Adjoint chaos in wall turbulence

Inferring past flow events in wall turbulence from surface measurements is an ill-posed inverse problem where chaos amplifies uncertainties exponentially in backward time. The adjoint Navier-Stokes equations, when solved in backward time, quantify the sensitivity of a wall measurement to earlier flow states and provide the measurement spatial-temporal domain of dependence (DOD). In turbulent channel flow, individual adjoint realizations exhibit exponential energy amplification in backward time, which indicates that two measurements that are infinitesimally close can be due to exponentially different earlier events. This phenomenon has been termed the dual butterfly effect [T.A. Zaki, Annu. Rev. Fluid Mech., Vol. 57 (2025), pp. 311–33]. In contrast to a single adjoint realization, the energy of the ensemble-averaged adjoint field decays. Computing this mean sensitivity for long backward time is, however, challenging due to the required number of realizations. We therefore introduce a linear eddy-viscosity closure in the adjoint equations to directly compute the mean DOD. Our results agree with ensemble statistics and reveal that both the DOD of wall-shear measurements, and display Reynolds-number-invariant behavior. The DOD exhibits a two-part structure: one tied to the reverse Orr mechanism, and another to self-similar streaky structures expanding near the wall. Together, these two mechanisms govern the sensitivity of wall measurements to past flow events.