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Gill's Chapter 4
1. What is a material element? [p.63]. [material element = infinitesimally small fluid sample that retains its identity.]

2. Why is it that if molecular diffusion is =0, the material element will consist of the same particles? What are "particles" in this case? [p.64]. [If diffusion=0, then molecules (=particles) cannot spread, so material element consists of the same particles.]

3. Why is (4.2.1) valid (i.e. why can the LHS be broken up into the 3 terms on the RHS)?
[Use d(ab)/dk = adb/dk+bda/dk).]

4. Derive (4.2.3), (4.2.4) & (4.2.5).
5. What is the difference between advection and convection? [p.66]
6. Derive (4.1.8) from (4.3.2) and (4.2.4) (== (4.2.3)).
7. Derive (4.2.5) [Use Fig.4.2].
8. Explain the flux vector F in (4.3.3) - as advective + diffusive fluxes.
9. Note how (4.3.4) [or 4.3.7] is similar to the Salinity equation used in POM. Is it identical? Why (yes or not)?
10. The entropy of a material element is fixed if the element is moved
(a) without exchanging heat with its surrounding and if
(b) it retains its fixed composition.
The motion is then called isentropic.
11. So, the salinity or 'salt' equation is fairly straight-forward, since the Qv on p.67 = rho*s = mass of salt per unit volume; and the "mass of salt" is a 'measurable' scalar quantity - i.e. a "mass."
12. Temperature is somewhat different however. It is still a scalar of course, but it is closely tied to the agitation of molecules and therefore also to the entropy or "heat content." Temperature is derived from the "heat equation." To understand this, we need to explore a little thermodynamics.
13. The "heat" equation is obtained from the 2nd law of thermodynamics:

dE = Tdpdvs(3.2.1)

E = internal energy per unit mass

Vs = specific volume (volume per unit mass)

T and p = temperature and pressure respectively

 = specific entropy (entropy per unit mass)

Energy increases because of (i) agitation (molecules) d, and (ii) compression dvs0. We assume no phase change and salinity is fixed – i.e. the fluid is of fixed composition; so  = (p, T).

Tdchange in heat content per unit mass

pdvs = work done by p in compressing fluid’s volume per unit mass.

14. Eqn.(3.2.6) is used often later – so we will derive it. From (3.2.1):

Td = dE + pdvs(N4.1)

So, increase in heat content for a change in T while keeping p constant is then, from (N4.1):

CpT(/T)p= (E/T)p + p(vs/T)p (N4.2)

The Cp is called the specific heat at constant pressure.

Similarly, we obtain the following for increase in heat content for a change in p while keeping Tconstant:

T(/p)T = (E/p)T + p(vs/p)T(N4.3)

Now, recall that  = (p, T), the Td (i.e.LHS of (3.2.6)) is:

Td = T(/T)pdT + T(/p)Tdp

= CpdT + T(/p)Tdp(N4.4)

where the definition of Cp from (N4.2) has been used for the first term. For the second term,T(/p)T, we may use (N4.3), but we can get rid of the “E” as follows. We can [T(N4.3)p(N4.2)]:

(/p)T +T(/pT)T(/Tp)= (E/pT)(E/Tp)+p(vs/pT)p(vs/Tp)(vs/T)p

so that,

(/p)T= (vs/T)p (3.2.5)

which can be used in (N4.4) to give (3.2.6):

Td = CpdTT(vs/T)pdp.(3.2.6)

15. A process is reversible if it changes very, very slowly due to infinitesimal changes in some property of the system, so that no dissipation occurs. For example, if a gas in an insulated cylinder is compressed by pushing a piston (see Fig.N4.1) and there is no friction between the piston and the cylinder wall, then if the piston is let go the compressed gas would expand back so the piston is reversed to its original position. If there is friction, then heat is permanently loss and the process is irreversible.

In practice, no process is reversible. But if the applied changes are slow and/or if the system’s response is fast, then the process may be considered as being reversible. Gill assumes this [p.42, approx. line 6].

Fig.N4.1

16. Since Td = increase in heat content (of the system), then if the process is isentropic and reversible, then it is also called adiabatic. In our case, we assume “reversible processes,” so isentropic = adiabatic.

Thermodynamic Definition: An adiabatic process a process in which no heat is transferred to or from the working fluid.

17a. When a parcel of fluid sinks adiabatically, say into the deep ocean, it experiences increasedpressure and therefore warms up. Similarly, when air parcel rises (to mountain top) it experiences less pressure and cools down. In this case, “adiabatic” means that there is no turbulent or molecular diffusion and no radiation across the parcel’s boundary, and also no change in composition or phase.

