a. Idealized large-scale conditions

We use the LES and bulk models presented in the next subsections integrated in the idealized framework developed by Bellon and Stevens (2012). This framework formulates the large-scale conditions and forcings in terms of a minimum set of parameters. The subsidence profile follows an exponential of altitude:

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where w(z) is the large-scale vertical velocity at altitude z, w₀ is the high-altitude asymptotic value of subsidence (w₀ > 0 for subsidence), and z_w is a typical vertical scale of large-scale dynamics. The divergence at the surface is D₀ = w₀/z_w and it decreases exponentially with altitude.

The large-scale horizontal advection of humidity is neglected and radiation and large-scale horizontal advection of energy are modeled by a prescribed, altitude-independent cooling R (R > 0 for cooling). The vertical gradients of free-tropospheric humidity q_ft and potential temperature θ_ft are constrained by the water balance w∂_zq_ft = 0 and energy balance w∂_zθ_ft = −R in the nonturbulent troposphere, so that the free-tropospheric humidity and potential temperature profiles are determined by a reference humidity and temperature:

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with q₀ and θ₀ two parameters that prescribe the uniform free-tropospheric specific humidity and the reference temperature at z = z_w ln(2). Equation (3) describes a profile that asymptotically approaches a linear profile at high altitude (∂_zθ_ft → R/w₀), and tends toward infinitely cold temperatures at the surface.

The SST is set to a constant, T_s, and a constant geostrophic zonal wind U is imposed (the latitude is set to 20°N). The parameters describing the large-scale conditions and forcings are summarized in Table 1.

Table 1.

Large-scale conditions and forcings.

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Using this framework, Bellon and Stevens (2012) studied the sensitivity of the equilibrium boundary layer to SST and free-tropospheric conditions, and Nuijens and Stevens (2012) studied the CuBL sensitivity to perturbations in the geostrophic wind speed. Here, we focus on the transient behaviors of the model. We present results for simulations in which T_s is instantaneously increased from 298 to 299 K at t = 0. The final state of the stationary simulation with T_s = 298 K presented in Bellon and Stevens (2012) is used as initial conditions and T_s is set to 299 K. We also performed simulations with lower and higher values of T_s and found very similar behaviors. A few LES were performed in which the T_s was decreased, and it was found that if the abrupt change is large enough (e.g., from 300 to 299 K), the model behavior is highly nonlinear, going through a cloud-free regime before the shallow convection is reestablished. If the abrupt change is smaller (e.g., from 299.5 to 299 K), the model behavior is very similar to the results presented here.

b. LES

We use the University of California, Los Angeles (UCLA) LES in a nonprecipitating configuration. The basic code is described by Stevens et al. (2005), who solve the Ogura–Phillips anelastic equations using finite differences on a regular-horizontal, stretched-vertical mesh. Cloud and microphysical processes are represented following the procedures described by Savic-Jovcic and Stevens (2008).

Simulations are performed over a square horizontal domain of 6.4 km × 6.4 km, with a horizontal resolution of 50 m in both directions. The vertical grid consists of 120 vertical levels with a 10-m grid spacing at the surface increasing by 2% at each level so that the domain top is at 4830 m.

With the large-scale conditions used in the present work, the boundary layer is not very deep and the nonprecipitating assumption seems reasonable. But precipitation could have an influence on the transient behavior of the models, in particular via the precipitation flux at cloud base. We do not address this issue in the present work.

c. Mixing-line model

The mixing-line model (XLM) is fully described in Bellon and Stevens (2012). It is a bulk model of the trade wind boundary layer based on the work of Betts and Ridgway (1988). Profiles of conserved variables above the cloud base are described as linear mixtures between the subcloud and inversion-top air. The mixing line is defined as the height-dependent ratio of inversion-top air in an air parcel at a given altitude. (The rest of the parcel is considered to originate from the subcloud layer.) In the bulk formulation of the mixing-line model, only the cloud-layer vertical average of the mixing line, which we denote by α, and its divergence-weighted vertical average, which we denote by γ, appear in the bulk water and energy budgets. In Bellon and Stevens (2012), these parameters are tuned using the stationary states of LES with varying surface and free-tropospheric conditions; here we use the same values of these parameters: α = 0.35 and γ = 0.8. The XLM was also generalized to model cloud-free boundary layers as well as cloudy ones, but here we focus on the CuBL. We reformulate the XLM in this regime [Eqs. (15a) and (16)–(18) in Bellon and Stevens (2012)] to express the equations in terms of total water q_t and θ_υ = θ_l + ε₁θ_rq_t, where θ_l is the liquid water potential temperature, θ_r is a reference temperature (299 K), and ε₁ = R_υ/R_d − 1, with R_d (respectively, R_υ) the gas constant of dry air (respectively, water vapor). In the absence of condensate, θ_υ is the virtual potential temperature; in the cloud layer, it is a crude measure of buoyancy and we will refer to it as pseudobuoyancy.

In this formulation, the prognostic variables of the XLM are the altitude of the inversion-top h, also referred to as the CuBL depth, and two conserved variables in the subcloud well-mixed layer: the virtual potential temperature θ_υM and total water mixing ratio q_M. The altitude of cloud-base η is a fourth, diagnostic variable. The equations of the XLM in the shallow-convective regime are the water and energy budgets of the CuBL and two closures at cloud base. For the large-scale conditions described above, the vertically integrated water and pseudobuoyancy budgets are

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with the inversion-top subsidence w_h = −w(h) and the characteristic speed of surface fluxes w_s = C_dU, with C_d = 1.2 × 10⁻³. The saturation mixing ratio at temperature T and pressure p is noted q*(T, p); q_s = q*(T_s, p_s) is the saturated mixing water ratio at the surface, with p_s the surface pressure, and θ_υs is the surface virtual potential temperature: with p₀ the reference pressure and c_p the specific isobaric heat capacity of dry air. Angle brackets indicate the vertical average over the CuBL:

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where φ is either q_t or θ_υ.

In Eqs. (4) and (5), the left-hand side corresponds to the rate of change of the vertically integrated water or energy content of the CuBL. On the right-hand side, the first terms correspond to the vertical advections by the subsidence w_h, with the parameter γ describing the effect of the mixing line compared to a well-mixed boundary layer (which corresponds to γ = 1). The second terms correspond to the surface fluxes, and the third term in Eq. (5) is the prescribed horizontal advection and radiation.

The first closure at cloud base assumes that the turbulent buoyancy flux F^υ at cloud base is proportional to the surface buoyancy flux: F^υ(η) = −kF^υ(0), with a factor k = 0.2. This closure is not expected to be valid instantaneously after an abrupt change in T_s and associated surface fluxes but rather to adjust on a time scale characteristic of the eddy turnover. From our LES as well as the one documented in Bretherton and Park (2008), it appears that the turbulence and cloud mass flux adjusts within a few minutes of the SST change. This time scale is small compared to the time scales of at least a few hours under consideration here, and we apply the closure instantaneously. It can be used to simplify the buoyancy budget of the subcloud layer:

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The second closure assumes that the cloud base is at the lifting condensation level (LCL):

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d. Mixed-layer model

The mixed-layer model (MLM) is used in Bellon and Stevens (2012) as well. It models the boundary layer as a well-mixed layer capped by an infinitely thin inversion, following the work of Lilly (1968) and Schubert et al. (1979a,b). The equations of the MLM are based on the water and energy budgets of this layer, and one closure on the downward pseudobuoyancy flux at the base of the inversion, which is considered as proportional to the surface upward buoyancy flux (with a factor k = 0.2). This simplification is the main difference with previous mixed-layer models starting with that in Schubert et al. (1979a,b). With our large-scale conditions, the resulting equations are as follows:

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These equations are identical to the XLM equations with α = 0, γ = 1, and η = h instead of the closure on the LCL. This type of model was developed to simulate the SCuBL or the convective boundary layer. Although it is not capable of representing the dependence of the stationary states of the CuBL as a function of changes in the large-scale conditions, we use it here to explore the initial evolution of the subcloud layer, as it provides some insight into the behavior of the more complex models and the LES.

a. LES results

Figure 1 shows the time evolution of h, η, q_M, and θ_M, latent and sensible turbulent heat fluxes at the surface and at cloud base, and the cloud fraction for the experiment described in the previous section (T_s is increased from 298 to 299 K at t = 0).

