text:
Journal topic
Ocean Sci., 16, 135–148, 2020
https://doi.org/10.5194/os-16-135-2020
Ocean Sci., 16, 135–148, 2020
https://doi.org/10.5194/os-16-135-2020

Research article 23 Jan 2020

Research article | 23 Jan 2020

# Photoproduction of nitric oxide in seawater

Photoproduction of nitric oxide in seawater
Ye Tian1,2,3, Gui-Peng Yang1,2,3, Chun-Ying Liu1,2,3, Pei-Feng Li3, Hong-Tao Chen1,2,3, and Hermann W. Bange4 Ye Tian et al.
• 1Key Laboratory of Marine Chemistry Theory and Technology, Ministry of Education, Qingdao 266100, China
• 2Laboratory for Marine Ecology and Environmental Science, Qingdao National Laboratory for Marine Science and Technology, Qingdao 266071, China
• 3College of Chemistry and Chemical Engineering, Ocean University of China, Qingdao, 266100, China
• 4Chemical Oceanography, Division of Marine Biogeochemistry, GEOMAR – Helmholtz-Zentrum für Ozeanforschung Kiel, Kiel 24105, Germany

Correspondence: Chun-Ying Liu (roseliu@ouc.edu.cn) and Hong-Tao Chen (chenht@ouc.edu.cn)

Abstract

Nitric oxide (NO) is a short-lived intermediate of the oceanic nitrogen cycle. However, our knowledge about its production and consumption pathways in oceanic environments is rudimentary. In order to decipher the major factors affecting NO photochemical production, we irradiated several artificial seawater samples as well as 31 natural surface seawater samples in laboratory experiments. The seawater samples were collected during a cruise to the western tropical North Pacific Ocean (WTNP, a N–S section from 36 to 2 N along 146 to 143 E with 6 and 12 stations, respectively, and a W–E section from 137 to 161 E along the Equator with 13 stations) from November 2015 to January 2016. NO photoproduction rates from dissolved nitrite in artificial seawater showed increasing trends with decreasing pH, increasing temperature, and increasing salinity. In contrast, NO photoproduction rates (average: $\mathrm{0.5}±\mathrm{0.2}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1 s−1) in the natural seawater samples from the WTNP did not show any correlations with pH, water temperature, salinity, or dissolved inorganic nitrite concentrations. The flux induced by NO photoproduction in the WTNP (average: $\mathrm{13}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1) was significantly larger than the NO air–sea flux density (average: $\mathrm{1.8}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1), indicating a further NO loss process in the surface layer.

1 Introduction

Nitric oxide (NO) is a short-lived intermediate of the oceanic nitrogen cycle; see Bange (2008) and Kuypers et al. (2018). There are only a few reports about the oceanic NO determination method so far because of its high reactivity with other substances (Zafiriou et al., 1980; Lutterbeck and Bange, 2015; Liu et al., 2017). NO is produced and consumed during various microbial processes such as nitrification, denitrification, and anammox (Schreiber et al., 2012; Kuypers et al., 2018). Moreover, it is known that both phytoplankton and zooplankton can metabolize NO, and they are influenced by ambient (extracellular) NO concentrations (Singh and Lal, 2017; Wang et al., 2017; Astier et al., 2018).

Table 1Photoproduction rates (R), methods, average NO concentrations, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations, and average flux densities of NO in different regions.

Apart from (micro)biological processes, NO can be produced photochemically from dissolved nitrite (${\mathrm{NO}}_{\mathrm{2}}^{-}$) in the sunlit surface ocean (Zafiriou and True, 1979; Zafiriou and McFarland, 1981):

$\begin{array}{}\text{(R1)}& {\mathrm{NO}}_{\mathrm{2}}^{-}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\stackrel{h\mathit{\nu }}{⟶}\mathrm{NO}+{\mathrm{OH}}^{\mathrm{•}}+{\mathrm{OH}}^{-}.\end{array}$

Mack and Bolton (1999) reviewed the possible subsequent reactions of Reaction (1); for example, the produced NO and hydroxyl radical (OH) of Reaction (1) could react to produce HNO2 reversely (Reaction 2) and some reactions that consumed NO or its oxides like Reaction (3) to Reaction (8).

$\begin{array}{}\text{(R2)}& & \mathrm{NO}+\mathrm{OH}\to {\mathrm{HNO}}_{\mathrm{2}}\text{(R3)}& & \mathrm{NO}+{\mathrm{NO}}_{\mathrm{2}}\to {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{3}}\text{(R4)}& & {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{3}}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\to \mathrm{2}{\mathrm{H}}^{+}+\mathrm{2}{\mathrm{NO}}_{\mathrm{2}}^{-}\text{(R5)}& & \mathrm{NO}+\mathrm{NO}\to {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}\text{(R6)}& & {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}+{\mathrm{O}}_{\mathrm{2}}\to {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{4}}\text{(R7)}& & \mathrm{2}{\mathrm{NO}}_{\mathrm{2}}\to {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{4}}\text{(R8)}& & {\mathrm{N}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{4}}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\to \mathrm{2}{\mathrm{H}}^{+}+{\mathrm{NO}}_{\mathrm{2}}^{-}+{\mathrm{NO}}_{\mathrm{3}}^{-}\end{array}$

Besides, in natural sunlit seawater, photolyzed dissolved nitrate (${\mathrm{NO}}_{\mathrm{3}}^{-}$) could be a source of NO through ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (Reaction 9); during the process of ammonium (${\mathrm{NH}}_{\mathrm{4}}^{+}/{\mathrm{NH}}_{\mathrm{3}}$) oxidation into ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and ${\mathrm{NO}}_{\mathrm{3}}^{-}$, NO might be an intermediate (Joussotdubien and Kadiri, 1970), or NO could be produced through amino-peroxyl radicals (${\mathrm{NH}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}^{\mathrm{•}}$) through Reactions (10) to (14) (Laszlo et al., 1998; Clarke et al., 2008).

$\begin{array}{}\text{(R9)}& & {\mathrm{NO}}_{\mathrm{3}}^{-}\stackrel{h\mathit{\nu }}{⟶}{\mathrm{NO}}_{\mathrm{2}}^{-}+\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}\text{(R10)}& & {\mathrm{OH}}^{\mathrm{•}}+{\mathrm{HCO}}_{\mathrm{3}}^{-}/{\mathrm{CO}}_{\mathrm{3}}^{\mathrm{2}-}\to {\mathrm{CO}}_{\mathrm{3}}^{\mathrm{•}-}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}/{\mathrm{OH}}^{-}\text{(R11)}& & {\mathrm{OH}}^{\mathrm{•}}+{\mathrm{NH}}_{\mathrm{3}}\to {\mathrm{NH}}_{\mathrm{2}}^{\mathrm{•}}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\text{(R12)}& & {\mathrm{CO}}_{\mathrm{3}}^{\mathrm{•}-}+{\mathrm{NH}}_{\mathrm{3}}\to {\mathrm{NH}}_{\mathrm{2}}^{\mathrm{•}}+{\mathrm{HCO}}_{\mathrm{3}}^{-}\text{(R13)}& & {\mathrm{NH}}_{\mathrm{2}}^{\mathrm{•}}+{\mathrm{O}}_{\mathrm{2}}\to {\mathrm{NH}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}^{\mathrm{•}}\text{(R14)}& & {\mathrm{NH}}_{\mathrm{2}}{\mathrm{O}}_{\mathrm{2}}^{\mathrm{•}}\to {\mathrm{NO}}^{\mathrm{•}}+{\mathrm{H}}_{\mathrm{2}}\mathrm{O}\end{array}$

Table 1 summarizes studies about photochemical production of NO measured in the surface waters of the equatorial Pacific Ocean (Zafiriou et al., 1980; Zafiriou and McFarland, 1981), the Seto Inland Sea (Olasehinde et al., 2009, 2010; Anifowose and Sakugawa, 2017), the Bohai Sea and Yellow Sea (Liu et al., 2017; Tian et al., 2019), and the Kurose River (Japan) (Olasehinde et al., 2009; Anifowose et al., 2015). NO concentration was determined by the balance of the production and the removal process; thus, changes in NO production and removal rates could influence NO concentration in the seawater. In the surface seawater, the photochemical process was regarded as the main production process (Zafiriou and McFarland, 1981; Olasehinde et al., 2010; Anifowose et al., 2015). In Table 1, NO photoproduction rates varied among different seawater samples: the photoproduction rates in Kurose River (average: $\mathrm{499}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1 s−1) were the largest, which might be due to an increase in nitrite being released into the river because of agricultural activity during the study time. However, NO concentration was about $\mathrm{1.6}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1, at the lowest level, which was because of higher scavenging rate in river water. Anifowose et al. (2015) found that NO lifetime, which was defined as the reciprocal of first-order scavenging rate constant of NO (Olasehinde et al., 2010) in Kurose River, was only 0.25 s. The lifetime of NO showed an increasing trend from river (several seconds) via inland sea (dozens of seconds) to open sea (dozens to hundreds of seconds), reviewed in Anifowose and Sakugawa (2017). However, NO showed higher concentration levels in coastal waters than in open sea; higher photoproduction rates in coastal waters than open sea, or other production processes in coastal waters, might account for this.

