Use simulated lighting for the background density

rot[x_, y_] = Curl[{Cos[x + y ^ 3], Sin[y + x ^ 3]}, {x, y}];

StreamDensityPlot[{{Cos[x + y ^ 3], Sin[y + x ^ 3]}, Log[Abs[rot[x, y]] + 1]}, {x, -2, 2}, {y, -2, 2}, StreamPoints -> Fine, ColorFunction -> "RedBlueTones", MaxRecursion -> 4, LightingAngle -> Automatic, StreamStyle -> Black, StreamColorFunction -> None]

-Graphics-