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Topography-induced flows in a sidewall-modified, rapidly rotating and precessing cylinder
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XiuYu Chen1, 2, ChangShun Liu1, LiGang Li1, *, DaLi Kong1, *
Earth and Planetary Physics | 2026, 10(3) : 400 - 409
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Earth and Planetary Physics | 2026, 10(3): 400-409
• RESEARCH ARTICLE •
Topography-induced flows in a sidewall-modified, rapidly rotating and precessing cylinder
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XiuYu Chen1, 2, ChangShun Liu1, LiGang Li1, *, DaLi Kong1, *
Affiliations
  • 1Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
  • 2University of Chinese Academy of Sciences, Beijing 100049, China
Published: 2026-05-01 doi: 10.26464/epp2026038
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The influence of topography on rotating fluids may exceed conventional expectations. Here, we numerically examine viscous incompressible flows induced by sidewall topography, confined within a modified cylinder that rotates rapidly about its central vertical axis and precesses about another axis. To investigate specific flow patterns and boundary-interior correspondences, the cylindrical sidewall is modified by adding a vertical fin-type barrier extending all the way from the bottom to the top. The fully nonlinear Navier−Stokes equations with precessional forcing are solved in this modified cylindrical geometry, using a mixed finite element method. Numerical results show that the introduction of sidewall topography significantly alters the precessionally driven flow, particularly at high precession rates. While the primary dynamics associated with inertial wave propagation persist, rich vortical structures and turbulence emerge. Interestingly, the barrier does not invariably suppress the kinetic energy density; when its height approaches the cylinder radius under strong precession, the kinetic energy density even exceeds that of the cylinder case without a barrier. Such an anomalous enhancement of kinetic energy may offer new insights into how precession-driven flows over topography could contribute to sustaining long-lived planetary magnetic fields, including that of the early Moon.

topography  /  precession  /  rotating flow  /  cylindrical geometry  /  inertial waves  /  turbulence  /  numerical simulation
XiuYu Chen, ChangShun Liu, LiGang Li, DaLi Kong. Topography-induced flows in a sidewall-modified, rapidly rotating and precessing cylinder[J]. Earth and Planetary Physics, 2026 , 10 (3) : 400 -409 . DOI: 10.26464/epp2026038
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1　Introduction
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The influence of boundary topography on fluid dynamics has attracted considerable attention across a wide range of scientific and engineering disciplines. In geophysics, for instance, the Earth's core−mantle boundary (CMB) exhibits significant topographic irregularities, which play an essential role in the evolution of core and mantle dynamics (Lay et al., 1998). These irregularities can induce boundary-layer instabilities that are believed to cause various phenomena, such as enhanced core−mantle coupling, alterations in core convection, and the modulation of dynamo processes (Garcia and Souriau, 2000; Vidal et al., 2024). In atmospheric and oceanic dynamics, it is well known that certain equatorially trapped shallow-water waves, e.g. Kelvin modes and Yanai modes, are protected by the underlying topography (Delplace et al., 2017). In industrial applications, such as the design of fuel tanks for highly maneuverable vehicles, aircraft, and spacecraft, the deliberate placement of internal obstacles (i.e., artificial topography) is used to mitigate unwanted fluid motions (Whitford, 2004). Collectively, these examples highlight the pervasive influence and practical importance of boundary topography in shaping fluid behavior.
In response to these diverse applications, many fundamental studies have been conducted to explore how topography influences fluid motion. To study the dynamo transition of an incompressible electrically conducting fluid in a Couette system, numerical simulations are performed with a rough inner boundary (Finke and Tilgner, 2012). The presence of boundary roughness enhances the boundary forcing and reduces the critical rotation rate for dynamo onset. Such a scenario is currently realized in experiments by means of baffles attached to the inner sphere of a spherical Couette device (Rojas et al., 2021). Inspired by the topologically protected states found in two-dimensional semiconductors, studies of three-dimensional viscous rotating convection in cylindrical containers have shown that nonlinear wall states can persist even in the presence of bulk turbulence, and remain robust when the cylinder's sidewall is partially truncated by a vertical rectangular barrier (Favier and Knobloch, 2020). In the above-mentioned numerical and experimental studies, boundary topography does not seem to change the primary characteristics of the bulk fluid motion. However, the latitudinal libration experiment (Chen XY et al., 2025) in a triaxial ellipsoid indicates that the turbulent boundary layer may be responsible for the onset of bulk turbulence, and that the influence of boundary topography should be considered more carefully.
