Hydrosphere

1)Thermal inertia for a mixed fluid layer (7.4)

  1. Energy balance – change in temperature over time controlled by temporal radiative input S(t) minus the non temporal radiative output, F(T) = σ T4
  2. d/dt (ρ Cp H T) = (1-α) S(t) – F(T), H ≡ depth of mixed layer
  3. Units (1/s, kg/m3, J/kg/K, m, K → W/m2) = (W/m2 – W/m2/K4 K4)
  4. Ocean mixed layer is 50 m (Pierrehumbert) - 150 m (winter)
  5. Atmosphere mixed layer – troposphere
  6. Hydrostatic eq’n → z = R T/g • ln(po/p), <T> = radiative equil T
  7. z = R <T>/g • ln(po/p) = 287 * 255/9.8 • ln(po/p) ≈ 7,500 • ln(po/p)
  8. ≈ 7,500 • ln(105/104) = 17 km

2)Compare thermal inertia of ocean to atmosphere

  1. (ρ Cp H)ocean/(ρ Cp H)air = (1000 • 4186 • 50) / (0.5 • 1000• 17,000) = 2e8 / 8.5e6 = 24
  2. Units [ kg/m3 • J/(kg K) • m = J/(m2 K)]
  3. Depth of ocean to account for atmosphere = Hsw = (ρ Cp H)air / (ρ Cp)ocean
  4. = (0.5 • 1000• 17,000) / (1000 • 4186) = 8,500/4186 ≈ 2 m

3)d/dt (ρ Cp H T) = (1-α) S(t) – F(T)

  1. µ = ρ Cp H (J/m2/K)
  2. dT/dt = 1/µ [(1-α) S(t) – F(T)]
  3. Let S = So + S’(t), T=To + T’(t)
  4. Expand F(T) in Taylor series keeping only first term,
  5. F(T) = F(To + T’(t)) = F(To) + dF(To)/dT T’(t) = Fo + b T’(t)
  6. Let To be mean radiative equilibrium temperature → F(To) = (1-α) So
  7. dT/dt = 1/µ [(1-α) (So + S’) – (Fo + b T’(t))]
  8. = 1/µ [(1-α) S’ –b T’(t)]
  9. = (1-α) S’/µ –T’(t)/(µ/b)= (1-α) S’/µ –T’(t)/τ
  10. µ/b = τ – a relaxation time for the mixed layer to accommodate to dF/dT, change in cooling due to change in T.
  11. dF/dT = 4 σ T3, for F ≡ radiative cooling, [W/m2/K4 K3 = J/s/m2/K]
  12. air – dF/dT = 4 • 5.67e-8 • (255)3 ≈ 3.8
  13. τair = µ/b = 8.5e6/3.8 s = 25 days
  14. oceans - dF/dT = 4 • 5.67e-8 • (290)3 ≈ 5.5
  15. τocean= µ/b (s) = 2e8/5.5 s = 420 days
  16. S’(t) has a diurnal and seasonal cycle and is thus comparable to the time scale for air, but much faster than τocean
  17. Thus for air dT/dt ≈ 0 for air and T’= (1-α) S’(t)/b
  18. T is nearly in equilibrium with S’(t) weighted by the change of blackbody cooling with temperature change.
  19. For ocean, S’/µ > T’/τ and so dT/dt = (1-α) S’(t)/µ
  20. T’ = 1/µ ∫0t (1-α) S’(t’) dt’
  21. Temperature will be out of phase with solar terms – hurricane season when?

4)Land vs ocean (Land/ocean absorb ≈ 50% of incoming solar insolation)(VG)

  1. ∆ E = cp m ∆T = cp ρ V ∆T [J/kg/K kg/m3 m3 K = J]
  2. Land: cp = 800 J/kg/K, ρ=3000 kg/m3, V = 1m m2), ∆T = 20 K (annual cycle)
  3. Ocean:cp = 4186 J/kg/K, ρ=1000 kg/m3, V = 50 m m2), ∆T = 10 K (annual cycle)
  4. ∆Eocean/∆Eland = 4.2/0.8 • 1/3 • 50/1 • 1/2 ≈ 50
  5. Again, ocean dominate heat reservoir as obvious from global distributions of temperature.
  6. In addition ocean a fluid, able to transport energy globally.

