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Research article Special Issues

Reconstruction of initial heat distribution via Green function method

  • Received: 25 November 2021 Revised: 18 April 2022 Accepted: 19 April 2022 Published: 08 June 2022
  • In this paper, layer potential techniques are investigated for solving the thermal diffusion problem. We construct the Green function to get the analytic solution. Moreover, by combining Fourier transform some attractive relation between initial heat distribution and the final observation is obtained. Finally iteration scheme is developed to solve the inverse heat conduction problem and convergence results are presented.

    Citation: Xiaoping Fang, Youjun Deng, Zaiyun Zhang. Reconstruction of initial heat distribution via Green function method[J]. Electronic Research Archive, 2022, 30(8): 3071-3086. doi: 10.3934/era.2022156

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  • In this paper, layer potential techniques are investigated for solving the thermal diffusion problem. We construct the Green function to get the analytic solution. Moreover, by combining Fourier transform some attractive relation between initial heat distribution and the final observation is obtained. Finally iteration scheme is developed to solve the inverse heat conduction problem and convergence results are presented.



    1. Introduction

    Diel migrations are a common behavior pattern in a wide variety of marine species, from zooplankton to apex predators. Throughout the world's oceans, many micronekton undertake diel vertical migrations, coming to shallower, near-surface waters at night to forage, and retreating to deeper, darker waters during the day to avoid visual predators [1,2,3]. Some micronekton may also focus their spawning in the warmer, shallower portion of their diel migration [4,5]. The timing of vertical migrations and the related changes in bulk acoustic backscatter are correlated with times of sunset and sunrise [3], and micronekton have been shown to adjust their depth to stay within a preferred range of light intensity [6].

    Near the slopes of islands and seamounts, some micronekton associate with the benthos throughout their diel migrations [7,8], resulting in longer migrations because there is an added horizontal travel component [9]. In the Hawaiian Islands, this layer has been termed the "mesopelagic boundary community, " and organisms in this slope-associated community include micronektonic fish, squid, and shrimp [7,9]. The majority of this community is made up of multiple species of myctophid fishes [9]. Observations of organism density by stereo-video camera have shown a maximum density of 1,800 organisms m-3 on western Oahu and 700 organisms m-3 on the Big Island of Hawaii, with averages of 15–23 organisms m-3[9]. These organisms are a key trophic link between zooplankton and larger predators in slope ecosystems [10], and due to their significant horizontal migrations, they may also facilitate nutrient transport between different regions of the slope. The diel migrations of this slope-associated community may be influenced by the amount of light in the water column, as has been observed in pelagic zooplankton and micronekton vertical migrations elsewhere in the world [6,11,12,13].

    As the Hawaiian mesopelagic boundary community undertakes these daily migrations along the slope of the islands, the community travels from depths of > 400 m in the daytime to as shallow as 10–100 m during the night [7,14]. The distance of this migration depends on the steepness of the island slope, but round-trips of up to 11 km have been observed on the western slope of Oahu, Hawaii [9]. Moon phase, but not sunset time, has previously been found to correlate with arrival time of the mesopelagic boundary community in the shallow nighttime habitat [15]. If light-mediated visual predation is an important factor driving migrations, it follows that smaller organisms should arrive before larger ones since they are harder to see [9]. However, larger organisms have actually been observed arriving before smaller organisms [9], indicating that the faster swimming speed of larger organisms may be more important than avoidance of light. Overall, the relationship of water-column light to mesopelagic boundary community horizontal migration timing is unclear.

    In this contribution, we present novel observations of the migration of the mesopelagic boundary community between the daytime and nighttime habitats on the southern slope of Oahu, Hawaii, in the region of a proposed seawater air conditioning system. We investigate the hypothesis that ambient light is an important factor in the timing of mesopelagic boundary layer migrations. Additionally, we demonstrate how the estimated backscatter data from uncalibrated ADCPs can be useful to investigate biological questions regarding behavior of the mesopelagic sound scattering community.


    2. Methods


    2.1. Study site

    The study area was offshore of Honolulu, Oahu, Hawaii, USA. Mamala Bay is an open embayment spanning from Barber's Point (21.29 °N, 158.11 °W) to Diamond Head (21.25 °N, 157.81 °W). Moorings were deployed in an area within Mamala Bay at approximately 21.28 °N and 157.87 °W and an area outside Mamala Bay near 21.24 °N and 157.89 °W. The instruments were deployed for multiple periods between March 1,2013 and November 7,2016.


    2.2. Field deployments

    ADCPs were deployed between March 2013 and November 2016 (Figure 1; S1–S8) as part of an ongoing environmental assessment for a proposed seawater air conditioning plant [20] and one ADCP was deployed in a separate physical oceanography study further offshore (Table 1). The shallow deployments were within Mamala Bay about 1.7 km from shore. For these eight shallow deployments, an upward-looking 300 kHz ADCP was situated in a frame on the seafloor. These deployments occurred sequentially in time (Table 1). The deployments were between 106–152 m, within the deeper part of the typical nighttime residence depth of the mesopelagic boundary community [9].

