The fundamental equations of motion
This begins with a perfect gas, for which the equation of state is
p = rRTorpv = RT ,(1)
where p is pressure (force per unit area), r is density (mass per unit volume),
v (=1/r) is specific-volume (volume per unit mass), T is
absolute temperature (Kelvin or Rankine degrees), and R is a constant that depends on the
composition (molecular weight) of the gas. (See thermo for a brief
discussion of the thermodynamics of a perfect gas.)
For such a gas the first law of thermodynamics is
CvdT = TdS - pdv ,(2)
where 'd' stands for (d/dt), Cv, the specific heat at constant volume,
is a constant, and entropy is defined as
S = S0 +
óõt
t0Q/T dt'. (3)
In eq(3)
Q(t') contains the net rate of absorbtion (possibly negative) of radiant
energy, effects of friction (necessarily positive), and other heat sources
(e.g. latent heats).
(Note: coordinates, 'x,y,z', will be added presently.)
From eqs(2&3) (and p/T = R/v for a perfect gas) it follows that
Cv ln(T/T0) + R ln(v/v0) = S - S0 .(4)
Another form of the first law is
CpdT = (Cv + R)dT = TdS + vdp ,(5)
where Cp is the specific heat
at constant pressure. From eqs(5,3&1) it follows that
Cp ln(T/T0) - R ln(p/p0) = S - S0 .(6)
In the fluid mechanics of a perfect gas (to follow next) the 'd' in eq(2) appears in the energy
equation, where it stands for (d/dt). Eq(5) is sometimes used in the momentum equation(s),
and a 'd' there stands for (¶/¶x), (¶/¶y) or (¶/¶z).
Air, a mixture of several important components (N2, O2,
H2O in at least three phases, CO2, CH4, etc.),
is not a perfect gas, and this is definitely physical science fiction, but
it is not qualitatively wrong in the troposphere if the absence of H2O
is provided for with one model or another.
More will be said about the perfect gas in a perfect atmosphere later; for now it need only be mentioned
that the molecular weight for this model of air is near M = 29,
R = R/M
where R is the universal gas constant, and the specific
heat at constant volume is near Cv = (5/2)R.
Cartesian coordinates, 'x,y,z' will now be introduced, but a few important details that figure in
the mechanics of the atmosphere will be saved for later. Ultimately, the plan is
to come up with a numerical model that doesn't have systematic errors that lead to
qualitatively incorrect results. So we shall consider quite a few simpler fluid models, with known solutions
that can be used to test the numerical analysis. Alas, it's a long-winded process -- but fun.
For three-dimensional, inviscid gasdynamics, the essential variables are: density, r;
pressure, p; temperature, T; and velocity components, u, v
and w; all functions of (x,y,z,t). (Note: 'v' will no longer be used for '1/r', except
in Cv.) The equations that govern the flow occur in
several equivalent forms, and numerical models can be based upon various combinations of them. Here is
the form that is usually presented first:
| the fundamental conservation laws |
| entity | conservation law |
| 1 | mass | rt
+ (ru)x
+ (rv)y
+ (rw)z = 0 |
| 2 | momentum | (ru)t
+ (ruu)x + (ruv)y
+ (ruw)z + px = 0 |
| 3 | momentum | (rv)t
+ (rvu)x + (rvv)y
+ (rvw)z + py = 0 |
| 4 | momentum | (rw)t
+ (rwu)x + (rwv)y
+ (rww)z + pz + rg = 0 |
| 5 | energy | to be discussed later |
- note 1: The subscripts, 'x,y,z,t' denote partial derivatives of the various fields
with respect to the indicated arguments. Other subscripts will not denote derivatives
(e.g. Cv).
- note 2: If the gas has a component (e.g. H2O) that undergoes phase changes,
the right-hand-sides will no longer be 'zero's.
The (five) conservation laws all contain the identity,
(rq)t + (rqu)x
+ (rqv)y + (rqw)z
= r(qt + uqx + vqy + wqz).
(7)
where q includes '1,u,v,w', energy densities, and sometimes things like 'x,y,z'. The identity follows
from the first row, which can also be written as
rt + urx
+ vry + wrz
+ r(ux + vy + wz) = 0. (8)
Now qt + uqx + vqy + wqz is a directional
derivative that can either be written out as is, or it can be thought of as dQ/dt, where
Q(t) = q(X(t),Y(t),Z(t),t). Then, by the chain rule, the directions are given by
dX/dt = u, dY/dt = v and dZ/dt = w. Two alternate forms of the first
four conservation laws are:
| transport equations -- two ways |
| entity | Eulerian form | semi-Lagrangian form |
| 1 | mass |
rt + urx + vry
+ wrz + r(ux + vy + wz) = 0 |
dr/dt + r(ux + vy + wz) = 0 |
| 2 | momentum |
ut + uux + vuy
+ wuz + px/r = 0 |
dX/dt = u , du/dt + px/r = 0 |
| 3 | momentum |
vt + uvx + vvy
+ wvz + py/r = 0 |
dY/dt = v , dv/dt + py/r = 0 |
| 4 | momentum |
wt + uwx + vwy
+ wwz + pz/r + g = 0 |
dZ/dt = w , dw/dt + pz/r + g = 0 |
The energy equation can be done the long way, with a conservation law for the
sum of kinetic, gravitational and thermal energy densities, or the short way,
which is to equate the 'dq's in eqs(2&5) to qt
+ uqx + vqy + wqz (Eulerian) or to
dq/dt (semi-Lagrangian).
So
CvdT/dt + RT(ux +
vy + wz) - TdS/dt =0
or (9)
CpdT/dt - (1/r)dp/dt - TdS/dt = 0
and
dS/dt - Q/T = 0 , (10)
where Q (this time) contains the net rate of absorbtion of radiant energy
and the net contribution of some other heat sources, such as latent heats
and heats of fusion.
What has been left out so far is:
- water vapor, though latent heats may still be modeled in Q,
- viscosity, in the momentum eqs and in Q,
- diffusion of components of different molecular weights,
- heat conduction (intimately related to diffusion).