  The study of $P$-polynomial association schemes (distance-regular graphs) and
$Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial
association schemes, has been a central theme not only in the theory of
association schemes but also in the whole study of algebraic combinatorics in
general. Leonard's theorem (1982) says that the spherical functions (or the
character tables) of $P$- and $Q$-polynomial association schemes are described
by Askey-Wilson orthogonal polynomials or their relatives. These polynomials
are one-variable orthogonal polynomials. It seems that the new attempt to
define and study higher rank $P$- and $Q$-polynomial association schemes had
been hoped for, but had gotten only limited success. The first very successful
attempt was initiated recently by Bernard-Cramp\'{e}-d'Andecy-Vinet-Zaimi
[arXiv:2212.10824], and then followed by Bannai-Kurihara-Zhao-Zhu
[arXiv:2305.00707]. The general theory and some explicit examples of families
of higher rank (multivariate) $P$- and/or $Q$-polynomial association schemes
have been obtained there. The main purpose of the present paper is to prove
that some important families of association schemes are shown to be bivariate
$Q$-polynomial. Namely, we show that all the nonbinary Johnson association
schemes and all the attenuated space association schemes are bivariate
$Q$-polynomial. It should be noted that the parameter restrictions needed in
the previous papers are completely lifted in this paper. Our proofs are done by
explicitly calculating the Krein parameters of these association schemes. At
the end, we mention some speculations and indications of what we can expect in
the future study.
