!
Revised Thursday, March 28, 2013!
1
Eddy Kinetic Energy and Zonal Kinetic Energy
David Randall
The derivations of the eddy kinetic energy, zonal kinetic energy, and zonally averaged
total kinetic energy equation follow methods similar to those used to derive the conservation
equation for the potential energy variance, and so will be omitted here for brevity.
We define the eddy kinetic energy per unit mass by
ΚE ≡
1
⎡( u *)2 + ( v *)2 ⎤ ,
⎦
2⎣
(1)
and the zonal kinetic energy by
ΚZ ≡
(
)
1
[u ]2 + [ v ]2 .
2
(2)
It follows that
[Κ ] = Κ Z + Κ E .
(3)
All three quantities in Eq. (98) are independent of longitude.
To derive equations that govern Κ E and Κ Z , we start from the zonal and meridional
equations of motion in flux form:
∂u
1
∂
1
∂
∂
uv tan ϕ
1 ∂φ
∂F
+
+ fv −
+g u ,
(uu ) +
( vu cosϕ ) + (ω u ) =
∂t a cosϕ ∂λ
a cosϕ ∂ϕ
∂p
a
a cosϕ ∂λ
∂p
(4)
∂v
1
∂
1
∂
∂
u tan ϕ
1 ∂φ
∂F
+
− fu −
+g v .
(uv ) +
( vv cosϕ ) + (ω v ) = −
∂t a cosϕ ∂λ
a cosϕ ∂ϕ
∂p
a
a ∂ϕ
∂p
2
(5)
We also need mass continuity:
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Copyright David Randall, 2013
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Revised Thursday, March 28, 2013!
2
1 ∂u
1
∂
∂ω
+
( v cos ϕ ) + = 0 .
a cos ϕ ∂λ a cos ϕ ∂ϕ
∂p
(6)
Zonal averaging (4) - (6) gives
∂[ u ]
1
∂
+
∂t
a cosϕ ∂ϕ
{([v][u ] + ⎡⎣v u ⎤⎦) cosϕ } + ∂∂p ([ω ][u ] + ⎡⎣ω u ⎤⎦)
* *
* *
([ v][u ] + ⎡⎣v u ⎤⎦) tanϕ + f [ v] + g ∂[ F ] ,
=
* *
u
∂p
a
∂[ v ]
1
∂
+
∂t
a cosϕ ∂ϕ
(7)
{([v][v] + ⎡⎣v v ⎤⎦) cosϕ } + ∂∂p ([ω ][v] + ⎡⎣ω v ⎤⎦)
* *
* *
([u ][u ] + ⎡⎣u u ⎤⎦) tanϕ − f [u ] − 1 ∂[φ ] + g ∂[ F ] .
=−
* *
v
a ∂ϕ
a
∂p
(8)
and
1
∂
∂
[ v ] cos ϕ ) + [ω ] = 0
(
a cos ϕ ∂ϕ
∂p
(9)
Use the zonally averaged continuity equation, (9), to convert (7) and (8) to advective
form:
∂[ u ] [ v ] ∂[ u ]
∂[ u ]
1
∂
∂
⎡⎣ v*u * ⎤⎦ cosϕ +
⎡ω *u * ⎤⎦
+
+ [ω ]
+
∂t
a ∂ϕ
∂ p a cosϕ ∂ϕ
∂p ⎣
(
)
([ v][u ] + ⎡⎣v u ⎤⎦) tanϕ + f [ v] + g ∂[ F ] ,
=
* *
u
∂p
a
∂[ v ] [ v ] ∂[ v ]
∂[ v ]
1
∂
∂
⎡⎣ v*v* ⎤⎦ cosϕ +
⎡⎣ω *v* ⎤⎦
+
+ [ω ]
+
∂t
a ∂ϕ
∂ p a cosϕ ∂ϕ
∂p
(
(10)
)
([u ][u ] + ⎡⎣u u ⎤⎦) tanϕ − f [u ] − 1 ∂[φ ] + g ∂[ F ] .
