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18 pages, 2104 KiB  
Article
Effect of High-Temperature Stress on Plant Physiological Traits and Mycorrhizal Symbiosis in Maize Plants
by Sonal Mathur, Richa Agnihotri, Mahaveer P. Sharma, Vangimalla R. Reddy and Anjana Jajoo
J. Fungi 2021, 7(10), 867; https://doi.org/10.3390/jof7100867 - 16 Oct 2021
Cited by 20 | Viewed by 3801
Abstract
Increasing high temperature (HT) has a deleterious effect on plant growth. Earlier works reported the protective role of arbuscular mycorrhizal fungi (AMF) under stress conditions, particularly influencing the physiological parameters. However, the protective role of AMF under high-temperature stress examining physiological parameters with [...] Read more.
Increasing high temperature (HT) has a deleterious effect on plant growth. Earlier works reported the protective role of arbuscular mycorrhizal fungi (AMF) under stress conditions, particularly influencing the physiological parameters. However, the protective role of AMF under high-temperature stress examining physiological parameters with characteristic phospholipid fatty acids (PLFA) of soil microbial communities including AMF has not been studied. This work aims to study how high-temperature stress affects photosynthetic and below-ground traits in maize plants with and without AMF. Photosynthetic parameters like quantum yield of photosystem (PS) II, PSI, electron transport, and fractions of open reaction centers decreased in HT exposed plants, but recovered in AMF + HT plants. AMF + HT plants had significantly higher AM-signature 16:1ω5cis neutral lipid fatty acid (NLFA), spore density in soil, and root colonization with lower lipid peroxidation than non-mycorrhizal HT plants. As a result, enriched plants had more active living biomass, which improved photosynthetic efficiency when exposed to heat. This study provides an understanding of how AM-mediated plants can tolerate high temperatures while maintaining the stability of their photosynthetic apparatus. This is the first study to combine above- and below-ground traits, which could lead to a new understanding of plant and rhizosphere stress. Full article
(This article belongs to the Special Issue The Impact of Climate Change on Plant–Fungal Interactions)
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Figure 1

Figure 1
<p>Quantum yields of energy conversion in PSII in control, AMF, high temperature (HT), AMF + HT for (<b>A</b>) Y(II) = quantum yield of PSII, Y(NO) = the yield of non-regulated energy dissipation, Y(NPQ) = the yield of regulated energy dissipation, (<b>B</b>) Chlorophyll fluorescence quenching coefficients (q<sub>P</sub>, q<sub>L</sub>, q<sub>N</sub>) in maize leaves under high temperature. The data are the mean values of three replicates ± standard deviation, treatment means followed with different letters vary significantly at <span class="html-italic">p</span> = 0.05 in compliance with Fisher least significant differences (LSD) and Duncan multiple range test (DMRT) for multiple comparisons.</p>
Full article ">Figure 2
<p>Response of (<b>A</b>) electron transport rates in PSI (ETR<sub>I</sub>) and PSII (ETR<sub>II</sub>), (<b>B</b>) maximal change in P700 (Pm) and maximum fluorescence (F<sub>m</sub>) signal, assessed in maize leaves. The data are the mean values of three replicates ± standard deviation, treatment means followed with different letters vary significantly at <span class="html-italic">p</span> = 0.05 in compliance with Fisher least significant differences (LSD) and Duncan multiple range test (DMRT) for multiple comparisons.</p>
Full article ">Figure 3
<p>Quantum yields of energy conversion in PSI in control, AMF, high temperature (HT), AMF + HT maize leaves under high temperature, for Y(I) = quantum yield of PSI, Y(NA) = the quantum yield of non-photochemical energy dissipation caused by acceptor-side limitation, Y(ND) = is the quantum yield of non-photochemical energy dissipation caused by donor side limitation. The data are the mean values of three replicates ± standard deviation, treatment means followed with different letters vary significantly at <span class="html-italic">p</span> = 0.05 in compliance with Fisher least significant differences (LSD) and Duncan multiple range test (DMRT) for multiple comparisons.</p>
Full article ">Figure 4
<p>(<b>A</b>) PCA score plot for the treatments and parameters (AMF biomass and microbial communities). The percent values that indicate the variation contributed by each PC are displayed in parentheses. The highest contribution was made by the parameters present in the right (<a href="#app1-jof-07-00867" class="html-app">Table S1</a>). (<b>B</b>) Graphical illustration of individual contribution of the treatments to PCs. Control = maize plants grown in normal soil. AMF = maize plants grown in AMF enriched soil, HT = maize plants grown in normal soil under higher temperature (natural temperature during summer 43 °C). AMF + HT = maize plants grown in AMF enriched soil under high-temperature stress. Fungi = fungal biomass, AM = PLFA = 16:1ω5cis (AM signature fatty acid biomarker for hyphal biomass), SC = spore count, RC = root colonization, NLFA = 16:1ω5cis neutral lipid fatty acid (AM signature fatty acid biomarker for storage lipids), GN = Gram-negative bacteria, GP GN ratio = Gram-positive/Gram-negative ratio, Acti = actinomycetes, GP = Gram-positive, FB ratio = Fungi/Bacteria ratio.</p>
Full article ">
73 pages, 23934 KiB  
Article
Nonlinear Fourier Analysis: Rogue Waves in Numerical Modeling and Data Analysis
by Alfred R. Osborne
J. Mar. Sci. Eng. 2020, 8(12), 1005; https://doi.org/10.3390/jmse8121005 - 9 Dec 2020
Cited by 8 | Viewed by 3635
Abstract
Nonlinear Fourier Analysis (NLFA) as developed herein begins with the nonlinear Schrödinger equation in two-space and one-time dimensions (the 2+1 NLS equation). The integrability of the simpler nonlinear Schrödinger equation in one-space and one-time dimensions (1+1 NLS) is an important tool in this [...] Read more.
