Advanced Ocean Modelling: Using Open-Source Software

3.7 Exercise 4: Density-Driven Flows 41criteria for both external gravity waves (Eq. 3.35) and advection (Eq. 3.43). Parametersof the S.O.R. scheme are kept at the same values as in the previous exercise.3.7.3 TheoryBaroclinic pressure gradients associated with horizontal density differences willproduce a bottom-arrested density-driven flow. The resultant speed of the flow canbe estimated from energy conservation principles. In this process, potential energyavailable from the initial density field is converted into kinetic energy of the flow.This energy conversion can be quantified by the Bernoulli equation. Provided thatthe plume density does not change, the Bernoulli equation reads:0.5u 2 + g ′ h = g ′ h o (3.47)where g ′ is reduced gravity, and h o and h, respectively, are initial and final plumethicknesses. The Bernoulli equation is named after Daniel Bernoulli (Bernoulli,1738) and his father Johann. The first true Bernoulli equation, however, was derivedby Euler (1755). The latter equation can be rewritten as:u = √ 2g ′ (h o − h) (3.48)3.7.4 ResultsAs anticipated, the initial density anomaly produces a density-driven flow spreadingtoward the right-hand side of the model domain (Fig. 3.15). This flow forms anisolated plume head. Counter-clockwise vortices forming in the lee of this headinduce vigorous mixing.With g ′ = 0.0095 m s −2 , h o = 100 m, and h ≈ 40 m, the Bernoulli equation(Eq. 3.48) suggests a plume speed of u = 1.1m/s. In good agreement with theory,maximum speeds of the simulated plume vary by ±0.2m/s around this value.It should be highlighted that the model appears capable of simulating turbulenceinitiated by vertical shear of the horizontal flow and the breaking of internal waves.Exercise 6 will explore this feature in greater detail.3.7.5 Can Reduced-Gravity Plumes Jump?The author decided to repeat the latter experiment with inclusion of variable bottomtopography with a ramp and a vertical bar (Fig. 3.16). Will the reduced-gravityplume make it over these obstacles?Results show that the density-driven current is energetic enough to pass the ramp.It shoots upward as it meets the vertical bar where it forms a counter-rotating vortex