}^{\rm {\bf U}} =10^{-4}$m$^2$.s$^{-1}$, the friction coefficient  
    681 is only $r=10^{-6}$m.s$^{-1}$. 
     684When \np{nn\_botfr}=1, the value of $r$ used is \np{rn\_bfric1}.  
     685Setting \np{nn\_botfr}=0 is equivalent to setting $r=0$ and leads to a free-slip  
     686bottom boundary condition. These values are assigned in \mdl{zdfbfr}.  
     687From v3.2 onwards there is support for local enhancement of these values  
     688via an externally defined 2D mask array (\np{ln\_bfr2d}=\np{.true.}) given 
     689in the \ifile{bfr\_coef} input NetCDF file. The mask values should vary from 0 to 1.  
     690Locations with a non-zero mask value will have the friction coefficient increased  
     691by \np{ mask\_value}$*$\np{rn\_bfrien}$*$\np{r}. 
    682692 
    683693% ------------------------------------------------------------------------------------------------------------- 
    684694%       Non-Linear Bottom Friction 
    685695% ------------------------------------------------------------------------------------------------------------- 
    686 \subsection{Non-Linear Bottom Friction (\np{nbotfr} = 2)} 
     696\subsection{Non-Linear Bottom Friction (\np{nn\_botfr} = 2)} 
    687697\label{ZDF_bfr_nonlinear} 
    688698 
     
    690700friction is quadratic:  
    691701\begin{equation} \label{Eq_zdfbfr_nonlinear} 
    692 \textbf {F}_h = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h  
     702{\cal F}_h^\textbf{U} = \frac{A^{vm}}{e_3 }\frac{\partial \textbf {U}_h  
    693703}{\partial k}=C_D \;\sqrt {u_b ^2+v_b ^2+e_b } \;\; \textbf {U}_h^b  
    694704\end{equation} 
    695  
    696 with $\textbf{U}_h^b = (u_b\;,\;v_b)$ the horizontal interior velocity ($i.e.$  
    697 the horizontal velocity of the bottom ocean layer), $C_D$ a drag coefficient,  
    698 and $e_b $ a bottom turbulent kinetic energy due to tides, internal waves  
    699 breaking and other short time scale currents. A typical value of the drag  
    700 coefficient is $C_D = 10^{-3} $. As an example, the CME experiment  
    701 \citep{Treguier_JGR92} uses $C_D = 10^{-3}$ and $e_b = 2.5\;10^{-3}$m$^2$.s$^{-2}$,  
    702 while the FRAM experiment \citep{Killworth1992} uses $e_b =0$  
    703 and $e_b =2.5\;\;10^{-3}$m$^2$.s$^{-2}$. The FRAM choices have been  
    704 set as default values (\np{bfric2} and \np{bfeb2} namelist parameters). 
     705where $C_D$ is a drag coefficient, and $e_b $ a bottom turbulent kinetic energy  
     706due to tides, internal waves breaking and other short time scale currents.  
     707A typical value of the drag coefficient is $C_D = 10^{-3} $. As an example,  
     708the CME experiment \citep{Treguier1992} uses $C_D = 10^{-3}$ and  
     709$e_b = 2.5\;10^{-3}$m$^2$\;s$^{-2}$, while the FRAM experiment \citep{Killworth1992}  
     710uses $C_D = 1.4\;10^{-3}$ and $e_b =2.5\;\;10^{-3}$m$^2$\;s$^{-2}$.  
     711The CME choices have been set as default values (\np{rn\_bfric2} and \np{rn\_bfeb2}  
     712namelist parameters). 
    705713 
    706714As for the linear case, the bottom friction is imposed in the code by  
    707 updating the value of the vertical eddy coefficient at the bottom level: 
    708 \begin{equation} \label{Eq_zdfbfr_nonlinKz} 
     715adding the trend due to the bottom friction to the general momentum trend  
     716in \mdl{dynbfr}. 