=−
* *
v
a
a ∂ϕ
∂p
Next, multiply (10) by [ u ] , and (11) by [ v ] , and add the results, to obtain
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3
∂
[ v ] ∂ KZ + [ω ] ∂ KZ
KZ +
∂t
a ∂ϕ
∂p
+
[u ]
∂
[ v ] ∂ ⎡ v*v* ⎤ cosϕ + [u ] ∂ ⎡ω *u * ⎤ + [ v ] ∂ ⎡ω *v* ⎤
⎡⎣ v*u * ⎤⎦ cosϕ +
⎦
⎦
⎦
a cosϕ ∂ϕ
a cosϕ ∂ϕ ⎣
∂p ⎣
∂p ⎣
(
)
(
)
[u ]( [ v ][u ] + ⎡⎣ v*u * ⎤⎦ ) tanϕ [ v ]( [u ][u ] + ⎡⎣u *u * ⎤⎦ ) tanϕ [ v ] ∂[φ ]
∂[ Fu ]
∂[ Fv ]
=
−
−
+ [u ] g
+ [v] g
.
a
a ∂ϕ
a
∂p
∂p
(12)
Use (9) to go back to flux form, and cancel the indicated terms to obtain
∂
1
∂
∂
KZ +
v ] KZ cosϕ ) +
[
(
([ω ] KZ )
∂t
a cosϕ ∂ϕ
∂p
+
[u ]
∂
[ v ] ∂ ⎡ v*v* ⎤ cosϕ + [u ] ∂ ⎡ω *u * ⎤ + [ v ] ∂ ⎡ω *v* ⎤
⎡⎣ v*u * ⎤⎦ cosϕ +
⎦
⎦
⎦
a cosϕ ∂ϕ
a cosϕ ∂ϕ ⎣
∂p ⎣
∂p ⎣
(
)
(
= [ u ] ⎡⎣ v*u * ⎤⎦ − [ v ] ⎡⎣u *u * ⎤⎦
(
)
) tanaϕ − [av] ∂∂[ϕφ ] + [u ] g ∂[∂Fp ] + [ v] g ∂[∂Fp ] .
u
v
(13)
The terms on the second line of (13) can be manipulated as follows:
[u ]
∂
[ v ] ∂ ⎡ v*v* ⎤ cosϕ + [u ] ∂ ⎡ω *u * ⎤ + [ v ] ∂ ⎡ω *v* ⎤
⎡⎣ v*u * ⎤⎦ cosϕ +
⎦
⎦
⎦
a cosϕ ∂ϕ
a cosϕ ∂ϕ ⎣
∂p ⎣
∂p ⎣
(
)
(
)
⎡⎣ v*u * ⎤⎦ ∂[ u ] ⎡⎣ v*v* ⎤⎦ ∂[ v ]
1
∂
* *
* *
=
[u ] ⎡⎣ v u ⎤⎦ + [ v ] ⎡⎣ v v ⎤⎦ cosϕ − a ∂ϕ − a ∂ϕ
a cosϕ ∂ϕ
∂
∂
∂
+
u ] ⎡⎣ω *u * ⎤⎦ + [ v ] ⎡⎣ω *v* ⎤⎦ − ⎡⎣ω *u * ⎤⎦ [ u ] − ⎡⎣ω *v* ⎤⎦ [ v ] .
