Prograde and retrograde precession of a fluid-filled cylinder

Precession denotes the motion of an object rotating around one axis which, in turn, rotates around a second axis fixed in a Galilean reference frame. Precession driven flows are ubiquitous natural phenomena occurring, e.g. in the cyclonic structures of the Earth's atmosphere or the fluid layers in the cores of planets, moons and asteroids [1]. Fluids enclosed in precessing containers are also interesting in a wide range of technical applications, in particular in spacecrafts, rockets and satellites [2, 3] where the stability of the payloads is of primary importance.

The theoretical investigation of precession-driven flows started with the case of an inviscid fluid in a spheroidal cavity [4, 5] whose analytical solution was obtained assuming a uniform vorticity. This Poincaré solution was later extended by Busse [6] to the weakly nonlinear regime including the viscous effects in boundary layers. Meanwhile, several experiments have shown that precession is an efficient mechanism to drive flows without making use of any propellers or pumps [7, 8]. With view on that, precessing-driven flows were also proposed as alternative or at least complementary energy sources for magnetic field self-excitation in the Earth's core [9, 10], the ancient Moon [11–13], and even asteroids [14]. Indeed, numerical works have demonstrated that precession is able to generate a magnetic field in spherical shells [15, 16], full spheres [17], spheroids [18], cylinders [19–21], and cubes [22, 23]. The amplification of an applied magnetic field by a factor of 3 has been observed previously in the precession experiment of Gans [24].

Further laboratory experiments in different geometries were dedicated to understand particular features of precession-driven flows such as flow instabilities [25–28], large scale vortex formation [29], transitions to and sustaining mechanisms of turbulence [30–32].

In frame of the project DRESDYN (DREsden Sodium facility for DYNamo and thermohydraulic studies) at Helmholtz-Zentrum Dresden-Rossendorf, a large-scale liquid sodium experiment is under construction whose final purpose is to show dynamo action in a precession driven flow [33–35]. The core of the experiment consists of a cylinder with radius R = 1 m and height H = 2 m that can achieve a rotation frequency f_c = 10 Hz and a precession frequency f_p = 1 Hz. In the first instance, the cylindrical geometry has been chosen because the presence of corners allows a more efficient injection of kinetic energy into the fluid than a corresponding spherical shell. Even if the cylindrical geometry seems to be far from a geophysical application, remarkable commonalities have been revealed between the characteristic flow transition in cylinders [32], spheroids [30, 31] and ellipsoids [36] while the spherical case is in some sense exceptional.

Precession-driven flows in cylinders show a non-trivial behavior already in the weakly forced regime being governed by a 3D flow field consisting of inertial waves which are particularly excited when approaching the resonance condition. While outside the resonance the amplitudes of the inertial waves can be predicted by a linear-inviscid model [37], at resonance the viscosity must be taken into account in order to find the saturated amplitude. Various models have been proposed which include viscous effects [38, 39] and even weakly nonlinear interactions [40]. At resonance the flow is highly unstable, tending to degenerate into a state of chaotic and fine-scale motion called 'resonant collapse' [25, 26]. Hysteresis phenomena, in terms of relaminarization-breakdown cycles, have been observed [25, 26, 32], too. Several instability mechanisms were identified [40–43] and particular attention was laid on the emergence of a geostrophic motion when the nonlinear effects become important. Both numerical [40, 44–47] and experimental [27] studies have revealed a geostrophic flow whose structure is prevalently axisymmetric and dominated by an azimuthal velocity opposite to the container's rotation. The generation of an axisymmetric geostrophic mode in a rotating fluid has been reported for various forcing mechanisms, like libration [48] or tidal forces as well as in case of instabilities (e.g. elliptical instability [49]). Although its impact for rotating turbulence is well established [49, 50] there is no unique scenario to explain its generation. Since Greenspan's theorem [51] forbids the possibility of wave-to-geostrophic transfer (at least for a first order nonlinear problem) several authors [40, 52] argued that the nonlinear interactions in the viscous boundary layers are able to trigger it. However, experimental works have evidenced that also interior shear layers could be a source for geostrophic flows [53]. More recent models take into account the possibility of instabilities as driving mechanisms for the geostrophic flow, e.g. in form of quartic wave interaction [54], or as a sequence of consecutive destabilization mechanisms [55]. The emergence of a geostrophic azimuthal circulation opposite to the cylinder rotation eventually goes along with the braking of solid body rotation (SBR) for large enough forcing and could result in a centrifugal unstable flow as proposed by Kobine [27].

