!
Revised Thursday, May 2, 2013!
1
A Non-Hydrostatic Model
Using Isentropic Coordinates
David A. Randall
Basic equations
The basic equations in height coordinates, without rotation and friction, are
DVh
1
= − ∇z p ,
Dt
ρ
(1)
Dw
1 ∂p
=−
−g,
Dt
ρ ∂z
(2)
∂
⎛ ∂ρ ⎞
⎜⎝ ⎟⎠ + ∇ z ⋅( ρ Vh ) + ( ρ w ) = 0 ,
∂t z
∂z
(3)
Dθ Q
θ ≡
= .
Dt Π
(4)
Here
D
is the Lagrangian time derivative, Vh is the horizontal velocity, ρ is density, p is
Dt
Dz
is the vertical velocity, g is the acceleration of gravity, θ is the
Dt
potential temperature, Q is the heating rate per unit mass, and Π is the Exner function, which
pressure, z is height, w ≡
satisfies
c pT = Πθ ,
(5)
where c p is the heat capacity of air at constant pressure, T is temperature, and
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κ
⎛ p⎞
Π = cp ⎜ ⎟ ,
⎝ p0 ⎠
(6)
where
κ≡
R
,
cp
(7)
and R is the specific gas constant. Finally, we include the prognostic equation for an arbitrary
scalar A , which is
∂
⎡∂
⎤
⎢⎣ ∂t ( ρ A ) ⎥⎦ + ∇ z ⋅( ρ Vh A) + ∂z ( ρ wA ) = ρ SA ,
z
(8)
where SA is the source of A per unit mass. Finally, we will need the ideal gas law, which is
p = ρ RT .
(9)
As shown in the QuickStudy on coordinate transformations, the horizontal pressure
gradient can be transformed to isentropic coordinates as follows:
1
1
1 ∂p
∇ z p = ∇θ p −
∇ z.
ρ
ρ
ρ ∂z θ
(10)
From (5), (6), and (9), we can show that
1 ∂p
∂Π
=θ
.
ρ ∂z
∂z
(11)
Eq. (11) immediately gives one form of the hydrostatic equation in θ coordinates, i.e.,
θ
∂Π
+g=0.
∂z
(12)
Use of (12 allows us to rewrite (10) as
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1
⎛ ∂Π
⎞
∇ z p = ∇θ M − ⎜ θ
+ g ⎟ ∇θ z .
⎝
⎠
ρ
∂z
(13)
where
M ≡ c pT + gz
= Πθ + gz
(14)
is the Montgomery potential. Eq. (13) shows that, for the hydrostatic case the horizontal
pressure-gradient force is a gradient, when expressed in θ -coordinates.
With the use of (11) and (13), we can now rewrite (1) and (2) as
DVh
⎡
⎤
⎛ ∂Π
⎞
= − ⎢∇θ M − ⎜ θ
+ g ⎟ ∇θ z ⎥ ,
⎝ ∂z
⎠
Dt
⎣
⎦
(15)
Dw
⎛ ∂Π
⎞
= − ⎜θ
+ g⎟ .
⎝
⎠
Dt
∂z
(16)
Using methods discussed in the QuickStudy on coordinate transformations, the continuity
equation, (3), can be written as
∂θ ∂ ρ ⎛ ∂z ⎞
∂θ ⎡ ∂
∂θ ∂
⎛ ∂ρ ⎞
⎤
ρ
V
⋅∇
z
+
( ρw ) = 0 ,
(
)
⎜⎝ ⎟⎠ −
⎜⎝ ⎟⎠ + ∇θ ⋅( ρ Vh ) −
h
θ
⎥⎦
∂t θ ∂z ∂θ ∂t θ
∂z ⎢⎣ ∂θ
∂z ∂θ
(17)
or
∂z ⎛ ∂ ρ ⎞
∂ ρ ⎛ ∂z ⎞
∂
⎡ ∂
⎤
⎜⎝ ⎟⎠ −
⎜⎝ ⎟⎠ + ∇θ ⋅( ρ Vh ) − ⎢ ( ρ Vh ) ⎥ ⋅∇θ z + ( ρ w ) = 0 .
∂θ ∂t θ ∂θ ∂t θ
∂θ
⎣ ∂θ
⎦
(18)
Note that
∂z ⎛ ∂ ρ ⎞
∂ ⎛ ∂z ⎞
⎡ ∂ ⎛ ∂z ⎞ ⎤
⎢ ∂t ⎜⎝ ρ ∂θ ⎟⎠ ⎥ = ∂θ ⎜⎝ ∂t ⎟⎠ + ρ ∂θ ⎜⎝ ∂t ⎟⎠ ,
⎣
⎦θ
θ
θ
(19)
and
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∇θ ⋅( ρ
4
∂z
∂z
⎛ ∂z ⎞
Vh ) =
∇θ ⋅( ρ Vh ) + ρ Vh ⋅∇θ ⎜ ⎟ .
⎝ ∂θ ⎠
∂θ
∂θ
(20)
Substituting (19) and (20) into (18), we obtain
⎤⎫
∂ ⎧ ⎡⎛ ∂z ⎞
⎛ ∂ ρθ ⎞
⎨ ρ ⎢⎜⎝ ⎟⎠ + Vh ⋅∇θ z − w ⎥ ⎬ = 0 ,
⎜⎝
⎟⎠ + ∇θ ⋅ ( ρθ Vh ) −
∂t θ
∂θ ⎩⎪ ⎣ ∂t θ
⎦ ⎭⎪
(21)
where
ρθ ≡ ρ
∂z
∂θ
(22)
is the pseudo-density. We recognize
⎡⎛ ∂z ⎞
⎤
µ ≡ − ρ ⎢⎜ ⎟ + Vh ⋅∇θ z − w ⎥
⎣⎝ ∂t ⎠ θ
⎦
(23)
as the upward mass flux across an isentropic surface. This allows us to rewrite (21) as
∂µ
⎛ ∂ ρθ ⎞
=0.
