# Richard Katz

I do research with the aim of **understanding the physics of liquid–solid systems in terrestrial and planetary settings**. "Understanding" means having the ability to make quantitative predictions of system behaviour that are consistent with available measurements or testable by future measurements. "Liquid–solid systems" means regions of space in which liquids and solids coexist and interact mechanically and/or thermochemically. And "terrestrial and planetary" means that the applications can be to anything from individual mineral grains, laboratory experiments, or natural settings on Earth or other planetary bodies. I lead a research group that uses methods of applied mathematics (formulating and analysing theory, developing numerical simulations, deriving scaling analysis) in collaboration with observationalists and experimentalists. Below are some current areas of interest within the group.

*Physics of natural granular media. *Rock is composed of solid mineral grains and, when partially molten, an interstitial liquid. Grains can slide along their boundaries, making deformation of a partially molten rock a bit like that of wet sand. Much is known about the physics of saturated granular suspensions, and some of the theory should be relevant for partially molten rock. I have recently considered how dilatancy and non-local fluidity can be incorporated into poro-viscous theory and applied to understanding laboratory experiments (*JFM* 2024). I am interested in development of microstructural models that can capture the rock mechanics of partially molten granular media across a range of deviatoric and confining stress states. I am also interested in developing applications of granular physics to other natural systems including faults and, potentially, to tidal deformation of the rubble core of Enceladus.

*Tidally driven flows in planetary bodies. *Tides are a ubiquitous process in the planetary systems of the universe that transfer rotational energy into variations in gravitational potential energy and, through irreversible viscous deformation, onward into dissipation of heat. For example, tidal heating is thought to be the driver of the extreme magmatism of Jupiter's moon Io (*JGR* 2020a, 2020b). Tides can drive flows in subsurface oceans, and I have worked with Hamish Hay and Prof Ian Hewitt on a theory for how such flows can have a non-periodic component — a mean circulation. This could drive asynchronous rotation of the ice shell (*JGR* 2024). I am interested in tidally driven porous flows of brine in the ice shells of water worlds, and how this might transport oxidants downward from the surface. Further, I am interested in whether tides can power a geodynamo.

*Mud volcanoes and their source regions.* There are thousands of mud volcanoes on land and probably tens of thousands on the sea floor. One mud volcano, Lusi in Indonesia, has caused billions of USD in damages and displaced thousands from their homes. There has been very little work to understand (quantitatively) how mud is mobilised from the subsurface. Working with Luke Kearney and Prof Joe Cartwright, we have developed a new theory for propagation of a mudstone liquefaction front. This follows from our work on fluid-escape pipes (*PRSA* 2023, *PNAS* 2024). I am interested in extending and applying this theory to explain observations from the Mediterranean basin and beyond.

*Flexure and fracture of ice shelves.* Supraglacial lakes and ocean tides cause ice shelves to flex and, at large enough stress, to fracture. These processes are important to our fundamental understanding of the cryosphere, which is a crucial part of the climate system. I am working with Hanwen Zhang and Prof Laura Stevens to understand how supraglacial lakes in the grounding zone of ice shelves drain by a hydrofracture under the influence of tidal stress (*ArXiv*). I am interested in extending this work to look at how hydrofracture affects other aspects of ice sheets and shelves, including stream margins and calving zones.

*Metasomatic zoning and formation of ore bodies by subsolidus reactive transport. *Trace metals essential for the modern technologies are concentrated in ore layers of igneous intrusions. One theory for how these form, proposed by Korzhinskii, is of transport of chemical species in an aqueous fluid coupled with metamorphic reactions. I am working with Rhiannon Ackland, Prof Jon Blundy and Dr Lena Melekhova to extend this theory and apply it to laboratory experiments and ore layers in the Bushveld complex. My interests in this area also include reactive flows in metal-rich volcanic sediments and their potential to harvest trace metals.

*Sea level, volcanism and faulting at mid-ocean ridges.* Observations show an intriguing fingerprint of Milankovitch cycles in the sea-floor topography adjacent to mid-ocean ridges. I have been involved in developing the hypothesis that variations in sea level associated with ice ages can drive variations in melt supply to the mid-ocean ridge system (*Science* 2015, *PNAS* 2022). Currently I am working on a model for how these variations are expressed in terms of the fault-bounded abyssal hills flanking the ridge axis at fast-spreading ridges. Until there is more data on petrological/geochemical variations on 10–100ka timescales, further theoretical work may not be justified.

*Fluid-driven fracture.* Fracture propagation driven by fluid pressure appears in many settings from magmatic to glacial. I am interested in the mechanics of this process in the natural environment, where other stresses are at play. Recent work with Tim Davis has been an effort to develop a predictive model for the runout of megadikes. We are finding that dynamic topography is required to explain lateral propagation of 100s to 1000s of km without eruption through the surface. I am interested in hydrofracture for production of natural hydrogen and for harvesting geothermal energy. I am also interested in magma-filled fractures in sedimentary basins and their thermochemical consequences.

