ATM 241, Spring 2020 Lecture 8 Ocean Dynamics
Paul A. Ullrich [email protected]
Marshall & Plumb Ch. 9
Paul Ullrich Ocean Dynamics Spring 2020 In this section…
Definitions Questions • Taylor-Proudman Theorem • How does the concept of balanced flow from • Steric effects atmospheric dynamics translate to the ocean? • How do the thermodynamic properties of the ocean determine its dynamical character? • What is the effect of temperature and salinity on sea surface height?
Paul Ullrich Ocean Dynamics Spring 2020 Balanced Flow in the Ocean
Paul Ullrich Ocean Dynamics Spring 2020 The Ocean Fluid Equations du uv tan uw 1 @p + = +2⌦v sin 2⌦w cos + ⌫ 2u dt r r ⇢r cos @ r dv u2 tan vw 1 @p + + = 2⌦u sin + ⌫ 2v dt r r ⇢r @ r dw u2 + v2 1 @p = g +2⌦u cos + ⌫ 2w dt r ⇢ @r r
u =0 (Incompressible fluid) r ·
dT d 1 c + p = J v dt dt ⇢ Material derivative: ✓ ◆ d @ dq = + u i = S dt @t · r dt i
Paul Ullrich Ocean Dynamics Spring 2020 Seawater Equation of State
Approximated equation of state by Taylor expansion (T,S) + ⇢ ↵ [T T ]+ [S S ] ⇡ 0 ref T 0 S 0 ⇣ ⌘ Thermal expansivity: Effect of salinity on density: 1 @⇢ 1 @⇢ ↵T = S = ⇢ref @T ⇢ref @S T =T0,S=S0 T =T0,S=S0 Surface 1km Depth o o o o o o T0 -1.5 C 5 C 15 C -1.5 C 3 C 13 C � (x 10-4 K-1) 0.3 1 2 0.65 1.1 2.2 Figure: Density anomalies plotted S0 (psu) 34 36 38 34 35 38 against salinity and temperature. -4 -1 �S (x 10 psu ) 7.8 7.8 7.6 7.1 7.7 7.4 -3 �0 (kg m ) 28 29 28 -3 0.6 6.9 Source: Marshall and Plumb Table 9.4
Paul Ullrich Ocean Dynamics Spring 2020 Hydrostatic Balance
From the vertical momentum equation: Dw 1 @p = g Dt ⇢ @r
Assuming vertically balanced flow:
@p = ⇢g @z
Writing this in terms of the density anomaly:
@p = g(⇢ + ) @z ref
Paul Ullrich Ocean Dynamics Spring 2020 Hydrostatic Balance @p = g(⇢ + ) @z ref
Neglecting the contribution from �, this equation can be integrated directly:
ps ⌘ dp = g⇢ dz ref Zp Zz Where � is the height of the Ocean’s free surface, where p(⌘)=ps
p(z)=p g⇢ (z ⌘) s ref
Observe the linear dependency of pressure with height. How does this compare to the atmosphere?
Paul Ullrich Ocean Dynamics Spring 2020 Hydrostatic Balance
The pressure at a depth of 1km in the ocean is thus about 107 Pa or 100 times atmospheric pressure.
Note that the the g ⇢ ref z part of this pressure is dynamically inert (does not drive dynamical circulations at constant depth).
Horizontal pressure gradients are generally only associated with:
• Variations in the free surface height
• Time-mean horizontal variations in surface atmospheric pressure (small)
• Interior density anomalies (associated with T and S variations, neglected above)
Paul Ullrich Ocean Dynamics Spring 2020 Buoyancy in the Ocean
Recall our derivation of the Brunt-Väisälä Frequency in the atmosphere:
D2 z g 2 + ( d ) z =0 Dt T (z0)
g( ) d > 0 Stable atmosphere T (z0)
Brunt-Vaisala Frequency
2 g( d ) z = z sin ( t + ) = 0 N N T (z0) Oscillatory solutions (magnitude of 2 Units of seconds oscillation depends on initial velocity)
Paul Ullrich Ocean Dynamics Spring 2020 Buoyancy in the Ocean
Figure: Consider a fluid parcel initially located at height z1 in an environment whose density is �(z). The fluid parcel
has density �1 = �(z1). Depth
It is now displaced adiabatically a small vertical distance z2 = z1 + δz. By Taylor series: @⇢ ⇢ = ⇢(z ) ⇢ + z E 2 ⇡ 1 @z ✓ ◆z=z1 Stable Unstable (⇢p ⇢E) abuoyant = g ⇢p Figure: Convectively stable and unstable vertical density profiles in the ocean. g @⇢ abuoyant = z ⇢1 @z ✓ ◆z=z1
Paul Ullrich Ocean Dynamics Spring 2020 Buoyancy in the Ocean
How about in the ocean?