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Time evolution of (a) the altitudes of the inversion top (solid line) and of the cloud base (dashed), (b) subcloud water ratio, (c) subcloud potential temperature, (d) latent and (e) sensible heat fluxes at the surface and cloud base, and (f) cloud fraction for the first 8 days of the LES with T_s = 299 K, using the stationary state with T_s = 298 K as initial conditions. Note that the horizontal time scale changes after 2 days.

Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-0219.1

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The adjustment of θ_M appears to be much faster than the other variables, with an adjustment time of a couple of hours (Fig. 1c). The sensible heat flux (SHF) at the surface and at cloud base and cloud-base latent heat flux (LHF) adjust fast as well (Figs. 1d,e). This adjustment happens on time scales that are significantly shorter than the time scales τ_s, τ_D, and τ_R associated with the physical processes, or a simple combination of them such the fast time scale τ_th identified in earlier studies (Schubert et al. 1979b; Bretherton et al. 2010), or even the surface-flux time scale of the subcloud layer (Betts 1993).

The subcloud mixing ratio and the LHF at the surface adjust somewhat slower (Figs. 1b,d) but still faster than h (which is not yet at its stationary value of 1370 m after 8 days; see Fig. 1a); the time evolutions of q_M and of the surface LHF are also nonmonotonic.

The initial decrease in q_M results from the initial increase in cloud-base LHF that is stronger than the initial increase in the surface LHF (Fig. 1d); the surface LHF increases at t = 0 because of the abrupt change in T_s, and it then increases further because of the decrease in q_M. The increase in the cloud-base LHF results from an increase in turbulence associated with the increase in surface buoyancy flux, which drives more cloud mixing as suggested by the increase in cloud fraction (Fig. 1f). The increased turbulence at cloud base warms and dries the subcloud layer.

b. XLM results

The XLM exhibits the same behavior as the LES. This is shown in Fig. 2, which shows the same simulation as was performed by the LES, but with the XLM (thick lines in Figs. 2a–c, all lines in Figs. 2d,e).

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Time evolution of (a) the altitudes of the inversion top and of the cloud base, (b) subcloud water ratio, (c) subcloud potential temperature, and (d) latent and (e) sensible heat fluxes at the surface (solid lines) and cloud base (dashed) for the first 8 days of the XLM experiment with T_s = 299 K, using the stationary state with T_s = 298 K as initial conditions. In (a)–(c), the nonlinear experiment is shown by the thick solid lines, the linear time evolution is shown by the thin solid lines, and the latter’s projections on mode 1, mode 2, and mode 3 are shown, respectively, by the dashed, dashed–dotted, and dotted lines. Note that the horizontal time scale changes after 2 days.

Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-0219.1

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As was also evident in the LES, the LHF increases more at cloud base than at the surface for complex reasons. One key element is that the strong increase in surface buoyancy flux is driven by the increase in surface SHF (a buoyancy flux increase of 12 W m⁻²) rather than by the surface LHF (a buoyancy flux increase of 3 W m⁻²). This buoyancy flux results in strong mixing at cloud base.

c. MLM interpretation of the initial tendencies of temperature and humidity

We can try to understand the initial warming and drying of the CuBL with the MLM. Our approach is to explore the initial response of the MLM about the initial state, which is the equilibrium state before the abrupt change in T_s. To do so we first solve for the equilibrium state before the abrupt change in T_s, and then use this to derive an equation for the initial tendencies of the boundary layer temperature and moisture, to see if the counterintuitive drying can be represented by this model.

If, at t = 0, the model is at equilibrium with T_s = 298 K, we can write Eq. (9) and the difference between Eqs. (10) and (11) (which is actually the closure at the inversion that simplifies the buoyancy budget of the infinitely thin inversion layer):

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with the overbar indicating the equilibrium with T_s = 298 K.

If at t = 0, T_s is increased by an increment , the initial tendency of the boundary layer virtual potential temperature can be obtained from Eq. (11):

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where is the perturbation of surface virtual potential temperature resulting from , with the surface saturation mixing ratio perturbation resulting from (which is of the same sign as ). For a warming, the initial tendency of θ_υM is positive.

Additionally, Eq. (9) can be rewritten introducing the CuBL-top entrainment speed w_e = ∂_th + w_h to express the initial tendency in water mixing ratio:

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The closure at the inversion provides an expression of w_e:

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which can be introduced in Eq. (15), while using Eqs. (12) and (13) to replace the ratio of the equilibrium jumps at the inversion by the ratio of the equilibrium sea–air contrasts at the surface:

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The sign of the initial tendency in humidity depends on the competition between the perturbations of evaporation and the perturbation of surface buoyancy flux weighted by the ratio of air–sea humidity contrast to air–sea virtual potential temperature contrast. This can be written as a simple dependence on the Bowen ratio B′ of the initial perturbation (defined as the ratio of the sensible flux anomaly to the latent flux anomaly) and on the equilibrium Bowen ratio :

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with the constant , L_υ being the latent heat of vaporization.

Through the closure on the buoyancy flux, the equilibrium surface sensible heat flux is constrained by the ratio of the mixed-layer radiative cooling over the surface latent heat flux: , with ΔF^R the divergence of the radiative flux across the layer. Since the radiative cooling is about 30 W m⁻² compared to a large evaporation of about 150 W m⁻², is about 0.1, which is a typical value for the tropical boundary layer. The Bowen ratio of the initial perturbation associated with a change of T_s at constant pressure is typically 0.4, so that the term in parentheses in Eq. (18) is negative. The initial tendency of q_M therefore has the opposite sign to the initial perturbation of evaporation and to the initial tendency of θ_M. Ultimately, these opposite signs result from the fact that the initial surface SHF perturbation associated with a change in surface temperature is comparable to the corresponding LHF perturbation, but the equilibrium SHF is small compared to the equilibrium LHF because of the constraints of the energy and water balances of the mixed layer. As a result, an increase in SST increases evaporation less than it increases entrainment drying through the increase in inversion-base buoyancy flux. This result is dependent on the closure used in the MLM: Schubert et al. (1979b), using a more refined closure, did not obtain such opposite tendencies.

Although the vertical structure and closures are different in the XLM and LES, the initial drying of the subcloud layer in our experiment suggests that a similar mechanism is at play. We can consider that the relevant processes in the cloud and inversion layers of the XLM and LES behave similarly to the processes controlling the infinitely thin inversion layer in the MLM. In the XLM and LES, the SST increase causes an increase in buoyancy flux at cloud base (in absolute value); the resulting increase in cloud turbulence increases entrainment at the inversion, and entrainment drying is efficiently communicated to the subcloud layer by the enhanced cloud turbulence. For the XLM, we have not found simple analytical expressions for the initial tendencies because of the increased complexity in the vertical structure, so in the next section we approach this through a more formal linearization.

d. Linear XLM results

We can linearize the system of Eqs. (4)–(8) around a reference equilibrium state and replace η in Eqs. (4)–(7) using Eq. (8), and nondimensionalize the system using the following scales:

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These scalings are dependent on the reference state indicated by the overbar. For a reference state that corresponds to the model equilibrium for T_s = 299 K, [z] ≈ 1 km is the equivalent height of the CuBL that characterizes the change of the CuBL vertically integrated heat or water content associated with a change in the subcloud variable, [q] ≈ 12 g kg⁻¹ and [θ_υ] ≈ 8 K characterize the stratification accross the CuBL, and [t] ≈ 22 h is the time scale associated with surface fluxes.

The linearized, nondimensionalized system can be written in a matrix formulation:

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with the state vector

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where the circumflex indicates the nondimensional perturbation. Using Eqs. (4)–(8), we can write the matrices and as follows:

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with the following notations: is the normalized reference depth of the subcloud layer, is the normalized reference gradient of potential temperature at the top of the inversion, is the nondimensional rate of subsidence there, ε_R = R/w_s · [z]/[θ] is the nondimensional rate of radiation, and is the nondimensional ratio between divergence and subsidence at the inversion top. Here Θ (Q) is the nondimensional rate of increase (decrease) of η with θ_υM (q_M):

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and can be computed by differentiating Eq. (8) above [see, for example, appendix 3 in Bretherton and Park (2008)].

Table 2 lists the nondimensional parameters for the T_s = 299 K equilibrium as reference state.

Table 2.

Nondimensional parameters for a reference equilibrium state with T_s = 299 K.

