Nonrotating models
The
conservation laws for a perfect atmosphere, where R and
cv are constants, are here in
eqs1.
The formal conservation law,
(rz)t | + |
(ruz)x | + |
(rwz)z | = |
rw | , |
is the one that leads to a very convenient way to find an equation for the
rate of change of the mean height of the atmosphere. The temperature has been
written as J(x,z,t), so that T(x,t) will be free for use as the mean value of it.
After the introduction of the material derivative,
the conservation laws are rewritten as the transport equations in
eqs2.
The last four of the equations are different ways to express the rate of change
of the specific internal energy density, e=cvJ, and the relation,
p=arg, is usually called the caloric equation of state.
Mean values of the fields, q(x,z,t), and equations to govern them are defined
in terms of:
áqñ is the integral, òq(x,z,t)dz, from the base of the atmosphere,
z=B(x,t) (usually independent of t), to the upper edge, z=H(x,t) (possibly
infinite), where p, r and J are all zero.
qH(x,t)=q(x,H,t) and qB(x,t)=q(x,B,t)
m(x,t) (not a mean value) is árñ.
Q(x,t)=árqñ/m is the mass-weighted mean value.
~q(x,z,t)=q-Q is the deviation.
T(x,t)=árJñ/m is the exceptional notation.
The identity that leads to equations of motion for mean
values is
á(rq)t | + |
(ruq)x | + |
(rwq)zñ | = |
árqñt | + |
áruqñx |
The result follows from
rH(wH-Ht-uHHx)=0 and
rB(wB-Bt-uBBx)=0.
The exact conservation laws for mean values (q=1,z,u,w,e) are:
eqs3.
(Note that the lumped heat source Qr contains both the mean
value of qr and whatever model may be adopted for
nonradiative heat transport.)
The one-dimensional material derivative is,
and the transport equations for the mean values are these, in
eqs4.
The quantity that acts like pressure in equations for mean values is
ápñ, and the analog of the perfect gas law is
where T(x,t) is the mean value of the temperature, J(x,z,t).
And now it gets messy: The transport equations for momentum components are
to be used to derive equations for analogs of specific internal energy and entropy,
but the influence of gravitational potential energy has to be included in the process.
An unusual connection between thermal and mechanical properties of the layer comes from
the first moment of the vertical momentum equation. With
(rzq)t | + |
(ruzq)x | + |
(rwzq)z | = |
rz(qt+uqx+wqz) |
+ | rqw , |
it follows directly from the equation for w that
árzwñt | + |
áruzwñx | + |
ázpzñ | + |
árgzñ | = | árw2ñ . |
After the introduction of q = Q + ~q and an
integration by parts in which HpH = 0, comes
where
From the expression for E,
and after the substitutions for material derivatives of U, W and Z have been made, the equation for T is
where
and
At this point no approximation has been made in the derivation of these equations from
eqs1
and/or
eqs2.
The simplest approximate result follows from the very crudest approximation where the
deviations, ~u and ~w, are both neglected.
Then
implies
The further relation, Dápñ/ápñDt=Dm/mDt+DT/TDt, then gives the analog
of the caloric equation of state,
with a=2-(1/g), d=1-(1/g) and g=cp/cv.
For a diatomic gas, g is 7/5, a is 9/7 and d is 2/7.
The further relation,
R=dcp, has been used
in the collected equations of the crudest approximation for a nonrotating
atmosphere,
the MD approximation.
The last parameter is the variable,
w2=gG2/RT=gmG2/ápñ,
and w is this theory's
version of the Brunt-Väisälä frequency. The coresponding, rotating version is
here.
The linearized version of this approximation is here.
The nonlinear version with first moments is here.
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