From a9d0ba4d95914f82e21a16d1cb6cf100e25200f6 Mon Sep 17 00:00:00 2001
From: HartmutBorth <hartmut.borth@uni-hamburg.de>
Date: Fri, 31 Mar 2017 11:32:22 +0200
Subject: [PATCH] centered spectral plot

---
 Cat_UG_00/cat_ug.pdf          | Bin 333110 -> 341139 bytes
 Cat_UG_00/chap_evoleq.tex     | 258 +++++++++++++++++++++++++---------
 Cat_UG_00/chap_pseudospec.tex | 107 ++++++++------
 Cat_UG_00/ref.bib             |  25 ++++
 cat/src/cat.f90               |   4 +-
 5 files changed, 287 insertions(+), 107 deletions(-)

diff --git a/Cat_UG_00/cat_ug.pdf b/Cat_UG_00/cat_ug.pdf
index e0b5abfa052065778bc25bae76ef8efa88bdc216..db104446bd4f258cd645d56e05395ec9d43e823c 100644
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diff --git a/Cat_UG_00/chap_evoleq.tex b/Cat_UG_00/chap_evoleq.tex
index 12305be..567c079 100644
--- a/Cat_UG_00/chap_evoleq.tex
+++ b/Cat_UG_00/chap_evoleq.tex
@@ -1,4 +1,4 @@
-%
+
 \chapter{Evolution equations of incompressible 2D fluids} 
 %
 %
@@ -20,7 +20,7 @@ From homogeneity and incompressibility of the fluid we get
   \rho \mathbf{\nabla \cdot u} = 0,
 \end{equation} 
 with $\rho$ the fluid density. Using this property we can introduce 
-a volume (mass) stream function measuring the volume (mass) flux  
+a volume (mass) streamfunction measuring the volume (mass) flux  
 across an arbitrary line from the point $(x_{0},y_{0})$ to a point $(x,y)$ 
 via the path integral
 \begin{equation} \label{eq_psizetadef}
@@ -45,10 +45,10 @@ via the path integral
 \end{equation}
 The minus sign in front of the integral just changes the direction
 of positive massflux across the line and is chosen to make 
-the classical stream function of 2D fluids compatible to 
-the stream function (see e.g.\ \cite{danilovandgurarie2000}) 
+the classical streamfunction of 2D fluids compatible to 
+the streamfunction (see e.g.\ \cite{danilovandgurarie2000}) 
 typically used for 2D rotating fluids in geosciences. 
-In terms of the stream function $\psi$ the integrated mass flux 
+In terms of the streamfunction $\psi$ the integrated mass flux 
 $\mathcal{M}$ across a line joining the points $(x_{0},y_{0})$ and 
 $(x,y)$ is given by
 \begin{equation} \label{eq_calM}
@@ -57,109 +57,235 @@ $(x,y)$ is given by
        \left[\psi(x,y) -  \psi(x_{0},y_{0}) \right],
 \end{equation}
 with $H_{0}$ the depth of the fluid. At the same time
-the stream function $\psi$ is connected to the velocity 
+the streamfunction $\psi$ is connected to the velocity 
 field $(u,v)$ and the (relative) 
 vorticity $\zeta = v_{x} - u_{y}$ via
 \begin{equation} \label{eq_psiuv}
   (u,v) = (- \psi_{y},\psi_{x}) \ \mbox{and} \ \ \zeta = \Delta \psi.
 \end{equation}
-Using the stream function $\psi$, the evolution equation (\ref{eq_2Dvortvel}
-can be also written in the form
+Using the streamfunction $\psi$, the evolution equation (\ref{eq_2Dvortvel})
+can be written in the form
 \begin{equation} \label{eq_2Dvortstream}
-  \zeta_{t} + J(\psi,\zeta) = F + D.
+  \zeta_{t} + J(\psi,\zeta) = F + D,
 \end{equation}
+with $J(\psi,\zeta) = \psi_{x} \zeta_{y} - \psi_{y} \zeta_{x}$ the Jacobian.
 Since the velocity field is divergence free (equation \ref{eq_conti})
-we can further derive the equivalent equation in flux form
+different representations of the Jacobian $J$ can be found. Using 
+appropriate superpositions of different representations one can derive 
+discrete analogs of the Jacobian $J$ which keep the symmetry 
+property $J(\psi,\zeta) = -J(\zeta,\psi)$ and conserve the integrated 
+vorticity $\zeta$, kinetic energy $\nabla \psi \cdot \nabla \psi $  and
+enstrophy $\zeta^{2}$. In \cite{arakawa1966} such a discrete analog is 
+derived for a finite difference numerical model. Keeping the notation of 
+\cite{arakawa1966} the Jacobian $J$ of equation (\ref{eq_2Dvortstream}) 
+is denoted by $J_{1}$. Starting from $J_{1}$ we can derive a second form 
+of the Jacobian 
+\begin{equation} \label{eq_Jacobi2} 
+  J_{2}(\psi,\zeta) 
+   = 
+  J_{1}(\psi,\zeta) + \psi \left( \zeta_{xy} - \zeta_{yx} \right)
+   = 
+  \left(\psi  \zeta_{y} \right)_{x}
