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Hi,

I have a question about the possibility to reproduce a vorticity balance from Telemac-2D results. This is a technical question that is related to the way that Telemac solves the shallow water equations.

First, as far as I know Telemac uses linear interpolation of velocity components u and v within an element (I will also use the term ‘cell’ hereafter) such that these components are continuous at cell edges between adjacent elements. However, the derivative across (i.e. perpendicular to) cell edges is not continuous. The story below assumes that this is correct.

Now vorticity W in 2DH is given by
W=v_x-u_y , (1)
where _x and _y are partial derivatives with respect to x and y, respectively. By default u and v vary linearly within an element, i.e. u=a*x + b*y + c (and likewise for v). This means that W is constant within an element. This way vorticity can be computed for, say, visualization purposes.

However, W is not continuous across cell edges: adjacent cells 1 and 2 in general have different vorticity values W1 and W2. This means that the perpendicular component of vorticity flux F=(u,v)W at a cell edge is not continuous: when evaluated from cell 1 and and cell 2, their values differ. In other words: vorticity spontaneously disappears / appears at edges of an element. This is a numerical (i.e. non-physical) effect as it is directly related to the way Telemac solves the shallow water equations.

This has - I think - profound consequences for the vorticity balance. From the equation of motion one can derive an equation for W that takes the form:
W_t + div(F)=P-D , (2)
where t_ the partial time derivative and div denotes the horizontal divergence. Note that the second term involved the vorticity flux I previously mentioned. The production term P is the vorticity production term that basically stems from torques that arise because friction parameter, water depth and velocity magnitude vary perpendicular to the flow direction. The dissipative term D contains contributions that destroy vorticity (bottom friction and viscosity).

One would like to study the different terms in Eq. (2) as function of space (i.e. per cell) and time in order to identify which mechanisms are responsible for the generation of gyres (which are regions of enhanced vorticity). However, when integrating (2) per element the abovementioned discontinuity of F is expected to prevent the formulation of a proper vorticity balance. Most likely, the rhs and lhs of Eq. (2) may not balance.

As a result, I would think that it is not possible to study a correct vorticity balance using Telemac.

My question is: is the reasoning above (and thus the conclusion) correct?

I look forward to your comments.

Kind regards,

George

Hello George,

Technical question that I am not sure to answer perfectly.
Anyway, I try to give a quick answer. Any other comments from you or other people are welcome as it it is not a usual issue that is studied with TELEMAC-2D.
You are right for Finite Elements.

What should be kept in mind is that TELEMAC-2D does not solve SWE at cells but at the vertices of the triangles. So to my point of view, you cannot think by integrating the equations by element (cell or triangle).

Saying that, it should be more thought deeper to conclude.

Chi-Tuan

Hi Chi-Tuan,

Thank you for your reply.

Whilst Telemac computes quantities at vertices, I think it does (by default) assume linear interpolation within a cell, see Ch. 7.1 of the manual. This is also what other users told me privately. As far as I have understood, one needs to make such assumptions in order to formulate the finite element problem. So I think using the presumed spatial variation within a cell is a consistent way to evaluate derivatives.

Alternatively, if it is possible to compute all derivatives (du/dx, du/dy, dv/dx and dv/dy) at vertices directly do point this out to me.

In the end, one should (a) obtain all terms in Eq. (2) that I listed in my question in such a way that (b) these terms balance in the sense that lhs of (2) = rhs of (2), apart from some small residual. My question basically is if (and if so: how) this can be done.

Kind regards,

George