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Geophysical Fluid Dynamics I 22 Jan 2004 Lab 3 2004 Coriolis – II, , change, latent , thermal equation, pδv .

1. Coriolis: We want to see geostrophic balance. Use the free-surface ‘ gauge’ to see the pressure variation in eddies on the big rotating table. Using reflected light, viewed with the video camera overhead, quite small deflections of the free surface can be seen. The appearance is not unlike the TOPEX/Poseidon satellite altimetry images of the world ocean.

2. Equation of state: water The curved function ρ(T,S,p), density as a function of , salinity and pressure can be explored. In Lab I one of the experiments measured the buoyancy force on a glass bulb immersed in water: ice was added to the water and the bulb then weighed less! Today we take an Erlenmayer flask (sloping sides) with a ‘chimney’, that is, a glass tube sticking out the top. The system is filled with water, initially in two layers of differing temperature. The height of the water in the chimney is marked, and then the two layers are mixed together with a magnetic stir-bar. The height of the fluid in the chimney would stay the same if the equation of state were linear, but (see plots in previous handout) it is not. The resulting change is a very sensitive measure of the curvature of the equation of state. The effect of the nonlinear equation of state in Nature can be quite important.

Now the experiment is repeated for salt stratification, a fresh layer floats on a salty layer, and the change in of the entire fluid is observed when the two are mixed…same idea. So the equation of state is curved (nonlinear) in both directions on the ρ(T,S) plane. There are also pressure nonlinearities. A fascinating form of convection occurs in the ocean: if two water parcels with the same density, yet different T and S, are mixed, the mixture will be more dense than either parcel was initially, so this can produce vertical convection.

3. Equation of state: gas The perfect gas equation of state is p = ρRT where R = 287.04 Joule kg-1 0C-1 for dry air. For water vapor, R = 461.5 Joule kg-1 0C-1. The state of an is determined by two state variables, and if we introduce , η, the equation of state becomes γ p =ηρ exp( /Cv ) (1) where γ is the ratio of specific , γ = Cp/Cv = 1.4 for a diatomic gas (also, Cp-Cv=R). For a gentle ‘reversible’ process, entropy is defined by δ 'qT=ηδ So equation (1) is very useful, because often we have no heat-source or heat-sink in a moving fluid, so that η is constant following a fluid parcel. In this case, p=× const .ρ γ Compare this with an isothermal (T=const) process where p = const x ρ.

Adiabatic heating/cooling occurs when we squeeze/expand a gas in an insulated container, and is a dominant atmospheric process. We explore this with a glass vessel with a rubber bulb that can be squeezed, a pressure gauge and a crude thermometer.

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4. Phase change, : In the same apparatus we can witness phase change due to adiabatic cooling: formation of a cloud. Do this by inserting a drop of water in the glass vessel, sealing it up, and changing the pressure. Initially you will not see any result. This is because condensation requires a nucleus on which to form a water droplet. So, if you add some nuclei, the result should be visible. Note how rapid the phase change occurs. Sometimes you will see a cloud form above the wing of an airplane as you are landing in moist air: this is because of the rapid decrease in pressure in the air flowing over the wing (which lifts the plane). The phase change is nearly instantaneous.

The of a fluid is the heating in Joules required to raise 1 kg. of fluid 10 C. For water at 280 C this is 4000 J kg-1 K-1 (see Gill, Appendix 1). For dry air it is

-1 -1 cp = 7/2 R = 1004.6 J kg K ; 3 the number 7/2 is for diatomic molecules. For a polyatomic molecule, with more than 3 atoms, the number is 4. This leads to a specific heat coefficient for moist air of

cp = 7/2 R(1 - q +8q/7ε)

(Gill, 1982, p43.). These are measured at constant pressure; at constant volume, the specific heat cv = cp - R. The specific heat capacity (at constant pressure) of fluids and solids normally exceed that for gases; for air cp is about 1/4 that of water, With its density some 800 times less, the heat capacity of the entire column of atmosphere is equivalent to that of just 5m of ocean beneath. For this reason the ocean is an active heat source and sink for the atmosphere, and generally more exchange occurs as latent heat of evaporation than as ‘sensible’ conducted heat

It is difficult to have an intuitive quantitative sense of heating. Perhaps the best visualization is the heating of a tea-kettle, where a 500W stove-top heating unit (500 J sec-1) takes about 300,000 J. to boil 1 kg. of water initially at 250C, and 10 minutes to do it. As a liquid is heated, it develops an increasing vapor pressure. At at a given temperature and pressure, a vessel of liquid has an equilibrium vapor pressure which describes the escape of water molecules from the surface. At the boiling point, the vapor pressure has risen to equal the atmospheric pressure. Raising the water to the higher energy level requires a great deal of heat: the latent heat of evaporation is

6 3 -1 Lv = 2.50 x 10 -2.3 x 10 T J kg , where T is the Celsius temperature. For this reason, it takes longer to boil a tea kettle dry, than the time initially to bring it to a boil.

Energy units are all round us; caloric value of foods, horsepower of automobiles, heating of a liquid, radiation of an electric heater. It takes about 1.5 horsepower to take a hot shower, and a small (250 food ‘calories’) candy bar eaten in 1 minute represents about a million J. of energy/60 seconds, or 22.3 horsepower! (our conversion of food into mechanical work is far from perfectly efficient). (1 hp. = 745.7watts = 745.7J sec -1). Note that food ‘calories’ are actually kilocalories; 1 thermodynamic calorie = 4.184 J, yet 1 food ‘calorie’ is 4186.8 J, which is strange. How much power output can the human body produce for a short time? Running upstairs, you can readily calculate your change in gravitational potential energy, and it’s difficult to do much better than 1 horsepower.

