Latex, 26K BODY \section{Introduction} We consider the 2-D surface quasi-geostrophic (QGS) equations $$\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta + \kappa (-\Delta)^{\alpha}\theta=0,\quad 0\le \alpha\le 1$$ where $\kappa>0$ in the case of the dissipative equations and $\kappa=0$ for the non-dissipative equations. Here the velocity $u=(u_1, u_2)$ is determined from $\theta$ by a stream function $\psi$: $$(u_1,u_2)=\left (-\frac{\partial \psi}{\partial x_2},\frac{ \partial \psi}{\partial x_1}\right)$$ where $\psi$ satisfies $${(-\Delta)^{\frac{1}{2}}}\psi=-\theta$$ \vspace{.1in} These equations have been under intensive investigations because of mathematical importance and potential applications in meteorology and oceanography (\cite{CMT},\cite{HPGS},\cite{Pe}). As pointed out in \cite{CMT}, the 2-D dissipative and non-dissipative QGS equations are strikingly analogous to the 3-D Navier-Stokes and the Euler equations. Inviscid limit results for the Navier-Stokes equations with smooth and rough initial data have been established (\cite{bm},\cite{c}, \cite{CW1},\cite{CW2},\cite{CW3}). Naturally the inviscid limit problem for the solutions of the dissipative QGS equations rises and so far we have seen no work in this direction. \vspace{.15in} We know from \cite{CMT} that the QGS equations (defined on $\Bbb R^2$ or $\Bbb T^2$) with smooth initial data admit unique classical solutions for short times and the quantity $$\int_{0}^{T}\|\nabla\theta(\cdot,s)\|_{L^\infty}ds$$ is responsible for possible singularity formation. The control of this quantity is also important in our proof of the inviscid limit results for smooth solutions. \vspace{.1in} Both dissipative and non-dissipative QGS equations on $\Bbb T^2$ with $L^2$ initial data have weak solutions in the distribution sense (\cite{Re}). Inviscid limits for weak solutions are in general hard to obtain and in this case we only obtain a weak $L^2$ result without a rate. We believe that explicit rates can be given in certain negative Sobolev norms. \vspace{.15in} The second part is devoted to regularity estimates for the dissipative QGS equations. Although we use the ideas of Foias-Temam \cite{FT} and Doering-Titi \cite{DT} developed for the Navier-Stokes equations, our methods of estimating the non-linear terms is significantly different from theirs because of the special structure of the QGS equations. We treat the norms $\|e^{\gamma t\Lambda^\alpha}\Lambda^{\beta}\theta\|_{L^2}$ (see notations in Section 3) and a special consequence of our estimates for $\|e^{\gamma t\Lambda^\alpha}\Lambda^{\beta}\theta\|_{L^2}$ is that as long as $\|\Lambda^{\beta}\theta\|_{L^2}^{2}$ remains bounded, the Fourier spectrum decays exponentially at high wave numbers. In a similar fashion as Doering and Titi \cite{DT} argue for the Navier-Stokes equations, these exponential decay estimates can be used to obtain bounds on small length scale defined through the exponential decay rate. \vspace{.1in} Finally, I would like to thank Professor Peter Constantin for his suggestions and help. \newpage \section {Inviscid Limits} We consider the 2-D