Introduction to Mesoscale Modelling Kazuo Saito - - PowerPoint PPT Presentation
WSN16 Evening school, 29 July 2016, Chinese University of Hong Kong Introduction to Mesoscale Modelling Kazuo Saito ksaito@mri-jma.go.jp Meteorological Research Institute 1. Fundamental equations for atmosphere 2. Waves in mesoscale atmosphere
WSN16 Evening school, 29 July 2016, Chinese University of Hong Kong
ksaito@mri-jma.go.jp
Meteorological Research Institute
In the case of moist atmosphere, preservation of water substances and the phase change must be considered (cloud micro-physics).
w dif g z p t d dw v dif y p t d dv u dif x p t d du . 1 . 1 . 1 = + + = + = + ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ
∂: partial derivative symbol
w dif g z p t d dw v dif y p t d dv u dif x p t d du . 1 . 1 . 1 = + + = + = + ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ
∂: partial derivative symbol
地軸 Universal gravitation Centrifugal force Gravity force Rotating spheroid
w dif g z p t d dw . 1 = + + ∂ ∂ ρ In case the aspect ratio of the atmospheric motion is much smaller than unity, the equation for vertical motion 1 = + g z p ∂ ∂ ρ can be replaced by hydrostatic equilibrium This relation equilibrium (balance) of forces between vertical pressure gradient force and gravity force.
Pressure gradient force Gravity force Isobaric plain high pressure low pressure
Vertical pressure gradient force usually balances with the gravity force (hydrostatic equilibrium)
acceleration
Earth rotation Earth rotation North Pole Low latitude High latitude
Northward motion shifts eastward in Northern hemisphere due to the difference
= + + + z w y v x u t ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ
Mass flux (density x wind speed)
) ( = + + + + + + z w y v x u z w y v x u t ∂ ∂ ∂ ∂ ∂ ρ ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ
1 = + + ∂ + z w y v x u t d d ∂ ∂ ∂ ∂ ∂ ρ ρ
T V R m M m R T
* *
RT m M T V R m M T V R p p
d d i i i i i i i d
ρ = = = =
* *
=
i i i i i d
m M M m
（ p0=1000hPa, Cp is the specific heat of dry air in constant pressure; 7R/2=1004.7J/Kg/K） State equation for dry air Using the specific volume α (inverse of density ρ)
π θ π T p p
p
C R
= = , ) (
/
p v
C C
p p R p
/
) ( θ ρ =
where Cv is the specific heat of dry air in constant volume; Cv = Cp –R=5R/2=717.6J/Kg/K If we define the non-dimensional pressure (Exner function) π and the potential temperature θ
Pressure gradient force Gravity force Isobaric plain high pressure low pressure
Veridical pressure gradient force usually balances with the gravity force (hydrostatic equilibrium)
acceleration
Applying the state equation to hydrostatic equilibrium
m
RT gz
−
We obtain well-known barometric height formula (pressure-height equation)
α pd dI dQ + =
‥change in the internal energy of a closed system is equal to the amount of heat supplied to the system minus the amount of work done by the system on its surroundings Let non-adiabatic heating rate Q,
θ π α α d C dp dT R C pd dT C Qdt
P v v
= − + = + = ) (
π θ π θπ α α d C p d p R dp RT p
p R C p
= = = ) ( ,
Thus,
π θ
P
C Q t d d =
‥Conservation law (prognostic equation) of potential temperature
ρa：density of moist air, Tv：virtual temperature, qv: Specific humidity Virtual potential temperature is defined by replacing temperature by virtual temperature in definition of potential temperature Partial pressure of moist air
v
RT RT q RT RT T V R M M T V R m m M M T V R m M m M p p p
