Session G35: Geophysical Fluid Dynamics: Rotating Flows

Helicity segregation by Ekman pumping in laminar rotating flows with gravity orthogonal to the rotation

Kinetic helicity in most geodynamo simulations has a distinct distribution above and below the equator -- it is negative in the north and positive in the south. Using direct numerical simulations of rotating convection, with the rotation axis perpendicular to gravity, we investigate the role of Ekman pumping as a possible mechanism behind this helicity segregation. For a fixed Rayleigh number $Ra=10^5$ and Prandtl number $Pr=1$, we consider two values of Taylor number, $Ta=10^5$ and $10^6$, and two different boundary conditions -- no-slip and fixed stress at the walls normal to the rotation axis.

                We observe cyclonic flow in bulk and helical flow driven by Ekman pumping near the boundaries normal to the rotation axis. This leads to helicity being negative (positive) near the wall with outward normal parallel (anti-parallel) to the rotation vector. The peak in helicity (averaged in the plane normal to the rotation axis) occurs inside the Ekman layer in all simulations. The helicity magnitude is smaller for the fixed stress boundary condition as compared to no-slip. We also observe that the buoyancy causes the large helicity regions inside the Ekman boundary layer to migrate away from the rotation axis. We also present the helicity budget analysis, with the contribution of Coriolis force, pressure-vorticity, buoyancy force, and viscosity in the helicity segregation. We find that both pressure-vorticity and Coriolis forces contribute significantly to the helicity segregation. Moreover, from modal energy analysis, we find that the mode (2, 2), which represents the secondary flows, is important for helicity segregation.   

Numerical study of rotating convection dynamics in a non-uniformly heated annulus

A new configuration for rotating convection is considered that mimics the localized heating of the earth’s equator for better modeling of the baroclinic waves and secondary structures in the atmosphere. Such a configuration is shown to overcome the limitation of the classical differentially heated rotating annulus by providing heating in both horizontal (meridional) and vertical directions using a localized heating strip near the outer bottom periphery and uniform cooling on the inner wall of the annulus. A finite volume method-based open-source CFD toolkit OpenFOAM is used to unravel the flow dynamics. The numerical solver is validated using the results from Williams (1971) and Ayan et. al. (2021). Numerical simulations are performed for a fixed Rayleigh number, Ra = 4.76×10⁸, and varying Taylor numbers, Ta = 6.5×10⁸, 1.5×10⁹, and 2.7×10⁹. The results show the co-existence of the baroclinic wave, meandering in the annulus, and convective columnar plume (CCP), present over the heating strip, in the annulus. For modal analysis, a Complex Empirical Orthogonal Function (CEOF) analysis of velocity and temperature data is performed which shows that the baroclinic wave is in a steady wave regime with wave mode, m = 4 at the lowest Ta. Further, with an increase in the Ta number, the baroclinic wave moves to the transition regime at Ta = 1.5×10⁹, and at Ta = 2.7×10⁹ wave moves to the irregular regime and completely breaks down into smaller eddies. Furthermore, we find that the CCP structure grows in number at Ta = 1.5×10⁹ and eventually disintegrates at Ta = 2.7×10⁹. In addition, the 1^st dominant mode of the baroclinic wave and CCP, with near 70% of the total variance, is present for Ta = 6.5×10⁸ and 1.5×10⁹. However, for the highest Ta, the dominant mode is absent due to the breakdown of the flow structures. In totality, the flow dynamics and heat transport in a non-homogenously heated annulus are governed by the non-linear interaction between the baroclinic waves and CCP.   

Multiphase plumes in a stratified rotating environment

The Deepwater Horizon oil rig explosion in 2010 led to a formation of a long-lived turbulent multiphase plume consisting of a mixture of saltwater, oil droplets and gas bubbles. The plume was discharged into a stratified rotating environment and persisted for several months. Thus, its dynamics were likely affected by the Earth's rotation. Additionally, gas bubbles and oil droplets possessed the so-called slip velocity by means of which they could separate from the entrained seawater flow. Thus, modelling the dynamics of a multiphase plume in a stratified rotating environment is important for understanding and predicting the transport and the ultimate fate of oil released into the ocean. In this talk, we present results from small-scale laboratory experiments on bubble plumes in a stratified rotating environment. We use sugar instead of salt to create background stratification, which allows us to control independently the main parameters of the problem: the stratification frequency, the Rossby number and the slip velocity of the bubbles. In our data analysis, we focus in particular on the vertical distribution of the multiphase effluent within the water column and discuss the changes in the structure of lateral intrusions of the multiphase plume compared to the non-rotating case.   

Focusing of inertial waves by a vertically oscillating annular forcing

We investigate the focusing phenomenon of inertial waves generated by a vertically oscillating slender torus in a uniformly rotating fluid. Based on the linearized Navier-Stokes equations, we obtain the analytical solution of the velocity field. It is shown that, under the axisymmetric annular forcing, the wave rays form a double cone symmetric about the plane on which the torus is located. At the vertex of the cone, the waves are focused in a shock-like manner, causing localized energy surges. After focusing, the waves continue their propagation and form a new inverted cone with the same cone angle. These purely theoretical results are in good agreement with the experimental and numerical study by M. Duran-Matute et al. (Phys. Rev. E87, 041001(R) 2013). Furthermore, when friction effects occur, the wave away from the focal point is significantly attenuated so that the symmetry about the focal point is broken. Under the same viscous condition, the amplitude at the focal point reaches its maximum at a forcing frequency close to the angular velocity of the rotating flow. Additionally, a preliminary analysis of the nonlinear problem shows that inertial waves with higher frequency appear and affect the energy transfer at the focal point.   

Zonal flows driven by energy and zonostrophy conservations in Charney-Hasegawa-Mima turbulence

Although zonal flows are ubiquitous in planetary systems, driving mechanisms of them in turbulence still remain poorly understood. Our present investigation focuses on the relations between zonal flows and inviscid invariants. Zonal flows owe their existence to the anisotropisation of energy transfer. Rotation facilitates two-dimensionalisation of turbulence and leads to the inverse energy cascade, driving the formations of large-scale structures.

The Charney-Hasegawa-Mima (CHM) equation is one of the simplest useful nonlinear models describing the interaction between zonal flows and turbulence. We performed DNSs of the CHM equation. Zonal structures in the real space have different appearance among vortex, stream, and pressure fields. The 2D energy spectra show dumbbell structure in low-k range and peaks on/around k_y-axis, corresponding to zonal flows. The local-flux vectors proposed in [1] [2]show consistent directional structures with the energy-enstrophy double cascade, critical balance and weak turbulence theory. The convergence points of these vector fields suggest there exist two kind of driving mechanisms for zonal flows: those owing to energy and zonostrophy conservations. The effect of finite Rossby deformation radius on the driving mechanisms will also be discussed.[3]