Fast Stratocumulus Timescale in Mixed Layer Model and Large Eddy Simulation

Transcription

1 JAMES, VOL.???, XXXX, DOI: /, 1 2 Fast Stratocumulus Timescale in Mixed Layer Model and Large Eddy Simulation C. R. Jones, 1 C. S. Bretherton, 1 and P. N. Blossey 1 Corresponding author: C. R. Jones, Department of Atmospheric Sciences, University of Washington, Seattle, WA , USA. (crj6@uw.edu) 1 Department of Atmospheric Sciences, University of Washington, Seattle, Washington, USA.

2 3 X - 2 Abstract. JONES ET AL.: FAST STRATOCUMULUS TIMESCALE A mixed layer model (MLM) and large eddy simulation (LES) are used to analyze the internal response timescales of a stratocumulus-topped boundary layer (STBL). Three separate timescales are identified: a slow timescale associated with boundary layer deepening (several days), an intermediate thermodynamic timescale (approximately 1 day), and a fast timescale (6-12 hours) for cloud water path adjustment associated with an internal entrainmentliquid flux (ELF) feedback. The nocturnal DYCOMSII-RF01 case study is used to establish and interpret the previously unidentified fast STBL adjustment timescale with the MLM. The role of the entrainment closure is investigated by repeating the analysis with several different closures. Nearly every closure considered exhibits a fast timescale. Perturbations are applied to the well-mixed CGILS stratocumulus case in both MLM and LES in order to elicit a short timescale response. Purely radiative perturbations do not project strongly onto the fast scale, while perturbations to the free tropospheric humidity do. A 2K surface and atmospheric temperature perturbation also projects strongly onto the fast scale. We show that the ELF adjustment mechanism behind the fast timescale is responsible for much of the steady state liquid water path response in the perturbed case, acting as a cloud-thinning feedback mechanism in a uniformly warmed climate.

3 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X Introduction The stratocumulus-topped boundary layer (STBL) is subject to large scale forcings applied across a range of time scales: mesoscale gravity waves passing through in minutes to hours; the diurnal cycle of insolation; synoptic, seasonal, and interannual variability; and climate change. To fully understand the response of an STBL to changing conditions, it behooves us to consider the characteristic timescales of STBL adjustment and the physical mechanisms responsible for them. Mixed layer models (MLMs) of the STBL have traditionally been interpreted to admit only two internal adjustment timescales. Schubert et al. [1979a] used a mixed layer model [Schubert et al., 1979b] to analyze step perturbations in sea surface temperature (SST) and large scale subsidence to a shallow steady state STBL. They observed that the thermodynamic variables (i.e., the moist static energy, h, and total water mixing ratio, q t ) adjusted with an e-folding time of approximately 4 hours, while the inversion approached steady state on a much slower timescale of approximately 80 hours. They found that the thermodynamic adjustment was driven primarily by surface flux feedbacks, and argued that the thermodynamic adjustment timescale could be approximated as z i /C T V 1 day, where z i is the height of the capping temperature inversion, C T is a surface transfer coefficient, and V is the surface wind speed. They also identified a longer inversion adjustment timescale determined by the large scale divergence D, 42 τ i = D 1. (1)

4 X - 4 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE The cloud base adjustment time of 4 hours observed by these authors was quite a bit faster than their thermodynamic adjustment timescale, hinting already at the existence of a yet faster timescale. Bretherton et al. [2010] expanded on this notion of a thermodynamic timescale, τ th = z i w e + C T V, (2) where w e is the entrainment rate. The scale separation between the thermodynamic timescale and the slow inversion-adjustment timescale was exploited to perform a slow manifold analysis in both MLM and large eddy simulation (LES). After an initial fast transient period, the dynamics are constrained to the slow manifold, wherein the ther- modynamic variables remain in local quasi-equilibrium while the trajectory is controlled primarily by the evolution of z i. Recently, further attention has been given to boundary layer adjustment timescales, particularly the fast timescale. A timescale analysis of a dry convective boundary layer driven by a specified surface flux was performed by van Driel and Jonker [2010]. Their analysis is similar in spirit to that undertaken in this manuscript. They used a MLM of a steadily forced dry convective boundary layer and extracted internal timescales by solving for the steady state, linearizing the system about that steady state, and calculating 60 its eigenvalues. They then subjected the linearized equations, an MLM, and an LES to sinusoidally-varying perturbations in the surface flux, and compared the resulting z i amplitude and phase lag relative to the input forcing. They noted that the timescales were dependent on the entrainment efficiency, and showed good agreement between the predicted linearized analysis and the LES provided the forcing frequency wasn t too high. When they coupled an equation for turbulent kinetic energy (TKE) into the MLM and

