DIFFERENTIAL ROTATION AND TURBULENT CONVECTION - A NEW REYNOLDS STRESS MODEL AND COMPARISON WITH SOLAR DATA

In most hydrodynamic cases, the existence of a turbulent flow superimp osed on a mean flow is caused by a shear instability in the latter. Bo ussinesq suggested the first model for the turbulent Reynolds stresses u(i)u(j)BAR in the form u(i)u(j)BAR = - 2v(t)S(ij) which physically i mplies that the mean shear S(ij) is the cause (or source) of turbulenc e represented by the stress u(i)u(j)BAR. In the case of solar differen tial rotation, exactly the reverse physical process occurs: turbulence (which must pre-exist) generates a mean flow which manifests itself i n the form of differential rotation. Thus, the Boussinesq model is who lly inadequate because in the solar case, cause and effect are reverse d. One should envisage the sequence of cause and effect relationships as follows: Buoyancy --> Turbulence --> Mean Flow (Differential Rotati on) where the source of turbulence has been identified with buoyancy w hich is present in stars for reasons unrelated to the fact that it may ultimately generate a differential rotation. An alternative way of in terpreting the sequence above is by saying that small scales (buoyancy ) have more energy than large scales (mean flow, differential rotation ), quite contrary to most situations usually encountered in turbulence studies. Thus, the relation between buoyancy, Reynolds stresses and d ifferential rotation must be viewed in a fundamentally different physi cal light from most standard hydrodynamic flows in which either the me an flow is the cause of turbulence (most laboratory and engineering ca ses) or both mean flow and buoyancy conspire to generate turbulence (t he boundary layer of the Earth's atmosphere). Since the Boussinesq mod el is inadequate, one needs an alternative model for the Reynolds stre sses. We present a new dynamical model for the Reynolds stresses, conv ective fluxes, turbulent kinetic energy, and temperature fluctuations. The complete model requires the solution of 11 differential equations . We then introduce a set of simplifying assumptions which reduce the full dynamical model to a set of algebraic Reynolds stress models. We explicity solve one of these models that entails only one differential equation. The main results are 1. Shear alone, namely the Boussinesq formula, u(i)u(j)BAR = -2v(t)S(ij), cannot give the expected result si nce it describes a flow in which turbulence is generated by shear, whi le in the solar case shear is generated by turbulence. 2. Shear and bu oyancy alone do not yield acceptable results. 3. Agreement with the da ta requires the nonlinear interaction between vorticity and buoyancy. 4. The predicted u(theta)u(phi)BAR agrees very closely with observatio nal data (Gilman & Howard 1984; Virtanen 1989). 5. The model predicts the magnitude and latitudinal behavior of the three components of the turbulent kinetic energy, two of which (u(theta)2BAR and u(theta)2BAR) could be compared to existing data. 6. The maximum production of shea r by buoyancy is predicted to occur at a latitude of approximately 40- degrees. 7. The model predicts that 2.5% of the buoyant production rat e is required to generate and maintain solar differential rotation. 8. The model predicts four independent anisotropic (turbulent) viscositi es v(vv), v(hh), v(vh), and v(hv) which depend on latitude, as well as three independent anisotropic (turbulent) conductivities, chi(rr), ch i(phir), and chi(thetar) which also depend on latitude (the present nu merical results are restricted to radial temperature gradients). 9. Th e degree of anisotropy in the turbulent viscosities, measured by the p arameter s, is found to depend on latitude and its value is in accorda nce with the empirical value of approximately 1.3. 10. The buoyancy ti mescale tau(b) = [(g/H(p)(del - del(ad)]-1/2 predicted by the model is in agreement with the results of stellar structure models. 11. The so -called LAMBDA-effect is naturally (and unavoidably) predicted by the model as a result of the presence of vorticity: while shear depends on ly on the derivatives of OMEGA, vorticity also depends on OMEGA itself . The overall agreement with the data is obtained with a model that is neither phenomenological nor one that requires a full numerical simul ation, since it is algebraic in nature. The new model can play an impo rtant role in understanding the complex physics underlying the interpl ay between solar differential rotation and convection, as many physica l processes can naturally be incorporated into the model.