Warning signs for boundary noise and their application to an ocean Boussinesq model

In this paper, we construct and discuss early-warning signs of the approach of a parameter to a deterministic bifurcation on a stochastic partial differential equation (SPDE) model with Gaussian white-noise on the boundary. We specifically focus on the qualitative behaviour of the time-asymptotic autocovariance and autocorrelation of the solutions of the linearized system. We also discuss the reliability of the tools from an analytic perspective and through various examples. Among those, the application of the early-warning signs for an ocean Boussinesq model is explored through numerical simulations. The analytic results obtained expand on previous work and present valuable early-warning signs for various applications. •Construction of early-warning signs on stochastic systems with boundary noise.•Explanation of the correct use of the observables in theoretical and practical models.•Application of the tools on an ocean Boussinesq model and stochastic heat equation.