In the present study, we investigate precessionally driven flows in a modified cylindrical geometry, where a vertical fin-type barrier is set up on the sidewall and its height is varied across simulations. Precession occurs when a rapidly rotating body experiences an external torque that perturbs its spin, causing its angular momentum to precess in the inertial frame of reference. Under weak precessional forcing, inertial modes can be excited in the rotating frame of reference (Wood, 1966; Manasseh, 1992; Meunier et al., 2008; Liao XH and Zhang K, 2012). In certain geometries such as spheres, cylinders, and ellipsoids, these precessionally-driven flows can become particularly strong when an inertial mode resonates with the precessional forcing (Kong DL et al., 2014, 2015; Jiang JF et al., 2015). The cylinder used in this study is deliberately chosen with an aspect ratio $ \Gamma= 0.502559 $, defined as the ratio of the cylinder radius to its height, which corresponds to the primary resonance of the inertial mode $ {{\boldsymbol{u}}}_{111} $. The geometry is a circular cylinder equipped with a single sidewall-mounted vertical fin; its height is varied systematically across simulations, in a configuration similar to Favier and Knobloch (2020). Geophysically speaking, a circular cylinder mimics the Earth's tangent cylinder, an imaginary cylindrical region tangent to the inner-core equator and aligned with the Earth's rotation axis (Zhang K and Liao XH, 2017). A series of numerical simulations are performed to examine the evolution of precessionally driven flows within these sidewall-distorted cylinders, spanning regimes from quasi-linear to strongly nonlinear.
It should be emphasized that topographic coupling is not exclusively introduced by artificial barriers. Even in a smooth circular cylinder without internal obstacles, precessionally driven flows are constrained by the container geometry (Kong DL et al., 2015). The introduction of barriers disrupts this geometric smoothness by adding discontinuities (sharp edges and vertical walls), thereby strengthening boundary-driven effects. In this sense, fin-type barriers amplify geometry-driven boundary effects that are already present in the unmodified cylinder.
The paper is organized as follows. Section 2 describes the geometry and the dimensionless governing equations. Section 3 presents the finite-element method and the three-dimensional discretization of a topography-modified cylinder. Section 4 discusses the numerical results and phenomena. And the concluding remarks and future perspectives are given in the final section 5.
2　Governing Equations
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A cylindrical container filled with viscous incompressible fluid is considered, which has a radius $ \Gamma d $ and height $ d $. The aspect ratio $ \Gamma=0.502559 $ is chosen to be at the primary resonance of the inertial mode $ {{\boldsymbol{u}}}_{111} $, where the triple subscripts $ (m,n,k) $ in $ {{\boldsymbol{u}}}_{mnk} $ specify the azimuthal ($ m $), vertical ($ n $) and radial ($ k $) wavenumbers, respectively (Zhang K and Liao XH, 2017; Kong DL et al., 2014). In a horizontal cross-section perpendicular to the central vertical axis, the sidewall topography of the cylinder is marked by a mountain-shaped smooth curvature that can be described by
$ {\cal{R}}(\phi) = \left\{\begin{array}{*{20}{l}} {\Gamma - H \,\exp \left[ 1 + \dfrac{1}{\left(\dfrac{\phi - \phi_c}{\Delta \phi} \right)^2 - 1} \right], }& {\text{if } \phi_c - \Delta \phi \lt \phi \lt \phi_c + \Delta \phi,} \\{\Gamma,} &\;\;\; {\text{elsewhere}} \end{array}\right. $
and illustrated in Figure 1, in which $ H $ is the height of the barrier, $ \phi_c $ is the central location of the barrier in the azimuthal coordinate, and $ \Delta \phi $ is the half-angle width of the barrier. In a three-dimensional view, the sidewall topography extends from the bottom to the top of the cylinder, forming a fin-type vertical barrier. The modified cylindrical container rotates rapidly with an angular velocity $ { {\text{Ω}}}_0 = \Omega_0 \hat{{{\boldsymbol{z}}}} $ along its positive $ z $ axis and precesses slowly about a different axis that is fixed in an inertial frame. The angle between the two axes is $ \alpha_p $.