⎜⎝
⎟⎠ + ∇θ ⋅ ( ρθ Vh ) +
∂t θ
∂θ
(24)
In a similar way, the conservation equation for an intensive scalar, (8), becomes
∂
⎡∂
⎤
⎢⎣ ∂t ( ρθ A ) ⎥⎦ + ∇θ ⋅( ρθ Vh A) + ∂θ ( µ A ) = ρθ SA .
θ
(25)
The advective form can be obtained by combining (39) with (38):
µ ∂A
⎛ ∂A ⎞
= SA .
⎜⎝
⎟⎠ + Vh ⋅∇θ A +
∂t θ
ρθ ∂θ
(26)
As a special case of (26), the advective form of the potential temperature equation is
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µ = ρθθ ,
(27)
where we have used θ ≡ Sθ . By using (27), we can rewrite (25) as
∂
⎡∂
⎤

⎢⎣ ∂t ( ρθ A ) ⎥⎦ + ∇θ ⋅( ρθ Vh A) + ∂θ ( ρθθ A ) = ρθ SA .
θ
(28)
The total time derivative is given by
D
(
Dt
) = ⎛⎜⎝
∂⎞
⎟ (
∂t ⎠ θ
) + Vh ⋅∇θ ( ) + θ
∂
( ).
∂θ
(29)
The prognostic equations needed to describe non-hydrostatic motions in θ -coordinates
can now be summarized as follows:
DVh
⎡
⎤
⎛ ∂Π
⎞
= − ⎢∇θ M − ⎜ θ
+ g ⎟ ∇θ z ⎥ ,
⎝ ∂z
⎠
Dt
⎣
⎦
(30)
Dw
⎛ ∂Π
⎞
= − ⎜θ
+ g⎟ .
⎝
⎠
Dt
∂z
(31)
∂
⎛ ∂ ρθ ⎞
⎜⎝
⎟⎠ + ∇θ ⋅ ( ρθ Vh ) + ( ρθθ ) = 0 ,
∂t θ
∂θ
(32)
∂z
⎛ ∂z ⎞
⎜⎝ ⎟⎠ = −Vh ⋅∇θ z + w − θ ,
∂t θ
∂θ
(33)
∂
⎡∂
⎤

⎢⎣ ∂t ( ρθ A ) ⎥⎦ + ∇θ ⋅( ρθ Vh A) + ∂θ ( ρθθ A ) = ρθ SA .
θ
(34)
Not counting the scalar A , the prognostic variables of the θ -coordinate model are Vh , w , ρθ ,
and z . The corresponding prognostic variables of the z -coordinate model are Vh , w , ρ , and θ .
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Finally, we have to determine Π . From ρθ and z , we can find the density ρ . Then the
equation of state in the form
κ
⎛ ρ Rθ ⎞ 1−κ
Π = cp ⎜
⎝ p0 ⎟⎠
(35)
can be used to diagnose Π .
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References and Bibliography
Bleck, Rainer, 1973: Numerical Forecasting Experiments Based on the Conservation of Potential
Vorticity on Isentropic Surfaces. J. Appl. Meteor., 12, 737–752. doi:
http://dx.doi.org/10.1175/1520-0450(1973)012<0737:NFEBOT>2.0.CO;2
Hsu, Y.-J. G., and A. Arakawa, 1990: Numerical Modeling of the Atmosphere with an Isentropic
Vertical Coordinate. Mon. Wea. Rev., 118, 1933-1959. doi:
http://dx.doi.org/10.1175/1520-0493(1990)118<1933:NMOTAW>2.0.CO;2
Kasahara, Akira, 1974: Various Vertical Coordinate Systems Used for Numerical Weather Prediction. Mon. Wea. Rev., 102, 509–522. doi:
http://dx.doi.org/10.1175/1520-0493(1974)102<0509:VVCSUF>2.0.CO;2
Konor, Celal S., Akio Arakawa, 2000: Choice of a Vertical Grid in Incorporating Condensation
Heating into an Isentropic Vertical Coordinate Model. Mon. Wea. Rev., 128, 3901-3910.
doi: http://dx.doi.org/10.1175/1520-0493(2001)129<3901:COAVGI>2.0.CO;2
Rossby, C. G., and collaborators, 1937:Isentropic Analysis. Bull. Amer. Meteor. Soc., 18, 201209.
Shapiro, M. A., 1975: Simulation of Upper-Level Frontogenesis with a 20-Level Isentropic Coordinate Primitive Equation Model. Mon. Wea. Rev., 103, 591–604. doi:
http://dx.doi.org/10.1175/1520-0493(1975)103<0591:SOULFW>2.0.CO;2
Starr, Victor P., 1945: A QUASI-LAGRANGIAN SYSTEM OF HYDRODYNAMICAL EQUATIONS. J. Meteor., 2, 227–237. doi:
http://dx.doi.org/10.1175/1520-0469(1945)002<0227:AQLSOH>2.0.CO;2
Trevisan, Anna, 1976: Numerical Experiments on the Influence of Orography on Cyclone Formation with an Isentropic Primitive Equation Model. J. Atmos. Sci., 33, 768–780. doi:
http://dx.doi.org/10.1175/1520-0469(1976)033<0768:NEOTIO>2.0.CO;2
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