Buckingham (1914) On physically similar systems—illustrations of the use of dimensional equations ,” Phys. Rev. 4, 345– 376 (1914). 1. Buckingham, E., “,” Phys. Rev., 345–(1914). https://doi.org/10.1103/PhysRev.4.345 geophysical processes has been studied by many scientists and engineers. Scaling transformations stretch or contract processes or objects in time and/or space. Understanding the effect of scaling transformations on mostly nonlinear geophysical processes that take place over a wide range of time and space scales is necessary to better understand and model these processes. A process or an object is self-similar if it is invariant under a subset of scaling transformations. Within the framework of a physical process, self-similarity is the case where the solution of the process at a specified time-space scale can be related to the solution of the process at another time-space scale. Meanwhile, a fractal may be thought as a geometric object that is self-similar at various scales. About three decades ago, Mandelbrot (1983) 9. Mandelbrot, B., The Fractal Geometry of Nature ( W.H. Freeman and Co. , San Francisco , 1983). fractal geometry of nature, the degree of irregularity or fragmentation of which is identical at all geometric scales. Similar to fractals, geophysical processes could be self-similar under certain scaling transformations at various time and space dimensions. Since the work ofabout a century ago on dimensional analysis, the role of scales inprocesses has been studied by many scientists and engineers.transformations stretch or contract processes or objects in time and/or space. Understanding the effect oftransformations on mostly nonlinearprocesses that take place over a wide range of time and space scales is necessary to better understand andthese processes. A process or an object is self-similar if it is invariant under a subset oftransformations. Within the framework of a physical process, self-similarity is the case where the solution of the process at a specified time-space scale can be related to the solution of the process at another time-space scale. Meanwhile, amay be thought as a geometric object that is self-similar at various scales. About three decades ago,introduced thegeometry of nature, the degree of irregularity or fragmentation of which is identical at all geometric scales. Similar toprocesses could be self-similar under certaintransformations at various time and space dimensions.

In the modeling of nonlinear hydrologic dynamics, hydrodynamics, or climate, the scaling of the governing equations and geometries with time-space scales is yet to be determined definitively. This lack of clear guidelines as to what governing equation or what scaling relationship will be applicable at what time-space scale, or whether there are universally applicable governing equations/models or scaling relationships for a process at various scales, creates serious modeling issues, yet to be resolved. Meanwhile, there have been recent advances in understanding the physics of the scaling relationships in hydrologic dynamics, hydrodynamics, and climate. Accordingly, the purpose of this focus issue is to report on these recent advances.

Dascaliuc et al. (2015) Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations ,” Chaos 25, 075402 (2015). 2. Dascaliuc, R., Mlchalowski, N., Thomann, E., and Waymire, E., “,” Chaos, 075402 (2015). https://doi.org/10.1063/1.4913236 Ercan and Kavvas (2015) Scaling and self-similarity in two-dimensional hydrodynamics ,” Chaos 25, 075404 (2015). 4. Ercan, A. and Kavvas, M. L., “,” Chaos, 075404 (2015). https://doi.org/10.1063/1.4913852 Haltas and Ulusoy (2015) Scaling and scale invariance of conservation laws in Reynolds transport theorem framework ,” Chaos 25, 075406 (2015). 6. Haltas, I. and Ulusoy, S., “,” Chaos, 075406 (2015). https://doi.org/10.1063/1.4917246 scaling issues in nonlinear hydrodynamics. Dascaliuc et al. (2015) Symmetry breaking and uniqueness for the incompressible Navier-Stokes equations ,” Chaos 25, 075402 (2015). 2. Dascaliuc, R., Mlchalowski, N., Thomann, E., and Waymire, E., “,” Chaos, 075402 (2015). https://doi.org/10.1063/1.4913236 Navier-Stokes equations, it is still unknown whether smooth initial data lead to the existence of unique smooth solutions that are valid for all times. Toward the solution of this problem, they provide a new mathematical approach that establishes connections between the structure of the deterministic Navier-Stokes equations and the structure of (similarity) equations that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result of this new approach is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respective equations coincide. Ercan and Kavvas (2015) Scaling and self-similarity in two-dimensional hydrodynamics ,” Chaos 25, 075404 (2015). 4. Ercan, A. and Kavvas, M. L., “,” Chaos, 075404 (2015). https://doi.org/10.1063/1.4913852 scaling transformations the conditions under which the set of depth-averaged two-dimensional (2D) hydrodynamic equations (depth-averaged Navier-Stokes equations), as an initial-boundary value (IBVP) problem, becomes self-similar. They also investigate the 2D k-ε turbulence model. They derive the corresponding self-similarity conditions for all the flow variables in the system and show by numerical experiments that the IBVP of depth-averaged 2D hydrodynamic flow process in a prototype domain can be self-similar with that of a scaled domain. In fact, by changing the scaling parameter and the scaling exponents of the length dimensions, one can obtain several different scaled domains. Haltas and Ulusoy (2015) Scaling and scale invariance of conservation laws in Reynolds transport theorem framework ,” Chaos 25, 075406 (2015). 6. Haltas, I. and Ulusoy, S., “,” Chaos, 075406 (2015). https://doi.org/10.1063/1.4917246 hydrodynamics are formulated by means of the Reynolds Transport Theorem (RTT). Accordingly, they develop the scaling behavior and scale invariance conditions of the hydrodynamic processes by applying the one-parameter Lie scaling transformation to the conservation laws within the RTT framework. The set of three papers by, andexplores variousissues in nonlinearreport that while the three dimensional incompressible fluid flow is mathematically described byit is still unknown whether smooth initial data lead to the existence of unique smooth solutions that are valid for all times. Toward the solution of this problem, they provide a new mathematical approach that establishes connections between the structure of the deterministicand the structure of (similarity)that govern self-similar solutions as expected values of certain naturally associated stochastic cascades. A principle result of this new approach is that explosion criteria for the stochastic cascades involved in the probabilistic representations of solutions to the respectivecoincide.investigate by means of a one-parameter Lie group of pointtransformations the conditions under which the set of depth-averaged two-dimensional (2D)(depth-averagedas an initial-boundary value (IBVP) problem, becomes self-similar. They also investigate the 2D k-ε turbulenceThey derive the corresponding self-similarity conditions for all the flow variables in the system and show by numerical experiments that the IBVP of depth-averaged 2Dflow process in a prototype domain can be self-similar with that of a scaled domain. In fact, by changing theparameter and theexponents of the length dimensions, one can obtain several different scaled domains.point out that the conservation laws ofare formulated by means of the Reynolds Transport Theorem (RTT). Accordingly, they develop thebehavior and scale invariance conditions of theprocesses by applying the one-parameter Lietransformation to the conservation laws within the RTT framework.

