13. As shown in Figure 2, in the regular quadrilateral prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, it is known that the tangent value of the angle formed by $A B_{1}$ and the base $A_{1} B_{1} C_{1} D_{1}$ is $a$. Then the tangent value of the dihedral angle $A-B_{1} D_{1}-A_{1}$ is $\qquad$
13. $\sqrt{2} a$.

Let the side length of the base and the height of a regular quadrilateral prism be $x$ and $y$, respectively. Then $\frac{y}{x}=a$. Let the intersection point of the diagonals of the base be O. Then the plane angle of the dihedral angle $A-B_{1} D_{1}-A_{1}$ is $\angle A O A_{1}$.
Thus, $\tan \angle A O A_{1}=\frac{\sqrt{2} y}{x}=\sqrt{2} a$.