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Equatorial wave attractors and inertial oscillations
- Leo R. M. Maas, Uwe Harlander
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- Journal:
- Journal of Fluid Mechanics / Volume 570 / 10 January 2007
- Published online by Cambridge University Press:
- 14 October 2021, pp. 47-67
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Three different approximations to the axisymmetric small-disturbance dynamics of a uniformly rotating thin spherical shell are studied for the equatorial region assuming time-harmonic motion. The first is the standard β-plane model. The second is Stern's (Tellus, vol. 15, 1963, p. 246) homogeneous, equatorial β-plane model of inertial waves (that includes all Coriolis terms). The third is a version of Stern's equation extended to include uniform stratification. It is recalled that the boundary value problem (BVP) that governs the streamfunction of zonally symmetric waves in the meridional plane becomes separable only for special geometries. These separable BVPs allow us to make a connection between the streamfunction field and the underlying geometry of characteristics of the governing equation. In these cases characteristics are each seen to trace a purely periodic path. For most geometries, however, the BVP is non-separable and characteristics and therefore wave energy converge towards a limit cycle, referred to as an equatorial wave attractor. For Stern's model we compute exact solutions for wave attractor regimes. These solutions show that wave attractors correspond to singularities in the velocity field, indicating an infinite magnification of kinetic energy density along the attractor. The instability that arises occurs without the necessity of any ambient shear flow and is referred to as geometric instability.
For application to ocean and atmosphere, Stern's model is extended to include uniform stratification. Owing to the stratification, characteristics are trapped near the equator by turning surfaces. Characteristics approach either equatorial wave attractors, or point attractors situated at the intersections of turning surfaces and the bottom. At these locations, trapped inertia–gravity waves are perceived as near-inertial oscillations. It is shown that trapping of inertia–gravity waves occurs for any monochromatic frequency within the allowed range, while equatorial wave attractors exist in a denumerable, infinite set of finite-sized continuous frequency intervals. It is also shown that the separable Stern equation, obtained as an approximate equation for waves in a homogeneous fluid confined to the equatorial part of a spherical shell, gives an exact description for buoyancy waves in uniformly but radially stratified fluids in such shells.
Stewartson-layer instability in a wide-gap spherical Couette experiment: Rossby number dependence
- Michael Hoff, Uwe Harlander
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- Journal:
- Journal of Fluid Mechanics / Volume 878 / 10 November 2019
- Published online by Cambridge University Press:
- 17 September 2019, pp. 522-543
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Instabilities of a viscous fluid between two fast but differentially rotating concentric spheres, the so-called spherical Couette flow, with a fixed radius ratio of $\unicode[STIX]{x1D702}=r_{i}/r_{o}=1/3$ are studied, where $r_{i}$ is the inner and $r_{o}$ the outer radius of the spherical shell. Of particular interest is the difference between cases where the Rossby number $Ro=(\unicode[STIX]{x1D6FA}_{i}-\unicode[STIX]{x1D6FA}_{o})/\unicode[STIX]{x1D6FA}_{o}>0$ and cases with $Ro<0$, where $\unicode[STIX]{x1D6FA}_{i}$ and $\unicode[STIX]{x1D6FA}_{o}$ are the inner- and outer-sphere angular velocities. The basic state in both situations is an axisymmetric shear flow with a Stewartson layer situated on the tangent cylinder. The tangent cylinder is given by a cylinder that touches the equator of the inner sphere with an axis parallel to the axis of rotation. The experimental results presented fully confirm earlier numerical results obtained by Hollerbach (J. Fluid Mech., vol. 492, 2003, pp. 289–302) showing that for $Ro>0$ a progression to higher azimuthal wavenumbers $m$ can be seen as the rotation rate $\unicode[STIX]{x1D6FA}_{0}$ increases, but $Ro<0$ gives $m=1$ over a large range of rotation rates. It is further found that in the former case the modes have spiral structures radiating away from Stewartson layer towards the outer shell whereas for $Ro<0$ the modes are trapped in the vicinity of the Stewartson layer. Further, the mean flow excited by inertial mode self-interaction and its correlation with the mode’s amplitudes are investigated. The scaling of the critical $Ro$ with Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{o}\,d^{2})$, where $\unicode[STIX]{x1D708}$ is the kinematic viscosity and $d$ the gap width, is well within the bounds that have been established in a number of experimental studies using cylindrical geometries and numerical studies using spherical cavities. However, the present work is the first that experimentally examines Stewartson-layer instabilities as a function of the sign of $Ro$ for the true spherical-shell geometry.
