The “second Kleene algebra” $\mathcal{K}_2$ is defined, e.g. here on nLab, or in section 1.4.3 of van Oosten's book *Realizability: an Introduction to its Categorical Side* (2008), or as example 3.4 of the notes “Realizability” by Thomas Streicher (2017–2018), or (implicitly) in ¶1.9.12 of Troelstra's *Metamathematical Investigations of Intuitionistic Arithmetic and Analysis* (1973), or in various other places. Let me reproduce the essential part of the definition for readers' convenience:

Let $\mathcal{B} = \mathbb{N}^{\mathbb{N}}$ denote Baire space, endowed with its usual product topology. Given $\alpha \in \mathcal{B}$, define $F_\alpha$ a partial function from $\mathcal{B}$ to $\mathbb{N}$ by $$ F_\alpha(\beta) = n\quad\text{iff}\quad\exists k\in\mathbb{N}.(\alpha(\bar\beta\upharpoonright k)=n+1 \land \forall \ell<k.(\alpha(\bar\beta\upharpoonright k)=0)) $$ (undefined if no such $n$ exists) where $\bar\beta \upharpoonright k$ denotes an integer encoding the finite sequence $\langle\beta(0),\ldots,\beta(k-1)\rangle$. This defines a continuous function $U \to \mathbb{N}$ with $U \subseteq \mathcal{B}$ open, and any such function is of the form in question.

Now let $$ \alpha\bullet\beta = (n \mapsto F_\alpha(\langle n\rangle \smallfrown \beta)) $$ provided $F_\alpha(\langle n\rangle \smallfrown \beta)$ is defined for every $n$, undefined otherwise, where $\langle n\rangle \smallfrown \beta$ denotes the function $0 \mapsto n$ and $k+1 \mapsto \beta(k)$. Then $\mathcal{K}_2$ is $\mathcal{B}$ endowed with this partial operation $\bullet$.

This definition is not complicated, but it is… extremely opaque, and none of the sources I've cited bother to explain why we define things in this particular way ($F_\alpha(\beta)$ searches for the first nonzero value among the values returned by $\alpha$ on the finite subsequences of $\beta$ and subtracts one to it: why do we want to do precisely this? rather than, say, return the index $k$ where this nonzero value was found), apart from the fact that “it works” (it gives a partial combinatory algebra).

Kleene's *first* algebra $\mathcal{K}_1$ (namely $\mathbb{N}$ with the operation $e\bullet n = \varphi_e(n)$ for some standard enumeration $\varphi$ of partial recursive functions) is easy to develop an intuition for: it's the set of computable functions, i.e., computer programs, and the operation is that of taking data and feeding it to a computer program. I understand that the idea behind $\mathcal{K}_2$ is to do something similar for continuous functions on Baire space (and that $F_\alpha$ must play a role analogous to $\varphi_e$), but this doesn't really help me explain the particulars of the definition above (I feel like if in the case of $\mathcal{K}_1$ I'd been described what a Turing machine was without any explanation as to how this definition was reached). So:

**Question:** What's the intuition behind the definition of $\mathcal{K}_2$ and the reason for this particular choice of definition? Has anyone written an introduction to $\mathcal{K}_2$ analogous to what a first course on computability theory would be on $\mathcal{K}_1$?