2 D z g @⇢E 2 = z Dt ⇢E @z
g @⇢E Stable ocean < 0 ⇢E = ⇢ref + ⇢E @z
Brunt-Vaisala Frequency
2 g @ z = z sin ( t + ) = 0 N N ⇢E @z
Oscillatory solutions (magnitude of -2 Units of seconds oscillation depends on initial velocity)
Paul Ullrich Ocean Dynamics Spring 2020 Buoyancy in the Ocean
Brunt-Vaisala Frequency g @ 2 = N ⇢E @z
Near the thermocline temperature gradients are much larger than salinity gradients, and so @T 2 g↵ N ⇡ T @z ↵T Thermal expansion coefficient of water Using T 15 C 4 1 ⇡ In surface layer ↵T 2 10 K z 1000 m ⇡ ⇥ ⇡ 3 1 5 10 s Or about 20 min. period of oscillation N ⇡ ⇥
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Balance Recall the horizontal momentum equation: Du 1 = p fk u + Dt ⇢r ⇥ F
Acceleration Pressure Grad. Friction (curvature) Coriolis
In the ocean, the length scale is determined by the size of typical ocean gyres (about 2000 km). Hence, the Rossby number is
U (0.1 m/s) 3 Ro = = 10 fL (10 4 s 1)(2 106 m) ⇠ ⇥ Compare: The atmosphere has a Rossby number of 0.1
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Balance For small Rossby number, curvature of the flow can be neglected: Du 1 = p fk u + Dt ⇢r ⇥ F
Acceleration Pressure Grad. Friction (curvature) Coriolis
This implies that geostrophic balance is an excellent approximation for the interior of the ocean away from the equator (where Coriolis force is large) and away from the surface, bottom and lateral boundaries (where friction is large).
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Balance
In the ocean the geostrophic wind is given by 1 @p 1 @p u = v = g f⇢ @y g f⇢ @x ref ✓ ◆z ref ✓ ◆z
Since density varies little throughout the ocean, one can replace � with �ref in the denominator.
Paul Ullrich Ocean Dynamics Spring 2020 Thermal “Wind” in the Ocean
In the ocean the geostrophic wind is given by 1 @p 1 @p u = v = g f⇢ @y g f⇢ @x ref ✓ ◆z ref ✓ ◆z Differentiate with respect to z (approximate f as constant): @v 1 @ @p @ug 1 @ @p g = = @z f⇢ @x @z @z f⇢ref @y @z ref @p Use hydrostatic relationship = g(⇢ + ) @z ref
Thermal wind @ug g @ @v g @ = g = relationship: @z f⇢ref @y @z f⇢ref @x
Paul Ullrich Ocean Dynamics Spring 2020 Thermal “Wind” in the Ocean
Figure: Annual-mean cross section of zonal average potential density σ.
Paul Ullrich Ocean Dynamics Spring 2020 Thermal “Wind” in the Ocean
?
z
u 0 g ⇡
@u g @ Question: Where are the major east-west g = oceanic currents? @z f⇢ref @y
Paul Ullrich Ocean Dynamics Spring 2020 Thermal “Wind” in the Ocean
? − + − + −
z
u 0 g ⇡
@u g @ Question: Where are the major east-west g = oceanic currents? @z f⇢ref @y
Paul Ullrich Ocean Dynamics Spring 2020 Thermal “Wind” in the Ocean
? + − − + − − + − + −
? z
u 0 g ⇡
@u g @ Question: Where are the major east-west g = oceanic currents? @z f⇢ref @y
Paul Ullrich Ocean Dynamics Spring 2020 Global Ocean Currents
Figure: Subtropical gyres and associated ocean currents (Source: NOAA) Recall that the direction of shallow ocean currents is largely independent of depth.