Nonlinear Fourier Analysis (NLFA) as developed herein begins with the nonlinear Schrödinger equation in two-space and one-time dimensions (the 2+1 NLS equation). The integrability of the simpler nonlinear Schrödinger equation in one-space and one-time dimensions (1+1 NLS) is an important tool in this analysis. We demonstrate that small-time asymptotic spectral solutions of the 2+1 NLS equation can be constructed as the nonlinear superposition of many 1+1 NLS equations, each corresponding to a particular radial direction in the directional spectrum of the waves. The radial 1+1 NLS equations interact nonlinearly with one another. We determine practical asymptotic spectral solutions of the 2+1 NLS equation that are formed from the ratio of two phase-lagged Riemann theta functions: Surprisingly this construction can be written in terms of generalizations of periodic Fourier series called (1) quasiperiodic Fourier (QPF) series and (2) almost periodic Fourier (APF) series (with appropriate limits in space and time). To simplify the discourse with regard to QPF and APF Fourier series, we call them NLF series herein. The NLF series are the solutions or approximate solutions of the nonlinear dynamics of water waves. These series are indistinguishable in many ways from the linear superposition of sine waves introduced theoretically by Paley and Weiner, and exploited experimentally and theoretically by Barber and Longuet-Higgins assuming random phases. Generally speaking NLF series do not have random phases, but instead employ phase locking. We construct the asymptotic NLF series spectral solutions of 2+1 NLS as a linear superposition of sine waves, with particular amplitudes, frequencies and phases. Because of the phase locking the NLF basis functions consist not only of sine waves, but also of Stokes waves, breather trains, and superbreathers, all of which undergo complex pair-wise nonlinear interactions. Breather trains are known to be associated with rogue waves in solutions of nonlinear wave equations. It is remarkable that complex nonlinear dynamics can be represented as a generalized, linear superposition of sine waves. NLF series that solve nonlinear wave equations offer a significant advantage over traditional periodic Fourier series. We show how NLFA can be applied to numerically model nonlinear wave motions and to analyze experimentally measured wave data. Applications to the analysis of SINTEF wave tank data, measurements from Currituck Sound, North Carolina and to shipboard radar data taken by the U. S. Navy are discussed. The ubiquitous presence of coherent breather packets in many data sets, as analyzed by NLFA methods, has recently led to the discovery of breather turbulence in the ocean: In this case, nonlinear Fourier components occur as strongly interacting, phase locked, densely packed breather modes, in contrast to the previously held incorrect belief that ocean waves are weakly interacting sine waves. Full article
Show Figures

Figure 1

Figure 1
<p>A typical nonlinear ocean wave spectrum has sine waves (in the right- and left-hand tails), Stokes waves (at intermediate amplitudes) and breathers (clustered about the peak of the spectrum). The black dots are “simple point eigenvalues” and the black lines are “spines” (Osborne [<a href="#B1-jmse-08-01005" class="html-bibr">1</a>]). When the spines connect an eigenvalue to the frequency axis one has sine or Stokes wave components (Osborne et al. [<a href="#B5-jmse-08-01005" class="html-bibr">5</a>]). When one has spines that directly connect two points of spectrum above the frequency axis, one has a breather component. The large peak of the spectrum indicates high nonlinearity (large BF parameter), and is dominated by five breathers.</p>
Full article ">Figure 2
<p>Evolution of a breather packet, or nonlinear beat. A small amplitude modulation of a 3 m sine wave (upper panel) evolves over two kilometers to form a well-defined wave packet (central panel). After a total of four kilometers the wave train evolves into a robust, localized wave packet with a maximum amplitude of 8 m (lower panel). The evolution from a low amplitude sine wave to a large amplitude packet is due to the Benjamin–Feir instability. This maximal state generates a central wave that is 8/3 = 2.67 times the initial amplitude. This behavior has tempted many investigators to refer to the central wave as a “rogue wave.” After the full localization of the wave packet in the lower panel, one finds that the packet then begins decreasing in amplitude once again and returns to the initial state in the upper panel, in exact reverse order. Such a dynamical motion is one “breath” of the nonlinear “dynamical breathing” of the modulational instability and constitutes a “breather cycle time.” See Osborne et al. [<a href="#B5-jmse-08-01005" class="html-bibr">5</a>] for more details.</p>
Full article ">Figure 3
<p>The smallest incommensurable frequencies in water waves <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mi>j</mi> </msub> <mo>·</mo> <mi mathvariant="bold-sans-serif">ω</mi> </mrow> </semantics></math> in a nonlinear wave equation can never be so small that they create singularities, i.e., <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mi>j</mi> </msub> <mo>·</mo> <mi mathvariant="bold-sans-serif">ω</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math> for all <math display="inline"><semantics> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mo>≠</mo> <mn>0</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mi mathvariant="bold-sans-serif">ω</mi> </semantics></math>. This occurs thanks to a “geometric repulsion” that pushes <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mi>j</mi> </msub> <mo>·</mo> <mi mathvariant="bold-sans-serif">ω</mi> </mrow> </semantics></math> away from the zero state. Such a consequence of nonlinearity is important because it guarantees that the quasiperiodic series (9), and their integrals with respect to time t, converge, a requirement referred to as the Diophantine condition, <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mi>j</mi> </msub> <mo>·</mo> <mi