    Figure 1. Map of the study location off the island of Oahu, HI. The shallow and deep deployments reported in this study are shown as circles labeled "S" and "D". Box 1 is enlarged in the inset and shows a close-up of the locations of the eight shallow deployments (S1–S8). The circle in box 2, labeled "Echo, " shows the location of a 200 kHz echosounder and 300 kHz ADCP deployed in 2005 in a separate study and used here for method validation purposes [15]. Areas where the mesopelagic boundary layer and plankton layer dynamics have been previously studied are indicated by box 2 [15], box 3 [9], and box 4 [15,16,17,18,19].
    Table 1. Instrumentation, depths, locations, and deployment dates of 300 kHz ADCPs.
    Mooring ID Water depth (m) Instrument Instrument depth (m) Orientation Latitude (°N) Longitude (°W) Deployed Recovered
    S1 120 RDI Workhorse 119 Up 21.2802 157.8721 04/22/13 07/10/13
    S2 145 RDI Workhorse 144 Up 21.2791 157.8722 08/06/13 10/14/13
    S3 130 RDI Sentinel V 129 Up 21.2797 157.8723 04/24/14 07/25/14
    S4 152 RDI Sentinel V 151 Up 21.2790 157.8720 04/20/15 07/23/15
    S5 106 RDI Sentinel V 105 Up 212793 157.8698 08/18/15 11/19/15
    S6 127 RDI Sentinel V 127 Up 21.2791 157.8713 12/11/15 03/18/16
    S7 120 RDI Sentinel V 119 Up 21.2791 157.8713 04/14/16 07/11/16
    S8 150 RDI Sentinel V 149 Up 27.2792 157.8719 07/18/16 11/07/16
    D 517 RDI Workhorse 426 Down 21.2353 157.8896 03/01/13 08/30/13
     | Show Table
    DownLoad: CSV

    One instrument was deployed downslope, outside Mamala Bay and about 6.8 km from shore. The downslope deployment (Figure 1, D) was located within the daytime residence depths of the mesopelagic boundary community; here, a downward-looking 300 kHz ADCP was moored 91 m above bottom in a water depth of 517 m. The deep ADCP was deployed from March 2013 through August 2013 (Table 1).

    For the shallow deployments, the ADCPs sampled at a ping rate of 12 s and data were averaged over 10-min intervals and 4 m depth bins. For the deep deployment, the ADCP sampled at a ping rate of 4.8 s and data were averaged over 2 min intervals and 2 m depth bins. For all deployments, the profiling range was ~100 m.


    2.3. General flow patterns

    Previous work has described the general flow at the shallow site. When integrated throughout the water column, flow is typically higher in the alongshore direction, but near the benthos bathymetric steering of a small canyon drives a dominant across-shore current [20]. The current flow direction primarily varies at the time scale of the M2 tidal constituent throughout the water column, including near the benthos [20]. The M2 constituent is the principal lunar semidiurnal tide, which has a period of 12.42 h and is the dominant tidal constituent in most of the world's oceans. At both shallow and deep sites, the alongshore and across-shore velocities were calculated based on the major and minor axes for each deployment. The major axes of the current ellipses ranged from 103–119, which correspond to an alongshore direction. The depth-integrated alongshore and across-shore flow components were calculated for the full range of ADCP data and for the near-bottom portion of the water column (8–16 meters above bottom). Mean current magnitudes in the across-shore direction were compared to calculated community movement speeds to ascertain the possible impact of advection on the observed migratory phenomenon.


    2.4. Backscatter estimation and general calculations

    The acoustic volume scattering strength (Sv, dB re 1 m-1), or backscatter, was estimated from ADCP echo intensity data following the methods of [21]. Geometric means and standard deviations were used because the data are on a logarithmic scale. The geometric means and geometric standard deviations of estimated backscatter data were calculated for each deployment during daytime and nighttime periods. Daytime was defined as the time between local sunrise and sunset, and nighttime was defined as the time between local sunset and sunrise. Sunrise and sunset data for Honolulu, HI were obtained from the U.S. Naval Observatory (http://aa.usno.navy.mil).

    The ADCPs in this study were deployed with the primary goal of investigating flow and were not calibrated to allow derivation of absolute backscatter strength, in contrast to a previous study on Oahu [22]. Therefore, all backscatter (Sv) values reported here are estimated, and they should not be directly compared between sites or instruments. However, the important factor in identifying micronekton diel migration is the relative change in estimated backscatter over time at each individual ADCP; therefore, when comparing two deployments, the estimated backscatter was normalized on a 0–1 scale, with 0 being the lowest estimated backscatter and 1 being the highest estimated backscatter within the time frame of interest.


    2.5. Identifying migrations using diel changes in estimated backscatter

    The mesopelagic boundary layer moves between deep (~400–600 m) and shallow (0–400 m) waters along the slope during the diel migrations [9]. In a full diel migration cycle, there is an evening arrival at the shallow habitat, a morning departure from the shallow habitat, a morning arrival at the deep habitat, and an evening departure from the deep habitat. The times of these events each day were identified using changes in the estimated backscatter at the deep and shallow ADCPs, and the community movement speed was estimated using departure and arrival times at deep and shallow moorings during the evening migration. It is important to note that the same individuals may not be detected at both ADCP sites, despite their positioning as an across-shore transect; however, the community as a whole leaves a deep habitat and migrates to a shallower habitat, so we can use these two instruments to estimate community arrival and departure times and community movement speeds between ~520 m and ~100–150 m isobaths. We do not attempt to resolve individual swimming speeds.

    We focused on the deepest part of the water column at each site, corresponding to a range of 8–16 m above the benthos, where the daily variation in backscatter was most pronounced (Figure 2). The mean of estimated backscatter was taken over this depth range at each time point to produce a single time-series of estimated backscatter, and the time-series was then filtered using a 1-hour low-pass Butterworth filter to remove high-frequency variation. Using this filtered time-series, we calculated the mean estimated backscatter for each non-migration period (day and night). For the shallow sites, the "non-migration periods" were defined as 00:00–02:00 and 10:00–18:00, based on previous observations of the community behavior in a shallow nighttime habitat in Hawaii's waters [15]. For the deep site, published observations were not available, and so the non-migration periods were defined as 01:00–04:00 and 10:00–15:00 based on observed patterns in the backscatter time series.

