Microphysics

As done in the RE87 model, cloud water and rain water are not considered separately.

When ql ≤ 1g kg⁻¹, all liquid water is in the form of cloud droplets with a terminal velocity of zero. On the other hand, when ql > 1g kg⁻¹, all liquid water is converted

−0.2 0 0.2 0.4 0.6

Figure 4-3: The difference in (a) temperature and (b) water vapor mixing ratio be-tween two initially identical parcels lifted pseudoadiabatically using the current for-mulation (4.21) and the full forfor-mulation (4.11).

to rainwater and falls at a terminal velocity of -7 m s⁻¹. Although this is a drastic simplification, it eliminates a prognostic variable and the need to include a micro-physics parameterization that converts cloud to rain water and vice versa. Thus, all condensation and evaporation occurs at the grid scale. The condensation algorithm is similar to that outlined in Klemp and Wilhelmson (1978), with the requirement that sp be conserved.

The saturation vapor pressure (e^∗) is approximated using Tetens’ formula, given by

e^∗ = 6.112exp 17.67(T − 273.15) T − 29.65



. (4.22)

Equation (4.22) is linearized about the initial temperature, Tⁱ, before any phase change takes place. Upon keeping the first two terms in the Taylor expansion and evaluating at the final state after the phase change, given by the superscript “f”,

e^∗f = e^∗i



1 + 4302.645

(Tⁱ− 29.65)² T^f − Tⁱ


. (4.23)

Assuming the phase change occurs isobarically, δT = Πδθ. Additionally, q_v^∗ can be substituted wherever e^∗ appears:

q_v^∗f = q_v^∗i



1 + 4302.645

(θⁱΠ − 29.65)²Π θ^f − θⁱ


. (4.24)

Next, the change in potential temperature can be related to the change in water vapor mixing ratio by using (4.21) for constant sp:

θ^f − θⁱ = −L^voθ( ˆTL− A) cpdTˆL

2 q_v^∗f − qvⁱ

. (4.25)

Substituting (4.25) into (4.24) and rearranging results in an expression for the con-densation/evaporation, Mql∆t = qⁱ_v− q^∗fv : and ∆t is the model time step. The physical effect of (4.26) is to form liquid water at supersaturated locations and evaporate rain in unsaturated air. In doing so, sensible heat is added to or subtracted from the local environment in order to conserve s_p. In the RE87 model, evaporation is assumed to occur just as rapidly as condensation and is limited only by the amount of liquid water present, i.e. the relative humidity is constrained to be 100% in the presence of liquid water. While this assumption is good for cloud droplets, it overestimates the rate of evaporation of raindrops falling through unsaturated air, which can have an evaporation timescale on the order of tens of minutes.

As a correction to the evaporation overestimation, an evaporation limiter (∆t/ˆτevap)

is included when q_l exceeds 1 g kg⁻¹. The evaporation limiter can be derived by con-sidering the change in mass of a raindrop falling freely through the air, which is governed by the equation

1 m

dm

dt = −3cvD∆ρv

a²ρw

, (4.28)

where m is the mass of the droplet, cv is the “ventilation coefficient” that takes into account air moving around the drop³, D is the diffusion coefficient, ∆ρv is the change in water vapor density from the surface of the drop to the surrounding environment, a is the radius of the drop, and ρw is the density of liquid water (Kinzer and Gunn, 1951). By assuming a homogeneous number of drops per unit volume and constant dry density of the local environment during the evaporation process, m can be replaced by ql. Moreover, if one assumes the right hand side of (4.28) is constant, then its inverse represents an e-folding timescale for the change in ql due to evaporation:

τevap = a²ρw

3cvD∆ρv

. (4.29)

The parameters in (4.29) are empirically estimated in Tab. 1 and Tab. 2 of Kinzer and Gunn (1951). Assuming a rain drop diameter of 2.2 mm, corresponding roughly to a terminal velocity of -7 m s⁻¹ (Gunn and Kinzer, 1949), and a temperature of 20^◦C, τevap becomes only a function of the relative humidity, as shown in Fig. 4-4 by the crosses. The evaporation timescale increases with relative humidity, especially as the environment approaches saturation. For a relative humidity of 10%, the timescale is about 6 minutes and increases to about an hour when the relative humidity is 90%.

Hence, it becomes progressively harder to saturate the atmosphere solely from the evaporation of rain unless it is raining in the same location for a long period of time.

Since the data from Kinzer and Gunn (1951) is discrete, it is fitted to a tangent curve, as shown by the blue line in Fig. 4-4, with the equation of the tangent curve being

3This is not to be confused with the ventilation (V) defined in previous chapters.

0 10 20 30 40 50 60 70 80 90 100 0

1000 2000 3000 4000 5000 6000 7000

Relative Humidity [%]

Evaporation Timescale [s]

Figure 4-4: The evaporation timescale, τevap, as a function of the relative humidity.

The timescale is derived from Kinzer and Gunn (1951) data for a drop diameter of 22 mm and a temperature of 20^◦C (crosses). The least squares tangent curve fit to the data is given by the blue line.

ˆ

τ_evap= 519.59tan πH 2



+ 231.83. (4.30)

This timescale is used to reduce the evaporation rate to more reasonable levels in (4.26) when rain falls through unsaturated air. The main effect of the evaporation limiter is to reduce the evaporation in the boundary layer below the eyewall, thereby increasing rain rates and decreasing relative humidities, as inflow is not instanta-neously saturated as it crosses into the heavily precipitating eyewall annulus.

In practice, the condensation/evaporation algorithm should be iterated because (4.27) is modified by phase changes. Condensation, for instance, results in an increase in the temperature and a corresponding small decrease in χ. However, only a few iterations are needed to achieve convergence. The evaporation limiter should only be applied once after the iteration is complete.

Rain falling and evaporating in to unsaturated air is an irreversible process, and in reality, there should be a slight increase in entropy. If evaporation is required to conserve enthalpy, instead of entropy, it can be shown that the change in entropy due

to evaporation is in a gain of entropy as evaporation takes place. The model ignores the contribution of this irreversible source of entropy (along with other irreversible sources) in order to be consistent with the complimenting theoretical framework. The effect of irreversible sources of entropy on TC evolution in this model will be looked at in the future.