2 Weak non-stationary rippling: a model

In the absence of a good theory we model rippling as spatial and temporal dependence localized within the ramp. We start with a monotonic magnetic profile of a low-Mach-number shock ramp which we model using the following expressions (Gedalin et al. Reference Gedalin, Friedman and Balikhin2015):

(2.1)\begin{gather} \frac{B_z}{B_u\sin\theta}=\frac{R+1}{2}+\frac{R-1}{2}\tanh\frac{3x}{D}, \end{gather}

(2.2)\begin{gather}B_x=B_u\cos\theta, \end{gather}

(2.3)\begin{gather}B_y=k_B\frac{{\rm d}B_z}{{\rm d} x}, \end{gather}

(2.4a,b)\begin{gather}E_y=V_uB_u\sin\theta, \quad E_z=0, \end{gather}

(2.5)\begin{gather}E_x={-}\frac{{\rm d}\phi_{{\rm NIF}}}{{\rm d} x}={-}k_E \frac{{\rm d}B_z}{{\rm d} x}. \end{gather}

Here, subscript $u$ refers to the upstream region, $\theta$ is the angle between the shock normal and the upstream magnetic field vector, $B_u$ is the upstream magnetic field magnitude, $E_y$ is the motional electric field, $E_x$ is the cross-shock electric field, $B_y$ is the non-coplanar component of the magnetic field, $D$ is the ramp width and $R$ is the ratio of the downstream to upstream $B_z$. The shock normal is along the $x$-direction and the non-coplanarity direction is $y$. The analysis is done in the normal incidence frame (NIF), where the upstream plasma flow is along the shock normal. The coefficients $k_E,k_B$ are obtained from

(2.6)\begin{gather} -\int E_x\,{\rm d}x=\phi_{{\rm NIF}}\equiv s_{{\rm NIF}}(m_pV_u^2/2e), \end{gather}

(2.7)\begin{gather}-\int (E_x+V_u\tan\theta B_y)\,{{\rm d} x}=\phi_{HT}\equiv s_{HT}(m_pV_u^2/2e), \end{gather}

where $\phi _{{\rm NIF}}$ and $\phi _{HT}$ are the cross-shock potentials in the NIF and the de Hoffman–Teller frame (HT), respectively. In the HT frame the upstream plasma flow is along the upstream magnetic field. The relations for $B_y$ and $E_x$ are approximations derived from the two-fluid plasma model for low-Mach-number shocks, both subcritical and supercritical (Goodrich & Scudder Reference Goodrich and Scudder1984; Gosling et al. Reference Gosling, Winske and Thomsen1988; Gedalin Reference Gedalin1996b). Within this approximation $k_B=c\cos \theta /M\omega _{pi}$ is the whistler length. Here, $M=V_u/V_A$ is the Alfvén Mach number, $c/\omega _{pi}$ is the ion inertial length, $\omega _p=\sqrt {4{\rm \pi} n_ue^2/m_p}$, $V_A=B_u/\sqrt {4{\rm \pi} n_um_p}$ is the Alfvén speed, $n_u$ is the upstream proton number density and $m_p$ is the proton mass. For simplicity, the plasma is assumed to consist of protons and electrons only. For brevity, in what follows we write down expressions for the fields using $c\equiv 1$. The speed of light may be easily restored at the end using dimension arguments. Let us introduce the vector and scalar potentials, as follows:

(2.8)\begin{gather} A_y=A(x)+B_u\cos\theta z-V_uB_u\sin\theta t, \end{gather}

(2.9)\begin{gather}A_z=k_B\frac{{\rm d}A}{{\rm d} x}, \end{gather}

(2.10)\begin{gather}\phi=k_E\frac{{\rm d}A}{{\rm d} x}, \end{gather}

so that

(2.11)\begin{gather} B_z=\frac{\partial A_y}{\partial x}, \end{gather}

(2.12)\begin{gather}B_x=\frac{\partial A_y}{\partial z}=B_u\cos\theta, \end{gather}