    Figure 2. Sample mean-day composites for March 3–May 2,2013 at the deep site. (A) High-pass filtered cross-shore velocity, (B) high-pass filtered vertical velocity, (C) estimated backscatter. The intersection of the two black rectangles represents the mean time of departure from deep site, and the boundaries of the rectangles represent one standard deviation. Note the difference in scale between A and B.

    Each migration event was then bounded by a "high" and "low" estimated backscatter mean. We estimated the times of daily migration events by identifying when the estimated backscatter signal rose or fell past certain percentage thresholds between consecutive low and high periods. We used a 50% threshold for statistical analyses (midway between day and night average backscatter readings), and used 20% and 80% thresholds to visualize the time-span over which migrations occurred. The times that the thresholds are crossed are referred to as the "arrival" and "departure" times of the community.

    Most of the time, the change in estimated backscatter during the migration periods was clear and abrupt. However, sometimes the signal was noisy or sloped slowly, which resulted in reduced confidence in the identification of the migration time. To control for this, the slopes of the filtered data for the one hour surrounding the identified migration times were calculated for each day. The observations associated with the lowest 20th percentile of slopes were removed from analysis.


    2.6. Validation: Relating ADCP observations to mesopelagic boundary community migration

    To provide confidence that observed changes in estimated backscatter were associated with the diel migration of the mesopelagic boundary layer, we examined data from two instruments deployed simultaneously [15]. One instrument was a 200 kHz echosounder calibrated to observe the presence of myctophid fishes [14], and the second was a 300 kHz ADCP. The instruments were deployed together at 40 m water depth off the southwest side of Oahu during April 2005 (Figure 1) [15]. By applying the method described in section 2.5 to the 2005 ADCP data, we found that estimated dusk arrival times ranged from 18:15–20:42. The echosounder recorded arrival times of 18:15–21:00. Estimated dawn departure times using the ADCP data were 4:54–7:00, but the echosounder showed that the departure actually happened between 02:15–4:30.

    At the deep site, the diel change in the near-bottom backscatter signal could be due to migration to a shallower pelagic habitat (vertical), slope-associated migration that moves inshore and shallower at night (horizontal), or both. We explored the data for evidence of horizontal and vertical components of the migration by examining "mean-day" composites of vertical and cross-shore velocities at the deep site [23]. ADCPs determine water velocity by measuring the velocity of suspended sound-scattering particles in the water column, such as sediment and plankton. These velocity measurements can be biased by nektonic organisms swimming through the water rather than being advected [23,24]. If a large-scale migration in a unified direction occurs at a certain time each day, these animal movements can manifest as a noticeable signal in a mean-day composite [23]. In the mean-day analysis, velocity data were first filtered with a 4th-order high-pass Butterworth filter to remove changes occurring with periods greater than 5 hours. This was done to remove bias in the means due to tidal effects.

    Distinct vertical and horizontal signals were visible in mean-day visualizations of filtered data (Figure 2). The mean-day composite of vertical velocity showed a short period of downward velocity centered near 06:30 and a short period of upward velocity centered near 19:00. The mean-day composite of cross-shore velocity showed a period of shoreward velocity centered slightly earlier than 18:00, concentrated mainly in the bottom 30 m. The shoreward pulse corresponded to the period of time and depth where estimated backscatter was decreasing. These results support the assertion that the daily fluctuations in the near-bottom estimated backscatter were significantly due to the presence/absence of the horizontally migrating micronekton community. Note that there was no significant offshore velocity in the morning that would correspond to a horizontal return to deep water. This observation indicates that the horizontal morning migration was less concentrated in time than the horizontal evening migration.

    Since the algorithm was more accurate for the dusk event than the dawn event, and the mean-day composites also showed a stronger signal at dusk than at dawn, we focused primarily on the dusk migration for statistical modeling and other analyses.


    2.7. Generalized additive modeling

    We tested the hypothesis that evening arrival time of the mesopelagic boundary community at the shallow site varied with sunset time, lunar illumination, cloud cover, and water column depth. Lunar illumination was defined as the percent of moon illuminated if the moon was up in the sky, but if the moon was not up between sunset and the arrival time or set/rose within one hour of arrival time (meaning moon was low on the horizon), the illumination was set to zero. Sun and moon data were obtained from the US Naval Observatory (http://aa.usno.navy.mil). Cloud cover was the mean percentage of the sky covered in clouds for each evening (18:00–00:00). Cloud data were obtained from the National Climate Data Center's Honolulu International Airport station at one hour intervals (http://www.ncdc.noaa.gov).

    Arrival times of the mesopelagic boundary layer at the shallow site were hypothesized to be due to environmental factors, namely sunset, cloud cover, lunar illumination, and water column depth. We employed a generalized additive model (GAM) to model this research hypothesis, as the error distribution of our response (and thus likely our residuals) was non-normal (approximately gamma) and relationships between response and predictor variables were assumed to be non-linear. The inverse (vs. log) link function resulted in the best-behaved residuals. Co-linearity among predictors was assessed both by calculating Pearson's correlation values and variance inflation factors (VIF, for multicollinearity) [25]. Correlation among predictors was found to be low (i.e., all VIFs < 3) [25] and all predictors were maintained in the model fit. Model selection continued by comparing models representing all possible combinations of predictors. The best-specified model was chosen based on corrected Akaike information criterion (AICc) and comparisons of model complexity (number of predictors). In addition, the assumption of non-linear relationships was tested by refitting the model with a Generalized Linear Model (GLM, with inverse link function) including main effects and all-way interactions of the predictor variables.

