To associate the particle distribution with dissolved gas content, we latch a particle mass Γζ to each seeded particle, which explicitly corresponds to the number of moles each particle represents (this mass has no influence on the particle buoyancy). Each particle is thus interpreted as a virtual single-point representation of some local spatial distribution of Γζ moles of dissolved gas molecules.

The initial mass, i.e. mass at release, of an arbitrary particle ζ is scaled such that the total released particle mass from all modeled seeps combined at timestep n approximates the total number of moles of gas dissolved in the water column during the time interval Δt centered on tn. In practice, we distribute the integrated sum of modeled (using the gas phase model) injected gas molecules from tn-Δt/2 to tn+Δt/2 evenly over the seeded particles. The mass of particle Γζ seeded at time-step n is then obtained by 3Γζ[n]=Δt∑p=0PΥp[n]S[n] where Υ1[n],Υ2[n],…,ΥP[n] [mol s⁻¹] are total injected dissolved gas from all P modeled seeps. Approximation to the modeled dissolved gas release profiles at each modeled seep is achieved by seeding different amount of particles at different depths. Particle masses are then subsequently individually adjusted at each time-step to simulate processes affecting gas content. Each particle thus has a successively constructed mass time-series, where the current mass Γζ[n] is determined by the previous mass Γζ[n-1] and selected mass modification functions.

Our framework must be able to model an extensive 3-dimensional domain (e.g. larger ocean regions), making computational complexity a challenge. Both computation time and estimation quality increase with the number of active particles present in the domain. This makes it crucial to be able to strike a decent compromise between the two, which typically involves removal of particles that have been present in the domain for a certain number of time-steps. Total particle count Λ[n] in the domain can be expressed as 4Λ[n]=Λ[n-1]+S[n]-L[n]-℘[n], where L[n] is the number of particles leaving the modelled geographical domain, and ℘[n] the number of removed particles. A constant particle count is obtained when S[n]∼L[n]+℘[n]. Since ℘[n] represents non-physical loss of gas, the model simulation would ideally run with a spin-up time that ensures S[n]∼L[n]. Unfortunately, this typically results in unreasonable computation times and/or spin-up periods, making particle removal necessary. To limit errors caused by removed particles, we apply a function that redistributes mass from all removed particles to nearby non-removed particles. The redistribution is weighted according to the inverse distance from the removed particle within a user defined distance limit dmax⁡, giving a non-removed particle τ an added mass of 5γτ=Γθ||ητ-ηθ||2-1∑ζ∈T||ηζ-ηθ||2-1 from the removed particle θ. Here, τ∈T and T is the set of non-removed particles with indices ζ satisfying ‖ηζ-ηθ‖2≤dmax⁡, and ‖⋅‖2 denotes the Euclidean norm. This solution changes the problem of non-physical loss of dissolved gas to one of non-physical redistribution. This can affect model results by shifting particle mass towards the seed location, since the density of particles are in general higher closer to the release point. However, we consider this artifact less problematic than mass simply disappearing.

To improve computational times, we have implemented a grid-projected estimator . This involves obtaining a preliminary density ϕ̃[i,j] using the histogram estimator via Eq. (), i.e. calculate the accumulated particle mass of all particles within each grid cell. All mass then belongs to a discrete grid, where any difference in position r between two locations of interest [i,j] and [i0,j0] can be expressed as (i-i0,j-j0)Δλ.

Furthermore, we pre-compute a set of normalized kernels with fixed, discrete bandwidths given by: 7h[ω]=ωΔλ3,whereω=0,1,2,…Ω. Each initial non-discrete bandwidth estimate h′ from the data-driven bandwidth algorithm is then mapped to the nearest candidate in h[ω]. Kernel support is set to ±ωΔλ beyond which the kernel contribution is set to zero. Leaked kernel mass is added back across the kernel domain using the Kernel function.

These simplifications make complexity scale with particle-containing cells instead of particles. It also allows for fast vectorized operations which drastically reduce computation time while giving negligible errors for large grids .