Another remarkable phenomenon which has gained attention in recent theoretical [40, 56] and numerical [57] works is the appearance of a zonal flow (also called streaming flow) which raises from nonlinear interaction of inertial modes and which may not be invariant along the axial direction (it is non-geostrophic).

Precession driven flows are governed by four key parameters: (i) the Reynolds number Re, i.e. the ratio of Coriolis force to viscosity; the precession ratio (also called Poincaré number) Po, i.e. the ratio of precession frequency to rotation frequency; the geometric aspect ratio Γ, i.e. the ratio of height to radius of the container and finally the nutation angle α defined as the angle between the precession and the rotation axis. So far, most studies of fluid-filled precessing cylinders focused on the impact of Reynolds number and/or precession ratio, were carried out for small nutation angles. While for cylindrical geometry the prograde precession (i.e. when the container and the turntable rotate in the same direction) [26, 40, 44, 46] and retrograde [57–59] precession were investigated separately, a direct comparison between these two motions is still elusive. The present numerical investigation aims at assessing the role of prograde and/or retrograde motion for comparably large nutation angles for the future large scale precession experiment in frame of the DRESDYN project. In particular, we will extend the work of Giesecke et al. [21, 60] where precession ratios for efficient dynamo action were found only for perpendicular nutation angle, by identifying the optimal parameters in terms of nutation angle and pro-or retrograde precession. The main idea is that the optimal range is characterized by the emergence of axisymmetric large scale rolls which resemble the flow structure studied in a spherical kinematic dynamo model by Dudley and James [61] and in a cylindrical model by Xu et al [62].

The paper is organized as follows: in section 2 we formulate the hydrodynamic problem and give a brief description of the numerical method. The results concerning the different flow responses to the nutation angle and the difference between prograde and retrograde motion in terms of inertial modes and flow structures are presented in section 3. The stability of the flow is analyzed in detail in section 3.2, and the poloidal structures emerging for certain precession ratios are discussed in section 3.3. All results of this work are summarized, together with their implications, in section 4.

We consider an incompressible fluid of kinematic viscosity ν enclosed in a cylinder of radius R and height H. The container rotates and precesses with angular velocities Ω_c and Ω_p (−Ω_p ) for prograde (retrograde) motion, with α denoting the nutation angle, as illustrated in figure 1(a). While α could also be considered to run between 0° and 180°, we restrict it here to the range between 0° and 90°, and differentiate instead between pro- and retrograde precession.

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The fluid motion inside the precessing cylinder is governed by the Navier–Stokes equation [57]

$$\begin{equation}\frac{\partial \boldsymbol{u}}{\partial t}+\boldsymbol{u}\cdot \nabla \boldsymbol{u}=-\nabla P+\frac{1}{\mathrm{R}\mathrm{e}}{\nabla }^{2}\boldsymbol{u}-2\mathbf{\Omega }\times \boldsymbol{u}+\frac{\mathrm{d}\mathbf{\Omega }}{\mathrm{d}t}\times \boldsymbol{r},\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad \nabla \cdot \boldsymbol{u}=0,\end{equation} \tag{ 1 }$$

which obeys no-slip boundary conditions u = 0 at all walls. Here u is the velocity flow field, Ω = Ω_c + Ω_p is the total rotation vector and r is the position vector with respect to the origin O. P is the reduced pressure which includes the hydrostatic pressure and the gradient terms (including the centrifugal force) that do not change the dynamical behavior of the flow. The last two terms on the right-hand side are the Coriolis and the Poincaré force, respectively. The cylindrical coordinates, fixed in the origin O, are the axial (z), radial (r) and azimuthal (φ) ones, respectively, as shown in figure 1(b). In order to non-dimensionalize the Navier–Stokes equation we use the radius R as length scale and ${\left\vert {\mathbf{\Omega }}_{\mathbf{c}}+{\mathbf{\Omega }}_{\mathbf{p}}\enspace \mathrm{cos}\enspace \alpha \right\vert }^{-1}$ as time scale, which represents the projection of total angular velocity on the cylinder axis, i.e. $({\mathbf{\Omega }}_{\mathbf{c}}+{\mathbf{\Omega }}_{\mathbf{p}})\cdot \hat{\boldsymbol{z}}$.