a v a v a v d v d v d v d v v d d v d
ρ ρ ρ ρ ρ ρ = + + + + + = + = + =
) 61 . 1 ( ) 61 . ( ) 61 . 1 ( ) 61 . 1 ( ) ( * ) (
θ π π θ ) 61 . 1 ( ) 61 . 1 (
v v v v
q T q T + = + = ≡
dT q C dT r rC C dQ
v p pv p
) 85 . 1 ( 1 ) ( + ≅ + + = First law of thermodynamics for moist air is given by The difference between θ and θmoist is less than 0.1K, and can be ignored as
p v
C C
p p v R p a
/
) ( θ ρ =
) 24 . 1 ( ) 85 . 1 ( ) 61 . 1 ( /
) ( ) ( ) (
v q C R v q C v q R pm C m R
p p
p p T p p T p p T moist
− + +
≅ = = θ
where r is the mixing ratio. Likewise, specific heat of moist air in constant volume is ) 94 . 1 ( 1 ) (
v v vv v vv
q C r rC C C + ≅ + + = Specific heat of water vapor in constant pressure Cpv =1854J/Kg/K Specific heat of water vapor in constant volume Cvv =Cpv –R*/mv=1390J/Kg/K Potential temperature of moist air may be modified as
= + + + z w y v x u t ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ
‥local tie tendency of q =advection + moisture source Combining with the continuity equation, M z q w y q v x q u t q = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ We obtain the following flux form equation,
M z wq y vq x uq t q ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ = + + +
where From state equation, perturbation of density can be divided into the following two terms:
) 11 . ( ' ' ' ' ' ) ( ) ' ( ) ( ' } ) ( { ' ) ( '
2 1 2
A C p p p C C p p p p C C R p p p R p p p R p R p
S p v C C p v C C C C
p v p v p v
+ − = + − = + − = = =
−
θ θ ρ ρ θ θ ρ θ θ θ θ θπ ρ
) 12 . (A RT C C C
v p S =
Since potential temperature is invariant for total derivation,
dt d C dt dp
S
ρ
2
' =
Starting from the following 2-dimensional basic equations, we derivate three wave solutions in mesoscale atmosphere.
If we linearize the equation (1)-(3) by
The system becomes Time tendency of the density is replaced by the pressure tendency as
) ' ' 3 . ( ) ' ' ( ' ) ' ' 2 . ( ' ' '
1 ' 1
) ' ' 1 . (
A z w x u t A z p t w t u
A
x p
= + + = + +
=
∂ ∂ ∂ ∂ ρ ∂ ∂ρ ∂ ∂ ∂ ∂ ∂ ∂
ρ ∂ ∂ ρ
) 13 . ( ) ' ' ( '
2
A z w x u C t p
S
= + + ∂ ∂ ∂ ∂ ρ ∂ ∂
) 14 . (
2 2 2 2 2 2
2
A
S
Eliminating u’ and w’ from (A.1’’),(A.2’’),(A.13), we obtain Since phase speed normal to wave plain is
S
m k c = + =
2 2
ω
Cs means the sound wave speed Assuming the solution of p’ as
We obtain the following dispersion relation:
2 2
2 , , 2 , , 2 m k m c m k c k
z z x x
+ = = = = = π λ ω π λ ω π λ ) 15 . ( )
2 2 2
2
A m k
S
= + − ω
Relationships between the wave number and wave length and phase speed:
If we linearize the equation (1)-(3) by
The system becomes
) ' 4 . ( ' ' ) ' 3 . ( ' ' ) ' 2 . ( ' ' ' '
2
1 ' 1
) ' 1 . (
A N w t b A z w x u A b z p t w t u
A
x p
= + = + = + +
=
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ σ ∂ ∂
ρ ∂ ∂ ρ
where
Eliminating u’ from (A.1’) and (A.3’), and eliminating b’ from (A.2’) and (A.4’), we obtain the following equations for p’ and w’
2 2 2 2 2 2 2
1
ρ Further eliminating p’ we obtain the following elliptic equation for w’
) 7 . ( ' ) ' ' (
2 2 2 2 2 2 2 2 2
A x w N z w x w t = + + ∂ ∂ ∂ ∂ ∂ ∂ σ ∂ ∂
If we assume solution of w’ as
) 9 . ( , 2 , 2 , , 2
2 2 2 2
A m k c m k m c m k c k
z z x x
+ = + = = = = = ω π λ ω π λ ω π λ
Relationships between the wave number and wave length and phase speed are: Dispersion relation for gravitiy waves is
2 2 2 2 2
