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 φ uw 1 @p + = +2⌦v sin φ 2⌦w cos φ + ⌫ 2u dt − r r −⇢r cos φ @ − r dv u2 tan φ vw 1 @p + + = 2⌦u sin φ + ⌫ 2v dt r r −⇢r @ − r dw u2 + v2 1 @p = g +2⌦u cos φ + ⌫ 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 σ(T,S) σ + ⇢ ↵ [T T ]+β [S S ] ⇡ 0 ref − T − 0 S − 0 ⇣ ⌘ Thermal expansivity: Effect of salinity on density: 1 @⇢ 1 @⇢ ↵T = βS = −⇢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(⇢ + σ) @z − ref Paul Ullrich Ocean Dynamics Spring 2020 Hydrostatic Balance @p = g(⇢ + σ) @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 + (Γd Γ)∆z =0 Dt T (z0) − g(Γ Γ) d − > 0 Stable atmosphere T (z0) Brunt-Vaisala Frequency 2 g(Γd Γ) ∆z = ∆z sin ( t + φ) = − 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 ) ⇢ + δz 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 = δz ⇢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 + σ ⇢E @z Brunt-Vaisala Frequency 2 g @ ∆z = ∆z sin ( t + φ) = 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(⇢ + σ) @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∆σ 1 usurface 8 cm s− ⇠ f⇢ref L ⇠ 3 ∆σ 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.