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'''Q-vectors''' are used in atmospheric dynamics to understand physical processes such as vertical motion and [[frontogenesis]]. Q-vectors are not physical quantities that can be measured in the atmosphere but are derived from the quasi-geostrophic equations and can be used in the previous diagnostic situations. On meteorological charts, Q-vectors point toward upward motion and away from downward motion. Q-vectors are an alternative to the [[omega equation]] for diagnosing vertical motion in the quasi-geostrophic equations.

==Derivation==
First derived in 1978,<ref name=autogenerated1>{{cite journal|last=Hoskins|first=B. J.|coauthors=I. Draghici and H. C. Davies|title=A new look at the ω-equation|journal=Quart. J. R. Met. Soc|year=1978|volume=104|pages=31–38}}</ref> Q-vector derivation can be simplified for the midlatitudes, using the midlatitude β-plane quasi-geostrophic prediction equations:<ref>{{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}}</ref>

# <math> \frac{D_g u_g}{Dt} - f_{0}v_a - \beta y v_g = 0 </math> (x component of quasi-geostrophic momentum equation)
# <math> \frac{D_g v_g}{Dt} + f_{0}u_a + \beta y u_g = 0 </math> (y component of quasi-geostrophic momentum equation)
# <math> \frac{D_g T}{Dt} - \frac{\sigma p}{R} \omega = \frac{J}{c_p} </math> (quasi-geostrophic thermodynamic equation)

And the [[thermal wind]] equations:

<math> f_{0} \frac{\partial u_g}{\partial p} = \frac{R}{p} \frac{\partial T}{\partial y} </math> (x component of thermal wind equation)

<math> f_{0} \frac{\partial v_g}{\partial p} = - \frac{R}{p} \frac{\partial T}{\partial x} </math> (y component of thermal wind equation)

where <math>f_0</math> is the [[Coriolis parameter]], approximated by the constant 1e<sup>−4</sup> s<sup>−1</sup>; <math>R</math> is the atmospheric [[ideal gas constant]]; <math> \beta </math> is the latitudinal change in the Coriolis parameter <math> \beta = \frac{\partial f} {\partial y} </math>; <math> \sigma </math> is a static stability parameter; <math>c_p</math> is the [[specific heat]] at constant pressure; <math>p</math> is pressure; <math>T</math> is temperature; anything with a subscript <math>g</math> indicates [[geostrophic]]; anything with a subscript <math>a</math> indicates [[ageostrophic]]; <math>J</math> is a diabatic heating rate; and <math>\omega</math> is the Lagrangian rate change of pressure with time. <math>\omega = \frac{Dp}{Dt}</math>. Note that because pressure decreases with height in the atmosphere, a <math> - \omega </math> is upward vertical motion, analogous to <math>+w=\frac{Dz}{Dt}</math>.

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

<math> Q_1 = - \frac{R}{p} \left[ \frac{\partial u_g}{\partial x} \frac{\partial T}{\partial x} + \frac{\partial v_g}{\partial x} \frac{\partial T}{\partial y} \right] </math>

<math> Q_2 = - \frac{R}{p} \left[ \frac{\partial u_g}{\partial y} \frac{\partial T}{\partial x} + \frac{\partial v_g}{\partial y} \frac{\partial T}{\partial y} \right] </math>

And in vector form:

<math> Q_1 = - \frac{R}{p} \frac{\partial \vec{V_g}}{\partial x} \cdot \vec{\nabla} T </math>

<math> Q_2 = - \frac{R}{p} \frac{\partial \vec{V_g}}{\partial y} \cdot \vec{\nabla} T </math>

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

<math> \left(\sigma \overrightarrow{\nabla^2} + f_{\circ}^2 \frac{\partial ^2}{\partial p^2} \right) \omega = -2 \vec{\nabla} \cdot \vec{Q} + f_{\circ} \beta \frac{\partial v_g}{\partial p} - \frac{\kappa}{p} \overrightarrow{\nabla^2} J </math>

Which in an adiabatic setting gives:

<math> -\omega \propto -2 \vec{\nabla} \cdot \vec{Q} </math>

Expanding the left-hand side of the quasi-geostrophic omega equation in a [[Fourier Series]] gives the <math> -\omega </math> above, implying that a <math> -\omega </math> relationship with the right-hand side of the [[omega equation|quasi-geostrophic omega equation]] can be assumed.

This expression shows that the divergence of the Q-vector (<math> \vec{\nabla} \cdot \vec{Q} </math>) is associated with downward motion. Therefore, convergent <math> \vec{Q} </math> forces ascend and divergent <math> \vec{Q} </math> forces descend.<ref>{{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}}</ref> 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.<ref>{{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}}</ref>

==Applications==
Q-vectors can be determined wholly with: [[geopotential height]] (<math> \Phi </math>) 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 <math> \frac{\partial T} {\partial y} <0 </math>), cyclones have Q-vectors which point parallel to the thermal wind and anticyclones have Q-vectors that point antiparallel to the thermal wind.<ref>{{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}}</ref> 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.<ref>{{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|accessdate=15 March 2012}}</ref> In areas of convergent Q-vectors, cyclonic vorticity is created, and in divergent areas, anticyclonic vorticity is created.<ref name=autogenerated1 />

==References==
{{Reflist}}

[[Category:Meteorology]]

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