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I'm reading the following paper:

Intercomparison of Bulk Aerodynamic Algorithms for the Computation of Sea Surface Fluxes Using TOGA COARE and TAO Data

and am having trouble working through some of the equations, I have copied these equations below:

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Would someone be able to show me the process of how integrating (1) to (6) gives (7)?

Any help would be appreciated.

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1 Answer

up vote 3 down vote accepted

Just to clarify the text:

  • Integrating (5) gives (7) (the very unstable conditions)
  • Integrating (3) gives (8) (moderately unstable to marginally unstable)
  • Integrating (2) gives (9) (marginally to moderately stable)
  • Integrating (6) gives (10) (very stable).

Equations (1) and (4) are definitions that are used. So for example, to get (9):

$$\phi_m = \frac{k z}{u_*}\frac{du}{dz} = 1 + 5\zeta = 1 + 5\frac{z}{L}$$

which is setting the definition given by (1) to the empirical correlation given by (6) and substituting in (4) for $\zeta$. Rearranging gives:

$$\frac{du}{dz} = \frac{u_*}{kz} + 5 \frac{u_*}{kL}$$

Integrating from $z_0$ to $\xi$ (a dummy variable of integration that will take the value $z$ when done):

$$u(z) = \frac{u_*}{k}\ln \xi |^{z}_{z_0} + 5 \frac{u_* \xi}{L} |^{z}_{z_0}$$

which simplifies to:

$$u(z) = \frac{u_*}{k}\left(\ln \frac{z}{z_0} + 5 \frac{z}{L}\right)$$

Then you would use (4) to convert that last term back into $\zeta$ which gives (9).

A similar process is used for the rest.

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good explanation, although could you please elaborate on the point where you mention a dummy variable of integration... I cant see how this step works. –  Kate Jun 27 '13 at 11:25
    
@Kate All I mean there is mathematicians don't like integrating something like $\int_{t_0}^{t} f(t)dt$, it's bad notation. So they will put in a dummy variable, like $\int_{t_0}^{t} f(\tau)d\tau$ So all that step did was change $du/dz$ into $du/d\xi$ so I could integrate from $z_0$ to $z$ without getting downvoted for poor notation :) –  tpg2114 Jun 27 '13 at 13:20
    
where does the k disappear to in kL, third equation shown? –  Kate Jul 3 '13 at 5:28
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