Exam Questions & Problems

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1 1 Exam Questions & Problems Summer School on Dynamics of the North Indian Ocean National Institute of Oceanography, Dona Paula, Goa General topics that have been considered during this course are indicated by the boldface headings. Each topic contains a list of specific questions or problems that were discussed in lectures or during the tutorials. If this course actually had mid-term or final examinations, good choices for exam questions would come from this list. Please read over the list. How many of these questions can you now answer that you would not have been able to do before the course? We (lecturers) hope and expect that you can now answer a lot of them, and so learned a lot during this course. History 1) When did awareness for the large-scale seasonal change of currents over the North Indian Ocean (NIO) first arise? 2) When was the link between reversing monsoon winds and reversing currents first established? How did our ideas regarding this link evolve over time? 3) What were the major, modern-day, oceanographic explorations of the Indian Ocean, and when did they take place? Atmospheric forcing 4) What processes determine the basic properties of the monsoon winds? In particular, what are the dynamical impacts of the seasonal migration of the Intertropical Convergence Zone (ITCZ) and of the Himalaya mountains? Observed annual cycle 5) What are the major NIO currents, and how do they vary seasonally? 6) How does sea level vary annually in the NIO, and how is its variability related to the annual cycle of the currents? Analytic methods 7) What are the eigenfunctions and eigenvalues of a differential operator? What is the normalization of a set of eigenfunctions, and why is it necessary? What is orthogonality of a set of eigenfunctions? 8) Write down the forced wave equation for non-dispersive waves: q xx c 2 q tt = X(x)T(t). Use the method of Laplace transforms to solve the system for a switched-on forcing. Why

2 2 is the method useful? 9) Repeat the previous problem, but also use the method of Fourier transforms to obtain the x-dependence of the response. Why is the method useful? Obtain the response for both oscillatory and switch-on forcing. Numerical methods 10) Numerical solutions are obtained on a spatial grid. How are the variables of a typical ocean model defined on the grid? How are differential operators defined on the grid? 11) What is numerical instability? One requirement for numerical stability is that the CFL condition is satisfied. What is that condition? Interpret it in physical terms. How does horizontal and vertical mixing help to eliminate numerical instability? Models 12) Write down the equations of motion for a typical ocean general circulation model (OGCM). Discuss the impact of each of the terms. 13) What simplifications are involved in reducing the OGCM equations to those of the linear, continuously stratified (LCS) model? 14) The baroclinic and barotropic modes of the LCS model are eigenfunctions of a differential operator. What equation and boundary conditions do the modes satisfy? What is the characteristic speed, c n, associated the nth mode? What approximations are required in order for these modes to exist? 15) Assume that N b is constant and that the ocean depth is D. Solve the mode equation for the baroclinic and barotropic modes, and evaluate the values of c n for realistic choices of model parameters. 16) Prove that the set of baroclinic and barotropic modes are orthogonal. 17) How do you obtain a three-dimensional (x,y,z) flow field from the set of twodimensional (x,y) responses for each mode? 18) What is a 1½-layer, reduced-gravity model? To what does the ½ layer correspond? Why is it also called a reduced-gravity model? How is the 1½-layer model related to the equations for a single baroclinic mode of the LCS model? Midlatitude (off-equatorial) gravity, Rossby, and Kelvin waves 19) The dispersion relations of the waves in an ocean model define all the basic properties of the system. Derive the dispersion relation for the waves of the LCS model. First, obtain an equation in v alone, adopt the classical β-plane approximation, and look for plane-wave solutions of the form exp(ikx + ily iσt). 20) Plot the dispersion in σ,k,l space. Argue that solutions exist in two regions, an upper

3 3 (gravity-wave) bowl and a lower, inverted (Rossby-wave) bowl. 21) What are the critical frequencies for each of the bowls? For realistic parameters, what are their values? Argue that Rossby (gravity) waves exist only at frequencies lower (higher) than the critical frequencies. 22) Derive analytic solutions for the coastal Kelvin waves that exist along basin boundaries on the f-plane. Consider a fixed value of σ. Argue that there is a critical latitude above which Rossby waves don t exist. Similarly, argue that there is another critical latitude poleward of which gravity waves don t exist. (Critical latitudes are just another way of interpreting the critical frequencies.) 23) Design a set of numerical experiments for a baroclinic mode that illustrate all the wave types. For example, obtain the response to an initial δ-function-like sea-level anomaly on the f-plane. How does the response differ on the β-plane? How do both responses differ if the sea-level anomaly is large scale in x and y? Ekman drift and inertial oscillations 24) Derive the time-dependent response of a 1½-layer model to a switched on, spatially uniform, zonal wind on the f-plane. Show that inertial oscillations are generated, as well as a steady Ekman drift directed 90º to the right of the wind. 25) Consider the LCS equations without expanding into modes. Derive the steady-state Ekman spiral. Argue that the surface current flows 45º to the right of the wind (in the northern hemisphere). Integrate u(z) and v(z) for the spiral from the bottom to the top of the ocean, and show that the net Ekman transport is directed 90º to the right of the wind, just as in the 1½-layer model. 26) Obtain numerical solutions to a baroclinic mode that illustrate the responses to a switched-on wind on the f- and β-planes. In the latter case, why does the steady-state response develop much more quickly? Ekman pumping and adjustment to Sverdrup balance 27) It is difficult (impossible) to solve the general v-equation for a baroclinic mode analytically. A simplification that does allow simple analytic solutions neglects the u t and v t terms in the equations of motion. Discuss how the dispersion relation is changed in the simplified system. Under what conditions is this approximate set of equations valid? 28) Find an analytic solution on the f-plane for the spin-up of a baroclinic mode in response to a switched-on patch of wind. Discuss how this solution illustrates Ekman pumping, one of the most important processes in the ocean. Discuss the initial and longterm responses of the solution. In particular, show that in the absence of damping, there is a response that grows in time. 29) Repeat the previous problem on the β-plane. Show that the initial response is Ekman pumping, followed by the adjustment to Sverdrup balance by the radiation of long-

