From b0e3012de711bee75b882eddf021514af8b095b2 Mon Sep 17 00:00:00 2001 From: ibidyouadu Date: Wed, 24 Jun 2020 02:02:13 -0400 Subject: [PATCH] summary procedure --- .../gravity_current_umana_anderson.pdf | Bin 261113 -> 276103 bytes .../gravity_current_umana_anderson.tex | 168 ++++++++++++++++-- 2 files changed, 156 insertions(+), 12 deletions(-) diff --git a/documentation/gravity_current_umana_anderson.pdf b/documentation/gravity_current_umana_anderson.pdf index f48b5f820f8393915158817a32ca25c5aec6ce59..9706db21bf85c8dd6e4f54560cf35d53d82671f7 100644 GIT binary patch delta 120079 zcmV)KK)S#A_Ya5d5RfASH#s?z!7V6%l~_%a+%yoq@2_xivWtvjec5s-st5(42!twX zH*g>vv}5gzDUZF_GucgkJ+CEs#xv^#I9ZaV{&@YmTXvpol05n0D*wdu+ncNBuXvI0 zk_oO=a&wo6oHL_&QV7G8lF3b-{E@!tvy~{)+OCZO9%KJ=<)P|LRS8Y#A z4>aTHLzAs|y5EM`b=%x$f8M-?wld+IRay(uCRxQ*f-OvP9(29zHm=1fh`BvyXiudt z1iZ=&P-})e`}i!J{Vn5pmA(uzg~(+dU%b73>7}d?4KCr9VzLrJ%+Gd=&K&L$ACuM@ zPwn2l$W}@g>FYxmjpWs{C^;87iF-nX=dYwmDpnQn6=6|^Nf^6QITNLSigCFe_k&&U zKXHD3e8sh9TH|RLH{#;_N=lh-dT%jM##E(SH@NVEF&2m-&m}6r2c`&OOceO$8b^#w z$HQ&5DpdNH^U{q<*R37L=3UO!8jzSFi@caldZ2%PyJ2#!Olbq>%1g-Ri6LYecwyyh*2n>`+~|E9gFa=seVi-^4P~xz9ZC|Cy6I3wm-L~CYL+X15%G+Geq2ynp=}(s25(7m z5qT&Joo*-oH=AutB6@FcgBd>Gd_TE37m~+=hvo5rHnNU;cFtlxy^72o;s?( z>(9g<1tF*!K}CyFfFfi@aAwY-E~GVe6?#M~#f#CxG|c zFVqd%^$6a7gsyDTv0o56xVacZW`)!4=rgtQ@N(+YM`QNeW~dhuC9PQ{0cJ13CIwF6m+Hs85TOA5kOs-mhG69AKUTq+NQZQC!xk^^cI2jcO&q2Gm3 z@C1Xgsa<^Q?=|yxv!cMl#o$<+NcR-4JQgP(a1C#Nbal>hUvvZ@?pAQX?wcbyAW$@q09-;?*6`QLWL8t!pW&RiiPfpw`YxqcG>uBLkfqc zzZgAKFsX}wDHyoEOgp}YU|pqoz(onpGys{iB@~i!$|MZ#1{X)C{bq%!!meKyG#Qo`SU7F^PeEG>7I2o;JXQgNPv7Dyro?B-1VZw85{lBK zIg9bdM^M4ld3ie0g0P=x;R9-8tWFd-+zw5w5n=!R{ouROW3xLhD_Ta2MsbQCDZV&= zoj4-TuWp!+{Y)pB%JC5`!sXo*5#&5uRa_n;E6#9{*f-0ed@&pt=2{}$GJa2fLz$8K z5A3V&Z?68n;t+w)2;7h8vq>ivnif8|*So7f{>+ma@6huYdO`BwckU>?3rR6yH=28!VPGqc(IBWUSr9WiT@CPzWjtM7|!nVg~zRACE%Jn82%lYGmhiZM$FR@ zSg}mq68FQ^Y}?@}OlW@0l>Z4P*i`Dz!6Y#jNXDlyeOz$D{|bQtXgU1Mi;F}p;3_@< zrT8CMO0iiAjP&G*yT)d*O>(;Ah_lZuDsNw%efwVuRwzrwy#8}v36_2rAa!#S!&E7K zYG^%u*3S1LLjM51LX@SK5&HoXvu6h^WdbxXlff-0v%PFH2!Am9Z5Z&#+9R|#cUc(x z`-IXKOrhd1>Cd8BykGe7-4CyVK|ZwgW)rNo3mJ|-@n;X+k0Yv8-F9}Y54{iL93kYo zkbskSi!L}Y+iN(ulcxU4)Zaygh5zCR(>1oN>bc!;$!|3`>5OtirQ@0@NN z5Vi@={|RB{oPV*Y?-L<|{Xng(ilyq!wLwilK1j|(c(}nZ&W+pKO?Gka?)&}3eX~c+ z*oKF$p81sk+#0Alk?OA`98~JOT6*;Lix+-Au1O=La3y|t&B=Kh(CD$7cU)a<}b%8Fk{_B|;73Fncz-p(FWm1U`LT-=Upw|2km~U4 zt*8f9bdwXbI6++3NsGBZq_ODVscQ<+M}eBZ?A4PsXt2u6Vv3tf{g8&E3Jop+zE54a zP~U9MJcM}W!X*ehJ;Y=3(nW#&{=d8nfBKgxCcyVa!o&-+^DQihQR2^g^&A-8lkc7t zVTMoMJvu*z5C5c-uN8zLO3b1(_D`WS&k8yFKG353AK%@SCwDi;I~D_e(Yff&$&P z0TQIiATLRPddHfv0be%0?9BfAeiSKN)NmvX?Af+JUo2Uo4teh1IV3Ia3tIf`ipJ|- zZm!xVqwIR{T2oVY9EUOCiK$KcDXXoy1*yv{+VzUnaexr&yj$anOm@b!_Yi=hGyYT+7ivNMthZc^bTjBxmiw-GHko<+UR9wW;fB6G&*AQuL`!%T!| z+$yKlh8)Nq^ngoS^3v31NmRW7Y%YlyJVe_OeR0O3}eDgqzu5r-y5>QovjlMN{aE);K8#-yLL zfc7ke$?!_)|29__ToK_Q1SW{3;v8Q;lA;hhlb7>#JK*a&_&U3&uZt1BZeX)qk1!y{ z8BEA^OemT3L=3B>?h-*9;lM`;2jQF-!y7`BXci|FXY17{GcJaY(W?~lf1DWJnH@

