Emanuel et al. (2014) develop an elegant theory that explains why a radiative-convective equilibrium (RCE) system develops into unstable states. They identify the sea surface temperature (SST) as an important player: the instability is more likely to occur in high SST. This conclusion is based on three steps of physical reasoning: 

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(1) In high SST, a negative anomaly in atmospheric moisture causes stronger radiative cooling (this is based on fundamental physics of radiative transfer).

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(2) Downward motions must spontaneously develop to balance the cooling by adiabatic warming (this is based on weak-temperature-gradient, WTG, assumption).

 

(3) Downward motions transport air with low moist static energy (MSE) to the boundary layer. This causes a decrease in upward mass fluxes to relax the boundary layer MSE reduction (this is based on boundary layer MSE quasi-equilibrium).

 

(4) Reduced upward mass fluxes (also reduced net upward mass fluxes) weaken the convective heating, cooling the atmosphere. The positive loop continues.  

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(5) As a result, both reduced convection and enhanced large-scale subsidence desiccate the troposphere, amplifying the initial negative moisture anomaly.

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In this framework, higher precipitation efficiency (PE), defined as the fraction of water that condenses in a cloud that reaches the surface as precipitation, enhances the positive loops. Why?

 

The PE enters the picture via an empirical parameterization: 

 

Md = -(1 - PE)*Mu   (1)

 

, in which the Mu and Md are the convective updraft and downdraft mass fluxes. With the Eq. (1), the net convective mass flux is:

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M = Mu - Md = PE*Mu  (2)

Eq. (2) here suggests that the net mass flux M (could be considered as convection) changes more rapidly with Mu for higher PE.

 

I remind you of the Step (4). It's important to note that a decrease in Mu does not directly cause a decrease in convective heating. M does. For a given reduction in Mu, a higher PE causes a larger reduction in M. This exaggerates the positive loops, thus promoting the Radiative-Convective Instability.

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Physical validity of Eq. (1)

It's obvious that the Eq. (1) is a key for understanding the relationship between PE and Radiative-Convective Instability. Here I discuss its physical validity. Eq. (1)  states two things:

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First, stronger convective updrafts drive stronger downdrafts.  Second, a higher PE buffers the first process.

 

The first argument is a common one that has been used extensively in mass flux parameterization in climate models. It is straightforward to understand. 

 

The second argument is relatively trickier. High PE means that more precipitation reaches the ground and, therefore, less precipitation and/or cloud water are evaporated into the environment. Less evaporation of precipitation means less unsaturated downdraft outside of clouds. Less evaporation of cloud condensates means less saturated penetrative downdrafts in the clouds (see note 1). Both correspond to weaker downdrafts and their combined response is consistent with the PE buffering effect.  

 

I use the MIT single-column model to examine its validity. The strategy is I run the model into RCE at SST = 30 deg C. Then I modify the autoconversion rate by changing the threshold of cloud water converting to precipitation (noted as l_cri), the default value of which is L = 1.1 g/kg. I run RCE simulations by multiplying the L by factors of 0.1, 0.3, 0.5, 1, 1.5, 2, 3, and 5. This gives us a composite of RCE states with different PEs. Unfortunately, diagnosing the "truth" PE in models has been challenging because of the difficulty in diagnosing the condensation rate that is the denominator of the PE formula. Here, I assume that the condensation rate is the same among all RCE states so that the simulated surface rain rate is used to represent the PE. The figure below shows the 1 - Md/Mu versus the surface rain rate. There is a nice correspondence between the two, suggesting that Eq. (1) is a good approximation. When l_crit is higher, less cloud water is converted to precipitation, causing less precipitation reaching the ground, if the condensation rate is assumed to be unchanged, smaller PE. In the meantime, higher l_crit causes more condensates detrained into the environment, resulting in a more humid climate. This is seen from the high relative humidity (RH) in high l_crit simulations in the figure.  

 

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The above analysis supports the validity of Eq. (1). This enables me to use the Eq. (1) as a physical foundation to investigate if higher PE favors the occurrence of Radiative-Convective instability.  This will be done in part 2.    

 

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Note 1:

Penetrative downdrafts are caused by the turbulent mixing of dry environmental air producing negative buoyant mixtures that penetrate downward (Emanuel et al., 1991). The strength of the penetrative downdraft depends not only on the cloud-base mass flux, but also details of mixing scheme and the environmental profiles of moisture and temperature.     

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