# [aspect-devel] compressibility in 2d

cedric thieulot c.thieulot at uu.nl
Fri Feb 10 13:18:31 PST 2017

So we’ve been talking about it with Menno today and it looks indeed that the literature is unanimous on this.
However, if \dot{\bm \epsilon} is a 2x2 tensor/matrix the formula with 1/3 does not make it traceless. I/we suppose that there is a somewhat subtle argument here,
but so far it has eluded us.
Furthermore, if one considers an infinite domain in, say, the z direction , dv_z/dz will be for sure nul, so that \dot{epsilon}_{zz} is nul but if we now look at the deviatoric
strainrate, \dot{epsilon}_{zz}^{dev} will receive the value -1/3 (e_xx+e_yy). What does this mean ?

Ce/

> On 10 Feb 2017, at 22:08, Juliane Dannberg <dannberg at gfz-potsdam.de> wrote:
>
> Hi Menno,
>
> we discussed this today (with Wolfgang and Rene), and one argument that came up was the one that Jonathan just mentioned.
>
> To make sure, we also checked how other people do this (like how the equations are written down for example in Turcotte, Schubert and Olson), and everywhere we looked, the 3 was also written down as a 3 (and not in dependence of the dimension).
>
> As we also have a 2D compressible benchmark case (the Tan Gurnis '07 benchmark), which has an analytical solution, and we converge to that solution with the expected rate, we can also be reasonably sure that we have implemented this in a way that other people think is correct.
>
> But we also had to think about this, and found no other explanation than interpreting a 2D model as a model that actually is 3D, but infinitely long in the third dimension and without any changes in properties in that direction.
>
> Cheers,
> Juliane
>
>
> On 02/10/2017 01:41 PM, Jonathan Perry-Houts wrote:
>> Hey Menno,
>>
>> I'm pretty sure the reason is that 2d models are supposed to represent a
>> slice through an infinite-depth 3d model, so the equations do not
>> change. This bothered me at first too, and that's how I justified it to
>> myself. Hopefully someone else can confirm or deny that justification.
>>
>> -JPH
>>
>> On 02/10/2017 06:04 AM, Menno Fraters wrote:
>>> Hello all,
>>>
>>>
>>> I have been looking at the formulation of the compressibility in aspect
>>> and in the manual there is a 1/3 in front of the div(u)*1* in equation
>>> 1. I would expect that this is only valid in 3d, because I think it
>>> stems from the deviator which is deviator(*epsilon*) = *epsilon* - 1/dim
>>> * trace(*epsilon*)*1*. So I would expect it to be 1/2 in 2d. But I found
>>> that in the code it is hardwired as 2/3 (where the 2 comes from I
>>> understand).
>>>
>>>
>>> Can someone explain me why this is 2/3 and not 2/2 = 1?
>>>
>>>
>>> Many thanks!
>>>
>>>
>>> Cheers,
>>>
>>>
>>> Menno**
>>>
>>>
>>>
>>> _______________________________________________
>>> Aspect-devel mailing list
>>> Aspect-devel at geodynamics.org
>>> http://lists.geodynamics.org/cgi-bin/mailman/listinfo/aspect-devel
>>>
>> _______________________________________________
>> Aspect-devel mailing list
>> Aspect-devel at geodynamics.org
>> http://lists.geodynamics.org/cgi-bin/mailman/listinfo/aspect-devel
>
> _______________________________________________
> Aspect-devel mailing list
> Aspect-devel at geodynamics.org
> http://lists.geodynamics.org/cgi-bin/mailman/listinfo/aspect-devel