Precessing cylinders at the second and third resonance: Turbulence controlled by geostrophic flow

We investigate, via both asymptotic analysis and direct numerical simulation, precessionally driven flow of a homogeneous fluid confined in fluid-filled circular cylinders that rotate rapidly about their symmetry axis and precess about a different axis and that are marked by radius-height aspect ratios $\mathrm{\Gamma }=1.045945$ and $\mathrm{\Gamma }=1.611089$. At these radius-height aspect ratios, the Poincaré force resonates directly with the two special inertial modes that have the simplest vertical structure. An asymptotic analytical solution in closed form describing weakly precessing flow is derived in the mantle frame of reference for asymptotically small Ekman numbers, showing quantitative agreement with the result of direct nonlinear numerical simulation. Our numerical simulation makes use of a finite-element method with the three-dimensional tetrahedralization of a cylindrical cavity that allows the construction of dense nodes in the vicinity of the bounding surface of the cavity for resolving the thin viscous boundary layer. It is found that axisymmetric geostrophic flow in the alternating eastward and westward direction can be generated and maintained by nonlinear and viscous effects in the viscous boundary layer. It is also found that, when the precessing rate is moderate and, consequently, the geostrophic flow is weak, nonlinear interaction between the resonant inertial mode and the nonesonant inertial modes driven by the Poincaré force and the boundary-layer influx leads to strongly turbulent flow with irregular temporal-spatial fluctuation. When the cylinders are strongly precessing such that the geostrophic flow becomes predominant, however, the effect of the geostrophic flow controls/stabilizes its nonlinear dynamics, leading to weakly turbulent flow that can be largely described by a dominant quasisteady geostrophic component and a weak nonaxisymmetric component localized in the region where the geostrophic flow is weak.

Figure 1

Contours of $u_z$ of precessing flow at an instant in the mantle frame at the $z=1/2$ plane for $\text{Po}={10}^{-4}$ and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$: (a) analytical solution computed from expression (16) and (b) corresponding finite-element solution from nonlinear numerical simulation. It should be noted that, since both solutions are in the form of an azimuthally traveling wave, the difference in the azimuthal phase between (a) and (b) has no mathematical or physical significance.

Figure 2

Kinetic energy densities $E_{\mathrm{kin}}$ as a function of $\text{Po}$ for ${\alpha }_p=\pi /4$ and $E={10}^{-4}$ at the second resonance in the precessing cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$. The asymptotic solution (dashed line) is computed from the analytical expression, (17), while circles represent the solutions of finite element simulation.

Figure 3

Scaled axisymmetric, steady geostrophic flow $U_G\left(s\right)/{\text{Po}}^2$ as a function of $s$ for $\text{Po}={10}^{-4},{10}^{-3}$, and ${10}^{-2}$ at $E={10}^{-4}$ in the precessing cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$.

Figure 4

Kinetic energy densities $E_{\mathrm{kin}}$ of the precessing flow as a function of time for ${\alpha }_p=\pi /4$ at $E={10}^{-4}$ in the resonant cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$ for $\text{Po}=0.1$. Direct numerical simulation lasts about 223 revolutions.

Figure 5

Contours of $u_z$ of strongly turbulent precessing flow with highly irregular fluctuation at four instants on the $z=1/2$ plane for $\text{Po}=0.1$ and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$.

Figure 6

Kinetic energy densities $E_{\mathrm{kin}}$ of strongly turbulent precessing flow as a function of time for $\text{Po}=0.04$ and $E={10}^{-4}$ in the resonant cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$.

Figure 7

Contours of $u_z$ in the mantle frame on the $z=1/2$ plane for $\text{Po}=0.04$ and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$ at four instants, $t=85$, 112, 127, and 140, during its transition to turbulence.

Figure 8

Contours of $u_z$ in the mantle frame on the $z=1/2$ plane for $\text{Po}=0.04$ and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$ at four instants after reaching nonlinear equilibrium: (a) $t=979$, (b) $t=990$, (c) $t=998.5$, and (d) $t=1000$.

Figure 9

Kinetic energy densities $E_{\mathrm{kin}}$ of the weakly turbulent flow as a function of time for $E={10}^{-4}$ in the resonant cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$ for $\text{Po}=0.25$ and $\text{Po}=0.5$.

Figure 10

Contours of $u_z$ of the weakly turbulent flow in the mantle frame on the $z=1/2$ plane for ${\alpha }_p=\pi /4,\text{Po}=0.25$, and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$.

Figure 11

Averaged axisymmetric geostrophic flow, ${\overline{U}}_G\left(s\right)$, as a function of $s$ for $\text{Po}=0.25$ and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_2$.

Figure 12

Kinetic energy densities $E_{\mathrm{kin}}$ plotted as a function of $\text{Po}$ for ${\alpha }_p=\pi /4$ and $E={10}^{-4}$ at the third resonance in the precessing cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$. The dashed line represents the asymptotic solution, while circles denote the results of direct nonlinear numerical simulation.

Figure 13

Contours of $u_z$ of the laminar precessing flow with a constant amplitude computed from direct nonlinear numerical simulation in the mantle frame on the $z=1/2$ plane for $E={10}^{-4}$ in the precessing cylinder at the third resonance with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$: (a) for $\text{Po}={10}^{-3}$ and (b) for $\text{Po}={10}^{-2}$.

Figure 14

Scaled axisymmetric, steady geostrophic flow, $U_G\left(s\right)/{\text{Po}}^2$, as a function of $s$ for $\text{Po}={10}^{-3}$ and ${10}^{-2}$ with ${\alpha }_p=\pi /4$ and $E={10}^{-4}$ at the third resonance in the precessing cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$.

Figure 15

Kinetic energy densities $E_{\mathrm{kin}}$ of the strongly turbulent precessing flow as a function of time for $\text{Po}=0.1$ and $E={10}^{-4}$ in the resonant cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$.

Figure 16

Contours of $u_z$ of the strongly turbulent precessing flow in the mantle frame on the $z=1/2$ plane for ${\alpha }_p=\pi /4,\text{Po}=0.1$, and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$.

Figure 17

Kinetic energy densities $E_{\mathrm{kin}}$ of the weakly turbulent precessing flow as a function of time for $\text{Po}=0.25$ and $\text{Po}=0.5$ at $E={10}^{-4}$ in the third resonant cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$.

Figure 18

Averaged geostrophic flow, ${\overline{U}}_G\left(s\right)$, as a function of $s$ for $\text{Po}=0.5$ for ${\alpha }_p=\pi /4$ and $E={10}^{-4}$ in a precessing cylinder with aspect ratio $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$.

Figure 19

Contours of $u_z$ of the weakly turbulent precessing flow in the mantle frame on the $z=1/2$ plane for ${\alpha }_p=\pi /4,\text{Po}=0.5$, and $E={10}^{-4}$ in the precessing cylinder with $\mathrm{\Gamma }={\mathrm{\Gamma }}_3$.