In this study, we present the results of our measurements of NO photoproduction in laboratory experiments using artificial and natural seawater samples. The major objectives of our studies were (i) to decipher the factors affecting NO photoproduction in seawater, (ii) to determine the photoproduction rates of NO from samples collected during a cruise to the western tropical North Pacific Ocean (WTNP), and (iii) to quantify the role of photoproduction as a source of NO in the surface waters of the WTNP.

2 Methods

## 2.1 Determination of dissolved NO in aqueous samples

For the measurements of dissolved NO, we applied the method described by Olasehinde et al. (2009). In brief, NO in the aqueous samples was determined by trapping it with added 4,5-diaminofluorescein (DAF-2, chromatographic grade from Sigma-Aldrich, USA) and measuring the reaction product triazolofluorescein (DAF-2T) with a high-performance liquid chromatography system (HPLC). We used an Agilent 1260 Infinity HPLC (Agilent Technologies Inc., USA) system equipped with a Venusil XBP-C18 column (5.0 µm; 4.6 mm × 250 mm i.d.). The column temperature was set to 25 C and the mobile phase was comprised of acetonitrile (HPLC grade from Merck, Germany) and phosphate buffer (disodium hydrogen phosphate heptahydrate, guaranteed reagent from Sinopharm Chemical Reagent Co., Ltd, China) solution (10 mmol L−1 at pH 7.4) with a ratio of 8:92 (v:v) and a flow rate of 1 mL min−1 in the isocratic mode.

The injected sample volume was 5.0 µL. The eluate was analyzed with a fluorescence diode array detector at wavelengths of 495 and 515 nm for excitation and emission, respectively. The retention time of DAF-2T was about 5.5 min.

An aliquot of 10 mL artificial seawater was bubbled with N2 gas at a flow of 10 mL min−1 for 2 h to remove O2 after 10 min of ultrasonic and heat degassing. The solution was then bubbled with high-purity NO gas (99.9 %, Dalian Date Gas Ltd., China) for 30 min. The concentration of the saturated NO stock solution was 1.4 mmol L−1, which could be used within 3 h (Lantoine et al., 1995). A series of diluted NO solutions were prepared in N2-purged water from the NO stock solution using a microsyringe (Xing et al., 2005; Liu et al., 2017). And the series samples were trapped by DAF-2 solution.

The detection limit of dissolved NO in Milli-Q water was $\mathrm{9.0}×{\mathrm{10}}^{-\mathrm{11}}$ mol L−1, which was determined by $S/N=\mathrm{3}$ (3×0.03) with the blank samples (n=7) and the slope (0.101) in the low concentration range (3.3–33 × 10−10 mol L−1). Average relative standard error of the NO measurements was ±5.7 % at a concentration of $\mathrm{3.0}×{\mathrm{10}}^{-\mathrm{9}}$ mol L−1.

## 2.2 Set up of irradiation experiments

We performed irradiation experiments with Milli-Q water (18.2 MΩcm, Millipore Company, USA), artificial seawater, and natural seawater samples. Artificial seawater was prepared by dissolving 23.96 g NaCl, 5.08 g MgCl2, 3.99 g Na2SO4, 1.12 g CaCl2, 0.67 g KCl, 0.20 g NaHCO3, 0.10 g KBr, 0.03 g H3BO3, and 0.03 g NaF in 1 L of Milli-Q water (Bajt et al., 1997) and filtered by 0.2 µm polyethersulfone membrane (Pall, USA) before the experiments.

All irradiation experiments (except the experiments for the temperature dependence, see section below) were conducted at a constant temperature of 20 C by controlling the temperature of a thermostat water bath (LAUDA Dr. R. Wobser GmbH & Co. KG, Germany). The height of the cylindrical quartz cuvette used for irradiation was 70 mm and the inner diameter was 14 mm with the volume about 10 mL. The optical pathlength was 70±1 mm. During the experiment, the quartz cuvette, filled with 10 mL sample and blocked by a PTFE stopper, was installed in the simulator and a little higher than the water bath surface. All quartz cuvettes were treated in the same manner except the cuvettes wrapped in aluminum foil, which served as dark control.

Milli-Q water and artificial seawater samples were spiked with varying amounts of NaNO2 (puriss. p.a. ACS grade from Sigma-Aldrich, USA; for details, see sections below). All other chemicals were of analytical grade from Tianjin Kemiou Chemical Reagent Co., Ltd. or Shanghai Sinopharm Chemical Reagent Co., Ltd.

Triplicate samples from each treatment were collected every 0.5 h with an entire irradiation time of 2 h. At the sampling time, the SUNTEST CPS+ was turned off and triplicate subsamples were collected from each sample in the dark with microsyringe (50 µL), and then the cuvettes were quickly put back into the water bath to continue the experiment until 2 h. The results showed that both in Milli-Q and artificial seawater samples, the photoproduced NO showed linear relationship with time (see below). However, for the natural seawater samples, a linear relationship was only found in the irradiation time range of 30 min, while the relationship was not found after 30 min. Therefore, we decided to choose 30 min as the total experimental time for natural seawater samples. Statistical analyses were done using SPSS v.16.0 or Origin 9.0 and results were considered significant at p≤0.05.

The artificial light source was a 1.5 kW xenon lamp, which provided a light intensity of 765 W m−2. The lamp was installed in an immersion well photochemical reactor called SUNTEST CPS+ solar simulator produced by ATLAS, Germany. The solar simulator employed in this study has been demonstrated to produce a spectrum which mimics that of solar radiation and emits radiation of wavelengths from 300 to 800 nm (Wu et al., 2015).

## 2.3 Experimental outline

### 2.3.1 Optimal DAF-2 concentration and storage time

In order to find out the optimal DAF-2 concentration, 10 mL of artificial seawater containing 0.5 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ was irradiated with various concentrations of DAF-2 ranging from 0.7 to 4.8 µmol L−1 for 2 h.

To ascertain the sample storage time, 10 mL with artificial seawater samples containing 5.0 or 0.5 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ were irradiated with various concentrations of DAF-2 for 2 h. After irradiation, samples were kept in the dark and measured every 2 h.

### 2.3.2 Influence of pH, temperature, salinity, and wavelengths

The influence of the pH was assessed by adjusting artificial seawater samples to pH levels of 7.1, 7.6, and 8.1 by adding appropriate amounts of hydrochloric acid (2 mol L−1) or caustic soda solution (2 mol L−1).

To assess the influence of the temperature, artificial seawater samples were adjusted to temperatures of 10, 20, and 30 C by controlling the temperature of the thermostat water bath.

To assess the influence of the salinity on the photoproduction of NO from dissolved ${\mathrm{NO}}_{\mathrm{2}}^{-}$, artificial seawater samples were adjusted to different salinity of 20 ‰, 30 ‰, and 35 ‰ by adding Milli-Q water or NaCl to the stock solution of artificial seawater.

In order to compare the contributions of ultraviolet A (UVA), ultraviolet B (UVB), and visible light to the NO photoproduction, two kinds of light filter film were used (wrapped around the quartz cuvette tubes): (i) a Mylar plastic film (from United States Plastic Corp., Lima, Ohio) which can only shield UVB and (ii) a film, usually used as a car insulation film (from CPFilms Inc., USA), shielding both UVA and UVB (Li et al., 2010; Wu et al., 2015).

## 2.4 Calculations of photoproduction rates (RNO), photoproduction rate constant (JNO), and reaction yield

For the artificial seawater experiments determining the generation of NO from the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ photochemical degradation, the data were fitted with a simple linear regression with the form $y={R}_{\mathrm{NO}}×t+b$, where y is the NO concentration which was calculated by the signal intensity of DAF-2T at time t, b is the intercept of the fitting equation, and RNO is the photoproduction rate.

The photoproduction rate constant of NO from nitrite (JNO) was determined by preparing different concentrations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (0.5, 2.0, and 5.0 µmol L−1) in Milli-Q water and artificial seawater. The slope of the linear correlation between photoproduction rates and concentrations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ represents JNO (Anifowose et al., 2015).

The yield of NO formation (%fNO) from the photodegradation via ${\mathrm{NO}}_{\mathrm{2}}^{-}$ was estimated according to Anifowose et al. (2015):

$\begin{array}{}\text{(1)}& \mathit{%}{f}_{\mathrm{NO}}=\mathrm{100}×{J}_{\mathrm{NO}}×c\left({\mathrm{NO}}_{\mathrm{2}}^{-}\right)×\left({R}_{\mathrm{NO}}{\right)}^{-\mathrm{1}},\end{array}$

where c(${\mathrm{NO}}_{\mathrm{2}}^{-}$) is the initial concentration of ${\mathrm{NO}}_{\mathrm{2}}^{-}$.

## 2.5 Seawater samples

Surface seawater samples were collected form a water depth of 1 m during a ship campaign to the western tropical North Pacific Ocean on board the R/V Dong Fang Hong 2 from 13 November 2015 to 5 January 2016. This cruise covered two sections: a N–S section from 36 to 2 N along 146 to 143 E with 6 and 12 stations, respectively, and a W–E section from 137 to 161 E along the Equator with 13 stations (Fig. 1). Stations S0701–S0723 were sampled between 11 and 28 November (i.e., the first part of the N–S section), followed by sampling of the W–E section between 16 and 27 December and finally sampling of stations S0725–S0735 between 30 December 2015 and 5 January 2016 (i.e., second part of the N–S section). In addition, relevant surface currents are indicated in Fig. 1 (Fine et al., 1994; Zhao et al., 2016; Zhang et al., 2018). The location of the Kuroshio Current on 15 November 2015 was referenced from https://www1.kaiho.mlit.go.jp/ (last access: 1 April 2018).

