We study the global attractor A(epsilon) of the non-autonomous 2D Navier-Stokes (N.-S.) system with singularly oscillating external force of the form g(0)(x,t)+ 1/epsilon(rho))g(1) (x/epsilon,t), x epsilon Omega subset of subset of R-2, t epsilon R, 0 <= rho <= 1. If the functions go(x, t) and g(1) (z, t) are translation bounded in the corresponding spaces, then it is known that the global attractor A(epsilon) is bounded in the space H, however, its norm ||A(epsilon) ||(H) may be unbounded as epsilon -> 0+ since the magnitude of the external force is growing. Assuming that the function g(1) (z, t) has a divergence representation of the form g(1) (z, t) = partial derivative(z1) (z, t) + partial derivative(z2) G(2)(z, t), z = (z(1), z(2)) epsilon R-2, where the functions G(j)(z, t) epsilon L-2(b) (R; Z) (see Section 3), we prove that the global attractors A(epsilon) of the N.-S. equations are uniformly bounded with respect to epsilon : ||A(epsilon) || H <= C for all 0 < epsilon <= 1. We also consider the "limiting" 2D N.-S. system with external force g(0)(x, t). We have found an estimate for the deviation of a solution u(epsilon)(x, t) of the original N.-S. system from a solution u(0)(x, t) of the "limiting" N.-S. system with the same initial data. If the function g(1)(z, t) admits the divergence representation, the functions g(0)(x, t) and g(1)(z, t) are translation compact in the corresponding spaces, and 0 <= p < 1, then we prove that the global attractors A(epsilon) converges to the global attractor A(0) of the "limiting" system as epsilon -> 0+ in the norm of H. In the last section, we present an estimate I, or the Hausdorff deviation of A(0) from A(0) of the form:dist(H) (A(epsilon), A(0)) <= C(rho)epsilon(1-rho) in the case, when the global attractor A(0) is exponential (the Grashof number of the "limiting" 2D N.-S. system is small).
