tokens: induced by turbulence (e.g. Raettig, Klahr Lyra 2015 ; Manger Klahr 2018 ) can concentrate dust particles (Barge Sommeria 1995 ; Klahr Henning 1997 ; Cuzzi, Hogan Shariff 2008 ) and seed planetesimal formation through streaming instability or gravitational instability (Youdin Goodman 2005 ; Johansen et al. 2007 ; Chiang Youdin 2010 ), whereas the growth of streaming instability can be substantially diminished by a moderate level of turbulent viscosity (Chen Lin 2020 ; Umurhan, Estrada Cuzzi 2020 ). Turbulence also influences the radial migration of planets, as well as the flow morphology and gap formation around them (Nelson Papaloizou 2004 ; Papaloizou, Nelson Snellgrove 2004 ). Therefore, understanding the origin and characteristics of turbulence in PPDs is essential to many aspects of planet formation and evolution. The lack of magnetohydrodynamic (MHD) turbulence in a PPD led to a surge in the interest of purely hydrodynamic instabilities (Lyra Umurhan 2019 ; Weiss, Bai Fu 2021 ). Among the most explored are the vertical shear instability (VSI; Nelson, Gressel Umurhan 2013 , hereafter N13 ; Lin Youdin 2015 ; Latter Papaloizou 2018 , hereafter LP18 ; Cui Bai 2020 ), the convective overstability in its linear (Klahr Hubbard 2014 ; Lyra 2014 ; Latter 2016 ) and non-linear (subcritical baroclinic instability; Klahr Bodenheimer 2003 ; Petersen, Julien Stewart 2007a ; Petersen, Stewart Julien 2007b ; Lesur Papaloizou 2010 ) phases, and the zombie vortex instability (Marcus et al. 2013 , 2015 ; Lesur Latter 2016 ; Umurhan, Shariff Cuzzi 2016b ). These instabilities set in under certain thermodynamic and structural conditions, thereby operating at distinct regions of PPDs (Malygin et al. 2017 ; Lyra Umurhan 2019 ; Pfeil Klahr 2019 ). The VSI is of particular interest as it extends a large portion of the disc (e.g. Lin Youdin 2015 ; Lyra Umurhan 2019 ). The VSI is inherited from the Goldreich–Schubert–Fricke instability (Goldreich Schubert 1967 ; Fricke 1968 ) and is initially discovered in the context of differentially rotating stars. Its importance to accretion discs was later explored by Urpin Brandenburg ( 1998 ), Urpin ( 2003 ), and Arlt Urpin ( 2004 ). The applicability of the VSI to PPDs has been demonstrated only recently in N13 . It quickly drew intensive interests (e.g. Stoll Kley 2014 ; Barker Latter 2015 ; Umurhan, Nelson Gressel 2016a ; LP18 ; Lin 2019 ; Cui Bai 2020 ; Flock et al. 2020 ; Schäfer, Johansen Banerjee 2020 ; Flores-Rivera et. al. 2020 ) and is considered to be a promising hydrodynamic mechanism in driving turbulence in PPDs. A differentially rotating disc with Keplerian profile is stable according to the Rayleigh criterion (Chandrasekhar 1961 ). The presence of vertical shear can destabilize inertial waves in a vertically global disc model (Barker Latter 2015 ). Nevertheless, a fluid element also experiences stabilizing effects from vertical buoyancy, which impedes the VSI growth. This can be overcome by
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