Figure 1Locations of the sampling stations in the western tropical North Pacific Ocean. The acronyms NGCC, SEC, NECC, NEC, and STCC stand for New Guinea Coastal Current, South Equatorial Current, North Equatorial Countercurrent, North Equatorial Current, and Subtropical Countercurrent, respectively.

Seawater samples were collected using 8 L Niskin bottles equipped with silicon O-rings and Teflon-coated springs and mounted on a Sea-Bird CTD (conductivity, temperature, depth) instrument (Sea-Bird Electronics, Inc., USA). A 750 mL black glass bottle was rinsed with in situ seawater three times, and then it was filled with seawater quickly through a siphon. When the overflowed sample reached the half volume of the bottle, the siphon was withdrawn rapidly, and the bottle was sealed quickly. Samples were filtered through 0.45 and 0.2 µm polyethersulfone membranes (Pall, USA) to minimize microbial influence (Kieber et al., 1996; Yang et al., 2011). Then the filtered seawater was transferred in the dark into acid-cleaned and precombusted amber glass bottles, stored in darkness at 4 C and brought back to the laboratory on land. Samples were refiltered with 0.2 µm polyethersulfone membranes (Pall, USA) before the irradiation experiments. DAF-2 solutions were added in the dark. The irradiation experiments were conducted within 2 weeks after the samples arrived in the land laboratory, and the maximum storage time was about 2 months.

## 2.6 Dissolved inorganic nitrogen (DIN) and pH measurements

The concentrations of dissolved inorganic nitrogen (DIN = nitrate, nitrite, and ammonium) from the cruise were analyzed using an automated nutrient analyzer (SKALAR San$++$ system, SKAlAR, the Netherlands) on board. The detection limit was 0.05 µmol L−1 for nitrate, nitrite, and ammonium. When the concentration was below detection limit, 1∕2 of the detection limit (0.025 rounded off to 0.02) was used.

The pH values were measured just before the experiments by using a benchtop pH meter (Orion Star A211, Thermo Scientific, USA) which was equipped with an Orion 8102 Ross combination pH electrode (Thermo Scientific, USA). In order to ensure comparability with the temperature in the irradiation experiments, pH values of the natural seawater samples were measured at 20 C. The pH meter was calibrated with three NIST-traceable pH buffers (pH = 4.01, 7.00, and 10.01 at 20 C). The precision of pH measurements was ±0.01.

Figure 2Changes in NO concentrations with initial DAF-2 concentrations of 0, 0.7, 1.4, 2.1, 2.8, 3.5, and 4.2 µmol L−1 after an irradiation time of 2 h (a) and changes in different NO concentrations with storage time monitored at about 2 h time intervals (b).

3 Results and discussion

## 3.1 Optimal DAF-2 concentration and storage time

NO concentrations generated from photolysis of artificial seawater samples with an initial ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration of 0.5 µmol L−1 increased with increasing DAF-2 concentrations, and the reached a maximum at a DAF-2 concentration of 1.4 µmol L−1 (Fig. 2a). At DAF-2 concentrations > 1.4 µmol L−1 no further increase in the NO concentrations was observed. Thus, we used a DAF-2 concentration of 1.4 µmol L−1 for all experiments.

Samples after reaction with DAF-2 and stored at 4 C in the dark were stable for at least 28 h with the measurement interval about 2 h (Fig. 2b). The relative standard deviations of the resulting NO concentrations after irradiating samples containing 0.5 and 5.0 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ were ±13 % and ±7 %, respectively. This demonstrated that photolysis samples with NO, which were allowed to react with DAF-2, could be stored for at least 1 d at 4 C in the dark.

Figure 3Photoproduction rates of NO with 0.5, 2, and 5.0 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (a) and the calculated JNO values in Milli-Q water and artificial seawater (b); red symbols represent the artificial seawater samples, and black symbols represent the Milli-Q water samples.

## 3.2 Photoproduction of NO in Milli-Q water and artificial seawater

The photoproduction rates of NO in samples with ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations of 0.5, 2.0, and 5.0 µmol L−1 were generally higher in artificial seawater than in Milli-Q water (Fig. 3a and b).

The resulting JNO values were $\mathrm{5.6}±\mathrm{0.9}×{\mathrm{10}}^{-\mathrm{4}}$ and $\mathrm{9.4}±\mathrm{1.4}×{\mathrm{10}}^{-\mathrm{4}}$ min−1 for Milli-Q water and artificial seawater, respectively. They are lower than the JNO of $\mathrm{34.2}×{\mathrm{10}}^{-\mathrm{4}}$ min−1 for Milli-Q water reported by Anifowose et al. (2015). The difference might be explained by a higher solar radiation flux in their study, which was about 1055 W m−2.

Figure 4NO concentration changes with irradiation time at different pH, salinity, temperature, and illumination conditions (a, c, e, and g for 0.5 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and b, d, f, and h for 5.0 µmol L−1 NO${}_{\mathrm{2}}^{-}$).

Figure 5Changes in NO photoproduction rates with irradiation time at different pH, salinity, temperature, and illumination conditions (a, c, e, and g for 0.5 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and b, d, f, and h for 5.0 µmol L−1 NO${}_{\mathrm{2}}^{-}$).

## 3.3 Influence of pH, temperature, salinity, and wavelengths

All irradiation experiments conducted in artificial seawater had two different ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations of 0.5 and 5.0 µmol L−1 added. The resulting NO concentrations were generally higher when irradiating the samples with the initial ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration of 5.0 µmol L−1. NO photoproduction rates showed increasing trends with decreasing pH, increasing temperature, and increasing salinity; the relationships between rates with salinity and temperature were significant (p<0.5) (Figs. 4 and 5).

Reaction (1) indicates that decreasing pH results in lower concentrations of OH, which in turn will promote NO formation via ${\mathrm{NO}}_{\mathrm{2}}^{-}$. This is in line with the finding of Li et al. (2011), who found that the photodegradation rate of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in Milli-Q water was higher at pH = 6.5 than at pH = 9.5. Tugaoen et al. (2018) also found the effect of lowering pH to conjugate ${\mathrm{NO}}_{\mathrm{2}}^{-}$ to HONO, which allowed for HONO photolysis (pH = 2.5). Besides, higher pH could also inhibit N2O4 and N2O3 hydrolysis reaction (Reactions 4 and 8) as reviewed by Mack and Bolton (1999). However, in a previous study of Chu and Anastasio (2007) and Zellner et al. (1990), the quantum yield of OH (which equals to the quantum yield of NO) was constant at the pH ranges from 6.0 to 8.0 and 5.0 to 9.0 under single-wavelength light in nitrite solution. This might indicate that decreasing pH in our study mainly reduced NO consumption rather than increased NO production.

Higher temperatures led to increasing NO photoproduction rates according to the temperature dependence of chemical reactions given by the Arrhenius formula:

$\begin{array}{}\text{(2)}& {R}_{T}=A×\mathrm{exp}\left(-\frac{E}{R×T}\right),\end{array}$

where A is an Arrhenius prefactor and T is the temperature (K). This indicates that an increasing temperature results in a higher rate; Chu and Anastasio (2007) also found that quantum yield of OH (or NO) showed a decreasing trend from 295 via 263 to 240 K. Moreover, this equation can be used to consider the difference in the rates at two temperatures T1 and T2:

$\begin{array}{}\text{(3)}& {R}_{T\mathrm{2}}={R}_{T\mathrm{1}}×\mathrm{exp}\left(\frac{E}{R}×\left(\frac{\mathrm{1}}{T\mathrm{1}}-\frac{\mathrm{1}}{T\mathrm{2}}\right)\right).\end{array}$

If we assume that E is a constant in the temperature range of 10 to 30 C when ${\mathrm{NO}}_{\mathrm{2}}^{-}=\mathrm{0.5}$µmol L−1 and we plot ln R against 1∕T, we would get the E value as 57.5 kJ mol−1 K−1. Using the photoproduction rate at 20 C (293.15 K) as our reference point (T1), an expression of RT with the temperature was as follows:

$\begin{array}{}\text{(4)}& {R}_{T}=\mathrm{2.7}×{\mathrm{10}}^{-\mathrm{10}}×\mathrm{exp}\left(\mathrm{6920}×\left(\frac{\mathrm{1}}{\mathrm{293.15}}-\frac{\mathrm{1}}{T\mathrm{2}}\right)\right).\end{array}$

Similarly, we could conclude that the expression of RT with the temperature when ${\mathrm{NO}}_{\mathrm{2}}^{-}$ =5.0µmol L−1 is

$\begin{array}{}\text{(5)}& {R}_{T}=\phantom{\rule{0.125em}{0ex}}\mathrm{7}\phantom{\rule{0.125em}{0ex}}×\phantom{\rule{0.125em}{0ex}}{\mathrm{10}}^{-\mathrm{10}}\phantom{\rule{0.125em}{0ex}}×\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\phantom{\rule{0.125em}{0ex}}\left(\mathrm{11}\phantom{\rule{0.125em}{0ex}}\mathrm{026}\phantom{\rule{0.125em}{0ex}}×\phantom{\rule{0.125em}{0ex}}\left(\frac{\mathrm{1}}{\mathrm{293.15}}-\frac{\mathrm{1}}{T\mathrm{2}}\right)\right).\end{array}$

However, the NO production rate at 0.5 µmol L−1 nitrite did not increase from 20 to 30 C; the plausible explanation is that the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration here was the main influencing factor and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ might run out at 30 C. If the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration increased to 5.0 µmol L−1, the temperature could make a noticeable difference.