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a/documentation/gravity_current_umana_anderson.tex +++ b/documentation/gravity_current_umana_anderson.tex @@ -565,21 +565,22 @@ Note that the inner solution~(\ref{eq:inner_solution}) allows us to write & = & - \frac{1}{\beta} \frac{H_M \left( \frac{1}{9} H_M^2 + 1 \right) }{ \left( \frac{1}{3} H_M^2 + 1 \right) \frac{\partial h}{\partial x}(x=x_M,t) } \eea -{\bf Summary of Procedure:} In summary, we generate a numerical `composite' solution by solving numerically the outer problem +{\bf Summary of Procedure (Possible Version 1):} In summary, we generate a numerical `composite' solution by solving numerically the outer problem \bea \label{eq:outer_h} \frac{\partial h}{\partial t} & = & \frac{\partial}{\partial x} \left[ f(h)\frac{\partial h}{\partial x} \right], \hspace{0.25in} \mbox{on $0 < x < x_M(t)$},\\ \label{eq:outer_bc1} \frac{\partial h}{\partial x}(x=0,t) & = & 0,\\ \label{eq:outer_bc2} -h(x=x_M(t)) & = & h_M, +h(x=x_M(t)) & = & h_M(t), \eea where the leading-edge of the gravity current, $x_N(t)$, satisfies \bea \label{eq:outer_bc3} -\frac{dx_N}{dt} & = & - \beta \left( \frac{1}{3} H^2_M + 1 \right) \left. \frac{\partial h}{\partial x} \right|_{x=x_M}, +\frac{dx_N}{dt} & = & - \beta \left( \frac{1}{3} H^2_M + 1 \right) \left. \frac{\partial h}{\partial x} \right|_{x=x_M}. \eea -and the values $x_M$, $\zeta_M$, $h_M$ and $H_M$ are obtained by fixing one of them (e.g.~fixing $H_M$) and determining the other three using the equations +Recall in the present context that $f(h) = \frac{1}{3} h^3 + \beta h$. +The values $x_M$, $\zeta_M$, $h_M$ and $H_M$ are obtained by fixing one of them (e.g.~fixing $\zeta_M$ appears to have some advantages but perhaps there are advantages to fixing a different one, such as $H_M$) and determining the other three using the equations \bea \label{eq:hM} h_M & = & \beta^{\frac{1}{2}} H_M, \\ @@ -588,7 +589,8 @@ x_M(t) & = & x_N(t) - \beta^{\frac{3}{2}} \zeta_M,\\ \label{eq:zM} \zeta_M & = & - \frac{1}{\beta} \frac{H_M \left( \frac{1}{9} H_M^2 + 1 \right) }{ \left( \frac{1}{3} H_M^2 + 1 \right) \frac{\partial h}{\partial x}(x=x_M,t) }. \eea -The inner solution on the region $x_M < x < x_N$ is determined by the inner solution formula~(\ref{eq:inner_solution}). +The inner solution on the region $x_M < x < x_N$ is determined by the inner solution formula~(\ref{eq:inner_solution}). Note that unless one of these quantities is fixed in time, +in general each of $x_M$, $\zeta_M$, $h_M$ and $H_M$ are functions of time. The outer solution applies on $0 < x < x_M(t)$ and it will be convenient to map this domain to a fixed domain using the spatial coordinate $s=x/x_M(t)$ where $s \in [0,1]$. In terms of this fixed domain we have \bea @@ -642,10 +644,9 @@ $s$ (and $x= s x_M(t)$) which give more resolution near the leading-edge of the {\bf Outline of algorithm: {\color{red}{DMA still thinking about some of these ideas ...