[
∂p
∂p
∂p
{(
(
)
}
)
(14)
Finally, we write
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[ v ] ∂[φ ] = [ v ] cosϕ ∂[φ ]
a ∂ϕ
a cosϕ ∂ϕ
=
=
1
∂
[φ ] ∂ ([ v ] cosϕ )
v ][φ ] cosϕ ) −
[
(
a cosϕ ∂ϕ
a cosϕ ∂ϕ
∂[ω ]
1
∂
v ][φ ] cosϕ ) + [φ ]
[
(
a cosϕ ∂ϕ
∂p
1
a cosϕ
1
=
a cosϕ
=
∂[φ ]
∂
∂
v ][φ ] cosϕ ) +
ω ][φ ]) − [ω ]
[
[
(
(
∂ϕ
∂p
∂p
∂
∂
v ][φ ] cosϕ ) +
[
(
([ω ][φ ]) + [ω ][α ] .
∂ϕ
∂p
(15)
Substituting back into (30), and combining terms, we find that
∂
KZ
∂t
1
∂
+
[ v ] KZ + [u ] ⎡⎣ v*u * ⎤⎦ + [ v ] ⎡⎣ v*v* ⎤⎦ + [ v ][φ ] cosϕ
a cosϕ ∂ϕ
∂
+
ω ] KZ + [ u ] ⎡⎣ω *u * ⎤⎦ + [ v ] ⎡⎣ω *v* ⎤⎦ + [ω ][φ ]
[
∂p
{(
)
(
}
)
⎡⎣ v*u * ⎤⎦ ∂[ u ] ⎡⎣ v*v* ⎤⎦ ∂[ v ]
∂[ u ]
∂[ v ]
=
+
+ ⎡⎣ω *u * ⎤⎦
+ ⎡⎣ω *v* ⎤⎦
a
∂ϕ
a
∂ϕ
∂p
∂p
tan ϕ
+ [ u ] ⎡⎣ v*u * ⎤⎦ − [ v ] ⎡⎣u *u * ⎤⎦
a
− [ω ][α ]
(
+ [u ] g
)
∂[ Fu ]
∂[ F ]
+ [v] v .
∂p
∂p
(16)
This is the zonal kinetic energy equation.
To derive the eddy kinetic energy equation, we return to the equations of motion. Use the
continuity equation, (6), to convert (4) - (5) to advective form:
∂u
u ∂u v ∂u
∂u uv tan ϕ
1 ∂φ
∂F
+
+
+ω
=
+ fv −
+g u ,
∂t a cosϕ ∂λ a ∂ϕ
∂p
a
a cosϕ ∂λ
∂p
(17)
∂v
u ∂v v ∂v
∂v
u tan ϕ
1 ∂φ
∂F
+
+
+ω
=−
− fu −
+g v .
∂t a cosϕ ∂λ a ∂ϕ
∂p
a
a ∂ϕ
∂p
2
(18)
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Subtract (10) and (11) from (17) and (18), respectively, to obtain:
∂[ u ] ⎞
∂u *
u ∂u * ⎛ v ∂u [ v ] ∂[ u ] ⎞ ⎛ ∂u
+
+⎜
−
+ ⎜ω
− [ω ]
⎟
∂t a cosϕ ∂λ ⎝ a ∂ϕ a ∂ϕ ⎠ ⎝ ∂ p
∂ p ⎟⎠
⎧ 1
⎫
∂
∂
⎡⎣ v*u * ⎤⎦ cosϕ +
⎡⎣ω *u * ⎤⎦ ⎬
−⎨
∂p
⎩ a cosϕ ∂ϕ
⎭
(
)
(
)
* *
uv tan ϕ [ v ][ u ] + ⎡⎣ v u ⎤⎦ tan ϕ
1 ∂φ *
∂F *
=
−
+ fv* −
+g u ,
a
a
a cosϕ ∂λ
∂p
(19)
∂[ v ] ⎞
∂v*
u ∂v* ⎛ v ∂v [ v ] ∂[ v ] ⎞ ⎛ ∂v
+
+⎜
−
+ ⎜ω
− [ω ]
⎟
∂t a cosϕ ∂λ ⎝ a ∂ϕ a ∂ϕ ⎠ ⎝ ∂ p
∂ p ⎟⎠
⎧ 1
⎫
∂
∂
⎡⎣ v*v* ⎤⎦ cosϕ +
⎡⎣ω *v* ⎤⎦ ⎬
−⎨
∂p
⎩ a cosϕ ∂ϕ
⎭
=−
u2
(
)
tan ϕ ([ u ][ u ] + ⎡⎣u u ⎤⎦ ) tan ϕ
+
− fu
* *
a
a
*
1 ∂φ *
∂Fv*
−
+g
.
