Q-Vectors

First derived in 1978,{{cite journal|last=Hoskins|first=B. J. |author2=I. Draghici |author3=H. C. Davies|title=A new look at the ω-equation|journal=Q. J. R. Meteorol. Soc.|year=1978|volume=104|issue=439 |pages=31–38|doi=10.1002/qj.49710443903 |bibcode=1978QJRMS.104...31H }} Q-vector derivation can be simplified for the midlatitudes, using the midlatitude β-plane quasi-geostrophic prediction equations:{{cite book|last=Holton|first=James R.|title=An Introduction to Dynamic Meteorology|year=2004|publisher=Elsevier Academic|location=New York|isbn=0-12-354015-1|pages=168–72}}

1. $\backslash frac\{D\_g\; u\_g\}\{Dt\}\; -\; f\_\{0\}v\_a\; -\; \backslash beta\; y\; v\_g\; =\; 0$ (x component of quasi-geostrophic momentum equation)
2. $\backslash frac\{D\_g\; v\_g\}\{Dt\}\; +\; f\_\{0\}u\_a\; +\; \backslash beta\; y\; u\_g\; =\; 0$ (y component of quasi-geostrophic momentum equation)
3. $\backslash frac\{D\_g\; T\}\{Dt\}\; -\; \backslash frac\{\backslash sigma\; p\}\{R\}\; \backslash omega\; =\; \backslash frac\{J\}\{c\_p\}$ (quasi-geostrophic thermodynamic equation)

And the thermal wind equations:

$f\_\{0\}\; \backslash frac\{\backslash partial\; u\_g\}\{\backslash partial\; p\}\; =\; \backslash frac\{R\}\{p\}\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; y\}$ (x component of thermal wind equation)

$f\_\{0\}\; \backslash frac\{\backslash partial\; v\_g\}\{\backslash partial\; p\}\; =\; -\; \backslash frac\{R\}\{p\}\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; x\}$ (y component of thermal wind equation)

where $f\_0$ is the Coriolis parameter, approximated by the constant 1e⁻⁴ s⁻¹; $R$ is the atmospheric ideal gas constant; $\backslash beta$ is the latitudinal change in the Coriolis parameter $\backslash beta\; =\; \backslash frac\{\backslash partial\; f\}\; \{\backslash partial\; y\}$; $\backslash sigma$ is a static stability parameter; $c\_p$ is the specific heat at constant pressure; $p$ is pressure; $T$ is temperature; anything with a subscript $g$ indicates geostrophic; anything with a subscript $a$ indicates ageostrophic; $J$ is a diabatic heating rate; and $\backslash omega$ is the Lagrangian rate change of pressure with time. $\backslash omega\; =\; \backslash frac\{Dp\}\{Dt\}$. Note that because pressure decreases with height in the atmosphere, a negative value of $\backslash omega$ is upward vertical motion, analogous to $+w=\backslash frac\{Dz\}\{Dt\}$.

From these equations we can get expressions for the Q-vector:

$Q\_i\; =\; -\; \backslash frac\{R\}\{\backslash sigma\; p\}\; \backslash left[\; \backslash frac\{\backslash partial\; u\_g\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; v\_g\}\{\backslash partial\; x\}\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; y\}\; \backslash right]$

$Q\_j\; =\; -\; \backslash frac\{R\}\{\backslash sigma\; p\}\; \backslash left[\; \backslash frac\{\backslash partial\; u\_g\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; v\_g\}\{\backslash partial\; y\}\; \backslash frac\{\backslash partial\; T\}\{\backslash partial\; y\}\; \backslash right]$

And in vector form:

$Q\_i\; =\; -\; \backslash frac\{R\}\{\backslash sigma\; p\}\; \backslash frac\{\backslash partial\; \backslash vec\{V\_g\}\}\{\backslash partial\; x\}\; \backslash cdot\; \backslash vec\{\backslash nabla\}\; T$

$Q\_j\; =\; -\; \backslash frac\{R\}\{\backslash sigma\; p\}\; \backslash frac\{\backslash partial\; \backslash vec\{V\_g\}\}\{\backslash partial\; y\}\; \backslash cdot\; \backslash vec\{\backslash nabla\}\; T$

Plugging these Q-vector equations into the quasi-geostrophic omega equation gives:

$\backslash left(\backslash sigma\; \backslash overrightarrow\{\backslash nabla^2\}\; +\; f\_\{\backslash circ\}^2\; \backslash frac\{\backslash partial\; ^2\}\{\backslash partial\; p^2\}\; \backslash right)\; \backslash omega\; =\; -2\; \backslash vec\{\backslash nabla\}\; \backslash cdot\; \backslash vec\{Q\}\; +\; f\_\{\backslash circ\}\; \backslash beta\; \backslash frac\{\backslash partial\; v\_g\}\{\backslash partial\; p\}\; -\; \backslash frac\{\backslash kappa\}\{p\}\; \backslash overrightarrow\{\backslash nabla^2\}\; J$

If second derivatives are approximated as a negative sign, as is true for a sinusoidal function, the above in an adiabatic setting may be viewed as a statement about upward motion:

$-\backslash omega\; \backslash propto\; -2\; \backslash vec\{\backslash nabla\}\; \backslash cdot\; \backslash vec\{Q\}$

Expanding the left-hand side of the quasi-geostrophic omega equation in a Fourier Series gives the $-\backslash omega$ above, implying that a $-\backslash omega$ relationship with the right-hand side of the quasi-geostrophic omega equation can be assumed.

This expression shows that the divergence of the Q-vector ($\backslash vec\{\backslash nabla\}\; \backslash cdot\; \backslash vec\{Q\}$) is associated with downward motion. Therefore, convergent $\backslash vec\{Q\}$ forces ascent and divergent $\backslash vec\{Q\}$ forces descend.{{cite book|last=Holton|first=James R.|title=An Introduction to Dynamic Meteorology|year=2004|publisher=Elsevier Academic|location=New York|isbn=0-12-354015-1|pages=170}} Q-vectors and all ageostrophic flow exist to preserve thermal wind balance. Therefore, low level Q-vectors tend to point in the direction of low-level ageostrophic winds.{{cite book|last=Hewitt|first=C. N.|title=Handbook of atmospheric science: principles and applications|year=2003|publisher=John Wiley & Sons|location=New York|isbn=0-632-05286-4|pages=286}}

Q-vectors can be determined wholly with: geopotential height ($\backslash Phi$) and temperature on a constant pressure surface. Q-vectors always point in the direction of ascending air. For an idealized cyclone and anticyclone in the Northern Hemisphere (where ), cyclones have Q-vectors which point parallel to the thermal wind and anticyclones have Q-vectors that point antiparallel to the thermal wind.{{cite book|last=Holton|first=James R.|title=An Introduction to Dynamic Meteorology|year=2004|publisher=Elsevier Academic|location=New York|isbn=0-12-354015-1|pages=171}} This means upward motion in the area of warm air advection and downward motion in the area of cold air advection.

In frontogenesis, temperature gradients need to tighten for initiation. For those situations Q-vectors point toward ascending air and the tightening thermal gradients.{{cite web|last=National Weather Service|first=Jet Stream - Online School for Weather|title=Glossary: Q's|url=http://www.srh.weather.gov/jetstream/append/glossary_q.htm|work=NOAA - NWS|access-date=15 March 2012}} In areas of convergent Q-vectors, cyclonic vorticity is created, and in divergent areas, anticyclonic vorticity is created.

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