    Statistical modeling was performed in R [26] using the mcgv, car, AICcmodavg, and MuMIn packages [27,28,29,30]. Other calculations were performed in MATLAB [31].


    3. Results


    3.1. General flow patterns

    In general, the current magnitude in the alongshore direction was greater than the current magnitude in the across-shore direction, and the current magnitudes at the deep site were higher than the current magnitudes at the shallow site (Table 2; Figure 3). However, in the bottom layer of the shallow site (8–16 m above bottom), we observed higher magnitudes in the across-shore direction than the alongshore direction (Table 2; Figure 3). Variation in alongshore and across-shore velocities were a function of the M2 tidal constituent (12.42 h) and centered around zero, meaning that when organisms migrated upslope and downslope, they sometimes swam with the current and sometimes against it.

    Table 2. Measured current magnitudes in the alongshore and across-shore direction at shallow and deep sites.
    Mean current speed (cm s-1) Mean current magnitude (cm s-1)
    8-16 m above seafloor Alongshore Cross-shore Alongshore Cross-shore
    Shallow site 0 ± 4.46 0 ± 4.34 3.33 ± 2.97 4.75 ± 6.44
    Deep site 0 ± 13.93 0 ± 7.54 10.98 ± 8.58 5.94 ± 4.65
    Full ADCP range
    Shallow site 0 ± 5.93 0 ± 2.99 4.50 ± 3.85 2.29 ± 1.92
    Deep site 0 ± 12.20 0 ± 5.05 9.67 ± 7.39 3.97 ± 3.12
     | Show Table
    DownLoad: CSV
    Figure 3. Visualizations of alongshore and across-shore current magnitudes. Arrow size represents the relative magnitude of the currents, all of which vary tidally around zero. (A) Visualization of currents observed at 8-16 m above the bottom. (B) Visualization of currents integrated throughout the full range of the ADCP (~100 m above bottom).

    3.2. Estimated Acoustic Volume Scattering Strength (Sv): Results and observed patterns

    Estimated Sv was calculated from echo intensity data for all ADCP deployments. For the six-month deployment of the deep ADCP (D), estimated Sv was typically higher during the day (-59.4 ± 1.1 dB) than at night (-68.1 ± 1.0 dB) (Table 3). In contrast, each of the shallow deployments had higher estimated Sv during the night than during the day (Table 3). Moorings S1 and D were deployed simultaneously and high estimated Sv occurred during the day at the deep site (D) and during the night at the shallow site (S1) (Figure 4).

    Table 3. Estimated backscatter statistics for each deployment. Note: These ADCPs were not calibrated, so estimated backscatter should only be compared within deployments, not between deployments. The same ADCP was used from deployments S4-S8.
    Estimated Backscatter (Sv) Statistics Day (geomean ± geostd) Night (geomean ± geostd)
    Mean
    D -59.4 ± 1.1 -68.1 ±1.0
    S1 -64.0 ± 1.1 -52.8 ± 1.1
    S2 -75.4 ± 1.0 -64.0 ± 1.1
    S3 -69.4 ± 1.1 -58.6 ± 1.1
    S4 -65.4 ± 1.1 -58.4 ± 1.1
    S5 -68.0 ± 1.1 -58.5 ± 1.1
    S6 -67.5 ± 1.1 -54.3 ± 1.1
    S7 -62.2 ± 1.1 -54.6 ± 1.1
    S8 -68.7 ± 1.1 -58.3 ± 1.1
     | Show Table
    DownLoad: CSV
    Figure 4. Contour plots of normalized estimated Sv (scale: 0–1) for a sample 10-day period. (A) Deployment S1 (bottom depth 120 m), where estimated backscatter was generally higher at night. (B) Deployment D (bottom depth 517 m), where estimated backscatter was generally higher during the day. (C) Close-up of the bottom layer for two days highlights the diel patterns (upper panel is from deployment S1; lower panel is from deployment D).

    3.3. Arrivals and departures of the scattering community

    Arrival and departure times relative to sunrise and sunset were calculated for each event in the diel cycle: arrival at the shallow site, departure from the shallow site, arrival at the deep site, and departure from the deep site. Estimated mean arrival time at the shallow site was 38 ± 57 min after sunset. Departure from the shallow site was 14 ± 89 min before sunrise. Arrival at the deep site was 54 ± 39 min after sunrise, and departure from the deep site was 81 ± 43 min prior to sunset (Figure 5).

    Figure 5. Departure and arrival times calculated from the D (black) and S1 (grey) deployments, shown for the span of time when deep and shallow instruments were simultaneously deployed. The vertical lines span the 20% to 80% thresholds for departures and arrivals, and the horizontal black lines represent a running mean of the deep sensor results (20% and 80%.) The horizontal grey lines represent a running means of the shallow sensor results (20 and 80%). The dashed lines indicate sunrise and sunset times.

    Using the 20% threshold for departure from the deep site and the 80% threshold for arrival at the shallow site (that is, the time that the migration away from the deep site was beginning to the time that the arrival at the shallow site was nearly complete), the migration time was estimated to be 4 h 13 min ± 1 h 25 min. Using 50% departure and 50% arrival thresholds, the migration time was estimated to be 2 h 38 min ± 1 h 18 min. The deep and shallow ADCPs were located 5.25 km apart, so assuming zero net cross-shore current, these elapsed time measurements correspond to estimated mean community movement speeds of 1.99 km h-1 (55 cm s-1) using the 50% to 50% thresholds, or 1.25 km h-1(35 cm s-1) using the 20% to 80% thresholds. However, we did measure cross-shore current velocities, and 95% of cross-shore current velocity measurements in the water column were less than 10.1 cm s-1 (Table 2). Additionally, the variability in cross-shore current direction occurs at an M2 tidal frequency, so different days would have different current directions during the migration time frame. Therefore, we show that this observed migration was not driven solely by advection, but more likely by swimming organisms, as was observed by [15].