The key parameters governing precession-driven flows, the Reynolds number Re, the Poincaré number Po and the aspect ratio of the container Γ are defined as

$$\begin{equation}\mathrm{R}\mathrm{e}=\frac{{R}^{2}\left\vert {{\Omega}}_{\mathrm{c}}+{{\Omega}}_{\mathrm{p}}\enspace \mathrm{cos}\enspace \alpha \right\vert }{\nu },\hspace{5.0pt}\mathrm{P}\mathrm{o}=\frac{{{\Omega}}_{\mathrm{p}}}{{{\Omega}}_{\mathrm{c}}},\hspace{5.0pt}{\Gamma}=\frac{H}{R}.\end{equation} \tag{ 2 }$$

For studying precession-driven flows in cylinders, two frames of reference are usually employed: the mantle frame (attached to the cylinder wall) and the turntable frame in which the cylinder walls rotate at Ω_c and the total vector Ω is fixed. We perform our simulations, using the DNS code SEMTEX [63], in the turntable frame where ∂Ω/∂t = 0 so that in equation (1) the Poincaré force disappears and both the rotation vector Ω_c as well the precession vector Ω_p are stationary. For the numerical simulations, we use 300 quadrilateral elements to mesh the meridional half plane (see figure 1(b)) and 128 Fourier modes in azimuthal direction. The initial conditions at t = 0 correspond to a pure SBR state, given by u = (Ω_c r)φ.

The parameter space to be investigated in this work is the following: the Reynolds number varies in the range [5 × 10², 10⁴] and the Poincaré number in the range ±[10⁻³, 3.5 × 10⁻¹]. The nutation angles are α = 60°, α = 75° (in appendix A we show results also for other angles), both for prograde and retrograde precession, and α = 90° for which there is no difference between prograde and retrograde precession. The aspect ratio will be fixed at Γ = 2 which is quite close to the resonance point Γ = 1.989 of the first inertial mode with (m, n, k) = (1, 1, 1) which represents a gyroscopic motion resulting from the tendency of the fluid flow to align the flow rotation and the precession axis. A summary of all simulations in the parameter space (Re, Po) is shown in the stability diagram of figure 5(a).

Although the simulations are performed in the precessing (i.e. turntable) frame of reference, almost all results will be presented and discussed in the mantle frame in which the inertial and geostrophic modes are more intuitive (the only exception are the plots of the angular momentum and the Rayleigh criterion shown in figures 4 and 5). The inertial modes are the eigenfunctions u_mnk of the inviscid-linearized form of (1), explicitly given in the appendix B, which are characterized by three integers (m, k, n) indicating azimuthal, axial and radial wave numbers.

A useful tool to analyze the numerical results is the projection of the DNS flow field onto a basis given by inertial modes which are solutions of the inviscid linear problem. Following the approach of Kong et al [44], we use the decomposition

$$\begin{align}\boldsymbol{u}(\boldsymbol{r},t)& =\sum\limits _{n=1}^{N}{A}_{00n}(t){\boldsymbol{u}}_{00n}\left(r\right)+\sum\limits _{m=1}^{M}\sum\limits _{n=1}^{N}\frac{1}{2}\left[{A}_{m0n}(t){\boldsymbol{u}}_{m0n}\left(r,\varphi \right)+\mathrm{c}.\mathrm{c}.\right]\\\ & \quad +\sum\limits _{k=1}^{K}\sum\limits _{n=1}^{N}\frac{1}{2}\left[{A}_{0kn}(t){\boldsymbol{u}}_{0kn}\left(z,r\right)+\mathrm{c}.\mathrm{c}.\right]\\\ & \quad +\sum\limits _{m=1}^{M}\sum\limits _{k=1}^{K}\sum\limits _{n=1}^{2N}\frac{1}{2}\left[{A}_{mkn}(t){\boldsymbol{u}}_{mkn}\left(z,r,\varphi \right)+\mathrm{c}.\mathrm{c}.\right]\\ & \quad +{\tilde{\boldsymbol{u}}}^{\text{bl}}(\boldsymbol{r})\end{align},$ \tag{ 3 }$$

