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For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
in which the constant A is a vertical stretching factor, B is a horizontal shift (so that the curve has a y-axis intercept at a finite value), and k is the decay power. It can take other forms such as negative exponential, [2] i.e. .
In an analogous way, one can obtain finite difference approximations to higher order derivatives and differential operators. For example, by using the above central difference formula for f ′(x + h / 2 ) and f ′(x − h / 2 ) and applying a central difference formula for the derivative of f ′ at x, we obtain the central difference approximation of the second derivative of f:
Stream power is the rate of energy dissipation against the bed and banks of a river or stream per unit downstream length. It is given by the equation: = where Ω is the stream power, ρ is the density of water (1000 kg/m 3), g is acceleration due to gravity (9.8 m/s 2), Q is discharge (m 3 /s), and S is the channel slope.
The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (x, f(x)) and (x + h, f(x + h)). [1] Choosing a small number h, h represents a small change in x, and it can be either positive or negative.
Using the heat transfer equation of Fourier as template W.E.H. Culling reasoned the flux of mass across the height gradient of a slope would could be described in a similar fashion as: [7] [8] Equation (1) q̃ = −K∇z. On the left-hand side is sediment flux which is the volume of the mass that passes a line each time unit (L 3 /LT).
In geodynamics, dynamic topography refers to topography generated by the motion of zones of differing degrees of buoyancy (convection) in Earth's mantle. [1] It is also seen as the residual topography obtained by removing the isostatic contribution from the observed topography (i.e., the topography that cannot be explained by an isostatic equilibrium of the crust or the lithosphere resting on ...
An example of a geostrophic flow in the Northern Hemisphere. A northern-hemisphere gyre in geostrophic balance; paler water is less dense than dark water, but more dense than air; the outwards pressure gradient is balanced by the 90 degrees-right-of-flow coriolis force The structure will eventually dissipate due to friction and mixing of water properties.