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In this paper, we consider the 2-D generalized surface quasi-geostrophicequation with the velocity $v$ determined by$v=\mathcal{R}^{\perp}\Lambda^{\gamma-1}\theta$. It is shown that the $L^p$type energy norm of weak solutions is conserved provided $\theta\inL^{p+1}(0,T; {B}^{\frac{\gamma}{3}}_{p+1, c(\mathbb{N})})$ for$0<\gamma<\frac32$ or $\theta\in L^{p+1}(0,T;{{B}}^{\alpha}_{p+1,\infty})~\text{for any}~\gamma-1<\alpha<1 \text{ with}~\frac{3}{2}\leq \gamma <2$. Moreover, we also prove that the helicity of weaksolutions satisfying $\nabla\theta \inL^{3}(0,T;\dot{B}_{3,c(\mathbb{N})}^{\frac{\gamma}{3}})$ for $0<\gamma<\frac32$or $\nabla\theta\in L^{3}(0,T; \dot{B}^{\alpha}_{3,\infty})~\text{forany}~\gamma-1<\alpha<1 \text{ with} ~\frac{3}{2}\leq \gamma <2$ is invariant.Therefore, the accurate relationships between the critical regularity for theenergy (helicity) conservation of the weak solutions and the regularity ofvelocity in 2-D generalized quasi-geostrophic equation are presented.

Energy and helicity conservation for the generalized quasi-geostrophic equation