Displacements due to surface temperature variation on a uniform elastic sphere with its centre of mass stationary

Geophysical Journal International Geophys. J. Int. (2014) 196, 194–203 Advance Access publication 2013 October 22 doi: 10.1093/gji/ggt335 Displacements due to surface temperature variation on a uniform elastic sphere with its centre of mass stationary Ming Fang,1 Danan Dong2 and Bradford H. Hager1 1 Department of Earth Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. E-mail: 2 East China Normal University, Shanghai, China SUMMARY We investigate the displacement field induced by temperature variation within a spherical thermal boundary layer under an Earth-like condition of surface heating by deriving analytical solutions on a uniform elastic sphere under the constraint that its centre of mass remains stationary in space. Similar to strain solutions, our displacement solution consists of spectra of two distinctive modes: an exponential mode relating to the thermal body force and a powerlaw mode relating to the (equivalent) thermal surface loading. The exponential modes of the thermal body force in our solution turn out to be identical to that in a classic half-space solution, while the effect of thermal loading by the power-law modes in our spherical solution is different from the exponential modes of thermal loading in the half-space solution. The thermal surface loading is found, by analytical and numerical analyses, equally important in order of magnitude as the thermal body force in producing the radial displacement at the surface throughout the entire harmonic spectrum. The transverse displacement arises mainly from the power-law modes of thermal surface loading. Numerical simulations, based on NASA’s space-borne observation of the global land surface temperature (ocean is masked out), have shown unique patterns in the annual variation of the global displacement field that fits the climatological and geographical settings. The predicted amplitude of the thermally induced surface deformation in global scale is at the millimetre level with the largest ∼2 mm for radial displacement and ∼1 mm for transverse displacement. Comparative analysis shows that the radial displacement field is asymptotically proportional to the surface temperature distribution, which justifies the use of the half-space solution as a good approximation for modelling the global radial displacement. The transverse displacement obtained by patched half-space solution fails to capture the long-range transverse variations on a spherical surface, and thus, is inadequate for modelling and synthesizing the global transverse displacement. Key words: Reference systems; Space geodetic surveys; Global change from geodesy; Mechanics, theory, and modelling. 1 I N T RO D U C T I O N Monitoring sea level variation and crustal deformation at millimetre level on a global scale requires the stability of a terrestrial reference frame at submillimetre level. Thermal elastic response to periodic solar radiative heating may produce surface displacement at such small amplitude. Results on crustal motions in regional GPS networks have shown distinctive seasonal variability at millimetre level (e.g. Prawirodirdjo et al. 2006; Yan et al. 2009; Tsai 2011), and further analysis suggests that such seasonal variations also exist in global GPS networks (Dong et al. 2002; Fritsche et al. 2012). There are three potential non-tectonic sources for the observed seasonal 194  C variations: (i) instrument changes, including monument settings, in response to local seasonality; (ii) incomplete removal of the effects of surface mass loading and (iii) global scale thermoelastic and poroelastic deformations in response to surface temperature and hydrological variations. Global scale displacement due to surface loading on an elastic spherical Earth has been solved for 50 yr (Longman 1962; Farrell 1972). In contrast, solution for the displacement due to surface heating on an elastic spherical Earth has not been available. There seems to be only one solution in geoscience given by Berger (1975) for thermoelastic strains on 2-D uniform half-space. Although applied in local and regional studies The Authors 2013. Published by Oxford University Press on behalf of The Royal Astronomical Society. GJI Gravity, geodesy and tides Accepted 2013 August 20. Received 2013 July 1; in original form 2013 January 9 Displacements due to surface heating 2 A REVIEW OF THE THERMAL E L A S T I C P RO B L E M We begin with a brief review of the Berger’s (1975) problem with more abstract formulations in terms of displacement. An isotropic thermal expansion by temperature variation, T, gives rise to the thermal strain tensor, eT (Love 1944; Berger 1975) eT = βT I, (1) where β is the coefficient of thermal expansion and I the identity tensor. In an elastic medium of Lamé moduli λ and μ, the relationship between the elastically deformed strain, ed , and the isotropic stress, τ , is ed = λ 1 τ− tr(τ )I, 2μ 2μ(3λ + 2μ) (2) where tr() stands for the trace of a tensor. Adding (1) and (2) yields the total strain e e = eT + ed = 1 λ τ− tr(τ )I + βT I. 2μ 2μ(3λ + 2μ) (3) The total displacement, u, is the sum of the partial displacement uT corresponding to eT and the partial displacement ud corresponding to ed : u = uT + ud . It follows from (3) and the classic linear relation between the strain and displacement that τ = τ̃ − (3λ + 2μ)βT I, τ̃ = λ(∇ · u) + μ(∇u + u∇), (4) where τ̃ is the so-called associated stress explicitly relating to the total displacement, u. The quasi-static equation of force balance is simply ∇ · τ . Using the notations in (4), we have for a uniform elastic medium of constant β, λ and μ λ∇(∇ · u) + μ∇ · (∇u + u∇) − (3λ + 2μ)β∇T = 0. (5) Apparently, the term (3λ + 2μ)βT serves as the potential of the thermal body force that balances the associated stress τ̃ . Berger (1975) interprets this term as the ‘added surface traction’ by the thermal effect. This is inaccurate. We will show in eq. (27) below that the real added surface traction, that is, the thermal surface loading arises from the surface boundary conditions by different terms. 3 HARMONIC DECOMPOSITION FOR THE FIELDS The temperature distribution, T, in the Earth’s interior of uniform thermal diffusivity, η, is governed by the conduction equation 1 ∂T = ∇ 2 T. η ∂t (6) In a geocentre-based spherical coordinate of radius r, colatitude, θ and longitude ϕ, the periodic temperature variation at frequency, ω, can be written in the form of standing waves with locally variable amplitude T (t, r, θ, φ) = eiωt T (r, θ, φ) r ≤ a. (7) The classic solution to eq. (6) in the form of (7) is a summation of spherical Bessel functions modulated by the spherical harmonic decomposition of the surface temperature amplitude (e.g. (e.g. Ben-Zion & Leary 1986; Prawirodirdjo et al. 2006; Yan et al. 2009; Tsai 2011), the half-space solution may not be adequate for analysing and maintaining a global geodetic network on a unified terr (...truncated)