Ohkitani, Koji

EPrint Series of Department of Mathematics, Hokkaido University

Family of two-dimensional ideal fluid dynamics related to surface quasi-
geostrophic equation
Koji Ohkitani (University of Sheffield)
Abstract
We study 2D surface quasi-geostrophic (SQG) equation numerically and theoretically. After
reviewing recent results, we consider a generalised class of equations of ideal fluid, where
the active scalar is a fractional power α of Laplacian applied to the stream function. This
includes 2D SQG and 2D Euler equations as special cases. We present some numerical
results of the generalised system and compare them for some different values of α. In an
attempt to unify the whole family systematically, a successive approximation is introduced
to treat the SQG equation.
I. INTRODUCTION
Mathematical study on the SQG equation was initiated in [1, 2]. Since then many papers
have been published regarding the analyses of this equation, which are too numerous to cite
here. Numerical studies have been done, e.g. in [1–6]. Mathematically, the following is
the best result known for its regularity. We consider the SQG equation with hypo-viscous
dissipativity either in R2 or in T2
∂θ
∂t
+ (u · ∇)θ = −ν(−4)γθ (0 ≤ γ ≤ 1),
with an initial datum θ(x, 0) = θ0(x). The velocity u = −∇
⊥(−4)−1/2θ is a skewed Riesz
transform of θ, where ∇⊥ = (∂y,−∂x). It has been proved that when γ ≥
1
2
we have no
blow-up [7, 8]. The hypo-viscous equation has been studied numerically in [9]. See also [10]
for more related works.
II. GENERALISED SQG EQUATION FOR INVISCID FLUIDS
We consider a generalised version of SQG equation [3, 11] for inviscid fluids
∂θ
∂t
+ (u · ∇)θ = 0, (1)
1
with θ(x, 0) = θ0(x). Here the velocity u is given by
u = ∇⊥ψ, Λαψ = θ (0 ≤ α ≤ 2).
Here Λ ≡ (−4)1/2 is Zygmund operator defined by Fourier transform Λ˜ = |k|. The system
reduces to the 2D Euler equations if α = 2, to the 2D SQG equation if α = 1, and to a
trivially steady state if α = 0.
III. PERTURBATION THEORY: ODE ANALOGY
We recall a perturbation theory a` la Poincare´ of an ordinary differential equation (ODE)
which depends upon a parameter µ, see e.g. [12, 13]. (We note that notations used in this
section are independent from those in the rest of the extended abstract.)
Consider an ODE
dy
dx
= f(x, y, µ), with an initial datum y(x0, µ) = y0,
which is assumed to be solvable for µ = µ0. If we consider a variation
z(x, µ) =
∂y(x, µ)
∂µ
, with an initial datum z(x0, µ) = 0,
it satisfies
dz
dx
=
∂f(x, Y, µ)
∂Y
∣∣∣∣
Y =y(x,µ)
z +
∂f(x, Y, µ)
∂µ
∣∣∣∣
Y =y(x,µ)
,
which is called an equation of variation.
An approximation for y(x, µ) for small |µ− µ0| may be written
y(x, µ)− y(x, µ0) =
∞∑
n=1
(µ− µ0)
nCn(x),
where Cn(x) are suitable coefficients, e.g.
z(x, µ0) = lim
µ→µ0
y(x, µ)− y(x, µ0)
µ− µ0
= C1(x).
IV. SUCCESSIVE APPROXIMATIONS
We apply the above idea to the generalised SQG equation. We illustrate how this is done
for the first variation. If we take the variation of (1) with respect to α, we find
D
Dt
∂θ
∂α
≡
∂
∂t
∂θ
∂α
+ u · ∇
∂θ
∂α
= −
∂u
∂α
· ∇θ.
2
In R2, we find more explicitly after straightforward manipulations [14]
D
Dt
∂θ
∂α
=
1
2pi
∫
R2
(x− y)⊥
|x− y|2
∂θ(y)
∂α
dy · ∇θ(x) +
1
4pi
∫
R2
(log |x− y|)2∇⊥θ(y)dy · ∇θ(x).
In principle, the equations for higher-order variations may be obtained by successive
differentiations. Given these, we may write, for example, near the 2D Euler limit α = 2
θ(x, t, α) = θ(x, t, 2) +
∞∑
n=1
(α− 2)nθn(x, t),
where θn(x, t) ≡
∂nθ
∂αn
(x, t).
V. CONCLUSION
In fact, under periodic boundary conditions we can carry out the analyses more system-
atically. A formal analysis in this case indicates that all the members in the family behave
similarly with respect to a ’new time variable’ ξ = αt. We discuss the implications of this
scaling, in connection with numerical simulations. These are to be reported in detail in [14].
Acknowledgments
It has been partially supported by an EPSRC grant EP/F009267/1. The author has been
supported by Royal Society Wolfson Research Merit Award.
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geostrophic flow, Phys. Fluids 6 (1994) 9–11.
[2] P. Constantin, A.J. Majda and E. Tabak, Formation of strong fronts in the 2-D quasi-
geostrophic thermal active scalar Nonlinearity 7 (1994) 1495–1533.
[3] I.M. Held, R.T. Pierrehumbert, S.T. Garner and K.L. Swanson, Surface quasi-geostrophic
dynamics, J. Fluid Mech. 282 (1995) 1–20
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flow, Phys. Fluids, 9 (1997) 876–882.
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Letters A 241 (1998) 168–172.
3
[6] J. Deng, T.Y. Hou, R. Li and X. Yu, Level Set Dynamics and the Non-blowup of the 2D
Quasi-geostrophic Equation Methods Appl. Anal. 13 (2006) 157–180.
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quasi-geostrophic equation, Invent. Math. 167 (2007) 445–453.
[8] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-
geostrophic equation, to appear in Acta Math.
[9] K. Ohkitani and T. Sakajo, Oscillatory damping in long-time evolution of the surface quasi-
geostrophic equations with generalised viscosity: a numerical study, Hokkaido University
Preprint Series in Mathematics # 959, April 2010.
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[14] K. Ohkitani, manuscript in preparation (2010).
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Family of two-dimensional ideal fluid dynamics related to surface quasi-geostrophic equation

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EPrint Series of Department of Mathematics, Hokkaido University