PRESSURE EQUAT ION S

The pressure has been written in several different ways, e.g.

p → p_{H} + G(H − z) + \frac{1}{2} (z̃^{2} − \frac{F^{2}}{4} ) p̂_{2} + \frac{1}{6}z̃ (z̃^{2} − \frac{F^{2}}{4} ) p̂_{3} + p̀,

and p → p_{0} + z̃p_{1} + \frac{1}{2}z̃^{2}p_{2} + \frac{1}{6}z̃^{3}p_{3} + p̀,

where F = H − B, Z = (H + B)/2 and z̃ = z − Z. In general, the coefficients {p̂_{k}} and {p_{k}} are different from one another, but either set can be expressed in terms of the other when the degree of the polynomial approximations has been specified. (It is three in the examples.) The functions p̀(x, y, z̃, t) are different too, depending on what is intended for further approximations: functions with p̀ = p̀z̃ = 0 at z̃ = ±F/2 is a possible choice for the first example, while further terms of the Taylor series is implied in the second. The first example, or any other that consistently employs p̀_{H} = p̀_{B} = 0, is useful for the writing of a gravity equation that defines p_{B}, albeit implicitly, and the second is far more suitable for use with the Poisson equation,

p_{xx} + p_{yy} + p_{zz} + ϕ = 0 with ϕ = u_{x}^{2} + v_{y}^{2} + w_{z}^{2} + 2(u_{y}v_{x} + v_{z}w_{y} + w_{x}u_{z}) .

The change of variables is

q(x, y, z, t) → q(x, y, z̃, t) , q_{t} → q_{t}−Z_{t}qz̃ and q_{tt} → q_{tt}−Z_{tt}qz̃−2Z_{t}q_{z̃t}+Z_{t}^{2}q_{z̃z̃} ,

with similar expressions for x− and y−derivatives. The advantage of the representation of p(z̃) as a Taylor series is that the recurrence relation for pressure coefficients is fixed. After ϕ has been written in the form ϕ = ∑ ϕ_{k}z̃^{k}/k! ,

Δp_{k} − \frac{1}{p_{k+1}} ∇^{⊤}p_{k+1}^{2}∇Z + (1 + |∇Z|^{2})p_{k+2} + ϕ_{k} = 0,

where ∇ = (∂_{x}, ∂_{y})^{⊤} and Δ = ∇^{⊤}∇. The result looks like a sequence of evaluations of p_{k+2}, but that is not quite true because of the way coefficients have been rearranged to formulate the gravity equation.

For the treatment of relatively high orders of approximation, it may be preferable to write the entire series as

p = p_{H} + G(H − z) + ∑_{ℓ=1}^{∞} \frac{1}{(2ℓ)!} (z̃^{2ℓ} − (F/2)^{2ℓ}) (p_{2ℓ} + \frac{p_{2ℓ+1}}{2ℓ + 1}) .

Then

p_{0} = p_{H} + \frac{1}{2}GF − ∑_{ℓ=1}^{∞} \frac{p_{2ℓ}}{(2ℓ)!}(F/2)^{2ℓ} and p_{1} = −G − \frac{2}{F} ∑_{ℓ=1}^{∞} \frac{p_{2ℓ+1}}{(2ℓ + 1)!}(F/2)^{2ℓ+1}

Take note: There are quite a few ways to derive different equations for {p_{k}}. One example of another way will be discussed in the section on Green-Naghdi theory.

The related problem, to find the Taylor series for ϕ, is most easily done by first changing the description of u from coefficients û_{k} or ũ_{k} (or others) to the coefficients u_{k}, For example, to the fourth order in orthogonal polynomials,

u → U + z̃ũ_{1} + \frac{1}{2} (z̃^{2} − \frac{F^{2}}{12} ) ũ_{2} + \frac{1}{6}z̃ (z̃^{2} − \frac{3F^{2}}{20} ) ũ_{3}
= (U − \frac{F^{2}}{24} ũ_{2}) + z̃ (ũ_{1} − \frac{F^{2}}{40} ũ_{3}) + \frac{1}{2}z̃^{2}ũ_{2} + \frac{1}{6}z̃^{3}ũ_{3} .

Given the u_{k}’s, the w’s after w_{0} are defined by the fixed recurrence relation

w_{k+1} = u_{k}^{⊤}∇Z − ∇^{⊤}u_{k} ,

and the rest is just more virtual algebra.