NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-25-201-2018Brief communication: A nonlinear self-similar solution to barotropic flow over varying topographyIbanezRuyKuehlJosephjkuehl@udel.eduhttps://orcid.org/0000-0001-9912-8597ShresthaKalyanAndersonWilliamMechanical Engineering Department, University of Rochester, Rochester, NY 14627, USAMechanical Engineering Department, University of Delaware, Newark, DE 19716, USAMechanical Engineering Department, University of Texas Dallas, Dallas, TX 75080, USAJoseph Kuehl (jkuehl@udel.edu)6March20182512012051November201714November201725January201831January2018This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/This article is available from https://npg.copernicus.org/articles/25/201/2018/npg-25-201-2018.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/25/201/2018/npg-25-201-2018.pdf
Beginning from the shallow water equations (SWEs), a nonlinear self-similar
analytic solution is derived for barotropic flow over varying topography. We
study conditions relevant to the ocean slope where the flow is dominated by
Earth's rotation and topography. The solution is found to extend the
topographic β-plume solution of Kuehl (2014) in two ways. (1) The
solution is valid for intensifying jets. (2) The influence of nonlinear
advection is included. The SWEs are scaled to the case of a topographically
controlled jet, and then solved by introducing a similarity variable, η=cxnxyny. The nonlinear solution, valid for topographies h=h0-αxy3, takes the form of the Lambert W-function for pseudo
velocity. The linear solution, valid for topographies h=h0-αxy-γ, takes the form of the error function for transport. Kuehl's
results considered the case -1≤γ<1 which admits expanding jets,
while the new result considers the case γ<-1 which admits
intensifying jets and a nonlinear case with γ=-3.
Introduction
Slope topography represents both a barrier to large-scale geophysical fluid
transport as well as an important location of mesoscale feature generation.
Standard quasi-geostrophic theory (Pedlosky, 1987) indicates that large-scale
circulation features act in such a way as to conserve their potential
vorticity, leading to the standard result of flow along (as opposed to
across) topographic contours. Thus, slope topography creates a barrier
between the open and coastal oceans, often inhibiting the transport of
nutrient-rich waters into the coastal zone and at the same time trapping
pollutants in the coastal zone.
As both numerical and observational approaches have limitations with respect
to modeling the slope region, the objective of this brief communication is to
provide an analytic framework for flow along slope topographies. Such a
framework will serve as an idealized backbone upon which observational,
numerical, experimental and further theoretical work can build and provide a
point of comparison for better interpretation of the respective dynamics. In
particular, the results presented have implications for cross-topography
exchange and provide significant insight into the coupling between the slope
bottom boundary layer and interior water column dynamics.
Problem formulation
The problem formulation considered in this work follows that of Sansón
and van Heijst (2002), Kuehl (2014) and Kuehl and Sheremet (2014). A
rotating, single fluid layer is considered which flows along a sloping bottom
topography (i.e., along slope barotropic flow). The momentum equations and
continuity Eq. () for this situation are
ut-(f+ω)v=-(p+e)x+ν∇2u,vt+(f+ω)u=-(p+e)y+ν∇2v,ht+(hu)x+(hv)y+∇⋅ΠE=0,
(Pedlosky, 1987; Cushman-Roisin, 1994, provide the scaling which leads to
these equations) where u,v are the across and along slope flow velocities,
respectively, h is the fluid depth of the at rest state, p is the
pressure anomaly divided by water density (ρ), e=(u2+v2)/2 is
kinetic energy per unit mass, ν is the viscosity and f is the Coriolis
parameter. The effect of the viscous bottom boundary layer is accounted for
by a small correction term ΠE=12hEk^×u, the Ekman flux. Its
divergence, ∇⋅(ΠE)=-12hEω, represents first-order Ekman suction at a
solid boundary with Ekman layer depth, hE=2ν/f.
Taking the curl of the momentum equations, defining the vorticity as ω=vx-uy, defining an interior transport function ψ through hu=k^×∇ψ+∇ϕ (where ∇2ϕ=-∇⋅ΠE=12hEω represents Ekman
divergence), and simplifying by letting q=f+ωh, gives us
the vorticity-transport Eq. (),
ωt+J(ψ,q)=ν∇2ω-hE2qω.
The divergent component caused by the Ekman suction is small (ϕ/ψ=hE/h=O(10-2), so ϕ can be neglected in the vorticity
advection terms. It is standard to expand the Jacobian
Jψ,q=ψxqy-ψyqx, Jψ,q=1hJψ,ω-fh2Jψ,η+β(x)hψy-β(y)hψx, where β(x)=(hxf)/h and
β(y)=(hyf)/h are the average topographic beta-effects and η
is a small free-surface displacement and thus the total water column depth is
h+η.
Kuehl (2014) provides a scaling analysis which justified Ekman dissipation
being the dominant dissipative term and relative vorticity being dominated by
cross-stream shear, ω≈1hψxx. These assumptions
are valid for flows which exhibit scale separation between the along and
cross flow (topography) directions and are thus valid for flow along the
oceanic slope. These assumptions, along with the steady flow assumption,
truncate a Taylor expansion in 1h at leading order (neglecting
terms of O(1h2) ) and assume f≫ω are again made and
result in a leading-order governing equation of the form
ψxψxxy-ψyψxxx+fhxψy-fhyψx=-fhE2ψxx.
