3.1. Fast and Slow Timescales in the MLM
 Figure 1 shows LES-tuned MLM solutions for N = 150,50 and 30 cm−3 starting with the specified RF01 initial conditions, for which zi(t = 0) = 840 m, the cloud base zb(0) ≈ 640 m, and the air-sea virtual temperature difference ΔTv0(0) ≈ −2.9 K (i. e. an unstable boundary layer with positive surface buoyancy flux). In all three cases, the evolution is similar. We see a roughly 12-hour period of rapid thermodynamic adjustment during which zb rises to nearly 800 m, LWP drops to 20 g m−2, and we decreases to 4 mm s−1, while inversion height changes by a relatively modest 50 m. After that, zi slowly increases, along with all the thermodynamic variables. Even after 5 days, zi is far from a steady state. This shows the two adjustment timescales in the MLM.
Figure 1. LES-tuned MLM time series plots for DYCOMS-RF01 case with droplet concentrations N = 150,50,30 cm−3 and control zi(0) = 840 m.
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 As explained by Uchida et al. , the differences between the simulations are mainly due to the parameterized effect of droplet sedimentation on MLM entrainment efficiency, which feeds back on liquid water path; drizzle plays a negligible role for these thin cloud layers. Until Section 5, we will restrict ourselves to the case N = 150 cm−3.
3.2. MLM Dependence on zi(0)
 We compare the MLM approach to equilibrium starting with different initial inversion heights zi(0), with the initial moist-conserved variables (and hence the cloud base) unchanged from the control case. The range of zi(0) was restricted to exceed zb(0) (to support a cloud) but thin enough that the initial BIR < 0.15, so the MLM predicts a well-mixed boundary layer. Figure 2 shows time series plots for these simulations. Figure 2a shows that if the initial cloud layer is thick enough (zi(0) 750 m), the cloud layer deepens and slowly reaches a steady state with zi ≈ 1250 m and LWP ≈ 70 g m−2. When the cloud is initially thin (zi(0) < 710 m), LWP and entrainment rate remain low, and the inversion slowly sinks, never achieving a steady state.
Figure 2. MLM time series for N = 150 cm−3 with initial zi = 710 m (green), 750 m (blue), 800 m (light blue), 840 m (red), 900 m (black) and 950 m (magenta): (a) Cloud top (solid) and base (dashed), (b) buoyancy integral ratio, with a dashed line showing a decoupling threshold of 0.15, (c) liquid water path, (d) precipitation rate at cloud base (solid) and surface (dashed) using a stretched time scale for the first day, (e) entrainment rate, and (f) convective velocity.
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 This bifurcation in behavior is due to a very thin cloud generating insufficient longwave radiative cooling to sustain adequate turbulence (Figure 2f) and entrainment (Figure 2e) to maintain the boundary layer against subsidence. Figure 2d shows that cloud-base drizzle remains negligibly small in all these simulations.
3.3. Slow Manifold Analysis of the MLM
 The MLM is a 3-variable nonlinear autonomous system of ordinary differential equations so it can be studied using a phase portrait. We construct a phase space using inversion height zi, cloud base height zb and surface virtual temperature difference ΔTv0, instead of the usual MLM prognostic variables zi, h and qt. Figure 3 shows the hourly evolution of the simulations in Figure 2 in this phase space. During the first day (the thermodynamic adjustment phase), the simulations collapse onto a single curve in zi – zb space, along which they then slowly evolve.
Figure 3. MLM phase portrait in zi – zb – ΔTv0 phase space, showing the trajectories in Figure 2 with various initial inversion heights. For each simulation, points colored with the air-sea virtual temperature difference ΔTv0 are plotted at every hour (negative values correspond to an unstable boundary layer). Black arrows indicate time evolution of trajectories.
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 This curve is what we call a slow manifold, defined as an invariant manifold of a set of autonomous differential equations that evolves slowly as scaled by some dimensionless small parameter, in our case the timescale ratio τth/τi. An invariant manifold is a family of initial states such each state always evolves into other states in the manifold.
