Stability Hydrostatic equation

1. Stability – Hydrostatic equation A hydrostatic fluid is assumed to be at rest and thus subject only to its internal pressure force (due to molecular motion) and the force of gravity (weight). For constant density with respect to height or depth we see that

2. Hydrostatic equation- effects If we assume Hydrostatic stability in the ocean or atmosphere and the density surfaces are slanted, the hydrostatic equation provides a means to deduce the orientation of the pressure surfaces By definition, dp1=dp2. But due to variation in density surfaces, Dz1<Dz2 due to smaller density at position 2 relative to position 1 thus we obtain a baroclinic relationship

3. Static stability To examine the stability of a fluid medium we must assume our object is a parcel of fluid and then displace that object by a small amount dz. We must then examine how the displaced parcel compares to the surrounding fluid using equation (1). From the figure, we can see that rparcel,1=rfluid(zo)

4. Static stability If then we can see that the system is stable since a parcel will sink back towards the equilibrium point so we require . If then the system is unstable since the parcel will continue to move further away from the equilibrium point so we require .

5. Stability (or lack of) in the atmosphere Image from http://www.floridalightning.com/files/Supercell_Thunderstorm.jpg

6. Stability in the atmosphere For the atmosphere, we have an analytic expression for the equation of state Then we know that density is a function of the state variables of virtual temperature and pressure. The differential of density, r(T*,p) is then

7. Stability in the atmosphere We know stability requires This results in the following inequality

8. Assumptions 1. Assume that variations of density with pressure is the same for the parcel and the surrounding environment 2. Assume that diffusion of temperature between the parcel and environment is instantaneous Where is the thermal expansion coefficient

9. Stability in the atmosphere The inequality then simplifies to Divide through by dz and taking the limit dz approaches zero, we obtain Where is the lapse rate of the parcel

10. Stability in the atmosphere There are three possibilities - Unstable - Environmental temperature has a steeper decrease with height - Neutral - Stable - Environmental temperature has a more gradual decrease with height as compared to parcel temperature

11. Stability in the atmosphereExample As stated earlier, in practice we are given a particular, adiabatic lapse rate for the parcel of air and observe the environmental conditions to determine if the environment is stable. The Dry Adiabatic Lapse rate is The moist Adiabatic Lapse rate is approximately On the next slide is an upper air sounding for the Gran Cayman. Determine if the atmosphere is stable or unstable between 1500-2500m for both a dry and saturated atmosphere. What can you say about the effects of moisture on stability?

12. Stability in the atmosphereExample Modified Upper air sounding for the Gran Cayman Owen Roberts Airport

13. (in)Stability in the Ocean Minimum value of Simpson-Hunter parameter during 30 day model-run. The colour scale is in dimensionless units of log10[h/u3]. A value below 2.7 indicates complete vertical mixing. Image from http://www.scielo.cl/scielo.php?pid=S0717-65382004000200051&script=sci_arttext

14. Stability in the Ocean For the Ocean, we have no analytic form of the equation of state but we do know that the density varies with temperature, pressure and salinity so, as before, let us consider the differential of density The requirement for stability of the system is

15. Assumptions Two of the assumptions are the same as before 1. Assume that variations of density with pressure is the same for the parcel and the surrounding environment 2. Assume that diffusion of temperature between the parcel and environment is instantaneous 3. We have an addition assumption that salinity diffusion is slow enough so that there is no transfer of salts between the parcel and the environment

16. Stability in the Ocean The requirement for stability simplifies to As a reminder: - Thermal expansion coefficient - Haline contraction coefficient

17. An aside- Thermodynamics To go any further we have to think about how the thermodynamic properties of a transported parcel will change with height. The most simple process is an isentropic process which means no change in entropy, h, between the parcel and the environment. An isentropic process is Adiabatic = no heat transfer Reversible = no friction How do we describe this process mathematically?

18. An aside- Thermodynamics Consider the differential of entropy, which depends on the state variables of temperature and pressure: The first term is related to the specific heat capacity The second term is related to volumetric changes with respect to temperature by one of the Maxwell relations

19. An aside- Thermodynamics Substitution into the differential of entropy, we obtain Now if the process is isentropic then dh =0 and we have an isentropic relationship between temperature and pressure

20. Stability in the Ocean Using the assumption that the system is isentropic The requirement for stability Takes the form Divide the above by dz and take the limit as dz approaches zero to obtain the result

21. Stability parameter Recall from the previous chapter the linear representation of density If we require dz<<1km, we can neglect the pressure terms. Take the derivative of density to obtain Substitution into the stability requirement, we obtain where for a given dz<<1km

22. Stability parameter Introduce the stability parameter So the requirement for stability is Note: if we consider variations in the first 1km of the water column (such as in the Chesapeake bay) then the parcel lapse rate will be much smaller then variations of density within the environment. The stability parameter then takes the form As a general rule of thumb, this form of the stability parameter is fine provided

23. Justification of simplified E

24. Stability in the Ocean Just like in the atmosphere, there are three possible scenarios Common Values of E

25. Example Evaluate the Stability parameter for the following water column between 300 and 400m and determine if it is stable or unstable. (This might be a trick question)

26. Brunt-Vaisala frequency Recall the stability equation: In the stable case, E>0 and a displaced parcel will oscillate about its neutral about due to its inertial properties (just like a spring or pendulum). We can see dimensionally that if we multiply E, which has units of m-1, by gravity with units of m/sec2 , we obtain units of sec-2. Taking the square root of this quantity gives us a value for the oscillation frequency of these stable displaced particles. Ng is called the Brunt-Vaisala frequency.

27. Double diffusive convection{A qualitative look} We know that density increases with an increase in salinity and a decrease in temperature. This leads to two straightforward cases of stratification: Warm/fresh over cold/salty – stable Cold/salty over warm/fresh – Unstable What about Warm/salty over old/fresh or cold/fresh over warm/salty? We need to examine the salinity and temperature diffusion effects.

28. Double diffusive convection{A qualitative look} Normally, thermal diffusion occurs much more rapidly than salinity transport. Warm/salty over cold/fresh – Displace a parcel upward, temperature will cause the parcel to heat up and become more buoyant then surroundings and rise. Conversely, particles displaced downward will become negatively buoyant and sink. This leads to “salt fingers” Occurs in outside the strong where warm Mediterranean waters flow over the cold Atlantic. Link

29. Double diffusive convection{A qualitative look} Cold/fresh over warm/salty – Displace a parcel upward and the parcel will naturally rise. It won’t take long for the heat transfer to cause the parcel to return to neutral buoyancy again and displace laterally (spread out). The overall effect leads to horizontal layering and making the ocean medium more continuous. This type of instability is why we cannot transport icebergs as a source of fresh water. (images from http://www.kranenborg.org/doublediff/)