# 5. Geostrophic balance

Now that we have an understanding of the main forces acting on fluid parcels moving through the ocean and the atmosphere, we focus on how these forces impact the flows in the atmosphere and the ocean. In the horizontal momentum balance equations \((1.2a)\) and \((1.2b)\) in Section 1, the pressure gradient and Coriolis terms are almost always much larger than the other terms. Therefore, to a good approximation, there exists a balance between the pressure gradient and Coriolis forces named geostrophic balance:

\[\dfrac{\left(\frac{dp}{dx}\right)}{\rho}=f \times v \tag{5.1a}\]

\[\dfrac{\left(\frac{dp}{dy}\right)}{\rho}=-f \times u \tag{5.1b}\]

Geostrophic balance is arguably the most central concept in physical oceanography and dynamical meteorology. Although higher-order processes are responsible for all the interesting dynamics, it is important to realize that almost all large-scale flows in the atmosphere and ocean are in geostrophic balance to leading order. The only place where geostrophic balance never holds is at the Equator where there is no Coriolis force. A key feature of geostrophic balance is that rather than flowing from high to low pressure, the fluid actually moves parallel to lines of equal pressure (isobars). Viewing in the direction of the flow, low pressure is to the left and high pressure is to the right on the Northern Hemisphere (and vice versa in the Southern Hemisphere). This is illustrated in the weather map below (from the National Weather Service, Southern Region Headquarters).

To understand conceptually how geostrophic balance emerges, imagine a fluid parcel that starts accelerating due to a pressure gradient. As the parcel moves, it is deflected by the Coriolis force (see left panel below), until it flows parallel to the isobars and the pressure gradient and Coriolis force cancel each other (right panel below).

From the equations \((5.1a)\) and \((5.1b)\), it can easily be seen that \(u_g = -\dfrac{\left(\frac{dp}{dy}\right)}{f \times \rho}\) and \(v_g = \dfrac{\left(\frac{dp}{dx}\right)}{f \times \rho}\), with \(u_g\) and \(v_g\) the geostrophic velocities in the zonal and meridional directions, respectively. Thus, the geostrophic velocities increase with an increasing pressure gradient. Somewhat counterintuitively, geostrophic velocities decrease with increasing latitude, although the Coriolis force is stronger at higher latitudes. This is because with a weaker Coriolis force, the fluid is able to accelerate more under the influence of the pressure gradient force, before geostrophic equilibrium is reached.