a ∂ϕ
∂p
(20)
Expand the nonlinear terms:
*
*
∂u * ⎛ [ u ] + u * ⎞ ∂u * ([ v ] + v ) ∂([ u ] + u )
∂
+⎜
+
+ ([ω ] + ω * ) ([ u ] + u * )
⎟
∂t ⎝ a cosϕ ⎠ ∂λ
a
∂ϕ
∂p
⎛ [ v ] ∂[ u ]
⎫
∂[ u ] ⎞ ⎧ 1
∂
∂
⎡⎣ v*u * ⎤⎦ cosϕ +
⎡⎣ω *u * ⎤⎦ ⎬
−⎜
+ [ω ]
−⎨
⎟
⎝ a ∂ϕ
∂ p ⎠ ⎩ a cosϕ ∂ϕ
∂p
⎭
(
=
)
{([u ] + u )([v] + v ) − ([v][u ] + ⎡⎣v u ⎤⎦)} tanaϕ + fv
*
*
* *
*
−
1 ∂φ *
∂F *
+g u ,
a cosϕ ∂λ
∂p
∂v* ([ u ] + u ) ∂v* ([ v ] + v ) ∂
∂
+
+
v ] + v* ) + ([ω ] + ω * ) ([ v ] + v* )
[
(
∂t
a cosϕ ∂λ
a
∂ϕ
∂p
*
(21)
*
⎛ [ v ] ∂[ v ]
⎫
∂[ v ] ⎞ ⎧ 1
∂
∂
⎡⎣ v*v* ⎤⎦ cosϕ +
⎡⎣ω *v* ⎤⎦ ⎬
−⎜
+ [ω ]
−⎨
⎟
⎝ a ∂ϕ
∂ p ⎠ ⎩ a cosϕ ∂ϕ
∂p
⎭
(
{
(
= − ([ u ] + u * ) + [ u ][ u ] + ⎡⎣u *u * ⎤⎦
2
)}
)
tan ϕ
1 ∂φ *
∂F *
− fu * −
+g v .
a
a ∂ϕ
∂p
(22)
Rearrange and simplify:
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6
⎛ ∂
∂ ⎞ *
[u ] ∂ [ v ] ∂
⎜⎝ ∂t + a cosϕ ∂λ + a ∂ϕ + [ω ] ∂ p ⎟⎠ u
⎛ u*
∂[ u ]
∂ v* ∂
∂ ⎞
v* ∂[ u ]
+⎜
+
+ ω * ⎟ u* +
+ω*
∂p⎠
a ∂ϕ
∂p
⎝ a cosϕ ∂λ a ∂ϕ
⎧ 1
⎫
∂
∂
⎡⎣ v*u * ⎤⎦ cosϕ +
⎡⎣ω *u * ⎤⎦ ⎬
−⎨
∂p
⎩ a cosϕ ∂ϕ
⎭
tan ϕ
1 ∂φ *
∂F *
= ([ u ] v* + u * [ v ] + v*u * ) − ⎡⎣ v*u * ⎤⎦
+ fv* −
+g u ,
a
a cosϕ ∂λ
∂p
(
)
{
}
(23)
⎛ ∂
∂ ⎞ *
[u ] ∂ [ v ] ∂
⎜⎝ ∂t + a cosϕ ∂λ + a ∂ϕ + [ω ] ∂ p ⎟⎠ v
⎛ u*
∂[ v ]
∂ v* ∂
∂ ⎞
v* ∂[ v ]
+⎜
+
+ ω * ⎟ v* +
+ω*
∂p⎠
a ∂ϕ
∂p
⎝ a cosϕ ∂λ a ∂ϕ
⎧ 1
⎫
∂
∂
⎡⎣ v*v* ⎤⎦ cosϕ +
⎡⎣ω *v* ⎤⎦ ⎬
−⎨
∂p
⎩ a cosϕ ∂ϕ
⎭
tan ϕ
1 ∂φ *
∂F *
= − ( 2 [ u ] u * + u *u * ) + ⎡⎣u *u * ⎤⎦
− fu * −
+g v .
