%ParentFile :A00compgfdE.tex %\documentclass[11pt]{book} %\setcounter{chapter}{2} %\begin{document} %====================================================================== %\chapter{Deformation and Rotation---Lagrange Recognition} In this chapter, we study the advection of the fluid parcels with time in the steady two-dimensional flow. The process of deformation of fluid parcel is decided by {\it deformation rate tensor}. %====================================================================== \section{Theoretical Background} %====================================================================== \subsection{Deformation of fluid} In case of time-invariant steady flow with letting $\mathbf{u}=(u_1,u_2,u_3)$ be the velocity of the fluid, the equation of motion for fluid parcel is given by \begin{equation} \frac{d\mathbf{x}}{dt} = \mathbf{u}(\mathbf{x}). \label{eq:lag1} \end{equation} The motion of fluid parcel can be traced by integrating (\ref{eq:lag1}) in time. Assuming that $\mathbf{x}(t)$ is the trajectory of the fluid parcel, we consider the time development of the trajectory of the parcel which slightly deviated from $\mathbf{x}(t)$ (Mizuta, 1999; p.5-8). Letting $\mathbf{y}(t)$ be this small displacement, we obtain its development with time as \begin{eqnarray} \frac{d}{dt}(\mathbf{x}(t)+\mathbf{y}(t)) &\! \! \! = \! \! \! & \mathbf{u}(\mathbf{x}(t)+\mathbf{y}(t),t), \nonumber \\ &\! \! \! = \! \! \! & \mathbf{u}(\mathbf{x}(t),t) + (\nabla\mathbf{u})^T \mathbf{y}(t) + O(\mathbf{y}^2). \end{eqnarray} Subtracting (\ref{eq:lag1}) from this, ignoring the $O(\mathbf{y}^2)$-term and linearizing locally, we obtain \begin{equation} \frac{d\mathbf{y}(t)}{dt} = J(\mathbf{x}(t)) \mathbf{y}(t). \label{eq:lag2} \end{equation} Here $J$ is the Jacobian of $\mathbf{u}$ at $\mathbf{x}(t)$ given by \begin{equation} J = (\nabla\mathbf{u})^T = \left( \begin{array}{ccc} \frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2} & \frac{\partial u_1}{\partial x_3} \\ \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2} & \frac{\partial u_2}{\partial x_3} \\ \frac{\partial u_3}{\partial x_1} & \frac{\partial u_3}{\partial x_2} & \frac{\partial u_3}{\partial x_3} \end{array} \right), \label{eq:lag3} \end{equation} which is called {\it deformation rate tensor}. By investigating the characteristics of $J$, we can understand the deformation of the continuum. %====================================================================== \subsection{Deformation of two-dimensional fluid -- analysis with eigenvalue --} \label{ss:eigen} The deformation in the two-dimentional fluid can investigate by categorizing $J$. Generally, $2\times2$-matrix is categorized as the following three types in attempting diagonalization. \begin{equation} \mathbf{A}: \ \left( \begin{array}{rr} \lambda & 0 \\ 0 & \mu \\ \end{array} \right), \qquad \mathbf{B}: \ \left( \begin{array}{rr} a & -b \\ b & a \\ \end{array} \right), \qquad \mathbf{C}: \ \left( \begin{array}{rr} \lambda & 0 \\ 1 & \lambda \\ \end{array} \right). \label{eq:lag4} \end{equation} Based on (\ref{eq:lag4}), the flow field can be classified as follows. $\mathbf{A}$: In a case that $J$ can be diagonalized and has real eigenvalues, \begin{equation} \frac{d}{dt} \left( \begin{array}{r} y_1 \\ y_2 \\ \end{array} \right) = \left( \begin{array}{rr} \lambda & 0 \\ 0 & \mu \\ \end{array} \right) \left( \begin{array}{r} y_1 \\ y_2 \\ \end{array} \right), \label{eq:lag5} \end{equation} \begin{equation} y_1 = y_1(t_0) e^{\lambda t}, \qquad y_2 = y_2(t_0)e^{\mu t}.