Sketching a Probability Theory of Transition to Turbulence

In this short article, we wish to initiate a probability theory to describe the (in)famous transition to turbulence phenomena. For normal systems, the instability threshold can be predicted because the transition probability jumps from zero to $ P\rightarrow 1 $ when the control parameter exceeds the threshold. However, for non-normal systems, such a transition is probabilistic, which depends on the control parameter and initial energy. A low-dimensional reduction idea is proposed to give a detailed description of the transition probability of non-normal systems in the future. We also wish that such an idea could be transplanted for understanding such a transition in more complex flows, e.g., atmospheric flows, mantle convection, and ocean circulation.