Time Dependent Saddle Node Bifurcation: Breaking Time and the Point of No Return in a Non-Autonomous Model of Critical Transitions

There is a growing awareness that catastrophic phenomena in biology and medicine can be mathematically represented in terms of saddle-node bifurcations. In particular, the term `tipping', or critical transition has in recent years entered the discourse of the general public in relation to ecology, medicine, and public health. The saddle-node bifurcation and its associated theory of catastrophe as put forth by Thom and Zeeman has seen applications in a wide range of fields including molecular biophysics, mesoscopic physics, and climate science. In this paper, we investigate a simple model of a non-autonomous system with a time-dependent parameter $p(\tau)$ and its corresponding `dynamic' (time-dependent) saddle-node bifurcation by the modern theory of non-autonomous dynamical systems. We show that the actual point of no return for a system undergoing tipping can be significantly delayed in comparison to the {\em breaking time} $\hat{\tau}$ at which the corresponding autonomous system with a time-independent parameter $p_{a}= p(\hat{\tau})$ undergoes a bifurcation. A dimensionless parameter $\alpha=\lambda p_0^3V^{-2}$ is introduced, in which $\lambda$ is the curvature of the autonomous saddle-node bifurcation according to parameter $p(\tau)$, which has an initial value of $p_{0}$ and a constant rate of change $V$. We find that the breaking time $\hat{\tau}$ is always less than the actual point of no return $\tau^*$ after which the critical transition is irreversible; specifically, the relation $\tau^*-\hat{\tau}\simeq 2.338(\lambda V)^{-\frac{1}{3}}$ is analytically obtained. For a system with a small $\lambda V$, there exists a significant window of opportunity $(\hat{\tau},\tau^*)$ during which rapid reversal of the environment can save the system from catastrophe

Galilean invariance of lattice Boltzmann models

It is well-known that the original lattice Boltzmann (LB) equation deviates from the Navier-Stokes equations due to an unphysical velocity dependent viscosity. This unphysical dependency violates the Galilean invariance and limits the validation domain of the LB method to near incompressible flows. As previously shown, recovery of correct transport phenomena in kinetic equations depends on the higher hydrodynamic moments. In this Letter, we give specific criteria for recovery of various transport coefficients. The Galilean invariance of a general class of LB models is demonstrated via numerical experiments

A wall interference assessment/correction system

A Wall Signature method, the Hackett method, has been selected to be adapted for the 12-ft Wind Tunnel wall interference assessment/correction (WIAC) system in the present phase. This method uses limited measurements of the static pressure at the wall, in conjunction with the solid wall boundary condition, to determine the strength and distribution of singularities representing the test article. The singularities are used in turn for estimating wall interferences at the model location. The Wall Signature method will be formulated for application to the unique geometry of the 12-ft Tunnel. The development and implementation of a working prototype will be completed, delivered and documented with a software manual. The WIAC code will be validated by conducting numerically simulated experiments rather than actual wind tunnel experiments. The simulations will be used to generate both free-air and confined wind-tunnel flow fields for each of the test articles over a range of test configurations. Specifically, the pressure signature at the test section wall will be computed for the tunnel case to provide the simulated 'measured' data. These data will serve as the input for the WIAC method-Wall Signature method. The performance of the WIAC method then may be evaluated by comparing the corrected parameters with those for the free-air simulation. Each set of wind tunnel/test article numerical simulations provides data to validate the WIAC method. A numerical wind tunnel test simulation is initiated to validate the WIAC methods developed in the project. In the present reported period, the blockage correction has been developed and implemented for a rectangular tunnel as well as the 12-ft Pressure Tunnel. An improved wall interference assessment and correction method for three-dimensional wind tunnel testing is presented in the appendix