Changeset 1841 for branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex

Timestamp:

2010-04-15T09:32:00+02:00 (14 years ago)

Author:

gm

Message:

DEV_r1826_DOC : bottom friction (bfr), see ticket: #658

File:

• branches/DEV_r1826_DOC/DOC/TexFiles/Chapters/Chap_ZDF.tex (modified) (6 diffs)

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

@@ -394 +394 @@
 %--------------------------------------------------------------------------------------------------------------
 
-
 %>>>>>>>>>>>>>>>>>>>>>>>>>>>>
 \begin{figure}[!htb] \label{Fig_npc}   \begin{center}
@@ -520 +519 @@
 \label{ZDF_ddm}
 
-%-------------------------------------------namddm--------------------------------------------------------
+%-------------------------------------------namzdf_ddm-------------------------------------------------
 \namdisplay{namzdf_ddm}
 %--------------------------------------------------------------------------------------------------------------
@@ -595 +594 @@
 % 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}
 %--------------------------------------------------------------------------------------------------------------
 
@@ -607 +605 @@
 diffusive flux. For the bottom boundary layer, one has:
 \begin{equation} \label{Eq_zdfbfr_flux}
-A^{vm} \left( \partial \textbf{U}_h / \partial z \right) = \textbf{F}_h
-\end{equation}
-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}
+\end{equation}
+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 
@@ -623 +621 @@
 this, consider the equation for $u$ at $k$, the last ocean level:
 \begin{equation} \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}}
-\end{equation}
-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}}
+\end{equation}
+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:
+\begin{equation} \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
+\end{equation}
+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):
 \begin{equation} \label{Eq_zdfbfr_linear}
-\textbf{F}_h = \frac{A^{vm}}{e_3} \; \frac{\partial \textbf{U}_h}{\partial k} = r \; \textbf{U}_h^b
-\end{equation}
-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
+\end{equation}
+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
-\begin{equation} \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: 
+\begin{equation} \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}
 \end{equation}
-
-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}
 
@@ -690 +700 @@
 friction is quadratic: 
 \begin{equation} \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 
 \end{equation}
-
-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:
-\begin{equation} \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: 
+\begin{equation} \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}
 \end{equation}
 
-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:
+\begin{equation} \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}
+\end{equation}
+\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:
+\begin{equation}
+ |\Delta u| < \;|u| 
+\end{equation}
+\noindent which, using \eqref{Eqn_bfrstab}, gives:
+\begin{equation}
+\begin{split}
+r\frac{2\Delta t}{e_{3u}} &< 1 \\
+\Rightarrow r &< \frac{e_{3u}}{2\Delta t}\\
+\end{split}
+\end{equation}
+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:
+\begin{equation}
+e_{3u} > 2\;r\;\Delta t
+\end{equation}
+\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:
+\begin{equation} \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 ]
+\end{equation}
+
+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.
+
+% ================================================================