Energy in a fluid layer



The main purpose of this is to explore two different ways to find approximate equations for the mean value and moments of the temperature, J(x,y,z,t). Easier results from a relatively simple procedure will be followed by some of the corresponding results from a more difficult procedure. Both cases will be for a perfect gas, where R and cv are constants in the equation of state, p=rRJ, and in the thermal energy density.

It may be helpful here to make a distinction between densities of mechanical energy (kinetic plus potential) and total energy (mechanical plus thermal), according to

y(x,y,z,t)= 1
_
2		 (u2 + v2 + w2) + gz   and   e(x,y,z,t)= y + cvJ .


As in other parts of all this, J(x,y,z,t) is temperature, and its mean value is T(x,y,t). The dimensions of both y and e are energy per unit mass; the dimensions of ry and re are energy per unit volume. From

dz
__
dt		= w ,  rdu
__
dt		 + px = 0 ,  rdv
__
dt		 + py = 0  and   rdw
__
dt		 + rg + pz = 0 ,


it follows, as usual, that

rdy
__
dt		 + (up)x + (vp)y + (wp)z= p(ux+vy+wz)     (= - p
_
r		 dr
__
dt		 ).


In gasdynamics, the usual procedure is to introduce a radiant heat source, rqr, and a heat flux, j, in an equation for rde/dt, and then to use the result for rdy/dt to derive a thermal energy equation. For a perfect gas, cv is a constant, and the result is

rcvdJ
__
dt		 + pÑ·u + Ñ·j = rqr .


One of several different ways to go about deriving an approximate description of the vertical structure (z-dependence of r, J and p) that accompanies a circulation model is to use the equation displayed above to derive equations for DT/Dt and rates of change of a few other moments of J(x,y,z,t). The first two of those are

cv(m DT
__
Dt		 +ár~u~Jñx + ár~v~Jñy ) + ápÑ·uñ = mQr - áÑ·jñ

and
cv(m D
__
Dt		 ár~z~Jñ
____
m		 + ár~u~zJñx + ár~v~zJñy - árJ d~z
__
dt		 ñ) + á~zpÑ·uñ = ár~z~qrñ - á~zÑ·jñ .


(See The generic conservation law for an explanation of the notation and for other details.) The crucial issue of how the right-hand-sides of these two are to be evaluated remains to be addressed, but at least this is not an incorrect procedure. There is, however, another way to find equations for T and the moments of J, and it is not an incorrect procedure either.

Consider first the mean value of y, which is Y(x,y,t)=áryñ/m. The kinematic boundary conditions,

Bt + uBBx + vBBy = wB   and   Ht + uHHx + vHHy = wH ,


are both used twice in the derivation of

ár dy
__
dt		 + Ñ·(up)ñ = m DY
__
Dt		 + ár~u~y+upñx + ár~v~y+vpñy - BtpB + HtpH .


The term BtpB is the rate at which the prescribed motion of the "base or bottom" at B(x,y,t) does work on the fluid above it. In a one-layer model of an atmosphere, pH=0, and in hydraulics -HtpH is the rate at which the water layer does work on the atmosphere above it. From this it appears that there might be unexpected contributions to equations for the mean and moments of thermal energy if they were derived from the mean and moments of the total energy equation.

To make things as simple as possible, v and the coordinate y will be suppressed from here on, and pH will be set equal to zero. Then

mDY
__
Dt		 + (Uápñ)x + ár~u(~y+R~J)ñx = BtpB ,
where
Y(x,t)= 1
_
2		 (U2 + W2 + ár(~u2+~w2)ñ
________
m		 ) + gZ
and
~y(x,~z,t)= U~u + W~w + g~z .


Once more, a thermal energy density, rcvJ, a radiant heat source, rqr, and a heat flux, j are introduced, and the integral of the total energy equation is

m( DY
__
Dt		 + cv DT
__
Dt		 ) + (Uápñ)x + ár~u(~y+cp~J)ñx = BtpB + mQr - áÑ·jñ .


(Note cp=cv+R for a perfect gas.) The equation for T comes from the evaluation of DY/Dt by the use of

m DZ
__
Dt		 + ár~u~zñx = mW ,

m DU
__
Dt		 + ár~u2ñx + ápñx + pBBx = 0 ,

m DW
__
Dt		 + ár~u~wñx + mg = pB ,


and exact or approximate evaluations of ~u, ~w and ~J. More precisely, it is exact or approximate equations for d~q/dt that are needed to define DT/Dt, except in the simplest approximation where ~u=0 and ~w=0. (Incidentally, setting ~J=0 is every bit as unacceptable as setting ~z=0 — in a stationary atmosphere, temperature decreases more-or-less linearly with height, from JB to zero, and the effect is never negligible.)