At 0.5 and 5.0 µmol L−1 initial ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations of Milli-Q water and artificial seawater samples, respectively, higher salinity showed higher photoproduction rates of NO. The regression relationship is $y=\mathrm{0.37}x-\mathrm{4.55}$ for 0.5 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and $y=\mathrm{2.3}x-\mathrm{39.5}$ for 5.0 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$, respectively, where x is the salinity (‰) and y is the photoproduction rate ($×{\mathrm{10}}^{-\mathrm{10}}$ mol L−1 s−1). This result indicates that with increasing ion strength NO production is enhanced; however, the exact mechanism is unknown and needs further study. Zafiriou and McFarland (1980) demonstrated that artificial seawater comprised with major and minor salts showed complex interactions and the addition of ethylenediaminetetraacetic acid (EDTA) could diminish NO concentration, which means trace metals could keep the NO concentration at a higher level, which is similar to our results. But Chu and Anastasio (2007) reported that addition of Na2SO4 (4.0–7.0 mmol L−1) in solution had no effect on the quantum yield of OH, which might be because of the complexity of the natural seawater samples. Overall, in artificial seawater samples, photoproduction rates showed an increasing trend with salinity.

The highest NO photoproduction rates were observed with full wavelength irradiation, whereas the lowest NO rates were observed with UVB. The NO photoproduction rates approached zero at wavelengths in the visible band. The contributions of visible, UVA, and UVB bands were <1 %, 30.7 %, and 85.2 % and <1.0 %, 34.2 %, and 63.1 % for 0.5 and 5.0 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$, respectively. Our results are in line with the findings of Zafiriou and McFarland (1981), who found that samples exposed to (UV+visible) wavelengths lost ${\mathrm{NO}}_{\mathrm{2}}^{-}$ more rapidly than those exposed only to the visible wavelengths alone. In the study of Chu and Anastasio (2007), under single-wavelength light, the quantum yield of OH decreased with the wavelength (280 to 360 nm and plateau until 390 nm) which means that single-wavelength UVB light had a higher photoproduction rate than UVA. Compared with the results in our study, this might be because of the wide band of UVA (320–420 nm) that led to the total higher rates under UVA than UVB (in our system 300–320 nm). Moreover, according to the UV–visible absorption spectra of ${\mathrm{NO}}_{\mathrm{2}}^{-}$, λmax was 354 nm, which is in the range of UVA (320–420 nm) (Zuo and Deng, 1998; Zafiriou and McFarland, 1981). Thus, it seems reasonable that in our study the photoproduction rate under UVA was higher than UVB; with full wavelength illumination, the photoproduction rates are the highest; and in the visible band, the NO photoproduction rates approached zero.

Figure 6Seawater temperature; salinity; concentrations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, and ${\mathrm{NH}}_{\mathrm{4}}^{+}$; wind speed; light intensity; and photoproduction rates of NO (RNO) in the western tropical North Pacific Ocean (a W–E transect; b N–S transect).

## 3.4 Kinetics of the NO photoproduction

The yields of NO formation from ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (%fNO) in artificial seawater samples were about 70.1 % and 97.9 % for the initial ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations of 0.5 and 5.0 µmol L−1, respectively. The missing NO yield (29.9 % for 0.5 µmol L−1 and 2.1 % for 5.0 µmol L−1) might result from NO production via other (unknown) nitrogen-containing substrates (Anifowose et al., 2015). Another plausible explanation would be that during the process of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ photoproduction, some NO was oxidized into NO2, then NO2 dimerized (Reaction 6) and the dipolymer N2O4 would hydrolyze into ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and ${\mathrm{NO}}_{\mathrm{3}}^{-}$ (Reaction 7), which actually reduces the concentration of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (Mack and Bolton, 1999).

Assuming a 100 % yield from ${\mathrm{NO}}_{\mathrm{2}}^{-}$ degradation and a fast reaction of NO with DAF-2, the observed linear relationships during the various irradiation experiments (Fig. 6) indicate that NO photoproduction was following a pseudo zero-order reaction. However, the RNO ratios (average: 4.8) listed in Table 2 were not the same for the experiments despite the fact that the ratio of the initial ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations (10) was the same for all experiments. This result, however, does point to a reaction which is different from a zero-order reaction.

Table 2The ratios of photoproduction rates (R5.0/R0.5) in the different irradiation experiments.

## 3.5 Photoproduction rates of NO in the western tropical North Pacific Ocean

During the cruise, surface temperatures and salinities were in the range from 22.15 to 30.19 C and 34.57 to 35.05, respectively. The concentrations of ${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{NH}}_{\mathrm{4}}^{+}$, and ${\mathrm{NO}}_{\mathrm{2}}^{-}$ ranged from 0.03 to 1.6, 0.20 to 1.2, and 0.02 to 0.33 µmol L−1, respectively (Fig. 6). The measured photoproduction rates of NO ranged from $\mathrm{0.3}×{\mathrm{10}}^{-\mathrm{10}}$ (station S0711) to $\mathrm{2.9}×{\mathrm{10}}^{-\mathrm{10}}$ mol L−1 min−1 (station S0303) (Tian, 2020), with an average value of $\mathrm{13.0}±\mathrm{7.6}×{\mathrm{10}}^{-\mathrm{11}}$ mol L−1 min−1. Photoproduction rates did not show significant correlations with ${\mathrm{NO}}_{\mathrm{2}}^{-}$, ${\mathrm{NO}}_{\mathrm{3}}^{-}$, ${\mathrm{NH}}_{\mathrm{4}}^{+}$, pH, salinity, water temperature, or with colored dissolved organic matter (data not shown, the same method with Zhu et al., 2017, absorption coefficients at 355 nm) (SPSS v.16.0, Pearson correlation test).

There was no linear relationship found between RNO and dissolved ${\mathrm{NO}}_{\mathrm{2}}^{-}$ during our cruise, which is in contrast to the results of Olasehinde et al. (2010), Anifowose et al. (2015), and Anifowose and Sakugawa (2017), who observed positive linear relationships between NO photoproduction rates and the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations in the surface waters of the Seto Inland Sea and the Kurose River. This might be because other factors like pH and salinity were different between samples collected at different stations.

Figure 7The potential temperature–salinity (TS) diagram with NO photoproduction rates indicated in the color bar. Water mass characteristics of surface currents shown in Fig. 1 are indicated. The acronyms NGCC, SEC, NECC, NEC, and STCC stand for New Guinea Coastal Current, South Equatorial Current, North Equatorial Countercurrent, North Equatorial Current, and Subtropical Countercurrent, respectively.

In Table 1, we found that the average photoproduction rate of NO measured in our cruise is lower than that of the Seto Inland Sea and the Kurose River, which could be ascribed to higher background ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in the inland sea waters (Olasehinde et al., 2009, 2010). Our result is slightly lower than the RNO from the central equatorial Pacific Ocean (> 10−12 mol L−1 s−1), and the lower concentration of ${\mathrm{NO}}_{\mathrm{2}}^{-}$ (0.06 µmol L−1) in our study area might account for this (Zafiriou and McFarland, 1981). In Table 1, the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration of 0.06 µmol L−1 in our study was lower than most other study areas like Qingdao coastal waters (0.75 µmol L−1) and the Seto Inland Sea (0–0.4 or 0.5–2 µmol L−1). In the study of Anifowose et al. (2015), since the ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration of upstream K1 station was similar to ours (0.06 µmol L−1), the higher RNO might be attributed to lower pH (7.36) as mentioned above. Or it might be because the difference in the river water and the seawater, considering lower nitrite level of K1, dissolved organic matter might also account for the higher RNO. Because of its conservative mixing behavior with salinity, dissolved organic matter always showed a higher level in river than open sea (Zhu et al., 2017), which could photodegrade itself to produce ${\mathrm{NO}}_{\mathrm{2}}^{-}$ and finally to promote RNO. In our study, the rates were adjusted to the ambient conditions, which included nighttime samples when the rates were lower. From the TS diagram (Fig. 7), we found that higher photoproduction rates at stations S0701 and S0704 might result from the influence of the Kuroshio (see Fig. 1), with enhanced concentrations of ${\mathrm{NO}}_{\mathrm{2}}^{-}$. The higher NO production rates measured for stations S0303/S0307 and S0717–S0723 might have been influenced by the South Equatorial Current and North Equatorial Current, respectively, but were obviously not associated with enhanced ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentrations.