5 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X used a w -type entrainment closure, they found good agreement even when the forcing frequency was high. Bellon and Stevens [2012] analyzed the timescales in response to a sudden perturbation in SST for a trade-wind cumulus-topped boundary layer (CuBL) using LES, a MLM, and a mixing-line model (XML). They linearized both the MLM and XML in order to derive the eigenvalues (and thereby the characteristic timescales), and found that the XML admits three timescales, including a fast timescale, that also agrees well with LES. The MLM, however, showed no scale separation between the two fastest scales. The authors concluded from the XML that the three scales correspond to the adjustment of sub-cloud buoyancy, the thermodynamic adjustment, and boundary layer deepening. The goal of this manuscript is to address the transient response of a STBL to large scale forcing using both an MLM and an LES. We show there is a previously unnoticed fast timescale associated with cloud thickness-turbulence-entrainment feedbacks that we call entrainment-liquid flux (ELF) adjustment, following Bretherton and Blossey [2013]. We show that this timescale is not only relevant to sudden forcing changes but may also illuminate the long-term response of stratocumulus cloud thickness to global warming. The MLM and LES used in this study are described in Section 2. The setups of the case studies considered are summarized in Section 3. Section 4 presents the linearization approach that is used to extract and interpret the timescales in the MLM, and then applies this approach to an illustrative case. The impact of the MLM entrainment closure is also considered in this section. In Section 5, a comparison between MLM and LES is made to show that a fast scale operates in both models. The fast timescale features prominently in response to perturbations of the entrained free tropospheric humidity, as

6 X - 6 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE well as in an instantaneously warmed climate, but does not come into play for purely radiative perturbations. The findings are summarized in Section Description of models used 91 The LES and MLM that we use are described below. Our simulations are based on cases studied in previous LES and single column intercomparisons. setups are discussed in Section 3. Case-specific simulation 2.1. University of Washington Mixed Layer Model The formulation of the MLM equations follows Uchida et al. [2010] and Caldwell and Bretherton [2009]. It is described here briefly to introduce to aspects of the model that are directly relevant to the discussion that follows. The MLM consists of prognostic equations for the inversion height z i, the moist static energy h = c p T + gz + Lq v, and total water mixing ratio q t = q l + q v : dz i = w e + w s (z i ) (3) dt dh dt = 1 ( w e i h + C T V s h ) BLF R z i (u h) (4) z i ρ 0 dq t = 1 (w e i q t + C T V s q t + F P (0) z i (u q t )) (5) dt z i Here c p is the specific heat for dry air at constant pressure, T the temperature, g the acceleration due to gravity, z the altitude, L the latent heat of vaporization for water, q v the water vapor mixing ratio, q l the cloud liquid water mixing ratio, w e the entrainment rate, w s (z i ) the large scale subsidence, C T = is a surface transfer coefficient, V the surface wind speed, F R (z) is the radiation flux, BL F R = F R (z i + ) F R (0), ρ 0 a reference density, F P (z) the precipitation flux, (u φ) a specified horizontal advection tendency of the mixed layer scalar φ M (h or q t here), i φ = φ(z i + ) φ M the jump of φ across the

7 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X capping inversion, and s φ = φ s φ M, where φ s is φ evaluated at sea surface temperature (SST), sea level pressure (SLP), and saturation water vapor q s (SST, SLP ). The MLM uses the Nicholls and Turton [1986] entrainment parameterization (hereafter referred to as the NT entrainment closure) modified to include LES-tuned parameterization of sedimentation effects on entrainment efficiency [Bretherton et al., 2007]. Drizzle is parameterized as in Caldwell and Bretherton [2009]. The evaporative enhancement coefficient a 2 = 60, the cloud base drizzle rate power law, and the feedback coefficient a sed = 9 for cloud droplet sedimentation on entrainment have been adjusted to match the LES, as in Bretherton et al. [2013]. This allows the MLM to simulate boundary layer properties that are comparable to the LES, permitting more meaningful comparison of sensitivities of the MLM and LES to climate-relevant perturbations. The MLM assumes that the boundary layer remains well-mixed. We use the buoyancy integral ratio (BIR) of Bretherton and Wyant [1997] to diagnose the likelihood of decoupling in terms of the vertically-integrated negative subcloud buoyancy flux. We use a threshold of BIR < 0.15 to identify a well-mixed boundary layer SAMA LES 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] and Blossey et al. [2013]. The advection scheme of Blossey and Durran [2008] is used for the four advected scalars, liquid static energy s l = c p T + gz L(q l + q r ), total nonprecipitating water mixing ratio q t = q v + q l, rain water mixing ratio q r, and rain number concentration N r. As noted by Blossey et al. [2013], this model version, which we call SAMA, produces less numerical diffusion than the default SAM at the sharp,

8 X - 8 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE poorly resolved inversion that caps the stratocumulus cloud layers that we are simulating, resulting in higher and more realistic simulated stratocumulus liquid water paths. The cloud liquid water and temperature are diagnosed from the advected scalars using the assumption of exact grid-scale saturation in cloudy grid cells. The Khairoutdinov and Kogan [2000] scheme is used for conversion between cloud and rain water. Cloud droplet sedimentation is included following Eq. (7) of Ackerman et al. [2009], based on a lognormal droplet size distribution with fixed cloud droplet number concentration and a 138 geometric standard deviation σ g = 1.2. Following Blossey et al. [2013], the domain is doubly-periodic in the horizontal, the vertical grid spacing is 5 m near the trade inversion while the horizontal resolution is 25 m. 3. Case Studies 141 Our simulations are based on cases that have previously been the subject of LES and 142 single column model intercomparisons. We use MLM simulations of the first research flight of the Second Dynamics and Chemistry and Marine Stratocumulus (DYCOMSII- RF01) case study [Zhu et al., 2005; Stevens et al., 2005] to introduce and interpret the fast adjustment timescale. The Cloud Feedback Model Intercomparison Project - Global Atmospheric System Study Intercomparison of Large-Eddy Simulation and Single Column Models (CGILS) S12 stratocumulus case [Zhang et al., 2013; Blossey et al., 2013] is used in conjunction with additional perturbations described below to compare the fast timescale response in the MLM and LES. For completeness, relevant MLM steady state variables are provided in Table 1 for each of the cases discussed in the remainder of this section. We consider two separate case studies for a few reasons. DYCOMSII-RF01 is based on a well-observed nonprecipitating nocturnal stratocumulus-capped mixed layer, has been

