Physical oceanography, MSCI 3001
Oceanographic Processes, MSCI 5004
Dr. Katrin Meissner
[email protected]
Week 4
Stability in the water column
Gravity acts on vertical density gradients in the ocean
to either stabilise or destabilise the water column:
Three examples of ocean stability are
Ocean Dynamics
Stable Stratification
27.25
27.5
22.25
25.5
27.25
Neutral Stratification
Unstable Stratification
Archimedes Principle:
The upward force acting on a body within a fluid, is equal to the weight of the
fluid displaced by the body
Wood ρ=500kg/m3
26.75
Weight = 5000N
Net upwards force
(10000-5000=5000N)
Upthrust = 10000N
Weight = 5000N
What is the vertical density gradient across the top 200m at (i) the equator,
and (ii) 60S ?(iii) Estimate the meridional (north-south) gradient at 60S.
Weight = mg = ρVg
ρ=1000kg/m3
And a weight of 1000x10=10000N
Wood block will accelerate upwards!
What would happen with a concrete block
ρ=2400kg/m3
1
Archimedes Principle:
The upward force acting on a body within a fluid, is equal to the weight of the
fluid displaced by the body
PROOF
F1=P1A=Dρgh2
Pressure, P=hρg
P=F/A
m=ρV
D
P1=Dρg
ρ1
ρ1Vg
Negatively
Buoyant
Weight of displaced water
= mg = ρ1Vg
ρ2Vg
ρ2
A=h2
V=h3
ρ
h
Area=h2
Net Force on box, F = F1 – F2
F = Dρgh2 – (D+h)ρgh2
= -h3ρg = -Vρg
= -mg
= - weight of water displaced
depth
Stability
ρ2
ρ1
Depth
Stable
ρ1< ρ2
Stable
ρ1< ρ2
ρ2
Density
small
•  If the balloon moves into a
fluid of different density…
•  Buoyancy forces will try to
return the balloon to its
equilibrium depth
Time
Time
Less energy
required to mix
•  Imagine a balloon filled with
seawater at rest (equilibrium
depth) in two different density
gradients
•  When set in motion the
balloon will oscillate about its
equilibrium depth with a
frequency proportional to the
density gradient
large
Depth
Depth
ρ1
•  Imagine a balloon filled with
seawater at rest (equilibrium
depth)
Depth
Neutral
Depth
Density
Depth
P2=(D+h)ρg
•  The larger the density
gradient, the higher the
frequency (Brunt-Väisälä or
buoyancy frequency) and the
shorter the period
•  Large density gradient
therefore gives rise to stable
conditions - with little mixing
Depth
F2=P2
A=(D+h)ρgh2
More energy
required to mix
2
⎛ g dρ ⎞
d 2 z'
= −⎜
⎟z'
dt 2
⎝ ρ dz ⎠
Where does the buoyancy frequency come from?
ρ0
Density
⎛ g dρ ⎞
N 2 = ⎜
⎟
⎝ ρ dz ⎠
d 2z
2
2 +N z =0
dt
€ oscillator
simple harmonic
€
h
ρ1
z’
Solving the equation:
ρ2Vg
ρ2
ρ1Vg
Positively buoyant
€two cases:
• N2<0
Net Force, F= ρ1Vg- ρ2Vg
=Vg((ρ0+kh)-(ρ0+k(h+z’)))
=-Vgkz’
ρ1Va =-Vgkz’
a =-(gkz’)/ρ1
⎛ g dρ ⎞
d 2 z'
= −⎜
⎟z'
dt 2
⎝ ρ dz ⎠
Conservation of linear momentum: F = ma
F = ρ1Va
d 2 z'
a= 2
dt
z
m2+N2=0
m2=-N2
ρ2> ρ1
ρ (z)=ρ0+kz
m1/ 2 = ± −N 2
z(t) = Ae
• N2>0
−N 2 t
+ Be −
−N 2 t
m1/2 are complex solutions
€
z(t) = Acos(Nt) + Bsin(Nt)
frequency f=N/(2π)
period T=2π/N
€
€
€
Where does the buoyancy frequency come from?
ρ0
ρ0
Density
Solution 1: run away solution (if N2 is negative, if dρ/dz<0)
h
g dρ
g dρ
−
t
− −
t
ρ dz
ρρ dz
z(t) = Ae
z’
1
ρ2Vg
ρ2
€
ρ1Vg
F=ma
= ρ1Va
Density
Solution 2: simple harmonic oscillator (if N2 is positive, if dρ/dz>0)
h
⎛ g dρ ⎞
⎛ g dρ ⎞
z(t) = Acos⎜ ρ1 t ⎟+ Bsin⎜
t ⎟
⎝ ρ dz ⎠
⎝ ρ dz ⎠
ρ Vg
+ Be
z’
ρ2> ρ1
ρ (z)=ρ0+kz
z
Where does the buoyancy frequency come from?
2
ρ2> ρ1
ρ (z)=ρ0+kz
ρ2
Positively buoyant
€
Net Force, F= ρ1Vg- ρ2Vg
=Vg((ρ0+kh)-(ρ0+k(h+z’)))
