CSIR NET Dec 2015 Q133: Physical Oceanography

The 'closure problem' arises in the study of turbulent flows because the governing equations of fluid motion (the Navier-Stokes equations) contain a non-linear advection term (u.grad(u)). When these equations are averaged (e.g., via Reynolds decomposition into mean and fluctuating components) to make the turbulence problem tractable, the non-linearity of the advection term generates new unknown terms (such as Reynolds stresses, representing correlations of velocity fluctuations) in the averaged equations. Each time an attempt is made to write an equation for these new unknown correlation terms, the non-linearity of the underlying equations generates yet higher-order unknown correlations, so the system of averaged equations can never be closed (i.e., there are always more unknowns than equations) without introducing additional empirical or modeled closure assumptions. This fundamental issue stems directly from the non-linearity of the governing equations of motion, matching option 2.

Option 1 (fluid motion is described in a Lagrangian framework) is not the source of the closure problem; turbulence is typically analyzed using the Eulerian framework (fixed spatial points), and the closure issue arises regardless of the descriptive framework chosen, stemming instead from the equations' non-linearity. Option 2 is correct: it is specifically the non-linear (advective) term in the governing equations that, upon averaging, generates new unknown higher-order correlation terms faster than new equations can be derived for them, creating the unclosed system that defines the closure problem. Option 3 (governing equations are mathematical statements of conservation of mass, momentum, and energy) is a true general description of the Navier-Stokes/continuity/energy equations but does not by itself explain why averaging them produces an unclosed system; the specific culprit is the equations' non-linearity, not their conservation-law character per se. Option 4 (a system of coupled partial differential equations) is also a true general description of the governing equations but, like option 3, does not identify the specific mathematical reason (non-linearity) that generates the closure problem upon Reynolds averaging.