Quasi-geostrophic lissafi

Duk da yake motsi na geostrophic yana nufin iska wanda zai haifar da daidaitattun daidaito tsakanin Ƙarfin Coriolis da Ƙarfin matsin lamba na kwance,  motsi na kusan geostrophian (QG) yana nufin gudana inda ƙarfin Corioli da ƙarfin matsawa kusan suna cikin daidaito, amma tare da inertiya kuma yana da tasiri. 

 Asalin 
Gudun yanayi da na teku suna faruwa a kan ma'auni na tsawon kwance wanda yake da girma sosai idan aka kwatanta da ma'aunin tsayin daka, don haka ana iya bayyana su ta amfani da ma'anar ruwa mai zurfi. Lambar Rossby lambar ce mara girma wacce ke nuna ƙarfin inertia idan aka kwatanta da ƙarfin ƙarfin ƙarfin Coriolis. Ƙididdigar ƙididdigar ƙasa sune kusanci ga ƙididdigaren ruwa mai zurfi a cikin iyakar ƙaramin lambar Rossby, don haka dakarun inertial suna da tsari na girman ƙarami fiye da Coriolis da ƙarfin matsin lamba. Idan lambar Rossby ta yi daidai da sifili to muna dawo da kwararar geostrophic.

Jule Charney ne ya fara tsara daidaitattun ƙididdigar ƙasa.

 Rarraba daidaitattun QG guda ɗaya 
A cikin ma'aunin Cartesian, abubuwan da ke cikin iska na geostrophic sune

 f    0        v    g      =      ∂ Φ     ∂ x        {\displaystyle {f_{0}}{v_{g}}={\ bangare \Phi \over \ bangare x}}     (1a) 
 f    0        u    g      = -      ∂ Φ     ∂ y        {\displaystyle {f_{0}}{u_{g}}=-{\ bangare \Phi \over \ bangare y}}     (1b) 

inda 

 Φ 

 {\displaystyle {\Phi }} 

 shine geopotential.

Girman geostrophic



sabili da haka ana iya bayyana shi dangane da geopotential kamar yadda

 ζ    g      =       ∂    v    g       ∂ x      -      ∂    u    g       ∂ y      =     1    f    0         (Abin da ya fi dacewa da shi)         ∂    2     Φ     ∂    x    2        +       ∂    2     Φ     ∂    da kuma    2         )     =     1    f    0         Sanya    2     Φ       {\displaystyle {\zeta _{g}}={{\partial v_{g} \over \partial x}-{\partial u_{g}. \over \partal y}={1 \over f_{0}}\left ({{\partiel ^{2}\Phi \over \parcial y^{2}}}\right) ={1 \ over f_{2}}{2}    (2)

Ana iya amfani da daidaitattun (2) don samun 

 ζ 

 g 

  (x, y)  

 {\displaystyle {\zeta _{g} (x,y) }} 

 daga sanannen filin 

 Φ (x , y)  

 {\displaystyle {\Phi (x,y) }} 

 . A madadin haka, ana iya amfani da shi don tantancewa 

 Φ 

 {\displaystyle {\Phi }} 

 daga sanannen rarraba 

 ζ 

 g 

 {\displaystyle {\zeta _{g}}} 

 ta hanyar juyar da mai aiki na Laplacian.

Za'a iya samun daidaitattun vorticity na kusan-geostrophic daga 

 x 

 {\displaystyle {x}} 

 da kuma 

 da kuma 

 {\displaystyle {y}} 

 abubuwan da ke tattare da daidaitattun ma'auni na ma'aunin ma'aikatar ma'aunar ma'aanar ma'aunonin ma'a

 D    V      D t      + f       k    ^      ×    V    = − Sin     {\displaystyle {D\mathbf {V} \over Dt}+f{\hat {\mathbf{k} }}\times \mathbf {\V} =-\nabla \Phi }    (3)

Abubuwan da aka samo a cikin (3) an bayyana su ta hanyar

 D    D t      =       (Abin da ya fi dacewa da shi)      ∂    ∂ t      )      p      +       (Abin da ya fi dacewa da shi)      V    Ta yaya    )      p      +    ω     ∂    ∂ p          {\displaystyle {{D \over Dt}={\left ({\partial \over \partial t}\right) _{p}}+{\lept ({\mathbf {V} \cdot \nabla }\right) _{p}} +{\omega {\partial \ over \partial p}}}}    (4)
 inda       ω =      D p     D t         {\displaystyle {\omega ={Dp \over Dt}}}    shine canjin matsin lamba bayan motsi.

