Continuity Equation in Pressure Coordinates

Here we will derive the continuity equation from the principle that mass is conserved for a parcel following the fluid motion (i.e., there is no flow across the boundaries of the parcel). This implies that δxδyδp δM = ρ δV = ρ δxδyδz = − g

is conserved following the fluid motion: 1 d(δM ) = 0 δM dt

1 d()δM = 0 δM dt g d ⎛ δxδyδp ⎞ ⎜ ⎟ = 0 δxδyδp dt ⎝ g ⎠ 1 ⎛ d(δp) d(δy) d(δx)⎞ ⎜δxδy +δxδp +δyδp ⎟ = 0 δxδyδp ⎝ dt dt dt ⎠ 1 ⎛ dp ⎞ 1 ⎛ dy ⎞ 1 ⎛ dx ⎞ δ ⎜ ⎟ + δ ⎜ ⎟ + δ ⎜ ⎟ = 0 δp ⎝ dt ⎠ δy ⎝ dt ⎠ δx ⎝ dt ⎠

Taking the limit as δx, δy, δp → 0,

∂u ∂v ∂ω Continuity equation + + = 0 in pressure ∂x ∂y ∂p coordinates

1 Determining Vertical Velocities

• Typical large-scale vertical motions in the atmosphere are of the order of 0. 01-01m/s0.1 m/s. • Such motions are very difficult, if not impossible, to measure directly. Typical observational errors for wind measurements are ~1 m/s. • Quantitative estimates of vertical velocity must be inferred from quantities that can be directly measured with sufficient accuracy.

Vertical Velocity in P-Coordinates

The equivalent of the vertical velocity in p-coordinates is: dp ∂p r ∂p ω = = +V ⋅∇p + w dt ∂t ∂z Based on a scaling of the three terms on the r.h.s., the last term is at least an order of magnitude larger than the other two. Making the hydrostatic approximation yields ∂p ω ≈ w = −ρgw ∂z

Typical large-scale values: for w, 0.01 m/s = 1 cm/s for ω, 0.1 Pa/s = 1 μbar/s

2 The Kinematic Method

By integrating the continuity equation in (x,y,p) coordinates, ω can be obtained from the mean divergence in a layer:

⎛ ∂u ∂v ⎞ ∂ω ⎜ + ⎟ + = 0 continuity equation in (x,y,p) coordinates ⎝ ∂x ∂y ⎠ p ∂p

p2 p2 ⎛ ∂u ∂v ⎞ ∂ω = − ⎜ + ⎟ ∂p rearrange and integrate over the layer ∫p ∫ ⎜ ⎟ 1 ∂x ∂y p1⎝ ⎠ p ⎛ ∂u ∂v ⎞ ω(p )−ω(p ) = (p − p )⎜ + ⎟ overbar denotes pressure- 2 1 1 2 ⎜ ⎟ weighted vertical average ⎝ ∂x ∂y ⎠ p

To determine vertical motion at a pressure level p2, assume that p1 = surface pressure and there is no vertical motion at the surface.

⎛ ∂u ∂v ⎞ ω()p2 −ω ()(p1 = p1 − p2 )⎜ + ⎟ ⎝ ∂x ∂y ⎠ p No vertical motion at surface.

Estimating vertical motion at the ⎛ ∂u ∂v ⎞ ω ∝ ⎜ + ⎟ top of a surface-based layer top ⎜ ∂x ∂y ⎟ using the kinematic method: ⎝ ⎠ p

vertical motion layer-mean at top of layer divergence

⎛ ∂u ∂v ⎞ If ⎜ + ⎟ < 0, then ωtop < 0, which implies rising motion. ⎝ ∂x ∂y ⎠ p ⎛ ∂u ∂v ⎞ If ⎜ + ⎟ > 0, then ωtop > 0, which implies sinking motion. ⎝ ∂x ∂y ⎠ p

3 Vertical Profiles of Divergence and Vertical Motion

Maximum magnitude of vertical motion (upward or downward) occurs at level of nondivergence. level of nondivergence

- r + - ω + ∇ p ⋅V For well-developed cyclones and anticyclones in midlatitudes, the level of nondivergence is typically between 500 and 700 hPa.

Calculating Horizontal Divergence (x ,y +Δy) r ∂u ∂v 0 0 ∇ ⋅V = + H ∂x ∂y Δy horizontal (x -Δx,y ) divergence 0 0 (x0,y0) (x0+Δx,y0) Δx Δx To compute horizontal divergence at Δy (x0,y0), evaluate derivatives using centered finite differences: (x0,y0-Δy) ⎛ ∂u ⎞ u()x +Δx, y − u(x −Δx, y ) ⎜ ⎟ ≈ 0 0 0 0 ∂x 2Δx ⎝ ⎠x0 ,y0 ⎛ ∂v ⎞ v()x , y +Δy − v(x , y −Δy) ⎜ ⎟ ≈ 0 0 0 0 ⎜ ∂y ⎟ 2Δy ⎝ ⎠x0 ,y0

4 v = 3.5 m/s Example: Δx = Δy = 50 km

u = 7.7 m/s u = 8.3 m/s

Divergence = 1 x 10-6 s-1 v = 4.0 m/s ⎛ ∂u ⎞ 8.3m s−1 − 7.7 m s−1 0.6m s−1 ⎜ ⎟ ≈ = = 6×10−6 s−1 ∂x 2()50 km 105 m ⎝ ⎠x0 ,y0 ⎛ ∂v ⎞ 3.5m s−1 − 4.0m s−1 − 0.5m s−1 ⎜ ⎟ ≈ = = −5×10−6 s−1 ⎜ ∂y ⎟ 2()50 km 105 m ⎝ ⎠x0 ,y0 r −6 −1 −6 −1 −6 −1 ∇2 ⋅V = 6×10 s − 5×10 s =1×10 s

3.3 v = 3.5 m/s Now, we introduce some small errors (0.1-0.2 m/s) in the wind estimates: 7.6 8.1 u = 7.7 m/s u = 8.3 m/s

-1 x 10-6 s-1 Divergence = 1 x 10-6 s-1 3.9 v = 4.0 m/s ⎛ ∂u ⎞ 8.1m s−1 − 7.6m s−1 0.5m s−1 ⎜ ⎟ ≈ = = 5×10−6 s−1 ⎝ ∂x ⎠ 2()50 km 105 m x0 ,y0 Divergence estimates using ⎛ ∂v ⎞ 3.3m s−1 − 3.9m s−1 − 0.6m s−1 ⎜ ⎟ ≈ = = −6×10−6 s−1 the kinematic ⎜ ∂y ⎟ 2()50km 105 m method are very ⎝ ⎠x0 ,y0 sensitive to small errors. r −6 −1 −6 −1 −6 −1 ∇2 ⋅V = 5×10 s − 6×10 s = −1×10 s

5 Typical Values for Horizontal Divergence

• Atmosphere: 10-5 to 10-6 s-1 • Ocean: 10 -7 to 10-8 s-1 • For large-scale weather systems in midlatitudes, the horizontal velocity is close to geostrophic balance. If not for the small contribution due to the variation of the Coriolis parameter with latitude, the geostrophic wind would be nondivergent. • Horizontal divergence in the atmosphere arises primarily due to the small departures of the wind from geostrophic balance.

Divergence in Natural Coordinates

r ∂β ∂V ∇ ⋅V = −V + H ∂n ∂s

∂β β = angle between β > 0 β < 0 −V < 0 streamline and flow s ∂n direction n ”directional convergence”

∂V < 0 s ∂s n ”speed convergence”

6