If we take the missing 30 % of fNO in artificial seawater as the experimental error, then in our study, using the JNO in the artificial seawater, the average %fNO value in natural water was calculated to be 52 % (−30 %), indicating that there are other unknown nitrogenous compounds, for example, ${\mathrm{NO}}_{\mathrm{2}}^{-}$ produced from ${\mathrm{NO}}_{\mathrm{3}}^{-}$ photolysis (Reaction 9) or from other organic matter which could further lead to NO production (Kieber et al., 1999; Benedict et al., 2017; Goldstein and Rabani, 2007; Minero et al., 2007).

According to the photoproduction rates and the relevant ${\mathrm{NO}}_{\mathrm{2}}^{-}$ concentration in Olasehinde et al. (2010) and Anifowose and Sakugawa (2017) (Table 1), the photoproduction rates under 0.02 µmol L−1 ${\mathrm{NO}}_{\mathrm{2}}^{-}$ might not be determined in nearshore waters like the Seto Inland Sea.

## 3.6 Flux densities of NO in the surface layer of the WTNP

### 3.6.1 Air–sea flux density of NO

The NO flux densities were computed with (Eq. 7):

$\begin{array}{}\text{(6)}& F={k}_{\mathrm{sea}}\left(\left[\mathrm{NO}\right]-p{\mathrm{NO}}_{\mathrm{air}}×{H}^{\mathrm{cp}}\right),\text{(7)}& p{\mathrm{NO}}_{\mathrm{air}}={x}^{\prime }{\mathrm{NO}}_{\mathrm{air}}×\left({p}_{\mathrm{ss}}-{p}_{\mathrm{w}}\right),\end{array}$

where F stands for the flux density (mass per area per time) across the air–sea interface, ksea is the gas transfer velocity (length per time), [NO] is the measured concentration of NO in the surface seawater (mol per volume), x'NOair is the mixing ratio of atmosphere NO (dimensionless), and pss is the barometric pressure with pw calculated after Weiss and Price (1980):

$\begin{array}{}\text{(8)}& \begin{array}{rl}\mathrm{ln}{p}_{\mathrm{w}}& =\mathrm{24.4543}-\mathrm{6745.09}/\left(T+\mathrm{273.15}\right)-\mathrm{4.8489}\\ & ×\mathrm{ln}\left(T+\mathrm{273.15}\right)/\mathrm{100}\right)-\mathrm{0.000544}×S\right).\end{array}\end{array}$

Hcp is the Henry's law constant which is calculated after Sander (2015) as

$\begin{array}{}\text{(9)}& {H}^{\mathrm{cp}}\left(T\right)={H}^{\mathrm{\Theta }}×\mathrm{exp}\left(-\mathrm{\Delta }\mathrm{sol}H/R×\left(\mathrm{1}/T-\mathrm{1}/{T}^{\mathrm{\Theta }}\right)\right),\end{array}$

where $-\mathrm{\Delta }\mathrm{sol}\frac{H}{R}=\frac{\mathrm{d}\mathrm{ln}H}{\mathrm{d}\mathrm{ln}\left(\frac{\mathrm{1}}{T}\right)}$, HΘ and $-\mathrm{\Delta }\mathrm{sol}H/R$ are tabulated in Sander (2015) ($-\mathrm{\Delta }\mathrm{sol}H/R=\mathrm{1600}$ and ${H}^{\mathrm{\Theta }}=\mathrm{1.9}×{\mathrm{10}}^{-\mathrm{5}}$ mol m−3 Pa−1). The reviewed literature about NO, HΘ, and the values in different literature studies were similar (Sander, 2015). In our calculation, the values in Warneck and Williams (2012) were used.

Then ksea was calculated after Wanninkhof (2014) as Eq. (10):

$\begin{array}{}\text{(10)}& {k}_{\mathrm{sea}}={k}_{w}\left(\mathrm{1}-{\mathit{\gamma }}_{\mathrm{a}}\right),\end{array}$

where γa is the fraction of the entire gas concentration gradient across the air-side boundary layer as a fraction of the entire gradient from the bulk water to the bulk air (dimensionless), ka is the air-side air–sea gas transfer coefficient (length per time) according to Jähne et al. (1987); Mcgillis et al. (2000); and Sharqawy et al. (2010). For details about the calculation of kw and γa, see Tian et al. (2019).

Since onboard wind speeds were not available, ECMWF (European Centre for Medium-Range Weather Forecasts) reanalysis data sets (ERA-5 hourly data) were applied (Fig. 6). We used a value of 10−11 (vv) for atmospheric NO (Law, 2001). The atmosphere pressure was set to 101.325 kPa.

Since the measurements of [NO] were not available from the cruise, we estimated [NO] by assuming that (1) NO production is mainly resulting from ${\mathrm{NO}}_{\mathrm{2}}^{-}$ photodegradation, (2) the NO photoproduction RNO, as measured in our irradiation experiment, is balanced by the NO scavenging rate Rs, and (3) rates of nitrite photoproduction to NO were proportional to the irradiance flux in order to adjust the rates under solar simulator light into ambient light at the sampling time (Zafiriou and McFarland, 1981; Olasehinde et al., 2010):

$\begin{array}{}\text{(11)}& {R}_{\mathrm{NO}}×\frac{{I}_{\mathrm{ambient}}}{{I}_{\mathrm{simulator}}}=\left[\mathrm{NO}\right]×{R}_{\mathrm{s}},\end{array}$

where Rs represents the sum of the rate constants for the scavenging compounds reacting with NO times the concentrations of the scavenger compounds, Iambient and Isimulator denote the light intensity of the sampling station and the CPS+ simulator (765 W m−2). Iambient was ECMWF reanalysis data sets (ERA-5 hourly data, Fig. 6), which ranged from 0 to 762.9 W m−2 and the resulting $\frac{{I}_{\mathrm{ambient}}}{{I}_{\mathrm{simulator}}}$ ranged from 0 to 1.01 with an average of 0.35. In the studies of Zafiriou et al. (1980) and Anifowose and Sakugawa (2017), they reviewed the NO lifetime in different areas for the Kurose River (0.05–1.3 s), the Seto Inland sea (1.8–20 s), and the central equatorial Pacific (28–216 s, 170 E equatorial regions), which showed an increasing trend from river to open sea. It seemed that NO lifetime in our study area should be most similar to the central equatorial Pacific. Considering that part of our sampling stations were in open sea while some stations were close to the continent, like New Guinea island and Japan, average lifetimes of about 100 s (with an uncertainty factor of 2.5) were applied in our study. Tian et al. (2019) found that NO concentration in the surface water showed no significant difference with that in the bottom water (average depth: 43 m), so it seems reasonable to estimate the steady-state NO concentration with the NO concentration in the mixed layer. Then, [NO] was estimated to range from 0 to $\mathrm{292}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1 (0 means that sampling time during nighttime), with an average of $\mathrm{49}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1, which was consistent with previous results in central equatorial Pacific ($\mathrm{46}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1), while it was lower than near-continent seawater, like the Seto Inland Sea (up to $\mathrm{120}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1) and the Jiaozhou Bay ($\mathrm{157}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1), which might be because of the higher nitrite concentration. NO showed the lowest concentration in Kurose River; the shortest lifetime in river water than in seawater might account for this (Anifowose and Sakugawa, 2017).

In Table 1, the resulting flux density of NO for WTNP ranged from 0 to $\mathrm{13.9}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1, with an average of $\mathrm{1.8}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1, which is in good agreement with that in the central equatorial Pacific (see Table 1), while it was lower than that in coastal seawater such as the Seto Inland Sea or Jiaozhou Bay, consistent with the NO concentration distribution.

### 3.6.2 Oceanic photoproduction rates of NO

The photoproduction rates from our irradiation experiments were extrapolated to oceanic photoproduction in the WTNP with the equation from Uher and Andreae (1996) and Bange and Uher (2005):

$\begin{array}{}\text{(12)}& {R}_{\mathrm{ocean}}={R}_{\mathrm{NO}}×\left(\frac{{I}_{\mathrm{ocean}}\left(\mathrm{1}-\mathrm{exp}\left(-{K}_{D}×\mathrm{MLD}\right)}{{I}_{\mathrm{ss}}\phantom{\rule{0.125em}{0ex}}×\phantom{\rule{0.125em}{0ex}}{K}_{D}\phantom{\rule{0.125em}{0ex}}×\phantom{\rule{0.125em}{0ex}}\mathrm{MLD}}\right),\end{array}$

where Rocean and RNO are the photoproduction rates for the ocean mixed layer and seawater irradiation experiments, respectively; see Sect. 3.5. Iocean and Iss are the average global irradiance at the surface of the ocean mixed layer and the solar simulator used here, respectively. KD is the light attenuation coefficient and MLD is the estimated mixed layer depth at the sampled station.

Iocean was set to 185 W m−2 (Bange and Uher, 2005), while Iss was 765 W m−2 in our study (Wu et al., 2015). As described above, KD-354 was applied to estimate the MLD. In Smyth (2011), KD-340 to KD-380, derived from 10 % residual light level depths, ranged from 0.04 to 0.07 m−1 for our study area, and we used the average value of 0.05 m−1. The MLD was taken as the layer depth where the temperature was 0.2 C lower than the 10 m near-surface seawater layer (Montégut, 2004), ranging from 13–77 m with an average of 37 m. The resulting average Rocean was about $\mathrm{8.6}±\mathrm{4.9}×{\mathrm{10}}^{-\mathrm{12}}$ mol L−1 min−1 for the WTNP at the time of our cruise. Besides, the temperature at 20 C in our laboratory experiment would induce an error of about 10 % (Fig. 4e).