5)Thermohaline circulation (THC)- aka abyssal circulation, deep circulation, meridional overturning circulation, global conveyor –

  1. THC → circulation of temperature and salt, but the oceanic circulation is neither.
  2. It is the circulation of oceanic waters of varying densities and compositions. Water will move on an isopycnic surface (constant density)

6)Seawater density

  1. ρ(Salinity, S), Salinity ≡ mass of dissolved salts to mass of water usually given in parts per thousand %o.
  2. Since dissolved salts are heavier than H2O, ρ(S) > ρ(S’) when S > S’
  3. Salinity scale of 1978 – salinity defined by the conductivity of the water compared to a standard KCl solution.
  4. This works because the salt fraction and ion fraction in seawater are nearly identical for the major constituents of sea water: Cl, Na, S, Mg, Ca, K. Only Mg and S differ significantly from their ion concentration.
  5. Conductivity = Salinity (1 ± 0.003)
  1. ρ(temperature) (VGs)
  2. ρ1 = ρ0 / (1 + β (t1 - t0))
  3. β = volumetric temperature expansion coefficient (m3/m3 K) = 0.0002 water
  1. ρ(pressure)
  2. ρ1 = ρ0 / (1 - (p1 - p0) / E)
  3. E = bulk modulus fluid elasticity (N/m2) = 2.15e9 (N/m2) water.

7)Seawater density equation of state – three polynomials and 41 constants.

  1. ρ(T, S, p) = ρ(T, S, 0)/ [1-KT(T, S, p)] (Curry & Webster – pp21)
  2. KT ≡ mean bulk modulus α compressibility-1
  3. ρ(T, S, 0) = a + b S + c S3/2 + d S2
  4. KT(T, S, p) = e + f S + g S3/2 + (h + i S + j S3/2) p + (m + n S) p2
  5. a - n = f(∑ ai Ti, i=0,5)
  6. Density of seawater not measured but calculated form equation of state after measuring, T, S, and p.

8)To infer motion of water use

  1. Potential temperature ≡ temperature of a parcel of water at depth brought to surface
  2. θ(T, S, p;po) = To + ∫ppo Γad(θ(T, S, p;po), S, p) dp, To & po are the reference values, usually taken as the surface (Curry&Webster, pp69).
  3. (Jackett et al., 2006)
  4. Potential density, ρθ ≡ density of water parcel lifted adiabatically to surface w/o change in salinity. ρθ = ρ (θ, S, 0)
  5. Show graphs of ρ(T,p) and ρθ and θsw

9)Abyssal (Thermohaline) circulation - formation

  1. Wind and differential heating driven
  2. Densest surface water – cold air across ocean, high latitudes, winter
  3. Northern Atlantic – between Greenland and Norway
  4. Near Antarctica – Weddell and Ross seas
  5. Not northern Pacific not saline enough– too big a basin?
  6. Mid and low latitude winter water not dense enough to sink more than 100 m
  7. Mediterranean an exception if enough water evaporates without mixing, then this saline water is added to the deep water circulation bottom current in the Atlantic.
  1. Wind cools/evaporates water. Ice increases salinity. Surface waters sink and travel south.
  2. The loop is completed and deep water added in Antarctic circumpolar current.
  3. Strong westerlies around Antarctica cause strong circumpolar current - giant mixmaster
  4. Divergence (convergence) occurs south (north) of the maximum wind speed since Coriolis force depends on speed and thus varies across the wind speed max.
  5. South of jet water moves away from continent – upwelling around continent - enhanced fisheries – Ekman spiral.
  6. North of jet – convergence, cold air, evaporation, ice production→more deep water.

