Dynamics of shallow flows with and without background rotation

Abstract

Large-scale oceanic and atmospheric flows tend to behave in a two-dimensional way. To further understand such flows, a large scientific effort has been devoted to the study of perfect two-dimensional flows. For the last 30 years, there has been a large interest in experimentally validating the results from numerical and theoretical studies concerning two-dimensional flows, particularly twodimensional turbulence and spatially periodic two-dimensional flows. Inspired by geophysical flows, experimentalists have used stratification, shallow fluid layer configurations, and background rotation to enforce the two-dimensionality of flows in the laboratory. However, as all these methods have shortcomings, it is difficult to achieve a perfectly two-dimensional flow in the laboratory. The work presented in this thesis focuses on two of the common methods used to enforce the two-dimensionality of flows: the shallow layer configuration and background rotation. To further understand the effect of these methods on the two-dimensionality of flows, we studied the dynamics of generic elementary vortical structures in a shallow fluid layer with and without background rotation. Through the analytical and numerical study of a decaying axisymmetric monopolar vortex, we revised the usual argument for considering shallow flowsas two-dimensional. This argument is based on the continuity equation, and it states that the vertical velocity can be neglected if the ratio of vertical to horizontal length scales of the flow is small. By performing numerical simulations and a perturbation analysis for shallow flows, it was shown that this argument is not valid in general, and that the two-dimensionality of the flow does not depend exclusively on the aspect ratio. Instead, it also depends on the dynamics of the flow; particularly, a shallow flow behaves in a two-dimensional way if the flow evolution is dominated by bottom friction over the whole fluid depth. These results were supported by the numerical and experimental study of a more complex flow structure, namely a dipolar vortex, in a shallow fluid layer. For the study of decaying dipolar vortices, numerical simulations were performed using a finite element code. The flow was initialized with a Lamb–Chaplygin dipolar vortex with a Poiseuille-like vertical profile, after which it was left to evolve freely. The 3D structure of the vortex was obtained using the 2 vortex detection criterion. Using this tool, it was observed how the vortex is gradually distorted due to the secondary 3D motions. An experimental investigation of an electromagnetically forced dipolar vortex, where Particle Image Velocimetry (PIV) was used to calculate the velocity field in a horizontal cross-section of the flow, supports the numerically obtained results. It is assumed that flows subjected to strong background rotation behave like two-dimensional flows due to the reduction of gradients in the direction parallel to the rotation axis, as stated by the Taylor–Proudman theorem. This phenomenon results in the formation of columnar structures. In the current work, it was found that the flow can behave in a two-dimensional way as long as the rotation rate is fast enough, irrespective of the aspect ratio. In other words, this is true even if the fluid depth is of the same order as the thickness of the Ekman boundary layer, for which case no columnar structures are formed. This is attributed to the linear coupling between primary and secondary motions. From the study of decaying vortical structures, it was concluded that neither adding background rotation to a shallow flow nor decreasing the depth of a rotating flow necessarily increases the degree of two-dimensionality of the flow. The last two chapters of this thesis are dedicated to the study of a shallow dipolar structure that is continuously driven by time-independent electromagnetic forcing. For a shallow structure without background rotation, it was observed that for weak forcing the flow can be considered indeed as twodimensional. However, every shallow flow, even for very small fluid depths, becomes three-dimensional for a sufficiently high forcing magnitude. An equivalent result was obtained for a similar flow subjected to background rotation. The change in behavior is associated with a change in the vertical profile of the horizontal velocity, which is clearly absent in perfectly two-dimensional flow. The results presented in this thesis confirm that under certain conditions shallow flows and flows subjected to background rotation can behave as a twodimensional flow. However, more importantly, it is shown that there are clear limits to this behavior. This work presents a better understanding of the basic dynamics of shallow flows with and without background rotation and of the extent to which these flows can be considered as quasi-two-dimensional.