ロスビー波によるエネルギーフラックスのシームレス全球解析に向けて

…" " Rossby " (Hidenori"Aiki) JAMSTEC8APL" A"" / / "(H278H31:"15H02129)" A A0509 2015 6 9 ( ) 6 10 " " " 20 " •  " •  " " " •  β β β " •  " Chang"and"Orlanski"(1994)" •  " •  β β β " " •  Impulse8bolus " " "

! Aiki,!H.,!K.!Takaya!and!R.!J.!Greatbatch,!2015:! ! A!divergence>form!wave>induced!pressure!inherent!in!the!extension!of! the!Eliassen>Palm!theory!to!a!three>dimensional!framework! for!all!waves!at!all!laJtudes,! ! Journal!of!the!Atmospheric!Sciences,!72,!2822>2849! " " hPp://dx.doi.org/10.1175/JAS8D81480172.1 hPp://www.jamstec.go.jp/res/ress/aiki/"

•  •  •  •  •  • 

A draft manuscript in preparation for J. Fluid Mech. Rapid 1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A† = A − A ∇yz = ⟨⟨∂y, ∂z⟩⟩ Q† = ψxx + ψyy + (ψzf₀2/N2)z (∂t + u∂x)[ψxx + ψyy + (ψzf₀2/N2)z ! "# $ Q† ] + ψx(β − uyy) = 0 Q†ψx = −∇yz · ⟨⟨−ψxψy ! "# $ vg_ug ,_−ψzψxf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂t[(1/2)Q†2/(uyy − β) ! "# $ E/Cx p ] + ∇yz · ⟨⟨ −ψxψy ! "# $ Cgy(E/Cpx) ,_−ψzψxf₀2/N2 ! "# $ Cz g(E/Cpx) ⟩⟩ = 0 ∂tu − f0v∗ = −∇yz · ⟨⟨−ψxψy ! "# $ vg_ug ,_−ψzψxf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂t[u + (1/2)Q†2/(β − uyy) ! "# $ −E/Cx p ] − f0v∗ = 0 (0.1) A′ = A − A (0.2) ∇ = ⟨⟨∂x, ∂y, ∂z⟩⟩ (0.3)

⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨u}g, vg, 0_{⟩⟩ + ⟨⟨u}a, va, wa_⟩⟩ (0.4)

ug = −ψy, vg = ψx, ψ = p′/f0, (0.5) ζ′ = −ρ′/ρ_z = (g/ρ0)ρ′/N2 = −ψzf0/N2 (0.6) Q = ψxx + ψyy + (ψzf₀2/N2)z (0.7) ∂t[ψxx + ψyy + (ψzf₀2/N2)z ! "# $ Q ] + βψ_x = 0 (0.8) ψx = −Qt/β (0.9)

† Email address for correspondence: [email protected]

A draft manuscript in preparation for J. Fluid Mech. Rapid 1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A† = A − A ∇yz = ⟨⟨∂y, ∂z⟩⟩ Q† = ψxx + ψyy + (ψzf₀2/N2)z (∂t + u∂x)[ψxx + ψyy + (ψzf₀2/N2)z ! "# $ Q† ] + ψx(β − uyy) = 0 Q†_ψx = −∇_{yz · ⟨⟨−ψxψy} ! "# $ vg_ug ,_−ψzψ_xf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂t[(1/2)Q†2/(uyy − β) ! "# $ E/Cx p ] + ∇yz · ⟨⟨ −ψxψy ! "# $ Cgy(E/C_px) ,_−ψzψxf₀2/N2 ! "# $ Cz g(E/Cpx) ⟩⟩ = 0 ∂tu − f0v∗ = −∇yz · ⟨⟨−ψxψy ! "# $ vg_ug ,_−ψzψxf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂_t[u + (1/2)Q†2_{/(β − uyy}) ! "# $ −E/Cx p ] − f0v∗ = 0 (0.1) A′ = A − A (0.2) ∇ = ⟨⟨∂x, ∂y, ∂z⟩⟩ (0.3)

⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨u}g, vg, 0_{⟩⟩ + ⟨⟨u}a, va, wa_⟩⟩ (0.4)

ug = −ψy, vg = ψx, ψ = p′/f0, (0.5) ζ′ = −ρ′/ρ_z = (g/ρ0)ρ′/N2 = −ψzf0/N2 (0.6) Q = ψxx + ψyy + (ψzf₀2/N2)z (0.7) ∂_t[ψ_xx + ψ_yy + (ψ_zf₀2/N2)_z ! "# $ Q ] + βψx = 0 (0.8) ψx = −Qt/β (0.9)

† Email address for correspondence: [email protected]

A draft manuscript in preparation for J. Fluid Mech. Rapid 1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A† = A − A ∇yz = ⟨⟨∂y, ∂z⟩⟩ Q† = ψ_xx + ψ_yy + (ψ_zf₀2/N2)_z (∂t + u∂x)[ψxx + ψyy + (ψzf₀2/N2)z ! "# $ Q† ] + ψx(β − uyy) = 0 Q†ψx = −∇yz · ⟨⟨−ψxψy ! "# $ vg_ug ,_−ψzψxf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂_t[(1/2)Q†2_{/(uyy − β)} ! "# $ E/Cx p ] + ∇yz · ⟨⟨ −ψxψy ! "# $ Cgy(E/C_px) ,_−ψzψ_xf₀2/N2 ! "# $ Cz g(E/Cpx) ⟩⟩ = 0 ∂_tu _{− f0}v∗ = −∇_{yz · ⟨⟨−ψx}ψ_y ! "# $ vg_ug ,_−ψzψ_xf₀2/N2 ! "# $ −f0vgρ†/ρz ⟩⟩ ∂t[u + (1/2)Q†2/(β − uyy) ! "# $ −E/Cx p ] − f0v∗ = 0 (0.1) A′ = A − A (0.2) ∇ = ⟨⟨∂x, ∂y, ∂z⟩⟩ (0.3)

⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨u}g, vg, 0_{⟩⟩ + ⟨⟨u}a, va, wa_⟩⟩ (0.4)

ug = −ψy, vg = ψx, ψ = p′/f0, (0.5) ζ′ = −ρ′/ρ_z = (g/ρ0)ρ′/N2 = −ψzf0/N2 (0.6) Q = ψxx + ψyy + (ψzf₀2/N2)z (0.7) ∂t[ψxx + ψyy + (ψzf₀2/N2)z ! "# $ Q ] + βψx = 0 (0.8) ψ_x = −Q_t/β (0.9)

† Email address for correspondence: [email protected]

A draft manuscript in preparation for J. Fluid Mech. Rapid 1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A† = A − A ∇yz = ⟨⟨∂y, ∂z⟩⟩ Q† = ψxx + ψyy + (ψzf₀2/N2)z (∂t + u∂x)[ψxx + ψyy + (ψzf₀2/N2)z ! "# $ Q† ] + ψx(β − uyy) = 0 Q†_ψx = −∇_{yz · ⟨⟨−ψxψy} ! "# $ vg_ug ,_−ψzψ_xf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂t[(1/2)Q†2/(uyy − β) ! "# $ E/Cx p ] + ∇yz · ⟨⟨ −ψxψy ! "# $ Cgy(E/C_px) ,_−ψzψxf₀2/N2 ! "# $ Cz g(E/Cpx) ⟩⟩ = 0 ∂tu − f0v∗ = −∇yz · ⟨⟨−ψxψy ! "# $ vg_ug ,_−ψzψxf₀2/N2 ! "# $ −f0vgρ†/ρ_z ⟩⟩ ∂_t[u + (1/2)Q†2_{/(β − uyy}) ! "# $ −E/Cx p ] − f0v∗ = 0 (0.1) A′ = A − A (0.2) ∇ = ⟨⟨∂x, ∂y, ∂z⟩⟩ (0.3)

⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨u}g, vg, 0_{⟩⟩ + ⟨⟨u}a, va, wa_⟩⟩ (0.4)

ug = −ψ_y, vg = ψ_x, ψ = p′/f₀, (0.5) ζ′ = −ρ′/ρ_z = (g/ρ0)ρ′/N2 = −ψzf0/N2 (0.6) Q = ψxx + ψyy + (ψzf₀2/N2)z (0.7) ∂_t[ψ_xx + ψ_yy + (ψ_zf₀2/N2)_z ! "# $ Q ] + βψx = 0 (0.8) ψ_x = −Q_t/β (0.9)

