Slow Manifolds and Multiple Equilibria in Stratocumulus-Capped Boundary Layers

Abstract

[1] In marine stratocumulus-capped boundary layers under strong inversions, the timescale for thermodynamic adjustment is roughly a day, much shorter than the multiday timescale for inversion height adjustment. Slow-manifold analysis is introduced to exploit this timescale separation when boundary layer air columns experience only slow changes in their boundary conditions. Its essence is that the thermodynamic structure of the boundary layer remains approximately slaved to its inversion height and the instantaneous boundary conditions; this slaved structure determines the entrainment rate and hence the slow evolution of the inversion height and can be regarded as a one-dimensional slow manifold. Slow-manifold analysis is applied to mixed-layer model and large-eddy simulations of an idealized nocturnal stratocumulus-capped boundary layer. Both models are found to have multiple equilibria; depending on the initial inversion height, the simulations slowly evolve toward a shallow thin-cloud boundary layer or a deep, well-mixed thick cloud boundary layer. In the mixed-layer model, this can be described using a single slow manifold bifurcated by an unstable equilibrium inversion height which separates a branch that evolves toward a deep steady state from a branch which shallows indefinitely. In the large-eddy simulations, there are two separate slow manifolds (one of which becomes unstable if cloud droplet concentration is reduced). On one, the boundary layer is well-mixed and deepens to a thick-cloud steady state. On the other, the boundary layer is decoupled and shallows to a thin-cloud steady state. If the initial inversion height supports an optically thick but nearly nondrizzling cloud, it evolves onto the well-mixed manifold; if the initial cloud layer is either too thin to efficiently radiatively cool, or thick enough to heavily drizzle, it evolves onto the decoupled manifold. Applications to analysis of stratocumulus observations and to pockets of open cells and ship tracks are proposed.

1. Introduction

[2] Marine cloud-topped boundary layers are complex systems that generations of modelers have strived to simulate and understand. A key issue in this understanding is their characteristic response timescales. Given a boundary layer subject to time-varying boundary conditions, such timescales can be used to infer what aspects, if any, of the boundary layer structure are approximately locked to the instantaneous boundary conditions and what aspects of the structure carry substantial ‘memory’ of the prior boundary layer evolution. This issue is important for interpreting field observations and large-eddy simulations (LES) of cloud-topped boundary layers, in which the boundary layer evolution is usually followed in detail for only a few hours. It also bears on the predictability of boundary layer evolution – will two boundary layers with the same forcing history but a slightly different initial state tend to evolve similarly or diverge in structure?

[3] Lilly's [1968] seminal mixed layer model (MLM) of stratocumulus focused on steady state solutions in which the boundary layer had sufficient time to fully reach equilibrium with a fixed set of boundary conditions – seasurface temperature or SST, subsidence rate, above inversion temperature and moisture profiles, etc. These steady states were recognizably similar to observed stratocumulus off the California coast. Lilly identified cloud-top radiative cooling as a key process in maintaining the strong capping inversion and turbulent mixing in stratocumulus-capped boundary layers.

[4] Schubert et al. [1979] analyzed the response timescales of a stratocumulus-topped boundary layer. They ran an MLM to steady state, then suddenly increased SST by 2 K. For parameters typical of subtropical stratocumulus, their analysis indicated two time scales. Thermodynamic variables such as mixing ratio, temperature, or cloud base adjust on a ‘fast’ timescale

• 

(typically about 0.5 day), where z_i is inversion height, w_e is entrainment velocity, V is mixed-layer wind speed, and C_T ≈ 1.2 × 10⁻³ is a surface transfer coefficient. The inversion height (which is also the cloud top) adjusts more slowly on a timescale

• 

(about 2–5 days), where D is the horizontal wind divergence in the boundary layer. Bretherton and Park [2008] showed a similar separation of timescales in both an LES and a bulk model of a shallow-cumulus-capped boundary layer. This timescale separation depends on the entrainment rate w_e being much weaker than the surface exchange velocity C_TV, which is typically true in subtropical marine boundary layers but may not hold in other situations common in midlatitudes, such as cold-air outbreaks.

[5] In this paper, we develop a novel ‘slow-manifold’ analysis of MLM and LES of an idealized nocturnal stratocumulus-capped boundary layer that exploits this separation of timescales. Over the slow timescale τ_i, boundary layer air has sufficient time to advect thousands of kilometers into dramatically different SSTs and free-tropospheric conditions, and even in a fixed location, synoptic variability of large-scale subsidence and horizontal advection cannot be neglected. Putting aside the obvious wrinkle of the diurnal insolation cycle (neglected in our idealized case), the boundary conditions change relatively slowly compared to the fast timescale τ_th, allowing the thermodynamic structure of the boundary layer time to approximately slave to the instantaneous boundary layer depth. Section 6.1 discusses how to extend this view to include the diurnal cycle.

[6] After describing our MLM and LES and the case setup (Section 2), we demonstrate the utility of this slow-manifold perspective in MLM simulations (Section 3), then show how it applies to LES as well (Section 4). We show an LES example in which, depending on initial conditions, the cloud-topped boundary layer evolves toward one of two stable equilibria. Even after as little as three days, small differences in the initial boundary layer depth can lead to large differences in the simulated boundary layer structure as the solutions bifurcate onto ‘slow manifolds’ that evolve toward the two equilibria.

