Bioluminescence potential modeling during polar night in the Arctic: impact of advection versus local sources

Abstract

Bioluminescence (BL) potential observations registered high BL potential emissions during the polar night of January 2012 at the mouth of a fjord Rijpfjorden (northern Svalbard, Norway). Notably, observations of BL potential at this location were significantly higher in the upper 50 m than observed BL potential at offshore stations located on the shelf-slope areas and in the deeper water off northern Svalbard. In the present paper, we address questions as to why the values of BL potential in the fjord are higher than at offshore stations and what the role of advection is in the observed elevation of BL potential values in the fjord in comparison to offshore stations. We utilized the ensemble approach when changes in BL potential are modeled with the advection-diffusion-source (ADS) model, and the focus is on modeling and predictions of a large ensemble of averaged values of BL potential over a specific domain of interest at the fjord mouth. Results of the modeling have demonstrated increases in ensemble members with BL potential values larger than or equal to the specific threshold in comparison to corresponding ensemble members in the initial distribution. Even when we introduced mortality rates in our simulations (values from 0.1 to 0.3 day⁻¹), increases in ensemble members with higher than the threshold value were more than 3.7 times larger in comparison to ensemble members in the initial distribution. The interpretation of our results is that during the polar night, the fjord represents an area where bioluminescent organisms from offshore aggregate through advection and mixing.

Appendix

Consider the following variable λsatisfying the adjoint to (1) equation:

$$ -\frac{\partial \lambda }{\partial t}=U\nabla \lambda +\nabla \left(k\nabla \lambda \right) $$

(15)

with initial conditions at t = T:

$$ \lambda {\left|{}_{t=T}=1\kern0.24em \mathrm{in}\kern0.37em \Omega, \mathrm{and}\;\lambda \right|}_{t=T}=0\kern0.24em \mathrm{outside}\ \mathrm{of}\;\Omega\ \mathrm{in}\ \mathrm{D} $$

(16)

and boundary conditions:

$$ \lambda \left(s,t\right)=0\;\mathrm{and}\;k\left(s,t\right)=0\;\mathrm{for}\;s\in {S}_d-\mathrm{boundary}\ \mathrm{of}\ \mathrm{the}\ \mathrm{D}\ \mathrm{domain} $$

(17)

Boundary S_d includes all boundaries of D: lateral boundaries, surface, and bottom.

We multiply (1) by λ and subtract (15) multiplied by C and integrate over domain D and time [t₀, T]:

$$ \underset{t_0}{\overset{T}{\int }}\underset{D}{\int}\left(\lambda \frac{\partial C}{\partial t}+C\frac{\partial \lambda }{\partial t}\right) d\tau dt=\underset{t_0}{\overset{T}{\int }}\underset{D}{\int}\left(- U\lambda \nabla C- U C\nabla \lambda \right) d\tau dt+\underset{t_0}{\overset{T}{\int }}\underset{D}{\int}\nabla \left(k\nabla C\right)\lambda -\nabla \left(k\nabla \lambda \right) Cd\tau dt++\underset{t_0}{\overset{T}{\int }}\underset{D}{\int}\lambda \left(\tau, t\right)S\left(\tau, t\right) d td\tau $$

(18)

Using (17) and non-divergent condition for velocity, the first term on the right site of (18) is equal to zero:

$$ \underset{0}{\overset{T}{\int }}\underset{D}{\int }- U\lambda \nabla C- U C\nabla \lambda d\tau d t=\underset{0}{\overset{T}{\int }}\underset{D}{\int }-\nabla \left(\lambda C U\right)+\lambda C\nabla Ud\tau d t=-\underset{0}{\overset{T}{\int }}\underset{S}{\int}\lambda {CU}_n dsdt=0 $$

(19)

Because

$$ \nabla \left(k\nabla C\right)\lambda -\nabla \left(k\nabla \lambda \right)C=\nabla \left( k\lambda \nabla C\right)-\nabla \left( k C\nabla \lambda \right) $$

(20)

and boundary conditions (17), the second term on the right site of (18) is also equal to zero:

$$ \underset{0}{\overset{T}{\int }}\underset{D}{\int}\left[\nabla \left( k\lambda \nabla C\right)-\nabla \left( k C\nabla \lambda \right)\right] d\tau dt=\underset{0}{\overset{T}{\int }}\underset{S}{\int}\left( k\lambda \frac{\partial C}{\partial n}- kC\frac{\partial \lambda }{\partial n}\right) dsdt=0 $$

(21)

For left site of (18) we have:

$$ \underset{t_0}{\overset{T}{\int }}\underset{D}{\int}\left(\lambda \frac{\partial C}{\partial t}+C\frac{\partial \lambda }{\partial t}\right) d\tau dt=\underset{D}{\int}\left(C\left(\tau, T\right)\lambda \left(\tau, T\right)-{C}_0\left(\tau \right)\lambda \left(\tau, 0\right)\right) d\tau $$

(22)

Combining (19)–(22) together, we have:

$$ \underset{D}{\int}\left(C\left(\tau, T\right)\lambda \left(\tau, T\right)-{C}_0\left(\tau \right)\lambda \left(\tau, 0\right)\right) d\tau =\underset{D}{\int}\underset{t_0}{\overset{T}{\int }}\lambda \left(\tau, t\right)S\left(\tau, t\right) d td\tau $$

(23)

In accord to (23) and (16), we have:

$$ J=\frac{\mu }{V_{\Omega}}\underset{\Omega}{\int }C\left(\tau, T\right) d\tau =\frac{\mu }{V_{\Omega}}\underset{D}{\int }{C}_0\left(\tau \right)\lambda \left(\tau, 0\right) d\tau +\frac{\mu }{V_{\Omega}}\underset{D}{\int}\underset{t_0}{\overset{T}{\int }}\lambda \left(\tau, t\right)S\left(\tau, t\right) d td\tau $$

(24)