     717For the non-linear friction case the terms 
     718computed in \mdl{zdfbfr}  are:  
     719\begin{equation} \label{Eq_zdfbfr_nonlinbfr} 
    709720\begin{split} 
    710 A_u^{vm} &=C_D\; e_{3uw} \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^ 
    711 {1/2}\\ 
    712 A_v^{vm} &=C_D\; e_{3uw} \left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ 
     721 c_b^u &= - \; C_D\;  
     722    \left[ u^2 + \left(\bar{\bar{v}}^{i+1,j}\right)^2 + e_b \right]^{1/2}\\ 
     723 c_b^v &= - \; C_D\; 
     724    \left[  \left(\bar{\bar{u}}^{i,j+1}\right)^2 + v^2 + e_b \right]^{1/2}\\ 
    713725\end{split} 
    714726\end{equation} 
    715727 
    716 This update is done in \mdl{zdfbfr}. The coefficients that control the strength of the  
    717 non-linear bottom friction are initialized as namelist parameters: $C_D$= \np{bfri2},  
    718 and $e_b$ =\np{bfeb2}. 
    719  
    720 % ================================================================ 
     728The coefficients that control the strength of the non-linear bottom friction are  
     729initialised as namelist parameters: $C_D$= \np{rn\_bfri2}, and $e_b$ =\np{rn\_bfeb2}.  
     730Note for applications which treat tides explicitly a low or even zero value of  
     731\np{rn\_bfeb2} is recommended. From v3.2 onwards a local enhancement of $C_D$  
     732is possible via an externally defined 2D mask array (\np{ln\_bfr2d}=\np{.true.}).  
     733See previous section for details. 
     734 
     735% ------------------------------------------------------------------------------------------------------------- 
     736%       Bottom Friction stability 
     737% ------------------------------------------------------------------------------------------------------------- 
     738\subsection{Bottom Friction stability considerations} 
     739\label{ZDF_bfr_stability} 
     740 
     741Some care needs to exercised over the choice of parameters to ensure that the 
     742implementation of bottom friction does not induce numerical instability. For  
     743the purposes of stability analysis, an approximation to \eqref{Eq_zdfbfr_flux2} 
     744is: 
     745\begin{equation} \label{Eqn_bfrstab} 
     746\begin{split} 
     747 \Delta u &= -\frac{{{\cal F}_h}^u}{e_{3u}}\;2\Delta t\\ 
     748          &= -\frac{ru}{e_{3u}}\;2\Delta t\\ 
     749\end{split} 
     750\end{equation} 
     751\noindent where linear bottom friction and a leapfrog timestep have been assumed.  
     752To ensure that the bottom friction cannot reverse the direction of flow it is necessary to have: 
     753\begin{equation} 
     754 |\Delta u| < \;|u|  
     755\end{equation} 
     756\noindent which, using \eqref{Eqn_bfrstab}, gives: 
     757\begin{equation} 
     758\begin{split} 
     759r\frac{2\Delta t}{e_{3u}} &< 1 \\ 
     760\Rightarrow r &< \frac{e_{3u}}{2\Delta t}\\ 
     761\end{split} 
     762\end{equation} 
     763This same inequality can also be derived in the non-linear bottom friction case  
     764if a velocity of 1ms$^{-1}$ is assumed. Alternatively, this criterion can be  
     765rearranged to suggest a minimum bottom box thickness to ensure stability: 
     766\begin{equation} 
     767e_{3u} > 2\;r\;\Delta t 
     768\end{equation} 
     769\noindent which it may be necessary to impose if partial steps are being used.  
     770For example, if $|u| = 1$ms$^{-1}$, $\Delta t = 1800$s, $r = 10^{-3}$ then 
     771$e_{3u}$ should be greater than 3.6m. For most applications, with physically 
     772sensible parameters these restrictions should not be of concern. But  
     773caution may be necessary if attempts are made to locally enhance the bottom 
     774friction parameters.  
     775To ensure stability limits are imposed on the bottom friction coefficients both during  
     776initialisation and at each time step. Checks at initialisation are made in \mdl{zdfbfr}  
     777(assuming a 1ms$^{-1}$ velocity in the non-linear case). 