a
a ∂ϕ
∂p
(
)
{
}
(24)
Now multiply (23) by u , and (24) by v :
*
*
⎛ ∂
[u ] ∂ + [ v ] ∂ + [ω ] ∂ ⎞ u *2
+
⎜⎝ ∂t a cosϕ ∂λ a ∂ϕ
∂ p ⎟⎠ 2
⎛ u*
∂ v* ∂
∂ ⎞ u *2 u *v* ∂[ u ] * * ∂[ u ]
+⎜
+
+ω* ⎟
+
+u ω
∂p⎠ 2
a ∂ϕ
∂p
⎝ a cosϕ ∂λ a ∂ϕ
⎧ 1
⎫
∂
∂
⎡⎣ v*u * ⎤⎦ cosϕ +
⎡⎣ω *u * ⎤⎦ ⎬
−u * ⎨
∂p
⎩ a cosϕ ∂ϕ
⎭
tan ϕ
u * ∂φ *
∂F *
= u * ([ u ] v* + u * [ v ] + v*u * ) − ⎡⎣ v*u * ⎤⎦
+ fu *v* −
+ u *g u ,
a
a cosϕ ∂λ
∂p
(
{
)
}
(25)
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⎛ ∂
[u ] ∂ + [ v ] ∂ + [ω ] ∂ ⎞ v*2
+
⎜⎝ ∂t a cosϕ ∂λ a ∂ϕ
∂ p ⎟⎠ 2
⎛ u*
∂ v* ∂
∂ ⎞ v*2 v*v* ∂[ v ] * * ∂[ v ]
+⎜
+
+ω* ⎟
+
+v ω
∂p⎠ 2
a ∂ϕ
∂p
⎝ a cosϕ ∂λ a ∂ϕ
⎧ 1
⎫
∂
∂
⎡⎣ v*v* ⎤⎦ cosϕ +
⎡⎣ω *v* ⎤⎦ ⎬
−v* ⎨
∂p
⎩ a cosϕ ∂ϕ
⎭
tan ϕ
v* ∂φ *
∂F *
= v* − ( 2 [ u ] u * + u *u * ) + ⎡⎣u *u * ⎤⎦
− fv*u * −
+ v* g v .
