AVERAGES AN D DIFFEREN CES

This is entirely different from the section on power series, but results turn out to be very similar. Here, we work with

q_{H}(x, t) = q(x, H(x, t), t) and q_{B}(x, t) = q(x, B(x, t), t) , (evaluations)
q_{D}(x, t) = q_{H} − q_{B} with z_{D} = F (x, t) = H − B ,  (differences)
q_{A}(x, t) = \frac{1}{2}(q_{H} + q_{B}) with z_{A} = Z(x, t) = \frac{1}{2}(H + B) . (averages)

The notations F and Z (fatness and mean height) will generally be used in preference to z_{D} and z_{A}.

The kinematic conditions are

H_{t} + u_{H}H_{x} = w_{H} and B_{t} + u_{B}B_{x} = w_{B} ,

and the average and difference of them gives the exact results,

Z_{t} + u_{A}Z_{x} + \frac{1}{4}u_{D}F_{x} = w_{A} and F_{t} + u_{A}F_{x} + u_{D}Z_{x} = w_{D} .

The approximation here is that u = u_{A} + (z − Z)u_{D}/F, and the integral of the continuity equation then implies

{w_{D} = Z_{x}u_{D} − F u_{Ax} , F_{t} + u_{A}F_{x} + F u_{Ax} = 0 \atop and w_{A} = B_{t} + u_{A}B_{x} − \frac{1}{2}F u_{Ax} + \frac{1}{4}F_{x}u_{D} .}

From the chain rule and the kinematic conditions it follows that

(q_{t} + uq_{x} + wq_{z})_{H} = q_{Ht} + u_{H}q_{Hx} and (q_{t} + uq_{x} + wq_{z})_{B} = q_{Bt} + u_{B}q_{Bx} .

Next, for & = H or B, comes

u_{&t} + u_{&}u_{&x} + p_{&x} − &_{x}p_{z&} = 0 and w_{&t} + u_{&}w_{&x} + p_{z&} + g = 0 ,

and, finally,

u_{At} + u_{A}u_{Ax} + \frac{1}{4}u_{D}u_{Dx} + p_{Ax} − Z_{x}p_{zA} − \frac{1}{4}F_{x}p_{zD} = 0 , ; u_{Dt} + u_{A}u_{Dx} + u_{D}u_{Ax} + p_{Dx} − Z_{x}p_{zD} − F_{x}p_{zA} = 0 , ; w_{At} + u_{A}w_{Ax} + \frac{1}{4}u_{D}w_{Dx} + p_{zA} + g = 0 , ; and w_{Dt} + u_{A}w_{Dx} + \frac{1}{4}u_{D}w_{Ax} + p_{zD} = 0 .

Similarities between these equations and those derived from truncated Taylor series can be seen by the use of

u_{0} = u_{A} , u_{1} = \frac{u_{D}}{F} , w_{0} = w_{A} − \frac{1}{8}F^{2}w_{2} , w_{1} = \frac{w_{d}}{F} ,

p_{0} = p_{A} − \frac{1}{8}F^{2}p_{2} and p_{1} = \frac{p_{D}}{F} − \frac{1}{24}F^{2}p_{3} .

When the substitutions are used to transform the various equations from one form to the other, results are found to be very similar, but not identical! Other small differences in nonlinear terms follow from quite a few different ways to look for equations for p_{2} and p_{3}. Some of the possibilities are:

• Take p_{2} and p_{3} from the Poisson equation (section on pressure equations). This will be taken to be the method of choice, to be used until some compelling argument to the contrary is found.

• Take the equation for w_{2t} from the section on power series.

• Use u_{0} = u_{A}, w_{0} = w_{A} − F^{2}w_{2}/8 and the equations that contain w_{0t} and w_{At} to derive an equation for (F^{2}w_{2})_{t}. The result is another, slightly different, equation for w_{2t}.

• Use the z-derivative of the vertical momentum equation to derive equations for w_{zDt} and w_{zAt}. It’s not particularly difficult, and the result is yet another equation for w_{2t} and another equation for w_{1t}.

• Introduce results from the section on means and moments, where the mean value of q is Q = ∫ q dz/F, the deviation is q̃ = q − Q, and the second approximation is based upon

u → U + z̃ũ_{1} and w → W + z̃w̃_{1} + \frac{1}{2} (z̃^{2} − \frac{F^{2}}{12} ) w̃_{2} .

After that, counting the number of different equations for p_{2} and p_{3} becomes more of a challenge.

Oh treacherous shallow waters - treacherous!

Equations for pressure coefficients, including p_{2} and p_{3}, will be found in the section on pressure equations, and only the gravity equation will be discussed here. The representation of pressure is

p = p_{H} + G(H − z) + p̀(x, y, z̃, t) , p_{B} = p_{H} + GF and p̀_{H} = p̀_{B} = 0 ,

and the vector and matrix notations are

u = (u, v)^{⊤} , ∇ = (∂_{x}, ∂_{y})^{⊤} , Δ = ∇^{⊤}∇ and \frac{D}{Dt} = ∂_{t} + u_{A}^{⊤}∇ .

Again, there are five degrees of freedom in

\frac{DH}{Dt} + F ∇^{⊤}u_{A} = B_{t} + u_{A}^{⊤}∇B ,

\frac{Du_{A}}{Dt} + \frac{1}{4}u_{D}^{⊤}∇u_{D} + ∇p_{H} + G∇H + \frac{1}{2}F ∇G = p̀_{zA}∇Z + \frac{1}{4}p̀_{zD}∇F ,

\frac{Du_{D}}{Dt} + u_{D}^{⊤}∇u_{A} = F ∇G + p̀_{zD}∇Z + p̀_{zA}∇F .

The gravity equation comes from

w_{A} = \frac{DB}{Dt} − \frac{1}{2}F ∇^{⊤}u_{A} + \frac{1}{4}u_{D}^{⊤}∇F and

G = g + \frac{Dw_{A}}{Dt} + \frac{1}{4}u_{D}^{⊤}∇(u_{D}^{⊤}∇Z − F ∇^{⊤}u_{A}) + p̀_{zA} .

The process is the same as in the section on series: The evaluation of Dw_{A}/Dt contains quite a few parts that have no explicit reference to G, p_{H} or p̀, and three that do, namely (∇B)^{⊤}Du_{A}/Dt, F ∇^{⊤}Du_{A}/Dt and (∇F)^{⊤}Du_{D}/Dt. The result is

(1 + ∇H^{⊤}∇B − \frac{1}{2}F ΔH) G − \frac{1}{4F^{2}} ∇^{⊤} (F^{4}∇G) = g + Γ + T ,

where

Γ = (\frac{1}{2}F ∇ − (∇B))^{⊤} (∇p_{H} − p̀_{zA}∇Z − \frac{1}{4}p̀_{zD}∇F )
+ \frac{1}{4}p̀_{zD}(∇F)^{⊤}∇Z + p̀_{zA} (1 + \frac{|∇F |^{2}}{4} )

and T contains all the virtual algebra. Slightly different treatments of this can give different coefficients (not 1) for F (∇F)^{⊤}∇G, but the coefficient of F^{2}ΔG is always 1/4.