In hydrostatic system (σ=0 or m>>k), ω=kN/m, Cx= N/m Phase speed for horizontal direction is independent from the wave number, thus hydrostatic gravity waves do not have dispersibility horizontally. In nonhydrostatic case (σ=1 and m～k), ω becomes smaller than N. In the limit of k=∞, ( in case of vertical motion only) magnitude of ω becomes N
Steady state linear mountain wave is a special case where upward phase velocity of internal gravity wave is in equilibrium with the ambient wind U. If we linearize the equation (1)-(3) by
and assuming the steady state (time tendency is zero), the system becomes
2
1 ' 1
x p
=
ρ ∂ ∂ ρ
Eliminating u’ from (B1) and (B3), and b’ from (B2) and (B4), we
2 2 2 2
U
ρ
Further eliminating p’ we obtain the following elliptic equation for w’
) 7 . ( ' ) ' ' (
2 2 2 2 2 2
B w N z w x w U = + + ∂ ∂ ∂ ∂ σ
If we assume solution of w’ as in (A.8), we obtain the following (famous) Long’s (1952) equation:
) 8 . ( ' ) ( '
2 2 2 2
B w k l z w = − + σ ∂ ∂
where l=N/U is the scorer parameter.
In hydrostatic system (σ=0 or m>>k), the solution is periodic waves with the vertical wave length 2π/l without the dispersibility horizontally. In nonhydrostatic case (σ=1 and m～k), characteristic of the solution depends on the magnitude relation of l and k. In case k2 > l2, the wave amplitude decreases exponentially in vertical (external wave). In case k2 < l2, the waves become periodic ones with the following vertical wave length (internal wave).
) 8 . ( 2
2 2
B k l z = − = π λ
The real solution of mountain waves can be obtained by superposition of several waves with the corresponding wave numbers.
In case of 3-dimensional mountain waves, the set of equation becomes
2
1 ' 1 ' 1
y p x p
= =
ρ ∂ ∂ ρ ∂ ∂ ρ
) ' 7 . ( ) ' ' ( }] ' ) ' ' ( { [
2 2 2 2 2 2 2 2 2 2 2 2 2 2
B y w x w N z w y w x w x U = + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ σ ∂ ∂
and the resultant equation for w’ becomes
Example of linear mountain waves over a 3-dimensional ax-symmetric bell-shaped mountain (h=100 m) in the stable atmosphere (dθ/dz=3 K/km) of U=8m/s. Upper) hydrostatic case (half width a = 6 km), Lower) nonhydrostatic case a = 1.2 km Left) Vertical cross-section through mountain top, Right) Horizontal plain at 2.44 km AGL.
P
Substituting the reference density in the momentum and continuity equations, we obtain governing equations of the anelastic model
The anelastc (AE) model removes sound waves from solutions by a scale analysis. Field variables are divided into the time independent horizontal uniform reference state f (z) and its perturbation f' (x, y, z, t) as
2
s
Because the density perturbation can be divided into the perturbation
) 15 . 1 ( ' ' ' ' ' ) ( ) ' ( ) ( ' } ) ( { ' ) ( '
2 1 2 S p v C C p v C C C C
C p p p C C p p p p C C R p p p R p p p R p R p
p v p v p v
+ − = + − = + − = = =
−
θ θ ρ ρ θ θ ρ θ θ θ θ θπ ρ Momentum equation may be rewritten as
) 20 . 1 ( . ) ( = ⋅ ∇ + ∂ ∂ v ρ ρ t
The quasi-compressible model considers the compressibility of air and predicts the pressure from divergence, while the reference density is used for momentum equations. Using the relation of
, ' ' '
2 S
C p + − = θ θ ρ ρ
following pressure equation is obtained
) 21 . 1 ( }, ) ( {
2
t C t p
S
∂ ∂ − ⋅ ∇ − = ∂ ∂ θ θ ρ ρv
if the Exner function is used to represent the pressure, the pressure equation is given by ) ' 21 . 1 ( . ) (
2 2
dt d C C C R t d d
p S v
θ θ π π + ⋅ ∇ − = v Momentum equation is same as in the anelastic equation; (1.16).