where c.c. means complex conjugate, and ${\tilde{\boldsymbol{u}}}^{\mathrm{b}\mathrm{l}}$ is the boundary layer velocity. The amplitude of each inertial mode is computed as

$$\begin{equation}{A}_{mkn}=\frac{2{\int }_{V}{\enspace \boldsymbol{u}}_{mkn}^{\ast }\cdot \boldsymbol{u}\enspace \mathrm{d}V}{{\left({\int }_{V}{\enspace \boldsymbol{u}}_{mkn}\cdot {\boldsymbol{u}}_{mkn}^{\ast }\enspace \mathrm{d}V\right)}^{1/2}}.\end{equation} \tag{ 4 }$$

It is well-known that a precession-driven flow in cylindrical geometry changes with increasing forcing due to the emergence of an axisymmetric azimuthal flow and the related modification of the non-axisymmetric poloidal flow [40, 44–46]. Here we focus on the specific influence of the nutation angle on this behavior.

In figure 2, we show the time-averaged axial velocity u_z which is a good representative for the base state. The three selected configurations α = 60° (p), α = 90° and α = 60° (r), as examples for prograde, perpendicular and retrograde precession, respond in different ways to the increase of the precession ratio Po. Only the prograde and the perpendicular cases show a reduction of the axial flow to negligible values inside the bulk region whose extension increases with the precession ratio. By contrast, the retrograde case (bottom row) remains essentially unchanged in magnitude and shape, but with counterclockwise phase shift from low to large Po.

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For the smallest precession ratio Po = 0.010 (left column) the 3 configurations appear very similar. The flow magnitude is quite weak and the three-dimensional structures (indicated by the ±0.15 levels) are symmetrical with respect to φ (the shape for α = 90° is a little more elongated). The plots for Po = 0.100 (central column) prove that in the bulk u_z vanishes at lower Po for prograde precession. Finally, at Po = 0.200, u_z is confined close to the sidewall for α = 60° (p) and α = 90°, with this region being thinner for the latter case.

In order to quantify the flow response, we investigate the energy densities of the main inertial modes in dependence of the precession ratio. We define the energy densities of the geostrophic-axisymmetric, the directly forced mode and the SBR energy (as reference) as follows:

$$\begin{equation}{e}_{00}=\frac{1}{2V}\sum\limits _{n}{\left\vert {A}_{00n}\right\vert }^{2},\hspace{25.0pt}{e}_{11}=\frac{1}{2V}\sum\limits _{n}\frac{{\left\vert {A}_{11n}\right\vert }^{2}}{2},\hspace{25.0pt}{e}_{\mathrm{s}\mathrm{b}\mathrm{r}}=\frac{1}{2V}{\int }_{V}{\left({{\Omega}}_{\mathrm{c}}r\right)}^{2}\enspace \mathrm{d}V.\end{equation} \tag{ 5 }$$

The energies for the axisymmetric-poloidal structures are:

$$\begin{equation}{e}_{02}=\frac{1}{2V}\sum\limits _{n}{\left(\mathfrak{I}{A}_{02n}\right)}^{2},\hspace{25.0pt}{e}_{0k}=\frac{1}{2V}\sum\limits _{n}{\left(\mathfrak{I}{A}_{0kn}\right)}^{2}\end{equation} \tag{ 6 }$$

with $\mathfrak{I}$ denoting the imaginary part (see appendix B for descriptions of this type of inertial modes which are called axisymmetric oscillations [44, 52]).

Figure 3 shows the energies as defined in equations (5) and (6) versus the precession ratio Po. The differences between different nutation angles (and prograde/retrograde precession) become stronger with increasing Po. The prograde and α = 90° cases display an abrupt transition from a state with low energy in the geostrophic axisymmetric mode (low state) to a state with high energy (high state). The retrograde cases (orange and red curves) reveal a much smoother increase of e₀₀ whose final level is lower than for the prograde counterparts (figure 3(a)). While in the low state region (for Po < 0.075) the prograde curves are almost overlapping, from the transition onward to the maximum precession ratio they diverge, e.g. the blue and green curves (α = 60° and α = 75°) show the jump at lower Po with the saturated levels differing by 10%. Interestingly, the linear increase of α = 90° case in the large precession region indicates the growth of the geostrophic azimuthal circulation which illustrates why the axial velocity contour (in figure 2) is so thin and concentrated close to the sidewall boundary for Po = 0.200. This is consistent with the outcome of Kong et al [44] who found that the larger the nutation angle the larger the bulk region that is occupied by the geostrophic-axisymmetric flow. The behavior of the forced mode energy e₁₁ is closely related to that of e₀₀. In figure 3(b) the prograde curves increase until Po ≈ 0.08 followed by a sharp breakdown of more than 60%. We notice two particular aspects: (i) the breakdown occurs at the same Po as does the jump of e₀₀; (ii) the smaller the angle the earlier the transition occurs. After the breakdown the curves for α = 60°, 75° remain flat, while the curve for α = 90° continues to decrease. In accordance with the linear increase of e₀₀ as discussed above, the two retrograde curves do not show any sharp transition.

