E1 Dynamic Core

Topic

The dynamic core of the atmosphere consists of the equations of motion governing the behavior of the atmospheric fluid as it rotates with the Earth. These equations—known as the primitive equations—are derived from the general Navier-Stokes equations that describe any fluid, but they are simplified using two approximations specific to the atmosphere: the hydrostatic approximation, whereby pressure at any point equals the weight of the air column above it; and an approximation that simplifies the representation of Earth's curvature at the vast horizontal scales handled by the atmospheric dynamic core. With these two simplifications, the primitive equations describe the evolution over time of horizontal wind, temperature, and humidity, expressed in coordinates that use pressure rather than altitude to define atmospheric levels.

Within these primitive equations, the Coriolis force is what distinguishes the atmosphere from other fluids: it is a force dependent on both wind speed and the Earth's rotation speed, deflecting any movement to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. This systematic deflection causes air—instead of flowing in a straight line from high-pressure to low-pressure zones—to circulate around them; thus, the Coriolis force is responsible for the formation of cyclones and anticyclones.

When the pressure gradient driving air from high to low pressure is exactly balanced by the deflecting Coriolis force, the result is geostrophic wind—a wind that flows parallel to lines of constant pressure rather than toward the low-pressure area. This balance between the pressure gradient and the Coriolis force is a good approximation of the actual wind at the scales of thousands of kilometers handled by the atmosphere's dynamical core, although it ceases to be so at much smaller scales.

There is, however, a quantity that is conserved even better than the geostrophic wind itself: Ertel potential vorticity, which remains constant as long as the air moves without exchanging heat with its surroundings and without being slowed by friction. Precisely because it is conserved under these conditions, Ertel potential vorticity is the most fundamental dynamical variable for diagnosing and predicting large-scale atmospheric flow; simply tracking the movement of this conserved quantity allows one to reconstruct much of how the rest of the surrounding atmosphere moves.

Ertel potential vorticity primarily describes the horizontal motion of the atmosphere, but that same horizontal motion also determines vertical motion: when horizontal winds converge toward a single point at the surface, the accumulating air has nowhere to go but up, causing that point to experience rising motion; conversely, when horizontal winds diverge—moving away from a single point—the resulting deficit of air is replenished from above, causing that point to experience subsidence. Horizontal wind divergence is thus what produces vertical motion—referred to in these pressure coordinates as "omega."

The equation relating this vertical motion (omega) to the rest of the geostrophic flow is the quasi-geostrophic omega equation. According to this equation, vertical motion depends on how the geostrophic wind transports vorticity and temperature from one location to another; where this transport of vorticity and temperature combines in a specific way, the atmosphere responds by generating the rising motion or subsidence necessary to maintain geostrophic balance. This omega equation is precisely the basis for diagnosing weather systems: the very cyclones and anticyclones that the Coriolis force helps to form are ultimately explained by how the atmosphere's dynamic core combines the Coriolis force, the geostrophic wind, Ertel's potential vorticity, and vertical motion into a single, coherent description of the rotating atmospheric fluid.

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