a
a ∂ϕ
∂p
(
)
{
}
(26)
Next, use the zonal mean continuity equation, (4), to convert to flux form the terms of (25) and
(26) that represent advection by the zonally averaged winds, and use the eddy continuity
equation,
1 ∂u*
1
∂
∂ω
+
( v* cos ϕ ) + * = 0 ,
a cos ϕ ∂λ a cos ϕ ∂ϕ
∂p
(27)
to similarly rewrite the terms that represent advection by the eddy winds. Taking the zonal means
of the results, we get
{([v] ⎡⎣u
1 ∂ *2
1 1
∂
⎡⎣u ⎤⎦ +
2 ∂t
2 a cosϕ ∂ϕ
*2
}
1 ∂
⎤⎦ + ⎡⎣ v*u *2 ⎤⎦ cosϕ +
[ω ] ⎡⎣u *2 ⎤⎦ + ⎡⎣ω *u *2 ⎤⎦
2 ∂p
)
(
)
⎡⎣u *v* ⎤⎦ ∂[ u ]
∂[ u ]
=−
− ⎡⎣u *ω * ⎤⎦
a
∂ϕ
∂p
+
u
∂φ ⎤ ⎡
∂F ⎤
+ ⎢u g
,
{([u ] ⎡⎣u v ⎤⎦ + ⎡⎣u u ⎤⎦[v] + ⎡⎣u v u ⎤⎦)} tanaϕ + f ⎡⎣u v ⎤⎦ − ⎡⎢⎣ a cos
⎥
ϕ ∂λ ⎦ ⎣
∂ p ⎥⎦
*
* *
* *
* * *
*
* *
*
*
u
(28)
1 ∂ ⎡⎣ v ⎤⎦ 1 1
∂
+
2 ∂t
2 a cosϕ ∂ϕ
*2
{([v] ⎡⎣v
*2
}
1 ∂
⎤⎦ + ⎡⎣ v*v*v* ⎤⎦ cosϕ +
[ω ] ⎡⎣ v*2 ⎤⎦ + ⎡⎣ω *v*2 ⎤⎦
2 ∂p
)
(
)
⎡⎣ v*v* ⎤⎦ ∂[ v ]
∂[ v ]
=−
− ⎡⎣ v*ω * ⎤⎦
a
∂ϕ
∂p
(
− 2 [ u ] ⎡⎣u *v* ⎤⎦ + ⎡⎣u *u *v* ⎤⎦
) tanaϕ − f ⎡⎣v u ⎤⎦ − ⎡⎢⎣ va ∂∂φϕ ⎤⎥⎦ + ⎡⎢⎣v g ∂F∂ p ⎤⎥⎦ .
*
* *
*
*
*
v
(29)
Now add (28) and (29) to obtain:
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Revised Thursday, March 28, 2013!
8
∂
1
∂ ⎧⎛
1 * *2 *2 ⎞
1 * *2 *2 ⎞
⎫ ∂ ⎛
KE +
⎨⎜⎝ [ v ] KE + ⎡⎣ v ( u + v ) ⎤⎦⎟⎠ cosϕ ⎬ +
⎜⎝ [ω ] KE + ⎡⎣ω ( u + v ) ⎤⎦⎟⎠
∂t
a cosϕ ∂ϕ ⎩
2
2
⎭ ∂p
⎡⎣u *v* ⎤⎦ ∂[ u ] ⎡⎣ v*v* ⎤⎦ ∂[ v ]
∂[ u ]
∂[ v ]
=−
−
− ⎡⎣u *ω * ⎤⎦
− ⎡⎣ v*ω * ⎤⎦
a
∂ϕ
a
∂ϕ
∂p
∂p
tan ϕ ⎡ u * ∂φ * ⎤ ⎡ v* ∂φ * ⎤ ⎡ * ∂Fu * ⎤ ⎡ * ∂Fv* ⎤
+ − [ u ] ⎡⎣u v ⎤⎦ + ⎡⎣u u ⎤⎦ [ v ]
−⎢
⎥−⎢
⎥ + ⎢u g ∂ p ⎥ + ⎢ v g ∂ p ⎥ .
a
⎣ a cosϕ ∂λ ⎦ ⎣ a ∂ϕ ⎦ ⎣
⎦ ⎣
⎦
(
* *
* *
)
(30)
Finally, by analogy with (15), the pressure-gradient terms of (30) can be rewritten as
⎡ u * ∂φ * ⎤ ⎡ v* ∂φ * ⎤
1
∂
∂
* *
* *
* *
⎢ a cosϕ ∂λ ⎥ + ⎢ a ∂ϕ ⎥ = a cosϕ ∂ϕ ⎡⎣ v φ ⎤⎦ cosϕ + ∂ p ⎡⎣ω φ ⎤⎦ + ⎡⎣ω α ⎤⎦ .
