Chap 9: DynamicsChap 9: Dynamics Light TransmissionLight Transmission
Solar heating Solar heating supplies short-wave supplies short-wave radiation to ocean radiation to ocean surface;surface;
Energy (light) Energy (light) penetrates the water penetrates the water column to different column to different depths as a function depths as a function of its wavelength of its wavelength and the clarity of the and the clarity of the waterwater
Hydrostatic Equation is a Hydrostatic Equation is a simplification of the simplification of the vertical component of vertical component of the equations of motion the equations of motion (momentum equations)(momentum equations)
Assume, instead that Assume, instead that there is no motion in the there is no motion in the vertical direction; vertical direction; Balance of forces is Balance of forces is between weight of the between weight of the water and the pressure water and the pressure that builds with depththat builds with depth
P =
Force
Area
=
ma
A
=
ρ Va
A
=
ρ Aza
A
= ρ zg
Force balance at the bottom of this boxForce balance at the bottom of this box
Many “boxes” stacked up in the water column:Many “boxes” stacked up in the water column:
Generalizing:Generalizing:
dp = – ρ gdz . .This is the differential form of the hydro eqn
Chap 9: DynamicsChap 9: Dynamics Hydrostatic BalanceHydrostatic Balance
What if we want to use What if we want to use the hydrostatic equation the hydrostatic equation to compute pressure?to compute pressure?
Appropriate to static (not Appropriate to static (not moving) water columns moving) water columns but is also a good but is also a good approximation to moving approximation to moving situationssituations €
Chap 9: DynamicsChap 9: Dynamics Static StabilityStatic Stability
The notion of static stability assess whether the vertical The notion of static stability assess whether the vertical distribution of density is stable relative to the downward force distribution of density is stable relative to the downward force of gravity; i.e., is there light water over heavy water, which is of gravity; i.e., is there light water over heavy water, which is stable, or the opposite, which is unstable?stable, or the opposite, which is unstable?
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E = −1
ρ
∂ρ
∂z−
g
c 2≈ −
1
ρ o
∂ρ
∂z
Related to compres-Related to compres-sibility (sound speed);sibility (sound speed);often neglectedoften neglected
Chap 9: DynamicsChap 9: Dynamics Static StabilityStatic Stability
The notion of static stability assess whether the vertical The notion of static stability assess whether the vertical distribution of density is stable relative to the downward force distribution of density is stable relative to the downward force of gravity; i.e., is there light water over heavy water, which is of gravity; i.e., is there light water over heavy water, which is stable, or the opposite, which is unstable?stable, or the opposite, which is unstable?
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E = −1
ρ
∂ρ
∂z−
g
c 2≈ −
1
ρ o
∂ρ
∂z
Related to compres-Related to compres-sibility (sound speed);sibility (sound speed);often neglectedoften neglected
Buoyancy Frequency, NBuoyancy Frequency, N
Used to characterizeUsed to characterizeinternal wave oscillationsinternal wave oscillations
The notion of static stability assess whether the vertical The notion of static stability assess whether the vertical distribution of density is stable relative to the downward force distribution of density is stable relative to the downward force of gravity; i.e., is there light water over heavy water, which is of gravity; i.e., is there light water over heavy water, which is stable, or the opposite, which is unstable?stable, or the opposite, which is unstable?
Effect is due toEffect is due tothe differencethe differencebetween rate ofbetween rate ofdiffusion of saltdiffusion of saltversus heatversus heat
Heat:Heat: = 1.4 x 10 = 1.4 x 10-3-3 cm cm22 sec sec-1-1
Salt:Salt: = 1.1 x 10 = 1.1 x 10-5-5 cm cm22 sec sec-1-1
e.g.,:e.g.,:
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∂C
∂t= ε∇ 2C
WF/CS WS/CF CF/WS CS/WF
Standard Case Salt Fingers Salt Layers Unstable
dT/dz > 0 dT/dz > 0 dT/dz < 0 dT/dz < 0
dS/dz < 0 dS/dz > 0 dS/dz < 0 dS/dz > 0
This is the typical situation in the ocean. It is stable and it does not lead to mixing due to double diffusion because both heat and salt fluxes act to reduce gradient
Case can lead to salt fingers because a “blob” of water perturbed up into top layer will continue to be sent up by the quick flux of heat into it
Case can lead to layers because rapid flux of heat into upper layer causes it to rise away from interface between layers
This case is always unstable, hence it cannot exist in a still condition in which double diffusion could operate
Likelihood ofLikelihood ofdouble diffusiondouble diffusionis measuredis measuredby the densityby the densityratio, R:ratio, R:
R must be positiveR must be positivefor double diffusionfor double diffusion(dT/dz and dS/dz same sign)(dT/dz and dS/dz same sign)
R =
αdT
dz
βdS
dz
Static StabilityStatic Stability
WF/CS WS/CF CF/WS CS/WF
Standard Case Salt Fingers Salt Layers Unstable
dT/dz > 0 dT/dz > 0 dT/dz < 0 dT/dz < 0
dS/dz < 0 dS/dz > 0 dS/dz < 0 dS/dz > 0
This is the typical situation in the ocean. It is stable and it does not lead to mixing due to double diffusion because both heat and salt fluxes act to reduce gradient
Case can lead to salt fingers because a “blob” of water perturbed up into top layer will continue to be sent up by the quick flux of heat into it
Case can lead to layers because rapid flux of heat into upper layer causes it to rise away from interface between layers
This case is always unstable, hence it cannot exist in a still condition in which double diffusion could operate
Chap 9: DynamicsChap 9: Dynamics Rotation EffectsRotation Effects
““Even for those with Even for those with considerable sophistication considerable sophistication in physical concepts, one’s in physical concepts, one’s first introduction to the first introduction to the consequences of the consequences of the Coriolis force often Coriolis force often produces something produces something analogous to intellectual analogous to intellectual trauma.”trauma.”