}}} \begin{enumerate} -\item Suppose we have a solution at time $t=t_i$ so that $x_N(t_i)$ is known and $h(s,t=t_i)$ for $0 < s < 1$ is known. -At the beginning this would correspond to a given initial condition at $t=t_i$. Later, this will represent a solution obtained at a previous time step. +\item Suppose we have a solution at some initial time $t=t_i$ so that $x_N(t_i)$ is known and $h(x,t=t_i)$ for $0 < x < x_N(t_i)$ is known. % -\item Choose a value $\zeta_M$ to be fixed throughout the calculation. Then $x_M(t_i)$ can be obtained from equation~(\ref{eq:xM}) since $x_N(t_i)$ is known. Further +\item Choose a value $\zeta_M$ that is meant to be fixed throughout the calculation. Then $x_M(t_i)$ can be obtained from equation~(\ref{eq:xM}) since $x_N(t_i)$ is known. Further observe that if $\zeta_M$ is fixed throughout the calculation it follows from equation~(\ref{eq:xM}) that $d x_M/dt = d x_N/dt$ for all time. % \item Next, we obtain $h_M(t_i)$ by computing $h(x_M(t_i),t_i)$ and subsequently can use equation~(\ref{eq:hM}) to obtain $H_M(t_i)$. @@ -655,14 +656,157 @@ observe that if $\zeta_M$ is fixed throughout the calculation it follows from eq \frac{\partial h}{\partial x}(x=x_M(t_i),t_i) = \frac{1}{x_M} \frac{\partial h}{\partial s}(s=1,t_i)& = & - \frac{1}{\beta} \frac{H_M \left( \frac{1}{9} H_M^2 + 1 \right) }{ \left( \frac{1}{3} H_M^2 + 1 \right) \zeta_M }. \eea At this point we must verify that the right-hand-side of this expression, which is known since values for $H_M$ and $\zeta_M$ are already specified, matches the left-hand-side. -That is, observe that the known profile $h(s,t_i)$ defines, say by a finite difference approximation, the derivative $\frac{\partial h}{\partial x}(x=x_M(t_i),t_i)$. Thus, we +That is, observe that the known profile $h(x,t_i)$ defines, say by a finite difference approximation, the derivative $\frac{\partial h}{\partial x}(x=x_M(t_i),t_i)$. Thus, we must specify a tolerance within which these two quantities must agree. If they fall outside of the tolerance, then some further iteration must be taken to -achieve the required agreement. +achieve the required agreement. Possibly the initial condition could be modified or the solution approach could be restarted with a different choice for $\zeta_M$. % -\item Once suitable $x_M$, $h_M$, $\zeta_M$ and $H_M$ are obtained, solve numerically (e.g. finite difference method, method of lines) the equations~(\ref{eq:outer_h_fixed_domain}) -subject to boundary conditions~(\ref{eq:outer_bc1_fixed_domain}), (\ref{eq:outer_bc2_fixed_domain}) and~(\ref{eq:outer_bc3_fixed_domain}) to advance the solution to the next time step and repeat the steps starting from the top. +\item Once suitable $x_M(t_i)$, $h_M(t_i)$, $\zeta_M(t_i)$ and $H_M(t_i)$ are obtained, solve numerically (e.g. finite difference method, method of lines) the equations~(\ref{eq:outer_h_fixed_domain_drds}) +subject to boundary conditions~(\ref{eq:outer_bc1_fixed_domain_drds}), (\ref{eq:outer_bc2_fixed_domain_drds}) and~(\ref{eq:outer_bc3_fixed_domain_drds}) to advance the solution to the next time step, $t=t_{i+1}$. Specifically advancing one time step would provide $x_N(t_{i+1})$, $h(x,t_{i+1})$ for all $x$ up to but (possibly?) not including $x_M$. Then, +determine the next round of values $h_M$, $H_M$, $x_M$ assuming $\zeta_M$ fixed??? \end{enumerate} +{\bf Summary Procedure (Version 2)}: As an alternate approach to that partially outlined above we can revisit the problem +\bea +\label{eq:outer_h_v2} +\frac{\partial h}{\partial t} & = & \frac{\partial}{\partial x} \left[ f(h)\frac{\partial h}{\partial x} \right], \hspace{0.25in} \mbox{on $0 < x < x_M(t)$},\\ +\label{eq:outer_bc1_v2} +\frac{\partial h}{\partial x}(x=0,t) & = & 0,\\ +\label{eq:outer_bc2_v2} +h(x=x_M(t)) & = & h_M(t), +\eea +where the leading-edge of the gravity current, $x_N(t)$, satisfies +\bea +\label{eq:outer_bc3_v2} +\frac{dx_N}{dt} & = & - \beta \left( \frac{1}{3} H^2_M + 1 \right) \left. \frac{\partial h}{\partial x} \right|_{x=x_M}. +\eea +Recall that in the present context $f(h) = \frac{1}{3} h^3 + \beta h$. +Suppose we fix the value $\zeta_M$. Recall that we have the relationships +\bea +\label{eq:hM_v2} +h_M & = & \beta^{\frac{1}{2}} H_M, \\ +\label{eq:xM_v2} +x_M(t) & = & x_N(t) - \beta^{\frac{3}{2}} \zeta_M,\\ +\label{eq:zM_v2} +\zeta_M & = & - \frac{1}{\beta} \frac{H_M \left( \frac{1}{9} H_M^2 + 1 \right) }{ \left( \frac{1}{3} H_M^2 + 1 \right) \frac{\partial h}{\partial x}(x=x_M,t) }. +\eea +Note that by fixing $\zeta_M$ it follows that $dx_M/dt = dx_N/dt$. +Again note that the inner solution on the region $x_M < x < x_N$ is determined by the inner solution formula~(\ref{eq:inner_solution}). + +In this context $H_M$ is a function of time for which we can obtain an expression for $dH_M/dt$. In particular, rearranging equation~(\ref{eq:zM_v2}) gives +\bea +\label{eq:zM_v2_rearranged} +-\beta \zeta_M \left( \frac{1}{3} H_M^2 + 1 \right) \frac{\partial h}{\partial x}(x=x_M,t) = H_M \left( \frac{1}{9} H_M^2 + 1 \right) +\eea +We can compute the time derivative of this quantity to obtain an expression for $dH_M/dt$. First note that +\bea +\frac{d}{dt} \left[ \frac{\partial h}{\partial x}(x=x_M,t) \right] = \left. \frac{\partial^2 h}{\partial x^2} \right|_{x=x_M} \frac{dx_M}{dt} + \left. \frac{\partial^2 h}{\partial t \partial x} \right|_{x=x_M}, +\eea +where +\bea +\left. \frac{\partial^2 h}{\partial t \partial x} \right|_{x=x_M} & = & \left. \frac{\partial^2}{\partial x^2} \left[ f(h) \frac{\partial h}{\partial x} \right] \right|_{x=x_M},\\ + & = & \left. \frac{\partial}{\partial x} \left[ f(h) \frac{\partial^2 h}{\partial x^2} + f'(h) \left( \frac{\partial h}{\partial x} \right)^2 \right]\right|_{x=x_M},\\ + & = & \left. \left[ f(h) \frac{\partial^3 h}{\partial x^3} + 3 f'(h) \frac{\partial h}{\partial x} \frac{\partial^2 h}{\partial x^2} + f''(h) \left( \frac{\partial h}{\partial x} \right)^3 \right]\right|_{x=x_M} +\eea +Remember that since $f(h) = \frac{1}{3} h^3 + \beta h$ we have $f'(h) = h^2 + \beta$ and $f''(h) = 2h$. +Returning to equation~(\ref{eq:zM_v2_rearranged}) and computing a time derivative of both sides gives +\bea +-\beta \zeta_M \left\{ \left( \frac{1}{3} H_M^2 + 1 \right) \frac{d}{dt} \left[ \frac{\partial h}{\partial x}(x=x_M,t) \right] + \frac{2}{3} H_M \frac{dH_M}{dt} \left. \frac{\partial h}{\partial x}\right|_{x=x_M} \right\} + = \left( \frac{1}{3} H_M^2 + 1 \right) \frac{dH_M}{dt} +\eea +Solving for $dH_M/dt$ gives +\bea +\frac{dH_M}{dt} & = & \frac{-\beta \zeta_M \left( \frac{1}{3} H_M^2 + 1 \right) \left[ \left. \frac{\partial^2 h}{\partial x^2} \right|_{x=x_M} \frac{dx_M}{dt} + \left. \frac{\partial^2 h}{\partial t \partial x} \right|_{x=x_M}\right] }{\left( \frac{1}{3} H_M^2 + 1 \right) + \frac{2}{3} \beta \zeta_M H_M \left. \frac{\partial h}{\partial x}\right|_{x=x_M} } +\eea +For the moment let's write this as +\bea +\frac{dH_M}{dt} & = & {\cal F}(\beta,\zeta_M,H_M,h,\partial h/\partial x, \partial^2 h/\partial x^2 , \partial^3 h/\partial x^3), +\eea +where $h$ and its derivatives are all evaluated at $x=x_M$. +Recall that +\bea +\frac{dx_M}{dt} & = & - \beta \left( \frac{1}{3} H^2_M + 1 \right) \left. \frac{\partial h}{\partial x} \right|_{x=x_M} +\eea + +The outer solution applies on $0 < x < x_M(t)$ and it will be convenient to map this domain to a fixed domain using the spatial coordinate $s=x/x_M(t)$ where $s \in [0,1]$. In terms of this fixed domain we have +\bea +\label{eq:outer_h_fixed_domain_v2} +\frac{\partial h}{\partial t} & = & \frac{s}{x_M} \frac{dx_M}{dt} \frac{\partial h}{\partial s} + \frac{1}{x_M^2} \frac{\partial}{\partial s} \left[ f(h)\frac{\partial h}{\partial s} \right], \hspace{0.25in} \mbox{on $0 < s < 1$},\\ +\label{eq:outer_bc1_fixed_domain_v2} +\frac{\partial h}{\partial s}(s=0,t) & = & 0,\\ +\label{eq:outer_bc2_fixed_domain_v2} +h(s=1) & = & h_M(t), +\eea +where +\bea +\label{eq:outer_bc3_fixed_domain_v2} +\frac{dx_N}{dt} & = & - \beta \left( \frac{1}{3} H^2_M + 1 \right) \frac{1}{x_M} \left. \frac{\partial h}{\partial s} \right|_{s=1}. +\eea +and +\bea +\label{eq:outer_bc_hm_ode} +\frac{1}{\beta^{1/2}} \frac{dh_M}{dt} = \frac{dH_M}{dt} & = & {\cal F}(\beta,\zeta_M,H_M,h,\partial h/\partial x, \partial^2 h/\partial x^2 , \partial^3 h/\partial x^3), +\eea +Note that the PDE in~(\ref{eq:outer_h_fixed_domain_v2}) requires that we have access to $x_M$, $dx_M/dt$ and $h_M(t)$. The boundary condition~(\ref{eq:outer_bc3_fixed_domain_v2}) +expresses the equation for the evolution of the actual leading-edge location $x_N(t)$ but with $\zeta_M$ assumed fixed we have $dx_M/dt = dx_N/dt$. Finally, the time dependent +$h_M(t)$ is obtained by solving simultaneously the ODE~(\ref{eq:outer_bc_hm_ode}) for $h_M$. Note that the coordinate transformation gives +\bea +\frac{\partial h}{\partial x} & = & \frac{1}{x_M} \frac{\partial h}{\partial s},\\ +\frac{\partial^2 h}{\partial x^2} & = & \frac{1}{x_M^2} \frac{\partial^2 h}{\partial s^2},\\ +\frac{\partial^3 h}{\partial x^3} & = & \frac{1}{x_M^3} \frac{\partial^3 h}{\partial s^3}. +\eea + +Converting this to a domain in which non-equally-spaced points can be implemented using the same relationship between $r$ and $s$ as used earlier gives +\bea +\label{eq:outer_h_fixed_domain_drds_v2} +\frac{\partial h}{\partial t} & = & \frac{s}{x_M} \frac{dx_M}{dt} \frac{dr}{ds} \frac{\partial h}{\partial r} + \frac{1}{x_M^2} \frac{dr}{ds} \frac{\partial}{\partial r} +\left[ f(h) \frac{dr}{ds} \frac{\partial h}{\partial r} \right], \hspace{0.25in} \mbox{on $0 < r < 1$},\\ +\label{eq:outer_bc1_fixed_domain_drds_v2} +\frac{\partial h}{\partial r}(r=0,t) & = & 0,\\ +\label{eq:outer_bc2_fixed_domain_drds_v2} +h(r=1) & = & h_M, +\eea +where +\bea +\label{eq:outer_bc3_fixed_domain_drds_v2} +\frac{dx_N}{dt} & = & - \beta \left( \frac{1}{3} H^2_M + 1 \right) \frac{1}{x_M} \frac{dr}{ds} \left. \frac{\partial h}{\partial r} \right|_{r=1}, +\eea +and +\bea +\label{eq:outer_bc_hm_ode_drds} +\frac{1}{\beta^{1/2}} \frac{dh_M}{dt} = \frac{dH_M}{dt} & = & {\cal F}(\beta,\zeta_M,H_M,h,\partial h/\partial x, \partial^2 h/\partial x^2 , \partial^3 h/\partial x^3), +\eea +where $h$ and its derivatives are evaluated at $r=1$. In terms of the new coordinate we have +\bea +\frac{\partial h}{\partial s} & = & \frac{\partial h}{\partial r} \frac{dr}{ds}, \\ +\frac{\partial^2 h}{\partial s^2} & = & \frac{\partial^2 h}{\partial r^2} \left( \frac{dr}{ds} \right)^2 + \frac{\partial h}{\partial r} \frac{d^2r}{ds^2},\\ +\frac{\partial^3 h}{\partial s^3} & = & \frac{\partial^3 h}{\partial r^3} \left( \frac{dr}{ds} \right)^3 + 3 \frac{\partial^2 h}{\partial r^2} \frac{dr}{ds} \frac{d^2 r}{ds^2} + \frac{\partial h}{\partial r} \frac{d^3r}{ds^3}. +\eea +{\color{red}{Double-check these!}} +Here, as before, we have +\bea +s & = & \frac{1- \exp (-\lambda r)}{1 - \exp (-\lambda)}, +\eea +or equivalently +\bea +r & = & - \frac{1}{\lambda} \ln \left[ 1 - s (1-e^{-\lambda}) \right], +\eea +and +\bea +\frac{dr}{ds} & = & \frac{ 1 - \exp(-\lambda) } {\lambda (1 - s(1 - \exp(-\lambda)))},\\ +\frac{d^2r}{ds^2} & = & \frac{ [1 - \exp(-\lambda) ]^2} {\lambda [1 - s(1 - \exp(-\lambda))]^2} = \lambda \left( \frac{dr}{ds} \right)^2,\\ +\frac{d^3r}{ds^3} & = & 2 \frac{ [1 - \exp(-\lambda) ]^3} {\lambda [1 - s(1 - \exp(-\lambda))]^3} = 2\lambda^2 \left( \frac{dr}{ds} \right)^3. +\eea +Again the idea is to implement this numerically using a uniform grid with respect to the variable $r$, which gives a nonuniformly distributed set of points in terms of +$s$ (and $x= s x_M(t)$) which give more resolution near the leading-edge of the gravity current as the parameter $\lambda$ is increased. + + +{\color{red}{Some comments: (1) With the need of higher derivatives of $r$ with respect to $s$ it seems likely that our choice of $\lambda$ will be much more limited than in the original version of our calculations but maybe we can get by with smaller $\lambda$ in this context? (2) We need one-sided derivative formulas for the second and third derivatives that appear in ${\cal F}$. (3) In principle this seems like it could be implemented in a pretty straightforward fashion to what we have done so far ...}} + + + + + + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\begin{figure}[h!] %\begin{center} -- 2.43.0