 This concept was introduced to the atmospheric sciences by Leith  in the context of large-scale dynamics of midlatitude flows. He considered linear and weakly nonlinear solutions of the Boussinesq primitive equations with constant nonzero Coriolis parameter f. In the linear limit, these solutions can be decomposed into normal modes which have either low frequencies proportional to f (‘slow’ modes) and ‘fast’ gravitational modes. The slow modes form a basis for a linear ‘slow manifold’; a linear solution that starts in the slow manifold stays there. Leith hypothesized that when the system was weakly nonlinear, it still possesses a slightly distorted version of this slow manifold, and he used this idea as the basis of ‘nonlinear normal mode initialization’ for numerical weather prediction models.
 We use the ‘slow manifold’ descriptor loosely in this paper, because there is not an asymptotically small parameter in the MLM or an LES with which we can separate the slow and fast timescale behavior. However, the separation of timescales in our MLM for the given boundary conditions is good enough to warrant our terminology. For our simulations, CTV ≈ 10−2 m s−1, entrainment we ≈ 3 × 10−3 m s−1, D = 3.75 × 10−6 s−1, and zi ≈ 103 m, so τth ≈ 0.8 d and τi ≈ 3 d, giving a timescale ratio τth/τi = 1/4. As can be seen in Figure 3, this ratio is large enough to produce a tight collapse of trajectories onto the slow manifold within approximately 1 day. In further MLM plots, the first day of evolution will not be included to accentuate the slow-manifold component of the multiday evolution of the MLM simulations.
 The thermodynamic adjustment in ΔTv0 (colors in Figure 3) takes a little longer than for zi and zb. Because the simulations are initialized from an observational case in a cold advection regime, but no cold advection was included in the forcings, ΔTv0(0) (dark blue) is about 2 K colder than the adjusted ΔTv0 (red) along the slow manifold. Although the e-folding time τth for this adjustment is roughly one day, it takes two days for ΔTv0 to visually settle into its slow manifold behavior. In the MLM this slower adjustment of ΔTv0 only has a minor impact on the cloud base and inversion evolution, visible as a kinked structure in the uppermost trajectories in Figure 3 as they approach the slow manifold.
 The bifurcation point (unstable equilibrium) near (zi, zb) = (720, 680) m divides slow manifold trajectories for which the cloud layer thickens and the inversion slowly deepens to the steady state and those for which the cloud layer remains very thin and the inversion slowly collapses.
 Since the slow manifold is one-dimensional, evolution along the slow manifold can be represented as a one-dimensional dynamical system. It is convenient to describe the position along the slow manifold in terms of zi, so that zb, LWP, qt, we, etc. are all functions of zi, for the given boundary conditions of the problem. Figure 4 illustrates this for LWP. For all the simulations in Figure 2, LWP(t) has been plotted every 6 hours against zi(t), after discarding the first day as a thermodynamic adjustment period. All the simulations nicely collapse onto the same curve of LWP vs. zi.
Figure 4. Scatterplot of zi against LWP for time > 1 day from all MLM simulations shown in Figure 2. Legend shows symbols corresponding to initial inversion height used in each simulation. Points are plotted at every 6 hour of simulation time.
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 For this system, the slow manifold dynamics reduce to a competition between entrainment and subsidence,
The complexities of the MLM have all been wrapped into the functional dependence we(zi). Figure 5 shows we plotted vs. zi for all the MLM simulations for t > 1 d. The slow manifold we(zi) onto which the simulations collapse lies to the right of the dashed line Dzi for 720 m< zi < 1250 m, indicating dzi/dt > 0 (deepening) over this range of zi, while we(zi) < Dzi (slow shallowing) for zi < 720 m. Because we has a tendency to increase with zi, the slow inversion height adjustment timescale τi is even longer than Schubert et al.'s  estimate D−1 suggests. In fact, near the unstable equilibrium and for zi < 1000 m, dwe/dzi > D, i. e. as the boundary layer deepens, its net rise rate increases yet further. We attribute this behavior to the dependence of cloud-top radiative cooling on LWP in this thin-cloud regime (LWP < 20 g m−2). As the inversion deepens, the LWP on the slow manifold increases, allowing stronger radiative driving of turbulence and entrainment, i. e. dwe/dzi > 0.
 Unlike the complete MLM phase plane analysis, this slow manifold analysis generalizes to LES, even when the simulated cloud-topped boundary layer has more complex vertical structure than a mixed layer. We look for collapse of solutions with different initial conditions onto a single relationship vs. zi. Such a behavior indicates a slow manifold; if it exists we can use (3.1) to describe the evolution along the slow manifold.