(2.13)\begin{gather}B_y={-}\frac{\partial A_z}{\partial x}, \end{gather}

(2.14)\begin{gather}E_x={-}\frac{\partial\phi}{\partial x}, \end{gather}

(2.15)\begin{gather}E_y={-}\frac{\partial A_y}{\partial t}=V_uB_u\sin\theta, \end{gather}

(2.16)\begin{gather}E_z={-}\frac{\partial A_z}{\partial t}=0. \end{gather}

Let us now introduce rippling, as follows. Let $X=x+f$, $f(x,y,z,t)=a\psi (y,z,t)g(x)$, $g(x\rightarrow \pm \infty )=0$, $({\rm d}g/{{\rm d} x})(x\rightarrow \pm \infty )=0$, where $a$ is the amplitude (dimensions of length), while $\psi$ and $g$ are dimensionless. Consider a vector potential and a scalar potential

(2.17)\begin{gather} A_y=A(X)+B_u\cos\theta z-V_uB_u\sin\theta t, \end{gather}

(2.18)\begin{gather}A_z=k_B\frac{\partial A}{\partial X}, \end{gather}

(2.19)\begin{gather}\phi=k_E\frac{\partial A}{\partial X}. \end{gather}

Note that it is always possible to choose the gauge where one of $\boldsymbol {A}$ components vanishes. The fields are now

(2.20)\begin{gather} B_z=B_u\sin\theta B(1+f_x), \end{gather}

(2.21)\begin{gather}B_x=B_u\sin\theta\left(\cot\theta -Bf_z+k_B B_Xf_y\right), \end{gather}

(2.22)\begin{gather}B_y={-}k_BB_u\sin\theta B_X (1+f_x), \end{gather}

(2.23)\begin{gather}E_x={-}k_EB_u\sin\theta B_X(1+f_x), \end{gather}

(2.24)\begin{gather}E_y=V_uB_u\sin\theta-B_u\sin\theta Bf_t-k_EB_u\sin\theta B_X f_y, \end{gather}

(2.25)\begin{gather}E_z={-}B_u\sin\theta k_BB_X f_t-k_E B_u\sin\theta B_Xf_z. \end{gather}

The details of the derivation and the definitions of $B_X$, $f_t$, $f_x$, $f_y$ and $f_z$ are given in the Appendix (A).

3 Implications for the fields in the ramp

In what follows we analyse what could be the signatures of the weak rippling in observations and simulations. In the first order on derivatives $f_t,f_x,f_y,f_z,f_{xx}$ one has

(3.1)\begin{gather} \frac{\partial B_z}{\partial x}=B_u\sin\theta \left(B_X(1+2f_x)+ B f_{xx}\right), \end{gather}

(3.2)\begin{gather}\frac{\partial B_z}{\partial y}=B_u\sin\theta B_Xf_y, \end{gather}

(3.3)\begin{gather}\frac{\partial B_x}{\partial y}=\frac{\partial B_x}{\partial z}=0, \end{gather}

(3.4)\begin{gather}\frac{\partial B_y}{\partial x}={-}k_BB_u\sin\theta\left( B_{XX} (1+2f_x)+ B_X f_{xx}\right), \end{gather}

(3.5)\begin{gather}\frac{\partial B_y}{\partial z}={-}k_BB_u\sin\theta B_{XX} f_z. \end{gather}

The most notable distinction from the stationary planar shock is that $B_x$ and $E_y$ are no longer constant throughout the ramp, and $E_z\ne 0$. If $k_B$ is also small, the major deviations are

(3.6)\begin{gather} \delta B_x\approx{-}B_u\sin\theta Bf_z, \end{gather}

(3.7)\begin{gather}\delta E_y\approx{-}B_u\sin\theta Bf_t-k_EB_u\sin\theta B_X f_y, \end{gather}

(3.8)\begin{gather}\delta E_z\approx{-}k_E B_u\sin\theta B_Xf_z. \end{gather}

As an example, consider ripples localized within the ramp and propagating along the shock front, of the form $\psi =\sin (k_yy+k_zz-\omega t)$. Then there is no phase difference between $\delta B_x$, $\delta E_y$ and $\delta E_z$, and

