( ) = 1005 J kg 1 K 1 ;

Transcription

1 Problem Set 3 1. A parcel of water is added to the ocean surface that is denser (heavier) than any of the waters in the ocean. Suppose the parcel sinks to the ocean bottom; estimate the change in temperature that parcel undergoes and also state explicitly the assumption you make. 2. Consider an ocean at rest with known vertical profiles of potential temperature and salinity θ(z) and S(z). Using the equation of state ρ =ρ (p, θ, S), obtain an expression for the buoyancy frequency. Also obtain the expression for buoyancy frequency in the atmosphere using the equation of state for an ideal gas. 3. Consider a liquid sitting in a container with a free surface at the top (z = H). The liquid obeys the equation of state ρ = [1 α T (T T )] and its internal energy is given as e = C p T per unit mass. Suppose the liquid is heated and its temperature rises by ΔT and the free surface rises by ΔH (small). Obtain an expression for the ratio of change in e to the change in gravitational potential energy (GPE) of the ocean and show that it is related to the temperature scale height H T (= C p α T g ) ; C p = C w = 3986 J kg 1 K 1 specific heat of seawater. 4. If global warming increases ocean temperature by 4 K, what is the value of Δ GPE? Estimate the Δ e average sea level rise due to thermal expansion of seawater. 5. Considering an ocean of uniform density = 1 kg / m 3, determine the pressure at a depth of 1 km and 5 km using the hydrostatic relation. 6. Consider an ocean of uniform density = 1 kg / m 3. Given the heat content of a mass dm of dry air at temperature T as C p (air)t dm, find the vertically integrated heat capacity (heat content per 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. Take C p air C p ( seawater) = 3986 J kg 1 K 1. ( ) = 15 J kg 1 K 1 ; 7. Derive an expression for potential density for dry air. Potential density of air is defined as the density which dry air would have if it were transformed reversibly and brought adiabatically from a pressure level p to the surface pressure p. 8. A hot meteorite falls with velocity 2 km/hr into the Indian Ocean. The meteorite, originally at a temperature of 1 C, weighs 1. kg and has a heat capacity of.82 J g 1 K 1. If the ocean temperature is 15 C, calculate the change in entropy of the universe as a result of this event. 9. Assume that in the surface mixed layer of the ocean, mixing maintains a vertically uniform temperature. A heat flux of 25W m 2 is lost to atmosphere at the ocean surface. Assume that the mixed layer depth does not change in time and there is no entrainment from its base (i.e. below the mixed layer ocean is stably stratified). Determine how long does it take for the mixed layer of 1 m ( ) = 385 J kg 1 K 1. depth to cool down by 1 C? Take C p seawater 1. The salinity at 57.5 N is observed to be much less than the salinity at 27.5 N; what reason do you ascribe to this observation? 11. The surface salinity of the Arctic Ocean waters is 28.8 psu, while the salinity at a depth of 1 m in this region is 33.6 psu. What depth of seawater needs to freeze in the Arctic Ocean to increase the average salinity by 4.6 psu in the top 1 m? [Hint : Salt content of the layer before and after the freezing is conserved]. 12. The Greenland ice sheet covers.35% of the surface area of earth and constitutes 8.6% of the total ice mass, which is equivalent to 6 m of depth of liquid water if spread over the entire surface of earth. Find the mass of the Greenland ice sheet. What will be the resultant salinity of the mixed layer if the 1

2 entire ice sheet melts due to global warming? [Take average salinity of the mixed layer as 35 psu and its depth as 1 m; and density of ice = 917 kg m 3 ]. 13. The potential density σ t = ρ t 1 (kg m 3 ) at depth of 1 m in the Southern Ocean is 28.5 kg m 3 and the salinity at this depth is found to be 34.9 psu. Estimate how much the salinity of the surface water of Southern Ocean would need to increase so that σ t at surface becomes equal to σ t at 1 m depth. [Hint: Use the T-S diagram with σ t contours and surface salinity 38.8 psu). 14. A convective oceanic mixed layer in winter develops as a result of heat loss at the surface. Take the initial temperature profile as T ( z) = T s + Λ z, where the gradient of temperature Λ >, z is depth (which is zero at the surface and increases upwards) and loss of heat from the surface at a rate Q W m 2. As the surface cools, convection sets in and mixes the developing, cold, mixed layer of depth h t ( ), which has a uniform temperature T m ( t). By matching the heat lost through the surface ( ) and T m ( t) to the changing heat content of the water column, determine the time evolution of h t during the winter period. negligible. Assume temperature continuous across the base and salinity effects [ ] ) (Hint: If salinity effects are negligible, ρ = 1 α T (T T ref ) 15. In the preceding exercise take Q = 25 W m 2 and Λ = 1 C km 1. How long will it take for the mixed layer to reach a depth of 1 meters? 16. If the top 1 m of the ocean warms by 5 C during a 3-month summer period, what is the average rate of net energy flow ( W m 2 ) in to the ocean during this period? If atmosphere warm up by 2 C during the same period, what is the average rate of net energy flow in to the atmosphere? 17. If top 1 m of ocean (σ t =27, C p =3986 J kg -1 K -1 ) warms by 5 C during a 3 months summer period, what is the average rate of net energy flow into the ocean during this period in units of W/m 2? What will be the warming ( C) of a relatively fresh 1 m deep ocean layer (σ t =8, C p =421 J kg -1 K -1 ), if this average rate of net energy flows into this layer during the same 3 months period? 18. The warming due to greenhouse gases is calculated as 1.6 Wm -2. If this heat is absorbed in the top 5 m layer of the ocean, calculate the temperature change in this layer of the global ocean in a century using the equation ρ m C w h m (t) dt m = Q dt Estimate the temperature rise over a century if this heat was mixed only in the mixed layer of depth 1 m. 19. Calculate the evaporation flux E (kg / m 2 / sec) under the following conditions: u a = 5 m / sec, q a =2 g/kg, T a = 3 C. Compare E for (i) a freshwater lake ; and (ii) the ocean with salinity S = 35 psu. [Hint: Compute saturation on vapour pressure using, e s = exp { } (hpa); Specific humidity q = ε e s, p e s ε =.622, q sea =.989 and finally compute E ] 2. At two locations P 1 and P 2, the following observations were made for temperature (T) and salinity (S) at the surface and at a depth of 1 m in the ocean: Location Surface 1 m depth P 1 T = 27 C ; S = 22.5 psu T=22 C ; S = 24.5 psu P 2 T=28 C ; S = 22 psu T = 24 C ; S = 23 psu Find the density of seawater at the two levels for these locations using the simplified form of the equation state for seawater 2

