A new parameterization of gravity waves foratmospheric circulation models based onthe radiative transfer equation

MATTHÄUS MAI, Erich BeckerLeibniz-Institute of Atmospheric Physics

LEIBNIZ-INSTITUTE OF ATMOSPHERIC PHYSICS | TRR 181 1

Solving the radiative transfer equation for Gravity Waves (GW) usingthe Gaussian variation principle

Radiative transfer equation [Olbers, Eden 2013 Journal of Physical Oceanography ]:

∂tE + ∂z (zE) + ∂m (mE) + ∂ωI(ωIE) =

ωI

ωI

E − m2DE + S (1)

Decomposition of spectral energy density: E(z, t, ωI ,m) = E0(z, t)A(m,m∗)B(ωI )

Assumptions/approximations:

Single columntextk = const

k = 0ω2

I = k2N2

m2ω2

I = k2N2

m2

Desaubies spectrumtextA(m,m∗) = NA

m/m∗1+(m/m∗)4

B(ωI ) = NBω−2

I

Mid-frequency, Boussinesq for GWtextω2

I = k2N2

m2

ω2

I = k2N2

m2

textDefinition of error squared:

χ2(z, t) :=

∫ ∫A(m,m∗)B(ω)

(radiative transfer equation

A(m,m∗)B(ω)

)2

dmdω

Variation ofχ2 with respect to ∂tE0(z, t) and ∂tm∗(z, t) yields the best approximate solution to the radiative transfer equation:

∂tE0 = +CE0∂zE0 + CE1∂zm∗E0 − CE2E0

∂zU

N+ CE3E0

∂zN

N− CE4DE0 + CE5S (2)

∂tm∗ = +Cm0

∂zE0E0

+ Cm1∂zm∗ − Cm2

∂zU

N+ Cm3

∂zN

N− Cm4D + Cm5

SE0

(3)

Ci = C∫∫

fi (m,m∗, ωI )dmdωI

LEIBNIZ-INSTITUTE OF ATMOSPHERIC PHYSICS | TRR 181 2

The Setting for the offline simulationWave field parameters: Background parameters: Dissipation by saturation:

Initial energy density:

0.0 0.1 0.2 0.30 in kg(ms2) 1

20

40

60

80

heig

ht in

km

Initial characteristic vertical wave number:

m∗(z, 0) = − 2π3000

m−1

Spectral parameters:

ωI ∈ [0.001826, 0.001916]s−1

m ∈[− 2π

10,− 2π

80000

]m−1

Source parameter:

S(z, t) = 0

Eastward mesospheric wind jet:

0 5 10 15 20U in ms 1

20

40

60

80

heig

ht in

km

Brunt-Väisälä frequency:

N(z, t) = 0.0187s−1

Density:

ρ(z) = 1.2 exp(−z/7000m)kgm−3

Static instability if:

1

ρN2

E0∫∫

m2A(m)B(ωI )dmdωI > α2

CDE0 > α2

When the lhs is bigger thanα2 the saturation isreached and dissipation sets in:

D = D0(CDE0 − α2)

case 1:

D0 = 0

case 2:

D0 = 100m2s−1

α2 = 0.5

LEIBNIZ-INSTITUTE OF ATMOSPHERIC PHYSICS | TRR 181 3

The scheme describes wave refraction and even critical layer (case1)

Horizontal wave propagation against the windjet, k < 0text text texttext text text

0 200 400 600 800time in min

20

40

60

80

heig

ht in

km

0 in 10 2kg(ms2) 1

0.01.22.43.64.86.07.28.49.6

0.020 0.015 0.010 0.005 0.000wave number in m 1

20

40

60

80

heig

ht in

km

m * at 500 min

Horizontal wave propagation in the direction of the windjet, k > 0

(critical layer)

0 200 400 600 800time in min

20

40

60

80

heig

ht in

km

0 in 10 2kg(ms2) 1

0.01.22.43.64.86.07.28.49.6

0.020 0.015 0.010 0.005 0.000wave number in m 1

20

40

60

80

heig

ht in

km

m * at 500 min

LEIBNIZ-INSTITUTE OF ATMOSPHERIC PHYSICS | TRR 181 4

The scheme describes wave refraction and even critical layer (case2)

Horizontal wave propagation against the windjet, k < 0text text texttext text text

0 200 400 600 800time in min

20

40

60

80

heig

ht in

km

0 in 10 2kg(ms2) 1

0.01.22.43.64.86.07.28.49.6

0.020 0.015 0.010 0.005 0.000wave number in m 1

20

40

60

80

heig

ht in

km

m * at 500 min

Horizontal wave propagation in the direction of the windjet, k > 0

(critical layer)

0 200 400 600 800time in min

20

40

60

80

heig

ht in

km

0 in 10 2kg(ms2) 1

0.01.22.43.64.86.07.28.49.6

0.020 0.015 0.010 0.005 0.000wave number in m 1

20

40

60

80

heig

ht in

km

m * at 500 min

LEIBNIZ-INSTITUTE OF ATMOSPHERIC PHYSICS | TRR 181 5