1. The Hydrostatic EquationThe concept of hydrostatic equilibrium
(in which the vertical pressure gradient force is equal and opposite to
the gravitational force) can be succinctly summarized by the hydrostatic
equation:...By convention, the Greek letter delta (Δ) stands for
“change in.” In this case, Δp refers to a change in pressure, while Δz
refers to a change in altitude. Thus, Δp/Δz on the left side of the
equation refers to the change in pressure for a unit of increase in
altitude.We have met the symbols ρ and g before as density and the
acceleration of gravity, respectively. The negative sign on the
right-hand side accounts for the fact that pressure decreases with
height; that is, the left-hand side is always negative. For the two
sides to balance, the right-hand side must also be negative.Thus, the
hydrostatic equation states that the rate at which pressure decreases
with height equals the product of the air density and the acceleration
of gravity. But because the acceleration of gravity is virtually
constant, the rate at which pressure declines with altitude is
determined almost completely by the density of the atmosphere. In
particular, higher-density air has a greater vertical pressure
gradient.As an example, let us compare the two columns of air in Figure
4–8b, supposing that their temperatures are 0 °C and 40 °C. Using the
surface pressure of 1000 mb, the equation of state gives the density of
the warm air as 1.1 kg per cubic meter. At the same pressure, the cool
air must have higher density, in this case 1.3 kg per cubic meter.
Assuming hydrostatic equilibrium, the corresponding vertical pressure
gradients at the surface are as follows:...This confirms our earlier
reasoning, where we concluded that pressure declines more rapidly in a
cool, dense air column than in a warm air column. As we discuss in the
body of the text, this sets up an upper-level horizontal pressure
gradient between warm and cool air.Why does a negative sign appear in
the hydrostatic equation? Get solution
2. The Hydrostatic EquationThe concept of hydrostatic equilibrium (in which the vertical pressure gradient force is equal and opposite to the gravitational force) can be succinctly summarized by the hydrostatic equation:...By convention, the Greek letter delta (Δ) stands for “change in.” In this case, Δp refers to a change in pressure, while Δz refers to a change in altitude. Thus, Δp/Δz on the left side of the equation refers to the change in pressure for a unit of increase in altitude.We have met the symbols ρ and g before as density and the acceleration of gravity, respectively. The negative sign on the right-hand side accounts for the fact that pressure decreases with height; that is, the left-hand side is always negative. For the two sides to balance, the right-hand side must also be negative.Thus, the hydrostatic equation states that the rate at which pressure decreases with height equals the product of the air density and the acceleration of gravity. But because the acceleration of gravity is virtually constant, the rate at which pressure declines with altitude is determined almost completely by the density of the atmosphere. In particular, higher-density air has a greater vertical pressure gradient.As an example, let us compare the two columns of air in Figure 4–8b, supposing that their temperatures are 0 °C and 40 °C. Using the surface pressure of 1000 mb, the equation of state gives the density of the warm air as 1.1 kg per cubic meter. At the same pressure, the cool air must have higher density, in this case 1.3 kg per cubic meter. Assuming hydrostatic equilibrium, the corresponding vertical pressure gradients at the surface are as follows:...This confirms our earlier reasoning, where we concluded that pressure declines more rapidly in a cool, dense air column than in a warm air column. As we discuss in the body of the text, this sets up an upper-level horizontal pressure gradient between warm and cool air.On a clear night with no wind, an air column cools. Does the vertical pressure gradient increase, decrease, or remain unchanged? Get solution
2. The Hydrostatic EquationThe concept of hydrostatic equilibrium (in which the vertical pressure gradient force is equal and opposite to the gravitational force) can be succinctly summarized by the hydrostatic equation:...By convention, the Greek letter delta (Δ) stands for “change in.” In this case, Δp refers to a change in pressure, while Δz refers to a change in altitude. Thus, Δp/Δz on the left side of the equation refers to the change in pressure for a unit of increase in altitude.We have met the symbols ρ and g before as density and the acceleration of gravity, respectively. The negative sign on the right-hand side accounts for the fact that pressure decreases with height; that is, the left-hand side is always negative. For the two sides to balance, the right-hand side must also be negative.Thus, the hydrostatic equation states that the rate at which pressure decreases with height equals the product of the air density and the acceleration of gravity. But because the acceleration of gravity is virtually constant, the rate at which pressure declines with altitude is determined almost completely by the density of the atmosphere. In particular, higher-density air has a greater vertical pressure gradient.As an example, let us compare the two columns of air in Figure 4–8b, supposing that their temperatures are 0 °C and 40 °C. Using the surface pressure of 1000 mb, the equation of state gives the density of the warm air as 1.1 kg per cubic meter. At the same pressure, the cool air must have higher density, in this case 1.3 kg per cubic meter. Assuming hydrostatic equilibrium, the corresponding vertical pressure gradients at the surface are as follows:...This confirms our earlier reasoning, where we concluded that pressure declines more rapidly in a cool, dense air column than in a warm air column. As we discuss in the body of the text, this sets up an upper-level horizontal pressure gradient between warm and cool air.On a clear night with no wind, an air column cools. Does the vertical pressure gradient increase, decrease, or remain unchanged? Get solution
