for the continental shelf areas along the eastern seaboard of the North America \cite{Cai_2010}. A similar salinity-based linear regression parameterization is adopted in the current generation of ecosystem models \cite{Wakelin_2012}. Differentiating equation (13) gives:

\begin{equation}
\frac{\partial ALK}{\partial z} = b \frac{\partial S}{\partial z}
\tag{14}
\end{equation}

One can therefore reduce the term describing the relative contributions of the diapycnal salt and alkalinity fluxes to:

\begin{equation}
\left( \beta_s \frac{\partial S}{\partial z} + \frac{\gamma_A}{ALK} \frac{\partial ALK}{\partial z} \right) = \frac{\partial S}{\partial z} \left( \beta_s + b \cdot \frac{\gamma_A}{ALK} \right)
\tag{15}
\end{equation}

Assuming typical English Channel values for ALK = 2400 \cite{Kitidis_2012}, $\gamma_A = -10$ and $b = 30$--$60$ indicate that the impact of the alkalinity flux will partly compensate for the impact of the salt flux accounting for between 12\% and 24\% of the impact of the salt flux on $pCO_2$.

Due to the low resolution of nutrient measurements from bottle samples, the nutrient gradient is often estimated through correlation with the temperature gradient \cite{Rippeth_2009, Williams_2013a}. It is therefore useful to compare the impact of the diapycnal nitrate flux (as a $CO_2$ sink) on $pCO_2$ relative to the impact of the cooling of the SML, i.e., balancing the terms:

\begin{equation}
\left( \frac{\gamma}{DIC} R_{pp} \frac{\partial NO_3^-}{\partial z} \right) = \left( \beta \frac{\partial T}{\partial z} \right)
\end{equation}

i.e.,

\begin{equation}
\left( \frac{\partial NO_3^-}{\partial z} \right) = \frac{DIC \cdot \beta}{\gamma R_{pp}} \left( \frac{\partial T}{\partial z} \right)
\tag{16}
\end{equation}

Assuming the phytoplankton C:N uptake ratio, $R_{pp}$ equivalent to the Redfield ratio (6.625), a reported deep water $DIC = 2100$ mmol m$^{-3}$ and $\gamma = 10$, equation (15) thus yields a ratio of:

\begin{equation}
\frac{DIC \cdot \beta}{\gamma R_{C:N}} \approx 1.34
\end{equation}

This may be compared to the typically observed ratios of nitrate to temperature gradients from Northwest European seasonally-stratified shelf seas of 2.1--3.3 (examples of fits are given in Figure~\ref{figure_5}). In other words, the impact of diapycnal nitrate flux alone on $pCO_2$, through the DIC sink created by the primary production it fuels, significantly exceeds the impact of the change in solubility driven by the cooling of the SML from the diapycnal heat flux, in these oceanographic settings.

The combined impacts of the diapycnal fluxes of the different parameters on $pCO_2$ are shown for the stations listed in Table~\ref{table_1} (Figure~\ref{figure_6}). Here, it is assumed that $R_{pp} = 6.625$ and that

\begin{figure}
\centering
\includegraphics{figure_4.pdf}
\caption{The ratio of $-\Delta S/\Delta T$ for stations across the Northwest European shelf seas. The geographic distribution of the stations is shown in the insert map. The diagonal line represents the ratio of $\beta/\beta_T$. For points lying above the line, the impact on sea surface $pCO_2$ of the diapycnal salt flux into the surface-mixed layer will exceed the cooling of the surface layer which results from the same turbulent mixing event.}
\label{figure_4}
\end{figure}