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We study the long time behaviour of solutions for the weakly damped forcedKawahara equation on the torus. More precisely, we prove the existence of aglobal attractor in $L^2$, to which as time passes all solutions draw closer.In fact, we show that the global attractor turns out to lie in a smoother space$H^2$ and be bounded therein. Further, we give an upper bound of the size ofthe attractor in $H^2$ that depends only on the damping parameter and the normof the forcing term.

Global attractor for the weakly damped forced Kawahara equation on the torus