Global mode

In mathematics and physics, a global mode of a system is one in which the system executes coherent oscillations in time. Suppose a quantity $y(x,t)$ which depends on space $x$ and time $t$ is governed by some partial differential equation which does not have an explicit dependence on $t$. Then a global mode is a solution of this PDE of the form $y(x,t)={\hat {y}}(x)e^{{i\omega t}}$, for some frequency $\omega$. If $\omega$ is complex, then the imaginary part corresponds to the mode exhibiting exponential growth or exponential decay.

The concept of a global mode can be compared to that of a normal mode; the PDE may be thought of as a dynamical system of infinitely many equations coupled together. Global modes are used in the stability analysis of hydrodynamical systems. Philip Drazin introduced the concept of a global mode in his 1974 paper, and gave a technique for finding the normal modes of a linear PDE problem in which the coefficients or geometry vary slowly in $x$. This technique is based on the WKBJ approximation, which is a special case of multiple-scale analysis.^[1] His method extends the Briggs–Bers technique, which gives a stability analysis for linear PDEs with constant coefficients.^[2]

In practice

Since Drazin's 1974 paper, other authors have studied more realistic problems in fluid dynamics using a global mode analysis. Such problems are often highly nonlinear, and attempts to analyse them have often relied on laboratory or numerical experiment.^[2] Examples of global modes in practice include the oscillatory wakes produced when fluid flows past an object, such as a vortex street.

References

1. Drazin, Philip (1974). "On a model of instability of a slowly-varying flow". Q J Mechanics Appl Math.
2. Huerre, Patrick; Monkewitz, Peter (1990). "Local and global instabilities in spatially developing flows.". Annu. Rev. Fluid Mech.