Meteo 422 – Lectures 14 – Quasi-geostrophic omega equation
The derivations below generally follow those in the course text: Holton’s “An Introduction to Dynamic Meteorology”
Goal: To understand how the quasi-geostrophic approximation can be used to derive a forecast equation for vertical velocity.
· Why quasi-geostrophic? – Same reasons as always
o Much simpler than using the full equations of motion
o Makes subsequent derivations easier
o Highlights those processes that are important on the synoptic scale
· Why adiabatic? – Same reasons as always
o It makes the derivation a bit easier if we drop the diabatic heating term at the outset.
· Why worry? – Same reasons as always – this time in spades
o Are these two approximations REALLY valid for the weather phenomenon you’re interested in?
o What happens to that adiabatic assumption when it starts raining heavily?
o Omega is the vertical velocity in pressure coordinates
o It is thus the total derivative of pressure with time following a parcel
o On the synoptic scale it is well approximated by density times gravity times the vertical velocity in height coordinates. This approximation is not nearly so good for smaller scale weather systems with large vertical velocities.
o Thus, from the hydrostatic approximation, have
· Starting point – Same as for the tendency equation
o Quasi-geostrophic vorticity equation
o Quasi-geostrophic thermodynamic equation
· Insights – similar to those used in deriving the tendency equation
o Both horizontal wind components have been combined into the vorticity equation
o So of the variables found in the primitive equations only vertical velocity, geopotential, and temperature remain.
o T can be restated in terms of geopotential, eliminating it.
o After doing this only two variables remain, vertical velocity and geopotential.
· Temperature and vorticity budgets – in the same form as we used to derive the tendency equation
o The temperature budget, rewritten as the thermodynamic energy equation, relates local rate of change in thickness (a measure of temperature) to its horizontal and vertical advection.
o The vorticity budget has been similarly rewritten in terms of geopotential tendency χ.
· Derivation of the omega equation
While the approach is identical to that we used to derive the tendency equation, we’ll make a different decision at one key step so as to eliminate geopotential tendency instead of omega.
o Guiding insights
· Our first task in doing a “non-universal” derivation like this is to determine which of the variables are so easy to determine by other means that we don’t need to eliminate then. As we discussed two lectures ago, the Coriolis parameter certainly falls into this category. Likewise, we can easily determine the geostrophic wind from the geopotential field so we don’t need to eliminate that variable. Similarly, if we know the current geopotential field and we have an equation for its tendency, we can forecast the future geopotential field. Thus, the only two “real” unknowns are the geopotential tendency and the vertical velocity. This leaves us with two equations and two unknowns. Since we have the same number of equations and unknowns, all we have to do is solve the equations for the unknowns. A piece of cake, right?
· Now, we face the choice of which unknown two solve for. Since we solved for geopotential tendency last time we’ll change course and solve for omega this time.
· Since we’re solving for omega, the other unknown (geopotential tendency) must be our designated “victim”. Thus, we need to figure out how to eliminate it when we combine the two equations listed above into one equation for omega. Because geopotential tendency appears in only one simple term in each equation, figuring out how to get it to cancel when the equations are combined is rather easy as these things go. The geopotential tendency appears in a Laplacian in one equation and in a vertical gradient in the other. By examining these differences, we can determine how to get rid of them. If we take the Laplacian of the equation where geopotential tendency occurs in a vertical gradient, and vice versa, we end up with two equations wherein geopotential tendency appears in the same form, the Laplacian of a vertical gradient.
· The last step is trivial, just subtract the two equations.
· The omega equation
o Note that as with the geopotential tendency equation, all the derivatives are in space rather than in time.
· Thus, it is a diagnostic equation
· It means we can deduce what is happening at a particular time instead of predicting how what it will change over time
o The first term is just a three-dimensional laplacian of omega.
· Its value is large when the vertical velocity is concentrated in a small volume of the atmosphere.
· Thus, it is harder to force rapid up or downdrafts in a small, sharply defined region of the atmosphere than it is to do so over a broader, more smoothly defined region.
· This laplacian applies in three dimensions. So if you try to force a up or downdraft of small horizontal extent, you also get small vertical extent.
o The second term is proportional to the vertical gradient of geostrophic vorticity advection at the level where you’re calculating omega.
· If a cyclone blows in over you, the air rises where you are.
· Likewise updrafts occur if you blow an anticyclone in under you.
· Downdrafts occur if you blow an anticyclone in over you or a cyclone in under you.
· These vertical motions are required to provide the adiabatic heating or cooling needed to ensure that this newly arrived cyclone/anticyclone does indeed change intensity with height.
· This term just says that if you move in a vortex in at one level but not another, you need to use adiabatic cooling/heating to adjust the thickness field appropriately..
· This term tends to dominate in the upper troposphere because of the strong winds and open waves there.
o The third term depends on the horizontal Laplacian of geostrophic advection of thickness.
· It says that if you blow cold (low-thickness) air at your level the sky falls, but if you blow in warm (high-thickness) air at your level the air rises.
· Because of the Laplacian, compact areas of heating give more effect than broad diffuse areas of heating..
· One important implication of this term is that the effect depends inversely on stability. Thus, thermal advection causes more vertical velocity if the atmosphere is near neutral than if it is stable. This is true because the vertical velocity is required for adiabatic temperature change. If the sounding is near neutral, you have to move the air a long way in the vertical to get much temperature change via adiabatic motion.
· This term is more important in the lower troposphere because of the large temperature gradients there.
· It becomes really important if diabatic processes decrease the stability to near zero.
o Thus, the equation makes a certain amount of hand waving physical sense.
· The only real surprise is that sharply defined localized highs and lows are so hard to form. That results goes a long way to explain why meteorologists focus so much on large weather systems (synoptic scale lows). The strong lows tend to be big.
· Perhaps not quite so surprising is the damper that stability can put on vertical motion. This makes any diabatic process that can reduce stability an important aspect of heavy precipitation events. Think about the role of heat and moisture from the Gulf Stream in East Coast snowstorms.
o All of these hand waving results can be quantified by plugging in our “standard” sinusoidal weather system as Holton does on pages 167 and 168.
o Holton’s figure 6.11 shows how the differential vorticity advection term works in a typical wave cyclone while his figure 6.12 shows the effects of thermal advection.