1 JNTS No 46 / 2018 (XXIII) PUBLISHED BY THE ACADEMY OF SCIENCES OF ALBANIA JNTS JOURNAL OF NATURAL AND TECHNICAL SCIENCES 2018 (1) XXIII (46) 2 JNTS No 46 / 2018 (XXIII) 3 JNTS No 46 / 2018 (XXIII) CORE-MANTLE BOUNDARY VELOCITY FIELD RECOVERING FROM A FOUR-CENTURY GEOMAGNETIC FIELD MODEL Klaudio PEQINI Department of Physics, Faculty of Natural Sciences, University of Tirana, Albania Epoka University, Faculty of Architecture and Engineering, Tirana, Albania Bejo DUKA Department of Physics, Faculty of Natural Sciences, University of Tirana, Albania ____________________________________________________________ ABSTRACT The present paper provides information about a new method of the velocity field calculation at the Core-Mantle Boundary (CMB) of the Earth when both the radial geomagnetic field and its secular variation (SV) are known. The fluid flow in the outer core is considered to be under the frozen-flux approximation. The simplified induction equation for the radial component is here involved. The velocity field is separated into a toroidal and poloidal part. The radial geomagnetic field and its secular variation and the toroidal and poloidal ingredients of the velocity field are expanded in spherical harmonics series. The maximum degrees of these expansions are chosen in such a way to match the number of unknown coefficients of the velocity field expansion with the number of algebraic equations. The Gauss coefficients of the geomagnetic field spherical harmonic expansion and their SV are taken from the gufm1 model. The system of equations is then inverted by a standard matrix inversion procedure. The recovered maps of the velocity field show a pronounced feature under the South Atlantic and Indonesian archipelago and other features that agree with those reported by other authors. Keywords: core-mantle boundary, velocity field, frozen flux approximation, Gauss coefficients, secular variation 1. INTRODUCTION Currently, it is well confirmed that the structure and evolution of the geomagnetic field are governed by the fluid convective motion in the 4 JNTS No 46 / 2018 (XXIII) terrestrial outer core. This motion is responsible for the generation and maintenance of the planetary magnetic field, known as main field, through a dynamo mechanism (Moffatt et al, 1978; Backus et al., 1996; Duka, 1998; Rüdiger and Hollerbach, 2004). This field is practically dipolar if measured at the Earth’s surface. However, inside the outer core the picture is rather complicated because the higher moments of the magnetic field become important (Backus et al., 1996). There has been extensive research activity on studying the patterns of fluid convection that could emerge in the outer core from experiments (Petrelis et al., 2009) as well as from numerical simulations (Kageyama and Sato, 1997). The large-scale fluid flow in the outer core and the patterns that emerge are considered to be the main drivers of the geomagnetic field evolution and, especially, of the polarity reversals (Schmitt et al., 2001). Simple stochastic models have been also constructed to model the geomagnetic reversals of the dipolar geomagnetic field (Duka et al., 2015; Peqini et al, 2015). It is strongly believed that the small-scale fluid convective motion affects and drives the temporal evolution of the main field, known as secular variation, SV (Backus et al., 1996; Dormy and Soward, 2007). Furthermore, probably the fluid motion in the core-mantle boundary (CMB) which is the physical boundary between the liquid outer core and rocky lower mantle, is crucial for the determination of the SV patterns observed on the Earth’s surface (Bloxham and Gubbins, 1991; Whaler et al., 2016). Recently, there have been some successful efforts to describe the SV of the dipolar field even with stochastic models which offer valuable insights (Schmitt et al., 2001; Duka et al., 2015; Peqini et al, 2015). The fluid flow that governs this process is concentrated in a very thin layer located in the uppermost part of the core, just beneath the CMB (Backus and Le Mouël, 1986). The thickness of the layer is much smaller compared to the other dimensions (this layer is extended to the whole CMB) and allows us to consider the flow to be 2- dimensional. Consequently, the velocity of the fluid, which characterizes the fluid flow, has only two components, vθ and vϕ . It is very important to recover this velocity to provide insights on the fluid convection and dynamo mechanisms (Bloxham and Gubbins, 1991). The corresponding information so obtained from these velocity field maps can provide short-term predictions of the SV (Beggan and Whaler, 2010) and important insights in the changes of Length of Day, LoD (Hide, 1993). In addition, some studies show that the velocity field at the CMB can be used to study the lateral variations of the electrical conductivity of the mantle (Holme, 2000). In principle, the velocity map enables also the calculation of various torques exerted on the fluid into the core by the surrounding mantle (Deleplace and Cardin, 2006). 