(3.9)\begin{gather} \delta E_y={-}\frac{\omega}{k_z} \delta B_x + \frac{k_z}{k_y}\delta E_z, \end{gather}

(3.10)\begin{gather}\frac{\delta E_z}{\delta B_x}=k_E\frac{B_X}{B}. \end{gather}

Figure 1 illustrates the effect of rippling on the components of the fields. Figure 1(a,b) provides a visual comparison of the main magnetic component $B_z$ for a stationary shock and its rippled counterpart. Figure 1(c,d) shows the field components for which the effect is especially pronounced. For this visualization the following parameters were used: $M=2.5$, $\theta =65^\circ$, $B_d/B_u=2.2$, $s_{{\rm NIF}}=0.5$, $s_{HT}=0.1$, $D=c/\omega _{pi}$, $k_zV_u/\varOmega _u=2{\rm \pi}$, $k_yV_u/\varOmega _u={\rm \pi} /2$, $\omega =\varOmega _u$, $a=0.3(c/\omega _{pi})$, where $\varOmega _u=eB_u/m_pc$ and $(B_d/B_u)^2=R^2\sin ^2\theta +\cos ^2\theta$. The localizing function is $g(x)=\cosh ^{-2}(x/D)$. The rippling amplitude, $a$ and the wavelength along the $z$ direction $2{\rm \pi} /k_z$ are taken to be similar to what was found numerically by Ofman & Gedalin (Reference Ofman and Gedalin2013), albeit for slightly higher Mach numbers. The profiles are shown for $y=0$ and $t=0$. Note that the rippling amplitude is rather small but the effect is quite noticeable, especially in the components $B_x$ and $E_y$. Thus, even if rippling may be difficult to recognize by the main magnetic component or the magnetic field magnitude, the two mentioned components easily disclose non-planarity and/or time dependence. For the chosen model of rippling $B_x$ deviates from the constant value mainly because of the spatial dependence on $z$, while $E_y$ deviates from the constant value due to the temporal dependence and the spatial dependence on $y$. Figure 1(c,d) suggest that the rippling parameters can be estimated from observations by comparing the variations of the components of the fields. For example, $\delta E_y+({\omega }/{k_z}) \delta B_x$ and $\delta B_x$ seem to not overlap, which suggests that minimization of $\int (\delta E_y+ \lambda \delta B_x)\delta B_x \,{{\rm d} x}$, where $\lambda$ is a variable parameter, may provide an estimate of $\omega /k_z$.

Figure 1.

A visual comparison of the main magnetic component (a) $B_z$ for a stationary planar shock, (b) $B_z$ for a rippled shock, (c) $B_x$ for a rippled shock and (d) $E_y$ for a rippled shock. Parameters are given in text.

There is an overshoot with $\max (|B|/B_u)=2.73$, and the magnetic field magnitude drops to below the upstream value, $\min (|B|/B_u)=0.76$. The normal component of the magnetic field varies in the range $-0.99\leq B_x/B_u<1.84$ while without rippling one has $B_x/B_u=0.42$. For all practical purposes the local normal can be defined as the direction of $\boldsymbol {\nabla }|\boldsymbol {B}|$ in the region where the magnitude of the gradient is maximum. For the chosen model rippled profile the deviations of this direction from $\hat {x}$ reach values $> 30^\circ$, as can be seen from figure 2. Thus, for a weakly rippled shock the observational determination of the shock normal using magnetic coplanarity or minimum variance might have a $\pm 30^\circ$ error depending on the spacecraft trajectory across the shock.

Figure 2.

Close-up on the magnetic field magnitude illustrating changes of the local normal direction along the rippled shock front.

Shocks which exhibit rippling typically have overshoots. An overshoot is not included in (2.1) and in the visualization. The generality of the expressions, however, is limited only by the model (2.17)–(2.19). An overshoot can be easily incorporated in the model profile, for example, as in Gedalin, Pogorelov & Roytershteyn (Reference Gedalin, Pogorelov and Roytershteyn2021).

