[MITgcm-support] GMREDI without the small slope approximation

[Paola Cessi]

Hi Flynn,

the reason why the off-diagonal k12 and k21 are set to 0 is because the antisymmetric component of that tensor is associated with the “bolus-velocity”, which needs to be incompressible. This point is lucidly explain in Vallis textbook (2017 edition) in chapter 13.6 (pages 490-493). In Vallis’ example the component set to zero is k31 and k13, but this is irrelevant: you can set any of the three antisymmetric components to zero, and guarantee that the resulting “bolus-velocity” is incompressible in 3-D.

In the implementation of Kgm in the MITgcm that I am familiar with, the antisymmetric component is explicitly determined in terms of a streamfuction associated with the bolus velocity, making sure that it also satisfies the boundary conditions on solid boundaries and on the free surface. You can read about it here: https://www.sciencedirect.com/science/article/pii/S1463500310000065 <https://www.sciencedirect.com/science/article/pii/S1463500310000065>

In summary, I recommend that you implement your large slope scenario on the existing components of knm, without adding a third antisymmetric part which would mess up incompressibility of the bolus velocity. The diffusive part of the tensor is in the diagonal components.

Cheers,
Paola

> On Apr 26, 2023, at 5:21 AM, Flynn Ames <f.ames at pgr.reading.ac.uk> wrote:
> 
> Hi Martin,
> 
> Thanks for getting back to me on this. This is helpful!
> 
> Flynn
> From: MITgcm-support <mitgcm-support-bounces at mitgcm.org <mailto:mitgcm-support-bounces at mitgcm.org>> on behalf of Martin Losch <Martin.Losch at awi.de <mailto:Martin.Losch at awi.de>>
> Sent: 25 April 2023 9:44 AM
> To: MITgcm Support <mitgcm-support at mitgcm.org <mailto:mitgcm-support at mitgcm.org>>
> Subject: Re: [MITgcm-support] GMREDI without the small slope approximation
>  
> Hi Flynn, 
> 
> I am not really qualified to answer this, so take my comments with a grain of salt.
> 
> As far as I can see, the missing tensor components  (basically k12=Kuy and k21=Kvx) need to be defined and computed and you are right that these are small and straightforward changes to the code (gmredi_calc_tensor.F). But you will also need to code the action of these tensor components on the \nabla(u) in the right place (k12*dv/dy in xtranport and k21*du/dy in ytransport?, involves some averaging of the derivatives). I find this hard to figure out, because there are so many different options in pkg/gmredi, but it should be possible.
> 
> Martin
> 
>> On 21. Apr 2023, at 18:38, Flynn Ames <f.ames at pgr.reading.ac.uk <mailto:f.ames at pgr.reading.ac.uk>> wrote:
>> 
>> Dear MITgcm community,
>> 
>> Hello - I'm wondering if anyone has ever tried implementing (or has implemented) GMREDI into the MITgcm without making the small slope approximation?
>> 
>> This is something I would like to do as I'm modelling a deep, ice-covered ocean where the steep isopycnals that develop violate the small slope approximation over much of the domain.
>> 
>> I may be being naive here, but looking at the MITgcm documentation, the terms that are neglected in the REDI projection tensor when making the small slope approximation appear to consist of quantities that are already computed in the model (Sx, Sy etc.).
>> I'm hoping to ask: Does anyone know if it would just be a case of putting these extra terms into the REDI tensor in the model? or is there a lot that I'm missing here? - I'm guessing the latter :)
>> 
>> Thank you for your time!
>> Flynn
>> 
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