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]​

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 negative value of ${\displaystyle \omega }$ 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:

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

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:

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

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 ascent 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]​

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 ${\displaystyle {\frac {\partial T}{\partial y}}<0}$), 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