Moshe, Aharon, Miriam

Question

Our sages say that there are three crowns - Keser Torah, Keser Kehunah, Keser Malchus - and that Moshe, Aharon, and Miriam represented these three. Moshe gave us the Torah, Aharon was the first Kohen Gadol and the father of all of them, Miriam was the ancestress of all the kings of Malchus Beis David, through her husband from Yehudah.

Our sages also say that the three great miracles in the Midbar were in the merit of these three: Moshe, the manna; Aharon, the ענני הכבוד, the clouds of glory that surrounded and protected the camp; Miriam, the באר מרים, the well of water used by the camp.

[I don't have references for all this handy but can probably find them if people need.]

Question: What do these have to do with one another? That is, can one explain why Torah = manna, Kehunah = clouds of glory, Malchus = water? I'm assuming it's not a coincidence that these three people are associated with these two different lists.

Update: Actually, I see that the miraculous third thing is not really the water - water is always the same - it's the well, the source, that followed them around. Whereas the first two, mann is itself miraculous and so are the "clouds".
Now I think about it, rain can follow people around, no problem. Somehow a well is different.

Answer 1

There seem to be evident strong connections, I do not know if there is a midrash, which directly answers the question. For example, Torah provides nourishment to the soul, just like manna provided nuorishment to the body. For 40 days that Moshe was on mt. Sinai he did not eat or drink, yet remained nourished from learning the Torah (more on this in sefer Maalos haTorah from r. Avraham Ragaler, brother of Vilna Gaon).

Kohanim serve to protect the faith of the Jewish people by maintaining kosher service, giving blessings, teaching the Torah, etc., while the clouds of glory physically protected the Jewish camp. As we know, a proper service by a Kohen Gadol ensured that there were no accidental deaths, which can break people's faith. Aharon accompanied Moshe to speak with pharao, as a guard. Aharon himself pursued peace (Pirkei Avos 1:12), and there are many statements in HaZaL on how peace protects faith (Yebamos 65b, Vayikra Rabba 9:9, Bamidbar Rabba 11:7, Devarim Rabba 5:15, Sifrei Bamidbar 42:2, etc.).

Finally, the king is supposed to draw the nation towards proper behavior and solidify connection with G-d (source of life), just like water is a medium that draws nutrients to every limb of the body and plants. The life sprouted after Adam prayed for water (i.e. everything was ready, but not yet in the right place, until water drew nutrients to the seeds). Similarly, we learn from Miriam / Puah who drew other midwifes to protect Jewish boys, bring her mother to nurse Moshe in pharaoh's palace, and after going through the sea she drew other women in song and dance as stated in the book of Shemos.

Answer 2

Possibly, one has to start with Avraham to answer the question, as in gemara Bava Metzia 86b: the school of Rabbi Yishmael taught: In reward for three acts of hospitality that Abraham performed for the angels, his descendants merited three rewards. The Gemara elaborates: In reward for providing them with curd and milk (Bereshis 18:8), the Jewish people merited the manna; in reward for: “And he stood [omed] by them (Bereshis 18:8),” the Jews merited the pillar [amud] of cloud; in reward for Abraham saying: “Let now a little water be fetched (Bereshis 18:4),” they merited the well of Miriam.

We know, that Moshe, Aharon, and Miriam excerted themselves in similar 3 ways. Moshe went to first to Mitzrayim, then to pharaoh to secure the Jewish people's geula and afterwards he went to Shamayim to provide Jewish people with the Torah, like Avraham exerted himself with getting curd and milk. Aharon stood by Moshe when bringing news of geulah to Jewish people, then accompanying him to pharaoh, and then during the war with Amalek, like Avraham stood by his guests. Miriam as Puah convinced Jewish midwives to save the boys from being thrown into the water of Nile, she convinced the daughter of pharaoh that her brother Moshe be nursed by his mother after he was saved from the river, and she drew women for a thanksgiving song and dance after splitting of the sea, i.e. she showed power over water, like Avraham did providing it to guests on a hot day (gemara has more details on all of these).