Vectores Q

Los vectores Q se utilizan en dinámica atmosférica para comprender procesos físicos como el movimiento vertical y la frontogénesis. Los vectores Q no son cantidades físicas que se puedan medir en la atmósfera, sino que se derivan de las ecuaciones cuasi-geostróficas y se pueden usar en las situaciones de diagnóstico anteriores. En los gráficos meteorológicos, los vectores Q apuntan hacia el movimiento hacia arriba y hacia el lado opuesto al movimiento hacia abajo. Los vectores Q son una alternativa a la ecuación omega para diagnosticar el movimiento vertical en las ecuaciones cuasi-geostróficas.

Derivaciones

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

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

And the thermal wind equations:

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

$f_{0}{\frac {\partial v_{g}}{\partial p}}=-{\frac {R}{p}}{\frac {\partial T}{\partial x}}$ (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; $\beta$ is the latitudinal change in the Coriolis parameter $\beta ={\frac {\partial f}{\partial y}}$ ; $\sigma$ 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 $\omega$ is the Lagrangian rate change of pressure with time. $\omega ={\frac {Dp}{Dt}}$ . Note that because pressure decreases with height in the atmosphere, a negative value of $\omega$ is upward vertical motion, analogous to $+w={\frac {Dz}{Dt}}$ .

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

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

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

And in vector form:

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

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

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

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

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:

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

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

Aplicaciones

Los vectores Q se pueden determinar completamente con: altura geopotencial ({\ estilo de visualización \ Phi}\Fi) y la temperatura en una superficie de presión constante. Los vectores Q siempre apuntan en la dirección del aire ascendente. Para un ciclón y anticiclón idealizados en el hemisferio norte (donde ${\frac {\partial T}{\partial y}}<0$ ), los ciclones tienen vectores Q que apuntan paralelos al viento térmico y los anticiclones tienen vectores Q que apuntan antiparalelos al viento térmico.^[5] Esto significa movimiento hacia arriba en el área de advección de aire cálido y movimiento hacia abajo en el área de advección de aire frío.

En la frontogénesis, los gradientes de temperatura deben ajustarse para la iniciación. Para esas situaciones, los vectores Q apuntan hacia el aire ascendente y los gradientes térmicos cada vez más estrechos.^[6] En áreas de vectores Q convergentes, se crea vorticidad ciclónica, y en áreas divergentes, se crea vorticidad anticiclónica.^[1]

Referencias

↑ ^a ^b Hoskins, B. J.; I. Draghici; H. C. Davies (1978). «A new look at the ω-equation». Quart. J. R. Met. Soc 104: 31-38.
↑ Holton, James R. (2004). An Introduction to Dynamic Meteorology. New York: Elsevier Academic. pp. 168-72. ISBN 0-12-354015-1.
↑ Holton, James R. (2004). An Introduction to Dynamic Meteorology. New York: Elsevier Academic. p. 170. ISBN 0-12-354015-1.
↑ Hewitt, C. N. (2003). Handbook of atmospheric science: principles and applications. New York: John Wiley & Sons. pp. 286. ISBN 0-632-05286-4.
↑ Holton, James R. (2004). An Introduction to Dynamic Meteorology. New York: Elsevier Academic. p. 171. ISBN 0-12-354015-1.
↑ National Weather Service, Jet Stream - Online School for Weather. «Glossary: Q's». NOAA - NWS. Consultado el 15 de marzo de 2012.

Datos: Q7265294

[autogenerated1-1] Hoskins, B. J.; I. Draghici; H. C. Davies (1978). «A new look at the ω-equation». Quart. J. R. Met. Soc 104: 31-38.

[2] Holton, James R. (2004). An Introduction to Dynamic Meteorology. New York: Elsevier Academic. pp. 168-72. ISBN 0-12-354015-1.

[3] Holton, James R. (2004). An Introduction to Dynamic Meteorology. New York: Elsevier Academic. p. 170. ISBN 0-12-354015-1.

[4] Hewitt, C. N. (2003). Handbook of atmospheric science: principles and applications. New York: John Wiley & Sons. pp. 286. ISBN 0-632-05286-4.

[5] Holton, James R. (2004). An Introduction to Dynamic Meteorology. New York: Elsevier Academic. p. 171. ISBN 0-12-354015-1.

[6] National Weather Service, Jet Stream - Online School for Weather. «Glossary: Q's». NOAA - NWS. Consultado el 15 de marzo de 2012.

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