Tides of global ice-covered oceans

doi:10.1016/j.icarus.2016.03.026

Icarus

Volume 274, August 2016, Pages 122-130

https://doi.org/10.1016/j.icarus.2016.03.026 Get rights and content

Highlights

•
Tides of an ice covered ocean can be larger than modern ones.
•
Tidal forcing could be competitive in driving a mean ocean circulation during the Neoproterozoic.
•
Some speculations on applicability to our Solar System satellites and exoplanets.

Abstract

The tides of an ice-covered ocean are examined using a Cartesian representation of the elastic and fluid equations. Although unconstrained by any observations, the ocean tides of a Neoproterozoic “snowball” Earth could have been significantly larger than they are today. Time-mean tidal-residual circulations would then have been set up that are competitive with the circulation driven by geothermal heating. In any realistic configuration, the snowball Earth would have had an ice cover that is in the thin shell limit, but by permitting the ice thickness to become large, more interesting ice tidal response can be found, ones conceivably of application to bodies in the outer Solar System or hypothetical exoplanets. Little can be said concerning a reduction in tidal dissipation necessary to avoid a crisis in the history of the lunar orbit.

Introduction

Several reasons exist for an exploration of the tides occurring in and under ice sheets, whether floating or land-confined. One motive arises from evidence that approximately 600 million years ago, during the Neoproterozoic, the entire Earth may have frozen, being everywhere covered with ice. Over the ocean, a floating ice sheet may have existed with an estimated thickness of several kilometers (the “hard snowball Earth”). Discussion of the evidence, primarily geological in nature, can be found in Hoffman and Schrag (2002). Ashkenazy et al. (2014, hereafter, A14), describe a theoretical/modeling study of the oceanic circulation that might exist under an oceanic ice cover of order of several kilometers. The forcing they assume is purely geothermal, at the average modern rate of roughly 0.1 W/m² (Davies, 2013, Pollack, Hurtrer, Johnson, 1993), with some localized maxima over ridge-crests. They find an equatorially enhanced meridional overturning circulation, with transports up to 30 × 10⁶m³/s (30 Sverdrups; Sv) with a nearly homogeneous ocean, both in temperature and salinity. Some account is taken of the oceanic interaction with the overlying ice sheet. Jansen (2016) has in turn suggested that the resulting flow would be a turbulent one.

Whether or not a complete snowball Earth actually existed, the question of what the ocean might be like under such circumstances is an interesting theoretical problem. A modern analogue may lie in the outer Solar System satellites Enceladus and Europa, which have been inferred to contain fluid oceans covered by multi-kilometer thick ice sheets. In contrast to the A14 solution, discussion of behavior of those oceans has centered on tidal forcing (e.g., Beuthe, 2015, Greenberg, et al., 1998, Tyler, 2008, Vance, Goodman, 2009).

Another motivation arises from the known difficulties in accounting for the history of the lunar orbit. The existing rate of tidal dissipation, if constant through time, would have brought the Moon catastrophically close to the Earth about 1 billion years ago (e.g., Goldreich, 1966, Macdonald, 1964, Munk, 1968). Munk called the catastrophe the “Gerstenkorn event,” and which is known not to have occurred. The conventional interpretation is that lunar tidal dissipation must have been greatly reduced some hundreds of millions of years in the past (see Bills and Ray, 1999 for discussion). Should tidal dissipation have been much reduced during the approximately 200MY of the Neoproterozoic, it would be a significant contribution to explaining how the reduction occurred.¹

A comparatively large literature exists on tides induced in ice sheets by the oceanic tidal forcing at the outflow (e.g., Arbic, Mitrovica, MacAyeal, et al., 2008, Reeh, Christensen, Mayer, et al., 2003, Thomas, 2007). These effects are of at least tangential interest here, but where the focus is instead on the directly driven tidal motions within the ice. Some of the parameter ranges used here are far beyond anything reasonable for the Earth. Perhaps they have some relevance for another planet or satellite.

Section snippets

A Cartesian configuration

Because of all of the uncertainties of the physical setting of the Neoproterozoic Earth, the restricted goals here are to understand the basic physics and to find orders of magnitude of the effects. Only a two-dimensional Cartesian system, as in the Airy “canal theory” of water tides (Lamb, 1932), is used. Consider the situation in Fig. 1, in which an ice sheet of uniform thickness $\bar{d}$ overlies an ocean of constant depth d; on the Earth, $d, \bar{d}$ would have an inverse relationship over time. Below

Ice alone

Consider an elastic ice sheet subject to tidal forcing in which both upper and lower boundaries are free. Absent any $y -$ dependence—as is being assumed here—the non-dimensional displacements in the ice can be written generally as, $\bar{u} = \frac{\partial \bar{φ}}{\partial x} + \frac{\partial \bar{ψ}}{\partial z}, \bar{w} = \frac{\partial \bar{φ}}{\partial z} - \frac{\partial \bar{ψ}}{\partial x}$ that is as the gradient of a potential and the curl of a stream function and whose solutions are coupled through the boundary conditions. By Eq. (7c), $\nabla^{2} \bar{φ} = 0 .$ Eq. (10) is the seismological $P -$ (compressional, acoustic) wave equation in the

Ocean alone

With tidal potentials having no curl, the water velocity can be written $u = \frac{\partial ϕ}{\partial x}, w = \frac{\partial ϕ}{\partial z},$ and the boundary condition at $z = - d$ is satisfied by taking $ϕ = F \cosh (k (z + d)) .$ The linearized non-dimensional Bernoulli equation is $p = - ϕ + \frac{1}{σ^{2}} \frac{\partial ϕ}{\partial z} - \frac{1}{i σ}$ where the last term arises from the tidal potential. Evaluating Eq. (16) at $z = 0,$ produces $F {i σ \cosh (k d) + \frac{k}{i σ} \sinh (k d)} = 1$ and which would be resonant if, $σ^{2} = k \tanh (k d),$ the conventional non-dimensional dispersion relationship for water waves. The surface elevation is then $η = \frac{F k}{i}$

