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 which depends on space and time is governed by some partial differential equation which does not have an explicit dependence on . Then a global mode is a solution of this PDE of the form , for some frequency . If 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 . 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
[edit]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
[edit]- ^ Drazin, Philip (1974). "On a model of instability of a slowly-varying flow". Q J Mechanics Appl Math. 27: 69–86. doi:10.1093/qjmam/27.1.69.
- ^ a b Huerre, Patrick; Monkewitz, Peter (1990). "Local and global instabilities in spatially developing flows". Annu. Rev. Fluid Mech. 22: 473. Bibcode:1990AnRFM..22..473H. doi:10.1146/annurev.fl.22.010190.002353.