Spectral Energetics Analysis of theGeneral Circulation of the Atmosphere inthe Vertical Wavenumber Domain

Koji Terasaki 1 and Hiroshi L. Tanaka2
1: Graduate School of Life and EnvironmentalSciences, University of Tsukuba
2: Center for Computational Sciences,
University of Tsukuba
External Review on Center for Computational Sciences, University of Tsukuba. 30 Oct, 2007

INTRODUCTION
Energetics in zonal wavenumber domain
Lorenz Energy Cycle
Available potential
energyKinetic Energy

zonaleddy(Lorenz, 1954)

Saltzman Cycle

G: Generation of APE
R: zonal-wave interaction ofzonal

APE
P: Available Potential Energy(APE)
S: wave-wave interaction ofAPE

eddy

C: Conversion from APE to KE
L: wave-wave interaction of KE
K: Kinetic Energy (KE)
M: zonal-wave interaction of KE
D: Dissipation(Saltzman, 1957 & 1970) Lorenz and Saltzman investigated the energy
flow of the atmospheric general circulation in
the zonal wavenumber domain.
(Fourier transform)

Purpose of this study
We can investigate the energy flow between
baroclinic component of the atmosphere, if
vertical transform is applied to the primitiveequation.
(Vertical structure function)

METHOD
Vertical structure function
Gm : vertical structure function

Vertical structure equation∂ ⎛ 2 ∂Gm ⎞ Rγ
Gm = 0⎟+⎜σ∂σ ⎝∂σ ⎠ ghm

Constant

Euler Equation

R : gas constantconstanthm : equivalent heightγ : static stability parameter

constant

Eigenvalue problem of the SturmLiouville type

∂Gm= 0,∂σ∂Gm+ αGm = 0,
∂σσ =εσ =1

Obtain the vertical structure function analytically …

Analytical Solution ofthe Vertical Structure Equation
9Barotropic Component

G0 (σ ) = C1σ + C2σr1r2

9Baroclinic Component

Gm (σ ) = σ−12{C1 cos(μ m ln σ ) + C2 sin (μ m ln σ )}Primitive equations

Kinetic Energy Equation∂K m∂U n∂Vn ⎞ps ⎡ rnlm ⎛= −∑+ VmU l⎜U mU l⎟⎢∂λ ⎠∂t∂λl , n g ⎣ a cos θ ⎝Kinetic energy∂U n∂Vn ⎞rnlm ⎛++ VmVl
⎜U mVl⎟interaction∂θ∂θ ⎠a ⎝tan θrnlm (U mU lVn − VmU lU n ) + rnl ′m (U m Ω lU n + Vm Ω lVn )]−apsps∂Am∂AmUmVm−−a cos θa∂λ∂θ− DmDissipation

Generation of kinetic energy（conversion from APE）

Available potential energy (APE)equationgps ⎡ AmAm∂Pm∂An∂An ⎤R1 ⎢UlVl=∑+⎥Raaγ
θλθ
cos∂t∂∂⎣⎦
l ,ngpsR2 Ω l An Am+∑
l , n Rγ

Available potential
energy interaction

psps
∂Am
∂AmUm
Vm++aa cos θ∂λ∂θps⎛⎞ Conversion toAm ⎜ H m + ∑ rpn′m H n ⎟ Kinetic energy+C pγn⎝⎠Generation of APE andradiative cooling

RESULTS

Available potentialenergy

Barotropic
component

Baroclinic
component

Kinetic energy

Barotropic

G: Generation of APE
R: barotropic-baroclinic
interactions of APE
P: Available Potential Energy(APE)
S: baroclinic-baroclinic
interactions of APEBaroclinic

C: Baroclinic conversion
L: baroclinic-baroclinic
interactions of KE
K: Kinetic Energy (KE)
M: barotropic-baroclinic
interactions of KE
D: Dissipation

CONCLUSIONSThe atmospheric energy flow in the verticalwavenumber domain was calculated, with expanding
the primitive equations using analytical verticalstructure functions.

9The energy interactions between available potential
energy in the vertical wavenumber domain arecomparatively small.
The kinetic energy converted from available potential
energy is interacting among baroclinic modes.

99

Future study
More analysis about baroclinic-baroclinic interactions ofkinetic energy.

Thank you forlistening!

Numerical Solution

Vertical expansion
M

東西風

:

u ( p ) = ∑ U mGm ( p )
m =0
M

南北風

:

v( p) = ∑ VmGm ( p )
m =0
M

気温:

T ( p ) = ∑ BmGm ( p )
m =0
M

高度:

z ( p ) = ∑ AmGm ( p )
m =0
M

鉛直風:

ω ( p) = ∑ Ω mGm ( p )m =0¾Vertical expansion∞F:Coefficien
tofGmmf ( p ) = ∑ FmGm ( p )m =0psps :Surface pressure (1000hPa )1Fm =f ( p )Gm ( p )dp∫ps 0¾Orthogonality of vertical structure function
ps1GmGn dp = δ mn∫
ps 0

Kinetic energy
1 ps u 2 + v 2
K= ∫dp02g
1 ps 1 ⎡⎤U mGm ∑ U nGn + ∑ VmGm ∑ VnGn ⎥dp= ∫∑⎢02⎣mgnmn⎦
1U mU n + VmVn ps= ∑∑GmGn dp∫02g m npsU m + Vm=
∑ 2g m22∂Vm ⎞∂U m∂K mps ⎛+ Vm=⎜U m⎟∂t ⎠∂t∂tg ⎝Available potential energy2⎛ ∂z ′ ⎞⎜⎜⎟⎟ dp⎝ ∂p ⎠1 ps 1
∂ ⎛ g 2 p 2 ∂z ′ ⎞⎟⎟dp⎜⎜= ∫ − z′02 ∂p ⎝ Rγ ∂p ⎠gps
1=∫z ′2 dp02hm
ps2=∑Amm 2hm

P=22g p1g ∫0 2 Rγps

∂Am ⎞∂Pmps ⎛=⎜ Am⎟∂t ⎠∂thm ⎝