mathvariant="bold-sans-serif">ω</mi> </mrow> <mo>|</mo> </mrow> <mo>≥</mo> <mi>γ</mi> <msup> <mrow> <mrow> <mo>|</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mo>|</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mi>r</mi> </mrow> </msup> </mrow> </semantics></math> in KAM theory, for <math display="inline"><semantics> <mrow> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mo>≠</mo> <mn>0</mn> <mo>∈</mo> <msup> <mi mathvariant="double-struck">Z</mi> <mi>N</mi> </msup> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mo>|</mo> </mrow> <mo>=</mo> <mstyle displaystyle="true"> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi>n</mi> <mi>i</mi> </msub> </mrow> <mo>|</mo> </mrow> </mrow> </mstyle> </mrow> </semantics></math>. The repulsion occurs below the diagonal line in the figure at very small frequencies <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> </mrow> <mo>|</mo> </mrow> </mrow> </semantics></math> and small moduli of the summation indices <math display="inline"><semantics> <mrow> <mrow> <mo>|</mo> <mstyle mathvariant="bold" mathsize="normal"> <mi>n</mi> </mstyle> <mo>|</mo> </mrow> </mrow> </semantics></math>.</p>
Full article ">Figure 4
<p>Boundary of modulation dispersion relation for the 2+1 NLS equation. The cyan region is where the unstable modes occur in the spectrum (this is where the actual breathers of the quasiperiodic solutions of the equation reside). The figure eight shows where the Phillips four-wave interactions occur.</p>
Full article ">Figure 5
<p>The wavenumber domain with the boundaries of the modulational dispersion relation for several nonlinear wave equations. The 2+1 NLS equation boundary is shown by two crossed black lines and associated hyperbolae. The boundary of the Dysthe equation is shown in green and the boundary of the Trulsen–Dysthe equation is shown in blue. The Zakharov equation boundary is shown in red and the Phillips figure eight is represented as a dotted line. For ocean waves, the central lines cross at the peak of the directional spectrum. The horizontal wavenumber coordinate lies along the dominant direction of the wave motion and the peak has been shifted to zero as seen on the horizontal axis. Additional information about coordinate definitions is shown in <a href="#jmse-08-01005-f004" class="html-fig">Figure 4</a>.</p>
Full article ">Figure 6
<p>The wavenumber domain for the 2+1 NLS equation <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>k</mi> <mi>y</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> is relative to the origin <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>k</mi> <mi>y</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>. The wave spectrum is indicated by the contours: Note that the peak of the spectrum lies at <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>k</mi> <mi>y</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>p</mi> </msub> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> and the dominant direction lies along <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>. The modulational wavenumbers <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>x</mi> </msub> <mo>,</mo> <msub> <mi>K</mi> <mi>y</mi> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>x</mi> </msub> <mo>−</mo> <msub> <mi>k</mi> <mi>p</mi> </msub> <mo>,</mo> <msub> <mi>k</mi> <mi>y</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> are centered about the peak of the spectrum. Each radial line in the spectrum is shown in the text to represent a solution of the “rotated” 1+1 NLS equation.</p>
Full article ">Figure 7
<p>The carrier coordinate frame for the 2+1 NLS equation is given by the coordinate pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. The rotated coordinate frame is defined by the coordinate pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. The particular rotation angle <math display="inline"><semantics> <mi>α</mi> </semantics></math> is such that <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, i.e., there is a plane wave solution of the modulation moving in the direction <math display="inline"><semantics> <mi>X</mi> </semantics></math>.</p>
Full article ">Figure 8
<p>The group speed <math display="inline"><semantics> <mrow> <msub> <mi>C</mi> <mi>g</mi> </msub> </mrow> </semantics></math> of the rotated 1+1 NLS solution is reduced with respect to that for the unrotated 1+1 NLS equation as the rotation angle varies between 0 and 35.2644 degrees.</p>
Full article ">Figure 9
<p>The dispersion coefficient <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> of the rotated 1+1 NLS solution is reduced with respect to that for the unrotated 1+1 NLS equation as the rotation angle varies between 0 and 35.2644 degrees.</p>
Full article ">Figure 10
<p>The nonlinearity parameter <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> of the rotated 1+1 NLS solution is increased with respect to that for the unrotated. 1+1 NLS equation as the rotation angle varies between 0 and 35.2644 degrees.</p>
Full article ">Figure 11
<p>Instability diagram for the rotated 1+1 NLS equation as a function of the modulational wavenumber. Several curves are given for different rotation angles. Instability diagram for the rotated 1+1 NLS equation as a function of the modulational wavenumber and the rotation angle.</p>
Full article ">Figure 12
<p>Instability diagram surface for the rotated 1+1 NLS equation as a function of the modulational wavenumber and the rotation angle.</p>
Full article ">Figure 13
<p>Oblique breather train moving at <math display="inline"><semantics> <mrow> <mn>30</mn> <mo>°</mo> </mrow> </semantics></math> relative to the dominant wave direction. This is a breather wave train formed by two phase-locked Stokes waves (Equations (78)–(84) below with rotation). It is the analogue of a breather in 1+1 dimensions and has a 2 × 2 Riemann spectrum from which this graph was made. This solution is a single nonlinear Fourier component in the finite gap theory for the unstable modes in the cyan region of <a href="#jmse-08-01005-f004" class="html-fig">Figure 4</a>.</p>