dissipative ($\kappa>0$) and the non-dissipative ($\kappa=0$) quasi-geostrophic equations on $\Bbb R^2$ or $\Bbb T^2$: $$\frac{\partial \theta}{\partial t}+u\cdot\nabla\theta + \kappa \Lambda^{2\alpha}\theta=0,\qquad 0\le \alpha\le 1$$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$ is the Riesz potential operator and the velocity $u$ is defined from the stream function $\psi=-\Lambda^{-1}\theta$ by $$u=(u_1,u_2)=\left (-\frac{\partial \psi}{\partial x_2},\frac{ \partial \psi}{\partial x_1}\right)$$ \vspace{.1in} We use $\Bbb D^2$ to denote either $\Bbb R^2$ or $\Bbb T^2$. We first consider the smooth initial data case: $\theta_0\in H^k( \Bbb D^2)(k\ge 3)$. As shown by Constantin, Majda and Tabak \cite{CMT}, the QGS equations with smooth initial data have local (in time) smooth solutions and the Beale-Kato-Majda type blowup conditions have been obtained. More precisely, \begin{prop}\label{prop:2.1} If the initial data $\theta|_{t=0}=\theta_0\in H^k(\Bbb D^2)$ for some $k\ge 3$, then both the QGS and the dissipative QGS equations have a unique smooth solution for a small time interval, respectively. Furthermore, the solution $\theta^{QG}$ of the QGS equations satisfies $$\int_{0}^{t}\|\nabla \theta^{QG}(\cdot,s)\|_{L^\infty}ds< \infty,$$ $$\int_{0}^{t}\|\theta^{QG}(\cdot,s)\|_{k}^{2}ds< \infty$$ for any $t$ belonging to the existence interval $[0,T^*)$. \end{prop} This proposition admits the possibility of finite-time singularity formation and consequently, the inviscid limit results for the smooth solutions are valid only for the time period before the possible breakdown. We need further estimates on the solutions. \begin{prop}\label{prop:33.1.2} Let $\theta^{QG}$ and $\theta^{DQG}$ be the smooth solutions of the QGS and the dissipative QGS equations with the same initial data $\theta_0\in H^k(\Bbb D^2)$$(k\ge 3). Respectively, u^{QG} and u^{DQG} are the corresponding velocities and \psi^{QG} and \psi^{DQG} are the stream functions. Then for any t in the maximal time interval [0,T^*) (when the smooth solutions exist), \begin{enumerate} \item The solution \theta^{QG} of the QGS equation satisfies:$$\int_{\Bbb D^2}G(\theta^{QG}(x,t))dx=\int_{\Bbb D^2} G(\theta_0)dx, $$where G is a continuous function with G(0)=0. Especially,$$\|\theta^{QG}(\cdot,t)\|_{L^p}=\|\theta_0\|_{L^p},\quad 1\le p\le\infty,$$furthermore,$$ \|u^{QG}(\cdot,t)\|_{L^2}= \|\theta^{QG}(\cdot,t)\|_{L^2}=\|\theta_0\|_{L^2}, \|u^{QG}(\cdot,t)\|_{L^q}\le C_q\|\theta^{QG}(\cdot,t)\|_{L^q},\qquad 12$ and $\chi_r=\chi(\frac{x}{r})$ for $r>0$. Using the Dominated Convergence Theorem and the divergence theorem, $$I=\lim_{r\to \infty}\int (u^{DQG}\cdot\nabla\theta)\theta\chi_r(x)dx =-\lim_{r\to \infty}\frac{1}{2r}\int\chi'\cdot u^{DQG}\theta^2 dx$$ since the last integral is bounded, $$|\int \chi'\cdot u^{DQG}\theta^2 dx|\le \int |u^{DQG}|\theta^2 dx \le \|\theta^{DQG}\|_{L^2}\|\theta\|^{2}_{L^4}\le 4\|\theta_0\|_{L^2} \|\theta_0\|^{2}_{L^4}$$ we obtain $I=0$. $II$ and $III$ can be estimated by using Proposition \ref{prop:33.1.2}, $$|II|\le \|\nabla\theta^{QG}\|_{L^\infty}\|u\|_{L^2}\|\theta\|_{L^2} =\|\nabla\theta^{QG}\|_{L^\infty}\|\theta\|_{L^2}^{2}$$ $$|III|\le \frac{\kappa^2}{2}\int (\Lambda^{2\alpha}\theta^{QG})^2dx +\frac{1}{2}\int\theta^2 dx$$ \vspace{.15in} Collecting these estimates, $$\frac{d}{dt}\int \theta^2dx+\kappa \|\Lambda^{\alpha} \theta\|_{L^2}^{2} dx \le \cal{P}(t)\|\theta\|_{L^2}^{2} +\kappa^2 \|\theta^{QG}\|_{2\alpha}^{2}$$ where $$\cal{P}(t)=2\|\nabla\theta^{QG}(\cdot,t)\|_{L^\infty}+1$$ By Gronwall's inequality, $$\|\theta\|_{L^2}^{2}\le e^{\int_{0}^{t}\cal{P}(s)ds} \|\theta_0\|_{L^2}^{2}+\kappa^2 \int_{0}^{t} e^{\int_{\tau}^{t}\cal{P}(s)ds} \|\theta^{QG}\|^{2}_{2\alpha} d\tau$$ Noting that $\theta_0=0$ and using the result of Proposition \ref{prop:2.1}, especially, $$\int_{0}^{t}\|\nabla\theta^{QG}(\cdot,s)\|_{L^\infty}ds< \infty,\quad \int_{0}^{t}\|\theta^{QG}(\cdot,s) \|_{k}^{2}ds <\infty$$ we obtain $$\|\theta\|_{L^2}\le C\kappa ,$$ which completes the proof of Theorem \ref{thm:chapter4}. \vspace{.24in} We now turn to weak solutions of these equations corresponding to $L^2$ initial data. We restrict ourselves to the periodic domain $\Bbb T^2=[0,L]\times[0,L]$. We quote the result of Resnick \cite{Re} on the existence of weak solutions. \begin{prop} Let $\theta_0\in L^2(\Bbb T^2)$ and $T>0$ be arbitrarily fixed. Then there exist weak solutions $\theta^{QG}\in L^\infty([0,T];L^2(\Bbb T^2))$ and $\theta^{DQG}\in L^\infty([0,T];L^2(\Bbb T^2))\cap \newline L^2([0,T];H^\alpha( \Bbb T^2))$ of the QGS and the dissipative QGS equations, respectively. That is, for each test function $\phi\in C^\infty(\Bbb T^2)$, $$\int \theta^{QG}\phi dx-\int\theta_0\phi dx-\int_{0}^{T} \int_{\Bbb T^2}\theta^{QG}(u^{QG}\cdot\nabla \phi)dxdt=0,$$ $$\int \theta^{DQG}\phi dx -\int \theta_0\phi dx- \int_{0}^{T}\int_{\Bbb T^2}\theta^{DQG}(u^{DQG}\cdot\nabla \phi)dxdt=0,$$ where $u^{QG}$ and $u^{DQG}$ are the velocities corresponding to $\theta^{QG}$ and $\theta^{DQG}$. \end{prop} \vspace{.1in} These weak solutions are constructed by using classical Galerkin approximations. The weak $L^2$ inviscid limit result is an easy consequence of this construction method. \begin{thm} Let $\theta_0\in L^2(\Bbb T^2)$ and $\theta^{QG}$ and $\theta^{DQG}$ be the weak solutions of the QGS and the dissipative QGS equations with the same initial data $\theta_0$. Then for any arbitrarily fixed $T>0$ and any $\phi\in L^2(\Bbb T^2)$, $$\label{eq:33.2.1} \limsup_{\kappa\to 0}\left(\theta^{DQG}(\cdot,t)-\theta^{QG}(\cdot,t),\phi\right)=0, \qquad \mbox{for any t\le T},$$ \end{thm} {\bf Proof}\quad Consider the $n$-th Galerkin approximation $\{\theta^{QG}_{n}\}$ and $\{\theta^{DQG}_{n}\}$, which are in the space $S_n$ spanned by the Fourier modes $e^{imx}$ with $0<|m|\le n$ and satisfy $$\frac{\partial \theta_n}{\partial t}+P_n(u_n\cdot\nabla\theta_n)+ \kappa\Lambda^{2\alpha}\theta_n=0,$$ $$\theta_n|_{t=0}=P_n\theta_0,$$ where $P_n$ is the orthogonal projection from $L^2$ onto $S_n$ and $\kappa=0$ in the case of $\theta^{QG}$. As we know from \cite{Re}, for some subsequences $$\theta^{DQG}_{n}\rightharpoonup\theta^{DQG},\quad\theta^{QG}_{n} \rightharpoonup\theta^{QG} \quad \mbox{weakly in L^2(\Bbb T^2)},$$ So we