Model description of AGCM5 of GFD-Dennou-Club edition

SWAMP project, GFD-Dennou-Club Mar 01, 2006

AGCM5 of the GFD-DENNOU CLUB edition is a three-dimensional primitive system on a sphere (Swamp Project, 1998). The model is basically the same as that used in Numaguti (1993), and its details are described in Numaguti(1992). In the following subsections, we briefly describe the model system and its implementation.

1 Basic equations

The dynamical part of the model consists of the vorticity, divergence, hydrostatic, humidity and thermodynamic equations as follows.

\frac{dζ}{dt} = \frac{1}{a cos φ} \frac{∂}{∂λ} \{−(ζ + f)u − σ̇ \frac{∂u}{∂σ} − \frac{RT′}{a cos φ} \frac{∂π}{∂λ} \}

−\frac{1}{a cos φ} \frac{∂}{∂φ} [\{(ζ + f)v − σ̇ \frac{∂u}{∂σ} − \frac{RT′}{a} \frac{∂π}{∂φ} \} cos φ] − F_{ζ}^{diff}, (1)

\frac{dD}{dt} = \frac{1}{a cos φ} \frac{∂}{∂λ} \{(ζ + f)v − σ̇ \frac{∂u}{∂σ} − x\frac{RT′}{a} \frac{∂π}{∂φ} \}

+\frac{1}{a cos φ} \frac{∂}{∂φ} [\{(ζ + f)v − σ̇ \frac{∂u}{∂σ} − \frac{RT′}{a} \frac{∂π}{∂φ} \} cos φ]

−∇^{2}(Φ + R\overline{T} π + E) − F_{D}^{diff},  (2)
\frac{dπ}{dt} = −\frac{1}{a cos φ} \frac{∂u}{∂λ} − \frac{1}{a cos φ} \frac{∂}{∂φ} (v cos φ) − \frac{∂σ̇}{∂σ} ,  (3)
\frac{∂Φ}{∂σ} = −\frac{RT}{σ} ,  (4)
\frac{dq}{dt} = \frac{g}{p_{s}} \frac{∂F_{q}^{vdf}}{∂σ} + F_{q}^{vdf} + S_{q}^{cond},  (5)

\frac{dT}{dt} = \frac{RT}{c_{p}} \{\frac{∂π}{∂t} + \frac{u}{a cos φ} \frac{∂π}{∂λ} + \frac{v}{a} \frac{∂π}{∂φ} + \frac{σ̇}{σ} \}

+ \frac{1}{c_{p}} (\frac{g}{p_{s}} \frac{∂F_{T}^{vdf}}{∂σ} + \frac{g}{p_{s}} \frac{∂F_{rad}^{vdf}}{∂σ} ) + F_{T}^{diff} + LS_{q}^{cond}.  (6)

In these equations,

\frac{d}{dt} ≡ \frac{∂}{∂t} + \frac{u}{a cos φ} \frac{∂}{∂λ} + \frac{v}{a} \frac{∂}{∂φ} + σ̇ \frac{∂}{∂σ} ,  (7)
ζ ≡ \frac{1}{a cos φ} \frac{∂v}{∂λ} − \frac{1}{a cos φ} \frac{∂}{∂φ} (u cos φ),  (8)

D ≡ \frac{1}{a cos φ} \frac{∂u}{∂λ} + \frac{1}{a cos φ} \frac{∂}{∂φ} (v cos φ),  (9)
Φ ≡ gz,  (10)
π ≡ ln p_{s},  (11)

where, (λ, φ) are the latitudinal and longitudinal coordinates; σ is the vertical coordinate; u, v are the horizontal wind components; σ̇ is the vertical wind in σ coordinate; T is temperature; q is specific humidity; p_{s} is surface pressure; T_{g} is surface temperature; Φ is geopotential; f is the Coriolis parameter; a is the radius of the earth; g is the acceleration of gravity; R is the gas constant of dry air; c_{p} is the atmospheric specific heat at constant pressure; L is the latent heat of water; F_{ζ}^{diff}, F_{D}^{diff}, F_{T}^{diff} F_{q}^{diff} are the horizontal diffusion terms; F_{u}^{vdf}, F_{v}^{vdf}, F_{T}^{vdf}, F_{q}^{vdf} are the vertical diffusion terms; and S_{q}^{cond} is the condensation source (loss) of the specific humidity.

