How can one explain lift generation in this case with the help of circulation theory. In circulation theory, the trailing edge vortex is necessary to generate circulation around airfoil and hence lift. In this case, vortex can not be produced because of water-air interface so how lift is generated
Lift is generated by deviation of the fluid, that implies that a force is applied to it in a direction orthogonal to the flow speed, and then by reaction that the fluid applies the opposite force to the airfoil.
So as long as any object can deviate a flow it creates lift.
Of course it is the case of an airfoil partially submerged.
I think that you may be confusing circulation theory (which is compatible with the present case) and the lifting line trailing vortices analogy.
How circulation theory is compatible with the present case?
In circulation theory, lift is proportional to circulation around airfoil and the circulation around airfoil is developed when the trailing edge vortex forms.