[7] Although the possibility of multiple equilibria for cloud-topped boundary layers for the same set of boundary conditions has been suspected in past, no prior LES case has clearly shown this behavior. Randall and Suarez [1984] proposed a MLM which for some parameter choices could have two stable steady states, a shear-driven shallow cloud-free boundary layer and a radiation-driven deeper stratocumulus-capped boundary layer, somewhat comparable to our case.

[8] Because slow manifolds are approached within a day or two, they may be more useful objects of study for stratocumulus than steady states, both for analyzing models and observations. In section 5 we examine the sensitivity of the slow manifold behavior to an important control parameter, the cloud droplet concentration. In the conclusion, we suggest some further applications of slow manifold thinking to the study of cloud-topped boundary layers.

2. Models and Case Setup

[9] The MLM and LES models and the case setup follow Uchida et al. [2010], to which the reader is referred for details beyond the synopsis presented here. We use a version of the GEWEX Cloud System Study (GCSS) nocturnal nonprecipitating stratocumulus case specifications for single column models [Zhu et al., 2005], slightly modified to use a linear temperature profile in the free troposphere. This case was idealized from Research Flight 1 (RF01) of the Second Dynamics and Chemistry of Marine Stratocumulus Experiment [Stevens et al., 2003].

[10] Table 1 gives key parameters describing the case setup, as follows. The specified above-inversion profiles are superscripted with a ‘+’. The case is initialized with a well-mixed boundary layer; the subscript ‘M’ denotes the initial boundary-layer values of the moist conserved variables. In the LES, the wind profile is forced by a vertically-uniform geostrophic wind (u_g, v_g) and surface fluxes are derived via Monin-Obukhov theory; the MLM uses a specified surface transfer velocity C_TV to derive surface heat and moisture fluxes. The radiation scheme, described below, involves some additional constants.

Table 1. Modified GCSS RF01 Case Specification Parameters

D	SST			slM	qtM	ug, vg	CTV
s^−1	K	K	g kg^−1	K	g kg^−1	m s^−1	m s^−1
3.75×10^−6	292.5	300.17 + 0.006z	1.5	294.53	9	7, −5.5	0.00735

[11] For this paper, we will vary two control parameters, the initial inversion height z_i(0) and the cloud droplet number concentration N. All runs are 15 days long.

2.1. Radiation Parameterization

[12] In the boundary layer, we use the idealized GCSS radiative heating parameterization from Eq. (3) of Stevens et al. [2005]. The net upward radiative flux F_r(z) depends on the liquid water paths L⁺(z) and L⁻(z) (g m⁻²) above and below the level z:

• 

where F₀ = 70 W m⁻² and F₁ = 22 W m⁻². For a thick cloud with high column liquid water path (LWP), for which 0.085 LWP ≫1, F₀ is the cloud-top longwave radiative flux divergence (cooling), F₁ is the cloud-base radiative flux convergence (warming), and F₀ – F₁ is the net radiative cooling of the boundary layer; for thinner clouds these quantities are all reduced.

[13] Following Uchida et al. [2010], we modified the GCSS radiation parameterization for our LES runs to better simulate partly cloudy boundary layers by enforcing a minimum boundary-layer average radiative cooling rate of 2 K d⁻¹ in each grid column, representing the clear-sky cooling due to water vapor.

[14] We specify the radiative cooling above the inversion so as to balance the subsidence warming of our linear free-tropospheric temperature profile. In Uchida et al. [2010] and for most of the LES runs presented here, we followed the GCSS algorithm and used the height where the mean q_t = 8 g kg⁻¹ to transition from the boundary layer to the free tropospheric radiative cooling parameterization. Unfortunately, early in some of our runs with deep initial z_i this level dropped below the inversion, leading to unphysical radiative cooling profiles. We redid those runs (z_i(0) = 1300 m for N = 10, 30 cm⁻³, and z_i(0) 1100 m for N = 50, 150 cm⁻³) with a radiative transition at the mean height of the q_t = 5.25 g kg⁻¹ isosurface instead. Sensitivity tests showed this change did not affect the evolution of our other runs, for which q_t(z_i − 50 m) remained larger than 8.2 g kg⁻¹.

[15] These considerations do not affect MLM runs, for which the inversion is a prognostic variable and time independent thermodynamic profiles are specified above the inversion.

2.2. LES

[16] All LES runs use a uniform 25 m horizontal grid spacing over a 2.4×2.4 km horizontal domain with doubly-periodic boundary conditions. Since entrainment rate is found to increase if the vertical grid spacing is increased, it is vital to use a uniform vertical grid over the large range of z through which the inversion heights evolve in our runs, even though this adds to the computational expense. With this in mind, a 5 m grid spacing is used up to 2000 m for runs with initial z_i = 1200 m or higher, while other simulations use the 5 m grid only up to 1500 m. Above the uniform-grid layer, the grid slowly stretches and reaches into an overlying sponge layer between 2 km and the 3 km domain top.

[17] The LES used in this study is version 6.7 of the System for Atmospheric Modeling (SAM), kindly supplied by Marat Khairoutdinov and documented by Khairoutdinov and Randall [2003]. Water is partitioned into a vapor mixing ratio q_v, a cloud liquid water mixing ratio q_l and a rain water mixing ratio q_r. The monotonicity-preserving scheme of Smolarkiewicz and Grabowski [1990] is used for advecting three thermodynamic scalars: q_r, the total water mixing ratio q_t = q_v + q_l, and the liquid static energy s_l = c_pT + gz − Lq_l (where z is height, c_p is the specific heat of dry air at constant pressure, g is gravity, and L is the latent heat of vaporization). The cloud liquid water and temperature are diagnosed from these scalars by assuming exact grid-scale saturation in cloudy grid cells. A single-moment bulk microphysics scheme dependent on the cloud droplet concentration N predicts sources and sinks of q_t and q_r, including cloud droplet sedimentation. A Deardorff sub-grid turbulent diffusivity with prognostic subgrid TKE is applied to moist-conserved variables and velocity components.