Full article ">Figure 14
<p>Determine the Riemann spectrum from measured data in one space and one-time dimensions.</p>
Full article ">Figure 15
<p>Determine Riemann spectrum <math display="inline"><semantics> <mrow> <mi mathvariant="bold-sans-serif">τ</mi> <mo>,</mo> <mo> </mo> <msup> <mi mathvariant="bold-sans-serif">ϕ</mi> <mo>±</mo> </msup> </mrow> </semantics></math> from a wave field <math display="inline"><semantics> <mrow> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>. Notice that in the box labeled “Compute Riemann Spectrum,” the frequencies are the usual ones determined with the modulational dispersion relation <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> <mo>=</mo> <msub> <mi mathvariant="sans-serif">Ω</mi> <mi>j</mi> </msub> <mo stretchy="false">(</mo> <msub> <mi>K</mi> <mi>j</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math>, which are incommensurable, and therefore require the usual limits over time.</p>
Full article ">Figure 16
<p>Two-dimensional wave model based on the Riemann spectrum.</p>
Full article ">Figure 17
<p>The period matrix for asymptotically solving the 2+1 NLS equation corresponds to diagonal block matrices for each rotated 1+1 NLS equations for each radial direction in the NLFA spectrum. The off-diagonal elements outside these matrices describe the interactions among the directional 1+1 NLS equations on the block diagonal and are assumed to be given by rescaled and rotated forms of the first equation in Equation (82) and the second equation in Equation (83).</p>
Full article ">Figure 18
<p>Ocean surface wave elevation in two dimensions at an instant in time, <math display="inline"><semantics> <mrow> <mi>η</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, on a grid of coordinate points <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>y</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math>, with spatial resolution <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mo>Δ</mo> <mi>x</mi> <mo>,</mo> <mo> </mo> <mo> </mo> <mo>Δ</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 19
<p>Two-dimensional ocean surface wave spectrum at an instant in time, <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math>, on a commensurable wavenumber grid <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> with wavenumber resolution <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mo>Δ</mo> <mi>k</mi> <mo>,</mo> <mo> </mo> <mo> </mo> <mo>Δ</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math>.</p>
Full article ">Figure 20
<p>Two-dimensional ocean surface wave spectrum at an instant in time, <math display="inline"><semantics> <mrow> <mi>F</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo>,</mo> <mi>t</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>η</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> on a wavenumber grid <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math>, which is commensurable with wavenumber resolution <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mo>Δ</mo> <mi>k</mi> <mo>,</mo> <mo> </mo> <mo> </mo> <mo>Δ</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> (upper panel) and the associated frequency spectrum with incommensurable frequencies (lower panel). Every wavenumber pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> (upper panel) maps to the frequency <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> (via the dispersion relation, lower panel): Therefore, the frequencies <math display="inline"><semantics> <mrow> <msub> <mi>ω</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mi>m</mi> </msub> <mo>,</mo> <msub> <mi>l</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> are incommensurable and each is uniquely associated with the spectral amplitudes <math display="inline"><semantics> <mrow> <msub> <mi>η</mi> <mrow> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mrow> </semantics></math> in Equations (165), (166) and (170).</p>
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<p>Comparison of linear, deep-water dispersion <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>=</mo> <msqrt> <mrow> <mi>g</mi> <mi>k</mi> </mrow> </msqrt> </mrow> </semantics></math>, (blue) with the dispersion relation of the 1+1 NLS Equation (181) (brown). The two agree rather well with each other near the peak of the spectrum.</p>
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<p>Incommensurable frequencies as a function of wavenumber for the 1+1 NLS Euqation (175).</p>
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<p>A Fourier spectrum of the 1+1 LSE equation with incommensurable frequencies given by the dispersion relation (181).</p>
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<p>A 150 s section of an experiment conducted at SINTEF in the long wave flume. Here a sine wave (carrier wave period of 1.3 sec) is given a small-amplitude modulation (period 19.5 sec) at the wave maker at 0 m. Here 13 wave probes are shown ordered from the bottom to the top of the figure: See probe numbers and distances from wave maker on the right. About eight carrier modulations are shown. From the figure, we see that the sine wave created by the wave maker is modulationally unstable because the modulation has grown as it propagates from the wave maker to the first probe at 10 m. The modulation continues to grow and focus as the wave train propagates down the tank to a distance of 85 m. The smooth modulations near the wave maker are transformed during their propagation down the tank: Beyond about 70 m, the wave amplitudes consists of seven localized packets that we call breathers. This is one of the best experimental evidences of the long-time evolution of the Benjamin-Feir instability.</p>
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<p>A 100 s section of a random wave experiment conducted at SINTEF. We used the value <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math> for the JONSWAP power spectrum. A total of 19 probes were placed along the tank, see probe numbers and distances down the right-hand side of the figure. The probe at 10 m has long and low packets corresponding to a high and narrow JONSWAP spectrum. Self-focusing occurs at later probes, leading to the formation of breather packets.</p>
