![]() ![]() However, it can lead us to identify physically meaningful scales because it is applied to equations that govern a given system physically: see e.g., discussions concerning Eq. The procedure is totally formal and abstract. In a certain sense, nondimensionalization is a brute-force approach without relying on any physical intuition, even observation or modelling. Yano and Tsujimura ( 1987), and Yano and Bonazzola ( 2009) systematically apply this methodology for their scale analysis. The depth of the Ekman layer is another such example (Sect. 4.3, Pedlosky 1987). This scale characterizes the quasi-geostrophic system (Sect. 3.12, Pedlosky 1987). The Rossby radius of deformation is a classical example of a characteristic scale identified by nondimensionalization. An advantage of the nondimensionalization procedure is that these scales are defined not only by dimensional consistencies, but also by requirements of balance by order of magnitude between the terms in a system. These conditions, in turn, constrain these characteristic scales in a natural manner. These scales cannot be arbitrary, because we expect that terms in an equation to balance each other, thus their orders of magnitudes must match. No doubt, this procedure is more straightforward and formal: characteristic scales are introduced for all the variables of a system for nondimensionalizing them. In atmospheric large-scale dynamics, these scales are typically derived by nondimensionalizations. However, dimensional analysis is not a sole possibility of defining the characteristic scales of a system. 2.4 and 6.1, but the theory based on gradient-based scales (Sorbjan 2006, 2010, 2016) is probably the most notable. There are various efforts to introduce alternative scales, as further discussed in Sects. ![]() As a result, the Obukhov length is hardly a unique choice. With the absence of an analytical solution to turbulent flows as well as difficulties in observations and numerical modelling, the usefulness of those proposed scales is often hard to judge. A wrong choice of controlling variables can lead to totally meaningless results (cf., Batchelor 1954). No systematic methodology exists for choosing them, and the choice is solely based on physical intuition. In performing a dimensional analysis correctly, a certain ingenuity is required for choosing the proper controlling variables of a given system. Resulting theories from these analyses are called “similarity theories”, because they remain similar regardless of specific cases by rescaling the relevant variables by characteristic scales. For example, the Obukhov length (Obukhov 1948) follows from dimensional analysis of the frictional velocity and the buoyancy flux. ![]() Some key variables controlling a given regime of turbulence are first identified, then various characteristic scales of the system (e.g., length, velocity, temperature) are determined from these key controlling variables by ensuring dimensional consistency. This methodology is alternatively called scaling in atmospheric boundary-layer studies, as reviewed by e.g., Holtslag and Nieuwstadt ( 1986), and Foken ( 2006). Theories of atmospheric boundary-layer turbulence have been developed by heavily relying on so-called dimensional analysis (Barenblatt 1996). A heuristic, but deductive, derivation of Monin–Obukhov similarity theory is also outlined based on the obtained nondimensionalization results. This rescaling factor increases with increasing stable stratification of the boundary layer, in which flows tend to be more horizontal and gentler thus the Obukhov length increasingly loses its relevance. A nondimensionalization length scale for a full set of equations for boundary-layer flow formally reduces to the Obukhov length by dividing this scale by a rescaling factor. Physical implications of the Obukhov length are inferred by nondimensionalizing the turbulence-kinetic-energy equation for the horizontally homogeneous boundary layer. Here, its derivation is pursued by the nondimensionalization method in the same manner as for the Rossby deformation radius and the Ekman-layer depth. The Obukhov length, although often adopted as a characteristic scale of the atmospheric boundary layer, has been introduced purely based on a dimensional argument without a deductive derivation from the governing equations. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |