Improving the parametrisation of transport and mixing processes: the importance of implementing the full Coriolis acceleration

<latexit sha1_base64="5bCyd0gYv8CeZMXZW/avxKBuZUk=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi1WPRi8cK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVqyppUCaU7ITFMcMmayFGwTqIZiUPB2uH4dua3n5g2XMkHnCQsiMlQ8ohTglZq9XDEkPTLFa/qzeGuEj8nFcjR6Je/egNF05hJpIIY0/W9BIOMaORUsGmplxqWEDomQ9a1VJKYmSCbXzt1z6wycCOlbUl05+rviYzExkzi0HbGBEdm2ZuJ/3ndFKPrIOMySZFJulgUpcJF5c5edwdcM4piYgmhmttbXToimlC0AZVsCP7yy6ukdVH1a9Xa/WWlfpPHUYQTOIVz8OEK6nAHDWgChUd4hld4c5Tz4rw7H4vWgpPPHMMfOJ8/p4mPMw==</latexit> ω <latexit sha1_base64="lxsCryPN6dXNkcOitfoyG04f9Bs=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BIvgqeyKVI9FL96sYD+gu5RsOtuGJtk1yRbK0t/hxYMiXv0x3vw3pu0etPXBwOO9GWbmhQln2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nikKTxjxWnZBo4ExC0zDDoZMoICLk0A5HtzO/PQalWSwfzSSBQJCBZBGjxFgp8MMo8+8FDEjPnfbKFbfqzoFXiZeTCsrR6JW//H5MUwHSUE607npuYoKMKMMoh2nJTzUkhI7IALqWSiJAB9n86Ck+s0ofR7GyJQ2eq78nMiK0nojQdgpihnrZm4n/ed3URNdBxmSSGpB0sShKOTYxniWA+0wBNXxiCaGK2VsxHRJFqLE5lWwI3vLLq6R1UfVq1drDZaV+k8dRRCfoFJ0jD12hOrpDDdREFD2hZ/SK3pyx8+K8Ox+L1oKTzxyjP3A+fwCJeJH3</latexit> !0 Equator North Pole <latexit sha1_base64="UfZPnkdQ2H0hCi1VgUSDGo/voto=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqSQi1WPRi8cK9gPaUDabSbt0sxt2J0IJ/RlePCji1V/jzX/jts1Bqw8GHu/NMDMvTAU36HlfTmltfWNzq7xd2dnd2z+oHh51jMo0gzZTQuleSA0ILqGNHAX0Ug00CQV0w8nt3O8+gjZcyQecphAkdCR5zBlFK/UHyEUEeTwbesNqzat7C7h/iV+QGinQGlY/B5FiWQISmaDG9H0vxSCnGjkTMKsMMgMpZRM6gr6lkiZggnxx8sw9s0rkxkrbkugu1J8TOU2MmSah7Uwojs2qNxf/8/oZxtdBzmWaIUi2XBRnwkXlzv93I66BoZhaQpnm9laXjammDG1KFRuCv/ryX9K5qPuNeuP+sta8KeIokxNySs6JT65Ik9yRFmkTRhR5Ii/k1UHn2Xlz3petJaeYOSa/4Hx8A1ZBkU4=</latexit> ˜ f0 <latexit sha1_base64="6An7g+WjSTtPZUVSkJZ7tmBR4Ds=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz0EPbdfrniVt05yCrxclKBHI1++as3iFkaoTRMUK27npsYP6PKcCZwWuqlGhPKxnSIXUsljVD72fzUKTmzyoCEsbIlDZmrvycyGmk9iQLbGVEz0sveTPzP66YmvPYzLpPUoGSLRWEqiInJ7G8y4AqZERNLKFPc3krYiCrKjE2nZEPwll9eJa2Lqler1u4vK/WbPI4inMApnIMHV1CHO2hAExgM4Rle4c0Rzovz7nwsWgtOPnMMf+B8/gDyQ42Y</latexit> f0 Neglected with TA Improving the parametrization of transport and mixing processes: the importance of implementing the full Coriolis acceleration 1Département d’Astrophysique-AIM, CEA/DRF/IRFU, CNRS/INSU, Université Paris-Saclay, Université de Paris-Cité, F-91191 Gif-sur-Yvette, France ([email protected]) 2LadHyX, Ecole Polytechnique/ CNRS/Institut Polytechnique de Paris, 91120 Palaiseau, France 3Centre for Fluid and Complex Systems, Coventry University, Priory Street, Coventry CV15FB, UK INTRODUCTION TWO-DIMENSIONAL NON-LINEAR INSTABILITY OF HORIZONTALLY SHEARED ZONAL FLOWS C. Moisset1, S. Mathis1, P. Billant2, J. Park3 INERTIA-GRAVITY WAVE DAMPING The Traditional Approximation does not hold for environments where the rotation competes with the stratification. As a result: CONCLUSIONS NON-LINEAR INERTIA-GRAVITY WAVE BREAKING REFERENCES Stellar interiors and planetary atmospheres and oceans are stratified and rotating fluids that we model using stellar evolutionary codes and General Circulation Models (GCM) for their long-term evolution. In those, the rotation is only partially implemented, using the Traditionnal Approximation (TA) of the rotation. •Linear analysis: •Non-linear analysis: Moisset et al. in prep. Fig. 2: Map of the different non-linear evolutions in the parameter space . Black dashed line corresponds to the explored limit . (Moisset et al. in prep.) <latexit sha1_base64="XX576VtK+S7KMCWujK5wopirOog=">AAAB9XicbVDLSgNBEOz1GeMr6tHLYBAiSNgViR6DXjxJBPOAZA2zk9lkyOyDmV4lLPkPLx4U8eq/ePNvnCR70MSChqKqm+4uL5ZCo21/W0vLK6tr67mN/ObW9s5uYW+/oaNEMV5nkYxUy6OaSxHyOgqUvBUrTgNP8qY3vJ74zUeutIjCexzF3A1oPxS+YBSN9FC6PSUdFLLHU3980i0U7bI9BVkkTkaKkKHWLXx1ehFLAh4ik1TrtmPH6KZUoWCSj/OdRPOYsiHt87ahIQ24dtPp1WNybJQe8SNlKkQyVX9PpDTQehR4pjOgONDz3kT8z2sn6F+6qQjjBHnIZov8RBKMyCQC0hOKM5QjQyhTwtxK2IAqytAElTchOPMvL5LGWdmplCt358XqVRZHDg7hCErgwAVU4QZqUAcGCp7hFd6sJ+vFerc+Zq1LVjZzAH9gff4AVVeRyA==</latexit> (N, ˜ f) <latexit sha1_base64="o7zsyAP2xM6wnR9K3RGtDssXZj4=">AAAB8nicbVBNS8NAEN3Ur1q/qh69LBbBU0lEqheh6MWTVLAfkIay2WzapZvdsDsRSujP8OJBEa/+Gm/+G7dtDlp9MPB4b4aZeWEquAHX/XJKK6tr6xvlzcrW9s7uXnX/oGNUpilrUyWU7oXEMMElawMHwXqpZiQJBeuG45uZ331k2nAlH2CSsiAhQ8ljTglYye8DFxHL4+nV3aBac+vuHPgv8QpSQwVag+pnP1I0S5gEKogxvuemEOREA6eCTSv9zLCU0DEZMt9SSRJmgnx+8hSfWCXCsdK2JOC5+nMiJ4kxkyS0nQmBkVn2ZuJ/np9BfBnkXKYZMEkXi+JMYFB49j+OuGYUxMQSQjW3t2I6IppQsClVbAje8st/Sees7jXqjfvzWvO6iKOMjtAxOkUeukBNdItaqI0oUugJvaBXB5xn5815X7SWnGLmEP2C8/ENUA+RSg==</latexit> ˜ f=N •Adiabatic propagation: Variable (see fig. 2) <latexit sha1_base64="SUzAsEu3u8JrOL4MNaVtn/ZeVrU=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJy2IRPJVEpHoRil48VrAf0IQy2W7apZtN2N0IJQT8K148KOLV3+HNf+O2zUFbHww83pthZl6QcKa043xbpZXVtfWN8mZla3tnd8/eP2irOJWEtkjMY9kNQFHOBG1ppjntJpJCFHDaCca3U7/zSKVisXjQk4T6EQwFCxkBbaS+fdSU114ogWSeSPPMG0OSQN63q07NmQEvE7cgVVSg2be/vEFM0ogKTTgo1XOdRPsZSM0Ip3nFSxVNgIxhSHuGCoio8rPZ+Tk+NcoAh7E0JTSeqb8nMoiUmkSB6YxAj9SiNxX/83qpDq/8jIkk1VSQ+aIw5VjHeJoFHjBJieYTQ4BIZm7FZAQmDG0Sq5gQ3MWXl0n7vObWa/X7i2rjpoijjI7RCTpDLrpEDXSHmqiFCMrQM3pFb9aT9WK9Wx/z1pJVzByiP7A+fwDbi5YX</latexit> Pr = ω ε <latexit sha1_base64="LM96RY/BTVMn/EKQp5ZKu08KIoM=">AAAB/nicbVBNS8NAEN3Ur1q/ouLJS7AInkoiUr0IRS8ePFSxH9CEsNlO2qWbTdjdCCUE/CtePCji1d/hzX/jts1BWx8MPN6bYWZekDAqlW1/G6Wl5ZXVtfJ6ZWNza3vH3N1ryzgVBFokZrHoBlgCoxxaiioG3UQAjgIGnWB0PfE7jyAkjfmDGifgRXjAaUgJVlryzYN7uHRDgUnW8u1b384zl6e5b1btmj2FtUicglRRgaZvfrn9mKQRcEUYlrLn2InyMiwUJQzyiptKSDAZ4QH0NOU4Aull0/Nz61grfSuMhS6urKn6eyLDkZTjKNCdEVZDOe9NxP+8XqrCCy+jPEkVcDJbFKbMUrE1ycLqUwFEsbEmmAiqb7XIEOswlE6sokNw5l9eJO3TmlOv1e/Oqo2rIo4yOkRH6AQ56Bw10A1qohYiKEPP6BW9GU/Gi/FufMxaS0Yxs4/+wPj8AcJLlWI=</latexit> Re = U0L0 ω <latexit sha1_base64="Vd4dszg2msWezJsRU85LgbGo2hw=">AAAB/XicbVDLSsNAFL2pr1pf8bFzM1gEVyURqW6EohsXUiqYttCGMJlO2qGTBzMToYbir7hxoYhb/8Odf+O0zUJbD9zL4Zx7mTvHTziTyrK+jcLS8srqWnG9tLG5tb1j7u41ZZwKQh0S81i0fSwpZxF1FFOcthNBcehz2vKH1xO/9UCFZHF0r0YJdUPcj1jACFZa8syD+mU3EJhkdc+69axx5ujmmWWrYk2BFomdkzLkaHjmV7cXkzSkkSIcS9mxrUS5GRaKEU7HpW4qaYLJEPdpR9MIh1S62fT6MTrWSg8FsdAVKTRVf29kOJRyFPp6MsRqIOe9ifif10lVcOFmLEpSRSMyeyhIOVIxmkSBekxQovhIE0wE07ciMsA6DKUDK+kQ7PkvL5LmacWuVqp3Z+XaVR5HEQ7hCE7AhnOowQ00wAECj/AMr/BmPBkvxrvxMRstGPnOPvyB8fkDYIWUjQ==</latexit> N= N0L0 U0 <latexit sha1_base64="O0kMjriihEtLfbdb+LOuDzjuJlg=">AAACDXicbVBNS8MwGE79nPOr6tFLcAqeRisyvQhDLx48TLDbYC0lTdMtLE1Lkgqj9A948a948aCIV+/e/DdmWxHdfCDhyfO8L2/eJ0gZlcqyvoyFxaXlldXKWnV9Y3Nr29zZbcskE5g4OGGJ6AZIEkY5cRRVjHRTQVAcMNIJhldjv3NPhKQJv1OjlHgx6nMaUYyUlnzz0FWUhSSPigs3EgjnP2/fuvGtInf05Zs1q25NAOeJXZIaKNHyzU83THAWE64wQ1L2bCtVXo6EopiRoupmkqQID1Gf9DTlKCbSyyfbFPBIKyGMEqEPV3Ci/u7IUSzlKA50ZYzUQM56Y/E/r5ep6NzLKU8zRTieDooyBlUCx9HAkAqCFRtpgrCg+q8QD5AORekAqzoEe3bledI+qduNeuP2tNa8LOOogH1wAI6BDc5AE1yDFnAABg/gCbyAV+PReDbejPdp6YJR9uyBPzA+vgEE3Zwp</latexit> ˜ f= ˜ f0L0 U0 <latexit sha1_base64="nsTxaIq5vmy6wXKy8FYwHthy0cY=">AAAB7nicbVDLSgNBEOyNrxhfUY9eBoPgKcwGiV6EoBePUcwDkiXMTnqTIbMPZmaFEPIRXjwo4tXv8ebfOEn2oIkFDUVVN91dfiKFNpR+O7m19Y3Nrfx2YWd3b/+geHjU1HGqODZ4LGPV9plGKSJsGGEkthOFLPQltvzR7cxvPaHSIo4ezThBL2SDSASCM2Ol1gNeVyilvWKJlukcZJW4GSlBhnqv+NXtxzwNMTJcMq07Lk2MN2HKCC5xWuimGhPGR2yAHUsjFqL2JvNzp+TMKn0SxMpWZMhc/T0xYaHW49C3nSEzQ73szcT/vE5qgitvIqIkNRjxxaIglcTEZPY76QuF3MixJYwrYW8lfMgU48YmVLAhuMsvr5JmpexWy9X7i1LtJosjDydwCufgwiXU4A7q0AAOI3iGV3hzEufFeXc+Fq05J5s5hj9wPn8Ar2WOgQ==</latexit> Re = 2000 <latexit sha1_base64="Sztly+ojGL44M0HKPMb1mrXnPmk=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqheh6MVjBPsBbSib7aZdursJuxuhhP4FLx4U8eof8ua/cdPmoK0PBh7vzTAzL0w408Z1v53S2vrG5lZ5u7Kzu7d/UD08aus4VYS2SMxj1Q2xppxJ2jLMcNpNFMUi5LQTTu5yv/NElWaxfDTThAYCjySLGMEml3x14w2qNbfuzoFWiVeQGhTwB9Wv/jAmqaDSEI617nluYoIMK8MIp7NKP9U0wWSCR7RnqcSC6iCb3zpDZ1YZoihWtqRBc/X3RIaF1lMR2k6BzVgve7n4n9dLTXQdZEwmqaGSLBZFKUcmRvnjaMgUJYZPLcFEMXsrImOsMDE2nooNwVt+eZW0L+peo954uKw1b4s4ynACp3AOHlxBE+7BhxYQGMMzvMKbI5wX5935WLSWnGLmGP7A+fwBcYyN3Q==</latexit> Pr =1 Fixed ( , ) Fig. 4: Total vertical vorticity at the maximum of transverse extension of the vortex during the evolution, for (a): and (b): (Moisset et al. in prep.). <latexit sha1_base64="x6G59qgjWWzWiQv+F1AJITTW97E=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQgslJKLVi1D04kkqWFtoQ9lsNu3SzSbsboQScvGvePGgiFd/hjf/jds2B219MPB4b4aZeV7MqFS2/W0UlpZXVteK66WNza3tHXN370FGicCkhSMWiY6HJGGUk5aiipFOLAgKPUba3uh64rcfiZA04vdqHBM3RANOA4qR0lLfPKjc1mBPUeaTNMiqlxXbOqtBu9o3y7ZlTwEXiZOTMsjR7JtfPT/CSUi4wgxJ2XXsWLkpEopiRrJSL5EkRniEBqSrKUchkW46fSCDx1rxYRAJXVzBqfp7IkWhlOPQ050hUkM5703E/7xuooILN6U8ThTheLYoSBhUEZykAX0qCFZsrAnCgupbIR4igbDSmZV0CM78y4vk4cRy6lb97rTcuMrjKIJDcAQqwAHnoAFuQBO0AAYZeAav4M14Ml6Md+Nj1low8pl98AfG5w++p5Pw</latexit> (N, ˜ f)=(0.5,0) <latexit sha1_base64="hR0O9gC2nQZC/kaL92WprCQBthA=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1iEFkpIRKsboejGlVSwD2hDmUwm7dDJJMxMhBKKG3/FjQtF3PoV7vwbp20W2nrgXg7n3MvMPV7MqFS2/W3klpZXVtfy64WNza3tHXN3rymjRGDSwBGLRNtDkjDKSUNRxUg7FgSFHiMtb3g98VsPREga8Xs1iokboj6nAcVIaalnHpRuK7CrKPNJGozLlyXbOqtA3co9s2hb9hRwkTgZKYIM9Z751fUjnISEK8yQlB3HjpWbIqEoZmRc6CaSxAgPUZ90NOUoJNJNpyeM4bFWfBhEQhdXcKr+3khRKOUo9PRkiNRAznsT8T+vk6jgwk0pjxNFOJ49FCQMqghO8oA+FQQrNtIEYUH1XyEeIIGw0qkVdAjO/MmLpHliOVWrendarF1lceTBITgCJeCAc1ADN6AOGgCDR/AMXsGb8WS8GO/Gx2w0Z2Q7++APjM8frAmUZw==</latexit> (N, ˜ f)=(0.5,0.5) <latexit sha1_base64="SmQ3iOpZqwc7eP7/2dQWFItosRc=">AAAB+nicbVDLSsNAFJ3UV62vVJduBovgQkpSpLosunGlFewDmhAm00k6dGYSZiZKqf0UNy4UceuXuPNvnLZZaOuBC4dz7uXee8KUUaUd59sqrKyurW8UN0tb2zu7e3Z5v62STGLSwglLZDdEijAqSEtTzUg3lQTxkJFOOLya+p0HIhVNxL0epcTnKBY0ohhpIwV2+SZwvDiGNe+WkxgFzmlgV5yqMwNcJm5OKiBHM7C/vH6CM06Exgwp1XOdVPtjJDXFjExKXqZIivAQxaRnqECcKH88O30Cj43Sh1EiTQkNZ+rviTHiSo14aDo50gO16E3F/7xepqMLf0xFmmki8HxRlDGoEzjNAfapJFizkSEIS2puhXiAJMLapFUyIbiLLy+Tdq3q1qv1u7NK4zKPowgOwRE4AS44Bw1wDZqgBTB4BM/gFbxZT9aL9W59zFsLVj5zAP7A+vwBQdSSuQ==</latexit> N0→2!0, is the Brunt-Väisälä frequency, and the rotation rate. <latexit sha1_base64="unkaspxngRC3ZZgBwY2McHv7roI=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF09S0X5AG8pmO2mXbjZhdyOU0J/gxYMiXv1F3vw3btsctPpg4PHeDDPzgkRwbVz3yymsrK6tbxQ3S1vbO7t75f2Dlo5TxbDJYhGrTkA1Ci6xabgR2EkU0igQ2A7G1zO//YhK81g+mEmCfkSHkoecUWOl+9u+2y9X3Ko7B/lLvJxUIEejX/7sDWKWRigNE1Trrucmxs+oMpwJnJZ6qcaEsjEdYtdSSSPUfjY/dUpOrDIgYaxsSUPm6s+JjEZaT6LAdkbUjPSyNxP/87qpCS/9jMskNSjZYlGYCmJiMvubDLhCZsTEEsoUt7cSNqKKMmPTKdkQvOWX/5LWWdWrVWt355X6VR5HEY7gGE7Bgwuoww00oAkMhvAEL/DqCOfZeXPeF60FJ585hF9wPr4BzbONgA==</latexit> N0 <latexit sha1_base64="xMSDfcMS2Q0MHyj212B3HGJtlqs=">AAAB73icbVBNS8NAEJ34WetX1aOXYBE8lUSkeix68WYF+wFtKJvtpF26u4m7G6GE/gkvHhTx6t/x5r9x2+agrQ8GHu/NMDMvTDjTxvO+nZXVtfWNzcJWcXtnd2+/dHDY1HGqKDZozGPVDolGziQ2DDMc24lCIkKOrXB0M/VbT6g0i+WDGScYCDKQLGKUGCu1u3cCB6Tn9Uplr+LN4C4TPydlyFHvlb66/ZimAqWhnGjd8b3EBBlRhlGOk2I31ZgQOiID7FgqiUAdZLN7J+6pVfpuFCtb0rgz9fdERoTWYxHaTkHMUC96U/E/r5Oa6CrImExSg5LOF0Upd03sTp93+0whNXxsCaGK2VtdOiSKUGMjKtoQ/MWXl0nzvOJXK9X7i3LtOo+jAMdwAmfgwyXU4Bbq0AAKHJ7hFd6cR+fFeXc+5q0rTj5zBH/gfP4AilWPqQ==</latexit> !0 Where: and are the characteristic length and velocity of the tanh profile respectively. is the kinematic viscosity and the thermal diffusivity. <latexit sha1_base64="VjCfIhy9PL0APGqDc+EpVkci0X8=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48VTVtoQ9lsJ+3SzSbsboRS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8MBVcG9f9dgpr6xubW8Xt0s7u3v5B+fCoqZNMMfRZIhLVDqlGwSX6hhuB7VQhjUOBrXB0O/NbT6g0T+SjGacYxHQgecQZNVZ68Htur1xxq+4cZJV4OalAjkav/NXtJyyLURomqNYdz01NMKHKcCZwWupmGlPKRnSAHUsljVEHk/mpU3JmlT6JEmVLGjJXf09MaKz1OA5tZ0zNUC97M/E/r5OZ6DqYcJlmBiVbLIoyQUxCZn+TPlfIjBhbQpni9lbChlRRZmw6JRuCt/zyKmleVL1atXZ/Wanf5HEU4QRO4Rw8uII63EEDfGAwgGd4hTdHOC/Ou/OxaC04+cwx/IHz+QPYXY2H</latexit> U0 <latexit sha1_base64="BmmZABQC9TfIxCI05dNHv9QYk90=">AAAB6nicbVA9SwNBEJ2LXzF+RS1tFoNgFe5EomXQxsIiovmA5Ah7m7lkyd7esbsnhCM/wcZCEVt/kZ3/xk1yhUYfDDzem2FmXpAIro3rfjmFldW19Y3iZmlre2d3r7x/0NJxqhg2WSxi1QmoRsElNg03AjuJQhoFAtvB+Hrmtx9RaR7LBzNJ0I/oUPKQM2qsdH/bd/vlilt15yB/iZeTCuRo9MufvUHM0gilYYJq3fXcxPgZVYYzgdNSL9WYUDamQ+xaKmmE2s/mp07JiVUGJIyVLWnIXP05kdFI60kU2M6ImpFe9mbif143NeGln3GZpAYlWywKU0FMTGZ/kwFXyIyYWEKZ4vZWwkZUUWZsOiUbgrf88l/SOqt6tWrt7rxSv8rjKMIRHMMpeHABdbiBBjSBwRCe4AVeHeE8O2/O+6K14OQzh/ALzsc3yqeNfg==</latexit> L0 <latexit sha1_base64="fEYB18708xNLzyXDtxlc6Ix7+eA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz00JNpv1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vktZF1atVa/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwBidY3i</latexit> ω <latexit sha1_base64="n6m7Oxf66ELcdLLGSqWs+mPTk3c=">AAAB7XicbVBNSwMxEM3Wr1q/qh69BIvgqeyKVI9FLx4r2A9olzKbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCAzlTNCmZZbTjtIUkojTdjS+nfntJ6oNk+LBThQNExgKFjMC1kmt3hiUgn654lf9OfAqCXJSQTka/fJXbyBJmlBhCQdjuoGvbJiBtoxwOi31UkMVkDEMaddRAQk1YTa/dorPnDLAsdSuhMVz9fdEBokxkyRynQnYkVn2ZuJ/Xje18XWYMaFSSwVZLIpTjq3Es9fxgGlKLJ84AkQzdysmI9BArAuo5EIIll9eJa2LalCr1u4vK/WbPI4iOkGn6BwF6ArV0R1qoCYi6BE9o1f05knvxXv3PhatBS+fOUZ/4H3+AJm+jyo=</latexit> ω Fig. 3: Total vertical vorticity in the non-linear regime of the instability, with the velocity in the horizontal plane, at t = 145, for (a): , (b): , (c): and (d): . (Moisset et al. in prep.). <latexit sha1_base64="V4berBHMOwibAO08tUUQKAv7MuY=">AAACBnicbVBNS8MwGE7n15xfVY8iBIfgxdGKTC/C0IvHCe4D1lLSLNvC0rQk6VgpO3nxr3jxoIhXf4M3/43pVkQ3Hwg8eZ73fZP38SNGpbKsL6OwtLyyulZcL21sbm3vmLt7TRnGApMGDlko2j6ShFFOGooqRtqRICjwGWn5w5vMb42IkDTk9yqJiBugPqc9ipHSkmceOmN65URIKIqYN4aj059LAmPPLFsVawq4SOyclEGOumd+Ot0QxwHhCjMkZce2IuWm2UTMyKTkxJJECA9Rn3Q05Sgg0k2na0zgsVa6sBcKfbiCU/V3R4oCKZPA15UBUgM572Xif14nVr1LN6U8ihXhePZQL2ZQhTDLBHapIFixRBOEBdV/hXiABMJKJ1fSIdjzKy+S5lnFrlaqd+fl2nUeRxEcgCNwAmxwAWrgFtRBA2DwAJ7AC3g1Ho1n4814n5UWjLxnH/yB8fENH2GY6w==</latexit> ω=εxv→εyu <latexit sha1_base64="x6G59qgjWWzWiQv+F1AJITTW97E=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQgslJKLVi1D04kkqWFtoQ9lsNu3SzSbsboQScvGvePGgiFd/hjf/jds2B219MPB4b4aZeV7MqFS2/W0UlpZXVteK66WNza3tHXN370FGicCkhSMWiY6HJGGUk5aiipFOLAgKPUba3uh64rcfiZA04vdqHBM3RANOA4qR0lLfPKjc1mBPUeaTNMiqlxXbOqtBu9o3y7ZlTwEXiZOTMsjR7JtfPT/CSUi4wgxJ2XXsWLkpEopiRrJSL5EkRniEBqSrKUchkW46fSCDx1rxYRAJXVzBqfp7IkWhlOPQ050hUkM5703E/7xuooILN6U8ThTheLYoSBhUEZykAX0qCFZsrAnCgupbIR4igbDSmZV0CM78y4vk4cRy6lb97rTcuMrjKIJDcAQqwAHnoAFuQBO0AAYZeAav4M14Ml6Md+Nj1low8pl98AfG5w++p5Pw</latexit> (N, ˜ f)=(0.5,0) <latexit sha1_base64="tUiQRUF5wSHW30qaEeKR0dPWodc=">AAACAHicbVDLSsNAFJ3UV62vqAsXbgaL0EIpSdHqRii6cSUV7APaUCaTSTt0MgkzE6GEbPwVNy4UcetnuPNvnLZZaOuBC4dz7uXee9yIUaks69vIrayurW/kNwtb2zu7e+b+QVuGscCkhUMWiq6LJGGUk5aiipFuJAgKXEY67vhm6nceiZA05A9qEhEnQENOfYqR0tLAPCrdVWBfUeaRxE/LV6VaBdrV8/LALFpVawa4TOyMFEGG5sD86nshjgPCFWZIyp5tRcpJkFAUM5IW+rEkEcJjNCQ9TTkKiHSS2QMpPNWKB/1Q6OIKztTfEwkKpJwEru4MkBrJRW8q/uf1YuVfOgnlUawIx/NFfsygCuE0DehRQbBiE00QFlTfCvEICYSVzqygQ7AXX14m7VrVrlfr92fFxnUWRx4cgxNQAja4AA1wC5qgBTBIwTN4BW/Gk/FivBsf89ackc0cgj8wPn8AwxGT8w==</latexit> (N, ˜ f)=(2,1.5) <latexit sha1_base64="nHfMxj63M7bNYvzx9HgIXesa9Ks=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksQgslJKLVi1D04kkqWFtoQ9lsNu3SzSbsboQScvGvePGgiFd/hjf/jds2B219MPB4b4aZeV7MqFS2/W0UlpZXVteK66WNza3tHXN370FGicCkhSMWiY6HJGGUk5aiipFOLAgKPUba3uh64rcfiZA04vdqHBM3RANOA4qR0lLfPKjc1mBPUeaTNMiqlxWnBm3rrNo3y7ZlTwEXiZOTMsjR7JtfPT/CSUi4wgxJ2XXsWLkpEopiRrJSL5EkRniEBqSrKUchkW46fSCDx1rxYRAJXVzBqfp7IkWhlOPQ050hUkM5703E/7xuooILN6U8ThTheLYoSBhUEZykAX0qCFZsrAnCgupbIR4igbDSmZV0CM78y4vk4cRy6lb97rTcuMrjKIJDcAQqwAHnoAFuQBO0AAYZeAav4M14Ml6Md+Nj1low8pl98AfG5w/AAJPx</latexit> (N, ˜ f)=(1,0.5) <latexit sha1_base64="hR0O9gC2nQZC/kaL92WprCQBthA=">AAACAnicbVDLSsNAFJ3UV62vqCtxM1iEFkpIRKsboejGlVSwD2hDmUwm7dDJJMxMhBKKG3/FjQtF3PoV7vwbp20W2nrgXg7n3MvMPV7MqFS2/W3klpZXVtfy64WNza3tHXN3rymjRGDSwBGLRNtDkjDKSUNRxUg7FgSFHiMtb3g98VsPREga8Xs1iokboj6nAcVIaalnHpRuK7CrKPNJGozLlyXbOqtA3co9s2hb9hRwkTgZKYIM9Z751fUjnISEK8yQlB3HjpWbIqEoZmRc6CaSxAgPUZ90NOUoJNJNpyeM4bFWfBhEQhdXcKr+3khRKOUo9PRkiNRAznsT8T+vk6jgwk0pjxNFOJ49FCQMqghO8oA+FQQrNtIEYUH1XyEeIIGw0qkVdAjO/MmLpHliOVWrendarF1lceTBITgCJeCAc1ADN6AOGgCDR/AMXsGb8WS8GO/Gx2w0Z2Q7++APjM8frAmUZw==</latexit> (N, ˜ f)=(0.5,0.5) <latexit sha1_base64="MOOUuLVAYu3/3z0qko+otViRIVc=">AAAB7HicbVBNSwMxEJ34WetX1aOXYBEqSNkVqR6LXjxWcNtCu5Rsmm1Ds9klyRbK0t/gxYMiXv1B3vw3pu0etPXBwOO9GWbmBYng2jjON1pb39jc2i7sFHf39g8OS0fHTR2nijKPxiJW7YBoJrhknuFGsHaiGIkCwVrB6H7mt8ZMaR7LJzNJmB+RgeQhp8RYyaukl+OLXqnsVJ058Cpxc1KGHI1e6avbj2kaMWmoIFp3XCcxfkaU4VSwabGbapYQOiID1rFUkohpP5sfO8XnVunjMFa2pMFz9fdERiKtJ1FgOyNihnrZm4n/eZ3UhLd+xmWSGibpYlGYCmxiPPsc97li1IiJJYQqbm/FdEgUocbmU7QhuMsvr5LmVdWtVWuP1+X6XR5HAU7hDCrgwg3U4QEa4AEFDs/wCm9Iohf0jj4WrWsonzmBP0CfP+wSjh8=</latexit> (u, v) •Non-adiabatic propagation: Moisset & Mathis in prep. Fig. 5: Spectrum of available frequencies for IGW at a given colatitude. 2Ω0N0σ Super-inertial Inertia-gravity waves (NTA) Sub-inertial σ-σ+ Inertia-gravity waves (TA) ØTA: most unstable mode of inflectional instability is 2D and independent of both stratification and rotation (Yanase et al. 1993, Deloncle et al. 2007, Arobone & Sarkar 2012). ØNTA introduces a dependency of the 2D fastest growing mode to both these parameters (Park et al. 2021). ØNS3D: pseudo-spectral direct numerical simulation (DNS) code solving NavierStokes equations with the Boussinesq approximation (Deloncle 2014). ØDNS is used to study the non-linear evolution of a horizontal hyperbolic tangent velocity profile in a linearly stratified rotating background. ØFull Coriolis acceleration (NTA) leads to a 2D configuration with 3 velocity components (Toghraei & Billant 2022). ØFour non-dimensional parameters drive the dynamics: In such cases, small-scale processes are affected by non-traditional effects : e.g. inertia-gravity waves (Gerkema & Shrira 2005) and their excitation (Mathis et al. 2014), the dynamics of vortices (Toghraei & Billant 2022) or the inertial instability (Tort et al. 2016, Zeitlin 2018, Park et al. 2021). The TA generally holds as long as the stratification dominates the rotation, i.e. for: ØNT effects widen the range of possible frequencies from [ 2Ω0;N0 ] ' with the TA to [ σ-;σ+ ]' with the NTA at a given colatitude for IGW, see fig. 5, (Gerkema & Shrira 2005). •Convective wave breaking: Moisset & Mathis in prep., Mathis in prep. Non-traditional effect should be taken into account in parametrization of transport and mixing processes. ØInertia-gravity waves (IGW) are a prime actor of angular momentum transport, interacting with numerous other transport processes (Schatzman 1993, Zahn et al. 1997, Lindzen 1981, Lott et al. 2013). ØTheir action is parametrized in stellar evolutionary codes and GCM (Talon & Charbonnel 2005, Achatz et al. 2024). ØIGW breaking: main candidate to explain Quasi-Biennial Oscillation in the equatorial stratosphere (Lott et al. 2023). ØConvective instability: principal mechanism of IGW breaking with the shear instability. (Lindzen 1981, Sutherland 2001). ØLocal, plane-wave model in the anelastic approximation, non-traditional and non-adiabatic, including the large-scale shears. ØBreaking criterion coming from the convective overturning of stable stratification impose a saturation condition on the vertical velocity (Lindzen 1981, Lott & Guez 2013, Mathis 2025). ØShear instabilities are ubiquitous in stars and in planetary atmospheres and oceans and are one of the main source of turbulent transport (Zahn 1992, Carpenter et al. 2012, Lecoanet et al. 2016, Garaud et al. 2020) ØViscous and radiative damping is one of the main channel of transport by IGW. ØLocal, plane-wave model in the Boussinesq approximation, non-traditional and non-adiabatic. ØQuasi-adiabaticity: and are treated as perturbations of order 1 of the adiabatic propagation (Vadas & Fritts 2005). <latexit sha1_base64="fEYB18708xNLzyXDtxlc6Ix7+eA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEqseiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+O4W19Y3NreJ2aWd3b/+gfHjU0nGqGDZZLGLVCahGwSU2DTcCO4lCGgUC28H4dua3n1BpHstHM0nQj+hQ8pAzaqz00JNpv1xxq+4cZJV4OalAjka//NUbxCyNUBomqNZdz02Mn1FlOBM4LfVSjQllYzrErqWSRqj9bH7qlJxZZUDCWNmShszV3xMZjbSeRIHtjKgZ6WVvJv7ndVMTXvsZl0lqULLFojAVxMRk9jcZcIXMiIkllClubyVsRBVlxqZTsiF4yy+vktZF1atVa/eXlfpNHkcRTuAUzsGDK6jDHTSgCQyG8Ayv8OYI58V5dz4WrQUnnzmGP3A+fwBidY3i</latexit> ω <latexit sha1_base64="n6m7Oxf66ELcdLLGSqWs+mPTk3c=">AAAB7XicbVBNSwMxEM3Wr1q/qh69BIvgqeyKVI9FLx4r2A9olzKbZtvYbBKSrFCW/gcvHhTx6v/x5r8xbfegrQ8GHu/NMDMvUpwZ6/vfXmFtfWNzq7hd2tnd2z8oHx61jEw1oU0iudSdCAzlTNCmZZbTjtIUkojTdjS+nfntJ6oNk+LBThQNExgKFjMC1kmt3hiUgn654lf9OfAqCXJSQTka/fJXbyBJmlBhCQdjuoGvbJiBtoxwOi31UkMVkDEMaddRAQk1YTa/dorPnDLAsdSuhMVz9fdEBokxkyRynQnYkVn2ZuJ/Xje18XWYMaFSSwVZLIpTjq3Es9fxgGlKLJ84AkQzdysmI9BArAuo5EIIll9eJa2LalCr1u4vK/WbPI4iOkGn6BwF6ArV0R1qoCYi6BE9o1f05knvxXv3PhatBS+fOUZ/4H3+AJm+jyo=</latexit> ω <latexit sha1_base64="EdTZAOQk8x8CioQ4t//xIEZAX20=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDbbTbt0swm7E6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGt1O/9cS1EbF6xHHC/YgOlAgFo2ilh+C8Vyq7FXcGsky8nJQhR71X+ur2Y5ZGXCGT1JiO5yboZ1SjYJJPit3U8ISyER3wjqWKRtz42ezSCTm1Sp+EsbalkMzU3xMZjYwZR4HtjCgOzaI3Ff/zOimG134mVJIiV2y+KEwlwZhM3yZ9oTlDObaEMi3srYQNqaYMbThFG4K3+PIyaV5UvGqlen9Zrt3kcRTgGE7gDDy4ghrcQR0awCCEZ3iFN2fkvDjvzse8dcXJZ47gD5zPHyvsjSQ=</latexit> b) <latexit sha1_base64="5bCyd0gYv8CeZMXZW/avxKBuZUk=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi1WPRi8cK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVqyppUCaU7ITFMcMmayFGwTqIZiUPB2uH4dua3n5g2XMkHnCQsiMlQ8ohTglZq9XDEkPTLFa/qzeGuEj8nFcjR6Je/egNF05hJpIIY0/W9BIOMaORUsGmplxqWEDomQ9a1VJKYmSCbXzt1z6wycCOlbUl05+rviYzExkzi0HbGBEdm2ZuJ/3ndFKPrIOMySZFJulgUpcJF5c5edwdcM4piYgmhmttbXToimlC0AZVsCP7yy6ukdVH1a9Xa/WWlfpPHUYQTOIVz8OEK6nAHDWgChUd4hld4c5Tz4rw7H4vWgpPPHMMfOJ8/p4mPMw==</latexit> ω <latexit sha1_base64="O2vVvv4bAUfEcNNFmgS/FouRACE=">AAAB+XicbZDLSsNAFIZP6q3WW9Slm2ARXNVEpLosCuJKKvQGbQiT6aQdOpmEmUmhhLyJGxeKuPVN3Pk2TtostPWHgY//nMM58/sxo1LZ9rdRWlvf2Nwqb1d2dvf2D8zDo46MEoFJG0csEj0fScIoJ21FFSO9WBAU+ox0/cldXu9OiZA04i01i4kbohGnAcVIacszzXsv7XjpYyvLLnLMPLNq1+y5rFVwCqhCoaZnfg2GEU5CwhVmSMq+Y8fKTZFQFDOSVQaJJDHCEzQifY0chUS66fzyzDrTztAKIqEfV9bc/T2RolDKWejrzhCpsVyu5eZ/tX6ighs3pTxOFOF4sShImKUiK4/BGlJBsGIzDQgLqm+18BgJhJUOq6JDcJa/vAqdy5pTr9WfrqqN2yKOMpzAKZyDA9fQgAdoQhswTOEZXuHNSI0X4934WLSWjGLmGP7I+PwBMHKTZA==</latexit> FVNT /FV <latexit sha1_base64="cdtRlBi4oIOba2pMRDIG6e65NA4=">AAAB/3icbVDLSsNAFJ3UV62vqODGTbAIdROTItWNUHTjzgr2AU0Ik+kkHTozCTMTocQu/BU3LhRx62+482+cPhbaeuDC4Zx7ufeeMKVEKsf5NgpLyyura8X10sbm1vaOubvXkkkmEG6ihCaiE0KJKeG4qYiiuJMKDFlIcTscXI/99gMWkiT8Xg1T7DMYcxIRBJWWAvPAkyRm8LRS9W4ZjmHgnFw6thuYZcd2JrAWiTsjZTBDIzC/vF6CMoa5QhRK2XWdVPk5FIogikclL5M4hWgAY9zVlEOGpZ9P7h9Zx1rpWVEidHFlTdTfEzlkUg5ZqDsZVH05743F/7xupqILPyc8zRTmaLooyqilEmschtUjAiNFh5pAJIi+1UJ9KCBSOrKSDsGdf3mRtKq2W7Nrd2fl+tUsjiI4BEegAlxwDurgBjRAEyDwCJ7BK3gznowX4934mLYWjNnMPvgD4/MHPe6UUQ==</latexit> ω/(2!0)=0.1 <latexit sha1_base64="aAEl/DYtIxjoGM+hOIRsI16Q1tA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0hEq8eiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1Bqw8GHu/NMDMvTAXXxvO+nMLK6tr6RnGztLW9s7tX3j9o6iRTDBssEYlqh1Sj4BIbhhuB7VQhjUOBrXB0M/Nbj6g0T+SDGacYxHQgecQZNVa699yLXrniud4c5C/xc1KBHPVe+bPbT1gWozRMUK07vpeaYEKV4UzgtNTNNKaUjegAO5ZKGqMOJvNTp+TEKn0SJcqWNGSu/pyY0FjrcRzazpiaoV72ZuJ/Xicz0VUw4TLNDEq2WBRlgpiEzP4mfa6QGTG2hDLF7a2EDamizNh0SjYEf/nlv6R55vpVt3p3Xqld53EU4QiO4RR8uIQa3EIdGsBgAE/wAq+OcJ6dN+d90Vpw8plD+AXn4xtdHo02</latexit> 0.5 <latexit 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sha1_base64="aAEl/DYtIxjoGM+hOIRsI16Q1tA=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0hEq8eiF48V7Qe0oWy2k3bpZhN2N0Ip/QlePCji1V/kzX/jts1Bqw8GHu/NMDMvTAXXxvO+nMLK6tr6RnGztLW9s7tX3j9o6iRTDBssEYlqh1Sj4BIbhhuB7VQhjUOBrXB0M/Nbj6g0T+SDGacYxHQgecQZNVa699yLXrniud4c5C/xc1KBHPVe+bPbT1gWozRMUK07vpeaYEKV4UzgtNTNNKaUjegAO5ZKGqMOJvNTp+TEKn0SJcqWNGSu/pyY0FjrcRzazpiaoV72ZuJ/Xicz0VUw4TLNDEq2WBRlgpiEzP4mfa6QGTG2hDLF7a2EDamizNh0SjYEf/nlv6R55vpVt3p3Xqld53EU4QiO4RR8uIQa3EIdGsBgAE/wAq+OcJ6dN+d90Vpw8plD+AXn4xtdHo02</latexit> 0.5 <latexit 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sha1_base64="9g2n7pKArsdMxGHIGDoQc4XyODc=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0hEqseiF48V7Ae0oWy2k3bp7ibsboRS+he8eFDEq3/Im//GpM1Bqw8GHu/NMDMvTAQ31vO+nNLa+sbmVnm7srO7t39QPTxqmzjVDFssFrHuhtSg4ApblluB3UQjlaHATji5zf3OI2rDY/VgpwkGko4UjzijNpc8168MqjXP9RYgf4lfkBoUaA6qn/1hzFKJyjJBjen5XmKDGdWWM4HzSj81mFA2oSPsZVRRiSaYLW6dk7NMGZIo1lkpSxbqz4kZlcZMZZh1SmrHZtXLxf+8Xmqj62DGVZJaVGy5KEoFsTHJHydDrpFZMc0IZZpntxI2ppoym8WTh+CvvvyXtC9cv+7W7y9rjZsijjKcwCmcgw9X0IA7aEILGIzhCV7g1ZHOs/PmvC9bS04xcwy/4Hx8A4utjUY=</latexit> 0.1 Fig. 6: Comparison of traditional (T) and non-traditional (NT) prescriptions for (solid lines): the damping factor and (dashed lines): the squared vertical wavenumber, at different colatitudes and relative frequencies compared to the rotation rate, with and . (Mathis in prep, Moisset & Mathis in prep.). <latexit sha1_base64="QkhD37n6/LlNZYF8w0sIAdS3Hcc=">AAAB/HicbVDLSgMxFM34rPU12qWbYBHqps4UrW6EohtXWsE+oB2GTJppQ5PMkGSEodRfceNCEbd+iDv/xrSdhbYeuHA4517uvSeIGVXacb6tpeWV1bX13EZ+c2t7Z9fe22+qKJGYNHDEItkOkCKMCtLQVDPSjiVBPGCkFQyvJ37rkUhFI/Gg05h4HPUFDSlG2ki+Xbj1HXgCS5XuHSd95DvHl2e+XXTKzhRwkbgZKYIMdd/+6vYinHAiNGZIqY7rxNobIakpZmSc7yaKxAgPUZ90DBWIE+WNpseP4ZFRejCMpCmh4VT9PTFCXKmUB6aTIz1Q895E/M/rJDq88EZUxIkmAs8WhQmDOoKTJGCPSoI1Sw1BWFJzK8QDJBHWJq+8CcGdf3mRNCtlt1qu3p8Wa1dZHDlwAA5BCbjgHNTADaiDBsAgBc/gFbxZT9aL9W59zFqXrGymAP7A+vwBJiaSiQ==</latexit> N0/(2!0)=5 <latexit