Meteorological measurements within Hurricane Art reveal the x direction pressure gradient acceleration apgx to be 0.0021 ms-2 and a y-direction pressure gradient acceleration apgy of 0.0026 ms-2. If the hurricane is located at 28 degrees latitude, compute the geostrophic wind velocity in ms-1数值题
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Question

Meteorological measurements within Hurricane Art reveal the x direction pressure gradient acceleration apgx to be 0.0021 ms-2 and a y-direction pressure gradient acceleration apgy of 0.0026 ms-2. If the hurricane is located at 28 degrees latitude, compute the geostrophic wind velocity in ms-1数值题

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BlackTom AI · Accepted Answer

We start by restating the given data and what is being asked.
- Given: apgx = 0.0021 m/s^2 and apgy = 0.0026 m/s^2 at latitude 28°. We are asked to compute the geostrophic wind velocity.
- In geostrophic balance, the Coriolis force balances the horizontal pressure-gradient force. Using the standard relationship, the geostrophic wind components satisfy:
  v_g = (1 / (ρ f)) ∂p/∂x and u_g = - (1 / (ρ f)) ∂p/∂y,
  where f is the Coriolis parameter and ∂p/∂x, ∂p/∂y are the pressure gradient components.
- The problem defines apgx and apgy as accelerations due to the pressure gradient per unit mass, i.e., apgx = (1/ρ) ∂p/∂x and apgy = (1/ρ) ∂p/∂y. This makes the expressions simplify to:
  v_g = apgx / f and u_g = - apgy / f.
- The Coriolis parameter f at latitude φ is f = 2 Ω sin φ, where Ω ≈ 7.2921 × 10^−5 s^−1 (Earth’s rotation rate) and φ = 28°. Computing f:
  sin(28°) ≈ 0.4695, so f ≈ 2 × (7.2921×10^−5) × 0.4695 ≈ 6.84×10^−5 s^−1.
Now we plug in the numbers:
- v_g = apgx / f = 0.0021 / (6.84×10^−5) ≈ 30.7 m/s.
- u_g = - apgy / f = -0.0026 / (6.84×10^−5) ≈ -38.0 m/s.
The geostrophic wind is a vector with components (u_g, v_g) ≈ (−38.0, 30.7) m/s. To obtain its speed (magnitude), compute:
  |v_g| = sqrt(u_g^2 + v_g^2) = sqrt( (−38.0)^2 + (30.7)^2 ) ≈ sqrt(1444 + 942) ≈ sqrt(2386) ≈ 48.8 m/s.
Thus, the geostrophic wind speed is approximately 48.8 m/s, and its direction corresponds to the negative x-component (westward) and positive y-component (northward) in the chosen coordinate frame. The key steps are using the geostrophic balance relations, substituting the given pressure-gradient accelerations, and evaluating f at 28° latitude.

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