A recent innovation by the Empire added weather observation sensors to their small AT-ST walkers (now called AT-MTs for All Terrain Meteorological Transports). In particular, the walkers, which are roughly
tall, can measure temperature (in
), pressure (in mb), and a velocity vector (in
). Each of the 3 walkers made atmospheric measurements that were oriented as depicted below. Assume 3 more walkers made observations
to the west and
to the east of these 3 and found identical observations. Use the data measured by the walkers and quasigeostrophic theory to answer the questions. a. Is there rising or sinking motion at the middle walker’s location? Explain your reasoning. b. What is the geostrophic wind vector at the middle walker’s location? c. What will the geostrophic relative vorticity at the central walker be in 30 minutes
The Correct Answer and Explanation is:
Here are the answers to the questions based on the provided data and quasigeostrophic theory.
a. Is there rising or sinking motion at the middle walker’s location? Explain your reasoning.
There is sinking motion at the middle walker’s location.
Reasoning:
Vertical motion in the atmosphere is directly related to the convergence or divergence of horizontal wind. According to the principle of mass continuity, where horizontal winds diverge (spread apart) near the surface, air from above must sink to fill the space. Conversely, where winds converge, air is forced upward.
We can determine the horizontal divergence (∇·V) at the middle walker’s location using a finite difference approximation with the data from the walkers to the north and south. The divergence is given by:
∇·V = ∂u/∂x + ∂v/∂y
From the problem statement, the conditions are identical 25 km to the east and west, which means there is no change in the zonal (east-west) direction. Therefore, ∂u/∂x = 0.
We calculate the meridional (north-south) component of divergence, ∂v/∂y, using the v-components of the wind vectors from the north and south walkers:
- North walker’s velocity (V_N): v_N = -1.6 m/s
- South walker’s velocity (V_S): v_S = -1.8 m/s
- Distance between them (Δy): 50 km = 50,000 m
∂v/∂y ≈ (v_N – v_S) / Δy = (-1.6 m/s – (-1.8 m/s)) / 50,000 m = 0.2 m/s / 50,000 m = 4 x 10⁻⁶ s⁻¹
Since ∂u/∂x = 0 and ∂v/∂y > 0, the total horizontal divergence ∇·V is positive (4 x 10⁻⁶ s⁻¹). This low-level divergence implies sinking motion from above to conserve mass.
b. What is the geostrophic wind vector at the middle walker’s location?
The geostrophic wind vector is Vg = 1.52i m/s (or 1.52 m/s directed due east).
Explanation:
The geostrophic wind (Vg) is a theoretical wind that results from the balance between the Pressure Gradient Force and the Coriolis force. Its components (ug, vg) are calculated as:
ug = -(1/fρ) * (∂p/∂y)
vg = (1/fρ) * (∂p/∂x)
First, we determine the necessary parameters for the middle walker’s location (Φ = 50° N):
- Coriolis parameter (f): f = 2Ωsin(Φ) = 2 * (7.292 x 10⁻⁵ rad/s) * sin(50°) ≈ 1.117 x 10⁻⁴ s⁻¹
- Air density (ρ): Using the ideal gas law (ρ = p/RT), with p = 1000.05 mb = 100005 Pa, T = 23.2°C = 296.35 K, and R_d = 287 J/(kg·K).
ρ = 100005 / (287 * 296.35) ≈ 1.176 kg/m ³ - Pressure gradients (∂p/∂x, ∂p/∂y):
- Since conditions are uniform east to west, ∂p/∂x = 0, which means the meridional geostrophic wind, vg, is 0.
- ∂p/∂y ≈ (p_N – p_S) / Δy = (1000 mb – 1000.1 mb) / 50 km = (-10 Pa) / (50,000 m) = -2 x 10⁻⁴ Pa/m
Now, we can calculate the zonal geostrophic wind, ug:
ug = – (1 / (1.117 x 10⁻⁴ s⁻¹ * 1.176 kg/m ³)) * (-2 x 10⁻⁴ Pa/m) ≈ 1.52 m/s
Thus, the geostrophic wind vector Vg is (1.52i + 0j) m/s.
c. What will the geostrophic relative vorticity at the central walker be in 30 minutes?
The geostrophic relative vorticity will be approximately 4.4 x 10⁻⁸ s⁻¹ in 30 minutes.
Explanation:
This requires the quasigeostrophic vorticity equation, which describes the time evolution of geostrophic vorticity. The local rate of change of geostrophic vorticity (∂ζg/∂t) is primarily due to the advection of absolute vorticity (ζg + f) by the horizontal wind (V).
∂ζg/∂t = -V · ∇(ζg + f)
- Calculate the initial geostrophic relative vorticity (ζg):
ζg = ∂vg/∂x – ∂ug/∂y. Since vg = 0, we have ζg = -∂ug/∂y. The pressure decreases linearly from south to north, meaning the pressure gradient (∂p/∂y) is constant. Since ug depends on this constant gradient, ug is also constant with respect to y. Therefore, ∂ug/∂y = 0, and the initial geostrophic relative vorticity ζg is 0. - Calculate the vorticity tendency (∂ζg/∂t):
With ζg = 0, the vorticity equation simplifies to the advection of planetary vorticity (f) by the wind:
∂ζg/∂t = -V · ∇f
The gradient of the Coriolis parameter, ∇f, is known as the beta parameter (β), which points north: ∇f = βj.
β = (2ΩcosΦ) / R_earth = (2 * 7.292 x 10⁻⁵ * cos(50°)) / 6.371×10⁶ m ≈ 1.47 x 10⁻¹¹ m⁻¹s⁻¹
The wind vector at the central walker is V = 2.4i – 1.7j m/s.
∂ζg/∂t = -(2.4i – 1.7j) · (βj) = -(-1.7 * β) = 1.7 * β
∂ζg/∂t = 1.7 m/s * (1.47 x 10⁻¹¹ m⁻¹s⁻¹) ≈ 2.499 x 10⁻¹¹ s⁻² - Calculate the future vorticity:
The change in vorticity (Δζg) over a time period (Δt = 30 min = 1800 s) is:
Δζg = (∂ζg/∂t) * Δt = (2.499 x 10⁻¹¹ s⁻²) * (1800 s) ≈ 4.498 x 10⁻⁸ s⁻¹
The final vorticity is the initial vorticity plus the change:
ζg(final) = ζg(initial) + Δζg = 0 + 4.498 x 10⁻⁸ s⁻¹ ≈ 4.5 x 10⁻⁸ s⁻¹
