# [aspect-devel] geoid

Wolfgang Bangerth bangerth at tamu.edu
Mon Apr 20 12:41:58 PDT 2015

```Scott,

> Your point about the gravity vector not being vertical but relative to the
> geoid is an interesting one.  I think we have always considered that second
> order, but it would be interesting to know.  Actually most of my career I
> thought of the earth as a 2D box!!!

At least you considered depth -- much of humanity thought of the earth as a 2d
sheet for a long time.

I've similarly never put things together. The perceived gravity vector in a
rotation coordinate frame would be

g = hat g - omega x omega x r

if I recall correctly, where hat g is due to gravity and omega is the rotation
vector. On earth, omega = 2*pi/(3600*24 s) = 7e-5, so

| omega x omega x r |  <= 0.034 m/s^2

or 0.34% of perceived gravity at the equator. I have no idea whether that's a
relevant quantity to include (that's for you geodynamicists to determine :-),
but it's responsible for a "dynamic" topography of 20km.

> So this is all ancient history and somewhat esoteric geodynamics, but it is
> part of why I'm interested not only in the dynamic topography but also the
> geoid anomalies as output.  It seems like Ian and Rene are working on this.
> I'm glad to know that.  I will make sure Shangxin hooks up with them.   He
> has been looking at dynamic topography as a function of element type.
> I've always wondered, since the usually computation of dynamic topography
> depends on stress and stress is less accurate than velocity, just how does
> the low order that we use affect the result.   As you can see from the
> plot, since dynamic topography and internal mass anomalies are of opposite
> sign, a smallish error in one can lead to a big error in the total geoid
> anomaly.

I think it's an interesting question. When Jacky and I implemented the dynamic
topography postprocessor, we noticed that it is definitely subject to a lot of
numerical noise. Consequently, we use an averaged velocity gradient/strain
rate tensor on cells at the surface, rather than one evaluated on the faces at
the surface. It would be interesting to see Shangxin quantify this.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth               email:            bangerth at math.tamu.edu
www: http://www.math.tamu.edu/~bangerth/

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