9 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X studied extensively using LES, and a previous MLM implementation is well-documented [Uchida et al., 2010]. However, it is not in steady state, and multi-day steadily-forced simulations with the SAMA LES evolve to a slightly decoupled steady state less amenable to close comparison with MLM. The CGILS S12 case can only be compared with climatological satellite observations, but does reach a well-mixed steady state with LES [Blossey et al., 2013], and also has been subjected to a variety of forcing perturbations relevant to climate change that may project onto the fast time scale [Bretherton et al., 2013]. For both cases these models use a bulk surface flux formulation, fixed cloud droplet concentration N d, and cloud-interactive longwave and shortwave radiation. A simplified parameterization of the radiation flux is employed in some configurations. This idealized radiative cooling has the form 164 F R (z) = F 0 e κ LQ(z, ) + F 1 e κ LQ(0,z) + F SW e κ SQ(z, ) + F s0 + F s1 z (6) where κ L, κ S, F 0, F 1, F SW, F s0, and F s1 are parameters, Q(a, b) = b a ρq l dz, (7) and ρ is the density of dry air. The first two terms in (6) represent longwave radiative cooling, and is the same expression that appears in Zhu et al. [2005], neglecting the term accounting for enhanced radiative cooling above the boundary layer. The remaining terms represent the diurnally-averaged shortwave contribution. For z > z i, F R (z) = F R (z i ) in (6).

10 X - 10 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE 3.1. DYCOMSII-RF The DYCOMSII-RF01 case study derives from the first research flight of Stevens et al. [2003] and is based on the SCM intercomparison specifications of Zhu et al. [2005]. The MLM implentation follows Uchida et al. [2010]. Salient details are summarized here. The subsidence is specified in terms of the large scale divergence, w s (z i ) = Dz i, with D = s 1. There is no horizontal advective tendency in q t or h. The free troposphere profile of h increases with a constant lapse rate of dh + (z i )/dz i = 6 kj kg 1, 178 while q + t = 1.5 g kg 1 is constant in the free troposphere. Fixed droplet concentration of N d = 150 cm 3 is used throughout. Longwave radiation is specified following (6), with κ L = 85 m 2 kg 1, F 0 = 70 W m 2, and F 1 = 22 W m 2. The shortwave parameters 181 F SW = F s1 = F s0 = 0 reflect that this case is nocturnal. The MLM is used to evolve 182 (3) (5) from the initial condition z i (0) = 840 m, z b (0) = 610 m, and q t (0) = 9 g kg CGILS S An overview of CGILS is provided by Zhang et al. [2013]. The goal of CGILS was to develop prototype cases for comparing the response of subtropical cloud topped boundary layers to idealized climate perturbations in both single-column models and LES. Cases were set up for three different cloud regimes. In the present study, we focus only on the well-mixed stratocumulus regime, with forcings corresponding to typical summer conditions at the S12 location along a transect across the northeast Pacific Ocean extending from near San Francisco past Hawaii. We use this case to compare the fast timescale response in LES and MLM, and also to demonstrate the relevance of the fast timescale for the STBL response to climate change. The LES and MLM are configured as described in Bretherton et al. [2013], except as described below.

11 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X In the MLM of Bretherton et al. [2013], the boundary layer evolution was coupled to a fully-interactive free troposphere whose temperature and moisture evolve in time to achieve radiative equilibrium. This facilitated a better quantitative comparison with the LES. However, the evolution of the free troposphere somewhat obscures the extraction of the boundary layer timescales. We therefore neglect the evolution of the free troposphere in the current MLM simulations, and hold the free troposphere moisture and temperature profiles fixed to the initial CGILS-specified profiles shown in Figure 1 of Blossey et al. [2013]. From Blossey et al. [2013], we consider the control (CTL) simulation, corresponding to July, 2003, monthly-mean boundary conditions and idealized advective forcings. Following Bretherton et al. [2013], we also consider the P2 and 4CO2 perturbations. The P2 simulation represents an idealized large scale climate change nominally corresponding to a uniform 2K warming of the sea-surface temperature (SST) locally and over the entire tropics, along with a moist-adiabatic increase of the free tropospheric temperature profile that might accompany such an SST increase, and a moistening of the free troposphere to maintain constant relative humidity. The 4CO2 simulation follows the CTL setup, but with 4 times the concentration of carbon dioxide. In the above-described S12 simulations, radiation is calculated using the Rapid Radiative Transfer Model - GCM version (RRTMG) scheme [Mlawer et al., 1997] by extending the model grid up into the stratosphere to allow specification of thermodynamic profiles throughout the entire atmospheric column. We also construct simulations and perturbations, denoted as S12R, that use the idealized radiative cooling of (6), with F 0 = 82 W m 2, F 1 = 15.8 W m 2, κ L = m 2 kg 1, F SW = 15.5 W m 2,