10)Importance of deep circulation

  1. Carries heat, Salt, O2, CO2, meridionally
  2. Determines ocean stratification
  3. Abyssal water volume > surface water volume so although currents weak their impact is not.
  4. Fluxes of constituents of deep water vary from decades – millennia. This variability influences climate on same time scales → climate changes from years – decades – millennia moderated by oceans, and may have modulated previous ice ages.
  5. Abyssal (colder) waters higher capacity for dissolved CO2
  6. Carbon reservoir comparisons
  7. Oceans 40,000 GtC, land 2200 GtC, air 750 GtC, 150 GtC released since industrial revolution.
  8. Future climate strongly dependent on the CO2 storage in deep water
  9. Deep circulation modulates meridional heat transport.
  10. Oceans responsible for about ½ the meriodonal heat transport which arises from differential heating of eq – pole.
  11. Gulf stream keeps north Atlantic ice free and Europe warm.
  12. So much energy is transported north in formation of Atlantic bottom water that heat transport in Atlantic is entirely north, even in southern hemisphere.(VGs)

11)Sensitivity of deep circulation (Sv = Sverdrup ≡ 106 m3/s of flow).

  1. Sinking of bottom water dependent primarily on salinity.
  2. Temperature all high latitude waters about the same, 271 K.
  3. Production bottom water highly sensitive to small changes in:
  1. Salinity – a change of ±0.1 Sv of flow of fresh water into N. Atlantic can shut off 14 Sv of deep circulation. Water would not be salty enough.
  2. Heat transport: gulf stream carries 40 Sv of 291 K water northward. 14 Sv returns south in deep circulation at 275 K. ∆E = cp ρ V ∆T = 4186 • 1000• 14e6 • 16 = 0.9 e15 J/s.
  3. Thus a significant freshening of gulf stream could shut off a petawatt of heat transported into the Arctic.
  4. Mixing – 2 TW of energy required to drive deep mechanical mixing – and this drives 2000 TW of poleward heat flux. Energy from wind and gravity.

Figure 13.3 The meridional-overturning circulation is part of a non-linear system. The circulation has two stable states near 2 and 4. The switching of north Atlantic from a warm, salty regime to a cold, fresh regime and back has hysteresis. This means that as the warm salty ocean in an initial state 1 freshens, and becomes more fresh than 2 it quickly switches to a cold, fresh state 3. When the area again becomes salty, it must move past state 4 before it can switch back to 1.

12)Climatic variations possibly relatedto deep circulation last 0.4 Ma based on ice cores.(VGs)

  1. Many times warming up to 8 K over < 50 years
  2. Warm Greenland → ice bergs → freshening → shut off deep circulation → reduce transport of energy north → cold NH → heat going north now goes south → Antarctic warms
  3. Switching on and off has large hysteresis and two stable states, see figure above.
  4. To reinitiate deep circulation requires waters saltier than available today, but once this occurs the warming in north quite rapid and then Antarctica cools.
  5. Could a minor circulation change of this nature → little ice age.
  6. 1000 years to transport water – N Atlantic→Antarctic circumpolar current → N Pacific.
  7. Knowledge of these circulations from long period observations of various tracers – temp, salinity, silicate, tritium, fluorocarbons. Temperature and salinity are formed at surface → below mixed layer these properties are conserved unless changed by mixing with adjacent water. No sources of heat or salt in deep ocean.

13)Abrupt climatic variations past 0.1 Ma–DO, H, and YD events(VGs)

  1. Dansgaard-Oeschger events – abrupt warming to interglacial levels in decades then gradual cooling.
  2. Latest event 11.5 kya →+8 K in 40 years. Led to current 8,000 year climatically quiescent/optimum period.
  3. 25 events over past 110 kya.SH slow warming and less δT.
  4. Causes uncertain – best candidate increased freshwater to N Atlantic.
  5. Heinrich events – ocean cores show ice rafted debris in NH.
  6. 6 events in past 110 kya. Events last ~750 years.
  7. Laurentide ice sheet prime suspect as source of ice bergs.
  8. Ice berg melting → fresh water N Atlantic → MTHC interrupted.
  9. Heinrich events may be tied to Dansgaard-Oeschger events, but still ???
  10. Younger Dryas 11.5-12.8 kya
  11. Significant cooling interrupted warming to interglacial levels.
  12. Greenland temps dropped 15 K, UK annual T of 5ºC.
  13. End occurred with warming of 10 K in 10 to 50 years.
  14. Causes still debated
  15. Most likely draining of LakeAgassiz and shutting down THC
  16. Bolide impact → end of Clovis culture (13 kya) extinction of large NH mammals. Geologic evidence for impact not found.

