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Long linear wave transformation in the basin of varying depth is studied for a case of a convex bottom profile in the framework of one-dimensional shallow water equation. The existence of travelling wave solutions in this geometry and the uniqueness of this wave class is established through construction of a 1:1 transformation of the general 1D wave equation to the analogous wave equation with constant coefficients. The general solution of the Cauchy problem consists of two travelling waves propagating in opposite directions. It is found that generally a zone of a weak current is formed between these two waves. Waves are reflected from the coastline so that their profile is inverted with respect to the calm water surface. Long wave runup on a beach with this profile is studied for sine pulse, KdV soliton and N-wave. Shown is that in certain cases the runup height along the convex profile is considerably larger than for beaches with a linear slope. The analysis of wave reflection from the bottom containing a shallow coastal area of constant depth and a section with the convex profile shows that a transmitted wave always has a sign-variable shape.
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We obtained the first experimental evidence for the magnetohydrodynamic (MHD) nature of ionospheric medium-scale travelling wave packets (MSTWP). We used data on total electron content (TEC) measurements obtained at the dense Japanese network GPS/GEONET (1220 stations) in 2008-2009. We found that the diurnal, seasonal and spectral MSTWP characteristics are specified by the solar terminator (ST) dynamics. MSTWPs are the chains of narrow-band TEC oscillations with single packets duration of about 1-2 hours and oscillation periods of 10-20 minutes. Their total duration is about 4--6 hours. The MSTWP spatial structure is characterized by a high degree of anisotropy and coherence at the distance of more than 10 wavelengths. The MSTWP direction of travelling is characterized by a high directivity regardless of seasons. Occurrence rate of daytime MSTWPs is high in winter and during equinoxes. Occurrence rate of nighttime MSTIDs has its peak in summer. These features are consistent with previous MS travelling ionosphere disturbance (TID) statistics obtained from 630-nm airglow imaging observations in Japan. In winter, MSTWPs in the northern hemisphere are observed 3-4 hours after the morning ST passage. In summer, MSTWPs are detected 1.5-2 hours before the evening ST occurrence at the point of observations, at the moment of the evening ST passage in the magneto-conjugate point. Both the high Q-factor of oscillatory system and synchronization of MSTWP occurrence with the solar terminator passage at the point of observations and in the magneto-conjugate area testify the MHD nature of ST-excited MSTWP generation. The obtained results are the first experimental evidence for the hypothesis of the ST-generated ion sound waves.
The group focused on a model problem of idealised moist air convection in a single column of atmosphere. Height, temperature and moisture variables were chosen to simplify the mathematical representation (along the lines of the Boussinesq approximation in a height variable defined in terms of pressure). This allowed exact simple solutions of the numerical and partial differential equation problems to be found. By examining these, we identify column behaviour, stability issues and explore the feasibility of a more general solution process.
Knowledge about the characteristics of the atmospheric boundary layer are vital for the redistribution of air and suspended contents that are particularly driven by turbulent motions. Despite many modelling studies, detailed observations are still demanded of the development of turbulent exchange under stable and unstable conditions. In this paper we present an attempt to observationally detail atmospheric internal waves, under stable conditions, and associated turbulent overturning, under quasi-stable and unstable conditions. Therefore, we mounted 198 high-resolution temperature T-sensors on a cable. The instrumented cable was attached along the 213 m tall mast of Cabauw, the Netherlands, during late-summer 2017. The mast has standard and special meteorological equipment at extendable booms every 20 m in height. A sonic turbulence anemometer is at 60 m above ground. The extra, originally underwater-, T-sensor cable was suspended down from the 206-m level, temporarily for about 3 months. While in water the sensors have a response time of tw=0.4 s and drift of 0.001 degC per month, in air the response time ta=3 s is relatively slow and the apparent drift of about 0.1 degC per month relatively large. Least performance is during daytime. These T-sensor characteristics hamper quantitative atmospheric turbulence research, as it results in a relatively narrow inertial subrange of only one order of magnitude. Nevertheless, height-time images from two contrasting days show common nocturnal marginally stable density stratification supporting internal waves up to the buoyancy period of about 300 s, shear and convective deformation of the stratification over the entire 197 m range of observations.
Tropical precipitation clusters exhibit power-law frequency distributions in area and volume (integrated precipitation), implying a lack of characteristic scale in tropical convective organization. However, it remains unknown what gives rise to the power laws and how the power-law exponents for area and volume are related to one another. Here, we explore the perspective that precipitation clusters are islands above a convective threshold on a rough column-water-vapor (CWV) topography. This perspective is supported by the agreement between the precipitation clusters and CWV islands in their frequency distributions as well as fractal dimensions. Power laws exist for CWV islands at different thresholds through the CWV topography, suggesting that the existence of power-laws is not specifically related to local precipitation dynamics, but is rather a general feature of CWV islands. Furthermore, the frequency distributions and fractal dimensions of the clusters can be reproduced when the CWV field is modeled to be self-affine with a roughness exponent of 0.3. Self-affine scaling theory relates the statistics of precipitation clusters to the roughness exponent; it also relates the power-law slopes for area and volume without involving the roughness exponent. Thus, the perspective of precipitation clusters as CWV islands provides a useful framework to consider many statistical properties of the precipitation clusters, particularly given that CWV is well-observed over a wide range of length scales in the tropics. However, the statistics of CWV islands at the convective threshold imply a smaller roughness than is inferred from the power spectrum of the bulk CWV field, and further work is needed to understand the scaling of the CWV field.
Tipping elements in the climate system are large-scale subregions of the Earth that might possess threshold behavior under global warming with large potential impacts on human societies. Here, we study a subset of five tipping elements and their interactions in a conceptual and easily extendable framework: the Greenland and West Antarctic Ice Sheets, the Atlantic Meridional Overturning Circulation (AMOC), the El-Nino Southern Oscillation (ENSO) and the Amazon rainforest. In this nonlinear and multistable system, we perform a basin stability analysis to detect its stable states and their associated Earth system resilience. Using this approach, we perform a system-wide and comprehensive robustness analysis with more than 3.5 billion ensemble members. Further, we investigate dynamic regimes where some of the states lose stability and oscillations appear using a newly developed basin bifurcation analysis methodology. Our results reveal that the state of four or five tipped elements has the largest basin volume for large levels of global warming beyond 4 {deg}C above pre-industrial climate conditions. For lower levels of warming, states including disintegrated ice sheets on West Antarctica and Greenland have higher basin volume than other state configurations. Therefore in our model, we find that the large ice sheets are of particular importance for Earth system resilience. We also detect the emergence of limit cycles for 0.6% of all ensemble members at rare parameter combinations. Such limit cycle oscillations mainly occur between the Greenland Ice Sheet and AMOC (86%), due to their negative feedback coupling. These limit cycles point to possibly dangerous internal modes of variability in the climate system that could have played a role in paleoclimatic dynamics such as those unfolding during the Pleistocene ice age cycles.
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