\label{eq:lag6} \end{equation} \begin{itemize} \item $\mathbf{A1}$: when $\lambda = \mu$, the point $\mathbf{x}$ is a {\it focus}. \item $\mathbf{A2}$: when $\lambda$ and $\mu$ have the same sign, $\mathbf{x}$ is a {\it node}. \item $\mathbf{A3}$: when $\lambda$ and $\mu$ have the different sign, $\mathbf{x}$ is a {\it saddle}. \end{itemize} $\mathbf{B}$: In a case that the eigenvalues are complex conjugates, \begin{equation} \frac{d}{dt} \left( \begin{array}{r} y_1 \\ y_2 \\ \end{array} \right) = \left( \begin{array}{rr} a & -b \\ b & a \\ \end{array} \right) \left( \begin{array}{r} y_1 \\ y_2 \\ \end{array} \right), \label{eq:lag7} \end{equation} \begin{equation} (y_1,y_2)^T = (y_1(t_0),y_2(t_0))^T e^{(a+ib)t}. \label{eq:lag8} \end{equation} \begin{itemize} \item $\mathbf{B1}$: when $a \neq 0$, $\mathbf{x}$ is a {\it spiral}. \item $\mathbf{B2}$: when $a=0$, $\mathbf{x}$ is a {\it center}. \end{itemize} $\mathbf{C}$: In a case that $J$ can not be diagonalized, \begin{equation} \frac{d}{dt} \left( \begin{array}{r} y_1 \\ y_2 \\ \end{array} \right) = \left( \begin{array}{rr} \lambda & 0 \\ 1 & \lambda \\ \end{array} \right) \left( \begin{array}{r} y_1 \\ y_2 \\ \end{array} \right). \label{eq:lag9} \end{equation} \begin{itemize} \item $\mathbf{C1}$: when $\lambda \neq 0$, $\mathbf{x}$ is an {\it improper node}. \item $\mathbf{C2}$: when $\lambda = 0$, $\mathbf{x}$ is a {\it simple shear}. \end{itemize} Based on the discriminant of the eigenvalue equation $\Delta=(\mbox{tr}J)^2 - 4 \mbox{det}J$, we can classify the flow field, i.e., $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ correspond to the cases that $\Delta \geq 0$, $\Delta < 0$ and $\Delta = 0$. Figure~\ref{fig:1} shows the relationship on the two-dimensional plain with tr$J$ and det$J$ axes. Since tr$J=\sum \partial u_i / \partial x_i$, the flow is non-divergent over $y$-axis, while the flow is divergent and convergent on the right and left side, respectively. The parabola through the origin indicates the curve where $\Delta=0$. Over this line $\mathbf{x}$ is a focus ($\mathbf{A}$) or an improper node ($\mathbf{C}$), and at the origin $\mathbf{x}$ is a simple shear ($\mathbf{C}$). Above this line, $\Delta < 0$ where $\mathbf{x}$ is a spiral or a center. On the other hand, since the discriminant $\Delta > 0$ below this line, in this domain $\mathbf{x}$ is a node and a saddle above (det$J>0$) and below $x$-axis (det$J<0$), respectively. \begin{figure} %\vspace{-1em} \begin{center} \includegraphics[height=8cm,width=12cm,clip]{fig/fig3-1.eps} \end{center} \caption{Classification of flow by the deformation rate tensor. After Ottino (1989; Fig.~E2.5.2, p.27).} \label{fig:1} \end{figure} %====================================================================== \subsection{Strain rate tensor and tensor for the rotation} Based on the text of Tatsumi (1995; p.43-48), we continue considering the deformation rate tensor. As in previous subsection, the effect of deformation by fluid motion is described by the deformation rate tensor $\nabla \mathbf{u}=\partial u_i /\partial x_j$. Dividing this into symmetric and asymmetric components, we obtain \begin{eqnarray} \frac{\partial u_i}{\partial x_j} & \! \! \! = \! \! \! & e_{ij} + \omega_{ij}, \label{eq:lag10} \\ e_{ij} & \! \! \! = \! \! \! & \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right), \label{eq:lag11} \\ \omega_{ij} & \! \! \! = \! \! \! & \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right), \label{eq:lag12} \end{eqnarray} where $e_{ij}$ represents local deformation, and is called {\it strain rate tensor}. On the other hand, $\omega_{ij}$ represents rotating rigid motion, and is called {\it rotation displacement}. The diagonal component of strain rate tensor $e_{ii}=\partial u_i /\partial x_i \quad (i=1,2,3) $ represents the velocity of elongation-contraction along $x_i$ axis. In addition, sum of the diagonal components, \begin{equation} \sum_{i} e_{ii} = \sum_{i}\frac{\partial u_i}{\partial x_i} = \mathit{div}\mathbf{u}, \label{eq:lag13} \end{equation} represents {\it volume dilatation rate}. This is the variation ratio of the volume by elongation or contraction. In case of two-dimensional motion, it