We　present　a　numerical　method　for　approximating　the　two-dimensional　surface　quasi-geostrophic　equation.　We　first　reformulate　the　equation　into　an　equivalent　system　by　using　the　scalar　auxiliary　variable　approach.　Then　we　propose　first-order　and　second-order　discretization　schemes　for　solving　the　new　surface　quasi-geostrophic　system　in　time.　Furthermore,　we　prove　that　both　of　these　schemes　based　on　the　scalar　auxiliary　variable　approach　satisfy　an　unconditionally　energy　stability　property.　Since　the　method　gives　two　linear　equations　with　constant　coefficients　in　each　time　step.　We　have　an　efficient　approach　and　accurate　for　solving　these　spacial　fractional　diffusion　equations.　Ample　numerical　experiments　are　carried　out　to　validate　the　correctness　of　these　schemes　and　their　accuracy　for　inviscid　problems　and　the　problems　with　small　viscosity.　©　The　Author(s)　2023.