Geostrophic Wind Speed from the Pressure Gradient

The geostrophic wind is the speed at which two forces cancel. The pressure-gradient force pushes air from high pressure toward low, and the Coriolis force, which comes from the Earth's rotation, turns moving air aside. When the two balance exactly, the air has no net push left, so it blows along the isobars instead of across them, with low pressure on its left in the northern hemisphere. This is a balance of forces, the geostrophic approximation of Holton and Hakim (chapter 3), and not a fitted curve. It says how fast the wind would blow if the balance were perfect, which turns a map of isobars into a map of wind.

Read the formula as a force per unit mass. The ratio $\Delta p/\Delta n$ is the pressure gradient: the pressure change per distance, measured at right angles to the isobars. Dividing by the air density $\rho$ gives the push on each kilogram of air, and denser air is pushed less by the same gradient. The Coriolis parameter $f = 2\Omega\sin(\text{latitude})$, with $\Omega$ the Earth's rotation rate, sets how strongly rotation turns the wind; enter it as a positive number, about $1.03\times10^{-4}\ \text{s}^{-1}$ at 45°. A larger $f$ needs less speed to balance a given push, so $V_g$ is larger where the isobars are closer together (small $\Delta n$) and at lower latitude (small $f$). Typical densities are about 1.2 kg/m³ near sea level and 0.74 at 500 hPa.

For the worked example, take 4 hPa across 200 km of isobar spacing at 45° N, with $\rho = 1.25$ kg/m³. In pascals and metres the pressure difference is 400 Pa and the spacing is 200 000 m, so $V_g = 400 / (1.25 \times 1.03\times10^{-4} \times 200\,000) = 400 / 25.75 = 15.53$ m/s. Run backward, a 15.5 m/s wind under the same conditions implies isobars about 200.6 km apart, so the two directions agree. The calculator handles the unit conversions; the arithmetic above shows why the answer lands in metres per second.

The limits matter more than the arithmetic. The relation describes the free atmosphere above the friction layer. A surface wind is slower and turned toward low pressure, because friction acts on the air near the ground; the wind-speed power law gives the surface wind that friction leaves. Whether rotation controls the flow enough for the balance to hold at all is a question for the Rossby number, and the Coriolis parameter that this page asks for comes from the latitude.