) 4 . 1 ( , ) 2 . 1 ( , ) 1 . 1 ( , 1 RT p t d d g p t d d ρ ρ ρ ρ = ⋅ ∇ − = − ∇ − = v k v
The fully compressible model uses basic equations without the linearization by the reference atmosphere Since the fully compressible model includes sound waves in its solutions and allows the time change of the density, careful attention must be paid on computation of sound waves and computational accuracy in the finite discretization.
Table 1. Classification of nonhydrostatic models and their behaviors.
Adiabatic expansion Constant volume heating Constant pressure heating Classification Pressure Density Volume Pressure Density Volume Density Anelastic (AE) impossible invariant invariant invariant invariant Quasi-compressible without the thermal expansion term decrease invariant increase invariant invariant invariant invariant Quasi-compressible with the thermal expansion term decrease invariant increase increase invariant increase invariant Fully-compressible without the thermal expansion term decrease decrease increase invariant decrease invariant decrease Fully-compressible with the thermal expansion term decrease decrease increase increase invariant increase decrease
2
S
thermal expansion term
) 25 . 1 ( , . ) (
2
AE FP hP z P = ∂ ∂ + ∇ Nonhydrostatic models can be classified further by the treatment of sound waves. Taking total divergence of momentum equations, the following 3- dimensional Poisson-type pressure diagnostic equation is obtained. where P=p’, and h=g/Cs
2 , and r.h.s. is the forcing term by
divergence of advection and buoyancy terms. ) 25 . 1 ( 2 1 ) . . ' ( ) . . ( ) . . ( .
t t
DIV t w Adv w Dif g z v Adv v Dif y u Adv u Dif x AE FP
∆ −
∆ + − + + − + − = ρ ρ θ θ ρ ∂ ∂ ρ ρ ∂ ∂ ρ ρ ∂ ∂ Here, the last term of r.h.s. (DIV) is the divergence at the time step t-∆t, which is required for computational stability to adjust the divergence zero at next time step (Clark, 1977).
) 44 . 1 ( , . ' ) (
2 2
HE FP P e P h z z P = + ∂ ∂ + ∂ ∂
τ τ τ
The HE-VI (horizontally explicit vertically implicit) scheme treats sound waves implicitly only for the vertical direction. This scheme is often referred as the split-explicit method because sound waves are treated in a short time step ∆τ while low frequency modes and physical processes are treated in a long time step ∆t. Following 1-dimensional Helmholtz pressure equation is obtained The pressure equation (1.44) is formally similar to the pressure equation of the HI-VI scheme (1.36) except the Laplacian in the pressure equation is vertically 1-dimensional.
) ' 29 . 1 ( . ) ( ) ' 28 . 1 ( ) ( ) ' 27 . 1 ( ) ' 26 . 1 (
2 2
E FP z W y V x U C P FW P C g z W FV y P V FU x P U
S S
= + + + = + + = + = +
∆ + ∆ + τ τ τ τ τ τ τ τ
∂ ∂ ∂ ∂ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ
) 34 . 1 ( τ δ
τ τ τ τ
∆ − ≡
∆ +
A A A
For example, if we define time tendency term of A as Momentum equations and the pressure equations may be written as
) 38 . 1 ( } ) ( . { ) 37 . 1 ( } ) ( { ) 36 . 1 ( ) ( ) 35 . 1 ( ) (
2 2 τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ
∂ ∂ ∂ ∂ ∂ ∂ τ ∂ ∂ τ ∂ ∂ τ ∂ ∂ τ
∆ + ∆ + ∆ + ∆ + ∆ + ∆ +
+ + − ∆ + = + − ∆ + = − ∆ + = − ∆ + = z W y V x U C E FP P P P C g z FW W W y P FV V V x P FU U U
S S
Solving for Pτ+∆t ) 39 . 1 ( . ) ( 1 ) (
2 2 2 2
HE FP P C P C g z P z
S S
= ∆ − +
∆ + ∆ + ∆ + τ τ τ τ τ τ
τ ∂ ∂ ∂ ∂
) 40 . 1 ( ) ( 1 } ) {( 1 . 1 .