The flux induced by NO photoproduction in the WTNP (NO photoproduction rates divide by MLD, average: $\mathrm{13}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1) was significantly larger than the NO air–sea flux densities (average: $\mathrm{1.8}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1), indicating a further NO loss process in the surface layer.

4 Conclusions

The results of our irradiation experiments showed that NO photoproduction from ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in artificial seawater is significantly affected by changes in pH, temperature, and salinity. We found increasing NO production rates from dissolved ${\mathrm{NO}}_{\mathrm{2}}^{-}$ with decreasing pH, increasing temperature, and increasing salinity. In contrast, we did not find any correlations for NO photoproduction with pH, salinity, water temperature, or dissolved ${\mathrm{NO}}_{\mathrm{2}}^{-}$ in natural surface seawater samples from a cruise to the western tropical North Pacific Ocean (November 2015–January 2016). We conclude that the trends observed in our irradiation experiments with artificial seawater do not seem to be representative of WTNP because of the complex settings of open ocean environments. Moreover, we conclude that future changes in NO photoproduction due to ongoing environmental changes such as ocean warming and acidification are, therefore, difficult to predict and need to be tested by irradiation experiments of natural seawater samples under varying conditions. The flux induced by NO photoproduction in the WTNP (average: $\mathrm{13}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1) was significantly larger than the NO air–sea flux densities (average: $\mathrm{1.8}×{\mathrm{10}}^{-\mathrm{12}}$ mol m−2 s−1), indicating a further NO loss process in the surface layer. In order to decipher and to quantify the NO production and consumption pathways in the oceanic surface layer, more comprehensive laboratory and onboard measurements are required.

Data availability
Data availability.

The data can be accessed by email request to the corresponding authors, or they can be downloaded from https://pan.baidu.com/s/19LQIPADW5KDhONQuX9ub5g, (Tian, 2020).

Author contributions
Author contributions.

YT, GPY, CYL, HTC, and PFL prepared the original manuscript and designed the experiments; HWB made many modifications and gave a lot of suggestions on design of figures and the computing method. All authors contributed to the analysis of the data and discussed the results.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

We thank the captain and crew of R/V Dong Fang Hong 2 for their support and help during the cruise. This is MCTL contribution no. 223. We thank the editor and two reviewers for their thoughtful feedback on the article.

Financial support
Financial support.

This research has been supported by the National Natural Science Foundation of China (grant no. 41676065), the National Key Research and Development Program of China (grant no. 2016YFA0601301), and the Fundamental Research Funds for the Central Universities (grant no. 201762032).

Review statement
Review statement.

This paper was edited by Mario Hoppema and reviewed by Oliver Zafiriou and one anonymous referee.

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# A Continuity Argument in the Proof of Rouche's Theorem

In Greene and Krantz's Function Theory of One Complex Variable, the proof of Rouche's theorem involves the following continuity argument.

Let $f,g\colon U \to \mathbb{C}$ be holomorphic from an open set $U$. Let $\bar{D}(p,r) \subseteq U$, and $$\vert f(\zeta) - g(\zeta) \vert \le \vert f(\zeta) + g(\zeta) \vert$$ for $\zeta \in \partial D(p,r).$

Define $$f_t (\zeta) = tf(\zeta) + (1-t)\,g(\zeta),$$ for $t\in [0,1]$. Then the integral $$I_t = \frac{1}{2\pi i} \oint_{\partial D(p,r)} \frac{f'_t (\zeta)}{f_t (\zeta)}\,d\zeta$$ is a continuous function of $t\in [0,1]$.

They add in parenthases that the denominator does not vanish and the integrand depends continuously on $t$.

Question: Would anyone be kind enough to supply the details behind this continuity of $I_t$ ?

-

Let $$F(t,\zeta)={tf'(\zeta)+(1-t)g'(\zeta)\over tf(\zeta)+(1-t)g(\zeta)}$$ Since the denominator does not vanish, the function $F$ is continuous on $[0,1]\times \partial D(p,r)$. This is a compact set, so $F$ is in fact uniformly continuous in $(t,\zeta)$.
Now, let $\epsilon>0$ be given, and let $t_0 \in [0,1]$. By the uniform continuity, there is some $\delta>0$ such that $$|F(t,\zeta)-F(t_0,\zeta)|<\epsilon/r$$for all $t$ with $|t-t_0| < \delta$. It follows that $$|I_t-I_{t_0}|\leq {1\over 2\pi}\Bigl|\int_{\partial D(p,r)}\bigl(F(t,\zeta)-F(t_0,\zeta)\bigr){\rm d}\zeta\Bigr| \leq {1\over 2\pi}\cdot {\epsilon\over r}\cdot 2\pi r=\epsilon$$ This means that $I_t$ is continuous with respect to $t$.

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# Seminars

There are three series of seminars: Imperial pure analysis and PDE, London analysis and probability, and Paris-London analysis seminar. The first two altenate weekly, and are listed in the green and blue boxes below, the Paris-London series meets four times per year.

Our PhD students also jointly organise the Junior Analysis seminar; which consists of informal talks by students and visitors.

## 2018-2019 - Summer term programme

Mira Shamis (Queen Mary), Applications of the Ky Fan inequality to random (and almost periodic) operators
Thursday 16 May, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar

Abstract: We shall discuss the Ky Fan inequality for the eigenvalues of the sum of two Hermitian matrices. As an application, we shall derive a sharp version of a recent result of Hislop and Marx pertaining to the dependence of the integrated density of states of random Schroedinger operators on the distribution of the potential. Time permitting, we shall also discuss an application to quasiperiodic operators.

Jonathan Fraser (St. Andrews), Approximate arithmetic structure in large sets of integers
Thursday 23 May, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar

Abstract: The celebrated Erdos conjecture on arithmetic progressions states that sets of positive integers whose reciprocals form a divergent series should contain arbitrarily long arithmetic progressions. I will discuss recent joint work with Han Yu, where we prove that such sets must at least approximate arithmetic progressions in a quantifiable sense.

## 2018-2019 - Spring term programme

Anders Hansen (Cambridge), On non-computable problems in computer assisted proofs - Why foundations of computations may interest pure mathematicians
Thursday 10 January, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar

Abstract: Computer assisted proofs have become increasingly popular over the last decades and turn out to be instrumental when proving many long standing conjectures. The recent computer assisted proof, led by T. Hales, of the more than four century old conjecture of Kepler (Hilbert’s 18th problem) on optimal packing of 3-spheres, is a striking example. Another fascinating case is the computer assisted proof of the Dirac-Schwinger conjecture, by C. Fefferman and L. Seco, on the asymptotic behaviour of the ground-state energy of certain Schrodinger operators. What may be surprising is that the computational problems used in these proofs are non-computable according to Turing. In this talk we will discuss this paradoxical phenomenon: Not only can non-computable problems be used in computer assisted proofs, they are crucial for proving important conjectures. A key tool for understanding this phenomenon is the Solvability Complexity Index (SCI) hierarchy, which allows for a classification theory for all types of computational problems. This classification theory may be of use to pure mathematicians for determining which computational problems that may be used in computer assisted proofs. In particular, there are non-computable problems that can be used and there are non-computable problems that are so difficult that they can never be used in computer assisted proofs. The question is: which ones are safe to utilise? Examples from mathematical physics and spectral theory will be highlighted.

Leonid Parnovski (UCL), Floating mats and sloping beaches: spectral asymptotics
Wednesday 23 January , 1.30 - 2.30pm, King's College London, King's Building, room K3.11, London Analysis and Probability seminar,

Abstract: I will discuss the asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.

Valery Smyshlyaev (UCL), Two-scale homogenisation for a general class of high-contrast PDE systems
Wednesday 23 January, 3.00 - 4.00pm, King's College London, King's Building, room K3.11, London Analysis and Probability seminar,

Abstract: I present recent work with Ilia Kamotski [1] on two-scale homogenisation of general PDE systems with periodic coefficients with a critically scaled high contrast, which reflects certain interesting effects due to underlying micro-resonancesâ€™â€™. It appears that a strong two-scale resolvent convergence of associated high-contrast elliptic operators holds under a rather generic decomposition assumption. This implies in particular (two-scale) convergence of parabolic and hyperbolic semigroups with applications to a wide class of initial value problems. In the end I briefly discuss most recent stronger results with operator-type error estimates for high-contrast problems (with Shane Cooper and I. Kamotski), as well as situations where the micro-resonances display certain randomness. In simplest cases, the resulting two-scale limit behaviour appears to be rather explicit and the macroscopic equations display a form of wave trapping by the micro-resonances due to their randomness.

[1] I.V. Kamotski, V.P. Smyshlyaev, Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients, {\it Applicable Analysis} 98 (1-2), 64--90 (2019).

Dmitry Jakobson (McGill), Quantum ergodicity for ray-splitting (branching) billiards
Wednesday 23 January, 4.30 - 5.30pm, King's College London, King's Building, room K3.11, London Analysis and Probability seminar.