The horizontal diffusion terms are represented by a hyper-viscosity type formula,

F^{diff} = −(−1)^{n}K∇^{2n}  (12)

for which n = 8 is adopted. The value of K is determined such that the e-folding time of the component with wavenumber 42 is four hours.

2 Discretization

The equations are discretized horizontally by a non-alias latitude-longitude grid and are integrated by the pseudo-spectral method with a triangular truncation. The scheme of Arakawa and Suarez (1983) is utilized for discretizing in the vertical direction. The time integration is performed by the use of a leap-frog scheme for the terms of dynamical processes other than dissipative processes, a backward scheme for the terms of physical processes, and an Euler scheme for the horizontal diffusion terms. The time filter of Asselin (1972) is used to remove the computational mode caused by the leap-frog scheme.

3 Physical processes

The model contains the following simplified physical processes.

3.1 Cumulus parameterization

The cumulus parameterization schemes implemented are the adjustment scheme (Manabe et al., 1965) and Kuo scheme (Kuo, 1974). In the followings, the description of adjustment schmes is given. Condensed water is removed from the system immediately after condensation and hence there are neither rain drops nor clouds.

Moist convective adjustment occurs when the following conditions are satisfied:

• In two neighboring levels, the atmosphere becomes moist unstable, that is , the temperature lapse rate exceeds the moist adiabatic lapse rate.

• Both of the neighboring levels are saturated or oversaturated.

In the process of adjustment, the temperature and specific humidity of the two neighboring levels are instantaneously adjusted to values so that the levels are saturated and the temperature lapse rate between them coincides with the moist adiabatic lapse rate. In due course, moist static energy has to be conserved:

∑_{l=k−1}^{k}(C_{p}T_{l} + Lq_{l})Δp_{l} = ∑_{l=k−1}^{k}(C_{p}T̂_{l} + Lq̂_{l})Δp_{l},  (13)

where k is the index of the vertical level. Here, T̂, q̂ denote the values before adjustment, while T, q denote the values after the adjustment.

The moist unstable condition for layer (k, k − 1) is evaluated by

ST_{k−1/2} ≡ T̂_{k} − T̂_{k−1} + \frac{L}{C_{p}} (q^{∗}(T̂_{k}, p_{k}) − q^{∗}(T̂_{k−1}, p_{k−1})) − \frac{R}{C_{p}} \frac{Δp_{k−1/2}}{p_{k−1/2}} \frac{T̂_{k} + T̂_{k−1}}{2}
> 0.  (14)

The saturation condition of two neighboring levels is simply expressed by

q̂_{k−1} > q^{∗}(T̂_{k−1}, p_{k−1}), q̂_{k} > q^{∗}(T̂_{k}, p_{k}).  (15)

The adjusted values (T_{k}, q_{k}) are determined as

T_{k−1} = T̂_{k−1} + ΔT_{k−1},  (16)
T_{k} = T̂_{k} + ΔT_{k},  (17)

q_{k−1} = q^{∗}(T̂_{k−1}, p_{k−1}) + \frac{∂q^{∗}}{∂T} |_{k−1} ΔT_{k−1},  (18)
q_{k} = q^{∗}(T̂_{k}, p_{k}) + \frac{∂q^{∗}}{∂T} |_{k} ΔT_{k},  (19)

ΔT_{k} = \frac{(1 + γ_{k−1})Δp_{k−1}S_{k−1/2} + [1 + γ_{k−1} − κ\frac{Δp_{k−1/2}}{2p_{k−1/2}} ]\frac{L}{C_{p}} ΔQ̂}{(1 + γ_{k−1})(1 + γ_{k})(Δp_{k−1} + Δp_{k}) + κ\frac{Δp_{k−1/2}}{2p_{k−1/2}} [(1 + γ_{k−1})Δp_{k−1} − (1 + γ_{k})Δp_{k}]},

