Chapter 7  Some Mathematics: 7.7 Conservation of Mass: The Continuity Equation Now lets derive the equation for the conservation of mass in a fluid. We begin by writing down the flow of mass into and out of a small box (Figure 7.5).
The mass flux into the volume must be (mass flow out)  (mass flow in) The third term inside the parentheses becomes much smaller than the first two terms as δx approaches 0; and
In three dimensions:
The mass flux must be balanced by a change of mass inside the volume, which is: and conservation of mass requires:
This is the continuity equation for compressible flow, first derived by Leonhard Euler (17071783). The equation can be put in an alternate form by expanding the derivatives of products and rearranging terms to obtain: The first four terms constitute the total derivative of density Dρ/Dt from (7.7), and we can write (7.17) as:
This is the alternate form for the continuity equation for a compressible fluid. The Boussinesq Approximation
Boussinesq's assumption requires that:
The approximations are true for oceanic flows, and they ensure that oceanic flows are incompressible. See Kundu (1990: 79 and 112), Gill (1982: 85), Batchelor (1967: 167), or other texts on fluid dynamics for a more complete description of the approximation. Compressibility
where V is volume, and p is pressure. For incompressible flows, β = 0, and: because dp/dt does not equal 0. Remembering that density is mass m per unit volume V, and that mass is constant: If the flow is incompressible, (7.18) becomes:
This is the Continuity Equation for Incompressible Flows.


Department of Oceanography, Texas A&M University Robert H. Stewart, stewart@ocean.tamu.edu All contents copyright © 2005 Robert H. Stewart, All rights reserved Updated on September 24, 2008 