+   - 
+  \left(\psi  \zeta_{x} \right)_{y},
+\end{equation}
+which gives the evolution equation
+\begin{equation} \label{eq_2DJacobi2} 
+  \zeta_{t} + \left(\psi  \zeta_{y} \right)_{x}  
+            - \left(\psi  \zeta_{x} \right)_{y} = F + D.
+\end{equation}
+Further we can derive a third form of Jacobian $J_{3}$ as follows 
+\begin{equation} \label{eq_Jacobi3}
+  J_{3}(\psi,\zeta) 
+   = 
+  J_{1}(\psi,\zeta) + \zeta \left( \psi_{xy} - \psi_{yx} \right)
+   = 
+  \left(\zeta  \psi_{x} \right)_{y}
+   - 
+  \left(\zeta  \psi_{y} \right)_{x},
+\end{equation}
+which inserted in the evolution equation yields the 2D fluid equations
+in flux form
 \begin{equation} \label{eq_2Dflux}
-  \zeta_{t} + \nabla \cdot \left(\mathbf{u} \zeta \right) = F + D.
+  \zeta_{t} + 
+  \left(\zeta  \psi_{x} \right)_{y}
+   - 
+  \left(\zeta  \psi_{y} \right)_{x},
+   = 
+  \zeta_{t} + \nabla \cdot \left(\mathbf{u} \zeta \right) 
+   = F + D.
 \end{equation}
-Starting from the flux form one can after repeated application of the
-continuity equation (\ref{eq_conti}) finally derive an equation where
-the Jacobian only depends on the two velocity fields $(u,v)$
-\begin{equation} \label{eq_2Duv}
-  \zeta_{t} 
-   + \partial_{x} \partial_{y} \left( v^{2} - u^{2} \right) 
-   + \left(\partial^{2}_{x} - \partial^{2}_{y} \right) uv
+It is possible to derive a fourth form of the Jacobian $J_{4}$
+which is a funciton of the field $v^{2} - u^{2})$ and 
+$uv$. We start again from the first Jacobian $J_{1}$
+\begin{equation} \label{eq_Jacobi4}
+  J_{4}(\psi,\zeta) 
+   = 
+  J_{1}(\psi,\zeta) + 
+   \left(2 \zeta - \psi_{x} \right) \ \left( \psi_{xy} - \psi_{yx} \right)
+   = 
+  \left(v^{2} - u^{2} \right)_{xy}  
+   + 
+  \left(uv \right)_{xx-yy},
+\end{equation}
+which leads to the evolution equation
+\begin{equation} \label{eq_2DJacobi4}
+  \zeta_{t}+ \left( v^{2} - u^{2} \right)_{xy} + \left( uv \right)_{xx - yy} 
    = 
   F + D.
 \end{equation}
-This form allows in the pseudo-spectral method (see \ref{sec_evolfourier}) 
-to reduce the number of Fourier transforms per time-step from 
-$3$, i.e.\ ($u,v,\zeta$) to $2$, i.e.\ ($u,v$).  
-
+All four forms given above are equivalent so that mathemtically one form is
+sufficient to catch all properties of the equations. Of course also other
+forms can be derived and it is also possible to superpose serveral forms.   
+The different forms get important if one tries to find discrete 
+representations of the Jacobian in the numerical scheme. Each form leads 
+to a discrete representation with different conservation and symmetry 
+properties. For more details see the chapters on the conservation 
+properties (\ref{sec_conprops}) and on the pseudo-spectral method 
+(\ref{sec_evolfourier}). 
 %
 \section{Quasi-two-dimensional rotating case} \label{sec_quasi2Dcase}
 %
 Starting from the shallow water equation on the $\beta$-plane one can
 derive (see e.g. \cite{danilovandgurarie2000}) an equation
 describing a rotating barotropic quasi-two-dimensional fluid which is a
-generalization of the 2D-equation (\ref{eq_2Dvortvel}). For small Rossby 
-numbers $\mathrm{Ro} = U/Lf$, with $U$ a typical horizontal fluid velocity
-at the length scale $L$ considered and $f$ the local coriolis paramter 
-we get the potential vorticity (PV) $q$ in quasi-geostrophic (QG) approximation 
+generalization of the 2D-equation (\ref{eq_2Dvortvel}). The shallow
+water potential vorticity is defined by 
+\begin{equation} \label{eq_qshallow}
+  q_{s} = \frac{\zeta + f}{H},
+\end{equation}
+where $H(x,y,t)$ is the fluid depth and $f$ the planetary vorticity. 
+Decomposing the fluid depth into a mean depth $H_{0}$, a constant 
+bottom topography B(x,y) and a time-dependent
+depth deviation $h(x,y,t)$ we can write $H(x,y,t) = H_{0} + h(x,y,t) - B(x,y)$. 
+Inserting the decomposition of the fluid depth into the equation 
+(\ref{eq_qshallow}) we can expand the potential vorticity
+\begin{equation} \label{eq_qshallowdecomp}
+  q_{s} 
+   =
+  \frac{\zeta + f}{H_{0} + h - B}
+   =
+  \frac{1}{H_{0}} \ \frac{\zeta + f}{1 + \Delta H / H_{0}}