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<p>A 4096-point time series from probe 8 at 70 m from the wave maker in the experiments of <a href="#jmse-08-01005-f025" class="html-fig">Figure 25</a>. Three extreme waves have amplitudes that are greater than three standard deviations. The largest packets (associated with their NLFA components, see <a href="#jmse-08-01005-f020" class="html-fig">Figure 20</a>) are indicated by <b><span style="color:red">red</span></b> numbers. The largest packet is greater than twice the significant wave height.</p>
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<p>The NLFA spectrum for the time series in <a href="#jmse-08-01005-f026" class="html-fig">Figure 26</a>. Sine wave and Stokes wave components are shown by <b>black</b> crosses connected to the frequency axis by a “spine.” See discussion with regard to <a href="#jmse-08-01005-f001" class="html-fig">Figure 1</a>. Breathers are colored <b><span style="color:red">red</span></b> and consist of two points of simple spectrum (<b><span style="color:red">red</span></b> crosses) connected by a spine (a <b><span style="color:red">red</span></b> line). When the two points of simple spectrum are too close, the spine is not visible and only a single cross is seen. This happens for the largest breathers near the peak of the spectrum. The numbers near the largest breathers coincide with those in the measured time series in <a href="#jmse-08-01005-f026" class="html-fig">Figure 26</a>. This emphasizes the one to one nature of the nonlinear spectral components of breathers (this figure) and the breathers themselves (<a href="#jmse-08-01005-f026" class="html-fig">Figure 26</a>) in the time domain.</p>
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<p>Time series of the Currituck Sound data after the removal of the low frequency part of the spectrum. This series is consistent with the known physics of the 1+1 NLS equation. The standard deviation of the filtered time series is 13.45 cm and significant wave height is 53.8 cm. The largest wave height is 111 cm, corresponding to <math display="inline"><semantics> <mrow> <mn>2.06</mn> <mo> </mo> <msub> <mi>H</mi> <mi>s</mi> </msub> </mrow> </semantics></math>.</p>
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<p>The measured time series of <a href="#jmse-08-01005-f028" class="html-fig">Figure 28</a> has been processed by the inverse nonlinear Stokes operator (Osborne et al. [<a href="#B5-jmse-08-01005" class="html-bibr">5</a>]) to obtain the present time series. As a consequence, the time series is up/down symmetric, the Stokes wave contribution has been removed, a condition consistent with further analysis using NLFA for the 1+1 NLS equation (see <a href="#jmse-08-01005-f030" class="html-fig">Figure 30</a>). Also shown (<b><span style="color:red">red</span></b>) is the modulational envelope <math display="inline"><semantics> <mrow> <mi>A</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mi>η</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mo>+</mo> <msup> <mover accent="true"> <mi>η</mi> <mo>˜</mo> </mover> <mn>2</mn> </msup> <mo stretchy="false">]</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> found by the Hilbert transform of the above time series for which the carrier amplitude is <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>o</mi> </msub> <mo>=</mo> <mn>16</mn> <mo> </mo> <mi>cm</mi> </mrow> </semantics></math>. The significant wave height of this time series is 52.0 cm. The height of the largest wave (labeled 1) is <math display="inline"><semantics> <mrow> <mn>2.04</mn> <mo> </mo> <msub> <mi>H</mi> <mi>s</mi> </msub> </mrow> </semantics></math>. The numbers labeling the largest ten packets coincide with the largest packets in the NLFA spectrum shown in <a href="#jmse-08-01005-f030" class="html-fig">Figure 30</a>. The largest three packets have central waves that are bigger than <math display="inline"><semantics> <mrow> <mn>2.2</mn> <mo> </mo> <msub> <mi>H</mi> <mi>s</mi> </msub> </mrow> </semantics></math> and are therefore classifiable as rogue waves.</p>
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<p>The nonlinear Fourier (NLFA) spectrum for the complex modulation function obtained from the time series in <a href="#jmse-08-01005-f029" class="html-fig">Figure 29</a>. Note that there are two vertical axes. The one on the left is the actual eigenvalue obtained in the computation of the NLFT spectrum and has units of meters. The right-hand vertical axis is dimensionless, and it is effectively the maximum amplitude of a breather normalized by the carrier amplitude. This means that the actual carrier amplitude <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>o</mi> </msub> <mo>=</mo> <mn>16</mn> <mo> </mo> <mi>cm</mi> </mrow> </semantics></math> (left vertical axis, see the horizontal red line <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>o</mi> </msub> </mrow> </semantics></math> and see horizontal red line in <a href="#jmse-08-01005-f029" class="html-fig">Figure 29</a>) and the normalized amplitude is 3 (see the right-hand vertical axis). The actual positions of the three theoretical breathers most often cited (Akhmediev, Peregrine, and Ma–Kuznetsov) are shown in blue. Spectral data points above <math display="inline"><semantics> <mrow> <mn>2.2</mn> <mo> </mo> <msub> <mi>H</mi> <mi>s</mi> </msub> </mrow> </semantics></math> (there are three of them) are considered to be rogue waves. The <b><span style="color:red">red</span></b> numbers labeling the ten largest breather packets coincide with the ten largest packets of the time series in <a href="#jmse-08-01005-f029" class="html-fig">Figure 29</a>.</p>
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<p>Radar measurements reported by Sutherland and Melville [<a href="#B62-jmse-08-01005" class="html-bibr">62</a>,<a href="#B63-jmse-08-01005" class="html-bibr">63</a>].</p>
