Increasing the Interoperability of an Earth System Model:

Atmospheric-Ocean Dynamics and Tracer Transports

NCC4-624

Milestone F

ÒUpgrade the parameterization of the planetary boundary layer (PBL) used in the AGCM to a version with multiple layers without loss of performance compared to the one in Milestone E. Provide scaling curves.

Documented source code made publicly available via the WebÓ

1. Introduction

In the UCLA Earth System Model (ESM) version with which we started this project, the atmospheric component (UCLA atmospheric general circulation model, AGCM) includes a variant of the bulk parameterization for the variable-depth PBL processes proposed by Deardorff (1972). This is relatively simpler and physically realistic. In the model, the layer next to the lower boundary is designated as the variable-depth PBL and is assumed to act as a well-mixed layer (Suarez et al. 1983). The PBL top is represented by a coordinate surface, whose height is predicted from the mass budget using a parameterized formulation of entrainment (detrainment). The PBL temperature, moisture and wind fields are predicted using the parameterized surface fluxes and the fluxes associated with entrainment (or detrainment) through the PBL top. A stratocumulus cloud sub-layer can develop along the PBL top if this is higher than the condensation level. In this case turbulence is primarily driven by the downward buoyancy due to the radiative cooling near the cloud top (Lilly, 1968), and the cloud-topped PBL can be maintained even without positive buoyancy due to the surface heating. The parameterization also incorporates the process that can destroy the cloud sub-layer by entrainment of drier air into the PBL. The merits of this unique approach to PBL parameterization are demonstrated by the highly successful performance of the UCLA AGCM in the simulation of formation and seasonal variations of marine stratocumulus (Li et al. 1999; Mechoso et al 2000). The AGCM is also fairly successful in capturing the diurnal cycle of precipitation over regions of strong monsoonal circulations (Fig. 1).

The approach to modeling the PBL as a single well-mixed layer is not without demerits. For example, it can provide a poor representation of low-level vertical shear when the PBL is deep. Also, there is no room for multiple cloud layers within the PBL. Of higher relevance to tropical convection simulated by the UCLA AGCM is that using such an approach, cloud base conditions in the Arakawa-Schubert cumulus parameterization (Arakawa and Schubert 1974) are determined by the PBL mean quantities only. Under separate funding we are implementing in the AGCM a new formulation (Konor and Arakawa 2001) that introduces multiple layers within the PBL in the framework of the modified sigma-coordinate (i.e., the upper and lower boundaries of the PBL are still coordinate surfaces). The PBL processes are then formulated following a hybrid approach, in which the effects of large-scale eddies and small-scale eddies are formulated separately. For the large-scale eddies, a relatively well-mixed vertical structure is assumed for conservative thermodynamic variables and a bulk approach is applied to the properties vertically averaged over the entire PBL so that the formulation is non-local. For the small-scale eddies, a K-closure formulation, which tends to produce a well-mixed vertical structure, is applied between the multiple layers within the PBL. Unlike the single-layer approach, the hybrid multi-layer parameterization allows for vertical shears and deviations from well-mixed profiles within the PBL. These deviations are expected to be small for thermodynamic conservative variables on a convectively active PBL, while they can be significantly large for a convectively inactive PBL. The formulation of processes highly concentrated near the PBL top remains tractable in this approach, while it is more directly applicable to a variety of PBL regimes, particularly the diurnally changing PBL over land.

The work to be performed in order to achieve Milestone F consists of upgrading the single-layer PBL parameterization used in the AGCM to the multiple-layer version without loss of performance. We have completed this work interpreting ÒupgradingÓ as the changes in the code that allow for a single or multi layer PBL within the same framework. Namely, the single and multiple layer PBL parameterizations appear as options to be set up at run time. The component of the code directly affected by the changes is AGCM-Dynamics. The AGCM-Physics part of the code is unchanged. A composite single layer PBL is computed from the multilayer PBL, and computation of physical column processes other than those of the PBL takes place as before.

The dynamical equations integrated in this new formulation (Konor and Arakawa 2001) are presented in section 2. Simulation of the effects of turbulent diffusion due to small scale eddies and to convective eddies inside the PBL on the prognostic variables is described in section 3. The diffusion coefficient for the small scale eddies is computed through a local Richardson number formulation, following Louis (1979) and Holstlag and Boville (1993), (section 3.3). Section 4 presents a selection of preliminary results.

2. Governing equations

2.2 Continuous case.

This section describes the prognostic equations using the sigma vertical coordinate system, defined in Arakawa Suarez (1983).

We define p_B as the pressure at the top of the PBL, and p_S as the pressure at the Earth surface. Then, inside the PBL, the sigma vertical coordinate is defined as

for (2.1.1a)

In the free atmosphere we consider two regions. One, between the PBL top and an intermediate pressure p = p_I, and the other, between p_I and the atmosphere top, that is considered to be at the level p = p_T. The definitions of the sigma coordinate in these regions are

for (2.1.1b)

and

for (2.1.1c)

We currently choose p_T = 1 hPa and p_I = 100 hPa.

