Global mode explained

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} , 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. On a model of instability of a slowly-varying flow. Q J Mechanics Appl Math. 1974. 27. 69–86. 10.1093/qjmam/27.1.69.
2. Huerre. Patrick. Monkewitz. Peter. Local and global instabilities in spatially developing flows.. Annu. Rev. Fluid Mech.. 1990. 22. 473. 10.1146/annurev.fl.22.010190.002353. 1990AnRFM..22..473H.