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3 u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′, −u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f₀ + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w′_z = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f0} + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )t + ∇ · ⟨⟨u′p′, v′p′, w′p′⟩⟩ = 0, cp ≡ ω/k, cg ≡ ∂ω/∂k (E/cp)t + ∇ · ⟨⟨u′p′/cp, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/cp)t + ∇ · ⟨⟨u′p′/cp, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞

−∞ cg(E/cp) dy, u′p′/cp ̸= cg(E/cp)

%+_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

A draft manuscript in preparation for J. Fluid Mech. Rapid

1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A

′

= A − A

(0.1)

∇ = ⟨⟨∂

x

, ∂

y

, ∂

z

⟩⟩

(0.2)

⟨⟨u

′

, v

′

, w

′

_{⟩⟩ = ⟨⟨u}

g

, v

g

, 0

_{⟩⟩ + ⟨⟨u}

a

, v

a

, w

a

_⟩⟩

(0.3)

u

g

= −ψ

y

,

v

g

= ψ

x

,

ψ = p

′

/f

0

,

(0.4)

ζ

′

= −ρ

′

/ρ

_z

= (g/ρ

0

)ρ

′

/N

2

= −ψ

z

f

0

/N

2

(0.5)

Q = v

_x′

_{− u}

′_y

_{− f}

0

ζ

_z′

= ψ

xx

+ ψ

yy

+ (ψ

z

f

₀2

/N

2

)

z

(0.6)

∂

t

[ψ

xx

+ ψ

yy

+ (ψ

z

f

₀2

/N

2

)

z

!

"#

$

Q

] + βψ

x

= 0

(0.7)

ψ

x

= −Q

t

/β

(0.8)

∂

t

[

E

#

$!

"

(ψ

2 x

+ ψ

y2

)/2 + ψ

z2

f

02

/(2N

2

)]+

∇ · ⟨⟨−ψ

xt

ψ

− βψ

2

/2

!

"#

$

Cx gE

,

_−ψ

yt

ψ

! "# $

CgyE

,

_−ψ

zt

ψf

₀2

/N

2

!

"#

$

Cz gE

⟩⟩ = 0

(0.9)

∂

t

[

E

#

$!

"

(ψ

2 x

+ ψ

y2

)/2 + ψ

z2

f

02

/(2N

2

)]+

∇ · ⟨⟨u

′

p

′

+ [(f

0

− βy)ψ

2

/2]

y

!

"#

$

Cx gE

, v

′

p

′

_{− [(f}

0

− βy)ψ

2

/2]

x

!

"#

$

CgyE

, w

′

p

′

!"#$

Cz gE

⟩⟩

= 0

(0.10)

Qψ

x

= −∇ · ⟨⟨−ψ

x

ψ

x

+ K + G

!

"#

$

ug_ug_−K+G

,

_−ψ

x

ψ

y

! "# $

vg_ug

,

_−ψ

z

ψ

x

f

₀2

/N

2

!

"#

$

ζ′_p′ x

⟩⟩

(0.11)

K

_{≡ (ψ}

_x2

+ ψ

_y2

)/2, G ≡ ψ

_z2

f

₀2

/(2N

2

)

(0.12)

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3 u′_{t − (f}0 + βy)v′ = −p′x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )t + ∇ · ⟨⟨u′p′, v′p′, w′p′⟩⟩ = 0, cp ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, %+_∞ −∞ u′p′/cp dy = %+_∞

−∞ cg(E/cp) dy, u′p′/cp ̸= cg(E/cp)

