7.6 Primitive Equation Models
375
neglected in the horizontal-momentum equations, and the radial distance between
any point within the atmosphere and the center of the Earth is approximated by
the mean radius of the Earth. Taken together, these approximations yield a system
that conserves both energy and angular momentum (Lorenz 1967, p. 16).
The primitive equations goveming inviscid adiabatic atmospheric motion may
be expressed using height as the vertical coordinate as folIows. Let x, y, and z be
spatial coordinates that increase eastward, northward, and upward, respectively.
Let u = (dx/dt , dy/dt) be the horizontal velocity vector, f the Coriolis parameter, k an upward-directed unit vector parallel to the z-axis, and Vz the gradient
with respect to x and y along surfaces of constant z. Then the rate of change of
horizontal momentum in the primitive equation system is govemed by
du
-
I
+ fk x u + - Vzp = 0,
dt
P
The continuity equation is
dO
ßO + u . V 0 + W - .
ßO
-
= -
dt
ßt
z
ßz
ßp
a; + V z(pu) +
ßpw = 0,
where
(7.97)
and the thermodynamic equation may be written
dT _
dt
_ 0
P - ,
(7.98)
where w = dpfdt is the change in pressure following a fluid parcel. The preceding system of equations for the unknown variables u, w, p, co, p, and T may be
closed using the hydrostatic relation (7.91) and the equation of state p = pRT.
7.6.1 Pressure and a Coordinates
The primitive equations are often solved in a coordinate system in which geometric height is replaced by a new vertical coordinate sex,y, z, t) . Simple functions
that have been used to define s include the hydrostatic pressure and the potential
temperature. The most commonly used vertical coordinates in current operational
models are generalized functions of the hydrostatic pressure .
The primitive equations may be expressed with respect to a different vertical
coordinate as folIows. Suppose that sex, y,z, t) is the new vertical coordinate and
that s is a monotone function of z for all fixed x, y , and t with a unique inverse
z(x, y , S, t) . Defining VI; as the gradient operator with respect to x and y along
surfaces of constant sand applying the chain rule to the identity
p[x, y, z(x , y, S, t) , tl = p(x , y, S, t)
yields
375
neglected in the horizontal-momentum equations, and the radial distance between
any point within the atmosphere and the center of the Earth is approximated by
the mean radius of the Earth. Taken together, these approximations yield a system
that conserves both energy and angular momentum (Lorenz 1967, p. 16).
The primitive equations goveming inviscid adiabatic atmospheric motion may
be expressed using height as the vertical coordinate as folIows. Let x, y, and z be
spatial coordinates that increase eastward, northward, and upward, respectively.
Let u = (dx/dt , dy/dt) be the horizontal velocity vector, f the Coriolis parameter, k an upward-directed unit vector parallel to the z-axis, and Vz the gradient
with respect to x and y along surfaces of constant z. Then the rate of change of
horizontal momentum in the primitive equation system is govemed by
du
-
I
+ fk x u + - Vzp = 0,
dt
P
The continuity equation is
dO
ßO + u . V 0 + W - .
ßO
-
= -
dt
ßt
z
ßz
ßp
a; + V z(pu) +
ßpw = 0,
where
(7.97)
and the thermodynamic equation may be written
dT _
dt
_ 0
P - ,
(7.98)
where w = dpfdt is the change in pressure following a fluid parcel. The preceding system of equations for the unknown variables u, w, p, co, p, and T may be
closed using the hydrostatic relation (7.91) and the equation of state p = pRT.
7.6.1 Pressure and a Coordinates
The primitive equations are often solved in a coordinate system in which geometric height is replaced by a new vertical coordinate sex,y, z, t) . Simple functions
that have been used to define s include the hydrostatic pressure and the potential
temperature. The most commonly used vertical coordinates in current operational
models are generalized functions of the hydrostatic pressure .
The primitive equations may be expressed with respect to a different vertical
coordinate as folIows. Suppose that sex, y,z, t) is the new vertical coordinate and
that s is a monotone function of z for all fixed x, y , and t with a unique inverse
z(x, y , S, t) . Defining VI; as the gradient operator with respect to x and y along
surfaces of constant sand applying the chain rule to the identity
p[x, y, z(x , y, S, t) , tl = p(x , y, S, t)
yields