+   =
+  \frac{1}{H_{0}} \ 
+  \left( \zeta + f \right) \left(1 - \frac{\Delta H}{H_{0}} + \dots \right),
+\end{equation}
+with $\Delta H = h - B$. Keeping only linear terms we finally get the barotopic
+vorticity
+\begin{equation} \label{eq_qbaro}
+  q = H_{0} q_{s} = \zeta - \frac{f}{H_{0}} \left(h - B\right) + f.
+\end{equation}
+Here we assume that the depth deviations $\Delta H = h - B$ are much smaller
+than the mean depth $H_{0}$. For small Rossby numbers $\mathrm{Ro} = U/Lf$, 
+with $U$ a typical horizontal velocity scale, $L$ a typical horizontal 
+length scale of the fluid motion and $f$ the local coriolis parameter 
+we can (see again e.g.\ \cite{danilovandgurarie2000}) introduce 
+the streamfunction 
+\begin{equation} \label{eq_psibaro}
+  \psi(x,y,t) = \frac{g}{f} \ h(x,y,t),    
+\end{equation}
+with $g$ the gravity. Using this streamfunction and expanding the 
+coriolis parameter linarly we can write the barotropic 
+quasi-geostrophic (QG) potential vorticity (PV) 
+on the $\beta$-plane (\ref{eq_qbaro}) as 
 \begin{equation} \label{eq_qdef}
-  q = \left( \nabla^{2}- \alpha^{2} \right) \psi + f,
+  q = \left( \nabla^{2}- \alpha^{2} \right) \psi 
+   + f_{0} + \beta y + \frac{f_{0}}{H_{0}} \ B.
+\end{equation}
+The linear expansion of the Coriolis parameter 
+$f(\varphi) = 2 \Omega \sin \varphi$ which is defined on a sphere 
+with radius $a$ and rotation rate $\Omega$ leads to
+\begin{equation} \label{eq_fbeta}
+  f(\varphi_{0} +  \Delta \varphi)
+   =
+  f(\varphi_{0}) + f^{\prime}(\varphi_{0}) \Delta \ \varphi   
+   = 
+  f_{0} + \beta  y,  
+\end{equation}
+with $\varphi_{0}$ the central latitude and the meridional 
+coordinate $y = a \Delta \varphi$ which is the linearization of 
+the projection $y = a \sin(\Delta \varphi)$. The $\beta$-parameter 
+is defined by $\beta = 2 \Omega \cos(\varphi_{0})/a$. The modification 
+parameter $\alpha = 1/L_{\mathrm{R}}$ is connected to the Rossby-Obukhov 
+radius of deformation $L_{\mathrm{R}} = \sqrt{g H_{0}}/f$. 
+As in the $2$-dimensional case the streamfunction $\psi$ 
+(remind the different definition) is related to the velocity field 
+$(u,v)$ again (see also equation \ref{eq_psiuv})  via
+\begin{equation} \label{eq_upsibaro}
+  u = -\psi_{y} \ \ \ \mbox{and} \ \ \ v = \psi_{x}.
 \end{equation}
-where we have the modification parameter $\alpha = 1/L^{2}_{\mathrm{R}}$ 
-with the Rossby-Obukhov radius of deformation 
-$L_{\mathrm{R}} = \sqrt{g H_{0}}/f$ and the stream function 
-$\psi = gh/f$. Here $g$ is the gravitational acceleration and $h(x,y)$ 
-the deviation of the mean fluid depth $H_{0}$ of the original 
-shallow water layer. As in the $2$-dimensional case the stream function $\psi$ 
-(remind the different definition) is related to the velocity fields $(u,v)$ 
-as defined in equation (\ref{eq_psiuv}). In an unforced non-dissipative fluid
-the QG PV is materially conserved
+In an unforced and non-dissipative fluid the QG PV is materially conserved
 \begin{equation} 
   q_{t} + J(\psi,q) = 0. 
 \end{equation}  
-Using the linear approximation of the coriolis parameter $f = f_{0} + \beta y$
+Using the linear approximation of the coriolis parameter (\ref{eq_fbeta})
 and introducing again forcing and dissipation we can write the evolution
-equation in the form
+equation for a barotropic fluid on the $\beta$-plane in the form
 \begin{equation} \label{eq_quasi2Dbaro}
-  q_{t} + J(\psi,q) + \beta \psi_{x} = F + D,
+  q_{t} + J(\psi,q + \frac{f_{0}}{H_{0}} \ B) + \beta \psi_{x} = F + D,
 \end{equation}
-with the vorticity $q$ given by
+with the vorticity $q$ again given by
 \begin{equation} \label{eq_vortquasi2Dbaro}
   q = \left(\nabla^2 -\alpha^{2} \right) \psi 
     = \zeta - \alpha^{2} \psi.
 \end{equation}
-Further one can introduce a variable mean fluid depth $H(x,y)$, 
-which in the simple case of a linear slope in $y$-direction leads 
-to a topographic $\beta$-effect (see e.g.\ \cite{vanheist1994}). 
+Here we used the property of the Jacobian $J(f,f) = 0$ for all fields
+$f(x,y)$ on the fluid domain.
 
