Maximal transport in the Lorenz equations

• New rigorous bounds on solutions of the Lorenz equations are derived.
• The analysis is shown to be sharp: it is saturated by steady convection solutions.
• Optimal control methods are brought to bear on the Lorenz system for the first time.

We derive rigorous upper bounds on the transport 〈XY〉〈XY〉 where 〈⋅〉〈⋅〉 indicates time average, for solutions of the Lorenz equations without assuming statistical stationarity. The bounds are saturated by nontrivial steady (albeit often unstable) states, and hence they are sharp. Moreover, using an optimal control formulation we prove that no other flow protocol of the same strength, i.e., no other function of time X(t)X(t) driving the Y(t)Y(t) and Z(t)Z(t) variables while satisfying the basic balance 〈X2〉=〈XY〉〈X2〉=〈XY〉, produces higher transport.