The ratio of heat flux carried by water vapor and that carried directly as ‘sensible’ heat (ρcpT) is a crucial part of atmospheric dynamics. Generally in the tropics the latent heat flux upward from the heated sea surface greatly exceeds the sensible: if it were not so, clouds would contain be the dominant vertical motion. The delayed dynamical impact of the latent heating (which appears when the water vapor condenses) makes a peculiar forced fluid dynamics problem (see, e.g., the book Atmospheric Convection by Kerry Emmanuel). At higher latitude, for example in the Labrador Sea, cold, dry winds from the continents flow over the ocean and are warmed; the sensible and latent heat fluxes there are more comparable. Visit the Clausius-Clapyron equation in Gill to learn more.

The saturation vapor pressure is the contribution of gaseous water to the pressure, and it is a function of temperature of the water. This is famous Clausius-Clapyron equation. We can measure the s.v.p. using the two-glass-bulb apparatus in the lab, which has almost no air in it. Swirl the water around so that the 4 inside of one bulb is wet. Warm that bulb in your hand with the glass tube downward. Watch the water column rise, and measure the hydrostatic pressure difference. This should give a point on the C.C. curve below!

The cooling by evaporation is used in the sling-psychrometer to measure humidity of the air: two thermometers, one with a moistened fabric surrounding the bulb, are whirled in the air. The ordinary thermometer comes to equilibrium with the atmosphere (for small Mach number!), this is the ‘dry bulb’ temperature. The other thermometer cools due to evaporation. Due to a complex process of evaporative cooling yet conductive warming, this thermometer reaches a stable ‘wet bulb’ temperature. The difference indicates how much water the air can hold…the humidity.

5. Thermal energy equation: The first law of thermodynamics for a parcel of fluid is δ E =−δδ'qpv where δE is a small change in thermal , δ’q is a heating or cooling, p is pressure and δv is a small volume change. This internal energy equation combines with the ‘mechanical energy’ equation derived from the momentum equation, to describe the total energy balance of a fluid. pδv is the work done by the fluid on its surroundings. For a perfect gas the internal thermal energy is simply E = Cv T where T is temperature and Cv is specific heat capacity at constant volume. This makes sense because if we heat a gas while keeping its volume constant (δv = 0) then δE = δ’q = CvδT which is really a definition of Cv . If we plot the pressure and volume on the p-v plane, and follow the fluid through some heating/cooling events, a curve will be traced out. The area under the curve is the work done by the fluid; in a repeated cycle, the curve is closed and the area inside it is the net work done during one cycle.

The Stirling engine is a that converts thermal energy into mechanical work. It is a very efficient cycle and can be used in practical machines (though it is rare). Engines have a couple of important properties: first, they use the p-v curve by being heated when the pressure is high, and cooled 5 when it is low. As an example, keep the volume of the gas constant while heating, then let it expand adiabatically. Now cool it at constant volume, then compress it adiabatically. The most efficient heat engine cycle is the , consisting of two isothermal processes and two adiabatic processes. The Carnot cycle can be thought of as the most efficient heat engine cycle allowed by physical laws. When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used.

In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy.

http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/carnot.html The conceptual value of the Carnot cycle is that it establishes the maximum possible efficiency for an engine cycle operating between TH and TC. It is not a practical engine cycle because the heat transfer into the engine in the is too slow to be of practical value. As Schroeder puts it "So don't bother installing a Carnot engine in your car; while it would increase your gas mileage, you would be passed on the highway by pedestrians."

The entropy change in a Carnot cycle is zero…it is the most efficient possible engine. The process described above (constant volume heating followed by adiabatic expansion) involves a net entropy change δ 'q during a cycle. The entropy change is round the closed curve. v∫ T

The Clausius Theorem and Inequality 6

The equality above represents the Clausius Theorem and applies only the the ideal or Carnot cycle. Since the integral represents the net change in entropy in one complete cycle, it attributes a zero entropy change to the most efficient engine cycle.

The Clausius Inequality applies to any real engine cycle and implies a negative change in entropy on the cycle. That is, the entropy given to the environment during the cycle is larger than the entropy transferred to the engine by heat from the hot reservoir. In the simplified heat engine where the heat QH is all added at temperature TH, then an amount of entropy ∆S = QH/TH is added to the system and must be removed to the environment to complete the cycle. In general, the engine temperature will be less than TH for at least part of the time when heat is being added, and any temperature difference implies an . Excess entropy is created in any irreversible process, and therefore more heat must be dumped to the cold reservoir to get rid of this entropy. This leaves less energy to do work.

A second property of most engines is that they are ‘oscillators’. They have a basic configuration that can change back and forth, for example with a flywheel in a gasoline engine. In the Stirling engine a glass tube rocks back and forth, and some marbles flop from one end to the other. The tube sits on a pivot and the air in the glass tube is connected to a small balloon which the pivot rests on. This is a feedback loop. The air is heated by a candle, expanding the balloon which tips the glass tube: it flops over and the marbles displace the air, so that the air is moved away from the heat source. It cools, the balloon contracts and it flops back so that the air is once again heated. Mechanical work is done (we could lift weights, etc.; actually the marbles’ motion is mechanical work which is lost again when they hit the end of the glass tube and come to rest).

We have discussed p-v work and heat engines to provide a basis for understanding energetics of ocean and atmosphere: particularly convection and circulation driven by buoyancy. This is really the heart of much of GFD. Picture parcels of fluid in a hula-hoop (a plastic loop) which is heated at one place and cooled at another: think of the way the energetics works when we have gravity involved.