Introduction
Two-dimensional steady-state flow of an isothermal, incompressible stratified
fluid over topography is modeled by Long's equation (Long, 1953, 1954, 1955,
1959). A generalization of this equation to three-dimensional flows has
appeared in the literature (Akylas and Davis, 2001). However, in the
following we restrict our discussion to two dimensions.
Numerical solutions of Long's equation for base flow without shear over
simple terrain, which consists of one hill, were derived and analyzed in the
literature by several authors (Drazin, 1961; Yih, 1967; Drazin and Moore,
1967; Lily and Klemp, 1979; Smith, 1980, 1989; Peltier and Clark, 1983;
Durran, 1992; Smith and Kruse, 2017).
In these studies it was usual to approximate the Brunt–Väisälä
frequency by a constant or a step function. In addition, two physical
parameters which control the stratification and dispersive effects of the
atmosphere were set to zero. Under these approximations, one of the leading
second-order derivatives in Long's equation drops out. Moreover, the equation
becomes linear (the nonlinear terms disappear). In this singular limit Long's
equation reduces to that of a linear harmonic oscillator over the
computational domain. The impact of these approximations on the validity of
the solution was analyzed in depth in the literature (Smith, 1980, 1989;
Peltier and Clark, 1983). These studies demonstrated that these
approximations set limits on the physical applicability of these solutions.
Solutions of Long's equation were also used as a framework for the
examination and study of experimental data on gravity waves (Shutts et al.,
1988, 1994; Fritts and Alexander, 2003; Jumper et al., 2004; Vernin et al.,
2007; Richter et al., 2010; Geller et al., 2013). In all of these studies it
was assumed that the base flow is shearless. However, this assumption is
incorrect, in general, and is not justified by the experimental data. (For a
comprehensive list of references, see Yih, 1980, Baines, 1995, and Nappo,
2012.)
A new method to derive analytic solutions of Long's equation was initiated by
the present author in Humi (2004, 2007, 2009, 2010, 2015). It was
demonstrated that Long's equation can be simplified for shearless base flow
with mild assumptions about the nonlinear terms. In this framework we were
able to identify the “slow variable” in Long's equation. This variable
controls the emergence of nonlinear oscillations in this equation. In
addition we proved the existence of self-similar solutions and derived a
formula for the attenuation of the gravity wave amplitude with height. These
results follow from the general properties of Long's equation and the
nonlinear terms present in this equation.
We considered the effect that shear in the base flow has on the generation of
gravity waves and their amplitude in Humi (2006). A new form of Long's
equation in which the stream function is replaced by the atmospheric density
was derived in Humi (2007). Finally a generalization of Long's equation to
time-dependent flows appeared in Humi (2015).
It obvious however that atmospheric flows over topography are not isothermal
in general (see Miglietta and Rotunno, 2014; Richter et al., 2010; Smith and
Kruse, 2017, and their bibliography). With this motivation we derive, in the
first part of this paper, an extension of this equation to include
non-isothermal flows with free convection. This extension of Long's equation is new.
The novel part of the current paper consists of a sequence of
transformations which linearize Long's equation and lead to an analytic form
of the solution without scarifying any of the physical contents of
this equation. In particular, we demonstrate that there exist
“solitonic-type solutions” to this equation in addition to regular gravity
waves. These types of solutions have never appeared in the
literature before. The solutions presented also show how the amplitudes of
the gravity waves depend on the height. The presentations in Sect. 2.1 and
2.3 are made in order to put the new novel aspects of this paper in context
and to give the reader a sense of their importance. The bulk of the paper, which comprises Sects. 2.2, 3 and 4, presents completely new results which have never appeared in the literature before.
The plan of the paper is as follows: in Sects. 2.1 and 2.3 we present an
overview of the derivation of the isothermal Long equation and the
approximations that are made for its numerical solutions. In Sect. 2.2 we
derive the corresponding Long equation for flows with free convection. In
Sect. 3 we introduce a transformation which (essentially) linearizes the
equation for the perturbation from the base flow. Section 4 discusses the
application of this transformation to a flow with shear and presents an
analytic solution for this flow. We end with some conclusions in Sect. 5.
