A discussion of different approaches and solutions of the basic tensor equation within the Dynamic Smagorinsky Model (DSM) suitable for General Circulation Models (GCM) is presented. Particular interest is dedicated to the relationship between various approaches (i.e., the specific formulation of the tensor equation), namely a least-square approach, a time lag approach, and a simple tensor contraction approach, and the impact of the specific solution (i.e., how to solve the equation) on the Smagorinsky parameter cS2$c_S^2$. In addition to the standard solutions, clipped solutions, absolute solutions, and tensor norm solutions are examined.The numerical results are based on calculations from a general circulation model, where the different approaches are applied to provide the turbulent horizontal momentum diffusion. Here, they are examined with focus on two issues: 1) At the beginning of the simulations, the different choices for the tensor equation result in different values for the locally distributed and zonally averaged values of the Smagorinsky parameter. These values show that for the standard solutions almost half of the values of cS2$c_S^2$ are negative, in accordance with known results from isotropic turbulence and leads to unstable simulations. In addition, the tensor norm is related to the absolute solution via the Cauchy-Schwarz inequality. 2) As the simulations proceed, the differences of the Smagorinsky parameter values diminish except for the tensor norm solutions while evolving to a stationary state in a process of self-organization such that they form a group with values comparable to isotropic three-dimensional simulations. In summary, the least-squares and time lag approaches provide reasonable results, while the simple contraction approach fluctuates more. For the solutions, it is discussed whether the clipped or the tensor norm solution is more reasonable.
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