The conventional Silverman's rule of thumb selects the optimal bandwidth h that minimizes the integrated mean square error under the assumption of Gaussian distributed data with variance σ2. For a d-dimensional Gaussian kernel, one obtains 8h=4d+21d+4N-1d+4σ, where N is the number of data samples, i.e., the number of particles. For d=2, the expression simplifies to 9h=N-16σ. Here we modify Silverman's rule to yield reasonable estimates for our multi-modal, correlated, non-homogeneous, weighted data set by: (i) adapting h locally for a square shaped horizontal “adaptation” grid of size P×P surrounding each particle containing grid cell where we assume near normal, unimodal distribution, (ii) estimating the effective (uncorrelated) sample size Neff and (iii) implementing bias corrections to a σ-estimator for weighted data-sets. The size P is determined by an integral length scale estimate (as outlined below) of the entire I×J 2-D grid. For an arbitrary cell [i,j], the adaptation grid is defined by letting l and m be discrete indices on the grid such that 1≤l, m≤P with step size Δλ in both directions. The number of particles contained in the local grid (prior to grid projection) is denoted as Ng, and the pre-computed histogram density ϕ̃ serves as the underlying data for obtaining estimates of the local h. We will now describe the procedure of obtaining estimates of Neff and σ to get the local h for an arbitrary adaptation grid.

Spatial correlations present in environmental data decrease the effective degrees of freedom in the sample set and we must therefore estimate and use the effective sample size Neff to obtain a reasonable h for the local grid (e.g. Larsen et al., 2002). To estimate the effective number of spatially uncorrelated samples, a measure of correlation length is employed. We use the so-called integral length scale known from turbulence theory and statistical physics, see e.g., , , and as our objective measure of the correlation length. The correlation length Lc in terms of the integral length scale is formally defined by 10Lc=∫0∞|R(ϱ)|dϱR(0), where R(ϱ)=Eϕ(x)ϕ(x+ϱ) is the spatial autocorrelation function (ACF) of a continuous univariate spatial random process ϕ(x), ϱ is a spatial lag coordinate, and E⋅ is the statistical expectation operator.

We now proceed by defining a local integral length scale Lc for the binned adaptation window. First, we estimate the local one-dimensional ACF of ϕ̃ along each row l and column m of the square grid by using the standard unbiased ACF estimator e.g. 11R^lrow[λ]=1P-λ∑m=1P-λϕ̃l,mϕ̃l,m+λforl=1,2,…,P12R^mcol[λ]=1P-λ∑l=1P-λϕ̃l,mϕ̃l+λ,mform=1,2,…,P where ϕ̃l,m is the binned particle density in grid cell [l,m] and λ=0,1,…,P-1 is the discrete horizontal spatial lag index. Assuming local spatial homogeneity, we then arithmetically average the ACF estimates over all P rows and columns, respectively, to yield two one-dimensional ACF estimates as 13R^row[λ]=1P∑l=1PR^lrow[λ]andR^col[λ]=1P∑m=1PR^mcol[λ]. We now assume local spatial isotropy and let the arithmetic average of the two perpendicular ACFs serve as a representative single ACF for the adaptation window 14R^[λ]=12R^row[λ]+R^col[λ]. Using the estimated ACF, we can finally estimate the local one-dimensional integral length scale L^c for the adaptation window by discretizing Eq. () as 15L^c=∑λ=0P-1|R^[λ]|ΔλR^[0]. We now express the correlation length in terms of the associated number of samples as Nc=L^c/Δλ. It is easy to show that 1≤Nc≤P. We then define the number of effectively uncorrelated particles Neff as 16Neff=NgNc, and it directly follows that Ng/P≤Neff≤Ng. The interpretation of Neff is straightforward: if all particles are spatially uncorrelated, then Neff=Ng, and if all particles are fully correlated (e.g., if they are all trapped in a coherent structure), then Neff attains its lower limit Neff=Ng/P.

To obtain an estimate σ2^ of the variance σ2 in the two-dimensional binned data, we need to account for the loss of degrees of freedom due to shortening of the residual vector (Bessel's correction) and weighting as well as increased variance due to the binning process itself. The estimate of variance for the binned particle density ϕ̃l,m, can then be expressed as 17σ2^=∑l,mϕ̃l,m‖rl,m-μ‖22∑l,mϕ̃l,m11-B+σb2, where rl,m denotes the grid cell center point position vectors, ∑l,m≡∑l=1P∑m=1P, and 18μ=∑l,mϕ̃l,mrl,m∑l,mϕ̃l,m is the weighted mean position vector, and 11-B, where 19B=∑l,mϕ̃l,m2∑l,mϕ̃l,m2, is a bias correction term that accounts for Bessel's correction and the reduced degrees of freedom due to uneven sample weights pp. 86–88. The variance increase due to the binning process Sheppard's correction, see e.g. is included through the correction term σb2=Δλ2/12.