12 X - 12 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE κ S = 39 m 2 kg 1, F s0 = W m 2, and F s1 = W m 3. The primary reason for this is to allow the MLM to be linearized using the approach presented in Section 4. In the LES, the S12R simulations also apply a horizontal advective temperature tendency of K day 1 and a horizontal advective q t tendency of day 1 up to the inversion, defined as the height at which q t = 8 g kg 1. We also consider the perturbations DR, with F 0 increased to 86 W m 2, and DQ, in which the free troposphere moisture, q t +, is increased by 1.5 g kg Linearization and Timescales in MLM Consider a generic system of ordinary differential equations given by dx dt = f(x; α), (8) where x = (x 1, x 2,..., x n ) T are n prognostic variables, α = (α 1, α 2,..., α m ) T are m pa- rameters, t is the independent variable, and f is the specified right hand side function governing the evolution of x. The internal adjustment timescales on which this system adjusts to a perturbation in either the forcing or the mean state are determined by lin- earizing this system of equations. In particular, if x 0 (t) satisfies (8), a small perturbation applied at t = 0 to either the trajectory (x = x 0 + δx) or the forcing parameters (δα) evolves approximately according to the linearization 232 dδx dt J(x 0 (0); α)δx + δf, (9) 233 where J is the Jacobian matrix whose i th row and j th column is given by 234 J ij = f i x j, (10)

13 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X evaluated at x 0 (0), and 236 δf = k f α k δα k (11) is the perturbation to the forcing. Provided J is diagonalizable, the solution to (9) can be given in terms of the eigenvalues λ i and eigenvectors v i of J as for suitable constants a j and b j. δx(t) = n ( aj + b j e ) λ jt v j (12) j=1 In the case that x 0 (t) is a fixed point (i.e., 0 = f(x 0, α)), the eigenvalues of J determine the linear stability of the fixed point to perturbations in the state variables x. If the real component of all eigenvalues is negative, then small perturbations about x 0 decay in time and the fixed point is asymptotically stable. The time scale τ j for the decay of eigenvector v j is then given by τ j = 1 R[λ j ]. (13) Thus, we need only know the steady state (which can be obtained numerically by evolving the model to equilibrium or using an appropriate nonlinear solver) and the specified form of f to extract the timescales on which a perturbation will decay to the steady state. This approach, which we use here to determine the timescales in the MLM, is the prototypical way to analyze fixed point stability [Strogatz, 2001]. For the remainder of this section, we will apply this formalism to the linearization of the DYCOMSII-RF01 case about its steady state. By convention, we order eigenvalues and eigenvectors such that λ 1 corresponds to the fastest timescale and λ 3 the slowest. It is convenient to decompose the Jacobian into a sum of terms 256 J = J 0 + J e + J R + J P, (14)

14 X - 14 where JONES ET AL.: FAST STRATOCUMULUS TIMESCALE J 0 = D 0 0 D dh+ dz i (D + C T V/z i ) (D + C T V/z i ) (15) 259 represents the essential Jacobian matrix in the absence of additional internal feedbacks, and J e = 1 i h z i i q t z i J R = 1 ρ 0 z i J P = z i 1 [ we z i w e h [ BL F R z i ] w e q t [ FP (0) F P (0) z i h BL F R h ] F P (0) q t ] BL F R q t represent entrainment, radiative, and surface precipitation feedbacks, respectively. (16) (17) (18) The eigenvalues associated with the traditional inversion-deepening timescale τ i and thermodynamic timescale τ th appear on the diagonal of J 0. Thus, these timescales are exact in the absence of internal feedbacks, and any steady state solution is a stable node. 268 Using the steady state z i from the top row of Table 1 indicates these two timescales 269 would be τ 0 i = 74.1 h and τ 0 th = 28.9 h. The feedback contributions from (16) (18) act to perturb these fundamental timescales. Contributions from at least one of these feedbacks is necessary for a fast timescale to arise. To extract the timescales, the MLM is run for 10 days of simulation time to approach steady state, and then an iterative nonlinear solver is used to determine the steady state. The Jacobian matrix is evaluated numerically at steady state using a second order accurate finite difference approximation. The eigenvalues and associated timescales are shown in the top row of Table 2. The second and third columns show that τ 2 τth 0 and τ 3 τi 0. However, a faster timescale (τ 1 ) of 7.4 hours is also present.

15 4.1. Origin of the Fast Timescale JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X We claim that the fast timescale is due to entrainment-lwp feedbacks. To support this, we run the MLM in two alternate configurations that yield nearly identical steady state solutions. In the first configuration, denoted hereafter as RP, we assume there is no precipitation or cloud droplet sedimentation, and concentrate the full 48 W m 2 radiative cooling at the cloud top. This has the effect of completely eliminating J R and J P in (14), leaving J = J 0 + J e. As shown in Table 2, this still produces a fast timescale, indicating that entrainment feedbacks alone are sufficient to exhibit a fast response timescale. In the constant entrainment configuration, the MLM is evolved assuming constant, steady state entrainment rate throughout while using the full parameterization for radiation and precipitation i.e., J = J 0 + J R + J P. Table 2 shows that this eliminates the scale separation between the fast timescale (τ 1 ) and the intermediate timescale (τ 2 ). Thus, entrainment feedbacks are also necessary for the fast timescale to arise. For this reason, we will refer to the fast timescale also as the entrainment timescale τ e. Time series of cloud top and cloud base are shown in Figure 1. To emphasize the short timescale adjustment, only the first day of evolution is shown. Both the default and RP configurations evolve similarly, though the RP configuration exhibits a somewhat higher entrainment rate throughout the first day. Using constant entrainment, z i evolves similarly to the other configurations, but the evolution of the cloud base during the first day differs markedly. In particular, both the RP and default configurations exhibit an initial rapid increase in z b that reduces substantially within the first day, whereas the cloud base rises more steadily for fixed w e. This suggests that the entrainment rate feedbacks are tied to changes in the LWP that operate by rapidly changing the cloud base. We provide a

