Similarities between grand circulation solvers and mantle convection solvers

My impression is that both ocean grand circulation models (e.g. MITgcm), and Mantle Convection models (e.g. CitcomS), both use Navier-Stoke's as the governing equation. What are the other major similarities between these two types of models?

Should there be more shared between these two modeling communities since they both rely on fluid dynamics?

Edit: For those wondering why this might be important: You can use mantle convection to calculate dynamic topography, and then couple it with heatflow models to predict sea level rise. Of course, you also need to couple that with glacial melting dynamics and GCMs. See Muller et al., 2008 for more information. I know there are a few geodynamicsts now working on this problem in general, but no substantial work has been published . (May 2015)

• In my u/g days, the mantle convection guys used a lot more physical lab models using fluids with scalable properties (Tate&Lyle Golden Syrup seemed popular!). Real world observations in those days were virtually nil, and even today it must still be an observation-poor environment. – winwaed Apr 15 '14 at 22:03
• This is quite a cool question, but I doubt there are many similarities, because you're dealing with things at such different scales (vertical scales, resolution, and very different viscosities), the approximations needed for the gridded models would probably be quite different. – naught101 Apr 16 '14 at 2:37
• The problem with getting this answered is that it probably needs somebody who is an expert in ocean modelling and mantle convection modelling... which is not a criticism of the question - I am now curious too! – Semidiurnal Simon Apr 16 '14 at 16:16
• I would be surprised if these two systems (Mantle and Oceans) were truly dimensionally similar. That is you should show that the appropriate dimensionless numbers in these two cases are of similar magnitude. Are they? – Mark Rovetta Apr 22 '14 at 17:38
• The issue here is the dominant process and the scales are different. The heat dynamics are fundamental in Mantle Convection and the Nusselt Number (relation between conductive and convective heat) is critical. In ocean dynamics, the Rossby (rotation versus inertia) and Ekman (friction versus rotation) numbers are much more important and they are usually taken as close to zero in mantle convection. – arkaia Apr 29 '14 at 15:52

Disclaimer: This is a partial answer given that my background is modeling of the ocean. I hope that some mantle convection modelers can complement this answer.

The question is good but the answer is complex. The short answer is:

No, they are not the same. Simply because computationally it would not make sense.

I will try my best to break it apart and make it as concise as possible.

Preface

As many people have pointed out scales are key. The environmental fluid dynamics problems we try to solve range enormous scales. However, every single motion is described by the Navier-Stokes (NS) equation, from the simplest flow you can think of all the way to the most complex - that includes turbulence (the continuum hypothesis says that the NS equations are valid when the Knudsen number $K_n \ll 1$).

Take a look at the chart below for oceanic processes alone. Temporal scales span 10 orders of magnitude, while spatial scales span 12 orders of magnitude. Presumably mantle solvers would extend the upper bounds on each of these scales.

The question you ask is specifically regarding oceanic grand circulation (OGC) models and mantle convection (MC) models. So according to the chart below, out of all oceanic models, the OGC and the MC models are the closest as far as temporal scales and spatial scales go.

The complexity of the Navier-Stokes equations and the difficulty in solving them

The Navier-Stokes system can be classified as a hybrid elliptic-hyperbolic type for steady flows and a hybrid parabolic-hyperbolic type for unsteady flows (the hyperbolic character comes from the continuity equation).

Where the nature of the equations say the following about each one and their respective numerical difficulties:

Hyperbolic nature is associated with wave phenomena and advective transport:

• Fast waves lead to numerical stability restrictions

• Nonlinear part of NS is hyperbolic ($\mathbf{u} \cdot \nabla \mathbf{u}$), which is the part of the equation that leads to turbulence.

• Possibly one of the most difficult aspect of CFD is the propagation of sharp density fronts which are hyperbolic.

Parabolic nature is associated with diffusion and mass transport:

• Boundary layers are governed by parabolic phenomena and are very thin compared to the environment that drives it. Notice the large scale disparity and associated numerical difficulty.

• Turbulence, can be modeled from a parabolic perspective and this typically results in stability concerns on the numerical method employed.

Elliptic nature implies instantaneous propagation of information:

• For environmental fluid dynamics, the nonhydrostatic pressure is of elliptic nature.

• Although, theoretically, any disturbance propagates at infinite speed throughout the domain, numerical iteration sets a finite speed at which information can propagate.

• Non-hydrostatic solvers have to invert a Poisson equation which is very computationally expen sive. In general, for the nonhydrostatic pressure, the 2-d problem requires the solution of a pentadiagonal, while the 3-d problem requires the solution of a septadiagonal (7 diagonals) (not all near the main diagonal!).

Scales and numerical solvers

So as one can see by now, solving the NS equations numerically is not a trivial matter. Numerical solvers must face concerns regarding accuracy, stability and consistency, which pose constraints on the timestep and grid resolution that one can employ. See this answer regarding different approaches to numerical solvers. Scales are important for numerical solvers because of the nature of the system of the NS equations (described above) and the analytical mathematical techniques available to us to transcribe those equations to computational mathematical language. As it stands, it is impossible to resolve all temporal and spatial scales, so modelers resort to specific techniques (solvers) that apply to the problem (the scales) that they are interested in.

Conclusion

From their website:

The MITgcm (MIT General Circulation Model) is a numerical model designed for study of the atmosphere, ocean, and climate. Its non-hydrostatic formulation enables it to simulate fluid phenomena over a wide range of scales; its adjoint capability enables it to be applied to parameter and state estimation problems. By employing fluid isomorphisms, one hydrodynamical kernel can be used to simulate flow in both the atmosphere and ocean.

and

CitcomS is a finite element code designed to solve compressible thermochemical convection problems relevant to Earth's mantle.

My guess is they both use different numerical techniques to solve different versions of the Navier-Stokes equations that make the most sense given the scales of the problem that each one aims to resolve.

The only similarity that they are fluids and therefore NS applies. Actually, to be fair mantle is solid as it allows shear waves to propagate through. However, on geologic time scales it behaves like a viscous fluid and can be modeled as such.

Circulation models solve compressible (non-hydrostatic) Euler equations where as mantle convection is governed by incompressible Stokes flow. The types of core numerical solvers and numerical schemes used by the two communities are very different (e.g., explicit for circulation models and implicit for convection models). The only common aspect is that both use a spherical geometry/meshes to solve the equations. Circulation models also have to account for topography but most mantle convection models neglect it.

Having said that both require some basic background in CFD so in that sense they are similar. Some of the earliest mantle convection models were written by aerospace engineers.