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Abstract Icy satellites host topography at many length scales, from rifts and craters on the small end to equatorial‐pole shell thickness differences that are comparable to these bodies' circumference. The current paradigm is that icy satellites should not host stable small‐scale topography. This idea comes from previous work using a “shallow”‐shell model (i.e., ice shell circumference much larger than shell thickness) with a rigid outer crust. In this limit, large‐scale topography relaxes over a longer time scale than small‐scale features. Here we revisit this paradigm and analyze relaxation of topography starting from the Stokes equations for viscous fluid flow. For a shell with a viscosity that decreases exponentially with depth, we show numerically that there is a regime where the larger viscosity outer crust acts as a nearly rigid boundary. In this case, the relaxation time scale depends on the wavelength. For the largest spatial scales, however, the time scale becomes independent of wavelength and the value is set by the average shell viscosity. However, the spatial scale that this transition occurs at becomes larger as the viscosity contrast increases, limiting the applicability of the scale‐independent relaxation rate. These results for the relaxation of topography have implications for interpreting relaxed crater profiles, inferences of ice shell thickness from topography, and upcoming observations from missions to the outer solar system.
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Finite boundary effects on the spherical Rayleigh–Taylor instability between viscous fluids
For the Rayleigh–Taylor unstable arrangement of a viscous fluid sphere embedded in a finite viscous fluid spherical shell with a rigid boundary and a radially directed acceleration, a dispersion relation is developed from a linear stability analysis using the method of normal modes. aR1 is the radially directed acceleration at the interface. ρi denotes the density, μi is the viscosity, and Ri is the radius, where i = 1 is the inner sphere and i = 2 is the outer sphere. The dispersion relation is a function of the following dimensionless variables: viscosity ratio s=μ1μ2, density ratio d=ρ1ρ2, spherical harmonic mode n, B=R1aR1ρ22μ221/3, H=R2R1, and the dimensionless growth rate α=σμ2aR12ρ21/3, where σ is the exponential growth rate. We show that the boundedness provided by the outer spherical shell has a strong influence on the instability behavior, which is reflected not only in the modulation of the growth rate but also in the selection of the most unstable modes that are physically possible. This outer boundary effect is quantified by the relative magnitude of the radius ratio H. We find that when H is close to unity, lower order harmonics are excluded from becoming the most unstable within a vast region of the parameter space. In other words, the effect of H has precedence over the other controlling parameters d, B, and a wide range of s in establishing what the lowest most unstable mode can be. When H ∼ 1, low order harmonics can become the most unstable only for s ≫ 1. However, in the limit when s → ∞, we show that the most unstable mode is n = 1 and derive the dispersion relation in this limit. The exclusion of most unstable low order harmonics caused by a finite outer boundary is not realized when the outer boundary extends beyond a certain threshold length-scale in which case all modes are equally possible depending on the value of B.
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- Award ID(s):
- 1748883
- PAR ID:
- 10597334
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- AIP Advances
- Volume:
- 12
- Issue:
- 4
- ISSN:
- 2158-3226
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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