16 X - 16 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE simple, physically-motivated argument for this in the following section by considering the fixed boundary layer depth limit Fixed Boundary Layer Depth Limit In order to understand the mechanism behind the fast timescale, it is useful to consider the limit where z i is held fixed. Such a simplification is reasonable because z i adjusts on a much slower timescale than the fast scale being sought here. This two-dimensional (2D) system also exhibits a fast scale associated with entrainment feedbacks. Figure 2a shows the evolution of the 2D system defined by the thermodynamic MLM equations (4) (5), with z i = 840 m held fixed. Phase plane trajectories, starting from the filled black circle and sampled at two hour intervals, are shown for both the default configuration (in blue) and the RP configuration (in red). Instead of using h and q t for the thermodynamic variables, we have transformed to a coordinate system defined by the cloud base z b and the air-sea virtual temperature difference T v0 = (s vl s vl,0 )/c p as in Bretherton et al. [2010]. Here s vl is the virtual liquid static energy, s vl = c p T v +gz Lq l h µlq t, T v is the virtual temperature, and µ 0.9 is a thermodynamic variable defined 314 in Bretherton and Wyant [1997]. Contours show the entrainment rate for the default 315 configuration. The eigenvectors of the steady state Jacobian matrix are also shown, along with the vector field denoting the contribution of entrainment to the tendency of ( T v0, z b ). The evolution is qualitatively similar between the two model configurations, though the steady state differs somewhat. From the w e contours in Fig. 2a, we see that w e varies most strongly with z b. This is due to the NT entrainment closure that scales w e proportional

17 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X to the integrated buoyancy flux, w e = 2.5A z i 0 w b dz. (19) z i i b Here b = s v /s v0 is a measure of parcel buoyancy, with s v = c p T v + gz the virtual static energy, s v0 a reference virtual static energy, and w b is the horizontally averaged turbulent buoyancy flux. Since there is a large jump in buoyancy flux proportional to the upward flux of liquid water at cloud base [Bretherton and Wyant, 1997], a relatively modest change in z b can have a large impact on the integrated buoyancy flux. On physical grounds, a cloud that is thicker generates more vigorous turbulent kinetic energy (TKE) and drives higher entrainment. Higher entrainment, in turn, dries and warms the boundary layer, causing the cloud base to rise, decreasing the LWP and thereby acting as a strong negative feedback on the entrainment increase. Figure 2 confirms that during the first several hours of evolution there is substantial change in the cloud base that rapidly adjusts the entrainment rate along the fast direction. Once this fast eigenvector has decayed, the slower thermodynamic evolution that projects primarily in the T v0 direction dominates the evolution Quantitative Fast Timescale Estimate We now construct a simplified argument that allows us to quantitatively estimate the fast timescale. The entrainment closure (19) can be written as w e = 2.5A b { w b SC + ( w b SC + w b C ) ( 1 z b z i )}, (20) where w b SC denotes the average of w b across the subcloud layer, and w b C the average over the cloud layer. Bretherton and Wyant [1997] showed that the cloud base jump in buoyancy flux is proportional to the turbulent liquid water flux at the cloud base,

18 X - 18 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE which in turn is proportional to the latent heat flux in a well-mixed STBL, yielding w e 2.5A b ( w b SC + ˆσ g s v0 L w q l ( 1 z b z i )), (21) where ˆσ 0.8 is a temperature-dependent scaling factor [Bretherton and Wyant, 1997]. For a sufficiently short period of time, provided the coefficients in (21) are slowly varying, ( w e w 0 + c 1 z b z i ), (22) where w 0 and c are constants. Following Schubert et al. [1979b], the cloud base can be written as 349 z b (h, q t ) = 1 (dq s /dz) u ( (1 + γ) s q t γ ) L sh, (23) where (dq s /dz) u is the rate of change of q s along a dry adiabat and γ = L/c p ( q s / T ) 1.7. Differentiating (23) and using (22) allows us to rewrite (4) and (5) in terms of z b and s vl instead of h and q t : dz b dt ds vl dt = 1 (w e + C T V + cz b + z /z i) z b + F B /z i (24) } i {{} λ b = 1 (w e + C T V ) s vl + F S /z i, (25) z } i {{} λ th where all source and sink terms that are not directly proportional to the state variables 356 are grouped together into F B and F S, and z + b = z b (h(z + i ), q t (z + i )). In the RP configuration these equations can be solved analytically, but for the purpose of showing the entrainment response provides a fast response timescale for cloud base adjustment, it suffices to note that c > 0 implies λ b > λ th. Thus, the cloud base, which is a function of the thermodynamic moisture and temperature variables, adjusts on a faster timescale than any other individual thermodynamic variable does because the entrainment closure is sensitive to the cloud thickness.

