## Problems

1. Consider an ocean of uniform density pref = 1000 kg m-3.

(a) From the hydrostatic relationship, determine the pressure at a depth of 1 km and at 5 km. Express your answer in units of atmospheric surface pressure, Ps = 1000 mbar = 105Pa.

Sea Surface Temperature (June 20, 2003)

Sea Surface Temperature (June 20, 2003)

longitude

FIGURE 9.23. A satellite-derived Sea Surface Temperature map (in °C) over the Gulf Stream, from June 20,2003. Note the advection of warm tropical waters northwards by the strong western boundary current and the presence of strong meanders and eddies in the seaward extension of the Gulf Stream.

longitude

FIGURE 9.23. A satellite-derived Sea Surface Temperature map (in °C) over the Gulf Stream, from June 20,2003. Note the advection of warm tropical waters northwards by the strong western boundary current and the presence of strong meanders and eddies in the seaward extension of the Gulf Stream.

(b) Given that the heat content of an elementary mass dm of dry air at temperature T is cpT dm (where cp is the specific heat of air at constant pressure), find a relationship for, and evaluate, the (vertically integrated) heat capacity (heat content per degree Kelvin) of the atmosphere per unit horizontal area. Find how deep an ocean would have to be to have the same heat capacity per unit horizontal area. (You will need information in Tables 1.4 and 9.3.)

### 2. Simple models of mixed layers.

(a) Assume that in the surface mixed layer of the ocean, mixing maintains a vertically uniform temperature. A heat flux of 25 W m-2 is lost at the ocean surface. If the mixed layer depth does not change and there is no entrainment from its base, determine how long it takes for the mixed layer to cool down by 1°C. [Assume the mixed layer has a depth of 100 m, and use data in Table 9.3.]

(b) Consider the development of a simplified, convective, oceanic mixed layer in winter. Initially, at the start of winter, the temperature profile is given by

T(z) = Ts + Az where z is depth (which is zero at the sea surface and increases upwards) is depth, and the gradient A > 0.

During the winter heat is lost from the surface at a rate Q W m-2.

FIGURE 9.24. Instantaneous map of surface current speed from a global ''eddy-resolving'' numerical model of ocean circulation. The scale is in units of m s-1. Modified from Menemenlis et al (2005).

As the surface cools, convection sets in and mixes the developing, cold, mixed layer of depth h(t), which has uniform temperature Tm(t). (Recall GFD Lab II, Section 4.2.4.) Assume that temperature is continuous across the base. By matching the heat lost through the surface to the changing heat content of the water column, determine how h(t) and Tm (t) evolve in time over the winter period. Salinity effects should be assumed negligible, so density is related to temperature through Eq. 4-4.

(c) If Q = 25Wm-2 and A = 10°C per kilometer, how long will it take the mixed layer to reach a depth of 100 m?

FIGURE 9.25. Schematic of ocean surface for Problem 3.

3. Consider an ocean of uniform density Pref = 1000 kg m-3, as sketched in Fig. 9.25. The ocean surface, which is flat in the longitudinal direction (x), slopes linearly with latitude (y ) from n = 0.1 m above mean sea level (MSL) at 40° N to n = 0.1 m below MSL at 50° N. Using hydrostatic balance, find the pressure at depth H below MSL. Hence show that the latitudinal pressure gradient dp/dy and the geostrophic flow are independent of depth. Determine the magnitude and direction of the geostrophic flow at 45° N.

4. Consider a straight, parallel, oceanic current at 45° N. For convenience, we define the x- and y-directions to be along and across the current, respectively. In the region -L < y < L, the flow velocity is i ny \ z u = Uo cos — j exp(d)

where z is height (note that z = 0 at mean sea level and decreases downward), L = 100 km, d = 400 m, and Uo = 1.5 m s-1. In the region | y | > L, u = 0.

The surface current is plotted in Fig. 9.26:

Using the geostrophic, hydrostatic, and thermal wind relations:

(a) Determine and sketch the profile of surface elevation as a function of y across the current.

(b) Determine and sketch the density difference, p(y,z) - p(0,z).

(c) Assuming the density is related to temperature by Eq. 4-4, determine the temperature difference,

T(L, z) - T(-L, z), as a function of z. Evaluate this difference at a depth of 500 m. Compare with Fig. 9.21.

5. Figure 9.21 is a section across the Gulf Stream at 38° N (in a plane normal to the flow), showing the distribution of temperature as a function of depth and of horizontal distance across the flow. Assume for the purposes of this question (all parts) that the flow is geostrophic.

(a) Using hydrostatic balance, and assuming that atmospheric pressure is uniform and that horizontal pressure gradients vanish in the deep ocean, estimate the differences in surface elevation across the Gulf Stream (i.e., between 70° W and 72° W ). Neglect the effect of salinity on density, and assume that the dependence of density p on temperature T is adequately described by Eq. 4-4.

at the surface, z = 0, against y/L.

(b) The near-surface geostrophic flow u is related to surface elevation n by