[Astrokiwi]

I just want to check I'm interpreting the notation correctly here: http://www.cfd-online.com/Wiki/Smagorinsky-Lilly_model

Firstly, the S_ij S_ij term - I'm interpreting this as Einstein notation, and so the repeated indices means it's a sum. So basically I square every cell of the tensor and add them all up.

Also, I notice that wikipedia uses a different notation - the bars are in opposite places ( http://en.wikipedia.org/wiki/Large_e....93Lilly_model ). I assume which way around you have the bars is arbitrary and it doesn't matter? i.e. these bars don't represent a complex conjugate or a mean or anything like that.

The context is that I'm trying to implement the turbulent metallicity diffusion algorithm described in section 2.2 of this paper: http://arxiv.org/pdf/0910.5956v2.pdf , and equation 6 mentions |S_ij| without defining what that operation is meant to mean, and I have to assume that this is the same |S_ij|=sqrt(2*S_ij*S_ij) because they refer to the Smagorinsky paper.

[Ahmed]

Originally Posted by Astrokiwi

I just want to check I'm interpreting the notation correctly here: http://www.cfd-online.com/Wiki/Smagorinsky-Lilly_model

Firstly, the S_ij S_ij term - I'm interpreting this as Einstein notation, and so the repeated indices means it's a sum. So basically I square every cell of the tensor and add them all up.

Also, I notice that wikipedia uses a different notation - the bars are in opposite places ( http://en.wikipedia.org/wiki/Large_e....93Lilly_model ). I assume which way around you have the bars is arbitrary and it doesn't matter? i.e. these bars don't represent a complex conjugate or a mean or anything like that.

The context is that I'm trying to implement the turbulent metallicity diffusion algorithm described in section 2.2 of this paper: http://arxiv.org/pdf/0910.5956v2.pdf , and equation 6 mentions |S_ij| without defining what that operation is meant to mean, and I have to assume that this is the same |S_ij|=sqrt(2*S_ij*S_ij) because they refer to the Smagorinsky paper.

D = C |Sij | h2, (6)
where A is any scalar, we use the trace-free shear tensor
for Sij and h is the measurement scale (here  hSPH).

The definition is given the next line to equation 6

[Astrokiwi]

Originally Posted by Ahmed

The definition is given the next line to equation 6

That's the definition for S_ij, not the definition for |S_ij|. Equation 7 has D=C|S_ij|h^2 but doesn't define the || operators. Normally these indicate some sort of absolute value or norm, but here this is clearly meant to represent some operation that transforms the tensor into a single scalar quantity.

I'm *guessing* that this is |S_ij|=sqrt(2*S_ij*S_ij) where S_ij*S_ij represents a sum over all i,j, but it doesn't specifically say what it's referring to in the paper so I'm trying to confirm if this is indeed the standard way you'd interpret those symbols.

[Ahmed]

Originally Posted by Astrokiwi

That's the definition for S_ij, not the definition for |S_ij|. Equation 7 has D=C|S_ij|h^2 but doesn't define the || operators. Normally these indicate some sort of absolute value or norm, but here this is clearly meant to represent some operation that transforms the tensor into a single scalar quantity.

I'm *guessing* that this is |S_ij|=sqrt(2*S_ij*S_ij) where S_ij*S_ij represents a sum over all i,j, but it doesn't specifically say what it's referring to in the paper so I'm trying to confirm if this is indeed the standard way you'd interpret those symbols.

In linear algebra, the two vertical bars containing a tensor, means the magnitude of that tensor, remember the dot and cross products of vectors (tensors of rank 1)

If you have doubts, you can email the authors of the paper.