Uwe Harlander

Equatorial wave attractors and inertial oscillations

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 /3-plane model. The second is Sterns (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 Sterns 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 Sterns 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, Sterns 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.

Inertial waves in a spherical shell induced by librations of the inner sphere: experimental and numerical results

We begin with an experimental investigation of the flow induced in a rotating spherical shell. The shell globally rotates with angular velocity Ω. A further periodic oscillation with angular velocity 0 ≤ ω ≤ 2Ω, a so-called longitudinal libration, is added on the inner spheres rotation. The primary response is inertial waves spawned at the critical latitudes on the inner sphere, and propagating throughout the shell along inclined characteristics. For sufficiently large libration amplitudes, the higher harmonics also become important. Those harmonics whose frequencies are still less than 2Ω behave as inertial waves themselves, propagating along their own characteristics. The steady component of the flow consists of a prograde zonal jet on the cylinder tangent to the inner sphere and parallel to the axis of rotation, and increases with decreasing Ekman number. The jet becomes unstable for larger forcing amplitudes as can be deduced from the preliminary particle image velocimetry observations. Finally, a wave attractor is experimentally detected in the spherical shell as the pattern of largest variance. These findings are reproduced in a two-dimensional numerical investigation of the flow, and certain aspects can be studied numerically in greater detail. One aspect is the scaling of the width of the inertial shear layers and the width of the steady jet. Another is the partitioning of the kinetic energy between the forced wave, its harmonics and the mean flow. Finally, the numerical simulations allow for an investigation of instabilities, too local to be found experimentally. For strong libration amplitudes, the boundary layer on the inner sphere becomes unstable, triggering localized Gortler vortices during the prograde phase of the forcing. This instability is important for the transition to turbulence of the spherical shell flow.

Characteristics and energy rays of equatorially trapped, zonally symmetric internal waves

Characteristic curves of partial differential equations (PDEs) in general differ from short wave energy rays. We give conditions for linear, two dimensional second order PDEs that guarantee an exact correspondence between characteristics and energy rays. The findings are applied to time-harmonic, zonally-symmetric small-amplitude equatorial internal waves. It is shown that for a fairly general model of inertia-gravity waves, an exact characteristic-ray correspondence holds. When characteristics of internal waves, trapped in a meridional plane, are followed over several boundary reflections, a convergence towards a limit cycle (called equatorial wave attractor) can generally be found. The results on the characteristic-ray correspondence help to interpret physically equatorial wave attractors. Recent ideas on energy accumulation by near inertial waves, trapped on wave attractors at deep ocean sites are confirmed by our results, at least in the short wave (WKB) sense.

An experimental study of regime transitions in a differentially heated baroclinic annulus with flat and sloping bottom topographies

Abstract. A series of laboratory experiments has been carried out in a thermally driven rotating annulus to study the onset of baroclinic instability, using horizontal and uniformly sloping bottom topographies. Different wave flow regimes have been identified and their phase boundaries – expressed in terms of appropriate non-dimensional parameters – have been compared to the recent numerical linear stability analysis of von Larcher et al. (2013). In the flat bottom case, the numerically predicted alignment of the boundary between the axisymmetric and the regular wave flow regime was found to be consistent with the experimental results. However, once the sloping bottom end wall was introduced, the detected behaviour was qualitatively different from that of the simulations. This disagreement is thought to be the consequence of nonlinear wave–wave interactions that could not be resolved in the framework of the numerical study. This argument is supported by the observed development of interference vacillation in the runs with sloping bottom, a mixed flow state in which baroclinic wave modes exhibiting different drift rates and amplitudes can co-exist.

Temperature fluctuations in a changing climate: an ensemble-based experimental approach

There is an ongoing debate in the literature about whether the present global warming is increasing local and global temperature variability. The central methodological issues of this debate relate to the proper treatment of normalised temperature anomalies and trends in the studied time series which may be difficult to separate from time-evolving fluctuations. Some argue that temperature variability is indeed increasing globally, whereas others conclude it is decreasing or remains practically unchanged. Meanwhile, a consensus appears to emerge that local variability in certain regions (e.g. Western Europe and North America) has indeed been increasing in the past 40 years. Here we investigate the nature of connections between external forcing and climate variability conceptually by using a laboratory-scale minimal model of mid-latitude atmospheric thermal convection subject to continuously decreasing ‘equator-to-pole’ temperature contrast ΔT, mimicking climate change. The analysis of temperature records from an ensemble of experimental runs (‘realisations’) all driven by identical time-dependent external forcing reveals that the collective variability of the ensemble and that of individual realisations may be markedly different – a property to be considered when interpreting climate records.

Taylor–Couette turbulence at radius ratio η=0.5: scaling, flow structures and plumes

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 6:2 109. The extracted angular velocity profiles follow a log law more closely than the azimuthal velocity profiles due to the strong curvature of this D 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 wellexplored D 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 counterrotation 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.

Study of turbulence and interacting inertial modes in a differentially rotating spherical shell experiment

We present a study of inertial modes in a differentially rotating spherical shell (spherical Couette flow) experiment with a radius ratio of

Double-diffusive convection and baroclinic instability in a differentially heated and initially stratified rotating system: the barostrat instability

\eta = 1/3

Inertial wave mode excitation in a rotating annulus with partially librating boundaries

. Inertial modes are Coriolis-restored linear wave modes which often arise in rapidly rotating fluids. Recent experimental work has shown that inertial modes exist in a spherical Couette flow for

On quasigeostrophic normal modes in ocean models: Weakly nonseparable situation

\Omega_{i}<\Omega_{o}