2.3. MLM

[18] We use the ‘LES-tuned’ MLM described in Uchida et al. [2010], in which the entrainment and drizzle parameterizations have been modified from default observationally based choices to better match the LES behavior. This allows us to meaningfully compare the sensitivity to initial conditions and slow evolution of the MLM and LES.

[19] The MLM predicts the mixed layer moist static energy h = c_pT + gz + Lq_v = s_l + Lq_t, total (vapor plus cloud liquid) water mixing ratio q_t and inversion height z_i. It allows for continuously varying profiles of radiative heating (using our modified GCSS specification) and precipitation flux within the mixed layer. The precipitation flux includes sedimentation and drizzle components. The sedimentation flux is related to the cloud liquid water content exactly as in the LES. The drizzle flux profile includes a simple representation of subcloud drizzle evaporation and is anchored to an estimate of cloud base drizzle flux as a function of cloud liquid water path. In the LES-tuned version of the MLM, this estimate is a power law fit to the LES behavior.

[20] The MLM entrainment parameterization is based on Nicholls and Turton [1986]:

• 

where w_e is the entrainment rate, w_* is a convective velocity computed as the cube root of 2.5 times the vertical integral of the buoyancy flux, and δb is the inversion jump of virtual potential temperature expressed in buoyancy units. A is a nondimensional entrainment efficiency given in Uchida et al. [2010], which includes an evaporative enhancement term related to the inversion temperature and humidity jumps and the cloud top liquid water content q_li. Sedimentation of cloud droplets out of the inversion zone, assuming a fall speed dependent on the mean cloud droplet radius r_d ∝ (q_li/N)^1/3, is assumed to reduce the evaporative enhancement [Bretherton et al., 2007]. In the LES-tuned MLM, the evaporative enhancement term is increased by a factor 4.4 compared to the observational tuning.

[21] The self-consistency of the MLM simulations is tested by computing a decoupling indicator, the buoyancy integral ratio or BIR [Bretherton and Wyant, 1997], defined as the vertical integral of the negative buoyancy flux in sub-cloud layer to the vertical integral of the positive buoyancy flux over the rest of the mixed layer. In this paper, a BIR exceeding 0.15 will be used as an indicator that the boundary layer will not remain well-mixed and the MLM is no longer appropriate [Bretherton and Wyant, 1997; Stevens, 2000].

3. Analysis of MLM Simulations

3.1. Fast and Slow Timescales in the MLM

[22] Figure 1 shows LES-tuned MLM solutions for N = 150,50 and 30 cm⁻³ starting with the specified RF01 initial conditions, for which z_i(t = 0) = 840 m, the cloud base z_b(0) ≈ 640 m, and the air-sea virtual temperature difference ΔT_v0(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 z_b rises to nearly 800 m, LWP drops to 20 g m⁻², and w_e decreases to 4 mm s⁻¹, while inversion height changes by a relatively modest 50 m. After that, z_i slowly increases, along with all the thermodynamic variables. Even after 5 days, z_i 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 z_i(0) = 840 m.

[23] As explained by Uchida et al. [2010], 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 z_i(0)

[24] We compare the MLM approach to equilibrium starting with different initial inversion heights z_i(0), with the initial moist-conserved variables (and hence the cloud base) unchanged from the control case. The range of z_i(0) was restricted to exceed z_b(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 (z_i(0) 750 m), the cloud layer deepens and slowly reaches a steady state with z_i ≈ 1250 m and LWP ≈ 70 g m⁻². When the cloud is initially thin (z_i(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 z_i = 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.

[25] 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

[26] 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 z_i, cloud base height z_b and surface virtual temperature difference ΔT_v0, instead of the usual MLM prognostic variables z_i, h and q_t. 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 z_i – z_b space, along which they then slowly evolve.

Figure 3. MLM phase portrait in z_i – z_b – ΔT_v0 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 ΔT_v0 are plotted at every hour (negative values correspond to an unstable boundary layer). Black arrows indicate time evolution of trajectories.

[27] 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 τ_th/τ_i. An invariant manifold is a family of initial states such each state always evolves into other states in the manifold.

[28] This concept was introduced to the atmospheric sciences by Leith [1980] 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.

[29] 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, C_TV ≈ 10⁻² m s⁻¹, entrainment w_e ≈ 3 × 10⁻³ m s⁻¹, D = 3.75 × 10⁻⁶ s⁻¹, and z_i ≈ 10³ m, so τ_th ≈ 0.8 d and τ_i ≈ 3 d, giving a timescale ratio τ_th/τ_i = 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.

[30] The thermodynamic adjustment in ΔT_v0 (colors in Figure 3) takes a little longer than for z_i and z_b. Because the simulations are initialized from an observational case in a cold advection regime, but no cold advection was included in the forcings, ΔT_v0(0) (dark blue) is about 2 K colder than the adjusted ΔT_v0 (red) along the slow manifold. Although the e-folding time τ_th for this adjustment is roughly one day, it takes two days for ΔT_v0 to visually settle into its slow manifold behavior. In the MLM this slower adjustment of ΔT_v0 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.

