On the inviscid flow in the earth's liquid core

Abstract

The momentum equation for the flow in the Earth's liquid core,which depends on two small parameters (the Ekman, E, and the Rossby, R o , numbers), is transformed to a form which depends on two other parameters: E 1/2 and λ = E 1/2 R o . The first of them is still small, but the other one exceeds E 1/2approximately by six orders of magnitude. The Ekman suction boundary conditions for inviscid flow in the bulk ofthe Earth's core also depend only on the first parameter. Therefore, we seek the solution of the hydrodynamic problem in the form of decomposition in the small parameter E 1/2. The equations obtained show that the flow of the leading approximation in the E 1/2 decomposition depends only on parameter λ. This means that for fast rotating fluids λ = (2Ων)1/2 V 1 becomes the universal parameter (as, e.g., the Reynolds number R = V 1 L 1/ν in hydrodynamics) and all flows can only be classified by it. The whole flow is presented as the sum of the force and geostrophic flows. The former can be directly determined in terms of integrals of force. The latter obeys the equations which depend on the time derivative of the geostrophic velocity. Thus the geostrophic flow is the only part of the whole flow which has its own time behavior. It evolves together with the magnetic field and temperature distributions. On the contrary, the force flow fits the force instantly and its time behavior is defined entirely by that of the force distribution. There are two (the inviscid λ → 0 and the viscous controlled λ → ∞) limits within which the whole flow does not depend on λ and hence on E. Therefore, the flow converts into the Taylor state within these limits. Are parameters of the Earth's core are suitable for them? Depending on the viscosity, parameter λ for the Earth 's core changes from 2.3×10−2 for kinematic viscosity to 33 for the largest value of turbulent viscosity ν T = η, where η is the magnetic diffusivity. Hence both types of the Taylor state flow can be realized in the Earth's core. From the point of view of computer simulations the viscous controlled case is especially important, because all the simulations use implicitly λ≫1. For example, Glatzmaier and Roberts (1995, 1996) in fact carried out their simulations for λ∼103. Thus their asymptotic (at λ → ∞) solution is approximately suitable for the Earth's core conditions with the “turbulent” λ = 33. This is not trivial, since if we adopt the same values of ν (= ν T = η) and V 1 for Venus, where Ω is two orders of magnitude smaller, then λ becomes of the order of unity and the viscous controlled numerical results become unapplicable to this planet. Moreover, being independent of λ (and, respectively, of E), the viscous controlled computer simulated flows convert into the Taylor state and thus describe adequately the flow in the Earth's core for turbulent viscosity. This is also a non-trivial conclusion, taking into account that the typical values of the Ekman numbers in computer simulations typically exceed those in the Earth's core by ten orders! Though the computer simulations describe the flows with large λ satisfactorily, they cannot be applied to solving the problem with smaller values of λ and especially in the inviscid limit λ → 0, because of the necessity of resolution of the thin boundary layers. We believe that this difficulty can be overcome by creating computer codes on the basis of the equations presented here which do not require this.