Conservation of Mass The Continuity Equation

The equations of motion describe the “conservation of momentum” in the atmosphere. We now turn our attention to another conservation principle, the “conservation of mass”. Mass cannot be created nor destroyed. We have to account for where the mass goes in the atmosphere. Otherwise, our numerical weather prediction models will work as well as a 1985 Chevy Cavalier.

Holton derives the continuity equation in two ways: Eulerian and Lagrangian. We will consider the Eulerian derivation. Let’s say we have a box in a Cartesian coordinate frame:

We want to know how much mass is coming into the box and how much is leaving. The amount of mass entering the shaded side (side B) over time is:

ρ δ δ δtzyu (1)

where u = wind velocity in the x-direction, δy is the width, δz is the height, and δt is the amount of time elapsed. The flow of mass out of the opposite side (A) is:

⎡ ∂(ρu) ⎤ ()ρu + δδδδ tzyx (2) ⎣⎢ ∂x ⎦⎥

∂()ρu The first term (ρu) is just the amount that came in face B. The second term ( δx ) is ∂x how much the mass flow has changed in the x-direction over a distance of δx . And again, we multiply by the area of face B. So to find out what the net flow is, simply subtract (2) from (1) (how much came in minus how much left):

⎡ ∂(ρu) ⎤ ()ρδδδρ utzyu +− δδδδ tzyx (3) ⎣⎢ ∂x ⎦⎥

After multiplying out and simplifying this yields:

∂()ρu − δδδδ tzyx (4) ∂x

Now that we have an expression for the net mass flow in the x-direction, we can use similar methods to get the y and z-direction net mass flow:

∂()ρu − δδδδ tzyx (5) ∂y

∂()ρu − δδδδ tzyx (6) ∂z

The conservation of mass says that the change of mass inside the cube must be equal to the sum of all the net mass flows coming from all 3 directions (equations 4,5, and 6):

∂ρ ∂()ρu ∂(ρu) ∂(ρu) δδδδ tzyx −= δδδδ tzyx − δδδδ tzyx − δδδδ tzyx (7) ∂t ∂x ∂y ∂z

Equation 7 is the general form of the continuity equation.

Mass Divergence Form A more common form of the continuity equation, called the “mass divergence form”, is found by dividing both sides of the equation by δ δ δ δtzyx to yield:

∂ρ ∂()ρu ∂(ρv) ∂(ρw) −= − − (8) ∂t ∂x ∂y ∂z

Equation 8 is commonly expressed in vector notation:

∂ρ v •−∇= (ρV ) (9) ∂t 3

v where ∇3 is the 3-dimensional gradient operator and V is the 3-D wind. Recall from the v math review that the expression •∇ V is the divergence of the wind. The negative of that expression is the convergence of the wind. The fact that it is multiplied by a scalar density means that the expression in (9) is the convergence of mass. So the change of mass with time is equal to the net convergence of mass. Equation 9 is equivalent to Equation 2.30 in Holton.

Velocity Divergence Form An alternative form of the continuity equation removes density from the RHS of (8), leaving only the 3-D velocity divergence on the RHS. Starting with (8), we arrive at the following after using the product rule and some manipulation:

dρ ⎡∂u ∂v ∂w⎤ α −= ⎢ + + ⎥ (10) dt ⎣ ∂x ∂y ∂z ⎦

Numerical models use one form or another of the continuity equation to predict the change in density with time.

Pressure Coordinate Form

So far we have been dealing with equations in which height (z) was the vertical coordinate. There are other vertical coordinates that can be used which may make certain equations and relationships easier to use and/or understand. Here we will introduce the use of pressure (p) as the vertical coordinate as it relates to the continuity equation.

See the “Isobaric Coordinates” lecture to see how the Equations of Motion and the Thermodynamic Equation are transferred to pressure coordinates.

First, we introduce ω, which is the vertical velocity in pressure coordinates:

dp ω = dt

For sinking motion, ω>0 and for rising motion ω<0. Note that in x,y,z,t coordinates the vertical velocity was given by w = dz/dt.

Starting from Equation (10), a series of steps can be performed (not shown) that will yield the continuity equation in pressure coordinates:

∂u ∂v ∂ω + + = 0 (11) ∂x ∂ ∂py

Some notes: 1) Note that density has disappeared from (10) to (11). This is nice because it is very difficult to measure it. 2) Eq. 11 is frequently used to compute the vertical velocity between two pressure surfaces. Moving the vertical velocity term to one side yields:

∂ω ⎛ ∂u ∂v ⎞ −= ⎜ + ⎟ (12) ∂p ⎝ ∂x ∂y ⎠

This can also be written as:

∂ω v (•∇−= V ) (13) ∂p H (a) (b)

where (a) is the rate of change of vertical velocity with pressure level and (b) is the horizontal convergence. This is a very important relationship. It says that the vertical velocity and horizontal convergence fields are intimately linked. A convergence or divergence of mass will induce vertical airflow in the atmosphere.

Real World Example

Let’s say we have an area of convergence in the lower troposphere:

v ∂ω Since convergence = ( •∇− V ), positive convergence means that > 0 . This requires H ∂p that ω increases from negative values in the middle troposphere to near 0 at the ground:

The continuity equation can also be applied to explaining vertical motions from synoptic scale systems. For example, low pressure centers experience a frictionally induced net convergence of mass into the center. This will induce upward motion.