This equation (with appropriate boundary conditions) describes the linear and
first-order nonlinear dynamics of a barotropic flow along the oceanic slope.
It is upon this equation that several analytic solutions will be presented.
Linear solutionsExpanding jet
Kuehl (2014) considered the linear case of Eq. (),
fhxψy-fhyψx=-fhE2ψxx.
Noting its similarity to the heat equation, which has been pointed out by
others (in particular Csanady, 1978), Kuehl attempted to find a similarity
solution. The solution derivation will be sketched through here for
completeness (details in Kuehl, 2014). Assuming
topography of the form h=h0-αxy-γ,
similarity variable ζ=xkyn,
boundary conditions ψ(-∞,y)=0 and ψ(∞,y)=Q,
initial condition ψ(x,0)=Qsgn(x),
Eq. () reduces to -2ζg′=g′′, where g=ψ/Q, with
conditions n=-1+γ2 and k=α2hE1-γ12n. This equation
has a well-known solution, ψ=Qerf(ζ)+12, and parameters α and γ may be set to mimic the
desired topography. The “topographic β-plume” solution is valid in
the parameter range -1≤γ<1. For the solution to be real, we must
have γ<1 and, for γ<-1, the jet would be compressing, which
does not satisfy the initial conditions. Physically, the Ekman pumping in the
bottom boundary layer relaxes the topographic vorticity control, allowing the
jet to spread across isobaths.
Compressing jet
In nature, compressing (or intensifying) jets are often observed and an
analysis of ocean slope topography finds many locations where γ<-1
is relevant (Ibanez, 2016). To extend the above result to the case of
compressing jets, the initial condition used above must be revisited.
Similarity solutions require one point of reference to tether the solution.
It is most common to place this singularity at the origin, as is done above
and in many other classical cases such as the Blasius boundary layer
(Blasius, 1908; Rogers, 1992). However, in the present case, we choose to
relocate the singularity to y=∞. Upon relocation, the solution given
above is still valid, but the domain of physical relevance of the solution
has a slightly altered interpretation.
For the expanding jet case, the analytical solution is valid over the domain
y=[0:∞]. However, the physical relevance of the solution demands the
neglect of the region near y=0, due to the singularity, as well as the
region near y=∞, as this region violates the across and along jet
scale separation assumption, though the interior solution is indeed a
physically relevant description of geophysical systems. For the compressing
jet case, the situation is simply reversed. In this case, the analytical
solution is still valid over the domain y=[0:∞]. However, the
physical relevance of the solution demands the neglect of the region near y=0, as this region violates the across and along jet scale separation
assumption, and the region near y=∞, due to the singularity, but
again the interior solution is a physically relevant description of
geophysical systems. The region of applicability is ultimately governed by
the assumption ω≈1hψxx (i.e., ψxx≫ψyy), which is reasonable but should be checked in each particular
application. Thus, we have adopted the terminology that expanding jets are
those with a singularity at the upstream source region (y=0) and
compressing jets as those with the singularity at the downstream exit region
(y=∞).
Nonlinear solution
Motivated by the success and utility of the linear solutions provided above,
we seek a similarity solution for the nonlinear case (Eq. ).
Again, consider the normalized transport function, g=ψQ, and
introduce a similarity variable of the form η=cxnxyny, where
c,nx,ny are constants. Note that from this point on η will refer
to the similarity variable and not surface displacement. The relevant
derivatives take the forms
gy=g′∂η∂y=nycxnxyny-1g′,gx=g′∂η∂x=nxcxnx-1ynyg′,gxx=∂∂xg′∂η∂x=g′′∂η∂x2+g′∂2η∂x2=nx2c2x2(nx-1)y2nyg′′+nx(nx-1)cxnx-2ynyg′,gxxx=g′′′nx3c3x3(nx-1)y3ny+3g′′nxcxnx-1ynynx(nx-1)cxnx-2yny+g′nx(nx-1)(nx-2)cxnx-3yny,gxxy=g′′′nycxnxyny-1c2nx2x2(nx-1)y2ny+g′′2nxcxnx-1ynynxnycxnx-1yny-1+g′′nycxnxyny-1nx(nx-1)xnx-2yny+g′cnx(nx-1)nyxnx-2yny-1.
In this work, we are interested in straight slope topographies. Upon setting
nx=1, it is seen that the nonlinear terms simplify significantly.
Specifically, all g′g′ terms are set to zero. Also, it is found that the
g′g′′′ terms cancel. Thus, the only remaining nonlinear term is the g′g′′ term, which in Eq. () takes the form Q2g′g′′c3nyy3ny-1. Ultimately, Eq. () becomes
Qg′hxfcnyxyny-1-hyfcyny︸1+Qg′′fhe2c2y2ny︸2+Q2g′g′′c3nyy3ny-1︸3=0.