Saurin kwance 

 V 

 {\displaystyle {\mathbf {V} }} 

 za a iya raba shi cikin geostrophic 

 V 

 g 

 {\displaystyle {\mathbf {V_{g}} }} 

 da kuma ageostrophic 

 V 

 a 

 {\displaystyle {\mathbf {V_{a}} }} 

 wani bangare

 V    =     V    g      +     V    a         {\displaystyle {\mathbf {V} =\mathbf{V_{g}} +\mathbbf { V_{a}} }}    (5)

Muhimman zato guda biyu na kusanci na kusan-geostrophic sune

 1.          V    g      ≫     V    a         {\displaystyle {\mathbf {V_{g}} \gg \mathbf{V_{a}} }}    , ko, mafi mahimmanci          Sanyawa      V    a       Sanyawa       Sanyawa      V    g       Sanyawa       ∼ O (    Lambar Rossby    )      {\displaystyle { avg_{a}} da kuma \over\mathbf {V_{g}} da kuma\sim O ({\text{Rossby number}}) }    .
 2. kusanci na beta-plane       f =    f    0     + β da      {\displaystyle {f=f_{0}+\beta y}}    tare da          β da     f    0       ∼ O (    Lambar Rossby    )       {\displaystyle {{\frac {\beta y}{f_{0}}}\sim O ({\text{Rossby number}}) }} 

Tunanin na biyu ya tabbatar da barin siginar Coriolis ta sami darajar dindindin f 0 {\displaystyle {f_{0}}} a cikin kusanci na geostrophic da kusanci da bambancinsa a cikin kalmar ƙarfin Coriolis ta hanyar f 0 + β da {\displaystyle {f_{0}+\beta y}} .[1] Koyaya, saboda hanzari bayan motsi, wanda aka ba shi a cikin (1) a matsayin bambanci tsakanin ƙarfin Coriolis da ƙarfin ƙarfin matsa lamba, ya dogara da tashiwar ainihin iska daga iska ta geostrophic, ba a yarda a sauya saurin ta hanyar saurin geostrophian a cikin kalmar Coriolis ba.[1] Za'a iya sake rubuta hanzari a cikin (3) kamar yadda

 f       k    ^      ×    V    + Sin    =     (Abin da ya fi dacewa da shi)     f    0     + β y)        k    ^      × (     V    g      +     V    a      ) -    f    0         k    ^      ×     V    g       =     f    0         k    ^      ×     V    a      + β da       k    ^      ×     V    g         {\displaystyle {f{\hat {\mathbf {k} }}\times \mathbf{V} +\nabla \Phi }={ (f_{0}+\beta y) {\hat {\mathf {k}}}}\times (\mathbf {\V_{a}} +\mathbbf {f_{\mathbhat}\mathb_{V_{\\mathb}}}\mathf}\mathg_{\    (6)

Kimanin daidaitattun ma'auni na kwance yana da nau'in

 D    g       V    g        D t      =    -    f    0         k    ^      ×     V    a      − β da       k    ^      ×     V    g         {\displaystyle {D_{g}\mathbf {V_{g}} \over Dt}={-f_{0}{\hat {\mathbf{k} }}\times \mathbf {\V_{a}} -\beta y{\hat{\mathbbf {k} }},\times \Mathbf { V_{g}},    (7)

Bayyana daidaitattun (7) dangane da abubuwan da ke ciki,

 D    g      u    g       D t      -     f    0      v    a      -    β da    v    g      = 0      {\displaystyle {{D_{g}u_{g}} \over Dt}-{f_{0}v_{a}}-{\beta yv_{g}},=0}}     (8a) 