+Idealizing the bottom topography to a linear slope in y-direction 
+$B(x,y) = B_{y} y$ the fluid experiences in addition to the ambient
+planetary $\beta$-effect a so called topographic $\beta$-effect 
+(see e.g.\ \cite{vanheist1994}) and the evolution equation simplifies to
+\begin{equation} \label{eq_qbarotopobeta} 
+  q_{t} + J(\psi,q) + \left(\beta + \frac{f_{0}}{H_{0}} B_{y} \right) \psi_{x} 
+   = F + D.
+\end{equation}
 In the form (\ref{eq_quasi2Dbaro}) and (\ref{eq_vortquasi2Dbaro}) 
-one can simulate incompressible 2D fluids and rotating quasi-2D fluids 
-with the same set of equations using different parameters. 
+one can simulate incompressible 2D fluids and rotating barotropic 
+quasi-2D fluids with the same set of equations using different parameters. 
 In this more general frame the simplest case of a non-rotating
 2D incompressible fluid is characterized by a vanishing 
 ambient vorticity gradient, i.e.\ $\beta = 0$, and the limit of
 an infinite Rossby radius $L_{\mathrm{R}} \longrightarrow \infty$
 or a vanishing modification parameter $\alpha \longrightarrow 0$.
+One has to keep in mind that the streamfunctions are different in 
+the two cases (see e.g.\ \cite{johnstonandliu2004}) and that there 
+are more subtle differences between 2D and QG Turbulence 
+(see e.g.\ \cite{tungandorlando2003}). 
+%
+\section{Multi-layer quasi-geostrophic case} \label{sec_multilayerqg}
+%
+%
+\section{The surface geostrophic case} \label{sec_sqg}
+%
+%
+\section{Conservation properties} \label{sec_conprops}
+%
+The properties of the Jacobian and the conditions at the fluid boundaries
+are at the base of the conservation properties of the fluid motions.
+Given two functions $g(x,y)$ and $h(x,y)$ defined on the fluid domains
+then the following integrals
+\begin{equation} \label{eq_intjacobian01}
+  \int_{x=0}^{X} \int_{x=0}^{Y} J(A,B) \ dxdy 
+  , \
+  \int_{x=0}^{X} \int_{x=0}^{Y} A J(A,B) \ dxdy 
+   \ \ \mbox{and} \ \    
+  \int_{x=0}^{X} \int_{x=0}^{Y} B J(A,B) \ dxdy 
+\end{equation}
+can be transformed through partial integration.
 
