Stuart–Landau equation

The Stuart–Landau equation describes the behavior of a nonlinear oscillating system near the Hopf bifurcation, named after John Trevor Stuart and Lev Landau. In 1944, Landau proposed an equation for the evolution of the magnitude of the disturbance, which is now called as the Landau equation, to explain the transition to turbulence without providing a formal derivation[1] and an attempt to derive this equation from hydrodynamic equations was done by Stuart for Plane Poiseuille flow in 1958.[2] The formal derivation to derive the Landau equation was given by Stuart, Watson and Palm in 1960.[3][4][5] The perturbation in the vicinity of bifurcation is governed by the following equation

{\frac {dA}{dt}}=\sigma A-{\frac {l}{2}}A|A|^{2}.

where

$A=|A|e^{i\phi }$ is a complex quantity describing the disturbance,
$\sigma =\sigma _{r}+i\sigma _{i}$ is the complex growth rate,
$l=l_{r}+il_{i}$ is a complex number and $l_{r}$ is the Landau constant.

The evolution of the actual disturbance is given by the real part of $A(t)$ i.e., by $|A|\cos \phi$ . Here the real part of the growth rate is taken to be positive, i.e., $\sigma _{r}>0$ because otherwise the system is stable in the linear sense, that is to say, for infinitesimal disturbances ( $|A|$ is a small number), the nonlinear term in the above equation is negligible in comparison to the other two terms in which case the amplitude grows in time only if $\sigma _{r}>0$ . The Landau constant is also taken to be positive, $l_{r}>0$ because otherwise the amplitude will grow indefinitely (see below equations and the general solution in the next section). The Landau equation is the equation for the magnitude of the disturbance,

{\frac {d|A|^{2}}{dt}}=2\sigma _{r}|A|^{2}-l_{r}|A|^{4},

which can also be re-written as[6]

{\frac {d|A|}{dt}}=\sigma _{r}|A|-{\frac {l_{r}}{2}}|A|^{3}.

Similarly, the equation for the phase is given by

{\frac {d\phi }{dt}}=\sigma _{i}-{\frac {l_{i}}{2}}|A|^{2}.

Due to the universality of the equation, the equation finds its application in many fields such as hydrodynamic stability,[7][8] Belousov–Zhabotinsky reaction,[9] etc.

General solution

The Landau equation is linear when it is written for the dependent variable $|A|^{-2}$ ,

{\frac {d|A|^{-2}}{dt}}+2\sigma _{r}|A|^{-2}=l_{r}.

The general solution for $\sigma _{r}\neq 0$ of the above equation is

|A(t)|^{-2}={\frac {l_{r}}{2\sigma _{r}}}+\left(|A(0)|^{-2}-{\frac {l_{r}}{2\sigma _{r}}}\right)e^{-2\sigma _{r}t}.

As $t\rightarrow \infty$ , the magnitude of the disturbance $|A|$ approaches a constant value that is independent of its initial value, i.e., $|A|\rightarrow (2\sigma _{r}/l_{r})^{1/2}$ when $t\gg 1/\sigma _{r}$ . The above solution implies that $|A|$ does not have a real solution if $l_{r}<0$ and $\sigma _{r}>0$ . The associated solution for the phase function $\phi (t)$ is given by

\phi (t)-\phi (0)=\sigma _{i}t-{\frac {l_{i}}{2l_{r}}}\ln \left[1+{\frac {|A(0)|^{2}l_{r}}{2\sigma _{r}}}(e^{2\sigma _{r}t}-1)\right].

As $t\gg 1/\sigma _{r}$ , the phase varies linearly with time, $\phi \sim (\sigma _{i}/\sigma _{r}-l_{i}/l_{r})\sigma _{r}t.$

References

Landau, L. D. (1944). On the problem of turbulence. In Dokl. Akad. Nauk SSSR (Vol. 44, No. 8, pp. 339-349).
Stuart, J. T. (1958). On the non-linear mechanics of hydrodynamic stability. Journal of Fluid Mechanics, 4(1), 1-21.
Stuart, J. T. (1960). On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow. Journal of Fluid Mechanics, 9(3), 353-370.
Watson, J. (1960). On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. Journal of Fluid Mechanics, 9(3), 371-389.
Palm, E. (1960). On the tendency towards hexagonal cells in steady convection. Journal of Fluid Mechanics, 8(2), 183-192.
Provansal, M., Mathis, C., & Boyer, L. (1987). Bénard-von Kármán instability: transient and forced regimes. Journal of Fluid Mechanics, 182, 1-22.
Landau, L. D. (1959). EM Lifshitz, Fluid Mechanics. Course of Theoretical Physics, 6.
Drazin, P. G., & Reid, W. H. (2004). Hydrodynamic stability. Cambridge university press.
Kuramoto, Y. (2012). Chemical oscillations, waves, and turbulence (Vol. 19). Springer Science & Business Media.

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[1] Landau, L. D. (1944). On the problem of turbulence. In Dokl. Akad. Nauk SSSR (Vol. 44, No. 8, pp. 339-349).

[2] Stuart, J. T. (1958). On the non-linear mechanics of hydrodynamic stability. Journal of Fluid Mechanics, 4(1), 1-21.

[3] Stuart, J. T. (1960). On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 1. The basic behaviour in plane Poiseuille flow. Journal of Fluid Mechanics, 9(3), 353-370.

[4] Watson, J. (1960). On the non-linear mechanics of wave disturbances in stable and unstable parallel flows Part 2. The development of a solution for plane Poiseuille flow and for plane Couette flow. Journal of Fluid Mechanics, 9(3), 371-389.

[5] Palm, E. (1960). On the tendency towards hexagonal cells in steady convection. Journal of Fluid Mechanics, 8(2), 183-192.

[6] Provansal, M., Mathis, C., & Boyer, L. (1987). Bénard-von Kármán instability: transient and forced regimes. Journal of Fluid Mechanics, 182, 1-22.

[7] Landau, L. D. (1959). EM Lifshitz, Fluid Mechanics. Course of Theoretical Physics, 6.

[8] Drazin, P. G., & Reid, W. H. (2004). Hydrodynamic stability. Cambridge university press.

[9] Kuramoto, Y. (2012). Chemical oscillations, waves, and turbulence (Vol. 19). Springer Science & Business Media.