Derivation of Long's equation
In the first part of this section we provide a short overview of the
(classical) isothermal Long equation and in the second part we generalize
this equation to include free convection.
Isothermal Long equation
In two dimensions (x,z) the flow of a steady isothermal, inviscid
and incompressible stratified fluid is modeled
by the following equations:
ux+wz=0,uρx+wρz=0,ρ(uux+wuz)=-px,ρ(uwx+wwz)=-pz-ρg.
In these equations, subscripts denote differentiation with respect to the
subscripted variable, u=(u,w) is the fluid velocity, p denotes the
pressure, ρ denotes the density and g is the acceleration of gravity.
To non-dimensionalize Eqs. ()–(), we introduce the
following scaled variables:
x‾=xL,z‾=N0U0z,u‾=uU0,w‾=LN0U02w,ρ‾=ρρ0‾,p‾=N0gU0ρ0‾p.
In these equations L represents a characteristic length, U0 is the free
stream velocity, and ρ0‾ is the averaged base density which
is considered to be a constant. N02 represents an averaged value of the
Brunt–Väisälä frequency which is defined as
N2=-gρ0dρ0dz
where ρ0(z) is the base density.
Using these new variables, Eqs. ()–() take the following
form (the bars were dropped for brevity):
ux+wz=0,uρx+wρz=0,βρ(uux+wuz)=-px,βρ(uwx+wwz)=-μ-2(pz+ρ),
where
μ=U0N0L,β=N0U0g.
In these equations μ is the longwave parameter which controls dispersive
effects or equivalently the deviation from the hydrostatic approximation.
When μ=0 the hydrostatic approximation is fully satisfied (Smith, 1980,
1989). The coefficient β is the Boussinesq parameter (Baines, 1995;
Nappo, 2012), which controls stratification effects (assuming U0≠0).
Equation () implies that it is possible to introduce a stream
function ψ so that
u=ψz,w=-ψx.
Using this definition of ψ, it is possible to rewrite Eq. () as
J{ρ,ψ}=0.
The symbol J(f,g) is defined for any two smooth functions f and g as
J{f,g}=∂f∂x∂g∂z-∂f∂z∂g∂x.
It is easy to show that when J(f,g)=0 it is possible to express each of
these functions in terms of the other (Yih, 1980). It follows then from
Eq. () that the functions ρ and ψ are dependent on each
other. This means that one can express ρ as ρ(ψ) or ψ as
ψ(ρ).
Using Eq. () one can rewrite the momentum Eqs. () and
() in terms of ψ.
βρ(ψzψzx-ψxψzz)=-pxβρ(-ψzψxx+ψxψxz)=-μ-2(pz+ρ)
To eliminate p from Eqs. () and (), we multiply
Eq. () by μ2 and then differentiate Eqs. () and
() with respect to z and x, respectively, and subtract. We
obtain
ρz(ψzψzx-ψxψzz)+ρ(ψzψzx-ψxψzz)z-βμ2ρx(-ψzψxx+ψxψxz)-βμ2ρ(-ψzψxx+ψxψxz)x=ρx.
Using Eq. () and the fact that
ρx=ρψψx,ρz=ρψψz,
we can eliminate ρ from Eq. () and obtain after some algebra
J{ψzz+μ2ψxx,ψ}-N2(ψ)Jβ2(ψz2+μ2ψx2),ψ=N2J{z,ψ}
where
N2(ψ)=-ρψβρ
is the non-dimensional Brunt–Väisälä frequency which is (by
definition) a function of ψ.
As a result we obtain the following equation for ψ (Baines, 1995; Nappo,
2012):
ψzz+μ2ψxx-N2(ψ)z+β2(ψz2+μ2ψx2)=G(ψ).
Equation () is referred to in the literature as “Long's
equation”, but it was derived first by Dubreil-Jacotin (Dubreil-Jacotin,
1935).
