Convective Instability

Figure 2.4: Convection.

If water is heated from the bottom and temperature difference between the top and the bottom exceeds a limit, convection is driven. Water heated from the bottom climbes and cool water on the top descends. This transfers the thermal energy from the bottom to the top. In this section, we describe the condition in which convection is driven.

Consider a hydrostatic balanced atmosphere in which the hydrostatic balance equation is satisfied:

$$-\frac{d p}{d z}-\rho\frac{d \phi}{d z}=-\frac{d p}{d z}+\rho g=0,$$ (2.78)

where we assumed $\mbox{\boldmath${v}$}=0$ in equation (2.2) and the gravity is working downwards in $z$-direction ($g < 0$) The pressure and density of the atmosphere are $p(z)$ and $\rho(z)$. We consider a gas element (hatched region in Fig.2.4), whose density $\rho_*$ and $p_*$ are equal to those of the atmosphere $\rho(z)$ and $p(z)$ as

$$\rho_*(z)=\rho(z), p_*(z)=p(z).$$ (2.79)

Further, we assume this gas element to move from $z$ to $z+\Delta z$ adiabatically, that is,

$$\frac{p_*(z+\Delta z)}{\left[\rho_*(z+\Delta z)\right]^\gamma} =\frac{p_*(z)}{\left[\rho_*(z)\right]^\gamma}.$$ (2.80)

Pressure balance is required between the pressures of the gas element at $z+\Delta z$, $p_*(z+\Delta z)$, and the atmosphere $p(z+\Delta z)$. If the density of the gas element at $z+\Delta z$, $\rho_*(z+\Delta z)$, is smaller than that of the atmosphere, the gravity force in equation (2.78) is weaker than the pressure force and the element keeps climbing. Thus the condition for the convective instability is written as

$$\rho_*(z+\Delta z) < \rho(z+\Delta z).$$ (2.81)

Using

$$p(z+\Delta z)=p(z)+\frac{d p}{d z}\Delta z {\rm and} \ \rho(z+\Delta z)=\rho(z)+\frac{d \rho}{d z}\Delta z,$$ (2.82)

we can rewrite equation (2.81) into the relation in the variables of the atmosphere (variables without *) as

$$\frac{d \ln p}{d z}<\gamma \frac{d \ln \rho}{d z}$$ (2.83)

or

$$\frac{d \ln (p/\rho^\gamma)}{d z}<0$$ (2.84)

This means the specific entropy $s = \ln (p/\rho^\gamma)+K$ decreases upwardly. Thus, if we consider the adiabatic process, the atmosphere in which a specific entropy decreases upwardly is unstable for the convection.

Problem
Obtain equation (2.83) from equation (2.81) using equation (2.82).