Assume the atmosphere to be an ideal gas of constant...

Assume the atmosphere to be an ideal gas of constant specific heat ratio $\gamma=C_p / C_v$. Also assume the acceleration due to gravity, $g$, to be constant over the range of the atmosphere. Let $z=0$ at sea level, $T_0, p_0$, $\rho_0$ be the absolute temperature, pressure, and density of the gas at $z=0$. (a) Assuming that the thermodynamic variables of the gas are related in the same way they would be for an adiabatic process, find $p(z)$ and $\rho(z)$. (b) Show that for this case no atmosphere exists above a $z_{\max }$ given by $z_{\max }=\frac{\gamma}{\gamma-1}\left(\frac{R T_0}{g}\right)$, where $R$ is the universal gas constant per gram.

Assume the atmosphere to be an ideal gas of constant specific heat ratio $\gamma=C_p / C_v$. Also assume the acceleration due to gravity, $g$, to be constant over the range of the atmosphere. Let $z=0$ at sea level, $T_0, p_0$, $\rho_0$ be the absolute temperature, pressure, and density of the gas at $z=0$.
(a) Assuming that the thermodynamic variables of the gas are related in the same way they would be for an adiabatic process, find $p(z)$ and $\rho(z)$.
(b) Show that for this case no atmosphere exists above a $z_{\max }$ given by $z_{\max }=\frac{\gamma}{\gamma-1}\left(\frac{R T_0}{g}\right)$, where $R$ is the universal gas constant per gram.

Key Concepts

Ideal Gas Law

The ideal gas law, given by p = ?RT, relates the pressure (p), density (?), and temperature (T) of a gas using the specific gas constant (R). This law is fundamental in atmospheric science as it provides the basic connection between these state variables and is the starting point for analyzing the behavior of a gaseous atmosphere under different thermodynamic conditions.

Hydrostatic Equilibrium

Hydrostatic equilibrium is the state in which the vertical pressure gradient in a fluid balances the gravitational force acting on it. In the context of an atmosphere, this equilibrium is expressed mathematically as dp/dz = -?g, where g is the acceleration due to gravity. It governs how pressure changes with altitude and is essential for deriving the vertical structure of the atmosphere.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings, meaning the process is thermally isolated. In the atmosphere, adiabatic processes are significant because air parcels that rise or sink do so without heat transfer, leading to temperature changes that follow a specific relationship with pressure and density. This concept underpins the assumption used to relate the thermodynamic variables in the problem.

Polytropic Process Relationship

When a gas undergoes an adiabatic or polytropic process, the pressure and density are related by a power law of the form p = K?^?, where K is a constant and ? is the specific heat ratio. This relation is critical in the analysis of the atmosphere where the temperature, pressure, and density are interlinked, and it is used to derive how these quantities vary with altitude under adiabatic conditions.

Maximum Atmospheric Height Concept

In certain atmospheric models, particularly those that assume adiabatic conditions, the mathematical solutions for pressure and density can lead to a cut-off altitude above which the atmosphere effectively ceases to exist. This maximum height arises because the derived expressions imply that beyond a certain altitude, the temperature would drop to zero, a result that highlights the limits imposed by the thermal structure in an atmosphere impacted by gravity.

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