Amplitude Vacillation in Baroclinic Flows Chapter 3 in T. von Larcher and P. Williams: Modeling Atmospheric and Oceanic Flows: Insights from Laboratory Experiments and Numerical Simulations

This chapter will introduce the phenomenology of vacillation in baroclinic flows based on experimental data, CFD and low-order numerical models. The processes leading to vacillation of a steady baroclinic wave will be discussed in terms of nonlinear interactions between different wave modes and between waves and the azimuthally or longitudinally averaged baroclinic flow. Complementing the review of the literature on amplitude vacillation, some new material will be presented to discuss the effect of the presence of lateral boundary layers in laboratory experiments and of the Prandtl number on feedback between the vacillating waves and the Ekman transport in the boundary layers. In the later sections, other forms of vacillation will be discussed and, finally, the role of vacillation in the transition to chaos and turbulence will be briefly addressed. 1. Phenomenology of Amplitude Vacillation The first reference to the term ‘vacillation’ in reference to baroclinic flows, with a qualitative description, can be found in a brief note from January 1953 by Hide [28] on observations in a rotating baroclinic annulus: ‘One cycle of this phenomenon, which has been termed ‘vacillation’ begins (say) with a symmetrical wave pattern with its continuous ‘jet’. Some seconds later there is a distinct leaning backward of the troughs and a decrease of their width. This is followed by the troughs returning to N.-S. orientation and then leaning forward in preparation of the stage when the ‘jet’ stream is actually interrupted and intense cyclones are formed in the position of the wave troughs. The cyclones decay and the ‘jet’ is re-established; the wave pattern returns to the initial stage and the cycle starts again. The period corresponds to a few ‘weeks’.’. A full description of his observations can be found in Hide [29] and two typical snapshots of a vacillating wave 3 are shown in Figure 1. While it will become apparent in this chapter that this excerpt describes structural vacillation, rather than amplitude vacillation, it initiated detailed research into vacillating flows in many places. Email address: [email protected], (Wolf-Gerrit Früh) Preprint submitted to Wiley, 2014, ISBN 978-1-118-85593-5 September 11, 2014 This term ‘vacillation’ was subsequently taken up as a technical term and its definition refined, distinguishing between amplitude vacillation and shape vacillation [32]. Fowlis and Pfeffer [15] characterised amplitude vacillation based on an array of thermistors in a large baroclinic annulus. Since then, amplitude vacillation has been investigated in laboratory experiments with a range of thermally-driven baroclinic rotating annulus experiments, for example by White and Koschmieder [86], Tamaki and Ukaji [78], and Sitte and Egbers [74] in addition to those by Raymond Hide and Richard Pfeffer. While these experiments generated the baroclinic flow by thermal forcing of the side-walls, it can also be mechanically forced by a differentially rotating lid in contact with the upper layer of a two-layer fluid. This system has also been investigated in detail by Hart [22] and successors. Like the thermally driven annulus, this two-layer experiment has also shown amplitude vacillation both, in experiments with two immiscible fluids [23, 24] and with salt-stratified water [14]. Another example of flow observations referred to as amplitude vacillation is from a thermally driven annulus which is rotated so rapidly that the centrifugal term outweighs terrestrial gravity [2, 56, 73]. In that case, the fluid is no longer stably stratified and the resulting flow is closer to rotationally constrained Rayleigh-Bénard convection than to baroclinic instability. The term vacillation is also used to describe atmospheric phenomena, such as tropospheric wave-zonal flow fluctuations [40, 12], the stratospheric vacillation [10, 67, 77], or sea-surface temperature (SST) fluctuations [82], and for climate fluctuations [76]. A widely accepted definition of amplitude vacillation (AV) is now that it is a (fairly) regular oscillation of the magnitude of a well-defined wave mode while the spatial structure remains (essentially) unchanged. A more complex form of vacillation is modulated amplitude vacillation (MAV) [71, 17] which frequently displays chaotic oscillations and usually involves fluctuations of several wave modes of different spatial structure. A periodic amplitude vacillation of a wave number 2 and a chaotic modulated amplitude vacillation are illustrated in Figure 2, both taken from the Direct Numerical Simulations of an air-filled annulus described by Randriamampianina et al. [69] and Randriamampianina and Crespo Del Arco [68] in this book. A special case of a flow which appears like a modulated amplitude vacillation is the superposition of two steady wave modes; this has been termed interference vacillation [55, 44]. Figure 1: Illustration of two flow stages within an amplitude vacillation cycle (Peter Read, published in [41], c ©John Wiley & Sons, used with permission).