19 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X Figure 2b shows w e vs 1 z b /z i for the trajectories plotted in Figure 2a. During the fast adjustment period, the entrainment rate scales linearly with cloud thickness. A leastsquares linear fit over the first 12 hours of the 2D RP simulation provides an estimate of c = 14.9 mm s 1. Inserting this into (24) yields λ b = s 1, corresponding to a timescale of 3.3 h, which is within about 10% of the fast scale λ 1 = s 1 (3.5 h timescale) calculated using finite differences. Thus, the crude approximation of (22) largely explains the fast scale in the 2D system. The same essential reasoning translates to the full MLM. Provided w e w 0 +c(1 z b /z i ) and z i varies slowly relative to z b during the short timescale, the expression derived in (24) applies and a similar interpretation holds. Figure 3 plots a bundle of trajectories in the full MLM starting from randomly chosen initial conditions. Each of the trajectories approaches the asymptotically stable fixed point in a qualitatively similar way. Initially there is a nearly linear fast response of w e to cloud thickness, which eventually collapses onto a curve corresponding to the slow manifold introduced by Bretherton et al. [2010]. This suggests the linear relationship for the fast scale is robust and motivates us to call this fast scale the entrainment timescale τ e. Also shown in Figure 3 is a linear fit of the form (22), where c = 29.5 mm s 1 was determined by taking the mean of the slopes from linear least squares fits over the fast adjustment portion of each trajectory in Figure 3. We do not expect the 3D MLM to have the same value of c as the 2D MLM, because the trajectories are randomly initialized with a substantially deeper inversion and thicker cloud layer than the 2D MLM steady state, both of which can be expected to change c following (20). This value of c corresponds to τ e 3.4 h. Considering the simplifying

20 X - 20 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE approximations made in the calculation, we still take this to be in reasonable agreement with the fast timescale determined numerically Sample Trajectory in Full MLM To illustrate the role of each timescale, consider a perturbation to the DYCOMSII-RF01 default MLM steady state of the form z i (t) = zi + δz i (t), where the asterisk denotes the steady state value, δz i (0) = 50 m, and the thermodynamic variables are left untouched. Figure 4 shows the MLM evolution of this system. Figure 4a overlays contours of constant entrainment with the trajectory in the (z b, z i ) phase plane. Each square represents the state at six hour intervals and the color indicates T v0. The three eigenvectors are shown as colored line segments. For this case, the slow eigenvector lies along a line of constant entrainment rate, while the fast eigenvector closely approximates the effect of a sudden pulse of entrainment on the state variables. The area of each colored circle in the bottom left corresponds to the magnitude of the projection of that eigenvector in the T v0 direction; only the intermediate eigenvector is seen to have a strong projection onto changes in T v0. Figure 4b show the evolution of z i and z b perturbations expanded into the basis defined by the eigenvectors of the steady state. The timescales associated with each eigenvector are given in the legend. In what follows, we call the three eigenmodes the fast, intermediate, and slow modes. The perturbation increases the cloud thickness so the entrainment rate initially increases. Thus, a pure δz i perturbation projects into the fast direction as well as the slow direction, resulting in a small additional increase in z i over the first 12 hours. Entrainment warming and drying also raises the cloud base by nearly 20 meters over this time through fast-mode adjustment, reducing the entrainment rate back nearly to its steady

21 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X state value. From this point, z i and z b evolve approximately along a curve aligned with the slow eigenvector s projection into this plane, even as the intermediate timescale operates to adjust T v0. Initial entrainment warming increases T v0, then it plateaus for a few days while the 411 intermediate component decays, before T v0 slowly decreases back to its equilibrium value. The slow mode component of T v0 is due to the free tropospheric lapse rate of h. As the inversion adjusts on the slow scale it determines the temperature and humidity of the free tropospheric air entrained into the boundary layer, resulting in a thermodynamic component to the slow eigenvector. This can be verified by considering the components of the eigenvectors of (15), in which the slow eigenvector has nonzero projection into the δh direction that is proportional to dh + /dz i. By contrast, were we to consider the evolution of δq t (t), we would see it projects almost entirely into the fast and intermediate modes 419 because q + t is uniform in the free troposphere Whether or not each of the eigenvectors would tend to increase or decrease the cloud LWP for a given increase in z i depends on whether the cloud top rises faster than the cloud base. Any eigenvector with slope greater than 1 in the (z b, z i ) plane will be associated with an increase in LWP, while any eigenvector with slope less than one is associated with decreasing LWP. Because the entrainment rate is quite sensitive to the LWP, the resulting fast eigenvector projects strongly in the v entr direction, δz i δh = δq t entr z i i h i q t, (26) shown by the orange dashed line in Figure 4a that is nearly indistinguishable from the 428 fast eigenvector. Provided there are no strong radiative or precipitation feedbacks to 429 perturb v 1 far from (26), the LWP response along the fast scale is determined by the

