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A process for using curvature invariants is applied to evaluate the accelerating Natario warp drive. Curvature invariants are independent of coordinate bases and plotting the invariants is free of coordinate mapping distortions. While previous works focus mainly on the mathematical description of the warp bubble, plotting curvature invariants provides a novel pathway to investigate the Natário spacetime and its characteristics. For warp drive spacetimes, there are four independent curvature invariants the Ricci scalar, r_1, r_2, and w_2. The invariant plots demonstrate how each curvature invariant evolves over the parameters of time, acceleration, skin depth and radius of the warp bubble. They show that the Ricci scalar has the greatest impact of the invariants on the surrounding spacetime. They also reveal key features of the Natario warp bubble such as a flat harbor in the center of it, a dynamic wake, and the internal structures of the warp bubble.