Paul Ullrich Ocean Dynamics Spring 2020 Thermal “Wind” in the Ocean
Thermal wind @ug g @ @v g @ = g = relationship: @z f⇢ref @y @z f⇢ref @x
Assuming that the Abyssal currents are weak, the mean surface geostrophic flow should be
g H 1 usurface 8 cm s ⇠ f⇢ref L ⇠
3 1.5kgm (between 20N and 40N) ⇠ L 2000 km Horizontal length scale ⇠ H 1000 m Vertical length scale ⇠
Paul Ullrich Ocean Dynamics Spring 2020 Taylor-Proudman Theory
Paul Ullrich Ocean Dynamics Spring 2020 Taylor-Proudman
Taylor-Proudman Theorem:
If flow is geostrophic and homogeneous (� = constant) then the flow is two dimensional and does not vary in the direction of the rotation vector.
1 @p 1 @p Proof: From u = v = g ⇢f @y g ⇢f @x ✓ ◆z ✓ ◆z Assume � and f constant. Then: @u 1 @ @p 1 @ g = = ( ⇢g)=0 @z ⇢f @y @z ⇢f @y @v 1 @ @p 1 @ g = = ( ⇢g)=0 @z ⇢f @x @z ⇢f @x
So the geostrophic flow does not vary in the direction z.
Paul Ullrich Ocean Dynamics Spring 2020 Taylor-Proudman
Taylor-Proudman Theorem:
If flow is geostrophic and homogeneous (� = constant) then the flow is two dimensional and does not vary in the direction of the rotation vector.
Proof (more generally): For low Rossby number flows (low curvature), acceleration can be neglected and so the dynamic primitive equations read 1 2⌦ u + p (gz)=0 ⇥ ⇢r r
Taking the curl of this expression, and using the fact that ug =0 and f =0 then yields r · r⇥r (⌦ )u =0 · r In other words, u does not vary in the direction �.
Paul Ullrich Ocean Dynamics Spring 2020 Taylor Columns
Taylor-Proudman Theorem:
If flow is geostrophic and homogeneous (� = constant) then the flow is two dimensional and does not vary in the direction of the rotation vector.
Figure: For slow, steady, frictionless flows the wind vectors do not vary along the direction of rotation.
Vertical columns of fluid (Taylor columns) remain vertical and undisturbed. These columns cannot be tilted.
Paul Ullrich Ocean Dynamics Spring 2020 Taylor Columns
Figure: Under the Taylor- Proudman theorem flows must be along contours of constant fluid depth (otherwise the flow will lose its 2D character).
Placing an obstacle on the base of the tank in a rotating environment will reveal that Rotating fluid tank the fluid above the obstacle will not mix with the environment.
Cylindrical obstacle
Paul Ullrich Ocean Dynamics Spring 2020 Taylor Columns
Figure: Taylor columns generated at the UCLA spinlab using a rotating turntable. https://www.youtube.com/watch?v=7GGfsW7gOLI
Paul Ullrich Ocean Dynamics Spring 2020 Near-Surface Flow and Flow at Depth
Paul Ullrich Ocean Dynamics Spring 2020 Near-Surface Flow 1 p ⇢r
u Figure: Consider a fluid parcel in the Gulf Stream being fk u transported eastward. ⇥
To compensate for the southward-directed Coriolis force there must be a pressure gradient force directed northward.
Paul Ullrich Ocean Dynamics Spring 2020 Hydrostatic Balance @p = g(⇢ + ) @z ref
This equation can be integrated as:
ps ⌘ dp = g (⇢ + )dz z depth ref Zp Zz Where η is the height of the Ocean’s free surface, where p(⌘)=ps
p(z)=p + g ⇢ (⌘ z) s h i
⌘ Mean layer density ⇢ = (⇢ + )dz h i ref Zz
Paul Ullrich Ocean Dynamics Spring 2020 Near-Surface Geostrophic Flow
From hydrostatic balance we have
p(z)=p + g ⇢ (⌘ z) Free surface s h i In the near-surface, fractional variations in column depth are much greater than those of density (so we neglect the latter):
p(z)=p g⇢ (z ⌘) s ref Figure: Schematic showing the height of the free surface of the ocean and two reference horizontal Horizontal variations in pressure depend on (1) variations in surfaces. One at the near surface atmospheric pressure and (2) (z = z0) and one at depth (z = z1). variations in free-surface height.
