Conservation of Tracers (Salt, Temperature) Chapter 4 – Knauss Chapter 5 – Talley et al.

Outline Conservation of salt and heat Molecular diffusion Turbulent diffusion Box models Residence time

The Continuity Equation A divergence in the flow in one direction, must be balanced by a convergence in the flow in another direction. Last class - derived the continuity equation based on the concept of conservation of mass Same approach can be used to derive conservation equations for any conservative material. Conservative materials have no sources or sinks.

Conservation of Salt or Heat or … change in mass = flux in – flux out Δx Δy Δz u1S1u1S1 u2S2u2S2 Consider flow in x-direction only Water with salinity S 1 is flowing into the box with velocity u 1 Water with salinity S 2 is flowing out of the box with velocity u 2 Flux of salinity into the box = [ S 1 × u 1 × Δy × Δz ] Flux of salinity out of the box = [ S 2 × u 2 × Δy × Δz ]

Δx Δy Δz u1S1u1S1 u2S2u2S2 This can be written more generally as: In the x-direction we have: Δx Δy Δz v1S1v1S1 v2S2v2S2 The same holds true in the y - direction giving: Δx Δy Δz w1S1w1S1 w2S2w2S2 And in the z -direction giving:

Δx Δy Δz Putting this all together gives the changes in salt due to advection Change in salinity in the box Convergence or divergence in advective flux

Remember from Calculus that This can be simplified further But this equals zero from continuity equation But this only accounts for advection

Flux driven by Molecular Diffusion Always goes from High Concentration to Low Concentration where κ c is the diffusion coefficient for the substance C Diffusion coefficients change from substance to substance, but for any given substance they are essentially constant. Thermal diffusivity κ H ~ 1.5 × m 2 /s and salinity diffusivity κ S ~ 1.5 × m 2 /s Flux due to diffusion is given as: x

Concentration only changes when there is a gradient in flux [flux in ≠ flux out] Flux due to diffusion is given as: F1F1 F2F2 x Diffusive flux of salinity into the box = [ F 1 × Δy × Δz ] Diffusive flux of salinity out of the box = [ F 2 × Δy × Δz ]

Just like advection, diffusion can occur in all three directions (x,y,z) So including both advection and diffusion, the conservation of salt equation becomes:

However many flows in the ocean are turbulent, so we also must account for turbulence In addition to molecular diffusion, mixing can also occur through turbulent diffusion: Turbulence is chaotic or unstable eddying motion in a fluid.

Flow Can be Either Laminar or Turbulent Turbulent Diffusion is much more effective at mixing than molecular diffusion. Molecular diffusion coefficient for salt = 1.5 ×10 -9 m 2 /s Turbulent diffusion coefficient for salt can be as big as 1×10 -2 m 2 /s Reynolds number (Re) is ratio of inertial to viscous forces Low Re High Re Velocity scale Length scale Kinematic viscosity (10 -6 m 2 /s for water)

Laminar flow does not result in any turbulent mixing (only molecular diffusion). Laminar Flow True mixing is an irreversible process. Turbulent Flow

More heat Less heat What is responsible for the deepening of the mixed layer? Pycnocline before wind

Just like molecular diffusion, turbulent diffusion can be represented as a gradient flux. Instead of molecular diffusion, now use a turbulent diffusion coefficient sometimes called an “eddy diffusivity” Just like the molecular diffusion coefficient, it has units of m 2 /s. It is convenient to think about this a the product of a velocity scale times a length scale. Eddy has diameter L (m) with a velocity of U (m/s). so A c ~ U × L

Final conservation of salt equation for a turbulent flow is: Because eddy coefficients are properties of the flow, they vary in both time and space and orientation. In many situations, A x = A y ≠ A z For instance, in environments the vertical turbulent flux is much larger than the horizontal so we represent the conservation of salt as:

Conservation Equations are Basic Idea Behind a Box Model Calculate the net flow in and out of the Mediterranean Sea through the Straight of Gibraltar. Evaporation – (Precipitation + Rivers) = 70 × 10 3 m 3 /s Atlantic Ocean Mediterranean Sea S = S = Straight of Gibraltar

Box Model Let (R+P) – E = X Conservation of volumeConservation of salt Combine above equation with Conservation of volume equation Knudsen’s Relationships 1900

Simple box models can also help define the residence time Evaporation Atlantic Ocean Mediterranean Sea Flux in Flux out Straight of Gibraltar Simplest definition is the time it would take to completely replace all the water in a basin given the exchange rate. What is the residence time of the Mediterranean if its volume is approximately 4×10 15 m 3 ?

Consider Black Sea Example Calculate the net flow in and out of the Black Sea through the Bosphorus Strait Evaporation – (Precipitation + Rivers) = -6.5 × 10 3 m 3 /s Aegean Sea and Sea of Marmara Black Sea S = 17 S = 35 Bosphorus Strait What is the residence time of the Black Sea if its volume is approximately 0.6×10 6 km 3 ?

Next Class Equations of Motion – Chapter 5 – Knauss – Chapter 7 – Talley et al.