Zeros of successive higher order derivatives of the Riemann zeta function seem to cluster along roughly horizontal lines.

Is there a heuristic explanation of why this happens (especially inside the critical strip)?

R. Spira, in Zero free regions of $\zeta^{(k)}(s)$ wrote "The zeros of $\zeta^{\prime\prime}$ have imaginary part almost exactly equal to those of $\zeta^\prime$, and lie to the right of them." Figure 1 from his paper shows zeros of $\zeta^\prime$ marked with triangles, and zeros of $\zeta^{\prime\prime}$ marked with squares.

Below is a *Mathematica* plot of the argument of $\zeta^{\prime\prime}/\zeta^\prime(s)$ for $1/2\le \sigma\le 1$ and $10^6\le t\le 10^6+20$ (Five strips of height $4$.). The poles at the zeros of $\zeta^\prime(s)$ have to opposite orientation of colors that the zeros of $\zeta^{\prime\prime}(s)$ have.

S.L Skorokhodov, in "Pade Approximants and Numerical Analysis of the Riemann zeta function", Computational Mathematics and Mathematical Physics vol 43 (2003) pp. 1277-1299, formula (7.7) differentiates the Dirichlet series expansion for $\zeta(s)$ term by term (for $\sigma>1$), and deduces "Therefore, it is again natural to expect a zero of the derivative $\zeta^\prime(s)$ to be located on the right [sic] of each finite zero of $\zeta^{\prime\prime}(s)$."

I don't understand this, particularly in regards to zeros in the critical strip.

Binder, Pauli, and Saidak, in "Zeros of High Derivatives of the Riemann Zeta Function", Rocky Mountain J. Math, vol 45 (2015), pp. 903-926 have similar results (Theorem 2.3), again using the Dirichlet series. They see chains of zeros up to order 90:

From Theorem 3 in the seminal paper of Levinson and Montgomery, we can see this happens *on average* over an interval of size, say, $U=\log T$:
$$
\frac{2\pi}{\log T}\left(\sum_{T<\gamma^{(k+1)}<T+\log T}(\beta^{(k+1)}-1/2)-\sum_{T<\gamma^{(k)}<T+\log T}(\beta^{(k)}-1/2)\right)
=\log\log T/2\pi+O(1).
$$
The right side is independent of $k$.

Edit/Correction: The corollary of the Levinson-Montgomery result speaks to the *horizontal* spacing of consecutive higher derivatives, on average independent of $k$. But it does not address Spira's observation about the *vertical* spacing: consecutive higher derivatives at very nearly the same height.