  Two-particle, pair-number correlation distributions on two-dimensional
transverse momentum ($p_{t1},p_{t2}$) constructed from the particle production
in relativistic heavy-ion collisions allow access to dynamical processes in
these systems beyond what can be studied with angular correlations alone. Only
a few measurements of this type have been reported in the literature and
phenomenological models, which facilitate physical interpretation of the
correlation structures, are non-existent. On-going effort at the Relativistic
Heavy-Ion Collider (RHIC) will provide a significant volume of these
correlation measurements in the future. In anticipation of these new data two
phenomenological models are developed which describe two-dimensional 2D
correlation distributions on transverse momentum. One model is based on a
collision event-by-event fluctuating blast wave. The other is based on
event-by-event fluctuations in fragmenting color-flux tubes and in jets. Both
models are shown to be capable of accurately describing the measured
single-particle $p_t$ distributions for minimum-bias Au+Au collisions at
$\sqrt{s_{\rm NN}} = 200$~GeV. Both models are then applied to preliminary,
charged-particle correlation measurements on 2D transverse momentum. The
capabilities of the two models for describing the overall structure of these
correlations, the stability of the fitting results with respect to collision
centrality, and the resulting trends of the dynamical fluctuations are
evaluated. In general, both phenomenological models are capable of
qualitatively describing the major correlation structures on transverse
momentum and can be used to establish the required magnitudes and centrality
trends of the fluctuations. Both models will be useful for interpreting the
forthcoming correlation data from the RHIC.
