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Rotational mixing, the key process in stellar evolution, transports angularmomentum and chemical elements in stellar radiative zones. In the past twodecades, an emphasis has been placed on the turbulent transport induced by thevertical shear instability. However, instabilities arising from horizontalshear and the strength of the anisotropic turbulent transport that they maytrigger remain relatively unexplored. This paper investigates the combinedeffects of stable stratification, rotation, and thermal diffusion on thehorizontal shear instabilities in the context of stellar radiative zones. Theeigenvalue problem describing linear instabilities of a flow with ahyperbolic-tangent horizontal shear profile was solved numerically for a widerange of parameters. As a first step, we consider a polar $f$-plane where thegravity and rotation vector are aligned. Two types of instabilities areidentified: the inflectional and inertial instabilities. The inflectionalinstability that arises from the inflection point is the most unstable when ata zero vertical wavenumber and a finite wavenumber in the streamwise directionalong the imposed-flow direction. The three-dimensional inflectionalinstability is destabilized by stratification, while it is stabilized bythermal diffusion. The inertial instability is rotationally driven, and a WKBJanalysis reveals that its growth rate reaches the maximum $\sqrt{f(1-f)}$ inthe inviscid limit as the vertical wavenumber goes to infinity, where $f$ isthe dimensionless Coriolis parameter. The inertial instability for a finitevertical wavenumber is stabilized as the stratification increases, whereas itis destabilized by the thermal diffusion. Furthermore, we found aself-similarity in both the inflectional and inertial instabilities based onthe rescaled parameter $PeN^2$ with the P\'{e}clet number $Pe$ and theBrunt-V\"{a}is\"{a}l\"{a} frequency $N$.

Horizontal shear instabilities in rotating stellar radiation zones: I. Inflectional and inertial instabilities and the effects of thermal diffusion