[31] The bifurcation point (unstable equilibrium) near (z_i, z_b) = (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.

[32] 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 z_i, so that z_b, LWP, q_t, w_e, etc. are all functions of z_i, 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 z_i(t), after discarding the first day as a thermodynamic adjustment period. All the simulations nicely collapse onto the same curve of LWP vs. z_i.

Figure 4. Scatterplot of z_i 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.

[33] 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 w_e(z_i). Figure 5 shows w_e plotted vs. z_i for all the MLM simulations for t > 1 d. The slow manifold w_e(z_i) onto which the simulations collapse lies to the right of the dashed line Dz_i for 720 m< z_i < 1250 m, indicating dz_i/dt > 0 (deepening) over this range of z_i, while w_e(z_i) < Dz_i (slow shallowing) for z_i < 720 m. Because w_e has a tendency to increase with z_i, the slow inversion height adjustment timescale τ_i is even longer than Schubert et al.'s [1979] estimate D⁻¹ suggests. In fact, near the unstable equilibrium and for z_i < 1000 m, dw_e/dz_i > 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. dw_e/dz_i > 0.

Figure 5. Same as Figure 4, for MLM-simulated w_e.

[34] 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. z_i. Such a behavior indicates a slow manifold; if it exists we can use (3.1) to describe the evolution along the slow manifold.

4. Analysis of LES Sensitivity to z_i(0)

[35] The MLM entrainment and precipitation parameterizations were retuned using LES model output. Hence, we might anticipate that for a given N, the LES would have similar dependence on initial z_i(0) as the MLM, including comparable slow-manifold behavior, at least along trajectories in which the boundary layer remains well-mixed.

[36] To test this prediction, we did a series of 15-day N = 150 cm⁻³ LES runs with the same range of z_i(0) used for the MLM. Figure 6 shows salient time series from these runs; plotted values are hourly averages, with the first two hours omitted as a spin up period. As anticipated, the LES time series have many similarities to the MLM time series shown in Figure 2. For both models, the trajectories split into two long-term evolutions, either a shallow boundary layer with very thin cloud for smaller z_i(0) ≤ 750 m, or deepening toward a deep well-mixed stratocumulus layer for larger z_i(0) > 750 m. As with the MLM, cloud base drizzle plays a negligible role in these LES runs.

Figure 6. Hourly-averaged time series for LES runs with N = 150 cm⁻³ and initial z_i ≤ 950 m; first hour (spin-up period) not plotted. (a) Cloud top (solid) and base (dashed), (b) cloud fraction, (c) liquid water path, (d) precipitation rate at cloud base (solid) and surface (dashed) using a stretched timescale for the first day, (e) entrainment rate and (f) decoupling index Δq_t.

[37] There are some differences that could probably be reduced by slight adjustment of the MLM entrainment efficiency parameter or thermodynamic reference height, despite the prior tuning of the MLM entrainment and precipitation parameterizations to be consistent with the LES. The bifurcation point in the initial inversion height is slightly higher for the LES than for the MLM, such that the MLM z_i(0) = 750 m run evolves to a deep steady-state, while the comparable LES run does not. The steady-state value of LWP is 32 g m⁻² for the LES, less than half as large as for the MLM.

[38] More important, there are also fundamental differences that are associated with the LES permitting solutions with decoupling boundary layers and partial cloud cover, features that cannot be represented by the MLM. The LES does not simulate a continuing collapse of the boundary layer as in the MLM; instead the LES reaches a steady state with z_i = 700 m and shallow patchy clouds with low cloud fraction of about 0.2 (Figure 6b). In Figure 6f, we plot a measure of decoupling, the total mixing ratio difference between the top and bottom of the boundary layer Δq_t = q_t(z_i − 50 m) − q_t(50 m). In an ideal mixed layer this would be zero. In the deep stratocumulus steady state, Δq_t ≈ −0.6 g kg⁻¹ over a depth of z_i − 100 = 1150 m. In the shallow steady state, Δq_t ≈ −1.1 g kg⁻¹ over a depth of 600 m, a vertical gradient four times as large that is indicative of much more pronounced decoupling.

[39] In contrast with the MLM, the LES can simulate all kinds of boundary layer structure so we also consider thicker initial cloud layers for which the MLM BIR criterion predicts immediate decoupling. Figure 7 shows LES runs with z_i(0) = 1000 m, 1100 m and 1200 m, in addition to the previously-shown cases 840 m, 710 m and 950 m for comparison. When z_i(0) is increased from 950 m to 1000 m, the boundary layer becomes a bit more decoupled during spinup (Δq_t < −0.7 g kg⁻¹) and ultimately approaches the same shallow steady state with thin patchy cloud as for z_i(0) = 710 m. Similar behavior is seen for z_i(0) = 1100 m and 1200 m, except the thicker initial cloud layer leads to more pronounced decoupling within the first few hours. This decoupling may be due to strong entrainment by the deep cloud layer. Figure 7d shows that cloud base drizzle is weak even in this decoupling period, so the drizzle does not seem to be the main cause of the initial decoupling.

Figure 7. As in Figure 6, except for z_i(0) = 710, 840, 950, 1000, 1100 and 1200 m.

4.1. Slow Manifold Analysis of LES

[40] In Figure 7a, the curves of inversion height and cloud base from the z_i(0) 1000 m runs cross over those from the z_i(0) = 840 m and z_i(0) = 950 m runs. This is inconsistent with all runs collapsing to a single one-dimensional slow manifold which can be uniquely characterized as a function of z_i; this result contrasts with the MLM. Nevertheless, we can characterize this behavior by plotting the slow evolution of the LES runs vs. the inversion height.