 D    g      v    g       D t      +     f    0      u    a      +    β da    u    g      = 0      {\displaystyle {{D_{g}v_{g}} \over Dt}+{f_{0}u_{a}}+{\beta yu_{g}},=0}}     (8b) 

Samun abubuwa 

 ∂ (8 b)  

 ∂ x 

 - 

 ∂ (8 a)  

 ∂ y 

 {\displaystyle {{\ bangare (8b) \over \ bangare x}-{\ bangare 8a) \over\ bangare y}}} 

 , da kuma lura da cewa iska ta geostrophic ba ta bambanta ba (watau, 

 Sanya ⋅ 

 V 

 =
0 

 {\displaystyle {\nabla \cdot \mathbf {V} =0}} 

 ), ma'aunin vorticity shine

 D    g      ζ    g       D t      = -    f    0       (Abin da ya fi dacewa da shi)        ∂    u    a       ∂ x      +      ∂    v    a       ∂ y       )     - β    v    g        {\displaystyle {{D_{g}\zeta _{g} \over Dt}=-f_{0}\left ({{\partial u_{a} \over \partial x}+{\partial v_{a}} \over \partal y}}\right) -\beta v_{g}}}    (9)

Saboda haka 

 f 

 {\displaystyle {f}} 

 ya dogara ne kawai akan 

 da kuma 

 {\displaystyle {y}} 

  (watau, 

 D 

 g 

 f 

 D t 

 = 

 V 

 g 

 ⋅ ̆ f = β 

 v 

 g 

 {\displaystyle {{D_{g}f \over Dt}=\mathbf {V_{g}} \cdot \nabla f=\beta v_{g}}} 

 ) kuma cewa bambancin iska ta ageostrophic za a iya rubuta shi dangane da 

 ω 

 {\displaystyle {\omega }} 

 bisa ga daidaitattun ci gaba



daidaitattun (9) sabili da haka ana iya rubuta su kamar yadda

 ∂    ζ    g       ∂ t      =    -     V    g      ⋅ ̆ (     ζ    g     + f    )     +     f    0        ∂ ω     ∂ p          {\displaystyle {{\partial \zeta _{g} \over \partial t}={-\mathbf {V_{g}} \cdot \nabla ({\zeta _}+f}) }+{f_{0}{\partial\omega \over \partal p}}    (10)

 Wannan ainihi ta amfani da geopotential 
Bayyana yanayin geopotential 

 χ = 

 ∂ Φ 

 ∂ t 

 {\displaystyle {\chi ={\ bangare \Phi \over \ bangare t}}} 

 kuma lura da cewa za'a iya juyar da bambancin ra'ayi, za'a mën a sake rubuta ma'auni (10) dangane da 

 χ 

 {\displaystyle {\chi }} 

 kamar yadda

 1    f    0         Sanya    2     χ    =    -     V    g      Ta yaya     (Abin da ya fi dacewa da shi)       1    f    0         Sanya    2     Φ    + f    )      +     f    0        ∂ ω     ∂ p          {\displaystyle {{1 \over f_{0}}{\nabla ^{2}\chi }={-\mathbf {V_{g}} \cdot \nabla \left ({{1 \over F_{0}},{\nablà ^{2}}\Phi }+f}\right) }+{f_{0}{\partial \omega \over \partial p}}}}    (11)

Hagu na dama na daidaitattun (11) ya dogara da masu canji 

 Φ 

 {\displaystyle {\Phi }} 

 da kuma 

 ω 

 {\displaystyle {\omega }} 

 . Ana iya samun daidaitattun daidaitattun da suka dogara da waɗannan masu canji guda biyu daga daidaitattun makamashi na thermodynamic