-One has to keep in mind that the stream functions are different in 
-the two cases. In the non-rotating case $\psi$ is defined by 
-equation (\ref{eq_calM}). In the rotating case we get 
-\begin{equation} \label{eq_hquasi2Dbaropsi}
-  h(x,y) = \frac{f}{g} \ \psi(x,y),    
-\end{equation} 
-so that $\psi$ is proportional to pressure deviations, which is not 
-the case in the non-rotating 2D case where the relation is more
-complex, see e.g.\ \cite{johnstonandliu2004}.
 
-Using the property of the Jacobian $J(f,f) = 0$ for all fields
-$f(x,y)$ on the fluid domain, equation (\ref{eq_quasi2Dbaro})
-is equivalent to
-\begin{equation} \label{eq_quasi2Dbarozeta}
-  q_{t} + J(\psi,\zeta) + \beta \psi_{x} = F + D,
-\end{equation}
-where the vorticity $q$ is still defined by equation 
-(\ref{eq_vortquasi2Dbaro}). From this from it directly follows that
-that the form of the 2D Jacobian in equations 
-(\ref{eq_2Dflux}) and (\ref{eq_2Duv}) can be also applied in the 
-quasi-two-dimensional rotating case.
 
-\section{Non-adiabatic terms}
+Above results can be generalized to more general fluid domains 
+(see e.g.\ \cite{salmonandtalley1989}).
+
+
+     
 
+
+%
+\section{Non-adiabatic terms}
+%
 \subsection{Laplacian based Viscosity and friction}
 
 Internal viscosity and external friction of the fluid are described by 
diff --git a/Cat_UG_00/chap_pseudospec.tex b/Cat_UG_00/chap_pseudospec.tex
index 330129d..259dee3 100644
--- a/Cat_UG_00/chap_pseudospec.tex
+++ b/Cat_UG_00/chap_pseudospec.tex
@@ -1079,11 +1079,11 @@ purely spectral and is called pseudo-spectral method, see e.g.\
 the higher wave numbers have to be filtered out. In CAT trunction is 
 used, see above. 
 
