# vertical wind gradients in the atmospheric boundary layer

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:

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

Any help would be appreciated.

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.

• 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. – KatyB 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? – KatyB Jul 3 '13 at 5:28