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<p>Conversion of radar measurements of <a href="#jmse-08-01005-f019" class="html-fig">Figure 19</a> to surface waves, see, for example, [Cheng and Chien, 17] [Lund et al., 45].</p>
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<p>Wave simulation using input radar data and the algorithm in the flow chart in <a href="#jmse-08-01005-f016" class="html-fig">Figure 16</a> extended to two dimensions. See Osborne [<a href="#B1-jmse-08-01005" class="html-bibr">1</a>] for additional numerical methods.</p>
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15 pages, 1306 KiB  
Article
The Short-Term Effects of Mineral- and Plant-Derived Fulvic Acids on Some Selected Soil Properties: Improvement in the Growth, Yield, and Mineral Nutritional Status of Wheat (Triticum aestivum L.) under Soils of Contrasting Textures
by Mahendar Kumar Sootahar, Xibai Zeng, Yanan Wang, Shiming Su, Permanand Soothar, Lingyu Bai, Mukesh Kumar, Yang Zhang, Adnan Mustafa and Ning Ye
Plants 2020, 9(2), 205; https://doi.org/10.3390/plants9020205 - 6 Feb 2020
Cited by 22 | Viewed by 4998
Abstract
Fulvic acids (FAs) improve the structure and fertility of soils with varying textures and also play a crucial role in increasing crop production. The pot experiment was carried out using wheat grown on three soils with a silty clay, sandy loam, and clay [...] Read more.
Fulvic acids (FAs) improve the structure and fertility of soils with varying textures and also play a crucial role in increasing crop production. The pot experiment was carried out using wheat grown on three soils with a silty clay, sandy loam, and clay loam texture, respectively. The soils were treated with FAs derived from plant and mineral materials. Plant-derived solid (PSFA), mineral-derived liquid (NLFA), and plant-derived liquid (PLFA) were applied at a rate of 2.5, 5, and 5 g kg−1 and control applied at 0 g kg−1. The results showed that in treated soils, the heavy fraction C was higher by 10%–60%, and the light fraction C increased by 30%–60%. Similarly, the available N content significantly increased in treated soils by 30%–70% and the available K content increased by 20%–45%, while P content significantly increased by 80%–90% in Aridisols and Vertisols and decreased by 60%–70% in Mollisols. In contrast, for P, the organic–inorganic compounds were greater in Aridisols and Vertisols and lower in Mollisols. However, organic–inorganic composites decreased in Vertisols relative to the other two soils. Further results showed that PSFA and NLFA accelerated the plant growth parameters in Mollisols and Aridisols, respectively. Our study demonstrates that the application of PSFA and NLFA had a positive effect on the physical and chemical properties and plant growth characteristics of Mollisol and Vertisol soils. Moreover, the application of solid-state FA yields better results in Mollisols. However, liquid FA increases the nutrient availability and the effects on the chemical, biological, and physical properties of Aridisol and Vertisol soils. Full article
(This article belongs to the Special Issue Soil Nutrition and Plants Growth)
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Figure 1

Figure 1
<p>The effect of plant-derived solid fulvic acid (PSFA), mineral-derived liquid fulvic acid (NLFA), and plant-derived liquid fulvic acid (PLFA) on (<b>A</b>) electrical conductivity and (<b>B</b>) soil pH of Mollisol, Aridisol, and Vertisol soils. The means ± standard errors are shown (<span class="html-italic">n</span> = 4). Different letters (a, b, and c) indicate significant differences between the initial and final values of the soils based on a least significant difference test at a 5% significance level.</p>
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<p>The effect of plant-derived solid fulvic acid (PSFA), mineral-derived liquid fulvic acid (NLFA), and plant-derived liquid fulvic acid (PLFA) on (<b>A</b>) heavy fraction C and (<b>B</b>) light fraction C content of Mollisols, Aridisols, and Vertisols soils. The means ± standard errors are shown (<span class="html-italic">n</span> = 4). Different letters (a, b, and c) indicate significant differences between the initial and final values of the soils based on a least significant difference test at a 5% significance level.</p>
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<p>The effect of plant-derived solid fulvic acid (PSFA), mineral-derived liquid fulvic acid (NLFA), and plant-derived liquid fulvic acid (PLFA) on (<b>A</b>) soil organic carbon, (<b>B</b>) available nitrogen, (<b>C</b>) available phosphorus and (<b>D</b>) available potassium content of Mollisols, Aridisols, and Vertisols soils. The means ± standard errors are shown (<span class="html-italic">n</span> = 4). Different letters (a, b, and c) indicate significant differences between the initial and final values of the soils based on a least significant difference test at a 5% significance level.</p>
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<p>The effect of plant-derived solid fulvic acid (PSFA), mineral-derived liquid fulvic acid (NLFA), and plant-derived liquid fulvic acid (PLFA) on (<b>A</b>) nitrogen, (<b>B</b>) phosphorus and (<b>C</b>) potassium uptake of wheat grown in Mollisols, Aridisols, and Vertisols soils. The means ± standard errors are shown (<span class="html-italic">n</span> = 4). Different letters (a, b, and c) indicate significant differences between the initial and final values of the soils based on a least significant difference test at a 5% significance level.</p>
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<p>The map showing the purple (Mollisols), yellow (Vertisols), and green (Aridisols) soils of the Heilongjiang, Gansu and Anhui Provinces.</p>
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31 pages, 1048 KiB  
Article
Breather Turbulence: Exact Spectral and Stochastic Solutions of the Nonlinear Schrödinger Equation
by Alfred R. Osborne
Fluids 2019, 4(2), 72; https://doi.org/10.3390/fluids4020072 - 15 Apr 2019
Cited by 20 | Viewed by 3652
Abstract
I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also [...] Read more.
