Q-Vectors

First derived in 1978,[1] Q-vector derivation can be simplified for the midlatitudes, using the midlatitude β-plane quasi-geostrophic prediction equations:[2]

1. ${\displaystyle {\frac {D_{g}u_{g}}{Dt}}-f_{0}v_{a}-\beta yv_{g}=0}$ (x component of quasi-geostrophic momentum equation)
2. ${\displaystyle {\frac {D_{g}v_{g}}{Dt}}+f_{0}u_{a}+\beta yu_{g}=0}$ (y component of quasi-geostrophic momentum equation)
3. ${\displaystyle {\frac {D_{g}T}{Dt}}-{\frac {\sigma p}{R}}\omega ={\frac {J}{c_{p}}}}$ (quasi-geostrophic thermodynamic equation)

And the thermal wind equations:

${\displaystyle f_{0}{\frac {\partial u_{g}}{\partial p}}={\frac {R}{p}}{\frac {\partial T}{\partial y}}}$ (x component of thermal wind equation)

${\displaystyle f_{0}{\frac {\partial v_{g}}{\partial p}}=-{\frac {R}{p}}{\frac {\partial T}{\partial x}}}$ (y component of thermal wind equation)

where ${\displaystyle f_{0}}$ is the Coriolis parameter, approximated by the constant 1e⁻⁴ s⁻¹; ${\displaystyle R}$ is the atmospheric ideal gas constant; ${\displaystyle \beta }$ is the latitudinal change in the Coriolis parameter ${\displaystyle \beta ={\frac {\partial f}{\partial y}}}$; ${\displaystyle \sigma }$ is a static stability parameter; ${\displaystyle c_{p}}$ is the specific heat at constant pressure; ${\displaystyle p}$ is pressure; ${\displaystyle T}$ is temperature; anything with a subscript ${\displaystyle g}$ indicates geostrophic; anything with a subscript ${\displaystyle a}$ indicates ageostrophic; ${\displaystyle J}$ is a diabatic heating rate; and ${\displaystyle \omega }$ is the Lagrangian rate change of pressure with time. ${\displaystyle \omega ={\frac {Dp}{Dt}}}$. Note that because pressure decreases with height in the atmosphere, a ${\displaystyle -\omega }$ is upward vertical motion, analogous to ${\displaystyle +w={\frac {Dz}{Dt}}}$.

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

${\displaystyle Q_{i}=-{\frac {R}{\sigma p}}\left[{\frac {\partial u_{g}}{\partial x}}{\frac {\partial T}{\partial x}}+{\frac {\partial v_{g}}{\partial x}}{\frac {\partial T}{\partial y}}\right]}$

${\displaystyle Q_{j}=-{\frac {R}{\sigma p}}\left[{\frac {\partial u_{g}}{\partial y}}{\frac {\partial T}{\partial x}}+{\frac {\partial v_{g}}{\partial y}}{\frac {\partial T}{\partial y}}\right]}$

And in vector form:

${\displaystyle Q_{i}=-{\frac {R}{\sigma p}}{\frac {\partial {\vec {V_{g}}}}{\partial x}}\cdot {\vec {\nabla }}T}$

${\displaystyle Q_{j}=-{\frac {R}{\sigma p}}{\frac {\partial {\vec {V_{g}}}}{\partial y}}\cdot {\vec {\nabla }}T}$

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

${\displaystyle \left(\sigma {\overrightarrow {\nabla ^{2}}}+f_{\circ }^{2}{\frac {\partial ^{2}}{\partial p^{2}}}\right)\omega =-2{\vec {\nabla }}\cdot {\vec {Q}}+f_{\circ }\beta {\frac {\partial v_{g}}{\partial p}}-{\frac {\kappa }{p}}{\overrightarrow {\nabla ^{2}}}J}$

Which in an adiabatic setting gives:

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

Expanding the left-hand side of the quasi-geostrophic omega equation in a Fourier Series gives the ${\displaystyle -\omega }$ above, implying that a ${\displaystyle -\omega }$ 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 (${\displaystyle {\vec {\nabla }}\cdot {\vec {Q}}}$) is associated with downward motion. Therefore, convergent ${\displaystyle {\vec {Q}}}$ forces ascend and divergent ${\displaystyle {\vec {Q}}}$ forces descend.[3] 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.[4]

Q-vectors can be determined wholly with: geopotential height (${\displaystyle \Phi }$) 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 ${\displaystyle {\frac {\partial T}{\partial y}}<0}$), cyclones have Q-vectors which point parallel to the thermal wind and anticyclones have Q-vectors that point antiparallel to the thermal wind.[5] 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.[6] In areas of convergent Q-vectors, cyclonic vorticity is created, and in divergent areas, anticyclonic vorticity is created.[1]