Quasi-Geostrophic ω-equation For large-scale flow in the atmosphere, we have learned about two very important characteristics:. The atmosphere is approximately hydrostatic.. The atmosphere is approximately geostrophic. These two features are what we mean by Quasi-Geostrophic (I suppose we should really say "Quasi-Geostrophic/Hydrostatic"). Indeed, the atmosphere is not only close to being geostrophic and hydrostatic, it greatly resists any attempts to take it out of geostrophic or hydrostatic balance. A critical equation that one can derive assuming the following characteristics is the Quasi- Geostrophic ω-equation. It is beyond the scope of this course to derive this equation, but I will provide you with some guidance on how it is done (Holton derives this equation in his book). Almost everywhere you see V in an equation, you simply replace it with V g. The only important exception is that the divergence of the actual wind is poorly approximated by the divergence of the geostrophic wind (i.e., V g is a very poor approximation for V). Geostrophic vorticity ( V g ) and geostrophic advection (V g ) are good approximations for total vorticity ( V ) and advection (V ). That's really the major assumption/approximations, and then you can derive the following equation: f f ( ) [ V g ( f)] [ Vg ( )] p p f p where φ=gz (referred to as geopotential) and and is a stability parameter which is greater than when the atmosphere is stable (the atmosphere is generally stable on the large scale although there are regions where it is unstable). Note that g V f k z f k g p p So the ω-equation is such that basically all the terms on the right hand side (RHS) are functions of φ. Also note that is really the following: k V k g P f k ( ) g P p p f
So, the term with is really a measure of the geostrophic vorticity. Before we try to understand the terms of these equations, there are two very important observations that can be determined by examining its form. First this equation is critical for forecasting since it tells us how to estimate vertical velocity. We need to know this if we are going to predict where the precipitation is going to occur. Second, as I already mentioned, all of the terms on the RHS of the equation basically depend on φ and f (f only depends on the latitude; so it is a known quantity). This is far-reaching observation since it states that to predict the vertical velocity, all you need to measure is the height field. This is a shocking discovery since it says you don't need to measure the horizontal windfield to determine the vertical velocity. This also solves are initial worry that we had no equation to predict w (recall that the vertical component of the momentum equation just reduced to hydrostatic balance). Interpreting the Quasi-Geostrophic ω-equation Before any of you have a heart attack, let's try to interprete the ω-equation and discover that it really isn't sending us a confusing message. f ( ) Let's start with the term on the left hand side (LHS) of the equation: p. This is the second derivative of ω in the x, y and p directions (the operator is often referred to as the Laplacian). In the atmosphere we often assume wavelike solutions. This is a pretty good assumption if you have ever looked at a weather map and noticed all of the troughs and ridges. They look very much like sin and cos waves. One of the properties of trig functions is that if you take a second derivative of them, you just get the same function back with a minus sign in front of it. In other words- f ( ) w p f [ Vg ( f )] Now, let us examine the first term on the RHS of the equation: p f. The terms in the bracket is the advection of something by the geostrophic wind. The something is the geostrophic vertical vorticity ( f ) plus the Coriolis parameter.
3 Let's draw a typical 5-mb trough to understand what this term is trying to tell us. Z Z Z L height contours X Vorticity There is no advection of vorticity along this line since the winds are parallel to the vorticity lines. NVA or Negative Vorticity Advection - Lower values of vorticity are being advected into the region by the wind. PVA or Positive Vorticity Advection - Higher values of vorticity are being advected into the region by the wind. The typical trough on the map has vorticity contours such as illustrated above. A maximum in vorticity exists within the base of the trough. The reason is due largely to the curvature effect. The vorticity lines that we see on the maps are values of ζf, which is referred to as the absolute vorticity (the relative vorticity plus the planetary vorticity).
4 The following is a National Weather Service Chart highlighting a real situation. This is an actual National Weather Service chart that shows 5 mb heights and vorticity lines. The solid lines are contours in decameters. The dashed lines are "absolute" vorticity (relative vorticity f) and are in units of s. Notice that all the vorticity values are >. This is due to the fact that the typical value of is s. Recall from the scale analysis that f is O( s ). The Coriolis parameter is so large that even anticyclonic vorticity centers are positive numbers. Can you identify the areas of PVA and NVA?
5 In the region where there is PVA, the term in the bracket is positive. In other words, we have the following in the region of PVA: f w positive number p [ ]. The derivative outside of the bracket involves height We can see that if we want positive vertical motion we must have the PVA increase with height. In actual forecasts we usually just look at the PVA at the 5 mb level and not concern ourselves with what is going on at lower levels. In essence we are assuming that the advection at lower levels is negligible. Strictly speaking that is wrong. You should examine the vertical derivative. Note that NVA that decreases with height would also indicated positive vertical motions Let's now attempt to understand why PVA that increases with height leads to rising vertical motion. P z z P t later z w> Large PVA means that you advecting large positive values of vertical vorticity. When you do this, you form a cyclonic circulation at that level. Since the flow wants to be geostrophic, it will develop an area of low pressure. Note that z has shrunk because the isobars have changed their shape in response to the circulations. Recall that the atmosphere not only remains geostrophic but it also wants to remain hydrostatic. Accordingly, you cannot force two pressure levels to come together hydrostatically without changing something. To do so hydrostatically, requires you to cool the air. At this point, there is only one way to cool the air--positive vertical motion (upward). Positive vertical motion expands and cools the air adiabatically.
6 Let's now look at the second term on the RHS of the equation. w [ V ( )] g p. Once again the term in the bracket represents the advection of something by the geostrophic wind. What is that something? p g z z p g gz p RT p p or p using the equation of state Since we are in a pressure coordinate system (p is a constant), p is proportional to temperature. In other words, the term in the brackets is the horizontal advection of temperature. Since, once again, we assume wavelike solutions, w [ V ( )] g p or w [ temperature advection] The simplification of the equation states that areas of warm air advection (term in bracket >) should be subject to rising motion and cold air advection (term in bracket <) should subject to sinking motion.
7 Let us try to understand why this should be the case. H z z P P t later - z In this case we are advecting warm air into a layer. L If you advect warm air into the layer, the p- surfaces separate as shown so that the atmosphere can maintain hydrostatic balance. However, distorting the pressure surfaces as shown produces high and low pressure centers aloft and near the surface, respectively. The atmosphere attempts to maintain geostrophic balance by creating circulations as shown above. If you examine the diagram above you will note that we will have formed two areas of ζ. One is of negative vorticity (anticyclonic) at high levels and another area of positive vorticity (cyclonic) at low levels. The question is - how does the atmosphere in quasi-geostrophic balance form these two circulations? For large-scale flow, vorticity increases and decreases are nominally generated by horizontal convergence and divergence, respectively. This is often referred to as the ice skater effect. When ice skaters bring their arms and legs closer to together (convergence), they spin faster. The opposite occurs when they spread their arms and legs. The technical term for this process is Vortex Stretching. DIV z - W> CON When there is horizontal convergence at low levels and divergence aloft, we must create rising
8 motion. We now see that, w [ warm air advection]. WESTWARD TILT WITH INCREASING HEIGHT We discovered in the previous section that, typically there is rising motion ahead of a 5 mb trough (due to differential vorticity advection) and sinking motion to the west of the trough. This leads to the development of surface lows and highs as illustrated by Holton. The area ahead of the trough is also a region of warm air advection. This westward tilt of systems is well known in quasi-geostrophic theory. Indeed when the systems become vertically stacked, that is typically when the surface low begins to fill/weaken.