14)Theory for deep circulation

  1. Cold deep water supplied by convection at a few high latitude locations in Atlantic (Irminger and Greenland seas) and Weddell sea in south.
  2. Uniform mixing brings water back to surface
  3. Deep circulation geostrophic in ocean interior → potential vorticity conserved.
  4. Notes
  5. Mixing required to complete the cycle, 14 a) alone would lead to deep cold pool and shallow mixing.
  6. Upwelling required to pump water through the thermocline and drive deep circulation
  7. Winds and tides primary energy source driving the mixing.

15)Details

  1. Continuity equation, ∂ρ/∂t + del•(ρ U) = 0
  2. → ∂ρ/∂t + ρ del•U + U • del ρ = 0,
  3. Incompressible fluid → ∂ρ/∂t = 0, delρ = 0, →,
  4. del • U= ∂u/∂x + ∂v/∂y + ∂w/∂z = 0
  1. Barotropic potential vorticity – valid for incompressible fluids
  2. Vorticity ≡ measure of fluid circulation,
  3. ζ + f ≡ absolute vorticity = (relative + planetary) vorticity
  4. f = 2 Ω sin φ, φ ≡ latitude, Ω ≡ earth’s rotation
  5. From scale analysis of the vorticity equation Dh(ζ + f)/Dt = -f (∂u/∂x + ∂v/∂y). See Holton (1992), pp105 ff. Now use incompressability → Dh(ζ + f)/Dt = -f (∂u/∂x + ∂v/∂y = f ∂w/∂z. Then
  6. ∂u/∂x + ∂v/∂y > 0 → δu/δy > 0 in positive x/y direction → div → ∂w/∂z < 0 → sinking motion
  7. ∂u/∂x + ∂v/∂y < 0 → δu/δy < 0 in positive x/y direction → con → ∂w/∂z > 0 → rising motion
  8. Physically - increase (decrease) of vorticity following motion on horizontal scale = convergence (divergence), which is related to upwelling (sinking).
  9. Dh/Dt ≡ horizontal total derivative following the motion.
  10. = ∂/∂t + U • del = ∂/∂t + u ∂/∂x + v ∂/∂y
  1. Apply to ocean
  2. ζ ≈ 0 → Dh(f)/Dt = f ∂w/∂z
  3. Dh(f)/Dt = ∂f/∂t + u ∂f/∂x + v ∂f/∂y = 0 + 0 + v ∂f/∂y
  4. → Sverdrup equation v ∂f/∂y = v β = f ∂w/∂z
  5. β = ∂f/∂y = 2 Ω cos φ • dφ/dy = 2 Ω cos φ • (2π/2πR), R≡ Earth’s radius.
  6. β = 2 Ω cos φ / R
  7. β v = f ∂w/∂z → meridional velocity is driven by rising sinking motion.
  1. Integrate from ocean bottom to top of abyssal circulation (thermocline)
  2. V = ∫0H v dz = ∫0H f/β ∂w/∂z dz, V ≡ vertical integral of meridional velocity
  3. V = f/β wo = R/H tanφ wo, wo ≡ vertical velocity base of thermocline.
  4. If wo > 0 then V is poleward both hemispheres
  5. wo is from Ekman pumping, → wo = k • (del x ws)/f, ws≡wind stress vector
  6. Then V=k • (del x ws)/β, Sverdrup relation
  7. Ocean currents move at 90º to the right (left) N (S) hemisphere of the wind. First noted by Nansen, Norwegian Arctic explorer, noting the drift of ice.
  8. Ekman spiral each subsequent lower layer moves more right/ left until current dies away at depth. Upwelling replaces the surface layers.

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Terry Deshler, University of Wyoming, Notes on Hydrosphere, 10/23/2018