After reading A Guided Tour of Mathematical Methods for the Physical Sciences, not only will you be able to impress your friends with a back-of-the-envelope scaling analysis to debunk the myth that a bathtub drains in different directions in both hemispheres. You will also know whether life is possible in a five-dimensional world. In fact, you will understand a lot of mathematical methods and know how to apply them to real physical problems. This unique textbook presents all of its material in the form of problems that invite the reader to participate in the development of the theory Guided by Roel Sneider's textbook, learning mathematical methods can be fun.

The theory of first-passage time distribution functions and its extension to last-passage time distribution functions are applied to the problem of tracking the movement of water masses to and from the surface mixed layer in a global ocean general circulation model. The first-passage time distribution function is used to determine in a probabilistic sense when and where a fluid element will make its first contact with the surface as a function of its position in the ocean interior. The last-passage time distribution is used to determine when and where a fluid element made its last contact with the surface. A computationally efficient method is presented for recursively computing the first few moments of the first- and last-passage time distributions by directly inverting the forward and adjoint transport operator. This approach allows integrated transport information to be obtained directly from the differential form of the transport operator without the need to perform lengthy multitracer time integration of the transport equations. The method, which relies on the stationarity of the transport operator, is applied to the time-averaged transport operator obtained from a three-dimensional global ocean simulation performed with an OGCM. With this approach, the author (i) computes surface maps showing the fraction of the total ocean volume per unit area that ventilates at each point on the surface of the ocean, (ii) partitions interior water masses based on their formation region at the surface, and (iii) computes the three-dimensional spatial distribution of the mean and standard deviation of the age distribution of water.

This dissertation is a study of the climatological mean seasonal cycle of the ventilation of the ocean. Firstly, I have developed a computationally efficient 4-D variational assimilation system, called CYCLOCIM. It assimilates monthly mean temperature and salinity data from the World Ocean Atlas, transient CFC-11 and CFC-12, and natural radiocarbon measurements for the deep ocean from the GLODAPv2 database. CYCLOCIM is a new variational assimilation system that is specifically designed for the problem of estimating the climatological seasonal cycle of the residual mean circulation. I independently obtain the modeled overturning circulation, and the meridional heat and fresh water transport that largely agree with existing studies. Moreover, CYCLOCIM improves the model fit to the observation in the upper ocean compared to previous studies that ignored the seasonal cycle. The main product of the assimilation system is a set of 12 monthly data-constrained of tracer transport operators. I then calculate transit-time distribution (TTD) using the monthly transport operators to quantify ``Stommel's demon'', that is, the seasonality of the ventilation in the main thermocline. The study provides an accurate estimate of effective subduction and obduction months and maps of the water fraction in the main thermocline was last transported from or will be first transported back to the surface. For example, subduction occurs mainly in the Southern Ocean (64%), which lasts from June to November and peaks in September (21%) and October (20%). Moreover, the subduction happens in the North Pacific (12%) and the North Atlantic (16%), which lasts from January to April and peaks in March (15%). Obduction occurs mainly in ACC region (48%), which starts from April to November and peaks in August. The upwelling regions of the tropical ocean is the secondary primary obduction region, which upwell 35% of the water back to surface and lasts the whole year. Furthermore, I have applied the TTD method to estimate the ventilation of the deep water masses. The formation of NADW, AABW, PDW, IDW occurs mainly in the surface of the Southern Ocean (respectively, 15%, 69%, 56% and 53%) and the Atlantic Ocean (respectively, 84%, 39%, 37% and 41%) during winter and spring in both hemisphere.

A data-constrained ocean circulation model is used to characterize the distribution of water masses and their ages in the global ocean. The model is constrained by the time-averaged temperature, salinity, and radiocarbon distributions in the ocean, as well as independent estimates of the mean sea surface height and sea surface heat and freshwater fluxes. The data-constrained model suggests that the interior ocean is ventilated primarily by water masses forming in the Southern Ocean. Southern Ocean waters, including those waters forming in the Antarctic and subantarctic regions, make up about 55% of the interior ocean volume and an even larger percentage of the deep-ocean volume. In the deep North Pacific, the ratio of Southern Ocean to North Atlantic waters is almost 3:1. Approximately 65% of interior ocean waters make first contact with the atmosphere in the Southern Ocean, further emphasizing the central role played by the Southern Ocean in the regulation of the earth’s climate. Results of the age analysis suggest that the mean ventilation age of deep waters is greater than 1000 yr throughout most of the Indian and Pacific Oceans, reaching a maximum of about 1400–1500 yr in the middepth North Pacific. The mean time for deep waters to be reexposed at the surface also reaches a maximum of about 1400–1500 yr in the deep North Pacific. Together these findings suggest that the deep North Pacific can be characterized as a “holding pen” of stagnant and recirculating waters.

A conceptually new approach to diagnosing tracer-independent ventilation rates is developed. Tracer Green functions are exploited to partition ventilation rates according to the ventilated fluid’s residence time in the ocean interior and according to where this fluid enters and exits the interior. In the presence of mixing by mesoscale eddies, which are reasonably represented by diffusion, ventilation rates for overlapping entry and exit regions cannot meaningfully be characterized by a single rate. It is a physical consequence of diffusive transport that fluid elements that spend an infinitesimally short time in the interior cause singularly large ventilation rates for overlapping entry and exit regions. Therefore, ventilation must generally be characterized by a ventilation-rate distribution, , partitioned according to the time that the ventilated fluid spends in the interior between successive surface contacts. An offline forward and adjoint time-averaged OGCM is used to illustrate the rich detail that and the closely related probability density function of residence times provide on the way the ocean communicates with the surface. These diagnostics quantify the relative importance of various surface regions for ventilating the interior ocean by either exposing old water masses to the atmosphere or by forming newly ventilated ones. The model results suggest that the Southern Ocean plays a dominant role in ventilating the ocean, both as a region where new waters are ventilated into the interior and where old waters are first reexposed to the atmosphere.