Abstract: This is joint work with Yu. Safarov and A. Strohmaier.
After giving an overview of Quantum Ergodicity results on compact Riemannian manifolds with ergodic geodesic flow (due to Shnirelman, Zelditch, Colin de Verdiere and others), we discuss joint work with Yuri Safarov and Alex Strohmaier, which concerns the semiclassical limit of spectral theory on manifolds whose metrics have jump-like discontinuities. Such systems are quite different from manifolds with smooth Riemannian metrics because the semiclassical limit does not relate to a classical flow but rather to branching (ray-splitting) billiard dynamics. In order to describe this system we introduce a dynamical system on the space of functions on phase space. We prove a quantum ergodicity theorem for discontinuous systems. In order to do this we introduce a new notion of ergodicity for the ray-splitting dynamics. If time permits, we outline an example (provided by Y. Colin de Verdiere) of a system where the ergodicity assumption holds for the discontinuous system.

♠ Alisa Knizel (Columbia), Log-gases on a quadratic lattice via discrete loop equations
Thursday 31 January, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

Abstract: We study a general class of log-gas ensembles on a quadratic lattice.
Using a variational principle we prove that the corresponding
empirical measures satisfy a law of large numbers and that their
global fluctuations are Gaussian with a universal covariance. We apply
our general results to analyze the asymptotic behavior of a q-boxed
plane partition model introduced by Borodin, Gorin and Rains. In
particular, we show that the global fluctuations of the height
function on a fixed slice are described by a one-dimensional section
of a pullback of the two-dimensional Gaussian free field.

Our approach is based on a q-analogue of the Schwinger-Dyson (or
loop) equations, which originate in the work of Nekrasov and his
collaborators, and extends the methods developed by Borodin, Gorin and
Guionnet to a quadratic lattice.

The talk is based on a joint work with Evgeni Dimitrov (Columbia University).

Paul Bourgade (Courant), TBA
Thursday 7 February 2019 , 3-4pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

♣ Mikhail Menshikov (Durham)Non-homogeneous random walks in critical regimes
Thursday 7 February 2019 , 4:30-5:30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar.

Abstract: http://www.london-analysis-seminar.org.uk/abstracts/Menshikov_070219.pdf

Christopher Kauffman (Imperial), Global Stability for Charged Scalar Fields in Spacetimes close to Minkowski
Thursday 14 February, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

Abstract: We prove global stability for the Charge-Scalar Field system on a background metric close to 1+3-dimensional Minkowski space. In particular, we consider a class of background metrics which satisfy certain estimates consistent with small-data solutions to Einstein's Vacuum Equations in harmonic coordinates. Our results are analogous to results obtained in Minkowski space, with slightly higher restrictions on the decay of initial data. The proof relies on a single-parameter modification of the standard Lorentz fields which depends on the initial ADM mass"

Martin Hairer (Imperial), Reconciling Ito and Stratonovich
Thursday 21 February, 3-4pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

Vlad Vysotsky (Sussex), Large deviations of convex hulls of planar random walks
Thursday 21 February, 4:30-5.30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar.

♠ Disheng Xu (University of Chicago), Lyapunov exponents and spectral theory of Schrodinger operator.
Thursday 28 February, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

Abstract: In this talk we define Lyapunov exponents (LE) over ergodic systems, and explain why LE is useful in the study of dynamical systems and the spectral theory of Schrodinger operator. We present a brief introduction to Kotani theory and Avila's density result on positive LE, and our recent results for higher dimensional case. This will be an introductory talk.

♣ Bassam Fayad (Paris)On the stability of quasi-periodic motion for analytic Hamiltonian systems
Thursday 14 March, 3-4pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar,

Abstract: The stability of an elliptic fixed point of a Hamiltonian system is a central question in mathematical physics and is one of the founding problems of dynamical systems. We explore this question from three points of view : topological stability (Lyapunov stability), statistical stability (KAM theory), and effective stability (finite time stability). We introduce in particular new diffusion mechanisms that give the first examples of real analytic Hamiltonians (in three or more degrees of freedom) with unstable elliptic fixed points, or unstable invariant quasi-periodic tori. We produce examples with arbitrary (including Diophantine) frequency vectors and with divergent or convergent (in the case of tori) Birkhoff Normal Forms. We also give examples of analytic Hamiltonians that are integrable on half of the phase space and diffusive on the other half.

♣ Eero Saksman (Helsinki), Decompositions of log-correlated fields with applications
Thursday 14 March, 4:30-5.30pm, Imperial College London, Huxley Building, room 140, London Analysis and Probability seminar.

Abstract: We consider a simple idea to decompose of log-correlated Gaussian fields into two-parts, both of which behave well in suitable sense. Applications include Onsager type inequalities in all dimensions, analytic dependence and existence of critical chaos measures for a large class of log-correlated fields. Talk is based on Joint work with Janne Junnila (EPFL) and Christian Webb (Aalto University).

Diane Holcomb (KTH Stockholm); Random matrices through differential operators
Thursday 21 March, 3:00-4:00, Imperial College London, Huxley 140, Pure analysis and PDE seminar.

Abstract: In 1984 Trotter described a tridiagonal random matrix model that has the same eigenvalues as the Gaussian Orthogonal Ensemble. This model and a later generalization share many structural properties with discrete Schrödinger operators. This led to the conjecture that the largest eigenvalues of the ensemble converged to the the eigenvalues a certain random differential operator. We will give an overview of where the conjecture comes from and a bit of the proof. We will then look at a process that appears when looking at eigenvalues of submatrices of the tridiagonal model. This process is notably different than the one that appears when considering submatrices of the full matrix model. This talk will survey work by Dumitriu-Edelman, Edelman-Sutton, and Ramirez-Rider-Virag. It finishes up with work that is joint A. Gonzalez.

♦ Program: http://www.london-analysis-seminar.org.uk/Paris-London/;
Friday 22 March, All day, Kings College London, The Paris-London Analysis Seminar.

## 2018-2019 - Autumn term programme

Sandrine Grellier (Orléans), Generic colourful tori and inverse spectral transform for Hankel operators
5 October, 10:30-11:20, UCL (Room 706), Paris-London Analysis seminar

Tom Korner (Cambridge), Can we characterise sets of strong uniqueness
5 October, 11:30-12:20, UCL (Room 706), London Analysis and Probability seminar

Emmanuel Fricain (Lille), Multipliers between sub-Hardy Hilbert spaces
5 October, 14:00-14:50, UCL (Room 706), Paris-London Analysis seminar

♦ Tom Sanders (Oxford),The Erdös Moser sum-free set problem
5 October, 15:20-16:10, UCL (Room 706), London Analysis and Probability seminar,

For abstracts of the talks please visit here

♠ Andrzej Zuk (CNRS- Paris), From PDEs to groups
12 October, 3:00-4:00, Imperial College London, Huxley 140), Pure analysis and PDE seminar

Abstract: We present a construction which associates to a KdV equation the lamplighter group. At crucial steps of it appear automata and random walks on ultra discrete limits. It is also related to the L2 Betti numbers introduced by Atiyah which are homotopy invariants of closed manifolds.

Jürg Fröhlich (ETH), The Arrow of Time - Images of Irreversible Behavior
18 October, 3:00-4:00, UCL (Room 706), London Analysis and Probability seminar,

Abstract:I sketch various examples of physical systems with time-reversal invariant dynamics exhibiting irreversible behavior. I start with deriving the Second Law of Thermodynamics in the formulation of Clausius from the existence of quantum-mechanical heat baths and then derive the Carnot bound for the degree of efficiency of heat engines. I continue with the analysis of a quantum-mechanical model with unitary time evolution describing a particle that exhibits diffusive motion when coupled to a suitably chosen (non-interacting) heat bath. A classical model with a Hamiltonian time evolution describing a particle coupled to a wave medium exhibiting friction is sketched next. I conclude with an attempt to draw the attention of the audience to the fact that the dynamics of isolated, open quantum systems featuring events is fundamentally irreversible.

Thomas Spencer (IAS), Edge reinforced random walk as a toy model of localization
18 October, 4:30-5:30, UCL (Room 707), London Analysis and Probability seminar,

Abstract: I will present some results and conjectures about edge reinforced random walk (ERRW). This is a history dependent walk which favors edges it has visited in the past. In three dimensions the walk has a phase transition as the reinforcement is varied. The relation of ERRW to a toy model of quantum localization will also be discussed.

♠ Jan Sbierski (Oxford), Uniqueness & non-uniqueness results for wave equations
25 October, 3:00-4:00, Imperial College (Huxley 140), Pure analysis and PDE seminar

Abstract:A well-known theorem of Choquet-Bruhat and Geroch states that for given smooth initial data for the Einstein equations there exists a unique maximal globally hyperbolic development. In particular, time evolution of globally hyperbolic solutions is unique. This talk investigates whether the same result holds for quasilinear wave equations defined on a fixed background. After recalling the notion of global hyperbolicity, we first present an example of a quasilinear wave equation for which unique time evolution in fact fails and contrast this with the Einstein equations. We then proceed by presenting conditions on quasilinear wave equations which ensure uniqueness. This talk is based on joint work with Harvey Reall and Felicity Eperon.