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In [see Eq. (26)], the diagonal terms of the upper-right block correspond to the effect of the surface turbulent fluxes and advection of anomalous humidity and temperature. The first coefficient on the first line corresponds to the additional advection due to the vertical displacement of the inversion top. The first coefficient on the second line corresponds to a combination of this additional subsidence with the additional radiative cooling and the change in the potential temperature of the advected air.

The matrix admits the inverse

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where λ = 1 − α − αΓ, μ = 1 − α − α²ΓΘ, ν = 1 − α − αΓ(1 − αQ), and ψ = 1 − α(Q + Θ).

The system (23) can be rewritten

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Figures 2a–c show the time evolution of the linear XLM with T_s = 299 K and an initial state that is the equilibrium solution for T_s = 298 K (solid thin lines). It shows that, although the nonlinearities are not negligible, the response of the nonlinear XLM is dominated by its linear component.

The linear response can be further projected onto the three eigenmodes of . This decomposition is shown in Fig. 2 as well (dashed, dashed–dotted, and dotted lines). The three modes have distinct characteristics:

• The fastest-decaying mode (mode 1, dashed lines) is associated with a decay time scale of τ₁ ≈ 5 h. It accounts for the fast adjustment of θ_M, while it also accounts for the early decrease in q_M, because the potential temperature and total water perturbations associated with the corresponding eigenvector are of opposite sign. It is also associated with a deepening of the subcloud layer (owing to the change in LCL associated with the temperature and humidity perturbations) and a slight thinning of the CuBL.

• The second-fastest-decaying mode (mode 2, dashed–dotted lines) is associated with a decay time scale of τ₂ ≈ 20 h. It accounts for a residual warming of the subcloud layer that adjusts θ_M to its final equilibrium value as well as its moistening, which compensates the early drying and adjusts q_M to its final equilibrium value. It is also associated with a thinning of the subcloud layer (owing to its moistening) and a deepening of the CuBL.

• The slowest-decaying mode (mode 3, dotted lines) is associated with a time scale of τ₃ ≈ 108 h. It accounts for a large part of the growth of the CuBL. But it is not associated with any significant perturbation of the subcloud conserved variables. It accounts for the slow adjustment of the cloud and inversion layers.

The adjustment of the subcloud layer can also be described in the (θ_M, q_M) plan, as shown in Fig. 3. This figure shows that the nonlinearities of the XLM tend to limit the decrease in q_M and hasten the adjustment at the beginning of the simulation. It also highlights that mode 1 adjusts most of the perturbation in θ_M and is responsible for the initial drying. Since mode 3 has a negligible role in the adjustment of the subcloud layer, the time evolution of θ_M and q_M is essentially along mode 2 after 20 h of simulation, once the contribution of mode 1 to the perturbation has decayed. The isolines of η show that the adjustment of the cloud-base altitude is the result of a compensation between mode 1 and mode 2. The isolines of θ_υM show that its perturbation projects on mode 2 more than the θ_M perturbation does, so that the subcloud virtual temperature is adjusted slower than the temperature.

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Time evolution of the XLM model in the (θ_M, q_M) plan. Hourly results from the nonlinear and linear models are shown, respectively, by open and filled circles. The initial perturbation from the final equilibrium is shown as a solid arrow, and its decomposition along linear modes 1 and 2 are shown, respectively, by dashed and dashed–dotted arrows. The contribution of mode 3 is not shown because it is very small. Isolines of η and θ_υM are also drawn.

Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-0219.1

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To better understand the time evolution of the subcloud layer, Fig. 4 shows the contributions of modes 1 and 2 to the perturbations of the terms in the subcloud water and energy budgets. The sum of these terms cause the time evolution of the perturbation θ_M and q_M associated to these modes in Figs. 2a and 2b. It is striking that all the terms in mode 1 tend to damp the humidity or temperature perturbation, while some compensation between terms occurs in mode 2. This explains why mode 1 is damped faster than mode 2.