The final local bandwidth estimate (for each particle-containing grid cell) then follows from Eq. (): 20h^=Neff-16σ^.

We establish a boundary solution for the density estimator by interpolating bathymetry data onto the model grid across all predefined depth layers, creating a matrix of “permissible” and “impermissible” cells for gas. The boundary control is imposed at the kernel estimation stage before summation, by directly modifying kernels whose support contains impermissible cells (see Fig. ). While being computationally intensive, this greatly simplifies the boundary control and entirely omits the difficulties of finding a reliable boundary solution that handles the complex bathymetry and physical processes appropriately.

Impermissible cells are treated as impenetrable obstacles, reflecting that dissolved gas cannot cross land or shallow bathymetric boundaries. Any density within, or “blocked” by impermissible cells, is considered misplaced. A cell is defined as blocked if it lacks a clear line of sight to the kernel center. We determine line of sight using Bresenham's line algorithm . This is an efficient incremental algorithm relying solely on integer arithmetics that identify grid cells located between an origin cell (x0,y0) and a target cell (x1,y1) (Fig. ). The algorithm is implemented on a normalized grid with unit cell lengths and initialized by first defining the direction, or step coefficients sx and sy in the x and y directions, respectively: 21sx=1,if x0<x1-1,if x0>x1,sy=1,if y0<y1,-1,if y0>y1., and an error term ε=dx-dy, where dx=|x1-x0| and dy=|y1-y0|. The algorithm then iteratively updates (x0,y0), using ε to determine whether to step in x or y direction (sx, sy are not updated) via the following criteria: 222ε>-dy:x0⇒x0+sxand2ε<dx:y0⇒y0+sy. At each time step, (x0,y0) is added to the list of grid cells, thereby iteratively forming the line of sight to the target cell. Density in blocked or impermissible cells are redistributed to permissible cells according to the kernel function.

Temperature and salinity data were extracted from a Conductivity Temperature Depth (CTD) cast performed in 20 May 2018 (Figs.  and a). Seabed gas flux for each seep was estimated using single beam echosounder data (Simrad EK-60 scientific SBE splitbeam echosounder) obtained between 20 and 22 May 2018 and are presented in . All other input parameters had to be inferred as outlined in the following paragraphs since we lack observations.

M2PG1 requires an initial bubble size distribution, and we used the polynomial fit to visual observations of bubbles as presented in . Note that since M2PG1 takes into account the non-spherical shape of bubbles, the bubble size distribution is given using the effective radius rE=(a2b)1/3, where a and b are the major and minor axis of the spheroid, respectively (Fig. b). We used the bubble rising speed model from using their recommendation for bubble contamination, and the linear flatness parametrization from (see Fig. b and c). We describe and discuss the bubble rising speed model and deformation parametrization selection in Appendix .

The horizontal domain size was determined using Appendix and an assumed barotropic current of U‾=0.1 m s⁻¹, horizontal diffusivity Dh=0.01 m² s⁻¹, rising speed 〈w〉=0.25 m s⁻¹ and a H=200 m deep water column. We estimated σw using the bubble rise spectrum (Fig. ) to σw=0.025 m s⁻¹. This resulted in an estimated model area of AM∼88 m² and grid cell side-lengths 9.4 m. Vertical and temporal resolution does not affect grid cell concentration but must obey the Courant-Friedrichs-Lewy numerical stability condition. Here we use a grid cell height of 1 m to obtain AM⟂=9.4 m² and a time-step of 0.0625 s.

We assumed a constant vertical mixing coefficient of Dv=0.001 m² s⁻¹ and background dissolved gas concentrations were set to the default values from at 6.8×10-4 mol L⁻¹, 2.5×10-4 mol L⁻¹, 2.5×10-5 mol L⁻¹, 2×10-9 mol L⁻¹ and 1.5×10-8 mol L⁻¹ for N₂, O₂, CO₂, CH₄, and Ar, respectively.

We provide an overview of microbial oxidation rate coefficient (kox) observations presented in literature and their associated uncertainty in Appendix and Fig. . Here, we use the simple average kox=3.6×10-7 s⁻¹ of the full compiled dataset in Table which include cold seep environments, hydrothermal vents, and human-made releases.