3 1 dρ = α T dt + β ds T ref = 233 K, S ref = 35 psu, = 1 kg m -2 Also calculate the potential temperature at the depth of 1 m at these locations. Take α T = K -1 and β = psu Access the netcdf file (-.nc) of Indian Ocean ARGO data on your computer system, and obtain the profiles of temperature and salinity at a location (i) North of equator (ii) South of the equation and (iii) on the equator. Using the data on α o = α o (T, S) and K T = K T (T, S), obtain the profile of density at these locations. 22. Write a Fortran program to calculate the density of seawater. Access the ARGO data from its portal and choose locations in (i) Arctic sea, north (South Atlantic Ocean, (iii) Pacific and (iv) Indian ocean. From the Fortran Program, compute the density profiles and lapse rate (adiabatic) in the ocean at the locations Identify where instability is possible. 23. Two Argo Floats positioned at two locations in the Indian Ocean record temperature and salinity up to a depth (Z) of 1 m. The temperature (T) and salinity (S) measurements from these instrumented platforms have been given at different depths in the ocean at a location P 1 (7E, 5N) on July 2, 27 and at a location P 2 (45E, ) on Aug 1, 27. Plot these measurements on a Temperature-Salinity Sigma-T chart and join the data points with straight-line segments to draw their profile on the chart. Answer the following questions (a) What is depth of the mixed layer at P 1 and P 2? (b) From the plots of the two profiles on Temperature-Salinity Sigma-T chart, find the density of seawater at the depths Z=, Z=5 m, Z=2 m, Z=5 m and Z=9 m at the two locations P 1 and P 2. (c) Find potential temperature at a depth 8 m at these two locations. (d) Identify the regions where instability can occur at P 1 and P 2. (e) Plot the temperatures at P 1 and P 2 with depth and discuss the different regions you could identify from the temperature profiles whether thermocline is deeper at P 1 or P 2. Discuss the consequences of deepening and shallowing of the thermocline region at the two locations. (f) Identify the isothermal and barrier layers, if they exist, at P 1 and P 2. Depth Loc. P 1 (45E, ) Loc. P 2 (7E, 5N) Z (m) T ( o C) S (psu) T ( o C) S (psu) The sensible heat flux (SH) from a surface is related to the mean wind and temperature by the bulk aerodynamic formula SH = ρ a C p C H ( u a u o )( T s T a ) where ρa is mean density (1.28 kg m-3); u a, T a are mean wind velocity and temperature at 1-m height above the surface; u o, T s are velocity and temperature at the surface. Thus u o =.56 u a for ocean and u o = for a solid surface. The constant C p = 15 J kg 1 K 1 is the specific heat of air at constant pressure and C H = is the aerodynamic transfer coefficient for heat. Assuming a steady state with no heat storage and no radiative and latent heat losses. Given the heat conductivity 3

