Pressure variation with height - The Hydrostatic Equation Distance between two pressure levels - The Hypsometric Equation
We have learned that pressure decreases with elevation in the atmosphere. Today, we will learn about the hydrostatic equation which helps describe how fast pressure decreases as you move upward in the atmosphere. Then we will learn how to comp ute the vertical distance between two pressure levels in the atmosphere. Suppose I was interested in knowing how high vertically I would need to go to reach a pressure level of 500 mb. Today we will learn how that could be calculated.
Relationship between Force and Pressure
We have learned that pressure is a force divided by an area. Pressure is what is known as a compressive force. In other words, it always acts in a compressive nature. Suppose we are considering the force that the Earth's atmosphere is exertin g on a given surface. Suppose the surface has an area of 100 inches squared. Let's suppose the atmospheric pressure is near its average value for Earth's surface at 14.7 lbs/in^2. Since Pressure = Force / Area, we can solve this equation and express it as Force = Pressure * Area. Therefore, if we multiply the Pressure acting on the surface, times the area of the surface, we will obtain a value that expresses the net force acting on the surface by the atmosphere. This concept will be important to unde rstand when we consider our next subject, the hydrostatic equation.
In the above illustration, if the circle represents some solid surface, we can see that the Earth's atmosphere exerts a force of over 6600 pounds that is distributed evenly over the entire surface.
Force of Gravity
All objects on Earth are under the influence of the Earth's gravity. The force of gravity is equal to the mass of the object times something called the gravitational constant given the symbol g.
Hydrostatic Balance
We will continue expanding on the principles of forces and force balances we have been discussing. We have learned that when the net forces acting on an object is zero, that force either remains at rest or continues moving at its current velocity and does not accelerate or decelerate. Let us consider a static fluid, in other words, a fluid at rest such as a glass of water.
Consider a parcel of fluid somewhere in the middle of the glass. It is at rest. However, there are two forces acting in opposite directions at equal magnitudes. A gravitational force acting downward and a pressure gradient force acting upward.
We are going to apply this same principle of hydrostatic balance to the atmosphere and derive an expression showing how pressure decreases with elevation.
Deriving the Hydrostatic Equation
Now let's take our glass of water example from above and apply it to the open atmosphere. Let us consider a parcel of air at some level. Remember that pressure always acts as a compressive force, so the bottom of our parcel will experience a c ompressive force acting upward and the top of the parcel will experience a compressive force acting downward. The parcel as a whole will be subjected to a gravitational foce that will act downward. The sum of these 3 forces will be equal to zero, hence the parcel will remain static in hydrostatic balance.
What we have shown here is how pressure decreases with elevation. What this mathematical stuff is saying is that as you go up in elevation a distance Delta Z, pressure will drop by a factor of Delta P such that the ratio between Delta Z and Delta P wi ll equal - g * rho.
Determining the distance between two pressure levels in the atmosphere
We will now work with the hydrostatic equation that we have derived above to learn how it is possible to determine the vertical distance between two pressure levels. To do this, we will show the hypsometric equation.
Suppose our surface pressure is 1000 mb and we wish to know how high vertically in the atmosphere we need to travel to encounter a pressure of 500 mb. We can do this making use of a simple mathematical relationship called the hypsometric equation.
In the above illustration, we see that if we wanted to know the vertical distance between the surface where the pressure is 1000 mb and the level where pressure is 500 mb, we would take the average temperature in the layer multiplied by the Gas Constant, R, divided by the gravitational constant, g, times the natural logarithm of the ratio of the bottom pressure (1000 mb) divided by the upper-level pressure (500 mb).
The vertical distance in this scenario would be referred to as the 1000-500 mb thickness. Thickness means the distance between two pressure levels.
Since the pressure P(top) and P(bottom) are set and since R and g are constants, we see that the only thing that effects thickness is the average temperature between the two levels in question. Hence, thicknesses are often used by meteorologists to di splay the temperature distribution.
To find the average temperature in a layer of the atmosphere, the temperature readings must be taken at many levels, not just the surface. Temperatures at many levels are gathered by weather balloon data.
Critical Thickness
When looking at the 1000-500 mb thickness, it is the 5400 m thickness line that is referred to as the Critical Thickness. This is because 5400 m often denotes the rain-snow cutoff line.