sha1_base64="R8MMaMa/fQtfVZrpMiw3tP8HJIs=">AAAB83icbVDLSgNBEJyNrxhfUY9eBoPgKeyKRI9BLx4jmAdkl9A7mU2GzM4O8xDCkt/w4kERr/6MN//GSbIHjRY0FFXddHfFkjNtfP/LK62tb2xulbcrO7t7+wfVw6OOzqwitE0ynqleDJpyJmjbMMNpTyoKacxpN57czv3uI1WaZeLBTCWNUhgJljACxklhOAEpIRyNQmEH1Zpf9xfAf0lQkBoq0BpUP8NhRmxKhSEctO4HvjRRDsowwumsElpNJZAJjGjfUQEp1VG+uHmGz5wyxEmmXAmDF+rPiRxSradp7DpTMGO96s3F/7y+Ncl1lDMhraGCLBcllmOT4XkAeMgUJYZPHQGimLsVkzEoIMbFVHEhBKsv/yWdi3rQqDfuL2vNmyKOMjpBp+gcBegKNdEdaqE2IkiiJ/SCXj3rPXtv3vuyteQVM8foF7yPbzMYkc8=</latexit> ω→ε <latexit sha1_base64="npJK3KzReQ/Jlb8iHA5Cmuib7Ec=">AAAB7nicbVBNS8NAEJ34WetX1aOXxSLUS0lEqseiF48V7Ae0oWy2m3bpZhN2J0IJ/RFePCji1d/jzX/jts1BWx8MPN6bYWZekEhh0HW/nbX1jc2t7cJOcXdv/+CwdHTcMnGqGW+yWMa6E1DDpVC8iQIl7ySa0yiQvB2M72Z++4lrI2L1iJOE+xEdKhEKRtFK7UoPaXpR7JfKbtWdg6wSLydlyNHol756g5ilEVfIJDWm67kJ+hnVKJjk02IvNTyhbEyHvGupohE3fjY/d0rOrTIgYaxtKSRz9fdERiNjJlFgOyOKI7PszcT/vG6K4Y2fCZWkyBVbLApTSTAms9/JQGjOUE4soUwLeythI6opQ5vQLARv+eVV0rqserVq7eGqXL/N4yjAKZxBBTy4hjrcQwOawGAMz/AKb07ivDjvzseidc3JZ07gD5zPHyHjjsw=</latexit> (ω) <latexit sha1_base64="xtE7fYu2XKl7MVhob9jf5ZcXHhE=">AAAB83icbVBNS8NAEJ3Ur1q/qh69LBahXkpSpHosevFYwX5AE8tmu2mX7iZhd1MooX/DiwdFvPpnvPlv3LQ5aOuDgcd7M8zM82POlLbtb6uwsbm1vVPcLe3tHxwelY9POipKJKFtEvFI9nysKGchbWumOe3FkmLhc9r1J3eZ351SqVgUPupZTD2BRyELGMHaSG7VlSKdTOdP9cvSoFyxa/YCaJ04OalAjtag/OUOI5IIGmrCsVJ9x461l2KpGeF0XnITRWNMJnhE+4aGWFDlpYub5+jCKEMURNJUqNFC/T2RYqHUTPimU2A9VqteJv7n9RMd3HgpC+NE05AsFwUJRzpCWQBoyCQlms8MwUQycysiYywx0SamLARn9eV10qnXnEat8XBVad7mcRThDM6hCg5cQxPuoQVtIBDDM7zCm5VYL9a79bFsLVj5zCn8gfX5A/GMkPw=</latexit> (kv2) • Achatz, U., Alexander, M. J., Becker, E., et al. 2024, J. Atmos. Sciences, 81,237 •Arobone, E. & Sarkar, S. 2012, J. Fluid Mech., 703, 29 •Carpenter, J. R., Tedford, E. W., Heifetz, E., & Lawrence, G. A. 2012, Applied Mech. Rev., 64,061001 • Deloncle, A. 2014, Ecole Polytechnique • Deloncle, A., Chomaz, J.-M., & Billant, P. 2007, J. Fluid Mech., 570, 297 •Garaud, P. 2020, ApJ, 901, 146 • Gerkema, T. & Shrira, V. I. 2005, J. Fluid Mech., 529, 195 • Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M., & Van Haren, H. 2008, Reviews of Geophysics, 46,RG2004 • Lecoanet, D., McCourt, M., Quataert, E., et al. 2016, MNRAS, 455, 4274 • Lindzen, R. S. 1981, J. Geophys. Res., 86, 9707 •Lott, F. & Guez, L. 2013, J. Geophys. Res.(Atmospheres), 118,8897 •Lott, F., Rani, R., Podglajen, A., et al. 2023, in EGU General Assembly Conference Abstracts, EGU–2661 ØThe horizontal inflexional instability can lead to a transition to small-scales turbulence via secondary instabilities, not predicted with the TA. ØThe damping of sub-inertial IGW is largely underestimated with the TA, especially at the equator. ØThe convective wave breaking of sub-inertial IGW is overestimated with the TA, especially at the equator. • weakly stratified layers (e.g. deep ocean, mixed layer) •rapid rotations (e.g. young massive and intermediate-mass stars) •equatorial regions The non-traditional (NT) component can significantly alter the dynamics for (Gerkema et al. 2008): <latexit sha1_base64="UfZPnkdQ2H0hCi1VgUSDGo/voto=">AAAB8nicbVBNS8NAEN3Ur1q/qh69BIvgqSQi1WPRi8cK9gPaUDabSbt0sxt2J0IJ/RlePCji1V/jzX/jts1Bqw8GHu/NMDMvTAU36HlfTmltfWNzq7xd2dnd2z+oHh51jMo0gzZTQuleSA0ILqGNHAX0Ug00CQV0w8nt3O8+gjZcyQecphAkdCR5zBlFK/UHyEUEeTwbesNqzat7C7h/iV+QGinQGlY/B5FiWQISmaDG9H0vxSCnGjkTMKsMMgMpZRM6gr6lkiZggnxx8sw9s0rkxkrbkugu1J8TOU2MmSah7Uwojs2qNxf/8/oZxtdBzmWaIUi2XBRnwkXlzv93I66BoZhaQpnm9laXjammDG1KFRuCv/ryX9K5qPuNeuP+sta8KeIokxNySs6JT65Ik9yRFmkTRhR5Ii/k1UHn2Xlz3petJaeYOSa/4Hx8A1ZBkU4=</latexit> ˜ f0 This poster give new insights for the parametrizations of transport and mixing by horizontally sheared flows, radiative and viscous damping and convective breaking of inertia-gravity waves for stellar interiors and planetary atmospheres and oceans, by taking into account the full Coriolis acceleration. • Mathis, S. 2025, A&A, 694,A173 • Mathis, S., Neiner, C., & Tran Minh, N. 2014, A&A, 565, A47 • Park, J.,Prat, V., Mathis, S.,& Bugnet,L. 2021, A&A, 646, A64 •Prat, V., Mathis,S., Augustson, K., et al. 2018, A&A, 615, A106 • Schatzman, E. 1993, A&A, 279, 431 •Sutherland, B. R. 2001, J. Fluid Mech., 429, 343 •Talon, S. & Charbonnel, C. 2005, A&A, 440, 981 •Toghraei, I. & Billant, P. 2022, J. Fluid Mech., 950, A29 •Tort, M., Winters, K. B., Ribstein, B., & Zeitlin, V. 2016, AGU • Vadas,S.L.&Fritts,D.C.2005, J. Geophys. Res., (Atmospheres),110, D15103 • Yanase, S., Flores, C., Métais, O., & Riley, J. J. 1993, Phys. Fluids A, 5,2725 •Zahn, J. P. 1992, A&A, 265, 115 •Zahn, J. P., Talon, S., & Matias, J. 1997, A&A, 322, 320 •Zeitlin, V. 2018, Phys. Fluids, 30, 061701 Fig. 1: Traditional Approximation of the rotation (TA), with the colatitude, and . the traditional and nontraditional components of the rotation vector . in the tangent plane respectively. <latexit sha1_base64="5bCyd0gYv8CeZMXZW/avxKBuZUk=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQi1WPRi8cK9gPaUDbbTbt2sxt2J0IJ/Q9ePCji1f/jzX/jts1BWx8MPN6bYWZemAhu0PO+ncLa+sbmVnG7tLO7t39QPjxqGZVqyppUCaU7ITFMcMmayFGwTqIZiUPB2uH4dua3n5g2XMkHnCQsiMlQ8ohTglZq9XDEkPTLFa/qzeGuEj8nFcjR6Je/egNF05hJpIIY0/W9BIOMaORUsGmplxqWEDomQ9a1VJKYmSCbXzt1z6wycCOlbUl05+rviYzExkzi0HbGBEdm2ZuJ/3ndFKPrIOMySZFJulgUpcJF5c5edwdcM4piYgmhmttbXToimlC0AZVsCP7yy6ukdVH1a9Xa/WWlfpPHUYQTOIVz8OEK6nAHDWgChUd4hld4c5Tz4rw7H4vWgpPPHMMfOJ8/p4mPMw==</latexit> ω <latexit sha1_base64="lxsCryPN6dXNkcOitfoyG04f9Bs=">AAAB9HicbVBNSwMxEM3Wr1q/qh69BIvgqeyKVI9FL96sYD+gu5RsOtuGJtk1yRbK0t/hxYMiXv0x3vw3pu0etPXBwOO9GWbmhQln2rjut1NYW9/Y3Cpul3Z29/YPyodHLR2nikKTxjxWnZBo4ExC0zDDoZMoICLk0A5HtzO/PQalWSwfzSSBQJCBZBGjxFgp8MMo8+8FDEjPnfbKFbfqzoFXiZeTCsrR6JW//H5MUwHSUE607npuYoKMKMMoh2nJTzUkhI7IALqWSiJAB9n86Ck+s0ofR7GyJQ2eq78nMiK0nojQdgpihnrZm4n/ed3URNdBxmSSGpB0sShKOTYxniWA+0wBNXxiCaGK2VsxHRJFqLE5lWwI3vLLq6R1UfVq1drDZaV+k8dRRCfoFJ0jD12hOrpDDdREFD2hZ/SK3pyx8+K8Ox+L1oKTzxyjP3A+fwCJeJH3</latexit> !0 <latexit sha1_base64="WjizO1jcFfphjmbbQnZjH6kFuJo=">AAACBXicbVBNS8NAEN3U7/oV9aiHxSJ4KkmR6kUoevGmglWhCWGznbSLu0nYnQgl9OLFv+LFgyJe/Q/e/Dduaw9+PRh4vDfDzLw4l8Kg5304lanpmdm5+YXq4tLyyqq7tn5pskJzaPNMZvo6ZgakSKGNAiVc5xqYiiVcxTfHI//qFrQRWXqBgxxCxXqpSARnaKXI3Uoi77ARnCroscijgVYlz8wwwD4gi9yaV/fGoH+JPyE1MsFZ5L4H3YwXClLkkhnT8b0cw5JpFFzCsBoUBnLGb1gPOpamTIEJy/EXQ7pjlS5NMm0rRTpWv0+UTBkzULHtVAz75rc3Ev/zOgUmB2Ep0rxASPnXoqSQFDM6ioR2hQaOcmAJ41rYWynvM8042uCqNgT/98t/yWWj7jfrzfO9WutoEsc82STbZJf4ZJ+0yAk5I23CyR15IE/k2bl3Hp0X5/WrteJMZjbIDzhvn6GtmBA=</latexit> f0=2!0cosω <latexit sha1_base64="mBfaAkPRNjDvMFWSfj0UC7PIM78=">AAACDXicbVA9SwNBEN3z2/gVtbRZjIJVuAuiNoJoY6eC0UAuHHubuWTJ7t6xOyeEI3/Axr9iY6GIrb2d/8ZNTKHRBwOP92aYmRdnUlj0/U9vanpmdm5+YbG0tLyyulZe37ixaW441HkqU9OImQUpNNRRoIRGZoCpWMJt3Dsb+rd3YKxI9TX2M2gp1tEiEZyhk6LyTohCtqFIBpF/XAsvFHRY5NPQqMIKPQixC8iicsWv+iPQvyQYkwoZ4zIqf4TtlOcKNHLJrG0GfoatghkUXMKgFOYWMsZ7rANNRzVTYFvF6JsB3XVKmyapcaWRjtSfEwVT1vZV7DoVw66d9Ibif14zx+SoVQid5Qiafy9KckkxpcNoaFsY4Cj7jjBuhLuV8i4zjKMLsORCCCZf/ktuatXgoHpwtV85OR3HsUC2yDbZIwE5JCfknFySOuHknjySZ/LiPXhP3qv39t065Y1nNskveO9fZPqbyw==</latexit> ˜ f0=2!0sinω Fig. 6: Vertical flux of momentum ((a): with the TA; (b): with the NTA) as a function of the co-latitude for different frequencies normalised to the flux in the nonrotating case with and . (Mathis in prep., Moisset & Mathis. in prep.). <latexit sha1_base64="aRXRZTVhYfA/e/ALe33L5dvkTKs=">AAAB/HicbVDLSsNAFJ3UV62vaJdugkWom5oUqS6LbtxZwT6gCWEynbRDZyZhZiKEUH/FjQtF3Poh7vwbJ20W2nrgwuGce7n3niCmRCrb/jZKa+sbm1vl7crO7t7+gXl41JNRIhDuoohGYhBAiSnhuKuIongQCwxZQHE/mN7kfv8RC0ki/qDSGHsMjjkJCYJKS75ZdSUZM3heb7p3DI+hb59VfLNmN+w5rFXiFKQGCnR888sdRShhmCtEoZRDx46Vl0GhCKJ4VnETiWOIpnCMh5pyyLD0svnxM+tUKyMrjIQurqy5+nsig0zKlAW6k0E1kcteLv7nDRMVXnkZ4XGiMEeLRWFCLRVZeRLWiAiMFE01gUgQfauFJlBApHReeQjO8surpNdsOK1G6/6i1r4u4iiDY3AC6sABl6ANbkEHdAECKXgGr+DNeDJejHfjY9FaMoqZKvgD4/MHlS+TcQ==</latexit> ω/(2!0) <latexit sha1_base64="oc3jAtZcOhueh+83+e7AZ5PQE2s=">AAAB+HicbVDLSsNAFJ3UV62PRl26GSyCqzaRUt0IRTeupIJ9QBvCZDpph85MwsxEqKFf4saFIm79FHf+jZM2C209cOFwzr3ce08QM6q043xbhbX1jc2t4nZpZ3dvv2wfHHZUlEhM2jhikewFSBFGBWlrqhnpxZIgHjDSDSY3md99JFLRSDzoaUw8jkaChhQjbSTfLg8UHXFUu/OdK7dWL/l2xak6c8BV4uakAnK0fPtrMIxwwonQmCGl+q4Tay9FUlPMyKw0SBSJEZ6gEekbKhAnykvnh8/gqVGGMIykKaHhXP09kSKu1JQHppMjPVbLXib+5/UTHV56KRVxoonAi0VhwqCOYJYCHFJJsGZTQxCW1NwK8RhJhLXJKgvBXX55lXTOq26j2rivV5rXeRxFcAxOwBlwwQVoglvQAm2AQQKewSt4s56sF+vd+li0Fqx85gj8gfX5A2VlkaA=</latexit> ω/N0=1/4 <latexit sha1_base64="R8MMaMa/fQtfVZrpMiw3tP8HJIs=">AAAB83icbVDLSgNBEJyNrxhfUY9eBoPgKeyKRI9BLx4jmAdkl9A7mU2GzM4O8xDCkt/w4kERr/6MN//GSbIHjRY0FFXddHfFkjNtfP/LK62tb2xulbcrO7t7+wfVw6OOzqwitE0ynqleDJpyJmjbMMNpTyoKacxpN57czv3uI1WaZeLBTCWNUhgJljACxklhOAEpIRyNQmEH1Zpf9xfAf0lQkBoq0BpUP8NhRmxKhSEctO4HvjRRDsowwumsElpNJZAJjGjfUQEp1VG+uHmGz5wyxEmmXAmDF+rPiRxSradp7DpTMGO96s3F/7y+Ncl1lDMhraGCLBcllmOT4XkAeMgUJYZPHQGimLsVkzEoIMbFVHEhBKsv/yWdi3rQqDfuL2vNmyKOMjpBp+gcBegKNdEdaqE2IkiiJ/SCXj3rPXtv3vuyteQVM8foF7yPbzMYkc8=</latexit> ω→ε These non-traditional effects are not included in parametrizations for any value of N0 / ( 2Ω0 ), which can lead to incorrect prescriptions in term of transport and mixing. Four different non-linear evolutions have been identified (see figs. 2 and 3) •Traditional: quasi-steady Kelvin-Helmholtz (KH) vortex (fig. 3 (a)). •Traditional-like: quasi-steady KH vortex, slightly stretched in the transverse direction with generation of positive vorticity (fig. 3 (b)). •Mixed: localised and moderate secondary instabilities develop within the primary KH vortex (fig. 3 (c)). •Non-traditional: strong secondary instabilities destabilize the developing KH vortex into small-scale turbulence (fig. 3 (d)). ØWith increasing NT effects, the KH vortex is stretched in the y direction and thin positive vorticity layers develop (see fig. 4). ØStretched vorticity braids become thinner and new inflection points appear: secondary instabilities are more likely to develop and lead to turbulence. ØEnstrophy analysis show that the turbulence is as if it were threedimensional as NT effects develop a vertical dynamics even in a twodimensional horizontal configuration. ØTransport of heat and chemicals by horizontally sheared zonal flows is expected to be different from the traditional prediction when the turbulence arises. KEY RESULTS ØNT effect: enhanced damping and shorter vertical wavelength for IGW of frequency around and below , especially closer to the Equator (see fig. 6). ØDeposition of