Chap 9: DynamicsChap 9: Dynamics Rotation EffectsRotation Effects
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ra =
dr v
dt
⎛
⎝ ⎜
⎞
⎠ ⎟inertial
=d
r v
dt
⎛
⎝ ⎜
⎞
⎠ ⎟earth
+ 2r Ω ×
r v +
r Ω ×
r Ω ×
r R ( )
Coriolis Acceleration Coriolis Acceleration accounts for the fact that accounts for the fact that the rotating Earth the rotating Earth coordinate system is coordinate system is notnot an inertial reference an inertial reference systemsystem
CoriolisCoriolistermsterms
CentripetalCentripetalAccelAccel
v = u i + v j + w k ; Ω = ∅ i + Ω cos θ j + Ω sin θ k
θθ
ΩΩ
ΩΩθθ
ΩΩ is a vector parallel is a vector parallel to the axis of to the axis of rotation whose rotation whose magnitude is equal magnitude is equal to 1 cycle/dayto 1 cycle/day
Chap 9: DynamicsChap 9: Dynamics GeostrophyGeostrophy
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−fv = −1
ρ
∂p
∂x
+ fu = −1
ρ
∂p
∂y
x-equationx-equation
y-equationy-equation
CoriolisCoriolis PressPressGradientGradient
The (numerically) largest The (numerically) largest terms in the horizontal terms in the horizontal momentum equations for momentum equations for large-scale oceans large-scale oceans currents are the currents are the CoriolisCoriolis acceleration effect and acceleration effect and horiz. horiz. pressure gradientspressure gradients
Currents that follow this dynamical balance of forces Currents that follow this dynamical balance of forces (accelerations) are called (accelerations) are called GeostrophicGeostrophic, which means “earth , which means “earth turning”turning”
Note: these currents represent a Note: these currents represent a steady statesteady state balance balance
Initial movement is driven toward high pressure (or downhill) Initial movement is driven toward high pressure (or downhill) but Coriolis effect acts to turn track to the right (in the N.H.) and but Coriolis effect acts to turn track to the right (in the N.H.) and the final, steady state or equilibrium result is parallel to lines of the final, steady state or equilibrium result is parallel to lines of constant pressure (around the hill)constant pressure (around the hill)
Rule:Rule: with your left hand in the direction of with your left hand in the direction of low pressure, the wind/current will hit low pressure, the wind/current will hit you in the back (in the N.H.)you in the back (in the N.H.)
FFcc: Coriolis; : Coriolis; FFgg: Pressure Gradient (related to gravity): Pressure Gradient (related to gravity)
Rule:Rule: with your left hand in the direction of with your left hand in the direction of low pressure, the wind/current will hit low pressure, the wind/current will hit you in the back (in the N.H.)you in the back (in the N.H.)
Chap 9: DynamicsChap 9: Dynamics GeostrophyGeostrophy
The Dynamic Method (or Geostrophic Method) for computing The Dynamic Method (or Geostrophic Method) for computing ocean currents depends on density observations to estimate ocean currents depends on density observations to estimate pressure gradients and, finally, currentspressure gradients and, finally, currents
2 or more 2 or more ρρ profiles profiles pp
ΔΔ
u,vu,v
It is not possible to directly measure pressure because depth It is not possible to directly measure pressure because depth cannot be determined with enough accuracycannot be determined with enough accuracy
Recall the hydrostatic equation:Recall the hydrostatic equation:
Quantity gdz is related to amount of work to move a unit of Quantity gdz is related to amount of work to move a unit of mass a unit distance in the vertical direction; we define the mass a unit distance in the vertical direction; we define the geopotential, geopotential, , by: d, by: d = gdz = – = gdz = –ααdpdp
Dynamic Height, D = Dynamic Height, D = /10, is numerical equiv to height in m/10, is numerical equiv to height in m
Note: Note: ρρoo >> >> We usually can’t obtain the barotropic portion so we:We usually can’t obtain the barotropic portion so we:1) Measure sea surface using satellite altimetry1) Measure sea surface using satellite altimetry2) Map velocity at one depth directly (current meters, floats)2) Map velocity at one depth directly (current meters, floats)3) Assume a 3) Assume a level of no motionlevel of no motion at some depth at some depth
Use Use p(z)p(z) solution in solution in geostophic equationsgeostophic equations
Typically Typically assume a assume a level of no level of no motion and motion and compute the compute the required required surface slope surface slope (or dynamic (or dynamic height)height)