5 JNTS No 46 / 2018 (XXIII) An important topic is the relation between jerks and fluid flow at the CMB. Jerks are sudden changes in the trend of the SV of the geomagnetic field. Such phenomena consist in changes from a positive trend (increasing field magnitude) to a negative one (decreasing field magnitude) and vice versa and occur in less than one year, so more rapidly than usual secular variations (Duka et al., 2012). It is believed that these events are related with corresponding sharp changes in the pattern of fluid flow at the CMB (Moffatt, 1978; Backus et al., 1996). Through the method described below we aim to study such relationship. However, in this paper we report the first stage of our ongoing work. The relation between jerks and fluid flow at the CMB will be reported in a forthcoming paper. There is a well-known relationship between the time evolution of the magnetic field and the fluid flow as described by the magnetic induction equation (Backus etG al., 1996) ∂B G G G = ∇×vB × +η ∇2 B (1) ∂t () G G where B is the magnetic field, v is the velocity of the fluid and η is the magnetic diffusivity (Moffatt, 1978; Dormy and Soward, 2007). In principle, if the velocity field, i.e. the fluid flow, is known analytically or numerically, together with the boundary conditions, then the magnetic field and its rate of change or SV can be calculated. Theoretically, it is possible to pose the inverse problem, that is recovering the fluid velocity field from knowing the magnetic field and its SV. The fact that the velocity is 2-dimensional and there is only one available equation means that the flow is inherently non-unique (Bloxham and Gubbins, 1991; Hide et al., 1993). To solve this problem different constraints can be imposed. Most frequently it is invoked the assumption that the flow in the upper layers of the core is under the conditions of the geostrophic balance with the pressure field (Backus and Le Mouël, 1986; Hide et al., 1993). This assumption is quite strong, so other more relaxed constraints can be applied. Recent studies show that the application of constraints is not crucial to obtain the velocity field in the CMB (Aubert et al., 2013; Whaler et al., 2016). If the magnetic field, SV and velocity field are expressed in series of spherical harmonics and spherical coefficients (i.e., the Gauss coefficients), we obtain an algebraic system of equations where the Gauss coefficients of the velocity components can be determined. This procedure is detailed in section 2. The magnetic field and SV are assumed to be known. These quantities may be obtained directly from the measurements in the geomagnetic observatories (Whaler et al., 2016) or the results of simulations of numerical dynamo models (Aubert, 2013). In this paper we have explored another possibility: we make use of the Gauss coefficients as provided by the gufm1 model which is 6 JNTS No 46 / 2018 (XXIII) based on the best historical magnetic measurements and describes the magnetic field from 1590-1990 (Jackson et al., 2000). The structure of the present paper is as follows: in section 2 we describe the main equations and the technique used to obtain the Gauss coefficients of the velocity field at the CMB. In section 3 we give the maps of velocity obtained for several epochs and discuss the results. In section 4 we provide some conclusions. 2. MATERIALS AND METHODS A very important aspect of the flow at the CMB is that there is valid the frozen-flux approximation (Roberts and Scott, 1965; Kahle et al., 1967). Under this assumption, the fluid is considered to be a perfect conductor and any diffusion of the magnetic field, i.e. η is neglected (η is inversely proportional to the conductivity).G Thus equation (1) becomes: ∂B G G =∇×vB × . (2) ∂t () Roberts and Scott (1965) applied this approximation to the fluid convection in the outer core. Then their calculations were improved and extended by Kahle et al., (1967). Shortly afterwards the frozen-flux approximation was also applied to SV (Backus, 1968). In this paper we will work in the framework of the frozen-flux hypothesis and will start our analysis from equation (2). The components of the magnetic field are measured at the Earth’s surface by means of the geomagnetic observatories or repeat stations as well as satellites at their altitude. However, we can calculate the magnetic components at the CMB through the downward continuation. This procedure is possible because the mantle is considered to be a poor electrical conductor, i.e. there are not any magnetic field sources, and the dipolar character of the main field is nearly preserved even close to the CMB (Backus, 1986; Backus and Le Mouël, 1986; Backus et al., 1996).