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Also related to the breakdown of e₁₁ for the prograde and perpendicular cases, and the weaker decrease of e₁₁ for the retrograde case, we observe energy peaks for the poloidal modes. Figure 3(c) shows e₀₂ = f(Po) which is the energy associated with the axisymmetric vortex structure in the cylinder's meridional semi-plane (see also figure 6). The nutation angle and the prograde or retrograde configuration are decisive in this respect. The largest value of e₀₂, as found for the case α = 75° (r), is almost 50% larger than for α = 90° (and more than double than those of the other cases). It has a maximum around Po = 0.130 with a remarkable wide range of ΔPo ≈ 0.020. By contrast, the α = 60° (r) case (orange curve) shows the smallest magnitude whose maximum values are situated at still larger values of Po. Quite generally, the peaks for the retrograde profiles are shifted to larger precession ratios compared to the prograde ones. The prograde and perpendicular precession show curves with similar values for Po < 0.090 but while α = 90° has a clear peak followed by a steep decrease, the other cases remain at non-negligible level of e₀₂. Similar features are shown by the complementary part of the poloidal energy ∑_k≠2 e_0k , figure 3(d), where α = 90° and α = 75° retrograde have the peak at the same Po as e₀₂.

If we sum up the energies contained in the inertial modes considered in figure 3 we obtain that, for prograde and perpendicular precession, their summation contributes more than 90% of the total energy of the flow, whereas the summation for the retrograde cases stays around 60%. This fact indicates that the prograde precession is essentially characterized by these inertial modes while the retrograde cases are characterized by a more complex flow structure. However, other contributions of inertial modes are outside the scope of the present work because they are less relevant for possible dynamo action.

We have shown that for the prograde and perpendicular cases three main regions can be identified: (i) a low state dominated by the forced m = 1 Kelvin mode, (ii) a transition region and (iii) a high state dominated by an axisymmetric-geostrophic flow. The increase of the axisymmetric-geostrophic mode results in the dominance of an azimuthal circulation and the near-absence of axial flow in the bulk region as shown in figure 2. The strong azimuthal rotation is opposed to the container rotation leading to the braking of SBR and, for large enough precession ratio, possibly to a centrifugally unstable flow. This topic will be our next focus.

As a common discriminant to evaluate the hydrodynamic stability of rotating flows the centrifugal stability criterion, or Rayleigh criterion, defined as

$$\begin{equation}\frac{\partial {(L)}^{2}}{\partial r} > 0\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\enspace L=r{u}_{\varphi }^{tt},\end{equation} \tag{ 7 }$$

is employed. In equation (7), L is the angular momentum and ${u}_{\varphi }^{tt}={u}_{\varphi }+{{\Omega}}_{\mathrm{c}}r$ denotes the azimuthal velocity in the turntable reference frame. Strictly speaking this criterion holds only for purely rotational shear flows. Nevertheless, the application in the present case is supported by the prevalently azimuthal nature of the flow once the high state is achieved, and is also consistent with the description of the experimental observations by Kobine [27].

The radial profile of the angular momentum and the radial derivative of L², averaged both in azimuthal and axial direction, are shown in figure 4. The left column of figures 4(a1)–(e1) shows the impact of the precession ratio on L. With increasing Po the flow deviates more and more from the SBR profile, eventually developing rather flat profiles in the bulk region (for r < 0.8, say). For prograde precession and α = 90°, at large enough Po the angular momentum becomes negative, indicating that in this region the flow rotates opposite to the container. For large Po the deviation from SBR goes along with an emergence of a huge velocity gradient between 0.90 < r < 1.0, owing to a marked sidewall boundary layer [47]. The retrograde cases do not show any negative L for Po ⩽ 0.20.