⎣
⎦ ⎣
⎦
(
)
(31)
Substitution of (31) into (30) gives
∂
KE
∂t
1
∂ ⎧⎛
1 * *2 *2
⎫
* * ⎞
+
⎨⎜⎝ [ v ] KE + ⎡⎣ v ( u + v ) ⎤⎦ + ⎡⎣ v φ ⎤⎦⎟⎠ cosϕ ⎬
a cosϕ ∂ϕ ⎩
2
⎭
∂ ⎛
1 * *2 *2
* * ⎞
+
⎜⎝ [ω ] KE + ⎡⎣ω ( u + v ) ⎤⎦ + ⎡⎣ω φ ⎤⎦⎟⎠
∂p
2
⎡⎣u *v* ⎤⎦ ∂[ u ] ⎡⎣ v*v* ⎤⎦ ∂[ v ]
∂[ u ]
∂[ v ]
=−
−
− ⎡⎣u *ω * ⎤⎦
− ⎡⎣ v*ω * ⎤⎦
a
∂ϕ
a
∂ϕ
∂p
∂p
tan ϕ
+ − [ u ] ⎡⎣u *v* ⎤⎦ + ⎡⎣u *u * ⎤⎦ [ v ]
a
* *
− ⎡⎣ω α ⎤⎦
(
)
⎡
∂F * ⎤ ⎡
∂F * ⎤
+ ⎢ u * g u ⎥ + ⎢ v* g v ⎥ .
∂p ⎦ ⎣
∂p ⎦
⎣
(32)
All of the transport terms, including the pressure-work terms, are combined on the second and
third lines of (32). The terms on the fourth line represent gradient production, i.e., the conversion
between the kinetic energy of the mean flow and that of the eddies. This conversion increases the
eddy kinetic energy when the eddy momentum flux is “down the gradient,” i.e., when it is from
higher mean momentum to lower mean momentum. The ⎡⎣ω *α * ⎤⎦ term represents conversion
between eddy kinetic energy and eddy available potential energy.
The (possibly surprising) metric terms in the equations for KZ and KE arise because we
have defined “eddies” in terms of departures from the zonal mean, so that a particular latitudeQuickStudies in Atmospheric Science
Copyright David Randall, 2013
!
Revised Thursday, March 28, 2013!
9
longitude coordinate system is implicit in the very definition of Κ E . The metric terms cancel
when we add the equations for Κ Z and Κ E to obtain the equation for the zonally averaged total
kinetic energy, [ Κ ] :
∂
[Κ ]
∂t
1
∂ ⎧⎛
1 * *2 *2
⎫
* *
* *
* * ⎞
+
⎨⎜⎝ [ v ][ Κ ] + ⎡⎣ v ( u + v ) ⎤⎦ + [ u ] ⎡⎣ v u ⎤⎦ + [ v ] ⎡⎣ v v ⎤⎦ + [ v ][φ ] + ⎡⎣ v φ ⎤⎦⎟⎠ cosϕ ⎬
a cosϕ ∂ϕ ⎩
2
⎭
∂ ⎛
1
⎞
+ ⎜ [ω ][ Κ ] + ⎡⎣ω * ( u *2 + v*2 ) ⎤⎦ + [ u ] ⎡⎣ω *u * ⎤⎦ + [ v ] ⎡⎣ω *v* ⎤⎦ + [φ ][ω ] + ⎡⎣ω *φ * ⎤⎦⎟
⎝
⎠
∂p
2
= − [ω ][α ] − ⎡⎣ω *α * ⎤⎦
∂[ Fu ]
∂[ F ]
⎡
∂F * ⎤ ⎡
∂F * ⎤
+ ⎢ u * g u ⎥ + ⎢ v* g v ⎥ + [ u ] g
+ [v] v .
∂p ⎦ ⎣
∂p ⎦
∂p
∂p
⎣
(33)
QuickStudies in Atmospheric Science
Copyright David Randall, 2013