The cylindrical coordinate system fixed to the container is employed to describe the fluid motion, with the unit vectors $ (\hat{{{\boldsymbol{s}}}} $, $ \hat{{ {\text{ϕ}} }} $, $ \hat{{{\boldsymbol{z}}}}) $ defined in the container frame of reference. The origin of this cylindrical coordinate system is at the center of the bottom plane. The time-dependent precession vector $ {{ {\text{Ω}}}}_p $ can be expressed in the container frame by
$ {{ {\text{Ω}}}}_p = |{{ {\text{Ω}}}}_p| \left[\hat{{{\boldsymbol{s}}}}\; \sin\;{\alpha_{p}} \cos\;{(\phi + \Omega_{0}t)} - \hat{{ {\text{ϕ}} }}\;\sin\;{\alpha_{p}} \sin\;{(\phi + \Omega_{0}t)} + \hat{{{\boldsymbol{z}}}}\;\cos\;{\alpha_{p}}\right], $
and the overall angular velocity can be written as $ {{ {\text{Ω}}}} = \Omega_0\hat{{{\boldsymbol{z}}}} + {{ {\text{Ω}}}}_p $. As a result, the incompressible Navier−Stokes equations that describe the flow in the container are
$ \begin{split} &\frac{\partial{{\boldsymbol{u}}}}{\partial{t}} + {{\boldsymbol{u}}}\cdot{\nabla{{{\boldsymbol{u}}}}} + 2\Big\{ {\boldsymbol{\hat{z}}}\Omega_0 + |{{ {\text{Ω}}}}_p|\left[{\boldsymbol{\hat{s}}}\, \sin\,{\alpha_{p}} \cos\,{(\phi + \Omega_0 t)} \right.\\&\left.\quad \quad -{\hat{\text{ϕ}}}\, \sin\,{\alpha_{p}} \sin\,{(\phi + \Omega_0 t)} + {\boldsymbol{\hat{z}}} \,\cos\,{\alpha_{p}} \right]\Big\} \times{{{\boldsymbol{u}}}} \\ &\quad \quad = -\nabla{p} + \nu \nabla^2 {{\boldsymbol{u}}} - 2 {\boldsymbol{\hat{z}}} |{{ {\text{Ω}}}}_p| \Omega_0 s \,\sin\,{\alpha_p} \cos\,{(\phi + \Omega_0 t)}, \end{split} $
$ \nabla \cdot{{{\boldsymbol{u}}}} = 0. $
In Equation (3), the last term on the right-hand side represents the Poincaré force which drives flows against viscous dissipation. The parameter $ \nu $ denotes the kinematic viscosity. It should be noted that the reduced pressure $ p $ in this equation absorbs a time-dependent potential term $ -s z |{{ {\text{Ω}}}}_p|\Omega_0 \sin{\alpha_p}\cos{(\phi + \Omega_0 t)} $. Equation (4) is the incompressibility condition. By adopting $ \Omega_0^{-1} $ as the time scale, the height $ d $ as the length scale, and $ \rho_0 d^2 \Omega_0^2 $ as the unit of pressure, the dimensionless equations read
$ \begin{split} &\frac{\partial{{\boldsymbol{u}}}}{\partial{t}} + {{\boldsymbol{u}}}\cdot{\nabla{{{\boldsymbol{u}}}}} + 2\Big\{ \hat{{{\boldsymbol{z}}}} + Po\left[\hat{{{\boldsymbol{s}}}}\, \sin\,{\alpha_{p}} \cos\,{(\phi + t)} \right.\\&\left.\quad \quad- \hat{{\text{ϕ}}}\, \sin\,{\alpha_{p}} \sin\,{(\phi + t)} + \hat{{{\boldsymbol{z}}}}\, \cos\,{\alpha_{p}} \right]\Big\} \times{{{\boldsymbol{u}}}} \\ &\quad \quad = -\nabla{p} + E k \nabla^2 {{\boldsymbol{u}}} - 2 \hat{{{\boldsymbol{z}}}} s Po\,\sin\,{\alpha_p} \cos\,{(\phi + t)}, \end{split} $
$ \nabla \cdot{{{\boldsymbol{u}}}} = 0, $
where the Poincaré number $ Po = |{{ {\text{Ω}}}}_p|/|{{ {\text{Ω}}}}_0| $ represents the relative strength of precession with respect to the rotation rate, and the Ekman number $ E k = \nu / (\Omega_0 d^2) $ measures the relative importance of viscous force compared with Coriolis force. In the container frame, the flow on the bounding surfaces of the modified cylinder is at rest, imposing
$ {{\boldsymbol{u}}}={\bf{0}} $
on the bottom $ z=0 $, the top $ z=1 $, and the sidewall $ s={\cal{R}}(\phi) $.