Most of the CH₄ gas is dissolved in the water column, with concentration appearing to decrease near exponentially towards the sea surface (Fig. a). Hourly seabed gas flow rate was ∼97 mol of which 93 % dissolved below 100 m depth and only ∼0.28 % reaching the atmosphere. Integrated atmospheric release from free gas over the 1-month period was 183.1 mol. Free CH₄ gas content closely follows the total free gas content throughout the water column (Fig. a) and loss of total free gas volume (bubble shrinkage, collapse, and dissolution) dominates over other gases replacing CH₄ in bubbles. The resulting change in dissolved gas profiles for the four other gases (N₂, O₂, CO₂ and Ar) due to bubble transit, i.e. the transport of gas molecules by entering bubbles, rising, and subsequently dissolving at shallower depths, was therefore negligible (never exceeding 0.1 % of background values). Atmospheric flux from the 45 seeps varied considerably from <10-7 to ∼10-5 [mol s⁻¹] (Fig. c), mostly due to large variations in seabed fluxes . Dissolved gas injection rates, which are needed as input in the particle dispersion modeling step, were calculated using the modeled (by M2PG1) dissolved gas profiles (not shown) and Eq. () and are shown for the 45 seeps in Fig. a.

The averaged distribution pattern of CH₄ throughout the study period is strongly affected by generally northeastward currents that transports gas along the coast, following the shelf and shelf break. The gas enters the more open fjord systems, and to a lesser degree inner fjords. North of 70° the CH₄ plume disperses more, branching into a northward plume leaving the coast and a coastal plume that keeps following the coastline. The concentration anomaly is generally small, around 2–4 orders of magnitude lower than typical oceanic CH₄ background concentration values (∼3×10-6 mol m⁻³, Fig. ), due to the weak seabed release.

The 3D concentration field is very dynamic due to the energetic regional current regimes, and shows variability on ranging from tidal (∼ 12 h) to fortnightly periods. A visual representation of the temporal variability of the top 9 layers in the water column (down to 200 m depth) is shown in Video S1.

Most of the CH₄ is displaced upward relatively quickly from the trough and pushed on top of the shelf break (Fig. ). Vertical distribution of CH₄ is therefore characterized by a quick (a few days) shift from CH₄ being mainly located close to the seafloor at the release site (∼ 150–200 m depth) to shallower depths (Fig. ). A thorough analysis of mechanisms causing the rapid shift in location of CH₄ is outside the scope of this study, however, upwelling within troughs and along the shelf break is well documented in this region e.g..

The time-integrated 2D diffusive atmospheric CH₄ release over the study period is shown in Fig. and the complete time series can be found in Video S2. Within the model domain, most of the CH₄ remains in the water column or is consumed by microbes, with a total of ∼ 0.76 % (∼ 528 mol) being exchanged diffusively with the atmosphere. This diffusive release is roughly three times greater than the local free gas release (0.27 %) and does not account for any diffusive release occurring outside of the model domain. The diffusive flux extends across a broad area spanning several hundred kilometers and shows pronounced temporal variability that is strongly correlated with wind speed (Fig. a), although no clear effect of surface water depletion is observed after storm events. Microbial CH₄ consumption exceeds atmospheric flux by a factor one to two orders of magnitude, emphasizing its crucial role in regulating dissolved CH₄ levels (Fig. b). Loss from the domain due to particles leaving the model domain is also substantial (∼ 5 to ∼ 50 times the atmospheric loss) and shows clear tidal excursions patterns with periodic variability (Fig. c). Non-physical methane loss due to removed isolated particles (i.e. too far away to be re-distribution) is on the same order of magnitude as loss to the atmosphere via diffusive release (Fig. d).

We analyzed the vertical redistribution and partitioning of CH₄ among available sinks within the model domain over a four week period (the particle lifetime). This analysis is important not only for evaluating CH₄ molecules potential to reach the atmosphere, but also in cases impacts on water column and/or seafloor ecosystems are of interest. Excluding removed particles and particles leaving the model domain, the accumulated fractional water column CH₄ loss due to atmospheric exchange shows an exponential increase the first couple of days, with a subsequent near linear slope until the end of the four week period. The initial rapid gradient increase in atmospheric loss fraction corresponds to a vertical redistribution of dissolved CH₄, where the concentration maximum shifts from ∼ 200 to ∼ 10 m depth (Fig. ). After four weeks, around 0.7 % of dissolved CH₄ molecules had been transferred to the atmosphere (Fig. a and b). Microbial oxidation dominates over both atmospheric diffusive and free-gas fluxes transforming 65 %, while around 34 % of the CH₄ remains in the water column at the end of the particle lifetime.

