Conjugate flows for a three-layer fluid

A method for computing conjugate flows for a non-Boussinesq, three-layer fluid with arbitrary constant currents is developed. The general solution for a two-layer fluid is obtained as a special case. Symmetric stratifications at rest, with the upper and lower layer depths equal to h and identical density jumps across each interface, are considered in detail using the Boussinesq approximation. Mode-1 conjugate flows exist for $h<hc1.$ A symmetric mode-2 conjugate flow exists for all values of h, however for $h>hc2>hc1$ two additional asymmetric solutions exist. Comparisons with solutions for continuous stratifications with thin pycnoclines are made. Mode-2 solutions are more sensitive to the width of the pycnocline than are mode-1 solutions. Comparisons between three-layer non-Boussinesq and Boussinesq solutions are also made. For a total density variation of 4% of the mean value the two solutions are similar. For larger density variations the mode-2 solutions can be significantly different.