Spontaneous Stochasticity in Atmospheric Turbulence and Climate Dynamics

Lorenz^a argued that “formally deterministic fluid systems which possess many scales of motion are observationally indistinguishable from indeterministic systems.” Recently, Lorenz’ idea has been related to spontaneous stochasticity, or persistent randomness in solutions of singular deterministic dynamics for fixed initial data, as regularizations and stochastic perturbations are both taken to vanish.^b,c We study the effect of thermal noise on turbulent solutions of the incompressible Navier-Stokes equation and argue that, for fixed deterministic initial data, a stochastic ensemble of non-unique Euler solutions is obtained in the high Reynolds-number limit. Our arguments are supported by numerical simulations. Thus, not only the flap of a seagull wing but even the “swerve of the molecules” leads to intrinsic randomness in one eddy turnover time, which “cannot be lengthened by reducing the amplitude of the initial error”.^a Our results support arguments that climate models must be intrinsically stochastic, even if they are resolved to 1 km scales and below.^d
aE. N. Lorenz, Tellus 21, 289–307 (1969) ^bD. Bernard, K. Gawedzki, and A. Kupiainen, J. Stat. Phys. 90, 519–569 (1998) ^cA. Mailybaev, Nonlinearity 29, 2238 (2016) ^dT. N. Palmer, Nature Rev. 1, 463 (2019)