A case for variational geomagnetic data assimilation: insights from a one-dimensional, nonlinear, and sparsely observed MHD system

Abstract

Abstract. Secular variations of the geomagnetic field have been measured with a continuously improving accuracy during the last few hundred years, culminating nowadays with satellite data. It is however well known that the dynamics of the magnetic field is linked to that of the velocity field in the core and any attempt to model secular variations will involve a coupled dynamical system for magnetic field and core velocity. Unfortunately, there is no direct observation of the velocity. Independently of the exact nature of the abovementioned coupled system -some version being currently under construction -the question is debated in this paper whether good knowledge of the magnetic field can be translated into good knowledge of core dynamics. Furthermore, what will be the impact of the most recent and precise geomagnetic data on our knowledge of the geomagnetic field of the past and future? These questions are cast into the language of variational data assimilation, while the dynamical system considered in this paper consists in a set of two oversimplified one-dimensional equations for magnetic and velocity fields. This toy model retains important features inherited from the induction and Navier-Stokes equations: nonlinear magnetic and momentum terms are present and its linear response to small disturbances contains Alfvén waves. It is concluded that variational data assimilation is indeed appropriate in principle, even though the velocity field remains hidden at all times; it allows us to recover the entire evolution of both fields from partial and irregularly distributed information on the magnetic field. This work constitutes a first step on the way toward the reassimilation of historical geomagnetic data and geomagnetic forecast.

Earth's dynamo limit of predictability

“…Whether this trend will last for some time, enhance space weather hazards on the decade timescale and possibly lead to a reversal of the field on the millennium timescale, is a question that forecasting strategies of the type commonly used in meteorology [Kalnay, 2003] could possibly answer. Such strategies start developing in the context of geodynamo assimilation [Fournier et al, 2007;Kuang et al, 2008]. But just as for meteorology, forecasts will inevitably be limited in time by the nonlinear nature of the governing equations, as the geodynamo unfortunately belongs to the same class of non-periodic dynamical systems that can exhibit chaotic behavior.…”

Forecasting secular variation using core flows

“…Since 2005, a number of forecasting techniques have been developed to incorporate physical approximations. For example, Sun et al (2007) and Fournier et al (2007) served magnetic field data can be assimilated into physical magnetohydrodynamic models. More recently, Kuang et al (2009) have assimilated historical field data into numerical dynamo models to investigate if improvements can be made to forecast field models.…”

A Survey of Implicit Particle Filters for Data Assimilation

“…where, ν = 10 −3 , g u = 0.01, g b = 1 are scalars, and where W is a spatially smooth stochastic process [29,52]. We consider the above equations on the strip 0 ≤ t ≤ 0.2, −1 ≤ x ≤ 1 and with given boundary and initial conditions.…”

“…Physically, u represents the velocity field at the core, and b represents the magnetic field of the earth. The model is essentially the model proposed in [52], but with additive noise…”