-We present three different forms of the Jacobian $J$ in physical space, 
-see equations (\ref{eq_2Dvortstream}), (\ref{eq_2Dflux}) 
-and (\ref{eq_2Duv}). Using the pseudo-spectral method for each
-form we get a different representation of the Jacobian $\hat{J}$ 
-in spectral space.   
+We present four different forms of the Jacobian $J$ in physical space, 
+see equations (\ref{eq_2Dvortstream}), (\ref{eq_2Dflux}), 
+(\ref{eq_2DJacobi2}) and (\ref{eq_2DJacobi4}). Using the pseudo-spectral 
+method for each form we get a different representation of the 
+Jacobian $\hat{J}$ in spectral space.   
 
 For the Jacobian (\ref{eq_2Dvortstream}) of the first form
 \begin{equation} \label{eq_jacobian01}
@@ -1091,18 +1091,18 @@ For the Jacobian (\ref{eq_2Dvortstream}) of the first form
    = 
   J(\psi,q) 
    = 
-  \partial_{x} \psi \ \partial_{y} q 
+  \psi_{x} \ q_{y} 
    -   
-  \partial_{x} q \ \partial_{y} \psi
+  q_{x} \  \psi_{y}
    =
-  \left[ v \ \partial_{y} q  + u \ \partial_{x} q \right],
+   v \ q_{y}  + u \  q_{x},
 \end{equation}
 one proceeds as follows. First the individual differential terms 
 are determined in spectral space using the fourier transform of 
 vorticity, and the spectral representation of the differential 
-operators. We get
+operators. The basic operators are given by 
 \begin{eqnarray} \label{eq_jacobian01_termsa}
-  \mathcal{F}(\partial_{x} \psi)
+  \mathcal{F}(\psi_{x})
   &=& 
   i  k_{x} \ \mathcal{F}(\psi)
    =
@@ -1111,11 +1111,11 @@ operators. We get
   - \ i \ \frac{k_{x}}{k^{2}_{x} + k^{2}_{y} + \alpha} \ 
   \mathcal{F}(q),
   \ \ 
-  \mathcal{F}(\partial_{y} q)
+  \mathcal{F}(q_{y})
    = 
   i k_{y} \ \mathcal{F}(q) \ \ \
   \\ \label{eq_jacobian01_termsb}
-  \mathcal{F}(\partial_{y} \psi)
+  \mathcal{F}(\psi_{y})
   &=&
   i  k_{y} \ \mathcal{F}(\psi)
    =
@@ -1124,7 +1124,7 @@ operators. We get
   - \ i \ \frac{k_{y}}{k^{2}_{x} + k^{2}_{y} + \alpha} \ 
   \mathcal{F}(q),
   \ \
-  \mathcal{F}(\partial_{x} q)
+  \mathcal{F}(q_{x})
    = 
   i k_{x} \ \mathcal{F}(q). \ \ \ 
 \end{eqnarray}
@@ -1137,43 +1137,41 @@ $\hat{J}$ is given by
 \begin{equation} \label{eq_jacobian01_J}
   \hat{J} 
    = 
-  \mathcal{F}(v \partial_{y} q)
-   -
-  \mathcal{F}(u \partial_{x} q).
+  \mathcal{F}(v q_{y})
+   +
+  \mathcal{F}(u q_{x})
+   = 
+  \mathcal{F}(v q_{y} + u q_{x}).
 \end{equation}
 Combining all necessary steps we can write
-\begin{eqnarray} \nonumber
+\begin{equation} \label{eq_jacobian01_Jall}
   \hat{J} 
-   &=&
+   =
   \mathcal{F}
-   \left(
+   \left[
     \mathcal{F}^{-1} 
      \left(
-      \ - \ i \frac{k_{x}}{k^{2}_{x}+k^{2}_{y}+\alpha} \ 
+       - i \frac{k_{x}}{k^{2}_{x}+k^{2}_{y}+\alpha} \ 
       \hat{q}
      \right) 
     \mathcal{F}^{-1} 
      \left(
       ik_{y} \ \hat{q}
      \right)
-   \right)
-    \\ \label{eq_jacobian01_Jall}
-   &-&
-  \mathcal{F}
-   \left(
+       +
     \mathcal{F}^{-1} 
      \left(
-     \ i \frac{k_{y}}{k^{2}_{x}+k^{2}_{y}+\alpha} \ 
+      i \frac{k_{y}}{k^{2}_{x}+k^{2}_{y}+\alpha} \ 
       \hat{q}
      \right) 
     \mathcal{F}^{-1}
      \left(
       ik_{x} \ \hat{q}
      \right)
-   \right),
-\end{eqnarray}
+   \right],
+\end{equation}
 where $\hat{q}$ is the vorticity in spectral space, the starting point
-for a new time step. As can be seen one needs $6$ $2$-FFT operations
+for a new time step. As can be seen one needs $5$ $2D$-FFT operations
 to determine the Jacobian. The components of the Jacobian 
 $\hat{J}_{k}$ are then used to determine the time 
 evolution of the different wave number components of the vorticity 
@@ -1182,11 +1180,7 @@ $\hat{q}_{\mathbf{k}}$, see equation (\ref{eq_evolqhat}).
 In the flux form of the evolution equation ({\ref{eq_2Dflux}})
 the second form of the Jacobian arises
 \begin{equation} \label{eq_jacobian02}
-  J(\psi,q)
-   = 
-  \partial_{x} (u \ q) 
-   +
-  \partial_{x} (v \ q).
+  J(\psi,q) = (u \ q)_{x} + (v \ q)_{y}.
 \end{equation}
 Following again equations (\ref{eq_jacobian01_termsa}) and 
 (\ref{eq_jacobian01_termsb}) we determine $\mathcal{F}(u)$ and
@@ -1235,14 +1229,47 @@ Combining again all necessary steps we can write
 As we can see the Jacobian in spectral space $\hat{J}$ can now be 
 determined by $5$ FFT operations.
 
