Geodesy & Crustal Deformation Geology 6690/7690 Geodesy & Crustal Deformation 6 Dec 2017 Last time: Earth elastic response to loading • Deformation is described in the spherical harmonic domain as a linear function of degree l by the load Love numbers hl, kl, ll: Vertical: Horizontal: Gravitational potential: • Surface displacements response to a point load are down and toward the load; vertical is sensitive to integral whereas horizontal is sensitive to local gradient of mass • Variability in elastic properties contributes to ~±20% perturbations in surface vertical response © A.R. Lowry 2017

Solid Earth Tidal Deformation (cf John Wahr’s class notes pp 251–271) Earth tides are deformation of the solid Earth in direct response to the change in gravitational potential of (primarily) the sun & moon resulting from rotation of the Earth and variations in orbits. (This is distinct from the solid-Earth deformation in response to ocean tidal mass). • Can be several tens of cm peak-to-peak • Not observed by most ground-based instruments (with important exceptions of strain-gauges & tiltmeters!) • Is observed by space-based positioning systems (e.g. GPS/GNSS!)

Deformation results from differences in the gravitational attraction at different spots within the Earth… Complexity added by Earth rotation coupled with change in orientation of Earth rotational axis relative to lunar orbit (±23.5° from equator over a lunar cycle). This leads to dominant periods of 12, 24 and ∞ hours (modulated by other periods of 1/10 days up to 1/18.6 years!) Total gravitational attraction Difference from mean gravitational attraction

Interesting aside… ~5° of lunar orbit inclination and last ~1% of Earth mass from early bombard- ment by planetesimals … Cuk et al, Nature 2016!

Isostasy in a viscoelastic fluid (“Rebound”) Given some initial sinusoidal deflection of the Earth’s surface: where k is wavenumber = 2/ what is the evolution of w through time? Elevated beaches or “strandlines” in Hudson’s Bay region of Canada…

We impose: (1) Conservation of fluid: (2D) v x + x x z Where: u and v are flow in x, z directions... Note this implies the fluid is incompressible u z + z

We impose: (2) Elemental force balance: includes pressure forces, viscous forces, and body force (= gravity). For pressure forces, the imbalance is given by the pressure gradients: (2D) p(z)x x + x x z p(x+dx)z p(x)z z + z p(z+z)x

We impose: (2) Cont’d: viscous forces: The net viscous forces are given by (to first order). For a Newtonian fluid with viscosity , zz(z)x zx(z)x x + x x z xz(x)z xx(x+x)z xx(x)z xz(x+x)z z + z zx(z+z)x zz(z+z)x

The body force is simply xz in the z direction. Typically we simplify pressure by subtracting a hydrostatic: P = p – gz Then (after some algebra; see T&S) we have the force balance equations:

We can define a stream function (= potential of the flow field) such that (for an incompressible fluid): is flow velocity in the x-direction; is flow velocity in the z-direction (vertical). Solutions  must satisfy: Substituting for u and v, we find the biharmonic equation has (eigenfunction) solutions of the form: and

Solving (with boundary conditions) we find that: where the decay constant Note: This solution is specific to the case of a viscoelastic half- space! To zero-th order, would be better approximated by an elastic layer over a viscoelastic halfspace, in which case one would “filter” the viscoelastic response (multiply in the spectral domain) by the amplitude response for the elastic layer.

Calais et al. examined GPS velocities in the “stable” NoAm plate interior with main result: Patterns are broadly consistent with GIA. Calais et al., J. Geophys. Res. 111 2006

Looking closer though there are some interesting patterns