    The speeds of the shoreward and vertical signals shown in the mean-day plots are much slower (0.5–2.0 cm s-1), but these signals should only be interpreted as directional indicators. The values result from averaging both over two-minute intervals as recorded by the ADCP and over many days for the mean-day composite, during which current speeds near zero were also often observed. Therefore, the mean-day velocities can be expected to be lower than actual micronekton swimming speeds.


    3.4. Statistical modeling

    The best-specified GAM (assessed via corrected Akaike information criterion, AICc and number of predictors) included smoothed (non-linear) effects of sunset and lunar illumination. This GAM performed better than a GLM with predictors of sunset and lunar illuminations and their interaction (based on AICc; -1855 and -1869 respectively) and the GAM was chosen as the best-specified model. The GAM was significant (P < < 0.0001), explaining 39.3% of the deviance in the time of arrival. Sunset time explained 36.6% of the deviance, and the addition of lunar illumination to the model explained an additional 2.7%. The GAM model prediction is plotted over a scatterplot of sunset time versus arrival time in Figure 6.

    Figure 6. Arrival plotted as a function of sunset time. The dashed line is the 1:1 line for sunset time. The solid black line represents the predicted values of the best-fit GAM. The jagged fluctuations in the prediction line are due to lunar illumination, which was included in the model but is not on the plot axes.

    4. Discussion

    The mesopelagic boundary community in Hawaii has been previously studied using a variety of methods, including net tows [7], stereo-video cameras and standard video [9], and calibrated echosounders [9,14,15,16,19]. Many of these studies have taken place along the western coasts of Oahu and the Big Island of Hawaii. This study contributes to this growing body of literature by being the first to observe the full diel migration of the mesopelagic boundary community on the southern slope of Oahu from departure to arrival and back again. This study is also unique in that it uses long-term moorings at both shallow and deep habitats to better assess the effects of changing seasons and lunar illumination on micronekton behavior.


    4.1. Extraction of useful biological data from a physical oceanographic instrument

    Data collected with the uncalibrated ADCPs in this study were able to reveal valuable observations about the behavior of the community of scattering organisms known as the Hawaiian mesopelagic boundary community. Through our comparison of a calibrated 200 kHz echosounder and uncalibrated 300 kHz ADCP data from a 2005 deployment [15], we showed that uncalibrated 300 kHz ADCP data could be used to estimate the behavior of the mesopelagic boundary layer community via changes in estimated backscatter readings, particularly during the evening migration. Though calibrated echosounders and/or calibrated acoustic Doppler current profiles (ADCPs) would provide more refined data [14,32], valuable biological information was successfully extracted from uncalibrated ADCP data even though the deployments were designed for measuring ocean currents rather than biological signals. This analysis enabled us to maximize the usefulness of the ADCP data collected at these sites by observing patterns in micronekton behavior in addition to the intended observations of the current field.

    While many factors can affect the observed echo intensity (and therefore the estimated backscatter) including turbulence, suspended solids, bubbles, and changes in plankton and micronekton density [33], the time scales of variability observed in these deployments further supports the assertion that these changes in backscatter are due to migrating organisms. Some typical sources of variability in estimated backscatter, such as sediment suspension and turbulence, would be more likely to vary at tidal scales rather than a daily scale. There would be no mechanistic reason for sediment suspension or turbulence to vary on diel frequencies at > 100 m depth in the water column in Hawaii. Diel variability in estimated backscatter at these depths is much more likely to be related to the diel habitat shifts of mesopelagic scattering organisms.


    4.2. General observations

    The general flow and estimated backscatter patterns observed in this study are consistent with previous work. Flow on the south shore of Oahu is strongly influenced by internal tides [34], and the predominant flow direction is generally alongshore and tidally variable [20,35,36]. There was significant across-shore flow observed at the shallow moorings in the near-bottom layer, which is likely caused by bathymetric steering due to the moorings' location in a submerged channel [20].

    We observed higher estimated backscatter readings at night in shallow water and during the day in deeper water. These patterns are consistent with the known diel vertical and horizontal migration of the scattering community along the Oahu slope [7,14,32]. Additionally, the across-shore currents measured were lower than the movement speeds calculated for the mesopelagic boundary community and were sometimes flowing opposite the direction of movement, supporting previous observations that showed swimming behavior to be more important than current advection [15]. The unique discoveries from this work are summarized in the following sections.


    4.3. Environmental drivers of the timing of migration

    Changes in estimated backscatter readings over time allowed estimation of mesopelagic boundary community arrivals and departures at both shallow and deep sites, though this method was found to be more reliable during the evening migration than the morning migration. At the shallow site, the evening arrival time of the mesopelagic boundary community was correlated with sunset time and lunar illumination.

    Previous research on western Oahu showed no correlation between the onset of mesopelagic boundary community migration and sunset time [15]. In [15], moorings were deployed at 10–40 m depth, but due to the relatively short length of the deployments (~35 days), the sunset time only varied by 15 minutes. In the current study, 80–100 day deployments in multiple seasons resulted in higher variability in sunset time, spanning about 70 minutes. The greater variation in sunset time may have allowed this correlation to emerge from a variable data set where a host of other factors potentially influence micronekton behavior and estimated backscatter readings.