22 X - 22 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE thermodynamic inversion jumps and the depth of the boundary layer. For DYCOMSII- RF01, the fast scale response to is to decrease the LWP as the boundary layer deepens, whereas the LWP increases with boundary layer deepening on the slow timescale Sensitivity to Entrainment Closure The fast scale explored in the preceding section owes its existence to the entrainment response, and therefore is dependent on the entrainment closure used in the MLM. Despite the importance of entrainment in the dynamics of the STBL, there is no consensus about the best way to represent it, and inadequate observational evidence to choose between several proposed entrainment parameterizations [Stevens, 2002]. Previous MLM studies of Caldwell and Bretherton [2009] and Dal Gesso et al. [2013] investigated the sensitivity of the MLM-simulated steady state to the entrainment closure and found that the quantitative details of the boundary layer state (e.g., z i and LWP) vary across parameterizations, but the steady state response to climate change perturbations was qualitatively similar. Zhang et al. [2005] found that the LWP response to the diurnal cycle of insolation was sensitive to the choice of entrainment parameterization. In this section, we address the impact of the entrainment closure on the fast scale by repeating the DYCOMSII-RF01 steady state timescale analysis with a representative set of entrainment closures spanning a range of underlying assumptions. Our default implementation of the NT entrainment closure accounts for the role of evaporative enhancement 448 and cloud droplet sedimentation to the entrainment efficiency. In addition to the de fault configuration, we also consider the case where the entrainment efficiency A is held constant.

23 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X The closure of Schubert et al. [1979b] is a flux-partitioning closure that fixes the subcloud buoyancy flux at a particular point (in this case, just below the cloud base) to a fraction of the integrated buoyancy flux: w b (zb ) = k zi w b (z)dz, (27) where k = 0.5 is used. The closure of Lewellen and Lewellen [1998] (hereafter LL) is another flux-partitioning closure which deduces the entrainment rate from the assumption that η LL = zi 0 ( w b NE w b )dz zi, (28) 0 w b NE dz is constant (η LL = 0.35 is used). Here w b NE is the buoyancy flux from all processes except entrainment. The closure of Lock et al. [2000] relates the entrainment rate to a combination of velocity scales due to different sources of turbulence. The Lock closure for the DYCOMSII-RF01 case can be written as we L V = A sum/z 3 i + g β T α t BL F R /(c p ρ 0 ) 1 b + cv 2 sum/z i (29) where V 3 sum is a sum of contributions from surface heat flux, cloud-top radiative cooling, and evaporative cooling. We take A 1 = 0.23, β T K 1, α t = 0.2, and c = 1. The cloud thickness factors into (29) through the surface and radiative contributions to V 3 sum, as shown in the appendix of Lock et al. [2000]. Bretherton and Wyant [1997] argues that the steady-state entrainment rate must ul- timately balance the diabatic flux divergence of s vl across the cloud layer. They argued this to be an approximate, rather than an exact, balance, as it amounts to neglecting the buoyancy flux below the cloud layer. Neglecting the subcloud component to the buoyancy

24 X - 24 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE 473 flux leads to the energy balance entrainment closure: 474 w E e = cldf R+P ρ 0 i s vl, (30) where cld F R+P is the divergence of F R+P = F R µlf p across the cloud layer. This closure was originally introduced as the minimal entrainment closure by Lilly [1968], and is equivalent to the Schubert closure with k = 0. It is closely related to a similar closure used as a simplified entrainment closure in Zhang et al. [2005], van der Dussen et al. [2013], and Dal Gesso et al. [2013]. Table 3 shows the eigenvalues obtained from the steady state of DYCOMSII-RF01 MLM simulations performed with each of these closures in the RP configuration. Since the emphasis is on the role of the entrainment closure, we present only the RP simulations to avoid biases in the timescales due to precipitation or radiative feedbacks. Although we ve shown previously that these effects are minor with the NT closure, each entrainment closure settles into its own steady state entrainment rate, so the relative role of precipitation 486 and radiative feedbacks may differ across the simulations. For example, a large LWP would enhance the precipitation feedbacks, whereas a very small LWP may decrease the overall radiative flux divergence. Somewhat surprisingly, nearly all of the closures investigated include some form of fast timescale response. The only closure for which the fast timescale is not present is the energy balance closure. The reason for this is that each of these closures, with the exception of the energy balance closure, depends on an integral of the buoyancy flux. Since there is a substantial jump in buoyancy flux at the cloud base due to latent heating, each entrainment closure is sensitive to the cloud thickness, and thus also the level of the

25 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X cloud base. Although the magnitude of this fast scale response varies between closures, its existence does not. In the Lock, LL, NT, and Schubert closures the fast timescale ranged from 5.6 h 10.1 h, exhibiting separation from the intermediate scale ( h). Although the energy balance closure ultimately produces a reasonable steady state, it does not capture the fast ELF adjustment timescale because this closure does not depend explicitly on the integrated buoyancy flux. 5. Fast Timescale in MLM and LES In this section we use the CGILS S12 framework to show that the fast timescale also arises in the LES in a way that is consistent with the interpretation laid out in the previous sections S12R DQ and DR Perturbations We consider first the CGILS S12 case with idealized radiation (S12R), described in Section 3.2. The LES is evolved for one day starting from the steady state of a similarly forced CTL simulation run to steady state in a smaller domain. The LES inversion depth and LWP are still slowly evolving when the DQ and DR perturbations are applied at t = 0, but the formalism introduced in Section 4 still applies. Rather than perturbing the state variables, as in the MLM analysis of DYCOMSII-RF01, we perturb the forcing and observe the growth of the difference between perturbed and control trajectories. That is, δf 0, but δx(0) = 0 in (9). 513 We consider two separate perturbations. In the first, denoted hereafter as DQ, q + t is 514 increased by 1.5 g kg 1 in the free troposphere, using the nudging approach discussed