Thierry Lévy (Paris 6), Quantum spanning forests
1 November, 3:00-4:00, UCL (Room 706), London Analysis and Probability seminar,

Abstract: I will report on a work in progress with Adrien Kassel (ENS Lyon) about an extension of Kirchhoffâ€™s matrix-tree theorem and determinantal point processes, to the framework of vector bundles over graphs. While trying to understand in combinatorial terms the determinant of the covariant Laplacian on the space of sections of a vector bundle over a graph endowed with a connection, we were led to the definition of a family of probability measures on the Grassmannian of a Euclidean or Hermitian space, associated with an orthogonal splitting of this space and a self-adjoint contraction on it. This family of measures contains and extends the family of determinantal point processes.

♣ Thomas Bothner (KCL), When J. Ginibre met E. Schrödinger
1 November, 4:30-5:30, UCL (Room 706), London Analysis and Probability seminar,

Abstract: The real Ginibre ensemble consists of square real matrices whose entries are i.i.d. standard normal random variables. In sharp contrast to the complex and quaternion Ginibre ensemble, real eigenvalues in the real Ginibre ensemble attain positive likelihood. In turn, the spectral radius of a real Ginibe matrix follows a different limiting law for purely real eigenvalues than for non-real ones. Building on previous work by Rider, Sinclair and Poplavskyi, Tribe, Zaboronski, we will show that the limiting distribution of the largest real eigenvalue admits a closed form expression in terms of a distinguished solution to an inverse scattering problem for the Zakharov-Shabat system. This system is directly related to several of the most interesting nonlinear evolution equations in 1+1 dimensions which are solvable by the inverse scattering method, for instance the nonlinear Schro ̈dinger equation. The results of this talk are based on the recent preprint arXiv:1808.02419, joint with Jinho Baik.

Vedran Sohinger (Warwick) Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimension d <= 3
8 November, 3:00-4:00, Imperial College (Huxley 140), Pure Analysis and PDE seminar

Abstract: Gibbs measures of nonlinear Schrödinger equations are a fundamental object used to study low-regularity solutions with random initial data. In the dispersive PDE community, this point of view was pioneered by Bourgain in the 1990s. We prove that Gibbs measures of nonlinear Schrödinger equations arise as high-temperature limits of appropriately modified thermal states in many-body quantum mechanics. We consider bounded defocusing interaction potentials and work either on the d-dimensional torus or on R^d with a confining potential. The analogous problem for d=1 and in higher dimensions with smooth non translation-invariant interactions was previously studied by Lewin, Nam, and Rougerie by means of variational techniques.
In our work, we apply a perturbative expansion of the interaction, motivated by ideas from field theory. The terms of the expansion are analysed using a diagrammatic representation and their sum is controlled using Borel resummation techniques. When d=2,3, we apply a Wick ordering renormalisation procedure. Moreover, in the one-dimensional setting our methods allow us to obtain a microscopic derivation of time-dependent correlation functions for the cubic nonlinear Schrödinger equation. This is joint work with Jürg Fröhlich, Antti Knowles, and Benjamin Schlein.

♠ Edward Crane (Bristol), Circle Packing and Uniformizations
15 November, 3:00-4:00, Imperial College (Huxley 140), Pure analysis and PDE seminar

Abstract: Koebe discovered his circle packing theorem in the 1930s as a limiting case of his uniformization theorem for multiply-connected plane domains. After Thurston had interpreted circle packing as a discretization of conformal structure, Rodin and Sullivan showed how one could deduce the Riemann mapping theorem as a limiting case of the circle packing theorem. I will explain conformal welding and its circle packing analogue. I will show how this technique can be used to approximate an unusual uniformization of multiply-connected domains, in which each complementary component is a disc in the hyperbolic metric associated to the complement of all the other complementary components.

Jonathan Bennett (Birmingham) The nonlinear Brascamp-Lieb inequality and applications
22 November, 3:00-4:00, UCL (Room tba), London Analysis and Probability seminar,

Abstract: The Brascamp--Lieb inequality is a broad generalisation of many well-known multilinear inequalities in analysis, including the multilinear H\"older, Loomis--Whitney and sharp Young convolution inequalities. There is by now a rich theory surrounding this inequality, along with diverse applications in convex geometry, partial differential equations, number theory and beyond. Of particular importance is Lieb's Theorem (1990), which states that the best constant in this inequality is exhausted by centred gaussian functions. In this talk we present a recent "nonlinear" variant of the Brascamp--Lieb inequality, and describe some of its applications in harmonic analysis and PDE. A key ingredient in our proof is a certain effective version of Lieb's theorem, providing information about the shapes of gaussian near-extremisers for the classical Brascamp--Lieb inequality. This is joint work with Stefan Buschenhenke, Neal Bez, Michael Cowling and Taryn Flock.

♣ Herbert Koch (Bonn), A continuous family of conserved energies for the Gross-Pitaevskii equation,
22 November, 4:30-5:30, UCL (Room tba), London Analysis and Probability seminar,

Abstract: The Gross-Pitaevskii equation is the defocusing cubic nonlinear Schrödinger equation with the boundary conditions |u(t,x)| -> 1 at infinity. A difficulty in the study of the Gross-Pitaevskii equation is that the state space is nonlinear. In joint work with Xian Liao we study the equation in one space dimension, equip it with a new metric, and construct a continuous family of conserved energies.

♠ Benjamin Fahs (Imperial College), Toeplitz determinants with Fisher-Hartwig singularities
29 November, 3:00-4:00, Imperial College (Huxley 140), Pure analysis and PDE seminar

Abstract: We consider the large n asymptotics of n dimensional Toeplitz determinants with Fisher-Hartwig singularities, uniformly as the location of the singularities are allowed to merge with each other. We discuss applications to moments of averages of the characteristic polynomials of the Circular Unitary Ensemble.

Horst Knörrer (ETH) Construction of oscillatory singular homogenuous space times
6 December, 3:00-4:00, UCL (Room tba), London Analysis and Probability seminar,

Abstract: The vacuum Einstein equations for Bianchi space times (that is space times that can be foliated into three dimensional space like slices that are all homogenuous spaces) reduce to a system of ordinary differential equations. The conjectures of Belinskii, Khalatnikov and Lifshitz predict that for almost all initial data the solutions of these differential equation behave like trajectories of a billiard in a Farey triangle in the hyperbolic plane, that is, a triangle whose three vertices are ideal points. In joint work with M.Reiterer and E.Trubowitz we show that, for a set of initial data that has positive measure, this is indeed the case. We use ideas inspired by scattering theory for approximations of the system. The fact that billiard in a Farey triangle is chaotic leads us to small divisor problems similiar to those of KAM theory in Hamiltonian dynamics.

Tuomas Sahsten (Manchester), Delocalisation of waves under scaling limits
6 December, 4:30-5:30, UCL (Room tba), London Analysis and Probability seminar,

Abstract: We establish quantitative quantum ergodicity type delocalisation theorem for waves on hyperbolic surfaces of large genus. In the compact setting our assumptions hold for random surfaces in the sense of Weil-Petersson volume in the TeichmÃ¼ller space due to the work of Mirzakhani and in non-compact setting for arithmetic surfaces coming from congruence covers of the modular surface. The methods are based on Benjamini-Schramm scaling limits of metric measure spaces and Stein type harmonic analysis ergodic theorems, and are inspired by similar results on graphs. We plan to give a gentle introduction to the field before going to our results. Joint work with Etienne Le Masson (Cergy-Pontoise University, France).

♠ Matthew Jacques (Open University), Semigroups of hyperbolic isometries and their parameter spaces
13 December , 3:00-4:00, Imperial College London, (Huxley 140), Pure analysis and PDE seminar

Abstract: Let M denote the collection of orientation-preserving isometries of the hyperbolic plane. Given an n-tuple x=(x_1, ..., x_n) in M^n, let S(x) denote the semigroup generated by the ordinates of x under composition. In a 2010 paper of Avila, Bochi and Yoccoz the authors define the hyperbolic locus, H, as the set of points in M^n whose ordinates generate composition sequences that grow exponentially. They also define the elliptic locus, E, as the set of those x in M^n for which S(x) contains an elliptic isometry. They show that both H and E are open, and that the closure of E is equal to the complement of H. Motivated by this work and by the theory of discrete (Fuchsian) subgroups contained in M, we introduce the term semidiscrete to describe a semigroup that does not contain the identity within its closure. The semidiscreteness property on semigroups appears to be a good analogue of the discreteness property on groups, and we give theorems that have familiar counterparts in the theory of Fuchsian groups. For instance, we find that every semigroup is one of four standard types: elementary, exceptional, semidiscrete, or dense in M. We use these ideas to characterise the set H in terms of the semidiscreteness property. Finally, we give an example of a point on the boundary of E but not on the boundary of H, and an example of a point on the boundary of H that does not lie on the boundary of any of its connected components, answering two questions posed by Avila, Bochi and Yoccoz.

♠ Tom Claeys (Uni Louvain-la-Neuve), TBA
13 December , 4:30-5:30, Imperial College London, (Huxley 140), Pure analysis and PDE seminar

Abstract:

♠ Ivan Gentil (Lyon), Analytic point of view of the Schrödinger problem : a review on the subject.
20 December , 3:00-4:00, Imperial College London, (Huxley 140), Pure analysis and PDE seminar

Abstract:We are going to describe the Schrödinger problem as a minimisation of a cost along paths. This point of view allows us to simplify the problem and to see how the Schrödinger problem approaches the optimal transportation problem and also dual formulation. This is a joint work with C. Léonard and L. Ripani.