    Moon phase has previously been shown to have a significant positive correlation with arrival time at moorings in 10–40 m water depth [15]. In this study, we found that night sky brightness was a significant factor influencing the dusk arrival time, as measured by the percent of the moon illuminated in the sky. The deviance explained by the GAM (39.3%) indicates that other sources of variability exist. In situ surface irradiance measurements and light at depth measurements were not available in this study, but may be important explanatory factors for variability in micronekton behavior. Changes in backscatter could also be caused by physical processes such as tidal flow, eddies, and internal waves, thin layers of phytoplankton and zooplankton [16,19], or suspended sediment.

    The observations of later arrivals of the scattering community with later sunset times and brighter night skies indicate that the community is likely responding to proximate light cues in the environment. The competing interests of avoiding visual predators and foraging drive diel migrations in zooplankton and micronekton species in locations throughout the world [12,37,38,39,40,41,42,43]. In open ocean environments, the timing of diel vertical migration and depths of residence of zooplankton and micronekton scattering layers have been shown to be closely related to changing light levels, while other factors such as dissolved oxygen concentration are less important [6,43,44]. The moon phase has also been shown to influence the nighttime distribution of slope and seamount-associated micronekton [45,46]. Light changes unrelated to daily or lunar cycles also affect the behavior of pelagic micronekton. In a high-latitude environment, myctophid fishes increased their daytime residence depth and began to school (presumably as a predator avoidance mechanism) during lit summer nights [47], and pelagic crustaceans and gelatinous zooplankton were observed decreasing their depth by about 100 m during a daytime influx of turbid water [48]. Our observations of the migration of the slope-associated micronekton community on Oahu, HI align with these observations and continue to support light-based mediation of micronekton diel migrations and residence depths, both in the pelagic and in slope-associated habitats.

    The timing of the dawn downslope migration was variable and the onset of events was less defined in nature compared to that of the evening upslope migration. Migrating organisms may forage in the shallower, relatively food-rich environment until they are satiated and then begin their downslope migration well before any dawn light cues become apparent [40,49]. We also suggest that predation pressure could disperse the foraging community and drive individual organisms to depth throughout the nighttime period. This more gradual descent may have contributed to the uncertainty in identifying the dawn migration time using estimated backscatter data.


    4.4. Migration and swimming speed

    Micronekton on the Oahu slope have been previously shown to migrate between 5.5 and 11 km round-trip each day as part of their diel migration [9]. Assuming that the horizontal migrators observed at the deep site in this study were also detected at the shallow site, the micronekton community was undertaking diel migrations of at least 10.6 km (twice the distance between the deep and shallow mooring sites) and up to ~14 km if they continued to shallower depths nearer to shore. Estimated community movement speeds from this work ranged from 1.25–1.99 km h-1 (35–55 cm s-1). Previously observed swimming speeds of myctophids ranged from 0.4–2.8 km h-1 (11–78 cm s-1) [7,50], and previous studies of the mesopelagic boundary community on Oahu reported horizontal swimming speeds of 0.9 km h-1 (25 cm s-1) [32] to 1.8 km h-1 (50 cm s-1) [15]. Therefore, the community movement speed estimated here does align with previously observed myctophid swimming speeds.


    4.5. Ecological importance and vulnerability to a marine industrial development

    The mesopelagic boundary community is an important component of the Hawaiian slope ecosystem, as members of the community consume large quantities of copepods and euphausiids in Hawaii's near-shore waters [51]. Mesopelagic micronekton are also a primary food source for higher trophic level predators such as spinner dolphins Stenella longirostris [10] and large pelagic fishes such as bigeye tuna Thunnus obesus and swordfish Xiphias gladius [52,53].

    New coastal development may soon impact the Hawaiian mesopelagic boundary community on the southern slope of Oahu. A seawater air conditioning (SWAC) system is proposed for the area of this study. The SWAC system will pipe deep, cold ocean water from 500 m depth and use the cold water in a heat-exchange system to cool a freshwater loop, which will then be piped to the downtown business district to use as air conditioning coolant [54]. The warmed deep seawater will then be discharged via a diffuser spanning 100–140 m depth along the slope. The discharged water will have a higher concentration of nutrients and lower oxygen concentration compared to the ambient water, among other differences [20,54], which may result in local increases in phytoplankton production. Higher productivity and gradients in temperature, density, salinity, and oxygen associated with the plume may cause attraction or avoidance of mesopelagic fishes [55].

    Light penetration to depth is likely to be very important in driving mesopelagic boundary community migrations, and water column clarity may be affected by eutrophication from industrial operations like the proposed seawater air conditioning plant or future development of ocean thermal energy conversion, a similar technology based on artificial upwelling [56]. Changes in water clarity and the rate of light extinction with depth have the potential to influence the diel migratory patterns of both pelagic and slope-associated organisms, which may in turn lead to ecosystem impacts including changes in the transfer of nutrients between shallow and deep habitats and in the foraging patterns of predators. Additionally, the discharge will be into the pycnocline, which may cause spreading of the plume over a larger horizontal area and expand the geographic scale of any impacts [20].

