## Balanced flow

7.1. Geostrophic motion

7.1.1. The geostrophic wind in pressure coordinates

7.1.2. Highs and lows; synoptic charts

7.1.3. Balanced flow in the radial-inflow experiment

7.2. The Taylor-Proudman theorem 7.2.1. GFD Lab VII: Taylor columns

7.3. The thermal wind equation

7.3.1. GFD Lab VIII: The thermal wind relation

7.3.2. The thermal wind equation and the Taylor-Proudman theorem

7.3.3. GFD Lab IX: Cylinder ''collapse'' under gravity and rotation

7.3.4. Mutual adjustment of velocity and pressure

7.3.5. Thermal wind in pressure coordinates

7.4. Subgeostrophic flow: The Ekman layer

7.4.1. GFD Lab X: Ekman layers: frictionally-induced cross-isobaric flow

7.4.2. Ageostrophic flow in atmospheric highs and lows

7.4.3. Planetary-scale ageostrophic flow

### 7.5. Problems

In Chapter 6 we derived the equations that govern the evolution of the atmosphere and ocean, setting our discussion on a sound theoretical footing. However, these equations describe myriad phenomena, many of which are not central to our discussion of the large-scale circulation of the atmosphere and ocean. In this chapter, therefore, we focus on a subset of possible motions known as balanced flows, which are relevant to the general circulation.

We have already seen that large-scale flow in the atmosphere and ocean is hydro-statically balanced in the vertical, in the sense that gravitational and pressure gradient forces balance one another rather than induce accelerations. It turns out that the atmosphere and ocean are also close to balance in the horizontal, in the sense that Coriolis forces are balanced by horizontal pressure gradients in what is known as geostrophic motionâ€”from the Greek, geo for ''Earth,'' and strophe for ''turning.'' In this chapter we describe how the rather peculiar and counterintuitive properties of the geostrophic motion of a homogeneous fluid are encapsulated in the Taylor-Proudman theorem, which expresses in mathematical form the ''stiffness'' imparted to a fluid by rotation. This stiffness property will be repeatedly applied in later chapters to understand the large-scale circulation of the atmosphere and ocean. We go on to discuss how the Taylor-Proudman theorem is modified in a fluid in which the density is not homogeneous but varies from place to place, deriving the thermal wind equation. Finally we discuss so-called ageostrophic flow motion, which is not in geostrophic balance but is modified by friction in regions where the atmosphere and ocean rub against solid boundaries or at the atmosphere-ocean interface.

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