POWER SERIES

For 2-D flow between z = B(x, t) and z = H(x, t) we first introduce the fatness and the mean (or average) height,

F (x, t) = H − B and Z(x, t) = \frac{1}{2}(H + B) = B + \frac{1}{2}F .

This second approximation is based upon the substitutions,

u → u_{0} + z̃u_{1} , w → w_{0} + z̃w_{1} + \frac{1}{2}z̃^{2}w_{2} , and p → p_{0} + z̃p_{1} + \frac{1}{2}z̃^{2}p_{2} + \frac{1}{6}z̃^{3}p_{3} ,

where z̃ = z − Z and −F/2 ≤ z̃ ≤ F/2 . Note the use of → to signify an association, rather than q = q_{0} + · · · . The second notation is too suggestive of Taylor series expansions and final determination of coefficients.

The expressions above are to be substituted in the Euler equations, and for that purpose it is convenient to make the change of variable in

u_{x} − Z_{x}uz̃ + wz̃ = 0, ; u_{t} + uu_{x} + (w − Z_{t} − uZ_{x})uz̃ + p_{x} − Z_{x}pz̃ = 0, ; w_{t} + uw_{x} + (w − Z_{t} − uZ_{x})wz̃ + pz̃ + g = 0.

Then

w_{1} = Z_{x}u_{1} − u_{0x} , w_{2} = −u_{1x} ,

and from the kinematic conditions follows

w − Z_{t} − uZ_{x} = \frac{1}{8} (F^{2}u_{1})_{x} − z̃u_{0x} − \frac{1}{2}z̃^{2}u_{1x} .

The three essential time-derivatives are in

F_{t} + u_{0}F_{x} + F u_{0x} = 0 , ; u_{0t} + u_{0}u_{0x} + \frac{1}{8}u_{1} (F^{2}u_{1})_{x} + p_{0x} − Z_{x}p_{1} = 0 , ; u_{1t} + u_{0}u_{1x} + u_{1}u_{0x} − u_{1}u_{0x} + p_{1x} − Z_{x}p_{2} = 0 .

(The cancellation of terms shown above changes when y and v are restored.) From Z = B + \frac{1}{2}F follows

w_{0} = B_{t} + u_{0}B_{x} − \frac{1}{2}F u_{0x} + \frac{1}{8} (F^{2}u_{1})_{x} ,

and three conditions on the pressure coefficients follow from

w_{0t} + u_{0}w_{0x} + \frac{1}{8} (F^{2}u_{1})_{x} (Z_{x}u_{1} − u_{0x}) + p_{1} + g = 0 , ; w_{1t} + u_{0}w_{1x} + u_{1}w_{0x} − u_{0x}(Z_{x}u_{1} − u_{0x}) − \frac{1}{8} (F^{2}u_{1})_{x} u_{1x} + p_{2} = 0 , ; w_{2t} + u_{0}w_{2x} + 2u_{1}(Z_{x}u_{1} − u_{0x})_{x} + 2u_{0x}u_{1x} + p_{3} = 0 .

Conceptually, it looks like that’s it: w’s have been evaluated in terms of u’s and their x-derivatives. The w_{t}’s provide relations between p’s, u’s and first and second x-derivatives of both. But these are treacherous waters, and there are several entirely different ways to treat the pressure. A different way to find pressure coefficients will appear in the section on pressure equations.

The coordinate y and the velocity component v will now be reintroduced. Before writing the equations, the representation of pressure will first be rearranged as

p_{H} + G(H − z) + p̀ and p̀ = \frac{1}{2} (z̃^{2} − \frac{F^{2}}{4} ) (p_{2} + z̃\frac{p_{3}}{3} ) .

That incorporates p_{H} explicitly and makes p_{B} = p_{H} + GF. Also, the vector and matrix notations,

u = (u, v)^{⊤} , ∇ = (∂_{x}, ∂_{y})^{⊤} , Δ = ∇^{⊤}∇ and \frac{D}{Dt} = ∂_{t} + u_{0}^{⊤}∇ ,

will be used. The five degrees of freedom appear in

\frac{DH}{Dt} + F ∇^{⊤}u_{0} = B_{t} + u_{0}^{⊤}∇B ,

\frac{Du_{0}}{Dt} + \frac{1}{8}u_{1}∇^{⊤}F^{2}u_{1} + ∇p_{H} + G∇H + \frac{1}{2}F ∇G + (∇p̀)_{0} = 0 ,

\frac{Du_{1}}{Dt} + u_{1}^{⊤}∇u_{0} − u_{1}∇^{⊤}u_{0} + (∇p̀)_{1} = ∇G ,

The formulas for the vertical velocity components are

w_{0} = \frac{DB}{Dt} − \frac{1}{2}F ∇^{⊤}u_{0} + \frac{1}{8} (∇^{⊤}F^{2}u_{1}) ,

w_{1} = u_{1}^{⊤}∇Z − ∇^{⊤}u_{0} and w_{2} = −∇^{⊤}u_{1} ,

and the vertical momentum equations are,

{\frac{Dw_{1}}{Dt} + ⋯ + p_{2} = 0 , \frac{Dw_{2}}{Dt} + ⋯ + p_{3} = 0 , \atop and \frac{Dw_{0}}{Dt} + \frac{1}{8}w_{1}∇^{⊤}F^{2}u_{1} + p̀_{1} + g = G .}

As suggested by the last equation, G will be called effective gravity, and the equation that follows from it, the gravity equation. The use of all three vertical momentum equations gives a direct derivation of three conditions on the pressure coefficients, but that willl not be done here. The section on pressure equations contains results that depend only the gravity equation and the Poisson equation that follows directly from the Euler equations and the continuity equation. The gravity equation has references to p̀ in it, but there will be no equation for determination of p̀ in this section.

Typical steps in the derivation of the gravity equation are

\frac{D∇^{⊤}u_{k}}{Dt} = ∇^{⊤} \frac{Du_{k}}{Dt} − ((∇u_{0}^{⊤})∇)^{⊤} ũ_{k}

and

\frac{D}{Dt} \frac{DB}{Dt} = B_{tt} + 2u_{0}^{⊤}∇B_{t} + u_{0}^{⊤}(∇∇^{⊤})u_{0} + (∇B)^{⊤} \frac{Du_{0}}{Dt} .

Substitutions for Du_{0}/Dt and Du_{1}/Dt give the equation for G:

(1 + (∇H)^{⊤}∇B − \frac{1}{2}F ΔH) G − \frac{3}{8}∇^{⊤}F^{2}∇G = g + Γ + T ,

where

Γ = (\frac{1}{2}F ∇ − (∇B))^{⊤} (∇p_{H} − (∇p̀)_{0}) + (p̀)_{1} − \frac{1}{8}F^{2}∇^{⊤}(∇p̀)_{1} ,

and T contains all the terms that have no explicit reference to G, p_{H} or p̀ in them. Evaluation of T is virtual algebra that needn’t be performed until someone decides to write a numerical implementation of this approximation.

Caution: (and this is a really treacherous point!) - the coefficient of F (∇F)^{⊤}∇G in the gravity equation can be 3/4, 7/8 or 1, depending on whether Du_{1}/Dt, (D/Dt)F u_{1} or (D/Dt)F^{2}u_{1} is used in the derivation. The choice is largely a matter of convenience. Further comments on this and other, similar gravity equations are in the section named that.