(20)

ΔT_{k−1} = −\frac{(1 + γ_{k})Δp_{k}}{(1 + γ_{k−1})Δp_{k−1}} ΔT_{k} + \frac{1}{(1 + γ_{k−1})Δp_{k−1}} \frac{L}{C_{p}} ΔT_{k}, (21)

where

γ_{k} = \frac{L}{C_{p}} \frac{∂q^{∗}}{∂T} |_{k} , γ_{k−1} = \frac{L}{C_{p}} \frac{∂q^{∗}}{∂T} |_{k−1} ,  (22)
ΔQ̂ = (q̂_{k−1} − q^{∗}(T̂_{k−1}, p_{k−1}))Δp_{k−1} + (q̂_{k} − q^{∗}(T̂_{k}, p_{k}))Δp_{k}. (23)

Precipitation of one column is calculated as

P = ∑_{k=1}^{K}(q̂_{k} − q_{k})\frac{Δp_{k}}{g} .  (24)

These adjustment procedures are applied to all levels until the vertical temperature and moisture profiles converge.

3.2 Large scale condensation

The large scale condensation scheme is applied at each grid point when the value of the specific humidity exceeds its saturation value. The large scale condensation scheme demands that the value of the specific humidity be adjusted to the saturation value. Condensed water is assumed to fall out as rain and is removed from the system immediately. There are no rain drops or clouds, as in the case of the convective adjustment scheme. In the time integration procedure, the large scale condensation scheme is applied just after the convective adjustment scheme.

The condition for large scale condensation can be denoted by

q̂ > q^{∗}(T̂, p),  (25)

where T̂, q̂ are the values of temperature and specific humidity before the adjustment. After the large scale condensation adjustment, the values of the temperature and specific humidity become

q = q^{∗}(T, p)  (26)

and

C_{p}T + Lq = C_{p}T̂ + Lq̂.  (27)

The solutions to these are obtained by Newton method;

T = T̂ + \frac{L}{C_{p}} \frac{q̂ − q^{∗}(T̂, p)}{1 + \frac{L}{C_{p}} \frac{∂q^{∗}}{∂T}},  (28)

q = q̂ − \frac{L}{C_{p}} \frac{∂q^{∗}}{∂T} \frac{q̂ − q^{∗}(T̂, p)}{1 + \frac{L}{C_{p}} \frac{∂q^{∗}}{∂T}}.  (29)

The amount of precipitation is calculated by

P = \frac{Δp}{g}(q̂ − q).  (30)

3.3 Radiation

Radiative flux is calclated by band model. The effects of scattering of solar radiation is not considered. It is assumed that there are no cloud nor rain drops.

As for longwave radiation, we include only the absorption and emission of water and dry air; the effect of scattering is not included. The absorption coefficients of the longwave radiation are chosen so that the cooling profile of the atmosphere roughly resembles the observed one.

Radiation flux F_{R,i} is

F_{R}(z) = (πB(T_{g}) − πB(T_{s})) T^{f}(z, 0)+πB(T (z_{T}))T^{f}(z, z_{T})−∫_{0}^{z_{T}} \frac{dπB}{dξ} T^{f}(z, ξ)dξ,

(31)

where T^{f}(z_{1}, z_{2}) is the flux transfer function between z = z_{1}, z_{2} and πB ≡ σ_{SB}T^{4} is the source function.