It is now convenient to address the y dependences of the coefficients in
terms 2 and 3 of Eq. (). We require the y dependency of terms 2
and 3 to balance, i.e., 2ny=3ny-1, which gives the condition ny=1.
Thus, the similarity variable has the form η=cxy. Apply this
condition, and upon division by the coefficient of term 2, this yields
2hecg′hxxy-2-hyy-1︸1+g′′︸2+2Qcfheg′g′′︸3=0.
Next, the bracketed portion of Eq. () in term 1 is considered.
Recall that h=h0-αxy-γ, hx=-αy-γ
and hy=αγxy-γ-1. We anticipate that the x must
be absorbed into an η term, so the bracketed terms become
-αcηy-γ-3(1+γ).
The y dependence is removed with the condition γ=-3 and the terms
in Eq. () reduce to 2αcη. It is then found
that Eq. () reduces to
4αhec2ηg′+g′′+2Qcfheg′g′′=0.
Note that, as expected, the limit of Eq. (), as Q→0,
recovers the linear solutions provided above with 4αhec2=2.
Thus, for topography of the form h=h0-αxy3 and a similarity
variable of the form η=cxy, the nonlinear PDE, Eq. (),
reduces to a nonlinear ODE of the form
ηg′=-K1+K2g′g′′,
with K1=hec24α and K2=Qc32fα.
Equation () can be solved for g′ by using separation of
variables. Let g′(η)=u(η) (the “pseudo velocity”) so ηu=-(K1+K2u)dudη or ηdη=-(K1+K2u)duu. Integrating both sides yields
η22+m=-(K1lnu+K2u),
where m is an integration constant related to the total transport.
It is possible to solve Eq. () for u, by using the Lambert
W-function (W),
u(η)=K1WK2K1e-2m+η22K1K2.
The integral of u(η) is the analytic solution to the normalized
transport equation, whose boundary conditions are g(-∞,y)=0,
g(∞,y)=1 and g(x,∞)=Qsgn(x). However, the solution
to the derivative of the transport function (pseudo velocity, u) is
sufficient to calculate the flow field, as ψx=Qg′(η)dη∂x and ψy=Qg′(η)dη∂y.
Calculation
It can be seen that m is related to total transport by taking the analytic
limit of Eq. () as K2→0 (which is an error function)
and evaluating the transport boundary conditions. To complete the analytic
solution in the nonlinear case, Eq. () can be integrated and an
iterative method can be employed to determine m based on the transport
boundary condition. Alternatively, Eq. () can be solved
numerically. A fourth-order Runge–Kutta method coupled with a shooting
algorithm was applied to iteratively meet the total transport boundary
condition. It should be noted that the iterative numerical approach is based
on a very small and sensitive velocity boundary condition, which cannot be
taken at -∞ but must be approximated at a small finite value. In the
linear and moderate nonlinear regimes, the numerical and analytical solutions
show good agreement (Fig. ). However, as nonlinearity increases,
the velocity boundary condition becomes extremely sensitive and difficult to
iterate on. Thus, the great advantage of an analytical solution is that it is
easily applicable at any amplitude.
Comparison between linear (open circles), nonlinear numerical (thick
dashed) and nonlinear analytic (solid lines) normalized transport functions.
Plotted is the ratio K2/K1 (nonlinear coefficient over linear
coefficient) of 0.001 (a), 10 (b) and 100 (c) with
K1=0.5.
Discussion
The solutions presented above are relevant to barotropic, along slope flow
over generic topographies of the form h=h0-αxy-γ. For
the linear solution cases, the Ekman pumping relaxes the topographic
vorticity control via the bottom boundary layer. When -1≤γ<1,
the Ekman pumping out paces the topographic control and an expanding
topographic β-plume solution is found. This represents
cross-topographic transport due solely to bottom boundary layer processes.
When γ<-1, the Ekman pumping is not able to overcome the topographic
influence and a compressing topographic β-plume solution is found. Such
compressing solutions result in intense currents, which may be subject to
instability. For the special case, h=h0-αxy3, a nonlinear
solution is found. As seen in Fig. , the nonlinear solution
broadens compared to the linear solution. At first this may seem to be a
contradiction; however, one must remember that in this case the topographic
slope is rapidly increasing, with the influence to compress the jet. The
influence of the nonlinear terms is to resist this compression. This is
consistent with the expected tendency of flow inertia. The details of this
nonlinear tendency are then relevant to the onset of barotropic instability
(or other forms of instability, analysis of which is ongoing work). Note also
that the nonlinear solution limits to the linear solution (both analytically
and numerically), as it must.
This is an analytical paper: the codes described are
standard and easily reproduced from explanations provided in the text.
Preparation of this paper was led by RI and JK;
however, the ideas contained herein are the result of numerous discussions
between all the authors listed.
The authors declare that they have no conflict of
interest.
Acknowledgements
This work was supported by the Texas General Land Office, Oil Spill Program
(program manager: Steve Buschang) under TGLO contract no. 16-019-0009283 and
the National Science Foundation, Physical Oceanography Program (program
manager: Eric Itsweire) under grant no. 1823452. Edited by: Juan Restrepo Reviewed by: two
anonymous referees
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