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Alternatively, we can also use the harmonic decomposition of Endlich and Penco (2017) to discuss these three families of modes. In this case the angular harmonics Y ℓ , j ​ ( x ) Y^{\ell,j}(x) are described by two quantum numbers j j and ℓ \ell (and the radial function R ℓ , n ​ ( y ) R_{\ell,n}(y) is specified by ℓ \ell and the radial overtone n n ). The quantum number j j specifies the total angular momentum of the perturbation while ℓ \ell describes the orbital angular momentum. Regularity of these harmonics requires that the integer m m must be such that | m | ≤ j |m|\leq j and that ℓ ∈ ℕ 0 \ell\in\mathbb{N}_{0} is constrained to take one of three values, namely ℓ ∈ { j − 1 , j , j + 1 } \ell\in\{j-1,j,j+1\} . Each one of these three ℓ \ell ’s represents one of the 3 families of perturbations. Once we turn-on the background rotation a a , this separation ansatz and associated harmonic decomposition are no longer valid. However, we still have three sectors of perturbations that are continuously connected to the three ℓ = { j − 1 , j , j + 1 } \ell=\{j-1,j,j+1\} families when a → 0 a\to 0 . Accordingly, if we wish so, we can designate these three families by the ℓ = j − 1 \ell=j-1 , ℓ = j \ell=j and ℓ = j + 1 \ell=j+1 families

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