Paul Ullrich Ocean Dynamics Spring 2020 Near-Surface Geostrophic Flow
Ignoring gradients in atmospheric surface pressure, since these occur on much shorter timescales:
p(z)=ps g⇢ (z ⌘) p g⇢ ⌘ ref r ⇡ refr
And so, from geostrophic balance,
g (u ) = k ⌘ g surf f ⇥r
g @⌘ g @⌘ (u ) = (v ) = g surf f @y g surf f @x
Analogous to geostrophic wind on constant geopotential surfaces
Paul Ullrich Ocean Dynamics Spring 2020 Near-Surface Geostrophic Flow
g @⌘ (u ) = g surf f @y
In the atmosphere, geostrophic winds of 15 m/s are associated with tilts of pressure surfaces by about 800 m over a distance of 5000 km (verify).
Because oceanic flow is weaker than atmospheric flow, we expect to see much gentler tilt of the free surface.
1 fLU U =0.1ms ⌘ = 4 1 g f = 10 s L = 106 m
Answer: ⌘ 1m ⇡
Paul Ullrich Ocean Dynamics Spring 2020 Sea Surface Height
Figure: 10-year mean height of the sea surface relative to the geoid (contoured every 20 cm) as measured by satellite altimeter.
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Flow at Depth
At depth (at z = z1), variations in density can no longer be neglected compared to those of column depth. Free surface
From earlier, for a given oceanic column: p(z)=p + g ⇢ (⌘ z) s h i Again ignoring variations in atmospheric surface pressure, p g ⇢ ⌘ + g(⌘ z) ⇢ r ⇡ h ir rh i k p = g ⇢ k ⌘ + g(⌘ z)k ⇢ ⇥r h i ⇥r ⇥rh i
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Flow at Depth
k p = g ⇢ k ⌘ + g(⌘ z)k ⇢ ⇥r h i ⇥r ⇥rh i
From the geostrophic wind relationship: 1 ug = k p f⇢ref ⇥r 1 ug = ⇢ ⌘ + g(⌘ z) ⇢ f⇢ref h ir rh i h i ⇢ Then using h i 1 gives Geostrophic flow in ⇢ref ⇡ the interior ocean g g(⌘ z) ug k ⌘ + k ⇢ ⇡ f ⇥r f⇢ref ⇥rh i
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Flow at Depth
g g(⌘ z) Geostrophic flow in ug k ⌘ + k ⇢ ⇡ f ⇥r f⇢ref ⇥rh i the ocean
Observe that if density is uniform that the geostrophic velocity is purely a function of the free surface height and independent of depth.
In practice, geostrophic velocities are smaller at depth than at the surface, suggesting that these two terms balance each other.
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Flow at Depth g g(⌘ z) ug k ⌘ + k ⇢ ⇡ f ⇥r f⇢ref ⇥rh i If geostrophic flow is nearly zero (as observed in the deep ocean), we must have that these two terms roughly balance each other: H ⇢ ⌘ where H is the column height ⇢ref |rh i| ⇡ |r | g @ Divide through by 2 = N ⇢E @z
g z ⌘ |r⇢ | ⇡ 2H ⇥ |r | N Isopycnal (surfaces of Free surface slope constant density) slope
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Flow at Depth
g z ⌘ |r⇢ | ⇡ 2H ⇥ |r | N
3 1 Using 5 10 s g N ⇡ ⇥ 400 2H ⇡ H =1km N
For every meter the free surface tilts up, density surfaces must tilt down by about 400m.
Paul Ullrich Ocean Dynamics Spring 2020 Geostrophic Flow at Depth
Subpolar Free surface North Subtropical
isotherms COLD WARM
Figure: Warm subtropical columns of fluid expand relative to colder polar columns. Thus the sea surface (measured relative to the geoid) is higher, by about 1 m, in the subtropics than at the poles.
Pressure gradients associated with sea surface tilt are largely compensated almost exactly by vertical thermocline undulations. Consequently the abyssal circulation is much weaker than the surface.
Paul Ullrich Ocean Dynamics Spring 2020 Sea Surface Height
Figure: 10-year mean height of the sea surface relative to the geoid (contoured every 20 cm) as measured by satellite altimeter.
Paul Ullrich Ocean Dynamics Spring 2020 Potential Density
Figure: Zonal average annual- mean potential density anomaly in the world oceans. Note that darker colors indicate less dense fluid. Compare with zonal average annual-mean temperature.