Note that the value of Ekman number depends on the choice of length scale. In this paper, the length scale is the height $ d $ of the cylinder. However, another typical choice of the length scale (e.g., Giesecke et al., 2019; Pizzi et al., 2021a) can be the radius of the cylinder. Because the aspect ratio $ \Gamma $ in this paper is about $ 0.5 $, the difference in the value of Ekman number caused by different length scales is by a factor of about $ 4 $. As a result, the adopted Ekman number $ 10^{-4} $ in Giesecke et al. (2019) is equivalent to $ E k=2.5\times10^{-5} $ of the present study. Although the value of Ekman number is not our primary concern, it indeed affects the dimensionless amplitude of the resonant flow. The critical precession forcing (the critical $ Po $ number), which characterizes the transition from a more laminar flow to a turbulent flow, shifts to smaller values when decreasing $ E k $.
3　Mixed Finite-element Methods and Numerical Computations
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The time integration for the numerical model engages a semi-implicit, second-order, uniform time-step backward differentiation formula (Ascher and Petzold, 1998; Chan KH et al., 2006),
$ \left(\frac{{\partial}{{\boldsymbol{u}}}}{{\partial}{t}}\right)^{n+1} =\frac{3{{\boldsymbol{u}}}^{n+1}-4{{\boldsymbol{u}}}^{n}+{{\boldsymbol{u}}}^{n-1}}{2\Delta t} + O(\Delta t^2), $
for the time derivative and
$ {{\boldsymbol{u}}}^{n+1} \cdot \nabla {{\boldsymbol{u}}}^{n+1} = (2{{\boldsymbol{u}}}^n - {{\boldsymbol{u}}}^{n-1}) \cdot \nabla {{\boldsymbol{u}}}^{n+1} + O(\Delta t^2), $
for the nonlinear term. The superscript $ n $ marks the $ n $-th time step at which the time is $ t_n=n\Delta t $ for $ n=0,1,2,... $. The full temporal discretization of the dimensionless Navier−Stokes Equation (5)−(6) reads
$ \begin{split} &\frac{3{{\boldsymbol{u}}}^{n+1}-4{{\boldsymbol{u}}}^{n}+{{\boldsymbol{u}}}^{n-1}}{2\Delta t} + (2{{\boldsymbol{u}}}^n - {{\boldsymbol{u}}}^{n-1}) \cdot \nabla {{\boldsymbol{u}}}^{n+1} + 2(\hat{{{\boldsymbol{z}}}} + Po{{\hat{{\text{Ω}}}}}_p^{n+1}) \times{{{\boldsymbol{u}}}^{n+1}} \\ &\quad\quad = -\nabla{p^{n+1}} + E k \nabla^2 {{\boldsymbol{u}}}^{n+1} - 2Po \left[s \,\sin\,{\alpha} \cos\,{(\phi + t_{n+1})}\right]\hat{{{\boldsymbol{z}}}}, \end{split} $
$ \nabla \cdot{{{\boldsymbol{u}}}^{n+1}} = 0, $
where
$ {\hat{{\text{Ω}}}}_p^{n+1} = \sin\,{\alpha_{p}}\left[\hat{{{\boldsymbol{s}}}}\,\cos\,{(\phi+t_{n+1})}-\hat{{ {\text{ϕ}} }}\,\sin\,{(\phi+t_{n+1})}\right]+\hat{{{\boldsymbol{z}}}}\,\cos\,{\alpha_p}. $
Starting from arbitrary initial conditions, $ {{\boldsymbol{u}}}^{n+1} $ can be successively determined from the known $ {{\boldsymbol{u}}}^n $ and $ {{\boldsymbol{u}}}^{n-1} $.