*Population dynamics of dislocations in crystals as a control on response to stress. * Dislocation creep is an important mechanism for long-term deformation of polycrystalline substances including rocks. In this context, dislocations are the basic carriers of deformation. However, dislocations generate small-scale elastic stress that repel each other. Therefore a high density of dislocations can inhibit dislocation motion and harden a rock. The evolution of dislocation density within a crystal is therefore coupled with deformation, giving rise to transient creep (*PNAS* 2021). I worked with Tom Breithaupt on zero-dimensional theory for this coupling and its calibration by comparison with laboratory experiments on steady and transient creep of olivine. I am interested in extending it to a tensorial formulation that can be used in geodynamic models of transient phenomena including postglacial rebound, post-seismic stress relaxation, and slab bending.

*The dynamics of partially molten rock.* A longstanding interest of mine, on which I have written a textbook (PUP 2023). Most of our observations of the asthenosphere are sensitive to or controlled by magmatic processes, but most models exclude magmatism or incorporate it as an afterthought. I work with a two-phase, poro-viscous flow theory developed by Dan McKenzie and others in the 1980s. Recently, with Adina Pusok, I have been investigating two-phase flow and its role in small-scale sublithospheric convection (*GJI* 2022). With Yuan Li and Adina, I have been investigating melt transport across the lithosphere (*GJI* 2023). In the latter case, we are incorporating a poro-viscoelastic-viscoplastic rheological law to capture diking. I am interested in further research in this area where it links closely with existing observations.

My teaching supports the quantitative side of the undergraduate course in Earth Sciences.

I teach a 3rd-year course called *Vector Calculus & Continuum Mechanics*. This covers the fundamentals of vector calculus including the divergence, gradient and curl operators, and their manipulations using vector extensions of the Fundamental theorem of calculus. We cover stress and the Cauchy stress equation, kinematics of strain and strain rate, and viscous, elastic and viscoelastic theories of deformation. We consider simple geophysical applications of these physical/mathematical theories.

I co-teach a 3rd-year course called *Physics of the Solid Earth*. My part is on mantle convection. We cover Rayleigh-Benard convection, including the classical stability analysis, mantle convection and plate tectonics via scaling analysis, the plate and plume modes of convection, thermal structure of mid-ocean ridges and subduction zones, basics of dynamic topography and heat transport by convection.

In the past I have written and delivered courses on *time-series analysis* (esp. Fourier series) for 2nd-year students, and *physical thermodynamics* for 1st-year students. I have also given graduate lectures on *magma dynamics* based on my textbook (PUP 2022).

The teaching that I enjoy most is a course at the African Institution for Mathematical Sciences on *Fluid Dynamics*. This course was written by Prof Grae Worster, with whom I have co-taught the course in the past.

If you are a lecturer and interested in using my typeset course notes, please email me.

Please see this PDF version of my curriculum vitæ. If you are curious about the Latin æ then please see here and here.

I am the author of the textbook The Dynamics of Partially Molten Rock, published in 2022 by Princeton University Press. The book is accompanied by a set of Jupyter notebooks demonstrating calculations. Here's the blurb from the back cover:

Magma genesis and segregation have shaped Earth since its formation more than 4.5 billion years ago. Now, for the first time, the mathematical theory describing the physics of magmatism is presented in a single volume. *The Dynamics of Partially Molten Rock* offers a detailed overview that emphasizes the fundamental physical insights gained through an analysis of simplified problems. This textbook brings together such topics as fluid dynamics, rock mechanics, thermodynamics and petrology, geochemical transport, plate tectonics, and numerical modeling. End-of-chapter exercises and solutions as well as online Python notebooks provide material for courses at the advanced undergraduate or graduate level.

This book focuses on the partial melting of Earth’s asthenosphere, but the theory presented is also more broadly relevant to natural systems where partial melting occurs, including ice sheets and the deep crust, mantle, and core of Earth and other planetary bodies, as well as to rock-deformation experiments conducted in the laboratory. For students and researchers aiming to understand and advance the cutting edge, the work serves as an entrée into the field and a convenient means to access the research literature. Notes in each chapter reference both classic papers that shaped the field and newer ones that point the way forward.

*The Dynamics of Partially Molten Rock* requires a working knowledge of fluid mechanics and calculus, and for some chapters, readers will benefit from prior exposure to thermodynamics and igneous petrology.

- The first book to bring together in a unified way the theory for partially molten rocks
- End-of-chapter exercises with solutions and an online supplement of Jupyter notebooks
- Coverage of the mechanics, thermodynamics, and chemistry of magmatism, and their coupling in the context of plate tectonics and mantle convection
- Notes at the end of each chapter highlight key papers for further reading