Diffusive exchange of CH₄ exceeds the local free gas release and spreads over a large ocean region, making it almost impossible to detect and quantify using conventional measuring instrument. This also poses a challenge for atmospheric inversion models, since these are better at detecting point sources rather than weak releases over large regions . These limitations highlight the uncertainty in quantifying the impact of seabed seepage on the atmospheric CH₄ budget, particular when considering the potential increased seepage in recent decades due to e.g. thawing marine permafrost, hydrate dissociation e.g., and anthropogenic disturbances of the seafloor (e.g. drilling).

Although here, the estimated total atmospheric flux is small, the impact of more extensive seepage such as e.g. could be significant and at the same time difficult to observe and/or trace.

Even though the dissolved gas spreads out over cold water coral reef areas, the CO₂ generated by microbial oxidation is likely too small to have any measurable effect on the local ocean environment and cold water corals. This primarily reflects the weak seabed fluxes. For more intense and/or localized seepage, e.g. a leaking gas well, this might not be the case. It is also worth noting that the influence of seabed gas seepage on cold-water coral ecosystems remains a sparsely explored field of research.

An additional major caveat, both regarding atmospheric fluxes and potential impact on the ocean environment, is the uncertainty caused by the microbial oxidation rate coefficient kox assumption in methane oxidation rates (MOx), which vary by several orders of magnitude and correspond to half-lives for dissolved methane (or methane turnover) ranging from 5 d to nearly 2 years (Fig. , Table ). To examine sensitivity to kox, we conducted a limited coefficient-sweep experiment, rerunning the framework using the lowest and highest reported “cold seep” rate coefficients from Table : 0.02×10-6 s⁻¹ (low) and 0.98×10-6 s⁻¹ (high) , as well as two intermediate values.

The low and high rate coefficient runs resulted in a ∼ 34 % increase (705 mol) and ∼ 41 % decrease (309 mol), respectively, in atmospheric emissions during over the model domain and particle lifetime. The impact on the final fate of dissolved CH₄ molecules were also considerable (Table ). After 4 weeks, ∼ 5.5 % of dissolved CH₄ were consumed and 1.16 % released to the atmosphere in the low rate coefficient run, compared to ∼ 91 % consumed and 0.36 % released in the high rate coefficient run. These results highlights the importance of selecting a suitable MOx rate coefficient and illustrates the huge span of results one can obtain when coefficients in the modeling framework are poorly constrained. It is also important to note that since MOx is a biologically mediated process, it can vary substantially on relatively small spatiotemporal scales depending on a wide range of factors, including CH₄ concentration, water temperature, salinity , nutrient availability , and the presence of trace elements . Thus, assuming a constant rate coefficient is in itself a potentially problematic simplification, since it will most likely vary considerably across the model domain. One would, for instance, expect MOx to decrease with the distance from seep area due to rapid dilution of methane and varying environmental stress for the CH₄ oxidizing microbes. To reduce the uncertainty concerning MOx rates, future studies must therefore not only constrain rate coefficients, but also improve our ability to model MOx dynamics within modeling frameworks. Including more complex MOx parametrizations is possible in our framework since we allow explicit modeling of higher order fields at each time-step and location in the modeling domain.

Another important source for uncertainty in the modeled fate of CH₄ arise from the atmospheric bulk model. The atmospheric gas transfer coefficient function (Eq. ) was derived based on, and designed for global CO₂ estimates and has an estimated uncertainty of ∼20 %, even for its intended use. We must expect considerably higher uncertainties due to our application in a local coastal region (as opposed to an ocean basin), where wind speeds may exceed the validity range, and since the specific coefficient Cκ has been determined solely for CO₂.

Uncertainties in the eddy diffusivity, vertical transport and distribution are also expected to be large. The choice of grid cell thickness can also modify the end result. If the grid cells are too thin, and temporal resolution too coarse, there is a risk of depletion of the surface layer between the model output time-steps. On the other hand, if the grid cells are too thick, one would incorporate CH₄ from depths where exchange with the atmosphere is unrealistic, thereby violating the assumptions of the atmospheric exchange bulk model. One can evaluate whether the surface thickness is sufficiently thick by comparing typical values for Eq. () with the typical mass of surface layer particles and ensure that the atmospheric loss is considerably smaller than the surface layer gas content (i.e. that γα[n]≪Γτ[n] always).