4 4 wavelength Rossby waves. Discuss how the response differ for τ x and τ y winds. 30) Obtain numerical solutions for a baroclinic mode that illustrate the above analytic solutions. Boundary layers 31) Read Stommel s (1948) famous paper, which shows that the β-effect is the process that generates the strong western-boundary currents. Stommel essentially solves for the steady-state response of a mode of the LCS model to a zonal wind patch when there is momentum damping of the form, νu and νv. He obtained the solution exactly. Obtain the solution using boundary-layer techniques, arguing that the interior response is essentially in Sverdrup balance and the western-boundary current is a Stommel layer. 32) Repeat the previous problem when the horizontal viscosity is Laplacian. Argue that the solution is the same except that the western-boundary layer is a Munk layer. Show that the Munk layer reverses offshore. Hermite functions 33) The Hermite functions are eigenfunctions of a particular differential operator? What is the operator? What are its eigenvalues? 34) Write down the generating function for Hermite functions? Use the generating function to obtain analytic expressions for the first 5 Hermite functions, and plot them. Equatorial waves 35) What is the equatorial β-plane approximation? 36) Derive the dispersion relations for equatorially trapped waves. Discuss similarities to, and differences from, the dispersion relations on the classical β-plane. 37) What are the critical frequencies and wavenumbers for equatorial gravity and Rossby waves? What are their values for realistic parameters? 38) Derive the dispersion relation for the Yanai wave, and discuss its properties. Why is it sensible to refer to it as the mixed Rossby-gravity wave? 39) Set v = 0 in the equations for a baroclinic mode, and solve the resulting equations for an equatorial Kelvin wave. 40) Discuss the symmetry of equatorial trapped waves. 41) What are the expressions for the equatorial Rossby radius of deformation R 0 and the equatorial inertial frequency σ 0? Write the dispersion relations for equatorial gravity, Rossby, Yanai, and Kelvin waves in terms of the non-dimensional frequency and wavenumber, σ = σ/σ 0 and k = kr 0. Discuss how this single set of curves applies to all the baroclinic modes.

5 5 42) Design a set of numerical solutions for a baroclinic mode of the LCS model that illustrate the structures of several of the equatorial waves, including the Yanai and Kelvin waves. Yoshida Jet 43) Obtain an analytic solution to the equations for a baroclinic mode forced by a switched-on, x-independent zonal wind. Show that the response has two parts: one that oscillates (equatorial inertial oscillations) and another that grows steadily in time (the Yoshida Jet). 44) It is possible to extend the preceding solution when the forcing has a zonally bounded structure X(x). Initially, the response is an accelerating Yoshida Jet, as in the preceding x- independent case. After the passage of Rossby and Kelvin waves, however, a pressure gradient develops along the equator and the Yoshida Jet stops accelerating. Design a numerical experiment that illustrates this adjustment. Boundary reflections 45) Consider the reflection of an equatorial Kelvin wave at an eastern-ocean boundary. Show that a chain of equatorial waves is needed in order to satisfy the boundary condition u = 0 there (Moore s famous chain rule). Design a numerical experiment that illustrates this reflection process. 46) If the wind oscillates at a frequency σ, show that the reflected waves radiate offshore as Rossby waves equatorward of the critical latitude, and remain trapped to the coast to form a β-plane, coastal Kelvin wave poleward of it. Design a numerical experiment that illustrates these properties. 47) Discuss the analogous reflection process at a western-ocean boundary? In particular, argue that reflected Rossby waves at a western boundary generate a chain of boundary waves that extends toward (not away) from the equator, and ends in either an equatorial Kelvin wave, Yanai wave, or both. Design a numerical experiment that illustrates this reflection process. Coastal dynamics 48) A simplification that allows for simple analytic solutions along a north-south coast neglects the u t term in the equations of motion for a baroclinic mode. Discuss how the dispersion relation is changed when the u t term is dropped. Under what conditions is this approximate set of equations correct? 49) Consider the response of a baroclinic mode (or 1½-layer model) on the f-plane in an ocean basin with an eastern boundary, when the forcing is by a uniform (i.e., x- and y- independent) meridional wind. Show that when the wind is southward (in the northern hemisphere), the interface continues to rise at the coast. This is the simplest solution that illustrates the process of coastal upwelling. The accelerating coastal jet is analogous to the Yoshida Jet on the equator.