[41] As with the MLM, there is a thermodynamic adjustment period during which the LES approaches its slow evolution. The LES cloud characteristics are more sensitive to the sea-air virtual temperature difference ΔT_v0 than in the MLM. As in the MLM, ΔT_v0 takes about 2 days to adjust from its initial value of −2.9 K into the 0 to −0.7 K range characterizing its slow evolution.

[42] Figure 8 plots 6-hour average values of z_i vs. LWP, cloud fraction and Δq_t for all the LES runs, excluding times less than this 2-day thermodynamic adjustment period. All the panels show a collapse of the solutions onto two slow manifolds, which coexist over a range 950 < z_i < 1300 m. Along the right manifold in the plots, a well-mixed (Δq_t ≈ −0.5 g kg⁻¹) and nearly fully cloud covered stratocumulus regime slowly deepens into a steady state with z_i = 1300 m. Along the left manifold, a decoupled (Δq_t < −1 g kg⁻¹) and partly cloudy regime slowly shallows into a decoupled steady state with z_i ≈ 700 m. The initial conditions 800 m≤ z_i(0) ≤ 950 m evolve onto the well-mixed manifold. The runs with z_i(0) < 800 m start similarly by deepening for two days, but during the third day they quickly transition toward the decoupled manifold.

Figure 8. Scatterplots of 6-hour average values of LWP, cloud fraction and Δq_t against z_i for the N = 150 cm⁻³ LES runs shown in Figure 6 and Figure 7 and all times t > 2 d. ‘DE’ and ‘ME’ label the decoupled and well-mixed equilibrium states.

[43] This behavior is an unanticipated contrast to the single slow manifold with a bifurcation point that we obtained with the MLM. It is not surprising that the LES favors decoupling in boundary layers with initially very thick stratocumulus which induce both strong entrainment and drizzle, both of which promote negative buoyancy fluxes below cloud base. However, it is less obvious why the LES favors decoupling in boundary layers with very thin stratocumulus.

[44] One possibility is that the LES subgridscale turbulence scheme is favoring cloud breakup through a form of cloud-top entrainment instability. There is still controversy about whether this is physically realistic or an artifact of numerical simuations that inadequately resolve entrainment processes atop stratocumulus, but Randall [1980], MacVean and Mason [1990], Lock [2009], and others have suggested criteria based on a parameter κ = 1 + δs_l/Lδq_t dependent on the ratio of the jumps across the inversion (indicated by δ) of s_l and q_t. For the shallow decoupled steady-state, we estimate δs_l/c_p = 7 K and δq_t = −8 g kg⁻¹, for which κ ≈ 0.65. This value of κ considerably exceeds the Lock [2009] threshold of 0.4 and nearly attains the stringent MacVean-Mason threshold of 0.7, suggesting that grid-scale mixing at the inversion may not allow the sustenance of a solid stratocumulus layer in this case. For the drier, deeper boundary layers that characterize the well-mixed manifold, the inversion jump δs_l/c_p = 9 K, δq_t = −7 g kg⁻¹ and κ ≈ 0.45, which while still large does not indicate as strong a susceptibility to entrainment instability.

4.2. Slow-Manifold Thermodynamic Profiles

[45] Figure 9 shows vertical profiles of the moist conserved variables q_t and s_l/c_p on the two slow manifolds, as illustrated by the runs for z_i(0) = 800 m (well-mixed manifold) and 1200 m (decoupled manifold) at three times. The boundary layer values of q_t and s_l/c_p do not differ much between the two slow manifolds, but the profiles of liquid water content are quite different, and decoupling is obvious in the profiles of both conserved variables in the right panels.

Figure 9. Vertical structure of total water mixing ratio q_t and liquid water mixing ratio q_l (top panels) and scaled liquid static energy s_l/c_p (bottom panels), for z_i(0) = 800 m (left) and 1200 m (right) cases averaged over 2–2.25 days (green), 4–4.25 days (blue) and 14–14.25 days (red).

[46] Figure 10 plots 6-hour average values of z_i and w_e, discarding the first two days of simulation. Again, the results collapse nicely, with the well-mixed manifold lying to the right of the line (w_e > Dz_i), corresponding to boundary layer deepening, and the decoupled manifold lying to the left of the line (w_e < Dz_i), corresponding to a shallowing boundary layer.

Figure 10. Scatterplot of 6-hour averaged w_e against z_i, with first two days discarded.

[47] Figure 11 shows 6-hour average scatterplots of w_e vs. cloud fraction and ΔF_r, the boundary layer radiative flux difference between z_i + 50 m and the surface. This figure emphasizes the role of radiative driving on the slow-manifold evolution. If the boundary layer can stay 100% cloud-covered (as on the well-mixed manifold), it sustains strong cloudtop radiative cooling and a high entrainment rate, so it can become deep. If its cloud fraction is low, ΔF_r is much smaller and only sustains a low entrainment rate, forcing the inversion to shallow (as on the decoupled manifold). These relationships smoothly span both the well-mixed and decoupled slow manifolds; they are not sensitive to details of the internal boundary layer structure.

Figure 11. Scatterplots of 6-hour average w_e against cloud fraction (top) and boundary-layer radiative flux divergence ΔF_r (bottom) for all simulations, with first two days discarded.

