# CMS-Flow:Salinity Calculation

by Honghai Li, Christopher W. Reed, and Mitchell E. Brown

# Introduction

Salinity refers to the salt content of water, with values ranging from 0 for fresh water to 31-35 ppt (parts per thousand) for ocean water. In water bodies with poor mixing and limited water exchange, or those experiencing high evaporation, salinity can be greater and lead to formation of brine. Table 1 presents typical values and nomenclature for describing the degree of water salinity:

Table 1. Water salinity.

Fresh water Brackish water Saline water Brine
< 0.05 % 0.05 – 3 % 3 – 5 % > 5 %
< 0.5 ppt 0.5 – 30 ppt 30 – 50 ppt > 50 ppt

In coastal zones and estuaries, both temporal and spatial variations in salinity are controlled by changes in circulation, waves, tides, precipitation, evaporation, and freshwater inflows. These changes in salinity can have major effects on water density and water stratification, changing circulation patterns. Dynamic behavior of suspended sediment can be controlled by the salinity-driven flow and mixing. Any sustained changes to salinity can directly change the aggregation and consolidation of cohesive sediment as well (Nicholson and O’Connor 1986). Salinity can also alter the water chemistry that is closely related to marine organisms. Distribution and abundance of marine life will change water turbidity and define water quality in coastal and estuarine systems. Modifications of coastal inlets, such as channel deepening and widening and rehabilitation or extension of jetties may alter the salinity distribution within the estuary.

# Transport Equation

The CMS calculates the salinity field based on the following 2D salinity conservation equation:

 ${\displaystyle {\frac {\partial (Sd)}{\partial t}}+{\frac {\partial (S{{q}_{x}})}{\partial x}}+{\frac {\partial (S{{q}_{y}})}{\partial y}}={\frac {\partial }{\partial x}}\left[{{K}_{x}}d{\frac {\partial S}{\partial x}}\right]+{\frac {\partial }{\partial y}}\left[{{K}_{y}}d{\frac {\partial S}{\partial y}}\right]+(P-E)S}$ (1)

where S is depth-averaged salinity; d is total water depth, qx and qy are flow per unit width in the x- and y-axis direction, respectively; Kx and Ky are the salt diffusion coefficients in the corresponding x- and y-axis directions, and P and E are precipitation and evaporation in m/year, respectively. Equation (1) represents the horizontal fluxes of salt in water bodies and is balanced by exchanges of salt via diffusive fluxes. Major processes contributing to the salinity are freshwater inflows from rivers, vertical fluxes of freshwater by precipitation and evaporation at the water surface, and groundwater fluxes, which can be specified as the surface and bottom boundary conditions in the equation.

# Model Assumptions

CMS-Flow is presently capable of 2D salinity computations in both the explicit and implicit solvers. 3D representation of salinity in the CMS, discussed herein, is being tested. The simulation of salinity can often require a 3D solution due to the presence of vertical salinity gradients that can significantly influence flow. It is therefore important to understand the limitations of 2D salinity simulations, and apply them only when the assumptions inherent in 2D simulations are valid. Typically, 2D salinity calculations are valid when the salinity is well mixed over the water column. These conditions are usually met for shallow bays with open exchanges to the ocean or gulf, dominant tidal signals, and sufficient wind energy to provide the vertical mixing. Also, the assumption of sufficient energy to mix over the water column is valid under storm conditions, even for deeper water bodies. Finally, when the exchange with the open sea is restricted by an inlet, the tidal range is an important indicator of vertical mixing conditions. For low tide ranges, significant vertical stratification can occur, even in shallow bays and estuaries, especially when the winds are calm. Pritchard (1955) and Cameron and Pritchard (1963) have classified estuaries using stratification and salinity distribution as the governing criteria, and these classifications can be used for guidance in applying the 2D simulations.

The lateral mixing for salinity in the CMS Flow model is the same as the lateral mixing in the momentum equations.

# Initial and Boundary Conditions

The initial condition is either specified as a spatially constant value, a user specified data or a Laplace solution to user-specified salinity points. The Laplace equation is given by

 ${\displaystyle {\frac {\partial ^{2}S}{\partial x^{2}}}+{\frac {\partial ^{2}S}{\partial y^{2}}}=0}$ (2)

A zero-gradient (Neumann BC) boundary condition is applied when applying the Laplace equation at all boundaries.

The inflow boundary condition value must be specified (Dirichlet BC) by the user all all open boundaries. For outflow conditions, a zero-gradient is applied (Neumann BC).

This wiki technical note was prepared and funded under the Coastal Inlet Research Program (CIRP) and was written by the following:

• Dr. Honghai Li (Honghai.Li@usace.army.mil, voice: 601-634-2840, fax: 601-634-3080) of the U.S. Army Engineer Research and Development Center (ERDC), Coastal and Hydraulics Laboratory (CHL),
• Dr. Christopher W. Reed (Chris_Reed@URSCorp.com) of URS Corporation
• Mitchell E. Brown (Mitchell.E.Brown@usace.army.mil) of ERDC, CHL.

Alejandro Sanchez of CIRP provided hydrodynamic model information for the Humboldt Bay example. The CIRP Program Manager, Dr. Julie D. Rosati (Julie.D.Rosati@usace.army.mil), the assistant Program Manager, Dr. Nicholas C. Kraus, the Chief of the Coastal Engineering Branch at CHL, Dr. Jeffrey P. Waters, and Dr. Lihwa Lin, the Coastal Engineering Branch, reviewed this CHETN. Files for the example may be obtained by contacting the author.

# References

• Aquaveo. 2010. SMS: XY Series Files (*.xys). http://www.xmswiki.com/xms/SMS:XY_Series_Files_(*.xys).
• Cameron, W. M. and D. W. Pritchard. 1963. Estuaries. In M. N. Hill (editor): The Sea vol. 2, John Wiley and Sons, New York, 306 - 324.
• Demirbilek, Z., K. J. Connell, N. J. MacDonald, and A. K. Zundel. 2008. Particle Tracking Model in the SMS10: IV. Link to Coastal Modeling System. Coastal and Hydraulics Engineering Technical Note ERDC/CHL CHETN-IV-71. Vicksburg, MS: U.S. Army Engineer Research and Development Center.
• H. Li, C. W. Reed, and M. E. Brown. 2012. Salinity Calculations in the Coastal Modeling System. Coastal and Hydraulics Engineering Technical Note ERDC/CHL CHETN-IV-80. Vicksburg, MS: U.S. Army Engineer Research and Development Center. An electronic copy of this CHETN is available from or
• Pritchard, D. W. 1955. Estuarine circulation patterns. Proceedings of the American Society of Civil Engineers 81, no 717, 1 – 11.