jats:pWe consider numerically the transition to turbulence and associated mixing in stratified shear flows with initial velocity distribution jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline1" />jats:tex-math$U―(z,0)ex=U0ex\tanh (z/d)$</jats:tex-math></jats:alternatives></jats:inline-formula> and initial density distribution jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline2" />jats:tex-math$𝜌―(z,0)=𝜌0[1-\tanh (z/𝛿)]$</jats:tex-math></jats:alternatives></jats:inline-formula> away from a hydrostatic reference state jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline3" />jats:tex-math$𝜌r\gg 𝜌0$</jats:tex-math></jats:alternatives></jats:inline-formula>. When the ratio jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline4" />jats:tex-math$R=d/𝛿$</jats:tex-math></jats:alternatives></jats:inline-formula> of the characteristic length scales over which the velocity and density vary is equal to one, this flow is primarily susceptible to the classic well-known Kelvin–Helmholtz instability (KHI). This instability, which typically manifests at finite amplitude as an array of elliptical vortices, strongly ‘overturns’ the density interface of strong initial gradient, which nevertheless is the location of minimum initial gradient Richardson number jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline5" />jats:tex-math$Rig(0)=Rib=g𝜌0d/𝜌rU02$</jats:tex-math></jats:alternatives></jats:inline-formula>, where jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline6" />jats:tex-math$Rig(z)=-([g/𝜌r]d𝜌―/dz)/(dU―/dz)2$</jats:tex-math></jats:alternatives></jats:inline-formula> and jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline7" />jats:tex-math$Rib$</jats:tex-math></jats:alternatives></jats:inline-formula> is a bulk Richardson number. As is well known, at sufficiently high Reynolds numbers (jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline8" />jats:tex-math$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>), the primary KHI induces a vigorous but inherently transient burst of turbulence and associated irreversible mixing localised in the vicinity of the density interface, leading to a relatively well-mixed region bounded by stronger density gradients above and below. We explore the qualitatively different behaviour that arises when jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline9" />jats:tex-math$R\gg 1$</jats:tex-math></jats:alternatives></jats:inline-formula>, and so the density interface is sharp, with jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline10" />jats:tex-math$Rig(z)$</jats:tex-math></jats:alternatives></jats:inline-formula> now being maximum at the density interface jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline11" />jats:tex-math$Rig(0)=RRib$</jats:tex-math></jats:alternatives></jats:inline-formula>. This flow is primarily susceptible to Holmboe wave instability (HWI) (Holmboe, jats:italicGeophys. Publ.</jats:italic>, vol. 24, 1962, pp. 67–113), which manifests at finite amplitude in this symmetric flow as counter-propagating trains of elliptical vortices above and below the density interface, thus perturbing the interface so as to exhibit characteristically cusped interfacial waves which thereby ‘scour’ the density interface. Unlike previous lower-jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline12" />jats:tex-math$Re$</jats:tex-math></jats:alternatives></jats:inline-formula> experimental and numerical studies, when jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline13" />jats:tex-math$Re$</jats:tex-math></jats:alternatives></jats:inline-formula> is sufficiently high the primary HWI becomes increasingly more three-dimensional due to the emergence of shear-aligned secondary convective instabilities. As jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline14" />jats:tex-math$Re$</jats:tex-math></jats:alternatives></jats:inline-formula> increases, (i) the growth rate of secondary instabilities appears to saturate and (ii) the perturbation kinetic energy exhibits a jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline15" />jats:tex-math$k-5/3$</jats:tex-math></jats:alternatives></jats:inline-formula> spectrum for sufficiently large length scales that are influenced by anisotropic buoyancy effects. Therefore, at sufficiently high jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline16" />jats:tex-math$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>, vigorous turbulence is triggered that also significantly ‘scours’ the primary density interface, leading to substantial irreversible mixing and vertical transport of mass above and below the (robust) primary density interface. Furthermore, HWI produces a markedly more long-lived turbulence event compared to KHI at a similarly high jats:inline-formulajats:alternatives<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112016004882_inline17" />jats:tex-math$Re$</jats:tex-math></jats:alternatives></jats:inline-formula>. Despite their vastly different mechanics (i.e. ‘overturning’ versus ‘scouring’) and localisation, the mixing induced by KHI and HWI is comparable in both absolute terms and relative efficiency. Our results establish that, provided the flow Reynolds number is sufficiently high, shear layers with sharp density interfaces and associated locally high values of the gradient Richardson number may still be sites of substantial and efficient irreversible mixing.</jats:p>