26 X - 26 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE 515 in Blossey et al. [2013, Appendix A4] with a timescale of 10 min. In the MLM, the perturbation occurs instantaneously. The second perturbation, denoted as DR, increases the radiative flux divergence by instantaneously increasing F 0 by 4 W m 2 in (6). The MLM is evolved for 10 days to achieve an approximate steady state before applying the same perturbations. The use of a fixed free tropospheric profile in the MLM and idealized radiation in place of a more sophisticated radiative transfer model previously used in both MLM and LES [Bretherton et al., 2013] allows the system to be formally linearized as in the previous section. The idealized radiation scheme is independent of droplet size and free troposphere humidity, so there is no direct radiative impact from the DQ perturbation. Instead, the increased free troposphere humidity influences the cloud thickness primarily through the amount of moisture entrained into the boundary layer, and secondarily by altering the entrainment efficiency and buoyancy jump across the inversion. Figure 5 shows the evolution of the mean inversion and cloud base in the LES and MLM for the CTL, DQ, and DR cases. In the LES, the inversion is calculated as the mean level where relative humidity decreases below 50%, and the cloud base is identified as the level at which the mean cloud cover exceeds 50% of the maximum cloud cover. The MLM-simulated boundary layer is deeper and has a slightly higher LWP (50 g m 2 vs. 45 g m 2 ), but the response to the perturbations is similar. The DQ perturbation rapidly decreases the cloud base, with the bulk of the response occurring by 0.5 days in both LES and MLM. This thickens the cloud which, in turn, generates greater entrainment. The fast response of the LWP is more clearly visualized by considering its difference in between the perturbed and CTL runs, shown in Figure 6. In both the LES and the MLM the LWP increases rapidly after the DQ perturbation, but slowly in response to

27 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE X the DR perturbation. It is tempting to consider this rapid response in the DQ case as an expression of the fast timescale, but it is actually due to additional moisture being entrained into the boundary layer in DQ relative to CTL. Since the fast timescale is a result of the entrainment adjustment to LWP, the fast response is better understood as the decrease of dlwp/dt over the first half day from its initial fast rate to the slower LWP evolution that follows. If the entrainment rate was held constant at the moment the perturbation is applied, the LWP difference would continue to increase nearly linearly. However, the larger LWP in the DQ case drives increased entrainment, which curtails further LWP growth. By contrast, the DR perturbation drives greater entrainment from the outset, but is accompanied by a weak LWP response that evolves on a slower time scale. A more complete argument is formed by linearizing the MLM about the CTL steady state at t = 0, following (9). This linearization is represented by the dotted lines in Fig- ure 6. The evolution of the linearized system accurately matches the full MLM for the DR perturbation. It also matches fairly well for the DQ perturbation, though the linearized system underestimates the long-term LWP response. For each perturbation, the fast scale component is the projection of the associated fast scale eigenvector onto the overall per- turbation. Based on the form of (12), the perturbations δz i and δz b can be expressed as a linear combination of eigenvectors at each time t. Since LWP is approximately a quadratic function of cloud thickness, the perturbation to LWP is δlw P (t) LW P (0) 2δ(z i z b ) z i (0) z b (0). (31) The dotted line in Figure 6 shows the δlw P attributable to the fast contribution to δ(z i z b ). There is very little fast δlw P response for the DR case (only about 2 g m 2 ),

28 X - 28 JONES ET AL.: FAST STRATOCUMULUS TIMESCALE whereas nearly all of the much larger δlw P response in the DQ case projects onto the fast scale. Thus we see that a purely radiative perturbation does not elicit a significant fast scale response in either MLM or LES, but changing the overlying humidity does result in a rapid response due to entrainment-mediated feedbacks. Because purely radiative perturbations do not project strongly onto the fast scale, the fast scale is not expected to contribute substantially to the diurnal response of the STBL. This was verified through MLM simulations and explicit linearization (not shown) Fast scale in CGILS S12 climate-warming perturbation Fast ELF adjustment is unexpectedly relevant to understanding how stratocumulus may respond to warming of the climate. We now consider three climate change sensitivity studies based on the CGILS S12 case as presented by Bretherton et al. [2013]: the control (CTL) simulation, a +2K warming scenario (P2), and the control run with quadrupled CO 2 concentration (4CO2). Unlike the previous section, fully interactive RRTMG radiation is used to be consistent with this previous MLM and LES modeling work. All simulations were initialized with the same cloud base, top, and air-sea temperature difference, as described in Section 3.2. Figure 7 shows the difference in LWP between P2 and CTL, and between 4CO2 and 577 CTL. Consider first the LES (solid lines). There is a fast scale adjustment in the P LW P relative to the CTL, evidenced by the 5 g m 2 decrease in LWP within the first half day before levelling off for the remainder of the time plotted. The LWP perturbation in 4CO2, however, decreases nearly linearly throughout the first two days, and thus does not project onto the fast scale. This is consistent with the analysis of the previous section, in which a perturbation to the radiative flux divergence did not excite a fast scale response.