## Publication Abstracts

### Canuto et al. 1994

**425**, 303-325, doi:10.1086/173986.

In most hydrodynamic cases, the existence of a turbulent flow superimposed on a mean flow is caused by a shear instability in the latter. Boussinesq suggested the first model for the turbulent Reynolds stresses bar-(u_{i}u_{j}) in the form

bar-(u_{i}u_{j}) = -2ν_{t}S_{ij}

which physically implies that the mean shear S_{ij} is the cause (or source) of turbulence represented by the stress bar-((u_{i}u_{j}). In the case of solar differential rotation, exactly the reverse physical process occurs: turbulence (which must pre-exist) generates a mean flow which manifests itself in the form of differential rotation. Thus, the Boussinesq model is wholly inadequate because in the solar case, cause and effect are reversed. One should envisage the sequence of cause and effect relationships as follows:

Buoyancy → Turbulence → Mean Flow (Differential Rotation)

where the source of trubulence has been identified with buoyancy which is present in stars for reasons unrelated to the fact that it may ultimately generate a differential rotation. An alternative way of interpreting 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 differential rotation must be viewed in a fundamentally different physical light from most standard hydrodynamic flows in which either the mean flow is the cause of turbulence (most laboratory and engineering cases) or both mean flow and buoyancy conspire to generate turbulence (the boundary layer of the Earth's atmosphere). Since the Boussinesq model is inadequate, one needs an alternative model for the Reynolds stresses.

We present a new dynamical model for the Reynolds stresses, convective 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 explicitly solve one of these models that entails only one differential equation. The main results are:

1. Shear alone, namely the Boussinesq formula, bar-(u_{i}u_{j}) = -2ν_{t}S_{ij}, cannot give the expected result since it describes flow in which turbulence is generated by shear, while in the solar case shear is generated by turbulence.

2. Shear and buoyancy alone do not yield acceptable results.

3. Agreement with teh data requires the nonlinear interaction between vorticity and buoyancy.

4. The predicted bar-(u_{θ}u_{φ}) agrees very closely with observational data (Gilman & Howard 1984; Virtanen 1989).

5. The model predicts the magnitude and latitudinal behavior of the three components of the trubulent kinetic energy, two of which (u_{φ}^{2} and u_{θ}^{2}) could be compared to existing data.

6. The maximum production of shear by buoyancy is predicted to occur at a latitude of ∼40°. 7. The model predicts that 2.5% of the buoyant production rate is required to generate and maintain solar differential rotation.

8. The model predicts four independent anisotropic (turbulent) viscosities ν_{vv}, ν_{hh}, ν_{vh}, and ν_{hv}, which depend on latitude, as well as three independent anisotropic (turbulent) conductivities, χ_{rr}, χ_{φr}, and χ_{θr} which also depend on latitude (the present numerical results are restricted to radial temperature gradients).

9. The degree of anisotropy in the turbulent viscosities, measured by the parameter s, is found to depend on latitude and its value is in accordance with the empirical value of ∼1.3.

10. The buoyancy timescale τ_{b} = [(g/H_{p})(∇-∇_{ad})]^{-1/2} predicted by the model is in agreement with the results of stellar structure models.

11. The so-called Λ-effect is naturally (and unavoidably) predicted by the model as a result of the presence of vorticity; while shear depends only on the derivatives of Ω, vorticity also depends on Ω itself.

The overall agreement with the data is obtained with a model that is neither phenomenological nor one that requires a full numerical simulation, since it is algebraic in nature. The new model can play an important role in understanding the complex physics underlying the interplay between solar differential rotation and convection, as many physical processes can naturally be incorporated into the model.

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#### BibTeX Citation

@article{ca07010j, author={Canuto, V. M. and Minotti, F. O. and Schilling, O.}, title={Differential rotation and turbulent convection: A new Reynolds stress model and comparison with solar data}, year={1994}, journal={Astrophys. J.}, volume={425}, pages={303--325}, doi={10.1086/173986}, }

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#### RIS Citation

TY - JOUR ID - ca07010j AU - Canuto, V. M. AU - Minotti, F. O. AU - Schilling, O. PY - 1994 TI - Differential rotation and turbulent convection: A new Reynolds stress model and comparison with solar data JA - Astrophys. J. VL - 425 SP - 303 EP - 325 DO - 10.1086/173986 ER -

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