Potential vorticity

Potential vorticity (PV) is seen as one of the important theoretical successes of modern meteorology. It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean. Its development traces back to the circulation theorem by Bjerknes in 1898, which is a specialized form of Kelvin's circulation theorem. Starting from Hoskins et al., 1985, PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field. Even after detailed numerical weather forecasts on finer scales were made possible by increases in computational power, the PV view is still used in academia and routine weather forecasts, shedding light on the synoptic scale features for forecasters and researchers.(1) introduce balancing conditions of certain form. These conditions must be physically realizable and stable without instabilities such as static instability. Also, the space and time scales of the motion must be compatible with the assumed balance;(2) specify a certain reference state, such as distribution of temperature, potential temperature, or geopotential height;(3) assert proper boundary conditions and invert the PV field globally. Potential vorticity (PV) is seen as one of the important theoretical successes of modern meteorology. It is a simplified approach for understanding fluid motions in a rotating system such as the Earth's atmosphere and ocean. Its development traces back to the circulation theorem by Bjerknes in 1898, which is a specialized form of Kelvin's circulation theorem. Starting from Hoskins et al., 1985, PV has been more commonly used in operational weather diagnosis such as tracing dynamics of air parcels and inverting for the full flow field. Even after detailed numerical weather forecasts on finer scales were made possible by increases in computational power, the PV view is still used in academia and routine weather forecasts, shedding light on the synoptic scale features for forecasters and researchers. In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis (the birth and development of a cyclone), especially along the polar front, and in analyzing flow in the ocean. Baroclinic instability requires the presence of a potential vorticity gradient along which waves amplify during cyclogenesis. Vilhelm Bjerknes generalized Helmholtz's vorticity equation (1858) and Kelvin's circulation theorem (1869) to inviscid, geostrophic, and baroclinic fluids, i.e., fluids of varying density in a rotational frame which has a constant angular speed. If we define circulation as the integral of the tangent component of velocity around a closed fluid loop and take the integral of a closed chain of fluid parcels, we obtain D C D t = − ∮ ⁡ 1 ρ ∇ p ⋅ d r − 2 Ω D A e D t , {displaystyle {frac {DC}{Dt}}=-oint {frac {1}{ ho }} abla pcdot mathrm {d} mathbf {r} -2Omega {frac {DA_{e}}{Dt}},} (1) where D D t { extstyle {frac {D}{Dt}}} is the time derivative in the rotational frame (not inertial frame), C {displaystyle C} is the relative circulation, A e {displaystyle A_{e}} is projection of the area surrounded by the fluid loop on the equatorial plane, ρ {displaystyle ho } is density, p {displaystyle p} is pressure, and Ω {displaystyle Omega } is the frame's angular speed. With Stoke's Theorem, the first term on the right-hand-side can be rewritten as D C D t = ∫ A ∇ ρ × ∇ p ρ 2 ⋅ d A − 2 Ω D A e D t , {displaystyle {frac {DC}{Dt}}=int _{A}{frac { abla ho imes abla p}{ ho ^{2}}}cdot mathrm {d} mathbf {A} -2Omega {frac {DA_{e}}{Dt}},} (2) which states that the rate of the change of the circulation is governed by the variation of density in pressure coordinates and the equatorial projection of its area, corresponding to the first and second terms on the right hand side. The first term is also called the 'solenoid term'. Under the condition of a barotropic fluid with a constant projection area A e {displaystyle A_{e}} , the Bjerknes circulation theorem reduces to Kelvin's theorem. However, in the context of atmospheric dynamics, such conditions are not a good approximation: if the fluid circuit moves from the equatorial region to the extratropics, A e {displaystyle A_{e}} is not conserved. Furthermore, the complex geometry of the material circuit approach is not ideal for making an argument about fluid motions. Carl Rossby proposed in 1939 that, instead of the full three-dimensional vorticity vector, the local vertical component of the absolute vorticity ζ a {displaystyle zeta _{a}} is the most important component for large-scale atmospheric flow. Also, the large-scale structure of a two-dimensional non-divergent barotropic flow can be modeled by assuming that ζ a {displaystyle zeta _{a}} is conserved. His later paper in 1940 relaxed this theory from 2D flow to quasi-2D shallow water equations on a beta plane. In this system, the atmosphere is separated into several incompressible layers stacked upon each other, and the vertical velocity can be deduced from integrating the convergence of horizontal flow. For a one-layer shallow water system without external forces or diabatic heating, Rossby showed that