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Tropical Power

Let \(w\) be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map \(f: \mathbb C\rightarrow \mathbb C^n\) such that f is equivalent to \(w\). We give several characterizations of those \(w\) for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, and approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered.

Tropical Power

Hurricanes, tropical storms, and other intense rotating storms fall into a general category called cyclones. There are two main types of cyclones: tropical and extratropical (those that form outside the tropics). Tropical cyclones get their energy from warm tropical oceans. Extratropical cyclones get their energy from the jet stream and from temperature differences between cold, dry air masses from higher latitudes and warm, moist air masses from lower latitudes.

Climate change is expected to affect tropical cyclones by increasing sea surface temperatures, a key factor that influences cyclone formation and behavior. The U.S. Global Change Research Program and the Intergovernmental Panel on Climate Change project that tropical cyclones will become more intense over the 21st century, with higher wind speeds and heavier rains.1,2

Records of tropical cyclones in the Atlantic Ocean have been collected since the 1800s. The most reliable long-term records focus on hurricanes, which are the strongest category of tropical cyclones in the Atlantic, with wind speeds of at least 74 miles per hour. This indicator uses historical data from the National Oceanic and Atmospheric Administration to track the number of hurricanes per year in the North Atlantic (north of the equator) and the number reaching the United States since 1878. Some hurricanes over the ocean might have been missed before the start of aircraft and satellite observation, so scientists have used other evidence, such as ship traffic records, to estimate the actual number of hurricanes that might have formed in earlier years.

This indicator also looks at the Accumulated Cyclone Energy (ACE) Index and the Power Dissipation Index (PDI), which are two ways of monitoring the frequency, strength, and duration of tropical cyclones based on wind speed measurements.

While Figures 2 and 3 cover several different aspects of tropical cyclones, there are other important factors not covered here, including the size of each storm, the amount of rain, and the height of the storm surge. The reason for the recent divergence between cyclone activity and sea surface temperature in Figure 3 has not been identified conclusively, but it may relate to other factors that influence the formation of storms, such as the difference in wind speeds at different levels in the atmosphere (called vertical wind shear).8

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Power outages are expected across parts of Florida's east coast as Tropical Storm Nicole moves into the area. As of about 8 a.m. Thursday, power companies reported the following number of households without electricity:

Variability in the tropical atmosphere is concentrated at wavenumber-frequency combinations where linear theory indicates wave modes can freely propagate, but with substantial power in between. This study demonstrates that such a power spectrum can arise from small-scale convection triggering large-scale waves via wave-wave interactions in a moderately turbulent fluid. Two key pieces of evidence are provided for this interpretation of tropical dynamics using a nonlinear rotating shallow-water model: a parameter sweep experiment in which the amplitude of an external forcing is gradually ramped up, and also an external forcing in which only symmetric or only antisymmetric modes are forced. These experiments do not support a commonly accepted mechanism involving the forcing projecting directly onto the wave modes with a strong response, yet still simulate a power spectrum resembling that observed, though the linear projection mechanism could still complement the mechanism proposed here in observations. Interpreting the observed tropical power spectrum using turbulence offers a simple explanation as to why power should be concentrated at the theoretical wave modes, and also provides a solid footing for the common assumption that the background spectrum is red, even as it clarifies why there is no expectation for a turbulent cascade with a specific, theoretically derived slope such as -5/3. However, it does explain why the cascade should be toward lower wavenumbers, that is an inverse energy cascade, similar to the midlatitudes even as compressible wave modes are important for tropical dynamics.

Infrared imagery from NASA's Terra satellite revealed Tropical Depression 13W contained some powerful thunderstorms pushing high into the troposphere as it was moving west in the Philippine Sea toward the Philippines. googletag.cmd.push(function() googletag.display('div-gpt-ad-1449240174198-2'); ); Tropical Depression 13W has already triggered warnings in the Philippines because it is located just east of the country. Philippines warnings include Tropical cyclone wind signal #1 over the following Luzon provinces: Cagayan, Isabela, Quirino, Nueva Vizcaya, Apayao, Abra, Kalinga, Mountain Province, Ifugao, Benguet, Aurora, Nueva Ecija, eastern portion of Pangasinan, northern portion of Quezon including Polillo Island and Catanduanes.

NASA's Terra satellite used infrared light to analyze the strength of storms and found the most powerful thunderstorms stretching north over the center from west to east. Infrared data provides temperature information, and the strongest thunderstorms that reach high into the atmosphere have the coldest cloud top temperatures.

"I am always impressed by the selfless service of our district's emergency managers and the volunteers we have who support the temporary power needs of those impacted by storms," said COL Andrew (Coby) Short, district commander. "In the last few years alone, more than 250 employees from the Pittsburgh District team have deployed, on short notice, to assist those impacted by hurricanes in their recovery efforts. I'm absolutely proud of our team and their dedication to their fellow Americans."

Pittsburgh District houses the National Temporary Power Emergency Operation Center headquarters. District personnel, soldiers and representatives from other agencies coordinate seven power teams from across the nation. It also oversees the management and quality assurance of the contract.

Each tropical cyclone is unique because of the myriad of weather conditions each storm encounters throughout its duration. Each storm begins as a tropical depression which might or might not develop into a tropical storm, which then might or might not develop into a tropical cyclone. In the current satellite era, each tropical depression is identified and tracked throughout its duration.

The attribution studies for tropical cyclones focus on storm frequency, intensity, speed, associated rainfall and track. One recent example is the attribution of a 10% increase in rainfall associated with Hurricane Ian. This attribution was based on analysis of those characteristics for similar storms in the past. However, our limited understanding suggests that these attributions are premature.

While the focus of attribution studies has been on the damage caused by tropical cyclones, there has been little focus on other potential attribution issues related tropical cyclones. These issues include whether climate change has any impact on the frequency and timing of the formation of tropical depressions, or the frequency with which tropical depressions develop into tropical storms and tropical storms develop into tropical cyclones. There has also been little focus on the potential affects of climate change on the paths of tropical cyclones or the frequency with which tropical cyclones dissipate at sea.

The 2022 hurricane season is a case in point. The map below shows the tracks of all named tropical storms in the Atlantic Basin in 2022. The season was originally predicted to be above average, but was nearly average on most metrics, though accumulated cyclone energy was significantly lower. 041b061a72


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