represents the variation of the area by elongation or contraction. On the other hand, asymmetric component $e_{ij}$ represents the velocity of the variation of the angle between $x_i$ and $x_j$ axis, which is called {\it shear strain rate}. The asymmetric components of deformation rate tensor $\omega_{ij}$ can be written as with vector $\mathbf{\omega}=(\omega_1,\omega_2,\omega_3)$ and Eddington sign $\epsilon_{ijk}$ \begin{equation} \omega_{ij} = - \frac{1}{2} \sum_k \epsilon_{ijk} \omega_k, \qquad (i,j=1,2,3) \label{eq:lag14} \end{equation} \begin{equation} \omega_k = - \sum_i \sum_j \epsilon_{ijk} \omega_{ij}, \label{eq:lag15} \end{equation} This equation can be rewritten as \begin{equation} \omega_k = - \frac{1}{2} \sum_i \sum_j \epsilon_{ijk} \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right), \label{eq:lag16} \end{equation} or in the vector form, \begin{equation} \mathbf{\omega} = \mathit{rot}\mathbf{u}. \label{eq:lag17} \end{equation} $\mathbf{\omega}$ is the rotational component of the velocity $\mathbf{u}$ and is called {\it vorticity}. Let $\mathbf{\theta}$ be the rotation angle, and (\ref{eq:lag17}) becomes \begin{equation} \mathbf{\omega} = 2 \dot{\mathbf{\theta}}. \label{eq:lag18} \end{equation} Vorticity $\mathbf{\omega}$ is twice as large as the angular velocity of the displacement vector $\mathbf{r}$ and each coodinate component of $\mathbf{\omega}$ $(\omega_1,\omega_2,\omega_3)$ is twice as large as the angular velocity around each axis. %====================================================================== \section{Exercises} Observe the deformation of the fluid parcel in linear or axisymmetric flow field using the program "C4-1" (for Examples 1 $\sim$ 5), "C4-2" (for Examples 6 and 7), and "C4-3" (for Examples 8 $\sim$ 10) in GFD Menu. \begin{figure}[t] \begin{center} \includegraphics[width=10cm,keepaspectratio,clip]{fig/C03_01.epsf} \caption{Result of Example 1.} \label{fig:C03_01} \end{center} \end{figure} %====================================================================== \subsection{Linear flow} Consider the steady and linear flow field as \begin{equation} u(x,y) = u_0 + \frac{du}{dx} x + \frac{du}{dy} y, \qquad v(x,y) = v_0 + \frac{dv}{dx} x + \frac{dv}{dy} y, \label{eq:lag19} \end{equation} where $u_0, \frac{du}{dx}, \frac{du}{dy}, v_0, \frac{dv}{dx}$ and $\frac{dv}{dy}$ are constant. Since the flow field is linear, the characteristics of the deformation of the fluid is uniform and is independent of the size of the fluid parcel. That is, (\ref{eq:lag2}) is satisfied rigidly. Set each constant in the configuration window. When starting the program, the fluid parcels settled around the circle initially can be traced with time and the deformation of the fluid parcel can be observed. \begin{description} \item[Example 1. Homogeneous flow] \end{description} \begin{center} \begin{tabular}{cccccc} 10.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\ \verb@u0@ &\verb@du/dx@ &\verb@du/dy@ &\verb@v0@ &\verb@dv/dx@ &\verb@dv/dy@ \end{tabular} \end{center} In this case, all the components of deformation rate tensor are zero. There is neither the deformation or rotation of the fluid, and the area does not change because of non-divergent flow. The fluid parcel is advected along $x$-axis with constant speed $u_0$, conserving initial circular pattern. \begin{description} \item[Example 2. Divergent flow] \end{description} \begin{center} \begin{tabular}{cccccc} 0.0 & 0.05 & 0.0 & 0.0 & 0.0 & 0.05 \\ \verb@u0@ &\verb@du/dx@ &\verb@du/dy@ &\verb@v0@ &\verb@dv/dx@ &\verb@dv/dy@ \end{tabular} \end{center} Following the classification by eigenvalue in section \ref{ss:eigen}, the origin in this case becomes a focus. Dividing deformation rate tensor into strain rate tensor and tensor for rotation, there is no rotation since the latter is zero. By the diagonal component of the strain