  (Abin da ya fi dacewa da shi)       ∂    ∂ t      +      V    g      Ta yaya     )       (Abin da ya fi dacewa da shi)       − ∂ Φ     ∂ p      )      - σ ω    =      k J    p        {\displaystyle {{{\left ({{\partial \over \partial t}+{\mathbf {V_{g}} \cdot \nabla }}\right) \left ({-\partial\Phi \over \partal p}\right) }-\sigma \omega }={kJ \over p}}}    (12)

inda 

 σ = 

 - R 

 T 

 0 

 p 

 d log da kuma 

 Θ 

 0 

 d p 

 {\displaystyle {\sigma ={-RT_{0} \over p}{d\log \Theta _{0}\over dp}}} 

 da kuma 

 Θ 

 0 

 {\displaystyle {\Theta _{0}}} 

 shine yiwuwar zafin jiki wanda ya dace da yanayin zafin jiki na asali. A cikin tsakiyar sararin samaniya, 

 σ 

 {\displaystyle {\sigma }} 

 ≈ 

 2.5
× 

 10 

 −
6 

 m 

 2 

 P a 

 −
2 

 s 

 −
2 

 {\displaystyle {2.5\lokaci 10^{-6}\mathrm {m} {^{2}}\mathrrm {Pa} ^{-2}\mathm {s} ^{2}}} 

 .

Ƙara (12) ta hanyar 

 f 

 0 

 σ 

 {\displaystyle {f_{0} \over \sigma }} 

 da bambanci game da 

 p 

 {\displaystyle {p}} 

 da kuma amfani da ma'anar 

 χ 

 {\displaystyle {\chi }} 

 amfanin gona

 ∂    ∂ p        (Abin da ya fi dacewa da shi)        f    0     σ        ∂ χ     ∂ p       )      = -      ∂    ∂ p        (Abin da ya fi dacewa da shi)        f    0     σ        V    g      Ta yaya       ∂ Φ     ∂ p       )      -      f    0         ∂ ω     ∂ p       -      f    0        ∂    ∂ p        (Abin da ya fi dacewa da shi)       k J     σ p      )         {\displaystyle {{{\partial \over \partial p}\left ({{f_{0} \over \sigma }{\partiel \chi \over \partal p}}\right) }=-{{\partial\over \partiel \ma}}\left    (13)

Idan don sauƙi 

 J 

 {\displaystyle {J}} 

 an saita su zuwa 0, kawar da 

 ω 

 {\displaystyle {\omega }} 

 a cikin daidaitattun (11) da (13) suna samarwa 

  (Abin da ya fi dacewa da shi)      Sanya    2     +      ∂    ∂ p        (Abin da ya fi dacewa da shi)        f    0     2     σ       ∂    ∂ p       )       )      χ     = -      f    0         V    g      Ta yaya      (Abin da ya fi dacewa da shi)        1    f    0         Sanya    2     Φ     + f    )      -      ∂    ∂ p        (Abin da ya fi dacewa da shi)      -       f    0     2     σ        V    g      Ta yaya      (Abin da ya fi dacewa da shi)       ∂ Φ     ∂ p      )      )         {\displaystyle {{\nabla ^{2}+{{\partial \over \partial p}\left ({{f_{0}^{2} \over \sigma }{\partial\over \partiel\\right} \fi\right}\\\\fi_{0}}{\mathbf}\fi_\fi__{\fi}\fi} \fifi\fi} (\fi\fi\\fi}}\fifi}\\fififi} \\fi}    (14)

Daidaitawa (14) galibi ana kiranta da daidaitattun yanayin ƙasa. Yana danganta yanayin geopotential na gida (kalma A) zuwa rarrabawar advection na vorticity (kalma B) da kauri advection (kalma C).

 Irin wannan ainihi ta amfani da kusan-geostrophic yiwuwar vorticity 
Yin amfani da tsarin sarkar bambanci, ana iya rubuta kalmar C kamar haka

 -       V    g      Ta yaya      ∂    ∂ p        (Abin da ya fi dacewa da shi)        f    0     2     σ        ∂ Φ     ∂ p       )      -       f    0     2     σ        ∂     V    g        ∂ p       Ta yaya       ∂ Φ     ∂ p          {\displaystyle {-{{\mathbf {V_{g}} \cdot \nabla }{\partial \over \partial p}\left ({{f_{0}^{2} \over \sigma }{\partal \Phi \over \partal p}}\right) }-{{f_ {0}^ {2} \ Over \sigma}{\part \mathbf{V_{G}} \over \ Partial p}{\partiel \Phi    (15)

Amma bisa ga dangantakar iska mai zafi,

 f    0        ∂     V    g        ∂ p       =        k    ^      × Sanya     (Abin da ya fi dacewa da shi)       ∂ Φ     ∂ p      )         {\displaystyle {{f_{0}{\partial \mathbf {V_{g}} \over \partial p}}={{\hat {\mathbf{k} }}\times \nabla \left ({\partiel \Phi \over \partal p}\right) }}}    .

