Description
A dispersion relation is derived for driftwave instabilities to include the effect of profile curvature \emph{i.e.} the second order radial variation in pressure $d^2P/dr^2$. Typically in deriving local dispersion relations [S. Ichimaru, CRC Press, 1973], only the first order radial variation in pressure, \emph{i.e.} $dP/dr$ term, is considered. This is because usually fluctuations associated with the drift waves are of the ion Larmor radius $\rho_i$ scale in width radially, and the length-scale $L=(d{\rm ln}P/dr)^{-1}$ of background pressure variation is of the order of machine size ($\sim$minor radius). Given that $\eta=\rho_i/L<<1$ in this situation, only the first order variation in $\eta$ (\emph{i.e.} $dP/dr$ term) is usually considered. However, the situation changes near internal transport barriers and H-mode pedestals where the pressure changes quickly over the radius and $L$ is of the order of a few ion Larmor radii. In such cases, it becomes pertinent to keep higher order terms in $\eta$ such as $d^2P/dr^2$. This formalism remains local and captures the most fundamental effect of profile curvature, excluding effects such as $E\times B$ shear and eddy tilting that are found in fully global [T. Goerler et al., J. Com. Phys. 230 (18), 2011] and partly global treatments [J. Candy et al., Phys. Rev. E 111, L053201, 2025]. The result shows that growth rate depends on $d^2P/dr^2$, including its sign; the modes are damped for concave downwards pressure profiles. The possibility of such a stability dependence to predict the formation of H-mode pedestals is studied.