Investigating the Haines Index using parcel model theory

During the saturation adjustment,

The net condensation rate C is unknown, and is eliminated from this set of equations by forming the following set of equations,

which imply that

where

Here n is the previous time level, n + 1 the current time level, and Δp the change in the parcel’s pressure. The pⁿ⁺¹ is found by integrating the hydrostatic pressure equation, pⁿ⁺¹ = pⁿ−ρgΔz, for environmental air density ρ ≈ (pⁿ/R_d/T_v ⁿ⁺¹), computational height interval Δz, and virtual temperature T_v ≈T (1 + 0.61q_v). Note that there is no dependence on the time interval.

It is now assumed that the air is saturated so that q_vⁿ⁺¹ = q_s(T*, pⁿ⁺¹). Although this closes the set, it must be solved iteratively since q_s is a non-linear function of T. Instead, to obtain a direct solution, q_s is expanded in a Taylor series about q_s(T*, pⁿ⁺¹) and all terms of second and higher order are neglected. The result is

Equations (12), (13), (14), and (15) constitute a closed set of equations. To solve this set, A, B, C, and D are substituted for known terms or expressions at time level n and pressure p^{n + 1}:

where

The saturation water vapor mixing ratio q_s is determined by

where e_s is the saturation vapor pressure. The saturation vapor pressure e_s is found by

the integrated Clausius–Clapeyron equation, where R_v = 4.615 J kg⁻¹ K⁻¹ is the gas constant for moist air, T₀ = 273.15 K, and e₀ (e_s at T₀) = 0.611 kPa. The Clausius–Clapeyron equation,

is used to determine de_s/dT. Equations (16) and (18) are combined to eliminate q_vⁿ⁺¹ and to solve for θⁿ⁺¹:

Then θⁿ⁺¹ is substituted into equation (18) to solve for q_vⁿ⁺¹, and q_vⁿ⁺¹ is substituted into equation (17) to solve for q_cⁿ⁺¹. If q_cⁿ⁺¹ ≠ 0 then θⁿ⁺¹, q_vⁿ⁺¹, and q_cⁿ⁺¹ are the values for a saturated air parcel. If q_cⁿ⁺¹ = 0, then the ascending parcel is not saturated and

The horizontal momentum carried by rising thermals differs from the environmental momentum, and the differences in horizontal momentum create horizontal shears. The turbulence generated by the shear at the parcel’s surface causes mixing or entrainment across the convective cell or parcel boundary. To model entrainment, represented by D_i in equations (3) to (6), where entrainment is considered to take place during the first stage of ascent, the following assumptions are made: air is mixed instantaneously (continuous) and completely (homogeneous) across the parcel, and entrainment is lateral. Let β represent the value of any (arbitrary) conservable variable per unit mass of air. The change of β due to entrainment of environmental air of value β is

which can be written

where m is the mass of the entraining air parcel. Equation (26) can determine the change of β due to entrainment when the entrainment rate 1/m(dm/dt) is available. Conservation of mass for a spherical parcel of radius r implies that the entrainment rate is

where V is volume, and it is assumed that density is not changed by lateral entrainment. For dr/dt = (dz/dt)_ϵ = w_ϵ, where w_ϵ is the entrainment velocity normal to the parcel’s surface, and z is the height of the center of the thermal, equation (27) becomes

Laboratory thermals in a neutral environment are observed to expand along a cone, where the radius is proportional to z:

If dz = w dt and dr/dt = α_rw, then equation (28) becomes

where λ, the fractional rate of entrainment, depends only on r. Observations indicate that r is approximately constant for a given thermal, but varies widely among different thermals. Laboratory experiments suggest that α_r ~0.2. Substituting dz = w, dt and λ into equation (26) gives

Equation (30) forms the basis for incorporating entrainment effects into the parcel model. The fractional rate of entrainment λ is a specified input parameter. For cloud studies, values for λ can range from 0.05 km⁻¹ to 0.20 km⁻¹. Entrainment rates are large for fire convection and, in this study, λ = 0.20 km⁻¹, an entrainment rate that is valid for active cloud convection. Although not entirely realistic, equation (30) represents another way that the air parcel interacts with its surroundings. Therefore, for entrainment during the first stage of ascent (or descent), θ, q_v*, q_c*, and w* are modified using

Note that there is no dependence on the time interval. Turbulent mixing comes before the saturation adjustment process.

Up to now −gq_c, the drag force due to cloud water, has been neglected when calculating w in equation (6) and A_r has been set to zero when calculating q_c in equation (5). Following Lord and Arakawa (1980), dq_c/dz = 1/w dq_c/dt, where A_r/w ≡ c₀, and c₀ is known as the autoconversion parameter. The c₀ is simply specified as c₀ = 0.002 m⁻¹, the empirical value used by Lord and Arakawa (1980) in the Arakawa–Schubert cumulus parameterization. Lord and Arakawa (1980) assume that any rain water formed immediately falls out of the parcel and, after the first stage of ascent, the cloud water mixing ratio is calculated using

This calculation takes place before the second stage of saturation adjustment.

By introducing w = dz/dt, equation (32) can be written

which when integrated over height z₀ to z becomes

To compute parcel vertical velocity, equation (36) is used and written in the form

with no dependence on the time interval. For each Δz, the parcel undergoes adiabatic ascent, during which entrainment or turbulent mixing occurs. Parcel then undergoes saturation adjustment. Some of the condensation in the parcel gets converted into rain, which drops immediately out of the parcel. While the condensation remaining in the parcel acts as a drag force in equation (37).