## 2 Subgrid Stress Tensor Equation Closure and Calibration
### Calibration Method  
In each of the closures in (4), (5), (7), (8) and (9), the empirical coefficients enter linearly. The naive method of their calibration is to obtain coefficient values from an overdetermined linear equation system by the least squares method, with each equation corresponding to a single data point at a single time instance and a single tensor component. Suppose we have a one-dimensional data vector, \(\mathbf{b}\), formed in this way, and that its model is \(\mathbf{\tilde{b}}=C_{1}\mathbf{a}_{1}+\ldots+C_{n}\mathbf{a}_{n}\). Here, \(\mathbf{a}_{i}\) and \(\mathbf{b}\) are exact filtered DNS data. Then, the least squares solution of overdetermined linear system is as follows:  
\[C_{1}\mathbf{a}_{1}+\ldots+C_{n}\mathbf{a}_{n}=\mathbf{b}\]  
which, with respect to \(C_{1},\ldots,C_{n}\), gives the coefficients which minimize the Euclidean distance \(\parallel\mathbf{\tilde{b}}-\mathbf{b}\parallel_{2}\), where  
\[\parallel\mathbf{x}\parallel_{p}=\left(\sum_{i}\lvert x_{i}\rvert^{p}\right) ^{1/p}.\]  
The corresponding model \(\mathbf{\tilde{b}}\) can be interpreted as orthogonal projection of \(\mathbf{b}\) onto the linear span of \(\{\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\}\). However, a drawback follows: if the \"basis vectors\" \(\mathbf{a}_{1},\ldots,\mathbf{a}_{n}\) are inadequate to represent the data vector \(\mathbf{b}\), the model vector \(\mathbf{\tilde{b}}\) will have significantly smaller norm than that of \(\mathbf{b}\). In our preliminary tests, for example, we were systematically obtaining the models for \(\mathbf{\Phi}\) in the form of (7), such that \(\parallel\mathbf{\tilde{\Phi}}\parallel_{2}/\parallel\mathbf{\Phi}\parallel_{ 2}\approx 0.4\ldots 0.5\). Having faced this problem, we developed a calibration strategy, which is--to our knowledge--a new approach. Let us introduce a correlation coefficient between \(\mathbf{\tilde{b}}\) and \(\mathbf{b}\):  
\[\text{Corr}\left(\mathbf{\tilde{b}},\ \mathbf{b}\right)=\frac{\left( \widetilde{b}_{a}-(\widetilde{b}_{a})\right)\left(b_{a}-(b_{a})\right)}{ \parallel\mathbf{\tilde{b}}-(\mathbf{\tilde{b}})\parallel_{2}\parallel \mathbf{b}-(\mathbf{b})\parallel_{2}}\]  
where angle brackets denote averaging over whole filtered dataset. Then, we can perform an iteration process. Denoting the coefficient approximation at the \(m^{\text{th}}\) iteration by \(C_{1}^{m},\ldots,C_{n}^{m}\), at the first step, we update their values by moving along the correlation coefficient gradient, as follows:  
\[\widetilde{C}_{i}^{m+1}=C_{i}^{m}+K\frac{\partial\text{Corr}\left(\mathbf{ \tilde{b}},\ \mathbf{b}\right)}{\partial C_{i}}, \tag{11}\]  
choosing the constant \(K\) to empirically maximize the convergence rate by maintaining the stability of the process (typically, \(K\) ranges from 0.5 to 10). In (11), we calculate the correlation coefficient derivative on the RHS by the following central difference formula:  
\[\frac{\partial\text{Corr}\left(\mathbf{\tilde{b}},\ \mathbf{b}\right)}{ \partial C_{i}}\approx\frac{\text{Corr}(\ldots+(C_{i}+\Delta C)\mathbf{a}_{i} +\ldots,\ \mathbf{b})-\text{Corr}(\ldots+(C_{i}-\Delta C)\mathbf{a}_{i}+\ldots,\ \mathbf{b})}{2\Delta C}\]  
with coefficient increment, \(\Delta C\), equal to \(10^{-4}\). At the second step, we multiply the new coefficients by the norm ratio, \(C_{i}^{m+1}=N\widetilde{C}_{i}^{m+1}\), where \(N=\parallel\mathbf{b}\parallel_{1}/\parallel\widetilde{C}_{1}^{m+1}\mathbf{a}_ {1}+\ldots+\widetilde{C}_{n}^{m+1}\mathbf{a}_{n}\parallel_{1}\), to ensure the correct norm of the model. Here, we prefer the \(L_{1}\) norm to avoid giving large weights to large data components. This procedure is repeated until the coefficients converge to a stationary state. The idea of the method is visualized in Figure 2. A correlation coefficient contour is depicted as a function of two model coefficients, \(C_{1}\) and \(C_{2}\). The white solid line corresponds to pairs \(C=\{C_{1},C_{2}\}\), which yield a correct model norm, \(\parallel\widetilde{\mathbf{b}}\parallel_{1}=\parallel\mathbf{b}\parallel_{1}\).