River meanders: The meander length, meander width, and river width are empirically observed to vary roughly with which function of discharge?

Introduction / Context:
Alluvial rivers adjust their planform and cross section to water and sediment supply. Empirical regime relationships relate channel width, meander wavelength, and other geometric attributes to the bankfull discharge.

Given Data / Assumptions:

• We seek the approximate functional dependence of meander scale and channel width on discharge.
• Alluvial, freely meandering rivers with mobile bed and banks.

Concept / Approach:
Regime and hydraulic-geometry studies (e.g., Leopold–Maddock type relations) commonly show power-law trends where width and meander wavelength increase with discharge to a power less than 1. A frequently cited approximation is proportionality to Q^0.5 (square root dependence) for width and related meander metrics.

Step-by-Step Solution:
Recall empirical relations: width ∝ Q^b with b around 0.4–0.6.Meander wavelength and belt width scale with channel width and hence with Q^b.Approximate practical choice: square root (Q^0.5).

Verification / Alternative check:
Field datasets show scatter, but the square-root trend is commonly accepted for quick estimates, with constants fitted to local conditions.

Why Other Options Are Wrong:

• Linear, square, cube powers greatly overpredict size at high discharges.
• None of these: incorrect because the square-root relation is widely used as a rule of thumb.

Common Pitfalls:
Applying a single exponent universally; ignoring sediment size, bank strength, and vegetation which modify coefficients and exponents.

Final Answer:
Square root of the discharge