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The right column, figures 4(a2)–(e2), shows the radial derivative of L². For prograde precession and α = 90°, at large enough Po the slope of L² becomes negative indicating a centrifugal unstable flow. We find that for α = 60° (p) and α = 75° (p) the violation of Rayleigh's criterion occurs at Po ≈ 0.100 (green curve in (a2) and (b2)) while for α = 90° the flow becomes unstable above Po ≈ 0.125 developing a marked 'nose' shape with a positive peak at r ≈ 0.8 and negative peak at r ≈ 0.9.

Up to this point, the analysis has been carried out for Re = 6500, so our goal is to extend the range to other Reynolds number.

In figure 5, we present regime diagrams for the instabilities described so far, i.e. the breakdown of the directly forced mode and the violation of Rayleigh's criterion. Figure 5(a) shows the parameter space (Po, Re) that includes all simulated cases accessible in our numerical simulations. The flow is defined as stable (black symbols) or unstable (red symbols) according to the Rayleigh criterion (7). From the preponderance of the red symbols in the upper half plane, it is obvious that the prograde motion is substantially more prone to become unstable. It is noteworthy that in order to find an unstable flow for retrograde motion we must achieve more than twice the precession ratio of the prograde counterpart, e.g. the first unstable solution at Re = 6500 for α = 75° (r) is found at Po = 0.250 while for prograde it is at Po = 0.100 (see zoom plot). This discrepancy is even more pronounced for the case α = 60° where no centrifugal instability is found at all for retrograde precession. In the zoom panel of figure 5(a), we observe that for the prograde cases the unstable points appear at smaller Po for smaller α, (see for instance at Re = 6500 and Re = 10 000 the asterisks for α = 60° and the triangles α = 90°). However, we do not find a systematic dependence of the critical precession ratio for the onset of the instability with respect to α. In particular it is not possible to unify the precession ratio and the nutation angle into a general forcing parameter Po sin α.

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Figure 5(b) focuses on the case α = 90° for which we have simulations for several Reynolds numbers. In addition to the violation of Rayleigh's criterion we illustrate also the breakdown of the energy of the directly forced mode. For the sake of clarity we show only the last stable point where e₁₁ is maximum (green symbols), the points included between the two stability curves (black symbols), and then the first centrifugally unstable point (red symbols). The grey dashed-dotted curve marks the derived scaling law for the breakdown of e₁₁ whose expression is Po^(c1) = 0.025 + 0.40 Re^−1/5. Note the absence of points for Re = 500 since this Reynolds number does not show a clear breakdown of e₁₁ (see figure B1 in the appendix B). The blue curve is the fit marking the scaling for the critical threshold, Po^(c2) = 0.033 + 0.7 Re^−1/4, above which the flow is centrifugally unstable. This kind of scaling law is reminiscent of the instability threshold found by Lin et al [28] for the regime of strongly non-linear flow in a precessing cylindrical annulus.

We should remark that the parameter space studied is quite limited in terms of Reynolds numbers, therefore any extrapolation for geophysical phenomena and the DRESDYN experiment should be taken with a grain of salt. Formally, the expressions for Po^(c1) and Po^(c2) would cross around Re ∼ 10¹¹, but the applicability for such large Reynolds numbers is questionable. Comparing these results with figure 3(a), we see that, for the prograde cases, the centrifugal instability appears close to the value of Po where e₀₀ achieves the high state. We conclude that the emergence of the axisymmetric-geostrophic flow (essentially an azimuthal flow which counteracts the SBR) is responsible for the decrease of the angular momentum L and consequently its negative radial derivative (i.e. violation of Rayleigh criterion). If this is the case our results indicate a hierarchical relation between the centrifugal instability and the secondary-geostrophic instability theory proposed by Kerswell [55] whose occurrence would scale ∝Re^−1/4.

In this section we focus our attention on the poloidal flow structure whose energy was plotted in figures 3(c) and (d). The interest in this particular kind of inertial mode is mainly related to their suitability for dynamo action in a precessing cylinder, as discussed in [21, 60].