4 coefficients k i and k s for ice and snow respectively, obtain the formula to calculate the heat flux through an ice-snow composite layer using expressions of fluxes through ice and snow layers F i = k i T B T i h i T ( ice) ; F s = k i T s s h s ( snow). 25. Suppose that air temperature above the Arctic Ocean is -3 C and the water temperature is -2 C. Solve for the sensible heat flux to the atmosphere if the bulk aerodynamic formula applies with surface pressure 1 5 Pa, C H = 1 1-3, u a = 5m /sec. Assume a steady state with no heat storage and ignore radiative and latent heat fluxes. Consider three cases: (a) no sea ice (b) 1.5 meter of sea ice (c) 3 meter of sea ice If the flux through the ice / snow is given by (i) F i = k i T B T i h i (ice) ; k i = 2.W m 1 K 1 ; F s = k s T i T s h s (Snow) ; k s =.3W m 1 K 1. Solve for temperature of the surface where there is no snow in the cases (b) and (c). k i and k s are the thermal conductivities for ice and for snow respectively. (ii) Solve for surface temperature T s, when there is 1 cm of snow on sea ice in (b) and (c). 26. The temperature (T) and salinity (S) measurements have been given at different depths in the Indian Ocean at a location P (16 48 S; E). Plot these measurements on a Temperature-Salinity Sigma- T chart and join the data points with straight-line segments to draw profile on the chart. Answer the following questions (a) What is depth of the mixed layer from observations? (b) From the profile on Temperature-Salinity Sigma-T chart, find the density of seawater at the depths Z=, Z=2 m, Z=1 m, Z=3 m, Z=4 m and Z=5 m at location P. (c) Find the potential temperature at Z=4 m at this location. (d) Identify the regions where instability can occur. (e) Infer the thermocline region from the given data. Discuss the consequences of deepening and shallowing of the thermocline region. (f) Analyze the inflexion points in the profile, if any. 27. The equation of the state of seawater 1 Depth Loc S E Z (m) T ( o C) S (psu) dρ = α T dt + β ds involves coefficients of thermal expansion (α T = 1 ρ 1 ) and of salinity contraction (β = T ρ ) which can be approximated S 4

5 linearly above 5 o C as α T T (1 6 / o C) and β T ((1 6 psu 1 ). Find the values these coefficients at temperature and salinity measurements given in Problem The large-scale steady circulation in the ocean interior is governed by geostrophic and hydrostatic equations which are as follows: fv = 1 p x ; fu = 1 p y (geostrophy) and p + gρ = (hydrostatic eq.) Derive the thermal wind equations and prove that for a homogeneous fluid (ρ = const.) the fluid columns move as a rigid body. (This is known as the Taylor-Proudman Theorem for rotating fluids). 29. Consider motion of a stratified fluid in the ocean on a f-plane (i.e. the Coriolis parameter f=const.). The dynamics is governed by the following equations: fv = 1 p x ; fu = 1 p y (geostrophy) and p + gρ = (hydrostatic eq.) u x + v y + w = (continuity eq.) and ρ u x + v ρ y + w ρ = (density conservation) For the above system, show that w = Further, take the vertical velocity w = at z =, and find out w for the whole depth of the water column using the above equation. This is the Taylor-Proudman Theorem for stratified fluids). 3. Consider the modification of geostrophic relation by friction on a f-plane, which is given by fv = 1 p x k δ u ; fu = 1 p y k δ v ; u x + v y + w = where k is the drag coefficient and δ is the Ekman layer depth. Show that on combine the geostrophic equations by eliminate pressure and using the continuity equation, the resultant equation is f w = k v δ x u y. Further from the above equation, Discuss the following giving appropriate analysis (a) State the condition when the Taylor-Proudman Theorem is violated. (b) Western boundary currents (poleward flowing current with boundary on the right) are permitted in both northern and southern hemispheres. (c) Eastern boundary currents (poleward flowing current with boundary on the left) prohibited in both northern and southern hemispheres. 31. Consider the sea breeze circulation depicted in the following figure: The mean temperature T 1 over ocean is colder than the mean temperature T 2 over the adjoining land, and the dashed lines represent surfaces of constant density. Pressure at the surface is p and at a mean 5

6 height h it is p 1. The pressure at the surface is uniform and isobaric surface p 1 will slope towards the oceanic region. Let p = 1 hpa and p 1 = 85 hpa, height h = 1.5 km, length L = 2 km (as shown in the figure) and T 1 T 2 = 15 C. In the absence of frictional retarding forces, the mean tangential velocity can be obtained from the following equation: v = R ln( p / p ) 1 (T 2 T 1 ) t 2(h + L) Calculate the acceleration produced by the this temperature difference and find the wind speed after one hour taking initial velocity zero and neglecting frictional retardation. Compute the evaporation from the ocean surface having temperature 27 o C, and the convergence of the moisture flux as marine air moves over land. Evaporation over the ocean may be computed from the following relation: E = ρ a C E (v v )(q s q a ); C E = and q a =.5q s To compute the moisture convergence, assume v becomes steady after one hour and wind breaking law of the mean wind at the surface as 1/2 on the ocean surface and 1/3 on the land surface. 32. An important simplification for the dynamics of the mixed layer is to set the density constant in its entire depth h but varying horizontally; that is, ρ = ρ(x, y), p = p(x, y). Choose the vertical coordinate system with the origin z = set at the surface and z decreases downward. Thus, the bottom of the mixed layer is found at z = h. Define the Bernoulli function as B = p + ρgz which is vertically constant within the mixed layer and B s = B(x, y,). By calculating the horizontal gradients h p, obtain the expressions for the geostrophic flow components (ug, v g ) in the mixed layer in terms of the Bernoulli function and identify the barotropic and baroclinic terms in the horizontal gradient of p. By calculating volume flux in the mixed layer. udz and v dz, obtain the expressions for the vertically integrated h h 6