The generic conservation law
The atmosphere models are defined by conservation laws that govern
the evolution of a number of scalar and vector fields q(x,y,z,t).
In the simplified case where nothing depends upon the coordinate y,
the flow takes place in a layer where
and the generic quantities are q(x,z,t). Some results of the simplified case are
collected here; their generalizations for three-dimensional, unsteady flows
are usually pretty straightforward.
The fields that appear in all of the conservation laws are
- the density,
r(x,z,t),
- the horizontal component of velocity, u(x,z,t),
- the vertical component of velocity, w(x,z,t).
The generic conservation law is
(rq)t | + |
(ruq)x | + |
(rwq)z | + |
P | = | rs . |
In this
- the subscripts, t,x and z, denote partial derivatives,
P(x,z,t) is a rate of loss of rq — per unit volume,
s(x,z,t) is a rate of gain of rq — per unit mass.
A specific conservation law that will be needed is
| q = 1 | ® |
rt | + |
(ru)x | + |
(rw)z |
= | 0 . |
From this follows the generic transport equation,
Mass-weighted mean values and moments of q(x,z,t) are defined by various integrals
of the form
árqñ(x,t) | = |
H
ó
õ
B | rq(x,z,t)dz |
With the notation, m(x,t) = árñ (not a mean value), the mean
values are
Z(x,t) = árzñ/m and Q(x,t) = árqñ/m ,
and the deviations are
~z(x,z,t)=z-Z and
~q(x,z,t)=q-Q .
The conservation law for the generic mean value, Q(x,t), that follows from
ár~qñ = 0 is
The (kinematic) boundary conditions that are used in the derivation
of the conservation law for Q are
(rq)B(Bt |
+ | uBBx |
- | wB) | = | 0 |
with |
qB | = | q(x,B(x,t),t) , |
|
(rq)H(Ht |
+ | uHHx |
- | wH) | = | 0 |
with |
qH | = | q(x,H(x,t),t) . |
More general cases where there is transport of mass (e.g. rain) across the
boundaries at z=B and z=H can easily be concocted, but only the simpler
case will be considered here. The special case where q = Q = 1 now gives
and the transport equation for Q is
The simplest conservation laws and transport equations after the ones for q=1 are
and
The plan to be followed here is to use the powers of the vertical
deviation, ~z, to construct a set of orthogonal
polynomials, {jk(x,~z,t)}, for
which
árjjjkñ =
NjNkdjk
with Ni(x,t) =
Öárji2ñ.
Both the N's and the coefficients in
vary with x and t because of r(x,Z+~z,t), B(x,t) and H(x,t).
The process is just Gram-Schmidt,
but it is rather more complicated than usual, and carrying on beyond k=1 or k=2
may not be worth the effort. In any case, by the usual procedure for a Galerkin
method, the nth polynomial approximation of q(x,z,t) is
and ár(q-Qn)2ñ is minimized when
| qnj(x,t) | = |
qj(x,t) | = |
árqjjñ |
/ |
Nj2 . |
The only break here is that there is final determination of the
expansion coefficients, qk(x,t), but finding the N's
and C's is still a terrible mess. When the convention,
Ckk=-1 is adopted, the final, disgusting result is
To begin the task of finding transport equations for qk(x,t), consider
where jk' = djk/d~z .
From this it follows that
The relation that can now be used to find the transport equation for qk is
and the final result is
| = | árjk2ñsk
| + | árqjk'(w -Zt-uZx)ñ |
- | 2árjkjk'(w -Zt-uZx)ñqk . |
In summary, the ingredients of the nth polynomial approximation of
q(x,z,t) are:
The approximations depend ultimately on the moments,
ár~zkñ,
and those can be found in several different ways. The most direct procedure,
and in my opinion the most dangerous, is to use
to write conservation laws or transport equations for the moments of q=1.
A problem with this is that the artificial
conservation laws for ár~zkñ (after
árzñ=mZ and
ár~z2ñ, which are essential) are
extra sources
of truncation errors in numerical simulations of the circulations. One worries
that the introduction of too many artificial conservation laws may lead to serious degradation
of the quality of the simulations.
A procedure that may lead to more reliable results is to use the mean values and a
few moments of z, u, w and J to construct updated evaluations
of r(x,z,t) and p(x,z,t). In other words, the plan is to extend the usual notion of
the standard atmosphere, in which u=w=0 and pz=-rg, to a vertical
structure model that incorporates approximations of the vertical accelerations that are
present in the flows.
In one approach to finding models of vertical structure
there is the additional z-dependent field
l(x,z,t)=lnr, whose mean value is L(x,t) and whose
transport equation is
The companion that couples this with the rest of the flow is
The substitution of the evaluations of du/dt and dw/dt then gives a transport equation that is in
the generic form. In the simplest of all possible worlds, the familiar result is
A different approach to models of vertical structure is discussed here.