4 Implications for upstream and downstream waves close to the ramp

The rippled ramp is acting as a boundary which is perturbed according to $\psi =\sin (k_yy+k_zz-\omega t)$. These boundary perturbations should generate waves propagating towards upstream and downstream. The only low-frequency electromagnetic wave which can propagate towards upstream is the whistler wave which has the Doppler shifted dispersion relation

(4.1)\begin{equation} \tilde{\omega}=\omega-k_xV_u=\frac{\sqrt{k_x^2+k_y^2+k_z^2}(k_x\cos\theta+k_z\sin\theta)}{\omega_{pi} }v_A,\end{equation}

where $\tilde {\omega }$ is the whistler frequency in the frame of the upstream plasma flow and $k_x<0$. The relation (4.1) determines $k_x$. Note that for $k_y=k_z=0$ and $\omega =0$ this relation reduces to the phase-standing whistler for a planar stationary shock. In this interpretation the time-dependent rippled shock surface is the source of the whistlers propagating into the upstream region. The inverse should be also true: if whistlers are constantly escaping from the shock front they should leave a corresponding imprint at the front itself. Thus, the rippling pattern at the shock front and the whistler pattern in the upstream region can be expected to be mutually consistent. For the upstream whistler $\cos \theta _{\boldsymbol {k},\hat {n}}=|k_x|/k$ and $\cos \theta _{\boldsymbol {k},\boldsymbol {B}_u}=(k_x\cos \theta +k_z\sin \theta )/k$, where $\theta _{\boldsymbol {k},\hat {n}}$ and $\theta _{\boldsymbol {k},\boldsymbol {B}_u}$ are the angles between the propagation direction of the whistler and the shock normal and the upstream magnetic field, respectively. For the parameters chosen for the above visualization $k_xV_u/\varOmega _u\approx -3.86$, $\theta _{\boldsymbol {k},\hat {n}}\approx 57^\circ$, $\theta _{\boldsymbol {k},\boldsymbol {B}_u}\approx 57^\circ$.

In the downstream region the dispersion relation for the whistler changes accordingly,

(4.2)\begin{equation} \omega-k_{d,x}V_{d,x} - k_zV_{d,z}=\frac{\sqrt{k_{d,x}^2+k_y^2+k_z^2}(k_{d,x}\cos\theta_d+k_z\sin\theta_d)}{\omega_{pi,d} }v_{A,d}, \end{equation}

where $V_{d,x}$ and $V_{d,z}$ are the components of the downstream flow velocity, $\theta _d$ is the angle between the shock normal and the downstream magnetic field, $\omega _{pi,d}$ is the ion plasma frequency calculated with the downstream ion density and $v_{A,d}$ is the Alfvén speed in the downstream region. Note that in the downstream region waves propagate from the ramp into the downstream and the corresponding $k_{d,x}>0$. In numerical simulations capable of resolving whistlers two sets of fronts would be observed diverging from the ramp (Yuan et al. Reference Yuan, Cairns, Trichtchenko, Rankin and Danskin2009; Riquelme & Spitkovsky Reference Riquelme and Spitkovsky2011). If whistler waves are not resolved properly, small pieces of such diverging fronts may be still observed. Diverging waves should remove energy from the ramp, thus providing an additional channel of the redistribution of the energy of the directed flow of the incident ions. The amplitude of the diverging waves and the amplitude of the rippling depend on the mechanism which causes rippling and further sustains it. This mechanism is not known at present and is the subject of intensive studies (Lowe & Burgess Reference Lowe and Burgess2003; Burgess & Scholer Reference Burgess and Scholer2007; Johlander et al. Reference Johlander, Schwartz, Vaivads, Khotyaintsev, Gingell, Peng, Markidis, Lindqvist, Ergun and Marklund2016,  Reference Johlander, Vaivads, Khotyaintsev, Gingell, Schwartz, Giles, Torbert and Russell2018; Omidi et al. Reference Omidi, Desai, Russell and Howes2021).

