3. A cold-core ocean eddy located at 40 S has been investigated by satellite measure- ments...

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3. A cold-core ocean eddy located at 40 S has been investigated by satellite measure- ments of sea-surface height and ship-based observations of temperature, salinity, and density. (a) Start with the equation for hydrostatic balance dp = -pog. dz Integrate this equation from depth z < 0 to the sea surface z = n(x, y), taking the atmospheric pressure to be po = constant and the pressure at depth z to be p(x, y, z). Using the resulting expression, show that the geostrophic current at the surface is u surf (x, y) g fo ay' surf (x, y) = g fo . (You can take po, fo as constants.) Find an expression for the geostrophic streamfunction (x, y) in terms of the sea-surface height n(x, y). (b) Satellite measurements of sea-surface height are fit to the function n(x, y) == e " r = x + y where n is the surface height perturbation, r is the distance from the eddy centre, and no and L are constants. Calculate the surface geostrophic currents usurf, surf (c) Ship measurements of T and S were used to compute density from the equation of state. For z> -H the results were fitted to the radially symmetric function p = Po a z + ber/L e/H where z is the usual vertical coordinate (increasing upwards with value zero at the mean sea level height) and po, a, b, and H are constants. By integrating the thermal wind equation, g ap = z fo po ay' find the geostrophic currents ug (x, y, z) and integration using the constraints u(x, y, 0) z = 0.] = = z fo po ax vg(x, y, z). [Hint: find the constants of usurf (x, y) and v(x, y, 0) = surf (x, y) at Explain why the geostrophic current field does not depend on the variable a. = = (d) The constants in the above equations are found to have the following values: no = 0.225 m, po = 1025 kg/m, L 100 km, H 1200 m, a = 4 10 kg/m, b = 0.3 kg/m. Using your favourite plotting program (Matlab, Maple, Python etc), make a contour plot of the density anomaly = p-po, in the y - z plane, i.e., x = 0, y in the horizontal and z > -H in the vertical. (e) Make a corresponding contour plot of the geostrophic current field u, (0, y, z) in the y z plane. In what direction is the eddy rotating? 3. A cold-core ocean eddy located at 40 S has been investigated by satellite measure- ments of sea-surface height and ship-based observations of temperature, salinity, and density. (a) Start with the equation for hydrostatic balance dp = -pog. dz Integrate this equation from depth z < 0 to the sea surface z = n(x, y), taking the atmospheric pressure to be po = constant and the pressure at depth z to be p(x, y, z). Using the resulting expression, show that the geostrophic current at the surface is u surf (x, y) g fo ay' surf (x, y) = g fo . (You can take po, fo as constants.) Find an expression for the geostrophic streamfunction (x, y) in terms of the sea-surface height n(x, y). (b) Satellite measurements of sea-surface height are fit to the function n(x, y) == e " r = x + y where n is the surface height perturbation, r is the distance from the eddy centre, and no and L are constants. Calculate the surface geostrophic currents usurf, surf (c) Ship measurements of T and S were used to compute density from the equation of state. For z> -H the results were fitted to the radially symmetric function p = Po a z + ber/L e/H where z is the usual vertical coordinate (increasing upwards with value zero at the mean sea level height) and po, a, b, and H are constants. By integrating the thermal wind equation, g ap = z fo po ay' find the geostrophic currents ug (x, y, z) and integration using the constraints u(x, y, 0) z = 0.] = = z fo po ax vg(x, y, z). [Hint: find the constants of usurf (x, y) and v(x, y, 0) = surf (x, y) at Explain why the geostrophic current field does not depend on the variable a. = = (d) The constants in the above equations are found to have the following values: no = 0.225 m, po = 1025 kg/m, L 100 km, H 1200 m, a = 4 10 kg/m, b = 0.3 kg/m. Using your favourite plotting program (Matlab, Maple, Python etc), make a contour plot of the density anomaly = p-po, in the y - z plane, i.e., x = 0, y in the horizontal and z > -H in the vertical. (e) Make a corresponding contour plot of the geostrophic current field u, (0, y, z) in the y z plane. In what direction is the eddy rotating?