Figure 6 shows the poloidal flow structure for various nutation angles, taken at the respective values of Po where the energy density e₀₂ is maximum (see figure 3(c)). The vector field comprises the azimuthally averaged radial and axial velocities ${[{u}_{z},{u}_{r}]}^{m=0}$, while the color scale represents the magnitude of the azimuthal vorticity (again azimuthally averaged) which is a measure of the rotatory behavior of [u_r , u_z ] in the meridional half plane:

$$\begin{equation}{\left(\boldsymbol{\omega }\cdot \hat{\boldsymbol{\varphi }}\right)}^{m=0}={\omega }_{\varphi }^{m=0}=\frac{1}{2\pi }{\int }_{\varphi }\left(\frac{\partial {u}_{r}}{\partial z}-\frac{\partial {u}_{z}}{\partial r}\right)\mathrm{d}\varphi .\end{equation} \tag{ 8 }$$

The nutation angle plays a major role both for the topology and the magnitude of the poloidal flow field. Remarkably, the prograde case with α = 60° (figure 6(a)) shows an opposite orientation of the double rolls compared to those in the other cases and includes smaller vortices in the corners. The colors illustrate how the azimuthal vorticity is distributed in the plane, including zones characterized by an alternation of signs. The strength and the extension of the larger rolls present an increase from α = 75° (p) to α = 90°, finally achieving a maximum at α = 75° (r). The case α = 60° (r), figure 6(e), shows the weakest and smallest vortices which additionally are centered more towards the corners. This behavior is in accordance with the maximum level of e₀₂ shown in figure 3(c).

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In order to have a direct quantitative comparison, we plot also the axial velocity ${u}_{z}^{m=0}$ at r = 0.5 and the radial velocity profile ${u}_{r}^{m=0}$ at the equator z = 0, in figures 6(f) and (g), respectively. Again, the largest values are achieved for α = 75° (r) (red solid-dashed curve) both for radial and axial velocities, while α = 60° (r) is the weakest case. The case α = 60° (p) has opposite values with respect to all other cases, consistent with the inverse rotation of the vortices in the bulk.

In the following, we will discuss the time dependence of the poloidal rolls. Specifically, we will work out the difference between α = 90° (double rolls) and α = 60° (p) (quadruple rolls), i.e. the two paradigmatic cases of double and four vortices. We choose to present the amplitude since it is directly related to the velocity and we include also the dependence on the radial wave number because it is a useful characterization of the flow field's radial distribution in the half plane.

The two cases show a different evolution of the poloidal structure as we can see in figures 7 and 8 where snapshots of the instantaneous flow field at different time-steps are presented together with the corresponding amplitudes.

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The case α = 90° (figure 7) exhibits a double-vortex structure that remains rather stable during the entire period showing only a little enlargement from t = 38 to t = 145. The vortex close to the top endwall has a counterclockwise direction associated with a positive azimuthal vorticity and the opposite direction for the bottom vortex. This behavior is also reflected in the flow amplitude and indeed, the A₀₂₁ component is dominant and always positive while the other ones are of minor importance.

The case α = 60° (p) (figure 8) shows a much more complex evolution. The first part of the transient (panel 1) is characterized by a double vortex structure (green arrows) analogous to the case α = 90°. The time t = 38 coincides with a local peak of A₀₂₁, which in the subsequent period decreases while A₀₂₂ (red curve) increases till t = 79 when the original double rolls are confined in the corner and a radial inward flow appears around the equator (panel 2, purple arrows). The next step (panel 3) is characterized by the organization of vortical flow around z ≈ 0 which evolves into a clockwise vortex, denoted by the purple arrow in the blue central region, followed by the formation of another roll of opposite rotation (panel 4). At t = 90 (blue dashed line in the amplitude plot) we observe the crossing of A₀₂₁ and A₀₂₂, which is accompanied by the formation of 4 axisymmetric vortices. In the remaining time, the central vortices migrate towards the endwalls (panel 5): the large bottom roll has acquired the final extension and inclination while the top vortex is stretched. The last panel at t = 145 shows the final setup where the corner rolls and bulk rolls lie next to each other with opposite rotation in a sort of 'gear' interaction. The final level of A₀₂₁ < 0 denotes an opposite rotation of the larger vortices with respect to the case α = 90° shown in figure 7.