5 Two-fluid hydrodynamics within the rippled shock

The two-fluid approach was used in attempts to describe the shock front of a laminar (low Mach number, low $\beta$) oblique shock (see, e.g. Gedalin Reference Gedalin1998). Although a shock-like profile was not obtained, some useful estimates of the scales were derived. Here we outline a semiquantitative extension of the two-fluid description with the above modelled rippling. The two-fluid model has to be adapted separately to the different conditions inside the ramp and downstream of the ramp. There are no changes in the upstream region in comparison with the standard description (Gedalin Reference Gedalin1998). The ramp width is substantially smaller than the ion convective gyroradius in supercritical and even laminar subcritical shocks (Russell et al. Reference Russell, Hoppe, Livesey and Gosling1982; Mellott & Greenstadt Reference Mellott and Greenstadt1984; Farris et al. Reference Farris, Russell and Thomsen1993; Newbury & Russell Reference Newbury and Russell1996; Bale et al. Reference Bale, Balikhin, Horbury, Krasnoselskikh, Kucharek, Möbius, Walker, Balogh, Burgess and Lembège2005; Hobara et al. Reference Hobara, Balikhin, Krasnoselskikh, Gedalin and Yamagishi2010; Krasnoselskikh et al. Reference Krasnoselskikh, Balikhin, Walker, Schwartz, Sundkvist, Lobzin, Gedalin, Bale, Mozer and Soucek2013). Therefore, it is more appropriate to treat the ions kinetically. The collisionless Vlasov equation which simply states that the distribution function is constant along the particle trajectory, $f_i(\boldsymbol {r}_i,\boldsymbol {v}_i,t)=f_0(\boldsymbol {r}_0,\boldsymbol {v}_0,t_0)$, where $\boldsymbol {r}_i$ and $\boldsymbol {v}_i$ are the solutions of the equations of motion

(5.1a,b)\begin{equation} \frac{{\rm d}\boldsymbol{r}_i}{{\rm d}t}=\boldsymbol{v}_i, \quad \frac{{\rm d}\boldsymbol{v}_i}{{\rm d}t}=\frac{e}{m_p}\left(\boldsymbol{E}+\frac{\boldsymbol{v}_i}{c}\times \boldsymbol{B}\right), \end{equation}

with the initial conditions $\boldsymbol {r}_i(t=t_0)=\boldsymbol {r}_0$, $\boldsymbol {v}_i(t=t_0)=\boldsymbol {v}_0$. The equations of motion inside the ramp are not integrable even in the stationary planar case. It was shown that a reasonable solution can be derived in the following approximation: (a) the ramp is narrow; and (b) the ratio of the upstream thermal speed to the flow speed is small, $v_{iT}/V_u\ll 1$ (Gedalin Reference Gedalin1997,  Reference Gedalin2021). Here $v_{iT}=\sqrt {T_u/m_p}$, $T_u$ being the temperature of the incident ion distribution. In this approximation

(5.2a,b)\begin{gather} V_{i,x}=\sqrt{V_u^2-2e\phi_{{\rm NIF}}/m_m}, \quad V_{i,y}=V_{i,z}=0, \end{gather}

(5.3)\begin{gather}n_i=\frac{n_uV_u}{\sqrt{V_u^2-2e\phi_{{\rm NIF}}/m_p}}, \end{gather}

where $n_i$ is the ion number density and $\boldsymbol {V}_{i}$ is the bulk (hydrodynamical) velocity of the ions. The main effect is the deceleration by the cross-shock electric field as given by (5.2a,b)–(5.3).