     778The number of breaches of the stability criterion are reported as well as the minimum  
     779and maximum values that have been set. The criterion is also checked at each time step,  
     780using the actual velocity, in \mdl{dynbfr}. Values of the bottom friction coefficient are  
     781reduced as necessary to ensure stability; these changes are not reported. 
     782 
     783% ------------------------------------------------------------------------------------------------------------- 
     784%       Bottom Friction with split-explicit time splitting 
     785% ------------------------------------------------------------------------------------------------------------- 
     786\subsection{Bottom Friction with split-explicit time splitting} 
     787\label{ZDF_bfr_ts} 
     788 
     789When calculating the momentum trend due to bottom friction in \mdl{dynbfr}, the 
     790bottom velocity at the before time step is used. This velocity includes both the 
     791baroclinic and barotropic components which is appropriate when using either the 
     792explicit or filtered surface pressure gradient algorithms ({\bf key\_dynspg\_exp} or  
     793{\bf key\_dynspg\_flt}). Extra attention is required, however, when using  
     794split-explicit time stepping ({\bf key\_dynspg\_ts}). In this case the free  
     795surface equation is solved with a small time step \np{rdtbt}, while the three  
     796dimensional prognostic variables are solved with a longer time step that is a  
     797multiple of \np{rdtbt}. The trend in the barotropic momentum due to bottom  
     798friction appropriate to this  
     799method is that given by the selected parameterisation (i.e. linear or non-linear 
     800bottom friction) computed with the evolving velocities at each  
     801barotropic timestep.  
     802 
     803In the case of non-linear bottom friction, we have elected 
     804to partially linearise the problem by keeping the coefficients fixed throughout 
     805the barotropic time-stepping to those computed in \mdl{zdfbfr} using the now  
     806timestep. This decision allows an efficient use of the $c_b^{\textbf U}$  
     807coefficients to: 
     808 
     809\begin{enumerate} 
     810\item On entry to {\bf dyn\_spg\_ts}, remove the contribution of the before 
     811barotropic velocity to the bottom friction component of the vertically 
     812integrated momentum trend. Note the same stability check that is carried out  
     813on the bottom friction coefficient in \mdl{dynbfr} has to be applied here to 
     814ensure that the trend removed matches that which was added in \mdl{dynbfr}. 
     815\item At each barotropic step, compute the contribution of the current barotropic 
     816velocity to the trend due to bottom friction. Add this contribution to the 
     817vertically integrated momentum trend. This contribution is handled implicitly which 
     818eliminates the need to impose a stability criteria on the values of the bottom friction 
     819coefficient within the barotropic loop.  
     820\end{enumerate} 
     821 
     822Note that the use of an implicit formulation 
     823for the bottom friction trend means that any limiting of the bottom friction coefficient  
     824in \np{dynbfr} does not adversely affect the solution when using split-explicit time  
     825splitting. This is because the major contribution to bottom friction is likely to come from  
     826the barotropic component which uses the unrestricted value of the coefficient. 
     827 
     828The implicit formulation takes the form: 
     829\begin{equation} \label{Eq_zdfbfr_implicitts} 
     830 \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 ] 
     831\end{equation} 
     832 
     833where $\bar U$ is the barotropic velocity, $H_e$ is the full depth (including sea surface height),  
     834$c_b^u$ is the bottom friction coefficient as calculated in \np{zdf\_bfr} and $RHS$ represents  
     835all the components to the vertically integrated momentum trend except for that due to bottom friction. 
     836 
     837% ================================================================ 
  • branches/DEV_r1826_DOC/DOC/TexFiles/Namelist/nambfr

    r1831 r1841  
    77   rn_bfri2    =    1.e-3  !  bottom drag coefficient (non linear case) 
    88   rn_bfeb2    =    2.5e-3 !  bottom turbulent kinetic energy background  (m^2/s^2) 
     9   ln_bfr2d    =   .false. !  horizontal variation of the bottom friction coef (read a 2D mask file ) 
     10   rn_bfrien   =    50.    !  local multiplying factor of bfr (ln_bfr2d = .true.) 
    911/ 
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