In order to solve Equations (8)−(9) subject to the boundary conditions Equation (7) at the time step $ n+1 $, the computational domain is discretized into tetrahedral elements using Netgen, an open-source mesh generator (Schöberl, 1997). It is flexible to put more nodes near the boundary for resolving the thin viscous boundary layer while retaining fewer bulk nodes in order to reduce computing loads. The typical mesh size, as illustrated in Figure 2, is therefore refined near the bounding surfaces but coarse in the interior domain. The total number of nodes in the mesh is about 2 million and these nodes make up about 1.5 million tetrahedral elements, which are sufficient for the numerical simulations at the Ekman number $ 10^{-4} $.
A P2/P1 mixed finite element method (Stenberg, 1984) is employed, by which the velocity field is expanded into a piece-wise second-order polynomial by nodal quadratic bases defined on tetrahedral vertices and edge midpoints, while the pressure field is approximated by a piece-wise first-order function constructed by nodal linear bases defined on tetrahedral vertices. A saddle-point algebraic problem is obtained after the Galerkin weighted residual approach is applied (John, 2016). A stabilized Krylov subspace iterative solver (the stabilized BiCG) is used to solve the system of linear equations. The convergence property and numerical stability of the spatial discretization have been extensively demonstrated in Chan KH et al. (2010), Kong DL et al. (2015, 2014).
Without loss of generality, our simulations set the precession angle $ \alpha_p = \pi/4 $, the barrier location $ \phi_c = \pi $ and half-width $ \Delta \phi = \pi/20 $. The Ekman number in most laboratory experiments and numerical simulations typically falls within the range of $ 10^{-5} \le E k \le 10^{-4} $ (for example, Kobine, 1995; Meunier et al., 2008; Lin YF et al., 2014). As direct numerical simulations are computationally expensive, and since numerical solutions at $ E k=10^{-5} $ or smaller show similar behavior to those at $ E k=10^{-4} $, we choose a moderate value of $ E k=10^{-4} $ for all simulations presented below.
4　The Influences of Sidewall Topography
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We have carried out three groups of numerical simulations, demonstrating the influences of the sidewall topography on precessionally-driven flows. The Poincaré number increases from small, to moderate, to large values across different groups, resulting in progressively more turbulent flow. In each group of simulations, the topographical distortions are imposed stronger and stronger by increasing the height $ H $ of the barrier.
4.1　A Perfect Cylinder without Sidewall Topography
In order to clarify the influence of sidewall topography, we first review the fundamental precessionally-driven flows in a perfect circular cylinder without any topography.
When the precession rate is small, $ Po = O(10^{-4}) $, in the perfect cylinder with aspect ratio $ \Gamma=0.502559 $, the precessionally-driven flow can be decomposed into a series of inertial modes $ {{\boldsymbol{u}}}_{mnk} $. The Poincaré force can resonate directly with the inertial mode $ {{\boldsymbol{u}}}_{111} $ whose half-frequency is 0.5. Other modes are damped heavily by viscous dissipation (Kong DL et al., 2015). Figures 3a–3f show the retrograde wave in one propagating period by plotting the $ z $ component of the flow velocity in the $ z=1/2 $ plane. Figures 3g–3h show, respectively, the axial component of the flow velocity $ \hat{{{\boldsymbol{z}}}}\cdot {{\boldsymbol{u}}} $ and the helicity $ h={{\boldsymbol{u}}}\cdot {{ {\text{ω}}}} $ in the perfect circular cylinder. Helicity quantifies the degree of alignment between the velocity ($ {{\boldsymbol{u}}} $) and vorticity ($ {{ {\text{ω}}}}=\nabla\times{{\boldsymbol{u}}} $) fields, and is a key indicator for identifying the three-dimensional and complex structure of turbulent flows, such as helical motions and strong vortex interactions (Moffatt, 2014). In this study, helicity is used to assess the intensity and spatial distribution of turbulence under various precessional forcings and with barriers of different heights. The results shown in Figure 3h demonstrate that the interior flow primarily exhibits azimuthal motion, where the velocity is perpendicular to the vorticity ($ {{\boldsymbol{u}}}\cdot{{ {\text{ω}}}}=0 $). In contrast, at the top and bottom boundaries, and at the sidewall layer, the combined effects of precession and viscosity (Ekman boundary layers) introduce secondary flows with axial and radial velocity components. In these regions, enhanced alignment between velocity and vorticity leads to elevated local helicity, while the interior remains stable and nearly laminar.
