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 12305becf9fc6d36d9413c2740efda781081a17d..567c07980ed8da9c118f4a3d5b96b56570cb2113 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 330129dda1bf7a71169f35d37839aab187847f09..259dee39b709dc4c017d21b0fd71f616c99b1898 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 70185125b8b52045eff83b04b67552699d99f27d..76438b226f19a8e0c9d24c4bfa328debe57925b9 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 4545867e3b91fbe6ccfe131c5e90bee9591c99f5..a429db51b775cf006f7dfc3723be81ef6b3f0be6 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