Unbounded two-dimensional wall turbulence induced by inverse cascade

What would be the ultimate statistical invariance for wall turbulence? In 3D flows, whether turbulent fluctuations are bounded or unbounded for a series of quantities when the Reynolds number (Re) tends to infinity, is debated. Here, we focus on 2D wall turbulence and find that fluctuations in wall units such as root mean square of wall pressure, wall shear stress intensity, streamwise velocity fluctuation, etc., all exhibit a distinct Re1/3 scaling law. This scaling indicates an unbounded behavior for 2D flows due to the inverse energy cascade, which sheds light on the asymptotical behavior of 3D flows that fluctuations might be bounded because of the forward energy cascade.