ner_tags: O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation O B-Citation I-Citation I-Citation I-Citation O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O B-Model O O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O B-Model O O O B-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O B-Citation I-Citation I-Citation I-Citation I-Citation I-Citation O O O O O O O O
tokens: fluid with a special cooling function (Lin Youdin 2017 ; Lovascio Paardekooper 2019 ). Dropping the subscript “c” for clarity, our one-fluid model equations in the local shearing box to O ( t s ) are (Laibe Price 2014 ; Lin Youdin 2017 ; Lovascio Paardekooper 2019 ; Chen Lin 2020 ; Paardekooper et al. 2020 ), where f d ≡ ρ d / ρ is the dust fraction. Note that for the isothermal gas we consider, so . The second term on the rhs of Equation ( B5 ) vanishes in a particle disk where ρ g → 0, because solids do not feel pressure gradients. Conversely, for a dust-free gas disk ( ρ g / ρ → 1), we recover the full pressure support from the global disk. The second term in the parentheses on the rhs of Equation ( B6 ) arises from the contribution to the terminal velocity approximation from the large-scale radial pressure gradient in Equation ( B3 ). B.1. Approximate Equilibria As in the two-fluid model, we seek steady, horizontally uniform solutions. The mass, energy, and vertical momentum equations are then where in Equation ( B8 ) we ignore the term a posteriori for consistency with the small t s approximation used to derive the one-fluid model. These equations may then be solved for constant τ s = t s / f g = St/Ω to yield From here, it is clear that v z = O (St), so it is self-consistent to ignore the term. Equations ( B10 )–( B12 ) are in fact the same solutions as in the full two-fluid model in the limit St ≪ 1 (see Equations ( 19 )–( 21 ); recall β → St for St → 0 and note v z = f d v d z ). The horizontal momentum equations are where we ignore the quadratic term in Equation ( B13 ) to obtainThis is expected on physical grounds for tightly coupled dust (St → 0). In this limit, the mixture behaves close to a single fluid with orbital velocity depending on the level of dust enrichment. For ε → 0, we have a pressure-supported gas disk at sub-Keplerian velocity (assuming η > 0), while ε → ∞ corresponds to a particle disk on exactly Keplerian orbits, because then v y → 0. Next, we use Equations ( B15 ), ( B14 ), and ( B11 ) to obtainNotice the radial velocity of the dust–gas mixture’s center of mass depends on height. It is only zero at the midplane and for ∣ z ∣ → ∞ where ε → 0. For η > 0, the specific angular momentum decreases with increasing ∣ z ∣: the mixture gains pressure support as it becomes more gas-rich away from the midplane. This means that as a parcel of the mixture settles, it finds itself having an angular-momentum deficit compared to its surrounding; it thus drifts inwards ( v x 0), as indicated by Equation ( B16 ). B.2. Linearized Equations We linearize the one-fluid equations about the above basic state, with nonuniform v x ( z ), v y ( z ), and v z ( z ). As in the main text, we assume axisymmetric perturbations in the form of and similarly for other variables. The linearized equations are whereand recall Q = δ ε / ε and W = δ ρ g / ρ g = δ P / P . B.3. Mode Energetics Following Ishitsu et al. ( 2009 ), we multiply the x , y , and