The flux transfer function T^{f}(z_{1}, z_{2}) is assumed to be given as

T^{f}(z_{1}, z_{2}) = ∑_{i=1}^{N_{R}} b_{i} exp (−δ_{R}|τ_{R,i}(z_{1}) − τ_{R,i}(z_{2})|) .  (32)

Here, τ_{i}(z) is the optical depth whose value at the top of the atmosphere is 0 and

τ_{R,i}(z) = ∫_{z}^{∞} k_{R,i}ρqdz + ∫_{z}^{∞} k̄_{R,i}ρdz,  (33)

where k_{R,i} and k̄_{R,i} are the absorption coefficients of water vapor and dry air for wavenumber band i, respectively. The values of these absorption coefficients are assumed to be constant. b_{i} is the ratio of the energy for wavenumber band i to the total radiative energy and is also assumed to be constant. The default value for δ_{R} is 1.5. In most cases, N_{R} is set to be 4, three representing water vapor bands and one representing dry air band. The default values of k_{R,i}, k̄_{R,i} and b_{i} are summarized in Table 1.

Table 1: The values of constants used in the longwave radiation scheme.

Band Number 1 2 3 4
k_{R,i}  8.0 1.0 0.1 0.0
\overline{k}_{R,i}  0.0 0.0 0.0 5.0 × 10^{−5}
b_{i}  0.2 0.1 0.1 0.6

3.4 Vertical mixing

Vertical turbulent mixing is represented by the level II scheme of Mellor and Yamada (1974).

In this scheme of Mellor and Yamada (1974), diffusion coefficients K_{M}, K_{T}, and K_{q} depends on stability and are given as functions of u, v, and T:

K_{M} = l^{2} \frac{Δ|v|}{Δz}S_{M},  (34)
K_{T} = K_{q} = l^{2} \frac{Δ|v|}{Δz}S_{H}.  (35)

Following Blakadar(1962), the mixing length l is evaluated by

l = \frac{κz}{1 + κz/l_{0}} ,  (36)

where l_{0} = 300 m and κ = 0.4 is the Kárman constant.

In these expressions, S_{M} and S_{H} are given as functions of the Richardson number.

S_{M} = B_{1}^{1/2}(1 − R_{f})^{1/2}S̃_{M}^{3/2},  (37)
S_{H} = B_{1}^{1/2}(1 − R_{f})^{1/2}S̃_{M}^{1/2}S̃_{H},  (38)

where

S̃_{H} = \frac{α_{1} − α_{2}R_{f}}{1 − R_{f}} ,  (39)
S̃_{M} = \frac{β_{1} − β_{2}R_{f}}{β_{3} − β_{4}R_{f}} S̃_{H}.  (40)

R_{f} is the flux Richardson number, given as

R_{f} = \frac{1}{2β_{2}} [β_{1} + β_{4}R_{iB} − \sqrt{(β_{1} + β_{4}R_{iB})^{2} − 4β_{2}β_{3}R_{iB}}] ,  (41)

where R_{iB} is the bulk Richardson number and

α_{1} = 3A_{2}γ_{1},  (42)
α_{2} = 3A_{2}(γ_{1} + γ_{2}),  (43)
β_{1} = A_{1}B_{1}(γ_{1} − C_{1}),  (44)
β_{2} = A_{1}[B_{1}(γ_{1} − C_{1}) + 6A_{1} + 3A_{2}],  (45)
β_{3} = A_{2}B_{1},  (46)
β_{4} = A_{2}[B_{1}(γ_{1} + γ_{2}) − 3A_{1}],  (47)
(A_{1}, B_{1}, A_{2}, B_{2}, C_{1}) = (0.92, 16.6, 0.74, 10.1, 0.08),  (48)
γ_{1} = \frac{1}{3} − \frac{2A_{1}}{B_{1}} , γ_{2} = \frac{B_{2}}{B_{1}} + 6\frac{A_{1}}{B_{1}} .  (49)

R_{iB} is defined as

R_{iB} = \frac{\frac{q}{θ_{s}} \frac{Δθ}{Δz}}{(\frac{Δu}{Δz})^{2} + (\frac{Δv}{Δz} )^{2}} .  (50)

Here, the value of (Δ•) at the vertical level k − 1/2 is evaluated as •_{k} −•_{k−1}.

