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The <zenith> is the point on the <celestial sphere> in the direction of the observer's local upward vertical. In the <horizontal coordinate system>, its <unit vector> is $\mathbf e_U$ and its <altitude> is $90^\circ$. The <zenith> belongs to the observer's site; it is not generally the north <celestial pole>, whose <altitude> depends on <geographical latitude>. A direction's <zenith distance> is its nonnegative <angle> from the <zenith>, so $z=90^\circ-\gamma$. A direction of <altitude> $30^\circ$ therefore has <zenith distance> $60^\circ$. https://www.astro.uvic.ca/~tatum/celmechs/celm6.pdf[Jeremy Tatum: Celestial mechanics, section 6.2, zenith and complementary zenith distance]{ref}.

At the <zenith>, every value of <azimuth> produces $(E,N,U)=(0,0,1)$. There is no horizontal component to identify a bearing: $\operatorname{atan2}(0,0)$ cannot provide a geometrically meaningful <azimuth>, even if software returns a number. This <coordinate singularity> is a defect of the angular description; the viewing direction itself remains regular. The same ambiguity occurs at the opposite downward vertical. Near the <zenith>, two directions at fixed <altitude> separated by a small <azimuth> increment $d\eta$ have separation $\cos\gamma\,|d\eta|$ to first order, with <angles> in <radians>. At <altitude> $89^\circ$, a $1^\circ$ <azimuth> increment spans only about $0.01745^\circ$ on that small arc. An <optical telescope> can consequently require a large bearing change for a small change of pointing near the <zenith>; working with <unit vectors> avoids inventing a preferred bearing at the singular point.


 Zenith (source code)