• In mode 1, the negative perturbation of θ_M causes an increase in surface SHF that warms the subcloud layer and powers more turbulence and a larger (negative) cloud-base SHF and a larger (positive) cloud-base LHF, which warm and dry the subcloud layer. These mechanisms are similar to those explaining the initial tendencies in the MLM (see previous section). The positive perturbation of q_M also causes a negative anomalous surface LHF that dries the subcloud layer. The moist and cool subcloud anomalies also lower the LCL and shoal the subcloud layer, reducing the radiative cooling of the layer, so that the perturbation of the radiative contribution also warms the subcloud layer.

• In mode 2, the negative perturbations of both θ_M and q_M cause an increase in surface SHF and LHF that damp the temperature and humidity perturbations. These fluxes in turn increase the cloud-base fluxes that warm and dry the subcloud layer, damping the temperature anomaly and opposing the reduction of the humidity anomaly. The contribution of the humidity perturbation to the change in LCL is larger than that of the temperature perturbation, so that the perturbation of the cloud-base altitude is positive and the perturbation of the radiative cooling is negative (more cooling), opposing the return to equilibrium.

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Time evolution of the perturbations in the (a),(b) water and (c),(d) energy budgets associated with modes (a),(c) 1 and (b),(d) 2. Surface fluxes are shown by solid lines, the contribution of cloud-base fluxes are shown by dashed lines, and the net radiative contribution is shown by dashed–dotted lines.

Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-0219.1

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This shows that the adjustment time scales depend on a complex interaction (constructive or destructive) between the processes rather than one or two processes only. This is the main reason why the time scales τ₁, τ₂, and τ₃ do not have systematically the same order of magnitude as the process time scales τ_D, τ_s, and τ_R.

e. Sensitivity of the adjustment time scales to SST

Figure 5 shows the sensitivity of the time scales τ_n to the equilibrium basic state. The sensitivity of the XLM equilibrium to SST was presented in Bellon and Stevens (2012). Here, we compute the corresponding adjustment time scales; these are quite sensitive to the reference state, with τ₁ doubling from 3.5 to 7 h, τ₂ going from 13 to 32 h, and τ₃ growing tenfold from 54 to 578 h for a 3-K increase in T_s. For T_s slightly warmer than 300 K, τ₃ goes to infinity, which is consistent with the instability of the equilibrium described in Bellon and Stevens (2012) (the corresponding growth rate changes sign). Furthermore, it explains why the subcloud conserved variables are almost stationary in the unstable cases shown in Bellon and Stevens (2012): the subcloud layer adjusts rapidly thanks to the quick damping of modes 1 and 2, while the growth of the CuBL follows the pattern of mode 3 with little variation of the subcloud temperature and humidity.

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Adjustment time scales as a function of the SST used to compute the reference equilibrium state (solid line). First-order time scales and (open circles) as well as their approximations with χ = 0 (dotted lines) are also shown.

Citation: Journal of the Atmospheric Sciences 70, 4; 10.1175/JAS-D-12-0219.1

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a. Adjustment time scales of the MLM

First, as a reference, we compute the characteristic time scales of the MLM. Similarly to the linear system of equations [Eq. (23)], the MLM system of Eqs. (9)–(11) can be linearized and nondimensionalized, and written in the following matrix form:

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with

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Matrix _m can be inverted analytically:

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The system (29) can then be rewritten as follows:

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with

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The characteristic equation of _m can be written

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The characteristic time scales of the linear adjustment of the MLM are the absolute values of the inverses of the three eigenvalues of _m. They can be expressed as follows:

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with

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The characteristic scale corresponds to the adjustment by the combination of the surface fluxes (indicated by the 1 since the surface-flux time scale was used to nondimensionalize the equations) and the subsidence ε_w, and was identified as the “fast” time scale τ_th by Schubert et al. (1979b) and Bretherton et al. (2010).

The parameter χ_m is small compared to 1 + k and so that neglecting χ_m only introduces a small error on (6% for the reference equilibrium state with T_s = 299 K) and a slightly larger error on (15% for the reference equilibrium state with T_s = 299 K). We can write

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which shows that is the time scale of adjustment of the buoyancy through turbulent fluxes at the surface (indicated by the 1) and at cloud base (indicated by k), and that results mostly from the combination of radiation ε_R and divergence at the top of the layer δε_w. While is strongly dependent on the closure, ’s scaling is not; it is similar to the “slow” time scale τ_i in Schubert et al. (1979b) and Bretherton et al. (2010), but it combines the damping effect of radiation to that of the divergence.