A wasu kalmomi, 

 ∂ 

 V 

 g 

 ∂ p 

 {\displaystyle {\partial \mathbf {V_{g}} \over \partial p}} 

 yana tsaye ne zuwa 

 Sanya ( 

 ∂ Φ 

 ∂ p 

 )  

 {\displaystyle {\nabla ({\partial \Phi \over \partial p}) }} 

 kuma lokaci na biyu a cikin daidaituwa (15) ya ɓace.

  (Abin da ya fi dacewa da shi)       ∂    ∂ t      +      V    g      Ta yaya     )     q    =       D    g     q     D t      = 0      {\displaystyle {{\hagu ({{\partial \over \partial t}+{\mathbf {V_{g}} \cdot \nabla }}\right) q}={D_{g}q \over Dt}=0}}    (16)

inda 

 q 

 {\displaystyle {q}} 

 shine kusan-geostrophic yiwuwar vorticity wanda aka bayyana ta

 q =       1    f    0         Sanya    2     Φ     +    f    +      ∂    ∂ p        (Abin da ya fi dacewa da shi)        f    0     σ        ∂ Φ     ∂ p       )          {\displaystyle {q={{{1 \over f_{0}}{\nabla ^{2}\Phi }}+{f}+{{\partial \over \partial p}\left ({{f_{0} \over \sigma }{\partial\Phi \over \partal p}}\right) }}}}    (17)

Kalmomi uku na daidaitattun (17) sune, daga hagu zuwa dama, Dangi na geostrophic, vorticity na duniya da kuma stretching vorticity.

 Sakamakon 
Yayin da iska ke motsawa a cikin yanayi, dangi, duniya da shimfida vorticities na iya canzawa amma daidaituwa (17) ya nuna cewa dole ne a kiyaye jimlar uku bayan motsi na geostrophic.

Ana iya amfani da daidaitattun (17) don samun 

 q 

 {\displaystyle {q}} 

 daga sanannen filin 

 Φ 

 {\displaystyle {\Phi }} 

 . A madadin haka, ana iya amfani da shi don hango hasashen juyin halitta na filin geopotential da aka ba da rarraba na farko 

 Φ 

 {\displaystyle {\Phi }} 

 da kuma yanayin iyaka mai dacewa ta amfani da tsarin juyawa.

Mafi mahimmanci, tsarin kusan-geostrophic yana rage daidaitattun daidaitattun ƙididdiga guda biyar zuwa tsarin daidaitattun guda ɗaya inda duk masu canji kamar 

 u 

 g 

 {\displaystyle {u_{g}}} 

 , 

 v 

 g 

 {\displaystyle {v_{g}}} 

 da kuma 

 T 

 {\displaystyle {T}} 

 ana iya samunsa daga 

 q 

 {\displaystyle {q}} 

 ko tsawo 

 Φ 

 {\displaystyle {\Phi }} 

 .

Har ila yau, saboda 

 ζ 

 g 

 {\displaystyle {\zeta _{g}}} 

 da kuma 

 V 

 g 

 {\displaystyle {\mathbf {V_{g}} }} 

 dukansu an bayyana su ne dangane da 

 Φ (x , y , p , t)  

 {\displaystyle {\Phi (x,y,p,t) }} 

 , ana iya amfani da ma'aunin vorticity don gano motsi na tsaye idan har filayen duka biyun 

 Φ 

 {\displaystyle {\Phi }} 

 da kuma 

 ∂ Φ 

 ∂ t 

 {\displaystyle {\partial \Phi \over \partial t}} 

 sanannu ne.

 Manazarta 

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