Stability Hydrostatic equation

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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

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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, dp1dp2. But due to variation in
  density surfaces, Dz1ltDz2 due to smaller density
  at position 2 relative to position 1 thus we
  obtain a baroclinic relationship

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Increasing density
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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,1rfluid(zo)

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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 .

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Stability (or lack of) in the atmosphere
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• Image from http//www.floridalightning.com/files/S
  upercell_Thunderstorm.jpg

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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

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Stability in the atmosphere

• We know stability requires
• This results in the following inequality

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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

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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

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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

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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?

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

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(in)Stability in the Ocean
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• Minimum value of Simpson-Hunter parameter during
  30 day model-run. The colour scale is in
  dimensionless units of log10h/u3. A value
  below 2.7 indicates complete vertical mixing.
• Image from http//www.scielo.cl/scielo.php?pidS07
  17-65382004000200051scriptsci_arttext

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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

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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

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Stability in the Ocean

• The requirement for stability simplifies to
• As a reminder
• - Thermal expansion coefficient
• - Haline contraction coefficient

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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?

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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

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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

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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

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Stability parameter

• Recall from the previous chapter the linear
  representation of density
• If we require dzltlt1km, we can neglect the
  pressure terms. Take the derivative of density
  to obtain
• Substitution into the stability requirement, we
  obtain
• where for a given dzltlt1km

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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

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Justification of simplified E
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Stability in the Ocean

• Just like in the atmosphere, there are three
  possible scenarios
• Common Values of E

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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)

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Brunt-Vaisala frequency

• Recall the stability equation
• In the stable case, Egt0 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.

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Double diffusive convectionA 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.

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Double diffusive convectionA 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

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Double diffusive convectionA qualitative look

• Cold/fresh over warm/salty Displace a parcel
  upward and the parcel will naturally rise. It
  wont 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/
  )

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