rate tensor $e_{ii}$, elongation-contraction by {\tt dudx} and {\tt dvdy} occurs in $x$- and $y$-direction, respectively. Since $\mathit{div}\mathbf{u} \ne 0$, the area of the fluid parcel changes. If setting {\tt dudx} and {\tt dvdy} with the same sign and different value, the origin becomes a node. \begin{description} \item[Example 3. Deformation field] \end{description} \begin{center} \begin{tabular}{cccccc} 0.0 & 0.05 & 0.0 & 0.0 & 0.0 & -0.05 \\ \verb@u0@ &\verb@du/dx@ &\verb@du/dy@ &\verb@v0@ &\verb@dv/dx@ &\verb@dv/dy@ \end{tabular} \end{center} Following the classification by eigenvalue, the origin becomes a saddle. strain rate tensor and tensor for rotation are the same as {\bf Example 2}, while the changing rate of the area is zero because of non-divergent flow. elongation-contraction occurs in the direction of $x$- and $y$-axis, conserving the area of the fluid parcel. In addition, changing the value of {\tt dvdy}, observe saddles with divergence. \begin{description} \item[Example 4. Rotation displacement] \end{description} \begin{center} \begin{tabular}{cccccc} 0.0 & 0.05 & -0.15 & 0.0 & 0.15 & 0.05 \\ \verb@u0@ &\verb@du/dx@ &\verb@du/dy@ &\verb@v0@ &\verb@dv/dx@ &\verb@dv/dy@ \end{tabular} \end{center} Both strain rate tensor and tensor for rotation are non-zero, and the flow is divergent with rotation. If {\tt dudx} and {\tt dvdy} are zero, the flow is non-divergent and center-type. The initial circular pattern is converved. \begin{description} \item[Example 5. Simple shear] \end{description} \begin{center} \begin{tabular}{cccccc} 0.0 & 0.0 & 0.15 & 0.0 & 0.0 & 0.0 \\ \verb@u0@ &\verb@du/dx@ &\verb@du/dy@ &\verb@v0@ &\verb@dv/dx@ &\verb@dv/dy@ \end{tabular} \end{center} In case of simple shear, dividing the deformation rate tensor into strain rate tensor and tensor for rotation gives \begin{equation} \nabla\mathbf{u} = \left( \begin{array}{cc} 0 & \frac{1}{2} \frac{\partial u}{\partial y} \\ \frac{1}{2} \frac{\partial u}{\partial y} & 0 \\ \end{array} \right) + \left( \begin{array}{cc} 0 & \frac{1}{2} \frac{\partial u}{\partial y} \\ - \frac{1}{2} \frac{\partial u}{\partial y} & 0 \\ \end{array} \right). \label{eq:lag20} \end{equation} Non-diagonal component of strain rate tensor $e_{ij}$ represents the velocity of the variation of the angle between $x$-axis and $y$-axis, and a shear strain occurs around the perpendicular of $x$-$y$ plain ($z$-axis). On the other hand, non-diagonal component of the tensor for rotation $\omega_{ij}$, from (\ref{eq:lag19}), represents that the fluid parcel rotate around the perpendicular of $x$-$y$ plain with angular velocity $\frac{1}{2} \frac{\partial u}{\partial x}$. The area does not change because of non-divergent flow. %====================================================================== \subsection{Rotating axi-symmetric flow} Consider the situation `{\it Rankine's connected vortex}' where the flow is two-dimensional and unelastic and the vorticity distributes homogeneously in the circle (Tatsumi, 1982; p.188-199). Letting $a$ and $\omega$ be the radius of the vortex domain and the (vertical component of) vorticity respectively, given that the flow field is axisymmetric, and assume the flow function as the function of only the radius $r$. In this case, velocity field have only tangental component $v_\theta (r)$ and is written, dependent of the radius $r$, as \begin{equation} v_\theta = \left\{ \begin{array}{ll} \frac{\omega r}{2} & (r \le a), \\ \frac{\omega a^2}{2r} & (r > a). \end{array} \right. \label{eq:lag21} \end{equation} Set $\omega$ and $a$ at {\tt omega} and {\tt edge} in the configuration box, respectively. When starting the program, the fluid parcels settled initially in square can be traced with time and the deformation of the fluid parcel can be observed. In addition, the initial position $(x,y)$ of the fluid parcel (square) can be