Visualisation of the fluid displacement ξ (in green), Lagrangian velocity u_L (in red), and Lagrangian mean velocity ⟨u⟩_L (in blue), in the case where the mean ⟨.⟩ is defined as an average over a given axis (spanning horizontally in the figure). The grey volume V_L represents a thin tube of fluid initially lying along the horizontal direction. The fluid displacement and velocity ξ and u_L pertain to the deformation and local velocity of the tube, while ⟨u⟩_L refers to the velocity of the centre of mass of the tube, represented by the dashed horizontal line. This illustration is inspired by Andrews & McIntyre (1978) (see bottom panel of their Fig. 1).