%+∞ −∞ u′p′/cp dy = %+_−∞∞ cg(E/cp) dy = %+_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = %+_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ %tp′dt, u′ _{− fη}′ = −π′_x, v′ + fξ′ = −π_y′, E _{≡ K + G} = (u′2_{+ v}′2 _{+ N}2_ζ′2_)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 +∇ · ⟨⟨−v′η′, u′η′, ζ′πx′⟩⟩/2 (−u′_ξ′ y − v′η′y + ζ′πzy′ )/2 = ζz′v′− q′ξ′/2 +∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′π′y⟩⟩/2 (0.22) ∂t(ζz′u′) + q′v′ = −∇ · ⟨⟨u′u′− K + G, v′u′, ζ′p′x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩, " "

1980

Table 2: Characteristics of mid-latitude and equatorial waves. The third column (A′

yy ≃ −l2A′) indicates whether waves

are nearly-plane in the meridional direction or not, where A′ _{is an arbitrary quantity and l is the wavenumber in the}

meridional direction. Symbols in the last three columns are defined by q′ _{≡ v}′

x−u′y−fζz′, Λ ≡ [(ξ′p′)x+(η′p′)y+(ζ′p′)z]/2,

K _{≡ (u}′2 + v′2)/2, G ≡ (N2/2)ζ′2, and E ≡ K + G.

type of waves acronym A′

yy ≃ −l2A′ η′ = −q′/β Λ (v′ξ′ − u′η′)/2

mid-latitude inertia-gravity waves MIGWs yes no (37), 0 _{(B3), (K − G)/f}

mid-latitude Rossby waves MRWs yes yes _{(36), (37), −E (B2), (B3), 2K/f}

equatorial Rossby waves EQWs no yes (36), (37) (B2), (B3)

equatorial mixed Rossby-gravity waves EQWs no yes (36), (37) (B2), (B3)

equatorial Kelvin waves EQWs no yes (36), (37) (B2), (B3)

equatorial inertia-gravity waves EQWs no yes (36), (37) (B2), (B3)

Table 3: List of the ageostrophic versions of the Taylor-Bretherton identity (1) and the Eliassen-Palm relation (2) in previous studies and the present study.

Ageostrophic Taylor-Bretherton identity zonal flux vertical flux

Tung (1986) Eqs. (4.5) and (2.10) absent present

Hayashi and Young (1987) Eq. (2.28) absent absent

McPhaden and Ripa (1990) Eq. (22) absent absent

Takehiro and Hayashi (1992) Eq. (39) absent absent

this study Eq. (28a) present present

Ageostrophic Eliassen-Palm relation zonal flux vertical flux

Ripa (1982) Eq. (2.6d) present absent

Andrews (1983a) Eq. (4.1) absent present

Haynes (1988) Eqs. (3.12a,b) present present

Brunet and Haynes (1996) Eqs. (3.4a-d) present absent

this study Eq. (27a) present present

• 

•  •  • 

• 

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′, −u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3 u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w′_z = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ c_g(ζ′ zu′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂_t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ _{− K + G, ζ}′p′_y⟩⟩,

• 

•  •  • 

• 

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′, −u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3 u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w′_z = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ c_g(ζ′ zu′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂_t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ _{− K + G, ζ}′p′_y⟩⟩, Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′, −u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

A draft manuscript in preparation for J. Fluid Mech. Rapid

1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A

′

= A − A

(0.1)

∇ = ⟨⟨∂

x

, ∂

y

, ∂

z

⟩⟩

(0.2)

⟨⟨u

′

, v

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, w

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g

, v

g

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_{⟩⟩ + ⟨⟨u}

a

, v

a

, w

a

_⟩⟩

(0.3)

u

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y

,

v

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ψ = p

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0

,

(0.4)

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2

H. Aiki and K. Takaya

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A

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%

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2

H. Aiki and K. Takaya

Qψx

!

