The Equations of Motion The equations of motion describe the forces in the atmosphere that act on a parcel of air. We have already looked at these individually. For 3-dimensional flow, v dV =PGF + CF + F (1) dt v dV Where is the change in the 3-D wind vector, PGF = pressure gradient force, CF = dt Coriolis Force, and F = friction force. If we substitute in all of the terms for PGF, CF, and F, we get: du dv dw ∂p ∂p ∂p iˆ+ ˆj +kˆ = −αiˆ− α ˆj − α kˆ +2 u Ω cosφ kˆ −2 w Ω cosφ iˆ −2 u Ω sinφˆ j dt dt dt ∂x ∂y ∂z ˆ ˆ ) ˆ ˆ +2v Ω sinφ i − gk + Frx i + Fry j + Frz k (2) (2) is the equation of motion for all 3 directions. Normally we decompose (2) into the equation of motion for each u,v, and w direction. For example, to get the equation of motion in the x-direction, dot multiply (2) by i : du ∂p = −α −2w Ω cosφ + 2v Ω sinφ + F (3) dt ∂x rx And the other two: dv ∂p = −α −2u Ωsinφ + F (4) dt ∂y ry dw ∂p = −α −g +2 u Ωcosφ + F (5) dt ∂z rz This brings us to a useful technique for making our lives simpler. Frequently, equations like (3-5) contain terms that really aren’t too significant, at least compared to the other terms. We employ a technique called scale analysis to determine which, if any, terms we can neglect. To perform a scale analysis, we need to calculate the average magnitude that all of the terms will have. Some scale magnitudes for mid-latitude synoptic scale systems are: Horizontal velocity (U) ≈ 10 m/s (u,v) Vertical velocity (W) ≈ 10-2 m/s (w) ∂ ∂ Horizontal Length (L) ≈ 106 m ( ,) ∂x∂ y ∂ Vertical Height (H) ≈ 104 m ( ) ∂z Angular Velocity (Ω) ≈ 10-4 s-1 (Ω) d Time Scale (T) ≈ 105 s ( ) dt -12 -2 Frictional Acceleration (Fr) ≈ 10 ms (Frx, Fry, Frz) Gravitational Acceleration (G) ≈ 10 m/s (g) ∂p ∂p Horizontal Pressure Gradient (∆p) ≈ 10-3 Pa/m ( ,) ∂x ∂y 5 ∂p Vertical Pressure Gradient (Po) ≈ 10 Pa ( ) ∂z Specific Volume (α) ≈ 1 m3kg-1 (α) Coriolis Effect (C) ≈ 1 (2sinφ, 2cosφ) Using scale analysis, it can be shown that the vertical Coriolis Effect in (3) is three orders of magnitude smaller than the other terms. So we can throw it out. Similarly, the frictional terms and the Coriolis term in (5) are 4 orders of magnitude smaller than the other terms, and dw/dt is 8 orders of magnitude less. Simplifying, equations 3-5 become primitive equations of motion: du ∂p = −α +2v Ωsinφ + F (6) dt ∂x rx dv ∂p = −α −2u Ωsinφ + F (7) dt ∂y ry ∂p 0 = −α − g (8) ∂z Note that (8) simplifies to the hydrostatic approximation. This is justification for assuming that large-scale flow is in hydrostatic balance for mid-latitude, synoptic scale features. Note also that we can’t use (8) to predict vertical velocity in numerical models – other approaches must be used, and we will cover those later. .