When the Poincaré number gets large, e.g. $ Po=0.5 $, the retrograde traveling wave in the perfect cylinder is shown in Figures 4a–4f, where many modes are excited (Kong DL et al., 2014). The strongly forced flow is marked by a predominant axisymmetric geostrophic component (zonal flow) in the bulk volume of the cylinder plus a sidewall-localized shear layer. Figure 4g shows the $ z $ component of the flow velocity in the perfect cylinder, revealing clearly the presence of a sidewall localized shear layer. As shown in Figure 4h, the helicity is predominantly elevated in these sidewall regions, whereas the interior of the cylinder exhibits lower helicity. This distribution suggests that the flow near the sidewall is more complex, while the interior region remains comparatively quiescent and less structured. This numerical simulation is run for a sufficiently long time, and the overall structure of the strongly nonlinear flow appears to be highly stable.
4.2　Small Poincaré Number with Different Barrier Heights
When the Poincaré number is small, the flow is laminar, and the sidewall topography does not fundamentally alter the primary inertial mode. In Figures 5a–5c, three sub-figures are presented demonstrating the wave propagation with different barrier heights (multimedia views are available via the Supplementary Materials link). The barrier heights $ H $ are $ 0.1 $, $ 0.25 $ and $ 0.5 $ respectively. In the cases with small to moderate barrier heights, the overall fluid motion is not substantially affected. A barrier has a greater effect on the precessionally-driven flow only when the height almost reaches the radius of the cylinder container (as shown in Figure 5c (multimedia view)). The inertial mode is slightly distorted as it propagates around the barrier but largely returns to its undistorted pattern after it propagates away from the barrier. Boundary topography does not cause bulk turbulences. This can be seen from Figures 5d–5f (multimedia view), which demonstrate the helicity.
4.3　Moderate Poincaré Number with Different Barrier Heights
As the Poincaré number increases to $ Po = 0.1 $, the topographical influence of the boundary on precessionally-driven flow becomes more apparent. Even without a barrier, the original flow is moderately turbulent (Kong DL et al., 2015). The presence of the barrier, regardless of its height, intensifies the chaotic condition both near the boundary and in the bulk, as shown in Figures 6a–6c (multimedia view). However, the propagating wave amplitude depicted by the $ \hat{{{\boldsymbol{z}}}}\cdot{{\boldsymbol{u}}} $ component is suppressed at the leading side of the barrier (i.e., the side first encountered by the propagating wave). A major bulk vortex is formed at the trailing side of the barrier (i.e., the side downstream where the wave passes after interacting with the barrier), especially when the barrier height is larger than half of the radius of the cylinder, as shown in Figures 6d–6f (multimedia view). Through the animation of the helicity variations, it can be seen that the sign, strength and large-scale pattern of the trailing vortices are roughly stable and periodic in time.
4.4　Large Poincaré Number with Different Barrier Heights
As the precession rate gets really large to $ Po=0.5 $, the strongly nonlinear waves are changed dramatically by sidewall barriers. Even a very short barrier can induce significant turbulence, as can be seen in Figure 7a (multimedia view) and Figure 7d (multimedia view). When the barrier becomes even taller, namely $ H=0.25 $ and $ H=0.5 $, the sidewall-localized wave (see Figures 4a–4f) is totally replaced by bulk turbulent flows, which are displayed in Figures 7b–7c (multimedia view) and Figures 7e–7f (multimedia view).
4.5　Downstream Modification of the $ u_z $ Structure by the Barrier
We further examine the flow in a vertical cross-section located immediately downstream of the barrier (Figure 8). Figure 8a shows the reference structure given by the axial component of the $ {{\boldsymbol{u}}}_{111} $ mode, $ \hat{{{\boldsymbol{z}}}}\cdot{{\boldsymbol{u}}}_{111} $. For a small Poincaré number and a finite barrier height $ H=0.1 $, the vertical pattern $ u_z $ is only weakly modified (Figure 8b), and remains largely consistent with the $ {{\boldsymbol{u}}}_{111} $-dominated structure. This is also consistent with the Fourier-spectral analysis, in which the response is dominated by the forcing frequency.