"#

$

∂

t

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2

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−2β)] +

∇ · ⟨⟨−ψ

x

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2

),

(0.14)

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t

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gy

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′

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(0.16)

q

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q

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⟨⟨u

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′

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, η

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(0.17)

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[ζ

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′

+ η

′

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′

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′

u

′

_{− K + G, v}

′

u

′

, ζ

′

p

′_x

_⟩⟩,

K = (u

′2

+ v

′2

)/2, G = (N

2

/2)ζ

′2

,

(0.18)

A

L

= A + ξ

′

A

′_x

+ η

′

A

′_y

+ ζ

′

A

′_z

%

A = A + ζ

′

A

′_z

&

A = %

A + ζ

_z′

A

′

= A + (ζ

′

A

′

)

z

(0.19)

u

Stokes

_{≡ u}

L

_{− u = ξ}

′

u

′_x

+ η

′

u

′_y

+ ζ

′

u

′_z

u

qs

_{≡ &}

u

_{− u = (ζ}

′

u

′

)

z

u

bolus

_{≡ &}

u

_{− %}

u = ζ

_z′

u

′

(0.20)

Taylor

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

A draft manuscript in preparation for J. Fluid Mech. Rapid

1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A

′

= A − A

(0.1)

∇ = ⟨⟨∂

x

, ∂

y

, ∂

z

⟩⟩

(0.2)

⟨⟨u

′

, v

′

, w

′

_{⟩⟩ = ⟨⟨u}

g

, v

g

, 0

_{⟩⟩ + ⟨⟨u}

a

, v

a

, w

a

_⟩⟩

(0.3)

u

g

= −ψ

y

,

v

g

= ψ

x

,

ψ = p

′

/f

0

,

(0.4)

ζ

′

= −ρ

′

/ρ

_z

= (g/ρ

0

)ρ

′

/N

2

= −ψ

z

f

0

/N

2

(0.5)

Q = v

_x′

_{− u}

′_y

_{− f}

0

ζ

_z′

= ψ

xx

+ ψ

yy

+ (ψ

z

f

₀2

/N

2

)

z

(0.6)

∂

_t

[ψ

_xx

+ ψ

_yy

+ (ψ

_z

f

₀2

/N

2

)

_z

!

"#

$

Q

] + βψ

_x

= 0

(0.7)

ψ

_x

= −Q

_t

/β

(0.8)

∂

t

[

E

#

$!

"

(ψ

2 x

+ ψ

y2

)/2 + ψ

z2

f

02

/(2N

2

)]+

∇ · ⟨⟨−ψ

xt

ψ

− βψ

2

/2

!

"#

$

Cx gE

,

_−ψ

yt

ψ

! "# $

CgyE

,

_−ψ

zt

ψf

₀2

/N

2

!

"#

$

Cz gE

⟩⟩ = 0

(0.9)

∂

_t

[

E

#

$!

"

(ψ

2 x

+ ψ

y2

)/2 + ψ

z2

f

02

/(2N

2

)]+

∇ · ⟨⟨u

′

p

′

+ [(f

0

− βy)ψ

2

/2]

y

!

"#

$

Cx g E

, v

′

p

′

_{− [(f}

0

− βy)ψ

2

/2]

x

!

"#

$

CgyE

, w

′

p

′

!"#$

Cz gE

⟩⟩

= 0

(0.10)

Qψ

x

= −∇ · ⟨⟨−ψ

x

ψ

x

+ K + G

!

"#

$

ug_ug_−K+G

,

_−ψ

x

ψ

y

! "# $

vg_ug

,

_−ψ

z

ψ

x

f

₀2

/N

2

!

"#

$

ζ′p′_x

⟩⟩

(0.11)

K

_{≡ (ψ}

_x2

+ ψ

_y2

)/2, G ≡ ψ

_z2

f

₀2

/(2N

2

)

(0.12)

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3 u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′, −u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w′_z = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ c_g(ζ′ zu′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂_t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ _{− K + G, ζ}′p′_y⟩⟩, Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞ −∞ cg(E/cp) dy, % +_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′, −u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

A draft manuscript in preparation for J. Fluid Mech. Rapid

1

Toward a seamless global diagnosis for the

horizontal flux of Rossby wave energy

H I D E N O R I A I K I

1

_†

_{A N D}

_{K O U T A R O U T A K A Y A}

2

1_{Application Laboratory, Japan Agency for Marine-Earth Science and Technology, Yokohama}

236-0001, Japan

2_{Department of Physics, Faculty of Science, Kyoto Sangyo University, Kyoto, Japan}

(Received ?; draft version on January 15, 2015)

A

′

= A − A

(0.1)