As a consequence of these definitions, s=1 for the PBL top (p = pB), which then corresponds to a coordinate surface. Also the top of the atmosphere and the Earth surface are coordinate surfaces, with s = - 1 and s = 2 respectively. From these definitions the pressure can be obtained as

for , were (2.1.2a)

for , were (2.1.2b)

for , were (2.1.2c)

Note that p_strat is constant.

The continuity equation can be written as

(2.1.3)

where p is either p_surf, p_trop or p_strat.

The momentum equation, for layers within the PBL, is

(2.1.4a)

where the last term is the vertical convergence of the turbulent flux of momentum, which will be discussed in section 3.

In the free atmosphere (p > p_B), the momentum equation is

(2.1.4b)

The geopotential F = gz is diagnosed from the hydrostatic equation,

, (2.1.5)

where P is the Exner function, defined as

(2.1.6)

The thermodynamic equation in terms of the potential temperature q is, within the PBL,

(2.1.7a)

where G_q is the contribution of the turbulent fluxes to the tendency of q, and will be discussed in the next section, and Q is the heating rate.

In the free atmosphere, if ,

, (2.1.7b)

if ,

(2.1.7c)

The continuity equation for the water mixing ratio (r) inside the PBL combines water vapor mixing ratio (q) plus the liquid water mixing ratio (l). This is because we admit that PBL turbulence allows the air to hold both phases of water, and large-scale precipitation processes would not occur there unless the pressure at the condensation level is greater than the surface pressure (condensation level bellow the earth surface), then large scale precipitation is computed, in order to make the condensation level equal to the surface level.

The continuity equation for r within the PBL is

, with (2.1.8a)

where G_r is the contribution of the vertical convergence of the turbulent flux of r, and C is the condensation rate.

In the free atmosphere, the equation is formulated in terms of the water vapor mixing ratio (q), since if r > q_sat, (q_sat is the saturation water vapor contain) liquid water would precipitate (large scale precipitation processes). If

(2.1.8b)

and if ,

(2.1.8c)

2.2. Discretized equations

In this section we discuss the vertical discretization of the equations in section 2.1. A similar discussion for a hybrid vertical coordinate is found in Konor Arakawa 2001.

the atmosphere is vertically divided in layers (Fig. 2), from k = 1 (upper most layer) to k = k_max-1 (lower most layer). We use half integer indexes for labeling the layer interfaces; k+1/2 is the interface between layer k and layer k+1. The top of the atmosphere is the first layer interface, with index 1/2, and the earth surface, the last layer interface, with index k_max-1/2.

The level (atmosphere top) will have vertical index equal to 1/2, the level will correspond to the interface k_strat + 1/2, the level (PBL top) will correspond to the interface k_trop - 1/2, and the level (earth surface) will correspond to the interface k_max-1/2.

Atmopshere top

_______________________________ 1/2

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.. 1 first layer

_______________________________ 3/2 layer interface

_______________________________ k – 1/2

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.. k generic layer

_______________________________ k – 1/2

_______________________________ ktrop – 1/2

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.. ktrop uppermost PBL layer

_______________________________ ktrop + 1/2

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..

_______________________________

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ..

_______________________________ kmax – 3/2

ÉÉÉÉÉÉÉÉÉÉÉÉÉÉÉ.. kmax – 1

_______________________________ kmax – 1/2

Earth Surface

Figure 2: Scheme of the layers, the layer interfaces and the vertical indexes.

Horizontal velocity, temperature and water vapor mixing ratio will be computed at the layers, while vertical velocities (Ds/Dt) will be computed at the layer interfaces, this arrangement consists in the Lorenz vertical grid (Lorenz 1963).

We discretize the mass continuity equation by

for (2.2.1)

where p will be either p_surf, p_trop or p_strat, according to the vertical interval in which the layer is, and

(2.2.2)

In the interface between the PBL and the free atmosphere, we compute the vertical mass flux as

(2.2.3)

where we take E > 0 if there is mass entrainment, or D > 0 if there is detrainment. M_B is the cumulus mass detrained from the PBL to the cumulus clouds from their bases. Summation in the vertical of equation (2.2.1) for all the free atmosphere layers yields

(2.2.4a)

where p_strat = constant was used. Summation (2.2.1) for the PBL layers gives

(2.2.4b)

and summation for all the atmosphere layers gives

. (2.2.4c)

Equations (2.2.4a-c) allow to predict p_top , p_surf, p_S and therefore, also p_B. and p_top. On the other hand, partial summation gives, within the PBL,

(2.2.4c)

which allows to diagnose vertical velocity in the interfaces within the PBL.

The vertical momentum advection is

(2.2.5a,b,c)

The term v_B⁺ is linearly extrapolated from the layers above the PBL. In the lowest layer of the free atmosphere,

(2.2.5d)

and in the other layers, for .

(2.2.5e)

where p is either p_trop or p_strat, according to the vertical interval where the layer is.

The geopotential, (used in the momentum equation) is computed with the discrete form of the hydrostatic equation,