  In this paper, we consider the following modified quasi-geostrophic equations
$\partial_t\theta +\Lambda^\alpha\theta +u\vec\nabla\theta =0$, $u=\Lambda
^{\alpha-1}\mathcal{R}^\perp(\theta)$ where $\alpha \in ]0,1[$ is a fixed
parameter. This equation was recently introduced by P. Constantin, G. Iyer and
J. Wu in \cite{CIW} as a modification of the classical quasi-geostrophic
equation. In this paper, we prove that for any initial data $\theta_\ast$ in
the Sobolev space $H^1(\mathbb{R}^2),$ the equation (MQG) has a global and
smooth solution $\theta $ in $C(\mathbb{R}^{+},H^1(\mathbb{R}^2)) .$