2 2 τ τ τ τ
τ ∂ ∂ ∂ ∂ ∂ ∂ τ ∂ ∂ τ P C z W y V x U FW z C E FP HE FP
S S
∆ − + + ∆ + + ∆ − =
∆ +
) 36 . 1 ( , . ' ) ' ( '
2
HI FP eP hP z P = + ∂ ∂ + ∇
The HI-VI (horizontally implicit vertically implicit) scheme treats sound waves implicitly for both vertical and horizontal directions. This scheme,
The following 3-dimensional Helmholtz-type pressure equation is used
) 31 . 1 ( . '
t t
P P P − =
where P’ is defined by Above pressure equation (1.36) is formally similar to the anelastic pressure equation (1.25).
) ' ' 29 . 1 ( . ) ( ) ' ' 28 . 1 ( ) ( ) ' ' 27 . 1 ( ) ' ' 26 . 1 (
2 2
E FP z W y V x U C P FW P C g z W FV y P U FU x P U
t t t S t t S t t t t t
= + + + = + + = + = + ∂ ∂ ∂ ∂ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ ∂ ∂ δ where the term –t is weight average of A at t-∆t and t+∆t
) ' 41 . 1 ( 2 1 2 1
t t t t t
A A A
∆ − ∆ +
− + + = α α
) ' 34 . 1 ( ) 1 ( ) ( ) 1 ( 2
2
t A A t A t A A A
t t t t t t t t
∆ + − + ∆ + ∆ = ∆ − ≡
∆ − ∆ − ∆ +
α α δ For example, if we define time tendency term of A as Momentum equations and the pressure equations may be written as
2 t t
If we define ∆2 as an operator Momentum equations and the pressure equations (1.26’’) ～(1.29’’)may be written as
) 46 . 1 ( ) ( ) 1 ( . ) ( ) 1 ( ) 45 . 1 ( ) ( ) 1 ( ) ( ) 1 ( ) 44 . 1 ( ) 1 ( ) 1 ( ) 43 . 1 ( ) 1 ( ) 1 (
2 2 2 2 2 2 2 2 2 2 2 2 2 2
z W y V x U C t P P E FP z W y V x U C t P P C g z t W W FW P C g z t W y P t V V FV y P t V x P t U U FU x P t U
t t t S t t t S t S t t t S t t t t t t t t
∂ ∂ ∂ ∂ ∂ ∂ ∆ α ∂ ∂∆ ∂ ∂∆ ∂ ∂∆ ∆ α ∆ ∂ ∂ ∆ α ∆ ∂ ∂ ∆ α ∆ ∂ ∂ ∆ α ∂ ∂∆ ∆ α ∆ ∂ ∂ ∆ α ∂ ∂∆ ∆ α ∆ + + + + − + = + + + + + − + − − = + + + − + − − = + + − + − − = + +
∆ − ∆ − ∆ − ∆ −
We obtain the following elliptic tendency equation for ∆2P
) 47 . 1 ( .
2 2 2 2 2 2 2 2 2 2
) 1 ( ) ( 1 ) ( ) (
2 2
HI FP P P z y x
t C C g z P
S S
=
+ − + + +
∆ ∆ ∂ ∂ ∂ ∂ ∂ ∂
α ∆ ∂ ∂ ∆
2
) ' ' ' ( .