∇ = ⟨⟨∂

x

, ∂

y

, ∂

z

⟩⟩

(0.2)

⟨⟨u

′

, v

′

, w

′

_{⟩⟩ = ⟨⟨u}

g

, v

g

, 0

_{⟩⟩ + ⟨⟨u}

a

, v

a

, w

a

_⟩⟩

(0.3)

u

g

= −ψ

y

,

v

g

= ψ

x

,

ψ = p

′

/f

0

,

(0.4)

ζ

′

= −ρ

′

/ρ

_z

= (g/ρ

0

)ρ

′

/N

2

= −ψ

z

f

0

/N

2

(0.5)

Q = v

_x′

_{− u}

′_y

_{− f}

0

ζ

_z′

= ψ

xx

+ ψ

yy

+ (ψ

z

f

₀2

/N

2

)

z

(0.6)

∂

t

[ψ

xx

+ ψ

yy

+ (ψ

z

f

₀2

/N

2

)

z

!

"#

$

Q

] + βψ

_x

= 0

(0.7)

ψ

_x

= −Q

_t

/β

(0.8)

∂

t

[

E

#

$!

"

(ψ

2 x

+ ψ

y2

)/2 + ψ

z2

f

02

/(2N

2

)]+

∇ · ⟨⟨−ψ

xt

ψ

− βψ

2

/2

!

"#

$

Cx gE

,

_−ψ

yt

ψ

! "# $

CgyE

,

_−ψ

zt

ψf

₀2

/N

2

!

"#

$

Cz gE

⟩⟩ = 0

(0.9)

∂

_t

[

E

#

$!

"

(ψ

2 x

+ ψ

y2

)/2 + ψ

z2

f

02

/(2N

2

)]+

∇ · ⟨⟨u

′

p

′

+ [(f

0

− βy)ψ

2

/2]

y

!

"#

$

Cx gE

, v

′

p

′

_{− [(f}

0

− βy)ψ

2

/2]

x

!

"#

$

CgyE

, w

′

p

′

!"#$

Cz gE

⟩⟩

= 0

(0.10)

Qψ

x

= −∇ · ⟨⟨−ψ

x

ψ

x

+ K + G

!

"#

$

ug_ug_−K+G

,

_−ψ

x

ψ

y

! "# $

vg_ug

,

_−ψ

z

ψ

x

f

₀2

/N

2

!

"#

$

ζ′_p′ x

⟩⟩

(0.11)

K

_{≡ (ψ}

_x2

+ ψ

_y2

)/2, G ≡ ψ

_z2

f

₀2

/(2N

2

)

(0.12)

† Email address for correspondence: [email protected]

4 H. Aiki and K. Takaya

∂_t(ζ_z′u′ + q′η′/2) + _{∇ · ⟨⟨u}′u′ _{− K + G, v}′u′, ζ′p′_{x⟩⟩ = 0,} ∂_t(ζ_z′v′ _{− q}′ξ′/2) + _{∇ · ⟨⟨u}′v′, v′v′ _{− K + G, ζ}′p′_{y⟩⟩ = β(v}′ξ′ _{− u}′η′)/2, AL = A + ξ′A′_x + η′A′_y + ζ′A′_z ! A = A + ζ′A′_z " A = !A + ζ_z′A′ = A + (ζ′A′)_z (0.23) uStokes _{≡ u}L _{− u = ξ}′u′_x + η′u′_y + ζ′u′_z uqs _{≡ "}u _{− u = (ζ}′_u′)_z ubolus _{≡ "}u _{− !}u = ζ′ zu′ (0.24) ∂_tu + _{∇ · (Uu) − fv = −px − ∇ · ⟨⟨u}′_u′_{, v}′_u′_{, w}′_u′_⟩⟩, (0.25) ∂_t[u + (ζ′_u′)_z # $% & b u ] + ∇ · (Uu) − f[v + (ζ′_v′)_z # $% & b v ] = −∂xp − ∇ · ⟨⟨u′u′, v′u′, ζ′p′_x⟩⟩, (0.26) ∂_t[u + ζ′u′_z # $% & e u ] + ∇ · (Uu) − f[v + ζ′_v′ z # $% & e v ] − (v′ x − u′y)v′ = −∂x[p + ζ′p′_z + (ζ′2/2)p_zz # $% & −G # $% & e p + (u′2 + v′2)/2 # $% & K ] (0.27) ∂_t[ ζ_z′u′ #$%& ubolus ] + (v′ x − u′y − fζz′) # $% & q′ v′+ = −∇ · ⟨⟨u′_u′ _{− K + G, v}′_u′_{, ζ}′_p′ x⟩⟩ (0.28) v′ = −q_t′/β (0.29) Q† = ψ_xx + ψ_{yy − (ψz}/ρ_z)_zf₀2ρ₀/g v†_Q† = ⟨⟨∂_{y, ∂z⟩⟩ · ⟨⟨ψxψy, ⟩⟩} (0.30)

