Bioluminescence potential modeling with an ensemble approach

Abstract

The approach for modeling bioluminescence (BL) potential is proposed. The approach consists of (1) estimation of the adjoint (by backward in time integration of the adjoint model), (2) generation of ensembles of BL potential initial conditions and source minus sink term, and (3) estimation of two integrals: the integration of the adjoint with the BL potential initial conditions and the integration of the adjoint with the BL potential source minus sink term. In this case, the impact of physical processes on BL potential dynamics is introduced through the one backward integration of the adjoint. The proposed approach has been applied to modeling of the BL potential changes in the area of the submesoscale filament, which developed during the upwelling event in the Monterey Bay area, California. Comparisons of modeled histograms with those observed show that predicted changes in mean BL potential values in the area of filament very closely resemble the observed ones: increase over 5 days in mean BL potential values in the top 15 m and decrease in mean BL potential values between 30 and 45 m depth in the area of the filament. There is a similar good qualitative and quantitative agreement between observed and model-predicted differences in histograms when BL potential values in 30–45 m depth are compared to the BL potential values in the top 15 m.

Appendix

Consider the following variable λ satisfying the adjoint to (1) equation:

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

(13)

With initial conditions at t = T

$$ \lambda {\left|{}_{t=T}=1\ \mathrm{in}\ \varOmega, \mathrm{and}\ \lambda \right|}_{t=T}=0\ \mathrm{outside}\ \mathrm{of}\ \varOmega\ \mathrm{in}\ D $$

(14)

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

(15)

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

We multiply (1) by λ and subtract (13) multiplied by C and integrate over domain D and time [t₀, T]:

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

(16)

Using (15) and non-divergent condition for velocity, the first term on the right site of (16) is equal to zero:

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

(17)

Because

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

(18)

and boundary conditions (15), the second term on the right site of (16) is also equal to zero:

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

(19)

For the left site of (16), we have:

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

(20)

Combining (17)–(20) together, we have:

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

(21)

In accord to (21), (14), and (3), we have:

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

(22)