3.5 Surface flux

The surface fluxes of momentum, heat and water vapor are evaluated by the bulk formulae

(F_{λ})_{1/2} = −ρ_{s}C_{M}|v_{1}|u_{1},  (51)
(F_{φ})_{1/2} = −ρ_{s}C_{M}|v_{1}|u_{1},  (52)
(F_{T})_{1/2} = ρ_{s}C_{p}C_{H}|v_{1}|(T_{g} − (\frac{T}{Π}_{1})),  (53)
(F_{q})_{1/2} = ρ_{s}βC_{E}|v_{1}|(q^{∗}(T_{g}, p_{s}) − q_{1}),  (54)

where the suffix 1/2 indicates that these are the value at the vertical level 1/2.

Following Louis et al. (1982), the bulk coefficients C_{M}, C_{H}, and C_{E} are determined by

C_{M} = \{ \begin{matrix}C_{M0} (1 + 10R_{iB}/\sqrt{1 + 5R_{iB}})^{−1} & R_{iB} ≥ 0, \\ C_{M0} [1 − 10R_{iB} (1 + 75C_{M0}\sqrt{\frac{z_{1}}{z_{M0}} |R_{iB}|})^{−1}] & R_{iB} < 0, \end{matrix}  (55)

C_{H} = C_{E} = \{ \begin{matrix}C_{H0} (1 + 15R_{iB}/\sqrt{1 + 5R_{iB}})^{−1} & R_{iB} ≥ 0, \\ C_{H0} [1 − 15R_{iB} (1 + 75C_{H0}\sqrt{\frac{z_{1}}{z_{M0}} |R_{iB}|})^{−1}] & R_{iB} < 0, \end{matrix} (56)

where R_{iB} is the bulk Richardson number evaluated by

R_{iB} = \frac{\frac{g}{θ_{s}} (θ_{1} − θ_{g})/g}{(u_{1}/z_{1})^{2}} = \frac{g}{θ_{s}} \frac{T_{1}(p_{s}/p_{1})^{κ} − T_{g}}{u_{1}^{2}/z_{1}} .  (57)

C_{M0} and C_{H0} are the bulk coefficients for neutral case:

C_{M0} = (\frac{κ}{ln (\frac{z_{1}}{z_{M0}} )}) ,  (58)

C_{H0} = (\frac{κ}{ln (\frac{z_{1}}{z_{H0}} )}) ,  (59)

where z_{M0} and z_{H0} are the roughness parameters of the surface. The values are 10^{−4}m.

References

Arakawa, A. and M.J. Suarez, 1983: Vertical differencing of the primitive equations in sigma coordinates. J. Atmos. Sci., 111, 34-45.

Asselin, R. 1972: Frequency filter for time integrations. Mon. Wet. Rev., 100, 487-490.

Blakadar, A.K., 1962: The vertical distribution of wind and turbulent exchange in neutral atmosphere. J. Geophys. Res., 67, 3095-3102.

Kuo, H. L., 1974: Further studies of the parameterization of the influence of cumulus convection on large-scale flow. J. Atmos. Sci., 31, 1232–1240.

Louis, J.-F. and M. Tiedtke and J.-F. Geleyn, 1982: A short history of the PBL parameterization at ECMWF. Workshop on Planetary Boundary layer Parameterization, 59-80, ECMWF, Reading U.K.

Manabe, S., J. Smagorinsky and R.F. Strickler, 1965: Simulated climatology of a general circulation model with a hydrologic cycle. Mon. Wea. Rev., 93, 769-798.

Mellor, G.L. and T. Yamada, 1974: A hierarchy of turbulence closure models for the planetary boundary layers. J. Atmos. Sci., 31, 1791-1806.

Numaguti, A., 1992: Numerical experiments on the large scale structure of cumulus activity in the tropics. Ph.D. Thesis, University of Tokyo. (in Japanese).

Numaguti, A., 1993: Dynamics and energy balance of the Hadley circulation and the tropical precipitation zones: Significance of the distribution of evaporation. J. Atmos. Sci., 50, 1874-1887.

Swamp Project (1998) AGCM5. http://www.gfd-dennou.org/arch/agcm5, GFD Dennou Club.