Let $k$ be a number field (a finite extension of $\mathbb{Q}$). Let $p$ be a prime. By saying "$\mathbb{Z}_p$-extensions", we mean Galois extensions $K/k$ of Galois group isomorphic to $\mathbb{Z}_p$.
In R. Greenberg's article "The Iwasawa invariants of $\Gamma$-extensions of a fixed number field" (1973), he considered the set $\mathcal{E}$ of all $\mathbb{Z}_p$-extensions of $k$. Moreover, he put a topology on $\mathcal{E}$ by giving a topological basis as follows: Fix any $K \in \mathcal{E}$. For any integer $n \geq 0$, consider $$ \mathcal{E}(K,n) := \{ K^{\prime} \in \mathcal{E}: [K \cap K^{\prime}: k] \geq p^n \}. $$ One varies $K$ and $n$ as positive integers to get a topological basis for $\mathcal{E}$.
My question is: how to explicitly construct elements in $\mathcal{E}(K,n > 0)$?
Example: Let $k = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field. Then it has two "independent" $\mathbb{Z}_p$-extensions, namely the cyclotomic one $k_{\mathrm{cyc}}$ and the other is the anticyclotomic one $k_{\mathrm{ac}}$. We see $k_{\mathrm{cyc}} \cap k_{\mathrm{ac}} = k$, hence $$ k_{\mathrm{ac}} \in \mathcal{E}(k_{\mathrm{cyc}}, 0), \quad k_{\mathrm{ac}} \not\in \mathcal{E}(k_{\mathrm{cyc}}, n > 0). $$ Then in this case my question is that can we construct explicitly $\mathbb{Z}_p$-extensions in the "nontrivial case", i.e. in $\mathcal{E}(k_{\mathrm{cyc}}, n > 0)$? Here by saying "explicit", I mean besides proofs on existence, can we get an explicit construction?
"Example": For general number fields $k$, the only $\mathbb{Z}_p$-extension I know is the cyclotomic one, so I cannot give any further examples. Therefore,
Another question: Besides cyclotomic $\mathbb{Z}_p$-extensions, do we have other explicit examples of $\mathbb{Z}_p$-extensions that we familiar with? Both systematically constructed examples and ad-hoc examples are welcome!
Since the only "explicit" $\mathbb{Z}_p$-extensions I know is $\mathbb{Q}_{\mathrm{cyc}}$, the above $k_{\mathrm{cyc}}$ and $k_{\mathrm{ac}}$, I have no idea on how to construct elements in $\mathcal{E}(k_{\mathrm{cyc}}, n > 0)$.
I am so sorry if this post is too trivial for this site.