set explicitly at {\tt xoffst} and {\tt yoffst}. \begin{description} \item[Example 6. Rigid-rotating vortex] \end{description} \begin{center} \begin{tabular}{cccc} 0.15 & 200.0 & 50.0 & 0.0 \\ $\omega$ & Edge & X offset & Y offset \end{tabular} \end{center} In the vortex domain ($r0$ and $\Delta<0$, the origin in this case corresponds to the center. When dividing deformation rate tensor into strain rate tensor and tensor for rotation, all the components of strain rate tensor are zero and the non-diagonal components of $\omega_{ij}$ remain. There is no strain and the fluid percel do the rigid rotation around the perpendicular of $x$-$y$ plain ($z$-axis) with constant angular velocity $\frac{\omega}{2}$. \begin{description} \item[Example 7. Out of the votex] \end{description} \begin{center} \begin{tabular}{cccc} 0.15 & 50.0 & 50.0 & 0.0 \\ $\omega$ & Edge & X offset & Y offset \end{tabular} \end{center} Out of the vortex domain ($r>a$), the velocity can be rewritten in Cartesian coordinates as $\mathbf{u} = ( -\frac{\omega a^2}{2r} \sin \theta, \frac{\omega a^2}{2r} \cos \theta ) = ( -\frac{\omega a^2}{2r^2} y, \frac{\omega a^2}{2r^2} x)$. Since tr$J=0$, det$J<0$ and $\Delta>0$, the flow corresponds to the saddle. Deformation rate tensor can be rewritten as \begin{equation} \nabla \mathbf{u} = \left( \begin{array}{cc} \frac{\omega a^2}{r^4}xy & \frac{1}{2}\frac{\omega a^2}{r^4}(y^2-x^2) \\ \frac{1}{2}\frac{\omega a^2}{r^4}(y^2-x^2) & - \frac{\omega a^2}{r^4}xy \end{array} \right) + \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right). \label{eq:lag22} \end{equation} Each term in strain rate tensor have values dependent of $x$ and $y$. By the diagonal component $e_{ii}$, elongation-contraction occurs in the direction of $x$- and $y$-axis and the strain propotional to $r^{-2}$ exists in $x$-$y$ plain. By the component $e_{ij}$, the fluid rotates around $z$-axis with strain, while vorticity is zero except for $z$-axis because $\omega_{ij}=0$. The area does not change because of non-divergent flow. %====================================================================== \section{Application} In the preveous section, the deformation rate tensor had the constant value everywhere in the linear flow field. However, in the realistic flow, deformation rate tensor variate complicatedly in space and time. In this section, study the deformation of the fluid parcel, by using the two-dimensional non-divergent flow assuming Rossby wave taken up by Pierrehumbert (1991) in the paper on the caotic mixing. Given the flow function $\psi(x,y) = A \sin k_1 x \sin l_1 y - u_0 y$ (upper panel of Figure~\ref{fig:2}). In this case, the velocity is \begin{equation} \left\{ \begin{array}{lrl} u(x,y) = & \! \! \! \! - \frac{\partial \psi}{\partial y} = & \! \! \! \! - A l_1 \sin k_1 x \cos l_1 y + u_0, \\ v(x,y) = & \! \! \! \! \frac{\partial \psi}{\partial x} = & A k_1 \cos k_1 x \sin l_1 y. \end{array} \right. \end{equation} The value tr$J=0$ because of non-divergent. In addition, det$J= - \cos^2 x \cos^2 y + \sin^2 x \sin^2 y$ and the flow is a center (det$J>0$), a simple shear (det$J=0$) or a saddle (det$J<0$), dependent of the location (lower panel of Fig.~\ref{fig:2}). around the stagnation point over $y=0, \pi$, the deformation of the fluid parcel is dominant, while in the positive and negative vortex the rotation is dominant. \begin{figure}[tb] \vspace{-1em} \begin{center} \includegraphics[width=13cm]{fig/fig3-2.eps} \end{center} \caption{The distribution of two-dimensional non-divergent flow (upper panel) and det$J$ (lower panel) assuming Rossby wave. Hatched area indicates the domain for det$J<0$.