Turbulent convective length scale in planetary cores

Abstract

Convection is a fundamental physical process in the fluid cores of planets because it is the primary transport mechanism for heat and chemical species and the primary energy source for planetary magnetic fields. Key properties of convection, such as the characteristic flow velocity and lengthscale, are poorly quantified in planetary cores due to their strong dependence on planetary rotation, buoyancy driving and magnetic fields, which are all difficult to model under realistic conditions. In the absence of strong magnetic fields, the core convective flows are expected to be in a regime of rapidly-rotating turbulence, 1 which remains largely unexplored to date. Here we use a combination of numerical models designed to explore this low-viscosity regime to show that the convective lengthscale becomes independent of the viscosity and is entirely determined by the flow velocity and planetary rotation. For the Earth's core, we find that the characteristic convective lengthscale is approximately 30km and below this scale, motions are very weak. The 30-km cut-off scale rules out small-scale dynamo action and supports large-eddy simulations of core dynamics. Furthermore, it implies that our understanding of magnetic reversals from numerical geodynamo models does not relate to the Earth, because they require too intense flows. 2, 3 Our results also indicate that the liquid core of the Moon might still be in an active convective state despite the absence of a present-day dynamo. 4 Core convection is strongly affected by the rapid planetary rotation through the Proudman-Taylor constraint, 5 which obliges the fluid to move in columns with little variation along the rotation axis compared with the orthogonal directions. The very low fluid viscosity in liquid cores implies that the convective flows are turbulent, but this turbulence differs from both 3D turbulence due to the anisotropy imposed by the rotation and 2D turbulence due to the presence of Rossby waves. 6 Conditions in planetary cores correspond to small Ekman numbers (Ek = ν/ΩR 2 with viscosity ν, rotation rate Ω and core radius R), large Reynolds numbers (Re = UR/ν with flow speed U), and small Rossby numbers (Ro = U/ΩR = ReEk ), with, for instance, Ek ≈ 10 −15 , Re ≈ 10 9 and Ro ≈ 10 −6 in the Earth's core. 7 Numerical models of planetary cores must employ a fluid viscosity that is orders of magnitude larger than realistic values to keep the range of time and length scales involved in the dynamics manageable, with typically Ek ≥ 10 −7 and Re ≤ 10 4 . 8 Unfortunately this has the undesirable effect that convection properties are still controlled by the viscosity. 9, 10 To go beyond this range of parameters and into the rapidly-rotating turbulent convection regime, in which the fluid viscosity plays a sub-dominant role, we use a combination of a state-of-the-art 3D model 11 down to Ek = 10 −8 supplemented by a simplified model of rotating convection 12 down to Ek = 10 −11 using a quasi-geostrophic (QG) approximation. Here the QG approximation means that the axial vor...

Shape and size of large-scale vortices: A generic fluid pattern in geophysical fluid dynamics

“…2,20] km for N / f ∈ [1,100], which shows that surface eddies can penetrate relatively far into the thermocline and potentially reach the seabed, especially in weakly stratified waters on the continental shelf. In the Earth's core, if we take ∼ 30 km and Ro ∼ 10 −6 for the diameter and Rossby number of the most intense LSVs, as suggested in a recent study [14], we find h ∼ 2 m for N / f = 1, which is a typical value used in previous works, e.g., Ref. [27].…”

“…LSVs studied in this work may be considered a simplified model of cyclones in Earth's atmosphere [63], and in particular of warm-core tropospheric cyclones penetrating into the stratosphere [64], of eddies in Earth's oceans [65,66], and of LSVs in Earth's outer core [14] and stars. Earth's atmosphere, oceans, outer core and stars have different fluid properties, such that the aspect ratio of the stratified cap of LSVs, and the penetration depth, depend on the geophysical or astrophysical fluid of interest.…”

“…Planetary atmospheres showcase a wide range of LSVs [12], including long-lived large planetaryscale vortices that control the global circulation and climate (e.g., polar vortices on Earth and Jupiter's Great red spot) as well as smaller cyclones with O(100) km diameter on Earth [13] that can have devastating consequences. Earth's outer core, which is made of turbulent liquid iron that powers the Earth's dynamo, is also expected to feature numerous LSVs of various sizes [14], such as the high-latitude geomagnetic flux patches [15], as well as a large-scale north polar vortex [16]. LSVs are also found in the solar photosphere [17], and are expected to exist in accretion disks [18] and potentially play an important role in planet formation [19].…”

Efficiency of tidal dissipation in slowly rotating fully convective stars or planets

“…Yet, rapid rotation is known to strongly affect spherical convection (e.g. Guervilly et al 2019), and is therefore believed to strongly modify the effective viscosity when Ro c 1 (Mathis et al 2016). Another complication with incorporating rapid rotation in our model is that the tidal (elliptical) instability can be triggered for large enough β when −1 ≤ Ω or b /Ω s = E/E or b ≤ 3 (Barker et al 2016;Vidal & Cébron 2017).…”

Scaling Laws in Rayleigh‐Bénard Convection

“…Thus, the dynamo process is highly complex and the intimate details such as the impact of field strength remain to be elucidated (Fauve & Petrelis, ; Tobias, ). We note that alternative approaches for uncovering the scalings or balances are currently being pursued for the dynamo problem (e.g., Aubert, ; Calkins et al, ; Cattaneo & Hughes, ; Davidson, ; Gastine et al, ; Guervilly et al, ; Tilgner, ).…”