As the Poincaré number increases to moderate and large values, the sidewall-localized structure observed in the perfect cylinder becomes increasingly perturbed by the barrier (Figure 8c–8d). Compared with the smooth, coherent pattern in Figure 8c, Figure 8d exhibits a more irregular, patchy structure with small-scale fluctuations. Likewise, the spectra extracted in the vicinity of the barrier are still dominated by the forcing frequency, but numerous secondary peaks emerge as $ Po $ increases, indicating that the flow has become fully turbulent. This suggests that the excitation of inertial waves may be associated with the formation of large-scale vortices at moderate and large Poincaré numbers.
4.6　Azimuthal Flow and Kinetic Energy Density
It is also interesting to look at the impact of sidewall topography on the flows both from their horizontal velocity and the perspective of kinetic energy density. For a perfect circular cylinder of $ \Gamma=0.502559 $, increasing precession forcing leads to a strong westward ($ -\hat{{\text{ϕ}} } $ direction) axisymmetric geostrophic flow (Giesecke et al., 2019; Pizzi et al., 2021b; Gao DL et al., 2021). In this paper, the geometry is apparently non-axisymmetric, which prevents analysis based on inertial modes decomposition in the cylindrical volume (Kong DL et al., 2015). But because the geometric symmetry with respect to the middle plane ($ z=1/2 $) remains, it is still sound to examine the horizontal flow component on the plane $ z=1/2 $ for barrier-flow interplay. We focus on the azimuthal component of velocity, which represents the part of a flow that directly collides onto a sidewall barrier which should be strongly influenced by the existence of a barrier. Figure 9 plots the mean azimuthal flows that are defined as
$ U_\phi(s) = \frac{1}{\mu\left(A_s \right)} \int_{A_s} \hat{{ {\text{ϕ}} }} \cdot {{\boldsymbol{u}}}(s,\phi;z=1/2){\rm{d}}\phi, $
where
$ A_s = \left\{ \phi \in [0,2\pi): s \leq {\cal{R}}(\phi) \right\}. $
Here, $ A_s $ represents the set of all azimuthal angles $ \phi $ at which the point $ (s,\phi) $ remains inside the fluid domain, i.e., $ s $ does not exceed the local boundary $ {\cal{R}}(\phi) $. The measure function
$ \mu(A_s) = \int_{A_s} {\rm{d}}\phi $
gives the total angular span (in radians) where radius $ s $ lies inside the domain. In the case of a perfect circular cylinder without barrier, i.e., $ {\cal{R}}(\phi)=\Gamma $ for any $ \phi $, we simply have $ \mu(A_s)=2\pi $ for all $ s \leq \Gamma $. Here, $ \mu(A_s) $ denotes the total azimuthal length (in radians) for which points at radius $ s $ are within the cylinder.
It is apparent that a barrier impacts the generation of zonal circulation a lot. In the $ H=0.1 $ case (see Figure 9a), the main feature of westward zonal flow and the trend of zonal flow saturation still can be seen. But for $ Po=0.5 $, because the barrier is well higher than the thickness of the outer shear layer of the strongly forced flow, the weakening of the flow is obvious. With further increase of $ H $, namely Figure 9b–9c, the nature of mean circulation changes fundamentally. There seems to be competition between the topography and precession. Large $ Po $ forcing still works hard to drive westward zonal flow, despite being suppressed, against the topographical resistance. But smaller $ Po $ forcing is unable to outweigh the obstacle, such that mean circulation can be reversed to eastward ($ +\hat{{\text{ϕ}} } $ direction) in the outer cylindrical region where barriers place azimuthal truncation. This feature can be seen in Figures 6–7, where vortices form downstream of the barrier.