Thus, in the case of adiabatic sinking or rising fluid parcel, because of pressure compression (expansion), DT/Dt0; it would be nice to define a quantity such that the D(.)/Dt = 0.

Also, in both cases (sinking or rising parcel), the (vertical) column of fluid (air) is apparently unstable (i.e. “warm” is under “cool”). But this is not true – i.e. the fluid is actually stable. To avoid this apparent confusion, and to make D(.)/Dt = 0, the quantity potential temperature is introduced.

The potential temperature “” is the temperature of the parcel at pressure pwould have if the parcel were brought adiabatically (i.e. isentropically; i.e. without exchange of heat with its surrounding) to a reference (i.e. common) pressure pr.

By using , we eliminate the apparent warming or cooling due to changes in pressure. Since now the various temperatures at different depths have been brought to a common pressure, we can compare them without worrying that the ’s represent false temperatures. The variation of  with height can therefore be used to judge if the vertical column is statically stable or not.

The situation is a bit similar to trying to judge if person (A) on Jupiter is heavier than person (B) on Pluto (Fig.N4.2). It would appear that (A) is heavier than (B). But to more fairly judge them, we can move them (without loosing even a single hair; constant entropy) all to a reference place, say the earth, then re-weigh them. The weight measured on earth may then be called “potential weight.”

Fig.N4.2

17b. An example and a simple exercise will make the concept of potential temperature clearer. It is convenient to do this with air than with water because explicit formulae may then be obtained.

Suppose we follow an air parcel which sinks adiabatically (i.e. isentropically) from a high mountain to the ground. For isentropic process, we have from (3.2.6) with d = 0:

dT = T(vs/T)p,s dp/Cp = T(/T)p,s dp/(2Cp). (N4.5)

Assume that air obeys the ideal-gas law:

 = p/(RT)(N4.6)

[note that high p gives rise to high T and also smaller volume vs = -1]. Then

 = (/T)p,s /

where  = thermal expansion coefficient and  = adiabatic lapse rate:

 = -1 (/T)p,s0 (3.6.2)

18. Static Stability (section 3.6).

Suppose the fluid is at rest; to judge a parcel’s static stability, we move the parcel up or down isentropically, calculate its density change, d, then compare this change with the change in the surrounding (i.e. ambient) density, da. If the parcel is moved up, then for stability, d must > da so that the parcel can sink back to its original depth when let go.

The general formula for the change in densityd(p,T,s) of the fluid (ambient or parcel fluid) as a result of a change in height “dz”is, by chain-rule:

d(p,T,s) (d/dz).dz = [(/p)T,s (dp/dz) + (/T)p,s (dT/dz) + (/s)p,T (ds/dz) ].dz(3.6.8)

where s = salinity. For a parcel of fluid that is moved isentropically, with  = (p, T), and fixed s; see “13”, (3.6.8) becomes:

d = (/p)T,sdp + (/T)p,sdT (N4.6T)

We can replace “dp” and “dT” with “dz” as follows. From (3.2.6), for isentropic process (d = 0):

dT = T(vs/T)p,s dp/Cp = T(/T)p,s dp/(2Cp) = Tdp/(Cp) = dz (3.6.1)

where = thermal expansion coefficient and  = adiabatic lapse rate:

 = -1 (/T)p,s0 (3.6.2)

 = gT/Cp,(3.6.5)

and the last equality in (3.6.1) (involving ) is obtained by using the hydrostatic equation:

dp/dz = g(N4.7T)

Substituting “dT” from (3.6.1) and “dp” from (N4.7T) into (N4.6T), that equation becomes:

d =  [g(/p)T,s + ]dz(3.6.7)

Reminder: equation (3.6.7) is the change in density, dpar, of the fluid parcel as it is moved. On the other hand, equation (3.6.8) is general, and gives (for example) the change, damb,of the ambient (i.e. surrounding) fluid density with z. Suppose that the parcel is moved upward (i.e. dz0), we must have, for stability, that dpardamb, so that the parcel can move back (i.e. downward) to its original position when let go:

 [+ dT/dz] sds/dz > 0(3.6.9)

or

Cp-12gT + dT/dzsds/dz > 0(3.6.10)

where

s = -1 (/s)p,T0(3.6.11)

is the expansion coefficient for salinity (I use s to avoid possible confusion with the planetary beta ).

Exercise 18.1: Show that the same stability condition is obtained if similar arguments as above are applied but the parcel is moved downward instead of upward.

19. The buoyancy frequency (or BruntVaisala frequency; section 3.7) squared is:

N2 = g [( + dT/dz) sds/dz] = g[Cp-12gT + dT/dzsds/dz]

Suppose we move the parcel up by dz0.

18. Potential temperature  is defined as

The