You can’t find a weather map of winds at 5000 m, but you can find one for 500 mb, which is about the same altitude as 5000 m (see figure below).

We will learn why weather maps use pressure as the vertical coordinate, but for now, we will show that higher altitudes on a pressure surface are the same as higher pressures on an altitude surface.

If we look down on Earth instead of at the side view, then we can plot the isobars as a function of latitude (y) and longitude (x). We can make a second plot of height surfaces on a constant-pressure surface (see figure below). Generally the pressure on average is greater at the equator on a given height surface than it is at the poles. This tilt makes sense if you think about the hydrostatic equilibrium equation because the temperature is greater at the equator than at the poles. Therefore, the scale height is greater, and so pressure decreases with height more gradually at the equator than it does at the poles.

We can arbitrarily choose one height surface and see how the pressure changes as a function of latitude. We see that it increases from pole to equator (see figure below).

If we now arbitrarily choose a constant pressure surface, say 500 mb, then the change in the height on an x-y horizontal plot on the pressure surface also shows an increase from pole to equator (see figure below).

Thus, low pressure on constant-height surfaces is related to low heights on constant-pressure surfaces. As a result of the hydrostatic approximation, for every height there is a unique pressure, so we can replace *z* with *p* as the vertical coordinate. We can then look at changes in variables as a function of *x* and *y*, but instead of doing this on a constant-height surface, we can do it on a constant-pressure surface.

We can now show how the equations of motion change when the vertical coordinate is switched from the height, *z*, to the pressure, *p*.

Consider first the pressure gradient force (PGF). The figure below provides a schematic of the math.

The slope of the isobar is just the change in z divided by the change in *x* on an isobar:

$$\frac{\Delta z}{\Delta x}\to {\left(\frac{\partial z}{\partial x}\right)}_{p}$$

where the subscript “*p*” means “constant pressure.” Since *Δp* is the same in the vertical and the horizontal:

$$\begin{array}{l}{\left(\frac{\partial p}{\partial x}\right)}_{z}\Delta x=-{\left(\frac{\partial p}{\partial z}\right)}_{x}\Delta z\\ {\left(\frac{\partial p}{\partial x}\right)}_{z}=-{\left(\frac{\partial p}{\partial z}\right)}_{x}{\left(\frac{\partial z}{\partial x}\right)}_{p}\\ \end{array}$$

where the subscripts "*x*" and "*z*" mean constant eastward distance and constant height, respectively.

Multiplying both sides by *1/ρ* and using the hydrostatic equilibrium equation:

$$\begin{array}{l}-\frac{1}{\rho}{\left(\frac{\partial p}{\partial x}\right)}_{z}=\frac{1}{\rho}{\left(\frac{\partial p}{\partial z}\right)}_{x}{\left(\frac{\partial z}{\partial x}\right)}_{p}=-g{\left(\frac{\partial z}{\partial x}\right)}_{p}\\ or\\ -\frac{1}{\rho}{\left(\frac{\partial p}{\partial x}\right)}_{z}=-g{\left(\frac{\partial z}{\partial x}\right)}_{p}\end{array}$$

Same for the *y* direction:

$$-\frac{1}{\rho}{\left(\frac{\partial p}{\partial y}\right)}_{z}=-g{\left(\frac{\partial z}{\partial y}\right)}_{p}$$

So, the geostrophic balance (Equation [10.24], [10.25]) in pressure coordinates becomes:

$$\begin{array}{l}\text{xmomentumequation:}0=-g\frac{\partial z}{\partial x}+fv\\ \text{ymomentumequation:}0=-g\frac{\partial z}{\partial y}-fu\end{array}$$

These equations can be rearranged to give the horizontal velocity on the pressure surface and can be rewritten as one in vector form:

$$\overrightarrow{{v}_{g}}=\frac{g}{f}\overrightarrow{k}\times {\overrightarrow{\nabla}}_{p}z$$

where $\overrightarrow{{v}_{g}}$ is designated geostrophic velocity and ${\overrightarrow{\nabla}}_{p}z=\overrightarrow{i}\frac{\partial z}{\partial x}+\overrightarrow{j}\frac{\partial z}{\partial y}$ is on a constant pressure surface.

### Geopotential

We can write these equations a little differently by using the concept of geopotential. Geopotential, *Φ*, is the potential energy that an air mass has at a height *z* above the surface.

$\Phi =\underset{o}{\overset{z}{{\displaystyle \int}}}gdz$

$\text{d}\Phi =gdz$

In vector form, the velocities become:

$${\overrightarrow{v}}_{g}=\frac{1}{f}\overrightarrow{k}\times {\overrightarrow{\nabla}}_{p}\Phi $$

A major advantage of using pressure coordinates is that the gradient of *z* or *Φ* is proportional to $\overrightarrow{{v}_{g}}$ for all pressure levels. This statement is not true for pressure gradients on height levels because you must know the density, ρ, as in Equations [10.24] and [10.25], and it varies dramatically with height.

The following video (1:19) provides a good overview of pressure surfaces: