Time and length scales of buoyancy-driven flow structures in a rotating hydromagnetic fluid

Abstract

Length and time-scales of buoyancy-driven flow in a rotating hydromagnetic system are investigated to understand better the flow driven by small-scale localized buoyancy in the Earth's core where the Coriolis and Lorentz forces are dynamically important. Both forces introduce anisotropy in the system creating preferred directions which cause the flow to take spatially long (wake) or short (boundary layer) structures. In this paper results of a systematic study of possible quasi-steady flow structures, mainly large-scale ones (i.e., wake modes), in a rotating hydromagnetic flow and of the time scales to establish these modes in an infinite fluid are presented. The wake modes are classified as either primary or non-primary. In the primary modes the fluid velocity is similar in magnitude and direction to that of the buoyant region, while the non-primary modes serve to carry the return mass flux. Four types of primary modes can occur depending on the relative strengths of Lorentz (L), Coriolis (C) and viscous (V) forces: (1) L ⪡ V ⪡ C: well-known Taylor mode (or column) elongated in the direction of rotation; (2) V ⪡ L ⪡ C: a mode similar to the Taylor column elongated in the direction of rotation but foreshortened compared with the case (1) by the effect of the Lorentz force; (3) V ⪡ C ⪡ L: a mode elongated in the direction of magnetic field but foreshortened compared with the case (4) by the effect of the Coriolis force; (4) C ⪡ V ⪡ L: a mode elongated in the direction of magnetic field. Two types of non-primary modes occur. One of the mode is elongated and flattened in the plane of rotation and magnetic field while the other mode is elongated more in the direction of rotation than in the plane perpendicular. Only one primary and one non-primary mode occurs in a given physical situation. Within the Earth's core, two primary wake modes under the influence of the Coriolis and Lorentz forces (case (2) and (3) above) are possible, depending on the size of the magnetic field, which is poorly constrained. When driven by a buoyant region of scale 1 km, these modes have length-scales of order 5–15 km and respond to changes in buoyancy distribution on a timescale short compared with the local advection time. In contrast, the non-primary mode driven by a buoyancy of scale 1 km has a length scale much larger than the radius of the core and a response time long compared with the time of rise through the outer core of the Earth.