4 H. Aiki and K. Takaya

∂_t(ζ_z′u′ + q′η′/2) + _{∇ · ⟨⟨u}′u′ _{− K + G, v}′u′, ζ′p′_{x⟩⟩ = 0,} ∂_t(ζ_z′v′ _{− q}′ξ′/2) + _{∇ · ⟨⟨u}′v′, v′v′ _{− K + G, ζ}′p′_{y⟩⟩ = β(v}′ξ′ _{− u}′η′)/2, AL = A + ξ′_A′ x + η′A′y + ζ′A′z ! A = A + ζ′_A′ z " A = !A + ζ′ zA′ = A + (ζ′A′)z (0.23) uStokes _{≡ u}L _{− u = ξ}′_u′ x + η′u′y + ζ′u′z uqs _{≡ "}u _{− u = (ζ}′u′)_z ubolus _{≡ "}u _{− !}u = ζ_z′u′ (0.24) ∂_tu + _{∇ · (Uu) − fv = −px − ∇ · ⟨⟨u}′u′, v′u′, w′u′_⟩⟩, (0.25) ∂_t[u + (ζ′u′)_z # $% & b u ] + ∇ · (Uu) − f[v + (ζ′_v′)_z # $% & b v ] = −∂xp − ∇ · ⟨⟨u′u′, v′u′, ζ′p′x⟩⟩, (0.26) ∂_t[u + ζ′_u′ z # $% & e u ] + ∇ · (Uu) − f[v + ζ′_v′ z # $% & e v ] − (v′ x − u′y)v′ = −∂x[p + ζ′p′_z + (ζ′2/2)p_zz # $% & −G # $% & e p + (u′2 + v′2)/2 # $% & K ] (0.27) ∂_t[ ζ′ zu′ #$%& ubolus ] + (v′ x − u′y − fζz′) # $% & q′ v′+ = −∇ · ⟨⟨u′_u′ _{− K + G, v}′_u′_{, ζ}′_p′ x⟩⟩ (0.28) v′ = −q_t′/β (0.29) Q† = ψ_xx + ψ_{yy − (ψz}/ρ_z)_zf₀2ρ₀/g v†Q† = ⟨⟨∂_y, ∂_{z⟩⟩ · ⟨⟨ψx}ψ_y,_⟩⟩ (0.30) "

2 H. Aiki and K. Takaya

∂t[ E ! "# $ (ψ2 x + ψy2)/2 + ψz2f02/(2N2)]+ ∇ · ⟨⟨−ψxtψ − βψ2/2 # $! " Cx gE , _−ψytψ # $! " C_gyE , _−ψztψf₀2/N2 # $! " Cz gE ⟩⟩ = 0 (0.10) ∂t[ E ! "# $ (ψ2 x + ψy2)/2 + ψz2f02/(2N2)]+ ∇ · ⟨⟨u′p′ + [(f0 − βy)ψ2/2]y # $! " Cx gE , v′p′ _{− [(f}0 − βy)ψ2/2]x # $! " C_gyE , w′p′ #$!" Cz gE ⟩⟩ = 0 (0.11) Qψx #$!" Qvg = −∇ · ⟨⟨−ψxψx + K + G # $! " ug_ug_−K+G , _−ψxψy # $! " vg_ug , _−ψzψxf₀2/N2 # $! " ζ′_p′ x ⟩⟩ (0.12) K = (ψ_x2 + ψ_y2)/2, G = ψ_z2f₀2/(2N2) (0.13) Qψ_x ! "# $ ∂t[Q2/(−2β)] + ∇ · ⟨⟨−ψxψx + K + G # $! " ug_ug_−K+G , _−ψxψy # $! " vg_ug , _−ψzψxf₀2/N2 # $! " ζ′_p′ x ⟩⟩ = 0 (0.14) K = (ψ_x2 + ψ_y2)/2, G = ψ_z2f₀2/(2N2), (0.15) ∂t[ E/C_px ! "# $ Q2/(_−2β)]+ ∇ · ⟨⟨ugug _{− K + G} # $! " Cx g(E/Cpx) , v_#$!"gug C_gy(E/Cx p) , ζ′p′_x #$!" Cz g(E/Cpx) ⟩⟩ = 0 (0.16) K = (ψ_x2 + ψ_y2)/2, G = ψ_z2f₀2/(2N2), (0.17) ζ′(u′ z − wx′ ) (0.18) −ζ′(u′ z − wx′ ) (0.19) ∂t[E/cp]+

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3 u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w_z′ = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )t + ∇ · ⟨⟨u′p′, v′p′, w′p′⟩⟩ = 0, cp ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞

−∞ cg(E/cp) dy, u′p′/cp ̸= cg(E/cp)

%+_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ − K + G, v′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,

Toward a seamless global diagnosis for the horizontal flux of Rossby wave energy 3

u′_{t − (f}0 + βy)v′ = −p′_x, v_t′ + (f0 + βy)u′ = −p′_y, ρ′_t + w′ρ_z = 0, u′_x + v_y′ + w′_z = 0, ⟨⟨u′, v′, w′_{⟩⟩ = ⟨⟨ξt}′, η_t′, ζ_t′_⟩⟩, (0.21) ζ′ _{≡ −ρ}′/ρ_z = −p′_z/N2, K = (u′2 + v′2)/2, G = (N2/2)ζ′2, q′ _{≡ vx}′ _{− u}′_{y − (f}0 + βy)ζ_z′, q_t′ + βv′ = 0, η′ = −q′/β, (K + G ! "# $ E )_{t + ∇ · ⟨⟨u}′_p′_{, v}′_p′_{, w}′_p′_{⟩⟩ = 0, c} p ≡ ω/k, cg ≡ ∂ω/∂k (E/c_p)_{t + ∇ · ⟨⟨u}′_p′_/c p, v′p′/cp, w′p′/cp⟩⟩ = 0, % +_∞ −∞ u′p′/cp dy = % +_∞

−∞ cg(E/cp) dy, u′p′/cp ̸= cg(E/cp)

%+_∞ −∞ u′p′/cp dy = % +_−∞∞ cg(E/cp) dy = % +_−∞∞ cg(ζ_z′u′ + q′η′/2) dy = % +_−∞∞ (u′_u′ _{− K + G) dy} π′ _≡ % tp′dt, u′ _{− fη}′ = −π_x′ , v′ + fξ′ = −π_y′ , E _{≡ K + G} = (u′2 _{+ v}′2 _{+ N}2_ζ′2)/2 = (u′_ξ′ t + v′ηt′ − ζπzt′ )/2, (−u′_ξ′ x − v′ηx′ + ζ′πzx′ )/2 = ζz′u′ + q′η′/2 + ∇ · ⟨⟨−v′η′, u′η′, ζ′πx′ ⟩⟩/2 (−u′_ξ′ y − v′ηy′ + ζ′πzy′ )/2 = ζz′v′ − q′ξ′/2 + ∇ · ⟨⟨v′ξ′,−u′ξ′, ζ′πy′ ⟩⟩/2 (0.22) ∂_t(ζ_z′u′) + q′v′ = −∇ · ⟨⟨u′u′ _{− K + G, v}′u′, ζ′p′_x⟩⟩, ∂t(ζ_z′v′) − q′u′ = −∇ · ⟨⟨u′v′, v′v′ − K + G, ζ′p′_y⟩⟩,