Lovejoy and de Lima (2015) 8. Lovejoy, S. and de Lima, I. P., “ The joint space-time statistics of macroweather precipitation, space-time factorization and macroweather models ,” Chaos 25, 075410 (2015). Poveda and Salas (2015) Statistical scaling, Shannon entropy and generalized space-time q-entropy of rainfall fields in tropical South America ,” Chaos 25, 075409 (2015). 12. Poveda, G. and Salas, H. D., “,” Chaos, 075409 (2015). https://doi.org/10.1063/1.4922595 Devineni et al. (2015) Scaling of extreme rainfall areas at a planetary scale ,” Chaos 25, 075407 (2015). 3. Devineni, N., Lall, U., Xi, C., and Ward, P., “,” Chaos, 075407 (2015). https://doi.org/10.1063/1.4921719 scaling issues concerning the precipitation fields. Lovejoy and de Lima (2015) 8. Lovejoy, S. and de Lima, I. P., “ The joint space-time statistics of macroweather precipitation, space-time factorization and macroweather models ,” Chaos 25, 075410 (2015). scaling properties characterized by negative temporal fluctuation exponents. They discuss the empirical joint space-time scaling properties of macroweather precipitation and develop two stochastic models that can reproduce the observed space-time scaling from about ten days to at least 20–40 years. They then examine in detail an important statistical property: the space-time statistics can be expressed as the product of separate spatial and temporal factors. Poveda and Salas (2015) Statistical scaling, Shannon entropy and generalized space-time q-entropy of rainfall fields in tropical South America ,” Chaos 25, 075409 (2015). 12. Poveda, G. and Salas, H. D., “,” Chaos, 075409 (2015). https://doi.org/10.1063/1.4922595 scaling (bi-dimensional Fourier spectrum and moment-scaling function) and information theory, which are theoretically connected, to study the diverse scaling and information theory characteristics of Mesoscale Convective Systems (MCS) as seen by the Tropical Rainfall Measuring Mission (TRMM) over continental and oceanic regions of tropical South America and 2D radar rainfall fields from Amazonia. They introduce and quantify the spatial generalized q-entropy to the set of radar rainfall fields and provide initial results on linking their scaling analysis results to physical characteristics of rainfall over tropical South America. Devineni et al. (2015) Scaling of extreme rainfall areas at a planetary scale ,” Chaos 25, 075407 (2015). 3. Devineni, N., Lall, U., Xi, C., and Ward, P., “,” Chaos, 075407 (2015). https://doi.org/10.1063/1.4921719 scaling characteristics of extreme rainfall areas for durations ranging from 1 to 30 days. Their findings lead to the question as to how the climate system organizes over these scales, overcoming the substantial apparent heterogeneity in process dynamics. They fit the broken power law model to each case and report that their results suggest that power law scaling may also apply to planetary scale phenomenon, such as frontal and monsoonal systems, and their interaction with local moisture recycling. Such features may have persistence over large areas corresponding to extreme rain and regional flood events. In a set of three papers,, andexplore variousissues concerning the precipitation fields.point out that in addition to the familiar weather and climate regimes, over the range of time scales from about 10 days to 30–100 years, there is an intermediate "macroweather" regime that has uniqueproperties characterized by negative temporal fluctuation exponents. They discuss the empirical joint space-timeproperties of macroweather precipitation and develop two stochasticthat can reproduce the observed space-timefrom about ten days to at least 20–40 years. They then examine in detail an important statistical property: the space-time statistics can be expressed as the product of separate spatial and temporal factors.use diverse tools from statistical(bi-dimensional Fourier spectrum and moment-scaling function) and information theory, which are theoretically connected, to study the diverseand information theory characteristics of Mesoscale Convective Systems (MCS) as seen by the Tropical Rainfall Measuring Mission (TRMM) over continental and oceanic regions of tropical South America and 2D radar rainfall fields from Amazonia. They introduce and quantify the spatial generalized q-entropy to the set of radar rainfall fields and provide initial results on linking theiranalysis results to physical characteristics of rainfall over tropical South America.present the first ever results on a global analysis of thecharacteristics of extreme rainfall areas for durations ranging from 1 to 30 days. Their findings lead to the question as to how the climate system organizes over these scales, overcoming the substantial apparent heterogeneity in process dynamics. They fit the broken power lawto each case and report that their results suggest that power lawmay also apply to planetary scale phenomenon, such as frontal and monsoonal systems, and their interaction with local moisture recycling. Such features may have persistence over large areas corresponding to extreme rain and regionalevents.