In this crudest of approximations, where ~u=~w=0, the evaluation of DY/Dt is easy, and the result is

mcv DT
__
Dt		 + pB D
__
Dt		 (Z-B) + ápñUx = mQr - áÑ·jñ .


From the definition of G (=pB/m) and p=rRJ, the result can be written as

cv DT
__
Dt		 + G D
__
Dt		 (Z-B) - RT
__
m		 Dm
__
Dt		 = Qr - 1
_
m		 áÑ·jñ .


The further result, G(Z-B)=RT, which follows from á~zpzñ=0 in this approximation, then gives

cp DT
__
Dt		 - RT( DG
___
GDt		 + Dm
___
mDt		 ) = Qr - 1
_
m		 áÑ·jñ .


This is to be compared with the corresponding result from the other procedure,

cv DT
__
Dt		 - RT Dm
___
mDt		 = Qr - 1
_
m		 áÑ·jñ .


The idea that the effective gravity (pB/m) might have a place in equations for the mean and moments of thermal energy is certainly intriguing, perhaps even attractive, but this procedure is considerably more complicated than the other one.

A relation between the two procedures follows from a return to the case where ~u and ~w are not neglected after all. Then

m D
__
Dt		 ár(~u2+~w2)ñ
________
2m		 = ár(~u d~u
__
dt		 +~w d~w
__
dt		 )ñ - 1
_
2		 ár~u(~u2+~w2)ñx ,

r( d~u
__
dt		 + ~uUx - 1
_
m		 (ár~u2+pñx+pBBx)) + px = 0 ,
and
r( d~w
__
dt		 + ~uWx - 1
_
m		 (ár~u~wñx - pB)) + pz = 0 .


The result of the substitution of all that in the equation for D(Y+cvT)/Dt is a terrible mess, but finding that the coefficient of pB is Bt+uBBx-wB (=0) is a fairly simple part of it. Continuing on to recover

mcv DT
__
Dt		 + cvár~u~Jñx + ápÑ·uñ = mQr - áÑ·jñ


was worth doing once, just to verify that nothing had been missed in the equations for d~u/dt and d~w/dt. This indicates that the equations for DT/Dt agree in the ultimate approximation (none), where the exact equations for ~u and ~w are included. That was the first step in the proof that the two different sequences of approximations, are merely different. (The corresponding result for moments of J remains to be shown.)

And the results really are different. In the easy approximation there is a perfect analogy with gasdynamics in the analog of the caloric equation of state, which is

ápñ = mRT = Kmg ,

where g = cp/cv, and
DK
___
KDt		 = 1
___
cvT		 (Qr - 1
_
m		 áÑ·jñ).


The corresponding result from the more difficult approximation is

ápñ = mRT = KmaGd ,

where a = (cp+R)/cp, d = R/cp and

DK
___
KDt		 = 1
___
cpT		 (Qr - 1
_
m		 áÑ·jñ).


Another example of the differences between the two approximations where ~u=~w=0 is the linearized equations for waves that vary as cis(kx-wt) (cosine+isine, as I learned from Karl Menger so many years ago). In the easiest examples, B=0, Qr=0 and Ñ·j=0. Then for the case where ápñ=Kmg,

GZ = RT   and   cvTt + RTUx = 0  ® Wtt + G
_
Z		 W = - RG
__
cv		 Ux
and
ápñ = Kmg   and   Kt = 0  ®  Utt - gRTUxx = 0 .


The dispersion relation has the two branches: The second branch is a soundwave that will have to be filtered out of numerical simulations. On the first branch G/Z is the square of the Brunt-Väisälä frequency, and if the waves are not filtered out they appear in crude linear and nonlinear theories of mountain waves.

In the slightly more complicated approximation where ápñ=KmaGd, there is a term proportional to Wx in the equation for Utt, and the branches of the dispersion relation have the topology shown here. The outer branches are for a=9/7 (which goes with g=7/5), and the inner ones, originally included for another purpose, are very close to the preceding result. For now, the details can all be found here.