Mean flow generation by an intermittently unstable boundary layer over a sloping wall
- Abouzar Ghasemi, Marten Klein, Andreas Will, Uwe Harlander
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- Journal:
- Journal of Fluid Mechanics / Volume 853 / 25 October 2018
- Published online by Cambridge University Press:
- 22 August 2018, pp. 111-149
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Direct numerical simulations (DNS) of the flow in various rotating annular confinements have been conducted to investigate the effects of wall inclination on secondary fluid motions due to an unstable boundary layer. The inner wall resembles a truncated cone (frustum) whose apex half-angle is varied from $18^{\circ }$ to $0^{\circ }$ (straight cylinder). The large inner radius $r_{1}$, the mean rotation rate $\unicode[STIX]{x1D6FA}_{0}$ and the kinematic viscosity $\unicode[STIX]{x1D708}$ were kept constant resulting in the constant Ekman number $E=\unicode[STIX]{x1D708}/(\unicode[STIX]{x1D6FA}_{0}r_{1}^{2})=4\times 10^{-5}$. Flows were excited by time-harmonic modulation of the inner wall’s rotation rate (so-called longitudinal libration) by prescribing the amplitude $\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FA}_{0}$ and the forcing frequency $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D6FA}_{0}$. By steepening the inner wall and hence reducing the effect of the local Coriolis force in the boundary layer three different flow regimes can be realized: a rotation-dominated, a libration-dominated and an intermediate regime. In the present study we focus on the libration-dominated regime. For small libration amplitudes (here $\unicode[STIX]{x1D700}=0.2$), a laminar Ekman–Stokes boundary layer (ESBL) is realized at the librating wall. With the aid of laminar boundary layer theory and DNS we show that the ESBL exhibits an oscillatory mass flux along the librating wall (Ekman property) and an oscillatory azimuthal velocity, which resembles a radially damped wave (Stokes property). For large libration amplitudes (here $\unicode[STIX]{x1D700}=0.8$), the DNS results exhibit an intermittently unstable ESBL, which turns centrifugally unstable during the prograde (faster) part of a libration period. This instability is due to the Stokes property and gives rise to Görtler vortices, which are found to be tilted with respect to the azimuth when the librating wall is at a finite angle relative to the axis of rotation. We show that this tilt is related to the Ekman property of the ESBL. This suggests that linear and nonlinear dynamics are equally important for this intermittent instability. Our DNS results indicate further that the Görtler vortices propagate into the fluid bulk where they generate an azimuthal mean flow. This mean flow is notably different from the mean flow driven in the case of the stable ESBL. A diagnostic analysis of the Reynolds-averaged Navier–Stokes (RANS) equations in the unstable flow regime hints at a competition between the radial and axial turbulent transport terms which act as generating and destructing agents for the azimuthal mean flow, respectively. We show that the balance of both terms depends on the wall inclination, that is, on the wall-tangential component of the Coriolis force.
Instabilities and small-scale waves within the Stewartson layers of a thermally driven rotating annulus
- Thomas von Larcher, Stéphane Viazzo, Uwe Harlander, Miklos Vincze, Anthony Randriamampianina
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- Journal:
- Journal of Fluid Mechanics / Volume 841 / 25 April 2018
- Published online by Cambridge University Press:
- 21 February 2018, pp. 380-407
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We report on small-scale instabilities in a thermally driven rotating annulus filled with a liquid with moderate Prandtl number. The study is based on direct numerical simulations and an accompanying laboratory experiment. The computations are performed independently with two different flow solvers, that is, first, the non-oscillatory forward-in-time differencing flow solver EULAG and, second, a higher-order finite-difference compact scheme (HOC). Both branches consistently show the occurrence of small-scale patterns at both vertical sidewalls in the Stewartson layers of the annulus. Small-scale flow structures are known to exist at the inner sidewall. In contrast, short-period waves at the outer sidewall have not yet been reported. The physical mechanisms that possibly trigger these patterns are discussed. We also debate whether these small-scale structures are a gravity wave signal.
Taylor–Couette turbulence at radius ratio ${\it\eta}=0.5$: scaling, flow structures and plumes
- Roeland C. A. van der Veen, Sander G. Huisman, Sebastian Merbold, Uwe Harlander, Christoph Egbers, Detlef Lohse, Chao Sun
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- Journal:
- Journal of Fluid Mechanics / Volume 799 / 25 July 2016
- Published online by Cambridge University Press:
- 23 June 2016, pp. 334-351
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Using high-resolution particle image velocimetry, we measure velocity profiles, the wind Reynolds number and characteristics of turbulent plumes in Taylor–Couette flow for a radius ratio of 0.5 and Taylor number of up to $6.2\times 10^{9}$. The extracted angular velocity profiles follow a log law more closely than the azimuthal velocity profiles due to the strong curvature of this ${\it\eta}=0.5$ set-up. The scaling of the wind Reynolds number with the Taylor number agrees with the theoretically predicted $3/7$ scaling for the classical turbulent regime, which is much more pronounced than for the well-explored ${\it\eta}=0.71$ case, for which the ultimate regime sets in at much lower Taylor number. By measuring at varying axial positions, roll structures are found for counter-rotation while no clear coherent structures are seen for pure inner cylinder rotation. In addition, turbulent plumes coming from the inner and outer cylinders are investigated. For pure inner cylinder rotation, the plumes in the radial velocity move away from the inner cylinder, while the plumes in the azimuthal velocity mainly move away from the outer cylinder. For counter-rotation, the mean radial flow in the roll structures strongly affects the direction and intensity of the turbulent plumes. Furthermore, it is experimentally confirmed that, in regions where plumes are emitted, boundary layer profiles with a logarithmic signature are created.