In Eq. (), G(ψ) is a function that has to be determined from
the base flow. To do so we consider Eq. () at x=-∞ and
assume that the base flow is a function of z only. Then we express the
left-hand side of Eq. () in terms of ψ only to determine
G(ψ). (Here we assumed, following Yih, 1967, 1980, and Baines, 1995,
that the disturbances from the base flow do not propagate upstream.)
For example, if we consider a shearless base flow with u(-∞,z)=1,
then
ψ(-∞,z)=z
and
G(ψ)=-N2(ψ)β2+ψ.
Equation () becomes
ψzz+μ2ψxx-N2(ψ)z-ψ+β2ψz2+μ2ψx2-1=0.
It follows from this example that different base
flows at x=-∞ will yield different functional forms of G(ψ).
We consider now a perturbation η from a shearless base flow
u(-∞,z)=1, viz.
η=ψ-z.
Substituting this expression into Eq. () leads to
ηzz+μ2ηxx-N2β2(ηz2+μ2ηx2+2ηz)+N2η=0.
Long's equation with free convection
When the flow is not isothermal, Eq. () has to be modified as
follows:
ρ(uwx+wwz)=-pz-γTρg,
where T is the temperature and γ is the thermal expansion
coefficient of the fluid. Moreover, an equation for the temperature has to be
added:
u⋅∇T=χ∇2T,
where χ is its thermometric conductivity. These equations hold under
the assumption that
ghc2≪γT0
where h is the fluid column height, c is the velocity of sound in the
fluid and T0 is the characteristic temperature difference.
We can non-dimensionalize these equations using Eq. () with the
addition of
T‾=TT0
(as in the previous subsection we drop the bars). Equations () and
() become
βρ(uwx+wwz)=-μ-2(pz+γTρ),u⋅∇T=1Pe∇2T,
where Pe=U0Lχ is the Peclet number.
Using Eq. () to introduce a stream function ψ, the momentum
Eqs. () and () become
βρ(ψzψzx-ψxψzz)=-px,βρ(-ψzψxx+ψxψxz)=-μ-2(pz+γTρ).
Using the same strategy as in the previous subsection to eliminate p from
these equations leads to
ρz(ψzψzx-ψxψzz)+ρ(ψzψzx-ψxψzz)z-μ2ρx(-ψzψxx+ψxψxz)-μ2ρ(-ψzψxx+ψxψxz)x=γβ(Tρ)x.
If the diffusion processes in Eq. () can be ignored, i.e.,
|1Pe∇2T|≪1, then this equation can
approximated by
J{T,ψ}=0;
i.e., T=T(ψ). Furthermore, since ρ=ρ(ψ), it follows that
(Tρ)x=-J{z,Tρ}=-∂(Tρ)∂ψJ{z,ψ}.
Using Eqs. (), () and (), we can eliminate
ρ from Eq. () and obtain, after some algebra,
J{ψzz+μ2ψxx,ψ}-N2(ψ)Jβ2(ψz2+μ2ψx2),ψ=M2J{z,ψ}
where
M2=-γβρ(Tρ)ψ.
Using these definitions, it follows that
ψzz+μ2ψxx-N2(ψ)β2(ψz2+μ2ψx2)-M2(ψ)z=G(ψ).
Eq. () can be considered a generalized form of Long's equation
which includes the effects of free convection. It contains two parameters
N2 and M2. The additional parameter M2 controls the change in the
temperature profile in the flow.
The function G(ψ) in Eq. () can be determined using the same
strategy as before. Thus, if ψ(-∞,z) is given by Eq. (),
then
G(ψ)=-N2(ψ)β2-M2(ψ)ψ
and Eq. () becomes
ψzz+μ2ψxx-N2(ψ)β2(ψz2+μ2ψx2-1)-M(ψ)2(z-ψ)=0.
For a perturbation η=ψ-z, from a base flow u(-∞,z)=1, we
obtain from Eq. ()
ηzz+μ2ηxx-N2β2(ηz2+μ2ηx2+2ηz)+M2η=0.
