Theory: Salinity Variance Budget

This chapter describes the theoretical framework underlying pySVA, based on the work of Burchard et al. (2008) and Li et al. (2018).

Total Salinity Variance Equation

Burchard et al. (2008) derived the total salinity variance equation starting from the Reynolds-averaged salt conservation advection-diffusion equation in a three-dimensional domain:

\[\frac{\partial S}{\partial t} + \mathbf{u} \cdot \nabla S - \nabla \cdot (\mathbf{K} \nabla S) = 0\]

where:

  • \(\mathbf{u} = (u, v, w)\) is the three-dimensional velocity vector

  • \(\mathbf{K} = (K_{xx} + K_{yy} + K_{zz})\) is the diffusivity tensor

Applying Reynolds decomposition, the total salinity and velocity can be decomposed into a volume mean and a varying part:

\[S = [[S]] + S'_{tot}, \quad \mathbf{u} = [[\mathbf{u}]] + \mathbf{u}'\]

where \([[\cdot]]\) denotes the volume average.

Substituting this decomposition and manipulating the equations yields the salinity variance equation:

\[\frac{\partial(S'_{tot})^2}{\partial t} + \nabla\cdot[\mathbf{u}(S'_{tot})^2 - \mathbf{K}\nabla(S'_{tot})^2] - 2S'[[\mathbf{u}' \cdot \nabla S'_{tot}]] = -2(\mathbf{K}\nabla S'_{tot}) \cdot (\nabla S'_{tot})\]

Vertical Salinity Variance

Li et al. (2018) extended this framework by decomposing the total salinity variance into horizontal and vertical contributions. For a single vertical water column, decomposing salinity and velocity as:

\[S = \overline{S} + S'_v, \quad \mathbf{u} = \mathbf{\overline{u}} + \mathbf{u}'_v\]

where the overline denotes depth averaging, the vertical salinity variance equation becomes:

\[\begin{split}\begin{split} \underbrace{\frac{\partial}{\partial t}\int(S'_v)^2 \mathrm{d}z}_{\text{tendency}} = -&\underbrace{\nabla_h \cdot \int \mathbf{u_h}(S'_v)^2 \mathrm{d}z}_{\text{advection}} + \underbrace{\int -2\mathbf{u'_v}S'_v \cdot \nabla\overline{S} \mathrm{d}z}_{\text{straining}} \\ &- \underbrace{\int 2\kappa_{zz}\bigg(\frac{\partial S}{\partial z}\bigg)^2 \mathrm{d}z}_{\text{dissipation}} - \underbrace{\int\mathcal{M}_{\mathrm{num}} \mathrm{d}z}_{\text{numerical mixing}} \end{split}\end{split}\]

Variance Decomposition

The total salinity variance can be decomposed into horizontal and vertical components:

\[\begin{split}(S'_v)^2 &= (S - \overline{S})^2 \quad \text{(vertical variance)} \\ (S'_h)^2 &= (\overline{S} - [[S]])^2 \quad \text{(horizontal variance)} \\ (S'_{tot})^2 &= (S - [[S]])^2 \quad \text{(total variance)}\end{split}\]

Budget Terms

The variance budget equation contains four main terms:

  1. Tendency: \(\frac{\partial}{\partial t}\int(S'_v)^2 \mathrm{d}z\) - Temporal change of variance - Left-hand side of the equation

  2. Advection: \(\nabla_h \cdot \int \mathbf{u_h}(S'_v)^2 \mathrm{d}z\) - Transport of variance by mean flow - Quantifies horizontal redistribution of mixed water

  3. Straining: \(\int -2\mathbf{u'_v}S'_v \cdot \nabla\overline{S} \mathrm{d}z\) - Variance production/destruction by vertical mixing and shear - Related to internal wave activity

  4. Dissipation: \(\int 2\kappa_{zz}(\partial S/\partial z)^2 \mathrm{d}z\) - Variance reduction by turbulent diffusion - Related to molecular and turbulent mixing

Understanding these terms enables researchers to identify which processes dominate stratification patterns in coastal-delta systems.

References

[1] Burchard, H., & Rennau, H. (2008). Comparative quantification of physically and numerically induced mixing in ocean models. Ocean Modelling, 20(3), 293-311. https://doi.org/10.1016/j.ocemod.2007.10.003

[2] Li, X., Geyer, W. R., Zhu, J., & Wu, H. (2018). The Transformation of Salinity Variance: A New Approach to Quantifying the Influence of Straining and Mixing on Estuarine Stratification. Journal of Physical Oceanography, 48(3), 607-623. https://doi.org/10.1175/JPO-D-17-0189.1