Inertial wave excitation and focusing in a liquid bounded by a frustum and a cylinder
- Marten Klein, Torsten Seelig, Michael V. Kurgansky, Abouzar Ghasemi V., Ion Dan Borcia, Andreas Will, Eberhard Schaller, Christoph Egbers, Uwe Harlander
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- Journal:
- Journal of Fluid Mechanics / Volume 751 / 25 July 2014
- Published online by Cambridge University Press:
- 18 June 2014, pp. 255-297
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The mechanism of localized inertial wave excitation and its efficiency is investigated for an annular cavity rotating with $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Omega _0$. Meridional symmetry is broken by replacing the inner cylinder with a truncated cone (frustum). Waves are excited by individual longitudinal libration of the walls. The geometry is non-separable and exhibits wave focusing and wave attractors. We investigated laboratory and numerical results for the Ekman number $E\approx 10^{-6}$, inclination $\alpha =5.71^\circ $ and libration amplitudes $\varepsilon \leq 0.2$ within the inertial wave band $0 < \omega < 2\Omega _0$. Under the assumption that the inertial waves do not essentially affect the boundary-layer structure, we use classical boundary-layer analysis to study oscillating Ekman layers over a librating wall that is at an angle $\alpha \neq 0$ to the axis of rotation. The Ekman layer erupts at frequency $\omega =f_{*}$, where $f_{*}\equiv 2 \Omega _0 \sin \alpha $ is the effective Coriolis parameter in a plane tangential to the wall. For the selected inclination this eruption occurs for the forcing frequency $\omega /\Omega _0=0.2$. For the librating lids eruption occurs at $\omega /\Omega _0=2$. The study reveals that the frequency dependence of the total kinetic energy $K_{\omega }$ of the excited wave field is strongly connected to the square of the Ekman pumping velocity $w_{{E}}(\omega )$ that, in the linear limit, becomes singular when the boundary layer erupts. This explains the frequency dependence of non-resonantly excited waves. By the localization of the forcing, the two configurations investigated, (i) frustum libration and (ii) lids together with outer cylinder in libration, can be clearly distinguished by their response spectra. Good agreement was found for the spatial structure of low-order wave attractors and periodic orbits (both characterized by a small number of reflections) in the frequency windows predicted by geometric ray tracing. For ‘resonant’ frequencies a significantly increased total bulk energy was found, while the energy in the boundary layer remained nearly constant. Inertial wave energy enters the bulk flow via corner beams, which are parallel to the characteristics of the underlying Poincaré problem. Numerical simulations revealed a mismatch between the wall-parallel mass fluxes near the corners. This leads to boundary-layer eruption and the generation of inertial waves in the corners.
Two alternatives for solving hyperbolic boundary value problems of geophysical fluid dynamics
- UWE HARLANDER, LEO R. M. MAAS
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- Journal:
- Journal of Fluid Mechanics / Volume 588 / 10 October 2007
- Published online by Cambridge University Press:
- 24 September 2007, pp. 331-351
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Linear internal waves in inviscid bounded fluids generally give a mathematically ill-posed problem since hyperbolic equations are combined with elliptic boundary conditions. Such problems are difficult to solve. Two new approaches are added to the existing methods: the first solves the two-dimensional spatial wave equation by iteratively adjusting Cauchy data such that boundary conditions are satisfied along a predefined boundary. After specifying the data in this way, solutions can be computed using the d'Alembert formula.
The second new approach can numerically solve a wider class of two dimensional linear hyperbolic boundary value problems by using a ‘boundary collocation’ technique. This method gives solutions in the form of a partial sum of analytic functions that are, from a practical point of view, more easy to handle than solutions obtained from characteristics. Collocation points have to be prescribed along certain segments of the boundary but also in the so-called fundamental intervals, regions along the boundary where Cauchy data can be given arbitrarily without over-or under-determining the problem. Three prototypical hyperbolic boundary value problems are solved with this method: the Poincaré, the Telegraph, and the Tricomi boundary value problem. All solutions show boundary-detached internal shear layers, typical for hyperbolic boundary value problems. For the Tricomi problem it is found that the matrix that has to be inverted to find solutions from the collocation approach is ill-conditioned; thus solutions depend on the distribution of the collocation points and need to be regularized.