Boundary conditions and approximations
We consider here numerical solutions of Long's equation over an unbounded
domain with a general base flow. The topography of the domain is represented
by a function h(x) whose maximum height is H. The boundary conditions
that are imposed on the stream function ψ are
ψ(-∞,z)=ψ0(z),ψ(x,τh(x))=constant,τ=HN0U0.
The constant in Eq. () which represents the value of the stream
line over the topography h(x) is (usually) set to zero.
To determine the proper boundary condition on ψ(∞,z), we note that
Long's equation has no dissipation terms. Therefore radiation boundary
conditions have to be imposed on ψ in this limit. Similarly it is
appropriate to impose radiation boundary conditions on ψ(x,∞)
(Durran, 1992).
When |τ|≪1 the boundary condition () can be approximated
(using Eq. ) by
η(x,0)=-τh(x).
When N and M are set to a constant, Eqs. () and ()
become invariant with respect to translations in x,z. This implies that
these equations admit self-similar solutions in the form η=f(mx+nz)
(Humi, 2004). These solutions represent gravity waves that are generated by
the flow over the topography.
To compute numerical solutions for the perturbation η over topography,
it has been common in the literature to consider Eq. () in the
limits μ=0 and β=0 (Durran, 1992; Lily and Klemp, 1979). In
addition, N is set to a constant or a step function over the computational
domain.
In these limits Eq. () becomes a linear equation:
ηzz+N2η=0.
The limit β=0 can be obtained by letting either N0→0 or
U0→0. For the stratification to persist, one has to assume
that the limit β=0 is obtained as U0→0.
Equation () is a singular limit of Eq. (). This is due to
the fact that one of the leading second-order derivatives drops when μ=0.
Moreover, the nonlinear terms in this equation drop out when β=0. The
approximate solutions that are derived from Eq. () and their
physical limitations have been considered extensively in the literature
(Drazin and Moore, 1967; Durran, 1992; Humi, 2004, 2006). It was found that
strong restrictions have to be imposed on the validity of these solutions
even under the assumption that the base flow is shearless. However, these
approximations and the solutions that are derived from Eq. () are
used routinely in the analysis of experimental atmospheric data (Shutts et
al., 1988; Baines, 1995; Jumper et al., 2004; Vernin et al., 2007).
The general solution of Eq. () is of the form
η(x,z)=q(x)cos(Nz)+p(x)sin(Nz).
The functions p(x) and q(x) have to satisfy the boundary conditions
derived from Eq. () and the radiation boundary conditions. To
satisfy the radiation boundary conditions, p(x) and q(x) have to satisfy
(Baines, 1995; Nappo, 2012) p(x)=H[q(x)], where H[q(x)] is the Hilbert
transform of q(x).
To satisfy the boundary condition on the terrain, one has to solve the
following integral equation (Drazin, 1961; Lily and Klemp, 1979; Durran,
1992):
q(x)cos(τNf(x))+H[q(x)]sin(τNf(x))=-τh(x).
Reductions and transformations
To begin with we observe that in Eqs. (), (),
(), and () one can suppress the appearance of the
parameter μ2 (μ≠0) by applying the transformation x=μx‾. Performing this transformation and assuming that N and M
are constants, these equations become invariant with respect to translations
in x and z. As a result they have solutions of the form
η=f(kx‾+mz) (Humi, 2004). These are gravity waves that are
generated by the atmospheric flow over the terrain.
Equation () becomes
ηzz+ηxx-α2(ηz2+ηx2+2ηz)+N2η=0
where
α2=N2β2.
Similarly, Eq. () becomes
ηzz+ηxx-α2(ηz2+ηx2+2ηz)+M2η=0.
To these equations we apply the transformation
ϕ=e-α2η-1.
Remark: the mathematical “inspiration” for this transformation comes from
somewhat similar transformations which linearize the Ricatti and Burger
equations. From a physical point of view the motivation comes from the desire
to replace the nonlinearities due to the derivatives of η in
Eq. () with expressions that correspond to η itself. This
replacement will enable us to make approximations which are based on physical
insights.