The kinetic energy density of a flow can be defined by
$ E_{\text{kin}}=\frac{1}{2V}\int_V\left|{{\boldsymbol{u}}}\right|^2{\rm{d}}V. $
It measures the average strength of the flow. The dependence of $ E_{\text{kin}} $ on the Poincaré number $ Po $ in modified cylinders with different topography $ H $ is plotted in Figure 10 for the purpose of seeing the combined effect of precession and topography. The numerical simulation agrees well with the linear analytic results given by Zhang K and Liao XH (2017) in the perfect cylinder when $ Po \leq 0.01 $. For example, direct nonlinear simulation yields $ E_{\text{kin}}= 1.6\times 10^{-3} $ for $ Po=0.01 $ at $ E k=10^{-4} $, while the analytical expression gives $ E_{\text{kin}}= 1.9\times10^{-3} $ for $ Po=0.01 $. However, when $ Po \gt 0.01 $, the nonlinear simulations show that $ E_{\text{kin}} $ gradually saturates and deviates from the linear theoretical prediction, which indicates that nonlinear effects become significant at larger $ Po $.
Another interesting phenomenon is that, at weak to moderate precessional forcing (i.e., at the small $ Po $ end in Figure 10), increasing the height of the obstacle leads to a greater reduction in the flow strength. The trend becomes opposite when the precession gets strong. As demonstrated in Figure 10, towards the large $ Po $ end, higher barrier height $ H $ results in higher kinetic energy density. For example, at the large $ Po $ end, $ E_{\text{kin}} $ is $ 5.2\times10^{-2} $ for $ H=0 $, while it increases to $ 8.8\times10^{-2} $ when $ H=0.5 $. Under strong Poincaré forcing, topographical barriers on the cylindrical sidewall interact with the flow more vigorously, generating significant bulk turbulence through complex boundary processes (see the helicity's magnitude in Figure 7). In this regime, taller barriers enhance these turbulent interactions and drive more energetic flows, resulting in a noticeable increase in kinetic energy with barrier height. This suggests that, rather than damping the flow, high barriers can act as catalysts for turbulence and energy input when the precession is sufficiently strong.
5　Conclusions
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In this paper, we have investigated the topographical influences of boundaries on fluid motions in rapidly rotating and precessing modified cylindrical containers via numerical experiments using 3D finite-element method. Compared with the precessionally driven flow in a perfect cylinder at the primary inertial-mode-resonance, which is mainly marked by a retrograde $ m=1 $ wave propagation, the topographical effects resulting from modified cylindrical sidewalls are mainly found to be of three remarkable characteristics. First, a simple vertical barrier on the cylinder's sidewall will not fundamentally alter the large-scale $ m=1 $ wave structure. The resonance between the Poincaré forcing and the $ {{\boldsymbol{u}}}_{111} $ mode is robust under topographical perturbations. Second, the presence of a vertical barrier on the cylinder's sidewall can interact nonlinearly with precessionally-driven flows, causing vortices near the boundary and even in the bulk interior as $ Po $ increases, which is fundamentally different from the phenomenon discovered in the rotating convection (Favier and Knobloch, 2020). Third, the topography does not always dampen the flow intensity. Under extremely strong Poincaré forcing, a sufficiently large barrier leads to a higher kinetic energy density in the modified cylinder than in the perfect cylinder. This suggests that boundary topography may exert a stronger influence on long lived planetary magnetic field than previously assumed.
It is worth pointing out that this study paves the way for a series of future explorations. In the currently reported work, the aspect ratio of the cylinder, $ \Gamma $, is chosen to be at the primary resonance with the specific inertial mode $ {{\boldsymbol{u}}}_{111} $ whose radial wave number is unity. For other resonant aspect ratios, the radial structure of such excited inertial modes could bring different types of topography-flow interplay. Especially, if the cylindrical geometry is not exactly at resonance with any particular mode, a large number of inertial modes will be excited, which might make the topographical effects even more complex. A second open question concerns the azimuthal modulation between terrain and flow. We by far have only set up a single topographical barrier on the cylindrical sidewall. It is necessary to extend the study to multiple-barrier cases or even general sidewall-topography cases. Last but not least, in geophysical problems such as the core dynamics, thermal convection and precession jointly play roles in driving core flows. How the terrain on core−mantle boundary or inner-outer core boundary affects core dynamics needs more numerical and experimental research.
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doi: 10.26464/epp2026038
  • Receive Date：2025-11-26
  • Online Date：2026-06-05
  • Published：2026-05-01
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  • Received：2025-11-26
  • Accepted：2026-02-13
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    1Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
    2University of Chinese Academy of Sciences, Beijing 100049, China

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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