  It is proved that, for $T^\epsilon\le G = G(T) \le {1\over2}\sqrt{T}$, $$
\int_T^{2T}\Bigl(I_1(t+G)-I_1(t)\Bigr)^2 dt = TG\sum_{j=0}^3a_j\log^j
\Bigl({\sqrt{T}\over G}\Bigr) + O_\epsilon(T^{1+\epsilon}+ T^{1/2+\epsilon}G^2)
$$ with some explicitly computable constants $a_j (a_3>0)$ where, for a fixed
natural number $k$, $$I_k(t,G) = {1\over\sqrt{\pi}}\int_{-\infty}^\infty
|\zeta(1/2+it+iu)|^{2k} {\rm e}^{-(u/G)^2} du. $$ The generalizations to the
mean square of $I_1(t+U,G) - I_1(t,G)$ over $[T, T+H]$ and the estimation of
the mean square of $I_2(t+U,G)-I_2(t,G)$ are also discussed.
