Modified shallow water equations for inviscid gravity currents.

Area of Science:

• Fluid Dynamics
• Geophysics
• Environmental Science

Background:

• Gravity currents are analyzed using shallow water equations (SWE) with specific boundary conditions.
• Standard SWE do not inherently account for ambient fluid resistance, necessitating complex front conditions.
• Accurate modeling of gravity currents is crucial for understanding various geophysical and environmental phenomena.

Purpose of the Study:

• To rederive shallow water equations (SWE) incorporating ambient resistance, leading to modified shallow water equations (MSWE).
• To analytically solve the nonlinear MSWE using perturbation and inner-outer expansion methods.
• To validate the MSWE by comparing solutions with experimental data, including turbulent drag effects.

Main Methods:

• Derivation of modified shallow water equations (MSWE) by including a nonlinear ambient resistance term.
• Analytical solution of the MSWE using perturbation expansion and inner-outer expansion techniques.
• Incorporation of the semiempirical Chezy drag term to model turbulent drag.

Main Results:

• The MSWE analytically solves for gravity current motion without requiring explicit front conditions.
• Outer solutions match SWE solutions with front conditions; inner solutions describe the current head profile and velocity.
• The composite solution accurately models the entire gravity current, with added Chezy drag aligning well with experimental data.

Conclusions:

• The MSWE provides a more unified and accurate framework for analyzing gravity currents compared to traditional SWE.
• The analytical approach simplifies the complex task of modeling gravity current dynamics.
• The MSWE framework shows potential for extension to three-dimensional gravity currents, though analytical solutions may become challenging.