Coupled ice and ocean

Using Eq. (15) to satisfy the boundary condition at $z = - d,$ the non-dimensional system of boundary conditions can be written, $\begin{matrix} \underset{cont. vert. displ., z = 0}{- i k A - i k B + k C + k D + \frac{k}{i σ} \sinh (k d) E = 0,} \end{matrix}$ $\begin{matrix} \underset{}{(β_{1}^{2} k m - i k) A + (- β_{1}^{2} k m - i k) B + (2 β_{1}^{2} k^{2} + k) C + (2 β_{1}^{2} k^{2} - k) D +} \\ \underset{cont. normal stress, z = 0}{(i σ \cosh k d + \frac{k}{i σ} \sinh k d) E = 0,} \end{matrix}$ $\begin{matrix} \underset{no shear stress, z = 0}{(- m^{2} + k^{2}) A + (- m^{2} + k^{2}) B + 2 i k^{2} C - 2 i k^{2} D + 0 E = 0}, \end{matrix}$ $\begin{matrix} \underset{no shear stress, z = \bar{d}}{(- m^{2} + k^{2}) A e^{i m \bar{d}} + (- m^{2} + k^{2}) B e^{- i m \bar{d}} + 2 i k^{2} C e^{k \bar{d}} - 2 i k^{2} D e^{- k \bar{d}} + 0 E = 0}, \end{matrix}$ $\begin{matrix} \underset{}{(2 β_{1}^{2} k m - i k) e^{i m \bar{d}} A + (- 2 β_{1}^{2} k m - i k) e^{- i m \bar{d}} B + (2 β_{1}^{2} + k) e^{k \bar{d}} C +} \\ \underset{no normal stress, z = \bar{d}}{(2 β_{1}^{2} - k) e^{- k \bar{d}} D + 0 E = 1} \end{matrix}$

with the

Snowball Earth–ocean

The conclusions from the previous section support the idea that the presence of a global ocean ice cover of thickness of several kilometers does not lead to a significant reduction in tidal amplitudes relative to today. The reduction in water depth, and the likely parallel reduction of tidal motions in highly dissipative shallow water (continental margins) in practice, suggest an increase in tidal amplitudes. For the Earth, the straightforward inference is that ordinary tides of the

Other processes

None of the present results as applied to the snowball Earth are definitive, and many unknowns and complications intervene. Among other intriguing complications not discussed here are the role of the changed Earth rotation rate and length of the month at times approaching $-$ 1 GY when the day was probably about 22 h long, and with about 13 synodic months in the year (Bills, Ray, 1999, Williams, 2000). These changes are consequences of tidal friction and the resulting braking of Earth-spin over

Summary comments

From “canal-theory”-like calculations, the ordinary semi-diurnal and diurnal tides of a snowball-Earth are found likely to be at least as strong as those in the modern ocean. As with modern tides, details depend sensitively on bottom topography, continental configurations, dissipation mechanisms, and overall water depths. Consequently, tidal motions, and in particular rectified mean values, are not obviously negligible in the discussion of the oceanic general circulation at that time.

Acknowledgments

J. Rice and E. Tziperman provided useful comments on an early version of the manuscript. Supported in part by the National Science Foundation under grant OCE0961713 at MIT.

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View full text

Tides of global ice-covered oceans

Highlights

Abstract

Introduction

Section snippets

A Cartesian configuration

Ice alone

Ocean alone

Coupled ice and ocean

Snowball Earth–ocean

Other processes

Summary comments

Acknowledgments

Icarus

Icarus

Global Planet. Change

On the factors behind large Labrador sea tides during the last glacial cycle and the potential implications for Heinrich events

Paleoceanography

Ocean circulation under globally glaciated snowball Earth conditions: Steady-state solutions

J. Phys. Oceanogr.

Lunar orbital evolution: A synthesis of recent results

Geophys. Res. Letts.

Topographic rectification in a stratified ocean

J. Mar. Res.

On the influence of gravity on elastic waves, and, in particular on the vibrations of an elastic globe

Proc. London Math. Soc.

Ekman veering, internal waves, and turbulence observed under arctic sea ice

J. Phys. Oceanogr.

Global map of solid Earth surface heat flow

Geochem. Geophys. Geosyst.

Significant dissipation of tidal energy in the deep ocean inferred from satellite altimeter data

Nature

Estimates of M_2 tidal energy dissipation from TOPEX/Poseidon altimeter data

J. Geophys. Res. Oceans

Elastic Waves in Layered Media

History of the lunar orbit

Rev. Geophys.

Wave motion in hydrodynamics

Am. J. Maths.

Forcing of oceanic mean flows by dissipating internal tides

J. Fluid Mech.

Earth Tides

Long waves and ocean tides

The snowball Earth hypothesis: Testing the limits of global change

Terra Nova.

A suggested explanation of the present value of the velocity of rotation of the Earth

Month. Not. Roy. Astron. Soc.

On mass transports generated by tides and long waves

J. Fluid Mech.

Neoproterozoic ‘snowball Earth’ simulations with a coupled climate/ice-sheet model

Nature

The turbulent circulation of a snowball Earth ocean

J. Phys. Oc.

Tidal flow over three-dimensional topography in a stratified fluid

Phys. Fluids

Hydrodynamics