    Entrainment of organisms is another potential impact of the SWAC system on the mesopelagic boundary community. The intake site of the SWAC pipe will be located near the benthos at 500 m depth, where we observed both vertical and horizontal diel migrators residing during the day. The uncalibrated estimated backscatter data reported in this study do not allow estimates of the density of organisms, but high densities of micronekton in this environment are certainly possible. Submersible dives on other slope environments have revealed persistent and very dense aggregations of micronekton between 0–20 m from the benthos [57], and maximum observed densities of the mesopelagic boundary community in Hawaiian waters can exceed 1000 animals m-3, although mean densities are much lower [14]. The intake velocity of the deep pipe will be 5.47 km h-1(152 cm s-1). Burst swim speeds of Hawaiian myctophid fishes are not specifically known. If the burst speeds are lower than the flow rate, they will be at high risk of entrainment, and even if burst swim speeds exceed the intake flow rate some members of the community will be entrained. Mesopelagic shrimp have burst swim speeds of only 0.72 km hr-1 (20 cm s-1) [58]. Mortality rates associated with entrainment are not known, but it is likely that at least some organisms entrained in the flow will not survive the rapid temperature and pressure changes as they move through the system [59]. This will result in daytime deposition of deceased micronekton around the area of the diffuser system. Especially considering the potential high densities of micronekton at the SWAC intake site, entrainment and deposition could have a significant effect on the local benthic food web and be an additional food source for scavenging organisms.


    5. Conclusions

    The mesopelagic boundary community in Hawaii undertakes diel horizontal migrations that are driven by competing interests of foraging and predator avoidance. This study contributes to this growing body of literature by observing the full diel migration of the mesopelagic boundary community, from departure to arrival in both directions, on the southern slope of Oahu and by using long-term moorings at both shallow and deep habitats to better assess the effects of changing seasons and lunar illumination on micronekton behavior. The amount of ambient light plays a significant role in the timing of the migration, with later sunsets and brighter night skies corresponding to later arrival times of the mesopelagic boundary community in the food-rich shallow nighttime habitat. Even using uncalibrated estimated backscatter data, these patterns were observable and statistically significant. In light of increasing coastal development worldwide, it is important to understand how the slope habitat is being used by important intermediate trophic level organisms such as this migrating community. The enhanced understanding of the behavior of boundary layer organisms at this location will enable better assessment of anthropogenic impacts on this community and on the ecosystem as a whole.


    Acknowledgments

    Funding for this work was provided by the Office of Naval Research (Award number N0001414100054–Asia-Pacific Research Initiative for Sustainable Energy Systems (APRISES)), the State of Hawaii, the Joint Institute for Marine and Atmospheric Research, and the Office of the Dean of the School of Ocean and Earth Science and Technology. We would like to thank the following people for assistance with fieldwork and instrumentation: Douglas Luther, Mark Merrifield, Carly Quisenberry, Chris Kontoes, Gordon Walker, Sarah Searson, Conor Jerolman, and Paula Moehlencamp. We thank the University of Hawaii Marine Center, Parker Marine Corp., and Sea Engineering Inc. for vessel support.


    Conflict of interest

    Both authors declare no conflicts of interest in this paper.