-The third form of the Jacobian used in equation (\ref{eq_2Duv}) is
-given by
+The third form of the Jacobian in equation (\ref{eq_2DJacobi2})
+is given by
 \begin{equation} \label{eq_jacobian03}
+  J(\psi,q) = (\psi \ q_{y})_{x} - (\psi \ q_{x})_{y}.
+\end{equation}
+In spectral space we get 
+\begin{equation} \label{eq_jacobian03_J}
+  \hat{J} = -ik_{x} \ \mathcal{F}(\psi \ q_{y}) 
+            +ik_{y} \ \mathcal{F}(\psi \ q_{x}).
+\end{equation}
+Combining all necessary steps the Jacobian is determined by
+\begin{eqnarray} 
+  \hat{J} 
+   &=& 
+ -ik_{x} \ 
+  \mathcal{F}\left(
+     \mathcal{F}^{-1} 
+       \left(-\frac{1}{k_{x}^{2} + k_{y}^{2} + \alpha} \ \hat{q} \right) \
+     \mathcal{F}^{-1}
+       \left(
+         -ik_{y} \ \hat{q}
+       \right)
+             \right) 
+    \\ \label{eq_jacobian03_Jall}
+   &+&
+  ik_{y} \ 
+  \mathcal{F}\left(
+     \mathcal{F}^{-1} 
+       \left(-\frac{1}{k_{x}^{2} + k_{y}^{2} + \alpha} \ \hat{q} \right) \
+     \mathcal{F}^{-1}
+       \left(
+         -ik_{x} \ \hat{q}
+       \right)
+             \right).
+\end{eqnarray}
+The fourth form of the Jacobian used in equation (\ref{eq_2DJacobi4}) is \begin{equation} \label{eq_jacobian04}
   J(\psi,q)
    = 
   \partial_{x}\partial_{y} \left(v^{2} - u^{2} \right)
    +
-  \partial^{2}_{x} \partial^{2}_{y} uv.
+  \left( \partial^{2}_{x} - \partial^{2}_{y} \right)  uv.
 \end{equation}
 In this representation it is possible to reduce the number of FFT
 operations from $5$ to $4$. We first have to determine $\hat{u}$ 
@@ -1252,12 +1279,12 @@ space where the products $v^{2} - u^{2}$ and $uv$ are formed.
 Finally we have to transform them back to spectral space where they
 are differentiated. The Jacobian in spectral space $\hat{J}$ is 
 given by   
-\begin{equation} \label{eq_jacobian03_J}
+\begin{equation} \label{eq_jacobian04_J}
   \hat{J}
    = 
   -k_{x}k_{y} \ \mathcal{F}(v^{2} - u^{2})
    +
-   k^{2}_{x}k^{2}_{y} \mathcal{F}(uv)
+   \left( k^{2}_{x} - k^{2}_{y} \right) \mathcal{F}(uv)
 \end{equation} 
 or by combining all necessary steps
 \begin{eqnarray} \nonumber
@@ -1284,7 +1311,7 @@ or by combining all necessary steps
    \right)
     \\ \label{eq_jacobian02_Jall}
    &+&
-  k^{2}_{x} k^{2}_{y} \ 
+  \left( k^{2}_{x} - k^{2}_{y} \right) \ 
   \mathcal{F}
    \left(
     \mathcal{F}^{-1} 
diff --git a/Cat_UG_00/ref.bib b/Cat_UG_00/ref.bib
index 7018512..76438b2 100644
--- a/Cat_UG_00/ref.bib
+++ b/Cat_UG_00/ref.bib
@@ -65,6 +65,14 @@
 @string{ pss =  {\it Planet.\ Space Sci.}}
 
