In this paper we consider the following modified quasi-geostrophic equation
\partial_{t}\theta+u\cdot\nabla\theta+\nu |D|^{\alpha}\theta=0,
\quad u=|D|^{\alpha-1}\mathcal{R}^{\bot}\theta,\quad x\in\mathbb{R}^2 with
ν>0 and α∈]0,1[∪]1,2[. When α∈]0,1[, the
equation was firstly introduced by Constantin, Iyer and Wu in \cite{ref
ConstanIW}. Here, by using the modulus of continuity method, we prove the
global well-posedness of the system with the smooth initial data. As a
byproduct, we also show that for every α∈]0,2[, the Lipschitz norm of
the solution has a uniform exponential bound.Comment: In this version we extend the range of α from (0,1) to (0,2),
we also show that for every α∈(0,2), the Lipschitz norm of the
solution has a uniform exponential bound. 27page