S
z FW y FV x FU HI FP
∂ ∂ ∂ ∂ ∂ ∂
− + + =
and FU’,FV’,FW’,FP’ are r.h.s. of (1.43)～(1.46), respectively. where
Method Accuracy Comput ational robust- ness Pressure equation Scalability in parallel computati
Efficiency in large scale computati
Compatibi lity with semi- Lagrangia n scheme Compatibi lity with spectral method AE Anelastic approxi- mation Good 3D Poisson Depends
solver Depends
solver Good Good HI-VI Good Fair 3D Helmholtz Depends
solver Depends
solver Good Good HE-VI Good Fair 1D Helmholtz Good Good Sound waves exist Fair
, , ,
3 2 2 1 1
dz ds m d ds m d ds = = = η ξ
The governing basic equations of the model consist of the flux form equations on the spherical curvilinear orthogonal coordinate. Let m1 and m2 be the map factors in the ξ (x) and η (y) directions,
3 2 2 1 1
Momentum equation is , 1 2 F g p V dt V d + + ∇ − × Ω − = ρ ) ( ) ( ) ( , , η ξ η ξ η ξ η ξ η η ξ ξ η ξ ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + + + = + + + + + = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ k nv k mu w j nv j mu v i nv i mu u dt dw k dt dv j dt du i dt k d w dt j d v dt i d u dt dw k dt dv j dt du i dt V d z w nv mu t z dt dz dt d dt d t dt d z k n j m i where
Change of unit vectors along each coordinates are an j k am i k an k n m i j m n i j n m j i am k m n j i 1 , 1 , 1 ) 1 ( ), 1 ( ), 1 ( , 1 ) 1 ( − = ∂ ∂ − = ∂ ∂ − ∂ ∂ − = ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ = ∂ ∂ − ∂ ∂ − = ∂ ∂ η ξ ξ η η ξ ξ η η ξ where a is radius of earth. Curvature term is
a v u k a wv j n v m u mnu j a wu i n v m u mnv i k nv k mu w j nv j mu v i nv i mu u M
2 2
)} 1 ( ) 1 ( { )} 1 ( ) 1 ( { ) ( ) ( ) ( + − + ∂ ∂ + ∂ ∂ − + + ∂ ∂ − ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ξ η ξ η η ξ η ξ η ξ
a w z w m v n u mn z w a w n mnu v n a w m mnv u m z w k w i u n j v n k w j v m i u m w k v j u i z k n j m i V V dt d 2 )} ( ) ( { ) 1 ( ) 1 ( ) ( ) ( ) )( ( + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + + ∂ ∂ + ∂ ∂ + + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = + + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∇ = ∇ + η ξ ξ η η ξ η η η ξ ξ ξ η ξ ρ ρ Continuity equation is
, 1 , 1 , 1
3 3 3 2 2 2 1 1 1
Dif g z p Crv Cor dt dw Dif y p m Crv Cor dt dv Dif x p m Crv Cor dt du + − − + = + − + = + − + = ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ρ
where Set n=m in the curvilinear orthogonal coordinates equations , and notate ξ, η by x, y,
v c u c Cor u w c Cor w c v Cor λ ϕ λ ϕ ϕ λ ϕ λ ϕ ϕ ∆ Ω + ∆ Ω = Ω − ∆ Ω − = ∆ Ω − Ω = sin cos 2 cos cos 2 3 sin 2 sin cos 2 2 cos cos 2 sin 2 1
a w u Crv a vw m x v m y u u m Crv a uw m y u m x v v m Crv
2 2 3 2 2 2 1
)} 1 ( ) 1 ( { )} 1 ( ) 1 ( { + = − − = − − = ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂
, cos cos sin cos
, 1
∆ − ∆ + =
=
λ ϕ λ ϕ ma y ma x y x
p p
c
1) Polar stereo projection , sin 1 sin 1 ϕ ϕ + + = m
2) Lambert-conformal projection . ) 2 4 tan( ) 2 4 tan( ln{ / ) cos cos ln(
2 1 2 1
ϕ π ϕ π ϕ ϕ − − = c
p p
