Meteo 422 – Lectures 8 and 9 – Quasi-geostrophic Approximation

Dr. George S. Young

The derivations below generally follow those in the course text: Holton’s “An Introduction to Dynamic Meteorology”

Goal: To understand the quasi-geostrophic approximation to the primitive equations including how to apply it and what weather phenomena it is valid for.

• Why make the quasi-geostrophic approximation?
• Makes the equations of motion simpler
• Simplifies the subsequent derivations of forecast equations for vorticity, surface pressure, and vertical velocity
• Highlights those aspects of the physics that are important for synoptic scale weather
• Aids in developing a physical understanding of how and why shortwaves and surface cyclones interact

• Quasi-geostrophic approximation
• What all is assumed?

· The phenomena of interest are hydrostatic

· The phenomena of interest are “nearly” geostrophic

· Friction is small enough to be safely ignored

• What it REALLY means

· Updrafts and downdrafts are weak – i.e. the weather phenomena is “flat”

· Weather phenomena is so large and slowly evolving that it is in geostrophic balance

· The large-scale dynamics are so strong that they overwhelm smaller-scale processes

• Why worry?
• Is the approximation you just used REALLY valid for the weather phenomenon you’re interested in?
• The Q-G approximations tend to break down right where the weather becomes most interesting – i.e. where there is lots of precipitation or rapid changes in temperature.

· At fronts

· Because vertical advection of momentum was ignored

• The Q-G approximation also fails where friction is important

· In the boundary layer

· Wherever thunderstorms move a lot of momentum in the vertical

• Scale analysis
• Horizontal Equation of Motion

_{Start with the “inviscid” equation for horizontal motion – i.e. friction has already been left out}

_{Divide the wind vector into two parts: geostrophic and ageostrophic – the ageostrophic wind is just the difference between the true wind and what the geostrophic wind law would lead one to expect.}

_{where we’ve assumed the Coriolis parameter is a constant f0 in the geostrophic wind law}

_{We split the wind into these two parts because the ageostrophic part of the wind is much smaller than the geostrophic part for synoptic scale systems. Put formally, the ratio of the two is about the same size as the Rossby number.}

_{Thus, small Rossby number means the flow is close to balanced – i.e. the pressure gradient force just about balances the Coriolis force – i.e. the wind is nearly geostrophic, which is what we said in the first place}

_{So we can approximate the horizontal advection using the geostrophic wind. We can ignore the vertical advection entirely because vertical motion arises only from ageostrophic flow (because the geostrophic wind is non-divergent). Thus the vertical velocity is much weaker than the geostrophic wind speed. Of course the vertical gradients are much larger than the horizontal gradients so we have to do a bit of checking to make sure vertical advection really is small on the synoptic scale.}

_{but scaling the continuity equation we get a relationship for vertical motion to the ageostrophic wind}

_{which we can plug into the equation above and rearrange to get the ratio of vertical to horizontal advection}

_{This ratio also scales as the Rossby number so we know that we really can ignore vertical advection for synoptic scale flows. Thus}

_where

_{Note that advection by both the horizontal and vertical components of the ageostrophic wind is gone. So the horizontal momentum equation is approximated as}

_{Now we have do decide what to do about the Coriolis and pressure gradient term. If we carry on cheerfully assuming that the wind is geostrophic these two terms cancel. This results in a horizontal momentum equation that says there is no change in the geostrophic wind following a parcel. For real life synoptic weather patterns this result provides unacceptably wrong results. The geostrophic wind really does evolve as a parcel flows down stream. So we have to back off on the geostrophic assumption for the Coriolis term. Instead we use the real wind and approximate the Coriolis parameter as fo+βy.}

_so

_{All this means is that the local change in the geostrophic wind plus the advection of the geostrophic wind by itself is balanced by the Coriolis force caused by the ageostrophic wind and the gradient of the north-south variation of the Coriolis force caused by the geostrophic wind. In plain English: the geostrophic wind changes because of Coriolis accelerations associated with the ageostrophic wind and because the Coriolis parameter changes as you move north south. It doesn’t seem so bad if you put it that way.}

• Continuity Equation of Motion

Start with the continuity equation in pressure coordinates

_{Then recall that the geostrophic wind is non-divergent so}

_{Everything here works by definition so we’ve not limited ourselves any further by this simplification.}

• Temperature Equation

Start with the temperature equation in pressure coordinates and split the temperature into a pressure dependent mean profile (i.e. the base state) and a weather dependent deviation from this base state.

Because most of the vertical variation in temperature is captured in the base state, only it appears in the approximation to the vertical advection term. In contrast, the base state has no horizontal variation so only the temperature deviation appears in the horizontal advection term. Likewise, the base state has no temporal variation so only the temperature deviation appears in the local change term.

where J is the diabatic heating and sigma is the stability calculated from the base state.

• Phenomena of applicability

· Works well on large-scale weather systems, i.e. those much larger than Texas

· Boundary layer phenomena are out because we ignored friction

· Equatorial phenomena are out because we used the geostrophic wind law

· Convective phenomena are out because we ignored friction and vertical motions in the momentum equation

· Frontal phenomena are out because we assumed geostrophy and weak vertical motions

· This leaves synoptic scale cyclones and the associated Rossby waves as the primary phenomena for which the Q-G approximation is suited.