Equations () and (), respectively, become
∇2ϕ-2α2∂ϕ∂z+N2(1+ϕ)ln(1+ϕ)=0,∇2ϕ-2α2∂ϕ∂z+M2(1+ϕ)ln(1+ϕ)=0.
Since |α2η|≪1 it follows that |ϕ|≪1, and we can make
the approximation ln(1+ϕ)≈ϕ. Equations () and
() become
∇2ϕ-2α2∂ϕ∂z+N2(1+ϕ)ϕ=0,∇2ϕ-2α2∂ϕ∂z+M2(1+ϕ)ϕ=0.
To simplify Eqs. () and (), we introduce the
transformation
ϕ=eα2zy.
Equation () becomes
∇2y+(N2-α4)y+N2eα2zy2=0.
If |α2z|≪1 (in the domain of interest), we can approximate this
equation by
∇2y+(N2-α4)y+N2y2=0.
This equation has an analytic closed form solution
y=3(N2-α4)n2tanh2(C1+C2x-iνz)-1
where
ν2=N2-α4+4C22
and C1 and C2 are integration constants.
Equation () represents solutions to a nonlinear equation for y
(and hence η). Since there is no superposition principle for these
solutions, Eq. () represents a new “soliton-type solution” for
η (in Eq. ). Using the approximation
eα2z=1+α2z, this solution for ϕ (using Eq. )
satisfies Eq. () up to terms on order α2.
If α2z is not small, one can approximate eα2z by
1+α2z and use a perturbation expansion y=y0+α2y1 to
compute y1 (numerically).
Similar treatment can be applied to Eq. ().
Linearized equations and solutions
To obtain a real solution for ϕ, we neglect the ϕ2 term in
Eqs. () and () as being of second order. These
approximations linearize Eqs. () and () and yield
(respectively)
∇2ϕ-2α2∂ϕ∂z+N2ϕ=0,∇2ϕ-2α2∂ϕ∂z+M2ϕ=0.
These equations can be solved using separation of variables. Due to the
similarity between Eqs. () and () we discuss henceforth the
solution procedure for Eq. () only.
If we substitute ϕ=f(x)g(z) into Eq. () and perform separation
of variables, we obtain the following equations for f and g:
d2fdx2+ω2f=0,d2gdx2-2α2dgdz+(N2-ω2)g=0.
Hence,
fω=A(ω)eiωx+B(ω)e-iωx,gω=eα2zC1(ω)eiνz+C2(ω)e-iνz,
where C1 and C2 are constants and ν=N2-α4-ω2.
Hence for a wave to exist (in the z-direction) we must have N2≥α4+ω2. In addition the wave amplitude increases with height by a
factor of eα2z.
Similarly to Eq. () we obtain the same expression for f(x) and
gω=eα2z(C3(ω)eiλz+C4(ω)e-iλz)
where λ=M2-α4-ω2.
The general solution of Eq. () can be written as
ϕ=eα2z∫[(D1(ω)ei(νz+ωx)+D2(ω)e-i(νz+ωx)]dω+eα2z∫[D3(ω)ei(νz-ωx)+D4(ω)e-i(νz-ωx)]d.ω
Since the exponents multiplying D1 and D2 are conjugates, it follows
that for ϕ to be real we must have D1‾=D2 (where the bar
stands for complex conjugation). Similarly we must have D3‾=D4.
The radiation boundary condition at z→∞ requires that
the group velocity of the outgoing wave is positive. For a hydrostatic flow
the dispersion relation is given by
λ(ω)=ω-sgn(ν)Nων
and the group velocity is
vg=∂λ∂ν=sgn(ν)Nων2.
Hence vg>0 if νω>0.
Since the integration in Eq. () is over positive ω, it
follows then that the last two terms in this equation must be zero
(νω<0).
To satisfy boundary condition (), we observe (using Eq. )
that
η=-ln(1+ϕ)α2.
Hence the boundary condition () becomes
ϕ(x,0)=eα2τh(x)-1≈α2τh(x).