I address the problem of breather turbulence in ocean waves from the point of view of the exact spectral solutions of the nonlinear Schrödinger (NLS) equation using two tools of mathematical physics: (1) the inverse scattering transform (IST) for periodic/quasiperiodic boundary conditions (also referred to as finite gap theory (FGT) in the Russian literature) and (2) quasiperiodic Fourier series, both of which enhance the physical and mathematical understanding of complicated nonlinear phenomena in water waves. The basic approach I refer to is nonlinear Fourier analysis (NLFA). The formulation describes wave motion with spectral components consisting of sine waves, Stokes waves and breather packets that nonlinearly interact pair-wise with one another. This contrasts to the simpler picture of standard Fourier analysis in which one linearly superposes sine waves. Breather trains are coherent wave packets that “breath” up and down during their lifetime “cycle” as they propagate, a phenomenon related to Fermi-Pasta-Ulam (FPU) recurrence. The central wave of a breather, when the packet is at its maximum height of the FPU cycle, is often treated as a kind of rogue wave. Breather turbulence occurs when the number of breathers in a measured time series is large, typically several hundred per hour. Because of the prevalence of rogue waves in breather turbulence, I call this exceptional type of sea state a breather sea or rogue sea. Here I provide theoretical tools for a physical and dynamical understanding of the recent results of Osborne et al. (Ocean Dynamics, 2019, 69, pp. 187–219) in which dense breather turbulence was found in experimental surface wave data in Currituck Sound, North Carolina. Quasiperiodic Fourier series are important in the study of ocean waves because they provide a simpler theoretical interpretation and faster numerical implementation of the NLFA, with respect to the IST, particularly with regard to determination of the breather spectrum and their associated phases that are here treated in the so-called nonlinear random phase approximation. The actual material developed here focuses on results necessary for the analysis and interpretation of shipboard/offshore platform radar scans and for airborne lidar and synthetic aperture radar (SAR) measurements. Full article
(This article belongs to the Special Issue Nonlinear Wave Hydrodynamics)
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Figure 1

Figure 1
<p>Complex modulation Fourier spectrum for the nonlinear Schrödinger (NLS) equation in wavenumber space (left, centered about zero wave number) and the surface wave elevation spectrum (right, centered about <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>o</mi> </msub> </mrow> </semantics></math>). The example shown is intended to represent the case of an ocean wave Fourier spectrum such as that of Pierson-Moskowitz or JONSWAP (Joint North Sea Wave Project [<a href="#B14-fluids-04-00072" class="html-bibr">14</a>]). The surface wave spectrum is obtained from the NLS spectrum by shifting the NLS spectrum to the right by <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>o</mi> </msub> </mrow> </semantics></math> (see Equation (14)). We call <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>o</mi> </msub> </mrow> </semantics></math> the carrier wavenumber.</p>
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<p>Fourier wavenumber spectrum of the complex envelope of the nonlinear Schrödinger equation computed to leading order in the nome <span class="html-italic">q</span> for a small amplitude modulation (Equation (37)). Here we have the carrier wave of amplitude <math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>o</mi> </msub> </mrow> </semantics></math>, together with two small side bands.</p>
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<p>Fourier wavenumber spectrum of the surface wave elevation as found from the modulation spectrum. This case corresponds to the small modulation solution of the Schrödinger equation already shown in <a href="#fluids-04-00072-f002" class="html-fig">Figure 2</a>, where the Fourier modulation spectrum is centered about zero wavenumber. To obtain the Fourier spectrum for the surface wave elevation, one just shifts the modulational spectrum to the right by the carrier wavenumber <math display="inline"><semantics> <mrow> <msub> <mi>k</mi> <mi>o</mi> </msub> </mrow> </semantics></math>.</p>
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16 pages, 3435 KiB  
Article
Slash-and-Burn Practices Decrease Arbuscular Mycorrhizal Fungi Abundance in Soil and the Roots of Didierea madagascariensis in the Dry Tropical Forest of Madagascar
by Alícia Donnellan Barraclough and Pål Axel Olsson
Fire 2018, 1(3), 37; https://doi.org/10.3390/fire1030037 - 1 Oct 2018
Cited by 11 | Viewed by 5868
Abstract
Deforestation and the use of fire to clear land have drastic effects on ecosystem functioning and compromise essential ecosystem services, especially in low-income tropical countries such as Madagascar. We evaluated the effects of local slash-and-burn practices on soil nutrients and arbuscular mycorrhizal (AM) [...] Read more.
