ATM 241, Spring 2020 Lecture 8 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 and on 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 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:

D2z 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 temperature gradients are much larger than salinity gradients, and so @T 2 g↵ N ⇡ T @z ↵T Thermal expansion coefficient of water Using T 15C 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)

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 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 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 . Note that darker colors indicate less dense fluid. Compare with zonal average annual-mean temperature.

= ⇢ ⇢ ref

Paul Ullrich Ocean Dynamics Spring 2020 Depth of the 1026.5 kg/m3 Density Surface

Figure: Depth in meters of the annual mean σ = 26.5 kg m-3 surface over the global ocean.

Paul Ullrich Ocean Dynamics Spring 2020 Steric Effects

The spatial variations in the height of the ocean surface are controlled to a large degree by expansion of warm ocean water and contraction of cold ocean water (observe that sea surface heights are high over the tropics and low over the polar regions).

Further, salty columns of water are shorter than fresh columns (temperature and pressure being equal).

Definition: Expansion and contraction of water columns due to T and S anomalies is known as the steric effect.

Paul Ullrich Ocean Dynamics Spring 2020 Steric Effects H Using ⇢ ⌘ ⇢ref |rh i| ⇡ |r |

And averaged equation of state

(T,S) + ⇢ ↵ [T T ]+ [S S ] ⇡ 0 ref T 0 S 0 ⇣ ⌘ ⇢ ⇢ [1 ↵ T T + S S ] h i⇡ ref T h 0i Sh 0i

⇢ ⇢ [ ↵ T + S ] rh i⇡ ref T rh i Srh i

⌘ Taking into account the fact that ↵T T SS temperature increases thickness H ⇡ and salinity decreases thickness.

Paul Ullrich Ocean Dynamics Spring 2020 Steric Effects

Over the meridional extent of the subtropical ocean gyres, we estimate

T 10C ⇡

Paul Ullrich Ocean Dynamics Spring 2020 Steric Effects

Over the meridional extent of the subtropical ocean gyres, we estimate

S 0.5 psu ⇡

Paul Ullrich Ocean Dynamics Spring 2020 Steric Effects

⌘ ↵ T S H ⇡ T S

T 10C ⇡ ⌘ 3 S 0.5 psu 2+(0.38) 10 ⇡ H ⇡ ⇥ ⇣ ⌘ 4 1 ↵T 2.0 10 C ⇡ ⇥ Temperature Salinity expansion contraction 4 1 7.6 10 psu S ⇡ ⇥ So over the top kilometer of the column, height variations due to salt are more than offset by approximately 2m of thermal expansion.

Paul Ullrich Ocean Dynamics Spring 2020 Sea Surface Height

Figure: SODA3 report mean height of the sea surface relative to the geoid (contoured every 10 cm).

Paul Ullrich Ocean Dynamics Spring 2020 ATM 241 Climate Dynamics Lecture 8 Ocean Dynamics

Paul A. Ullrich [email protected]

Thank You!

Paul Ullrich Ocean Dynamics Spring 2020