angular momentum and interaction with mean-flows occur closer to the source of IGW than what is predicted using TA. <latexit sha1_base64="Fm857na2uLe92FBK5gF4KevIewM=">AAAB8HicbVDLSgNBEOyNrxhfUY9eFoPgKewGiR6DXrwZwTwkWcLspDcZMjO7zMwKIeQrvHhQxKuf482/cZLsQRMLGoqqbrq7woQzbTzv28mtrW9sbuW3Czu7e/sHxcOjpo5TRbFBYx6rdkg0ciaxYZjh2E4UEhFybIWjm5nfekKlWSwfzDjBQJCBZBGjxFjpsdK9EzggPa9XLHllbw53lfgZKUGGeq/41e3HNBUoDeVE647vJSaYEGUY5TgtdFONCaEjMsCOpZII1MFkfvDUPbNK341iZUsad67+npgQofVYhLZTEDPUy95M/M/rpCa6CiZMJqlBSReLopS7JnZn37t9ppAaPraEUMXsrS4dEkWosRkVbAj+8surpFkp+9Vy9f6iVLvO4sjDCZzCOfhwCTW4hTo0gIKAZ3iFN0c5L86787FozTnZzDH8gfP5A/yfj+U=</latexit> 2!0 KEY RESULTS ØNT effects: vertical flux of angular momentum by sub-inertial IGW is reduced compared to the TA, especially near the Equator, because of the stabilizing action of the rotation on the convective instability (see fig. 7). ØThe stabilizing action of the rotation over the convective instability is underestimated by the TA. ØDeposition of angular momentum and interactions with mean-flows should be weaker than what is expected with the TA. KEY RESULTS Sub-inertial regime <latexit sha1_base64="uL9fuNI4TBii4e2Lj/GPCiGBzKs=">AAAB+3icbVDLSsNAFJ3UV62vWJduBotQNzUpUl0W3bizgn1AE8JkOkmHzkzCzEQspb/ixoUibv0Rd/6N0zYLbT1w4XDOvdx7T5gyqrTjfFuFtfWNza3idmlnd2//wD4sd1SSSUzaOGGJ7IVIEUYFaWuqGemlkiAeMtINRzczv/tIpKKJeNDjlPgcxYJGFCNtpMAue4rGHJ1X694dJzEKnLPArjg1Zw64StycVECOVmB/eYMEZ5wIjRlSqu86qfYnSGqKGZmWvEyRFOERiknfUIE4Uf5kfvsUnhplAKNEmhIaztXfExPElRrz0HRypIdq2ZuJ/3n9TEdX/oSKNNNE4MWiKGNQJ3AWBBxQSbBmY0MQltTcCvEQSYS1iatkQnCXX14lnXrNbdQa9xeV5nUeRxEcgxNQBS64BE1wC1qgDTB4As/gFbxZU+vFerc+Fq0FK585An9gff4AWu+TXQ==</latexit> ω/(2!0) <latexit sha1_base64="+ELh3QdJ1JNOWbRX9Xn1ViSEDSU=">AAAB8nicbZDLSgMxFIYz9VbrrerSTbAIrsqMSHVZdOOyir3AdCiZNNOGJpMhOSOUoY/hxoUibn0ad76NmXYW2vpD4OM/55Bz/jAR3IDrfjultfWNza3ydmVnd2//oHp41DEq1ZS1qRJK90JimOAxawMHwXqJZkSGgnXDyW1e7z4xbbiKH2GasECSUcwjTglYy+9rmT3kaGaDas2tu3PhVfAKqKFCrUH1qz9UNJUsBiqIMb7nJhBkRAOngs0q/dSwhNAJGTHfYkwkM0E2X3mGz6wzxJHS9sWA5+7viYxIY6YytJ2SwNgs13Lzv5qfQnQdZDxOUmAxXXwUpQKDwvn9eMg1oyCmFgjV3O6K6ZhoQsGmVLEheMsnr0Lnou416o37y1rzpoijjE7QKTpHHrpCTXSHWqiNKFLoGb2iNwecF+fd+Vi0lpxi5hj9kfP5A8ZFkZg=</latexit> Ratios <latexit sha1_base64="mqy0RXK30FvfXpLsx14LtLwNGCI=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hoduq3nlBpHstHM07Qj+hA8pAzaqz0QM97pbJbcWcgy8TLSRly1Hulr24/ZmmE0jBBte54bmL8jCrDmcBJsZtqTCgb0QF2LJU0Qu1ns0sn5NQqfRLGypY0ZKb+nshopPU4CmxnRM1QL3pT8T+vk5rw2s+4TFKDks0XhakgJibTt0mfK2RGjC2hTHF7K2FDqigzNpyiDcFbfHmZNC8qXrVSvb8s127yOApwDCdwBh5cQQ3uoA4NYBDCM7zCmzNyXpx352PeuuLkM0fwB87nDypnjSM=</latexit> a) <latexit sha1_base64="EdTZAOQk8x8CioQ4t//xIEZAX20=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDbbTbt0swm7E6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGt1O/9cS1EbF6xHHC/YgOlAgFo2ilh+C8Vyq7FXcGsky8nJQhR71X+ur2Y5ZGXCGT1JiO5yboZ1SjYJJPit3U8ISyER3wjqWKRtz42ezSCTm1Sp+EsbalkMzU3xMZjYwZR4HtjCgOzaI3Ff/zOimG134mVJIiV2y+KEwlwZhM3yZ9oTlDObaEMi3srYQNqaYMbThFG4K3+PIyaV5UvGqlen9Zrt3kcRTgGE7gDDy4ghrcQR0awCCEZ3iFN2fkvDjvzse8dcXJZ47gD5zPHyvsjSQ=</latexit> b) <latexit sha1_base64="HHKZgNaJvp7r4jDMHrHEx5yBnjA=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDabTbt0swm7E6GU/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSKUw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmiTTjDdYIhPdDqjhUijeQIGSt1PNaRxI3gqGt1O/9cS1EYl6xFHK/Zj2lYgEo2ilh/C8Vyq7FXcGsky8nJQhR71X+uqGCctirpBJakzHc1P0x1SjYJJPit3M8JSyIe3zjqWKxtz449mlE3JqlZBEibalkMzU3xNjGhszigPbGVMcmEVvKv7ndTKMrv2xUGmGXLH5oiiTBBMyfZuEQnOGcmQJZVrYWwkbUE0Z2nCKNgRv8eVl0ryoeNVK9f6yXLvJ4yjAMZzAGXhwBTW4gzo0gEEEz/AKb87QeXHenY9564qTzxzBHzifPy72jSY=</latexit> d) <latexit sha1_base64="aKF2Utr9MWkaDZcislaSuFOxR0A=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hoduq3nlBpHstHM07Qj+hA8pAzaqz0wM57pbJbcWcgy8TLSRly1Hulr24/ZmmE0jBBte54bmL8jCrDmcBJsZtqTCgb0QF2LJU0Qu1ns0sn5NQqfRLGypY0ZKb+nshopPU4CmxnRM1QL3pT8T+vk5rw2s+4TFKDks0XhakgJibTt0mfK2RGjC2hTHF7K2FDqigzNpyiDcFbfHmZNC8qXrVSvb8s127yOApwDCdwBh5cQQ3uoA4NYBDCM7zCmzNyXpx352PeuuLkM0fwB87nDy1xjSU=</latexit> c) <latexit sha1_base64="mqy0RXK30FvfXpLsx14LtLwNGCI=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDbbSbt0swm7G6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSATXxnW/nZXVtfWNzcJWcXtnd2+/dHDY1HGqGDZYLGLVDqhGwSU2DDcC24lCGgUCW8Hoduq3nlBpHstHM07Qj+hA8pAzaqz0QM97pbJbcWcgy8TLSRly1Hulr24/ZmmE0jBBte54bmL8jCrDmcBJsZtqTCgb0QF2LJU0Qu1ns0sn5NQqfRLGypY0ZKb+nshopPU4CmxnRM1QL3pT8T+vk5rw2s+4TFKDks0XhakgJibTt0mfK2RGjC2hTHF7K2FDqigzNpyiDcFbfHmZNC8qXrVSvb8s127yOApwDCdwBh5cQQ3uoA4NYBDCM7zCmzNyXpx352PeuuLkM0fwB87nDypnjSM=</latexit> a) <latexit sha1_base64="EdTZAOQk8x8CioQ4t//xIEZAX20=">AAAB6XicbVBNS8NAEJ34WetX1aOXxSLopSQi1WPRi8cq9gPaUDbbTbt0swm7E6GE/gMvHhTx6j/y5r9x2+agrQ8GHu/NMDMvSKQw6Lrfzsrq2vrGZmGruL2zu7dfOjhsmjjVjDdYLGPdDqjhUijeQIGStxPNaRRI3gpGt1O/9cS1EbF6xHHC/YgOlAgFo2ilh+C8Vyq7FXcGsky8nJQhR71X+ur2Y5ZGXCGT1JiO5yboZ1SjYJJPit3U8ISyER3wjqWKRtz42ezSCTm1Sp+EsbalkMzU3xMZjYwZR4HtjCgOzaI3Ff/zOimG134mVJIiV2y+KEwlwZhM3yZ9oTlDObaEMi3srYQNqaYMbThFG4K3+PIyaV5UvGqlen9Zrt3kcRTgGE7gDDy4ghrcQR0awCCEZ3iFN2fkvDjvzse8dcXJZ47gD5zPHyvsjSQ=</latexit> b)