5. Sensitivity of Slow Manifolds to N

[48] The slow-manifold evolution of the stratocumulus-capped boundary layer depends on the boundary conditions (such as SST and free-tropospheric conditions) and other control parameters (such as droplet concentration N). In Figure 2, we saw that an MLM simulation of the boundary layer evolution with z_i(0) = 840 m was fairly insensitive to decreasing N from 150 cm⁻³ to 30 cm⁻³ except for a small increase in LWP, and that further decrease of N led the MLM to diagnose immediate decoupling. This suggests that over the MLM's limited range of validity, its slow manifold behavior varies only slightly with N.

[49] In this section, we examine how the more complex LES slow manifold behavior depends on N. As in the MLM, decreased N leads to more efficient droplet sedimentation and drizzle production. We performed simulations configured just like the previous N = 150 cm⁻³ runs with a range of initial inversion heights (710, 840, 900, 1100 and 1300 m) for N = 50, 30, 10 cm⁻³.

[50] Figure 12 shows LES time series plots for N = 50 cm⁻³ and all chosen z_i(0). In contrast to the N = 150 cm⁻³ case, all the runs converge to a single steady-state with deep, fairly well mixed stratocumulus. One might expect lower N to increase drizzle and favor decoupling, but surprisingly the decoupled steady-state we saw for N = 150 cm⁻³ has disappeared. Why?

Figure 12. LES time series as in Figure 6, for N= 50 cm⁻³ and z_i(0) = 710, 840, 900, 1100 and 1300 m.

[51] Figure 12a shows that the inversion and cloud base heights of the z_i(0) = 1100 and 1300 m cross those of the z_i(0) = 840 and 900 m runs. This suggests that as in the N = 150 cm⁻³, there are still multiple slow manifolds. In the N = 50 cm⁻³ runs, there is significant cloud base drizzle for the first few hours for the deepest z_i(0), but as the simulated boundary layers evolve toward the slow manifolds, LWP rapidly decreases and drizzle becomes negligible like for N = 150 cm⁻³. The key contrast is in the z_i(0) = 710 m case, for which cloud fraction decreases to 0.5 and LWP decreases to 10 g m⁻², but then the trends reverse and the cloud thickens into the deep steady state, unlike with N = 150 cm⁻³.

[52] Figure 13 plots selected variables vs. z_i for the N = 50 cm⁻³ runs after the initial 2-day thermodynamic adjustment phase. It suggests there are still two slow manifolds for 1200 m < z_i < 1350 m (consistent with the crossing behavior mentioned above). However, the decoupled manifold loses stability as z_i drops below 1200 m, and the solution quickly jumps to the well-mixed manifold. For lower z_i, it appears that only the well-mixed manifold is stable, in contrast to the N = 150 cm⁻³ case. The low z_i end of this manifold is only sampled by the z_i(0) = 710 m trajectory. In the absence of the collapse of multiple runs onto the same curve, we can hypothesize but cannot be sure that the entire z_i(0) = 710 m simulation actually has reached a slow manifold after two days (i. e. well before it reaches a minimum in cloud fraction). If so, we can see that along this ‘well-mixed’ slow manifold, for z_i < 1000 m the cloud layer becomes more broken and has lower LWP, reaching a minimum cloud fraction below 0.6 for z_i < 900 m while remaining fairly well-mixed (Δq_t ≈ −0.7 g kg⁻¹).

Figure 13. Scatterplots vs. z_i as in Figure 8, for N = 50 cm⁻³.

[53] We interpret the loss of the decoupled steady state and much of the decoupled slow manifold as follows. After the initial thermodynamic adjustment, the most important effect of lower N is to increase droplet sedimentation; the slow-manifold LWP is too low to support drizzle even for N = 50 cm⁻³. Enhanced sedimentation makes entrainment drying and evaporative cooling less efficient, which inhibits cloud-top entrainment instability that could breaks up the cloud layer. For an initially thin cloud, the decreased entrainment efficiency preserves enough LWP and cloud fraction and maintains enough cloud-top radiative cooling to keep turbulence vigorous and prevent decoupling.

[54] Figure 14 shows corresponding LES time series for N = 30 cm⁻³. For initially deeper cloud layers, significant cloud base drizzle from persists up to a day, and weak cloud base drizzle of 0.1 mm d⁻¹ redevelops toward the end of the integrations, but all simulations ultimately reach a deep fully cloud-covered steady state similar to the nearly nondrizzling N = 50 cm⁻³ case. Figure 15 shows the corresponding scatterplots vs. z_i. The z_i(0) = 1300 m evolves onto a decoupled slow manifold. This loses stability as the inversion subsides to z_i ≈ 1220 m, slightly higher than for N = 50 cm⁻³. The z_i(0) = 1100 m run also seems to evolve onto this manifold near z_i = 1220 m. Both runs then switch over to the ‘well-mixed’ slow manifold followed by the runs with lesser z_i(0); whereafter they slowly deepen into the steady state.

Figure 14. LES time series as in Figure 12, for N = 30 cm⁻³.

Figure 15. Scatterplots vs. z_i as in Figure 8, for N = 30 cm⁻³.