1 1 1 c c
−
Normally, ϕ1=π/6, ϕ2=π/3
3）Mercator projection
, ) cos sin 1 ( cos cos ,
ln
+ + ∆ + = =
ϕ ϕ ϕ ϕ
λ a y a x y x c
Continuity equation was ) 40 . 2 ( 2 )} ( ) ( {
2
prc a w z w m v y m u x m dt d = + + + + ρ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρ ρ
) 41 . 2 (
If we neglect the last term of l.h.s. of (2.40)by shallow assumption, we
) 42 . 2 ( ) ( )} ( ) ( { )} ( ) ( {
2 2
) (
prc w z m v y m u x m z w m v y m u x m
t z w y v x u m t
= + + = + + +
+ + + +
ρ ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρ
∂ ∂ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ ∂ ∂ρ
where
From (2.42)
) 45 . 2 ( ) ( )} ( ) ( {
2 2
= − + +
Prc m m w z m m v y m u x m
φ ρ ∂ ∂ φ ρ ∂ ∂ ρ ∂ ∂ φ φ
) 46 . 2 ( ) 1 ) ) ) 1
2 2 2 2
( ( ( (
Prc m w m m v m u m m
z y x t dt d
φ φ ρ φ ρ φ ρ ρφ ρ
∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ φ
−
+ + + = ) 47 . 2 ( ) )} ) )
Prc m m w m v m u m m
φ φ ρ φ ρ φ ρ ρφ ρ
−
thus
) 44 . 2 (
2 2 2
m w m v m u m m
ρ ρ ρ ρ ρ
For arbitrary variable φ, from (2.41)
Terms Adv.U, Adv.V, Adv.W：replace φ by u, v, w in Terms Crv.U, Crv.V, Crv.W , Cor.U, Cor.V, Cor.W：multiply ρ/m to the terms of the equations in conformal projection
) 50 . 2 ( ) 49 . 2 ( ) 48 . 2 (
. . . ) ( 1 . ) ( . . . . ) ( . . . . ) ( W Dif W Cor W Crv g z p m W Adv m w t V Dif V Cor V Crv y p V Adv m v t U Dif U Cor U Crv x p U Adv m u t + + = + + + + + = + + + + = + + ρ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρ ∂ ∂ ∂ ∂ ρ ∂ ∂
(2.31)～(2.33) become
Prognostic equation for potential temperature is where
) 51 . 2 (
θ
p
dt d
) 52 . 2 ( ) )} ) [ .
ρ ∂ ∂ρ θ θ ρ θ ρ θ ρ θ
m t Prc m m w m v m u ADV
− − =
underlined term is divergence.
From state equation
Here、Cm is the sound wave speed in case no liquid water Combining continuity equation Pressure equation is
p v C
C m
p p R p
/
) ( θ ρ=
t p C t t
m m m
∂ ∂ + ∂ ∂ − = ∂ ∂
2
1 θ θ ρ ρ
2
] ) ( )} ( ) ( { [
2 2
t prc w z m v y m u x m C t p
m m m
∂ ∂θ θ ρ ρ ∂ ∂ ρ ∂ ∂ ρ ∂ ∂ ∂ ∂ + − + + − =
In order to enhance the computational robustness, advection terms of momentum and potential temperature are split at small time step At the center of the Leapfrog time step, high-order advection terms are fully evaluated with the flux correction, and then second-order components are adjusted at each short time steps in the later half of the Leapfrog time integration
τ
small time step ∆τ large time step 2∆t ADV(kt) kt-1 (ns-1)/2+1 ns-1 Advection terms are fully evaluated by higher order difference with flux correction ∆t = 40sec for 10km model; corresponds to ∆t = 80 sec in RK2 Modified by lower order advection components
) 20 ( . ) 1 ( ) ( 1 1 ) 19 ( , . ) (
2 2 * 2 1 τ τ τ τ β β τ τ τ τ τ τ τ τ
σ ∂ ∂ τ θ θ θ θ π θ θ θ τ θ θ P mC g RW ADVLW ADVLW ADVW BUOY m P mC g z P mG W W t ADVL ADVL dif c Q ADVL ADVL ADV
m m p
− + − + − − = + + ∆ − ∂ ∂ + − = + + + − − = ∆ −
∆ + ∆ + ∆ +
The tendency term in (19) is given by a tentative time integration in the cloud microphysical process.