 %==========================================================================
+@article{arakawa1966,
+   author    = {A. Arakawa},
+   title     = {{C}omputational {D}esign for {L}ong-{T}erm {N}umerical {I}ntegration of the {E}quations of {F}luid {M}otion: {T}wo-{D}imensional {I}ncompressible {F}low. {P}art {I}},
+   journal   = {J. Comput. Phys.},
+   pages     = {119--143 he},
+   year      = {1966},
+   volume    = {1}
+}
 
 @book{batchelor1967,
    author    = {G. K. Batchelor},
@@ -160,3 +168,20 @@
   pages =        {253--259}
 }
 
+@article{salmonandtalley1989,
+   author    = {R. Salmon and D. Talley},
+   title     = {Generalizations of Arakawa's Jacobian},
+   journal   = {\it Journal of Computational Physics},
+   year      = {1989},
+   volume    = {83}
+   pages     = {247--259}
+}
+
+@article{tungandorlando2003,
+   author    = {K. K. Tung and W. W. Orlando},
+   title     = {{O}n the Differences between 2D and QG Turbulence},
+   journal   = {\it Discrete and Continuous Dynamical Systems-Series B},
+   year      = {2003},
+   volume    = {3},
+   pages     = {145--162}
+}
diff --git a/cat/src/cat.f90 b/cat/src/cat.f90
index 4545867..a429db5 100644
--- a/cat/src/cat.f90
+++ b/cat/src/cat.f90
@@ -1480,7 +1480,9 @@ implicit none
 ggui(:,:) = gq(:,:) ! double precision -> single
 call guiput("GQ" // char(0), ggui, ngx, ngy, 1)
 
-c4(:,:) = cq(:,:)
+! double -> single and center about ky = 0
+c4(:,0:nky-1)     = cq(:,nky+1:nfy)  
+c4(:,nky:nfy) = cq(:,0:nky)
 call guiput("C4" // char(0), c4, nkx+1, nfy+1, 2)   ! fc
 
 if (npost > 0) then
-- 
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