[55] Figures 16 and 17 show time series and z_i-scatterplots for N = 10 cm⁻³. Again there is only one slow manifold along which trajectories evolve toward one steady state. This steady state has a much lower z_i ≈ 870 m, a smaller cloud fraction of 0.4, LWP = 10 g m⁻² and more decoupling (Δq_t ≈ −1 g kg⁻¹) than for N = 30 cm⁻³. To gain further insight into it, we focus on the first two days of the z_i(0) = 1100 and 1300 m runs. Comparison of Figures 16c and d suggests that for LWP = 40 g m⁻² (the steady-state value for N = 30 cm⁻³), there is strong cloud base drizzle of 1–2 mm d⁻¹ when N = 10 cm⁻³. This drizzle evaporates in the subcloud layer, inhibiting deep vertical turbulent mixing and entrainment and promoting decoupling. The decoupling dries out the upper part of the boundary layer and keeps the cloud layer thin and patchy. The correspondingly weak radiative forcing reduces entrainment compared to the N = 30 cm⁻³ case, so the inversion sinks to a much lower steady state. In summary, the N = 10 cm⁻³ slow manifold seems analogous to the well-mixed slow manifold of the higher-N cases, but unlike in those cases, the cloud properties along the N = 10 cm⁻³ slow manifold are affected by drizzle-catalyzed decoupling.

Figure 16. LES time series as in Figure 12, for N = 10 cm⁻³.

Figure 17. Scatterplots vs. z_i as in Figure 8, for N = 10 cm⁻³.

[56] We see that the LES slow-manifold dynamics depend significantly on N, in a much more interesting way than for the MLM. A two-manifold structure is seen at high N, and a one-manifold structure at low N. As N decreases, the well-mixed manifold gradually morphs into a more decoupled manifold supporting less cloud cover. The decoupled manifold seen at high N loses stability and disappears as N decreases. Viewing N as a control parameter, this implies that we could get either a smooth or a rapid change in cloud cover as this parameter slowly changes, depending on whether the boundary layer structure stays on a single manifold or is forced to make a transition between the well-mixed or decoupled manifolds.

6. Discussion

[57] For an idealized nocturnal stratocumulus case, we have found that:

[58] 1. Slow-manifold dynamics is a useful organizing concept for analyzing simulations of multiday stratocumulus evolution.

[59] 2. For fixed boundary conditions and forcings, slow manifolds can be visualized by plotting cloud and boundary-layer quantities vs. inversion height after discarding a 1–2 day thermodynamic adjustment period.

[60] 3. In some cases, LES-simulated stratocumulus can evolve onto either a well-mixed or decoupled slow manifold, depending on initial conditions. These slow manifolds evolve toward different steady states with very different cloud fraction and thickness, and may lose stability at some z_i, producing a rapid change in boundary layer structure.

[61] 4. There is clear similarity between the well-mixed slow manifolds produced by an LES and that of a similarly-configured mixed layer model.

[62] 5. The slow manifold structure can be sensitive to boundary conditions, forcings, and parameters such as cloud droplet concentration N.

[63] Real-world cloud-topped boundary layers go through a diurnal cycle, are subject to time-varying boundary conditions and large-scale winds, and often exhibit sharp regime boundaries across which cloud structure changes drastically (such as the edge of a pocket of open cells or POC, e. g. Bretherton et al. [2004]). Parameters we have idealized to be external (such as N) co-evolve with the boundary layer. In this setting, of what use is the slow-manifold concept? In the remainder of this paper, we will discuss two possible applications.

6.1. Data Analysis

[64] Analyzing field and satellite observations of cloud-topped boundary layers, or even global model simulations, is challenging because of omnipresent variability on day-to-day timescales. This is often distilled by time-averaging into monthly, seasonal or annual means, but that washes out features such as the inversion sharpness and the internal boundary layer and cloud structure and its response to synoptic ‘weather’ variability.

[65] Slow-manifold thinking suggests that the day-to-day variability of the cloud-topped boundary layer can most efficiently be studied with a careful choice of variables. While our modeling example was for stratocumulus, the ideas should apply to most marine cloud-topped boundary layers in strong-inversion regimes in which the entrainment rate is significantly weaker than the wind-driven surface transfer velocity, so that there is a clear separation between a fast thermodynamic timescale and a slow inversion timescale.

[66] The diurnal cycle of radiative forcing presents an apparent challenge, but can be conceptually included in the slow manifold. Imagine a cloud-topped boundary layer with specified surface and free-tropospheric boundary conditions, and with a given z_i. We can imagine the boundary layer structure thermodynamically adjusting to a quasisteady diurnal cycle dependent on the chosen z_i; this is then the slow-manifold response. At the same time, the inversion height starts to drift due to imbalance of daily-average entrainment rate 〈w_e〉(z_i) and the specified mean subsidence rate; this gives the evolution along the slow manifold.

[67] The first key slow-manifold idea which we expect will prove fruitful is to use inversion height as an independent variable in place of large-scale vertical motion. The boundary layer takes time to respond to vertical motion changes, and while it does so, the thermodynamic variables can still stay in rough balance with the current z_i. Inversion height can be observed by satellite lidar or stereography or inferred from cloud-top temperature.

[68] A second key idea is to consider the Lagrangian time-dependent boundary conditions and forcings to which a moving boundary layer air column is subjected. For now we include the droplet concentration N among these, though it is partly determined internally and is prognosed in some models. At each time, the slow manifold will reflect the current values of these boundary conditions; most notably this includes DSST/Dt (the advective derivative of SST following the boundary-layer mean horizontal wind) as well as SST itself. We anticipate that day-to-day cloud properties at some fixed location should mainly depend on current z_i, N, free-tropospheric conditions, wind speed, and DSST/Dt, all of which can be expected to strongly covary. One or two dominant modes of covariability of these parameters probably explain most of the synoptic variability at any particular location and season; the utility of slow-manifold theory can be tested by how well the cloud properties collapse as a function of their modal amplitudes. Slow-manifold theory also predicts that there should be no significant day-to-day ‘memory’ in the cloud properties that is not captured by these parameters. Potentially, this could also allow a more systematic observational test of the response of cloud-topped boundary layers to the kinds of boundary condition changes associated with greenhouse warming, helping to address important uncertainties in the feedbacks of subtropical marine boundary layer clouds on climate [e. g. Bony et al., 2006].