} \label{fig:2} \end{figure} Deformation rate tensor is \begin{equation} \nabla \mathbf{u} = \left( \begin{array}{cc} - \cos x \cos y & 0 \\ 0 & \cos x \cos y \\ \end{array} \right) + \left( \begin{array}{cc} 0 & \sin x \sin y \\ - \sin x \sin y & 0 \\ \end{array} \right). \end{equation} Both the diagonal component of strain rate tensor $e_{ij}$ and the non-diagonal component of tensor for rotation $\omega_{ij}$ remain. An elongation and a contraction dependent of the location in the direction of $x$- and $y$-axis and a rotation with angular velocity $\sin x \sin y$ are observed. Set the location of initial circle (X,Y), the radius of initial circle (\verb@r@) and homogeneous background flow speed (\verb@u0@) in the configuration box. When starting the program, the fluid parcels settled around the circle initially can be traced with time and the deformation of the fluid parcel can be observed. \begin{description} \item[Example 8. Winding jet] \end{description} \begin{center} \begin{tabular}{cccc} 0.9 & 1.6 & 0.1 & 0.5 \\ X & Y & \verb@r@ & \verb@u0@ \end{tabular} \end{center} Initially the fluid parcel is advected by jet flow with little strain. When the fluid parcel enter into the area where the deformation is dominant, it is heavily strained. Locate the initial circle around the center of the jet or in the vortex, and observe the difference dependent of the location. \begin{figure}[tb] \vspace{-3em} \begin{center} \includegraphics[width=13cm]{fig/fig3-3.eps} \end{center} \vspace{-3em} \caption{The deformation of checker-flag pattern in two-dimensional non-divergent flow assuming Rossby wave.} \label{fig:3} \end{figure} \begin{description} \item[Example 9. Vortex street] \end{description} \begin{center} \begin{tabular}{cccc} 0.1 & 1.6 & 0.1 & 0.0 \\ X & Y & \verb@r@ & \verb@u0@ \end{tabular} \end{center} Make an experiment with setting the homogeneous background flow speed (\verb@u0@) to zero, and compare with previous example. \begin{description} \item[Example 10. Checker-flag pattern] \end{description} \begin{center} \begin{tabular}{cccc} 0.41 & 1.56 & 0.7854 & 0.5 \\ X & Y & \verb@r@ & \verb@u0@ \end{tabular} \end{center} Trace the fluid parcel having checker-flag pattern in the same two-dimensional non-divergent flow, and study the deformation (Figure~\ref{fig:3}). Modify the parameters and executes the program. By changing the condition, realize the relationship between the flow field and the deformation. %====================================================================== \section{Appendix} Weiss (1990) classified the flow field by tr$A^2$, where $A$ is deformation rate tensor $\nabla\mathbf{u}$. In case that tr$A^2>0$, the motion is hyperbolic and deformation of the fluid parcel is dominant. On the other hand, in case that tr$A^2<0$ the motion is elliptic and rotation is dominant. Since \begin{equation} \mbox{tr} A^2 = \frac{\partial^2 \psi}{\partial x \partial y} - \frac{\partial^2 \psi}{\partial x^2} \frac{\partial^2 \psi}{\partial y^2} = \cos^2 x \cos^2 y - \sin^2 x \sin^2 y = - \mbox{det} J, \end{equation} it just correspond with the classification at section \ref{ss:eigen}. That is, hyperbolic and elliptic motion corresponds to the saddle and center, respectively. %====================================================================== \section*{References} \begin{description} \itemsep -1.55mm \topsep 0mm \parsep 0mm \item Mizuta, R., 1999: Caotic mixing in quasi-periodic flow assuming the polar vortex in the stratosphere. Master thesis, Kyoto University, 60 pp. \item Ottino, J., 1989: {\it The kinematics of mixing: Stretching, chaos, and transport.} Cambridge University Press, 364 pp. \item Pierrehumbert, R.T., 1991: Large-scale horizontal mixing in planetary atmospheres. {\it Phys. Fluids A}, {\bf 3}, 1250-1260. \item Tatsumi, T., 1982: {\it Fluid dynamics.} Baifu-kan, 453 pp. \item Tatsumi, T., 1995: {\it Continuum dynamics.} Iwanami, 334 pp. \item Weiss, J., 1990: The dynamics of enstrophy transfer in two-dimensional hydrodynamics. {\it Physica D}, {\bf 48}, 273-294. \end{description} %\end{document}