For a thin, 500-m-deep boundary layer under a strong inversion, , , and correspond to dimensional time scales of 9, 9, and about 50 h. The MLM therefore provides a scale separation between the thermodynamic time scales and the dynamic (or mass) time scale, as pointed out by Schubert et al. (1979b) and Bretherton et al. (2010), and which can alternatively be thought of as the fast processes of turbulence and mean advection of the anomaly and the slow processes associated with radiation and anomalous advection. For a shallow-convective CuBL such as simulated by the XLM with T_s = 299 K, , , and correspond to dimensional time scales of 19, 16, and 37 h, so the MLM does not capture the scale separation between the two thermodynamic time scales as clearly as the LES. In the next section, we investigate how the introduction of a vertical structure in the CuBL and the use of the buoyancy closure at cloud base rather than at the inversion change these adjustment time scales in the XLM and explain the early time evolution of the thermodynamical variables.

b. Approximations to the XLM adjustment time scales

In the linear system of equations [Eq. (23)], the coefficient in the first column of , which are associated to the perturbations in , are small [o(1) while the other coefficients are O(1)]. For the reference equilibrium state at T_s = 299 K, δγε_w ≈ 0.10, and δγε_w − (1 − γ)ε_R ≈ 0.06. At first order, these coefficients can be neglected, and the matrices and can be approximated by the following matrices ^o and ^o:

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where ε_b = 1 + k + Θε_R is the characteristic response rate of the subcloud-layer buoyancy to a buoyancy perturbation, including the response of the turbulent fluxes at the surface (1) and at cloud base (k), and the radiative contribution associated to the change in LCL (Θε_R).

The characteristic equation of ^o can be written

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where x₃ = 0 is an eigenvalue of the matrix that corresponds to an infinite adjustment time scale , and the two other eigenvalues x₁ and x₂ are solutions of the quadratic equation in parentheses. Here and are first-order approximations of the adjustment time scales τ₁ and τ₂:

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with

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The first-order time scales and are shown in Fig. 5 (open circles): they match the exact time scales τ₁ and τ₂ very well. The parameter χ is small compared to νε_b + μQε_R so that neglecting χ only introduces a small error on (6% for the reference equilibrium state with T_s = 299 K). The error is more significant for (27% for the reference equilibrium state with T_s = 299 K). Figure 5 also shows the approximations of and computed with χ set to zero (dotted lines). This approximation is fair for τ₁, and although it is not as good an approximation for τ₂ as for τ₁, it scales fairly well with τ₂. Replacing ε_b by its expression, we can write the following approximations for τ₁ and τ₂:

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Equation (47) shows that the fastest adjustment time scale τ₁ results from the buoyancy adjustment of the subcloud layer [and scales with its depth similarly to the surface-flux time scale proposed by Betts (1993)] through turbulent fluxes (1 + k) and a radiative contribution associated to the perturbation of the subcloud-layer depth [(Θ + μ/νQ)ε_R]. Equation (48) shows that the second-fastest adjustment time scale τ₂ is essentially controlled by the combined damping effect of the surface fluxes (1) and vertical advection of the anomalies (γε_w) on the whole CuBL, with a multiplicative factor ν/λ resulting from the mixing-line hypothesis: −λ is the rate of increase of the CuBL heat content with h, and is the determinant of matrix .

Compared to the scalings of the MLM characteristic time scales and [see Eqs. (39) and (40)], the XLM τ₁ is shorter because the closure is applied to the top of the subcloud layer (thus the factor ) that follows the LCL (thus the terms in Q and Θ), and both effects reduce the time scale τ₁ compared to . The time scale τ₂ is lengthened because of the introduction of a vertical structure in the CuBL that appears through the ratio ν/λ > 1, which results from the redistribution of heat and humidity imposed by the mixing-line constraints on their profiles, and through the coefficient γ < 1, which reduces the effect of the vertical advection. As a result, the XLM provides an additional scale separation compared to the well-mixed case between the buoyancy adjustment of the subcloud layer and the thermodynamic adjustment of the whole CuBL. The residual adjustment of the CuBL depth is slower than in the well-mixed case so that its time scale is still separated from the thermodynamic adjustment time scales.