ATMS 310 Omega Equation

The calculation of vertical motion in the atmosphere is one of the most challenging aspects of numerical prediction. It is challenging because typical vertical motions are three orders of magnitude (0.01 m/s vs. 10 m/s) smaller than horizontal motions.

So far we have learned about the kinematic and adiabatic methods for determining the vertical velocity. Recall that the kinematic method involved integrating the continuity equation in the vertical, yielding (in isobaric coordinates):

⎛ ∂ u ∂ v ⎞ ωω −+= pppp )()()( ⎜ + ⎟ (1) s s ⎜ ∂x ∂y ⎟ ⎝ ⎠ p

This allowed us to calculate a vertical velocity by measuring just the large-scale wind and pressure fields (easy quantities to measure). But, the calculation is very sensitive to relatively small errors in the wind fields.

The adiabatic method was an alternate method based on the thermodynamic equation:

−1 ⎛ ∂T ∂T ∂T ⎞ ω = S p ⎜ + u + v ⎟ (2) ⎝ ∂t ∂x ∂y ⎠

It was an attractive method because only height and temperature data are required to ∂T calculate a vertical velocity. However, difficulties in accurately calculating , as well ∂t as concerns when there is strong diabatic heating, make this option less attractive.

We have now discussed a third option for calculating ω – the QG equation:

∂ζ g v ∂ω V ()ζ ++∇•−= ff (3) ∂t g g o ∂p

v Both the (Vg ) and the geostrophic vorticity (ζ g ) are functions of the geopotential (Ф) – for example, the geostrophic wind is (from Eq. 6 in the QG_theory notes): v −1 v og kfV Φ∇×≡ (4)

∂ω Therefore, we can find a vertical velocity field ( ) if the geopotential and local change ∂p Φ∂ of geopotential with time ( ) are known. These quantities are measured every 12- ∂t hours at widely spaced upper air stations around the world. The problem with this method is that the quantities are measured every 12 hours at widely spaced upper air stations around the world. The resolution both in space in time is very coarse; thus, calculations of vertical velocity are tenuous. It is generally thought, however, that this method is more accurate than the kinematic method.

We now introduce a new method for calculating the vertical velocity field – the omega equation. The very nice feature of the omega equation is that the vertical velocity field can be determined by the instantaneous geopotential (Ф) field. It does not require wind (as in the kinematic method), vorticity tendency (as in the (Eq. 3) method, or temperature tendency (as in the adiabatic method) observations. However it is a much more complicated equation to solve than the others.

The omega equation is derived from the QG vorticity equation and the thermodynamic equation:

∂ζ g v ∂ω ⎛ ∂ ⎞⎛ ∂φ ⎞ κJ ⎜ ⎟ Vg ()ζ g ++∇•−= ff o ⎜ Vg ∇•+ ⎟⎜− ⎟ σω =− ∂t ∂p ⎝ ∂T ⎠⎝ ∂p ⎠ p QG Vorticity Equation QG Thermodynamic Equation

See Holton (p. 165) for the derivation steps. Essentially, the horizontal Laplacian (∇ 2 ) is ∂ applied to the thermo equation and the partial with respect to p ( ) is applied to the ∂p vorticity equation. After subtracting the two and simplifying, the omega equation is found:

2 2 ⎛ 2 f ∂ ⎞ f ∂ ⎡ v ⎛ 1 2 ⎞⎤ 1 2 ⎡ v ⎛ Φ∂ ⎞⎤ κ 2 ⎜ +∇ o ⎟ω = o V ⎜ +Φ∇∇• f ⎟ V ⎜−∇•∇+ ⎟ ∇− J (5) ⎜ 2 ⎟ ⎢ g ⎜ ⎟⎥ ⎢ g ⎜ ⎟⎥ ⎝ σ ∂p ⎠ σ ∂p ⎣ ⎝ f o ⎠⎦ σ ⎣ ⎝ ∂ ⎠⎦ σpp A B C D

Some simplifications can be done. Some terms in B and C can be cancelled out. The simplified form of the omega equation is:

v ⎛ f 2 ∂ 2 ⎞ f ⎡∂V ⎛ 1 ⎞⎤ ⎜ 2 +∇ o ⎟ω ≈ o g ⎜ 2 +Φ∇∇• f ⎟ (6) ⎜ 2 ⎟ ⎢ ⎜ ⎟⎥ ⎝ σ ∂p ⎠ σ ⎣⎢ ∂ ⎝ fp o ⎠⎦⎥ A B

TERM A The forcing of the vertical velocity by the right hand side can be interpreted as follows. If the right hand side (B) of Eq. (6) is positive, upward motion is forced (and vice versa). This may seem backwards, since we normally think that upward motion occurs Dp when ω <≡ 0 . But the left hand side of (6) is actually proportional to –ω if certain Dt qualitative assumptions are made about the vertical velocity field (see Holton p. 166 for details).

TERM B v ∂V B is the of absolute vorticity by the (note the g term). This ∂p term can be estimated by figuring out how the vorticity changes along the isotherms. Upward motion occurs where the vorticity decreases moving left to right along an isotherm. This pattern shows that upward motion is expected downstream of the upper level trough – where cyclones are observed to form in the atmosphere. See the figure below, which illustrates idealized regions of rising (w>0) and sinking (w<0) air for a developing synoptic-scale system.