Uniform atmosphere with U=10m/s, N＝0.02/s Bell-shaped mountain h=100m、a=30km with horizontal resolution 10km
Cross-section of vertical velocity without time-splitting of advection. Left）∆t=30sec, t=10 hrs. Contour interval is 0.5 cm/s. Center） ∆t=40sec、t=2 hrs. Contour interval is1cm/s Right）∆t=50sec, t=10 hrs. Contour interval is 0.5 cm/s. With time-splitting of advection.
, } ) ( ) ( 1 { ) , (
2 3 2 2
a y a x h y x Z
m s
+ + =
Gravity waves sometimes become unstable when an inversion layer exists in strong wind environment. Vertical cross-section of 3 hrs forecast of NHM with ∆x=10km. 9 April 2001, 06UTC initial , ∆t=40 sec. Potential temperature and vertical wind.
Without time-splitting of advection With time-splitting of advection
, ) ( * , ) ( *
2 1 2 1 2 1 2 1
23 13
DIVT y RV ADVV z G P G G y P V V DIVT x RU ADVU z G P G G x P U U
H H
α ∂ ∂ ∂ ∂ τ α ∂ ∂ ∂ ∂ τ
τ τ τ τ τ τ τ τ τ τ
∂ ∂ + + − = + + ∆ − ∂ ∂ + + − = + + ∆ −
∆ + ∆ +
T DIV z RW ADVW BUOY m P mC g z P mG W W
Z m
α ∂ ∂ τ
β β τ τ τ
∂ ∂ + − − = + + ∆ −
∆ +
) ( 1 1
2 * 2 1
Based on the idea of Wicker and Skamarock (1992), while acts on the flux form total divergence where αH=0.06* (∆x)2/∆τ αZ((kz)=0.05* (∆z(kz))2/∆τ
) 24 ( ) ( ) 1 ( '
2 1
2 1
gG g G BUOY
m m
ρ ρ σ θ θ ρ σ − − + ≡
. 1 * 1
2
2 1
RW ADVW BUOY m mC P z P mG t W
m
+ − = + + σ ∂ ∂ ∂ ∂
σ=0: density perturbation, σ=1: potential temperature perturbation vertical momentum equation In case of σ = 0 , BUOY includes pressure perturbation, which has to be treated implicitly. The buoyancy term BUOY is defined by
Independent of σ, pressure perturbation term is treated implicitly as
) 29 . 2 . 2 ( . ) 1 ( ) ( 1 1
2 2 * 2 1
P mC g RW ADVW BUOY m P mC g z P mG W
m m
σ ∂ ∂ δ
β β τ
− + − − = + +
To split gravity waves, diagnosis of density is required in each short time step. ) 39 . 2 . 2 ( . } ) 1 ( { 1 ) ( ) 1 (
2 2 * 2 1
P C g BUOY m RW ADVW P mC g z mG
m m
σ ∂ ∂
β
− + + − − = + The last term in r.h.s. is evaluated in short time step. Upper and lower boundary condition of pressure equation is
– water vapor, cloud water, rain, cloud ice, and graupel (Qv, Qc, Qr, Qi, Qs, Qg) – number density of cloud ice, snow and graupel (Ni, Ns, Ng)
graupel rain snow cloud ice cloud water water vapor
Tetens’ formula for saturated vapor pressure [hPa]
t t
s
e
+ ×
× =
7 . 237 5 . 7
10 11 . 6
In bulk method, the size distribution function of water substance is expressed by the inverse exponential function of the particle diameter D.
D
λ −
Fall-out terminal velocity of particle is given as a power function of D by the Stokes’ law in the form of
b
The mass-weighted mean velocity is obtained by Here Γ(z) is the Gamma function
b D D b w w
b a dD e D dD e aD D dD D N D dD D N D V D V λ ρ π ρ π
λ λ
6 ) 4 ( ) ( 6 ) ( ) ( 6
3 3 3 3
+ Γ = = =
− −
2 1 1
∞ − − ∞ − − ∞ − −
t z t z t z
By Euler’s partial integration, the Gamma function has the following characteristic
t z
∞ − −
1