Transient growth, edge states, and repeller in rotating solid and fluid

For the classical problem of the rotation of a solid, we show a somehow surprising behavior involving large transient growth of perturbation energy that occurs when the moment of inertia associated to the unstable axis approaches the moment of inertia of one of the two stable axes. In that case, small but finite perturbations around this stable axis may induce a total transfer of energy to the unstable axis, leading to relaxation oscillations where the stable and unstable manifolds of the unstable axis play the role of a separatrix, an edge state. For a fluid in solid-body rotation, a similar linear and nonlinear dynamics apply to the transfer of energy between three inertial waves respecting the triadic resonance condition. We show that the existence of large transient energy growth and of relaxation oscillations may be physically interpreted as in the case of a solid by the existence of two quadratic invariants, the energy and the helicity in the case of a rotating fluid. They occur when two waves of the triad have helicities that tend towards each other, when their amplitudes are set such that they have the same energy. We show that this happens when the third wave has a vanishing frequency which corresponds to a nearly horizontal wave vector. An inertial wave, perturbed by a small-amplitude wave with a nearly horizontal wave vector, will then be periodically destroyed, its energy being transferred entirely to the unstable wave, although this perturbation is linearly stable, resulting in relaxation oscillations of wave amplitudes. In the general case we show that the dynamics described for particular triads of inertial waves is valid for a class of triadic interactions of waves in other physical problems, where the physical energy is conserved and is linked to the classical conservation of the so-called pseudomomentum, which singles out the role of waves with vanishing frequency.

Figure 1

Trajectories on the sphere of unity $\mathcal{E}=1$ in the $(M_0^{\prime },M_1^{\prime },M_2^{\prime })$ space with $J_2=(1-\varepsilon )J_0, \varepsilon =2\times {10}^{-2}, J_1=(1-\mathrm{\Sigma })J_0$, and $\mathrm{\Sigma }=0.9$. The point ${\mathbf{M}}_{\mathbf{1}}^{\prime }={(0,1,0)}^{\text{T}}$ is stable, and close trajectories around ${\mathbf{M}}_{\mathbf{1}}^{\prime }$ are plotted in blue; ${\mathbf{M}}_{\mathbf{0}}^{\prime }={(0,0,1)}^{\text{T}}$ is stable too and close trajectories around ${\mathbf{M}}_{\mathbf{0}}^{\prime }$ are in red.

Figure 2

Energy gain $G$ defined in (4) as a function of time $\tau$ for the linearized problem around the stable point ${\mathbf{M}}_{\mathbf{0}}^{\prime }={(0,0,1)}^{\text{T}}$ for $\mathrm{\Sigma }=0.9$ and $\varepsilon =2\times {10}^{-1}$ (continuous line), $\varepsilon =2\times {10}^{-2}$ (dashed line), and $\varepsilon =2\times {10}^{-3}$ (dotted line).

Figure 3

Resonant triads in the wave vector space ${\mathbf{k}}_1=(k_{x1},k_{z1})$ when the wave $({\omega }_0,{\mathbf{k}}_0)$ is fixed with $s_0=+1, k_0=1$, and ${\omega }_0/f=0.84$ (value taken from [5]), ${\mathbf{k}}_0$ being plotted in the figure with black arrows. The unstable branches $(+,-,-)$ are represented in bold red lines, whereas the stable branches $(+,-,+)$ and $(+,+,-)$ are represented in thin blue and green lines respectively. The unit circle centered on $-{\mathbf{k}}_0$ is plotted in dashed gray lines. The points $(E_1,D_1,A_1)$ marked with squares indicate triads where ${\sigma }_2{k}_2\to {\sigma }_0{k}_0=1$. $(E_2,D_2,A_2)$ are the symmetrics of $(E_1,D_1,A_1)$ by exchanging ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ such that ${\sigma }_1{k}_1\to {\sigma }_0{k}_0=1$. The exemplified triad $T$ [${\mathbf{k}}_0\sim {(0.54,0.84)}^{\text{T}}, {\mathbf{k}}_1\sim {(-0.83,0.1)}^{\text{T}}$, and ${\mathbf{k}}_2\sim {(0.29,-0.94)}^{\text{T}}$) is plotted in black.

Figure 4

Trajectories corresponding to the dynamics given by Eq. (8) for the triad $T$ plotted in Fig. 3, with $\varepsilon =1-{\sigma }_2{k}_2/\left({\sigma }_0{k}_0\right)=0.02$ and $\mathrm{\Sigma }=1-{\sigma }_1{k}_1/\left({\sigma }_0{k}_0\right)=0.16$. (a) For projection in the modulus space $\left(\right|{\mathrm{\Phi }}_0|,|{\mathrm{\Phi }}_1|,|{\mathrm{\Phi }}_2\left|\right)$, trajectories are on the sphere of constant energy $\mathcal{E}=1$ and correspond to lines of constant helicity $\mathcal{H}$. (b) Trajectories along the separatrix in black dashed lines on (a) in the plane $(\phi ,|{\mathrm{\Phi }}_2\left|\right)$. Since $\mathcal{K}$ is an invariant, contour plots of $\mathcal{K}$ for values given by the color bar define the trajectories. Two trajectories obtained by numerical integration of (8) for different initial conditions are also reported in heavy lines: $(\phi ,|{\mathrm{\Phi }}_2\left|\right)=(\pi /4,0.44)$ in copper, (0,0.99) in black, and $(\pi ,0.01)$ also in black.