T. Ii, Presentation of the various runs performed using the LPN equations All runs are in rectangular domains of size 10? × 2? × 2?. Those with Re = 10 4 have the same effective resolution as in Table 1. Those with Re = 100 have 96 meshpoints per interval of length 2? and those with

F. Richardson, The Supply of Energy from and to Atmospheric Eddies, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.97, issue.686, pp.354-373, 1920.
DOI : 10.1098/rspa.1920.0039

I. Taylor, Effect of Variation in Density on the Stability of Superposed Streams of Fluid, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.132, issue.820, pp.499-523, 1931.
DOI : 10.1098/rspa.1931.0115

. Goldstein, On the Stability of Superposed Streams of Fluids of Different Densities, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.132, issue.820, pp.524-548, 1931.
DOI : 10.1098/rspa.1931.0116

W. Miles, On the stability of heterogeneous shear flows, Journal of Fluid Mechanics, vol.132, issue.04, pp.496-508, 1961.
DOI : 10.1063/1.1706011

N. Howard, Note on a paper of John W. Miles, Journal of Fluid Mechanics, vol.10, issue.04, pp.509-512, 1961.
DOI : 10.1017/S0022112061000317

N. Howard and S. A. Maslowe, Stability of stratified shear flows, Boundary-Layer Meteorology, vol.36, issue.1-4, pp.511-523, 1973.
DOI : 10.1007/BF02265252

A. Townsend, The effects of radiative transfer on turbulent flow of a stratified fluid, Journal of Fluid Mechanics, vol.236, issue.04, pp.361-375, 1958.
DOI : 10.1088/0370-1301/63/2/305

J. Dudis, The stability of a thermally radiating stratified shear layer, including self-absorption, Journal of Fluid Mechanics, vol.1, issue.01, pp.65-83, 1974.
DOI : 10.1063/1.1693944

-. Zahn, Rotational Instabilities and Stellar Evolution, Stellar Instability and Evolution , IAU Symposium, pp.185-194, 1974.
DOI : 10.1007/978-94-010-9794-9_34

. Pinsonneault, MIXING IN STARS, Annual Review of Astronomy and Astrophysics, vol.35, issue.1, pp.557-605, 1997.
DOI : 10.1146/annurev.astro.35.1.557

M. J. Mosser, K. Goupil, J. P. Belkacem, P. G. Marques, S. Beck et al., Spin down of the core rotation in red giants Thermal turbulence at very small Prandtl number, A&A J. Geophys. Res, vol.548, issue.67, pp.10-3063, 1962.

. Thual, Zero-Prandtl-number convection, Journal of Fluid Mechanics, vol.4, issue.-1, pp.229-258, 1992.
DOI : 10.1017/S0022112081002541
URL : https://hal.archives-ouvertes.fr/jpa-00248343

. Lignì-eres, The small-Péclet-number approximation in stellar radiative zones, A&A, vol.348, pp.933-939, 1999.

A. Spiegel and G. Veronis, On the Boussinesq Approximation for a Compressible Fluid., The Astrophysical Journal, vol.131, p.442, 1960.
DOI : 10.1086/146849

J. Sommeria and R. Moreau, Why, how, and when, MHD turbulence becomes two-dimensional, Journal of Fluid Mechanics, vol.12, issue.-1, pp.507-518, 1982.
DOI : 10.1063/1.1692443

J. Balmforth, S. G. Smith, and W. R. Young, Dynamics of interfaces and layers in a stratified turbulent fluid, Journal of Fluid Mechanics, vol.355, pp.329-358, 1998.
DOI : 10.1017/S0022112097007970

M. Phillips, Turbulence in a strongly stratified fluid???is it unstable?, Deep Sea Research and Oceanographic Abstracts, vol.19, issue.1, pp.79-81, 1972.
DOI : 10.1016/0011-7471(72)90074-5

. Holmboe, On the behavior of symmetric waves in stratified shear layers, Geophys. Publ, vol.24, pp.67-113, 1962.

F. Lignì-eres, A. Califano, and . Mangeney, Shear layer instability in a highly diffusive stably stratified atmosphere, A&A, vol.349, pp.1027-1036, 1999.

V. Prat and F. Lignì-eres, Turbulent transport in radiative zones of stars The stability of spatially periodic flows, A&A J. Fluid Mech, vol.551, issue.108, pp.3-461, 1981.

B. Squire, On the Stability for Three-Dimensional Disturbances of Viscous Fluid Flow between Parallel Walls, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.142, issue.847, pp.621-628, 1933.
DOI : 10.1098/rspa.1933.0193

J. Balmforth and Y. Young, Stratified Kolmogorov flow, Journal of Fluid Mechanics, vol.450, pp.131-167, 2002.
DOI : 10.1017/S0022111002006371

S. Yih, Stability of two-dimensional parallel flows for three-dimensional disturbances, Quarterly of Applied Mathematics, vol.12, issue.4, pp.434-435, 1955.
DOI : 10.1090/qam/65330

D. Smyth, G. P. Klaassen, and W. R. Peltier, Finite amplitude holmboe waves, Geophysical & Astrophysical Fluid Dynamics, vol.20, issue.2, pp.181-222, 1988.
DOI : 10.1017/S0022112058000045

D. Smyth and W. R. Peltier, Three-dimensional primary instabilities of a stratified, dissipative, parallel flow, Geophysical & Astrophysical Fluid Dynamics, vol.12, issue.4, pp.249-261, 1990.
DOI : 10.1080/03091929008219506

M. Gotoh, J. Yamada, and . Mizushima, The theory of stability of spatially periodic parallel flows, Journal of Fluid Mechanics, vol.142, issue.-1, pp.45-58, 1983.
DOI : 10.1017/S0022112078002918

S. Gage and J. R. Miller, Linear viscous stability analysis of the stratified Bickley jet, Physics of Fluids, vol.17, issue.5, pp.883-887, 1974.
DOI : 10.1063/1.1694825

A. Jones, The onset of shear instability in stars, Geophysical & Astrophysical Fluid Dynamics, vol.4, issue.1, pp.165-184, 1977.
DOI : 10.1080/03091927708240377

D. Joseph, Stability of fluid motions. I, II, NASA STI, vol.28, p.12423, 1976.
DOI : 10.1007/978-3-642-80994-1

E. Doering, R. Spiegel, and . Worthing, Energy dissipation in a shear layer with suction, Physics of Fluids, vol.99, issue.8, 2000.
DOI : 10.2514/3.12363

A. Stellmach, P. Traxler, N. Garaud, T. Brummell, and . Radko, Dynamics of fingering convection. Part 2 The formation of thermohaline staircases, Journal of Fluid Mechanics, vol.14, pp.554-571, 2011.
DOI : 10.1016/0198-0149(87)90015-X

S. Traxler, P. Stellmach, T. Garaud, N. Radko, and . Brummell, Dynamics of fingering convection. Part 1 Small-scale fluxes and large-scale instabilities, Journal of Fluid Mechanics, vol.16, pp.530-553, 2011.
DOI : 10.1029/2007JC004455