6 6 50) Repeat the preceding problem when the forcing is has a y-structure, Y(y), of finite extent (i.e., forcing by wind band). Show that in response to a switched-on wind, coastal Kelvin waves radiate poleward along the coast, and after their passage the coastal jet stops accelerating. The resulting steady coastal jet is analogous to the bounded Yoshida Jet on the equator. 51) Repeat the preceding problem on the β-plane, setting u t = v t = 0 in the equations of motion. Show that the steady coastal jet of the preceding solution propagates offshore as a Rossby wave. 52) Design a set of numerical experiments that illustrate these analytic solutions. Equatorial and Coastal Undercurrents 53) Obtain numerical solutions to the LCS model forced by a switched-on patch of zonal wind centered on the equator. Find the responses for N = 25 baroclinic modes, and sum the responses. Make a movie that show the spin-up of the solution. The steady response has an Equatorial Undercurrent (EUC). Read McCreary (1981a), and discuss the dynamics of the EUC in this solution. 54) Repeat the previous problem for forcing by a midlatitude band of meridional wind that is independent of x. Make a movie that show the spin-up of the solution. The steady response has a Coastal Undercurrent (CUC). Read McCreary (1981b), and discuss the dynamics of the CUC in this solution. Intraseasonal variability 55) What types of intraseasonal variability (ISV) exist in the equatorial IO? What atmospheric forcing fields exist at intraseasonal frequencies? 56) What are the dynamical processes that generate IO ISV? For forcing by a specific σ, the response of an equatorial wave with wavenumber k is proportional to a zonal integral of X(x)exp(ikx). Under what conditions is this integral large, and hence the wave is strongly excited? What is a basin resonance? 57) What types of ISV are present in the EEZ surrounding India? Equatorial and coastal beams 58) For forcing at a specific σ, the energy of equatorial waves propagates vertically as well as horizontally. Derive the angles of energy propagation for long-wavelength Rossby waves, Yanai waves, and Kelvin waves. 59) Similarly, the energy of coastal waves also propagates downward. Derive the angle of energy propagation for coastal Kelvin waves. 60) Design a set of numerical experiments that illustrates vertical energy propagation.

7 7 Surface mixed layer and realistic basins 61) Describe the surface mixed layers that exist in NIO waters. What is a barrier layer? 62) A number of different models have been developed to simulate the surface mixed layer. They include the Kraus-Turner (KT), Price-Weller-Pinkel (PWP), KPP, and Mellor- Yamada models. Describe the basic physics of each of these models. Discuss the similarities and differences among them. Realistic forcing and OGCMs 63) The LCS model produces many observed features of the NIO climatological circulation when it is forced by realistic winds (e.g., NCEP winds). Design a set of numerical experiments that illustrate the importance of specific processes (e.g., equatorial zonal winds, coastal alongshore winds, interior Ekman pumping) in the overall response. 64) The dynamical core of all OGCMs is the essentially the same as in the LCS model. The OGCMs, however, include also include a representation of the ocean mixed layer and advection. What processes are OGCMs able to simulate that the LCS model cannot? Shallow overturning circulations 65) What are shallow overturning circulations in the world ocean? 66) The two primary shallow overturning cells in the Indian Ocean are the Crossequatorial Cell (CEC) and the Subtropical Cell (STC). Describe the complete flow paths of these cells. IO biophysics 67) What is the euphotic zone of the ocean? 68) Perhaps the most important way that ocean physics influences biology is through the thickness of the mixed layer, h m : Phytoplankton growth tends to occur (not occur) if h m is less (greater) than the depth of the euphotic zone Discuss the phytoplankton blooms that occur during upwelling, entrainment, and detrainment events. Interannual variability ENSO is the earth s most prominent mode of interannual variability. What are the impacts of ENSO in the Indian Ocean. In particular, how does ENSO generate warming in the 5 10ºS ridge, and the Southwest Monsoon during the following year. Another prominent type of interannual variability in the Indian Ocean is the Indian Ocean Dipole (IOD; also known as the IO Zonal Mode (IOZM), IODZM, and IOZDM). Is the IOD really dipole in temperature, as originally stated Saji et al. (1999)? Is the IOD really an distinct climatic mode, separate from ENSO?

8 8 Observing systems 69) How have oceanographic measuring techniques changed during the past 50 years? 70) When were the plans for the current Indian Ocean observing system (IndOOS) formulated? How close is the implementation of those plans to completion? What types of observing systems does IndOOS include?: Designing numerical experiments 71) How does a researcher decide what oceanic problem to investigate numerically? 72) A number of technical issues must be considered in designing numerical experiments. What are they?