In the physics of hydrodynamics, a '''global mode''' of a system is one in which the system executes coherent oscillations in time. Suppose a quantity <math>y(x,t)</math> which depends on space <math>x</math> and time <math>t</math> is governed by some partial differential equation which does not have an explicit dependence on <math>t</math>. Then a global mode is a solution of this PDE of the form <math>y(x,t) = \hat{y}(x) e^{i\omega t}</math>, for some frequency <math>\omega</math>. If <math>\omega</math> 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 <math>x</math>. This technique is based on the WKBJ approximation, which is a special case of multiple-scale analysis.<ref name="drazin-1974">{{cite journal|last1=Drazin|first1=Philip|title=On a model of instability of a slowly-varying flow|journal=Q J Mechanics Appl Math|date=1974|volume=27|pages=69–86|doi=10.1093/qjmam/27.1.69}}</ref> His method extends the Briggs–Bers technique, which gives a stability analysis for linear PDEs with constant coefficients.<ref name="huerre-monkewitz-1990">{{cite journal|last1=Huerre|first1=Patrick|last2=Monkewitz|first2=Peter|title=Local and global instabilities in spatially developing flows.|journal=Annu. Rev. Fluid Mech.|date=1990|volume=22|page=473|doi=10.1146/annurev.fl.22.010190.002353|bibcode=1990AnRFM..22..473H}}</ref>
==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.<ref name="huerre-monkewitz-1990"/> Examples of global modes in practice include the oscillatory wakes produced when fluid flows past an object, such as a vortex street.
==References== <references/>
Category:Partial differential equations