To this point we have discussed only linear stability theory, which considers infinitesimal perturbations and predicts exponential growth when the relevant parameter exceeds a critical value. The effect of the perturbations on the basic flow is neglected in the linear theory. An examination of
(11.88) shows that the perturbation field must be such that the average
uv (the average taken over a wavelength) must be non-zero for the perturbations to extract energy from the basic shear; similarly, the heat flux, the average of
uT′, must be non-zero in a thermal convection problem. These rectified fluxes of momentum and heat change the
basic velocity and temperature fields. Linear instability theory neglects these changes of the basic state. A consequence of the constancy of the basic state is that the growth rate of the perturbations is also constant, leading to predictions of exponential growth. However, after some time, the perturbations eventually become so large that the rectified fluxes of momentum and heat significantly change the basic state, which in turn alters the growth of the perturbations.
A frequent effect of nonlinearity is to change the basic state in such a way as to arrest the growth of the disturbances after they have reached significant amplitude via their initial exponential growth. (Note, however, that the effect of nonlinearity can sometimes be destabilizing; for example, the instability in a pipe flow may be a finite-amplitude effect because the flow is stable to infinitesimal disturbances.) Consider thermal convection in the annular space between two vertical cylinders rotating at the same speed. The outer wall of the annulus is heated and the inner wall is cooled. For small heating rates the flow is steady. For large heating rates a system of regularly spaced waves develop and progress azimuthally at a uniform speed without changing their shape. (This is the equilibrated form of baroclinic instability, discussed in Section 13.17.) At still larger heating rates an irregular, aperiodic, or chaotic flow develops. The chaotic response to constant forcing (in this case the heating rate) is an interesting nonlinear effect and is discussed further in
Section 11.14. Meanwhile, a brief description of the transition from laminar to turbulent flow is given in the next section.