The electron velocity $\boldsymbol {V}_e$ can be obtained from

(5.4)\begin{equation} \frac{4{\rm \pi} en}{c} (\boldsymbol{v}-\boldsymbol{V}_e)=\boldsymbol{\nabla}\times \boldsymbol{B}, \end{equation}

where quasineutrality $n_i=n_e=n$ is assumed. Neglecting ion velocity in the $y$ and $z$ directions inside the ramp one has

(5.5)\begin{gather} V_{ex}=V_{i,x}-\frac{B_u\sin\theta}{4{\rm \pi} ne}\left(B_Xf_y+k_BB_{XX}f_z\right), \end{gather}

(5.6)\begin{gather}V_{ey}=\frac{B_u\sin\theta}{4{\rm \pi} ne}\left(B_X(1+2f_x)+Bf_{xx}\right), \end{gather}

(5.7)\begin{gather}V_{ez}=\frac{B_u\sin\theta}{4{\rm \pi} ne}\left(k_BB_{XX}(1+2f_x)+k_BB_Xf_{xx}\right), \end{gather}

where $V_x=\sqrt {V_u^2-2e\phi /m}$ and we restricted ourselves with the lowest and first order only. In the lowest order the electron velocity is

(5.8)\begin{gather} V^{(0)}_{ex}=V_{i,x}, \end{gather}

(5.9)\begin{gather}V^{(0)}_{ey}=\frac{cB_u\sin\theta}{4{\rm \pi} ne}B_X, \end{gather}

(5.10)\begin{gather}V^{(0)}_{ez}=\frac{cB_u\sin\theta}{4{\rm \pi} ne}k_BB_{XX}, \end{gather}

as in the stationary planar case. In the approximation of massless cold electrons one has $\boldsymbol {E}+\boldsymbol {V}_e\times \boldsymbol {B}/c=0$ which eventually gives

(5.11)\begin{gather} \frac{e}{m_p}\left(\frac{{\rm d}\phi_{{\rm NIF}}}{{\rm d}X}\right)\frac{n_uV_u}{\sqrt{V_u^2-2e\phi_{{\rm NIF}}/m_p}}=\frac{1}{8{\rm \pi} m_p}\frac{{\rm d}}{{\rm d}X}B^2, \end{gather}

(5.12)\begin{gather}V_u-V_{i,x}=\frac{B^2-B_u^2}{8{\rm \pi} n_uV_um_p}, \end{gather}

(5.13)\begin{gather}\frac{V_{i,x}}{V_u}=1-\frac{b^2-1}{2M^2}, \quad b=B/B_u, \end{gather}

(5.14)\begin{gather}1-\frac{n_u}{n_i}=\frac{b^2-1}{2M^2} \rightarrow 1+\frac{1}{2M^2}=\frac{V_{i,x}}{V_u}+\frac{b^2}{2M^2}. \end{gather}

This expressions are identical to the expressions obtained for the stationary planar shock with the only replacement $x\rightarrow X$ (Gedalin Reference Gedalin2021). The above means that in the lowest-order approximation the ramp structure and the ion velocity remain the same as in the stationary planar case, only the position of the ramp edges depend on $y,z,t$. This has important implications for the ion motion and the ion distribution just behind the ramp.

The main correction to the electron velocity due to the rippling is that $V_{ex}\ne V_x$, which probably may be observable in spacecraft measurements. In a quasi-perpendicular shock, where $k_B$ is small, this velocity difference is directly related to the spatial variations in the $y$ direction. For simplicity, we write down the expression $\boldsymbol {E}+\boldsymbol {V}_e\times \boldsymbol {B}/c=0$ for $\cos \theta =0$ and $k_B=0$ as

(5.15)\begin{gather} -k_E B_X(1+f_x)+\frac{B_u}{4{\rm \pi} ne}B\left(B_X(1+3f_x)+Bf_{xx}\right)=0, \end{gather}

(5.16)\begin{gather}V_u- Bf_t-k_E B_X f_y-B\left(V_x-\frac{B_u}{4{\rm \pi} ne}\left(B_Xf_y+k_BB_{XX}f_z\right)\right)=0, \end{gather}

(5.17)\begin{gather}-k_E B_Xf_z -\frac{B_u}{4{\rm \pi} ne}BB_Xf_z=0. \end{gather}

In the lowest order we have

(5.18a,b)\begin{equation} -k_E +\frac{B_u}{4{\rm \pi} ne}B=0, \quad V_u-V_{i,x}B=0. \end{equation}

Both mean $n/B=\text {const.}$, as should occur in a perpendicular shock. The first-order corrections add the following constraints:

(5.19)\begin{gather} -k_E B_Xf_x+\frac{B_u}{4{\rm \pi} ne}B\left(3B_Xf_x)+Bf_{xx}\right)=0, \end{gather}

(5.20)\begin{gather}- Bf_t-k_E B_X f_y+B\frac{B_u}{4{\rm \pi} ne}\left(B_Xf_y+k_BB_{XX}f_z\right)=0. \end{gather}

For $f=g(x)\sin (k_yy+k_zz-\omega t)$ the equations take the form

(5.21)\begin{gather} -k_E B_Xg_x+\frac{B_u}{4{\rm \pi} ne}B\left(3B_Xg_x)+Bg_{xx}\right)=0, \end{gather}

(5.22)\begin{gather}\omega B-k_yk_E B_X +B\frac{B_u}{4{\rm \pi} ne}\left(k_yB_X+k_zk_BB_{XX}\right)=0. \end{gather}

Note that in (5.22) all variables depend only on $X$ and all derivatives are with respect to $X$. It is tempting to interpret (5.22) as a consistence condition for $k_y,k_z,k_E,k_B$ and $B=(X)$, while (5.21) may be interpreted as an equation for viable $g(x)$. However, at this stage using (5.21) and (5.22) to place restrictions on the model parameters is premature, given the number of approximations made to separate the first-order corrections. These two equations only show that there are no gross inconsistencies in the proposed model of the rippling.

Two-fluid hydrodynamics in the downstream region requires taking into account the non-gyrotropy of the ion distribution and its slow gyrotropization. It is more convenient to replace the two-fluid approach with the conservation laws together with the collisionless relaxation principles (Gedalin et al. Reference Gedalin, Friedman and Balikhin2015).

6 Implications for downstream collisionless relaxation

Upon crossing the ramp ions begin to gyrate. In a stationary planar shock the total downstream ion pressure $p_{ij}$ is a function of the distance from the ramp $L$. The total pressure includes the dynamic pressure $nmV_iV_j$ and the kinetic pressure $P_{ij}$, $p_{ij}=nmV_iV_j+P_{ij}$, where $V_i$ is the bulk flow velocity. As a result of the kinematic collisionless relaxation, the kinetic pressure $P_{ij}$ gradually gyrotropizes and further isotropizes, while the dynamic pressure tensor reduces to three components only: $nmV_x^2$; $nmV_z^2$ and $nmV_xV_z$. In a stationary planar shock the total downstream ion pressure depends on the distance $x_d$ from the downstream edge of the ramp, $p_{ij}=p_{ij}(x_d)$. In the approximation of a small amplitude rippling this dependence may be replaced with the dependence on $x_n=x_d+a\psi (y,z,t)$. In a more general way, the ion distribution function becomes dependent on $\psi$. Therefore, all other moments, such as the bulk velocity vector, are also functions of $\psi$. For $\psi =\sin (k_yy+k_zz-\omega t)$ this would mean a spatially periodic pattern propagating along the shock front. Such a pattern should be easily observed in simulations but is difficult to identify even with four-spacecraft measurements. One immediate implication of the new dependence is that weak spatial and/or temporal averaging results in smearing out the peak values of the moments. In particular, the minimum value of averaged $p_{xx}$ is larger than the minimum value of $p_{xx}$ in the case if the shock were stationary and planar. Accordingly, the maximum value of averaged $p_{xx}$ decreases. The conservation laws read

(6.1)\begin{gather} n_t+\sum_{j=x,y,z} (nV_j)_j=0, \end{gather}

(6.2)\begin{gather}(nmV_i)_t+\sum_{j=x,y,z}(p_{ij}+\varPi_{ij})_j=0, \end{gather}

(6.3)\begin{gather}\varPi_{ij}=\frac{1}{8{\rm \pi}} (B^2\delta_{ij}-2B_iB_j). \end{gather}