=-Vgkz
ρ1Va =-Vgkz
a =-1/ ρ(gkz)
ρ1Vg
F=ma
= ρ1Va
Positively buoyant
Net Force, F= ρ1Vg- ρ2Vg
=Vg((ρ0+kh)-(ρ0+k(h+z’)))
=-Vgkz
ρ1Va =-Vgkz
a =-1/ ρ(gkz)
z
3
Buoyancy frequency
The influence of stability is usually expressed by the
buoyancy frequency which is also referred to as the
Brunt-Väisälä Frequency :
Density
Gradient
• The buoyancy Frequency (N) quantifies the
importance of stability
Buoyancy frequency
•  The buoyancy frequency is the frequency at which a vertically
displaced parcel will oscillate within a stable environment.
•  Parcels that are vertically perturbed are accelerated back to their
initial position by a restoring buoyancy force.
•  In an unstable environment, vertically displaced fluid parcels do not
oscillate, but instead continue to move in the direction of their initial
displacement.
• Larger frequency (N) the more stable the water column
http://www.es.flinders.edu.au/~mattom/Utilities/bvf.html
Brunt-Väisälä Frequency
Brunt-Väisälä frequency (N)
Sir David Brunt
D. Brunt, The period of simple vertical oscillations in the
atmosphere, Quarterly Journal of the Royal Meteorological
Society 53 (1927), pp. 30–32.
Vilho Väisälä
V. Väisälä, Über die Wirkung der Windschwankungen auf
die Pilotbeobachtungen, Soc. Sci. Fenn. Commentat.
Phys.-Math. 2 19 (1925), pp. 1–46.
4
Brunt-Väisälä Oscillation or Buoyancy Oscillation
• The frequency with which a parcel of seawater will oscillate when displaced a small
vertical distance from its equilibrium position
Strait of Gibraltar
Internal Waves
Somalia
The period is given by T = 2 π /N
If the stratification is strong, then N is large and T is small
Fast oscillations in a strongly stratified ocean.
If the stratification is weak (deep ocean) then N is small and T
is large
• These oscillations are called internal waves – they occur along density gradients
An internal wave travelling up the
Derwent River estuary in Hobart,
Tasmania. The effect of the wave is
visible by streaks of smooth water
produced by the convergences above
the wave troughs.The wave seen
here is of the shortest possible
wavelength (compare the distance
between the streaks with the yacht)
and has a period of 10 - 20 minutes.
Brunt Väisälä frequency
100
200
?
?
Taken from Matthias Tomczak’s webpage:
http://www.es.flinders.edu.au/~mattom/IntroOc/
notes/lecture10.html
5
Classify these ocean scenarios:
Brunt Väisälä frequency
N=0.014s-1
T=?
N=0.003s-1
T=?
Are they stable, unstable or neutral?
Units: s-1
Increasing Property
T=2π/N
Depth (m)
The time taken to
make one oscillation
S
S
T
T
S
T
Acts as a barrier for mixing of nutrients up into photic zone
What happens if
Convective overturning
• Leads to the formation of winter mixed layers
• E.g. night time cooling during winter, cold
water overlies warm water… UNSTABLE
Depth
ρ1
Unstable
ρ1> ρ2
ρ2
6
Mixing
How are properties transferred in water?
• Water can be mixed through convective
overturning (previous examples…)
• Water can also be mixed through turbulent
diffusion (e.g. wind driven mixing)
Molecular diffusion is much slower than turbulent diffusion
How are properties transferred in water?
Molecular Diffusion
Turbulent Diffusion
clear fluid + red dye = pink water
Just like shaking a container will
speed up the mixing rate
7
The diffusive flux is proportional to
the gradient (and of opposite sign)
Salt transport and molecular diffusion