It follows then from Eq. () that
∫2ReD1(ω)cos(ωx)dω-∫2ImD1(ω)sin(ωx)dω=α2τh(x).
This can be satisfied by standard Fourier integral expansion of h(x).
The special case μ=0 was treated in detail in Humi (2004).
Application
To examine the application of the formulas derived above, we consider the
flow over a “witch of Agnesi” hill where the height of the topography is
given by
h(x)=a2(a2+x2).
The Fourier integral expansion of h(x) is
h(x)=∫0∞A(ω)cos(ωx)dω
where
A(ω)=ae-aω.
Using Eq. (), this implies that ImD1=0 and
D1(ω)=α2τA(ω)2.
Substituting this result in Eq. () yields
ϕ=eα2z∫[D1(ω)ei(νz+ωx)+D2(ω)e-i(νz+ωx)]dω.
Hence,
ϕ=α2τeα2z∫e-aωcos(νz+ωx)dω.
From this expression we can compute η using Eq. (). Figure 1
displays the solution for η for isothermal flow with N=1.5,
β=0.01, a=1, and τ=1. Figure 2 displays the solution for η
for non-isothermal flow with the same parameters as in Fig. 1 but with M=2.
These plots demonstrate the dependence of the gravity wave amplitude on the
height and the impact that non-isothermal flow might have on the direction
and amplitude of the wave.
Contour plot of η for isothermal flow over a topography.
Solutions with shear
We consider here a base flow with u=z, i.e., ψ(-∞,z)=z2. Using
Eq. () to compute G(ψ), we find that
G(ψ)=2-N2(ψ1/2+2βψ).
Long's equation () (with μ≠0) becomes
ψzz+μ2ψxx-N2(ψ)z+β2(ψz2+μ2ψx2)=2-N2(ψ1/2+2βψ).
Applying the transformation x‾=xμ, we obtain (after
dropping the bars)
(ψzz-α2ψz2)+(ψxx-α2ψx2)-N2z=2-N2(ψ1/2+2βψ).
For a perturbation η from the base flow, i.e., ψ=z2+η, we
obtain the following equation (where the square root was linearized assuming
|η|≪1)):
Contour plot of η for non-isothermal flow over a topography.
ηzz-4α2zηz-α2(ηz)2+ηxx-α2(ηx)2+4α2+N22zη=0.
We now introduce the transformation
ϕ=e-α2η-b
where b≠0 is a parameter to be determined later. Applying this
transformation to Eq. () and making the approximation
ln(b+ϕ)=ln(b)+ϕb (assuming |ϕ|≪b) leads to the
following:
2bzϕzz+2bzϕxx-8bα2z2ϕz+(8α2z+N2)[ϕ2+b(ln(b)+1)ϕ+b2ln(b)]=0.
Dropping the nonlinear term in ϕ2 and letting b=e-1 (to suppress
the term containing ϕ), Eq. () becomes
2zϕzz+2zϕxx-8α2z2ϕz-e-1(8α2z+N2)=0.
A particular solution ϕp of this (linear) equation is (Abramowitz and
Stegun, 1974)
ϕp=-14∫e2α2z2-1[-4α2πerf(2αz)+N2Γ(0,2α2z2)]dz.
The homogeneous part of Eq. () can be solved by separation of
variables, viz. ϕ=f(x)g(z), where f(x) satisfies Eq. (). The
resulting equation for g(z) has an analytic solution in terms of Kummer
functions (Abramowitz and Stegun, 1974).
g(z)=C1zKummerMν1,32,2α2z2+C2zKummerUν1,32,2α2z2
where ν1=4α2+ω28α2.
For μ=0 the equation for the perturbation η is
ηzz-4α2zηz-α2(ηz)2+ηN2z+4α2=0.
Applying the transformation Eq. () to Eq. () with
b=e-1 and omitting the nonlinear term in ϕ2, we obtain for ϕ
the same equation as Eq. () without the derivatives with respect to
x. A particular solution of this equation is given by Eq. (),
while the solution of the homogeneous equation is
ϕ(z)=c1erf(i2αz)+c2
where c1 and c2 are constants.