6.2. POCs and Localized Microphysical Gradients

[69] Slow-manifold theory also provides an intriguing perspective on POCs and other rapid horizontal changes in cloud properties. POCs are regions of broken, cumuliform cloud cover with very low concentrations of accumulation-mode aerosols and cloud droplets surrounded by nearly unbroken stratocumulus with much higher accumulation-mode aerosol and droplet concentrations [Bretherton et al., 2004; Stevens et al., 2005; Sharon et al., 2005; Wood et al., 2008]. They can persist for days in the southeast and northeast Pacific stratocumulus regions, seemingly maintained by feedbacks between aerosol, precipitation and cloud macrostructure.

[70] Figure 18 shows a schematic cross-section across a POC. We hypothesize that the broken cloud inside a POC induces less net radiative cooling of the boundary layer than in the surrounding unbroken cloud layer, and that this results in markedly less entrainment inside than outside the POC. Yet observations show that the inversion height is similar inside and outside the POC; indeed, the big inversion density jump strongly inhibits the development of substantial inversion topography. Together, these suggest that subsiding free-tropospheric air must be horizontally converged from above the POC toward the surrounding overcast clouds, as shown by the streamlines in Figure 18.

Figure 18. Schematic cross-section through a pocket of open cells (POC), with precipitating cumuli detraining into thin broken stratocumulus surrounded by thicker overcast stratocumulus. Inversion height z_inv is dashed. Streamlines descend from the mid-troposphere into the inversion, diverging from the POC into the overcast region, where air is entrained into the boundary layer. Wide arrow length indicates subsidence rate; eddy-like symbols indicate entrainment. Horizontal-mean divergence is assumed to be zero above the boundary layer for graphical simplicity.

[71] From the slow-manifold perspective, this would suggest that the POC and the surrounding cloud could be regarded as evolving on separate slow manifolds, but with their inversion heights ‘symbiotically’ locked to each other. The POC is evolving on a decoupled manifold; its surroundings are evolving on a well-mixed manifold. We envision that the cloud droplet/aerosol concentration evolves as an integral part of the slow-manifold structure; the POC maintains a lower droplet concentration than its surroundings through a more cumuliform precipitating cloud structure that efficiently scavenges aerosol from the updrafts. Just like in the fixed-N_d case, we suggest that both the decoupled and well-mixed slow manifolds can be characterized by one parameter, the inversion height. Because of the dynamical coupling between the POC and its surroundings, their inversion heights remain equal and slowly evolve driven by the difference between the mean combined entrainment rate and the mean subsidence rate averaged across the two regions. In particular, the inversion in the POC may deepen even if the local entrainment rate is too weak to balance the areamean subsidence rate; i. e. the inversion inside the POC can be ‘pulled’ upward by the more rapidly entraining surrounding cloud layer. In turn, the surrounding layer will not deepen as fast due to the presence of a neighboring POC. In summary, we suggest that the cloud evolution inside the POC cannot be treated as independent of that outside the POC; they are a system tightly coupled through inversion dynamics.

[72] Similar considerations may apply to ship tracks in regions of low, thin, aerosol-poor stratocumulus, in which ship-emitted gases and particles create a optically thick line of cloud [e. g. Conover, 1966; Taylor and Ackerman, 1999]. In that case, one might anticipate that the ship track radiatively cools more strongly and suppresses drizzle formation, both of which enhance turbulence and entrainment compared to the ambient clouds. This will cause a differential rise of cloud tops in the ship track that on at least one occasion has been directly observed [Taylor and Ackerman, 1999]. However, this is opposed by the strong inversion stability, which should create a mesoscale circulation with enhanced subsidence over the track and reduced subsidence in a surrounding region that broadens with time. Such ship tracks usually are distinct only for 12 hours or less [Durkee et al., 2000], but (1.1) suggests that the shallow boundary layers that favor such ship tracks should have fast thermodynamic adjustment timescales, so slow-manifold thinking may still be relevant to them.

7. Concluding Remarks

[73] LES is our most realistic simulation tool for stratocumulus-topped boundary layers. Thus it is noteworthy that for the first time, we have shown an LES case which, depending only on initial boundary layer depth, can evolve into one of two quite different cloud and boundary layer structures. It is also noteworthy that this behavior does not require inclusion of drizzle processes or two-way cloud-aerosol interaction. This suggests that transitions between well-mixed and decoupled stratocumulus-capped boundary layers may sometimes sensitively depend on the initial state of the boundary layer and may be inherently hard to predict. Further investigation of this behavior in other LES and in high-resolution regional models should bear out whether this is an occasional curiosity or a common issue for cloud-topped boundary layers. In either case, a slow-manifold perspective on the boundary layer evolution should be considered.

Acknowledgments

[74] We acknowledge support from NASA grant NNX09AH73G and NSF grant ATM-0745702. The speculations about ship tracks were inspired by discussions with Andrew Ackerman.