# Vertical pressure variation

Vertical pressure variation is the variation in pressure as a function of elevation. Depending on the fluid in question and the context being referred to, it may also vary significantly in dimensions perpendicular to elevation as well, and these variations have relevance in the context of pressure gradient force and its effects. However, the vertical variation is especially significant, as it results from the pull of gravity on the fluid; namely, for the same given fluid, a decrease in elevation within it corresponds to a taller column of fluid weighing down on that point.

## Basic formula

A relatively simple version [1] of the vertical fluid pressure variation is simply that the pressure difference between two elevations is the product of elevation change, gravity, and density. The equation is as follows:

$\Delta P = - \rho g \Delta h$, where
P is pressure,
ρ is density,
g is acceleration of gravity, and
h is height.

The delta symbol indicates a change in a given variable. Since g is negative, an increase in height will correspond to a decrease in pressure, which fits with the previously mentioned reasoning about the weight of a column of fluid.

When density and gravity are approximately constant, simply multiplying height difference, gravity, and density will yield a good approximation of pressure difference. Where different fluids are layered on top of one another, the total pressure difference would be obtained by adding the two pressure differences; the first being from point 1 to the boundary, the second being from the boundary to point 2; which would just involve substituting the ρ and (Δh) values for each fluid and taking the sum of the results. If the density of the fluid varies with height, mathematical integration would be required.

Whether or not density and gravity can be reasonably approximated as constant depends on the level of precision needed, but also on the length scale of height difference, as gravity and density also decrease with higher elevation. For density in particular, the fluid in question is also relevant; seawater, for example, is considered an incompressible fluid; its density can vary with height, but much less significantly than that of air, so given the same height difference, water's density can be more reasonably approximated as constant than that of air.

The barometric formula depends only on the height of the fluid chamber, and not on its width or length. Given a large enough height, any pressure may be attained. This feature of hydrostatics has been called the hydrostatic paradox. As expressed by W. H. Besant,[2]

Any quantity of liquid, however small, may be made to support any weight, however large.

In 1916 Richard Glazebrook mentioned the hydrostatic paradox as he described an arrangement he attributed to Pascal: a heavy weight W rests on a board with area A resting on a fluid bladder connected to a vertical tube with cross-sectional area α. Pouring water of weight w down the tube will eventually raise the heavy weight. Balance of forces leads to the equation

$W = \frac {w \ A}{\alpha} .$

Glazebrook says, "By making the area of the board considerable and that of the tube small, a large weight W can be supported by a small weight w of water. This fact is sometimes described as the hydrostatic paradox."[3]

Demonstrations of the hydrostatic paradox have been used in teaching.[4]

## In the context of Earth's atmosphere

If one is to analyze the vertical pressure variation of the Atmosphere of Earth, the length scale is very significant (troposphere alone being several kilometres tall; thermosphere being several hundred kilometres) and the involved fluid (air) is compressible. Gravity can still be reasonably approximated as constant, because length scales on the order of kilometres are still small in comparison to Earth's radius, which is, on average, about 6371 kilometres,[5] and gravity is a function of distance from Earth's core.[6]

Density, on the other hand, varies more significantly with height. It follows from the ideal gas law that:

$\rho = (m P)/(k T)$

Where

m is average mass per air molecule,
P is pressure at a given point,
k is the Boltzmann constant, and
T is the temperature in Kelvin.

Put more simply, air density depends on air pressure. Given that air pressure also depends on air density, it would be easy to get the impression that this was circular definition, but it is simply inter-dependency of different variables. Furthermore, one can use calculus to work with this, as is shown in a Georgia State University webpage on atmospheric pressure.[7] This then yields a more accurate formula, of the form:

$P_h = P_0 e^{(-mgh) / (kT)}$

Where

Ph is the pressure at point h,
P0 is the pressure at reference point 0, (typically referring to sea level)
e is Euler's number,
m is the mass per air molecule,
g is gravity,
h is height difference from reference point 0, and
k is the Boltzmann constant, and
T is the temperature in Kelvin.

And the superscript is used to indicate that e is raised to the power of the given ratio.

Therefore, instead of pressure being a linear function of height as one might expect from the more simple formula given in the "basic formula" section, it is more accurately represented as an exponential function of height.

Note that even that is a simplification, as temperature also varies with height. However, the temperature variation within the lower layers (troposphere, stratosphere) is only in the dozens of degrees, as opposed to difference between either and absolute zero, which is in the hundreds, so it is a reasonably small difference. For smaller height differences, including those from top to bottom of even the tallest of buildings, (like the CN tower) or for mountains of comparable size, the temperature variation will easily be within the single-digits. (See also lapse rate.)

An alternative derivation, shown by the Portland State Aerospace Society,[8] is used to give height as a function of pressure instead. This may seem counter-intuitive, as pressure results from height rather than vice versa, but such a formula can be useful in finding height based on pressure difference when one knows the latter and not the former. Different formulas are presented for different kinds of approximations; for comparison with the previous formula, the first referenced from the article will be the one applying the same constant-temperature approximation; in which case:

$z = (-RT/g) \ln (P/P_0)$

Where

z is the elevation,
R is the gas constant,
T is temperature in kelvin,
g is gravity,
P is pressure at a given point, and
P0 is pressure at the reference point.

And for the sake of comparison to the above, another formula derived in the same article shows a more complete picture for when constant temperature isn't assumed, and is also a formula for height as a function of pressure difference:

$z = (T_0/L)((P/P_0)^{-LR/g} - 1)$

Where

L is the atmospheric lapse rate, and
T0 is the temperature at the same reference point for which P=P0

Apart from that, the units are the same as those of the formula mentioned before it.

Basically, which formula is best to use depends on which variables are known, which are meant to be found, and which simplifying assumptions are valid to make.