Averaging over $y,z,t$ we arrive at the conservation laws in the form

(6.4)\begin{gather} \langle nV_x\rangle=\text{const.}, \end{gather}

(6.5)\begin{gather}\langle p_{ix}\rangle+\langle\varPi_{ix}\rangle=\text{const.}, \end{gather}

where $\langle \cdots \rangle$ means averaging. Smearing out the peak values of the pressure means a reduction of the amplitude of the downstream magnetic field oscillations. In this way rippling enhances kinematic collisionless relaxation. Figure 3 illustrates the dispersion of ion trajectories caused by the rippling. For this purpose figure 3(b) shows $x$ versus $v_x$ for an ion moving from upstream with the velocity of the flow, $\boldsymbol {v}_{\text {initial}}=(V_u,0,0)$, in the shock without rippling with the above chosen profile. Figure 3(a) shows $x$ versus $v_x$ for ions with $\boldsymbol {v}_{\text {initial}}=(V_u,0,0)$ starting at randomly chosen $0< z< V_u/\varOmega _u$ and the same $x_{\text {initial}}$. For figure 3(a) the shock is rippled with the parameters mentioned above but $k_y=0$ and $\omega =0$. The blue dotted line shows the magnetic field magnitudes corresponding to the positions $x$ of all ions, independently of $y$ and $z$.

Figure 3.

Trajectories, $x$ versus $v_x$, for ions entering the shock with the velocity of the upstream flow at different initial $z$: (a) the rippled shock; (b) no rippling. The blue dotted line shows the magnetic field magnitudes corresponding to the positions $x$ of all ions.

Since the incident ions cross the ramp in different positions because of rippling, it can be expected that the downstream heating parameters, averaged over $y$, $z$ and $t$, would differ from those which are achieved in a stationary planar shock with the same parameters. In order to compare the parallel and perpendicular heating in a rippled shock with its stationary counterpart, 40 000 initially Maxwellian distributed ions with $\beta _i=0.2$ were traced across the shock in both cases and parallel and perpendicular temperatures were calculated well downstream of the transition region. The normalized upstream temperature in both cases is $T_u/m_pV_u^2=0.016$. The distributions behind the shock are strongly anisotropic. Without rippling the temperatures are $T_{d,\parallel }/m_pV_u^2= 0.016$ and $T_{d,\perp }/m_pV_u^2= 0.178$. No parallel heating occurs. With rippling we obtained $T_{d,\parallel }/m_pV_u^2= 0.018$, $T_{d,\perp }/m_pV_u^2= 0.182$, which means weak parallel heating and lower anisotropy.

Corrections to the averaged conservation laws may be obtained by replacing

(6.6)\begin{equation} \partial_i=a\psi_i \partial_x, \quad i=t,y,z \end{equation}

which gives

(6.7)\begin{gather} \partial_x\left(nV_x+\sum_{i=t,y,z} a\psi_i nV_i\right)=0, \end{gather}

(6.8)\begin{gather}\partial_x\left[a\psi_t nmV_i +p_{xx}+\varPi_{xx} + \sum_{j=y,z} a\psi_j(p_{ij}+\varPi_{ij})\right]=0. \end{gather}

Thus, for example, $nV_x$ and $p_{xx}+\varPi _{xx}$ are no longer constant throughout the shock but the deviations are proportional to the small amplitude of the rippling.

The above analysis was done in the lowest-order approximation which describes the rippling as a simple shift of the position of the ramp of the kind $\Delta x=a\psi (y,z,t)$. Beyond this approximation, the maximum magnetic field and the cross-ramp potential also change with the shift, which further affects the ion motion and can be expected to cause more efficient relaxation. As a result, the spatial scale of the gyrotropization and isotropization downstream of the ramp should be smaller than the corresponding scale in a stationary planar shock. In the present study we model rippling with a monochromatic wave. Even multispacecraft observations provide information only about a small part of the shock surface. Simulations (Umeda & Daicho Reference Umeda and Daicho2018; Omidi et al. Reference Omidi, Desai, Russell and Howes2021) show that rippling is only approximately monochromatic. Finite width of the spectrum would further enhance gyrotropization and isotropization by adding randomness in the gyrophases of the ion which are mixed at a fixed spatial position behind the ramp.