Vertical pressure variation

Vertical pressure variation describes how pressure changes with height. While pressure can also shift horizontally due to pressure gradient forces, the vertical change is especially important because it is driven by gravity: lower points in a fluid support a taller column of fluid above them, so pressure increases as elevation decreases.

The basic relationship is that the pressure difference between two heights equals the product of the height change, gravity, and density. This is expressed as dP/dh = −ρg, where P is pressure, ρ is density, g is gravity, and h is height. Since g acts downward, an increase in height leads to a decrease in pressure, consistent with the idea of a fluid column’s weight. For small height changes where density and gravity are roughly constant, multiplying the height difference by ρ and g gives a good approximation of the pressure difference. If the pressure at one point in a uniform-density liquid is known as P₀, the pressure at another point is P₁ = P₀ − ρg(h₁ − h₀). When multiple fluids are layered, the total pressure difference is found by summing the differences across each layer, using each fluid’s density and height change. If density varies with height, integration is needed. Whether density and gravity can be treated as constant depends on the required accuracy and the height scale; gravity and density both decrease with altitude. For example, seawater is nearly incompressible, so its density changes much less with height than air does, making the constant-density approximation more valid for water over the same vertical distance.

The barometric formula depends only on the height of the fluid chamber, not its width or length. With sufficient height, any pressure can be achieved. This is known as the hydrostatic paradox, which W. H. Besant described as: any quantity of liquid, however small, may be made to support any weight, however large. Simon Stevin first explained this mathematically. In 1916, Richard Glazebrook described an arrangement attributed to Pascal: a heavy weight W rests on a board of area A atop a fluid bladder connected to a vertical tube of cross-sectional area α. Pouring water of weight w into the tube eventually lifts the weight, with the balance of forces giving W = wA/α. Glazebrook noted that by making the board area large and the tube area small, a small weight of water can support a large weight. Hydraulic machinery uses this principle to multiply force, and demonstrations of the paradox are common in teaching.

For Earth’s atmosphere, the vertical scale is large—the troposphere is several kilometers thick, and the thermosphere extends hundreds of kilometers—and air is compressible. Gravity can still be approximated as constant because these distances are small compared to Earth’s average radius of about 6371 km. Density, however, changes significantly with height. From the ideal gas law, ρ = mP/(kT), where m is the average mass per air molecule, P is pressure, k is Boltzmann’s constant, and T is temperature in kelvins. This shows that air density depends on air pressure, and since pressure also depends on density, the two are interdependent.