Inputs
Formula used
In natural coordinates, the gradient-wind balance along a curved streamline is:
- V: gradient-wind speed (m/s)
- Vg: geostrophic speed (m/s), from straight balance
- f = 2Ω sin(φ): Coriolis parameter (1/s)
- R: radius of curvature of the flow (m)
- s: +1 cyclonic, −1 anticyclonic curvature
When using pressure gradients: Vg = (1/(ρ|f|)) (∂p/∂n) using the cross-isobar pressure gradient magnitude.
How to use this calculator
- Select a hemisphere and enter latitude in degrees.
- Enter the flow radius of curvature and choose curvature type.
- Pick an input mode: provide Vg or provide ∂p/∂n with density.
- Press Calculate to view results above the form.
- Use CSV or PDF buttons to export the latest output.
Example data table
| Case | Hemisphere | Latitude (deg) | Radius (km) | Curvature | Vg (m/s) | Expected note |
|---|---|---|---|---|---|---|
| A | N | 45 | 500 | Cyclonic | 20 | Gradient wind typically exceeds geostrophic. |
| B | N | 45 | 500 | Anticyclonic | 20 | Solution may be weaker; watch stability. |
| C | S | 30 | 300 | Anticyclonic | 25 | Possible no-solution if inputs exceed limit. |
Gradient wind interpretation guide
1) What gradient wind represents
Gradient wind is the steady, frictionless flow that follows curved isobars when the pressure-gradient force is balanced by Coriolis force and the centrifugal term. It bridges straight-line geostrophic flow and strongly curved motion near synoptic highs and lows.
2) Why curvature changes the speed
When streamlines curve, centripetal acceleration requires an additional inward force. In cyclonic flow, this typically demands a stronger pressure gradient for a given speed, so the balanced speed often exceeds the geostrophic estimate. In anticyclonic flow, curvature can reduce the allowable balanced speed.
3) Latitude and the Coriolis parameter
The Coriolis parameter f = 2Ω sin(φ) scales the rotational influence. At higher latitudes, larger |f| strengthens rotational control and pushes the solution toward geostrophic behavior. Near the equator, |f| becomes small and balanced solutions become sensitive and less reliable.
4) Cyclonic versus anticyclonic branches
The sign of curvature determines whether the centrifugal term adds to or subtracts from the rotational balance. Cyclonic curvature (around lows) commonly yields a single practical positive root. Anticyclonic curvature (around highs) can produce weak-flow branches and can fail to produce any real solution when forcing is too strong.
5) The no-solution condition
The calculator evaluates the quadratic discriminant. A negative discriminant means there is no real gradient-wind speed that satisfies the steady balance for the chosen curvature and forcing. This is most common for anticyclonic cases with large pressure gradients or small radii, which can exceed the theoretical stability limit.
6) Using pressure-gradient inputs
If you enter \partial p/\partial n and air density, the tool first converts units to Pa/m and computes the implied geostrophic speed Vg = (\partial p/\partial n)/(ρ|f|). This is convenient for map-based estimates such as several hPa across 100 km in mid-latitudes.
7) Rossby number as a regime check
The Rossby number V/(|f|R) compares curvature-driven acceleration to rotational influence. Values well below 1 indicate strong rotational control and near-geostrophic behavior. Values approaching or exceeding 1 suggest curvature effects dominate and the assumptions behind large-scale balance should be reviewed carefully.
8) Practical ranges and quality control
For synoptic systems, radii of 200–1500 km and winds of 5–40 m/s are common. Always verify latitude, radius sign choice, and consistent units for the pressure gradient. If the tool flags no-solution, try increasing radius, reducing forcing, or reassessing whether friction or unsteady effects are important.
FAQs
1) Is gradient wind the same as geostrophic wind?
No. Geostrophic wind assumes straight flow with pressure gradient balanced only by Coriolis. Gradient wind includes curvature, adding the centrifugal term, so speeds can differ, especially near tight highs or lows.
2) Why does the calculator warn near the equator?
Near the equator, |f| becomes very small, so small input errors can cause large changes in computed wind. Large-scale balanced approximations are less stable there and other dynamics often dominate.
3) What does cyclonic versus anticyclonic mean here?
It refers to curvature around lows versus highs. In the Northern Hemisphere, cyclonic curvature corresponds to counterclockwise flow around lows. Anticyclonic curvature corresponds to clockwise flow around highs.
4) When can there be no real solution?
In anticyclonic curvature, strong forcing or very small radius can make the discriminant negative. Physically, the required balance would be unstable or impossible under the steady, frictionless assumptions.
5) How do I estimate radius of curvature from a weather map?
Trace an isobar near your location and approximate its local arc. Use the best-fit circle radius for that arc. Broad, gently curved isobars imply large radius; tight bends imply smaller radius.
6) Which pressure-gradient unit should I choose?
Use hPa per 100 km for typical synoptic map readings. Use Pa per m for numerical-model output or instrument-scale gradients. The tool converts all units internally to Pa/m for consistency.
7) Does this include friction or boundary-layer effects?
No. The calculation assumes steady, horizontal, frictionless balance. In the boundary layer, friction reduces wind speed and turns flow across isobars, so observed winds may be lower than the computed gradient wind.