Γ = ρsu

u
= velocity of the fluid (in m/s)
Fick’s Law of diffusion


Γ = −ρK D ∇s
ρ = density of fluid (in kg/m3)
€
€

u

Γ
= rate of transport of salt across
the unit area

normal to the direction of Γ
∂s
∂x
∂s
Γy = −ρK D
∂y
∂s
Γz = −ρK D
∂z
Γx = −ρK D

€
due to fluid motion
€ Γ is the flux2 of salinity per unit area €
(in kg/(m s)), it is also called the advective flux.
There is another means of transporting salt: molecular diffusion
€

u
Our total flux per unit area becomes:
€



Γ = ρsu − ρK D ∇s
KD is the molecular diffusivity of salt in water (~10-9 m2/s)
€
But do we really know
€

u?

We cannot calculate/measure changes in u on
very short timescales and very small spatial
scales…


Γ = ρsu = ? €
€
BUT we try to take their effects on the main
circulation into account!
€
In analogy with molecular diffusion (which is based on thermodynamics), we
define an “eddy viscosity” (eddy diffusion, which is based on nothing) to take
into account the mixing due to turbulence that we cannot resolve.
Flux of salinity per unit area
due to turbulence:
€


Γ = −ρK E ∇s
8
The full flux of salinity per unit area is therefore:
Diffusion - the flow from higher to lower concentrations




Γ = ρsu − ρK E ∇s − ρK D ∇s
Flux due to large scale fluid
motion (the mean flow)
If box 2 contains water containing 1kg of
salt per cubic meter and box 1 contains
fresh water. If L =1m and KD=10-9m2s-1
what will the initial flux be?
Flux due to molecular diffusion
Flux due to “eddy diffusion”
(= non-resolved fluid motion)
€
Γx = −K
KE >> KD
In oceanography we usually do not take molecular diffusion into account.



Γ ≈ ρsu − ρK E ∇s
What will the initial flux be if
we consider eddy diffusion
instead of molecular diffusion
(KE=10-1m2s-1)?
This transport equation can be written for every scalar in the fluid
(oxygen, nutrients, temperature, CFCs,…).
Fick’s Law of diffusion
K is the diffusion constant
SI unit m2s-1
€
€
Flows from high concentrations to low concentrations
 the flux is ‘down gradient’
Flux (kgm-2s-1)
dC
dx


Γ = −ρK E ∇s
∂s
Γx = −ρK E,h
∂x
∂s
Γy = −ρK E,h
∂y
∂s
Γz = −ρK E,v
∂z
Concentration gradient
(could be salinity,
temperature, oxygen,…)
Vertical or horizontal
diffusivity (m2/s)
Molecular versus Turbulent (Eddy) Diffusivity
Eddy (turbulent) diffusivities are usually
many orders of magnitude larger
than their molecular counterparts.
Example:
Molecular diffusivity of salt in water KD ~ 10-9 m2/s
Vertical diffusion in the ocean, KE,v ~ 10-5 - 10-3 m2/s
Horizontal diffusion in the ocean, KE,h ~ 1 - 1000 m2/s
• Deep ocean mixing occurs near rough topography
• Surface layer turbulence can be caused by wind mixing
Diffusion effects will therefore be strongest in
regions with high concentration gradients
€
This is very important in estuaries
where there is a SALT gradient….
More appropriate to consider mixing along
and across isopycnals
9
The Gulf Stream
Richardson Number:
Predicts the onset of turbulence
Remember
N2 can be <0
Where N is the Brunt Väisälä frequency, and u the horizontal velocity.
If Ri < 0 (i.e. N2 < 0) the water column is unstable – turbulent convective mixing
If Ri > 0 (i.e. stable stratification) then velocity shear (du/dz) must be large to
generate turbulence.
If Ri > ¼ no turbulence is found
If Ri < ¼ turbulence may be found.
•  Where will diffusive effects be strongest?
… In regions with high concentration
gradients
Determine if turbulence will occur in the
following examples:
What is du/dz?
Transport of Volume, Heat & Salt
Flux: Amount of heat, salt or volume that flows through a unit
area per unit time (e.g. in Jm-2s-1 (for heat), kgm-2s-1 (for salt)
or m3m-2s-1=ms-1 (for volume)).
Volume transport: The volume of moving water measured
between two points of reference and expressed in cubic
meters per second or in Sverdrup (Sv).
1 Sv = 1 million cubic meters per second
= 106 m3/s
Upper Ocean
Deep Ocean
Harald Sverdrup
10
The amount of salt in a parcel of water moving
through the ocean remains relatively constant
(unless the parcel is in contact with the
surface, then river discharge, evaporation,
precipitation, sea ice, will change it!)
 Salinity is conservative.
In most instances the advective flux will dominate, so the diffusive
part can be ignored. We are also usually interested in 1D flow
(e.g. flow through a channel or across a section)
Volume transport of an ocean
current with velocity ‘u’ through
an area A is given by:
Volume transport = u . A = u.H.L
The flux of salt into the cube includes an
advective contribution (ρsu) and a diffusive
contribution (-Kρds/dx)
Diffusive flux (turbulent and molecular) of salt
has a negative sign because it is directed in
the opposite direction to the salinity gradient
u
A
Volume
swept out
Advective Flux
Diffusive Flux
The conversation equation for salt per unit
volume is given by
But don’t panic...
Transport of Volume, Heat & Salt
Total Heat and Salt Fluxes
If we know the heat (or salt) per m3 we can convert
volume flux into heat (or salt) flux.
VT = u × L × H
QT = ρc p uT × L × H
Heat per m3 = ρcpT
Salt per m3 = ρS
S fT = ρuS × L × H
So, multiplying by uA
Q=ρcpTuA
Sf=ρSuA
ρ is density, cp ~ 4000 J/Ckg, salinity is in absolute units, 35kg/
1000kg
m3/s
J/s
Kg/s
T, S
€
UNITS heat flux – Watts (1 Watt = 1 J/s). Salt flux - kg/s
11
Example
Calculate the total volume, heat and salt fluxes
a)  Past Tasmania
b)  Through the Drake Passage.
Assume the ocean currents persist over the upper 1000m only.
c)  Why might the volume flux be different?
12