One of the strengths of the discontinuous Galerkin (DG) method has been its balance between accuracy and robustness, which stems from DG’s intrinsic (upwind) dissipation being biased towards high frequencies/wave numbers. This is particularly useful in high Reynolds-number flow simulations wherelimitations on mesh resolution typically lead to potentially unstable under-resolved scales. In continuous Galerkin (CG) discretisations, similar properties are achievable through the addition of artificial difusion, such as spectral vanishing viscosity (SVV). The latter, although recognised as very useful in CG-based high-fidelity turbulence simulations, has been observed to be sub-optimal when compared toDG at intermediate polynomials orders (P⇡≈3). In this paper we explore an alternative stabilisation approach by the introduction of a continuous interior penalty on the gradient discontinuity at elemental boundaries, which we refer to as a gradient jump penalisation (GJP). Analogous to DG methods, this introduces a penalisation at the elemental interfaces as opposed to the interior element stabilisation of SVV. Detailed eigen analysis of the GJP approach shows its potential as equivalent (sometimes superior) to DG dissipation and hence superior to previous SVV approaches. Through eigenanalysis, a judicious choice of GJP’sP-dependent scaling parameter is made and found to be consistent with previous a-priori error analysis. The favourable properties of the GJP stabilisation approach are also supported by turbulent flow simulations of the incompressible Navier-Stokes equation, as we achieve high-quality flow solutions atP= 3 using GJP, whereas SVV performs marginally worse atP= 5 with twice as many degrees of freedom in total.