Why are zero values added in the FFT of a concatenated noise signal?
You're running into a property of the DFT that is usually used in the opposite direction: stuffing zeros between samples in one domain results in replication of the entire sequence in the opposite domain. Let's start in the frequency domain with the signal you plotted, $X[k]$, and look at the inverse transform to see what time-domain signal it corresponds to.

$$
x[n] = \sum_{k=0}^{N-1}X[k] e^{j 2 \pi k n/N}
$$

$N$ is the length of the DFT that you apply to the data. Let's say that $X[k]$ consists of sparse values with $M$ zeros between each one, so that $X[k] = 0\ \forall\ k \neq kM$. The above then looks like:

$$
x[n] = \sum_{k=0}^{\left \lfloor{\frac{N-1}{M}}\right\rfloor}X[k] e^{j 2 \pi k M n/N}
$$

Do some rearranging within the exponent to get:

$$
x[n] = \sum_{k=0}^{\left \lfloor{\frac{N-1}{M}}\right\rfloor}X[k] e^{j 2 \pi \frac{k n}{\frac{N}{M}}}
$$

This is nothing but a $\frac{N}{M}$-point inverse DFT of the nonzero samples in your frequency-domain signal $X[k]$. Due to the periodicity property of the DFT, you can then show that $x[n]$ is periodic with period $\frac{N}{M}$, which is exactly the form of signal that you shoved into the DFT to begin with.

Summary: Replication of the signal in one domain results in a zero-stuffed signal in the opposite domain. This is used in the theory behind interpolating filters and other multirate processing. There is a dual property where if you downsample a signal (by removing $M$ out of every $N$ samples), the data in the opposite domain collapses on top of one another and adds (resulting in aliasing).