Deforestation and the use of fire to clear land have drastic effects on ecosystem functioning and compromise essential ecosystem services, especially in low-income tropical countries such as Madagascar. We evaluated the effects of local slash-and-burn practices on soil nutrients and arbuscular mycorrhizal (AM) fungi abundance in a southwestern Madagascar forest. Nine sampling plot pairs were established along the border of a reserve within the Fiherenana–Manombo (pk-32) complex, where soil and seedling root samples of the endemic tree Didierea madagascariensis were taken. We analysed soil extractable PO43−, NH4+, and NO3 as well as total soil carbon and nitrogen. We analysed AM fungal abundance in soil and roots through fatty acid marker analysis (NLFA and PLFA 16:1ω5), spore extraction, and root staining. Slash-and-burn caused an increase in pH and doubled the plant available nutrients (from 7.4 to 13.1 µg PO43− g−1 and from 6.9 to 13.2 µg NO3 g−1). Total C and total N increased in deforested soil, from 0.6% to 0.84% and from 0.06% to 0.08%, respectively. There was a significant decline in AM fungi abundance in soil, with a decrease in soil NLFA 16:1ω5 from 0.2 to 0.12 nmol/g. AM fungi abundance in D. madagascariensis roots was also negatively affected and colonization decreased from 27.7% to 16.9% and NLFA 16:1ω5 decreased from 75.7 to 19 nmol/g. Together with hyphal network disruption, increased nutrient availability caused by burning is proposed as an explanation behind AM decline in soil and roots of D. madagascariensis. This is the first study to report the effects of slash-and-burn on AM symbiosis in Madagascar’s dry forests, with likely implications for other tropical and subtropical dryland forests worldwide where slash-and-burn is practiced. Full article
(This article belongs to the Special Issue Land-Use and Fire around the World from the Past to the Present)
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Figure 1
<p>The sampling site situated 1 km from the coast town of Mangily (<b>indicated by a cross</b>). Dark areas indicate intact forest and light areas indicate burned deforested areas. The vertical line indicates the position of the first sampling pair transect (<b>a</b>) and the blue line indicates the position of the last transect (<b>i</b>).</p>
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<p>Linear regression showing changes in plant available nutrients with distance from the town of Mangily in meters. (<b>a</b>) Negative correlation between distance and extractable orthophosphate as measured by two different methods (Bray-1 and NaFS) in forested (Bray-1 y = −0.0053x + 16.25 <span class="html-italic">R</span><sup>2</sup> = 0.58, NaFS y = −0.001x + 2.65 <span class="html-italic">R</span><sup>2</sup> = 0.62 (p &lt; 0.05)) and deforested areas (Bray-1 y = −0.0094x + 28.84, <span class="html-italic">R</span><sup>2</sup> = 0.82, NaFS y = −0.0016x + 4.71 <span class="html-italic">R</span><sup>2</sup> = 0.63 (<span class="html-italic">p</span> &lt; 0.05)). (<b>b</b>) Extractable N in forested (NO<sub>3</sub><sup>−</sup> y = −0.0078x + 19.97 <span class="html-italic">R</span><sup>2</sup> = 0.53 (<span class="html-italic">p</span> &lt; 0.05), NH<sub>4</sub><sup>+</sup> y = −0.0006x + 2.99 <span class="html-italic">R</span><sup>2</sup> = 0.23 (<span class="html-italic">p</span> &gt; 0.05)) and deforested areas (NO<sub>3</sub><sup>−</sup> y = −0.0123x + 33.738 <span class="html-italic">R</span><sup>2</sup> = 0.54 (<span class="html-italic">p</span> &lt; 0.05), NH<sub>4</sub><sup>+</sup> y = −0.0004x + 2.81 <span class="html-italic">R</span><sup>2</sup> = 0.31 (<span class="html-italic">p</span> &gt; 0.05)).</p>
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<p>Effect of deforestation on arbuscular mycorrhizal (AM) fungi variables of <span class="html-italic">Didierea madagascariensis</span> in deforested (D) and forested (F) areas, fatty acid 16:1ω5 content in roots of <span class="html-italic">D. madagascariensis</span> (nmol/g dried root) in forested and deforested areas (upper panel), and fatty acid 16:1ω5 in soil (nmol/g dry soil) collected in forested and deforested areas (lower panel). Error bars indicate 95% confidence interval and different letters above the bars indicate a significant difference (paired <span class="html-italic">t</span>-test, n = 9, <span class="html-italic">p</span> &lt; 0.05).</p>
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<p>Effect of deforestation on AM fungi root colonization levels (%) in <span class="html-italic">D. madagascariensis</span> in deforested (D) and forested (F) areas. Error bars indicate 95% confidence interval and different letters above the bars indicate a significant difference (paired <span class="html-italic">t</span>-test, n = 9, <span class="html-italic">p</span> &lt; 0.05).</p>
Full article ">Figure 5
<p>Spore densities in 100 g of dry soil in deforested area (D) and forested area (F): (<b>a</b>) total spores, (<b>b</b>) Class A spores (<b>c</b>), Class B spores (<b>d</b>), and Class C spores. Error bars indicate 95% confidence interval. In all cases, differences were nonsignificant according to a paired <span class="html-italic">t</span>-test.</p>
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