By Dale R. Durran and Leonard W. Snellman

 “The physical reason for quasi-geostrophic vertical motion is reviewed. Various techniques for estimating synoptic-scale vertical motion are examined, and their utility (or lack thereof) is illustrated by a case study. The Q-vector approach appears to provide the best means of calculating vertical motions numerically. The vertical motion can be estimated by eye with reasonable accuracy by examining the advection of vorticity by the thermal wind or by examining the relative wind and the isobar field on an isentropic chart. The traditional form of the omega equation is not well suited for practical calculation.”

 Keeps geostrophic advection from changes in the hydrostatic and geostrophic balances Vertical velocity is present to control thermal wind balance  Simple for forecasting Easy to calculate and understand  In synoptic scales- Decent approximation of total vertical velocity

 The Q-G vertical velocity is calculated from the Q-G Omega equation  Left hand side ~ - ω

 Feb. 12, 1986 At 1200 UTC

 The second term on the RHS can be proportional to -1 times the temperature advection.

700 mb heights (solid) and mb thickness (dashed) -1 times 700 mb warm advection

Laplacian of the 700 mb warm advection

500 mb heights (solid) and absolute vorticity (dashed) 500 mb vorticity advection

Increase in vorticity advection with height at 500 mb

 Estimate the Q-G vertical velocity without numerical computation  Q-G omega equation can be written as: 1 st term (increase in vorticity advection with height) = advection of absolute vorticity by the thermal wind + advection of thermal vorticity by the wind 2 nd term (laplacian of warm advection) = advection of relative vorticity by the thermal wind – advection of thermal vorticity by the wind + terms involving the deformation of the wind field  Ascent should be located where there is advection of vorticity by the thermal wind

Advection of 500 mb vorticity by the mb thermal wind

 “Neither the Laplacian of the warm advection nor the rate of change of vorticity advection with height should be regarded as a cause of synoptic scale vertical motion. …quasi-geostrophic vertical motion is caused by the tendency for advection by the geostrophic wind to destroy thermal wind balance.”  Therefore instead of calculating the total forcing from the QG omega equation, Hoskins used Q vectors.

 Q vectors allow us to view the ageostrophic horizontal wind

 Hoskins et al showed that the RHS of equation 1 (the Q-G omega equation) goes as -2(the divergence of Q)  Divergence of the Q vector

Divergence of the Q-vectors at 500 mb Total forcing for omega at 500 mb from test 1

 Defined a 3D height field –  Numerical calculations with high horizontal and vertical resolution  Horizontal resolution 0.5 degree latitude by 0.5 degree longitude  Changes in vertical resolution First 2 images: 200 mb  Data from the 700, 500 and 300 mb levels used Second 2: 50 mb  Data from the 550, 500 and 450 mb levels used

Total forcing for ω with Δ p = 200 mb. (a) is calculated with traditional omega equation and (b) is calculated with divergence of Q vectors

Total forcing of ω with Δ p = 50 mb.

500 mb divergence of Q vectors

Total forcing for omega at 500 mb using Eq. 1

Divergence of the Q vectors at 500 mb Advection of 500 mb vorticity by the mb thermal wind from Eq. 3

 Q-G vertical motion is the result of keeping the balance between the hydrostatic and geostrophic balance  Q-G vertical motion at 500 mb calculated from the forcing terms in the omega equation matched up with the observed precipitation at the surface in the case study  Approximation of RHS of equation 1 is ~ to – ω holds true in the middle troposphere (Trenberth(1978))  Cannot estimate the total forcing using equation 1 (increase in vorticity advection with height + Laplacian of warm advection) without numerical calculations.

 Use Trenberth’s approximation (advection of vorticity by the thermal wind) if no access to numerical calculations  If calculating the Q-G omega equation numerically use Hoskins’ method of using Q-vectors  Errors are smaller in Trenberth and Hoskins’ methods due to the cancellation of the advection of thermal vorticity by the wind (a large term)

 175/ (1987)002%3C0017:TDOSSV%3E2.0.CO;2