Rossby number

The effect of fluid inertia manifests during abrupt change in velocity of the fluid mass. It is quantified by the Rossby number ... 

We expect an efficient a — Q-dynamo to be at work in the merger remnant. The differential rotation will wind up initial poloidal into a strong toroidal field ( Q-effect ), the fluid instabilities/convection will transform toroidal fields into poloidal ones and vice versa ( a—effect ). Usually, the Rossby number, Ro = is adopted as a measure of the efficiency of dynamo action in a star. In the central object we find Rossby numbers well below unity, 0.4, and therefore expect an efficient amplification of initial seed magnetic fields. A convective dynamo amplifies initial fields exponentially with an e-folding time given approximately by the convective overturn time, rc ss 3 ms the saturation field strength is thereby independent of the initial seed field (Nordlund et al. 1992). 

Later Mayle, 1970 [400] continued their research by performing measurements of velocity and pressure within the fire whirl. He found that the behavior of the plume was governed by dimensionless plume Froude, Rossby, second Damkohler Mixing Coefficient and Reaction Rate numbers. For plumes with a Rossby number less than one the plume is found to have a rapid rate of plume expansion with height. This phenomenon is sometimes called vortex breakdown , and it is a hydraulic jump like phenomena caused by the movement of surface waves up the surface of the fire plume that are greater than the speed of the fluid velocity. Unfortunately, even improved entrainment rate type models do not predict these phenomena very well. 

Fig. 18.2 Rotating liquid column jet profiles at the pinching condition for various Rossby numbers [57] (Courtesy of Elsevier)...

In representing the vertical structure of the continuously stratified atmosphere it is generally necessary to discretize the vertical dependence so that the semiinfinite atmosphere may be approximated by a finite number of parameters. Often finite differences at a prespecified number of levels are used to represent the vertical structure. Another method is to divide the atmosphere into a finite number of homogeneous layers, as shown in Fig. 2. This representation is often used in dynamic oceanography, which has many similarities to dynamic meteorology, essentially because the Rossby number is also small for large-scale motions in the ocean. Both disciplines may be considered subsets of geophysical fluid dynamics. 

The scaling law in Eq. (2) derives its simplicity, and hence its empirical tractability, from the neglect of a variety of other, potentially important, parameters. For instance, the formulation implicitly says that the structure of the near surface flow is independent of viscosity, v, the rotational frequency of the Earth, /, the depth of the turbulent boundary layer, h, and so on. These arguments are asymptotic rather than absolute statements. They effectively state that the Reynolds number (in this case the inverse of the nondimensional viscosity). Re = M z/v, is so large that the flow ceases to depend on it. Likewise for the Rossby number, Ro = ujfz. In these cases, we speak of the flow obeying Reynolds or Rossby number similarity. 

To motivate the geostrophic approximation, we invoke a scale analysis approach. Assume frictional forces can be neglected, and the atmospheric motions have a characteristic horizontal length-scale, L, and velocity scale, U. Recalling the definition of the advective derivative operator D/Dt, we find that the magnitude of the acceleration terms, Dw/Dt and Du/Dt in Eqs. (9.2.20) and (9.2.21), is 17 /L, provided the magnitude of the time scale of the motion is greater than or equal to the advective time, L/U. The terms proportional to tan 0 are of order U /a, and the Coriolis terms are of order /(/. If L < a, then the ratio of each of the terms to the Coriolis term is less than or comparable to the Rossby number, defined as... 

U, L, H are the characteristic velocity, length and layer thickness scales. L/j is the Rossby deformation radius. The Rossby number measures the importance of rotation on the flow, the Burger number measures the stratification in the atmosphere via the Brunt-Vaisala frequency. As it is known, the atmospheric structure in the giant planets in the solar system is in bands. This structure evolves from shallow-water turbulence. Two dimensional turbulence is characterized by an inverse energy cascade—this means a transfer from small to large scales. 

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