Abstract: The concept of similarity is central to nearly all studies of fluid dynamics because it provides a means of reducing a whole class of flows to one set of equations, after nondimensionalizing with carefully chosen scales. By studying one instantiation of the class (say, in a laboratory or wind tunnel setting), results can be applied to other examples, perhaps less amenable to direct measurement. Familiar applications include testing of scale models to evaluate aerodynamic drag or lift. In this chapter, similarity in planetary boundary layers is examined in some detail. Relatively well known concepts (Monin-Obukhov similarity for buoyancy effects in the atmospheric surface layer and Rossby similarity for the drag exerted by the atmosphere on the surface) are described and used to illustrate the similarity between the atmospheric and oceanic boundary layers. We then combine these into a similarity theory for the IOBL stabilized by positive buoyancy flux at the surface (melting). The crucial parameters identified in the exercise, including important turbulence scales, then provide the rationale for development of the local turbulence closure model described in subsequent chapters.

From the first studies of wind structure near the surface in the atmosphere and in wind tunnels, there was strong evidence that the vertical profile of wind velocity was often nearly logarithmic, i.e., that

U «log z which implies that the wind shear is nearly inversely proportional to distance from the surface:

dU 1

dz z

M. McPhee, Air-Ice-Ocean Interaction, 65-86. 65

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This was found to hold through the lower tens of meters in the atmosphere, where it was assumed (and verified by experiment) that the turbulent stress was nearly the same as the wind stress acting at the surface, so that friction speed is uo {u'w')\l/2, where the Reynolds stress is measured near the surface, typically at a standard level of 10 m.

If wind shear depends mainly on distance from the surface, on the local stress, and on viscosity of the fluid, then straightforward dimensional analysis (e.g., Barenblatt 1996) reveals that a dimensionless parameter including the dependent quantity wind shear and one or more of the independent governing parameters (z, u*o, v) will be a function of one other dimensionless group formed from the governing parameters since only two have independent dimensions. Consequently, u*0 v where Re is a Reynolds number formed with the friction velocity. Practically speaking, at the high Reynolds numbers typical of nearly all flows in the atmosphere or ocean, the Reynolds number dependence is minimal, and the dimensionless wind shear (for the neutrally stable) surface layer is

u*o where k(= O-1) is von Karman's constant, usually taken to be 0.4.

Note that in this development, there is no consideration of rotation, which manifests itself in the fluid equations in terms of the Coriolis parameter, f. If we accept that v is unimportant, then we can reformulate the dimensional analysis above by substituting f for v, where now the governing parameters have dimensions (following the notation of Barenblatt 1996): [z] = L, [u*0] = LT-1, and [f] = T-1. We again have a dimensionless group including the variable Uz that depends on one other dimensionless group (since there are three postulated governing parameters, two with independent dimensions):

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