have for any $\epsilon>0$ by taking large $n$, $$|(\theta^{DQG}(\cdot,t)-\theta^{QG}(\cdot,t),\phi)|\le \epsilon+ |(\theta^{DQG}_{n}-\theta^{QG}_{n},\phi)|$$ $$\label{eq:2} \le \epsilon +\|\phi\|_{L^2}\|\theta^{DQG}_{n}-\theta^{QG}_{n}\|_{L^2} \le \epsilon+C_n\kappa$$ which implies (\ref{eq:33.2.1}). Here we've applied the inviscid limit result for smooth solutions to $\theta^{DQG}_{n}-\theta^{QG}_{n}$. \vspace{.12in} \begin{rem} Since the constants $C_n$ in the inequality (\ref{eq:2}) depends on $n$, we obtain no convergence rate. An explicit rate may exist in negative Sobolev norms. \end{rem} \newpage \section {Regularity Estimates} We consider the 2-D dissipative QGS equations with smooth initial data on the torus $\Bbb T^2=[0,L]\times[0,L]$, which admits a unique classical local solution. Let $\theta$ be this solution. We will estimate the quantity $$\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta\|_{L^2}^{2}$$ where the operators $\Lambda^\beta$ and $e^{\lambda \Lambda^\alpha}$ are defined through the Fourier transform $$\Lambda^\beta f=L^{-2}\sum_{k}e^{ik\cdot x}|k|^{\beta}\widehat{f}(k)$$ $$e^{\lambda\Lambda^\alpha}f=L^{-2}\sum_{k}e^{ik\cdot x+\lambda|k|^\alpha} \widehat{f}(k)$$ with $\widehat{f}(k)$ being the k-th Fourier mode of $f$, $$\widehat{f}(k)=\int_{\Bbb T^2}e^{-ik\cdot x}f(x)dx.$$ It is easy to see from these notations that $e^{\lambda\Lambda^\alpha}$ commutes with $\Lambda^\beta$ and partial derivatives for periodic boundary conditions considered here. \vspace{.14in} We obtain bounds for the quantity $\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta\|_{L^2}^{2}$, which lead to the exponential decay of the Fourier spectrum of $\theta$. The precise estimates are \begin{thm} Consider the 2-D dissipative QGS equations $$\label{eq:3.1} \frac{\partial \theta}{\partial t}+u\cdot\nabla \theta +\kappa\Lambda^{2\alpha}\theta=0,\qquad \kappa>0,\quad \frac{1}{2}<\alpha\le 1$$ on the 2-D torus $\Bbb T^2=[0,L]\times[0,L]$. Let the initial data $\theta_0\in H^k(\Bbb T^2)$ and $\theta$ be the unique smooth solution. We take $\beta:$ $$\beta>0, \qquad\beta+2\alpha> 2.$$ Then $\theta$ satisfies for any $\gamma>0$ $$\label{eq:3.2} \|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta \|_{L^2}^{2}\le \frac{e^{\frac{2\gamma^2t}{\kappa}}\|\Lambda^\beta\theta_0\|_{ L^2}^{2}} {\left(1-C(\|\Lambda^\beta\theta_0\|_{L^2}^{2})^{N-1}\kappa^{-(M-1)}\gamma^{-2} (e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}},$$ which is finite for $t\in[0,t^*)$, $$t^*=\frac{\kappa}{2(N-1)\gamma^2}\log\left(1+\frac{\kappa^{M-1}\gamma^2}{ C\|\Lambda^\beta\theta_0\|^{2(N-1)}_{L^2}}\right)$$ Here $C$ is a constant and $M,N$ are given by $$M=\frac{\alpha+\sigma}{\alpha-\sigma},\quad N=1+\frac{\alpha}{\alpha-\sigma}, \quad \sigma=\left\{\begin{array}{ll}1-\alpha,&\quad \mbox{if \beta\ge 1} \\2-\beta-\alpha,&\quad \mbox{if \beta\le 1}\end{array}\right.$$ \end{thm} \vspace{.12in} A special consequence of this theorem is that each Fourier mode amplitude can be individually controlled. In fact, a rough estimate gives $$e^{2\gamma t |k|^{\alpha}}|k|^{2\beta}|\theta(k,t)|^2 \le \sum_{k}e^{2\gamma t |k|^{\alpha}}|k|^{2\beta}|\theta(k,t)|^2 =L^{2}\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta \|_{L^2}^{2}$$ Thus the k-th mode is bounded by $$|\theta(k,t)|^2\le \frac{L^2}{|k|^{2\beta}}\frac{e^{\frac{\gamma^2 t}{\kappa} -2\gamma t|k|^{\alpha}}\|\Lambda^\beta\theta_0\|_{L^2}^{2}} {\left(1-C(\|\Lambda^\beta\theta_0\|_{L^2}^{2})^{N-1}\kappa^{-(M-1)}\gamma^{-2} (e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}}$$ for $t\in[0,t^*).$ \vspace{.1in} Doering and Titi \cite{DT} establish the exponential decay of power spectrum for the flow field of the 3-D Navier-Stokes equations. Similar analysis based on these bounds can be made to conclude that if $\|\Lambda^\beta\theta\|^{2}_{L^2}$ is bounded uniformly in time, then after a transient time of length $t^*/2$ the Fourier spectrum of $\theta$ decays exponentially at high wave numbers. Furthermore, the associated decay length can be defined and estimated in terms of the dissipation rates. \vspace{.2in} The main difficulty in proving the estimate (\ref{eq:3.2}) is how to bound the non-linear term properly. We need the inequalities for the Calderon-Zygmund type singular integrals. We'll also use the following lemma concerning the operator $\Lambda^s$, which is proved in \cite{Re},\cite{Ta}. \begin{lemma} For $s>0$ and $11$ and $q=\frac{2}{\beta}$ if $\beta\le 1$. Choose $p$ such that $$\frac{2}{p}+\frac{2}{q}=1$$ and $\sigma=2-\frac{2}{q}-\alpha$. The condition that $\beta+2\alpha>2$ implies that $0<\sigma<\alpha$. We apply Lemma 2.2 to obtain $$|II|\le C\|\tilde{\theta}\|_{H^{\alpha+\beta}}\left(\|\tilde{u}\|_{L^p} \|\Lambda^{\beta+1-\alpha}\tilde{\theta}\|_{L^q} +\|\tilde{\theta}\|_{L^p}\|\Lambda^{\beta+1-\alpha}\tilde{u}\|_{L^q}\right) =II_1+II_2$$ \vspace{.1in} Using Sobolev imbeddings $$H^{\beta}\hookrightarrow H^{\frac{2}{q}}\hookrightarrow L^p,\qquad H^{\beta+\sigma}\hookrightarrow L^{p}_{\beta+1-\alpha}$$ and the Gagliado-Nirenberg interpolation (since $\sigma<\alpha$): $$\beta+\sigma=\frac{\sigma}{\alpha}(\beta+\alpha)+\left(1-\frac{\sigma} {\alpha}\right)\beta$$ we have $$II_1\le C\|\tilde{\theta}\|_{H^{\beta+\alpha}}\|\tilde{u}\|_{H^{\beta}} \|\tilde{\theta}\|_{H^{\beta+\sigma}}\le C\|\tilde{\theta}\|_{H^{\beta}} \|\tilde{\theta}\|_{H^{\beta+\alpha}}\|\tilde{\theta}\|_{H^{\beta+\sigma}}$$ $$\le C\|\tilde{\theta}\|_{H^{\beta+\alpha}}^{1+\frac{\sigma}{\alpha}} \|\tilde{\theta}\|_{H^{\beta}}^{2-\frac{\sigma}{\alpha}}$$ where $C$ is constant depending on $\alpha$ and $\beta$. By Young's inequality $$II_1\le \frac{\kappa}{8}\|\tilde{\theta}\|_{H^{\beta+\alpha}}^{2}+ \frac{C}{\kappa^{M}}\left(\|\tilde{\theta}\|_{\beta}^{2}\right)^{N}$$ where $N=\frac{2\alpha-\sigma}{\alpha-\sigma}$ and $M=\frac{\alpha+\sigma}{\alpha-\sigma}$. \vspace{.11in} A similar estimate results in the same bound for $II_2$. \vspace{.17in} Collecting the estimates for $I,II$ and $III$ and reintroducing $\tilde{\theta}=e^{\gamma t\Lambda^{\alpha}}\theta$, $$\frac{d}{dt}\|e^{\gamma t\Lambda^{\alpha}}\Lambda^{\beta}\theta\|_{L^2} ^{2}\le -\kappa \|e^{\gamma t\Lambda^{\alpha}}\Lambda^{\beta+\alpha} \theta\|_{L^2}^{2}$$ $$\qquad\qquad \qquad+\frac{2\gamma^2}{\kappa}\| e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\|_{L^2}^{2} +\frac{C}{\kappa^M}\|e^{\gamma t\Lambda^\alpha}\Lambda^\beta\theta\|_{L^2}^{2N}$$ Now we let $$Z(t)= \|e^{\gamma t\Lambda^{\alpha}}\Lambda^{\beta}\theta\|_{L^2} ^{2}$$ the differential inequality becomes $$\frac{dZ}{dt}\le \frac{2\gamma^2}{\kappa}Z+\frac{C}{\kappa^M}Z^N$$ An elementary algebra results $$\frac{dY}{dt}\le \frac{Ce^{2(N-1)\frac{\gamma^2t}{\kappa}}}{\kappa^M}Y^{N}$$ where $Y=e^\frac{-2\gamma^2t}{\kappa}Z$. After a simple calculation, $$Y\le \frac{Y_0}{\left(1-CY_{0}^{N-1}\kappa^{-(M-1)}\gamma^{-2}( e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}}$$ Reintroducing $Z(t)=\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta \|_{2}^{2}$ and noting that $Z(0)=\|\Lambda^\beta\theta_0\|_{2}^{2}$ that $$\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta \|_{2}^{2}\le \frac{e^\frac{2\gamma^2t}{\kappa}\|\Lambda^\beta\theta_0\|_{2}^{2}} {\left(1-C(\|\Lambda^\beta\theta_0\|_{2}^{2})^{N-1}\kappa^{-(M-1)}\gamma^{-2} (e^{\frac{2(N-1)\gamma^2t}{\kappa}}-1)\right)^{\frac{1}{N-1}}}$$ This means that $\|e^{\gamma t\Lambda^{\alpha}}\Lambda^\beta\theta \|_{2}^{2}$ is finite on the interval $[0,t^*)$, where $$t^*=\frac{\kappa}{2(N-1)\gamma^2}\log\left(1+\frac{\kappa^{M-1}\gamma^2}{ C\|\Lambda^\beta\theta_0\|^{2(N-1)}_{2}}\right)$$ The smaller the initial decay rate $\|\Lambda^\beta\theta_0\|_{2}$ is , the larger $t^*$. Similarly, the larger the parameter $\gamma$ , the shorter $t^*$. \newpage \begin{thebibliography}{99} \bibitem{bm} J. T. Beale, A. Majda, Rates of convergence for viscous splitting of the Navier-Stokes equations, Math. Comp. {\bf 37} (1981), 243-259. \bibitem{c} P. Constantin, Note on loss of regularity for solutions of the 3D incompressible Euler and related equations, Commun. Math. Phys., {\bf 104} (1986), 311-326. \bibitem{CMT} P. Constantin, A. Majda, E. Tabak, Formation of strong fronts in the 2-D quasi-geostrophic thermal active scalar, Nonlinearity, {\bf 7}(1994), 1495-1533 \bibitem{CW1} P. Constantin, J. Wu, Inviscid limit for vortex patches, Nonliearity, {\bf 8}(1995), 735-742 \bibitem{CW2} P. Constantin, J. Wu, Vanishing viscosity limit for vortex patches, Institut Mittag-Leffler, No. {\bf 26}(1995) \bibitem{CW3} P. Constantin, J. Wu, The inviscid limit for non-smooth vorticity, Indiana Univ. Math. J., {\bf 44}(1995) \bibitem{DT} Ch. Doering, E. Titi, Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations, Phys. of Fluids, {\bf 7} (1995), No.6, 1384-1390. \bibitem{FT} C. Foias, R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., {\bf 87}(1989), 359-369. \bibitem{HPGS} I. Held, R. Pierrehumbert, S. Garner, K. Swanson, Surface quasi-geostrophic dynamics, J. Fluid Mech., {\bf 282}(1995), 1-20 \bibitem{Pe} J. Pedlosky, Geophysical Fluid Dynamics", Springer, New York, 1987 \bibitem{Re} S. Resnick, Dynamical Problems in Non-linear Advective Partial Differential Equations", Ph.D. thesis, University of Chicago, 1995 \bibitem{Ta} M. Taylor, Pseudodifferential Operators and Nonliear PDE", Birkh\"{a}user, 1993 \end{thebibliography} \end{document}