    [1] J. Bear, Dynamics of Fluids in Porous Media, Elsevier, New York, 1972.
    [2] R. H. S. Winterton, Heat transfer, Oxford University Press, Oxford, 1997.
    [3] I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Probl., 11 (1995), L11–L16. https://doi.org/10.1088/0266-5611/11/4/001 doi: 10.1088/0266-5611/11/4/001
    [4] A. Hasanov, Simultaneous determination of source terms in a linear parabolic problem from the final overdetermination: Weak solution approach, J. Math. Anal. Appl., 330 (2007), 766–779. https://doi.org/10.1016/j.jmaa.2006.08.018 doi: 10.1016/j.jmaa.2006.08.018
    [5] Y. C. Hon, T. Wei, A Meshless Computational Method for Solving Inverse Heat Conduction Problem, Int. Ser. Adv. Bound. Elem., 13 (2002), 135–144.
    [6] Y. C. Hon, T. Wei, A fundamental solution method for inverse heat conduction problem, Eng. Anal. Bound. Elem., 28 (2004), 489–495. https://doi.org/10.1016/S0955-7997(03)00102-4 doi: 10.1016/S0955-7997(03)00102-4
    [7] Y. C. Hon, T. Wei, The method of fundamental solutions for solving multidimensional inverse heat conduction problems, Comput. Model. Eng. Sci., 7 (2005), 119–132.
    [8] V. Isakov, Inverse parabolic problems with the final overdetermination, Comun. Pure Appl. Math., 44 (1991), 185–209. https://doi.org/10.1002/cpa.3160440203 doi: 10.1002/cpa.3160440203
    [9] G. Nakamura, S. Saitoh, A. Syarif, Representations of initial heat distributions by means of their heat distributions as functions of time, Inverse Probl., 5 (1999), 1255–1261. https://doi.org/10.1088/0266-5611/15/5/310 doi: 10.1088/0266-5611/15/5/310
    [10] A. Shidfara, G. R. Karamalib, J. Damirchia, An inverse heat conduction problem with a nonlinear source term, Nonlinear Anal. Theory Methods Appl., 65 (2006), 615–621. https://doi.org/10.1016/j.na.2005.09.030 doi: 10.1016/j.na.2005.09.030
    [11] M. Yamamoto, J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Probl., 17 (2001), 1181–1202. https://doi.org/10.1088/0266-5611/17/4/340 doi: 10.1088/0266-5611/17/4/340
    [12] L. Ling, M. Yamamoto, Y. C. Hon, T. Takeuchi, Identification of source locations in two-dimensional heat equations, Inverse Probl., 22 (2006), 1289–1305. https://doi.org/10.1088/0266-5611/22/4/011 doi: 10.1088/0266-5611/22/4/011
    [13] Y. Deng, Z. Liu, Iteration methods on sideways parabolic equations, Inverse Probl., 25 (2009), 095004. https://doi.org/10.1088/0266-5611/25/9/095004 doi: 10.1088/0266-5611/25/9/095004
    [14] Y. Deng, Z. Liu, New fast iteration for determining surface temperature and heat flux of general sideways parabolic equation, Nonlinear Anal. Real World Appl., 12 (2011), 156–166. https://doi.org/10.1016/j.nonrwa.2010.06.005 doi: 10.1016/j.nonrwa.2010.06.005
    [15] C. L. Fu, Simplified Tikhonov and Fourier regularization methods on a general sideways parabolic equation, J. Comput. Appl. Math., 167 (2004), 449–463. https://doi.org/10.1016/j.cam.2003.10.011 doi: 10.1016/j.cam.2003.10.011
    [16] D. N. Ha´o, H-J. Reinhardt, On a sideways parabolic equation, Inverse Probl., 13 (1997), 297–309. https://doi.org/10.1088/0266-5611/13/2/007 doi: 10.1088/0266-5611/13/2/007
    [17] D. N. Ha´o, H-J. Reinhardt, A. Schneider, Numerical solution to a sideways parabolic equation, Int. J. Numer. Methods Eng., 50 (2001), 1253–1267. https://doi.org/10.1002/1097-0207(20010220)50:5<1253::AID-NME81>3.0.CO;2-6 doi: 10.1002/1097-0207(20010220)50:5<1253::AID-NME81>3.0.CO;2-6
    [18] H. Lobo, C. Cohen, Measurement of thermal conductivity of polymer melts by the line-source method, Polymer Engng. Sci., 30 (1990), 65–70. https://doi.org/10.1002/pen.760300202 doi: 10.1002/pen.760300202
    [19] R. Kato, A. Maesono, I. Hatta, Thermal diffusivity measurement af a thin film in the direction across the film by AC calorimetric method, Japan. J. Appl. Phys., 32 (1993), 6353–6355. https://doi.org/10.1143/JJAP.32.3653 doi: 10.1143/JJAP.32.3653
    [20] C. H. Huang, M. N. O¨zisik, A direct integration approach for simultaneously estimating temperature dependent thermal conductivity and heat capacity, Numer. Heat Transfer A, 20 (1991), 95–1l0. https://doi.org/10.1080/10407789108944811 doi: 10.1080/10407789108944811
    [21] H. Ammari, H. Kang, Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, Vol. 162, Springer-Verlag, New York, 2007.
    [22] H. Ammari, Y. Deng, H. Kang, H. Lee, Reconstruction of Inhomogeneous Conductivities via the concept of Generalized Polarization Tensors, Ann. I. H. Poincare-AN, 31 (2014), 877–897. https://doi.org/10.1016/j.anihpc.2013.07.008 doi: 10.1016/j.anihpc.2013.07.008
    [23] H. Ammari, Y. Deng, P. Millien, Surface plasmon resonance of nanoparticles and applications in imaging, Arch. Ration. Mech. Anal., 220 (2016), 109–153. https://doi.org/10.1007/s00205-015-0928-0 doi: 10.1007/s00205-015-0928-0
    [24] Y. Deng, H. Li, H. Liu, Analysis of surface polariton resonance for nanoparticles in elastic system, SIAM J. Math. Anal., 52 (2020), 1786–1805. https://doi.org/10.1137/18M1181067 doi: 10.1137/18M1181067
    [25] Y. Deng, J. Li, H. Liu, On identifying magnetized anomalies using geomagnetic monitoring, Arch. Ration. Mech. Anal., 231 (2019), 153–187. https://doi.org/10.1007/s00205-018-1276-7 doi: 10.1007/s00205-018-1276-7
    [26] Y. Deng, J. Li, H. Liu, On identifying magnetized anomalies using geomagnetic monitoring within a magnetohydrodynamic model, Arch. Ration. Mech. Anal., 235 (2020), 691–721. https://doi.org/10.1007/s00205-019-01429-x doi: 10.1007/s00205-019-01429-x
    [27] Y. Deng, H. Liu, X. Liu, Recovery of an embedded obstacle and the surrounding medium for Maxwell's system, J. Differ. Equ., 267 (2019), 2192–2209. https://doi.org/10.1016/j.jde.2019.03.009 doi: 10.1016/j.jde.2019.03.009
    [28] Y. Deng, H. Liu, G. Uhlmann, On an inverse boundary problem arising in brain imaging, J. Differ. Equ., 267 (2019), 2471–2502. https://doi.org/10.1016/j.jde.2019.03.019 doi: 10.1016/j.jde.2019.03.019
    [29] Y. Deng, H. Liu, X. Wang, W. Wu, Geometrical and topological properties of transmission resonance and artificial mirage, SIAM J. Appl. Math., 82 (2022), 1–24. https://doi.org/10.1137/21M1413547 doi: 10.1137/21M1413547
    [30] C. A. Brebbia, W. L. Wendland, G. Kuhn, Boundary elements IX.: Fluid flow and potential applications, vol.3, Computational Mechanics, Stuttgart, 1987,213–229.
    [31] L. Evans, Partial Differential Equations, Providence: American Mathematical Society, 1998.
    [32] A.Friedman, Partial diffrential equations of parabolic type, Prentice-Hall, Englewood-Cliff, NJ, 1964.
    [33] A.K. Louis, Inverse und schlecht gestellte Probleme. Stuttgart, Teubner, 1989.
    [34] V. A. Morozov, On the solution of functional equations by the method of regularization, Soviet Math. Dokl., 7 (1966), 414–417.
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